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You are an expert at summarizing long articles. Proceed to summarize the following text:
physics of the bilayer quantum hall ( qh ) system is enormously rich owing to the intralayer and interlayer phase coherence controlled by the interplay between the spin and the layer ( pseudospin ) degrees of freedom@xcite . at the filling factor @xmath2 there arises a unique phase , the spin - ferromagnet and pseudospin - ferromagnet phase , which has well been studied both theoretically and experimentally . one of the most intriguing phenomena is the josephson - like tunneling between the two layers predicted in refs.@xcite , whose first experimental indication was obtained in ref.@xcite . other examples are the anomalous behavior of the hall resistance reported in counterflow experiments@xcite and in drag experiments@xcite . they are triggered by the supercurrent within each layer@xcite . quite recently , careful experiments @xcite were performed to explore the condition for the tunneling current to be dissipationless . these phenomena are driven by the goldstone mode describing an interlayer phase coherence . it exhibits the linear dispersion relation in the zero tunneling - interaction limit ( @xmath1 ) . on the other hand , at @xmath0 the bilayer qh system has three phases , the spin - ferromagnet and pseudospin - singlet phase , the spin - singlet and pseudospin ferromagnet phase , and a canted antiferromagnetic phase@xcite ( abridged as the caf phase ) , depending on the relative strength between the zeeman energy @xmath3 and the tunneling energy @xmath4 . the pattern of the symmetry breaking is su(4)@xmath5u(1)@xmath6su(2)@xmath6su(2 ) , associated with which there appear four complex goldstone modes@xcite . a part of them has been studied in refs.@xcite . we have recently analyzed the full details of these goldstone modes in each phase@xcite . the caf phase is the most interesting , where the spins are canted coherently and making antiferromagnetic correlations between the two layers . moreover , one of the goldstone modes becomes gapless and has a linear dispersion relation@xcite as @xmath1 . it is an urgent and intriguing problem what kind of phase coherence this goldstone mode develops . in this paper , we show that it is the entangled spin - pseudospin phase coherence , and we explore associated phase coherent phenomena . we employ the grassmannian formalism@xcite , where the basic field is the grassmannian field consisting of two complex - projective ( @xmath7 ) fields . the cp@xmath8 field emerges when composite bosons undergo bose - einstein condensation@xcite . the formalism provides us with a clear physical picture of the spin - pseudospin phase coherence in the caf phase . furthermore , it enables us to analyze nonperturbative phase coherent phenomena , where the phase field @xmath9 is essentially classical and may become very large . we show that the supercurrent flows within the layer when there is inhomogeneity in @xmath9 . this is precisely the same as in the @xmath2 bilayer qh system . indeed , the supercurrent leads to the same formula@xcite of the anomalous hall resistivity for the counterflow and drag geometries as the one at @xmath2 . what is remarkable is that the total current flowing the bilayer system is a supercurrent carrying solely spins and not charges in the counterflow geometry . we note that the supercurrent flows both in the balanced and imbalanced systems at @xmath2 but only in imbalanced systems at @xmath0 . in the bilayer system an electron has two types of indices , the spin index @xmath10 and the layer index @xmath11 . they can be incorporated into 4 types of isospin index @xmath12 f@xmath13,f@xmath14,b@xmath13,b@xmath14 . the electron field @xmath15 has four components , and the bilayer system possesses the underlying algebra su(4 ) with the subalgebra @xmath16(2)@xmath17(2 ) . we denote the three generators of the @xmath16(2 ) by @xmath18 , and those of @xmath19(2 ) by @xmath20 . there are remaining nine generators @xmath21 , which are the generators of the r - spin operators . their explicit forms are given in appendix d in ref.@xcite . all the physical operators required for the description of the system are constructed as the bilinear combinations of @xmath22 and @xmath23 . they are 16 density operators @xmath24 , @xmath25 , @xmath26 , and @xmath27 , where @xmath28 describes the total spin , @xmath29 measures the electron - density difference between the two layers . the operator @xmath30 transforms as a spin under @xmath31 and as a pseudospin under @xmath32 . it is @xmath30 that plays the key role in the entangled spin - pseudospin phase coherence in the caf phase . the kinetic hamiltonian is quenched , since the kinetic energy is common to all states in the lowest landau level ( lll ) . the coulomb hamiltonian is decomposed into the su(4)-invariant term @xmath33 and the su(4)-noninvariant term @xmath34 . the additional potential terms are the zeeman , tunneling , and bias terms , @xmath35 , where @xmath36 is the bias voltage which controls the density imbalance between the two layers . the total hamiltonian is @xmath37 . we project the density operators to the lll . what are observed experimentally are the classical densities , which are expectation values such as @xmath38 , where @xmath39 represents a generic state in the lll . we may set @xmath40 , @xmath41 , @xmath42 , and @xmath43 for the study of goldstone modes , where @xmath44 is the density of states . taking the nontrivial lowest order terms in the derivative expansion , we obtain the su(4 ) effective hamiltonian density@xcite @xmath45 , \label{su4hamil}\end{aligned}\ ] ] where @xmath46 with @xmath47 and @xmath48 the intralayer and interlayer stiffness , @xmath49 the capacitance energy , @xmath50 the exchange coulomb energy due to @xmath34 : their explicit formulas are given in appendix a in ref.@xcite . this effective hamiltonian is valid at @xmath51 . the ground state is obtained by minimizing the effective hamiltonian ( [ su4hamil ] ) for homogeneous configurations of the classical densities . the order parameters are the classical densities for the ground state . they are explicitly given in ref.@xcite for the @xmath0 system . in the limit @xmath1 , they read @xmath52 and all others being zero . here , @xmath53 is the imbalance parameter with @xmath54 being the electron density in the front ( back ) layer . both the spin and the pseudospin are polarized into the @xmath55-axis in this limit . we have analyzed the excitations around the classical ground state@xcite . there emerge four complex goldstone modes associated with the spontaneous symmetry breaking su(4)@xmath5u(1)@xmath6su(2)@xmath56su(2 ) . when @xmath57 and @xmath58 , the su(4 ) symmetry is exact and all of them are gapless , but they get gapped by these interactions . we are interested in the limit @xmath1 since we expect the enhancement of the interlayer phase coherence just as in the @xmath59 system . we have already shown that there exists one gapless goldstone mode with a linear dispersion relation in a perturbation theory@xcite . in this paper we employ the grassmannian formalism@xcite to make the physical picture of this goldstone mode and its phase coherence clearer , and to construct a nonperturbative theory in terms of the density difference field @xmath60 and its conjugate phase field @xmath61 . the grassmannian field @xmath62 consists of two @xmath7 fields @xmath63 and @xmath64 at @xmath0 , since there are two electrons per one landau site . due to the pauli exclusion principle they should be orthogonal one to another . hence , we require @xmath65 with @xmath66 . using a set of two @xmath7 fields subject to this normalization condition we introduce a @xmath67 matrix field , the grassmannian field given by @xmath68 obeying @xmath69 . the dimensionless su(4 ) isospin densities are given by @xmath70 = \frac{1}{2}\sum_{i=1}^{2}\boldsymbol{n}_{i}^{\dagger}\tau_{a}^{\text{spin}}\boldsymbol{n}_{i } , \notag \\ \mathcal{p}_{a}(\boldsymbol{x } ) & = \frac{1}{2}\text{tr}\left [ z^{\dagger } \tau_{a}^{\text{ppin}}z\right ] = \frac{1}{2}\sum_{i=1}^{2 } \boldsymbol{n}_{i}^{\dagger}\tau_{a}^{\text{ppin}}\boldsymbol{n}_{i } , \label{su4isospin1 } \\ \mathcal{r}_{ab}(\boldsymbol{x } ) & = \frac{1}{2}\text{tr}\left [ z^{\dagger } \tau_{a}^{\text{spin}}\tau_{b}^{\text{ppin}}z\right ] = \frac{1}{2 } \sum_{i=1}^{2}\boldsymbol{n}_{i}^{\dagger}\tau_{a}^{\text{spin } } \tau_{b}^{\text{ppin}}\boldsymbol{n}_{i } , \notag\end{aligned}\ ] ] where @xmath71 consists of the basis @xmath72 . it is a straightforward task to carry out the perturbative analysis of the effective hamiltonian ( [ su4hamil ] ) in terms of the grassmannian field and obtain the same results as given in ref.@xcite . we concentrate solely on the gapless mode in the limit @xmath1 . we parametrize the @xmath7 fields as @xmath73 for @xmath74 , and @xmath75 for @xmath76 . the isospin density fields are expressed in terms of @xmath60 and @xmath61 , @xmath77 with all others being zero . the ground - state expectation values are @xmath78 , @xmath79 , with which the order parameters are reproduced from ( [ isospin ] ) . it is notable that the fluctuations of the phase field @xmath61 affect both spin and pseudospin components of the @xmath80-spin . this is very different from the spin wave in the monolayer qh system or the pseudospin wave in the bilayer qh system at @xmath2 . hence we call it the entangled spin - pseudospin phase field @xmath61 . by substituting ( [ isospin ] ) into ( [ su4hamil ] ) , apart from irrelevant constant terms the resulting effective hamiltonian is @xmath81 where @xmath82 , @xmath83 , and @xmath84 is the capacitance parameter at @xmath2 . the effective hamiltonian is correct up to @xmath85 as @xmath1 . when we require the equal - time commutation relation , @xmath86 = i\delta ( \boldsymbol{x}-\boldsymbol{y } ) , \label{cr}\ ] ] the hamiltonian ( [ effechamil ] ) is second quantized , and it has the linear dispersion relation , @xmath87 this agrees with eq.(136 ) of ref.@xcite . it should be emphasized that the effective hamiltonian ( [ effechamil ] ) is valid in all orders of the phase field @xmath61 . it may be regarded as a classical hamiltonian as well , where ( [ cr ] ) should be replaced with the corresponding poisson bracket . the effective hamiltonian ( [ effechamil ] ) for @xmath61 and @xmath60 reminds us of the one that governs the josephson - like effect at @xmath2 . the main difference is the absence of the tunneling term , as implies that there exists no josephson - like tunneling . nevertheless , the supercurrent is present within the layer , which is our main issue . -axis in the counterflow geometry . ( a ) all spins are polarized into the positive z axis due to the zeeman effect at @xmath88 . no spin current flows . ( b ) all electrons belong to the front layer at @xmath89 . no spin current flows . ( c ) in the caf phase for @xmath90 , some up - spin electrons are moved from the back layer to the front layer by flipping spins . there appears a goldstone mode associated with this charge - spin transfer . the interlayer phase difference @xmath91 is created by feeding a charge current @xmath92 to the front layer , which also drives the spin current . electrons flow in each layer as indicated by the dotted horizontal arrows , and the spin current flows as indicated by the solid horizontal arrow . ( d ) in the caf phase for @xmath93 , similar phenomena occur but the direction of the spin current becomes opposite.,scaledwidth=48.0% ] by using the hamiltonian ( [ effechamil ] ) and the commutation relation , we obtain the equations of motion , @xmath94 we now study the electric supercurrent carried by the gapless mode @xmath61 . the electron densities are @xmath95 on each layer . taking the time derivative and using we find @xmath96 the time derivative of the charge is associated with the current via the continuity equation , @xmath97 . we thus identify @xmath98constant , where @xmath99 consequently , the current @xmath100 flows when there exists inhomogeneity in the phase @xmath61 . it is a supercurrent because the coherent mode exhibits a linear dispersion relation . it is intriguing that the current does not flow in the balanced system since @xmath101 at @xmath102 . let us inject the current @xmath92 into the @xmath103 direction of the bilayer sample , and assume the system to be homogeneous in the @xmath104 direction ( fig.[spincurrentfigure ] ) . it creates the electric field @xmath105 so that the hall current flows into the @xmath103-direction . a bilayer system consists of the two layers and the volume between them . the coulomb energy in the volume is minimized@xcite by the condition @xmath106 . we thus impose @xmath107 . the current is the sum of the hall current and the supercurrent , @xmath108 with @xmath109 the von klitzing constant . we obtain the standard hall resistance when @xmath110 . namely , the emergence of the supercurrent ( @xmath111 ) is detected if the hall resistance becomes anomalous . we apply these formulas to analyze the counterflow and drag experiments since they occur without tunneling . in the counterflow experiment , the current @xmath92 is injected to the front layer and extracted from the back layer at the same edge . since there is no tunneling we have @xmath112 . hence , it follows from ( [ totalcurrent ] ) that @xmath113 , or @xmath114 all the input current is carried by the supercurrent , @xmath115 . it generates such an inhomogeneous phase field that @xmath116 . on the other hand , in the drag experiment , since the interlayer coherent tunneling is absent , no current flows on the back layer , or @xmath117 . hence , it follows from ( [ totalcurrent ] ) that @xmath118 , or @xmath119 a part of the input current is carried by the supercurrent , @xmath120 . the standard hall resistance is given by @xmath121 at @xmath0 . we thus predict the anomalous hall resistance ( [ counterflowanomalous ] ) and ( [ draganomalous ] ) in the caf phase at @xmath0 by carrying out similar experiments@xcite due to kellogg _ et al_. and tutuc _ et al_. in imbalanced configuration ( @xmath122 ) . the phase field @xmath61 describes the entangled spin - pseudospin coherence according to the basic formula ( [ isospin ] ) in the caf phase . the spin density in each layer is defined by @xmath123 , where @xmath124 for @xmath125 and @xmath126 for @xmath127 . we note the relation @xmath128 up to @xmath129 , we obtain @xmath130 , and @xmath131\partial_{x}^{2}\vartheta ( \boldsymbol{x } ) , \\ \partial_{t}\rho_{\text{f}\uparrow}^{\text{spin } } & = \partial_{t } \rho_{\text{b}\downarrow}^{\text{spin}}=-\frac{j_{\vartheta}}{4}[1-\text{sgn}(\sigma_{0})]\partial_{x}^{2}\vartheta ( \boldsymbol{x } ) . \label{rhotderivative}\end{aligned}\ ] ] the time derivative of the spin is associated with the spin current via the continuity equation , @xmath132 for each @xmath133 . we thus identify @xmath134 the spin current @xmath135 flows along the @xmath103-axis , when there exists an inhomogeneous phase difference @xmath61 . in the counterflow experiment , the total charge current along the @xmath103-axis is zero , @xmath136 . consequently , the input current generates a pure spin current , @xmath137 this current is dissipationless since the dispersion relation is linear . it is appropriate to call it a spin supercurrent . it is intriguing that the spin current flows in the opposite directions for @xmath138 and @xmath139 , as illustrated in fig.[spincurrentfigure ] . a comment is in order : the spin current only flows within the sample , since spins are scattered in the resistor @xmath80 and spin directions become random outside the sample . we have explored the entangled spin - pseudospin phase coherence in the caf phase , governed by the goldstone mode @xmath61 describing the @xmath80-spin according to the formula ( [ isospin ] ) . we have predicted anomalous hall resistivity in the counterflow and drag experiments in the imbalanced regime ( @xmath122 ) at @xmath0 . in particular , there flows a spin supercurrent in the counterflow geometry . this research was supported in part by jsps research fellowships for young scientists , and a grant - in - aid for scientific research from the ministry of education , culture , sports , science and technology ( mext ) of japan ( no . 21540254 ) . l. tiemann , w. dietsche , m. hauser , and k. von klitzing , new . j. phys . * 10 * , 045018 ( 2008 ) ; l. tiemann , y. yoon , w. dietsche , k. von klitzing , and w. wegscheider , phys . b * 80 * , 165120 ( 2009 ) ; y. yoon , l. tiemann , s. schmult , w. dietsche , k. von klitzing , and w. wegscheider phys . rev . * 104 * , 116802 ( 2010 ) . v. pellegrini , a. pinczuk , b. s. dennis , a. s. plaut , l. n. pfeiffer , and k. w. west , phys . lett . * 78 * , 310 ( 1997 ) ; v. pellegrini , a. pinczuk , b. s. dennis , a. s. plaut , l. n. pfeiffer , and k. w. west , science * 281 * , 779 ( 1998 ) . m. f. yang and m. c. chang , phys . b * 61 * , r2429 ( 2000 ) ; e. demler and s. das sarma , phys . rev . lett . * 82 * , 3895 ( 1999 ) ; k. yang , phys . rev . b * 60 * , 15578 ( 1999 ) ; e. demler , e. h. kim , and s. das sarma , phys.rev.b * 61 * , r10567 ( 2000 ) ; a. lopatnikova , s. h. simon , and e. demler phys . b * 70 * , 115326 ( 2004 ) .
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the spin and layer ( pseudospin ) degrees of freedom are entangled coherently in the canted antiferromagnetic phase of the bilayer quantum hall system at the filling factor @xmath0 .
there emerges a complex goldstone mode describing such a combined degree of freedom . in the zero tunneling - interaction limit ( @xmath1 )
, its phase field provokes a supercurrent carrying both spin and charge within each layer .
the hall resistance is predicted to become anomalous precisely as in the @xmath2 bilayer system in the counterflow and drag experiments .
furthermore , it is shown that the total current flowing in the bilayer system is a supercurrent carrying solely spins in the counterflow geometry .
it is intriguing that all these phenomena occur only in imbalanced bilayer systems .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
ensemble classifiers have become very popular for classification and regression tasks . they offer the potential advantages of robustness via bootstrapping , feature prioritization , and good out - of - sample performance characteristics ( @xcite ) . however , they suffer from lack of interpretability , and oftentimes features are reported as `` word bags '' - e.g. by feature importance ( @xcite ) . generalized linear models , a venerable statistical toolchest , offer good predictive performance across a range of prediction and classification tasks , well - understood theory ( advantages and modes of failure ) and implementation considerations and , most importantly , excellent interpretability . until recently , there has been little progress in bringing together ensemble learning and glms , but some recent work in this area ( e.g. @xcite ) has resulted in publicly - available implementations of glm ensembles . nevertheless , the resulting ensembles of glms remain difficult to interpret . meantime , human understanding of models is pivotal in some fields - e.g. in translational medicine , where machine learning influences drug positioning , clinical trial design , treatment guidelines , and other outcomes that directly influence people s lives . improvement in performance without interpretability can be useless in such context . to improve performance of maximum - likelihood models , @xcite proposed to learn multiple centroids of parameter space . built bottom - up , such ensembles would have only a limited number of models , keeping the ensemble interpretable . in this paper , we work from a model ensemble down . we demonstrate that minimum description length - motivated ensemble summarization can dramatically improve interpretability of model ensembles with little if any loss of predictive power , and outline some key directions in which these approaches may evolve in the future . the problem of ml estimators being drawn to dominant solutions is well understood . likewise , an ensemble consensus can be drawn to the ( possibly infeasible ) mode , despite potentially capturing the relevant variability in the parameter space . relevant observations on this issue are made in @xcite , who have proposed centroid estimators as a solution . working from the ensemble backwards , we use this idea as the inspiration to compress ensembles to their constituent centroids . in order to frame the problem of ensemble summarization as that of mdl - driven compression , we consider which requirements a glm ensemble must meet in order to be compressible , and what is required of the compression technique . to wit , these are : 1 . representation * the ensemble members needs to be representible as vectors in a cartesian space * the ensemble needs to be `` large enough '' with respect to its feature set * the ensemble needs to have a very non - uniform distribution over features 2 . compression : the compression technique needs to * capture ensemble as a number of overlapping or non - overlapping clusters * provide a loss measure * formulate a `` description length '' measure it is easy to see that glm ensembles can satisfy the representation requirement very directly . it is sufficient to view ensembles of _ regularized _ glms as low - dimensional vectors in a high - dimensional space . the dimensionality of the overall space will somewhat depend on the cardinality of the ensemble , on the strictness of regularization used , on the amount of signal in the data , on the order of interactions investigated , and on other factors influencing the search space of the optimizer generating the ensemble of glms . coordinates in this space can be alternately captured by ( ideally standardized ) coefficients or , perhaps more meaningfully , by some function of statistical significance of the terms . in this work , we apply the latter . for representation , we choose a basis vector of subnetworks . in order to identify this basis vector , we have experimented with gaussian mixture decomposition ( gmm ) ( finding clusters of vectors in model space ) and hierarchical clustering . for performance reasons , we present results using the latter technique , despite its shortcomings : instability and inability to fit overlapping clusters ( this may lead to overfitting ) . nevertheless , in practice we find that this latter technique performs reasonably well . optionally , to summarize the clusters , centroids can be fit _ de novo _ once these groups of models are identified , or medoids can be used , obviating the need for further fitting . here we use the first method , refitting centroids from training data on just the terms occurring in the models in a given cluster . lastly , bayesian information criterion ( _ bic _ ) satisfies the representation scoring requirement . the likelihood term serves as the loss function and the penalty term captures `` description length '' ( @xcite ) . the bic - regularized glm ensembles were fit for binary - outcome datasets used in @xcite and using the software from the same paper ( number of bags = = 100 , other settings left at defaults ) . the result of this step was an ensemble @xmath0 which , ignoring the outcome variable and the intercepts , could be captured via a non - sparse matrix as follows : @xmath1 where @xmath2 , the ensemble dimensionality , refers to the number of fitted models and @xmath3 to the number of terms found in the whole fitted ensemble . importantly , @xmath2 is always an arbitrary parameter - the fact that partially motivated our study . for each dataset , the fitted ensembles were then compressed using the following procedure . first of all , for each ensemble we created the significance matrix s : @xmath4 where @xmath5 , and the p - value is determined from the fit of the linear model @xmath6 of the glm ensemble ( s is the heatmap in figure [ figure1 ] ) . each row of @xmath7 projects every model @xmath6 into a multivariate cartesian space where each axis corresponds to model terms observed in the whole ensemble and each coordinate corresponds to log - scaled term significance . log - scaling is introduced to induce separability of models that share terms in the presence or absence of collinear covariates that would be expected to influence , but not obviate , shared terms significance . note that , in this representation , if @xmath8 is missing , @xmath9 . as an example of what the @xmath0 and @xmath7 matrices would look like after this step , for the * compressed * ensemble in figure [ figure2 ] , @xmath0 = @xmath10 and @xmath7 = @xmath11 of course , in practice it is the full ensemble we are interested in representing in this manner en route to compressing it , so @xmath0 and @xmath7 will have their dimensions , @xmath2 and @xmath3 , of @xmath12 for a computational - biological application using a regularized glm ensemble construction approach . having constructed the matrix s for each ensemble in this manner , we clustered its rows ( the models coefficients ) using ward s clustering criterion ( @xcite ) and euclidean distance metric . we then traversed the resulting dendrogram from left to right , using each cutpoint to perform model assignment to clusters implied by the leaves of the resulting dendrogram ( figure [ figure1 ] ) . we captured the assignment of models to k clusters produced by each step in this traversal as a column vector @xmath13 expressed as a categorical variable with @xmath14 unique values , such that @xmath15 . then , for each model ensemble term @xmath16 , we extracted the vector @xmath17 - a column slice through the matrix s for term @xmath16 . we next defined a linear model @xmath18 with @xmath19 parameters . the success of ensemble compression via the @xmath14 clusters implied by @xmath13 could then be assessed by defining @xmath20@xmath21 being the log - likelihood of the model . this evaluation of cost across clustering levels to find maximal average likelihood compression , @xmath22 , could be viewed as using the bayes factor as the loss function for optimization , and the process of describing the model ensemble by centroids ( or medoids ) of the clusters of models described by @xmath13 could be described as an mdl - driven compression of the ensemble , using the bic - penalized likelihood as the measure of optimal compression . it is worth adding that the aforementioned gmm approach for cluster membership assignment , which can also be driven by bic(@xcite , @xcite ) , does not imply a specific nested membership of models in clusters , but generally results in cluster membership strongly correlated with that identified via the hierarchical model clustering technique . while the gmm approach is more robust ( e.g. , it s not path - dependent , and does nt require specification of a linkage function ) , it scales worse when number of terms in the ensemble is large . using the datasets described above , we performed 3-fold cross - validation repeated three times , and for each training fold and repeat fitted medoid- and centroid - compressed model ensembles . for each held - out fold , we then computed out - of - sample auc for every method ( table [ table : table1 ] ) . additionally , we performed paired one - tailed t - tests comparing medoid and centroid compression strategies to uncompressed ensembles across folds . all aucs arising from repeats and folds were averaged prior to t - test to avoid pseudo - replication issues . while medoids performed slightly worse , on average , than uncompressed ensembles , centroids performance was degraded only in the statistically `` suggestive '' sense ( 0.05 < p < 0.1 ) . note that in our experience using this technique on real - world datasets , larger datasets and continuous outcomes result in even smaller , if any , degradation of performance , performance being especially degraded for logistic regression . in other words , we believe that these results , reported for binary outcomes , are essentially a lower bound . maximum - likelihood methods identify the most likely fit in the parameter space . however , unless the most likely fit is vastly superior to all others and is sharply defined , a rare scenario in practice , the total probability of this fit in the infinite model ensemble may be very small . for that reason , model ensembles are thought to be superior to individual models . ensemble construction gains power by sampling multiple models from the parameter space but , by so doing , loses interpretability by introducing alternative parameter configurations and values . we build on the understanding that model parameter space centroids should be sufficient to capture predictive power of large ensembles ( @xcite ) , while observing that such centroids exhibit better interpretability by having fewer parameters among the alternative models . working top - down , we demonstrate and validate on several datasets a novel approach to summarizing ensembles of glms . our data shows this approach can result in models nearly identical to full ensembles in performance and vastly superior in interpretability , owing to dramatically reduced ensemble sizes ( figure [ figure2 ] ) . in addition , since this approach can operate on any models that can be shoehorned into a cartesian space , it shows promise for compressing and thus summarizing ensembles of other types - for instance , causal model ensembles with individual models represented as orderings ( @xcite ) . we posit that our approach can alter applicability of ensemble methods in general , making their use possible for a wide range of applications where the bottleneck has been the interpretability of results . future directions of research may include multivariate classification methods beyond gmm and hierarchical clustering , as well as extension of this methodology beyond ensembles of glms to other types of predictive ensembles . presented at nips 2016 workshop on interpretable machine learning in complex systems .aucs by method and dataset . last column shows p - value of one - tailed paired t - test vs randomglm ( @xmath23 : the compressed ensemble performs as well as the full ensemble ) . values are averaged over 3 folds and 3 repeats . because of multiple repeats , standard errors are not shown . [ cols=">,<,<,<,<,<,<",options="header " , ] the authors would like to acknowledge leon furchtgott and fred gruber for their invaluable feedback on the manuscript , and fred gruber for his help with latex . song , l. , langfelder , p. , horvath , s. ( 2013 ) random generalized linear model : a highly accurate and interpretable ensemble predictor . _ bmc bioinformatics _ 14:5 pmid : 23323760 doi : 10.1186/1471 - 2105 - 14 - 5 teyssier , m. and koller , d ( 2005 ) . ordering - based search : a simple and effective algorithm for learning bayesian networks . _ proceedings of the twenty - first conference on uncertainty in ai ( uai ) _ ( pp . 584 - 590 ) . chitraa , v. , thanamani a.s . ( 2013 ) , review of ensemble classification . _ international journal of computer science and mobile computing a monthly journal of computer science and information technology _ , vol . 2 , issue . 5 , may 2013 , pg.307 312 liaw , a. , wiener , m. ( 2002 ) classification and regression by randomforest r news 1822 hansen , m.h . , and yu , b. model selection and the principle of minimum description length . _ journal of the american statistical association _ 96.454 ( 2001 ) : 746 - 774 . carvalho , l.e . , lawrence , c.e . , centroid estimators for inference in high - dimensional discrete spaces _ pnas _ , 105 : 32093214 ( 2008 ) ward , j. h. , jr . hierarchical grouping to optimize an objective function , _ journal of the american statistical association _ , 58 , 236244 ( 1963 ) fraley , c. , raftery , a.e . , murphy , t.b . , and scrucca , l. ( 2012 ) mclust version 4 for r : normal mixture modeling for model - based clustering , classification , and density estimation technical report no . 597 , department of statistics , university of washington fraley , c. and raftery , a.e . ( 2002 ) model - based clustering , discriminant analysis and density estimation journal of the american statistical association 97:611 - 631
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over the years , ensemble methods have become a staple of machine learning .
similarly , generalized linear models ( _ glms _ ) have become very popular for a wide variety of statistical inference tasks .
the former have been shown to enhance out - of - sample predictive power and the latter possess easy interpretability .
recently , ensembles of glms have been proposed as a possibility . on the downside
, this approach loses the interpretability that glms possess .
we show that minimum description length ( _ mdl_)-motivated compression of the inferred ensembles can be used to recover interpretability without much , if any , downside to performance and illustrate on a number of standard classification data sets .
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observations of our and outer galaxies allowed to identify various galactic populations : the halo , the thick disk , the thin disk , and the bulge . a model for the evolution of galaxies should explain the origin and properties of these populations , as well as other basic observations like e.g. the relation of hubble types with local environment , in a unifying scheme . current models for galaxy formation broadly divide into two families : those considering a dissipational collapse ( eggen et al . 1962 ; larson 1974 ) ; and those which consider galaxies as the results of the accretion of individual fragments undergoing ( some ) indipendent chemical and dynamical evolution ( toomre & toomre 1972 ; searle & zinn 1978 ) . the transition between the halo and disk phases is continuous in smooth _ dissipational collapse _ models , while disk formation is a secondary mechanism in _ accretion _ ones . separation between these two classes of models may be quite artificial : in fact various properties of galaxies , like e.g. the light distribution of ellipticals , are well reproduced by inhomogenous collapses leading to some kind of violent relaxation ( lynden - bell 1967 ) ; on the other side , simulations based on cosmologies dominated by cold dark matter predict that in high density regions galaxies form hierchically by merging of smaller subunits , while in low density ones they form more gradually by infall of diffuse matter ( frenk et al . 1985 ) . within this framework , the mechanisms of formation of our own galaxy ( the milky way ) could be determined by examining fossil remnants of the early phases represented by the old ( and often metal - poor ) stars . the interpretation of the large amount of data gathered in the last years on dynamics and metallicities ( as defined by the most easily observed element , fe ) of field stars is however still controversial , and while e.g. some authors consider the thick disk and the bulge ( gilmore et al . 1989 ) as distinct galactic components , others ( norris 1993 ) think they are simply the outer ( and oldest ) part of the disk and central part of the halo respectively . scenarios of galactic evolution including a hiatus between the formation of the halo and of a secondary disk ( ostriker & thuan 1975 ) , that were introduced to justify the rarity of metal - poor stars in the solar neighbourhood ( schmidt 1963 ) , are widely applied e.g. to explain the hot , metal - rich intergalactic gas seen in clusters ( berman & suchkov 1991 ) ; however , up to now the observational basis for this hiatus ( based on the age gap between open and globular clusters : demarque et al . 1992 , carraro et al . 1999 ; the white dwarf cooling sequence : wonget et al . 1987 , knox et al . 1999 ; and the th / nd ratio nucleo - chronometer for disk and halo stars : malaney & fowler 1989 , cowan et al . 1999 ) are rather weak and controversial . relative abundances of o and fe in stars of different overall metal abundance provide further constraints to the early evolution of the halo and the formation of the galactic disk ( wheeler et al . o is the main product of hydrostatic he - burning : hence the ejecta of core - collapse supernovae ( sne ) resulting from the evolution of massive stars , usually identified with type ii sne , are expected to be very rich in o ( woosley & weaver 1986 ; thielemann et al 1990 ) . on the other side , while a fraction of the fe presently observed in the interstellar medium was synthesized in massive stars ( thielemann et al 1990 ) , a large fraction of it was likely produced in explosive burning under degenerate conditions in type ia sne ( nomoto et al . 1984 ) . typical lifetimes of the progenitors of type ia sne ( @xmath4 yr ) are much longer than those of the progenitors of type ii sne ( @xmath5 yr ) , and they are actually longer than , or of the same order of , the free fall time in the galaxy ( @xmath6 yr ) ; for these reasons the production of the bulk of fe is expected to be delayed with respect to that of o ( matteucci & greggio 1986 ) . a clear break in the run of o abundances with overall metallicity [ fe / h]=log@xmath7(x)@xmath8log@xmath7(x)@xmath9 for any abundance ratio x. ] should signal the onset of the contribution by type ia sne , and the location of this break provides an independent estimate for the timescale of star formation during the early stages of galactic evolution ( matteucci & franois 1992 : hereinafter mf ) . it should be added that other @xmath10elements ( like mg , si , and ca ) are expected to behave similarly to o , although for si and ca a small contribution by type ia sne is also expected . in the last years various investigations have been devoted to the study of the run of [ o / fe ] with [ fe / h ] in halo and disk stars ( wheeler et al . 1989 , king 1994 , nissen & schuster 1997 , fuhrman 1998 , 1999 , israelian et al . 1998 , boesgaard et al . 1999 ) . however , a variety of basic questions still lacks of a clearcut answer . the [ o / fe ] ratio in the halo and the location of the change of slope in the run [ o / fe ] vs [ fe / h ] have been addressed by king ( 1994 ) , who concluded that this change may occur at any value in the range @xmath11[fe / h]@xmath12 , corresponding to timescales for the halo formation between @xmath13 and @xmath14 yr ( mf ) ; this range is large enough to accomodate both a fast , ordered dissipational collapse ( eggen et al . 1962 ) , or a much slower , accretion scenario ( searle & zinn 1978 ) . edvardsson et al . ( 1993 ) studied the [ o / fe ] run in disk stars ; they suggested that this ratio is constant for [ fe / h]@xmath15 , and argued that the spread in [ fe / h ] values at any age is an evidence for infall of metal - poor material . even less understood is the [ o / fe ] run at intermediate metallicities , corresponding to the thick disk phase ( gilmore et al . 1989 ; nissen & schuster 1997 ) . the main concerns in previous investigations on o abundances relate ( i ) to the paucity of samples of significant sizes studied in a homogeneous way , and then to the possible existence of systematic offsets between different sets of data ; and ( ii ) to the discrepancy between o abundances determined using high excitation permitted and low excitation forbidden lines ( the first usually observed in dwarfs , the second in giants ) . most of this discrepancy can be removed by adopting higher temperatures in the analysis of dwarfs ( king 1993 ) ; furthermore , the effects of departures from the local thermodynamic equilibrium ( lte ) assumption when considering the formation of high excitation permitted o i lines should also be considered , in order to provide abundances at the level of accuracy required for the present purposes . recent results based on the oh band at the extreme uv edge of ground - based observations have further complicated this issue , suggesting the presence of a quite strong slope in the [ fe / o ] run with [ fe / h ] amongst metal - poor stars ( israelian et al . 1998 , boesgaard et al . 1999 ) both a high temperature scale , and consideration of departures from lte were included in the new homogeneous determinations of abundances of light elements and fe for a large sample of stars we are presenting in this series of papers . a new , hopefully improved temperature scale based on irfm temperatures for population i stars and the new model atmospheres by kurucz ( 1993 , cd - rom 13 ) was obtained in gratton et al . ( 1996a , paper i ) . an extensive discussion of the effects of departures from the assumption of lte in line formation in the stellar atmospheres was given in gratton et al . ( 1999 , paper ii ) ; in that discussion , we exploited an empirical calibration of the poorly known cross sections for collisions with h i atoms drawn from a parallel analysis of the spectra of rr lyare variables at minimum light , where non - lte effects are much larger than in the stars here considered ( clementini et al . our final abundances for about 300 stars were presented and discussed in carretta et al . ( 2000 , paper iii ) . we found that most discrepancies present in earlier works have been removed , our results showing a high degree of internal consistency , at least for stars with effective temperature @xmath16 k , although our results are still not easy to be reconciled with the o abundances from the uv oh bands . however , in this paper we will show that once combined with stellar kinematics and compared with models of galactic chemical evolution , our results allow to throw new light into some of above mentioned questions : we find that in the framework of homogeneous models , the collapse of the halo and the formation of the thick disk occurred on a short timescale ( a few @xmath17 yr ) , although star formation in these environments likely lasted for @xmath18 gyr ; and that there was a sudden decrease in star formation between the thick and thin disk phases , which are then clearly distinct galactic components . it is worth noticing that a decrease in the star formation rate between the formation of the spheroidal components and the disc in disc dominated galaxies , had already been suggested by larson ( larson et al . he suggested that such a decrease in the star formation could be due either to the action of tidal forces inhibiting star formation during the later stages of the collapse or to a two - phase structure of the gas , with dense clouds forming rapidly in a spheroidal component and less dense intercloud gas not forming stars and settling to a disc . we argue that our results fit in a scenario in which both collapse and accretion were important in the formation of the milky way , these mechanisms having similar timescales ; the relative weights of the two contributions in other galaxies might explain the hubble sequence . an early , short presentation of the content of this paper was given as a talk at the conference on formation of the galactic halo ( gratton et al . 1996b ) . here we give a more complete presentation of our arguments . in the meanwhile , fuhrmann ( 1998 ) reached quite similar conclusions , based on an independent careful analysis of a smaller smaple of nearby stars . our results for fe , o and mg are displayed in fig . [ fig:1 ] and [ fig:2 ] . to further improve homogeneity , we only plotted data for dwarfs ( edvardsson et al . 1993 ; tomkin et al . 1992 ; nissen & edvardsson 1992 ; zhao & magain 1990 ) ; all these stars have @xmath16 k. inclusion of cooler and/or lower gravity stars in our sample does not change the present discussion . the original data ( equivalent widths ) used in these papers are very consistent with each other , and were measured on high signal - to - noise , high resolution spectra ; typical errors of individual equivalent widths determined from independent estimates for the same stars , are @xmath19 m , yielding errors of @xmath20 dex in abundances derived from individual lines . possible errors in the atmospheric parameters required in the analysis of individual stars ( @xmath21 k in the effective temperatures ; @xmath22 dex in the surface gravity ; @xmath23 in the microturbulent velocity ) cause uncertainties of @xmath24 dex in [ fe / h ] , @xmath25 dex in [ o / h ] , and @xmath25 dex in [ fe / o ] ; these are internal errors . analogous values for mg are @xmath26 dex in [ mg / h ] , and @xmath27 dex in [ fe / mg ] . systematic errors should be small since the present analysis is differential with respect to the sun . in total , we determined [ o / fe ] ratios for about 160 stars and [ mg / fe ] for 197 stars . following wheeler et al . ( 1989 ) we have chosen to use o and mg as reference elements , because they are almost uniquely produced in massive stars . the scatter of data for individual stars is small and compatible with observational errors for stars with [ o / h]@xmath28 and [ mg / h]@xmath29 , although we can not exclude a gentle ( @xmath30 ) slope in the run of [ fe / o ] with [ o / h ] ; the scatter we obtain for more metal - poor stars ( 0.10 dex in both [ fe / o ] and [ fe / mg ] ratios ; for o the peculiar n - rich dwarf hd 74000 was omitted ) may indicate that errors are larger in this abundance range ( perhaps due to the apparent star faintness ) . however , the composition of the interstellar matter could have been slightly inhomogeneous during the early phases of galactic evolution , as suggested by the large spread in the abundances of n - capture elements found by mcwilliam et al . ( 1995 ) for [ fe / h]@xmath31 ( i.e. [ o / h@xmath32 ) , and the scatter in the fe / o ratios amongst halo stars found in the careful analysis by nissen & schuster ( 1997 ) . the upper limit for intrinsic star - to - star variations derived from the spread in our data is 0.07 dex , once observational errors are taken into account . [ fig:1]a indicates that the run of [ fe / o ] with [ o / h ] is quite flat from [ o / h][email protected] , [ fe / o][email protected] ( these are the most metal - poor stars in our sample ) to [ o / h][email protected] , [ fe / o][email protected] , where it is possible to locate ( with an uncertainty of about @xmath34 dex ) the change of slope due to the onset of the contribution to nucleosynthesis by the bulk of type ia sne . a similar result is provided by fig . [ fig:2 ] for the run of [ fe / mg ] with [ fe / h ] . we remark that the values of [ o / h ] and [ mg / h ] at which the changes of slope occur is large ( [ o / h]@xmath35[mg / h]@xmath36 ) . a direct comparison with existing galactic evolution model is possible for o , for which the predictions of mf models are available . we overposed on fig . [ fig:1]a lines representing the predictions given by models 1 and 2 of mf ; these models were computed with an @xmath37folding time of 1 gyr for the infall ( halo collapse ) , and two different laws of star formation : the times required for [ fe / h ] to raise at [ fe / h]=@xmath38 ( roughly corresponding to the halo - disk transitions ) are @xmath39 and @xmath13 yr respectively . both models were arbitrarily scaled to match the [ fe / o ] ratio for halo stars ( [ fe / h]@xmath40 ) . these scalings only imply small changes in the adopted yields of fe from type ii sne , which on turn depend on the cut - off mass for the remnants for core - collapse sne , an ill - defined quantity at present ( timmes et al . while these scalings do not affect our conclusions , they help to see the main features we like to point out . insofar metal - poor stars are considered ( [ fe / h]@xmath41 ) , the r.m.s . values of the residuals of points for individual stars around the lines representing the mf models are 0.158 and 0.127 dex for model 1 and 2 respectively : data for metal - poor stars are then better represented by model 2 , which considers the production of fe from massive stars alone during this phase ( the shallow slope of [ fe / o ] with [ o / h ] in this model is due to the dependence of the o / fe abundance ratio in the ejecta of type ii sne with progenitor mass ) ; while model 1 is in clear disagreement with observations . the conclusion of this ( and other comparisons not shown in fig . [ fig:1 ] ) , is that the raise of o abundances up to [ o / h]=@xmath42 occurred on a timescale which is not much longer than the lifetime of type ia sne . this timescale will be better quantified later . while model 2 is able to better reproduce observations at low metallicities , it fails in the metal - rich range . in fact , @xmath43-tests show that a linear dependence of [ fe / o ] on [ o / h ] ( roughly similar to that given by both model 1 and 2 ) is not a good representation of the observed [ fe / o ] s for [ o / h]@xmath28 , since the scatter of [ fe / o ] values for [ o / h]@xmath44 is much larger than observational errors . a very similar result might be obtained for mg . the simplest interpretation is that the fe content suddenly increased at [ o / h]@xmath35[mg / h]@xmath36 ; the implication is that there was a phase in which the production of o and mg ( i.e. the formation of type ii sne ) felt down to small values , leaving only the fe producers in activity . this obviously means a sudden decrease in the formation of massive stars and , since we are moving in the framework of a constant initial mass function ( mf ) , in the star formation _ tout court_. we found this same feature when considering si and ca abundances rather than o ones ( these abundances are not discussed in paper iii , but they display trends similar to those found for o and mg : gratton et al . , in preparation ) . the synthesis of these elements is also likely related to massive stars ( thielemann et al . 1990 ) . the phase of low star formation must have lasted enough to allow the explosion of a large fraction of halo and thick disk type ia sne , since [ o / h ] and [ mg / h ] start increasing again from values of [ fe / o ] and [ fe / mg ] @xmath1 dex higher than that achieved in the previous phase . the simultaneous increase of both [ o / h ] and [ fe / o ] in stars with [ o / h]@xmath28 , [ o / fe]@xmath45 seems to require that both type ia and type ii sne contribute to the chemical enrichment during this phase . the regression line through data in this region is : @xmath46}=(0.34\pm 0.07){\rm [ o / h ] } + ( 0.04\pm 0.10),\ ] ] based on 71 stars . however , results for mg in the metal - rich regime are quite different ; the regression line through data in this region is : @xmath47}=(-0.055\pm 0.025){\rm [ mg / h ] } - ( 0.089\pm 0.076),\ ] ] based on 164 stars . if real , this result indicates that mg abundances do not increase at the same rate as o ones , suggesting that the o / mg ratio in the ejecta of type ii sne is a function metal abundance ; this could be understood if severe mass loss reduce production of mg in massive , metal rich stars . lrclrclrcl parameter & & & + @xmath48 ( ) & 21 & 48 & 100 & 21 & 144 & 52 & 164 & 206 & 24 + @xmath49 ( kpc ) & 21 & 2.1 & 2.3 & 21 & 0.6 & 1.1 & 164 & 0.18 & 0.20 + @xmath50 & 21 & 0.69 & 0.28 & 21 & 0.38 & 0.19 & 164 & 0.14 & 0.07 + @xmath51 ( gyr ) & 13 & 1.16 & 0.08 & 21 & 1.12 & 0.06 & 164 & 0.69 & 0.23 + @xmath52fe / h@xmath53 $ ] & 29&@xmath331.68 & 0.41 & 21 & @xmath330.63 & 0.15 & 164 & @xmath330.19 & 0.26 + @xmath52o / h@xmath53 $ ] & 29&@xmath331.23 & 0.40 & 19 & @xmath330.29 & 0.12 & 68 & @xmath330.18 & 0.18 + @xmath52fe / o@xmath53^a$ ] & 28&@xmath330.46 & 0.10 & 19 & @xmath330.36 & 0.06 & 67 & @xmath330.01 & 0.11 + @xmath52mg / h@xmath53 $ ] & 22&@xmath331.42 & 0.41 & 21 & @xmath330.29 & 0.13 & 164 & @xmath330.10 & 0.24 + @xmath52fe / mg@xmath53 $ ] & 22&@xmath330.30 & 0.10 & 21 & @xmath330.32 & 0.05 & 164 & @xmath330.08 & 0.08 + a further basic step can be done by combining information provided by the ( [ o / h ] vs [ fe / o ] ) and ( [ mg / h ] vs [ fe / mg ] ) diagrams with those obtained from the kinematics of stars in our sample . a caveat should be done here , since stars analyzed in the present paper were collected from various sources , and the ( sometimes not well defined ) selection criteria introduce important biases : the distribution of stars with o and mg abundances in our sample is very different from that obtained from a volume limited sample in the solar neighbourhood , and high velocity stars are likely overrepresented amongst the most metal poor ones . in this section we will use this comparison simply to identify stellar populations defined on chemical grounds with those defined from dynamics . the basic data for this comparison are the o , mg and fe abundances drawn from our analysis , and dynamical data determined by edvardsson et al . ( 1993 ) , for stars with [ o / h]@xmath28 ; for stars more metal - poor than this limit in the tomkin et al . ( 1992 ) and zhao & magain ( 1990 ) samples , similar data were obtained using parallaxes and proper motion from hipparcos ( perryman et al . 1997 ) , radial velocities from the literature ( carney et al . 1994 whenever possible ; else data were taken from the simbad database ) , and the code for galactic orbit calculations by aarseth as modified by carraro ( 1994 ) . both these samples only include dwarfs in the solar neighbourhood ; the edvardsson et al . sample is essentially a magnitude - limited sample ( although the magnitude limit is a - known - function of metallicity ) , with no kinematical bias ; while tomkin et al . ( 1992 ) and zhao & magain ( 1990 ) selected high proper motion stars , so that a bias toward high velocity stars is certainly present amongst the most metal - poor stars . the location of the stars with known dynamical data in the ( [ o / h ] vs [ fe / o ] ) diagram is shown in fig . [ fig:3 ] , where we plotted with different symbols stars in different regions of the diagram : metal - poor stars ( [ o / h]@xmath41 : group a ) ; fe - poor o - rich stars ( [ o / h]@xmath28 , [ fe / o]@xmath54 : group b ) ; and fe - rich o - rich stars ( [ o / h]@xmath28 , [ fe / o]@xmath45 : group c ) . the analogous diagram for mg is shown in fig . [ fig:4 ] , but in this case group a are stars with [ mg / h]@xmath55 ; group b are stars with @xmath56[mg / h]@xmath57and [ mg / fe]@xmath54 ; and group c are the remaining metal rich stars . a star - by - star comparison shows that stars are attributed to the same groups when using o and mg . in the four panels of fig . [ fig:5 ] and [ fig:6 ] [ fe / o ] and [ fe / mg ] ratios for these stars are plotted against the rotational velocity around the galactic center , the orbital eccentricity , the maximum height of the orbit over the galactic plane @xmath58 , and the age @xmath59 respectively . the age values are averages of two independent values : * those determined by edvardsson et al . ( 1993 ) and schuster & nissen ( 1989 ) derived using an homogenous procedure from the @xmath60 diagram , calibrated against age using standard isochrones ( _ i.e. _ not o - enhanced ) by vandenberg and bell ( 1985 ) * and those derived using absolute magnitudes from hipparcos parallaxes ( perryman et al . 1997 ) , interpolating within the grid of isochrones by padua group ( girardi et al . 2000 ) ; in this case the enhancement of o and the other @xmath10elements was considered by modifying the fe abundance according the procedure suggested by straniero et al . ( 1997 ) the two age scales are somewhat different ; they were homogeneized to an a common ( arbitrary ) scale before averaging them . absolute ages are affected by various uncertainties related to stellar models , while the relative ranking should be quite reliable , as indicated by the good agreement existing between the two set of age determinations ( see fig .. [ fig:10 ] ) . since basic data for the two age estimates are independent each other , internal uncertainties on the ages may be obtained by the r.m.s scatter of the differences ; in this way , we estimate that typical error bars in log(age ) for individual stars are @xmath61 dex ( i.e. @xmath62% ) . the internal scatter we get for group a and b is of this same order , indicating that small if any internal scatter exists for these groups . the average properties for the three chemical groups are listed in table [ tab:1 ] . the main conclusions we may draw from fig . 1 - 6 and table [ tab:1 ] are : 1 . though some bias may be present , present data support the identification of group a as the halo , of group b as the thick disk ( see e.g. robin et al . 1996 , norris , 1999 , buser et al . 1999 ) , and of group c as the thin disk . 2 . in the framework of homogeneous models for the galactic evolution , the formation of stars in the halo and the thick disk was fast ( a few @xmath17 yr ) , i.e. shorter than the typical timescale of evolution for the progenitors of type ia sne . 3 . given the lack of a clear selection criterion amongst metal - poor stars , our data can not be used to draw any conclusion about clear breaks between the halo and thick disk populations ; the age difference must be small ( i.e. the two populations are virtually coeval ) , else the [ o / fe ] and [ mg / fe ] ratios of the two populations should be different ; the ages derived for both groups are well in excess of 10 gyr . anyway , this discontinuity is supported by other studies ( see norris 1993 ) . as noticed by wyse and gilmore ( 1992 ) , the dynamical properties of the thick disk are clearly indicative of a flattened intermediate population supported by rotation ( these characteristics are drawn from the edvardsson et al . sample , which are not affected by kinematical biases ) ; while the halo space distribution is much wider , and the system appears to be supported by velocity dispersion rather than by rotation . in our analysis , this last result is likely enhanced by the selection biases in the carney et al . sample . further strong supports to the present conclusion that the thick disk is a homogeneous , chemically old population is given by the results by fuhrmann ( 1998 , 1999 ) , and nissen & schuster ( 1997 ) : these authors found that thick disk stars have low [ fe / o ] and [ fe / mg ] ratios ( equal or even lower than those found for halo stars of similar metallicity ) , with very small intrinsic scatter . 4 . there was a sudden decrease in star formation during the transition between the thick and thin disk phases , which are then clearly distinct at least from the chemical point of view . the phase of low star formation must have lasted at least 1 gyr in order to allow for the explosion of the bulk of type ia sne ; however , the hiatus was not longer than 3 gyr , else there would be obvious break in the [ fe / o ] and [ fe / mg ] vs age diagrams . we remark that the age difference between the thick disk and the oldest thin disk stars could have been overestimated in fig . [ fig:5]d and [ fig:6]d , since these ages were derived assuming solar ratios of o and mg to fe . again , results from fuhrmann ( 1998 , 1999 ) and nissen & schuster ( 1997 ) well support this picture . both the [ fe / o ] and [ fe / mg ] ratios for thin disk stars increase with time : in fact there is a significative correlation of these ratios with stellar ages ; the linear regression line are : @xmath46 } = -(0.19\pm 0.06 ) \log t + ( 0.12\pm 0.11),\ ] ] based on 68 stars for o , and : @xmath47 } = -(0.085\pm 0.026 ) \log t - ( 0.024\pm 0.075),\ ] ] based on 164 stars for mg . the persson correlation coefficients are 0.35 and 0.25 respectively : the probability of getting such high correlation coefficients by chance are @xmath63 and @xmath64 . the scatter around the mean lines are compatible with the error bars in ages , [ fe / o ] and [ fe / mg ] values . if the raise of [ fe / o ] at constant [ o / h ] is due to a sudden decrease in star formation during the transition between the thick and thin disk phases , then there should be a corresponding gap at [ fe / o]@xmath65 in the distribution of stars with [ fe / o ] . a cumulative diagram of the distribution of [ fe / o ] and [ fe / mg ] values for stars in the edvardsson et al . sample ( fig . [ fig:7 ] and [ fig:8 ] ) indeed supports this inference . in the rest of this section , we will evaluate the statistical significance of this gap for o ; similar ( though slightly less significative ) results were obtained for mg . to this purpose , we performed tests on the distribution of [ fe / o ] values in that sample , which consists of the brightest _ bona fide _ single stars with @xmath66@xmath67 k , having an absolute magnitude from 0.4 to 2 mag smaller than that of the zero age main sequence at the same @xmath68 colour , in equally spaced bins in [ fe / h ] . generally , populations of each bin were kept similar , but the two most metal - poor bins ( [ fe / h]@xmath55 ) have populations about half those of the others , due to the magnitude limit of the survey . however , the 85 stars for which o abundances are available are nearly uniformly distributed with [ fe / h ] , since we have these data for 42% of the stars with [ fe / h]@xmath29 , and for 79% of the stars more metal - poor than this limit . a @xmath43-test confirms that the distribution of stars having [ fe / o ] values can not be distinguished from a uniform distribution in [ fe / h ] ; anyway , we repeated the following analysis using both a uniform distribution , and distributions obtained by summing a random gaussian distributed term ( representing observational errors ) to the observed [ fe / h ] s . the results are very similar . we first tested the hypothesis that the [ fe / o ] values are distributed uniformly , as expected if a linear dependence of [ fe / o ] on [ fe / h ] holds . we found that this hypothesis can be rejected at a high level of confidence , using both a @xmath43-test and a serie of monte carlo simulations . each monte carlo simulation included 10,000 extractions of 85 [ fe / h ] values ( using both early described approaches ) ; individual [ fe / o ] s were the sum of the values deduced from these [ fe / h ] s using the [ fe / o]-[fe / h ] law , and of a random gaussian distributed term representing the spread due to observational errors and to the intrinsic star - to - star variations . the simulations were repeated for three values of the standard deviation for this term : 0.05 , 0.07 , and 0.10 dex ; the intermediate value was deduced from our error analysis , and it is equal to the standard deviation from the mean value for thick disk stars ; the last one is consistent with the residuals around the best fit relation for thin disk stars . most of the deviation from a uniform distribution in [ fe / o ] is due to the excess of stars with [ fe / o]@xmath69 ( corresponding to the thick disk population ) , and to the lack of stars with @xmath70[fe / o]@xmath71 ( the expected location of the gap ) . it can be noticed that a quadratic dependence of [ fe / o ] on [ fe / h ] gives a better fit to observational data . we then repeated the monte carlo simulations with a similar [ fe / o]-[fe / h ] law . we found that the probability that the very low number of stars in a bin of 0.1 dex centered at [ fe / o]=@xmath72 is due to chance is about 0.024 for a uniform distribution with [ fe / h ] , and 0.013 for a distribution function equal to the frequency distribution ( the exact values depend on the assumed observational errors ; the above mentioned values refer to the less significant cases obtained assuming that present [ fe / o ] values have errors of 0.10 dex ) . a good significance ( chance probability @xmath73 ) is achieved for 0.1 dex bins centered over the range @xmath74[o / fe]@xmath54 ; this suggests that the gap is broader than 0.1 dex . we conclude that present available data support the hypothesis that there is a gap in the distribution of [ fe / o ] as expected from a sudden decrease in star formation during the transition from the thick to thin disk phases ; however we think this test should be repeated using a larger and properly selected sample . the overall run of [ fe / o ] with [ fe / h ] is certainly related to the delayed fe synthesis : however , the interplay between star formation rate , progenitor lifetime , and infall can only be cleared out by detailed modelling of galactic chemical evolution . unfortunately , various aspects of star formation and evolution are still not well understood , so that these models have several free parameters . furthermore , a fully appropriate comparison between abundances in metal - poor stars and models of galactic evolution should require models which include consistent and detailed treatment of both chemistry and dynamics . however , this is still beyond current computational capabilities , and at present dynamics has to be introduced into chemical evolution models parametrically , increasing even more the number of free perameters . in order to explore allowed ranges for some of the relevant quantities , we then computed a large number of single - zone models ( with infall of original unprocessed material ) appropriate for the solar neighborhood ; the code we used is an improved version of that of mf . the main advantage of this approach is the reduction in the number of free - parameters , although results of our comparisons strictly apply only to homogeneous collapse models , and not e.g. to accretion scenarios . we will see in the next section that even with these limitations , results of our comparisons are enough to suggest that the best scenario required to explain the star formation history in the solar neighborhood should include both dissipational collapse and accretion . when comparing the predictions of our models with observations , we considered a wide range of constraints , including present abundances in the interstellar medium , current gas density , star formation and sn rates , the distribution of abundances among g - dwarfs , the run of [ fe / h ] with age , imf , etc ( matteucci 1991 ) , and retained only those models which give an overall match to all of them . we find that in the best cases , r.m.s . values of the residuals of [ fe / o ] s for individual stars around lines representing our models are @xmath75 dex for halo and thick disk stars , and @xmath76 dex for thin disk stars , in good agreement with observational errors , as shown by monte carlo simulations where both errors in [ fe / o ] and [ o / h ] were taken into account ( similar simulations were also used to reject those models providing poor fits ) . predictions about the [ fe / o ] ratios provided by some of these models are plotted in fig . [ fig:3]b and c , overposed to the same observational points shown in fig . [ fig:3]a . on the whole , we found that only models having two distinct infall episodes , the first connected to the halo ( and thick disk ) , and the second to the thin disk , fit observational data ( see also chiappini et al . 1997 for similar models ) ; while models with a single infall episode do not give a good match even when star formation law was changed with time , due to either large accumulation of gas during the phase corresponding to the raise of the [ fe / o ] ratio , or lack of gas at present ( depending on the rate of decay ) . decay of the halo infall rate should be fast in order to reproduce the almost flat run of [ fe / o ] with [ o / h ] in the halo and thick disk : the @xmath37folding time is shorter than or equal to the adopted time delay for type ia sne , i.e. @xmath77 gyr . the halo infall should have contributed no more than 20 - 30% of the present density in the solar neighborhood ( and the fraction in stars should be about half that ) . the decay of the thin disk infall should be rather slow ( @xmath78 gyr , which implies a present infall rate @xmath79pc@xmath80gyr@xmath81 ) , else there would be not enough gas at present , the present interstellar medium should be too metal rich , the observed roughly linear run of [ fe / o ] with [ o / h ] in the thin disk would not be reproduced , and there would be an excess of moderately metal - poor stars ( [ fe / h]@xmath82 ) . this value for the present infall rate is larger than the upper limit deduced from observations ( @xmath83pc@xmath80gyr@xmath81 : mirabel 1989 ) . this inconsistency might be removed by either assuming that star formation law changes with time ( the efficiency should then be larger in the halo than in the thin disk ) , or that the initial mass function ( imf ) is flatter at large masses ( slope @xmath84 ) than that adopted in most of our models ( scalo 1986 ) ( note that in order to avoid o overproduction with this flatter imf , an upper limit of @xmath85 has to be adopted for type ii sne ) . both these alternatives do not contradict basic costraints . models where both infall episodes start at high values at the beginning and then decay ( fig . [ fig:3]b ) , as well as models where the disk infall is delayed with respect to the halo one ( fig . [ fig:3]c ) , fits data quite well . it should be noticed that in the first class of models the star formation must be arbitrarily lowered at the thick - thin disk transition ( that should occur @xmath0 gyr after beginning ) , and there is no hiatus : the raise of the [ fe / o ] ratio at constant [ o / h ] is due to fe production by the large number of type ia sne from the halo - thick disk , while in the meantime only small amounts of o are produced , due to the decreased star formation rate , which barely compensate for the dilution by metal - poor infalling material ( the [ o / h ] ratio starts increasing again when most gas is consumed and the infall rate decays down to small values ) . the less appealing aspect of this class of models is the large fraction of metal - poor stars , that only can be reconciled with the small number of metal - poor g - dwarfs observed in the solar neighborhood by assuming that the scale height is a strong function of metallicity . on the other side , the second class of models naturally yield to a hiatus if a threshold ( surface ) gas density for star formation is adopted ( kennicutt 1989 ) . a threshold gas density might be expected if the processes involved in star formation are self - regulating , and for a low enough surface gas density the feedback mechanisms that regulate the star formation rate break down ( gallagher & hunter 1984 ; elmegreen 1992 ; burkert et al observationally , an approximate value of @xmath86 atoms @xmath87pc@xmath80 has been proposed for this threshold from observation of irregular and hi - rich spiral galaxies ( skillman 1986 ; van der hulst et al . 1987 ) ; this threshold value is close to the current gas density in the solar neighbourhood ( rana & basu 1991 ; kujiken & gilmore 1989 ) , as expected in a self - regulating mechanism . with the above mentioned infall rates , a star formation rate with threshold produces a hiatus in star formation at the end of the halo and thick disk phase ; in our models , star formation in this early phase lasts for 1.5 - 2.5 gyr ( inversely depending on the adopted threshold density ) , and for given yields the final [ o / h ] value depends on the ratio between the halo infall and the threshold density , because the early chemical evolution is essentially that of a closed box ( phillips et al 1990 ) . in these models , type ia sne begin to contribute to nucleosynthesis during the last phases of halo and thick disk evolution , raising the [ fe / o ] ratio ; this contribution continues during the hiatus , which must be at least as long as time delay of sn ia ( @xmath88 gyr ) , in order to raise the [ fe / o ] by at least 0.13 dex , that we estimate as the lower limit given by observations . however , the upper limit for hiatus duration can not be determined from the [ fe / o ] run due to a saturation effect . part of the gas might be lost at the thick - thin disk transition ( due e.g. to a galactic wind induced by sn explosions : silk 1985 ) ; if this occurs , this metal - rich gas will be replaced by more metal - poor infalling gas , and the starting value of [ o / h ] in the thin disk is lower than the maximum achieved during the thick disk evolution . models where up to 75% of the gas is lost at this epoch fit data quite well , but they predict the existence of some very o - poor , fe - rich thin disk stars , which are not present in the observed samples . we think that the overposition of the thick and thin disk sequences over some range in [ o / h ] can be better explained by noting that stars currently in the solar neighborhood likely formed over a range of galactocentric distances ( see franois and matteucci 1993 ) , where the halo surface density and the largest [ o / h ] values achieved during the halo and thick disk evolution were different . note that this is also the explanation favoured by other authors ( e.g. edvardsson et al . what our data tells us about scenarios of galaxy formation ? the fast rate of star formation for the halo and thick disk ( @xmath89 yr ) is consistent with a smooth dissipational collapse ( eggen et al . 1962 ; larson 1974 ) . however , in this case the transition from the thick to thin disk phases should be continuous : some heating mechanisms causing the observed sudden decrease in star formation during this transition is missing and should then be introduced . possible candidates are a high sn rate ( silk 1985 ; berman & suchkov 1991 ) , and merging of smaller galaxies with our own ; these mechanisms might also be acting simultaneously . our tests with models with a single infall episode ( but variable star formation rate ) suggest that a simple heating as that produced by sne can not reproduce the whole spectrum of observations . also , the discontinuity in specific angular momentum between halo and thick disk suggests that there is not a smooth transitions between these two populations ( wyse and gilmore 1992 ) . on the other side , merging with gas poor satellite(s ) having a mass larger than a few hundredths the disk mass may heat ( and destroys ) any pre - existing stellar thin disk ( quinn et al . 1993 , walker et al . 1996 ) and create a thick disk , although this result is not obtained in all simulations ( see e.g. huang & carlberg 1997 ) , so that it seems to depend on the initial conditions as well as on the properties of the satellite : merging of satellites on prograde orbits more likely produce thick disks , while those on retrograde orbits mainly produce disk tilts ( velazquez and white 1999 ) . the mass range is fixed by the amount of kinetic energy to be injected into the disk : merging with a companion of comparable mass would have transformed the milky way into an elliptical , while merging with a very small satellite has only minor effects . if the disk or the satellite contained gas ( as indicated by chemical evolution models ) , merging was likely accompanied by a burst in star formation ; a possible support in favour of such a burst is the presence of a numerous population of globular clusters likely connected to the thick disk , distinct from the population connected to the halo ( zinn , 1985 ) . perhaps this burst exhausted the existing gas , contributing to the present thick disk or it may have been concentrated in the bulge regions , far from the solar neighbourhood ( as suggested by some simulations : mihos & hernquist 1994 ) , or finally the high sn rate might have caused a galactic wind ( berman & suchkov 1991 ) . anyway , the present thin disk should have formed later by a secondary process ( ostriker & thuan 1975 ) ; models of disk evolution ( kennicutt 1989 ; burkert et al . 1992 ) then indicate that some time would be required before the critical density was reached and star formation started again . thick disk stars are then likely older than the oldest stars with thin - disk kinematics and a discontinuity would be expected between the thick and thin disk phases . this agrees with our fig . [ fig:5]d , which also indicates that a similar large merging could not have occurred during the last 10 gyrs , although observations of the sagittarius dwarf galaxy ( ibata et al . 1994 ) , presently merging with the milky way , indicate that minor episodes are still occurring . the n - body simulations ( quinn et al . 1993 ) indicate that if relatively large merging occurred , the present thick disk would be composed of both stars belonging to the merged satellite(s ) and to the original disk ; the last one should dominate due to its larger mass . a pure accretion scenario readily explains the lack of appreciable kinematic and chemical gradients in the outer halo ( carney 1993 ) , but it fails to explain the gradients observed in the inner regions . in a smooth dissipational collapse scenario , like that of eggen et al . , the proposed merging episode appears as an _ ad hoc _ hypothesis . however , more realistic simulations of galaxy formation by dissipational collapse which include dark matter , gas dynamics , star formation and sn feedback ( katz 1992 ) suggest that many stars form in the cores of dark matter clumps that form during the collapse : even within this scheme then a long - living thin disk could likely form only after the end of an early chaotic phase . the emerging favoured scenario considers then the inhomogeneous dissipational collapse of the protogalaxy with formation of a few secondary fragments ( having of the order of several hundredths or even a few tenths of the total mass ) which are accreted later , as proposed by norris ( 1994 ) . in this scenario , a significant fraction of the halo is due to accretion of fragments , explaining its low specific angular momentum ( wyse and gilmore 1992 ) . our contributions to this scheme is the consideration that the similar [ fe / o ] ratios for thick disk and halo stars can best be understood if timescales for both contraction and accretion are @xmath90 gyr ( although some later accretion of low - mass fragments is possible ) , and the suggestion that this scheme might naturally produce a discontinuity between the thick and thin disk phases . up to now , the proposed scenario refers to the milky way . however , it can be easily extended to other galaxies . a strong support to a scenario of spiral formation including both dissipational collapse and accretion is given by the observation that spirals without significant bulges do not have thick disks ( van der kruit & searle 1981a , 1981b ; morrison et al . 1994 , 1997 ; fry et al . 1999 ; matthews et al . 1999 ) . then ( i ) the presence of thick disks and bulges is not an obvious outcome of galactic formation , but rather depend on some mechanism ( e.g. accretion ) that may or may not be active ; and ( ii ) their origins are likely related ( although likely not on an evolutionary sequence , due to the very different specific angular momentum ) . this last assertion agrees with the low [ fe / mg ] ratios for stars in the bulge of our own galaxy ( mcwilliam & rich 1994 ) , which suggest that the difference between the ages of the bulk of stars in the bulge , halo and thick disk is @xmath90 gyr . the presence in the galactic bulge of stars much more metal - rich than the thick disk stars in the solar neighborhood might be understood by assuming either that these stars formed from metal - enriched material in the burst induced by the same merging episode(s ) causing the formation of the thick disk ( mihos & hernquist 1994 ) ; or simply by a pre - existing radial metallicity gradient in the early disk , which is predicted by dissipational collapse models ( larson 1974 ) and should not be canceled by later merging episode(s ) ( quinn et al . 1993 ) . within the mixed scenario , significant thick disks and bulges are related to accretion of satellites . currently available statistics ( zaritsky et al . 1993 ) indicate that there is about one satellite with @xmath91 per primary , though we notice that the milky way has a much larger number of faint companions and the statistics is based on regions where galaxy density is lower than in the local group ; furthermore , there is an excess of close satellites ( separation @xmath92 kpc ) near the minor axis of primaries ( holmberg 1969 ; zaritsky et al . 1997 ) , and it has been suggested that the present population of satellites to the milky way represents only a small fraction of the original population ( see e.g. klypin et al . 1999 ) . the excess of close satellite near the minor axis may be explained by assuming that close satellites on low inclination prograde orbits have smaller chance to survive , due to dynamical friction ; it has been suggested that these _ missing _ satellites have merged into the primary ( zaritsky & gonzalez 1999 ) ; their perturbation may have created the thick disk , and their gas may have fueled the bulge . given the small number of satellites for each primary galaxy , a strong stochastic variation from galaxy to galaxy is expected . the present scenario is then coherent to a picture where the entire hubble sequence from sc to ellipticals might be reproduced by assuming an increasing importance of accretion , which should be correlated with the density of galaxies in the local environment ( see e.g. schweizer 2000 ) . berman , b.g . , suchkov , a.a . 1991 , ass , 184 , 169 boesgaard , a.m. , king , j.r . , deliyannis , c.p . , vogt , s. 1999 , aj , 117 , 492 burkert , a. , truran , j.w . , hensler , g. 1992 , apj 391 , 651 buser , r. , rong , j. , karaali , s. 1999 , a&a 348 , 98 carney , b.w . 1993 , in the globular cluster - 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the accurate o , mg and fe abundances derived in previous papers of this series from a homogenous reanalysis of high quality data for a large sample of stars are combined with stellar kinematics in order to discuss the history of star formation in the solar neighborhood .
we found that the fe / o and fe / mg abundance ratios are roughly constant in the ( inner ) halo and the thick disk ; this means that the timescale of halo collapse was shorter than or of the same order of typical lifetime of progenitors of type ia sne ( @xmath0 gyr ) , this conclusion being somewhat relaxed ( referring to star formation in the individual fragments ) in an accretion model for the galaxy formation .
both fe / o and fe / mg ratios raised by @xmath1 dex while the o / h and mg / h ratios hold constant during the transition from the thick to thin disk phases , indicating a sudden decrease in star formation in the solar neighbourhood at that epoch .
these results are discussed in the framework of current views of galaxy formation ; they fit in a scenario where both dissipational collapse and accretions were active on a quite similar timescale .
12c13@xmath2c/@xmath3c enth
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there is currently much interest in performing ancestral inference from molecular population genetic data . to facilitate this inference , there has been an explosion of research in developing computationally efficient methods . these techniques are designed either to compute the likelihood , for maximum likelihood estimation , of a sample of genes or for deriving the posterior distribution on parameters in coalescent models , which describe the ancestry of the genes . broadly speaking there are three main approaches to inference in molecular population genetics : ( i ) importance sampling for likelihood evaluation , whose application in population genetics was pioneered by ( griffiths & tavar , 1994a , b , c ) ( ii ) markov chain monte carlo methods ( e.g. kuhner et al . ( 1995 ) , wilson & balding ( 1998 ) ) ( iii ) approximate bayesian computation ( abc ) ( del moral et al . ( 2009 ) , marjoram et al . ( 2003 ) ) . see stephens ( 2004 ) for a review . in this paper we concentrate on likelihood - based methods . molecular data have a sampling distribution which is a mixture over possible ancestries . the state space of the ancestries is huge and closed - form expressions are available only in the simplest cases . the objective is to calculate a parameter @xmath0 ( @xmath1 ) such that @xmath2 for some observed genetic data @xmath3 , parameter @xmath4 , probability density @xmath5 on @xmath6 and @xmath7 an integrable function . note that @xmath8 is typically the genetic types of a random sample of chromosomes . in addition , @xmath9 denotes the coalescent history , i.e. the set of ancestral configurations at the embedded events in a markov process where coalescence , mutations or other events take place . @xmath10 denotes the current state , while @xmath11 is the state when a singleton ancestor is reached . statistical inference associated to @xmath12 can be regarded as a missing data problem and could , in principle , be tackled by the em algorithm ( dempster et al . 1977 ) and its monte carlo extensions ( e.g. fort & moulins ( 2003 ) ) . however , @xmath13 , the stochastic tree , can be computationally expensive to simulate and such techniques are typically avoided . for example , for the coalescent ( kingman , 1982 ) and ancestral recombination graphs ( e.g. fearnhead & donelly ( 2001 ) ) , the standard approach is to use is ( de iorio & griffiths , 2004a ; griffiths & tavar , 1994a ; stephens & donelly , 2000 ) and smc methods ( chen et al . 2005 ) to approximate . these approximations are usually computed on a discrete grid @xmath14 and the estimate of @xmath15 corresponds to the largest approximated likelihood on @xmath16 . see also olsson & rydn ( 2008 ) for an alternative procedure for state - space models . techniques such as abc and composite likelihood ( wiuf , 2006 ) do not give solutions which are exact w.r.t . the original model whilst , when possible , exact inference is of interest . this is because , given a reasonable stochastic model , the approach allows investigators to exactly ( up - to a numerical error ) average over the uncertainty in the tree structure when estimating genetic parameters of interest . one of the main drawbacks of existing exact is / smc schemes is the simulation of the tree backward in time , from observed data , until the tree coalesces . in many scenarios , especially for large data sets , when getting close to the top of the tree , it often takes a long time to coalesce . this is due to genetic parameters ( e.g. mutation rates ) that can be very large relative to the size of the data . consequently , it can take a very long time to simulate the tree back to the mrca . as a result , the variance of the estimate of the likelihood can be higher than is desirable , along with long cpu times . it should be noted that the calculation of the likelihood at these points , @xmath4 , can be inferentially important . in addition , it is seldom possible to speed up the simulation via importance sampling as the variance of the weights can become too large . that is , by adapting the parameter of the proposal to lead to a fast coalescence , the discrepancy between the true process and the proposal leads to a very inefficient algorithm w.r.t . variance . the approach proposed in this paper is based on is . stephens & donelly ( 2000 ) proposed a way to use is efficiently to simulate ancestral trees by characterizing an optimal proposal distribution and similar methods have since been developed for a variety of genetic scenarios ( e.g. de iorio & griffiths ( 2004a , b ) ) . the basic idea is to define an efficient proposal distribution on ancestral histories which allows us to reconstruct markov histories backwards in time from the sample @xmath8 to an mrca . we introduce a stopping time in the is proposal , backward in time , to stop the simulation before the mrca is reached . then using a simple stopped identity , forward in time we are able to characterize the bias introduced in the evaluation of the likelihood due to stopping the simulation of the stochastic tree . the bias can be understood by considering two aspects : 1 . the underlying mixing of the evolutionary process[pt : evpr ] 2 . the last exit time distributions on the process . [ pt : lepr ] in the context of ( [ pt : evpr ] ) , the idea is that for many models , close to the top of the tree , the process is able to forget its initial condition . as a result , stopping the simulation is reasonable , because the place where it is stopped is forgotten by the process forward in time ; we formalize these ideas later on . in reference to ( [ pt : lepr ] ) , the more information there is on the true marginal distributions of the process , the more it is possible to reduce the bias . ideas from the theory of population genetics models ( ethier & griffiths 1987 ; ewens , 1972 ) will be used to achieve the latter . in reference to a comment of edwards ( 2000 ) , our method is termed the ` time machine ' . this is because , estimation is performed saving the simulation time of going all the way back in time to the mrca . a similar idea , in the context of filtering , can be found in the work of olsson et al . ( 2008 ) and also in option pricing avramidis & lcuyer ( 2006 ) . in our context , we have a simpler underlying process than in filtering , but the ergodicity conditions considered there do not apply here . the mixing conditions that they require only apply locally and thus the proofs have to be modified . recall that approximate tools for inference from stochastic trees ( e.g. del moral et al . ( 2009 ) , meligkotsidou & fearnhead ( 2007 ) , tavar et al . ( 2000 ) ) are available . however , our approach is ` less approximate ' , in that our point - wise estimate of the likelihood is significantly less - biased , but costing more in computational - time . this paper is structured as follows . in section [ sec : motex ] we introduce a motivating example , the coalescent model , which will help to illustrate our ideas . in section [ sec : stopsimos ] our methodology is described ; section [ sec : bias ] features an analysis of the bias of the approach ; section [ sec : simos ] presents a simulation study to demonstrate the performance of our algorithm and we conclude the paper in section [ sec : summary ] . appendix 1 contains some proofs , appendix 2 details of our numerical implementations . our ideas are illustrated in the context of the coalescent . however , the formulation is kept as general as possible , as the framework can be extended to other tree models , such as the infinite sites model . in appendix 3 we show how this can be done . the coalescent model is used as a motivating example for our work . some notations are first introduced . in particular , we consider the case in which the type space @xmath17 for the collection of the @xmath18 genes / chromosomes is finite and the only genetic process of interest is mutation . denote by @xmath19 a measurable space . for two @xmath20finite measures @xmath21 and @xmath22 mutual absolute continuity is written @xmath23 and the radon - nikodym derivative as @xmath24 . given a markov kernel @xmath25 $ ] , let @xmath26 , ( the dirac measure ) and write the composition for @xmath27 as @xmath28 , with a corresponding composition of inhomogeneous kernels as @xmath29 . write @xmath30 as the indicator of a set . for @xmath31 @xmath32 denotes the class of stochastic matrices for which there exist a stationary distribution @xmath33 . the collection of bounded and measurable function are denoted @xmath34 . the supremum norm is written @xmath35 . the total variation distance between two probability measures @xmath21 and @xmath22 on @xmath19 is @xmath36 . given a probability measure @xmath37 , and a @xmath38 , the product measure is written @xmath39 , @xmath40 . the vector notation @xmath41 is adopted . in addition , let the @xmath42-dimensional vector @xmath43 where the 1 is in the @xmath44 position . the @xmath45 norm of a vector is written @xmath46 . for @xmath47 , @xmath48 . define the tree model on the measurable space @xmath49 , with @xmath50 . let @xmath51 . the basic idea is to maximize , w.r.t @xmath4 , the quantity @xmath52 where the observed data is @xmath53 , @xmath54 , normally the identity , for @xmath55 @xmath56 and @xmath57 for some @xmath58 and @xmath59 depending upon the model under study . in all of our examples , @xmath60 corresponds to the density of a non - decreasing ( in some sense ) markov process in discrete time , stopped at a random time @xmath61 ; that is @xmath62 throughout the article it is assumed that @xmath63 , i.e. that the stopping time is a.s . finite w.r.t @xmath5 . the stopping time will be determined by the first time that the tree is of ` size ' @xmath64 . introduce an absolutely continuous distribution @xmath65 on @xmath6 and sample @xmath66 according to @xmath65 , then the is estimator of @xmath12 is @xmath67\ ] ] where @xmath68 and @xmath69 the empirical measure of the simulated samples . denote the number of genes of type @xmath70 at event @xmath71 of the process as @xmath72 , with @xmath73 . the objective is to find the genetic parameters @xmath74 where @xmath75 and @xmath76 , @xmath77 . @xmath78 is the mutation rate per chromosome per generation and mutations along the edges of the tree occur according to a markov chain with transition matrix @xmath79 . the various components of the identity ( [ eq : likelihood_id ] ) for the coalescent model are defined as : @xmath80 with @xmath81 the identity function , @xmath82 and finally , @xmath83 where @xmath84 @xmath85 and @xmath86 write @xmath87 ( here @xmath88 is counting measure ) . note that for any fixed @xmath51 , @xmath4 , @xmath89 . for simplicity of exposition , the results are given with only mutation . however , they can be easily extended to the case of migration as well ( e.g. de iorio & griffiths ( 2004b ) ) . to compute the likelihood , for a given @xmath4 , importance sampling is adopted . an importance distribution , @xmath65 , is introduced to simulate the tree backward in time to the mrca ; this ensures that the data is hit . in details , let @xmath90 denote the reverse chain backward in time and write @xmath91 instead of @xmath13 ( this convention is used throughout the article , see also figure [ fig : coalgraph ] ) . let : @xmath92 for some markov transition @xmath93 ; see stephens & donelly ( 2000 ) for the optimal @xmath65 . then the likelihood is @xmath94 the simulation proceeds by sampling from @xmath95 and computing the weight @xmath96 simulations backward in time are carried out until we reach the mrca , i.e. when there is only one individual in the sample . this procedure is repeated @xmath97 times to provide a monte carlo estimator for the likelihood @xmath98\ ] ] where @xmath99 are the simulated samples , @xmath100 for every @xmath101 and @xmath102 this can be repeated for many @xmath103 using a driving value ( griffiths & tavar , 1994 ) or bridge sampling ideas ( e.g. fearnhead & donelly ( 2001 ) ) . in addition , to deal with the problem of weight degeneracy ( e.g. doucet et al . ( 2001 ) ) resampling steps can be added . see , for example , chen et al . ( 2005 ) . ( 1,1 ) ( .5,.75)(0,1)0.25 ( 0.25,.75)(1,0).5 ( 0.75,.75)(0,-1)0.5 ( 0.25,.75)(0,-1)0.1 ( 0.125,.65)(1,0)0.25 ( 0.125,.65)(0,-1)0.65 ( 0.375,.65)(0,-1)0.15 ( 0.3125,.5)(1,0)0.125 ( 0.3125,.5)(0,-1)0.5 ( 0.4375,.5)(0,-1)0.5 ( 0.625,.25)(1,0).25 ( 0.625,.25)(0,-1).25 ( 0.875,.25)(0,-1).25 ( 0.1,0.5)(0.1,0)8(1,0)0.05 ( 0.95,.95)@xmath104 ( 0.01,.95)@xmath105 ( 0.95,.75)@xmath106 ( 0.95,.65)@xmath107 ( 0.95,.5)@xmath108 ( 0.95,.25)@xmath109 ( 0.95,.01)@xmath110 ( 0.01,.5)@xmath111 ( 0.01,.25)@xmath112 ( 0.01,.01)@xmath113 ( 0.25,.45)@xmath114 ( 0.7,.45)@xmath115 it is now detailed how we stop the simulation of the stochastic tree back in time before the mrca is reached . in the next section we provide theoretical results and connections to the theory of smc are established . for the purpose of stopping the simulation , introduce two stopping times ( forwards in time ) : the first hitting time of the set @xmath116 @xmath117 and some stopping time @xmath118 associated to the hitting of a set @xmath119 @xmath120 such that @xmath121 where @xmath122 is the @xmath123probability . for example , in the context of the coalescent , it is suggested to take , for @xmath124 @xmath125 let @xmath126 denote the expectations w.r.t the process @xmath127 . then the likelihood ( [ eq : likelihood_id ] ) can be written as @xmath128\ ] ] and applying the strong markov property we have @xmath129\;\ ] ] that is , @xmath130 dz_{\alpha } \label{eq : stoplike}\end{aligned}\ ] ] where @xmath131 the equation ( [ eq : stoplike ] ) will be the starting point for constructing our biased estimates of the likelihood function . consider the coalescent model . specifically , define , for @xmath132 the stopping time @xmath118 @xmath133 which is the first time the forward process has @xmath134 individuals . using equation ( [ eq : stoplike ] ) , we have @xmath135dz_{\alpha } \label{eq : coallikestp}.\ ] ] in words this means that to have @xmath134 chromosomes , we need a minimum of @xmath136 steps in the process and @xmath137 has to be at least @xmath138 steps . in this case , write @xmath139 note this is well - defined due to the fact that the size of the population is non - decreasing , and then , for any @xmath140 @xmath141 where @xmath142 that is , given @xmath118 , the distribution of the chromosome counts at the first entrance time of @xmath143 can be written as the composition of : * the distribution of the counts at the last exit time from @xmath144 * and the markov transition . returning to the likelihood ( [ eq : coallikestp ] ) and making the substitutions , @xmath145 , @xmath146 , it thus follows that @xmath147 here @xmath148 is the time from the last time there are @xmath136 chromosomes to @xmath64 chromosomes . now set @xmath149 in other words the simulation is stopped the first time there are @xmath136 chromosomes . our approximation of the likelihood is then @xmath150 on the basis of the above analysis , it is then clear that if @xmath151 then the approximation of the likelihood is exact . that is , to minimize the bias an approximation of the true distribution of the counts at the last time there are @xmath136 chromosomes should be used . the ideas and notation are clarified in figure [ fig : coalgraph ] . in our biased simulation , using the decomposition , the procedure will approximate @xmath152 where our notation is such that : * @xmath153 is the time reversed process * @xmath114 is a first hitting time associated to @xmath153 * @xmath154 an approximation of a marginal probability . we begin by giving a simple result on the error bounds for smc algorithms . the result applies to the standard is algorithms , for example in de iorio & griffiths ( 2004b ) , stephens & donelly ( 2000 ) , and for the smc algorithms as in chen et al . the simulation is to be performed backward in time , as in section [ sec : likelihood_computation ] . the ideas here are adapted from the theory of del moral ( 2004 ) . the biased estimates are denoted as @xmath155 , @xmath156 , where @xmath157 depends upon whether is or smc is implemented . for example , in the is case : @xmath158 where @xmath159 is such that @xmath160 is the set associated to @xmath161 ( @xmath162a.s . ) , @xmath163 and @xmath164 . below expectations w.r.t the stochastic process that is simulated by the algorithm are written as @xmath165 and it is assumed @xmath166 [ prop : lpbound ] for any @xmath51 , @xmath167 , @xmath4 , @xmath168 , there exists a @xmath169 such that : @xmath170^{1/p } \leq \frac{b_{p , n}(\theta)}{\sqrt{n } } + |l_b(y_{1:n};\theta ) - l(y_{1:n};\theta)|.\ ] ] * remark*. _ the result shows the standard variance - bias type decomposition . that is , @xmath171 can be thought of as a bound on the variance and @xmath172 is the bias . our estimate converges to @xmath173 , and it is sought to control the bias term , which , in our case can be approximately written in the form @xmath174(p_{1:k}(f))| \label{eq : biasdecomp}\ ] ] for @xmath175 two probability measures and @xmath176 a sequence of non - homogenous markov kernels ( @xmath103 is suppressed on the r.h.s ) . _ a simple technical result is now given which shows how to control the bias term ( [ eq : biasdecomp ] ) . some assumptions are now made , that can be satisfied by many stochastic tree models . introduce a sequence of time inhomogeneous markov kernels @xmath177 , on space @xmath178 and a sequence of sets @xmath179 . [ hyp : p_nassump ] _ stability of @xmath177 . _ * \(i ) * initial probability measures*. @xmath175 are concentrated on @xmath180 . * \(ii ) * absorption of @xmath177*. for every @xmath181 , @xmath182 we have @xmath183 * \(iii ) * local mixing of @xmath177*. for every @xmath181 , there exist @xmath184 , @xmath185 concentrated on @xmath186 , such that for all @xmath182 @xmath187 the assumption ( a[hyp : p_nassump ] ) ( which is comprised of ( i)-(iii ) ) will refer to the fast mixing of the process close to the top of the tree . the absorption type assumption refers to the birth process associated to coalescent type chains . [ prop : biascontrol ] assume ( a[hyp : p_nassump ] ) . then , for any @xmath188 , define : @xmath189 and we have @xmath190p_{1:k}\|_{tv } \leq \vartheta_k\|\lambda_1-\lambda_2\|_{tv}.\ ] ] * remark 1*. _ the result helps to bound the bias as @xmath191(p_{1:k}(f))| \leq \|f\|_{\infty}\|[\lambda_1-\lambda_2]p_{1:k}\|_{tv}.\ ] ] essentially , the fast mixing of @xmath192 within the domain it is constrained to allow the composition of kernels to forget its initial distribution at an exponential rate . in addition , as in olsson et al . ( 2008 ) , assuming @xmath193 is uniform in @xmath194 , the benefits of stopping , in terms of variance / bias trade off can be substantial . _ * remark 2*. _ one point of interest in the sequel is that , if the mixing condition ( a[hyp : p_nassump ] ) does not hold , it is possible to establish a similar bound when the initial measures @xmath21 and @xmath22 are similar . that is to say , when @xmath23 and @xmath195 such that @xmath196 this is unsurprising as it implies that if the kernels do not mix , we need to ` match ' @xmath21 and @xmath22 for the bias to be small . _ ( a[hyp : p_nassump ] ) is now discussed in the context of the coalescent . note that the results follow , with some extra work , for coalescent processes with migration . readers interested in how the method may be applied can skip to section [ sec : simos ] , with no loss in continuity . suppose that the transition matrix @xmath79 satisfies , for any @xmath197 , @xmath198 and probability @xmath199 , @xmath200 @xmath201 this condition implies that @xmath79 mixes extremely quickly . let @xmath202 ; this corresponds to the space of @xmath203 . also let @xmath204 . it is clear that @xmath205 : since we start with at most 3 chromosomes and the most possible after 3 steps is 6 . now it can be seen that , for any @xmath206 @xmath207 and @xmath208 with @xmath209\bigg(\frac{\mu}{\mu+2}\bigg)^2\bigg]^3.\ ] ] here the minorising probability @xmath210 puts all its probability on having 3 chromosomes . then it can be subsequently seen that @xmath211 satisfies condition ( [ eq : mixingcond ] ) , with @xmath212 and so fourth . in effect the condition ( [ eq : mixingcond ] ) holds with @xmath213 ; that is , the closer to the top of the tree we stop , the faster the process will mix forward in time . as a result , to bound the bias we can write it , approximately , in the form , for @xmath214 @xmath215\ ] ] with @xmath21 as in ( [ eq : lastexit ] ) , @xmath216 , @xmath217 , @xmath218 is associated to the fact that we need to iterate the kernels to satisfy ( [ eq : mixingcond ] ) and @xmath219 . @xmath220 is an integer big enough ( say @xmath221 ) where we suspect that the possibility of generating a tree of length @xmath194 and hitting the data is extremely small , so we can neglect the upper term . thus , approximately , the bound shows that the bias falls geometrically as we stop closer to the top of the tree . note , however , it can not go to zero unless @xmath21 and @xmath22 are equal . to an extent , finding good approximations is more difficult than being able to stop the tree , which is why we focus on this . * remark 1*. _ the result given here mirrors one proved by donelly & kurtz ( 1999 ) for fleming - viot models . in theorem 9.4 of that paper they show that the particle process is uniformly ergodic , if the mutation process is . this is very similar to the property established above . _ * remark 2*._the information , in terms of when to stop the simulation , that is contained in the bound on the bias is as follows . if the mutation process mixes quickly , as above , then the bias falls at a geometric rate : we should stop the simulation when the process starts to mutate many times . this could be measured in terms of the effective sample size ( e.g. liu ( 2001 ) ) , if trees are simulated in parallel , or alternatively , if @xmath222 , for @xmath223 a large multiple of the current size of the tree . _ * remark 3*. _ in terms of the expression @xmath224 , one could adopt a parent - independent mutation ( pim ) marginal . if we have @xmath225 where @xmath226 is the transition for the pim , and the mutation vector is @xmath227 , then ideas from perturbed markov chains ( e.g. mitrophanov ( 2005 ) ) can be adopted to determine a quantitative bound . we are currently investigating a meaningful bound . _ to illustrate our approach , we consider three simulation scenarios : two pim models and one parent dependent mutation model ( pdm ) . the two pim models , denoted pim 0.5 - 0.5 and pim 0.1 - 0.9 are based on the following per - locus transition matrices : @xmath228 while the per - locus mutation probability matrix underlying the pdm model is @xmath229 in all three scenarios , the initial population was set to 100 sequences and we considered a single - locus case ( @xmath230 with 2 possible types ) . for the pdm model only , we also considered the case of 10 loci ( @xmath230 @xmath231=1024 different types ) . irrespective of the number of loci considered , the distribution of the 100 initial sequences among the different types was sampled from a multinomial distribution with a probability vector @xmath79 defined as the invariant point , solution of equation , @xmath232 , where @xmath194 denotes the number of loci considered and @xmath233 is the full mutation probability matrix . the algorithm description is given in appendix 2 . for the function @xmath234 , we use the distribution of an un - ordered sample from a pim model ( which , even for the pim cases , is not the correct distribution in the bias term ) . simulations were carried out until there were @xmath235 sequences left in the population . @xmath236 exactly corresponds the approach in stephens & donelly ( 2000 ) and is subsequently referred to as @xmath237 . for each simulation , we examined 60 values for @xmath78 ranging from 0.1 to 30.1 . we report in figure [ graphe_distrib ] the estimated log - likelihood distribution , based on 100,000 samples for all four simulations scenarios ( presented in lines ) and three values of @xmath78 ( presented in columns ) . estimated distribution of the likelihood for the four simulation scenarios and for three values the mutation rate @xmath238 and @xmath239 . in each model , results are presented for the six stopping times in the simulation of the genealogical tree . plots are based on 100,000 samples . ] in figure [ graphe_distrib ] it is clear that as expected , uniformly across the values of @xmath78 , the closer to the mrca the algorithm is stopped , the more accurate the distribution of the likelihood is estimated . however , up to tm 10% ( and even tm 25% for @xmath240 ) our results suggest that the time machine approximation and correction provides an accurate estimate of the distribution of the likelihood . conversely , when the algorithm is stopped too early ( tm @xmath241 25% ) the biased estimator underlying the time machine approach leads to very inaccurate estimates of the likelihood . for even more extreme cases ( tm 50% for @xmath242 ) , this results in a highly shifted estimated distribution of the likelihood . the above observations are also reflected in the mean likelihood ( figure [ meanl ] ) . for every model considered here , the simulations of the time machine up to tm 25% seem to provide estimates of the mean likelihood that are similar to the sd approach , although for larger values of @xmath78 , tm 25% seems to overestimate the mean likelihood . furthermore , the time machine approach seems to accurately locate the value of @xmath78 maximizing the likelihood for tm@xmath243 10% , and to provide acceptable approximations for for this when tm@xmath244 10% , regardless of the simulation scenario . estimated likelihood for the four simulation scenarios as a function of the mutation rate @xmath78 . plots are based on 100,000 samples . ] in figure [ meant ] , the average computation time per iteration is plotted as a function of @xmath78 for the pdm-10 loci simulations . results for all other models led to the same conclusions and are therefore not shown . from this figure , the computation time appears to be a linearly increasing function of @xmath78 : increasing the mutation rate naturally decreases the probability of simulating a coalescent event and therefore tends to increase the time to reach the mrca ( or any population size ) . however , it seems that stopping the simulation when there are only more than 5 sequences left in the population drastically reduces the computation time : for tm 5% the simulation run is on average more than twice as fast as the sd simulation , and for tm 25% , the time machine is more than 3 times more time - efficient than the sd algorithm . it should also be noted that ` large ' values of @xmath78 ( around 10 ) , for which the time savings are most significant , also seem to be inferentially important ( see the fourth panel of figure [ meanl ] ) . in figure [ fig : relative_sd ] the relative standard deviation across our 100 repeats of the algorithm , of the time machine to sd are plotted for all the scenarios considered . it can be seen , as expected , that there is some variance reduction and , for example for the tm 5% pdm , the variance reduction is of the order 1.5 . average computation time as a function of the mutation rate @xmath78 . figures are based on 100,000 samples . ] relative standard deviation across 100 repeats of the time machine to sd . figures are based on 100,000 samples . ] on the basis of our experiments , combining both computational efficiency and the numerical accuracy , the use of the time machine with tm 5% is an efficient alternative to the sd algorithm . the c++ code is available upon request from the third author . in this paper we have considered a new approach for simulation of stochastic trees and likelihood calculation of sample probabilities in population genetics models . the approach consists in stopping the backward simulations before the top of the tree is reached . we have provided theoretical results on the bias introduced in the estimation of the likelihood . some extensions to our work are described below . firstly , to extend our analysis to different models . the paper has been written to facilitate such analysis and we believe it is rather simple to deal with other stochastic tree models . also , some further empirical investigations would help support the simulations and theoretical analyses presented here . our methodology would be further enhanced with gpu technology ( e.g. lee et al . ( 2010 ) ) , and this is one area that we are currently investigating . secondly , to look at the consistency ( in a likelihood sense ) of our biased monte carlo estimator . as we observed in section [ sec : simos ] , it appears that the time machine seems to recover the maximum likelihood estimator . therefore consistency , or potential asymptotic bias is of genuine interest . there are very few results in the context of consistency , due to the dependency in the data , after integrating out the tree . that is , it is difficult to apply uniform laws of large numbers to complex dependency structures . none - the - less , we suggest the work of douc et el . ( 2004 ) , fearnhead ( 2003 ) , olsson et al . ( 2008 ) , olsson & rydn ( 2008 ) as possible starting points for a proof . thirdly , the time machine can be used in the context of markov chain monte carlo ( mcmc ) . if one is interested in bayesian parameter inference , then a stopping - time smc algorithm can be used within an mcmc algorithm ( particle mcmc ( andrieu et al . significant time savings per iteration can be gained by using the time machine ; see jasra & kantas ( 2010 ) for some details . we thank prof . arnaud doucet for some valuable conversations related to this work . we give the proofs of propositions [ prop : lpbound ] and [ prop : biascontrol ] . in the case of is , the result follows by adding and subtracting @xmath173 applying minkoswki and the marincinkiewicz - zygmund inequality . in the case of the smc algorithm the proof follows from the fact that the algorithm approximates a multi - level feynman - kac formula ; see chapter 12 , proposition 12.2.3 del moral ( 2004 ) . note that this point is apparently over - looked in chen et al . ( 2005 ) , and such a result helps to verify the convergence of the algorithm . in addition , note that the proposition 12.2.3 of del moral ( 2004 ) does not depend on the importance weights being upper - bounded by 1 . hence , due to the boundedness of the weights , the same proof as for is applies , except the @xmath245 bound for particle approximations of feynman - kac formulae is used instead of the marincinkiewicz - zygmund inequality . the proof is fairly simple and combines the proof of lemma 3.9 and theorem 4.1 of le gland & oudjane ( 2004 ) ( see also theorem 3.1 of tadi & doucet ( 2005 ) ) . the idea is to use the contraction property of the total variation distance and hilbert metric , as well as the relation between the two ( see lemma 6.1 of tadi & doucet ( 2005 ) ) . the only real complication is using the local mixing condition ( [ eq : mixingcond ] ) to derive a bound on the radon - nikodym derivatives @xmath246 consider @xmath247 , clearly @xmath248 in addition @xmath249 since @xmath250 it follows that @xmath251 the proof can then be concluded by following the arguments of le gland & oudjane ( 2004 ) , lemma 3.9 and theorem 4.1 . let @xmath252 , @xmath253 , the population size within each of the @xmath42 states at time @xmath81 . the algorithm will simulate backward in time genealogical trees for an initial population , @xmath254 the ( @xmath255dimensional ) counts associated to the observed data , until there are @xmath256 sequences left in the population . the case where @xmath256=@xmath257 corresponds to ordinary coalescent and @xmath258 to the time machine . most of the notations can be found in sections [ sec : motivation ] and [ sec : id_interest ] . for any generation @xmath81 , there are @xmath259 sequences left in the population , the following steps will be iterated until @xmath259=@xmath256 : 1 . sampling the type of the offspring sequence ( @xmath70 ) with probability @xmath260 2 . getting the type of the ancestor sequence ( @xmath71 ) . + a sequence of a given type @xmath70 can have arisen from an ancestor sequence of type @xmath71 through : 1 . a coalescent event with a probability proportional to @xmath261 2 . a @xmath71 to @xmath70 mutation event ( inclusive of self mutations , from type @xmath70 to type @xmath70 ) , with probability proportional to : @xmath262 where @xmath263 3 . updating the population sizes within each type . + @xmath264 4 . calculate the contribution to the likelihood of the simulated event ( suppressing the subscript @xmath103 ) + @xmath265 where @xmath266 and @xmath267 5 . updating the log likelihood + @xmath268 6 . assessing the stopping criterion . + when the time machine is used ( @xmath230 @xmath258 ) , steps 1 to 5 are repeated until @xmath269 . otherwise , when the full tree is simulated ( @xmath256=@xmath257 ) , steps 1 to 5 are repeated until there are 2 sequences left in the population . then , mutations are simulated until both remaining sequences are of the same type , based on the following three steps : 1 . choose one of the two sequence , of type @xmath70 , with probability 0.5 . 2 . simulate the mutation event from an ancestor of type @xmath71 ( to type @xmath70 ) according to the probability defined in equation ( [ p_mut ] ) , and setting the coalescent probability to 0 . calculate the corresponding weight for the sampled @xmath71 to @xmath70 simulated transition . at this final stage there are only two individuals in the population ( @xmath270 ) , hence @xmath271 , and @xmath272 , then : @xmath273 when the final generation is reached @xmath274 , and @xmath275 for any other iterations in that step . the weight for each generation , derived from is then defined as : @xmath276 4 . update the likelihood according to equation ( [ update_l ] ) this step is specific to the time machine ( i.e. if @xmath277 ) . recalling that @xmath114 is the generation at which the iterative algorithm was stopped , the bias induced by stopping the simulation before reaching the mrca is estimated as : @xmath278 where @xmath279 denotes the gamma function . the likelihood of the tree is then updated @xmath280 the above algorithm is independently repeated @xmath97 times , the estimate of the log - likelihood is @xmath281 where @xmath282 is the value of the final weight for sample @xmath70 and @xmath283 . we now consider our results in the context of the infinite sites model . we concentrate upon likelihoods associated to rooted genealogical trees ; see ethier & griffiths ( 1987 ) or griffiths & tavar ( 1995 ) for more details . the model is based upon the simulation of distinct dna sequences , and the multiplicity of the sequences . in more details , the simulation begins with a single dna sequence @xmath284 , and counts @xmath285 . the process can then undergo a mutation ( rate @xmath78 ) or a split . if a mutation occurs ( to the first sequence say ) we have the new state @xmath286 , @xmath287 and @xmath288 , otherwise the new state is @xmath284 and @xmath289 . the key point is that new mutations introduce a new site ( that is a new integer number ( which is larger than all others currently present ) to the start of a selected sequence ) and hence dna sequence , whilst splits only increase the number of an existing sequence . the state - space consists of the @xmath255distinct sequences ( vectors of potentially different length sequences @xmath290 ) and the respective counts @xmath291 of the sequences that have been simulated . that is , in the previous notation @xmath292 the simulation stops , as before , when @xmath293 . in general , transitions are governed by the following markov kernel . a mutation ( rate @xmath78 ) , at time @xmath294 , of the @xmath295 sequence occurs with probability @xmath296 and a split of the @xmath295 sequence occurs with probability @xmath297 see ethier & griffiths ( 1987 ) and griffiths ( 1989 ) for details on the transition dynamics . in this scenario , the state - space is more complicated . let @xmath298 here @xmath299 are the lengths of the distinct sequences , and the ordering constraint notes that the discovery of a new site is added to the beginning of the segment vector . in addition , let @xmath300 then @xmath301 there are three trans - dimensional aspects to the state - space ; the time to simulate @xmath64 sequences ; the number of distinct sequences and the respective lengths of the distinct sequences ( which is determined in part by the first two aspects ) . for the infinitely - many - sites model , we will use the idea of the first time the number of segregating sites is @xmath136 ( or mutations here ) to stop the simulations backward in time . in a similar manner to section [ sec : stopcoal ] , it can be established that we want the approximating function @xmath302 to be the marginal of the process at the last time we have @xmath136 segregating sites . in the context of the infinitely - many - sites model , the bias is controlled by our ability to approximate this marginal ( see remark 2 in section [ sec : contrbias ] ) . this is because the markov transitions can only change the multiplicity of counts , or increase the number of distinct sequences ; we are unable to change the beginning of sequences . as a result , it is not possible to establish conditions such as ( a[hyp : p_nassump ] ) . we propose the following approximation of the marginal , based upon the theoretical properties of such models ( ethier & griffiths , 1987;griffiths , 1989 ) and the relation to the infinitely - many - alleles model ( e.g. griffiths ( 1979 ) and the references there - in ) . let us consider the marginal distribution , call it @xmath303 . we extend the state - space to include uncertainty on @xmath42 , the number of distinct types , @xmath304 and @xmath305 the number of segregating sites , and adopt the decomposition @xmath306 now , under certain conditions , there are results about the exact densities @xmath307 ( ewens , 1972 ) and @xmath308 ( watterson , 1975 ) . in the case @xmath309 , as noted by griffiths ( 1979 ) , for large populations ( such that diffusion results can apply ) the infinitely - many - sites and infinitely - many - allele frequencies are not too different . therefore , we propose to use the probability ( as in ewens ( 1972 ) ) @xmath310 with @xmath311 are stirling numbers of the first kind . for the quantities @xmath312 and @xmath313 , we use approximations . for the former , a uniform distribution is adopted @xmath314\bigg)^{-1}\ ] ] where @xmath315 that is , it is a simple task in combinatorics to show that if there are @xmath305 mutations with @xmath316 repetitions of mutations @xmath257 to @xmath305 , ( subject to the constraint that each mutation can only occur at most once in each sequence and that order of allocating a mutation does not matter ) then there are @xmath317 possible sequences ; summing over all the possible multiples yields the desired cardinality of the state - space . @xmath313 is not known ( except as the marginal of a recursion ( as in ethier & griffiths ( 1987 ) ) ) and is assigned @xmath318 ( poisson ) distribution ( at time @xmath71 ) . in practice , it may not be possible to evaluate some of these quantities and a further monte carlo simulation / numerical approximation ( for the integral over @xmath305 and the normalizing constant of @xmath319 ) will be required . that is to say , we set @xmath320 the approximation will be different for every simulated sample . lee , a. , yau , c. , giles , m. b. , doucet , a. & holmes , c.c . ( 2010 ) . on the utility of graphics cards to perform massively parallel simulation of advanced monte carlo methods , _ j. comp . graph . _ , ( to appear ) .
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in the following paper we consider a simulation technique for stochastic trees .
one of the most important areas in computational genetics is the calculation and subsequent maximization of the likelihood function associated to such models .
this typically consists of using importance sampling ( is ) and sequential monte carlo ( smc ) techniques .
the approach proceeds by simulating the tree , backward in time from observed data , to a most recent common ancestor ( mrca ) .
however , in many cases , the computational time and variance of estimators are often too high to make standard approaches useful . in this paper
we propose to stop the simulation , subsequently yielding biased estimates of the likelihood surface .
the bias is investigated from a theoretical point of view .
results from simulation studies are also given to investigate the balance between loss of accuracy , saving in computing time and variance reduction .
+ * key words * : stochastic trees , sequential monte carlo , coalescent .
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a basic prediction of @xmath5cdm galaxy - formation models is the existence of a hot ( @xmath6k ) halo of gas accreted from the intergalactic medium around milky way - sized galaxies ( extending to the virial radius ) , which forms as infalling gas is heated to the virial temperature at an accretion shock ( e.g. , * ? ? ? these halos may provide most of the fuel for long - term star formation in these galaxies @xcite , but their predicted properties are sensitive to the input physics , which can be constrained by the measurable properties of the gas . based on work over the past several years , we know that these extended halos exist , including around the milky way @xcite . the extent and luminosity of the hot gas implies that it has a similar mass to the stellar disk , and therefore could play an important role in galaxy evolution . thus , it is important to measure the properties of the hot gas beyond mass and temperature ( such as metallicity and density or velocity structure ) . however , hot halos are faint and the measurable x - ray luminosity can be dominated by stellar feedback ejecta near the disk @xcite , which makes these measurements difficult . only in the milky way can one measure the structure , temperature , metallicity , and kinematics of the hot gas through emission and absorption lines @xcite , but kinematic constraints from prior studies @xcite are weak . recent developments in the calibration of the x - ray grating spectrometers and the accumulation of multiple high quality data sets for individual objects have made it possible to determine line centroids to an accuracy of tens of kms@xmath4 , which enables us to improve the constraints on the kinematics of the gas by measuring doppler shifts in lines that trace the hot gas . the @xmath721.602 resonance absorption line @xcite is the best candidate , since it is sensitive to temperatures of @xmath8k ( which includes much of the galactic coronal gas ) and it is detected at zero redshift towards a large number of background continuum sources ( e.g. , * ? ? ? the _ emission _ lines produced by the same species are useful for determining the structure and temperature of the hot gas @xcite , but they are too faint for high resolution spectroscopy . in this paper we constrain , for the first time , the radial and azimuthal velocity of the hot gas by measuring the doppler shifts in lines detected towards bright sources outside the disk of the milky way . to measure the global velocity of the million - degree gas around the galaxy , one needs to measure doppler shifts towards a range of sources across the sky in lines sensitive to this temperature . this gas is detected in x - ray emission and absorption , but the emission lines are far too faint for a focused grating observation . x - ray imaging ccds measure the energies of incoming photons and are thus also low resolution spectrometers , but their spectral resolution is far too low to measure doppler shifts of tens of kms@xmath4 . the only instruments capable of this accuracy are the _ chandra _ low / high - energy transmission grating ( letg / hetg ) spectrometers and the _ xmm - newton _ reflection grating spectrometer ( rgs ) , and the 21.602 line is the only line that probes the relevant temperatures and is detected at @xmath9 towards many background continuum sources . the oviii line at 18.96 probes slightly hotter gas and is only detected towards a few objects . our initial sample included all archival letg and rgs data sets where the line has been detected in the literature . the letg has modestly better spectral resolution at 21.6 ( @xmath10 ) than the rgs ( @xmath11 ) , but it only has a third of the effective area at this wavelength ( 15@xmath12 for the letg , 45@xmath12 for the rgs ) . in addition , unlike the letg the rgs is always on ( it has a dedicated telescope , whereas the gratings must be moved into the focal plane on _ chandra _ ) , and thus has accumulated many more spectra . these factors lead to many more detected lines in the rgs , so we only use the letg data towards several calibration sources as a check on the wavelength solution ( see below ) . our analysis sample includes 37 known absorbers at @xmath9 with rgs data @xcite ( table [ table.velocities ] ) . these include agns as well as several x - ray binaries in the milky way s halo and magellanic clouds . we tried to include all sources known to be outside the disk with reported absorption lines , but we excluded three sources : ngc 3783 , pks @xmath13 , and swift j@xmath14 . ngc 3783 has an intrinsic oxygen line with a p cygni profile where we can not disentangle the galactic line , pks @xmath13 has a broad line that suggests blending or a non - galactic origin , and the line in swift j@xmath14 is only detected in two of four high @xmath15 exposures . we include ngc 5408 x-1 ( which is an x - ray binary , not an agn ) and ngc 4051 , but these systems have redshifts smaller than 1000kms@xmath4 so the lines may be intrinsic . ngc 4051 and mcg-6 - 30 - 15 also have known outflows ; as they have lines attributed to the galaxy in some prior studies ( e.g. , * ? ? ? * ) we include them here , but we show below that excluding them does not strongly change our results . the data for each target were reprocessed using standard methods in the _ xmm - newton _ science analysis software ( sas v14.0.0 ) with the appropriate calibration files . this included excluding hot , cold , and `` cool '' pixels , and data from periods when the background count rate exceeds 3@xmath16 from the mean . we applied the ( default ) empirical correction for the sun angle of the spacecraft and its heliocentric motion @xcite . we used the highest precision coordinates available rather than the proposal coordinates , which improves the accuracy of the wavelength scale . for each object , we merged the first - order rgs1 spectra and response matrices into a `` stacked '' spectrum . standard processing resamples the data from native bins ( about 0.011 at 21.6 ) into a user - specified bin size . we binned the data to 0.02 ( one resolution element is about 0.055 ) . this resampling causes small but stochastic changes in the bin assignment for some events , leading to variation under the same protocols , which we quantify by processing each object ten times in the same way . to measure the doppler shifts , we fit a model consisting of a power law and an absorption line to the spectrum in the 21 - 22 bandpass using xspec v12.9.0 @xcite . we exclude an instrumental artifact between 21.7521.85 in each spectrum , and in several spectra there are one or more bad channels in the bandpass that we also exclude . the parameters of the absorption - line model include the line centroid , the line width , and the line strength , but we fix the line width at the instrumental line - profile width because we do not expect detectable line broadening , an assumption we validated in the brightest sources . the best - fit centroid is converted to a velocity using the best - fit line energy from a stacked spectrum of capella as a reference , corrected for the radial velocity of the star ( described below ) . we measured the velocity for each of the ten stacked spectra per object , and we report the mean value with its 1@xmath16 uncertainty in table [ table.velocities ] , including the resampling uncertainty . the systematic uncertainty in the wavelength scale limits the accuracy of our measurements , and recent improvements in the calibration of the wavelength scale @xcite and multiple high @xmath15 spectra for the objects in our sample are largely what enable this study . here we show the accuracy of the wavelength solution for the protocols we adopt and briefly describe the sources of uncertainty and their magnitudes . we created spectra for the active stars capella and hr 1099 following the protocols above , then measured the ( emission ) line centroids for strong , mostly unblended lines , and compared them to their laboratory rest wavelengths ( figure [ figure.capella_wavelength ] ) . we accounted for the radial velocity of each star ( @xmath17kms@xmath4 for capella and @xmath18kms@xmath4 for hr 1099 ; * ? ? ? we find no systematic offset in the 5 - 30 bandpass or change over time . the wavelength offsets in @xmath721.602 for stacked spectra are @xmath19m ( capella ) and @xmath20m ( hr 1099 ) . these correspond to @xmath21 and @xmath22kms@xmath4 . to verify that stacking does not introduce artificial offsets , we also measured centroids in each individual exposure ( 16 for capella and 14 for hr 1099 ) and computed the weighted mean . the offsets are consistent with the stacked spectrum : @xmath23 and @xmath24kms@xmath4 for capella and hr 1099 . we also checked the wavelength solution against the letg and hetg data for these stars . we reduced te data using the chandra interactive analysis of observations software ( ciao v4.7 ) , and we found good agreement with spectra from the reduced data available through the tgcat project @xcite . we co - added the @xmath25first - order spectra and stacked all observations . the letg offsets are @xmath26kms@xmath4 ( capella ) and @xmath27kms@xmath4 ( hr 1099 ) , whereas the hetg medium energy grating offsets are @xmath28kms@xmath4 ( capella ) and @xmath29kms@xmath4 ( hr 1099 ) . these results agree with the rgs and the laboratory wavelength ( figure [ figure.capella_wavelength ] ) . we also compared velocity measurements between the rgs and letg in the four brightest quasars @xcite . the letg and rgs centroids agree to within the 1@xmath16 error bar . these results show that the wavelength scale is sufficiently accurate for our measurement , but it is important to note that there is a substantial systematic scatter that makes _ individual _ observations unreliable , and also that there will be systematic differences between our measurements and those reported for the same data using an earlier version of sas or using incorrect source coordinates . different observations of the same object have a systematic scatter of @xmath305m in the wavelength solution around the true mean value . this corresponds to 70kms@xmath4 at 21.602 , and leads to the standard quoted systematic uncertainty of 100kms@xmath4 . the reason for this scatter is not clear , but about 3m could be explained by limits in the pointing accuracy of the telescope @xcite . in any case , the scatter is normally distributed ( based on measurements from many exposures of calibration stars such as capella ) , and can thus be strongly mitigated with multiple observations of the same source . to reduce the scatter to within 20kms@xmath4 requires 15 - 20 independent spectra ( assuming equal @xmath15 and a @xmath31m ) . the sources in our sample have between 2 - 60 observations , and at the low end the reported 1@xmath16 statistical errors ( 200kms@xmath4 or more ) are much larger than a 70kms@xmath4 systematic error . for the bright quasars with many observations , we estimate a typical systematic error due to this scatter of @xmath32kms@xmath4 . . two models are shown for comparison with the data ( black points ) : a stationary model ( blue points ) and a corotating model ( red points ) . error bars are 1@xmath16 ( standard deviation ) . the four objects with the smallest error bars are labeled 1 - 4 . the velocities of the lines towards mrk 421 and ngc 4051 ( labeled ` 2 ' and ` 3 ' ) suggest some intrinsic scatter . , scaledwidth=50.0% ] in addition to the scatter , there are systematic offsets produced by the sun angle of the telescope and its projected heliocentric motion ( possibly an inaccuracy in the star - tracker ) that were measured and corrected by @xcite ; we refer the reader to their paper for a detailed description . the earliest version of sas that conained these corrections was v13.0.0 , and it was not enabled by default until v14.0.0 ; all prior absorption - line halo studies using the rgs used an earlier version or did not mention the correction . we measured line centroids with and without these corrections , and found centroid shifts in our sample with magnitudes between @xmath33kms@xmath4 ( higher for weak lines ) . to show the effect of not including them , we show the measured centroids without these corrections in table [ table.velocities ] . finally , if we used the default ( proposal ) coordinates instead of the simbad coordinates , we measured offsets of @xmath34kms@xmath4 . thus , we would expect our measured centroids to be correlated with prior results but perhaps significantly different relative to the statistical errors . for example , there is a systematic offset of about 60kms@xmath4 between our measurements and @xcite for the several systems where it can be measured . this offset can not be entirely explained by the sun angle and heliocentric motion corrections , but it is consistent with shifts seen relative to prior versions of sas @xcite . they do not report the line centroids for active stars , as they only need to measure the centroids to sufficient accuracy to identify halo absorbers . thus , we do not regard the apparent inconsistency as reflecting an inherent uncertainty in line centroid measurements . lcccccccccc mkn 421 & 179.832 & 65.031 & 0.0300 & @xmath35 & @xmath36 & @xmath37 & @xmath38 & @xmath38 & [email protected] & 232 + 3c 273 & 289.951 & 64.360 & 0.1580 & @xmath39 & @xmath40 & @xmath41 & @xmath42 & @xmath43 & [email protected] & 55 + pks 2155 - 304 & 17.730 & @xmath44 & 0.1160 & @xmath45 & @xmath46 & @xmath47 & @xmath48 & @xmath49 & [email protected] & 95 + mkn 509 & 35.971 & @xmath50 & 0.0340 & @xmath51 & @xmath52 & @xmath53 & @xmath38 & @xmath54 & [email protected] & 48 + ngc 4051 & 148.883 & 70.085 & 0.0023 & @xmath55 & @xmath56 & @xmath57 & @xmath58 & @xmath59 & [email protected] & 33 + mcg -06 - 30 - 15 & 313.292 & 27.680 & 0.0077 & @xmath60 & @xmath61 & @xmath62 & @xmath63 & @xmath64 & [email protected] & 38 + ark 564 & 92.138 & @xmath65 & 0.0247 & @xmath66 & @xmath67 & @xmath42 & @xmath68 & @xmath69 & [email protected] & 50 + eso 141 - 55 & 338.183 & @xmath70 & 0.0371 & @xmath71 & @xmath72 & @xmath73 & @xmath74 & @xmath75 & [email protected] & 18 + h1426 + 428 & 77.487 & 64.899 & 0.1290 & @xmath76 & @xmath77 & @xmath78 & @xmath79 & @xmath80 & [email protected] & 25 + 1h 0707 - 495 & 260.169 & @xmath81 & 0.0405 & @xmath82 & @xmath83 & @xmath84 & @xmath85 & @xmath86 & [email protected] & 22 + pks 0558 - 504 & 257.962 & @xmath87 & 0.1370 & @xmath88 & @xmath89 & @xmath57 & @xmath43 & @xmath90 & [email protected] & 32 + ngc 4593 & 297.483 & 57.403 & 0.0090 & @xmath91 & @xmath92 & @xmath93 & @xmath48 & @xmath94 & [email protected] & 13 + mrk 335 & 108.763 & @xmath95 & 0.0258 & @xmath96 & @xmath97 & @xmath98 & @xmath99 & @xmath100 & [email protected] & 22 + pg 1211 + 143 & 267.552 & 74.315 & 0.0809 & @xmath101 & @xmath102 & @xmath103 & @xmath104 & @xmath105 & 40.3@xmath2511 . & 7 + pg 1244 + 026 & 300.041 & 65.214 & 0.0482 & @xmath106 & @xmath107 & @xmath108 & @xmath109 & @xmath110 & 40.8@xmath2511 . & 10 + mkn 501 & 64.600 & 38.860 & 0.0337 & @xmath111 & @xmath112 & @xmath113 & @xmath114 & @xmath115 & 25.2@xmath2510 . & 22 + 1es 1028 + 511 & 161.439 & 54.439 & 0.3600 & @xmath116 & @xmath117 & @xmath118 & @xmath84 & @xmath119 & 36.4@xmath2511 . & 12 + eso 198 - 24 & 271.639 & @xmath120 & 0.0455 & @xmath121 & @xmath122 & @xmath123 & @xmath124 & @xmath125 & 62.2@xmath2519 . & 7 + ngc 2617 & 229.300 & 20.939 & 0.0142 & @xmath126 & @xmath127 & @xmath128 & @xmath129 & @xmath130 & [email protected] & 10 + pg 1553 + 113 & 21.909 & 43.964 & 0.3600 & @xmath131 & @xmath132 & @xmath133 & @xmath134 & @xmath135 & [email protected] & 13 + h1101 - 232 & 273.190 & @xmath136 & 0.1860 & @xmath137 & @xmath138 & @xmath139 & @xmath140 & @xmath141 & 45.5@xmath2517 . & 8 + 3c 390.3 & 111.438 & 27.074 & 0.0561 & @xmath142 & @xmath143 & @xmath84 & @xmath80 & @xmath144 & 25.0@xmath2510 . & 10 + 1h 0419 - 577 & 266.963 & @xmath145 & 0.1040 & @xmath146 & @xmath147 & @xmath148 & @xmath149 & @xmath150 & 39.9@xmath2516 . & 6 + fairall 9 & 295.073 & @xmath151 & 0.0470 & @xmath152 & @xmath153 & @xmath154 & @xmath63 & @xmath155 & 37.2@xmath2515 . & 6 + iras 13224 - 3809 & 310.189 & 23.979 & 0.0660 & @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 & 45.1@xmath2519 . & 7 + ngc 5408 x-1 & 317.149 & 19.496 & 0.0017 & @xmath161 & @xmath162 & @xmath68 & @xmath163 & @xmath64 & 30.4@xmath2514 . & 7 + pds 456 & 10.392 & 11.164 & 0.1840 & @xmath164 & @xmath165 & @xmath166 & @xmath167 & @xmath59 & 60.0@xmath2529 . & 7 + mrk 279 & 115.042 & 46.865 & 0.0305 & @xmath168 & @xmath169 & @xmath37 & @xmath113 & @xmath170 & [email protected] & 15 + e1821 + 643 & 94.003 & 27.417 & 0.2970 & @xmath171 & @xmath172 & @xmath173 & @xmath174 & @xmath175 & 35.7@xmath2518 . & 7 + ic 4329a & 317.496 & 30.920 & 0.0160 & @xmath176 & @xmath177 & @xmath178 & @xmath179 & @xmath180 & 38.3@xmath2520 . & 7 + pg 0804 + 761 & 138.279 & 31.033 & 0.1000 & @xmath181 & @xmath182 & @xmath43 & @xmath178 & @xmath183 & 28.5@xmath2515 . & 6 + ngc 5548 & 31.960 & 70.496 & 0.0172 & @xmath184 & @xmath185 & @xmath114 & @xmath186 & @xmath59 & 18.0@xmath2510 . & 13 + mrk 766 & 190.681 & 82.270 & 0.0129 & @xmath187 & @xmath188 & @xmath189 & @xmath38 & @xmath190 & [email protected] & 27 + lmc x-3 & 273.576 & @xmath191 & - & @xmath192 & @xmath193 & @xmath186 & @xmath194 & @xmath195 & [email protected] & 36 + 4u 1957 + 11 & 51.308 & @xmath196 & - & @xmath197 & @xmath198 & @xmath199 & @xmath200 & @xmath201 & [email protected] & 22 + maxi j0556 - 032 & 238.939 & @xmath202 & - & @xmath203 & @xmath204 & @xmath205 & @xmath206 & @xmath207 & [email protected] & 33 + smc x-1 & 300.414 & @xmath208 & - & @xmath209 & @xmath210 & @xmath211 & @xmath148 & @xmath212 & [email protected] & 13 + in addition to the wavelength grid , there are a few sources of systematic uncertainty and some fitting choices that affect our final results but where we believe there is a correct choice . we briefly describe these here . [ [ cool - pixels ] ] _ cool pixels _ + + + + + + + + + + + + + there are several `` cool '' pixels in the vicinity of 21.6 that have a lower than expected signal ( by about 20% ) . by default , these pixels are included in the spectrum . we exclude them because they can affect weak absorption features . the typical velocity shift between keeping and excluding them is @xmath213kms@xmath4 in bright sources . [ [ binning - and - resampling ] ] _ binning and resampling _ + + + + + + + + + + + + + + + + + + + + + + + + the native rgs binning is about 11m at 21.6 , but the default processing resamples the events onto a user - specified grid , with a default value of 10m . this resampling is probabilistic with a random element ( normally distributed ) , which means that running the same protocol on a given data set multiple times will result in slightly different spectra . the magnitude of the velocity shift is strongly dependent on the line strength and the continuum @xmath15 , so we reprocessed each data set ten times per reduction protocol set . we then added the standard deviation in the measured velocities in quadrature to the statistical error , since it behaves in essentially the same way . the bin sizes also affect the measured velocities , with a mean line centroid shift of @xmath214kms@xmath4 between bins of 10m and 20m in bright systems ( we use the latter ) . [ [ stacking ] ] _ stacking _ + + + + + + + + + + temporal changes in the instrumental response or changes in the spectral shape of the source can bias the results of stacked spectra . on the other hand , jointly fitting several low-@xmath15 spectra with the continuum shape as a free parameter leads to poorer constraints on the velocity of a line based on spectral bins far from the absorption line . we stack the spectra to improve the continuum @xmath15 , but to determine if this provides systematic bias we measured the line shift between joint fits and stacked spectra in our brightest sources and the calibration sources . we find a typical @xmath215kms@xmath4 because the instrumental response in these regions does not appear to have temporal changes . [ [ line - profile - and - fixed - line - width ] ] _ line profile and fixed line width _ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + for lines with doppler @xmath216kms@xmath4 , we do not expect to measure reliable line widths ( however , see the doppler @xmath217 measurements in some of the same lines in * ? ? ? since the rgs line - spread function is best described as a lorentzian near the core , we fit our spectra using a lorentzian profile with the width fixed at the instrumental line width . if we use a gaussian line profile instead but keep the width fixed , the line shift is consistent with zero in most cases but can be up to @xmath218kms@xmath4 in weak lines ( statistical errors greater than 200kms@xmath4 ) . when the line width is a free parameter , we find shifts in the lorentzian centroids of @xmath219kms@xmath4 ( the shift magnitude is negatively correlated with @xmath15 ) and @xmath220kms@xmath4 in the gaussian case . however , for the gaussian lines the best - fit line widths tend to be @xmath221 times the instrumental resolution , and the statistical error also increases . these fits are usually not significantly better than with a fixed line width , so in our view a non - zero line width is not required by the data , and these line widths reflect some small curvature in the continuum . [ [ bandpass ] ] _ bandpass _ + + + + + + + + + + the fitting bandpass is important because the continuum model needs to fit well even if the result is unphysical . typically , one fits the local continuum , but in the literature for the line this can vary from a fitting interval less than 1 wide to about 5 wide . our choice of fitting bandpass ( 21 - 22 ) is motivated by strong instrumental features below 21and above 22 , but if we ignore these features and expand our bandpass by @xmath251 , the typical velocity shift is @xmath222kms@xmath4 . [ [ bad - columns - near-21.6 ] ] _ bad columns near 21.6 _ + + + + + + + + + + + + + + + + + + + + + + + + there is asymmetry in the line - spread function ( lsf ) , which is the instrumental response to a @xmath223-function , near the line . this is not the same as the well known asymmetry in the lsf at the 1% level in the line wings ( which is not important ) , but rather due to bad columns that are not included in the cool pixel list ( figure [ figure.lsf_error ] ) . for an arbitrarily strong @xmath223-function line , the offset in the measured centroid from this defect can range from @xmath224kms@xmath4 depending on where the true line centroid is . however , if the line is unresolved but has some physical width ( so that incident photons would be dispersed over multiple bins even before the lsf spreads them out ) , the error is strongly mitigated . we used xspec simulations to determine the error as a function of doppler @xmath217 parameter , and for @xmath225kms@xmath4 ( about the thermal width even without turbulence ) , the typical error is reduced to 15kms@xmath4 ( figure [ figure.lsf_error ] ) . we expect the galactic halo lines to be in this regime . alternatively , one can ignore the columns , but because of their proximity to the region of interest , this will also bias the results . overall , the systematic error should allow measurements to better than 30kms@xmath4 accuracy with sufficient @xmath15 . for the high @xmath15 lines we estimate a typical systematic error of 15 - 20kms@xmath4 , and for lines weaker than about 4@xmath16 , the systematic error is dominated by the statistical error . we used a simple halo model for comparison to the data . we adopted an extended density profile @xcite in which @xmath226 where @xmath227@xmath228kpc@xmath229 and @xmath230 , computed on a grid with 0.05kpc cells . this model describes a large , all - sky sample of 649 and oviii emission lines @xcite well . we then imposed global bulk radial ( @xmath231 ) or azimuthal ( @xmath232 ) velocities , assuming that these are constant with radius . these are the free parameters in the model . we likewise assume a constant metallicity ( @xmath233 ) and a doppler @xmath217 parameter of 85kms@xmath4 due to random turbulent motion in each cell , based on hydrodynamic simulations @xcite . we also account for the possibility of intrinsic scatter ( resulting from hydrodynamic flows ) about any model by adding a velocity dispersion to the velocities when comparing to the models . this dispersion is not the same as line _ broadening _ , but refers to the typical value for a distribution of centroid shifts . we obtain model velocities for each object by integrating from the position of the sun outward along each line of sight , computing the line - of - sight velocity component and broadening for each cell and summing the resultant voigt profiles ( a more detailed description is given in * ? ? ? we convolve the result with the lsf , compute the centroid , and add the solar reflex motion for comparison to the observations . in other words , the velocities are compared in the frame of the local standard of rest . for this model , at least 50% of the total absorption comes from beyond 7kpc from the sun , and 90% from within 50kpc . in the simplest case of a stationary halo , the galactic rotation of the local standard of rest @xcite ( @xmath234kms@xmath4 ) is reflected in the measured doppler shifts : @xmath235 , where @xmath236 is the galactic longitude and @xmath217 the galactic latitude . the product @xmath237 corresponds to the sun moving directly towards or away from that direction , resulting in a doppler shift of @xmath238kms@xmath4 . another simple case is a corotating halo ( @xmath239 ) , in which case the doppler shifts will be closer to zero . figure [ figure.velocities ] shows the measured velocities with the stationary and corotating models . the best - fit @xmath231 and @xmath232 values were obtained from a markov - chain monte carlo ( mcmc ) approach , using the @xmath240 statistic as a goodness - of - fit parameter . for zero dispersion , the best model has a prograde rotation velocity of @xmath241kms@xmath4 and a global inflow of @xmath242kms@xmath4 , corresponding to a net accretion rate of @xmath243yr@xmath4 ( figure [ figure.results ] ) . this is a formally acceptable fit , whereas the stationary halo is rejected with 99.95% probability and the corotating halo is marginally rejected with 95% probability . the velocity dispersion for which the reduced @xmath240 is closest to 1.0 is about 50kms@xmath4 ( figure [ figure.results ] ) ; substantially more than this produces a @xmath240 that is too small for the observed line centroids . the apparent inflow is not statistically significant , and the suggestion of inflow primarily results from mrk 421 ( figure [ figure.velocities ] ) , which has a small uncertainty and is near @xmath244 . taking these results at face value , the large @xmath232 and extent of the halo imply that the total angular momentum of the hot gas within 50kpc is comparable to that in the stars and gas in the disk of the galaxy @xcite ( figure [ figure.momentum ] ) . the spread in recent measurements of @xmath245kms@xmath4 @xcite leads to @xmath246kms@xmath4 , which does not change the picture of a lagging halo with prograde rotation . in addition to parameter fitting , we used nonparametric statistical tests to test the hypothesis that some rotation is necessary . we tested the stationary and corotating models using the sign test and the kolmogorov - smirnov ( ks ) test , which compare distributions and are less sensitive to scatter . the sign test asks whether the medians of the measured and model velocities are consistent with each other ( assuming a binomial distribution and a 50% probability that the model velocity exceeds the measured velocity ) . the left panel of figure [ figure.nonpar ] shows the residuals from subtracting the data from the stationary model values , and the strong asymmetry rules out the stationary model with 99.87% probability . the ks test compares the cumulative distributions of the measured and model velocities , and this test rules out the stationary halo with 98.5% probability . the corotating model is acceptable in the sign test ( 43% rejection probability ) and not in the ks test ( 99.8% probability ) . the rejection of the stationary halo is model independent . if we exclude weak lines with @xmath247kms@xmath4 , a stationary model is still ruled out at more than 99.4% probability . if we exclude ambiguous lines ( such as those towards ngc 5408 x-1 , ngc 4051 , and mcg-6 - 30 - 15 ) , a stationary model is ruled out at about 98% probability in the ks test and 99.6% in the sign test . excluding ngc 4051 ( with its small error bars ) increases the range of acceptable @xmath232 in the parametric fits , but the best - fit @xmath231 does not change much as the error bars for mrk 421 are even smaller . we investigated the sensitivity of the best - fit parameters and the implied angular momentum to the model assumptions . first , the assumption of constant velocity must break down at some radius . an effort to measure the galactic rotation curve to 200kpc using disk and non - disk objects found @xcite that @xmath232 is flat to 80kpc , while a measurement from disk stars in the sloan digital sky survey found @xcite that @xmath232 in the disk declines by 15% from @xmath248 at 10 - 20kpc and remains constant from 20 - 55kpc ( the maximum probed ) . thus , the assumption of a constant @xmath249 or @xmath232 within @xmath250kpc appears to be reasonable , and more than 80% of the cumulative equivalent width comes from within this region ( figure [ figure.momentum ] ) . when we considered a model with a constant @xmath232 within 50kpc and @xmath251kms@xmath4 beyond , the best - fit @xmath232 is nearly identical to the reported value . for a 10kpc cutoff , the best - fit @xmath232 increases . second , there may be a metallicity gradient in the hot gas , in which case the cumulative equivalent width will be even more dominated by nearby gas . this would impact the inferred mass and total angular momentum of the halo gas . constraints from o vii and oviii and pulsar dispersion measures towards the large magellanic cloud are consistent @xcite with a gradient of @xmath252 with @xmath253 at 1kpc and @xmath254 at 10kpc . the uncertainty is large , but the gradient is shallow enough that gas beyond a few kpc contributes 50% or more of the equivalent width . finally , even a single - component hot halo is probably not isothermal and may also have flows arising from satellite galaxy motions or gaseous inflows and outflows . since the combined emission- and absorption - line data favor an extended halo and its temperature is near the virial temperature , the gas is probably volume filling . hydrodynamic flows in such a halo will primarily induce scatter of the type described above , which does not strongly affect our @xmath232 measurement ( figure [ figure.results ] ) . thus , the measured halo rotation probably extends to tens of kpc and possibly to 100kpc , and this gas will have a substantial angular momentum . however , the angular momentum values ( figure [ figure.momentum ] ) are strongly model dependent , especially due to the assumption of constant @xmath255 . the interpretation of the measured doppler shifts as a rotation signature depends on the validity of our single - component halo model . this depends on the following assumptions : ( 1 ) a volume - filling spherical halo is an approximately accurate representation of most of the hot gas around the galaxy ; ( 2 ) there is no strong local absorber that we have ignored ; ( 3 ) the spherical halo has bulk global motion ; ( 4 ) the assumption of @xmath256 is reasonable within about 50kpc ; and ( 5 ) the doppler shift measurements are accurate . the fourth and fifth assumptions were addressed above , so here we focus on the first three . two basic models are suggested in the literature for the structure of the hot halo : an extended , spherical distribution or an exponential disk with a scale height of a few kpc . when using individual sight lines or small samples @xcite , the observed and absorption and emission line strengths are consistent with an exponential disk model with a scale height of a few kpc . they are also consistent with a spherical model . however , larger samples of absorption lines ( @xmath3040 sight lines ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the all - sky emission - line intensities ( @xmath2571,000 sight lines * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) favor a spherical model . similar analyses on independent observables that also probe the hot gas , such as the pulsar dispersion measure towards the large magellanic cloud and the ram - pressure stripping of milky way satellites favor an extended halo @xcite . finally , a recent analysis of both galactic and extragalactic sightlines for @xmath258 galaxies finds that the traces hot gas @xcite . thus , we adopted the spherical density profile of @xcite . however , these analyses are based on single - component models , and from basic galaxy models we expect at least two x - ray absorbing components : infalling gas that is shock heated to the virial temperature ( @xmath259k for the milky way ) and forms an extended halo ( e.g. , * ? ? ? * ) , and supernova - driven outflows from the disk ( e.g. , * ? ? ? * ; * ? ? ? * ) . in the latter case , we expect an exponential disk of hot gas with a scale height set ( for a given galactocentric radius ) by the temperature at the midplane . it is worth noting that for @xmath260k , the scale height at the solar circle is larger than the galactocentric distance , so the distribution of outflowing gas could also be spherical , if not very extended ; the true shape depends on how widespread the outflows are in the disk , the actual midplane temperature , and the amount of radiative cooling . within the galaxy disk itself , supernovae contribute to the hot interstellar medium , much of which is confined within the disk . since x - ray absorption covers a wider temperature range than emission , we are also weakly sensitive to cooler gas ( described in the following subsection ) . these components may be kinematically distinct , as is seen in the warm gas by @xcite , but at the relatively low resolution of the rgs , we expect them to be blended . this complicates the interpretation of the measured centroids , which are the weighted average of the offsets for each component along that line of sight . at a qualitative level , we expect the gas confined in the galaxy disk to rotate with it ( although depending on the distances to the absorbers , there may not be any rotation signature ) and the gas in supernova - driven outflows to rotate in the same direction but lagging the disk as it reaches larger heights or radii . to constrain the column and thus the influence on the line centroids from a disk component , we extended the analysis of @xcite to fit a disk@xmath261halo model to the same data set they used : 648 emission lines from @xcite , which are filtered for contamination from solar - wind charge exchange and ignore most of the galactic plane . we refer the reader to @xcite for a more detailed explanation of the modeling procedure , but we summarize our model components here . the sun exists in a region known as the local bubble , which emits soft x - rays ( e.g. , * ? ? ? * ; * ? ? ? * ) , and we adopt for the bubble a temperature of @xmath262k , a variable path length between 100 - 300pc , and a density @xmath263@xmath228 . the nature of the bubble and the density remain debatable because of the contribution from solar - wind charge exchange ( e.g. , * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? we have ignored hot interstellar gas confined within the galaxy disk other than the local bubble . constraints from the intensity of the soft x - ray background imply that our sight lines will incur only a small column from this material , and the emission - line sample excludes the galactic plane . for the extended halo component , we assume an isothermal ( @xmath264k ) plasma with a density profile described by equation [ equation.beta_model ] . for the exponential disk component , we assume a form for the density of @xmath265 where @xmath266 is the scale height . the temperature declines in an analogous way , and we fix the midplane temperature and its scale height to @xmath267k and 5.6kpc , respectively @xcite because we do not model the line ratios as a temperature diagnostic . the free parameters are the normalizations , @xmath268 , and @xmath266 , which we constrain using the same mcmc method as @xcite . we assume that each component is optically thin and compute its contribution to the intensity along each line - of - sight @xmath269 : @xmath270 where @xmath271 is the volumetric line emissivity from the apec thermal plasma code @xcite . finally , the line intensities from @xcite do not account for absorption due to galactic hi , so to compare our model intensities to theirs we apply photoelectric absorption using the neutral hydrogen column from @xcite to the halo and disk components : @xmath272 where @xmath16 is the absorption cross - section and @xmath273 is the intensity from the local bubble . figure [ figure.disk_halo ] shows the results from the mcmc analysis as marginalized posterior probability distributions for each of the free parameters . the model is not a significant improvement on the pure halo model , and the best - fit parameters for the halo ( @xmath274@xmath228kpc@xmath229 , @xmath275 ) are consistent with the results in @xcite . the disk parameters are poorly constrained and indicate that _ the exponential disk contributes at most 10% of the total column density . _ this corroborates the @xcite result . in contrast , the @xcite disk model ( which assumes the halo gas can be described entirely by a disk and is based on one sightline ) finds @xmath276 and @xmath277kpc . @xcite , who include some sight lines near the galactic plane and some outside the plane , find that the column is consistent with @xmath278@xmath228 and a scale height of @xmath279kpc , but the authors acknowledge that mixing results from sight lines towards nearby x - ray binaries at low latitudes and those at high latitudes can lead to a strong bias ; the hot material confined to the galaxy disk is not part of the exponential disk structure that we have modeled , but it can contribute at low latitudes . thus , the impact of the disk component on the measured centroids must be small , since 80 - 90% of the column density will come from the spherical component . assuming that the two components are kinematically distinct , for the latter to be stationary and consistent with the measured centroids requires a disk speed much faster than co - rotation . we verified this by modeling a rotating disk and a stationary halo where the disk contributes 10% of the column . since this is inconsistent with models where the gas originates in the galaxy disk , the data probe motion in the spherical component . for reference , if the @xcite disk model is adopted instead of the @xcite model or our disk@xmath261halo model , the best - fit azimuthal and radial velocities are @xmath280kms@xmath4 and @xmath281kms@xmath4 . we have assumed that , to first order , the spherical component moves as a solid body with some @xmath232 and @xmath231 . there may be second - order effects such as the internal flows mentioned above ( perhaps due to satellite motions or infalling clouds ) or modes in the fluid , which would add scatter to the velocity measurements . however , if the halo is volume filling and in steady state , then large scale disturbances will tend to dissipate in a sound - crossing time ( which could be a long time if the halo is very extended ) , and the gas ( if stationary ) will tend towards hydrostatic equilibrium . multiple major kinematic components are therefore not expected , although we would expect differential rotation . in this case , the measured rotation velocity is some average but still provides useful information . also , if the material is fresh infall from the cosmic web then it likely accretes along filaments , which lead to a preferential orbital angular momentum axis . finally , gas ejected from the disk ( either cold or hot ) could spin up the halo @xcite , although perhaps not to the velocity that we infer . it is possible that there is a layer of warm - hot gas near the galactic plane ( in addition to the local bubble ) that affects every sight line out of the galaxy , but to which our models are not sensitive because we ignore data close to the galactic plane ( where dust , supernova remnants , and other features make modeling very difficult ) . characterizing the structure and filling factor of the hot interstellar medium has been a major effort by itself ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and is beyond the scope of this paper , but if there is such a layer ( possibly a disk with a scale height of a few hundred pc ) with a high density , the rotation signature in the rgs data could be misattributed to the halo . on the other hand , for plausible path lengths , oxygen column densities , and foreground absorption column densities this layer should also produce emission in excess of the local bubble contribution to the soft x - ray background . to summarize , the halo models that are based on many data points as opposed to a few sightlines favor a spherical halo ( especially in emission ) as the dominant component . any contribution to the column density from a kinematically distinct ( thick ) exponential disk is small in this scenario , so the doppler shifts support a non - stationary extended halo . reinterpreting these shifts will be necessary if future data or analyses rule out a spherical halo ( or at least a volume - filling one ) or a high resolution x - ray spectrometer resolves the lines into components inconsistent with rotation . also , we reiterate that the parametric fit strongly depends on the few agns with the highest @xmath15 , but the nonparametric tests indicate rotation at some level . the ion fraction is high between @xmath282k , so the cooler ( non - coronal ) gas seen in and around the galaxy in with _ fuse _ @xcite will also absorb . the lines towards background objects reveal galactic and high - velocity absorbers @xcite . the galactic absorbers ( also seen towards stars ) are consistent with an exponential disk of scale height 1 - 4kpc @xcite , whereas the high - velocity absorbers may come from the halo or outside the galaxy . if the column densities of these absorbers are high , then their lines could bias our results . since the disk component will rotate , we are most concerned about contamination from this gas . @xcite measured column densities for about 150 sight lines around the galaxy and found a typical column of @xmath283@xmath284 . for the sight lines in our sample , we expect @xmath285@xmath284 . if the absorbers are at @xmath286k and in collisional ionization equilibrium , the o vii/ ratio is 1.7 , which leads to an expected contribution of @xmath287@xmath284 . in contrast , the typical column ( assuming an optically thin plasma ) is @xmath288@xmath284 @xcite . hence , the contribution directly from absorbers in the galaxy is at most 15% . it is likely lower , since @xcite and @xcite argue that the lines are not optically thin . the bias from the absorbers also declines at higher latitude , since the column declines more slowly than the column with increasing @xmath217 ( corroborated in _ emission _ by * ? ? ? for most of our sight lines we estimate that the local and disk contribution to the column is less than 10% . even a 10% contribution is important for measuring the true halo velocity , but compared to the uncertainty in our best - fit velocity parameters , it is small . more importantly , as with the hot disk considered above , it can not by itself account for the measured velocities if we assume that the million - degree halo is stationary . we have measured a signature of rotation in the line around the milky way using 37 sight lines towards background agns and archival rgs data . the parametric fit strongly depends on the brightest agns , but nonparametric tests indicate that the absorbers are not stationary , even when removing suspect lines . from larger samples of emission and absorption lines , we believe that an extended halo is more consistent with the data than an exponential ( thick ) disk , and in this case this halo rotates at some velocity smaller than @xmath248 . taken at face value , this implies that the million - degree gas has a comparable angular momentum to the galaxy disk . it is possible that both components exist , in which case the rgs lines are blends of kinematically distinct components . we use a large sample of emission lines to constrain the contribution of each component to the column density , and find that an exponential disk accounts for no more than 10% of the column density . thus , the measured doppler shifts are dominated by the motion of the gas in the extended halo , which is consistent with prograde rotation . even if a disk were to dominate ( which , for our data , produces an unacceptable @xmath240 value ) , the best - fit azimuthal velocities imply that it is rotating at nearly the same speed and with a comparable amount of angular momentum to the spherical model . this conclusion depends on assumptions about the underlying model . measuring the true velocity and separating the x - ray absorbers into their various components requires a high - resolution x - ray instrument with a large effective area ( e.g. _ arcus _ , * ? ? ? * ) , which would also give important information on their line shapes that could constrain the optical depth and doppler @xmath217 parameters @xcite , as well as reveal the contribution to the total column from the disk and the coronal halo gas seen in x - ray emission . the recent work on calibrating the conversion of dispersion angle to a wavelength grid for the rgs @xcite and stacking multiple observations of the same object enables a wavelength accuracy of tens of kms@xmath4 for the first time ( figure [ figure.capella_wavelength ] ) . after investigating a variety of systematic issues , we find that the statistical uncertainty ( due to the low @xmath15 in many spectra and the relatively small sample ) remains the major source of error . the inferred halo velocity and angular momentum are strongly model dependent ( and the uncertainties that we report are large ) , but the basic conclusion that the hot gas distribution rotates is less so . several scenarios could produce a rotating halo that lags behind the disk , depending on the geometry . these include a galactic fountain of cool gas that spins up hot gas @xcite , a hot galactic fountain of superbubble ejecta that produces an exponential disk before cooling , or infall from the cosmic web with some preferential direction . our measurements can not , by themselves , distinguish between these models ( which may not be mutually exclusive ) , but they are an important kinematic constraint for future halo and galaxy formation models . we thank eric bell for analysis suggestions and constructive criticism , and oleg gnedin for information on the galaxy s angular momentum . frits paerels provided technical advice regarding the rgs . we thank the referees for substantial comments that significantly improved the manuscript . we gratefully acknowledge financial support from the nasa adap program , through grant nnx11ag55 g . , r. k. , ackermann , m. , allured , r. , et al . 2014 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 9144 , society of photo - optical instrumentation engineers ( spie ) conference series , 4
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the hot gaseous halos of galaxies likely contain a large amount of mass and are an integral part of galaxy formation and evolution .
the milky way has a @xmath0 k halo that is detected in emission and by absorption in the resonance line against bright background agns , and for which the best current model is an extended spherical distribution . using xmm - newton rgs data
, we measure the doppler shifts of the absorption - line centroids toward an ensemble of agns .
these doppler shifts constrain the dynamics of the hot halo , ruling out a stationary halo at about @xmath1 and a co - rotating halo at @xmath2 , and leading to a best - fit rotational velocity @xmath3kms@xmath4 for an extended halo model .
these results suggest that the hot gas rotates and that it contains an amount of angular momentum comparable to that in the stellar disk .
we examined the possibility of a model with a kinematically distinct disk and spherical halo . to be consistent with the emission - line x - ray data the disk must contribute less than 10% of the column density , implying that the doppler shifts probe motion in the extended hot halo .
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in recent years many attempts have been made to understand nucleon structure functions as measured in lepton deep - inelastic scattering ( dis ) . although perturbative qcd is successful in describing the dependence of structure functions on the squared momentum transfer , their magnitude is governed by the non - perturbative physics of composite particles , and is up to now not calculable directly from qcd . a variety of models have been invoked to describe nucleon structure functions . the so called `` spectator model '' is a typical covariant approach amongst them@xcite . in this approach the leading twist , non - singlet quark distributions are calculated from the process in which the target nucleon splits into a valence quark , which is scattered by the virtual photon , and a spectator system carrying baryon number @xmath4 . furthermore the spectrum of spectator states is assumed to be saturated through single scalar and vector diquarks . thus , the main ingredient of these models are covariant quark - diquark vertex functions . until now vertex functions have been merely parameterized such that the measured quark distributions are reproduced , and no attempts have been made to connect them to some dynamical models of the nucleon . in this work we construct the vertex functions from a model lagrangian by solving the bethe salpeter equation ( bse ) . however , we do not aim at a detailed , quantitative description of nucleon structure functions in the present work . rather we outline how to extract quark - diquark vertex functions from euclidean solutions of the bse . in this context several simplifications are made . we consider only scalar diquarks as spectators and restrict ourselves to the @xmath5 flavor group . the inclusion of vector diquarks and the generalization to @xmath6 flavor are relatively straightforward extensions and will be left for future work . the cross section for dis of leptons from a nucleon is characterized by the hadronic tensor @xmath7 where @xmath8 and @xmath9 are the four - momenta of the target and exchanged virtual photon respectively . for unpolarized dis , the hadronic tensor @xmath10 is conventionally parameterized by two scalar functions @xmath11 and @xmath12 . in the bjorken limit ( @xmath13 ; but finite @xmath14 ) in which we work throughout , both structure functions depend ( up to logarithmic corrections ) on @xmath15 only , and are related via the callan - gross relation : @xmath16 . in the scalar diquark spectator model within @xmath5 flavor , the valence quark distributions are extracted from the hadronic tensor ( fig . [ fig : diquarkspectatorfort ] ) : @xmath17 where the isospin matrix element @xmath18 has to be evaluated in the nucleon isospin space . we define @xmath19 as the target nucleon spinor and we use @xmath20 and @xmath21 to denote the propagators of the quark and scalar diquark , respectively . the integration runs over the quark momentum @xmath22 , subject to on mass shell conditions for the diquark and the struck quark . note that the vertex function @xmath23 and its pt conjugate @xmath24 consist of two lorentz scalar functions which depend on @xmath25 only because the diquark is on shell . in the next section we shall determine the vertex functions using a ladder bethe - salpeter equation . we consider the following model lagrangian : @xmath26 where we have explicitly indicated color @xmath6 indices only . the symmetric generator @xmath27 of the flavor @xmath5 group acts on the iso doublet field @xmath28 for the constituent quark carrying an invariant mass @xmath29 . the charged scalar field @xmath30 denotes the flavor singlet scalar diquark with invariant mass @xmath31 . in this model the nucleon with the momentum @xmath8 and spin @xmath32 is described by the bs vertex function @xmath23 ( fig . [ fig : bsvertex ] ) : @xmath33 where we set the weight factors to the classical values : @xmath34 and @xmath35 . then the vertex function @xmath23 obeys in the ladder approximation the following bse ( fig.[fig : ladderbse ] ) : @xmath36 to solve the bse for positive energy nucleon states we are free to choose the following dirac matrix structure : @xmath37\lambda^{(+)}(p ) , \label{define_f}\ ] ] with @xmath38 , the projector onto positive energy nucleon states . we assume that the diquark and the nucleon are stable , namely @xmath39 and @xmath40 . we then perform the wick rotation of the relative energy variable and choose the nucleon rest frame : @xmath41 . the `` euclidean '' functions @xmath42 in terms of the momentum @xmath43 for a real @xmath44 are then functions of @xmath45 and @xmath46 $ ] . we expand each of these functions ( @xmath47 ) as follows : @xmath48 where @xmath49 are gegenbauer polynomials and the phase @xmath50 is introduced for convenience since then the radial functions @xmath51 are real . the bse in eq.([ladderbse ] ) then reduces to the following system of one dimensional equations : @xmath52 where we have introduced the `` eigenvalue '' @xmath53 for the quark diquark coupling constant . the kernel function @xmath54 is a matrix whose elements are real and regular functions of @xmath55 and @xmath56 . by terminating the infinite series in eq.([expanf ] ) at sufficiently high order , we can easily solve eq.([fn_eq ] ) numerically as an `` eigenvalue '' problem for a fixed bound state mass , @xmath57 . to compare the magnitude of the radial functions , let us introduce the normalized @xmath58 radial functions @xmath59 and @xmath60 together with the @xmath58 spherical spinor harmonics @xmath61@xcite . in the conventional gamma matrix representation we can write the nucleon solution at rest as : @xmath62 where @xmath63 denotes angles for @xmath64 vector in the four dimentional polar coordinate system . the factor @xmath65 is introduced such that @xmath66 , while @xmath60 is expressed as a linear combination of @xmath67 and @xmath68 . thus @xmath69 and @xmath70 correspond to the `` upper '' and `` lower '' components of the nucleon dirac field , respectively . now , let us consider the analytic continuation of @xmath71 . since we are interested in applying the bs vertex function to the dis process , we need to rotate the relative energy variable from on the imaginary axis back to on the real one . recall that the sum over @xmath72 in eq.([expanf ] ) converges even for a complex @xmath73 , if @xmath74 . we can then analytically continue the gegenbauer polynomials rewriting the argument @xmath75 for the momenta satisfying @xmath76 . for the radial functions we introduce new functions @xmath77 and @xmath78 . and @xmath79 are functions of @xmath80 was confirmed numerically . ] we analytically continue these functions by changing the argument @xmath81 . we obtain the physical scalar functions @xmath82 : @xmath83 note that the gegenbauer polynomials together with the square root factors are polynomials of @xmath84 , @xmath57 , and @xmath85 , so that each term in the series ( [ f1minkowski ] ) and ( [ f2minkowski ] ) is regular and real as far as @xmath86 and @xmath87 are regular . we may then impose the on mass shell condition for the diquark and evaluate the sum over @xmath72 . the resulting @xmath88 and @xmath89 are then functions of the squared quark momentum @xmath25 and can be applied to the dis process . however , the expressions ( [ f1minkowski ] ) and ( [ f2minkowski ] ) are valid only for momenta satisfying @xmath90 . also we found that a naive numerical sum over @xmath72 , based on eqs.([f1minkowski ] ) and ( [ f2minkowski ] ) does not converge . nevertheless , it can be shown that the vertex function @xmath23 for any kinematically allowed @xmath25 is regular when the diquark is on mass shell . this suggests that one may be able to continue some appropriate linear combinations of @xmath88 and @xmath89 outside of this kinematical range . indeed , we found such a combination which we denote by @xmath91 . in terms of this on shell scalar functions and the quark momentum , @xmath22 , the vertex function @xmath23 together with the diquark on mass shell condition is given by @xmath92 with this on shell vertex function the valence contribution to the structure function @xmath93 can now be calculated from eq.([compton ] ) . in this section we present our numerical results . for simplicity we considered an equal mass system ; @xmath94 , and we shall use @xmath95 as a unit for dimensionful quantities . to solve the bse we used a @xmath96channel form factor . we replaced the quark diquark coupling constant such that @xmath97 with @xmath98 and @xmath96 is the usual mandelstam variable . this form factor weakens the short range interaction between the constituents and ensures the existence of a discrete bound state spectrum for a large range of @xmath57 . we solved eq.([fn_eq ] ) as follows . first we terminated the infinite series in eq.([expanf ] ) at some fixed value , @xmath99 . next we discretized the euclidean momentum @xmath55 and @xmath56 and performed the integration over @xmath56 numerically together with some initially assumed radial functions @xmath100 . this integral generated new radial functions and an `` eigenvalue '' @xmath101 associated with them . we then used these functions as an input and repeated the above procedure until the radial functions and @xmath101 converged . in fig . [ fig : fandg ] we plot the normalized @xmath58 radial functions , @xmath59 and @xmath60 , for the bound state mass @xmath102 as functions of @xmath55 . it is clear that the magnitude of the radial functions with higher @xmath58 angular momentum are strongly suppressed compared with the lowest ones . this justifies the truncation of the series in eq.([expanf ] ) . it also confirms that the contribution to the `` eigenvalue '' @xmath101 from the @xmath58 radial functions with @xmath103 is less than 1 % . the dominance of the lowest @xmath58 radial function has been also observed in the scalar scalar ladder model@xcite . in fig . [ fig : fon ] we plot the physical , on shell scalar functions @xmath104 for the bound state mass @xmath102 as functions of the squared quark momentum , @xmath25 . these functions are calculated with the maximum @xmath58 angular momentum @xmath105 . we found that the magnitude of @xmath106 and @xmath107 are almost the same even for weakly bound states . this result suggests that so called `` non relativistic '' approximations , in which one neglects the non leading components of the vertex function ( @xmath89 in our model ) are valid only for extremely weakly bound states : @xmath108 . we have also confirmed that for weakly bound states ( @xmath109 ) the dependence of @xmath104 on @xmath110 is negligible for a small spacelike @xmath25 , e.g. , @xmath111 . however , for large spacelike @xmath25 , the convergence of the sum over @xmath72 becomes slow for any value of @xmath57 and numerical results for fixed @xmath110 become less accurate . in fig.[fig : struct ] we plot the valence quark distribution @xmath93 for a weakly ( @xmath102 , solid line ) and for a strongly bound state ( @xmath112 , dashed line ) . we have used @xmath105 and the distributions are normalized such that the area below the curve is unity . for the weakly bound system , the valence quark distribution has a peak , if quark and diquark masses such as @xmath113 are used . ] around @xmath114 . on the other hand , the distribution becomes flat for the strongly bound system . this behavior however turns out to be mainly of kinematic origin , since the distribution function is given by an integral over the squared momentum @xmath25 carried by the struck quark with integration bounds from @xmath115 to @xmath116 , where @xmath117 this kinematical bound determines the global shape of @xmath93 to a large extent . we have solved a bse for a nucleon , described as a bound state of a quark and diquark within a covariant quark scalar - diquark model . we have extracted the physical quark - diquark vertex function when the diquark is on mass shell from the euclidean solution . the vertex function obtained was applied to a diquark spectator model for dis , and the valence quark contribution to the structure function @xmath93 was calculated . we found that the shape of the unpolarized valence quark distribution is mainly determined by relativistic kinematics and is independent of the detailed structure of the vertex function . h. meyer and p.j . mulders , _ nucl . phys . _ * a528 * , 589 ( 1991 ) ; w. melnitchouk , a.w . schreiber and a.w . thomas , _ phys . rev . _ * d49 * , 1183 ( 1994 ) ; w. melnitchouk , a.w . schreiber and a.w . thomas , _ phys . * b335 * , 11 ( 1994 ) ; w. melnitchouk , g. piller and a.w . thomas , _ phys . _ * b346 * , 165 ( 1995 ) . [ fig : fon ] : the on shell scalar functions @xmath106 ( solid ) and @xmath107 ( dashed ) as functions of the quark momentum @xmath25 . the mass of the bound sate is @xmath120 and the @xmath58 angular momentum is truncated at @xmath105 .
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nucleon structure functions , as measured in deep - inelastic lepton scattering , are studied within a covariant scalar diquark spectator model . regarding the nucleon as an approximate two - body bound state of a quark and diquark ,
the bethe - salpeter equation ( bse ) for the bound state vertex function is solved in the ladder approximation .
the valence quark distribution is discussed in terms of the solutions of the bse .
= 15.9 cm = -1 cm adp-96 - 5/t210 , hep - ph/9603260 + _ talk given at the joint japan - australia workshop on quarks , hadrons + and nuclei , adelaide , south australia , november 15 - 24 , 1995 + ( to appear in the conference proceedings ) _ * structure functions of the nucleon + in a covariant scalar spectator model * + k. kusaka@xmath0 , g. piller@xmath1 , a.w .
thomas@xmath2 and a.g .
williams@xmath2 + _ @xmath0department of physics and mathematical physics , university of adelaide , + s.a . 5005 , australia _ + + _
@xmath3institute for theoretical physics , university of adelaide , s.a . 5005 ,
australia _ + e - mail : _ kkusaka , awilliam , [email protected]_ +
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this paper reports on the still preliminary , but already satisfying results of the learning computational grammars ( lcg ) project , a postdoc network devoted to studying the application of machine learning techniques to grammars suitable for computational use . the member institutes are listed with the authors and also included issco at the university of geneva . we were impressed by early experiments applying learning to natural language , but dissatisfied with the concentration on a few techniques from the very rich area of machine learning . we were interested in a more systematic survey to understand the relevance of many factors to the success of learning , esp . the availability of annotated data , the kind of dependencies in the data , and the availability of knowledge bases ( grammars ) . we focused on syntax , esp . noun phrase ( np ) syntax from the beginning . the industrial partner , xerox , focused on more immediate applications @xcite . the network was focused not only by its scientific goal , the application and evaluation of machine - learning techniques as used to learn natural language syntax , and by the subarea of syntax chosen , np syntax , but also by the use of shared training and test material , in this case material drawn from the penn treebank . finally , we were curious about the possibility of combining different techniques , including those from statistical and symbolic machine learning . the network members played an important role in the organisation of three open workshops in which several external groups participated , sharing data and test materials . this section starts with a description of the three tasks that we have worked on in the framework of this project . after this we will describe the machine learning algorithms applied to this data and conclude with some notes about combining different system results . in the framework of this project , we have worked on the following three tasks : 1 . base phrase ( chunk ) identification 2 . base noun phrase recognition 3 . finding arbitrary noun phrases text chunks are non - overlapping phrases which contain syntactically related words . for example , the sentence : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath0 he @xmath1 $ ] @xmath2 reckons @xmath1 $ ] @xmath0 the current account deficit @xmath1 $ ] @xmath2 will narrow @xmath1 $ ] + @xmath3 to @xmath1 $ ] @xmath0 only @xmath4 1.8 billion @xmath1 $ ] + @xmath3 in @xmath1 $ ] @xmath0 september @xmath1 $ ] . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ contains eight chunks , four np chunks , two vp chunks and two pp chunks . the latter only contain prepositions rather than prepositions plus the noun phrase material because that has already been included in np chunks . the process of finding these phrases is called chunking . the project provided a data set for this task at the conll-2000 workshop @xcite . it consists of sections 15 - 18 of the wall street journal part of the penn treebank ii @xcite as training data ( 211727 tokens ) and section 20 as test data ( 47377 tokens ) . a specialised version of the chunking task is np chunking or basenp identification in which the goal is to identify the base noun phrases . the first work on this topic was done back in the eighties @xcite . the data set that has become standard for evaluation machine learning approaches is the one first used by ramshaw and marcus . it consists of the same training and test data segments of the penn treebank as the chunking task ( respectively sections 15 - 18 and section 20 ) . however , since the data sets have been generated with different software , the np boundaries in the np chunking data sets are slightly different from the np boundaries in the general chunking data . noun phrases are not restricted to the base levels of parse trees . for example , in the sentence _ in early trading in hong kong monday , gold was quoted at $ 366.50 an ounce . _ , the noun phrase @xmath0 $ 366.50 an ounce @xmath1 $ ] contains two embedded noun phrases @xmath0 $ 366.50 @xmath1 $ ] and @xmath0 an ounce @xmath1 $ ] . in the np bracketing task , the goal is to find all noun phrases in a sentence . data sets for this task were defined for conll-99 . the data consist of the same segments of the penn treebank as the previous two tasks ( sections 15 - 18 ) as training material and section 20 as test material . this material was extracted directly from the treebank and therefore the np boundaries at base levels are different from those in the previous two tasks . in the evaluation of all three tasks , the accuracy of the learners is measured with three rates . we compare the constituents postulated by the learners with those marked as correct by experts ( gold standard ) . first , the percentage of detected constituents that are correct ( precision ) . second , the percentage of correct constituents that are detected ( recall ) . and third , a combination of precision and recall , the f@xmath5 rate which is equal to ( 2*precision*recall)/(precision+recall ) . this section introduces the ten learning methods that have been applied by the project members to the three tasks : lscgs , allis , lsommbl , maximum entropy , aleph , mdl - based dcg learners , finite state transducers , ib1ig , igtree and c5.0 . * local structural context grammars * ( lscgs ) @xcite are situated between conventional probabilistic context - free production rule grammars and dop - grammars ( e.g. , bod and scha ) . lscgs outperform the former because they do not share their inherent independence assumptions , and are more computationally efficient than the latter , because they incorporate only subsets of the context included in dop - grammars . local structural context ( lsc ) is ( partial ) information about the immediate neighbourhood of a phrase in a parse . by conditioning bracketing probabilities on lsc , more fine - grained probability distributions can be achieved , and parsing performance increased . given corpora of parsed text such as the wsj , lscgs are used in automatic grammar construction as follows . an lscg is derived from the corpus by extracting production rules from bracketings and annotating the rules with the type(s ) of lsc to be incorporated in the lscg ( e.g. parent category information , depth of embedding , etc . ) . rule probabilities are derived from rule frequencies ( currently by maximum likelihood estimation ) . in a separate optimisation step , the resulting lscgs are optimised in terms of size and parsing performance for a given parsing task by an automatic method ( currently a version of beam search ) that searches the space of partitions of a grammar s set of nonterminals . the lscg research efforts differ from other approaches reported in this paper in two respects . firstly , no lexical information is used at any point , as the aim is to investigate the upper limit of parsing performance without lexicalisation . secondly , grammars are optimised for parsing performance _ and _ size , the aim being to improve performance but not at the price of arbitrary increases in grammar complexity ( hence the cost of parsing ) . the automatic optimisation of corpus - derived lscgs is the subject of ongoing research and the results reported here for this method are therefore preliminary . * theory refinement * ( allis ) . allis ( @xcite , @xcite ) is a inductive rule - based system using a traditional general - to - specific approach @xcite . after generating a default classification rule ( equivalent to the n - gram model ) , allis tries to refine it since the accuracy of these rules is usually not high enough . refinement is done by adding more premises ( contextual elements ) . allis uses data encoded in xml , and also learns rules in xml . from the perspective of the xml formalism , the initial rule can be viewed as a tree with only one leaf , and refinement is done by adding adjacent leaves until the accuracy of the rule is high enough ( a tuning threshold is used ) . these additional leaves correspond to more precise contextual elements . using the hierarchical structure of an xml document , refinement begins with the highest available hierarchical level and goes down in the hierarchy ( for example , starting at the chunk level and then word level ) . adding new low level elements makes the rules more specific , increasing their accuracy but decreasing their coverage . after the learning is completed , the set of rules is transformed into a proper formalism used by a given parser . * labelled som and memory based learning * ( lsommbl ) is a neurally inspired technique which incorporates a modified self - organising map ( som , also known as a ` kohonen map ' ) in memory - based learning to select a subset of the training data for comparison with novel items . the som is trained with labelled inputs . during training , each unit in the map acquires a label . when an input is presented , the node in the map with the highest activation ( the ` winner ' ) is identified . if the winner is unlabelled , then it acquires the label from its input . labelled units only respond to similarly labelled inputs . otherwise training proceeds as with the normal som . when training ends , all inputs are presented to the som , and the winning units for the inputs are noted . any unused units are then discarded . thus each remaining unit in the som is associated with the set of training inputs that are closest to it . this is used in mbl as follows . the labelled som is trained with inputs labelled with the output categories . when a novel item is presented , the winning unit for each category is found , the training items associated with the winning units are searched for the closest item to the novel item and the most frequent classification of that item is used as the classification for the novel item . * maximum entropy * when building a classifier , one must gather evidence for predicting the correct class of an item from its context . the maximum entropy ( maxent ) framework is especially suited for integrating evidence from various information sources . frequencies of evidence / class combinations ( called features ) are extracted from a sample corpus and considered to be properties of the classification process . attention is constrained to models with these properties . the maxent principle now demands that among all the probability distributions that obey these constraints , the most uniform is chosen . during training , features are assigned weights in such a way that , given the maxent principle , the training data is matched as well as possible . during evaluation it is tested which features are _ active _ ( i.e. , a feature is active when the context meets the requirements given by the feature ) . for every class the weights of the active features are combined and the best scoring class is chosen @xcite . for the classifier built here we use as evidence the surrounding words , their pos tags and basenp tags predicted for the previous words . a mixture of simple features ( consisting of one of the mentioned information sources ) and complex features ( combinations thereof ) were used . the left context never exceeded 3 words , the right context was maximally 2 words . the model was calculated using existing software @xcite . * inductive logic programming ( ilp ) * aleph is an ilp machine learning system that searches for a hypothesis , given positive ( and , if available , negative ) data in the form of ground prolog terms and background knowledge ( prior knowledge made available to the learning algorithm ) in the form of prolog predicates . the system , then , constructs a set of hypothesis clauses that fit the data and background as well as possible . in order to approach the problem of np chunking in this context of single - predicate learning , it was reformulated as a tagging task where each word was tagged as being ` inside ' or ` outside ' a basenp ( consecutive nps were treated appropriately ) . then , the target theory is a prolog program that correctly predicts a word s tag given its context . the context consisted of pos tagged words and syntactically tagged words to the left and pos tagged words to the right , so that the resulting tagger can be applied in the left - to - right pass over pos - tagged text . * minimum description length * ( mdl ) estimation using the minimum description length principle involves finding a model which not only ` explains ' the training material well , but also is compact . the basic idea is to balance the generality of a model ( roughly speaking , the more compact the model , the more general it is ) with its specialisation to the training material . we have applied mdl to the task of learning broad - covering definite - clause grammars from either raw text , or else from parsed corpora @xcite . preliminary results have shown that learning using just raw text is worse than learning with parsed corpora , and that learning using both parsed corpora and a compression - based prior is better than when learning using parsed corpora and a uniform prior . furthermore , we have noted that our instantiation of mdl does not capture dependencies which exist either in the grammar or else in preferred parses . ongoing work has focused on applying random field technology ( maximum entropy ) to mdl - based grammar learning ( see osborne for some of the issues involved ) . * finite state transducers * are built by interpreting probabilistic automata as transducers . we use a probabilistic grammatical algorithm , the ddsm algorithm @xcite , for learning automata that provide the probability of an item given the previous ones . the items are described by bigrams of the format feature : class . in the resulting automata we consider a transition labeled feature : class as the transducer transition that takes as input the first part ( feature ) of the bigram and outputs the second part ( class ) . by applying the viterbi algorithm on such a model , we can find out the most probable set of class values given an input set of feature values . as the ddsm algorithm has a tuning parameter , it can provide many different automata . we apply a majority vote over the propositions made by the so computed automata / transducers for obtaining the results mentioned in this paper . * memory - based learning * methods store all training data and classify test data items by giving them the classification of the training data items which are most similar . we have used three different algorithms : the nearest neighbour algorithm ib1ig , which is part of the timbl software package @xcite , the decision tree learner igtree , also from timbl , and c5.0 , a commercial version of the decision tree learner c4.5 @xcite . they are classifiers which means that they assign phrase classes such as i ( inside a phrase ) , b ( at the beginning of a phrase ) and o ( outside a phrase ) to words . in order to improve the classification process we provide the systems with extra information about the words such as the previous _ n _ words , the next _ n _ words , their part - of - speech tags and chunk tags estimated by an earlier classification process . we use the default settings of the software except for the number of examined nearest neighbourhood regions for ib1ig ( k , default is 1 ) which we set to 3 . when different systems are applied to the same problem , a clever combination of their results will outperform all of the individual results @xcite . the reason for this is that the systems often make different errors and some of these errors can be eliminated by examining the classifications of the others . the most simple combination method is majority voting . it examines the classifications of the test data item and for each item chooses the most frequently predicted classification . despite its simplicity , majority voting has found to be quite useful for boosting performance on the tasks that we are interested in . we have applied majority voting and nine other combination methods to the output of the learning systems that were applied to the three tasks . nine combination methods were originally suggested by van halteren et al . . five of them , including majority voting , are so - called voting methods . apart from majority voting , all assign weights to the predictions of the different systems based on their performance on non - used training data , the tuning data . totprecision uses classifier weights based on their accuracy . tagprecision applies classification weights based on the accuracy of the classifier for that classification . precision - recall uses classification weights that combine the precision of the classification with the recall of the competitors . and finally , tagpair uses classification pair weights based on the probability of a classification for some predicted classification pair @xcite . the remaining four combination methods are so - called stacked classifiers . the idea is to make a classifier process the output of the individual systems . we used the two memory - based learners ib1ig and igtree as stacked classifiers . like van halteren et al . , we evaluated two features combinations . the first consisted of the predictions of the individual systems and the second of the predictions plus one feature that described the data item . we used the feature that , according to the memory - based learning metrics , was most relevant to the tasks : the part - of - speech tag of the data item . in the course of this project we have evaluated another combination method : best - n majority voting @xcite . this is similar to majority voting except that instead of using the predictions of all systems , it uses only predictions from some of the systems for determining the most probable classifications . we have experienced that for different reasons some systems perform worse than others and including their results in the majority vote decreases the combined performance . therefore it is a good idea to evaluate majority voting on subsets of all systems rather than only on the combination of all systems . apart from standard majority voting , all combination methods require extra data for measuring their performance which is required for determining their weights , the tuning data . this data can be extracted from the training data or the training data can be processed in an n - fold cross - validation process after which the performance on the complete training data can be measured . although some work with individual systems in the project has been done with the goal of combining the results with other systems , tuning data is not always available for all results . therefore it will not always be possible to apply all ten combination methods to the results . in some cases we have to restrict ourselves to evaluating majority voting only . this sections presents the results of the different systems applied to the three tasks which were central to this this project : chunking , np chunking and np bracketing . chunking was the shared task of conll-2000 , the workshop on computational natural language learning , held in lisbon , portugal in 2000 @xcite . six members of the project have performed this task . the results of the six systems ( precision , recall and f@xmath5 can be found in table [ tab - chunking ] . belz used local structural context grammars for finding chunks . djean applied the theory refinement system allis to the shared task data . koeling evaluated a maximum entropy learner while using different feature combinations ( me ) . osborne used a maximum entropy - based part - of - speech tagger for assigning chunk tags to words ( me tag ) . thollard identified chunks with finite state transducers generated by a probabilistic grammar algorithm ( fst ) . tjong kim sang tested different configurations of combined memory - based learners ( mbl ) . the fst and the lscg results are lower than those of the other systems because they were obtained without using lexical information . the best result at the workshop was obtained with support vector machines @xcite . . the chunking results for the six systems associated with the project ( shared task conll-2000 ) . the baseline results have been obtained by selecting the most frequent chunk tag associated with each part - of - speech tag . the best results at conll-2000 were obtained by support vector machines . a majority vote of the six lcg systems does not perform much worse than this best result . a majority vote of the five best systems outperforms the best result slightly ( @xmath6 error reduction ) . [ cols="<,^,^,^",options="header " , ] finding arbitrary noun phrases was the shared task of conll-99 , held in bergen , norway in 1999 . three project members have performed this task . belz extracted noun phrases with local structural context grammars , a variant of data - oriented parsing ( lscg ) . osborne used a definite clause grammar learner based on minimum description length for finding noun phrases in samples of penn treebank material ( mdl ) . tjong kim sang detected noun phrases with a bottom - up cascade of combinations of memory - based classifiers ( mbl ) . the performance of the three systems can be found in table [ tab - npbracketing ] . for this task it was not possible to apply system combination to the output of the system . the mdl results have been obtained on a different data set and this left us with two remaining systems . a majority vote of the two will not improve on the best system and since there was no tuning data or development data available , other combination methods could not be applied . the project has proven to be successful in its results for applying machine learning techniques to all three of its selected tasks : chunking , np chunking and np bracketing . we are looking forward to applying these techniques to other nlp tasks . three of our project members will take part in the conll-2001 shared task , ` clausing ' , hopefully with good results . two more have started working on the challenging task of full parsing , in particular by starting with a chunker and building a bottom - up arbitrary phrase recogniser on top of that . the preliminary results are encouraging though not as good as advanced statistical parsers like those of charniak and collins . it is fair to characterise lcg s goals as primarily technical in the sense that we sought to maximise performance rates , esp . the recognition of different levels of np structure . our view in the project is certainly broader , and most project members would include learning as one of the language processes one ought to study from a computational perspective like parsing or generation . this suggest several further avenues , e.g. , one might compare the learning progress of simulations to humans ( mastery as a function of experience ) . one might also be interested in the exact role of supervision , in the behaviour ( and availability ) of incremental learning algorithms , and also in comparing the simulation s error functions to those of human learners ( wrt to phrase length or construction frequency or similarity ) . this would add an interesting cognitive perspective to the work , along the lines begun by brent , but we note it here only as a prospect for future work . r. bod and r. scha . data - oriented language processing . in s. young and g. bloothooft , editors , _ corpus - based methods in language and speech processing _ , pages 137173 . kluwer academic publishers , boston . erik f. tjong kim sang , walter daelemans , herv djean , rob koeling , yuval krymolowski , vasin punyakanok , and dan roth . 2000 . applying system combination to base noun phrase identification . in _ proceedings of the 18th international conference on computational linguistics ( coling 2000)_. saarbruecken , germany .
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this paper reports on the learning computational grammars ( lcg ) project , a postdoc network devoted to studying the application of machine learning techniques to grammars suitable for computational use .
we were interested in a more systematic survey to understand the relevance of many factors to the success of learning , esp .
the availability of annotated data , the kind of dependencies in the data , and the availability of knowledge bases ( grammars ) .
we focused on syntax , esp .
noun phrase ( np ) syntax .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in the present almost frenetic rate of advance of cosmology it is useful to be reminded that the big news this year is the establishment of evidence , by two groups ( @xcite , @xcite ) , of detection of the relativistic curvature of the redshift - magnitude relation . the measurement was proposed in the early 1930s . compare this to the change in the issues in particle physics since 1930 . the slow evolution of cosmology has allowed ample time for us to lose sight of which elements are reasonably well established and which have been adopted by default , for lack of more reasonable - looking alternatives . thus i think it is appropriate to devote a good part of my assigned space to a discussion of what might be included in the standard model for cosmology . i then comment on additions that may come out of work in progress . main elements of the model are easily listed : in the large - scale average the universe is close to homogeneous , and has expanded in a near homogeneous way from a denser hotter state when the 3 k cosmic background radiation was thermalized . the standard cosmology assumes conventional physics , including general relativity theory . this yields a successful account of the origin of the light elements , at expansion factor @xmath0 . light element formation tests the relativistic relation between expansion rate and mass density , but this is not a very searching probe . the cosmological tests discussed in 3 could considerably improve the tests of general relativity . the model for the light elements seems to require that the mass density in baryons is less than that needed to account for the peculiar motions of the galaxies . it is usually assumed that the remainder is nonbaryonic ( or acts that way ) . our reliance on hypothetical dark matter is an embarrassment ; a laboratory detection would be exceedingly welcome . in the past decade many discussions assumed the einstein - de sitter case , in which there are negligibly small values for the curvature of sections of constant world time and einstein s cosmological constant @xmath1 ( or a term in the stress - energy tensor that acts like one ) . this is what most of would have chosen if we were ordering . but the evidence from the relative velocities of the galaxies has long been that the mass density is less than the einstein - de sitter value @xcite , and other more recent observations , notably the curvature of the redshift - magnitude relation ( @xcite , @xcite ) , point in the same direction . now there is increasing interest in the idea that we live in a universe in which the dominant term in the stress - energy tensor acts like a decaying cosmological constant ( @xcite - @xcite ) . this is not part of the standard model , of course , but as discussed in 3 the observations seem to be getting close to useful constraints on space curvature and @xmath1 . we have good reason to think structure formation on the scale of galaxies and larger was a result of the gravitational growth of small primeval departures from homogeneity , as described by general relativity in linear perturbation theory . the adiabatic cold dark matter ( acdm ) model gives a fairly definite and strikingly successful prescription for the initial conditions for this gravitational instability picture , and the acdm model accordingly is widely used in analyses of structure formation . but we can not count it as part of the standard model because there is at least one viable alternative , the isocurvature model mentioned in 3.3 . observations in progress likely will eliminate at least one , perhaps establish the other as a good approximation to how the galaxies formed , or perhaps lead us to something better . the observational basis for this stripped - down standard model is reviewed in references @xcite and @xcite . here i comment on some issues now under discussion . pietronero @xcite argues that the evidence from redshift catalogs and deep galaxy counts is that the galaxy distribution is best described as a scale - invariant fractal with dimension @xmath2 . others disagree ( @xcite , @xcite ) . i am heavily influenced by another line of argument : it is difficult to reconcile a fractal universe with the isotropy observed in deep surveys ( examples of which are illustrated in figs . 3.7 to 3.11 in @xcite and are discussed in connection with the fractal universe in pp . 209 - 224 in @xcite ) . -1.0truecm fig . 1 shows angular positions of particles in three ranges of distance from a particle in a fractal realization with dimension @xmath3 in three dimensions . at @xmath3 the expected number of neighbors scales with distance @xmath4 as @xmath5 , and i have scaled the fraction of particles plotted as @xmath6 to get about the same number in each plot . the fractal is constructed by placing a stick of length @xmath7 , placing on either end the centers of sticks of length @xmath8 , where @xmath9 , with random orientation , and iterating to smaller and larger scales . the particles are placed on the ends of the shortest sticks in the clustering hierarchy . this construction with @xmath10 ( and some adjustments to fit the galaxy three- and four - point correlation functions ) gives a good description of the small - scale galaxy clustering @xcite . the fractal in fig . 1 , with @xmath3 , the dimension pietronero proposes , does not look at all like deep sky maps of galaxy distributions , which show an approach to isotropy with increasing depth . this can not happen in a scale - invariant fractal : it has no characteristic length . a characteristic clustering length for galaxies may be expressed in terms of the dimensionless two - point correlation function defined by the joint probability of finding galaxies centered in the volume elements @xmath11 and @xmath12 at separation @xmath13 , dp = n^2[1+_gg(r)]dv_1dv_2 . [ eq : xigg ] the galaxy two - point function is quite close to a power law , = ( r_o / r)^,= 1.77 , 10hr10 , [ eq : xiggparameters ] where the clustering length is r_o=4.50.5 , and the hubble parameter is h_o=100h^-1 ^ -1 . the rms fluctuation in galaxy counts in a randomly placed sphere is @xmath14 at sphere radius @xmath15 mpc , to be compared to the hubble distance ( at which the recession velocity approaches the velocity of light ) , @xmath16 mpc . the isotropy observed in deep sky maps is consistent with a universe that is inhomogeneous but spherically symmetric about our position . there are tests , as discussed by paczyski and piran @xcite . for example , we have a successful theory for the origin of the light elements as remnants of the expansion and cooling of the universe through @xmath17 mev @xcite . if there were a strong radial matter density gradient out to the hubble length we could be using the wrong local entropy per baryon , based on conditions at the hubble length where the cbr came from , yet the theory seems to be successful . but to most people the compelling argument is that distant galaxies look like equally good homes for observers like us : it would be startling if we lived in one of the very few close to the center of symmetry . mandelbrot @xcite points out that other fractal constructions could do better than the one in fig . 1 . his example does have more particles in the voids defined by the strongest concentrations in the sky , but it seems to me to share the distinctly clumpy character of fig . 1 . it would be interesting to see a statistical test . a common one expands the angular distribution in a given range of distances in spherical harmonics , a_l^m = d()y_l^m ( ) , [ eq : alm ] where @xmath18 is the surface mass density as a function of direction @xmath19 in the sky . the integral becomes a sum if the fractal is represented as a set of particles . a measure of the angular fluctuations is e_l = l_-l < m < l |a_l^m|^2/(a_0 ^ 0)^2 , where ^2/^2 - 1 = _ l 1 e_l / l . [ eq : variance ] in the approximation of the sum as an integral @xmath20 is the contribution to the variance of the angular distribution per logarithmic interval of @xmath21 . it will be recalled that the zeros of the real and imaginary parts of @xmath22 are at separation @xmath23 in the shorter direction , except where the zeros crowd together near the poles and @xmath22 is close to zero . thus @xmath20 is the variance of the fractional fluctuation in density across the sky on the angular scale @xmath24 and in the chosen range of distances from the observer . i can think of two ways to define the dimension of a fractal that produces a close to isotropic sky . first , each octant of a full sky sample has half the diameter of the full sample , so one might define @xmath25 by the fractional departure of the mean density within each octant from the mean in the full sample , ( e_2)^1/2~2 ^ 3-d - 1.[eq : e_2 ] thus in fig . 1 , with @xmath3 , the quadrupole anisotropy @xmath26 is on the order of unity . second , one can use the idea that the mean particle density varies with distance @xmath13 from a particle as @xmath27 . then the small angle ( large @xmath21 ) limber approximation to the angular correlation function @xmath28 is @xcite 1+w(=/l)~^l e_l dl / l _ 0 ^ 1du [ u^2 + ( /l)^2]^-(3-d)/2 . [ eq : e_l ] to find @xmath20 differentiate with respect to @xmath21 . at @xmath3 this gives @xmath29 : the surface density fluctuations are independent of scale . at @xmath30 , @xmath31 . the x - ray background fluctuates by about @xmath32 at @xmath33 , or @xmath34 . this is equivalent to @xmath35 in the fractal model in eq . ( [ eq : e_l ] ) . the universe is not exactly homogeneous , but it seems to be remarkably close to it on the scale of the hubble length . it would be interesting to know whether there is a fractal construction that allows a significantly larger value of @xmath36 for given @xmath20 than in this calculation . expansion that preserves homogeneity requires that the mean rate of change of separation of pairs of galaxies with separation @xmath4 varies as the hubble law , v = hr.[eq : hl ] the redshift - distance relation for type ia supernovae gives an elegant demonstration of this relation ( @xcite , @xcite ) . arp ( @xcite , @xcite ) points out that such precision tests do not directly apply to the quasars , and he finds fascinating evidence in sky maps for associations of quasars with galaxies at distinctly lower redshifts . but there is a counterargument , along lines pioneered by bergeron @xcite , as follows . a quasar spectrum may contain absorption lines characteristic of a cloud of neutral atomic hydrogen at surface density @xmath37 atoms @xmath38 . if this absorption system is at redshift @xmath39 a galaxy at the same redshift is close enough that there is a reasonable chance observing it , and with high probability an optical image does show a galaxy close to the quasar and at the redshift of the absorption lines ( @xcite , @xcite ) . also , when a galaxy image appears in the sky close to a quasar at higher redshift then with high probability the quasar spectrum has absorption lines at the redshift of the galaxy . we have good evidence the galaxy is at the distance indicated by its redshift . we can be sure the quasar is behind the galaxy : the quasar light had to have passed through the galaxy to have produced the absorption lines . if quasars were not at their cosmological distances we ought to have examples of a quasar appearing close to the line of sight to a lower redshift galaxy and without the characteristic absorption lines produced by the gas in and around the galaxy . arp s approach to this issue is important , but i am influenced by what seems to be this direct and clear interpretation of the bergeron effect , that indicates redshift is a good measure of distance for quasars as well as galaxies . in the relativistic friedmann - lematre cosmological model the wavelength of a freely propagating photon is stretched in proportion to the expansion factor from the epoch of emission to detection : 1+z=_obs_em = a_obsa_em . [ eq : redshift ] the first expression defines the redshift @xmath40 in terms of the ratio of observed wavelength to wavelength at emission . the cosmological expansion parameter @xmath41 is proportional to the mean distance between conserved particles . the most direct evidence that the redshift is a result of expansion is the thermal spectrum of the cbr @xcite . in a tired light model in a static universe the photons suffer a redshift that is proportional to the distance travelled , but in the absence of absorption or emission the photon number density remains constant . in this case a significant redshift makes an initially thermal spectrum distinctly not thermal and inconsistent with the measured cbr spectrum . one could avoid this by assuming the mean free path for absorption and emission of cbr photons is much shorter than the hubble length , so relaxation to thermal equilibrium is much faster than the rate of distortion of the spectrum by the redshift . but this opaque universe is quite inconsistent with the observation of radio galaxies at redshifts @xmath42 at cbr wavelengths . that is , the universe can not have an optical depth large enough to preserve a thermal cbr spectrum in a tired light model . in the standard world model the expansion has two effects : it redshifts the photons , as @xmath43 , and it dilutes the photon number density , as @xmath44 . the result is to cool the cbr while keeping its spectrum thermal . thus the expanding universe allows a self - consistent picture : the cbr was thermalized in the past , at a time when when the universe was denser , hotter , and optically thick . i have not encountered any serious objection to this argument ; the issue is the expansion factor . in the relativistic friedmann - lematre model the expansion of the universe traces back at least as far as redshift @xmath0 , when the light elements formed in observationally reasonable amounts @xcite . in the model of arp _ et al . _ @xcite the expansion and cooling traces back to a redshift only moderately greater than the largest observed values , @xmath45 , when there would have been a burst of creation of matter and radiation followed by rapid clearing of the dust that thermalized the radiation . the arp _ _ picture for the origin of the light elements has not been widely debated . if it were agreed that it is viable then a choice between this and the friedmann - lematre model would depend on other tests , such as the angular fluctuations in the cbr , as discussed next . -1.0truecm the tests in table 1 are organized in four categories : spacetime geometry , galaxy peculiar velocities , structure formation , and early universe physics . i offer grades for three sets of parameter choices . as the tests improve we may learn that one narrowly constrained set of values of the cosmological parameters receives consistent passing grades , or else that we have to cast our theoretical net more broadly . in the relativistic friedmann - lematre cosmological model the mean spacetime geometry ( ignoring curvature fluctuations produced by local mass concentrations in galaxies and systems of galaxies ) may be represented by the line element ds^2 = dt^2 - a(t)^2 , [ eq : lineelement ] where the expansion rate satisfies the equation h^2 = ( aa ) ^2 = 83 g + 3 , which might be approximated as h^2= h_o^2[(1+z)^3 + ( 1+z)^2 + ] . [ eq : cos_pars ] the last equation defines the fractional contributions to the square of the present hubble parameter @xmath46 by matter , space curvature , and the cosmological constant ( or a term in the stress - energy tensor that acts like one ) . the time - dependence assumes pressureless matter and constant @xmath1 . other notations are in the literature ; a common practice in the particle physics community to add the matter and @xmath1 terms in a new density parameter , @xmath47 . i prefer keeping them separate , because the observational signatures of @xmath19 and @xmath48 can be quite different . by 1930 people understood how one would test the space - time geometry in these equations , and as i mentioned there is at last direct evidence for the detection of one of the effects , the curvature of the relation between redshift and apparent magnitude ( @xcite , @xcite ) . as indicated in line 1b , the measured curvature is inconsistent with the einstein - de sitter model in which @xmath49 and @xmath50 . the measurements also disagree with a low density model with @xmath51 , though the size of the discrepancy approaches the size of the error flags , so i assign a weaker failing grade for this case . the measurements are magnificent . the issue yet to be thoroughly debated is whether the type ia supernovae observed at redshifts @xmath52 are drawn from essentially the same population as the nearer ones . in a previous volume in this series krauss @xcite discusses the time - scale issue . stellar evolution ages and radioactive decay ages do not rule out the einstein - de sitter model , within the still considerable uncertainties in the measurements , but the longer expansion time scales of the low @xmath19 models certainly relieve the problem of interpretation of the measurements . thus i enter a tentative negative grade for the einstein - de sitter model in line 1a . in the analysis by falco _ _ @xcite of the rate of lensing of quasars by foreground galaxies ( line 1d ) for a combined sample of lensing events detected in the optical and radio , the @xmath53 bound on the density parameter in a cosmologically flat ( @xmath54 ) universe is @xmath55 . the sneia redshift - magnitude relation seems best fit by @xmath56 , @xmath57 , a possibly significant discrepancy . a serious uncertainty in the analysis of the lensing rate is the number density of early - type galaxies in the high surface density branch of the fundamental plane at luminosities @xmath58 , the luminosity of the milky way . if further tests confirm an inconsistency of the lensing rate and the redshift - magnitude relation the lesson may be that @xmath48 is dynamical , rolling to zero , as ratra & quillen @xcite point out . the relation between the mass density parameter @xmath19 and the gravitational motions of the galaxies is an issue rich enough for a separate category in table 1 . it has been known for the past decade that if galaxies were fair tracer of mass then the small - scale relative velocities of the galaxies would imply that @xmath19 is well below unity @xcite . if the mass distribution were smoother than that of the galaxies , the smaller mass fluctuations would require a larger mean mass density to gravitationally produce the observed galaxy velocities . davis , efstathiou , frenk & white @xcite were the first to show that such a biased distribution of galaxies relative to mass readily follows in numerical n - body simulations of the growth of structure , and the demonstration has been repeated in considerable detail ( @xcite , @xcite , and references therein ) . this is a serious argument for the biasing effect . but here are three arguments for the proposition that galaxies are fair tracers of mass for the purpose of estimating @xmath19 . first , in many numerical simulations dwarf galaxies are less strongly clustered than giants . this is reasonable , for if much of the mass were in the voids defined by the giant galaxies , as required if @xmath59 , then surely there would be remnants of the suppressed galaxy formation in the voids , irregular galaxies that bear the stigmata of a hostile early environment . the first systematic redshift survey showed that the distributions of low and high luminosity galaxies are strikingly similar @xcite . no survey since , in 21-cm , infrared , ultraviolet , or low surface brightness optical , has revealed a void population . there is a straightforward interpretation : the voids are nearly empty because they contain little mass . second , one can use the galaxy two - point correlation function in eq . ( [ eq : xigg ] ) and the mass autocorrelation function @xmath60 from a numerical simulation of structure formation to define the bias function b(r , t ) = ^1/2 . [ eq : bias ] in numerical simulations @xmath61 typically varies quite significantly with separation and redshift @xcite . that is , the galaxies give a biased representation of the statistical character of the mass distribution in a typical numerical simulation . the issue is whether the galaxies , or the models , or both , are biased representations of the statistical character of the real mass distribution . what particularly strikes me is the observation that the low order galaxy correlation functions have some simple properties . the galaxy two - point function is close to a power law over some three orders of magnitude in separation ( eq . [ eq : xiggparameters ] ) . the value of the power law index @xmath62 changes little back to redshift @xmath63 . within the clustering length @xmath64 the higher order correlation functions are consistent with a power law fractal . a reasonable presumption is that the regularity exhibited by the galaxies reflects a like regularity in the mass , because galaxies trace mass . i am impressed by the power of the numerical simulations , and believe they reflect important aspects of reality , but do not think we should be surprised if they do not fully represent other aspects , such as relatively fine details of the mass distribution . -1.0truecm the third argument deals with the idea that blast waves or radiation from the formation of a galaxy may have affected the formation of nearby galaxies , producing scale - dependent bias . in this case the apparent value of the density parameter derived from gravitational motions within systems of galaxies on the assumption galaxies trace mass would be expected to vary with increasing scale , approaching the true value when derived from relative motions on scales larger than the range of influence of a forming galaxy . 2 shows a test . the abscissa at the entry for clusters of galaxies is the comoving radius of a sphere that contains the mass within the abell radius . the estimates at larger scales are plotted at approximate values of the radius of the sample . if it were not for the last two points at the right - hand side of fig . 2 , one might conclude that the apparent density parameter is increasing to the true value @xmath65 at @xmath66 mpc . but considering the last two points , and the sizes of the error flags , it is difficult to see any evidence for scale - dependent bias . i assign a strongly negative grade for the einstein - de sitter model in line 2a in table 1 , based on galaxy motions on relatively small scales , because biasing certainly is required if @xmath49 and i have argued there is no evidence for it . the more tentative grade in line 2b is based on fig . 2 : the apparent value of the density parameter does not seem to scale with depth . the friedmann - lematre model is unstable to the gravitational growth of departures from a homogeneous mass distribution . the present large - scale homogeneity could have grown out of primeval chaos , but the initial conditions would be absurdly special . that is , the friedmann - lematre model requires that the present structure the clustering of mass in galaxies and systems of galaxies grew out of small primeval departures from homogeneity . the consistency test for an acceptable set of cosmological parameters is that one has to be able to assign a physically sensible initial condition that evolves into the present structure of the universe . the constraint from this consideration in line 3c is discussed by white _ @xcite , and in line 3b by bahcall _ ( @xcite , @xcite ) . here i explain the cautious ratings in line 3a . as has been widely discussed , it may be possible to read the values of @xmath19 and other cosmological parameters from the spectrum of angular fluctuations of the cbr ( @xcite and references therein ) . this assumes nature has kept the evolution of the early universe simple , however , and we have hit on the right picture for its evolution . we may know in the next few years . if the precision measurements of the cbr anisotropy from the map and planck satellites match in all detail the prediction of one of the structure formation models now under discussion it will compel acceptance . but meanwhile we should bear in mind the possibility that nature was not kind enough to have presented us with a simple problem . -1.0truecm an example of the possible ambiguity in the interpretation of the present anisotropy measurements is shown in fig . the two models assume the same dynamical actors cold dark matter ( cdm ) , baryons , three families of massless neutrinos , and the cbr but different initial conditions . in the adiabatic model the primeval entropy per conserved particle number is homogeneous , the space distribution of the primeval mass density fluctuations is a stationary random process with the scale - invariant spectrum @xmath67 , and the cosmological parameters are @xmath68 , @xmath69 , and @xmath70 ( following @xcite ) . the isocurvature initial condition in the other model is that the primeval mass distribution is homogeneous there are no curvature fluctuations and structure formation is seeded by an inhomogeneous composition . in the model shown here the primeval entropy per baryon is homogeneous , to agree with the standard model for light element production , and the primeval distribution of the cdm has fluctuation spectrum p(k)k^m , m = -1.8 . [ eq : piso ] the cosmological parameters are @xmath71 , @xmath72 , and @xmath73 . the lower density parameter produces a more reasonable - looking cluster mass function for the isocurvature initial condition @xcite . in both models the density parameter in baryons is @xmath74 , the rest of @xmath19 is in cdm , and space sections are flat ( @xmath75 ) . both models are normalized to the large - scale galaxy distribution . the adiabatic initial condition follows naturally from inflation , as a remnant of the squeezed field that drove the rapid expansion . a model for the isocurvature condition assumes the cdm is ( or is the remnant of ) a massive scalar field that was in the ground level during inflation and became squeezed to a classical realization . in the simplest models for inflation this produces @xmath76 in eq . ( [ eq : piso ] ) . the tilt to @xmath77 requires only modest theoretical ingenuity @xcite . that is , both models have pedigrees from commonly discussed early universe physics . the lesson from fig . 3 is that at least two families of models , with different relations between @xmath19 and the value of @xmath21 at the peak , come close to the measurements of the cbr fluctuation spectrum , within the still substantial uncertainties . an estimate of @xmath19 from the cbr anisotropy measurements thus may depend on the choice of the model for structure formation . programs of measurement of @xmath78 in progress should be capable of distinguishing between the adiabatic and isocurvature models , even given the freedom to adjust the shape of @xmath79 . the interesting possibility is that some other model for structure formation with a very different value of @xmath19 may give an even better fit to the improved measurements . i assign a failing grade to the einstein - de sitter model in line 3a because the adiabatic and isocurvature models both prefer low @xmath19 ( @xcite , @xcite ) . i add question marks to indicate this still is a model - dependent result . in their version of table 1 dekel , burstein , & white @xcite give the einstein - de sitter model the highest grade on theoretical grounds , and a cosmologically flat model with @xmath1 the next highest grade . the point is well taken : this is the order most of us would choose . the issue is whether nature agrees with our ideas of elegance , or maybe prefers physics that produces an open universe ( @xcite - @xcite ) . full closure of cosmology may come with the discovery of physics that predicts the values of @xmath48 and space curvature ( eq . [ [ eq : cos_pars ] ] ) in terms of the expansion age of the universe , consistent with all the other constraints in table 1 . but since we seem to be far from that goal i am inclined to omit entries in line 4 . we have a secure if still schematic standard model for cosmology , and the prospect for considerable enlargement from the application of the cosmological tests . the theoretical basis for the tests was discovered seven decades ago . a significant application likely will take a lot less than seven more decades : the constraints in table 1 already are serious , if debatable , and people know how to do better . application of the tests could yield a set of tightly constrained values of the cosmological parameters and a clear characterization of the primeval departure from homogeneity . if so cosmology could divide at a fixed point , the situation at @xmath80 , say , when the universe is well described by a slightly perturbed friedmann - lematre model . one branch of research would analyze evolution from these initial conditions to the present complex structure of the universe . the other would search for the physics of the very early universe that produced these initial conditions . but before making any long - term plans based on this scenario i would wait to see whether the evidence really is that the early universe is simple enough to allow such a division of labor .
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we have a well - established standard model for cosmology and prospects for considerable additions from work in progress .
i offer a list of elements of the standard model , comments on controversies in the interpretation of the evidence in support of this model , and assessments of the directions extensions of the standard model seem to be taking .
psfig * the standard cosmological model * + + +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
as the only missing particle in the standard model ( sm ) , the higgs boson is being intensively hunted at the lhc . recently , both the atlas and cms experiments have revealed hints of a higgs particle around @xmath6 @xcite . while such a higgs mass can be well accommodated in the sm and the reported signal rates in several channels are also in agreement with the sm expectations after taking into account the large experimental uncertainty @xcite ( albeit the central value of the observed di - photon rate is somewhat above the sm prediction ) , the low energy supersymmetry ( susy ) seems to be a better framework to account for such a higgs . in low energy susy the sm - like higgs mass is theoretically restricted in a narrow range and its di - photon rate at the lhc may exceed the sm prediction @xcite , both of which are welcomed by the lhc results . however , as the most popular low energy susy model , the minimal supersymmetric standard model ( mssm ) @xcite may have some tension to accommodate such a 125 gev higgs . as is well known , in the mssm the sm - like higgs mass is upper bounded by @xmath7 at tree - level and to get a higgs around @xmath6 we need sizable top / stop loop contributions , which depend quartically on the top quark mass and logarithmically on the stop masses @xcite . this will , on the one hand , impose rather tight constraint on the mssm , and on the other hand , incur some fine - tuning @xcite . such a problem can be alleviated in the so - called next - to - minimal supersymmetric standard model ( nmssm ) @xcite , which is the simplest singlet extension of the mssm with a scale invariant superpotential . in the nmssm , due to the introduction of some new couplings in the superpotential , the sm - like higgs mass gets additional contribution at tree - level and also may be further pushed up by the mixing effect in diagonalizing the mass matrix of the cp - even higgs fields @xcite . as a result , a sm - like higgs around @xmath6 does not entail large loop contributions , which may thus ameliorate the fine - tuning problem @xcite . in this work , motivated by the recent lhc results , we assume a sm - like higgs boson in @xmath8 and study its implication in the mssm and nmssm . different from recent studies in this direction @xcite , we scan the model parameters by considering various experimental constraints and perform a comparative study for the mssm and nmssm . we will investigate the features of the allowed parameter space in each model and particularly we pay more attention to the space of the nmssm which may be distinct from the mssm . noting the lhc experiments utilize the channels @xmath9 , @xmath10 and @xmath11 in searching for the higgs boson @xcite , we also study their normalized production rates defined as @xmath12 where @xmath13 , and @xmath14 , @xmath15 and @xmath16 are respectively the rescaled couplings of the higgs to gluons , photons and weak gauge bosons by their sm values . we are more interested in the case with @xmath17 because it is favored by current atlas and cms results @xcite . as will be shown below , this case usually predicts a slepton or a chargino lighter than @xmath18 . this paper is organized as follows . in sec . ii , we recapitulate the characters of the higgs mass in the models for better understanding our numerical results . in sec . iii , we perform a comprehensive scan over the parameter space of each model by imposing current experimental constraints and also by requiring a sm - like higgs boson in @xmath19 . then we scrutinize the properties of the surviving parameter space . finally , we draw our conclusions in sec in the mssm , the higgs sector consists of two doublet fileds @xmath20 and @xmath21 , which after the electroweak symmetry breaking , result in five physical higgs bosons : two cp - even scalars @xmath22 and @xmath23 , one cp - odd pseudoscalar @xmath24 and a pair of charged scalars @xmath25@xcite . traditionally , such a higgs sector is described by the ratio of the higgs vacuum expectation values , @xmath26 , and the mass of the pseudoscalar @xmath27 . in most of the mssm parameter space , the lightest higgs boson @xmath22 has largest coupling to vector bosons ( i.e. the so - called sm - like higgs boson ) , and for moderate @xmath28 and large @xmath27 its mass is given by @xcite @xmath29 where @xmath30 , @xmath31 with @xmath32 and @xmath33 being the stop masses , @xmath34 with @xmath2 denoting the trilinear higgs - stop coupling and @xmath35 being the higgsino mass parameter . obviously , the larger @xmath28 or @xmath36 is , the heavier @xmath22 becomes , and for given @xmath36 , @xmath37 reaches its maximum when @xmath38 , which corresponds to the so - called @xmath39 scenario . about eq.([mh ] ) , three points should be noted @xcite . the first is this equation is only valid for small splitting between @xmath32 and @xmath33 . in case of large splitting , generally @xmath40 is needed to maximize @xmath37 . the second is @xmath41 in eq.([mh ] ) is symmetric with respect to the sign of @xmath42 . this behavior will be spoiled once higher order corrections are considered , and usually a larger @xmath37 is achieved for positive @xmath43 with @xmath44 being gluino soft breaking mass . and the last is in eq.([mh ] ) , we do not include the contributions from the sbottom and slepton sectors . such contributions are negative and become significant only for large @xmath28 . compared with the mssm , the higgs sector in the nmssm is rather complex , which can be seen from its superpotential and the corresponding soft - breaking terms given by @xcite @xmath45 here @xmath46 is the superpotential of the mssm without the @xmath35 term , the dimensionless parameters @xmath47 and @xmath48 are the coefficients of the higgs self couplings , and @xmath49 , @xmath50 , @xmath51 , @xmath52 and @xmath53 are the soft - breaking parameters . after the electroweak symmetry breaking , the three soft breaking masses squared for @xmath20 , @xmath21 and @xmath54 can be expressed in terms of their vevs ( i.e. @xmath55 , @xmath56 and @xmath57 ) through the minimization conditions of the scalar potential . so in contrast to the mssm where there are only two parameters in the higgs sector , the higgs sector of the nmssm is described by six parameters @xcite : @xmath58 the higgs fields can be written in the following form : @xmath59 where @xmath60 , @xmath61 with @xmath62 and @xmath63 , @xmath64 and @xmath65 are goldstone bosons and @xmath66 . in the cp - conserving nmssm , the fields @xmath67 , @xmath68 and @xmath69 mix to form three physical cp - even higgs bosons , and @xmath70 and @xmath71 mix to form two physical cp - odd higgs bosons . obviously , the field @xmath72 corresponds to the sm higgs field , and the scalar @xmath22 with largest @xmath68 component is called the sm - like higgs boson . under the basis ( @xmath67 , @xmath68 , @xmath69 ) , the elements of the mass matrix for @xmath73 fields at tree level are given by @xcite @xmath74,\\ { \cal m}^2_{33}&= & \frac{1}{4 } \lambda^2 v^2 ( \frac{m_a \sin 2\beta}{\mu})^2 + \frac{\kappa\mu}{\lambda } ( a_\kappa + \frac{4\kappa\mu}{\lambda } ) - \frac{1}{2 } \lambda \kappa v^2 \sin 2 \beta , \label{33}\end{aligned}\ ] ] where @xmath75 is nothing but @xmath41 at tree level without considering the mixing among @xmath73 , and its second term @xmath76 originates from the coupling @xmath77 in the superpotential . for such a complex matrix , it is useful to consider two scenarios for understanding the results : * scenario i : @xmath78 and @xmath35 is fixed . in this limit , since @xmath79 , the singlet field @xmath69 is decoupled from the doublet fields , and the mssm mass matrix is recovered for the ( @xmath67,@xmath68 ) system . this scenario indicates that , even for moderate @xmath47 and @xmath48 , @xmath37 should change little from its mssm prediction . so in order to show the difference of the two models in predicting @xmath37 , we are more interested in large @xmath47 case . especially we will mainly discuss @xmath80 case , where the tree level contributions to @xmath41 , i.e. @xmath75 , are maximized for moderate values of @xmath28 rather than by large values of @xmath28 as in the mssm . * scenario ii : @xmath81 and @xmath82 , which can be easily realized for a large @xmath83 . in this limit , @xmath67 is decoupled from the ( @xmath68,@xmath69 ) system and the properties of @xmath37 can be qualitatively understood by the @xmath84 matrix @xcite @xmath85 where @xmath86 denotes the radiative corrections to @xmath37 with its form given by the last two terms of eq.([mh ] ) , and @xmath87 represents the potentially important effect of ( @xmath67,@xmath69 ) mixing on @xmath88 . this matrix indicates that the ( @xmath68,@xmath69 ) mixing can push @xmath37 up once @xmath89 , and such effect is maximized for @xmath90 slightly larger than @xmath91 and at the same time @xmath92 slightly below @xmath93 ( larger @xmath94 will destabilize the vacuum ) @xcite . obviously , in this push - up case , @xmath22 is the next - to - lightest cp - even higgs boson and the larger the ( @xmath95 ) mixing is , the heavier @xmath22 becomes . alternatively , the mixing can pull @xmath37 down on the condition of @xmath96 , which occurs for large @xmath97 ( for @xmath98 , see discussion below ) as indicated by the expression of @xmath99 and the positiveness of @xmath87 . here we remind that , due to the extra contribution @xmath76 to @xmath41 at tree level , @xmath37 in the pull - down case may still be larger than its mssm prediction for a certain @xmath86 . + since our results presented below is approximately described by scenario ii , we now estimate the features of its favored region to predict @xmath100 . first , since @xmath101 , @xmath102 implies that @xmath103 . numerically , we find @xmath103 for the push - up case , and @xmath104 for the pull - down case ( see fig.[fig7 ] ) . second , @xmath105 must be relatively small , which implies that @xmath98 for @xmath106 . this can be understood as follows . in the push - up case , since @xmath107 , the condition @xmath108 ( for vacuum stability ) has limited the size of @xmath109 . while in the pull - down case , a very large @xmath109 will suppress greatly @xmath37 to make it difficult to reach @xmath6 , and this in return limits the size of @xmath109 . given that @xmath110 as required by the lep bound on chargino mass and that a larger @xmath35 is favored for the pull - down scenario , one can infer that the value of @xmath111 should be around 2 after considering that the third term in @xmath109 is less important . numerically speaking , we find @xmath112 and @xmath113 ( see fig.[fig7 ] ) . lastly , light stops may be possible in the nmssm with large @xmath47 to predict @xmath100 . to see this , we consider the parameters @xmath114 and @xmath115 , and we get @xmath116 without considering the mixing effect to predict @xmath100 . this is in sharp contrast with @xmath117 in mssm for @xmath118 . in this work we use the package nmssmtools @xcite to calculate the higgs masses and mixings , which includes the dominant one - loop and leading logarithmic two - loop corrections to @xmath119 . we checked our mssm results of @xmath37 by using the code feynhiggs@xcite and found the results given by nmssmtools and feynhiggs are in good agreement ( for @xmath120 they agree within @xmath121 for same mssm parameters ) . in this work , we scan the parameters of the models and investigate the samples that predict @xmath122 and at the same time survive the following constraints @xcite : ( 1 ) the constraint from the lhc search channel @xmath123 for non - standard higgs boson . ( 2 ) the limits from the lep and the tevatron on the masses of sparticles as well as on the neutralino pair productions . ( 3 ) the constraints from b - physics , such as @xmath124 , the latest experimental result of @xmath125 , @xmath126 , @xmath127 and the mass differences @xmath128 and @xmath129 . ( 4 ) the indirect constraints from the electroweak precision observables such as @xmath130 , @xmath131 and @xmath132 , and their combinations @xmath133 @xcite . we require @xmath134 to be compatible with the lep / sld data at @xmath135 confidence level . we also require the susy prediction of the observable @xmath136 ( @xmath137 ) to be within the @xmath138 range of its experimental value @xcite . ( 5 ) the constraints from the muon anomalous magnetic moment : @xmath139 @xcite . we require the susy effects to explain the discrepancy at @xmath0 level . ( 6 ) the dark matter constraints from wmap relic density ( 0.1053 @xmath140 0.1193 ) @xcite and the direct search result from xenon100 experiment ( at @xmath141 c.l . ) ( 7 ) for the nmssm , we also require the absence of a landau singularity below the gut scale , which implies @xmath142 for small @xmath48 and @xmath143 for @xmath144 at weak scale . in our calculation , we fix @xmath145 and @xmath146 @xcite ( @xmath147 denotes the strange quark fraction in the proton mass ) , and use the package nmssmtools to implement most of the constraints and to calculate the observables we are interested in . in our scan , we note that the soft parameters in the slepton sector can only affect significantly the muon anomalous magnetic moment @xmath148 , which will in return limit the important parameter @xmath28 , so we assume them a common value @xmath149 and treat it as a free parameter . for the soft parameters in the first two generation squark sector , due to their little effects on the properties of the higgs bosons , we fix them to be @xmath150 . as for the gaugino masses , we assume the grand unification relation , @xmath151 with @xmath152 being the fine structure constants of the different gauge groups , and treat @xmath153 as a free parameter . in order to reduce free parameters , we also assume the unimportant parameters @xmath154 and @xmath155 to satisfy @xmath156 and @xmath157 . in order to study the implication of @xmath158 in generic susy , we relax the soft mass parameters to 5 tev and perform an extensive random scan over the following parameter regions : @xmath159 for the mssm , and @xmath160 for the nmssm . in our scan , we only keep the samples satisfying the requirements listed in the text ( including @xmath161 ) . to show the differences between the mssm and the nmssm , we also perform a scan similar to eq.([nmssm - scan ] ) except that we require @xmath162 . ) , projected in the plane of @xmath32 versus @xmath163 with @xmath164 and @xmath34.,title="fig:",width=211 ] ) , projected in the plane of @xmath32 versus @xmath163 with @xmath164 and @xmath34.,title="fig:",width=211 ] ) , projected in the plane of @xmath32 versus @xmath163 with @xmath164 and @xmath34.,title="fig:",width=211 ] in fig.[fig1 ] , we show the correlation of the lighter top - squark mass ( @xmath165 ) with the ratio @xmath166 ( @xmath167 ) for the surviving samples in the mssm and nmssm . as expected from eq.([mh ] ) , in order to predict @xmath100 in the mssm , a large @xmath42 is needed for a moderate light @xmath168 , and with @xmath169 becoming heavy , the ratio @xmath166 decreases , but is unlikely to vanish for @xmath170 . these features are maintained for nmssm with @xmath171 ( see the middle panel ) but changed for nmssm with a large @xmath47 , where @xmath42 can possibly vanish even for @xmath172 . fig.[fig1 ] also shows that a @xmath169 as light as @xmath173 is still able to give the required @xmath37 . but in this case @xmath42 is large ( @xmath174 ) , which leads to a large mass splitting between two stops ( @xmath175 ) . note that a @xmath169 as light as @xmath176 does not contradict the recent susy search result of the lhc @xcite . since in heavy susy the radiative correction @xmath86 is usually very large , @xmath100 is unlikely to impose tight constraints on other parameters of the models . considering heavy susy is disfavored by naturalness , we in the following concentrate on the implication of @xmath100 in sub - tev susy . in this section , we study the implication of @xmath158 in low energy mssm and nmssm . in order to illustrate the new features of the nmssm , we only consider the case with @xmath106 . our scans over the parameter spaces are quite similar to those in eq.([mssm - scan ] ) and eq.([nmssm - scan ] ) except that we narrow the ranges of @xmath177 , @xmath178 and @xmath2 as follows : @xmath179 ) for both models and the requirement @xmath106 for the nmssm . here the samples are projected in the plane of @xmath32 versus @xmath2 . the upper ( lower ) panels correspond to @xmath180 ( @xmath17 ) with @xmath181 . for the nmssm results , the circles ( green ) denote the case of the lightest higgs boson being the sm - like higgs ( the so - called pull - down case ) , and the times ( red ) denotes the case of the next - to - lightest higgs boson being the sm - like higgs ( the so - called push - up case ) . , title="fig:",width=245 ] ) for both models and the requirement @xmath106 for the nmssm . here the samples are projected in the plane of @xmath32 versus @xmath2 . the upper ( lower ) panels correspond to @xmath180 ( @xmath17 ) with @xmath181 . for the nmssm results , the circles ( green ) denote the case of the lightest higgs boson being the sm - like higgs ( the so - called pull - down case ) , and the times ( red ) denotes the case of the next - to - lightest higgs boson being the sm - like higgs ( the so - called push - up case ) . , title="fig:",width=245 ] ) for both models and the requirement @xmath106 for the nmssm . here the samples are projected in the plane of @xmath32 versus @xmath2 . the upper ( lower ) panels correspond to @xmath180 ( @xmath17 ) with @xmath181 . for the nmssm results , the circles ( green ) denote the case of the lightest higgs boson being the sm - like higgs ( the so - called pull - down case ) , and the times ( red ) denotes the case of the next - to - lightest higgs boson being the sm - like higgs ( the so - called push - up case ) . , title="fig:",width=245 ] ) for both models and the requirement @xmath106 for the nmssm . here the samples are projected in the plane of @xmath32 versus @xmath2 . the upper ( lower ) panels correspond to @xmath180 ( @xmath17 ) with @xmath181 . for the nmssm results , the circles ( green ) denote the case of the lightest higgs boson being the sm - like higgs ( the so - called pull - down case ) , and the times ( red ) denotes the case of the next - to - lightest higgs boson being the sm - like higgs ( the so - called push - up case ) . , title="fig:",width=245 ] in fig.[fig2 ] we project the surviving samples of the models in the plane of @xmath32 versus @xmath2 , showing the results with @xmath180 and @xmath182 separately . as we analyzed in sec . ii , the sm - like higgs in the nmssm may be either the lightest higgs boson ( corresponding to the pull - down case ) or the next - to - lightest higgs boson ( the push - up case ) . in the figure we distinguished these two cases . we note that among the surviving samples the number of the pull - down case is about twice the push - up case . fig.[fig2 ] shows that , in order to get @xmath100 in the mssm , @xmath32 and @xmath183 must be larger than about @xmath184 and @xmath185 respectively , and the bounds are pushed up to @xmath186 and @xmath187 respectively for @xmath17 . while in the nmssm , a @xmath169 as light as about @xmath188 ( in either the pull - down or push - up case ) is still able to predict @xmath100 , and even if one require @xmath17 , a @xmath169 as light as about @xmath176 is allowed . the fact that the nmssm allows a lighter @xmath169 than the mssm indicates that the nmssm is more natural than the mssm in light of the lhc results . .the ranges of the rescaled couplings @xmath189 and @xmath190 predicted by the surviving samples of the two models . the region of @xmath14(@xmath191 ) is obtained by only considering the samples with @xmath192 . [ cols="^,^,^,^,^",options="header " , ] since a light @xmath169 may significantly change the effective couplings @xmath14 and @xmath193 , we present in table i their predicted ranges for the surviving samples . this table shows that @xmath14 is always reduced , and for @xmath194 in the nmssm , the reduction factor may reach @xmath195 . while @xmath193 exhibits quite strange behaviors : it is enhanced in the mssm , but may be either enhanced or suppressed in the nmssm . this is because , unlike @xmath14 which is affected only by squark loops , @xmath193 gets new physics contributions from loops mediated by charged higgs boson , charginos , sleptons and also squarks , and there exists cancelation among different loops . as will be shown below , the current experiments can not rule out light sparticles like @xmath196 and chargino . although the contributions of these particles to @xmath193 are far smaller than the @xmath197 loop contribution , they may still alter the coupling significantly . , but showing the dependence of the di - photon signal rate @xmath198 on the effective @xmath199 coupling @xmath200.,title="fig:",width=264 ] , but showing the dependence of the di - photon signal rate @xmath198 on the effective @xmath199 coupling @xmath200.,title="fig:",width=264 ] since the di - photon signal is the most important discovery channel for the higgs boson around @xmath6 , it is useful to study its rate carefully . from eq.([definition1 ] ) one can learn that the rate is affected by @xmath14 and @xmath193 discussed above , and also by the total width of @xmath22 ( or more basically by the @xmath201 coupling since @xmath202 is the dominant decay of @xmath22 ) . the importance of @xmath201 coupling on the di - photon rate was recently emphasized in @xcite . here we restrict our study to the @xmath100 case . in fig.[fig3 ] we show the dependence of the di - photon signal rate @xmath198 on the effective @xmath199 coupling ( including the potentially large susy corrections @xcite ) normalized by its sm value . this figure indicates that although the rate is suppressed for most of the surviving samples in both models , there still exist some samples with enhanced rate , especially the nmssm is more likely to push up the rate than the mssm . this feature can be understood as follows . in susy , @xmath17 requires approximately the combination @xmath203 to exceed 1 . for the mssm , given @xmath204 and @xmath205 for nearly all the cases , this condition is not easy to satisfy . while in the nmssm , due to the mixing between the doublet field @xmath68 and the singlet field @xmath69 , @xmath206 is possible once the singlet component in @xmath22 is significant , which is helpful to enhance the combination . in fact , we analyzed carefully the @xmath207 cases and found they are characterized by @xmath208 and @xmath209 in the mssm , and by @xmath210 in the nmssm . in other words , it is the enhanced @xmath211 coupling ( reduced total width of @xmath22 ) that mainly push up the di - photon rate in the mssm ( nmssm ) to exceed its sm prediction . fig.[fig3 ] also indicates that the pull - down case in the nmssm is less effective in reducing @xmath212 and thus can hardly enhance the di - photon rate . this is because in the push - up case , both @xmath213 and @xmath88 in eq.([re - m ] ) are moderate and often comparable , which are helpful to enhance the ( @xmath214 ) mixing . finally , we note that in some rare cases of the nmssm the ratio @xmath215 may be very small even for @xmath210 . this is because there exists very light higgs boson so that @xmath22 decays dominantly into them . , but only for the mssm , projected in the planes of @xmath216 versus @xmath35 and @xmath215 versus @xmath217.,title="fig:",width=264 ] , but only for the mssm , projected in the planes of @xmath216 versus @xmath35 and @xmath215 versus @xmath217.,title="fig:",width=264 ] in order to further clarify the reason for the enhancement of the di - photon rate in the mssm , we scrutinize carefully the parameters of the model and find that the samples with @xmath17 correspond to the case with a large @xmath218 and @xmath219 , which is illustrated in fig.[fig4 ] . this means that the stau loop plays an important role in enhancing @xmath220 . we note that similar conclusion was recently achieved in @xcite , but in that work the authors did not consider the tight experimental constraints . from fig.[fig4 ] we also note that @xmath221 is predicted in the mssm with @xmath222 . so future precise measurement of @xmath215 and @xmath223 may be utilized to verify the correctness of the mssm . , but showing the signal rate @xmath224 versus the coupling @xmath225 . , title="fig:",width=264 ] , but showing the signal rate @xmath224 versus the coupling @xmath225 . , title="fig:",width=264 ] considering the process @xmath226 ( @xmath4 ) is another important higgs search channel , we in fig.[fig5 ] show the signal rate versus the @xmath227 coupling . this figure shows that in the mssm , @xmath22 is highly sm - like , while in the nmssm , the singlet component in @xmath22 may be sizable , especially in the push - up case , so that @xmath16 is reduced significantly . the signal rate @xmath228 also behaves differently in the two models . in the mssm , because @xmath204 and in most cases @xmath229 , @xmath228 is always less than 1 ( for @xmath17 it varies between 0.7 and 0.95 ) . in the nmssm , however , @xmath228 may exceed 1 and in this case we find @xmath230 . the reason for such a correlation is the two quantities have the same origin for their enhancement , i.e. the suppression of the @xmath231 due to the presence of the singlet component in @xmath22 . versus @xmath216 and versus @xmath48 respectively . , title="fig:",width=245 ] versus @xmath216 and versus @xmath48 respectively . , title="fig:",width=245 ] versus @xmath216 and versus @xmath48 respectively . , title="fig:",width=245 ] versus @xmath216 and versus @xmath48 respectively . , title="fig:",width=245 ] next we investigate the favored parameter space of the nmssm to predict @xmath100 . as introduced in sec . ii , besides the soft parameters in the stop sector , the sensitive parameters include @xmath28 , @xmath35 , @xmath48 as well as @xmath232 . in fig.[fig6 ] , we project the surviving samples in the planes of @xmath35 versus @xmath216 and versus @xmath48 . this figure shows three distinctive characters for the allowed parameters . the first is that @xmath28 must be moderate , below 4 and 9 for the pull - down and the push - up case , respectively . two reasons can account for it . one is that in the nmssm with large @xmath47 , the precision electroweak data , i.e. the constraint ( 4 ) , strongly disfavor a large @xmath28 @xcite . the other reason is that , as far as @xmath106 is concerned , a moderate @xmath28 is welcomed to enhance the tree level value of @xmath41 ( i.e. @xmath233 ) so that even without heavy stops , @xmath37 can still reach @xmath6 . moreover , since the ( @xmath214 ) mixing is to reduce the value of @xmath213 in eq.([re - m ] ) in the pull - down case , a larger @xmath88 ( or equivalently a smaller @xmath28 ) is favored by the higgs mass . the second character is that @xmath97 in the push - up case is usually much smaller than that in the pull - down case . this is because , as we introduced in sec . ii , a large @xmath97 is needed by the pull - down case to enhance @xmath88 in eq.([re - m ] ) . the third character is obtained by comparing the parameter regions in the upper panels with those in the lower panels , which shows that @xmath17 puts a lower bound on @xmath48 , i.e. @xmath234 . the underlying reason is that for @xmath235 , the dark matter will be light and singlino - like , and to get its currently measured relic density , the dark matter must annihilate in the early universe by exchanging a light higgs boson @xcite . in this case , @xmath22 mainly decays into the light bosons , which in return will suppress the di - photon rate . , but showing the correlation between @xmath232 and @xmath236 . the dashed line denotes the relation @xmath237.,width=264 ] in fig.[fig7 ] we show the correlation of @xmath232 with @xmath236 . this figure indicates that @xmath238 for the push - up case and @xmath239 for the pull - down case , which is in agreement with our expectation . in fact , we checked each surviving sample and found it satisfies the condition : @xmath240 and @xmath241 , so the samples can be well described by scenario ii . we also checked that the mixing of the field @xmath67 with @xmath68/@xmath69 is small and @xmath232 is approximately the heaviest cp - even higgs boson mass . fig.[fig7 ] also indicates that the relation @xmath242 is maintained quite well in the push - up case , but is moderately spoiled in the pull - down case . the reason is , as we introduced in sec . ii , the requirement that @xmath109 should be moderately small actually implies @xmath243 with @xmath244 . in the push - up case the third term in @xmath245 is not important , while in the pull case , although it is several times smaller than the second term , it may not be negligible . we checked our results and found @xmath246 and @xmath247 for all the surviving samples . about the nmssm with @xmath106 , three points should be noted . the first is , from the results presented in fig.[fig6 ] , one may find the presence of a smuon and/or a chargino lighter than @xmath248 . this is because the surviving samples are characterized either by @xmath249 or @xmath250 or by both in the nmssm ( see fig.[fig6 ] ) . then to explain the discrepancy of muon anomalous magnetic moment @xmath251 is needed for a low @xmath216 , and @xmath252 implies @xmath253 . we numerically checked the validity of this conclusion . the second point is , although @xmath254 may be regarded as a new source of fine tuning in the nmssm , it is rather predictive to get the value of @xmath232 once @xmath35 and @xmath28 are experimental determined . finally , we note the favored region for @xmath35 and @xmath28 shown in fig.[fig6 ] does not overlap with that in fig.[fig4 ] . this may be used to discriminate the models . , but projected on the plane of the charged higgs boson mass and @xmath216.,title="fig:",width=264 ] , but projected on the plane of the charged higgs boson mass and @xmath216.,title="fig:",width=264 ] finally , we briefly describe other implications of @xmath100 in the susy models . in fig.[fig8 ] we project the surviving samples on the plane of @xmath28 versus @xmath255 with @xmath256 denoting the charged higgs boson . this figure shows that @xmath256 must be heavier than about @xmath176 in the mssm . this bound is much higher than the corresponding lep bound , which is about @xmath257 . for the nmssm with a large @xmath47 , the bound can be further pushed up to about @xmath184 . this figure also indicates that in the mssm , @xmath28 may reach 35 for @xmath258 . then based on the mc simulation by the atlas collaboration @xcite , one may expect that the charged higgs may be observable from the process @xmath259 at the early stage of the lhc . however , this may be impossible . the reason is , for relatively light @xmath256 and large @xmath28 , @xmath35 must be large to satisfy the constraint from dark matter direct detection experiments such as xenon100 . this will greatly suppress the @xmath260 coupling @xcite . for the nmssm , the hope to observe @xmath256 is also dim because @xmath28 is small . , but exhibiting the spin - independent @xmath261-nucleon scattering cross section as a function of the dark matter mass.,title="fig:",width=264 ] , but exhibiting the spin - independent @xmath261-nucleon scattering cross section as a function of the dark matter mass.,title="fig:",width=264 ] in fig.[fig9 ] we show the spin - independent elastic scattering between dark matter and nucleon . we use the formula presented in the appendix of @xcite to calculate the scattering rate . as expected , the xenon100 ( 2012 ) data to be released in near future will further exclude some samples , especially the pull - down case of the nmssm will be strongly disfavored if xenon100 ( 2012 ) fails to find any evidence of dark matter ( assuming the grand unification relation of the gaugino mass ) . from the left of fig.[fig9 ] one can learn that for the samples with @xmath17 in the mssm , the scattering rate is small , usually at least one order below than the sensitivity of the xenon100 ( 2012 ) . motivated by the recent lhc hints of a higgs boson around 125 gev , we assume a sm - like higgs with the mass 123 - 127 gev and study its implication in low energy susy by comparing the mssm and nmssm . under various experimental constraints at @xmath0 level ( including the muon @xmath1 and the dark matter relic density ) , we scanned over the parameter space of each model . then in the parameter space allowed by current experimental constraints and also predicting a sm - like higgs in 123 - 127 gev , we examined the properties of the sensitive parameters and calculated the rates of the di - photon signal and the @xmath3 ( @xmath4 ) signals at the lhc . among our various findings the typical ones are : ( i ) in the mssm the top squark and @xmath2 must be large and thus incur some fine - tuning , which can be much ameliorated in the nmssm ; ( ii ) in the mssm a light @xmath262 is needed to make the di - photon rate of the sm - like higgs exceed its sm prediction , while the nmssm has more ways in doing this ; ( iii ) in the mssm the signal rates of @xmath5 at the lhc are never enhanced compared with their sm predictions , while in the nmssm they may be enhanced ; ( iv ) a large part of the parameter space so far survived will be soon covered by the expected xenon100(2012 ) sensitivity ( especially for the nmssm ) . therefore , although the low energy susy can in general accommodate a sm - like higgs boson near 125 gev and enhance its di - photon signal rate at the lhc , not all models of low energy susy are equally competent if they are required to satisfy all current experimental constraints . from our present study and some other studies in the literature , we conclude : * the fancy cmssm / msugra is hard to give a 125 gev sm - like higgs boson @xcite . * the mssm can give such a 125 gev higgs and can also enhance its di - photon signal rate at the lhc , which , however , will incur some fine - tuning . * the nmssm ( the nearly minimal susy model ) can give a 125 gev sm - like higgs , but severely suppress its di - photon signal rate at the lhc @xcite . * the nmssm is so far the best model to accommodate such a 125 gev higgs ; it can naturally ( without fine - tuning ) predict such a sm - like higgs mass and readily enhance its di - photon signal rate at the lhc . at the same time , in a large part of its parameter space , this model can also enhance the signal rates @xmath5 ( @xmath263 ) at the lhc and predict a large scattering rate of dark matter and nucleon at the xenon100 . so the interplay of lhc and xenon100 will soon allow for a good test of this model . this work was supported in part by the national natural science foundation of china ( nnsfc ) under grant nos . 10821504 , 11135003 , 10775039 , 11075045 , by specialized research fund for the doctoral program of higher education with grant no . 20104104110001 , and by the project of knowledge innovation program ( pkip ) of chinese academy of sciences under grant no . kjcx2.yw.w10 . 99 atlas collaboration , arxiv:1202.1408 [ hep - 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motivated by the recent lhc hints of a higgs boson around 125 gev , we assume a sm - like higgs with the mass 123 - 127 gev and study its implication in low energy susy by comparing the mssm and nmssm .
we consider various experimental constraints at @xmath0 level ( including the muon @xmath1 and the dark matter relic density ) and perform a comprehensive scan over the parameter space of each model .
then in the parameter space which is allowed by current experimental constraints and also predicts a sm - like higgs in 123 - 127 gev , we examine the properties of the sensitive parameters ( like the top squark mass and the trilinear coupling @xmath2 ) and calculate the rates of the di - photon signal and the @xmath3 ( @xmath4 ) signals at the lhc .
our typical findings are : ( i ) in the mssm the top squark and @xmath2 must be large and thus incur some fine - tuning , which can be much ameliorated in the nmssm ; ( ii ) in the mssm a light stau is needed to enhance the di - photon rate of the sm - like higgs to exceed its sm prediction , while in the nmssm the di - photon rate can be readily enhanced in several ways ; ( iii ) in the mssm the signal rates of @xmath5 at the lhc are never enhanced compared with their sm predictions , while in the nmssm they may get enhanced significantly ; ( iv ) a large part of the parameter space so far survived will be soon covered by the expected xenon100(2012 ) sensitivity ( especially for the nmssm ) .
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[ sec : intro ] we show that a hamiltonian diffeomorphism of a closed symplectically aspherical manifold has infinitely many periodic points . more precisely , we prove that such a diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period . for tori , this fact , recently established by hingston , @xcite , was conjectured by conley , @xcite and is frequently referred to as the _ conley conjecture_. ( see also @xcite and references therein for similar results for hamiltonian diffeomorphisms and homeomorphisms of surfaces . ) the proof given here uses some crucial ideas from @xcite , but is completely self - contained . the main result of the paper is [ thm : main ] let @xmath0 be a hamiltonian diffeomorphism of a closed symplectically aspherical manifold @xmath1 . assume that the fixed points of @xmath2 are isolated . then @xmath2 has simple periodic points of arbitrarily large period . we refer the reader to section [ sec : general ] for the definitions . here we only point out that a hamiltonian diffeomorphism is the time - one map of a time - dependent hamiltonian flow and that the manifolds @xmath1 with @xmath3 ( e.g. , tori and surfaces of genus greater than zero ) are among symplectically aspherical manifolds . thus , theorem [ thm : main ] implies in particular the conley conjecture for tori , @xcite , and the results of @xcite on hamiltonian diffeomorphisms of such surfaces . [ cor : main ] a hamiltonian diffeomorphism @xmath2 of a closed symplectically aspherical manifold has infinitely many simple periodic points . the example of an irrational rotation of @xmath4 shows that in general the requirement that @xmath1 is symplectically aspherical can not be completely eliminated ; see , however , @xcite . let @xmath5 be a periodic in time hamiltonian giving rise to @xmath2 . since periodic points of @xmath2 are in one - to - one correspondence with periodic orbits of the time - dependent hamiltonian flow @xmath6 , theorem [ thm : main ] and corollary [ cor : main ] can be viewed as results about periodic orbits of @xmath5 . then , in both of the statements , the periodic orbits can be assumed to be contractible . ( it is not hard to see that contractibility is a property of a fixed point rather than of an orbit , independent of the choice of @xmath5 . ) finally note that , as simple examples show , the assumption of theorem [ thm : main ] that the fixed points of @xmath2 are isolated can not be dropped as long as the periodic orbits are required to be contractible . there are numerous parallels between the hamiltonian conley conjecture considered here and its lagrangian counterpart ; see , e.g. , @xcite and references therein . the similarity between the two problems goes beyond the obvious analogy of the statements and can also easily be seen on the level of the proofs , although the methods utilized in @xcite are quite different from the floer homological techniques used in the present paper . thus , for instance , our proposition [ prop2 ] plays the same role as bangert s homological vanishing method originating from @xcite in , e.g. , @xcite . in the framework of symplectic topology , there are two essentially different approaches to proving results along the lines of the conley conjecture . the first approach , due to conley and salamon and zehnder , @xcite , is based on an iteration formula for the conley zehnder index , asserting that the index of an isolated weakly non - degenerate orbit either grows linearly under iterations or its absolute value does not exceed @xmath7 , where @xmath8 . this , in particular , implies that the local floer homology of such an orbit eventually becomes zero in degree @xmath9 as the order of iteration grows , provided that the orbit remains isolated . ( we refer the reader to sections [ sec : prelim ] and [ sec : lfh ] for the definitions . the argument of salamon and zehnder , @xcite , does not rely on the notion of local floer homology , but this notion becomes indispensable in the proof of theorem [ thm : main ] . ) since the floer homology of @xmath1 in degree @xmath9 is non - zero , it follows that when all one - periodic orbits are weakly non - degenerate , new simple orbits must be created by large prime iterations to generate the floer homology in degree @xmath9 ; see @xcite for details . the second approach comprises a broad class of methods and is based on the idea that a hamiltonian @xmath5 with sufficiently large variation must have one - periodic orbits with non - vanishing action . since iterating a hamiltonian diffeomorphism @xmath2 has the same effect as , roughly speaking , increasing the variation of @xmath5 , one can expect @xmath2 to have infinitely many periodic points . when a sufficiently accurate upper or lower bound on the action is available , the orbits can be shown to be simple . the results obtained along these lines are numerous and use a variety of symplectic topological techniques and assumptions on @xmath1 and @xmath5 . for instance , if the support of @xmath5 is displaceable and the variation of @xmath5 is greater than the displacement energy @xmath10 of the support , one - periodic orbits with action in the range @xmath11 $ ] have been shown to exist for many classes of symplectic manifolds and hamiltonians ; see , e.g. , @xcite . then , the _ a priori _ bound on action implies the existence of simple periodic orbits with non - zero action and arbitrarily large period . these methods do not rely on particular requirements on the fixed points of @xmath2 , but the assumption that the support is displaceable appears at this moment to be crucial . within this broad class is also a group of methods applicable to hamiltonians @xmath5 with sufficiently degenerate large or `` flat '' maximum and detecting orbits with action slightly greater than the maximum of @xmath5 ; see , e.g. , @xcite . iterating @xmath2 can be viewed as stretching @xmath5 near its maximum , and thus increasing its variation . hence , methods from this group can also be used in some instances to prove the existence of simple periodic orbits of large period . here the condition that the maximum is in a certain sense flat is crucial , but the assumption that the support of the hamiltonian is displaceable is less important and , in some cases , not required at all . in fact , what appears to matter is that the set where the maximum is attained is relatively small ( e.g. , symplectic as in @xcite or displaceable as in @xcite or just isolated as in @xcite ) . it is one of these methods , combined with the conley zehnder approach , that we use in the proof of the conley conjecture . a work of hingston @xcite clearly suggests the idea , which is central to our proof , that the two approaches outlined above can be extended to cover the case of an arbitrary hamiltonian . namely , the method of @xcite detects infinitely many simple periodic points of arbitrarily large periods , unless there exists a strongly degenerate @xmath12-periodic point @xmath13 such that the local floer homology groups of the @xmath12-th iteration @xmath14 and of a large iteration of @xmath14 at @xmath13 are non - zero in degree @xmath9 ; see section [ sec : proof ] . then we show ( proposition [ prop1 ] ) that the @xmath15 is the time-@xmath12 flow of a @xmath12-periodic hamiltonian , say @xmath16 , such that @xmath13 is a ( constant ) local maximum of @xmath16 for all @xmath17 and this maximum is in a certain sense very degenerate ; cf . ( however , the hessian @xmath18 need not be identically zero . ) finally , we prove ( proposition [ prop2 ] ) that large iterations of @xmath5 have periodic orbits with actions arbitrarily close to the action of the iterated hamiltonian at @xmath13 . these orbits are necessarily simple due to the lower and upper bounds on the action . proposition [ prop2 ] is established by using a simple squeezing argument akin to the ones from @xcite . this concludes the proof of the theorem . this argument is extremely flexible and readily extends to manifolds convex at infinity or geometrically bounded and wide ; see @xcite and @xcite for the definitions . we will give a detailed proof of the conley conjecture for such manifolds elsewhere . in section [ sec : prelim ] , we set notation and conventions , briefly review elements of floer theory , and also discuss the properties of loops of hamiltonian diffeomorphisms relevant to the proof . local floer homology is the subject of section [ sec : lfh ] . in section [ sec : proof ] , we state propositions [ prop1 ] and [ prop2 ] mentioned above and derive theorem [ thm : main ] from these propositions . proposition [ prop1 ] reduces the problem to the case of a hamiltonian with strict , but `` flat '' , local maximum . this proposition is proved in sections [ sec : pr - prop1 ] and [ sec : gf ] by adapting an argument from @xcite . proposition [ prop2 ] asserting the existence of simple periodic orbits of large period for such a hamiltonian and completing the proof of theorem [ thm : main ] is established in section [ sec : prop2-pf ] . the author is deeply grateful to baak grel , doris hein , ely kerman , and felix schlenk for their numerous valuable remarks and suggestions . [ sec : prelim ] [ sec : notation ] [ sec : general ] throughout the paper , a smooth @xmath19-dimensional manifold is denoted by @xmath20 and @xmath1 stands for a symplectic manifold of dimension @xmath21 , equipped with a symplectic form @xmath22 . the manifold @xmath23 is always assumed to be closed and _ symplectically aspherical _ , i.e. , @xmath24 , where @xmath25 is the first chern class of @xmath1 ; see , e.g. , @xcite . an almost complex structure @xmath26 on @xmath1 is said to be _ compatible with @xmath22 _ if @xmath27 is a riemannian metric on @xmath1 . when @xmath28 depends on an extra parameter @xmath17 ( time ) , this condition is required to hold for every @xmath17 . a ( time - dependent ) metric of the form @xmath27 is said to be compatible with @xmath22 . the group of linear symplectic transformations of a finite - dimensional linear symplectic space @xmath29 is denoted by @xmath30 . we will also need the fact that @xmath31 ( see , e.g. , @xcite ) , and hence @xmath32 . to be more specific , fixing a linear complex structure @xmath26 on @xmath33 , compatible with @xmath22 , gives rise to an inclusion @xmath34 of the unitary group into the symplectic group . this inclusion is a homotopy equivalence . the isomorphism @xmath31 is the composition of the isomorphism @xmath35 , the isomorphism of the fundamental groups induced by @xmath36 ( the unit circle in @xmath37 ) , and the identification @xmath38 arising from fixing the counter clock - wise orientation of @xmath39 . note that the resulting isomorphism is independent of the choice of @xmath26 . the _ maslov index _ of a loop in @xmath30 is the class of this loop in @xmath40 . we use the notation @xmath39 for the circle @xmath41 and the circle @xmath42 of circumference @xmath43 is denoted by @xmath44 . all hamiltonians @xmath5 on @xmath1 considered in this paper are assumed to be @xmath45-periodic ( in time ) , i.e. , @xmath46 . we set @xmath47 for @xmath48 . the hamiltonian vector field @xmath49 of @xmath5 is defined by @xmath50 . let @xmath51 be a contractible loop . the action of @xmath5 on @xmath52 is defined by @xmath53 here @xmath54 is the negative symplectic area bounded by @xmath52 , i.e. , @xmath55 where @xmath56 is such that @xmath57 . the least action principle asserts that the critical points of @xmath58 on the space of all contractible maps @xmath51 are exactly the contractible @xmath45-periodic orbits of the time - dependent hamiltonian flow @xmath59 of @xmath5 . when the period @xmath45 is clear from the context and , in particular , if @xmath60 , we denote the time-@xmath45 map @xmath61 by @xmath62 . the action spectrum @xmath63 of @xmath5 is the set of critical values of @xmath58 . this is a zero measure , closed set ; see , e.g. , @xcite . in this paper we are only concerned with contractible periodic orbits . _ a periodic orbit is always assumed to be contractible , even if this is not explicitly stated_. a @xmath45-periodic orbit @xmath52 of @xmath5 is _ non - degenerate _ if the linearized return map @xmath64 has no eigenvalues equal to one . following @xcite , we call @xmath52 _ weakly non - degenerate _ if at least one of the eigenvalues is different from one . when all eigenvalues are equal to one , the orbit is said to be _ strongly degenerate_. when @xmath52 is non - degenerate or even weakly non - degenerate , the so - called _ conley zehnder index _ @xmath65 is defined , up to a sign , as in @xcite . more specifically , in this paper , the conley zehnder index is the negative of that of @xcite . in other words , we normalize @xmath66 so that @xmath67 when @xmath52 is a non - degenerate maximum of an autonomous hamiltonian with small hessian . more generally , when @xmath5 is autonomous and @xmath52 is a non - degenerate critical point of @xmath5 such that the eigenvalues of the hessian ( with respect to a metric compatible with @xmath22 ) are less than @xmath68 , the conley zehnder index of @xmath52 is equal to one half of the negative signature of the hessian . sometimes , the same hamiltonian can be treated as @xmath45-periodic for different values of @xmath43 . for instance , an autonomous hamiltonian is @xmath45-periodic for every @xmath43 and a @xmath45-periodic hamiltonian can also be viewed as @xmath69-periodic for any integer @xmath70 . in this paper , it will be essential to keep track of the period . unless specified otherwise , every hamiltonian @xmath5 considered here is originally one - periodic and @xmath45 is always an integer . when we wish to view @xmath5 as a @xmath45-periodic hamiltonian , we denote it by @xmath71 and refer to it as the _ @xmath45-th iteration _ of @xmath5 . ( the parentheses here are used to distinguish iterated hamiltonians from families of hamiltonians , say @xmath72 , parametrized by @xmath73 . ) since @xmath71 is regarded as a @xmath45-periodic hamiltonian , it makes sense to speak only about @xmath45-periodic ( or @xmath69-periodic ) orbits of @xmath71 . clearly , @xmath45-periodic orbits of @xmath71 are simply @xmath45-periodic orbits of @xmath5 . when @xmath74 is a one - periodic orbit of @xmath5 , its _ @xmath45-th iteration _ is the obviously defined map @xmath75 obtained by composing the @xmath45-fold covering map @xmath76 with @xmath52 . thus , @xmath77 is a @xmath45-periodic orbit of @xmath5 and @xmath71 . we call a @xmath45-periodic orbit _ simple _ if it is not an iteration of an orbit of a smaller period . as is well - known , the fixed points of @xmath78 are in one - to - one correspondence with ( not - necessarily contractible ) one - periodic orbits of @xmath5 . likewise , the @xmath45-periodic points of @xmath62 , i.e. , the fixed points of @xmath79 , are in one - to - one correspondence with ( not - necessarily contractible ) @xmath45-periodic orbits . in the proof of theorem [ thm : main ] , we will work with ( contractible ! ) periodic orbits of a hamiltonian @xmath5 whose time - one flow is @xmath2 . in fact , as is easy to see from section [ sec : loops ] , the free homotopy type of the one - periodic orbit of @xmath5 through a fixed point @xmath13 of @xmath2 is completely determined by @xmath2 and @xmath13 and is independent of the choice of @xmath5 . the same holds for @xmath45-periodic points and orbits . hence , `` contractible fixed points or periodic points '' of @xmath2 , i.e. , those with contractible orbits , are well defined and we will establish theorem [ thm : main ] for points in this class . when @xmath80 and @xmath5 are two ( say , one - periodic ) hamiltonians , the composition @xmath81 is defined by the formula @xmath82 the flow of @xmath81 is @xmath83 . in general , @xmath81 is not one - periodic . however , this will be the case if , for example , @xmath84 . another instance when the composition @xmath81 of two one - periodic hamiltonians is automatically one - periodic is when the flow @xmath85 is a loop of hamiltonian diffeomorphisms , i.e. , @xmath86 . [ sec : norms ] in what follows , it will be convenient to use a somewhat unconventional terminology and work with @xmath87-norms of functions , vector fields , etc . taken with respect to a coordinate system . let @xmath88 be a coordinate system on a neighborhood @xmath89 of a point @xmath90 , i.e. , @xmath88 is a diffeomorphism @xmath91 sending @xmath13 to the origin . ( thus , @xmath89 is a part of the data @xmath88 . ) let @xmath92 be a function on @xmath89 or on the entire manifold @xmath20 . the @xmath87-norm @xmath93 of @xmath92 with respect to @xmath88 is , by definition , the @xmath87-norm of @xmath92 on @xmath89 with respect to the flat metric associated with @xmath88 , i.e. , the pull - back by @xmath88 of the standard metric on @xmath94 . the @xmath87-norm with respect to @xmath88 of a vector field or a field of operators on @xmath89 is defined in a similar fashion . likewise , the norm @xmath95 of a vector @xmath96 in a finite - dimensional vector space @xmath33 with respect to a basis @xmath97 is the norm of @xmath96 with respect to the euclidean inner product for which @xmath97 is an orthonormal basis . the norm of an operator @xmath98 with respect to @xmath97 is defined in a similar way . when @xmath88 is a coordinate system near @xmath13 , we denote by @xmath99 the natural coordinate basis in @xmath100 arising from @xmath88 . [ exam : norm1 ] let @xmath101 be a linear map with all eigenvalues equal to zero . then @xmath102 can be made arbitrarily small by choosing a suitable basis @xmath97 . in other words , for any @xmath103 there exists @xmath97 such that @xmath104 . indeed , in some basis @xmath105 is given by an upper triangular matrix with zeros on the diagonal ; @xmath97 is then obtained by appropriately scaling the elements of this basis . [ exam : norm2 ] restricting @xmath88 to a smaller neighborhood of @xmath13 reduces the norm of @xmath92 . however , one can not make , say , @xmath106 arbitrarily small by shrinking @xmath89 unless @xmath107 and @xmath108 . indeed , @xmath109 . it is clear that for a fixed basis @xmath97 in @xmath100 and a function @xmath92 near @xmath13 there exists a coordinate system @xmath88 with @xmath110 , such that @xmath106 is arbitrarily close to @xmath111 . in this section , we briefly recall the notion of filtered floer homology for closed symplectically aspherical manifolds . all definitions and results mentioned here are quite standard and well known and we refer the reader to floer s papers @xcite , to @xcite , or to @xcite for introductory accounts of the construction of floer homology in this ( or more general ) setting . let us first focus on one - periodic hamiltonians . consider a hamiltonian @xmath5 such that all one - periodic orbits of @xmath5 are non - degenerate . this is a generic condition and we will call such hamiltonians _ non - degenerate_. let @xmath28 be a ( time - dependent : @xmath112 ) almost complex structure on @xmath1 compatible with @xmath22 . for two one - periodic orbits @xmath113 of @xmath5 denote by @xmath114 the space of solutions @xmath115 of the floer equation @xmath116 which are asymptotic to @xmath113 at @xmath117 , i.e. , @xmath118 point - wise as @xmath119 . the energy @xmath120 of a solution @xmath121 of the floer equation , , is @xmath122 where we set @xmath123 . every finite energy solution of is asymptotic to some @xmath113 and @xmath124 when @xmath26 meets certain standard regularity requirements that hold generically ( see , e.g. , @xcite ) , the space @xmath114 is a smooth manifold of dimension @xmath125 . this space carries a natural @xmath126-action @xmath127 and we denote by @xmath128 the quotient @xmath129 . when @xmath130 , the set @xmath128 is finite and we denote the number , @xmath131 , of points in this set by @xmath132 . let @xmath133 be two points outside @xmath63 . denote by @xmath134 the vector space over @xmath135 generated by one - periodic orbits of @xmath5 with @xmath136 and @xmath137 . the floer differential @xmath138 is defined by @xmath139 where the summation is over all @xmath140 such that @xmath141 and @xmath142 . as is well - known , @xmath143 . the homology @xmath144 of the resulting complex is called the _ filtered floer homology _ of @xmath5 for the interval @xmath145 . thus , @xmath146 is the ordinary floer homology . it is a standard fact that @xmath147 . in general , @xmath148 depends on the hamiltonian @xmath5 , but not on @xmath26 . the subcomplexes @xmath149 , where @xmath150 , form a filtration of the total floer complex @xmath151 , called the _ action filtration _ , and @xmath152 can be identified with the complex @xmath153 ; see , e.g. , @xcite . let now @xmath154 be three points outside @xmath63 . then , similarly , @xmath155 is a subcomplex of @xmath156 , and @xmath157 is naturally isomorphic to @xmath158 . as a result , we have the long exact sequence @xmath159 the filtered floer homology of @xmath5 is defined even when the periodic orbits of @xmath5 are not necessarily non - degenerate , provided that @xmath133 are outside @xmath63 . namely , let @xmath160 be a @xmath161-small perturbation , rather than the difference @xmath162 , as a perturbation of @xmath5 . however , it is the difference @xmath162 that is required to be @xmath161-small . ] of @xmath5 with non - degenerate one - periodic orbits . the filtered floer homology @xmath148 of @xmath5 is by definition @xmath163 . ( clearly , @xmath133 are still outside @xmath164 . ) these groups are canonically isomorphic for different choices of @xmath160 ( close to @xmath5 ) and the results discussed here hold for @xmath148 , @xcite . in fact , when an assertion concerns individual hamiltonians ( as opposed to families of hamiltonians ) , it is usually sufficient to prove the assertion in the non - degenerate case , for then it extends `` by continuity '' to all hamiltonians . these constructions carry over to @xmath45-periodic hamiltonians word - for - word by replacing one - periodic orbits with @xmath45-periodic ones . when @xmath5 is one - periodic , but we treat it as @xmath45-periodic for some integer @xmath43 , we denote the resulting floer homology groups @xmath165 . [ sec : homotopy ] consider two non - degenerate hamiltonians @xmath166 and @xmath167 . let @xmath72 be a homotopy from @xmath166 to @xmath167 . by definition , this is a family of hamiltonians parametrized by @xmath168 such that @xmath169 when @xmath73 is large negative and @xmath170 when @xmath73 is large positive . ( strictly speaking , the notion of homotopy includes also a family of almost complex structures @xmath171 ; see , e.g. , @xcite . we suppress this part of the homotopy structure in the notation . ) assume , in addition , that the homotopy is _ monotone decreasing _ , i.e. , @xmath172 is a decreasing function of @xmath73 for all @xmath173 and @xmath112 . ( thus , in particular , @xmath174 . ) then , whenever @xmath133 are outside @xmath175 and @xmath176 , the homotopy @xmath72 induces a homomorphism of complexes @xmath177 by the standard continuation construction ; see , e.g. , @xcite . namely , for a one - periodic orbit @xmath178 of @xmath166 and a one - periodic orbit @xmath140 of @xmath167 , let @xmath179 be the space of solutions of with @xmath72 on the right hand side , asymptotic to @xmath178 and , respectively , @xmath140 at @xmath117 . under the well - known regularity requirements on @xmath26 and @xmath72 , the space @xmath180 is a smooth manifold of dimension @xmath181 ; see , e.g. , @xcite . moreover , @xmath180 is a finite collection of points when @xmath182 . the map @xmath183 is defined by @xmath184 where the summation is over all @xmath140 such that @xmath182 and @xmath142 . the induced _ homotopy map _ in the filtered floer homology , also denoted by @xmath183 , is independent of the decreasing homotopy @xmath72 and commutes with the maps from the long exact sequence , see , e.g. , @xcite . by `` continuity '' in the hamiltonians and the homotopy , this construction extends to all ( not necessarily non - degenerate ) hamiltonians and all decreasing ( but not necessarily regular ) homotopies as long as @xmath133 are not in @xmath175 and @xmath176 . a ( non - monotone ) homotopy @xmath72 from @xmath166 to @xmath167 with @xmath185 and @xmath186 outside @xmath187 for all @xmath73 gives rise to an isomorphism between the groups @xmath188 , and hence , in particular , @xmath189 see @xcite . this isomorphism is defined by breaking the homotopy @xmath72 into a composition of homotopies @xmath190 close to the identity . each of the homotopies @xmath190 and its inverse homotopy increase action by no more than some small @xmath191 . then , it is shown that the map in @xmath192 induced by @xmath190 is an isomorphism . although this construction requires additional choices , it is not hard to see that the isomorphism is uniquely determined by the homotopy @xmath72 . furthermore , commutes with the maps from the long exact sequence , provided that all three points @xmath154 are outside @xmath187 for all @xmath73 ; @xcite . note also that when @xmath72 is a decreasing homotopy , the isomorphism coincides with @xmath183 . [ ex : isospec ] a homotopy @xmath72 is said to be _ isospectral _ if @xmath187 is independent of @xmath73 . in this case , the isomorphism is defined for any @xmath133 outside @xmath187 . for instance , let @xmath193 , where @xmath112 and @xmath194 $ ] , be a family of loops of hamiltonian diffeomorphisms based at @xmath195 , i.e. , @xmath196 for all @xmath73 . in other words , @xmath193 is a based homotopy from the loop @xmath197 to the loop @xmath198 . let @xmath199 be a family of one - periodic hamiltonians generating these loops and let @xmath5 be a fixed one - periodic hamiltonian . then @xmath200 is an isospectral homotopy , provided that @xmath201 are suitably normalized . ( namely , @xmath202 for all @xmath73 ; see section [ sec : loops ] . ) it is easy to see that if @xmath203 for all @xmath73 , the isomorphism intertwines monotone homotopy homomorphisms from @xmath80 to @xmath166 and to @xmath167 , i.e. , the diagram @xmath204_{\psi_{k , h^0}}\ar[rd]^{\psi_{k , h^1 } } & \\ { \hf^{(a,\,b)}_*(h^0 ) } \ar[r]^{\cong } & { \hf^{(a,\,b)}_*(h^1 ) } } \ ] ] is commutative . note that it is not at all clear whether the same is true if we only require that @xmath205 and @xmath206 . we conclude this section by establishing a technical result , used later on in the proof , giving a criterion for a monotone homotopy map to be non - zero . [ lemma : non - zero ] let @xmath72 be a monotone decreasing homotopy such that a point @xmath13 is a non - degenerate constant one - periodic orbit of @xmath72 and @xmath207 for all @xmath73 and @xmath17 . then the monotone homotopy map @xmath208 is non - trivial , provided that @xmath209 . [ rmk : non - zero ] in fact , we will prove a stronger result . let us perturb @xmath166 and @xmath167 away from @xmath13 , making these hamiltonians non - degenerate . then @xmath13 is a cycle in @xmath210 and @xmath211 and , moreover , this cycle is not homologous to any cycle that does not include @xmath13 . ( this is easy to see from the energy estimates ; see , e.g. , @xcite . ) in particular , @xmath212\neq 0 $ ] in @xmath213 and @xmath214 and we will show that @xmath183 sends @xmath212\in\hf^{(a,\,b)}_*(h^0)$ ] to @xmath212\in \hf^{(a,\,b)}_*(h^1)$ ] . moreover , a simple modification of our argument proves the following : there exist @xmath161-small non - degenerate perturbations @xmath215 of @xmath166 and @xmath216 of @xmath167 for which @xmath13 is still a non - degenerate constant one - periodic orbit , and a regular monotone decreasing homotopy @xmath217 from @xmath215 to @xmath216 such that the cycle @xmath13 for @xmath215 is connected to @xmath13 for @xmath216 by an odd number of homotopy trajectories and all such trajectories are contained in a small neighborhood of @xmath13 . ( note that we do not assume that @xmath13 is a non - degenerate constant one - periodic orbit of @xmath217 for all @xmath73 or that @xmath218 . of course , the lemma can be further generalized . for instance , the constant orbit @xmath13 can be replaced by a fixed one - periodic orbit . let us perturb the homotopy on the complement of a neighborhood @xmath89 of @xmath13 , keeping the homotopy monotone decreasing , to ensure that all but a finite number of the hamiltonians @xmath72 are non - degenerate . in particular , we will assume that @xmath166 and @xmath167 are such . this can be achieved by an arbitrarily small perturbation of @xmath72 . we keep the notation @xmath72 for the perturbed homotopy . if the homotopy @xmath72 were regular , we would simply argue that the constant connecting trajectory @xmath219 is the only connecting trajectory from @xmath13 for @xmath166 to @xmath13 for @xmath167 . indeed , @xmath220 for any such connecting trajectory @xmath96 , and thus @xmath96 must be constant . however , while it is easy to guarantee that @xmath72 is regular away from @xmath13 by reasoning as in , e.g. , @xcite , it is not _ a priori _ obvious that the transversality requirements can be satisfied for @xmath219 because of the constraint @xmath207 . rather than checking regularity of @xmath121 by a direct calculation , we chose to circumvent this difficulty . as in the proof of ( see , e.g. , @xcite ) , we can break the homotopy @xmath72 by reparametrization of @xmath73 into a composition of homotopies @xmath221 from @xmath222 to @xmath223 with @xmath224 and @xmath225 . these homotopies are monotone , since @xmath72 is monotone , and close to the identity homotopy . for every @xmath191 , this can be done so that the inverse homotopy @xmath226 from @xmath227 to @xmath228 increases the actions by no more than @xmath191 . without loss of generality , we may assume that the hamiltonians @xmath228 are non - degenerate . since all `` direct '' homotopies are monotone decreasing , we have @xmath229 observe that it suffices to establish the lemma for @xmath185 and @xmath186 arbitrarily close to @xmath230 . let @xmath89 be so small that @xmath13 is the only one - periodic trajectory of @xmath72 entering @xmath89 for all @xmath73 . ( since @xmath13 is isolated for all @xmath72 , such a neighborhood @xmath89 exists . ) there exists a constant @xmath231 such that every floer anti - gradient trajectory @xmath96 connecting @xmath13 with any other one - periodic orbit with action in the range @xmath145 has energy @xmath232 for any regular hamiltonian in the family @xmath72 ( cf . in particular , this holds for @xmath228 and @xmath227 and , moreover , for every regular hamiltonian in the homotopy @xmath221 . we will pick @xmath185 and @xmath186 so that @xmath233 and @xmath234 . then , for every @xmath228 the point @xmath13 is a cycle in @xmath235 and @xmath13 is not homologous to any cycle that does not include @xmath13 . now it is sufficient to prove that @xmath236 sends @xmath212\in\hf^{(a,\,b)}_*(k^i)$ ] to @xmath212\in\hf^{(a,\,b)}_*(k^{i+1})$ ] . to this end , let us first prove that @xmath237)\neq 0 $ ] . we may assume that none of the points @xmath185 , @xmath238 , @xmath186 , @xmath239 is in @xmath240 or in @xmath241 . it is easy to see ( see , e.g. , @xcite ) that @xmath242 is the natural `` quotient - inclusion '' map , i.e. , the composition of the `` quotient '' and `` inclusion '' maps @xmath243 note that @xmath244 is completely determined by @xmath72 and @xmath89 and is independent of how @xmath72 is broken into the homotopies @xmath221 , and thus of @xmath191 . pick @xmath191 so that @xmath245 and @xmath246 . then @xmath13 is a cycle in @xmath247 , which is not homologous to any cycle that does not include @xmath13 . as a consequence , @xmath212\neq 0 $ ] in both of the floer homology groups in and @xmath248)=[p]$ ] therefore , @xmath237)\neq 0 $ ] in @xmath249 . to show that @xmath237)=[p]$ ] , we need to refine our choice of @xmath244 . note that there exists @xmath231 such that , in addition to the above requirements , every @xmath221-homotopy trajectory starting at @xmath13 and leaving @xmath89 has energy greater than @xmath244 . then , clearly , the same is true for every sufficiently @xmath161-small perturbation @xmath250 of @xmath221 . again , @xmath244 depends only on @xmath72 and @xmath89 , but not on breaking @xmath72 into the homotopies @xmath221 . ( the existence of @xmath231 with these properties readily follows from energy estimates for connecting trajectories , cf . pick a @xmath161-small regular perturbation @xmath250 of @xmath221 . we may still assume that @xmath250 is monotone decreasing and @xmath13 is a non - degenerate constant orbit of @xmath251 and @xmath252 . however , @xmath13 is not required to be a constant one - periodic orbit of @xmath250 for all @xmath73 ; nor is @xmath253 constant as a function of @xmath73 and @xmath17 . clearly , the inverse homotopy to @xmath250 does not increase action by more than @xmath191 . hence , in the homological analysis of the previous paragraph we can replace @xmath228 and @xmath227 by @xmath254 and @xmath252 . in fact , the original and perturbed hamiltonians have equal filtered floer homology for relevant action intervals and the maps @xmath236 and @xmath226 are induced by the maps for @xmath250 acting on the level of complexes . therefore , @xmath255)=\psi_{k^i , k^{i+1}}([p])\neq 0 $ ] in @xmath256 , and thus @xmath257 in @xmath258 . since @xmath233 and @xmath259 and every connecting orbit leaving @xmath89 must have energy greater than @xmath244 , we conclude that @xmath260 in the floer complexes , and hence in the floer homology . [ sec : loops ] in this section , we recall a few well - known facts about loops of hamiltonian diffeomorphisms of @xmath1 . we will focus on loops parametrized by @xmath39 , but obviously all results discussed here hold for loops of any period . furthermore , throughout the paper all loops @xmath261 are assumed to be based at @xmath195 , i.e. , @xmath262 ; contractible loops are thus required to be contractible in this class . recall that , as is proved in @xcite , the filtered floer homology of the hamiltonian @xmath5 is determined , up to a shift of filtration , entirely by the time - one map @xmath62 and is independent of the hamiltonian @xmath5 . this fact translates to geometric properties of loops of hamiltonian diffeomorphisms , which are briefly reviewed below , and is actually proved by first establishing these properties . let @xmath263 , @xmath112 , be a loop generated by a periodic hamiltonian @xmath264 . then all orbits @xmath265 of @xmath266 with @xmath112 and @xmath173 are one - periodic and lie in the same homotopy class . hence , every orbit of @xmath264 is contractible by the arnold conjecture . the action @xmath267 is independent of @xmath173 ( see , e.g. , @xcite ) and @xmath268 , where @xmath269 is the symplectic volume of @xmath1 . the latter identity is easy to prove when the loop @xmath266 is contractible ( see , e.g. , @xcite ) ; in the general case , this is a non - trivial result , @xcite . for @xmath52 as above pick a trivialization of @xmath270 that extends to a trivialization of @xmath271 along a disk bounded by @xmath52 . using this trivialization , we can view the maps @xmath272 as a loop in @xmath273 . hence , the linearization @xmath274 along @xmath52 gives rise to an element in @xmath275 , which could be called the maslov index , @xmath276 , of the loop @xmath266 if it were non - trivial ; cf.@xcite . the maslov index is well defined : it is independent of @xmath52 , the trivialization and the disc . ( the latter follows from the fact that @xmath277 . ) however , as is well known and as we will soon reprove , @xmath278 ; see also , e.g. , @xcite . let @xmath5 be a periodic hamiltonian on @xmath1 . recall that @xmath279 is the hamiltonian generating the flow @xmath280 . this hamiltonian is automatically one - periodic and its time - one map is @xmath62 . we claim that there exists an isomorphism of filtered floer homology @xmath281 indeed , composition with @xmath263 sends one - periodic orbits of @xmath5 to one - periodic orbits of @xmath282 with shift of action by @xmath283 and shift of conley zehnder indices by @xmath284 . ( see , e.g. , @xcite ; the negative sign is a result of the difference in conventions . ) furthermore , let @xmath121 be a floer anti - gradient trajectory for @xmath5 and a time - dependent almost complex structure @xmath26 . then , as a straightforward calculation shows , @xmath285 is a floer anti - gradient trajectory for @xmath282 and the almost complex structure @xmath286 . furthermore , it is clear that the transversality requirements are satisfied for @xmath287 if and only if they are satisfied for @xmath288 . therefore , the composition with @xmath289 commutes with the floer differential and thus induces an isomorphism of floer complexes ( and hence homology groups ) shifting action by @xmath283 and grading by @xmath284 . applying this construction to the full floer homology @xmath290 , we see that the grading shift must be zero , i.e. , @xmath278 . when the loop @xmath289 is contractible , the existence of an isomorphism readily follows from ; see example [ ex : isospec ] . however , it is not clear whether this is the same isomorphism as constructed above . let now @xmath266 be a loop of ( the germs of ) hamiltonian diffeomorphisms at @xmath173 generated by @xmath264 . in other words , the maps @xmath266 and the hamiltonian @xmath264 are defined on a small neighborhood of @xmath13 and @xmath291 for all @xmath112 . then the action @xmath283 and the maslov index @xmath276 are introduced exactly as above with the orbit @xmath52 taken sufficiently close to @xmath13 . in fact , we can set @xmath292 and hence @xmath293 and @xmath276 is just the maslov index of the loop @xmath294 in @xmath295 . note that in this case @xmath276 need not be zero . we conclude this section by giving a necessary and sufficient condition , to be used later , for @xmath289 to extend to a loop of global hamiltonian diffeomorphisms of @xmath1 . [ lemma : loop - ext ] let @xmath266 , @xmath112 , be a loop of germs of hamiltonian diffeomorphisms at @xmath173 . the following conditions are equivalent : * the loop @xmath289 extends to a loop of global hamiltonian diffeomorphisms of @xmath1 , * the loop @xmath289 extends to a loop of global hamiltonian diffeomorphisms of @xmath1 , contractible in the class of loops fixing @xmath13 , * the loop @xmath289 is contractible in the group of germs of hamiltonian diffeomorphisms at @xmath13 , * @xmath278 . the implications ( ii)@xmath296(i ) and ( iii)@xmath296(iv ) are clear and ( i)@xmath296(iv ) is established above . to prove that ( iv)@xmath296(iii ) , we identify a neighborhood of @xmath13 in @xmath1 with a neighborhood of the origin in @xmath297 . then , as is easy to see , the loop @xmath266 is homotopy equivalent to its linearization @xmath274 , a loop of ( germs of ) linear maps . by the definition of the maslov index , @xmath274 is contractible in @xmath298 if and only if @xmath299 . to complete the proof of the lemma , it remains to show that ( iii)@xmath296(ii ) . to this end , let us first analyze the case where @xmath266 is @xmath300-close to the identity . fixing a small neighborhood @xmath89 of @xmath13 , we identify a neighborhood of the diagonal in @xmath301 with a neighborhood of the zero section in @xmath302 . then the graphs of @xmath266 in @xmath301 turn into lagrangian sections of @xmath302 . these sections are the graphs of exact forms @xmath303 on @xmath89 , where all @xmath304 are @xmath161-small and @xmath305 . then we extend ( the germs of ) the functions @xmath304 to @xmath161-small functions @xmath306 on @xmath1 such that @xmath307 . the graphs of @xmath308 in @xmath309 form a loop of exact lagrangian submanifolds which are @xmath300-close to the zero section . thus , this loop can be viewed as a loop of hamiltonian diffeomorphisms of @xmath1 . it is clear that the resulting loop is contractible in the class of loops fixing @xmath13 . to deal with the general case , consider a family @xmath310 , @xmath311 $ ] , of local loops with @xmath312 and @xmath313 . let @xmath314 be a partition of the interval @xmath315 $ ] such that the loops @xmath316 and @xmath317 are @xmath300-close for all @xmath318 . in particular , the loop @xmath319 is @xmath300-close to @xmath320 , and thus extends to a contractible loop @xmath321 on @xmath1 . arguing inductively , assume that a contractible extension @xmath322 of @xmath316 has been constructed . consider the loop @xmath323 defined near @xmath13 . this loop is @xmath300-close to the identity , for @xmath317 and @xmath316 are @xmath300-close . hence , @xmath324 extends to a contractible loop @xmath325 on @xmath1 . then @xmath326 is the required extension of @xmath317 , contractible in the class of loops fixing @xmath13 . [ rmk : loop - ext ] it is clear from the proof of lemma [ lemma : loop - ext ] that the extension of the germ of a loop near @xmath13 to a global loop fixing @xmath13 can be carried out with some degree of control of the @xmath87-norm and the support of the loop . we will need the following simple fact , which can be easily verified by adapting the proof of the implication ( iii)@xmath296(ii ) . assume that @xmath289 is the germ of a loop near @xmath13 and the linearization of @xmath289 at @xmath13 is equal to the identity : @xmath327 for all @xmath17 . then @xmath289 extends to a loop @xmath328 of global hamiltonian diffeomorphisms of @xmath1 such that @xmath328 is contractible in the class of loops fixing @xmath13 and having identity linearization at @xmath13 . [ sec : lfh ] [ sec : lmh ] let @xmath329 be a smooth function on a manifold @xmath20 and let @xmath330 be an isolated critical point of @xmath92 . fix a small neighborhood @xmath89 of @xmath13 containing no other critical points of @xmath92 and consider a small generic perturbation @xmath331 of @xmath92 in @xmath89 . to be more precise , @xmath331 is morse inside @xmath89 and @xmath300-close to @xmath92 . then , as is easy to see , for any two critical points of @xmath331 in @xmath89 , all anti - gradient trajectories connecting these two points are contained in @xmath89 . moreover , the same is true for broken trajectories connecting these two points . as a consequence , the vector space ( over @xmath135 ) generated by the critical points of @xmath331 in @xmath89 is a complex with ( morse ) differential defined in the standard way . ( see , e.g. , @xcite . ) furthermore , the continuation argument shows that the homology of this complex , denoted here by @xmath332 and referred to as the _ local morse homology _ of @xmath92 at @xmath13 , is independent of the choice of @xmath331 . this construction is a particular case of the one from @xcite . assume that @xmath13 is a non - degenerate critical point of @xmath92 of index @xmath70 . then @xmath333 when @xmath334 and @xmath335 otherwise . [ exam : morse - max ] when @xmath13 is a strict local maximum of @xmath92 , we have @xmath336 . indeed , in this case , as is easy to see from standard morse theory , @xmath337 where @xmath191 is assumed to be small and such that @xmath338 is a regular value of @xmath92 . we will need the following two properties of local morse homology : 1 . let @xmath339 , @xmath194 $ ] , be a family of smooth functions with _ uniformly isolated _ critical point @xmath13 , i.e. , @xmath13 is the only critical point of @xmath339 , for all @xmath73 , in some neighborhood of @xmath13 independent of @xmath73 . then @xmath340 is constant throughout the family , and hence @xmath341 ; cf . * lemma 4 ) . the function @xmath92 has a ( strict ) local maximum at @xmath13 if and only if @xmath342 , where @xmath343 . the first assertion , ( lm1 ) , is again established by the continuation argument ; cf . @xcite . we emphasize that here the assumption that @xmath13 is uniformly isolated is essential and can not be replaced by the weaker condition that @xmath13 is just an isolated critical point of @xmath339 for all @xmath73 . ( example : @xmath344 on @xmath126 with @xmath345 . the author is grateful to doris hein for this remark . ) regarding ( lm2 ) first note that , by example [ exam : morse - max ] , @xmath342 when @xmath92 has a strict local maximum at @xmath13 . the converse requires a proof although the argument is quite standard . denote by @xmath266 the anti - gradient flow of @xmath92 . let @xmath346 be a closed connected neighborhood of @xmath13 with piecewise smooth boundary @xmath347 such that whenever @xmath348 and @xmath349 the entire trajectory segment @xmath350 with @xmath351 $ ] is also in @xmath346 , and @xmath13 is the only critical point of @xmath92 contained in @xmath346 . we call @xmath346 a gromoll meyer neighborhood of @xmath13 . it is not hard to show that @xmath13 has an ( arbitrarily small ) gromoll meyer neighborhood ; see @xcite or @xcite . ( strictly speaking , the above definition is slightly different from the one used in @xcite . however , the existence proof given in @xcite goes through with no modifications . ) when @xmath331 is a @xmath161-small generic perturbation of @xmath92 supported in @xmath346 , the morse complex of @xmath352 is defined and its homology is equal to @xmath332 . ( the fact that @xmath143 follows from the requirements on @xmath346 . ) assume that @xmath342 . then there exists a non - zero cycle @xmath353 of degree @xmath19 in the morse complex of @xmath352 . let @xmath33 be the closure of the union of the unstable manifolds of @xmath352 for all local maxima entering @xmath353 . the set @xmath33 is the closure of a domain with piecewise smooth boundary . the condition that @xmath353 is a cycle implies that for every critical point @xmath178 of @xmath331 in @xmath33 , the intersection of the unstable manifold of @xmath178 with @xmath346 is contained entirely in the interior of @xmath33 . hence , @xmath354 , and thus @xmath355 . it follows that at every smooth point @xmath356 , the gradient @xmath357 either points inward or is tangent to @xmath347 . consider a gromoll meyer neighborhood @xmath358 of @xmath13 . note that for a small generic @xmath191 the connected component @xmath346 of @xmath359 containing @xmath13 is also a gromoll meyer neighborhood . clearly , when @xmath13 is not a local maximum of @xmath92 , there are smooth points on @xmath347 where @xmath360 points inward , provided that @xmath191 is small . as a consequence of the above analysis , @xmath361 if @xmath13 is not a local maximum . this completes the proof of the implication @xmath362 . generalizing example [ exam : morse - max ] and the proof of ( lm2 ) , it is not hard to relate local morse homology to local homology of a function , introduced in @xcite ; see also @xcite . however , we do not touch upon this question , for such a generalization is not necessary for the proof of theorem [ thm : main ] . in the setting of local homology , the analogues of ( lm1 ) and ( lm2 ) are established in @xcite and , respectively , in @xcite . [ sec : lfh2 ] let @xmath52 be an isolated one - periodic orbit of a hamiltonian @xmath363 . pick a sufficiently small tubular neighborhood @xmath89 of @xmath52 and consider a non - degenerate @xmath161-small perturbation @xmath160 of @xmath5 supported in @xmath89 . more specifically , let @xmath89 be a neighborhood of @xmath364 , where @xmath52 is viewed as a curve in the extended phase space @xmath365 , and let @xmath160 be a hamiltonian @xmath161-close to @xmath5 , equal to @xmath5 outside of @xmath89 , and such that all one - periodic orbits of @xmath160 that enter @xmath89 are non - degenerate . ( such perturbations @xmath160 do exist ; see ( * ? ? ? * theorem 9.1 ) . ) abusing notation , we will treat @xmath89 simultaneously as an open set in @xmath1 and in @xmath366 . consider one - periodic orbits of @xmath160 contained in @xmath89 . every anti - gradient trajectory @xmath121 connecting two such orbits is also contained in @xmath89 , provided that @xmath367 and @xmath368 are small enough . indeed , the energy @xmath120 is equal to the difference of action values on the periodic orbits , and thus is bounded from above by @xmath369 . the @xmath161-norm of @xmath160 is bounded from above by a constant independent of @xmath160 , say @xmath370 . therefore , @xmath371 is pointwise uniformly bounded by @xmath369 , and it follows that @xmath121 takes values in @xmath89 ; see @xcite . note also that for a suitable small perturbation of a fixed almost complex structure on @xmath1 the transversality requirements are satisfied for moduli spaces of floer anti - gradient trajectories connecting one - periodic orbits @xmath160 contained in @xmath89 , @xcite . by the compactness theorem , every broken anti - gradient trajectory @xmath121 connecting two one - periodic orbits in @xmath89 lies entirely in @xmath89 . hence , the vector space ( over @xmath135 ) generated by one - periodic orbits of @xmath160 in @xmath89 is a complex with ( floer ) differential defined in the standard way . the continuation argument ( see , e.g. , @xcite ) shows that the homology of this complex is independent of the choice of @xmath160 and of the almost complex structure . we refer to the resulting homology group @xmath372 as the _ local floer homology _ of @xmath5 at @xmath52 . homology groups of this type were first considered ( in a more general setting ) by floer in @xcite ; see also ( * ? ? ? * section 3.3.4 ) . in fact , an orbit @xmath52 can be replaced by a connected isolated set @xmath373 of one - periodic orbits of @xmath5 ; @xcite . ( note that @xmath374 is constant , for @xmath58 is continuous and @xmath63 is nowhere dense . ) assume that @xmath52 is non - degenerate and @xmath375 . then @xmath376 when @xmath334 and @xmath377 otherwise . we will need the following properties of local floer homology : 1 . let @xmath72 , @xmath194 $ ] , be a family of hamiltonians such that @xmath52 is a _ uniformly isolated _ one - periodic orbit for @xmath72 , i.e. , @xmath52 is the only periodic orbit of @xmath72 , for all @xmath73 , in some open set independent of @xmath73 . then @xmath378 is constant throughout the family , and hence @xmath379 . this is again an immediate consequence of the continuation argument . however , it is worth pointing out that unless @xmath72 is monotone decreasing , the isomorphism @xmath379 is not induced by the homotopy @xmath72 in the same sense as the homomorphism @xmath183 is induced by a monotone homotopy ; see . the isomorphism in question is constructed similarly to by breaking @xmath72 into a composition of homotopies close to the identity . local floer homology spaces are building blocks for filtered floer homology . namely , essentially by definition , we have the following 1 . let @xmath380 be such that all one - periodic orbits @xmath381 of @xmath5 with action @xmath230 are isolated . ( as a consequence , there are only finitely many such orbits . ) then , if @xmath191 is small enough , @xmath382 in particular , if all one - periodic orbits @xmath52 of @xmath5 are isolated and @xmath383 for some @xmath70 and all @xmath52 , we have @xmath384 by the long exact sequence of filtered floer homology . the effect on local floer homology of the composition of @xmath5 with a loop of hamiltonian diffeomorphisms is the same as in the global setting and is established in a similar fashion ; see section [ sec : loops ] . 1 . let @xmath263 be a loop of hamiltonian diffeomorphisms of @xmath1 . then @xmath385 for every isolated one - periodic orbit @xmath52 of @xmath5 , where @xmath386 stands for the one - periodic orbit @xmath387 of @xmath282 corresponding to @xmath52 ; see section [ sec : loops ] . as is clear from the definition of local floer homology , @xmath5 need not be a function on the entire manifold @xmath1 it is sufficient to consider hamiltonians defined only on a neighborhood of @xmath52 . for the sake of simplicity , we focus on the particular case , relevant here , where @xmath388 is a constant orbit , and hence @xmath389 for all @xmath112 . then ( lf1 ) still holds and ( lf3 ) takes the following form : 1 . let @xmath263 be a loop of hamiltonian diffeomorphisms defined on a neighborhood of @xmath13 and fixing @xmath13 ( i.e. , @xmath291 for all @xmath112 ) . then @xmath390 where @xmath391 is the maslov index of the loop @xmath392 . note that in ( lf3 ) , in contrast with ( lf4 ) , we _ a priori _ know that @xmath393 as is pointed out in section [ sec : loops ] . hence , the shift of degrees does not occur when @xmath266 is a global loop . in other words , comparing ( lf3 ) and ( lf4 ) , we can say that the group @xmath372 is completely determined by the hamiltonian diffeomorphism @xmath394 and its fixed point @xmath395 , while the germ of @xmath62 at @xmath13 determines @xmath396 only up to a shift in degree . the degree depends on the class of @xmath59 in the universal covering of the group of germs of hamiltonian diffeomorphisms . finally note that in the construction of local floer homology the hamiltonian @xmath5 need not have period one . the definitions and results above extend word - for - word to @xmath45-periodic hamiltonians and , in particular , to the @xmath45-th iteration @xmath71 of a one - periodic hamiltonian @xmath5 as long as the @xmath45-periodic orbit in question is isolated . [ sec : lfh - lmh ] a fundamental property of floer homology is that @xmath397 is equal to the morse homology of a smooth function on @xmath1 ( and thus to the homology of @xmath1 ) . the key to establishing this fact is identifying @xmath397 with @xmath398 , when the hamiltonian @xmath5 is autonomous and @xmath161-small ; see @xcite . a similar identification holds for local floer homology . we consider here the case of @xmath45-periodic hamiltonians , for this is the ( superficially more general ) situation where the results will be applied in the subsequent sections . [ ex : lfh - lmh ] assume that @xmath13 is an isolated critical point of an autonomous hamiltonian @xmath399 and @xmath400 then @xmath401 . indeed , when the condition is satisfied , the hamiltonians @xmath402 , @xmath403 $ ] , have no non - trivial @xmath45-periodic orbits ( uniformly ) near @xmath13 . ( see @xcite or the proof of lemma [ lemma : lfh - lmh ] below . ) thus , @xmath13 is a uniformly isolated @xmath45-periodic orbit of @xmath402 for @xmath404 $ ] when @xmath191 is small , and @xmath405 is constant throughout this family by ( lf1 ) . the argument of @xcite shows that the floer complex of @xmath406 is equal to the local morse complex of @xmath399 when @xmath73 is close to zero . in what follows , we will need a slightly more general version of this fact , where the hamiltonian is `` close '' to a function independent of time . [ lemma : lfh - lmh ] let @xmath399 be a smooth function and let @xmath80 be a @xmath45-periodic hamiltonian , both defined on a neighborhood of a point @xmath13 . assume that @xmath13 is an isolated critical point of @xmath399 , and the following conditions are satisfied : * the inequalities @xmath407 and @xmath408 hold pointwise near @xmath13 for all @xmath48 . ( the dot stands for the derivative with respect to time . ) * the hessians @xmath409 and @xmath410 and the constant @xmath191 are sufficiently small . namely , @xmath411 and @xmath412 then @xmath13 is an isolated @xmath45-periodic orbit of @xmath80 . furthermore , 1 . @xmath413 ; 2 . if @xmath414 , the functions @xmath415 for all @xmath17 and @xmath399 have a strict local maximum at @xmath13 . the requirement of this lemma , asserting that @xmath80 is in a certain sense close to @xmath399 , plays a crucial role in our proof of theorem [ thm : main ] ( cf . lemmas [ lemma : local ] and [ lemma : local2 ] ) and in the argument of @xcite . to the best of the author s knowledge , this requirement is originally introduced in ( * ? ? ? * lemma 4 ) as that @xmath80 is relatively autonomous . in what follows , we will sometimes call @xmath399 a _ reference function _ and say that the pair @xmath416 meets the requirements of lemma [ lemma : lfh - lmh ] . since the statement is local , we may assume that @xmath417 . consider the family of hamiltonians @xmath418 starting with @xmath419 and ending with @xmath420 . we claim that @xmath292 is a uniformly isolated @xmath45-periodic orbit of @xmath421 for @xmath194 $ ] . we show this by adapting the proof of ( * ? ? ? * proposition 17 , p.184 ) . fix @xmath422 and let @xmath423 be the ball of radius @xmath424 centered at @xmath13 . since @xmath13 is a constant @xmath45-periodic orbit of @xmath421 , every @xmath45-periodic orbit @xmath52 of @xmath421 with @xmath395 sufficiently close to @xmath13 is contained in @xmath423 . recall also that @xmath425 for any map @xmath426 with zero mean . applying this inequality to @xmath427 , we obtain @xmath428 furthermore , from the first requirement on @xmath429 and @xmath430 , it is easy to see that @xmath431 pointwise . hence , @xmath432 once holds and @xmath422 is small , we have @xmath433 therefore , @xmath434 . in other words , @xmath52 is a constant loop , and thus a critical point of @xmath435 for @xmath48 . then , by , @xmath436 . as a consequence , @xmath292 since @xmath13 is an isolated critical point of @xmath399 . this shows that @xmath13 is a uniformly isolated @xmath45-periodic orbit of @xmath421 . by ( lf1 ) , the local floer homology @xmath437 is constant throughout the family @xmath421 , and @xmath438 . as a consequence of , the condition of example [ ex : lfh - lmh ] is satisfied . applying this example , we conclude that @xmath439 . this proves ( a ) . by ( lm2 ) , @xmath13 is an isolated local maximum of @xmath440 , and hence , as is easy to see from the first condition of the lemma , @xmath13 is a uniformly isolated critical point of @xmath435 for @xmath441 $ ] and every fixed @xmath48 . now , by ( lm1 ) and ( lm2 ) applied to @xmath442 , all functions @xmath435 , and , in particular , @xmath443 , have a ( strict ) local maximum at @xmath13 . this proves ( b ) and concludes the proof of the lemma . [ sec : proof ] as has been pointed out above , it is sufficient to prove the theorem for ( contractible ! ) periodic orbits of a hamiltonian @xmath5 generating @xmath2 rather than for all periodic points of @xmath2 . let @xmath363 be a one - periodic hamiltonian with finitely many one - periodic orbits @xmath444 . then , these orbits are isolated and the action spectrum of @xmath5 is comprised of finitely many points . for every one - periodic orbit @xmath444 of @xmath5 denote by @xmath445 the degrees of roots of unity , different from 1 , among the floquet multipliers of @xmath444 . arguing by contradiction , assume that for every sufficiently large integer @xmath12 , all @xmath12-periodic orbits of @xmath5 are iterated or , in other words , @xmath62 has only finitely many _ simple periods _ , i.e. , periods of simple , non - iterated , orbits . in particular , every periodic orbit of @xmath5 with sufficiently large period is iterated . let @xmath446 be the finite collection of integers comprised of all simple periods ( greater than 1 ) and the degrees @xmath447 for all one - periodic orbits @xmath444 . then , in particular , every @xmath12-periodic orbit is an iterated one - periodic orbit when @xmath12 which is not divisible by any of the integers @xmath448 . moreover , all @xmath12-periodic orbits are isolated and @xmath449 . recall also that , when @xmath444 is a weakly non - degenerate one - periodic orbit of @xmath5 and @xmath12 is a sufficiently large integer , not divisible by @xmath445 , we have @xmath450 indeed , as is shown in @xcite , for a generic perturbation of @xmath5 supported near @xmath444 , the orbit @xmath451 splits into non - degenerate orbits with conley zehnder index different from @xmath9 . next observe that there exists a strongly degenerate one - periodic orbit @xmath52 of @xmath5 such that @xmath452 is an isolated @xmath453-periodic orbit for some sequence @xmath454 and @xmath455 where all @xmath453 are divisible by @xmath456 and none of @xmath453 is divisible by @xmath446 . to prove this , first note that by for any sufficiently large integer @xmath12 , not divisible by @xmath446 , there exists a totally degenerate one - periodic orbit @xmath324 such that @xmath457 . ( otherwise , held for all @xmath12-periodic orbits , and we would have @xmath458 by ( lf2 ) . ) pick an infinite sequence @xmath459 of such integers satisfying the additional requirement that @xmath460 is divisible by @xmath461 for all @xmath462 . ( for instance , we can take @xmath463 , where @xmath464 is a sufficiently large prime . ) as we have observed , for every @xmath461 there exists a totally degenerate one - periodic orbit @xmath465 such that @xmath466 . since there are only finitely many distinct one - periodic orbits , one of the orbits @xmath52 among the orbits @xmath465 and some infinite subsequence @xmath453 in @xmath461 satisfy . ( we also re - index the subsequence @xmath453 to make it begin with @xmath456 . ) let @xmath467 . we will use the orbit @xmath52 and the sequence @xmath453 to prove * _ for every @xmath191 there exists @xmath468 such that for any @xmath469 and some @xmath470 in the range @xmath471 , depending on @xmath45 , we have @xmath472 _ the theorem readily follows from the claim . indeed , set @xmath473 . then , if @xmath469 is such that @xmath474 is not divisible by @xmath446 , we have @xmath475 . thus , for any fixed @xmath191 and @xmath476 , the interval @xmath477 contains no action values of @xmath478 when @xmath45 is sufficiently large . this contradicts the claim . ( note that we have used the assumption that @xmath2 has finitely many simple periods twice : the first time to find the orbit @xmath52 and the sequence @xmath453 and the second time to arrive at the contradiction with the claim . ) to establish the claim , it is convenient to adopt the following [ def : sympl - deg ] a one - periodic orbit @xmath52 of a one - periodic hamiltonian @xmath5 is said to be a _ symplectically degenerate _ maximum if there exists a sequence of loops @xmath465 of hamiltonian diffeomorphisms such that @xmath479 , i.e. , @xmath465 sends @xmath13 to @xmath52 , for some point @xmath173 and all @xmath480 and @xmath17 , and such that the hamiltonians @xmath228 given by @xmath481 and the loops @xmath465 have the following properties : * the point @xmath13 is a strict local maximum of @xmath482 for all @xmath112 and all @xmath480 , * there exist symplectic bases @xmath483 in @xmath484 such that @xmath485 * the linearization of the loop @xmath486 at @xmath13 is the identity map for all @xmath480 and @xmath487 ( i.e. , @xmath488 for all @xmath112 ) and , moreover , the loop @xmath489 is contractible to @xmath195 in the class of loops fixing @xmath13 and having the identity linearization at @xmath13 . [ rmk : sympl - deg ] regarding ( k1 ) and ( k3 ) note that since @xmath479 , the point @xmath13 is a fixed point of the flow @xmath490 of @xmath228 , and thus a critical point of @xmath491 for all @xmath17 . furthermore , @xmath13 is also a fixed point of the loop @xmath486 for all @xmath480 and @xmath487 , for @xmath492 , and hence @xmath493 . we refer the reader to section [ sec : norms ] for the definition and discussion of the norm with respect to a basis , used in ( k2 ) . the hamiltonians @xmath228 and @xmath5 have the same time - one flow and there is a natural one - to - one correspondence between ( contractible ) one - periodic orbits of the hamiltonians . the hamiltonians @xmath228 can be chosen so that @xmath494 is constant and equal to @xmath495 . in what follows , we will always assume that @xmath228 is normalized in this way . then the corresponding orbits of @xmath228 and @xmath5 have equal actions and , in particular , all hamiltonians @xmath228 have the same action spectrum and action filtration ; see section [ sec : loops ] . symplectically degenerate maxima are further investigated in @xcite . in particular , it is shown there that condition ( k3 ) is superfluous ; see ( * ? ? ? * remark 5.5 ) . this fact is not used in the present paper . assume that @xmath16 has a strict local maximum at @xmath13 ( and @xmath496 ) and @xmath497 for all @xmath17 . then @xmath13 is a symplectically degenerate maximum of @xmath5 . indeed , we can take @xmath498 and @xmath499 and any fixed symplectic basis as @xmath483 . more generally , vanishing of the hessian may be replaced by the condition that @xmath500 can be made arbitrarily small by a suitable choice of @xmath97 , cf . this condition is satisfied , for instance , when @xmath5 is autonomous and all eigenvalues of the linearization of @xmath49 at @xmath13 are equal to zero ; see lemma [ lemma : la ] . [ ex : sympl - deg ] assume that @xmath52 is a symplectically degenerate maximum of @xmath5 . let @xmath160 be a hamiltonian generating the flow @xmath501 , where @xmath289 is a loop of hamiltonian diffeomorphisms . then , the periodic orbit @xmath502 of @xmath160 is a symplectically degenerate maximum of @xmath160 as is easy to verify . ( in other words , symplectic degeneracy is a property of the fixed point @xmath395 of the time - one map @xmath62 . ) for instance , in the notation of definition [ def : sympl - deg ] , the constant orbit @xmath13 is a symplectically degenerate maximum of each hamiltonian @xmath228 . now we are in a position to state the two results that we need to complete the proof of theorem [ thm : main ] . the first result gives a floer homological criterion for an isolated , strongly degenerate orbit @xmath52 to be a symplectically degenerate maximum , and thus translates local floer homological properties of @xmath52 to geometrical features of a constant orbit @xmath13 of hamiltonians @xmath228 . the second one asserts non - vanishing of the filtered floer homology of an iterated hamiltonian @xmath71 for an interval of actions just above the action @xmath503 , provided that @xmath52 is a symplectically non - degenerate maximum of @xmath5 . when applied to the hamiltonian @xmath504 in place of @xmath5 , where @xmath456 is as in the claim , these results will yield the claim . [ prop1 ] let @xmath52 be a strongly degenerate isolated one - periodic orbit of @xmath5 such that its @xmath505-th iteration @xmath506 is also isolated and @xmath507 then @xmath52 is a symplectically degenerate maximum of @xmath5 . [ rmk : comp ] note that , similarly to definition [ def : sympl - deg ] , requirement is a condition on the fixed point @xmath395 of @xmath62 , independent of a particular choice of @xmath5 . [ prop2 ] let @xmath52 be a symplectically degenerate maximum of @xmath5 and let @xmath495 . then for every @xmath191 there exists @xmath468 such that @xmath508 combining the propositions , we conclude that whenever a strongly degenerate one - periodic orbit @xmath52 of @xmath5 satisfies the hypotheses of proposition [ prop1 ] , for every @xmath191 there exists @xmath468 such that @xmath509 where @xmath495 and @xmath476 . to prove the claim , first note that although propositions [ prop1 ] and [ prop2 ] are stated for one - periodic hamiltonians , similar results hold , of course , for hamiltonians and orbits of any period . thus , consider the hamiltonian @xmath504 in place of @xmath5 and the isolated orbit @xmath510 in place of @xmath52 in and . then the requirement is met due to : @xmath511 if @xmath480 is large enough , since @xmath454 . furthermore , @xmath512 and follows immediate from . it remains to establish propositions [ prop1 ] and [ prop2 ] to complete the proof of the theorem . it is illuminating to compare the above proof with the argument due to salamon and zehnder from @xcite asserting that ever large prime is a simple period whenever all one - periodic orbits of @xmath5 are weakly non - degenerate . ( in particular , the number of simple periods less than or equal to @xmath70 is of order at least @xmath513 . ) in the context of the present paper relying , of course , on @xcite , this is an immediate consequence of . to be more specific , if @xmath12 is a large prime and all @xmath12-periodic orbits are iterated , holds for all weakly non - degenerate one - periodic orbits and @xmath458 by ( lf2 ) , if there are no totally degenerate one - periodic orbits . when such one - periodic orbits exist , we can no longer use the salamon zehnder argument to conclude that every large prime is a simple period or even to establish the existence of infinitely many simple periods . the reason is that in this case the argument implies that for every large prime @xmath12 there is a one - periodic orbit @xmath52 such that @xmath514 . it is unclear , however , if @xmath515 , and hence whether or not @xmath52 is a symplectically degenerate maximum . [ sec : pr - prop1 ] our goal in this section is to construct the hamiltonians @xmath228 and the loops @xmath465 meeting requirements ( k1k3 ) . this construction relies on two technical lemmas , proved in section [ sec : gf ] , and is carried out in several steps . in section [ sec : near - p ] , we construct the hamiltonians @xmath228 and the loops @xmath465 near @xmath13 . we begin by proving in section [ sec : step2 ] that the time one map @xmath516 can be made @xmath300-close to @xmath195 by an appropriate choice of a canonical coordinate system @xmath88 near @xmath13 . this is essentially an elementary linear algebra fact ( lemma [ lemma : la ] , proved in section [ sec : la ] ) , asserting that a strongly degenerate linear symplectomorphism can be made arbitrarily close to the identity by conjugation within the linear symplectic group . as a consequence , near @xmath13 , the map @xmath2 is given by a generating function @xmath399 in the coordinate system @xmath88 . in section [ sec : step3 ] , we show that on a neighborhood of @xmath13 there exists a hamiltonian @xmath80 with time - one flow @xmath2 , which is in a certain sense close to @xmath399 . here , the key result is lemma [ lemma : local ] spelling out the relation between @xmath399 and @xmath80 and established in section [ sec : gf ] . choosing a sequence of coordinate systems @xmath517 so that @xmath518 , we obtain a sequence of hamiltonians @xmath228 defined near @xmath13 and meeting requirement ( k2 ) . then , again near @xmath13 , the loop @xmath465 is defined by @xmath492 . utilizing condition , we show in section [ sec : step4 ] that the maslov index of @xmath465 is zero . this enables us to relate homological properties of @xmath292 to the geometrical properties of @xmath228 near @xmath13 and prove ( k1 ) as a consequence of lemma [ lemma : lfh - lmh ] . ( assertion ( k2 ) easily follows from the construction of @xmath228 . ) property ( k3 ) is proved in section [ sec : k3 ] . at this stage , we further specialize our choice of canonical coordinate systems @xmath517 to ensure that all flows @xmath519 have the same linearization at @xmath13 . then , the first part of assertion ( k3 ) is obvious . by lemma [ lemma : loop - ext ] , the loops @xmath465 extend to @xmath1 , for @xmath520 . this , in turn , gives an extension of @xmath228 to @xmath1 . carrying out these extensions with some care , we can guarantee that ( k3 ) holds in its entirety . first recall that for any contractible , closed curve @xmath523 there exists a contractible loop of hamiltonian symplectomorphisms @xmath266 for which @xmath52 is an integral curve , i.e. , @xmath524 ; cf . * section 9 ) . for the sake of completeness , let us outline a proof of this fact . consider a smooth family of closed curves @xmath525 , @xmath526 $ ] , connecting the constant loop @xmath527 to @xmath528 . it is easy to show that there exists a smooth family of hamiltonians @xmath199 such that for every @xmath17 , the curve @xmath529 is an integral curve of @xmath199 with respect to @xmath73 , i.e. , @xmath530 . let @xmath531 be the time - one flow ( in @xmath73 ) of this family , parametrized by @xmath112 . then @xmath524 . the family of hamiltonians @xmath199 can be chosen so that @xmath532 . then @xmath266 is a loop of hamiltonian diffeomorphisms with @xmath533 . as readily follows from the construction , the loop @xmath289 is contractible . composing @xmath6 with the loop @xmath534 and adding , if necessary , a time - dependent constant function to the resulting hamiltonian @xmath535 , we may assume without loss of generality that @xmath388 is a fixed point of the flow @xmath536 for all @xmath112 and @xmath537 . then @xmath535 has the same time - one map and the same filtered floer homology as @xmath5 . by example [ ex : sympl - deg ] and remark [ rmk : comp ] , it is sufficient to prove the proposition for @xmath535 . thus , we will assume from now on that @xmath292 and keep the notation @xmath5 for the modified hamiltonian @xmath535 . [ [ the - construction - of - the - hamiltonians - ki - and - the - loops - eta_i - near - p ] ] the construction of the hamiltonians @xmath228 and the loops @xmath465 near @xmath13 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ our main objective in this section is to show that for every @xmath103 , there exists a symplectic basis @xmath97 in @xmath484 and a hamiltonian @xmath80 on a neighborhood of @xmath13 such that the time - one flow of @xmath80 is @xmath2 , condition ( k1 ) is satisfied , and @xmath538 then , clearly , there exists a sequence of symplectic bases @xmath483 and a sequence of hamiltonians @xmath228 meeting requirements ( k1 ) and ( k2 ) . the loop @xmath465 is defined near @xmath13 by @xmath539 . * _ for any @xmath103 there exists a symplectic basis @xmath97 in @xmath484 such that @xmath540 . as a consequence ( cf . example [ exam : norm1 ] ) , for any @xmath103 there exists a canonical system of coordinates @xmath88 on a neighborhood @xmath89 of @xmath13 such that the @xmath541-distance from @xmath2 to the identity is less than @xmath542 . _ [ lemma : la ] let @xmath543 be a linear symplectic map of a finite dimensional symplectic vector space @xmath29 such that all eigenvalues of @xmath544 are equal to one . then @xmath544 is conjugate in @xmath545 to a linear map which is arbitrarily close to the identity . indeed , since @xmath13 is a strongly degenerate fixed point of @xmath5 , all eigenvalues of @xmath546 are equal to one . thus , the desired statement follows from this lemma applied to @xmath547 . the proof of the lemma is elementary and provided for the sake of completeness in section [ sec : la ] . here we only mention that @xmath544 is given by an upper triangular matrix in some basis @xmath97 and , by scaling the elements of @xmath97 appropriately , one can make @xmath544 arbitrarily close to the identity , cf . example [ exam : norm1 ] . hence , we only need to show that @xmath97 and the scaling can be made symplectic . [ sec : step3 ] pick a system @xmath88 of canonical coordinates near @xmath13 such that @xmath2 is @xmath541-close to the identity . in particular , @xmath548 is small . furthermore , the map @xmath2 is given , near @xmath13 , by a generating function @xmath399 . the precise definition of @xmath399 and the relation between @xmath399 and @xmath2 and @xmath88 are immaterial at the moment and these issues will be discussed in section [ sec : gf ] . at this stage , we only need to know that @xmath399 is defined on a neighborhood of @xmath13 and uniquely determined by @xmath88 and @xmath2 . ( to make this statement accurate , let us agree that a canonical coordinate system is comprised of _ ordered _ pairs of functions @xmath549 such that @xmath550 . thus , each coordinate function is assigned to either @xmath551- or @xmath552-group . ) moreover , @xmath399 has the following properties : the second item ( gf2 ) requires , perhaps , a clarification . first note that @xmath555 stands here for the @xmath541-distance from @xmath2 to @xmath195 ; see section [ sec : norms ] . furthermore , @xmath399 and @xmath555 depend on @xmath88 . thus , in ( gf2 ) , we view both @xmath556 and @xmath555 as functions of @xmath88 with @xmath2 fixed and the second item asserts that @xmath557 , where @xmath558 is independent of @xmath88 , provided that @xmath555 is small enough . more generally , let @xmath92 and @xmath559 be non - negative functions of @xmath88 and some ( numerical ) variables . we write @xmath560 , when @xmath561 pointwise , where @xmath558 is independent of @xmath88 . the notation @xmath562 will be used when @xmath563 pointwise as functions of other variables , with @xmath564 depending on @xmath88 and possibly becoming arbitrarily large . furthermore , we denote by @xmath565 the ball of radius @xmath424 with respect to @xmath88 centered at @xmath13 . [ lemma : local ] let @xmath88 be a coordinate system near @xmath13 such that @xmath555 is small . then for every sufficiently small @xmath422 ( depending on @xmath88 ) , there exists a one - periodic hamiltonian @xmath415 on @xmath565 such that 1 . the time one - flow @xmath566 of @xmath80 is @xmath2 , 2 . @xmath13 is an isolated critical point of @xmath415 and @xmath567 , 3 . @xmath568 , 4 . the following estimates hold pointwise near @xmath13 : @xmath569 and @xmath570 where the dot denotes the time derivative of a vector field . note that in ( iv ) we could have written @xmath548 in place of @xmath571 by ( gf2 ) . the important point here is that @xmath548 and @xmath572 can be made arbitrarily small by choosing an appropriate coordinate system @xmath88 . then , shrinking the domain of @xmath80 , we can also make the right hand sides in the estimates ( iii ) and ( iv ) arbitrarily small . a proof of lemma [ lemma : local ] can be extracted from @xcite . however , to make our proof of theorem [ thm : main ] self - contained , we provide a detailed argument . deferring this to section [ sec : gf ] , we proceed with the proof of proposition [ prop1 ] . [ sec : step4 ] let @xmath80 be a hamiltonian on a neighborhood of @xmath13 , such that ( i)(iv ) of lemma [ lemma : local ] are satisfied and @xmath555 is small . our first goal is to prove that @xmath80 meets requirements ( k1 ) and ( k2 ) . to establish ( k1 ) , consider the loop @xmath575 , where @xmath576 . thus , @xmath577 note that @xmath578 , i.e. , @xmath579 with @xmath112 is a loop of hamiltonian symplectomorphisms near @xmath13 . we denote this loop by @xmath580 . the @xmath45-th iteration @xmath581 of @xmath580 is simply @xmath579 with @xmath48 . let @xmath583 be the time - average of @xmath415 , i.e. , @xmath584 a straightforward calculation utilizing lemma [ lemma : local ] and and detailed in section [ sec : bark - k ] shows that the requirements of lemma [ lemma : lfh - lmh ] are met , for any fixed @xmath45 , by the pair @xmath585 , provided that @xmath555 is small enough . in other words , these requirements are satisfied when @xmath2 is @xmath541-close to the identity and @xmath583 is taken as the reference function in lemma [ lemma : lfh - lmh ] ( denoted there by @xmath399 ) . in particular , @xmath13 is an isolated @xmath505-periodic orbit . we set , @xmath586 , where @xmath505 is as in . next note that @xmath589 for any @xmath590 . by ( lf4 ) , @xmath591 as long as @xmath13 is an isolated one - periodic orbit of @xmath71 . applying this identity to @xmath586 , we conclude from that @xmath592 in particular , since @xmath593 , @xmath594 combining and , we conclude that @xmath393 . a different proof of this fact , relying on the properties of the mean conley zehnder index ( see @xcite ) , can be found in ( * ? ? ? * section 5.2 ) . furthermore , when @xmath2 is sufficiently @xmath541-close to the identity , the requirements of lemma [ lemma : lfh - lmh ] with @xmath60 and the hamiltonians @xmath80 and @xmath399 as in lemma [ lemma : local ] are obviously met due to ( gf2 ) , , and ( iv ) . thus , by lemma [ lemma : lfh - lmh](b ) , the function @xmath415 has strict local maximum at @xmath13 . this proves ( k1 ) . applying this construction to a sequence of symplectic bases @xmath483 in @xmath484 such that @xmath598 , we obtain a sequence of hamiltonians @xmath228 , meeting requirements ( k1 ) and ( k2 ) , and also the loops @xmath465 . we emphasize that @xmath228 and @xmath599 have so far been defined only on a neighborhood of @xmath13 . the goal of this auxiliary section , which is included for the sake of completeness , is to show that , as stated above , the pair @xmath585 satisfies the hypotheses of lemma [ lemma : lfh - lmh ] with @xmath45 fixed . to this end , note first that by lemma [ lemma : local](iv ) , we have @xmath600 pointwise near @xmath13 . ( here the coordinate system @xmath88 is suppressed in the notation . ) let us integrate the first of these inequalities with respect to @xmath17 over @xmath44 . then , since @xmath399 is independent of time , we have , again pointwise near @xmath13 , @xmath601 with @xmath602 . thus , @xmath603 and , as a consequence , @xmath604 then @xmath605 likewise , @xmath606 therefore , @xmath607 where @xmath608 and all inequalities are pointwise . recall now that we can make @xmath609 arbitrarily small ( with @xmath45 fixed ) by making a suitable choice of @xmath88 and then requiring @xmath422 to be sufficiently small . it follows that we can also make @xmath191 arbitrarily small . in the same vein , the left hand side of can be made arbitrarily small . furthermore , since @xmath13 is an isolated critical point of @xmath399 , it is also an isolated critical point of @xmath583 . therefore , the pair @xmath585 satisfies the hypotheses of lemma [ lemma : lfh - lmh ] . as has been pointed out in section [ sec : step4 ] , a pair of function satisfying the hypotheses of lemma [ lemma : local ] also satisfies the hypotheses of lemma [ lemma : lfh - lmh ] . we have shown that @xmath585 satisfies the conditions of lemma [ lemma : lfh - lmh ] whenever @xmath416 meets the requirements of lemma [ lemma : local ] . moreover , by arguing as in this section , it is not hard to show that @xmath585 satisfies the conditions of lemma [ lemma : local ] ( and hence of lemma [ lemma : lfh - lmh ] ) once @xmath416 does . we omit this ( straightforward ) calculation , for it is never used in the proof . here @xmath228 is the sequence of hamiltonians constructed in section [ sec : step4 ] using lemma [ lemma : local ] . we prove lemma [ lemma : special ] in section [ sec : gf ] along with lemma [ lemma : local ] . at this point , we only note that , as will become clear in section [ sec : gf ] , the linearized flow @xmath610 is completely determined by @xmath2 and the basis @xmath483 . in particular , the linearization is independent of the extension of @xmath483 to a canonical coordinate system @xmath517 near @xmath13 . ( here , we use a convention similar to that of section [ sec : step3 ] for canonical coordinate systems : a symplectic basis is divided into two groups of @xmath9 vectors spanning lagrangian subspaces and this division is a part of the structure of a symplectic basis . ) let us now extend the loops @xmath465 and the hamiltonians @xmath228 to @xmath1 so that the remaining part of requirement ( k3 ) is met : the loop @xmath486 is contractible to @xmath195 in the class of loops with identity linearization at @xmath13 . recall that the maslov index of the loop @xmath465 is zero , as is shown in section [ sec : step4 ] . hence , by lemma [ lemma : loop - ext ] , each of these loops extends to a loop of hamiltonian diffeomorphisms of @xmath1 , contractible in the class of loops fixing @xmath13 . let us fix such an extension for @xmath613 . for the sake of simplicity we denote this extension by @xmath613 again . consider now the loop @xmath614 . then @xmath615 . hence , by lemma [ lemma : loop - ext ] and remark [ rmk : loop - ext ] , @xmath616 extends to a loop of hamiltonian diffeomorphisms of @xmath1 , contractible in the class of loops with identity linearization at @xmath13 . keeping the notation @xmath616 for this extension , we set @xmath617 . it is clear that @xmath465 is contractible in the class of loops with identity linearization at @xmath13 . [ lemma : la2 ] let @xmath543 be a linear symplectic map of a finite dimensional symplectic vector space @xmath29 . assume that all eigenvalues of @xmath544 are equal to one . then @xmath33 can be decomposed as a direct sum of two lagrangian subspaces @xmath618 and @xmath619 with @xmath620 . moreover , by a suitable choice of @xmath621 preserving the subspaces @xmath618 and @xmath619 , the map @xmath622 can be made arbitrarily close to the identity . in the former case , we decompose @xmath33 as @xmath627 , where the superscript @xmath22 denotes the symplectic orthogonal . it is easy to see that this decomposition is preserved by @xmath544 and @xmath628 . now the assertion follows from the induction hypothesis applied to @xmath629 . in the latter case , pick a symplectic subspace @xmath626 complementary to @xmath625 in @xmath630 and an isotropic subspace @xmath358 complementary to @xmath630 in @xmath33 . ( we are assuming at the moment that @xmath631 , i.e. , @xmath618 is not lagrangian . ) thus , @xmath632 and @xmath633 . furthermore , @xmath80 and @xmath630 are preserved by @xmath544 ; the spaces @xmath626 and @xmath358 can be canonically identified with @xmath634 and @xmath635 , respectively ; and @xmath636 . note that @xmath544 induces a symplectic linear map @xmath637 with all eigenvalues equal to one and the identity map @xmath638 on @xmath639 . hence , using the decomposition @xmath640 we can write @xmath544 in the block upper - triangular form @xmath641 where @xmath642 and @xmath643 and @xmath644 . ( there are relations between these operators , resulting from the fact that @xmath544 is symplectic . ) consider a block - diagonal symplectic linear transformation of the form @xmath645 where @xmath646 is symplectic , @xmath647 is invertible , and we have identified @xmath358 with @xmath648 . then @xmath649 by the induction assumption , there exists a decomposition @xmath650 , where @xmath651 , and transformations @xmath652 preserving this decompositions and making @xmath653 arbitrarily close to @xmath654 . set @xmath655 and @xmath656 . then @xmath620 and the decomposition @xmath657 is preserved by @xmath658 . furthermore , noticing that @xmath659 is close to zero when @xmath660 is close to zero , we can pick @xmath660 to make the off - diagonal entries in @xmath544 arbitrarily small . with this choice of @xmath658 , the map @xmath622 is close to @xmath661 if @xmath662 is close to @xmath654 . when @xmath663 is lagrangian ( i.e. , @xmath664 ) , no induction reasoning is needed . we simply set @xmath665 and let @xmath666 be an arbitrary complementary lagrangian subspace . then the map @xmath544 is decomposed as a two - by - two block upper - triangular matrix , and , similarly to the argument above , @xmath660 is chosen to make the off - diagonal block arbitrarily small . indeed , @xmath13 is a symplectically degenerate maximum of @xmath825 as is pointed out in example [ ex : sympl - deg ] . the hamiltonians @xmath825 and @xmath5 have the same time-@xmath45 flow and there is a natural one - to - one correspondence between ( contractible ) @xmath45-periodic orbits of the hamiltonians , for @xmath826 with @xmath827 . due to our normalization of @xmath825 , the corresponding @xmath45-periodic orbits of @xmath825 and @xmath5 have equal actions and , in particular , @xmath828 has the same action spectrum and action filtration in the floer complex as @xmath71 ; see section [ sec : loops ] . as a consequence , @xmath829 thus , the proposition holds for @xmath5 if ( and only if ) it holds for @xmath825 . furthermore , when @xmath5 is replaced by @xmath825 , the loops @xmath830 get replaced by the loops @xmath831 which have the identity linearization at @xmath13 by ( k3 ) . with these observations in mind , we establish the proposition by using the squeezing method of @xcite . namely , closely following @xcite , we construct functions @xmath833 such that @xmath834 ( see fig . 1 ) and such that the map @xmath835 in the filtered floer homology for the interval @xmath836 induced by a monotone homotopy from @xmath837 to @xmath838 is non - zero . this map factors as @xmath839 and , therefore , @xmath840 as required . the hamiltonian @xmath837 depends only on @xmath5 and @xmath841 . outside a ball @xmath739 of radius @xmath736 , centered at @xmath13 , the function @xmath837 is constant and equal to @xmath842 . ( here the distance is taken with respect to some fixed metric compatible with @xmath22 . ) within @xmath739 , the hamiltonian @xmath837 is a function of the distance to @xmath13 , equal to @xmath843 when the distance is small , dropping to some constant @xmath844 , and then increasing to @xmath842 near the boundary of @xmath739 . the period @xmath45 is required to be large enough , i.e. , @xmath845 , where @xmath468 is determined by @xmath837 . the function @xmath838 and the constant @xmath846 depend on @xmath845 . the condition that @xmath13 is a symplectically degenerate maximum of @xmath5 is used in the construction of @xmath838 and also in proving that @xmath847 . the function @xmath838 is constructed as follows . pick @xmath480 so that @xmath848 is small . ( here , as above , @xmath849 is normalized by @xmath850 . ) there exists a bump function @xmath851 , supported near @xmath13 , with non - degenerate maximum at @xmath13 and @xmath852 and such that @xmath853 is also small . then @xmath854 . setting @xmath838 to be the hamiltonian generating the flow @xmath855 , normalized by @xmath856 , we note that @xmath857 . hence , @xmath834 . the hamiltonian @xmath838 has the same filtered floer homology as @xmath399 , and we show that @xmath858 using ( k3 ) . in the standard canonical coordinates @xmath741 on @xmath297 , set @xmath859 . all orbits of @xmath860 are closed and have period @xmath861 . fix a ball @xmath862 of radius @xmath422 , centered at the origin @xmath13 . consider a rotationally symmetric function @xmath399 on @xmath297 supported in @xmath863 . the function @xmath399 depends only on the distance to @xmath13 and it will be convenient in our analysis to also view @xmath399 as a function of @xmath864 . assume , in addition , that @xmath399 has the following properties ( see fig . 2 ) : * @xmath399 is decreasing as a function of @xmath864 ; * @xmath865 and @xmath866 is increasing on some closed ball @xmath867 ; and * on the shell @xmath868 , * * @xmath399 is concave , i.e. , @xmath869 , on @xmath870 $ ] , where @xmath871 , * * @xmath399 is convex ( @xmath872 ) on @xmath873 $ ] , where @xmath874 , * * @xmath399 has constant slope ( @xmath875 ) on the interval @xmath876 $ ] , where @xmath877 is irrational . we will refer to @xmath399 as a _ standard bump function _ on @xmath297 and we will call @xmath878 and @xmath879 and @xmath424 and other constants from the construction of @xmath399 the parameters of @xmath399 . the trivial one - periodic orbits of @xmath399 ( i.e. , its critical points ) are either contained in @xmath880 or in the complement to @xmath881 . the orbits from the first group form a closed ball ( possibly of zero radius ) centered at @xmath13 and have action @xmath353 ; the orbits from the second group are exactly the points where @xmath882 . let @xmath885 be the standard @xmath161-small ( periodic in time ) perturbation of @xmath399 as the ones considered in , e.g. , @xcite , and still supported in @xmath863 . for such a perturbation each sphere filled in by one - periodic orbits of @xmath399 breaks down into @xmath21 non - degenerate orbits . within @xmath886 , we may assume that @xmath885 is still autonomous and rotationally symmetric and @xmath887 . ( hence the only one - periodic orbit of @xmath885 in @xmath886 is the trivial orbit @xmath13 of conley - zehnder index @xmath9 . ) as is well known , the filtered floer complex of @xmath885 and the filtered floer homology of @xmath885 ( and @xmath399 ) are still defined , say , for any positive interval of actions @xmath888 even though @xmath297 is not compact ; see , e.g. , @xcite . here we adopt the conventions of @xcite . it is not hard to see that @xmath885 has no one - periodic orbits of index @xmath7 . furthermore , it has exactly two one - periodic orbits of index @xmath9 . one of these is the constant orbit @xmath13 . the second orbit @xmath140 arises from the sphere of periodic orbits farthest from the origin . this sphere has radius @xmath894 and @xmath895 , up to an error of order @xmath896 . by , @xmath140 is outside the range of action . finally , @xmath885 has two one - periodic orbits of index @xmath892 , but only one of them , @xmath178 , has action in @xmath893 . ( the action of the second orbit is approximately equal to @xmath897 . ) the orbit @xmath178 arises from the sphere of periodic orbits closest to the origin . this is the sphere of radius @xmath898 and @xmath899 , up to an error of order @xmath896 . ( see @xcite or section [ sec : k+ ] , where we analyze in detail periodic orbits of a function similar to @xmath399 . ) * @xmath885 has no one - periodic orbits of index @xmath7 ; * @xmath13 is the only one - periodic orbit of @xmath885 with index @xmath9 and action in @xmath893 ; * @xmath178 is the only one - periodic orbit of @xmath885 with index @xmath892 and action in @xmath893 ; * the connecting map from the long exact sequence @xmath902 is an isomorphism sending @xmath903 $ ] to @xmath212 $ ] , and hence @xmath904 . let @xmath89 be a small neighborhood of @xmath173 . fixing a canonical coordinate system on @xmath89 , denote the open ball of radius @xmath422 in @xmath89 , centered at @xmath13 , by @xmath423 and let @xmath905 be the boundary of @xmath423 . we define a bump function @xmath399 on @xmath1 exactly as for @xmath297 by using the coordinate system in @xmath89 . furthermore , since @xmath1 is compact , now we need not assume that @xmath399 is supported in @xmath863 . instead we just require @xmath399 to be constant outside @xmath863 . in other words , we allow @xmath399 to be shifted up and down . the description of periodic orbits of @xmath885 and the floer homology of @xmath399 given in section [ sec : bump - r2n ] extends word - for - word to this case , provided that @xmath863 is sufficiently small ( e.g. , displaceable ) and the variation @xmath906 is sufficiently large . the requirement @xmath891 is replaced by that @xmath907 , where @xmath908 depends only on @xmath863 and goes to zero as @xmath909 ; see , e.g. , @xcite . ( hypothetically , @xmath908 is equal to the displacement energy of @xmath863 , although the estimate we have been able to prove is somewhat weaker . ) the requirement carries over to this case unchanged when @xmath399 is supported in @xmath863 , and is , in general , replaced by @xmath910 [ sec : traj ] let us show that by making @xmath424 sufficiently small , we can ensure that the floer gradient trajectories of @xmath399 from @xmath178 to @xmath13 are close to @xmath13 . ( when @xmath399 is a bump function on @xmath297 , all such trajectories are contained in @xmath911 by the maximum principle . ) to this end , pick a ball @xmath912 contained in @xmath89 and fix once and forever a compatible with @xmath22 almost complex structure @xmath913 on @xmath1 coinciding with the standard complex structure on a neighborhood of @xmath914 . consider holomorphic curves @xmath96 in @xmath915 with boundary in @xmath916 and such that the part of the boundary of @xmath96 lying in @xmath917 is non - empty . ( then the part of the boundary of @xmath96 in @xmath918 is also non - empty due to the maximum principle . ) denote by @xmath919 the infimum of the areas of such curves @xmath96 . it is easy to see that @xmath920 remains separated from zero as @xmath909 . ( otherwise we would have @xmath921 for some fixed @xmath422 as is clear from considering the intersections with @xmath915 of holomorphic curves whose areas approach zero . ) in other words , @xmath922 . replacing @xmath793 by @xmath923 , we see that there exists @xmath924 such that @xmath925 let @xmath399 be an arbitrary bump function @xmath399 such that holds and @xmath926 . assume that @xmath191 and @xmath927 satisfy . then for a perturbation @xmath885 of @xmath399 as above and any regular perturbation of @xmath26 of @xmath913 all floer anti - gradient trajectories from @xmath178 to @xmath13 are contained in @xmath928 . assume the contrary . then for some @xmath885 close to @xmath399 and for some sequence of regular perturbations @xmath929 , there exists a sequence of connecting trajectories @xmath930 from @xmath178 to @xmath13 , leaving a neighborhood of @xmath931 . observe that the part of @xmath930 contained in @xmath932 is a @xmath933-holomorphic curve . by the compactness theorem , in the limit we have a @xmath913-holomorphic curve @xmath96 in @xmath932 with non - empty boundary in @xmath917 . by the definition of @xmath934 , the area of @xmath96 is greater than @xmath934 . therefore , the same is true for the part of @xmath930 contained in @xmath932 when @xmath933 is close to @xmath913 . thus , @xmath935 by . this is impossible , for @xmath936 [ sec : k+ ] without loss of generality , we may assume that @xmath937 . furthermore , throughout this section we will keep the notation and convention of section [ sec : bump ] . in particular , we fix a system of canonical coordinates on a neighborhood @xmath89 of @xmath13 and let , as in section [ sec : bump ] , the function @xmath864 on @xmath89 be one half of the square of the distance to @xmath13 with respect to this coordinate system . * @xmath939 ; * @xmath940 ; * on the shell @xmath868 the function @xmath837 is monotone decreasing , as a function of @xmath864 , and * * @xmath837 is concave ( @xmath941 ) on @xmath942 $ ] , where @xmath871 , * * @xmath837 is convex ( @xmath943 ) on @xmath873 $ ] , where @xmath874 , * * @xmath837 has constant slope ( @xmath944 ) on the interval @xmath945 $ ] , where @xmath877 is irrational ; * @xmath946 on the shell @xmath947 , where the constant @xmath185 is to be specified latter ; * @xmath837 is monotone increasing on the shell @xmath948 , and * * @xmath837 is convex ( @xmath943 ) on @xmath949 $ ] , where @xmath950 , * * @xmath837 is concave ( @xmath941 ) on @xmath951 $ ] , where @xmath952 , * * @xmath837 , as a function of @xmath864 , has constant slope ( @xmath944 ) on the interval @xmath953 $ ] , where @xmath877 is irrational ; * @xmath954 on @xmath955 , with the constant @xmath956 to be specified . furthermore , we extend @xmath837 to @xmath1 by setting it to be constant and equal to @xmath842 on the complement of @xmath89 . the constant @xmath842 is chosen so that @xmath957 on @xmath1 and @xmath956 . note that within @xmath739 , the function @xmath837 is a standard bump function of section [ sec : bump ] . this bump function has variation @xmath958 which , due to the requirement @xmath959 , may be very small . the neighborhood @xmath89 is chosen so that @xmath843 is a strict global maximum of @xmath5 on @xmath89 and @xmath89 is displaceable in @xmath1 by a hamiltonian diffeomorphism . the values @xmath960 are chosen arbitrarily , with the only restriction that @xmath961 . to pick @xmath424 , we fix @xmath191 and also fix a compatible with @xmath22 almost complex structure @xmath913 on @xmath1 coinciding with the standard complex structure on a neighborhood of the closed ball @xmath914 . the radius @xmath424 is chosen so that @xmath962 where the upper bound @xmath963 is as in section [ sec : traj ] . the radius @xmath964 is chosen arbitrarily with the only restriction that @xmath965 . the constants @xmath185 and @xmath842 are picked so that @xmath844 and @xmath939 on @xmath966 $ ] and on @xmath967 ( this may require @xmath185 to be very close to @xmath230 . ) likewise , on the intervals @xmath968 $ ] and @xmath969 $ ] , the behavior of @xmath837 is specified to guarantee that @xmath939 . finally , we will also impose the condition that @xmath970 [ sec : orbits ] in this section , we analyze the relevant @xmath45-periodic orbits of @xmath837 when @xmath45 is sufficiently large . since @xmath837 is autonomous , its @xmath45-periodic orbits can simply be treated as one - periodic orbits of @xmath972 . furthermore , it is clear that all @xmath45-periodic orbits of @xmath837 outside @xmath89 are trivial . those in @xmath89 are either trivial or fill in spheres of certain radii . replacing @xmath837 by its standard time - dependent @xmath161-small perturbation @xmath973 as in @xcite and section [ sec : bump ] results in each of these spheres splitting into @xmath21 non - degenerate orbits . here , as in section [ sec : bump ] , we are primarily interested in the orbits with index @xmath7 or @xmath9 or @xmath892 and action in the interval @xmath974 for some small @xmath927 . we will show that these orbits are essentially the same as for a bump function with ( large ) variation @xmath975 . the perturbation @xmath973 is similar to @xmath885 from section [ sec : bump ] . we emphasize that @xmath837 is perturbed not only within the shells @xmath976 and @xmath977 where non - trivial periodic orbits are , but also within the ball @xmath978 where @xmath979 . on this ball , @xmath973 is a monotone decreasing function of @xmath864 with a non - degenerate maximum at @xmath13 equal to @xmath230 . this function is @xmath161-close to the constant function @xmath837 so that ( for a fixed @xmath45 ) , the function @xmath980 is @xmath161-close to @xmath981 on @xmath978 . in particular , the eigenvalues of the hessian @xmath982 are close to zero and the conley zehnder index of the constant @xmath45-periodic orbit @xmath13 of @xmath973 is @xmath9 . in what follows , we will always assume that @xmath973 is as close to @xmath837 as necessary . in the shell @xmath947 and in the complement to @xmath983 we keep @xmath973 constant and equal to @xmath837 . the trivial @xmath45-periodic orbits of @xmath837 are the points of @xmath886 ( with action @xmath981 ) , the points of @xmath985 ( with action @xmath986 ) , and the points of @xmath987 ( with action @xmath988 ) . here , only the points of @xmath886 have action within the range in question . indeed , @xmath989 by and @xmath990 , for @xmath991 by . thus , @xmath13 is the only trivial @xmath45-periodic orbit of @xmath973 with action in @xmath974 ; it has index @xmath9 . the first group is formed by the @xmath45-periodic orbits in the shell @xmath992 . these orbits fill in a finite number of spheres @xmath993 of radii @xmath994 with @xmath995 the orbits on @xmath993 have action @xmath996 . once the hamiltonian @xmath837 is replaced by @xmath973 , a sphere @xmath993 breaks down into @xmath21 non - degenerate orbits . the conley zehnder indices of these orbits are @xmath997 as is proved in @xcite . only one of the orbits in this group has index from @xmath7 to @xmath892 and action in the range @xmath974 . this is a periodic orbit , denoted by @xmath178 , of index @xmath892 and action @xmath998 ( up to an error of order @xmath999 ) , arising from the sphere @xmath1000 . the conley zehnder indices of the remaining orbits are greater than @xmath892 although some of these orbits may have action within the range @xmath974 . the second group is comprised of the @xmath45-periodic orbits in the shell @xmath1001 . these orbits fill in the spheres @xmath1002 of radii @xmath1003 with @xmath1004 the orbits on @xmath1002 have action @xmath1005 . again , once the hamiltonian @xmath837 is replaced by @xmath973 , each sphere @xmath1002 breaks down into @xmath21 non - degenerate orbits . the conley zehnder indices of these orbits are @xmath1006 ; see @xcite . only the orbits arising from @xmath1007 and @xmath1008 can have index @xmath7 or @xmath9 or @xmath892 . ( other spheres give rise to orbits of index greater than @xmath1009 . ) however , the orbits coming from the spheres @xmath1007 and @xmath1008 have action not exceeding @xmath1010 if @xmath973 is close to @xmath837 . by , these orbits are outside the action range @xmath974 . the @xmath45-periodic orbits in the shell @xmath1011 are in the third ( possibly empty ) group . these orbits fill in the spheres @xmath1012 of radii @xmath1013 with @xmath1014 and the orbits on @xmath1012 have action @xmath1015 . hence , all of these orbits are outside of the range of action @xmath974 . the fourth group , which may also be empty , is formed by the @xmath45-periodic orbits in the shell @xmath1016 . these orbits fill in a finite collection of spheres @xmath1017 of radii @xmath1018 such that @xmath1019 and the orbits on @xmath1017 have action @xmath1020 , which can be in the interval @xmath974 . however , calculating the conley zehnder indices of the resulting orbits of @xmath973 as in @xcite , it is easy to see that the sphere @xmath1017 brakes down into non - degenerate orbits of @xmath973 with indices @xmath1021 . in particular , all resulting orbits have indices not exceeding @xmath1022 , and none of the orbits has index @xmath7 or @xmath9 or @xmath892 . to summarize , the perturbation @xmath973 has only one @xmath45-periodic orbit of index @xmath9 with action in @xmath974 this is the trivial orbit @xmath13 and only one orbit , namely @xmath178 , of index @xmath892 with action within this range . the action of @xmath13 is @xmath981 and the action of @xmath178 is @xmath998 up to an error of order @xmath999 . there are no orbits with index @xmath7 and action in the range @xmath974 . [ sec : fh - k+ ] as in the previous section , assume that @xmath45 is sufficiently large and @xmath927 is small ( independently of @xmath45 ) . explicitly , now we require in addition to that @xmath1023 where @xmath1024 is defined in section [ sec : bump - w ] . in this section we prove [ lemma : fh1 ] under the above assumptions on the function @xmath837 , the period @xmath45 , and @xmath841 and @xmath1025 , we have @xmath1026 with generators @xmath212 $ ] and , respectively , @xmath903 $ ] . moreover , the connecting map @xmath1027 is an isomorphism . since is satisfied , the results of the previous section apply , and @xmath178 and @xmath13 are the only @xmath45-periodic orbits of @xmath973 of index @xmath7 or @xmath9 on @xmath892 with action in the range @xmath974 . it is clear that @xmath212 $ ] is the generator of @xmath1028 . furthermore , @xmath178 is the only @xmath45-periodic orbit of index @xmath892 with action in @xmath1029 and there are no @xmath45-periodic orbits of index @xmath9 with action in this interval . hence , the homology @xmath1030 , generated by @xmath903 $ ] , is either zero or @xmath135 . ( the former is _ a priori _ possible , for in fact there exists a @xmath45-periodic orbit of index @xmath1031 with action in @xmath1029 . ) to finish the proof of the lemma , it is sufficient now to show that the connecting map is onto or , equivalently , @xmath1032 i.e. , the number of floer anti - gradient trajectories for @xmath1033 from @xmath178 to @xmath13 is odd . within @xmath739 , the hamiltonian @xmath972 coincides with a standard bump function @xmath399 whose variation @xmath1034 is greater than @xmath1035 by . thus , the assumptions of section [ sec : bump - w ] are satisfied , and @xmath1036 . furthermore , @xmath1033 agrees with @xmath885 on @xmath739 . due to our choice of @xmath424 , the condition holds and lemma [ lemma : traj ] is applicable . therefore , @xmath885 , and hence @xmath1033 , have an odd number of floer anti - gradient trajectories from @xmath178 to @xmath13 contained in @xmath739 . moreover , every floer anti - gradient trajectory for @xmath1033 from @xmath178 to @xmath13 is automatically in @xmath739 . this is established by arguing exactly as in the proof of lemma [ lemma : traj ] with @xmath399 replaced by @xmath1033 and using again . as a consequence , @xmath1033 and @xmath399 have the same floer anti - gradient trajectories from @xmath178 to @xmath13 , and the total number of such trajectories is odd . this concludes the proof of and of the lemma . recall that the function @xmath837 and the parameter @xmath191 were fixed above , while @xmath45 and @xmath1025 have been variable . at this point , we also fix a large period @xmath45 meeting the requirement and such that @xmath1037 . then , condition is satisfied if @xmath927 is small , and hence lemma [ lemma : fh1 ] applies . in this section , we construct a hamiltonian @xmath1038 , depending on @xmath45 , such that @xmath1039 and @xmath1040 , and the connecting map @xmath1041 is an isomorphism if @xmath846 is sufficiently small . such that the hamiltonian @xmath1044 generating the flow @xmath1045 has a strict local maximum at @xmath13 and @xmath1046 moreover , the loop @xmath324 has identity linearization at @xmath13 , i.e. , @xmath1047 for all @xmath112 , and is contractible to @xmath195 in the class of loops with identity linearization at @xmath13 . ( see ( k3 ) and section [ sec : prop2-outline ] . ) let @xmath1048 be a homotopy from @xmath324 to the identity such that @xmath1049 and let @xmath1050 be the one - periodic hamiltonian generating @xmath1051 and normalized by @xmath1052 . the condition @xmath1053 is equivalent to that @xmath1054 . as usual , we normalize @xmath80 by requiring that @xmath567 or , equivalently , by @xmath1055 . without loss of generality , we may also assume that @xmath1056 , where @xmath33 is the domain of the coordinate system @xmath88 and @xmath978 is the ball from the construction of @xmath1057 ; see section [ sec : k+ ] . note that this ball is taken with respect to the original metric and is not related to @xmath88 . let @xmath399 be a bump function , `` centered '' at @xmath13 , with respect to the coordinate system @xmath88 . as in section [ sec : bump - w ] , we do not require @xmath399 to be supported in @xmath33 , but only constant outside @xmath33 . thus , @xmath1058 on @xmath1059 . we may assume that @xmath1060 . it is also clear that @xmath399 can be chosen so that the variation of @xmath1071 , equal to @xmath1072 , is much larger than @xmath1073 . hence , as shown in section [ sec : bump ] , and we have the isomorphism @xmath1074 provided that @xmath846 is sufficiently small . finally note that @xmath202 for all @xmath73 , for @xmath1075 . therefore , the functions @xmath1066 have equal filtered floer homology for any period @xmath45 . in particular , the filtered floer homology of @xmath1076 and of @xmath1077 are the same and the latter is identical to the ( one - periodic ) filtered floer homology of @xmath1071 . thus , we obtain the desired isomorphism . by construction , @xmath1078 . a monotone decreasing homotopy from @xmath837 to @xmath838 induces maps of filtered floer homology , which , as is well known , commute with the maps from the long exact sequence . in particular , combining the monotone homotopy maps with the connecting maps for @xmath837 and for @xmath838 , we obtain the following commutative diagram : @xmath1079 to prove the theorem , it is sufficient to show that the right vertical arrow @xmath658 in the diagram , i.e. , the homotopy map @xmath1080 is an isomorphism . indeed , the rows of are isomorphisms , and hence the left vertical arrow is an isomorphism whenever @xmath658 is an isomorphism . since @xmath834 by , the left vertical arrow factors as @xmath1081 and , as a consequence , the middle group is non - zero as desired . to show that @xmath658 is an isomorphism , first observe that since @xmath1070 for all @xmath73 and @xmath1082 and @xmath1068 , the diagram @xmath1083_\psi\ar[rd ] & \\ { \hf_n^{(tc-\delta_t,\,tc+\delta_t)}\big(h_-^{(t)}\big ) } \ar[r]^{\cong } & { \hf_n^{(tc-\delta_t,\,tc+\delta_t)}\big(f^{(t)}\big ) } } \ ] ] is commutative , where the horizontal isomorphism is induced by the isospectral homotopy @xmath1066 and the remaining two arrows are monotone homotopy maps . ( see section [ sec : homotopy ] and , in particular , . ) recall also that @xmath837 and @xmath399 are autonomous . it remains to prove that the diagonal arrow , which can be identified with @xmath1084 is an isomorphism . consider a @xmath161-small autonomous perturbation @xmath1085 of @xmath837 such that @xmath1086 is negative definite and @xmath1087 . ( it is straightforward to construct @xmath1088 by modifying @xmath837 on a neighborhood of @xmath886 . ) then @xmath1089 and @xmath1090 can be made arbitrarily small , for @xmath1091 . we take @xmath1088 such that @xmath1092 is @xmath161-close to @xmath972 , and , in particular , @xmath1093 is small . essentially by definition , the filtered floer homology of @xmath1092 is isomorphic to the filtered floer homology of @xmath972 , and it suffices to show that @xmath1094 is an isomorphism for some small @xmath846 independent of the choice of @xmath1088 . recall that @xmath981 is an isolated action value of @xmath1071 and @xmath1095 , and hence of @xmath1096 , for a generic choice of parameters of these functions . fix @xmath846 meeting the requirement and such that @xmath981 is the only action value of these hamiltonians in @xmath1097 . consider the linear decreasing homotopy @xmath1098 . since both of the hessians @xmath1086 and @xmath410 are negative definite , @xmath1099 is also negative definite . thus , @xmath13 is a non - degenerate critical point of @xmath1100 for all @xmath194 $ ] with @xmath1101 . furthermore , by and since @xmath1102 is small , @xmath1103 . as a consequence , @xmath13 is a uniformly isolated one - periodic orbit of @xmath1104 ; see , e.g. , @xcite or section [ sec : lfh - lmh ] . by lemma [ lemma : non - zero ] , the homotopy map is non - zero , and hence an isomorphism . this concludes the proof of proposition [ prop2 ] and of theorem [ thm : main ] . a slightly different proof of the proposition , although based on the same ideas as the present argument and following the same line of reasoning , can be found in ( * ? ? ? * section 5 ) . m. poniak , floer homology , novikov rings and clean intersections , in _ northern california symplectic geometry seminar _ , 119181 , amer . ser . 2 , * 196 * , amer . math . soc . , providence , ri , 1999 . a. weinstein , _ lectures on symplectic manifolds . _ expository lectures from the cbms regional conference held at the university of north carolina , march 812 , 1976 . regional conference series in mathematics , no . american mathematical society , providence , r.i . , 1977 . in this section , we recall the definition of a generating function on @xmath297 and set the stage for proving lemma [ lemma : local ] . the material reviewed here is absolutely standard it goes back to poincar and we refer the reader to ( * ? ? ? * appendix 9 ) and @xcite for a more detailed discussion of generating functions . let us identify @xmath297 with the lagrangian diagonal @xmath667 via the projection to the first factor , where @xmath668 is equipped with the symplectic structure @xmath669 , and fix a lagrangian complement @xmath358 to @xmath670 . thus , @xmath668 can now be treated as @xmath671 . let @xmath2 be a hamiltonian diffeomorphism defined on a neighborhood of the origin @xmath13 in @xmath297 and such that @xmath672 is sufficiently small . then the graph @xmath373 of @xmath2 is close to @xmath670 , and hence @xmath373 can be viewed as the graph in @xmath671 of an exact form @xmath673 near @xmath674 . ( we normalize @xmath399 by @xmath675 . ) the function @xmath399 , called the generating function of @xmath2 , has the following properties : for instance , it is clear that the critical points of @xmath399 are in one - to - one correspondence with the fixed points of @xmath2 . if @xmath13 ( the origin ) is an isolated fixed point of @xmath2 , the origin is also an isolated critical point of @xmath399 . hence , ( gf1@xmath678 ) holds . the second property of @xmath399 , ( gf2@xmath678 ) , is also easy to check ; see the references above . the function @xmath399 depends on the choice of the lagrangian complement @xmath358 to @xmath670 . to be specific , we take as @xmath358 the linear subspace of vectors of the form @xmath679 in @xmath668 , where @xmath680 and @xmath681 are the standard canonical coordinates on @xmath297 , i.e. , @xmath682 . in the setting of section [ sec : step3 ] , let @xmath88 be a coordinate system near @xmath173 . using @xmath88 , we identify a neighborhood of @xmath13 in @xmath1 with a neighborhood of the origin in @xmath297 , keeping the notation @xmath13 for the origin . with this identification , @xmath2 defined near @xmath173 turns into a hamiltonian diffeomorphism @xmath683 defined near the origin @xmath684 . by definition , @xmath685 . abusing notation , we denote the resulting hamiltonian diffeomorphism @xmath683 near @xmath684 by @xmath2 again . by our background assumptions , @xmath13 is an isolated fixed point of @xmath2 , and thus ( gf1 ) and ( gf2 ) follow immediately from ( gf1@xmath678 ) and ( gf2@xmath678 ) , respectively . [ lemma : local2 ] let @xmath2 be a hamiltonian diffeomorphism of a neighborhood of the origin @xmath686 . assume that @xmath13 is an isolated fixed point of @xmath2 and @xmath672 is so small that the generating function @xmath399 is defined . then for every sufficiently small @xmath422 ( depending on @xmath2 ) , there exists a one - periodic hamiltonian @xmath415 on the ball @xmath423 of radius @xmath424 centered at @xmath13 such that 1 . the time one - flow @xmath566 of @xmath80 is @xmath2 , 2 . @xmath13 is an isolated critical point of @xmath415 and @xmath687 for all @xmath112 , 3 . @xmath688 , 4 . the upper bounds @xmath689 and @xmath690 hold pointwise near @xmath13 . the notation used here is similar to that of section [ sec : step3 ] . for instance , should be read as that its left hand side is pointwise bounded from above by @xmath691 , where @xmath692 is independent of @xmath2 and @xmath693 depends on @xmath2 and can be arbitrarily large . the proof of the lemma is organized as follows . first we consider the time - dependent hamiltonian @xmath694 generating the flow @xmath695 given by the family of generating functions @xmath696 , @xmath596 $ ] , and verify ( i)(iv ) for @xmath694 . the time - one flow of @xmath694 is @xmath2 . however , in general , the hamiltonian @xmath694 is _ not _ periodic in time . hence , as the next step , we modify @xmath694 to obtain the required periodic hamiltonian @xmath80 and then again check that the new hamiltonian @xmath80 satisfies ( i)(iv ) . consider the family of generating functions @xmath696 with @xmath697 $ ] . the family of graphs @xmath698 of @xmath699 in @xmath671 beginning with @xmath700 and ending with @xmath701 can be viewed as a family of graphs of hamiltonian diffeomorphisms @xmath695 near @xmath13 with @xmath702 and @xmath703 . thus , @xmath695 is a time - dependent hamiltonian flow with the time - one map @xmath2 , defined near @xmath13 . let @xmath704 be the hamiltonian generating this flow , normalized by @xmath705 . let @xmath709 . then , as is well known and can be checked by a simple calculation , we have @xmath710 differentiating with respect to time , we obtain the following expression for the hamiltonian @xmath706 ( cf . @xcite ) : @xmath711 where @xmath712 here @xmath713 stands for the matrix of partial derivatives @xmath714 and , similarly , @xmath715 is the matrix @xmath716 . clearly , @xmath722 uniformly in @xmath17 . in particular , @xmath717 is indeed a diffeomorphism , fixes @xmath13 , and is , moreover , @xmath300-close to the identity when @xmath2 is close to @xmath195 . furthermore , @xmath723 pointwise near @xmath13 and uniformly in @xmath17 . hence , @xmath105 is invertible near @xmath13 . since @xmath13 is an isolated critical point of @xmath399 by ( gf1@xmath678 ) , @xmath13 is also an isolated zero of @xmath429 , and thus an isolated zero of @xmath724 . this gives a direct proof of ( ii ) for @xmath694 . since @xmath725 , the linearization at @xmath13 yields @xmath726 here , @xmath727 and @xmath728 are the linear hamiltonian vector fields on @xmath729 with quadratic hamiltonians @xmath730 and , respectively , @xmath410 . furthermore , it is easy to see from and that @xmath731 and @xmath732 are both close to @xmath733 , with error @xmath734 , and hence are small . combining these observations , we conclude that @xmath735 proving ( iii ) for @xmath694 . [ sec : proofkf1 ] turning to the proof of ( iv ) for @xmath694 , observe that for every small @xmath736 there exists @xmath422 such that @xmath737 and @xmath738 are in @xmath739 for all @xmath740 $ ] and all @xmath741 in @xmath423 . furthermore , it is clear that @xmath742 to establish the upper bound of ( iv ) , let us first show that @xmath743 for every @xmath721 in @xmath423 and all @xmath740 $ ] . we have @xmath744 where in the last inequality we used the fact that , by , @xmath745 thus , we only need to show that @xmath746 by , we have @xmath747 since @xmath748 and @xmath725 , the first summand is obviously @xmath749 . ( when @xmath750 , both @xmath751 and @xmath752 are , by , in the ball of radius @xmath749 . ) the second summand is bounded as @xmath753 here , the first and the last factors are @xmath754 , and hence bounded from above by a constant independent of @xmath2 , when @xmath2 is sufficiently close to @xmath195 . the middle factor is @xmath755 , for @xmath756 and , as a consequence , @xmath757 thus , the second summand is @xmath755 , which completes the proof of . then @xmath758 by , the second term is bounded as @xmath759 and the first term is @xmath760 by . this proves the pointwise estimate at @xmath761 in place of @xmath721 . since @xmath717 is a diffeomorphism fixing @xmath13 , the upper bound in its original form ( at @xmath721 ) follows . arguing exactly as in the proof of , it is easy to show that @xmath762 and , as a consequence , @xmath763\cdot\| x_{f}(z)\|\leq \|x_f(\kappa^t(z))\|.\ ] ] therefore , to establish for @xmath694 , it is sufficient to prove the upper bound @xmath764 for @xmath750 . differentiating with respect to @xmath17 and setting @xmath751 , we obtain @xmath765 to prove , we will estimate all three terms in this identity . as a straightforward calculation shows , @xmath766 thus , @xmath767 and @xmath768 furthermore , @xmath769 as follows from the definition of @xmath717 and . using again the inequality @xmath770 , we see that @xmath771 where @xmath558 is independent of @xmath2 , when @xmath2 is sufficiently close to @xmath195 . then , since @xmath772 , we have @xmath773 and @xmath774 for @xmath775 . fix a monotone increasing function @xmath776 \to [ 0,\ , 1]$ ] such that @xmath777 when @xmath17 is near @xmath778 and @xmath779 when @xmath17 is near @xmath780 . this function is independent of @xmath2 , and hence @xmath781 and @xmath782 are bounded from above by constants independent of @xmath2 . the hamiltonian @xmath80 is the one generating the flow @xmath783 explicitly , since @xmath399 is autonomous , @xmath784 it is clear that @xmath785 when @xmath17 is close to @xmath778 and @xmath780 and hence @xmath80 can be viewed as a hamiltonian one - periodic in @xmath17 . also , @xmath786 , i.e. , requirement ( i ) is satisfied . as has been pointed out , the second condition , ( ii ) , follows from ( gf1@xmath678 ) and ( iv ) which is proved below . the hamiltonian vector field of @xmath80 is @xmath789 where @xmath790 . since @xmath399 is autonomous , @xmath791 . in particular , @xmath792 when @xmath721 is close to @xmath13 . also note that when @xmath750 , the point @xmath790 is in @xmath739 , where the radius @xmath793 satisfies . differentiating with respect to @xmath17 , we obtain @xmath795(w ) . \end{split}\ ] ] arguing as in section [ eq : kf1k ] , we see that the norm of the first term in this sum is @xmath796 . similarly , the same holds for the second term by for @xmath694 . ( in both cases we use to relate @xmath797 and @xmath798 and also the fact that @xmath799 when @xmath750 . ) to estimate the third term , it is sufficient to show that @xmath800(w)\|=\big ( o(\|d^2 f_p\|)+o_\varphi(r)\big)\|x_f(w)\|,\ ] ] for then , by , this term is @xmath796 when @xmath750 . to prove , observe that @xmath801(w)\big\|&=\big\|[x_f , x_{{\tilde{k}}_t}-x_f](w)\big\|\\ & \leq \alpha(w)\|x_f(w)\|+\beta(w)\|x_{{\tilde{k}}_t}(w)-x_f(w)\| , \end{split}\ ] ] where the functions @xmath802 and @xmath803 are bounded from above by the partial derivatives of @xmath804 and , respectively , @xmath429 at @xmath805 . hence , both of these functions are @xmath806 and follows from for @xmath694 . let @xmath547 and let @xmath657 be the decomposition of @xmath807 from lemma [ lemma : la2 ] . pick a linear canonical coordinate system @xmath741 on @xmath484 , which is compatible with the decomposition , i.e. , such that the @xmath178-coordinates span @xmath618 and the @xmath140-coordinates span @xmath619 . by lemma [ lemma : la2 ] , we can do this so that @xmath808 is small in this coordinate system , and thus @xmath2 is given by the generating function @xmath399 . denote by @xmath809 the hessian of @xmath399 at @xmath13 and let @xmath810 be the linear hamiltonian vector field of @xmath809 on @xmath33 . linearizing at @xmath13 , we see that @xmath544 and @xmath809 are related via the equation @xmath811 here @xmath812 is obtained from @xmath544 by replacing its @xmath140-component by the identity map , i.e. , @xmath813 in the decomposition @xmath657 , where @xmath814 . note that uniquely determines @xmath810 . it is clear from and that @xmath810 and @xmath815 depend only on the decomposition @xmath657 . hence , any other coordinate system compatible with this decomposition will give rise to the same quadratic form @xmath809 and the same maps @xmath815 . due to , the linearization @xmath818 is equal to @xmath819 hence , @xmath820 also depends only on the decomposition , but not on the coordinate system as long as the latter is compatible with the decomposition . in other words , all such coordinate systems result in the flows @xmath85 with linearization @xmath820 . lemma [ lemma : special ] follows now from lemma [ lemma : la2 ] , which guarantees that there exist symplectic bases @xmath97 ( or , equivalently , linear canonical coordinate systems ) compatible with @xmath657 and making @xmath821 arbitrarily small . recall that @xmath822 due to ( k1 ) and that all eigenvalues of @xmath544 are equal to one . combining these facts with the normal forms of quadratic hamiltonians ( see ( * ? ? ? * appendix 7 ) and @xcite ) , it is not hard to show that @xmath823 in some symplectic basis compatible with the decomposition @xmath824 . then it is straightforward to write down an explicit expression for @xmath815 and @xmath820 . this , however , does not lead to any simplification in the line of reasoning used here , for the required result readily follows from and .
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we prove the conley conjecture for a closed symplectically aspherical symplectic manifold : a hamiltonian diffeomorphism of a such a manifold has infinitely many periodic points .
more precisely , we show that a hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period .
this theorem generalizes , for instance , a recent result of hingston establishing the conley conjecture for tori .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of non - bps objects such as non - bps branes , brane - antibrane configurations and spacelike branes has recently attracted great attention given its implications for string / m - theory and cosmology . the tachyon field associated with unstable d - branes , might be responsible for cosmological inflation at early epochs due to tachyon condensation near the top of the effective potential @xcite , and could contribute to some new form of cosmological dark matter at late times @xcite . several authors have investigated the process of rolling of the tachyon in the cosmological background @xcite . in the slow roll limit in frw cosmology , the exact solution of tachyonic inflation with exponential potential is found @xcite . a question which has not yet been addressed in the literature on tachyonic inflation is the issue of constraints on the phase space of initial conditions for inflation which arise when one takes into account the fact that in the context of cosmology the momenta of the tachyon field can not be neglected in the early universe . for models of the type of chaotic inflation , the work of @xcite shows that most of the energetically accessible field value space give rise to a sufficiently long period of slow roll inflation . however , for models of the type of new inflation , allowing for non - vanishing initial field momenta may dramatically reduce the phase space of initial conditions for which successful inflation results , and the attractor is the slow rolling solution @xcite . in this paper we investigate the constrains on the initial conditions of inflation with tachyon rolling down an exponential potential in phase space required for successful inflation . we demonstrate the attractor behaviour of the tachyonic inflation using the hamilton - jacobi formalism . we else use an explicitly numerical computation of the phase space trajectories and obtain analytical approximations to the trajectories of the tachyon in different regions . we find that in phase space there exists a curve that attracts most of the solutions . according to sen @xcite , the effective action of the tachyon field in the born - infeld form can be written as @xmath0 where t is the tachyon field minimally coupled to gravity . the rolling tachyon in a spatially flat frw cosmological model can be described by a fluid with a positive energy density @xmath1 and a negative pressure @xmath2 given by @xmath3 thus @xmath4 note that @xmath5 , and a universe dominated by this rolling tachyonic matter will smoothly evolve from a phase of accelerating expansion to a phase dominated by a non - relativistic fluid @xcite . the evolution equation of the tachyon field minimally coupled to gravity , and the friedmann equation are @xmath6 where @xmath7 denotes the velocity of tachyon . a universe dominated by tachyon field would go under accelerating expansion as long as @xmath8 which is very different from the condition of inflation for non - tachyonic field , @xmath9 . the tachyon potential @xmath10 as @xmath11 , but its exact form is not known at present @xcite . sen has argued that the qualitative dynamics of string theory tachyons can be described by ( [ e1 ] ) with the exponential potential @xcite @xmath12 where @xmath13 is the tachyon mass . the cosmological aspects of rolling tachyon with exponential potential are investigated @xcite . in what follows we will consider ( [ e1 ] ) with exponential potential in purely phenomenological context without claiming any identification of @xmath14 with the string tachyon field . the hamilton - jacobi formulation @xcite is a powerful way of rewriting the equations of motion , which allows an easier derivation of many inflation results . we concentrate here on the homogeneous situation as applied to spatially flat cosmologies , and demonstrate the attractor behaviour of the tachyonic inflation using the hamilton - jacobi formalism @xcite . differentiating eq.([5f ] ) with respect to @xmath15 and substituting in eq.([4 t ] ) gives @xmath16 where primes denote derivatives with respect to the tachyon field @xmath14 , which gives the relation between @xmath14 and @xmath15 . this allows us to write the friedmann equation in the first - order form @xmath17 ^ 2-\frac{9}{4}h^4(t)=-\frac{\kappa ^4}{4}v^2(t)\ ] ] eq.([hj ] ) is the hamilton - jacobi equation . it allows us to consider @xmath18 , rather than @xmath19 , as the fundamental quantity to be specified . suppose @xmath20 is any solution to eq.([hj ] ) , which can be either inflationary or non - inflationary . add to this a linear homogeneous perturbation @xmath21 ; the attractor condition will be satisfied if it becomes small as @xmath14 increases . substituting @xmath22 into eq.([hj ] ) and linearizing , we find that the perturbation obeys @xmath23 which has the general solution @xmath24 where @xmath25 is the value at some initial point @xmath26 . because @xmath27 and @xmath28 have opposing signs , the integrand within the exponential term is negative definite , and hence all linear perturbations do indeed die away . if there is an inflationary solution , all linear perturbations approach it at least exponentially fast as the tachyon field rolls . we choose different initial conditions @xmath14 , in the range @xmath29 , and @xmath30 in the range @xmath31 and we obtain the phase portrait in the @xmath32 plane . figure 1 shows that there exists a curve that attracts most of the trajectories , in the @xmath33 plane where @xmath34 and @xmath35 are dimensionless coordinates . the initial kinetic term decays rapidly and does not prevent the onset of inflation . = 6.6 cm the behavior of the trajectories can be also analyzed analytically . to understand the evolution of the tachyon field we define two regions @xmath2 and @xmath36 in the @xmath32 plane as indicated in figure 2 . the region @xmath2 is the region where the potential dominates over the energy density , and the region @xmath36 is the region where the kinetic energy dominates . = 5.7 cm * curve @xmath37 . * this curve describes the slow rolling solution where the evolution eq . ( [ 4 t ] ) and the friedman eq.([5f ] ) in expanding universe can be approximated by @xmath38 from which it follows that @xmath39 where @xmath40 . the expression for the number @xmath41 of e - foldings of inflation can be written as @xmath42 where @xmath43 is the value of the field at the point where it reaches curve @xmath37 and @xmath44 is the value of the field at the end of the slow rolling phase . in order to obtain enough e - foldings of slow roll inflation the value of @xmath43 for such a trajectory must satisfy @xmath45 where @xmath46 is calculated from eq.([en ] ) with @xmath47 . * region @xmath2 . * in this region , the potential dominates over the energy density . the potential force @xmath48 is negligible compared to the friction term since the friction coefficient is proportional to the potential . the evolution eq.([4 t ] ) and friedman eq.([5f ] ) are approximately @xmath49 so that @xmath50}\end{aligned}\ ] ] where @xmath51 and @xmath52 are the values at the boundary between region @xmath2 and region @xmath36 . let us now denote by @xmath43 and @xmath53 the values of the inflaton and its momentum when the trajectory reaches the slow roll curve . @xmath54}\ ] ] where we have neglected @xmath53 since it is exponentially smaller than @xmath52 . in general the unconventional forms of the tachyonic energy density and pressure make the cosmology with tachyon field differ from that with a normal scalar field , and make it difficult to separate kinetic term from potential term . we assume that @xmath19 and @xmath55 are regarded as potential and kinetic term respectively . therefore , the boundary value between region @xmath2 and region @xmath56 is @xmath57 . so @xmath58}\ ] ] to lead to sufficient inflation , from ( [ tb ] ) such initial conditions must satisfy @xmath59}<t_b^{60}\ ] ] * region @xmath36 . * it is the region of kinetic energy domination where @xmath48 is negligible compared to the friction term and ( [ 4 t ] ) and ( [ 5f ] ) become @xmath60 these can be integrated and we find : @xmath61 @xmath62}\end{aligned}\ ] ] where @xmath63 is gauss hypergeometric function . to demonstrate the attractor behaviour of the tachyonic inflation , we use the hamilton - jacobi formalism , which greatly simplifies the analysis . adding to any solution to eq.([hj ] ) a linear homogeneous perturbation , we find the perturbation die away exponentially . the attractor behaviour indicates that , regardless of initial conditions , the late - time solutions are the same up to a time shift , which can not be measured @xcite . we else use an explicitly numerical computation of the phase space trajectories and obtain analytical approximations to the trajectories of the tachyon in different regions . one can easily verify from figure 1 and figure 2 that these approximations are in very good agreement with the numerical results , and that the slow rolling solution is the late - time attractor . although the initial kinetic term decays rapidly and does not prevent the onset of inflation , allowing for non - vanishing initial field momenta around @xmath64 may dramatically reduce the phase space of initial conditions for which successful inflation results . according to the picture of tachyonic inflation , the homogeneous tachyon field near the top of the potential rolls down towards the minimum of the potential at @xmath11 . tachyonic matter behaves at late time as a pressureless gas of massive particles . in most versions of the theory of reheating , production of particles occurs only when the inflaton field oscillates near the minimum of its effective potential . however , the effective potential of the rolling tachyon does not have any minimum at finite @xmath14 , so this mechanism does not work . it is unclear how the universe could be reheated in the framework of tachyon cosmology . recently , some studies pointed out that as the tachyon evolves into the late - time , the coupling to the closed string becomes more and more large @xcite . these results motivate us to expect that the tachyon could emit closed string radiation @xcite , such as graviton and dilation , into the bulk and eventually settles in the finite minimum . it is a pleasure to acknowledge helpful discussions with g.n.felder and r.s.tung . this project was in part supported by nnsfc under grant nos . 10175070 and 10047004 as well as also by nkbrsf g19990754 . 99 d.choudhury , d.ghoshal , d.p.jatkar and s.panda , hep - th/0204204 ; 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we study the complete evolution of a flat and homogeneous universe dominated by tachyonic matter .
we demonstrate the attractor behaviour of the tachyonic inflation using the hamilton - jacobi formalism .
we else obtain analytical approximations to the trajectories of the tachyon field in different regions .
the numerical calculation shows that an initial non - vanishing momentum does not prevent the onset of inflation .
the slow - rolling solution is an attractor .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in general , quantum field theories have been built in the context of local models . however , there exist physical situations that lead to non local interactions in a straightforward way . let us mention , for instance , wheeler and feynman s description of charged particles @xcite , string theories with non local vertices @xcite , and non local kinetic terms that appear when bosonizing fermions in ( 2 + 1 ) dimensions @xcite @xcite . as we shall see , several recently considered non local field theories are related to the study of electronic systems in one spatial dimension ( 1d ) @xcite @xcite . indeed , in recent years the physics of 1d systems of strongly correlated particles has become a very interesting subject since one can take advantage of the simplicity of the models at hand and , at the same time , expect to make contact with experiments . for instance , the recently discovered carbon nanotubes are perfect experimental realizations of 1d conductors @xcite . on the other hand , as the dimensionality of a system decreases , charge screening effects become less important and the long - range interaction between electrons is expected to play a central role in determining the properties of the system . in fact , from a theoretical point of view the effects of long - range interactions have been recently discussed in connection to several problems such as the fermi - edge singularity @xcite , the insulator - metal transition @xcite , the role of the lattice through umklapp scattering and size dependent effects @xcite , etc . in the specific context of carbon nanotubes , a low - energy theory including coulomb interactions has been also recently derived and analyzed @xcite . non local fermionic models have been also used in the study of fluctuation effects in low - dimensional spin - peierls systems @xcite . as shown in @xcite , starting from a non - local and non - covariant version of the thirring model @xcite , in which the fermionic densities and currents are coupled through bilocal , distance - dependent potentials , one can make direct contact with the `` g - ology '' model @xcite currently used to describe different scattering processes characterized by coupling functions @xmath3 , @xmath4 , @xmath5 and @xmath6 . when one bosonizes this theory by either operational or functional methods , due to the contributions of @xmath3 ( backscattering ) @xcite and @xmath5 ( umklapp ) @xcite one finds an even more drastic departure from the local case . indeed , instead of the well - known integrable sine - gordon model ( sg ) one gets a non local extension of it , which , as far as we know , is not exactly solvable . recently , in ref.@xcite , the physical content of this model was explored by using the self - consistent harmonic approximation ( scha ) @xcite . as it is well - known the scha is a non controlled approximation , i.e. there is no perturbative parameter involved . it is then desirable to have an alternative analysis of this problem . this is the main motivation of the present work . we will apply the renormalization group ( rg ) technique @xcite , usually employed in local cases , to the non local sine - gordon model ( nlsg ) mentioned above . for simplicity we shall assume that non locality plays a role only in umklapp interactions ( @xmath5 ) whereas all other potentials are local , i.e. proportional to delta functions . in section 2 we briefly show how the nlsg action is obtained from the non local thirring model . in section 3 we derive the rg equations and compute the gap in the charge - density spectrum . this allows us to determine the effect of non contact @xmath5 couplings . finally , in section 4 , we discuss our results . let us sketch the derivation of the nlsg action . we start from the fermionic ( 1 + 1)-dimensional quantum field theory with euclidean action given by @xmath7 where @xmath8 is the unperturbed action associated to a linearized free dispersion relation . the contributions of the different scattering processes can be written as @xmath9 and @xmath10 where the @xmath11 are the usual two - dimensional dirac matrices and @xmath12 , @xmath13 . the coupling potentials @xmath14 and @xmath15 are assumed to depend on the distance @xmath16 and can be expressed in terms of `` g - ology '' parameters as @xmath17 in the above equations @xmath18 and @xmath19 are just numerical constants that could be set equal to one . we keep them to facilitate comparison of our results with those corresponding to the usual thirring model . indeed , this case is obtained by choosing @xmath20 and @xmath21 . on the other hand , the non - covariant limit @xmath20 , @xmath22 gives one version ( @xmath23 ) of the tl model @xcite . the terms in the action containing @xmath4 and @xmath6 represent forward scattering events , in which the associated momentum transfer is small . in the @xmath4 processes the two branches ( left and right - moving particles ) are coupled , whereas in the @xmath6 processes all four participating electrons belong to the same branch . on the other hand , @xmath3 and @xmath5 are related to scattering diagrams with larger momentum transfers of the order of @xmath24 ( bs ) and @xmath25 ( us ) respectively ( this last contribution is important only if the band is half - filled ) . for simplicity , throughout this paper we will consider spinless electrons . the extension of our results to the spin-@xmath26 case with spin - flipping interactions , though not trivial , could be done by following the lines of ref . @xcite . at this point we consider the partition function @xmath27 expressed as a functional integral over fermionic variables . the implementation of a generalized hubbard - stratonovich identity @xcite allows to write @xmath27 in terms of a fermionic determinant . although this determinant is highly non trivial , one can combine a chiral change in the fermionic path - integral measure with a formal expansion in @xmath28 in order to obtain a bosonic representation ( see ref . @xcite for details ) . one thus establishes an equivalence between the original fermionic action and the following bosonic action depending on five scalars @xmath29 , @xmath30 , @xmath31 , @xmath32 and @xmath33 : @xmath34\\ + \frac{1}{2}{\int d^2x\,d^2y\,}c_{\mu } ( x ) u^{-1}_{(\mu)}(x , y ) c_{\mu } ( y ) + \frac{{g ' } \lambda c}{\pi } { \int d^2x\ , } [ c_{(0 ) } ( x)f_{0}(x ) + i c_{(1 ) } ( x)f_{1}(x ) ] \label{39}\end{gathered}\ ] ] where @xmath35 \\ b(p)&=\frac{1}{2}\left[p_0 ^2 \hat{v}_{(0)}^{-1 } ( p ) + p_1 ^2 ( \hat{v}_{(1)}^{-1 } ( p ) - \frac{g^2}{\pi})\right ] \\ c(p)&=p_0p_1\left(\hat{v}_{(0)}^{-1 } ( p)- \hat{v}_{(1)}^{-1 } ( p ) + \frac{g^2}{\pi}\right),\end{aligned}\ ] ] and @xmath36 since the integrals in @xmath31 and @xmath32 are quadratic these fields are easily integrated out and one gets @xmath37 } \label{41}\ ] ] with @xmath38 = s_{0 } + s_{int } \label{42}\ ] ] where @xmath39\ ] ] @xmath40\\ -\frac{(\lambda c)^2}{2\pi^2}{\int d^2x\,d^2y\,}g_{3 } ( x , y ) \cos\left[\sqrt{4 \pi}(\varphi(x ) + \varphi ( y ) ) -2i g ( \phi(x ) + \phi(y))\right].\end{gathered}\ ] ] it is now convenient to diagonalize the quadratic part of the effective action by introducing the fields @xmath41 , @xmath42 and @xmath43 : @xmath44 where we have defined @xmath45 and @xmath46 . we then obtain @xmath47 , \label{44}\ ] ] @xmath48\\-\frac{(\lambda c)^2}{2\pi^2}{\int d^2x\,d^2y\,}g_{3 } ( x , y ) \cos\sqrt{4\pi}[\phi(x ) + \phi ( y)].\label{45}\end{gathered}\ ] ] one can see that the @xmath41 and @xmath42 fields become completely decoupled from @xmath43 . moreover , it becomes apparent that the @xmath43-dependent piece of the action @xmath49 is the only one containing potentially relevant contributions ( i.e. gapped modes ) . in this section we shall focus our attention on the non local action derived above . for simplicity , from now on we will consider the case in which @xmath50 and @xmath51 are constants ( local forward scattering ) and @xmath52 , i.e. a pure non local umklapp interaction . thus , we start from the action @xmath53={\int\frac{d^2p}{(2\pi)^2}\,}\phi(p)\frac{f(p)}{2}\phi(-p ) -\frac{(\lambda c)^2}{2\pi^2}{\int d^2x\,d^2y\,}g_3(x , y)\cos\sqrt{4\pi}[\phi(x)+\phi(y)]\label{46}\end{gathered}\ ] ] with @xmath54 where we have now expressed all formulae in terms of @xmath18 coupling functions . in the local case ( @xmath55 ) the action ( [ 46 ] ) corresponds to the well - known sine - gordon model , which is an integrable , exactly solvable field theory . in particular , a rg analysis shows that the stiffness constant " , @xmath56 has to be lower than @xmath57 in order to have a relevant cosine interaction , i.e. to have a gap in the spectrum . recently , by reinterpreting bethe - ansatz results , zamolodchikov obtained the exact expression for this gap @xcite . unfortunately , as far as we know , the present non local version of the theory is not exactly soluble and one is then forced to consider an approximation . in ref . @xcite a scha was employed in order to estimate the energy gap . but this is a non controlled , non perturbative approximation . besides this general disadvantage , the implementation of the scha technique led to a set of coupled algebraic equations that could be numerically solved only for very weak non locality . it is then natural to try another method to attack the problem and eventually improve the approximation . let us consider the wilsonian approach to the rg ( see for instance @xcite ) . first of all we restrict our analysis to a non local interaction of the form : @xmath58 with @xmath59 where @xmath60 is the second derivative of the delta function with respect to @xmath61 . at this point it is worth mentioning that rg calculations involving fermionic non local interactions already exist in the literature . for instance , the authors of ref.@xcite used a jordan - wigner fermion representation for the 1d heisenberg - ising model which includes not only a non local fermion - fermion interaction but also a linear fermion - lattice coupling . due to the curvature of the band , the non trivial fermion - lattice coupling and the presence of both forward and umklapp scattering , comparison of their rg equations with ours is not straightforward . let us point out , however , that in @xcite fermions corresponding to nearest - neighbor lattice sites interact through a potential of the form @xmath62 , i.e. a unique constant @xmath63 is associated to @xmath64 . as shown in the above equation ( [ g3 ] ) , in the present work we are interested in a coupling which depends at least on two constants @xmath65 and @xmath66 . in fact , the derivation of rg equations for these coupling constants will be our next task . in condensed matter problems one is usually interested in the physics at long distances , compared to a lattice spacing of the order of @xmath67 . since , in momentum space , this corresponds to small @xmath68 , it is natural to consider correlations between fields with momenta @xmath69 , with s very large . these are the so called `` slow modes '' @xmath70 . on the other hand , the `` fast modes '' @xmath71 are those carrying momenta that satisfy @xmath72 . in the present approach to rg these fast modes are integrated in the path integral framework , giving rise to an effective theory depending only on slow modes . from this action , in turn , one can read the flow equations for the couplings . indeed , writing the initial action as @xmath73 , to first order , after a suitable rescaling of coordinates and momenta and a redefinition of the fields ( see the appendix for details ) we obtain the following relationship between the original and rg transformed actions : @xmath74\delta^2(x - y)-\frac{\epsilon - 4 k \epsilon t}{\lambda^2}\,\partial_1 ^ 2\delta^2(x - y ) \right)\times \\ \times \cos\sqrt{4\pi}(\phi(x ) + \phi ( y ) ) = \\ = \frac{(\lambda c)^2}{2\pi^2}\int d^{2}x\,d^{2}y\left(\lambda \delta^2 ( x - y)-\frac{\epsilon}{\lambda^2}\,\partial_1 ^ 2\delta^2(x - y)\right ) \cos\sqrt{4\pi}(\phi(x ) + \phi ( y ) ) = s_{int}^{'}.\end{gathered}\ ] ] as usual , imposing the invariance of the action under rg , we get the flow equations for the couplings @xmath65 and @xmath66 : @xmath75 @xmath76 where @xmath77 , with the initial conditions @xmath78 and @xmath79 . the solution of this system is elementary , yielding : @xmath80 + k \epsilon_{0}\exp[-4 k t],\ ] ] @xmath81.\ ] ] from the last equation one clearly sees that the non local piece of the interaction is irrelevant , as expected . concerning the local interaction one sees that it is relevant for @xmath82 , i.e. non locality does not modify this well - known condition already found for the local sg model . therefore , for @xmath82 , @xmath65 will grow with increasing t and there will be a gap in the cd spectrum which can be estimated by determining the value @xmath83 for which @xmath84 . from now on we shall restrict our analysis to the case @xmath82 . the gap is then given by @xmath85 , where @xmath86 . it is also convenient to define the dimensionless gap @xmath87 . from equation ( [ rg3 ] ) one can thus derive an equation for @xmath88 which gives the behavior of the energy gap as function of the forward scattering potentials ( @xmath56 ) and the non local contribution of the umklapp scattering ( @xmath89 ) . before analyzing this non trivial equation it seems reasonable to check if it predicts sensible results for the local case . to this end we set @xmath90 and @xmath91 in ( [ rg3 ] ) , obtaining : @xmath92 this result can be compared with the exact solution obtained by zamolodchikov @xcite and with the approximated result given by the scha method @xcite . the corresponding expressions for the gap are respectively given by : @xmath93 and @xmath94 in order to compare @xmath95 and @xmath96 with @xmath97 in an efficient and easy to visualize way , we have computed the relative error @xmath98 as function of @xmath56 for both approximations . the results are depicted in fig.1 where one sees that our rg computation gives values of the gap closer to the exact values for a wide range of the stiffness constant @xmath56 . interestingly , the scha result works well when one approaches the end points of the interval . going back to the case @xmath99 , by combining eqs.([rg1 ] ) and ( [ rg2 ] ) one readily gets a phase diagram in the @xmath100 plane ( see fig . 2 ) . there is a critical line given by @xmath101 . if the initial parameters are tuned to lie on this line , the system will flow to the tomonaga - luttinger fixed point , at the origin . in this case , of course , the system remains gapless . on the other hand , for initial conditions outside the critical line , the system flows to strong coupling , giving rise to a gap @xmath88 , as mentioned above . for simplicity let us consider the case @xmath102 and define the variable @xmath103 . the gap equation can then be written as @xmath104 where @xmath105 . this equation is one of our main results . for fixed values of @xmath56 and @xmath106 it gives the behaviour of the energy gap as function of the non local contribution to umklapp scattering , associated to non contact interactions . the form of this formula suggests that it could be easier to handle the inverted equation : @xmath107 where @xmath108 . in fig.3 we show the numerical solution of this last equation for @xmath109 and @xmath110 . we see that @xmath111 decreases for increasing @xmath89 , in qualitative agreement with the scha prediction @xcite . a quantitative comparison between both approximations is given in fig.4 , for the same fixed values of @xmath106 and @xmath56 . since the scha result obtained in @xcite is valid for small values of the coefficient associated to non locality , we have plotted the solutions in the interval @xmath112 . we find that the gap decay predicted by rg , for increasing non locality , is much slower than the one obtained through the gaussian approximation . as a final comment , we note that the rg treatment for non local interactions depicted in this work can be extended to a more general coupling function including an arbitrary number of even powers of @xmath113 . in coordinate space such an interaction can be written as @xmath114 where @xmath115 . the corresponding set of rg equations for the @xmath116 can be obtained from the expansion of the cosine integral function which appears when one integrates the fast modes after the mode separation ( see appendix ) . the result is @xmath117 the general solution of this system can be expressed as a combination of exponentials and the computation of the gap can not be done , in principle , in an analytical way . it is then illustrative to consider a particular case in which this calculation is simplified . indeed , for small @xmath118 , it can be proved by induction that the solutions of this system of equations are of the form : @xmath119.\ ] ] in this limit one obtains the following expression for the gap : @xmath120 which is consistent with the conditions @xmath121 and @xmath122 . in this paper we have considered a non local extension of the sine - gordon model . this theory is obtained when one bosonizes a non local and non covariant version of the thirring model used to describe certain 1d many - body systems . since the integrability of this non local sine - gordon model has not been proved , one needs to implement some approximation to study its physical content . we have performed a rg calculation up to first order in the coupling function @xmath5 , which in a condensed matter context is associated to the so called umklapp scattering . we obtained an expression for the energy gap as function of the non local piece of the interaction @xmath123 . for purely local interactions ( the exactly solvable sg ) our result seems to be a sensible approximation , improving the scha predictions for a wide range of forward interactions . in the non local case , in which no exact answer is known , we predict decreasing values for the gap for increasing values of @xmath123 , in qualitative agreement with a previous scha computation . we were able to give a precise comparison between both approximations in the interval @xmath124 , showing that the gap decrease , for increasing non locality , is much weaker according to the rg computation . since , as is well - known , the scha method is not a controlled approximation , the present results contribute to a better understanding of the physics of non local field theories . we think that our results are also of interest in the context of 1d many - body systems ( luttinger liquids ) in which most of the previous investigations involving umklapp scattering do not consider non local effects associated to long range interactions @xcite . this work was partially supported by universidad nacional de la plata and consejo nacional de investigaciones cientficas y tcnicas , conicet ( argentina ) . in order to illustrate the computation leading to the flow equations we first define the free bosonic propagator : the next step is the analysis of @xmath49 , as given by the second term of equation ( [ 46 ] ) . for simplicity , in this appendix we disregard the overall constant @xmath128 . going to momentum space and performing the separation in slow and fast modes @xmath70 and @xmath71 , according to : we obtain @xmath132 \\ \cos[\frac{\sqrt{4\pi}}{(2\pi)^{2}}\int\,d^{2}p\,\phi_{>}(p)\,f(p , x , y_{1 } ) ] - \sin[\frac{\sqrt{4\pi}}{(2\pi)^{2}}\int\,d^{2}p\,\phi_{<}(p)\,f(p , x , y_{1 } ) ] \\ \sin[\frac{\sqrt{4\pi}}{(2\pi)^{2}}\int\,d^{2}p\,\phi_{>}(p)\,f(p , x , y_{1})])\end{gathered}\ ] ] where @xmath133 @xmath135>_{0}\ , = \\ = \exp \left(-k \int\,dp_1 \,(\theta(p_1)-\theta(-p_1))\,\frac{1+\cos(p_1(x_1-y_1))}{p_1}\right)=\\ = \exp \left(-k \left[2\ln s + 2 ci\,(\lambda(x_1-y_1))-2 ci ( \frac{\lambda}{s}(x_1-y_1))\right]\right)\end{gathered}\ ] ] where @xmath136 is the cosine integral function and the free vacuum expectation values are , of course , taken with respect to fast modes . rescaling momenta , coordinates and fields such that the free piece of the action @xmath137 remains invariant : @xmath138 and using the fact that @xmath139 , @xmath49 can be written as @xmath140\times \\ \times\ , ( 1 + [ ( 3 - 4 k ) + k ( \lambda ( x_{1}^{'}- y_{1}^{'}))^{2 } - \frac{k}{12}(\lambda ( x_{1}^{'}- y_{1}^ { ' } ) ) ^{4 } ] t ) , \end{gathered}\ ] ] where we have used the power expansion of the function @xmath136 . finally , using the explicit expression for @xmath5 in terms of @xmath141 and @xmath123 we obtain @xmath142 t - 2 k \epsilon_0 t ) \delta^2(x_{1}^{'}- y_{1}^{'})-\\-\frac{\epsilon_0 - 4 k \epsilon_0 t}{\lambda^2}\partial_1 ^ 2\delta^2(x_{1}^{'}- y_{1}^ { ' } ) ] \cos\sqrt{4\pi}[\phi(x^ { ' } ) + \phi ( y^ { ' } ) ] = \\ = \int d^{2}x^{'}\,d^{2}y^{'}[\lambda^ { ' } \delta^2(x_{1}^{'}- y_{1}^{'})-\frac{\epsilon^{'}}{\lambda^2}\partial_1 ^ 2\delta^2(x_{1}^{'}- y_{1}^ { ' } ) ] \cos\sqrt{4\pi}[\phi(x^ { ' } ) + \phi ( y^ { ' } ) ] = s_{int}^{'},\end{gathered}\ ] ] 99 r.p.feynman and j.a.wheeler , rev . * 17 * ( 1945 ) 157 . d.a.eliezer and r.p.woodard , nucl . b * 325 * ( 1989 ) 389 . e.c.marino , phys . b * 263 * ( 1991 ) 63 . nucl . phys . b * 408 * ( 1993 ) 551 . d.g.barci , c.d.fosco and l.oxman , phys . b * 375 * ( 1996 ) 267 . d.g.barci and l.oxman , mod . a * 12 * ( 1997 ) 493 . i. v. krive et al . , phys . b * 52 * 10865 ( 1995 ) . c.nan , m.c.von.reichenbach and m.l . trobo , nucl . phys . * b 435 * [ fs ] 567 ( 1995);ibid . * b 485 * [ fs ] 665 ( 1997 ) . + m.manas , c.nan , m.l.trobo , nucl . b 525 [ fs ] , * 721 * ( 1998 ) . + v. i. fernndez and c. m. nan , theor . 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we study a non local version of the sine - gordon model connected to a many - body system with backward and umklapp scattering processes . using renormalization group methods
we derive the flow equations for the couplings and show how non locality affects the gap in the spectrum of charge - density excitations .
we compare our results with previous predictions obtained through the self - consistent harmonic approximation .
carlos m. nan@xmath0 and mariano j. salvay@xmath0 @xmath1 _ instituto de fsica la plata , departamento de fsica , facultad de ciencias exactas , universidad nacional de la plata .
cc 67 , 1900 la plata , argentina . _
@xmath2 _ consejo nacional de investigaciones cientficas y tcnicas , argentina . _ _ keywords : _ field theory , non local , renormalization group _ pacs : _
11.10.lm , 05.30.fk
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You are an expert at summarizing long articles. Proceed to summarize the following text:
at present information on the characteristics of hadronic interactions in fragmentation region is still scarce or missing and experiments with ` roman pots ' are anticipated to improve the situation . some of this information , in principle , could be obtained with the use of the data on cosmic ray ( cr ) muon and hadron spectra , provided primary spectra are known with high precision , but that is not the case . the obvious obstacle here is that at high energies primary cosmic ray ( pcr ) fluxes , measured in direct experiments , themselves are functionals of various interaction parameters plus their accuracy is appreciably affected by additional systematic effects @xcite . in the series of our papers @xcite we underlined , that these effects can lead to underestimation of light nuclei fluxes and thus can explain discrepancy between measured and calculated muon fluxes for @xmath0 gev . preliminary data of atic-2 @xcite , covering the gap between magnetic spectrometer and emulsion chamber experiments , seems to be in concordance with our conclusions , but situation is more complicated in fact , as further consideration will show . atic-2 slope of proton spectrum @xmath1 @xcite for primary energies below 10 tev is in remarkable contradiction with previously measured values @xmath2 and @xmath3 by runjob @xcite and jacee @xcite experiments correspondingly , but this discrepancy is removed by steepening in atic-2 proton spectrum at energies above 10 tev . these new data were already exploited in extensive calculations of muon and hadron fluxes with number of interaction models at different atmospheric depths and zenith angles in @xcite where it was shown , that their use allows to get reasonable agreement with the most of the data under appropriate choice of hadronic interaction parameters . in fact , it is not possible to obtain concordant conclusions on primary spectra or hadronic interactions parameters coming even from much smaller subset of the experimental data . in this paper we demonstrate this on the basis of analysis of the data on muon flux at vertical direction and only one set of the data on hadron flux of eas - top @xcite . first , we use the data of sea - level and underground experiments to obtain conclusions on behavior of muon spectrum at sea - level in the energy range 40 gev10 tev . further we analyze influence of uncertainties in the muon data and interaction parameters on properties of reconstructed primary proton fluxes . and finally we show , that such different in approach and characteristics interaction models as sibyll 2.1 @xcite and nexus 3.97 @xcite can bring to hardly distinguishable predictions on muon and hadron fluxes . depth - intensity relation , needed for reconstruction of sea - level muon spectrum , may be obtained via numerical solution of one - dimensional transport equation . in adjoint approach this equation has the following form @xcite @xmath4 here @xmath5 is survival probability of muon with energy @xmath6 , being born at the distance @xmath7 from detector , @xmath8 total interaction cross - section , @xmath9 differential cross - sections for processes of ionization , bremsstrahlung , pair production and photonuclear interaction correspondingly , @xmath10 detector sensitivity function . the numerical method , applied for solution of this equation @xcite , allows to avoid any approximations ( such as continuous losses one ) and to obtain muon intensities at large depths of matter with account of fluctuations in all muon interaction processes . accuracy of our calculations of muon survival probabilities and intensities was thoroughly examined in @xcite and comparison with the results of monte - carlo codes mum @xcite and music @xcite is presented in fig . [ fig : vsmum ] . anticipating further discussion of sea - level muon spectrum behavior it is necessary to note , that our calculations give upper estimate of muon flux at large depths in comparison with mum , because of use of @xmath111% lower muon energy losses @xcite . to describe the data on muon intensity underground and at sea - level , and to estimate influence of uncertainty in muon flux data on reconstruction of primary proton flux , we used two parameterizations in the simple form , proposed in work of bogdanova et al . original fit for the vertical from @xcite @xmath12 provides good agreement with the data at sea - level ( fig . [ fig : param ] ) , but leads to underestimation of the muon flux for the depths below 6 km w.e . [ fig : strock ] ) . to match better underground data for the depths 26 km w.e . we shall also apply modified fit with slightly @xmath13% increased intensity in multi - tev region @xmath14 as it is seen from fig . [ fig : strock ] , for depths from 4 km w.e . up to 8 km w.e . , corresponding to @xmath15 tev median muon energies at sea level , use of this spectrum provides good agreement with the data of lvd @xcite , bno @xcite and frejus @xcite collaborations and leads to underestimation of data of macro @xcite and soudan @xcite experiments . our consideration will touch muon energies only above 40 gev , to exclude different effects , not related to high - energy hadronic interactions features , such as geomagnetic effect , influence of uncertainties in low - energy interaction models , or even absorption of low energy muons in ground as in case of l3+c detector . % variation of total muon energy losses . standard rock ( muon spectrum at sea level from @xcite ) : dashed line present work , open circles music @xcite . ] ) , dashed line for muon spectrum ( [ eq : mbogdanova ] ) . neutrino induced muon contribution is taken from @xcite . ] average numbers of hadrons @xmath16 and muons @xmath17 with energies above @xmath18 in eas from primary nucleon of energy @xmath19 were obtained with the help of one - dimensional hybrid code conex @xcite in regime of cascade equations solution for interaction models qgsjet 01 @xcite , sibyll 2.1 @xcite , nexus 3.97 @xcite and qgsjet - ii-03 @xcite . to get differential spectrum of hadrons ( or in the same way of muons ) for some energy @xmath20 the following simple formula had been used : @xmath21/2\delta e.\ ] ] here @xmath22 is the integral spectrum of hadrons @xmath23 for primary nucleon spectrum @xmath24 . the interval width @xmath25 must be chosen to provide the difference between integral intensities in ( [ eq : diff ] ) to be much larger then calculations error . test computations for @xmath26 and observation levels 820 and 1030 g/@xmath27 for energy @xmath28 tev brought to differential flux values lying within @xmath29 from each other . change of upper integration limit @xmath30 in formula for @xmath22 from @xmath31 to @xmath32 gives less then 1% increase of differential flux . and , the last , increase of number of primary energy bins in @xmath33 from 10 to 20 per order introduces @xmath34 variation to differential flux value . the listed error sources partially compensate each other and the total error of our calculations does not exceed 3% . the calculation were performed for the set of energies , coinciding with the set from eas - top experiment paper @xcite . as a basic model of pcr nuclei spectra the parameterizations from @xcite were chosen . nuclei with @xmath35 were treated in the framework of the superposition model , high accuracy of this approach is well known and was checked by our calculations with the use of conex both for muons and hadrons once again . comparison of the calculated muon fluxes with the experimental data , presented in fig . [ fig : param ] , reveal familiar picture of high energy muon deficit . the reasons of its appearance were considered in our previous papers @xcite and they still hold true regardless of the fact , that three more interaction models were included in our analysis . all interaction codes , except qgsjet 01 , satisfactory describe data on muon flux only up to @xmath36100 gev and then one by one fail to do it . accounting that such muon energies correspond to primary energies above 1 tev , studied with balloon(satellite)-borne emulsion chambers , we related muon deficit to underestimation of primary light nuclei fluxes , taking place in these experiments @xcite . unfortunately , disagreement between the models in the muon fluxes also appears at energies around 100 gev , thus making impossible precise reconstruction of primary nucleon spectrum for @xmath37 tev . in fact , in such conditions there are no reasons to rule out any of the models , except qgsjet 01 , which leads to remarkable disagreement with the experiment even in the range of reliable magnetic spectrometers data on pcr and muon spectra . to find out , why the models differ in the predicted muon fluxes let us consider quite characteristic energy 1.29 tev , where discrepancies between the models reach appreciable values and the data on muons from underground installations are yet quite reliable . contributions of primary protons to the differential flux of muons of the given energy , presented in fig . [ fig : inclusmu ] show , that spread in muon fluxes between the interaction models is entirely due to uncertainties in the description of @xmath38-spectra in fragmentation region @xmath39 . since inclusive muon flux is sensitive nearly only to the characteristics of the very first primary particle interaction , hence , the harder these spectra are in the particular model , the larger muon intensity its use leads to . for the lower values of @xmath40 , i.e. for @xmath41 tev , all the models give practically the same muon yields . as noted above , in view of uncertain situation with primary spectra for @xmath37 tev , one can not give preference to any of the models in comparison with the others . if to demand the minimal disagreement with the direct measurements data on pcr spectra , then obviously sibyll 2.1 satisfies this requirement the best , or , on the other hand , one could say that it provides the most acceptable description of @xmath38 production spectra in @xmath42-air collisions in fragmentation region . we shall discuss this affirmation in detail below . from the previous consideration it is clear , that reconstructed fluxes of protons shall be higher than measured in emulsion chamber experiments , but can be comparable with recently obtained data of atic-2 group @xcite . we performed the reconstruction simply by picking up of appropriate primary proton flux parameters in order to minimize deviation of the obtained muon fluxes from the parameterizations ( [ eq : bogdanova ] ) and ( [ eq : mbogdanova ] ) . , to muon flux parameterizations ( [ eq : bogdanova ] ) and ( [ eq : mbogdanova]).,title="fig:",scaledwidth=48.0% ] , to muon flux parameterizations ( [ eq : bogdanova ] ) and ( [ eq : mbogdanova]).,title="fig:",scaledwidth=48.0% ] first , let us consider an attempt to minimize maximal deviation from spectra ( [ eq : bogdanova]),([eq : mbogdanova ] ) for @xmath43 gev10 tev in assumption , that primary proton spectrum can be described by single power law function @xmath44 in the entire energy range 100 gev500 tev . in fig . [ fig : primp ] and in table [ tab : pp ] one can find the results of this reconstruction , upper and lower lines for each muon spectrum parametrizations in the fig . [ fig : primp ] correspond to low and high helium flux fits @xcite . as expected , the obtained spectra are flatter , than measured by runjob and jacee groups , but , except for qgsjet 01 model , agree well with atic-2 results @xcite . figure [ fig : fitratio ] shows , that in this case it is possible to achieve agreement with the fitted muon spectrum within 10% , but only sibyll 2.1 reproduces its shape correctly , with other models it is not possible to get right muon spectrum slope variation . we should also note , that this behavior turned out to be insensitive to the choice of helium flux parametrization . as a result , there is a dip in the ratio of the muon flux , obtained from fitting of proton spectrum , to the parameterizations ( [ eq : bogdanova]),([eq : mbogdanova ] ) for energies around 1 tev , where the underground data were already underestimated ( overestimation of muon flux at higher energies does not compensate this effect completely ) , and growth in small energies range , which brings to contradiction with low energy @xmath45 gev data . parametrization ( [ eq : mbogdanova ] ) was also projected to make muon spectrum slope variation less sharp , but this lead to problems with its fitting in @xmath46 gev energy range , as it is seen from fig . [ fig : fitratio ] ( right panel ) for sibyll 2.1 and nexus 3.97 models . the problem with muon spectrum shape matching is better illustrated by upper left panel of fig . [ fig : primpbr ] , where it is shown , that correct reproduction of muon spectrum for energies below 1 tev with single power law proton spectrum leads to appreciable overestimation of muon flux at higher energies . there are three possible explanations or solutions of this problem . first , the discrepancy can be completely removed by choice of appropriate interaction parameters , e.g. similar to those in sibyll 2.1 . another argument , which can be given is that the data on muon flux for energies above 1 tev are not so definite to claim their inconsistency with the calculations , but it does not look well supported by underground data ( see fig . [ fig : strock ] and calculations @xcite ) . and the last possibility is to assume , that primary proton spectrum is not monotonous and either has sharp break or slowly changing exponent @xmath47 . let us consider the latter assumption , which finds experimental @xcite and theoretical @xcite justifications , in more detail . the results for the simple case with break ( fig . [ fig : primpbr ] ) , which allows to achieve correct description of muon spectrum shape with right asymptotic and deviation in flux value @xmath48% , show , that small difference between spectra ( [ eq : bogdanova ] ) and ( [ eq : mbogdanova ] ) results not only in different proton intensities , but also in break positions . the latter lies for parametrization ( [ eq : mbogdanova ] ) in the primary energy range 1015 tev , the change in power index reaches appreciable values up to @xmath49 for qgsjet 01 and qgsjet ii models ( table [ tab : pp ] ) . proton spectra , obtained from muon flux ( [ eq : mbogdanova ] ) , with qgsjet ii and nexus 3.97 models are in the best agreement with atic-2 data , while sibyll 2.1 provides intermediate between atic-2 and emulsion chambers experiments slope value . spectra , reconstructed from parametrization ( [ eq : bogdanova ] ) , have breaks at 36 tev and in case of qgsjet ii proton flux poorly agrees with experiments at primary energies around 100 gev . evidently , the latter problems are explained by too low , in comparison with underground data , muon flux and this parametrization is considered here mostly for estimation of sensitivity of primary spectrum features to the choice of reference muon flux . it is necessary to note , that due to low sensitivity of differential muon flux to helium and heavier groups of primary nuclei it is impossible to derive any conclusions on presence of the break in these pcr components . for illustration let us consider example of calculations for qgsjet 01 and high helium flux , where the break in proton spectrum is positioned at @xmath50 tev and change of power index is equal to 0.14 ( see table [ tab : pp ] ) . introduction of rigidity - dependent break in he spectrum at @xmath51 per nucleus of the same value @xmath52 gives remarkable discrepancy between calculated muon spectrum and parameterization ( [ eq : mbogdanova ] ) only for energies above 7 tev , which reaches 10% at 20 tev . to correct this asymptotic behavior it suffices to reduce @xmath53 to 0.11 simultaneously for protons and helium without change of the break position , and thus we get proton spectrum lying well within corridor between parameterizations for high and low helium fits , shown in fig . [ fig : primpbr ] . hence , this corridor covers all possible cases of he flux behavior ( with or without break ) , provided the helium flux stays within limits , given in @xcite . summarizing we can say , that primary proton spectrum shape turns out to be sensitive to the choice of interaction model and allows presence of break at 1015 tev with @xmath54 up to 0.15 , which can be slightly softened though , if to allow presence of the same break in other pcr components spectra . lccccccc + & & + model & & & break & + & @xmath55 & @xmath56 & @xmath55 & @xmath56 & ( tev ) & @xmath55 & @xmath56 + qgsjet 01 & 1.42 & 2.675 & 0.93 & 2.620 & 15 & 3.25 & 2.750 + qgsjet ii & 1.04 & 2.650 & 0.69 & 2.595 & 10 & 2.39 & 2.730 + nexus 3.97 & 1.22 & 2.680 & 1.01 & 2.655 & 13 & 1.78 & 2.715 + sibyll 2.1 & 1.29 & 2.695 & & & & & + + & & + model & & & break & + & @xmath55 & @xmath56 & @xmath55 & @xmath56 & ( tev ) & @xmath55 & @xmath56 + qgsjet 01 & 1.45 & 2.690 & 0.95 & 2.635 & 15 & 3.65 & 2.775 + qgsjet ii & 1.08 & 2.670 & 0.69 & 2.610 & 14 & 3.18 & 2.770 + nexus 3.97 & 1.25 & 2.700 & 1.01 & 2.670 & 12 & 2.14 & 2.750 + sibyll 2.1 & 1.37 & 2.720 & & & & & + comparison of our calculations of hadron spectrum for the primary spectra from @xcite ( high helium flux ) with the recent measurements , performed by eas - top collaboration @xcite , is presented in fig . [ fig : eastop ] ( left panel ) . first , let us note the following facts . below 100 gev all calculated spectra have breaks , caused by non - perfect matching of low - energy interaction model gheisha to the high - energy models . shape of the measured hadron spectra also breaks at energies above 4 tev and the data become less definite , thus in the forthcoming analysis we are going to consider the data for energies from 129 gev to 4 tev . for these energies qgsjet 01 , qgsjet ii and sibyll 2.1 quite reasonably reproduce the shape of the measured hadron spectrum , nexus 3.97 leads to spectrum with almost constant power index . one can see , that the most consistent description of the data for specified energies provide qgsjet 01 and sibyll 2.1 . in contrast with the muons there are no energy range , where the models agree on the hadron fluxes and the reasons of this disagreement are not as simply to point out as in the case with muons . the most important characteristics in this analysis are total inelastic cross section , determining chances of primary particle to survive , shapes of inclusive spectra @xmath57 , @xmath58 , @xmath59 in the very forward region , responsible for substantial process of leading particles production ( see fig . [ fig : hinclus ] for the listed spectra ) . let us briefly outline the major conclusions , which one may come to in the given situation . nexus 3.97 gives the lowest fluxes as of hadrons in total , so of nucleons and mesons ( fig . [ fig : mesnuc_sp ] ) , and this happens in spite of the lowest inelastic cross - section values . inclusive spectrum @xmath60 immediately helps to figure out , that incident protons in nexus 3.97 have comparably low chances to save most of their energy in collision and this leads to such low nucleon flux , the same can be said about meson flux and production of pions by pions . similarly , from comparison of the inclusive spectra , it can be easily understood , why qgsjet ii gives the highest hadron flux . note , that sibyll 2.1 concedes to qgsjet ii in hadron intensity mostly because of less effective production of leading neutrons in @xmath42-air collisions and due to the larger total interaction cross - section . thus , from analysis of the data on hadron flux it is difficult to imply any strict constraints on inclusive spectra shapes , since mechanism of hadron spectrum formation is more sophisticated than that in the case of muon spectrum . sibyll 2.1 and qgsjet 01 display quite different behaviors of the relevant inclusive spectra and total interaction cross - sections , but both models almost equally succeed in description of the eas - top data ( i.e. produce close hadron fluxes ) . the given standard approach to analysis of situation , in fact , is of little sense , since it is based on assumption about validity of primary spectra in form of fits from @xcite , which was called into question in our previous discussion . in this case it is logical to analyze interaction models self - consistency , i.e. their ability to give correct estimates of several cr components at once . provided we know behavior of primary proton spectra for every model , required to match the data on muon flux , we may check how these proton spectra agree with the data on hadrons . in fig . [ fig : eastop ] ( right panel ) we give hadron intensities , calculated for primary proton spectra with breaks from fig . [ fig : primpbr ] ( for sibyll 2.1 see fig . [ fig : primp ] ) , corresponding to muon spectrum parametrization ( [ eq : mbogdanova ] ) . after increase of primary nucleon flux , dictated by the data on muons , one can see , that the best agreement with eas - top measurement provide nexus 3.97 and sibyll 2.1 . it would be interesting to note that two models with different philosophies and inclusive spectra give the most self - consistent results on muons and hadrons , but , of course , this conclusion must be taken with much care , since it is based on the single set of data and we have only indirect indications on the accuracy of this set , e.g. such as agreement of primary proton fluxes , obtained by eas - top and kascade teams ( the latter is derived from flux of unaccompanied hadrons @xcite ) . if to try to perform the same analysis with the variety of the data , obtained at different atmospheric altitudes and zenith angles , no consistent notions of such kind will be obtained as can be easily seen from calculations , presented in @xcite . the progress in cr and high - energy physics , achieved during last 1015 years allowed to turn from statements about satisfactory ( qualitative ) concordance between different kinds of data to investigation of more fine effects . as an example , we managed to show that reconstructed from the data on vertical muon flux primary proton spectra have not only expected interaction model dependent intensities , but also model - dependent shapes . it is demonstrated , that application of qgsjet 01 , qgsjet ii and nexus 3.97 models brings to proton spectrum with break at 1015 tev and power index @xmath47 before break close to that , measured in atic-2 experiment . nevertheless one can see , that absolute proton flux for qgsjet 01 is hardly compatible with any data of direct experiments , and the break for all these three models is more moderate , compared to what can be inferred from atic-2 data , which though become less definite right in the break region . on the other hand sibyll 2.1 allows to reproduce shape of the muon spectrum with single power law proton spectrum , which is in reasonable agreement with both emulsion chamber and atic-2 data within experimental errors . further improvement of our understanding of the situation , which is of primary astrophysical interest , can be achieved via experimental study as of muon cr component characteristics with water and ice neutrino telescopes and so of inclusive spectra @xmath61 in fragmentation region . reduction of uncertainties in the latter component with the use of the data on primary spectra , hadron and muon components , does not look possible , because of 1 ) poor correlation between muon and hadron production mechanisms , 2 ) ambiguity of existing cr experimental data and 3 ) possibility to realize self - consistent description of the data on muons and hadrons with the models , having remarkably differing inclusive spectra and underlying philosophies . authors greatly acknowledge conex team and personally tanguy pierog for their kind permission to use conex cross - section tables and for technical support . we are grateful to anonymous referee for constructive remarks , which helped us to improve the manuscript . + this work was supported in part by rfbr grant 07 - 02 - 01154-a .
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it is shown , that primary proton spectrum , reconstructed from sea - level and underground data on muon spectrum with the use of qgsjet 01 , qgsjet ii , nexus 3.97 and sibyll 2.1 interaction models , demonstrates not only model - dependent intensity , but also model - dependent form . for correct reproduction of muon spectrum shape primary proton flux should have non - constant power index for all
considered models , except sibyll 2.1 , with break at energies around 1015 tev and value of exponent before break close to that obtained in atic-2 experiment . to validate presence of this break understanding of inclusive spectra behavior in fragmentation region in p - air collisions should be improved , but we show , that it is impossible to do on the basis of the existing experimental data on primary nuclei , atmospheric muon and hadron fluxes .
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the _ howard s algorithm _ ( also called _ policy iteration algorithm _ ) is a classical method for solving a discrete hamilton - jacobi equation . this technique , developed by bellman and howard @xcite , is of large use in applications , thanks to its good proprieties of efficiency and simplicity . it was clear from the beginning that in presence of a space of controls with infinite elements , the convergence of the algorithm is comparable to newton s method . this was shown under progressively more general assumptions @xcite until to @xcite , where using the concept of _ slant differentiability _ introduced in @xcite , the technique can be shown to be of semi - smooth newton s type , with all the good qualities in term of superlinear convergence and , in some cases of interest , even quadratic convergence . in this paper , we propose a parallel version of the policy iteration algorithm , discussing the advantages and the weak points of such proposal . in order to build such parallel algorithm , we will use a theoretical construction inspired by some recent results on domain decomposition ( for example @xcite ) . anyway , for our purposes , thanks to a greater regularity of the hamiltonian , the decomposition can be studied just using standard techniques . we will focus instead on convergence of the numerical iteration , discussing some sufficient conditions , the number of iteration necessary , the speed . parallel computing applied to hamilton jacobi equations is a subject of actual interest because of the strict limitation of classical techniques in real problems , where the memory storage restrictions and limits in the cpu speed , cause easily the infeasibility of the computation , even in cases relatively easy . with the purpose to build a parallel solver , the main problem to deal with is to manage the information passing through the threads . our analysis is not the first contribution on the topic , but it is an original study of the specific possibilities offered by the policy algorithm . in particular some non trivial questions are : is convergence always guaranteed ? in finite time ? with which rate ? which is the gain respect to ( the already efficient ) classical howard s algorithm ? in literature , at our knowledge , the first parallel algorithm proposed was by sun in 1993 @xcite on the numerical solution of the bellman equation related to an exit time problem for a diffusion process ( i.e. for second order elliptic problems ) ; an immediately successive work is @xcite by camilli , falcone , lanucara and seghini , here an operator of the semilagrangian kind is proposed and studied on the interfaces of splitting . more recently , the issue was discussed also by zhou and zhan @xcite where , passing to a quasi variational inequality formulation equivalent , there was possible a domain decomposition . our intention is to show a different way to approach the topic . decomposing the problem directly in its differential form , effectively , it is possible to give an easy and consistent interpretation to the condition to impose on the boundaries of the sub - domains . thereafter , passing to a discrete version of such decomposed problem it becomes relatively easy to show the convergence of the technique to the correct solution , avoiding the technical problems , elsewhere observed , about the manner to exchange information between the sub - domains . in our technique , as explained later , we will substitute it with the resolution of an auxiliary problem living in the interface of connection in the domain decomposition . in this way , data will be passed implicitly through the sub - problems . the paper is structured as follows : in section 2 we recall the classic howard s algorithm and the relation with the differential problem , focusing on the case of its control theory interpretation . in section 3 , after discussing briefly the strategy of decomposition , we present the algorithm , and we study the convergence . section 4 is dedicated to a presentation of the performances and to show the advantages with respect the non parallel version . we will end presenting some possible extensions of the technique to some problems of interest : reachability problems with obstacle avoidance , max - min problems . the problem considered is the following . let be @xmath0 bounded open domain of @xmath1 ( @xmath2 ) ; the steady , first order , _ hamilton - jacobi equation _ ( hj ) is : @xmath3 where , following its _ optimal control interpretation _ , @xmath4 is the _ discount factor _ , @xmath5 is the _ exit cost _ , and the _ hamiltonian _ @xmath6 is defined by : @xmath7 with @xmath8 ( _ dynamics _ ) and @xmath9 ( _ running cost _ ) . the choice of such hamiltonian is not restrictive but useful to simplify the presentation . as extension of the techniques we are going to present , it will be shown , in the dedicate section , as the same results can be obtained in presence of different kind of hamiltonians , as in obstacle problems or in differential games . under classical assumptions on the data ( for our purposes we can suppose @xmath10 and @xmath11 continuous , @xmath12 and @xmath13 lipschitz continuous for all @xmath14 and verified the _ soner s condition _ @xcite ) , it is known ( see also @xcite , @xcite ) that the equation admits a unique continuous solution @xmath15 in the _ viscosity solutions _ sense . the solution @xmath16 is the value function to the infinite horizon problem with exit cost , where @xmath17 is the _ first time of exit _ form @xmath0 : @xmath18 numerical schemes for approximation of such problem have been proposed from the early steps of the theory , let us mention the classical finite differences schemes @xcite , semilagrangian @xcite , discontinuous galerkin @xcite and many others . in this paper we will focus on a _ monotone _ , _ consistent _ and _ stable _ scheme ( class including the first two mentioned above ) , which will provide us the discrete problem where to apply the howard s algorithm . considered a discrete grid @xmath19 with @xmath20 points @xmath21 , @xmath22 on the domain @xmath23 , the finite @xmath20-dimensional approximation of @xmath16 , @xmath24 , will be the solution of the following discrete equation ( @xmath25 ) @xmath26 where @xmath27 , ( maximal diameter of the family of simplices @xmath28 built on @xmath19 ) is the discretization step , and related to a subset of the @xmath29 , there are included the dirichlet conditions following the obvious pattern @xmath30 we will assume on @xmath31 , some hypotheses sufficient to ensure the convergence of the discretization * _ monotony . _ for every choice of two vectors @xmath32 such that , @xmath33 ( component - wise ) then @xmath34 for all @xmath35 . * _ stability . _ if the data of the problem are finite , for every vector @xmath24 , there exists a @xmath36 such that @xmath24 , solution of , is bounded by @xmath37 i.e. @xmath38 independently from @xmath39 . * _ consistency . _ this hypothesis , not necessary in the analysis of the convergence of the scheme , is essential to guarantee that the numerical solution obtained approximates the continuous solution . it is assumed that @xmath40 for every @xmath41 , @xmath42 , with @xmath43 , @xmath44 , and @xmath45 . under these assumptions it has been discussed and proved @xcite that @xmath24 , solution of , converges to @xmath16 , viscosity solution of for @xmath46 . the special form of the hamiltonian @xmath47 gives us a correspondent special structure of the scheme @xmath31 , in particular , with a rearrangement of the terms , the discrete problem can be written as a resolution of a nonlinear system in the following form : @xmath48 where @xmath49 is a @xmath50 matrix and @xmath51 is a @xmath20 vector . the name @xmath51 is chosen to underline ( it will be important in the following ) that such vector there are contained information about the dirichlet conditions imposed on the boundaries . policy iteration algorithm _ ( or howard s algorithm ) consists in a two - steps iteration with an alternating improvement of the policy and the value function , as shown in table [ ha ] . it is by now known @xcite that under a monotonicity assumption on the matrices @xmath52 , ( we recall that a matrix is monotone if ans only if it is invertible and every element of its inverse are non negative ) , automatically derived from ( h1 * ) ( as shown below ) , the above algorithm is a non smooth newton method that converges superlinearly to the discrete solution of problem . the convergence of the algorithm is also discussed in the earlier work @xcite where the results are given in a more regular framework . additionally , if @xmath53 has a finite number of elements , and this is the standard case of a discretized space of the controls , then the algorithm converges in a finite number of iterations . let us state , for a fixed vector @xmath54 the subspace of controls @xmath55 [ p:1 ] let us assume the matrix @xmath52 is invertible . if ( h1 * ) holds true , then @xmath52 is monotone and not null for every @xmath56 with @xmath54 . for a positive vector @xmath24 , consider a vector @xmath57 such that @xmath58 componentwise , then for h1 * @xmath59 where @xmath60 , therefore @xmath61 suppose now that the @xmath62 column of @xmath63 has a negative entry : choosing @xmath64 ( @xmath65 @xmath62 column of the identity matrix ) multiplying the previous relation for @xmath63 we have a contradiction . then @xmath66 is monotone . ( 1,0)380 + howard s algorithm ( ha ) ( 1,0)380 + inputs : @xmath67 , @xmath68 . ( implicitly , the values of @xmath24 at the boundary points ) + initialize @xmath69 and @xmath70 + iterate @xmath71 : * find @xmath72 solution of @xmath73 . + if @xmath74 and @xmath75 , then stop . otherwise go to ( ii ) . . + set @xmath77 and go to ( i ) outputs : @xmath78 . ( 1,0)380 + it is useful to underline the conceptual distinction between the convergence of the algorithm and the convergence of the numerical approximation to the continuous function @xmath16 as discussed previously . in general , the howard s algorithm is an acceleration technique for the calculus of the approximate solution , the error with the analytic solution will be depending on the discretization scheme used . to conclude this introductory section let us make two monodimensional basic examples . [ ex1 ] an example for the matrix @xmath52 and the vector @xmath79 is the easy case of an upwind explicit euler scheme in dimension one @xmath80 where @xmath81 is a uniform discrete grid consisting in @xmath20 knots of distance @xmath39 . moreover , @xmath82 and @xmath83 . in this case the system is @xmath84 } { h\lambda } & -\frac{f^+_1}{h\lambda } & 0 & \cdots & 0 \\ \frac{f^-_2}{h\lambda } & 1+\frac{\left[f^+_2-f^-_2\right]}{h\lambda } & -\frac{f^+_2}{h\lambda } & \cdots & 0\\ 0 & \ddots & \ddots & \ddots & 0\\ 0 & \cdots & \cdots & \frac{f^-_n}{h\lambda } & 1+\frac{\left[f^+_n - f^-_n\right]}{h\lambda } \end{array } \right),\ ] ] and @xmath85 it is straightforward that the solution of howard s algorithm , verifying @xmath86 , is the solution of . [ ex2 ] if we consider the standard 1d semilagrangian scheme , the matrix @xmath52 and the vector @xmath79 are @xmath87 and @xmath88 where @xmath89 and the coefficients @xmath90 are the weights of a chosen interpolation @xmath91(x_i+h f(x_i,\alpha_j))=\sum_{i=0}^{n+1}b_i(\alpha_j)v_i$ ] . despite the good performances of the policy algorithm as a _ speeding up _ technique , in particular in presence of a convenient initialization ( as shown for example in @xcite ) an awkward limit appears naturally : the necessity to store data of very big size . just to give an idea of the dimensions of the data managed it is sufficient consider that for a 3d problem solved on a squared grid of side @xmath92 , for example , it would be necessary to manage a @xmath93 matrix , task which becomes soon infeasible , increasing @xmath92 . this give us an evident motivation to investigate the possibility to solve the problem in parallel , containing the complexity of the sub problems and the memory storage . the strict relation between various points of the domain displayed by equation , makes the problem to find a parallel version of the technique , not an easy task to accomplish . the main problem , in particular , will be about passing information between the threads , necessary without a prior knowledge of the characteristics of the problem . our idea is to combine the policy iteration algorithm with a domain decomposition principle for hj equations . using the theoretical framework of the resolution of partial differential equations on submanifolds , presented for example in @xcite , we consider a decomposition of @xmath0 on a collection of subdomains : @xmath94 where the interfaces @xmath95 , @xmath96 are some strata of dimension lower than @xmath97 defined as the intersection of two subdomains @xmath98 for @xmath99 . the notion of viscosity solution on the manifold , in this regular case , will be coherent with the definition elsewhere a upper semicontinuos function @xmath100 in @xmath101 is a subsolution on @xmath101 if for any @xmath102 , any @xmath103 sufficiently small and any maximum point @xmath104 of @xmath105 is verified @xmath106 where with @xmath107 we indicate the hamiltonian @xmath47 restricted on @xmath108 . + the definition of supersolution is made accordingly . it is useful to underline that , differently from multidomains problems ( like the already quoted @xcite ) there is no need to use a specific concept of solutions through the interfaces . thanks to the regularity of the hamiltonian , the simple definition of viscosity solution on an enlargement of @xmath101 ( called @xmath108 ) will be effective ; as described by the following result . [ t:1 ] let us consider a domain decomposition as stated in . the continuous function @xmath109 , verifying , for a @xmath103 , in the viscosity sense the system below @xmath110 is coincident with the viscosity solution @xmath111 of . it is necessary to prove the uniqueness of a continuous viscosity solution for . after that , just invoking the existence and uniqueness results for the solution @xmath16 ( solution of the original problem ) , and observing that it is also a continuous viscosity solution of the system , from coincidence on the boundary , we get thesis . to prove the uniqueness it is possible to use the classical argument of `` doubling of variables '' . we recall the main steps of the technique for the convenience of the reader . for two continuous viscosity solutions @xmath112 of using the auxiliary function @xmath113 which has a maximum point in @xmath114 , it is easy to see that @xmath115 now the limit @xmath116 is proved as usual deriving @xmath117 and using the properties of sub supersolution , ( for example , @xcite theo . ii.3.1 ) with the observation that no additional difficulty appears when a subsequence @xmath118 is definitely in @xmath101 because of the regularity of the hamiltonian through the same interface ; for the possibility to exchange the role between @xmath119 and @xmath120 ( both super and subsolutions ) we have uniqueness . in the following section we propose a parallel algorithm based on the numerical resolution of the decomposed system above . this technique consists of a two steps iteration : * use howard s algorithm to solve in parallel ( @xmath92 threads ) the nonlinear systems obtained after discretization of on the subdomains @xmath121 ( in this step the values of @xmath24 are fixed on the boundaries ) ; * update the values of @xmath24 on the interfaces of connection @xmath122 by using howard s algorithm on the nonlinear system obtained from the second equation of ( in this case the _ interior points _ of @xmath123 are constant ) . as it is shown later , this two - step iteration permits the transfer of information through the interfaces performed by the phase ( ii ) . this procedure , anyway , is not priceless , the number of the steps necessary for its resolution will be shown to be higher than the classic algorithm ; the advantage will be in the resolution of smaller problems and the possibility of a resolution in parallel . moreover , the coupling between phase ( i ) and ( ii ) produces a succession of results convergent in finite time , in the case of a finite space of controls . the good performances of the algorithm , benefits and weak points will be discussed in details in section [ s : test ] . to describe precisely the algorithm it is necessary to state the following . let us consider as before a uniform grid @xmath124 , the indices set @xmath125 , and a vector of all the controls on the knots @xmath126 . the domain @xmath0 is decomposed as @xmath127 , where , coherently with above @xmath128 ; this decomposition induces an similar structure in the indices set @xmath129 , where every point @xmath130 of index in @xmath131 is an `` _ _ interior point _ _ '' , in the sense that for every @xmath132 ( ball centred in @xmath130 of radius @xmath39 , defined as previously ) , @xmath133 , for every @xmath134 . the set @xmath135 is the set of all the `` _ _ border points _ _ '' , which means , for a @xmath136 we have that there exists at least two points @xmath21 , @xmath137 such that @xmath138 and @xmath139 with @xmath134 . we will build @xmath92 discrete subproblems on the subdomains @xmath121 using as described before a monotone , stable and consistent scheme . in this case a discretization of the hamiltonian provides , for every subdomain @xmath121 , related to points @xmath21 , @xmath140 , a matrix @xmath141 and a vector @xmath142 . we highlighted here , the dependance of @xmath143 from the border points which are , either , points where there are imposed the dirichlet conditions ( data of the problem ) or points on the interface @xmath101 which have to be estimed . assumed for simplicity that every @xmath131 has the same number of @xmath144 elements , called @xmath145 , we have @xmath146 , and @xmath147 , @xmath148 . in resolution over @xmath101 we will have a matrix @xmath149 and a relative vector @xmath150 , in the spaces , respectively , @xmath151 and @xmath152 . ( for the 1d case , e.g. , we can easily verify that @xmath153 ) . in this framework , the numerical problem after the discretization of equations is the following : find @xmath154 with @xmath155 for @xmath156 and @xmath157 , solution of the following system of nonlinear equations : @xmath158 the resolution of first and the second equation of will be called respectively _ parallel part _ and _ iterative part _ of the method . solving the parallel and the iterative part will be performed alternatively , as a double step solver . the iteration of the algorithm will generate a sequence @xmath159 solution of the two steps system @xmath160 where @xmath161 , @xmath162 are the matrices and vectors in @xmath163 , and @xmath164 , containing @xmath165 , @xmath166 and such to return as solution the argument of @xmath167 elsewhere . evidently , @xmath161 @xmath168 with @xmath169 are : equal to @xmath170 in the @xmath171 blocks , and equal to the rows @xmath172 of the identity matrix elsewhere , @xmath173 in the @xmath174 elements of the vector and @xmath175 elsewhere , ( we call these entries , in the following _ identical arguments _ ) ; the same , in the @xmath176 block , @xmath177 elements of the vector for @xmath178 . it is clear that , despite this formal presentation , made to simplify the notation in the following , each equation of , negletting the trivial relations , is a nonlinear system on the same dimension than . clearely , a solution of is the fixed point of . the convergence of the discrete problem above to the solution of equation , for a consistent , monotone and stable scheme was proved by souganidis in @xcite ) , other examples are @xcite . it is consequent then , the domain decomposition result stated before gives the theoretical justification to the transition . different issue will be to show the convergence of the method ; point discussed in the following . ( 1,0)380 + parallel howard s algorithm ( pha ) ( 1,0)380 + inputs : @xmath165 , @xmath179 for @xmath180 + initialize @xmath69 and @xmath181 . + iterate @xmath71 : 1 . _ ( parallel step ) _ for each @xmath182 + call ( ha ) with inputs @xmath183 and @xmath184 + get @xmath185 2 . _ ( sequential step ) _ + call ( ha ) with inputs @xmath186 and @xmath187 + get @xmath188 3 . compose the solution @xmath189 + if @xmath190 then _ exit _ , otherwise go to ( 1 ) . outputs : @xmath78 ( 1,0)380 + it is evident that such technique can be expressed as @xmath191 where , coherently with above @xmath192_j$ ] for @xmath140 . the hypotheses ( h1 * -h2 * ) will be naturally adapted to the new framework as below : * _ monotony . _ for every choice of two vectors @xmath32 such that , @xmath33 ( component - wise ) then @xmath193 for all @xmath194 , and @xmath180 . * _ stability . _ if the data of the problem are finite , for every vector @xmath24 , and every @xmath57 s.t . @xmath195 , there exists a @xmath36 such that @xmath24 , solution of @xmath196 with @xmath197 and @xmath198 , is bounded by @xmath37 independently from @xmath39 . this will be sufficient , thanks also to h3 , to ensure convergence of @xmath199 solution of @xmath200 to @xmath201 for @xmath43 . from the assumptions on the discretization scheme some specific properties of @xmath161 and @xmath168 can be derived [ pp ] let us assume @xmath202 . let state also * if @xmath203 then @xmath204 , for all @xmath180 , for all @xmath14 . then it holds true the following . 1 . if invertible , the matrices @xmath205 are monotone , not null for every @xmath198 , and for every @xmath206 with @xmath207 . if @xmath208 , we have that for all @xmath198 and for every @xmath209 , there exists a @xmath210 such that @xmath211 the same relation holds for @xmath212 . 3 . called @xmath213 the fixed point of , if we have @xmath214 ( resp . @xmath215 ) , then there exists a @xmath14 such that , for all @xmath216 , @xmath217 to prove @xmath218 let us just observing that the monotony of @xmath165 is sufficient end necessary for the monotony of @xmath161 , ( elsewhere @xmath161 is a diagonal block matrix with all the other blocks invertible ) , then the argument is the same of proposition [ p:1 ] , starting from two vectors @xmath219 with the only difference that we need assumption h4 to get @xmath220 or equivalently @xmath221 then the thesis . to prove 2 , it is sufficient to see @xmath222 , then for h2 the thesis . the proof of 3 is a direct consequence of monotony assumption h1 with the definition of @xmath213 as @xmath223 here we introduce a convergence result for the ( pha ) algorithm . [ t:1 ] assume that the function @xmath224 , with @xmath225 invertible , and @xmath226 are continuous on the variable @xmath227 for @xmath216 , @xmath53 is a compact set of @xmath1 , and @xmath228 hold . then there exists a unique @xmath213 in @xmath164 solution of . moreover , the sequence @xmath229 generated by the ( pha ) has the following properties : * every element of the sequence @xmath230 is bounded by a constant @xmath37 , i.e. @xmath231 . * if @xmath232 then @xmath233 for all @xmath71 , vice versa , if @xmath234 then @xmath235 . * @xmath236 when @xmath237 tends to @xmath238 . the existence of a solution comes directly from the monotonicity of the matrices @xmath239 , the existence of an inverse and then the existence of a solution of every system of . let us show that such solution is limited as limit of a sequence of vectors of bounded norm . observing that , @xmath240 without loss of generality we assume that @xmath241 . considering the problem @xmath242 we have for h2 that if @xmath243 is bounded then @xmath244 . adding that @xmath245 is chosen bounded , the thesis follows for induction . let us to pass now to prove the uniqueness : taken @xmath246 two solutions of , we define the vector @xmath247 equal to @xmath24 in the identical arguments of @xmath167 and equal to @xmath57 elsewhere , for a @xmath198 . we have that , for a control @xmath248 ( for proposition [ pp].3 ) , @xmath249 then @xmath250 and for monotonicity @xmath33 . exchanging the role of @xmath24 and @xmath57 , and for the arbitrary choice of @xmath251 ( in some arguments the relation above is trivial ) we get the thesis . \(i ) to prove that @xmath72 is an increasing sequence is sufficient to prove that taken @xmath252 solution of @xmath253 with ( the opposite case is analogue ) @xmath254 , for a choice of @xmath198 is such that @xmath255 . let us observe , for a choice of @xmath256 and using of prop . [ pp ] @xmath257 then @xmath258 then @xmath255 . + we need also to prove that @xmath259 : if it should not be true , then , with a similar argument than above @xmath260 then for h4 , @xmath261 which contradicts what stated previously . it is also possible to show that the method stops to the fixed point in a finite time . this is an excellent feature of the technique ; unfortunately , the estimate which is possible to guarantee is largely for excess and , although important from the theoretical point of view , not so effective to show the good qualities of the method . the performances will checked in the through some tests in the section [ s : test ] . if @xmath262 and convergence requests of theorem [ t:1 ] are verified , then @xmath263 converges to the solution in less than @xmath264 iterative steps . the proof is slightly similar to the classic howard s case ( cf . for example @xcite ) . let us consider the abstract formulation @xmath265 , where @xmath266 is determined by @xmath267 parameter in @xmath268 , and @xmath269 , where @xmath270 is determined by @xmath271 parameter in @xmath268 . then if we consider the iteration @xmath272 and we suppose ( theorem [ t:1 ] ) @xmath273 , @xmath274 ; than called @xmath275 the @xmath276 variables in @xmath268 associated to @xmath277 we know that there exist a @xmath144 and a @xmath278 where @xmath279 , such that @xmath280 , and again @xmath281 . afterwards @xmath277 is a fixed point of . + to restrict to our case is sufficient identify the process @xmath282 with the ( parallel ) resolution on the sub - domains and @xmath283 with the iteration on the interfaces between the sub domains . it is worth to notice that the above estimation is worse than the classical howard s case . in fact , the classical algorithm find the solution in @xmath264 , the @xmath263 will have the same number of iterative steps . this number has to be multiplied , called @xmath284 the maximum number of nodes in a sub - domain and @xmath285 the number of nodes belonging to the interface , for @xmath286 getting , at the end , a total number of simple steps equal to @xmath287 , much more than the classical case . in this analysis we do not consider anyway , the good point of the decomposition technique , the fact that any computational step is referred to a smaller and simpler problem , with the evident advantages in term of time elapsed in every thread and memory storage needed . the performances of the algorithm and its characteristics as speeding up technique will be tested in this section . we will use a standard academic example where , anyway , there are present all the main characteristics of our technique . [ [ d - problem ] ] 1d problem + + + + + + + + + + consider the monodimensional problem @xmath288 it is well known that this equation ( _ eikonal equation _ ) modelize the distance from the boundary of the domain , scaled by an exponential factor ( _ kruzkov transform _ , cf . @xcite ) . through a standard euler discretization is obtained the problem in the form . in table [ tt:2 ] is shown a comparison , in term of speed and efficacy , of our algorithm and the classic howard s one , in the case of a two thread resolution . it is possible appreciate as the parallel technique is not convenient in all the situations . this is due to the low number of parallel threads which are not sufficient to justify the construction . in the successive test , keeping fixed the parameter @xmath289 and tuning number of threads it is possible to notice how much influential is such variable in terms of efficacy and time necessary for the resolution . c|*2c|*4c & & + dx & time & it . & t. ( par . p. ) & it . p. ) & t. ( it . p. ) & total t. + * 0.1 & e-3 & 10 & 1e-4 & 4 & 1e-5 & 1e-3 + * 0.05 & 6e-3 & 20 & 8e-4 & 5 & e-5 & 3e-3 + * 0.025 & 0.09 & 40 & 7e-3 & 6 & 2e-5 & 0.04 + * 0.0125 & 0.32 & 80 & 0.048 & 8 & 1e-4 & 0.36 + * 0.00625 & 2.22 & 160 & 0.34 & 14 & 8e-4 & 3.26 + * * * * * c|*2c|*4c dx=0.0125 & & + threads & t. & it . & t. ( par . p. ) & it . ( par . ) & t. ( it . p. ) & total t. + * 2 & & & 0.48 & 4 & 1e-4 & 0.36 + * 4 & & & 8e-3 & 6 & 1e-4 & 0.086 + * 8 & 0.32 & 80 & 18e-4 & 7 & 6e-4 & 0.014 + * 16 & & & 7e-4 & 10 & 4e-4 & + * 32 & & & 2e-4 & 8 & 6e-3 & 0.011 + * * * * * in table [ tt:2 ] we compare the iterations and the time ( expressed in seconds as elsewhere in the paper ) necessary to reach the approximated solution , analysing in the various phases of the algorithm , time and iterations necessary to solve every sub - problem ( first two columns ) , time elapsed for the iterative part ( which passes the information through the threads , next column ) , finally the total time . it is highlighted the optimal choice of number of threads ( 16 thread ) ; it is evident as that number will change with the change of the discretization step @xmath289 . therefore it is useful to remark that an additional work will be necessary to tune the number of threads accordingly to the peculiarities of the problem ; otherwise the risk is to is to loose completely the gain obtained through parallel computing and to get worse performances even compared with the classical howard s algorithm . as in the rest of the paper all the codes are developed in mathworks matlaband performed on a processor 2,8 ghz intel core i7 ; in the tests the parallelization is simulated . c|*2c|*5c & & + dx & t. & it . & t. ( p.p . ) & it . & t. ( it.p . ) & it . ( it.p . ) & total t. + * 0.1 & 0.05 & 11 & 0.009 & 8 & 0.02 & 2 & 0.04 + * 0.05 & 2.41 & 21 & 0.05 & 13 & 0.03 & 2 & 0.14 + * 0.025 & 73.3 & 40 & 2.5 & 22 & 0.15 & 3 & 7.83 + * 0.0125 & @xmath290e5 & - & 76 & 40 & 1.293 & 5 & 383.3 + * * * * [ [ d - problem-1 ] ] 2d problem + + + + + + + + + + the next test is in a space of higher dimension . let us consider the approximation of the scaled distance function from the boundary of the square @xmath291 , solution of the eikonal equation @xmath292 where @xmath293 is the usual unit ball . for the discretization of the problem is used a standard euler discretization . similar tests than the 1d case are performed , confirming the good features of our technique and , as already shown , the necessity of an appropriate number of threads with respect to the complexity of the resolution . -norm ( left ) and distribution of the error @xmath294 , @xmath295 threads ( right ) . , title="fig:",height=170 ] -norm ( left ) and distribution of the error @xmath294 , @xmath295 threads ( right ) . , title="fig:",height=170 ] in table [ tt:3 ] performances of the classic howard s algorithm are compared with our technique . in this case the number of threads are fixed to 4 ; the parallel technique is evaluated in terms of : maximum time elapsed in one thread and max number of iterations necessary ( first and second columns ) , time and number of iterations of the iterative part ( third and fourth columns ) and total time . in both the cases the control set @xmath296 is substituted by a @xmath297points discrete version . it is evident , in the comparison , an improvement of the speed of the algorithm even larger than the simpler 1d case . this justifies , more than the 1d case , our proposal . c|*2c|*4c dx=0.025 & & + threads & t. & it . & t. ( par . p. ) & it . ( par . ) & t. ( it . p. ) & total t. + * 4 & & & 2.5 & 22 & 0.15 & 7.83 + * 9 & & & 0.9 & 18 & 0.5 & 5.08 + * 16 & 73.3 & 40 & 0.05 & 13 & 1.6 & + * 25 & & & 0.03 & 12 & 2.4 & 2.52 + * 36 & & & 0.016 & 11 & 6.04 & 6.11 + * * * * * in the table [ tt:4 ] are compared the performances for various choices of the number of threads , for a fixed @xmath298 . as in the 1d case is possible to see how an optimal choice of the number of threads can drastically strike down the time of convergence . in figure [ f : in ] is possible to see the distribution of the error . as is predictable , the highest concentration will correspond to the non - smooth points of the solution . it is possible to notice also how our technique apparently does not introduce any additional error in correspondence of the interfaces connecting the sub - domains . this is reasonable , although not evident theoretically . in fact , it is possible to prove the convergence of the scheme to the solution of using classical techniques @xcite but the rate of convergence could be different in the various subproblems , because of the ( possibly different ) local features of the problem . as shown in the tests , an important point of weakness of our technique is represented by the iterative part , which can be smaller and therefore easier than the ones solved in the parallel part , but it is highly influential in terms of general performances of the algorithm . in particular the number of the iterations of the coupling iterative - parallel part is sensible to a good initialization of the `` internal boundary '' points . as is shown in figure [ f : in ] a right initialization , even obtained on a very coarse grid , affects consistently the overall performances . in this section , all the tests are made with a initialization of the solution on a @xmath299-points grid , with @xmath97 dimension of the domain space . the time necessary to compute the initial solution is always negligeable with respect to the global procedure . ( left ) @xmath300 ( right ) ) of the approximated solution obtained with a @xmath301 and an @xmath302threads pha . , title="fig:",height=170 ] ( left ) @xmath300 ( right ) ) of the approximated solution obtained with a @xmath301 and an @xmath302threads pha . , title="fig:",height=170 ] c|*2c|*5c & & + dx & time & it . & t. ( p. p. ) & it . & t. ( it . p. ) & it . p. ) & total t. + * 0.4 & 0.004 & 4 & 0.003 & 4 & 0.002 & 1 & 0.05 + * 0.2 & 0.22 & 6 & 0.026 & 6 & 0.016 & 2 & 0.052 + * 0.1 & 164.2 & 11 & 1.102 & 8 & 2.1 & 4 & 6.78 + * 0.05 & @xmath290e5 & - & 164 & 10 & 4.98 & 3 & 494 + * * * * [ [ d - problem-2 ] ] 3d problem + + + + + + + + + + analogue results are obtained also in the approximation of a 3d problem . of course the effects of the increasing number of control points produces a greater complexity and will limit , for a same number of processors available , the possibility of a fine discretization of the domain . + let us consider the domain @xmath303 ^ 3 $ ] and the equation , where @xmath304 , unitary ball in @xmath305 . in figure [ fig3d ] there are shown two level sets of the solution obtained . a comparison with the performances of the classic howard s algorithm are shown in table [ tt:4 ] . [ multiloop ] with the growth of the dimensionality of the problem a special care should be dedicated to the resolution of the iterative step . suppose to simplify the procedure considering a square domain ( in dimension @xmath306 an interval , a square , a cube .. ) and a successive splitting in equal regular subdomains . calling @xmath20 the number of total variables and @xmath307 the number of the splitting ( which generates a division in @xmath308 subdomains ) the number of the elements in every thread of the parallel part is @xmath309 , and the number of the variables in the iterative part @xmath310{n}}(n_s-1)d$ ] . clearly the optimal choice of the number of threads is such that the elements of the iterative part are balanced with the nodes in each subdomain , so it is straight forward to find the following optimal relation between number of splitting and total elements @xmath311 it is evident that for a very high number of elements , ( figure [ fig4 ] ) , it is useless to use a great and non optimal number of threads . this contradiction comes from the bottleneck effect of the resolution on the interfaces of communication between the subdomains , indeed the complexity of such subproblem will grow with the number of threads instead to decrease , reducing our possibilities of resolution . the problem can be overcome with an additional parallel decomposition of the iterative pass , permitting us to decompose each subproblem to a complexity acceptable . imagine to be able to solve ( for computational reasons , memory storage , etc . ) only problem of dimension `` white square '' ( we refer to figure [ fig4 ] , right ) and to want to solve a bigger problem ( `` square 1 '' ) with an arbitrary number of processors available . through our technique we will decompose the problem in a finite number of subproblems `` white square '' and a ( possibly bigger than the others ) problem `` square 2 '' . we will replicate our parallel procedure for the `` square 2 '' obtaining a collection of manageable problems and a `` square 3 '' . through a reiteration of this idea we arrive to a decomposition in subproblems of dimension desired . in this section there are shown some non trivial extensions to more general situations of the method . we will discuss , in particular , how to adapt the parallelization procedure to the case of a target problem , an obstacle problem and max - min problems , where the special structure of the hamiltonian requires some cautions and remarks . an important class of problems where is useful to extend the techniques discussed is the target problems where a trajectory is driven to arrive in a _ target set _ @xmath312 optimizing a cost functional . a easy way to modify our algorithm to this case is to change the construction procedure for @xmath49 and @xmath37 : @xmath313_i:=\left\{\begin{array}{ll } \left[b(\alpha)\right]_i , & \hbox { if } x_i\notin { { \mathcal t } } , \\ \left[\mathbb{i}\right]_i , & \hbox { otherwise;}\end{array } \right . \ ; c'(\alpha)_i:=\left\{\begin{array}{ll } c(\alpha)_i , & \hbox { if } x_i\notin { { \mathcal t } } , \\ 0 , & \hbox { otherwise;}\end{array } \right.\ ] ] this , with the classical further construction of _ ghost nodes _ outside the domain @xmath0 to avoid the exit of the trajectories from @xmath0 , will solve this case . a question arises naturally in this modification : are the convergence results still valid ? the answer is not completely trivial because , for example , a monotone matrix modified as above is not automatically monotone ( the easiest counterexample is the identical matrix flipped vertically : it is monotone because invertible and equal to its inverse , but changing any row as in we get a non invertible matrix ) . to prove the convergence it is sufficient to start from the numerical scheme associated to such modified algorithm . it is quite direct to show verified the hypotheses ( h1-h4 ) getting as consequence the described properties of the algorithm . a well known benchmark in the field is the so - called zermelo s navigation problem , the main feature , in this case , is that the dynamic is driven by a force of comparable power with respect to our control . the target to reach will be a ball of radius equal to @xmath314 centred in the origin , the control is in @xmath315 . the other data are : @xmath316 ^ 2,\quad \lambda = 1 , \quad l(x , y , a)=1.\ ] ] .,title="fig:",height=170 ] .,title="fig:",height=170 ] in table [ tt:6 ] a comparison with the number of threads chosen is made . now we are in presence of characteristics not aligned with the grid , but the performances of the method are poorly effected . convergence is archived with performances comparable to the already described case of the eikonal equation . c|*2c|*4c dx=0.025 & & + threads & t. & it . & t. ( par . p. ) & it . ( par . ) & t. ( it . p. ) & total t. + * 4 & & & 1.31 & 11 & 0.13 & 5.4 + * 9 & & & 0.7 & 9 & 0.7 & 4.2 + * 16 & 37.9 & 20 & 0.031 & 7 & 1.38 & + * 25 & & & 0.02 & 7 & 2.7 & 3.9 + * 36 & & & 0.01 & 8 & 5.19 & 5.28 + * * * * * dealing with an optimal problem with constraints using the bellman s approach , various techniques have been proposed . in this section we will consider an implicit representation of the constraints through a level - set function . let us to consider the general single obstacle problem @xmath317 where the hamiltonian @xmath47 is of the form discussed in section [ s:1 ] and the standard hypothesis about regularity of the terms involved are verified . the distinctive trait of this formulation is about the term @xmath318 , assumed regular , typically stated as the opposite of the signed distance from the boudary of a subset @xmath319 . the solution of this problem is coincident , where defined , with the solution of the same problem in the space @xmath320 , explaining the name of `` obstacle problem '' ( cf . @xcite ) . through an approximation of the problem in a finite dimensional one , in a similar way as already explained , is found the following variation of the howard s problem @xmath321 where the term @xmath57 is a sampling of the function @xmath322 on the knot of the discretization grid . it is direct to show that changing the definition of the matrix @xmath49 and @xmath323 , is possible to come back to the problem . adding an auxiliary control to the set @xmath324 and re - defying the matrices @xmath49 and @xmath323 as @xmath325_i:=\left\{\begin{array}{ll } \left[b(\alpha)\right]_i , & \hbox { if } b(\alpha)v - c_g(\alpha)\geq v - w \\ \left[\mathbb{i}\right]_i , & \hbox { otherwise;}\end{array } \right . \\ c_g'(\alpha)_i:=\left\{\begin{array}{ll } c_g(\alpha)_i , & \hbox { if } b(\alpha)v - c_g(\alpha)\geq v - w \\ w_i , & \hbox { otherwise;}\end{array } \right . \end{split}\ ] ] @xmath326 ( where the @xmath327 is the @xmath328row if @xmath329 is a matrix , and the @xmath328 element if @xmath329 is a vector , and @xmath330 is the identity matrix ) , the problem becomes @xmath331 which is in the form . even in this case the verification of hypotheses ( h1-h4 ) by the numerical scheme associated to the transformation is sufficiently easy . it is in some cases also possible the direct verification of conditions of convergence in the obstacle problem deriving them from the free of constraints case . for example if we have that the matrix @xmath52 is strictly dominant ( i.e. @xmath332 for every @xmath333 , and there exists a @xmath103 such that for every @xmath251 , @xmath334 ) , then the properties of the terms are automatically verified , ( i.e. since all @xmath225 are strictly dominant and thus monotone ) . a classical problem of interest is the optimization of trajectories modelled by @xmath335 which produces a collection of curves in the plane @xmath336 with a constraint in the curvature of the path . typically this is a simplified model of a car of constant velocity @xmath323 with a control in the steering wheel . + the value function of the exit problem from the domain @xmath337 , @xmath338 $ ] discretized uniformly in 8 points is presented in figure [ f : dub ] . it is natural to imagine the same problem with the presence of constraints . such problem can be handled with the technique described above producing the results shown in the same figure [ f : dub ] , where there are presented some optimal trajectories ( in the space @xmath336 ) for the exit from @xmath337 in presence of some constraints . from the picture it is possible to notice also the constraint about the minimal radius of curvature contained in the dynamics . the last , more complicated extension of the howard s problem is about max - min problems of the form ( 1,0)380 + pha ( maxmin case ) ( 1,0)380 initialize @xmath69 @xmath181 for all @xmath198 . + k:=1 ; 1 . iterate _ ( parallel step ) _ for every @xmath182 do : + @xmath339 * find @xmath340 solution of @xmath341 . + if @xmath342 and @xmath343 , then @xmath344 , and exit ( from inner loop ) . + otherwise go to ( 1.ii ) . * @xmath345 . + set @xmath346 and go to ( 1.i ) 2 . ( sequential step ) _ for @xmath347 * find @xmath348 solution of @xmath349 . + if @xmath350 and @xmath351 , then @xmath352 , and go to ( 3 ) . + otherwise go to ( 2ii ) . * @xmath353 . + set @xmath354 and go to ( 2i ) 3 . compose the solution @xmath355 + k:=k+1 ; + if @xmath356 then _ exit _ , otherwise go to ( 1 ) . ( 1,0)380 @xmath357 such a non linear equations arises in various contexts , for example in differential games and in robust control . the convergence of a parallel algorithm for the resolution of such problem is also discussed in @xcite . also in this case , a modified version of the policy iteration algorithm can be shown to be convergent ( cf . our aim in this subsection is to give some hints to build a parallel version of such procedure . let us introduce the function @xmath358 , for @xmath359 and @xmath360 defined by @xmath361 the problem , in analogy with the previous case , is equivalent to solve the following system of nonlinear equations @xmath362 the parallel version of the howard algorithm in the case of a maxmin problem is summarized in table [ mm ] . it is worth to notice that at every call of the function @xmath363 is necessary to solve a minimization problem over the set @xmath53 , this can be performed in an approximated way , using , for instance , the classical howard s algorithm . this gives to the dimension of this set a big relevance on the performances of our technique . for this reason , if the cardinality of @xmath53 ( in the case of finite sets ) is bigger than @xmath364 , it is worth to pass to the alternative problem @xmath365 ( here there are used the isaacs conditions ) before the resolution , inverting in this way , the role of @xmath53 and @xmath364 in the resolution . one of the most known example of max - min problem is the pursuit evasion game ; where two agents have the opposite goal to reduce / postpone the time of capture . the simplest situation is related to a dynamic @xmath366 where controls are taken in the unit ball @xmath367 and capture happens when the trajectory is driven to touch the small ball @xmath368 , ( @xmath369 , in this case ) . the passage to a target problem is managed as described previously . , title="fig:",height=170 ] , title="fig:",height=170 ] in figure [ f : pe ] the approximated value function of that problem is shown . the main difficulty in the use of the howard s algorithm , i.e. the resolution of big linear systems can be overcome using parallel computing . this is important despite the fact that we must accept an important drawback : the double loop procedure ( or multi - loop procedure as sketched in remark [ multiloop ] ) does not permit to archive a superlinear convergence , as in the classical case ; we suspect ( as in figure [ f : in ] ) that such rate is preserved looking to the ( external ) iterative step , where we have to consider , anyway , that in every step of the algorithm a resolution of a reduced problem is needed . another point influential in the technique is the manner chosen to solve every linear problem which appears in the algorithm . in this paper , being not in our intentions to show a comparison with other competitor methods rather studying the properties of the algorithm in relation of the classical case , we preferred the simplicity , using a routine based on the exact inversion of the matrix . using of an iterative solver , with the due caution about the error introduced , better performances are expected ( cf . @xcite ) . through the paper we showed as some basic properties of the schemes used to discretized the problem bear to sufficient conditions for the convergence of the algorithm proposed , this choice was made to try to keep our analysis as general as possible . a special treatment about the possibility of a domain decomposition in presence of non monotone schemes is possible , although not investigated here . this work was supported by the european union under the 7th framework programme fp7-people-2010-itn sadco , sensitivity analysis for deterministic controller design . + the author thanks hasnaa zidani of the uma laboratory of ensta for the discussions and the support in developing the subject . 00 , _ an efficient policy iteration algorithm for dynamic programming equations _ , pamm 13 n.1 ( 2013 ) 467468 . , optimal control and viscosity solution of hamilton - jacobi - bellman equations . birkhauser , boston heidelberg , 1997 . , _ a bellman approach for two - domains optimal control problems in @xmath164 _ , esaim contr va . , 19 n. 3 ( 2013 ) 710739 . , _ a bellman approach for regional optimal control problems in @xmath370 _ , siam j. cont . , 52 no . 3 ( 2014 ) 17121744 . , stochastic and differential games : theory and numerical methods , birkhuser , boston , 1999 . , _ flow invariance on stratified domains _ , set - valued var . , 21 ( 2013 ) 377403 . , dynamic programming , princeton university press , princeton , nj , 1957 . , _ some convergence results for howard s algorithm _ , siam j. numer . , 47 n. 4 ( 2009 ) 30013026 . , _ a domain decomposition method for bellman equations _ , cont . , 180 ( 1994 ) 477483 . , _ systems of convex hamilton - jacobi equations with implicit obstacles and the obstacle problem _ , comm . , 8 ( 2009 ) 12911302 . , _ a discontinuous galerkin finite element method for directly solving the hamilton - jacobi equations _ , j. comput . phys . , 223 n. 1 ( 2007 ) 398415 . , two approximations of solutions of hamilton - jacobi equations , math . comp . , 43 n. 167 ( 1984 ) 119 . , partial differential equations : graduate studies in mathematics . american mathematical society 2 , 1998 . , _ semi - lagrangian approximation schemes for linear and hamilton - jacobi equations _ , applied mathematics series , siam , 2013 . , _ advances in parallel algorithms for the isaacs equation _ , in advances in dynamic games . birkhuser boston , 2005 . 515 - 544 . , dynamic programming and markov processes , the mit press , cambridge , ma , 1960 . , _ on the convergence of policy iteration in stationary dynamic programming _ , math . res . , 4 no.1 ( 1979 ) 60 - 69 . , _ convergence analysis of some algorithms for solving nonsmooth equations _ , res . , 18 ( 1993 ) 227244 . , _ a nonsmooth version of newton s method _ , math . , 58 ( 1993 ) 353367 . , _ hamilton - jacobi - bellman equations on multi - domains _ , in : _ control and optimization with pde constraints _ , birkhauser basel , 164 ( 2013 ) 93116 . , _ convergence properties of policy iteration _ , siam j. contr . opt . , 42 n. 6 ( 2004 ) 2094 - 2115 . , _ optimal control problems with state - space constraints _ , siam j. contr . opt . , 24 ( 1986 ) 552562 . , _ approximation schemes for viscosity solutions of hamilton - jacobi equations _ , j. differ . equations 59 n. 1 ( 1985 ) 143 . , _ domain decomposition algorithms for solving hamilton jacobi - bellman equations _ , num . analysis opt . , 14 ( 1993 ) 145166 . , _ on the convergence of policy iteration in stationary dynamic programming _ , math . res . , 4 n. 1 ( 1979 ) 6069 . , _ a new domain decomposition method for an hjb equation _ , j. comput . appl . math . , 159 n. 1 ( 2003 ) 195204 .
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the classic howard s algorithm , a technique of resolution for discrete hamilton - jacobi equations , is of large use in applications for its high efficiency and good performances .
a special beneficial characteristic of the method is the superlinear convergence which , in presence of a finite number of controls , is reached in finite time .
performances of the method can be significantly improved by using parallel computing ; how to build a parallel version of method is not a trivial point , the difficulties come from the strict relation between various values of the solution , even related to distant points of the domain . in this contribution
we propose a parallel version of the howard s algorithm driven by an idea of domain decomposition .
this permits to derive some important properties and to prove the convergence under quite standard assumptions .
the good features of the algorithm will be shown through some tests and examples .
* keywords : * howard s algorithm ( policy iterations ) , parallel computing , domain decomposition + * 2000 msc : * 49m15 , 65y05 , 65n55
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accurate thermodynamic measurements are essential to understand fundamental properties of materials in various fields of physics . in condensed matter , the measurement of specific heat is a central characterization applicable to all kind of materials . low - temperature calorimetry is particularly suited for the investigation of superconductors and other novel systems with electronic phase transitions.@xcite such measurements require high resolution , since the electronic contribution to the heat capacity is only a minor part of the total heat capacity , except at the very lowest temperatures . the high resolution can be achieved through differential calorimeter designs and various temperature - modulated techniques.@xcite good accuracy is also often needed . the temperature dependence of the specific heat may , for instance , reveal central aspects of the nature of the electronic system , including energy gap structure , anisotropy , and possible signatures of quantum phase transitions . measurements are , furthermore , often performed in magnetic fields of various direction , using single crystals of highest available quality . this requires both a small calorimeter and small samples . all this together puts high demands on the calorimeter setup , typically excluding commercially available calorimeters . to meet this demand , calorimetry development is going in the direction of nanocalorimetry . nanocalorimetry is a rapidly growing area of research , driven by several fields of physics . nanocalorimeters include absorption sensors,@xcite devices to study transition enthalpies,@xcite fast scanning calorimeters to study microscopic nanostructure ensembles and thin films@xcite and kinetics and glass transitions of polymers@xcite , and combinatorial calorimeters.@xcite there are also several low - temperature microcalorimeters for @xmath12 samples@xcite and nanocalorimeters for heat capacity measurements of thin films@xcite and very small ( @xmath1 ) samples at high@xcite , intermediate,@xcite and down to low temperatures.@xcite going down in sample size favors the use of temperature - modulated techniques with corresponding high resolution , but makes it harder to obtain good absolute accuracy . this is partly due to an increasing relative contribution of the device addenda , but also due to non - adiabatic conditions and practical design issues , such as thermometry , system complexity , and thermal links considerations . devices for general use that combine high - resolution measurements of small samples with good absolute accuracy over an extended temperature range are hard to find . the membrane - based nanocalorimeter presented here is intended to fill this gap . our nanocalorimeter is developed for measurements of the temperature- and field dependence of the absolute specific heat of samples with a typical mass around @xmath13@xmath14 g , and for angular - dependent studies in magnetic fields . the device is built onto a pair of silicon nitride membranes using thin film techniques that provide low background heat capacity , less than @xmath3 at @xmath4 , decreasing to @xmath5 at @xmath6 , and a low thermal conductance , going from about @xmath15 at @xmath4 to @xmath16 at @xmath6 . the thermal relaxation time of the device is long enough ( ms to s range ) to enable ac steady - state and relaxation methods to be used concurrently . the high resolution of the ac steady - state method allows small changes in the heat capacity , such as contributions from the electronic specific heat , to be accurately determined , and investigations of phase transitions and phase diagrams to be performed . absolute accuracy is obtained by a combination of low background addenda , a stacked calorimeter design , and extensive measurement electronics operated with self - regulation and frequency feedback . the compact format enables the calorimeter to be placed on sample holders for rotation in magnetic field . the calorimeter could even be used for studies of dynamic ( frequency dependent ) heat capacity.@xcite the device is thus a versatile tool for general thermodynamic studies of small samples . ( a ) cross - sectional schematic of the calorimeter on top of a copper base with vacuum channels . ( b ) top view layout of the calorimeter with the two membrane - based calorimeter cells surrounded by 20 bonding pads that connect the calorimeter to the measurement electronics . ( c ) schematic of one of the @xmath17 membrane cells , composed by ac heater , thin film geau thermometer , and offset heater . the sample is placed on the @xmath18 central thermometer area . on top of the stack , a thermalization layer made of au is deposited if smaller samples are to be used , to obtain good internal thermalization and a uniform temperature distribution over the whole thermometer . ( d ) illustration of the active layers . all active layers are electrically insulated from each other and the sample by sio@xmath19/alo@xmath20 layers ( not shown ) . ] the nanocalorimeter is built on top of two custom - designed , pre - fabricated silicon nitride membranes , @xmath21 in size and @xmath22 thick ( spi supplies , west chester , usa ) . the membranes are suspended by a si frame @xmath23 , which is attached by means of stycast to a copper base as illustrated in fig . [ fig1](a ) . figure [ fig1](b ) shows a top view of the calorimeter layout . each cell , shown in fig . [ fig1](c ) , is composed of a stack of ac heater , thermometer , and offset heater in the central area of the membrane . between each of these active layers there are electrical insulation layers . all layers are fabricated using photolithography , deposition ( e - beam evaporation or sputtering ) , and double - layer resist lift - off . the active layers are illustrated separately in fig . [ fig1](d ) . the ac heater , shown in fig . [ fig1](d ) , is a meander - shaped resistor made of e - beam evaporated titanium , @xmath24 wide and @xmath25 thick , which covers the central sample area . it is used to oscillate the sample temperature with a well - defined ac power . it has a four - point probe geometry to allow accurate determination of the power at the sample without contributions from lead and contact resistances . ti becomes superconducting below about @xmath26 , but since the superconductivity can be suppressed by relatively modest fields the heaters may still be used at even lower temperatures . if needed , the film thickness could also be decreased to suppress @xmath27 . the active part of the thermometer , shown in fig . [ fig1](d ) , is a @xmath28 square made of @xmath29 thick , sputtered @xmath7 alloy . it senses the sample area using a four - point probe configuration and is in direct thermal contact with the sample , thus probing the actual sample temperature . consequently , heat dissipation in the thermometer does not pose a problem after initial calibration so that the thermometer can be operated at rather high powers to achieve a high sensitivity . furthermore , there is no need to rely on measurements of the frame / base temperature , unlike when thermocouples are used.@xcite this eliminates hysteretic effects when sweeping temperature and results in a high reproducibility . the sensor layer is fabricated using rf magnetron sputtering from a 2-inch target made of cast @xmath7 with nominally 17 at% au.@xcite the chip is annealed at @xmath30 on a hotplate for at least 1 hour after deposition , resulting in a room temperature resistivity @xmath31 and dimensionless sensitivity @xmath32 between @xmath4 and @xmath33 , increasing to about @xmath34 at lower temperatures . the geau sensor layer is deposited on top of au leads as shown in fig . [ fig1](d ) . the leads are in turn connected to external leads in ti , as seen in fig . [ fig1](c ) . by combining metals with high ( au ) and low ( ti ) thermal conductivity a more well - defined isothermal area in the center of the membrane is obtained.@xcite the offset heater , shown in fig . [ fig1](d ) , is driven by a dc current and can locally increase the sample temperature up to at least @xmath35 above the base temperature . this heater is designed to give an isothermal interior area , but is not in direct contact with the active part of the thermometer . it is thus less suitable as an ac heater , since the thermal diffusion between thermometer and heater would have to be taken into account . titanium has good robustness so the deposited layer can be quite thin . its addition to the background heat capacity is rather insignificant , but to simplify the fabrication process the layer is sometimes skipped . the leads that connect the active layers to the external bonding pads are also in ti . because of the relatively low thermal conductance of ti compared to many other metals , the use of ti for the leads ensures a long relaxation time dominated by the membrane itself . outside the membrane area , a thick layer of au is deposited onto the leads to minimize lead resistance effects and to provide a suitable layer for bonding , see fig . [ fig1](a ) . as a last element of the calorimeter stack , a @xmath36 square may be deposited as a thermalization layer to distribute the temperature evenly over the sample area if small samples are to be studied . this layer may be deposited simultaneously with the outer au bonding pad layer . the electrical insulation layers , not shown in fig . [ fig1 ] , are made by thermally evaporated aluminum oxide in combination with sputtered sio@xmath19 . the sio@xmath19 is deposited directly following the deposition of heaters or geau layer , while the alo@xmath20 layers are patterned as separate layers . there are in total three alo@xmath20 layers in the shape of rounded squares of different sizes : those between heaters and thermometer measure @xmath37 while the last insulation between thermometer and thermalization layer is @xmath38 . care was taken to design the layers so that edges of different layers do not coincide . this greatly reduces the risk of shorts between layers , especially for thin insulation layers . ( a ) calorimeter bonded onto cryostat plug - in . ( b ) calorimeter cell with a sample covering ac heater and thermometer . the membrane and electrical insulation layers appear transparent . this particular calorimeter has no offset heater . ( c ) microscope image of the central part of a calorimeter cell , illustrating the active layers and insulation ( but without geau layer for clarity ) . ] figure [ fig2](a ) shows a picture of the calorimeter bonded onto a cryostat sample holder plug - in . the calorimeter requires up to 20 wires , but can otherwise be fitted onto most cryogenic sample holders . the sample is placed on one of the membrane cells by means of a simple micro - manipulator , or , with some practice , by hand . the other cell may either carry a reference sample or be left empty . figure [ fig2](b ) shows a calorimeter cell with a typical sample ( the pb sample discussed in section [ sec : meas ] ) . the central parts of the calorimeter are shown in fig . [ fig2](c ) , where also the larger electrical insulation layers can be seen . in the standard measurement mode , known ac and dc currents flow through the heaters and thermometers . to practically enable the measurements of temperatures and oscillation amplitudes , a set of time and phase synchronized lock - in amplifiers based on a field - programmable gate array ( fpga ) is used.@xcite we implemented such an instrument using the pxi-7854r card by national instruments . the card has integrated @xmath39 adcs and @xmath40 dacs , providing eight integrated , simultaneous - sampling analog inputs and outputs for ac and dc biasing . for each input , the first and/or second harmonic amplitudes with corresponding phases are extracted , as well as the dc component . a central phase generator delivers the digital reference for all inputs and outputs . the resulting instrument thus allows tuning of output voltages and working frequency during the measurements , without any loss of correlation between inputs and outputs . each signal , before being read , passes through a low - noise , custom - built preamplifier stage . in total eleven preamplifiers are used , eight of which have variable gain ( between 1 and 5000 ) , controlled by the fpga lock - in , the others with fixed gain ( 100 or 1000 ) . while measuring , the eight variable - gain preamplifiers are automatically adjusted to maximize their performance . the circuit scheme used for the current bias and voltage read - out of the thermometers is illustrated in fig . an automated process is implemented that auto - adjusts the current through the sample - side thermometer in order to regulate the voltage across the thermometer . the applied voltage @xmath41 is thus varied to keep the dc component @xmath42 of the sample voltage @xmath43 equal to a variable setpoint value that is typically around @xmath44 and almost constant over the full temperature range . when the resistance of the thermometer increases with decreasing temperature , the applied current decreases and a suitable power is still delivered . the reading of @xmath42 is made synchronously by the lock - in so that frequency - dependent signals are separated from the dc component . changes to @xmath41 are synchronized with the measurements as well . thermometer bias and read - out circuit . the sample and reference geau thermometer resistances @xmath45 and @xmath46 are measured in four - probe configurations with the current provided by synchronous voltage sources @xmath41 and @xmath47 . the currents through the thermometers are measured by the voltages over @xmath48 series resistors . optional , adjustable voltage dividers are used to avoid any electrical discharge from destroying the membranes during initial connection . ] since the sample and reference thermometers are fabricated together at the same time , they are quite well balanced and display the same temperature dependences , within measurement uncertainty . there may , however , still be a small imbalance in the resistance ratio of the order of 1% between the two sides , arising from the lithographic tolerance . to compensate for this , a reference adjustment system is applied to vary the reference output @xmath47 in fig . [ fig3 ] to keep @xmath49 so that @xmath50 , rather than setting the currents or powers equal on the two sides . by balancing the thermometers in this way , the time - varying temperature difference between sample and reference is given by the @xmath51 signal through the simple relation @xmath52 here @xmath53 is the absolute temperature , which is obtained from the dc component of the thermometer resistance . the additional temperature difference caused by the power difference is usually insignificant , but can be compensated for by a corresponding power from the offset heater . the absence of dc offsets in the bridge differential @xmath51 makes it possible to apply a high amplification to @xmath51 without overloading , which results in highest possible resolution . the close proximity of sample and reference thermometers further eliminates common noise sources such as electromagnetic interference and temperature variations of the base frame . it should be noted that eq . ( [ eqdeltatemp ] ) has to be modified when the thermometer resistance becomes comparable to the series resistance of fig . [ fig3 ] , to include the effect of a varying bias current . in the ac steady - state method @xcite the sample temperature is made to oscillate with an amplitude typically in the range @xmath54@xmath55 . this modulation is created by an ac power , which for resistive heating is given by @xmath56 . it is thus generated by an ac current with rms amplitude @xmath57 and angular frequency @xmath58 flowing through the ac heater resistor @xmath59 . the temperature response of the cell is given by @xmath60 . @xmath61 is the base temperature , @xmath62 is the dc offset due to the time - averaged power supplied by the heater resistance , where @xmath63 is the thermal conductance between sample and thermal bath ( si frame ) , and @xmath64 is the oscillating term whose steady - state amplitude @xmath65 is directly related to the heat capacity of the sample : @xmath66^{-1/2}.\ ] ] here @xmath67 is the external relaxation time between sample and external thermal bath and @xmath68 is a rather complicated function of the internal relaxation time between sample and calorimetric cell , @xmath69 , where @xmath70 is the sample heat capacity and @xmath71 is the thermal conductance between the sample and the central cell platform . the value of @xmath71 strongly depends on the agent used to attach the sample to the nanocalorimetric cell . equation ( [ eqtac ] ) is impractical to use to obtain @xmath72 from @xmath65 , due to the complications of @xmath68 . we have , however , previously shown@xcite that for a system with good thermal connection between the active layers , but with possibly significant @xmath73 , the thermometer temperature oscillation @xmath74 and corresponding phase @xmath75 between power and temperature oscillation can be found as [ eqtphi ] t _ & = + & = provided that @xmath72 and @xmath76 are taken as c & = c _ + ( 1-g ) c _ + k & = k_+g k _ + g & = [ eq_cktotal ] here , @xmath77 is the empty cell contribution to the heat capacity and @xmath78 is the effective thermal link of the membrane.@xcite equation ( [ eqtphi ] ) can be wrapped around into the practical , functional relations [ eqck ] c & = + k & = which form the basis of evaluating the measurements . note that the phase is carrying information that is needed to achieve good absolute accuracy . the phase can be used to verify that the frequency is selected correctly , i.e. , that @xmath79 so that @xmath80 . if the frequency is so high that @xmath81 in eq . ( [ eqck ] ) , it is very likely that @xmath82 contributes significantly in eq . ( [ eq_cktotal ] ) . a known phase is also necessary to accurately extract @xmath63 from ac steady - state measurements . when using the ac steady - state technique in practical terms , we apply an ac current to the heater(s ) and measure the thermometer and series resistance voltages @xmath43 , @xmath83 , @xmath84 , @xmath85 , and @xmath51 , defined in fig . [ fig3 ] , at the second harmonic of the heater current frequency ( simultaneously with the synchronous dc mean ) . the amplitudes of the sample and reference temperature oscillations @xmath86 and @xmath87 , are related to the corresponding measured voltages over the thermometers and currents through the reference resistors by @xmath88 here both @xmath53 and sensitivity @xmath89 are obtained from the dc measurement of the thermometer resistance @xmath90 and previous calibration of @xmath91 . for the temperature differential the corresponding expression is @xmath92 where @xmath93 and @xmath94 is the series resistance for current measurement . all ac signals in eqs . ( [ tacfromuac ] ) and ( [ tdifffromudiff ] ) refer to amplitudes in the steady state . note that @xmath95 in general , since the sample and reference temperature oscillations may have different phases . in the simplest measurement case , the reference heater power is kept off . this method is used when the heat capacity of the sample is fairly large as compared to the heat capacity of the empty cell and no reference sample is placed on the reference side . the sample temperature oscillation is then measured through the differential signal @xmath96 while @xmath97 . since @xmath51 has no dc offset , the differential signal yields a significantly higher resolution than @xmath98 . to subtract @xmath77 from @xmath72 , sample and empty cell are measured in separate runs . the measurement of @xmath65 is thus made differentially , but the measurement of @xmath72 is not . if the sample is small , or if a similar - size reference sample is added to the reference side , a truly differential measurement mode can be employed . in this case , the same power is applied to both sample and reference sides ( @xmath99 ) . the heat capacities @xmath100 and @xmath101 of the individual sides are still given by eq . ( [ eqck ] ) , but the differential heat capacity @xmath102 can be obtained as well from @xmath103 : @xmath104 equation ( [ c_diff ] ) is valid under the assumption that @xmath76 is the same for both sample and reference sides . since @xmath63 is given mainly by the membrane itself , and other contributing layers are manufactured lithographically , this requirement is easy to fulfill in normal cases . if a large sample is studied , the sample itself may enhance @xmath76 through a small , but nonzero @xmath82 . the absolute accuracy of eq . ( [ c_diff ] ) may in this case be tested by a separate verification measurement with @xmath105 . if the differential heat capacity mode is used with a large sample but without reference , the signal from the reference side will dominate @xmath51 , and the benefit of a differential measurement would be lost . an alternative to reverting to a single - side measurement can in this case be to adjust the amplitude and output phase of the reference heater power to maintain @xmath106 . this mode of measurement is , however , still unexplored . we have previously shown@xcite that good absolute accuracy can be obtained provided that one has good control of the phase in eq . ( [ eqck ] ) . in fig . [ fig4 ] , the frequency dependence of @xmath86 and @xmath75 is shown for a typical sample . measured frequency dependence of temperature oscillation amplitude @xmath65 and phase @xmath75 , expressed as @xmath107 and @xmath108 , respectively , with @xmath109 and @xmath72 constant ( @xmath110 ) . at low frequency @xmath111 and measurements yield good absolute accuracy . at too low frequencies , however , the signal no longer comes from the heat capacity but from the thermal link , and the resolution decreases . note that the middle of the adiabatic plateau ( where @xmath112 is constant ) is not corresponding to the best measurement frequency if good accuracy is required . ] good accuracy is found at low frequencies , where @xmath111 . at higher frequencies , the effect of @xmath113 is no longer negligible and the accuracy is quickly deteriorating . for the resolution , the conditions are the opposite ; at high frequencies the resolution is good , but at low frequencies , @xmath114 , eq . ( [ eqtphi ] ) is reduced to @xmath115 and heat capacity is no longer probed . by fixing the phase @xmath75 to a constant value in the range where measurements yield both good absolute accuracy and good resolution , optimal conditions are found . we do this by continuously adjusting the frequency @xmath116 during the measurement to keep @xmath108 constant by means of an auto - tuning routine in the fpga lock - in . by having a control loop time equal to a single sample of the adc , the frequency is smoothly adjusted without introducing additional noise to the measurements . the frequency is thus varying even during a single cycle of the output . this is possible thanks to the synchronous sampling . it should be noted that the middle of the adiabatic plateau ( with @xmath117 ) is typically at too high frequencies for good absolute accuracy , as seen in fig . [ fig4 ] . the thermal relaxation method @xcite consists of applying a known power to the heater to raise the sample temperature an amount @xmath118 above the base temperature @xmath61 . after a stable sample temperature has been reached , the heater power is turned off and the temperature is allowed to relax back to @xmath61 . the time dependence of the relaxation is exponential and depends on the external time constant of the system : @xmath119 when using the thermal relaxation method , we apply a square wave with sufficiently low repetition rate to the sample heater , while keeping the reference heater off . the induced temperature response is directly given by eq . ( [ eqdeltatemp ] ) , from which @xmath118 and @xmath120 can be obtained . the thermal conductance @xmath63 between sample and base frame is then found as @xmath121 where @xmath122 is given by the directly measured step in heater power . the sample heat capacity including cell addenda is finally given by @xmath120 and @xmath63 , according to @xmath123 to practically perform relaxation measurements , a routine was developed that automatically acquires the voltage pulses , fits the exponential decay to data within 10% and 90% of the relaxation , and determines @xmath118 . in this way , the heat capacity and device thermal conductance can be measured as a function of temperature or magnetic field with immediate results displayed on the screen . ( a ) temperature dependence of the geau thermometer resistance and dimensionless sensitivity @xmath124 . ( b ) thermometer magnetoresistance expressed as apparent relative temperature change in a magnetic field of @xmath125 . the inset shows the corresponding field dependence at different temperatures . the magnetoresistance is following a parabolic field dependence except for temperatures below @xmath126 at high fields.,title="fig : " ] + ( a ) temperature dependence of the geau thermometer resistance and dimensionless sensitivity @xmath124 . ( b ) thermometer magnetoresistance expressed as apparent relative temperature change in a magnetic field of @xmath125 . the inset shows the corresponding field dependence at different temperatures . the magnetoresistance is following a parabolic field dependence except for temperatures below @xmath126 at high fields.,title="fig : " ] before any measurements , the geau thermometers on sample and reference sides need to be calibrated against a known thermometer . we use a cernox sensor on the sample holder . the calibration is done in the presence of a few mbar helium gas to increase the thermal link between geau thermometers and si base frame , which quickly reduces @xmath120 and temperature offsets due to self - heating . the typical temperature dependence of the geau thermometer resistance is shown in fig . [ fig5](a ) . once the calibration curve has been obtained , one can attain the temperature dependence of the sensitivity @xmath89 which is approximately constant from @xmath4 down to about @xmath127 , below which it increases slowly as shown in fig . [ fig5](a ) . the temperature dependence of the sample to reference thermometer resistance ratio @xmath128 is constant within experimental uncertainty over the entire temperature range . the same calibration curve can thus be used for both sides , except for a scaling pre - factor . thermometers deposited in the same sputtering cycle display very similar temperature dependences , while thermometers fabricated at different times require individual calibrations , although differences are small enough for a standard calibration curve to be used initially in most cases . the thermometer resistance is stable over time provided that the thermometer is not heated excessively after the heat treatment at @xmath129 . to obtain @xmath89 and @xmath53 from @xmath90 while avoiding numerical noise and other artifacts , we fit an analytical expression to the calibration data . the thermometer @xmath130 relation is well described by @xmath131 here @xmath132 , @xmath133 , and @xmath134 are constants . remaining small deviations are fitted by a high - degree polynomial in @xmath135 . an analytical expression of @xmath89 is then directly found from the fit parameters , and is given by @xmath136,\ ] ] up to the contribution from the polynomial correction . this procedure to obtain temperature and sensitivity from resistance avoids interpolations , which , even in log - log scale , tend to give rise to artificial kinks that may become significant in certain cases such as when studying specific heat differences . resistive thermometers require corrections for magnetic - field - induced changes at low temperature . the magnetoresistance of the geau sensor was studied by sweeping a magnetic field between @xmath137 and @xmath138 at several fixed temperatures from @xmath139 to @xmath140 . the relative change in apparent temperature was then calculated as @xmath141 , where @xmath89 is the zero field sensitivity . the curves , shown in the inset of fig . [ fig5](b ) , can be fairly well approximated by a parabolic field dependence @xmath142 above @xmath143 . the magnetoresistance is positive and its magnitude decreases quite quickly with increasing temperature . the field - induced error at @xmath138 , if no correction is made , is @xmath144 at @xmath145 and @xmath146 at @xmath140 , as shown in fig . [ fig5](b ) . the nanocalorimeter was characterized thoroughly from room temperature down to @xmath2 . figure [ fig6](a ) shows the heat capacity of the empty cell as obtained from eq . ( [ eqck ] ) . ( a ) empty cell heat capacity @xmath147 as a function of temperature . ( b ) device thermal conductance @xmath63 . the insets show the low - temperature behavior in log - log scale . note that @xmath147 and @xmath63 display a fairly similar temperature dependence , resulting in a rather constant time constant @xmath148 as a function of temperature for the empty device.,title="fig : " ] + ( a ) empty cell heat capacity @xmath147 as a function of temperature . ( b ) device thermal conductance @xmath63 . the insets show the low - temperature behavior in log - log scale . note that @xmath147 and @xmath63 display a fairly similar temperature dependence , resulting in a rather constant time constant @xmath148 as a function of temperature for the empty device.,title="fig : " ] this heat capacity will , under certain conditions , depend on the choice of woking frequency ( i.e. @xmath108 ) , since the amount of membrane that is temperature - modulated decreases with increasing frequency.@xcite the characteristic frequency of the membrane is , however , typically higher than the frequency of normal measurements , making it possible to treat @xmath147 as a reproducible background addenda for a given calorimeter and to assume that @xmath77 is frequency independent for large samples ( i.e. with @xmath149 ) . in differential mode , the addenda heat capacity is normally less than 5% of @xmath147 . a typical noise level is @xmath150 at @xmath127 . the noise level expressed as @xmath151 is fairly constant over the entire temperature range . with a typical measurement time of @xmath152 and a moderately large @xmath153 , it is possible to reach @xmath154 . the thermal link @xmath63 of the calorimeter cell is shown in fig . [ fig6](b ) . it can be obtained from eq . ( [ eqck ] ) provided that the measurements are made at low enough frequency , @xmath155 , or through relaxation measurements . while the cell heat capacity is dominated by the membrane that has a fairly high debye temperature , the thermal link is given by a combination of membrane and metallic leads . the characteristic time constant @xmath156 is nevertheless fairly temperature independent . figure [ fig7](a ) shows the range of operational power of the ac heater as a function of temperature . the power required for a certain ratio @xmath153 between temperature oscillation amplitude and absolute temperature is not depending on @xmath72 but only @xmath63 , provided that the phase @xmath75 is kept constant . an easy way to adjust the power to the proper level is thus to maintain a constant ratio @xmath157 while adjusting the frequency so that @xmath75 is constant . in this way , the temperature offset due to the ac heater power is always a constant fraction of @xmath53 as well . to maintain a similar power for the dc bias of the thermometer , the @xmath42 should decrease from about @xmath44 at @xmath35 to @xmath158 at @xmath6 . in practice , @xmath42 can be kept almost constant , so that the relative temperature offset due to thermometer self - heating is somewhat higher at the lowest temperatures and somewhat lower at the highest . ( a ) range of operational ac heater power as a function of temperature for measurement frequency adjusted so that @xmath159 . ( b ) typical range of working frequency @xmath160 as a function of temperature , with and without sample . note that the sample specific heat controls the frequency range over which good absolute accuracy is obtained . different samples thus require different frequency adjustment.,title="fig : " ] + ( a ) range of operational ac heater power as a function of temperature for measurement frequency adjusted so that @xmath159 . ( b ) typical range of working frequency @xmath160 as a function of temperature , with and without sample . note that the sample specific heat controls the frequency range over which good absolute accuracy is obtained . different samples thus require different frequency adjustment.,title="fig : " ] figure [ fig7](b ) shows the typical range of working frequency @xmath160 as a function of temperature , with and without sample . for the empty device , the time constant @xmath148 is rather temperature independent , leading to a fairly constant @xmath160 . with a sample , the frequency @xmath161 is lowered , but the frequency also becomes more strongly temperature dependent . this variation depends on the sample heat capacity . it is thus clear that only narrow temperature ranges can be studied with combined good accuracy and resolution if a constant frequency is used as in the case of traditional ac calorimetry . as an initial test of the calorimeter , we measured a small gold sample from @xmath162 down to about @xmath33 to compare the relaxation and ac steady - state techniques . the temperature dependence of the heat capacity , shown in fig . [ fig8 ] , was also compared with available data to study the absolute accuracy . measured heat capacity versus temperature of a small gold grain . the contribution of the membrane heat capacity has been subtracted . there is also some contribution from the apiezon - n grease used to attach the sample . this addenda has not been accounted for , but is expected to be of the order of @xmath163@xmath164 . the heat capacity measured by geballe and giauque@xcite on a sample more than @xmath165 times larger is scaled to agree with our data at @xmath166 . ] it is seen that the over - all agreement between the two measurement methods is fairly good . the ac steady - state curve , however , lies somewhat above the relaxation curve for temperatures below @xmath127 . the heat capacity measurement by geballe and giauque@xcite on roughly @xmath167 ( ! ) au was scaled to agree with our data at @xmath166 , where the relaxation and ac steady - state measurements coincide . the relaxation curve follows the temperature dependence of the literature with deviations within 5% over the full temperature range . from the scaling , the sample mass is estimated to @xmath168 , which lies within the uncertainty of the volumetric measurement of the sample size , initially estimated to be @xmath169 using a simple microscope . the real sample mass is likely in between the two numbers , since the data of fig . [ fig8 ] includes a small contribution from the apiezon - n grease that was used to attach the sample . to illustrate the low - temperature capabilities of the calorimeter , we studied the heat capacity of a @xmath11 , 99.999@xmath170 pure pb sample , cut from a single crystal , as a function of temperature and in magnetic fields . figure [ fig9](a ) shows the measured specific heat in the superconducting and normal states , plotted as @xmath171 vs @xmath53 . ( a ) temperature dependence of the low - temperature specific heat expressed as @xmath171 of pb in the superconducting and normal state ( obtained by a @xmath172 field ) . the inset shows the measured heat capacity at higher temperatures before and after subtraction of addenda , and the contributions from membrane and apiezon - n grease . ( b ) specific heat difference @xmath173 , with the data provided by neighbor _ _ etal.__@xcite for comparison . the inset shows the entropy difference @xmath174 obtained by integrating @xmath175.,title="fig : " ] + ( a ) temperature dependence of the low - temperature specific heat expressed as @xmath171 of pb in the superconducting and normal state ( obtained by a @xmath172 field ) . the inset shows the measured heat capacity at higher temperatures before and after subtraction of addenda , and the contributions from membrane and apiezon - n grease . ( b ) specific heat difference @xmath173 , with the data provided by neighbor _ _ etal.__@xcite for comparison . the inset shows the entropy difference @xmath174 obtained by integrating @xmath175.,title="fig : " ] the specific heat jump in zero field at the transition temperature @xmath176 is seen clearly . the normal state curve was obtained by applying a @xmath172 magnetic field to suppress the superconductivity . the inset shows the heat capacity at higher temperatures . the addenda heat capacity from membrane and grease were measured in separate runs ( i.e. not in true differential mode ) . apiezon - n is typically used as a thermal contact agent for low - temperature experiments , but undergoes a glass transition at above @xmath177 , which leads to a somewhat irreproducible high - temperature addenda,@xcite decreasing the absolute accuracy at high temperatures . the membrane cell dominates the addenda at the absolute lowest temperatures , but already at about @xmath126 the apiezon contribution becomes the main background . the membrane and grease addenda are subtracted in the main figure [ fig9 ] . after subtraction , the superconducting state measurements still display a @xmath178 residual gamma term , i.e. , a remaining linear - in-@xmath53 contribution to the specific heat . this could possibly be due to an incompletely accounted background addenda . the ratio @xmath179 at @xmath27 is , however , only @xmath180 of expected . we therefore believe that the unaccounted addenda is a non - superconducting part of the sample entering @xmath72 but not @xmath181 , possibly arising from an oxidized surface layer , or from vacancies and dislocations that were not annealed away before the measurements . figure [ fig9](b ) shows the temperature dependence of the specific heat difference @xmath173 . this difference is insensitive to background addenda , which makes it a good probe of accuracy and reproducibility . from the temperature dependence , fundamental properties such as the superconducting gap energy and coupling strength can be obtained.@xcite the measurements provide a sommerfeld parameter @xmath182 and @xmath183 in good agreement with literature.@xcite the temperature dependence of @xmath175 is also following the expected behavior , as seen by comparing the measurements with polynomial - fit data provided by neighbor _ _ etal.__@xcite while the resolution remains good at all temperatures , the accuracy decreases somewhat at @xmath184 , which can be seen in fig . [ fig9](b ) as some wiggles in @xmath175 . this can be attributed to the difficulties in obtaining an accurate calibration and corresponding sensitivity of the thermometer using @xmath185 power levels . one way to overcome this problem may be to calibrate the thermometer simultaneously with the measurements , with a self - consistency requirement on the thermal link . such a method has been successfully tested , but requires further investigation . integrating @xmath175 gives the entropy difference @xmath186 , shown in the inset of fig . [ fig9](b ) . the entropy - conservation requirement is fulfilled within a @xmath187 uncertainty of @xmath188 , obtained from the low - temperature slope of @xmath189 . the free energy difference @xmath190 is then obtained as @xmath191 , where @xmath192 is found by integrating @xmath181 from @xmath27 to @xmath53 . from @xmath190 , the thermodynamic critical field @xmath193 is calculated from the relation @xmath194 , where @xmath195 is the molar volume ( or sample volume , if @xmath190 is given in units of energy ) . figure [ fig10](a ) shows @xmath193 obtained in this way using the data in fig . [ fig9](b ) . ( a ) thermodynamic critical field @xmath193 . the small , closely spaced symbols correspond to @xmath193 as obtained from the @xmath196 data of fig . [ fig9](b ) . they display the expected@xcite small deviation from the two - fluid expression , which is shown for reference . also shown is the directly measured location of the superconducting transition in various magnetic fields ( big circles ) . the inset shows normalized heat capacity measured on increasing fields at different temperatures . ( b ) temperature dependence of the specific heat in small magnetic fields near @xmath27 . ( c ) transition in a @xmath197 magnetic field measured with different @xmath65 . while improving the energy resolution , too high @xmath65 results in a pronounced @xmath53 smearing . ] the temperature dependence of @xmath198 for pb is expected to display a small , positive deviation from the two - fluid expression @xmath199 $ ] . a weak - coupling bcs superconductor , on the other hand , would display a negative deviation.@xcite the difference between @xmath193 , as obtained from the measured @xmath196 , and the two - fluid expression is clearly seen in fig . [ fig10](a ) . indeed , the temperature dependence of the deviations themselves are within a few % of the deviations obtained by decker _ _ etal.__@xcite the most uncertain factor in going from heat capacity to specific heat in nanocalorimetry is the determination of sample mass . it can be done through a careful measurement of volume and density , or from a known reference point at some temperature , such as room temperature or @xmath27 . since a microscopic volume measurement would require sub-@xmath200 resolution , which is difficult for soft materials such as pb , we used the measurement of @xmath72 around @xmath27 by shiffman _ _ etal.__@xcite to obtain the scale in fig . [ fig9 ] . for type - i superconductors , it is however also possible to directly measure @xmath193 by studying the superconducting transition in magnetic field . such measurements at various temperatures and magnetic fields are shown in the inset of fig . [ fig10](a ) and in fig . [ fig10](b ) . the resulting @xmath193 and @xmath201 transitions , shown as big , green circles in the main panel of fig . [ fig10](a ) , agree well with @xmath193 as obtained from the measurement of @xmath181 . the sample volume can thus be found directly from a comparison of @xmath190 ( measured in units of energy ) and @xmath202 ( having units of energy per volume ) . the measurements in magnetic field were made with @xmath203 perpendicular to the plate - like sample . ( the sample is shown in fig . [ fig2 ] ) . this causes @xmath72 to become field dependent in the superconducting state , due to the large demagnetization factor that drives the sample into the intermediate state . the effect is clearly seen in the field - dependence curves of the inset of fig . [ fig10](a ) . at low fields the sample is in the meissner state and @xmath70 is constant , but at higher fields @xmath70 starts to increase . one could interpret this effect as a distributed latent heat when normal domains enter the superconductor . in fig . [ fig10](b ) it is seen that the specific heat in small magnetic fields is higher than the zero field specific heat near @xmath198 . while the sample temperature oscillates , a small fraction of the sample is undergoing the transition back and forth between the meissner and normal states , with accompanying latent heat @xmath204 . this causes the @xmath65 amplitude to decrease , making the latent heat appear as an excess specific heat @xmath205 . the excess specific heat thus relates to the fraction of the sample that undergoes the superconducting transition at each temperature . it is tempting to quantify the latent heat from such measurements . however , only a fraction of the total latent heat is found in this way . this is due to possible hysteretic effects of the first - order transition in combination with an incomplete analysis of the temperature oscillation , which will display higher harmonics when latent heat is involved . as a final illustration of the capability of the calorimeter , the transition in a @xmath197 magnetic field is shown in fig . [ fig10](c ) for different temperature oscillation amplitudes @xmath65 . the sharp latent heat peak , which is not present in zero field [ cf . [ fig10](b ) ] , is a good probe of combined high resolution of both temperature and specific heat . the peak is only seen if @xmath65 is small enough . by increasing @xmath65 the specific heat resolution will increase , but the transition is then quickly smeared out . the latent heat involved in this transition is of the order of a few @xmath206 . in summary , we have developed a membrane - based nanocalorimeter for specific heat measurements of small samples and thin films over an extended temperature range from above room temperature down to below @xmath6 . our device has sub - pj / k resolution at low temperature , corresponding to @xmath207 for thin films and @xmath208 heat exchanges , and is capable of probing @xmath1-sized samples with combined high resolution and good absolute accuracy , thus exceeding the typical capability of commercial calorimeters by almost four orders in sample size . the calorimeter features a differential design with a variable - frequency technique where the measurement conditions are automatically maintained at optimal conditions . the versatility of the calorimeter invites the exploration of several novel ac measurement procedures in addition to the ones described here , including power - compensation and multi - frequency modes . the ultimate capability of the calorimeter is thus still an open question . we thank s. latos and p. a. favuzzi for assistance with calorimetry development and r. nilsson for contributing to the fpga - based lock - in amplifier . initial development of the nanocalorimeter was performed at argonne national laboratory in collaboration with u. welp , w .- k . kwok , and g. w. crabtree . financial support from the swedish research council and technical support from the su - core facility in nanotechnology is gratefully acknowledged . a. rydh , _ calorimetry of sub - microgram grains _ , in encyclopedia of materials : science and technology , 2006 online update , k. h. j. buschow , m. c. flemings , r. w. cahn , p. veyssire , e. j. kramer and s. mahajan ( eds ) . elsevier ltd . , oxford , 2006 .
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a differential , membrane - based nanocalorimeter for general specific heat studies of very small samples , ranging from @xmath0 to sub-@xmath1 in mass , is described .
the calorimeter operates over the temperature range from above room temperature down to @xmath2 .
it consists of a pair of cells , each of which is a stack of heaters and thermometer in the center of a silicon nitride membrane , in total giving a background heat capacity less than @xmath3 at @xmath4 , decreasing to @xmath5 at @xmath6 .
the device has several distinctive features : i ) the resistive thermometer , made of a @xmath7 alloy , displays a high dimensionless sensitivity @xmath8 over the entire temperature range .
ii ) the sample is placed in direct contact with the thermometer , which is allowed to self - heat .
the thermometer can thus be operated at high dc current to increase the resolution .
iii ) data are acquired with a set of eight synchronized lock - in amplifiers measuring dc , @xmath9 and @xmath10 harmonic signals of heaters and thermometer .
this gives high resolution and allows continuous output adjustments without additional noise .
iv ) absolute accuracy is achieved via a variable - frequency - fixed - phase technique in which the measurement frequency is automatically adjusted during the measurements to account for the temperature variation of the sample heat capacity and the device thermal conductance .
the performance of the calorimeter is illustrated by studying the heat capacity of a small au sample and the specific heat of a @xmath11 piece of superconducting pb in various magnetic fields .
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the spontaneous chiral symmetry breaking ( @xmath0sb ) is one of the most important features in low - energy qcd . this is considered to be the origin of several hadron masses , such as the lightest nucleon mass . however , there is a possibility that only a part of the lightest nucleon mass is generated by the spontaneous @xmath0sb and the remaining part is the chiral invariant mass . this structure is nicely expressed in so called parity doublet models ( see , e.g. refs . @xcite ) . it is an interesting question to ask how much amount of the nucleon mass is generated by the spontaneous @xmath0sb , or to investigate the origin of nucleon mass . studying dense baryonic matter would give some clues to understand the origin of our mass , since a partial restoration of chiral symmetry will occur at high density region . we expect that the mass generated by the spontaneous @xmath0sb will become small near the chiral phase transition point . it is not so an easy task to study the dense baryonic matter from the first principle , namely starting from qcd itself : it may not be good to use the perturbative analysis , and the lattice qcd is not applicable due to the sign problem at this moment . then , instead of the analysis from the first principle , it may be useful to make an analysis based on effective models , especially for qualitative understanding . holographic qcd ( hqcd ) models ( see , for reviews , e.g. refs . @xcite and references therein . ) are constructed based on the ads / cft correspondence @xcite and powerful tools to study the low - energy hadron physics . there exist several ways to apply hqcd models for dense baryonic matter ( see e.g. refs . recently the holographic mean field theory approach was proposed to study dense baryonic matter in ref . this approach allows us to predict the equation of state between the chemical potential and the baryon number density . in ref . @xcite , this approach was applied to a top - down model of hqcd @xcite including the baryon fields in the framework of the sakai - sugimoto ( ss ) model @xcite . it is known @xcite that the ss model provides the repulsive force mediated by iso - singlet mesons such as @xmath1 meson among nucleons , while the attractive force mediated by the scalar mesons are not generated . as a result ref . @xcite shows that the chemical potential increases monotonically with the baryon number density . on the other hand , when the attraction mediated by the scalar meson is appropriately included , the chemical potential is expect to decrease up until the normal nuclear matter density , and then turn to increase ( see e.g. ref . thus , it is interesting to study whether the chemical potential decreases with increasing density when the scalar degree of freedom is included . in this paper , for studying this , we adopt a bottom - up model given in ref . @xcite which includes five - dimensional baryon field included in the model proposed in refs . there the five dimensional scalar field @xmath2 is explicitly included to express the chiral symmetry breaking by its vacuum expectation value ( vev ) . yet another interest appears in a hqcd model of ref . since there is no chirality in five dimension , the hqcd model includes two baryon fields ; one transforms linearly under u(2)@xmath3 and another under u(2)@xmath4 . the existence of two baryon fields naturally generates the parity doublet structure mentioned above . in ref . @xcite , the boundary condition is adopted in such a way that all of the nucleon mass is generated by the chiral symmetry breaking . in the present analysis , we will show that we can adjust the amount of nucleon mass coming from the chiral symmetry breaking by changing the boundary value of the five - dimensional baryon fields : the percentages of the chiral invariant mass in the nucleon mass is controlled by changing the boundary value . we study how the equation of state in the dense baryonic matter depends on the percentage of the nucleon mass originated from the spontaneous @xmath0sb in the holographic mean field theory approach . our result shows that , larger the percentage of the mass coming from the spontaneous @xmath0sb is , more rapidly the effective nucleon mass , which is extracted from the equation of state by comparing it with the one obtained in a walecka type model given in ref . @xcite , with increasing baryon number density . this paper is organized as follows : in section [ sec : parity ] , we first review the model proposed in ref . @xcite , and then show the parity doubling structure . we study the equation of state at non - zero baryon density in the model in section [ holographic_mft ] . we also discuss the interpretation of our results in terms of a walecka - type model . finally , we give a summary and discussions in section [ sec : sd ] . we summarize several intricate formulas needed in this paper in appendix [ parity ] . in this subsection we briefly review the holographic qcd model including baryons given in ref . @xcite . the fields relevant to the present analysis are the scalar meson field @xmath2 and two baryon fields @xmath5 and @xmath6 , as well as the 5-dimensional gauge fields @xmath7 and @xmath8 , which transform under the 5-dimensional chiral symmetry as @xmath9 where @xmath10 denote the transformation matrix of chiral symmetry , and @xmath11 with @xmath12 . by using these fields , the bulk action is given as @xmath13 where @xmath14 with @xmath15 and @xmath16 being the bulk masses for baryons and mesons , @xmath17 the scalar - baryon coupling constant , @xmath18 the gauge coupling constant . the vielbein @xmath19 appearing in eqs . ( [ action_n1 ] ) and ( [ action_n2 ] ) satisfies @xmath20 where @xmath21 labels the general space - time coordinate and @xmath22 labels the local lorentz space - time , with @xmath23 . by fixing the gauge for the lorentz transformation , we take the vielbein as @xmath24 the dirac matrices @xmath25 are defined as @xmath26 and @xmath27 which satisfy the anti - commutation relation @xmath28 the covariant derivatives for baryon and scalar meson are defined as @xmath29 where @xmath30/(2i)$ ] . @xmath31 is the spin connection given by @xmath32 in this subsection , we study the parity doubling structure of baryons in the model described in the previous subsection . note that the analysis in this subsection is done for zero chemical potential , so that only the scalar field @xmath2 has a mean field part , or 4-dimensional vacuum expectation value ( vev ) , expressed by @xmath33 . the equation of motion ( eom ) for @xmath33 is read from the action @xmath34 in eq . ( [ s x ] ) as @xmath35 the solution for this eom is obtained as @xcite @xmath36 where @xmath21 is the current quark mass and @xmath37 is the quark condensate @xmath38 . by using this vacuum solution , the eoms for @xmath5 and @xmath6 are given by @xmath39 as done in ref . @xcite , we decompose the bulk fields @xmath5 and @xmath6 as @xmath40 where @xmath41 the mode expansions of @xmath42 and @xmath43 are performed as @xmath44 it is convenient to introduce @xmath45 and @xmath46 as @xmath47 which satisfy @xmath48 and @xmath49 with @xmath50 corresponding to mass eigenvalues . it should be noticed that eq . ( [ eomf- ] ) is rewritten as @xmath51 which is the same form as in eq . ( [ eomf+ ] ) . this implies that the solutions of eq . ( [ eomf+ ] ) and those of eq . ( [ eomf- ] ) are not independent with each other . for example , a solution of eq . ( [ eomf+ ] ) with negative energy eigenvalue is actually a solution of eq . ( [ eomf- ] ) with positive energy eigenvalue , which is the reflection of the charge conjugation invariance at zero density . for solving eq . ( [ eomf+ ] ) we need to fix the boundary conditions for @xmath52 and @xmath53 : at the uv boundary ( @xmath54 ) , @xmath52 and @xmath53 should be zero required by the normalizability . the value of @xmath55 at the ir boundary can be set @xmath56 without loss of generality since the coupled differential equations in eq . ( [ eomf+ ] ) are homogeneous equations . in ref . @xcite , the value of @xmath57 at the ir boundary was taken as @xmath58 in such a way that all of the mass of ground state baryon is generated by the chiral symmetry breaking expressed by the vev of @xmath33 . in the present analysis , we regard the ir value of @xmath57 , i.e. @xmath59 , as a parameter , which turns out to control the percentages of the chiral invariant mass included in the nucleon mass . we summarize the boundary condition in table [ tab:1 ] for a convenience . .boundary conditions for baryon fields [ cols="^,^,^",options="header " , ] we solve the equations of motion in eq . ( [ eomn+ ] ) for given values of @xmath60 and @xmath61 , with regarding @xmath62 in eq . ( [ def : mu ] ) as an eigenvalue . using the solutions for the baryonic mean fields @xmath63 and @xmath64 we calculate the baryon number density from eq . ( [ numberdensity_b ] ) . we first study the density dependence of the chiral condensate for checking the partial chiral restoration . here we define the in - medium condensate through the holographic mean field @xmath65 as @xmath66 we plot the density dependence of the @xmath37 normalized by the vacuum value @xmath67 in fig . [ fignd_sigma ] . for several choices of @xmath60 . ] this shows that the quark condensate @xmath37 decreases with the increasing number density , which can be regarded as a sign of the partial chiral symmetry restoration . when the value of @xmath60 is decreased , the corresponding value of @xmath17 becomes larger ( see fig . [ figb_g1 ] ) to reproduce the nucleon mass . since the larger @xmath17 implies the larger correction to the scalar from the nucleon matter , the smaller @xmath60 we choose , the more rapidly the condensate @xmath37 decreases . the degreasing property of the chiral condensate is similar to the one obtained in ref . @xcite . we next show the resultant equation of state , a relation between the chemical potential and the baryon number density in fig . [ fignd_mu ] . , and the vertical axis does the chemical potential by the nucleon mass of @xmath68gev . the dashed line shows the eos for @xmath69 , the solid line for @xmath70 and the dotted line for @xmath71 . ] this figure shows that the chemical potential increases with the increasing baryon number density . this does not agree with the nature , in which the chemical potential decreases against the density in the low density region below the normal nuclear matter density . this decreasing property is achieved by the subtle cancellation between the repulsive and attractive forces . so this increasing property indicates that , in the present model , the repulsive force mediated by the u(1 ) gauge field is stronger than the attractive force mediated by the scalar degree included in x field . for studying the attractive force mediated by the scalar fields , we extract the density dependence of the effective nucleon mass using the walecka type model ( see e.g. refs . @xcite ) , in which the chemical potential @xmath62 is expressed as @xmath72 where @xmath73 is the baryon number density , @xmath74 is the coupling for @xmath75th eigenstate of the omega mesons , @xmath76 is its mass , @xmath77 is the fermi momentum , and @xmath78 is the effective nucleon mass . note that , in the free fermi gas , @xmath77 is related to @xmath73 as @xmath79 , which leads to @xmath80 . in the present hqcd model , the @xmath81 coupling is calculated in vacuum as @xmath82 depending on the value of @xmath60 . using these couplings together with the masses of @xmath83 , we convert the density dependence of @xmath62 obtained above into the one of the effective nucleon mass @xmath78 through eq . ( [ walecka ] ) . we plot the density dependence of the effective mass @xmath78 in fig . [ fig : effective mass ] . . ] this shows that the effective mass decreases with increasing density . the decreasing rate is larger than the one obtained in ref . @xcite , which is the reflection of the iterative corrections included through the holographic mean field theory . it should be noted that the decreasing of @xmath78 is more rapid for smaller value of @xmath60 . in other word , the larger the percentage of the mass coming from the chiral symmetry breaking is , more rapidly the effective mass @xmath78 decreases with density . in fig . [ fig : distribution ] , we plot the baryon charge distribution @xmath84 defined in eq . ( [ numberdensity_b ] ) for @xmath85 , @xmath86 and @xmath87 with @xmath88 fixed . . ] this figure shows that the distribution is broader for larger value of @xmath89 . this indicates that the distribution becomes more important for larger density , as shown in ref . @xcite . we develope the holographic mean field approach in a bottom - up holographic qcd model proposed in ref . @xcite which includes five - dimensional baryon field in the model proposed in refs . we first study the mass spectrum of baryons with paying attention to the chiral invariant mass @xmath90 , which were formulated in parity doublet models ( see , e.g. refs . we found the parameter ( @xmath60 ) , which is one boundary value of two baryon fields , controls the percentage of the chiral invariant mass : for @xmath69 all of the mass of the ground - state nucleon is generated by the spontaneous chiral symmetry breaking , while for @xmath91 , more than half of the nucleon mass is actually the chiral invariant mass . we studied the density dependence of the chiral condensate . our result shows that the quark condensate @xmath37 decreases with the increasing number density , which is consistent with the analysis done in ref . furthermore , we found that the @xmath37 decreases more rapidly for smaller value of @xmath60 . this is because the sigma coupling to the nucleon is larger for smaller @xmath60 . we next calculated the equation of state between the baryon chemical potential and the baryon number density using the holographic mean field approach proposed in ref . the resultant equation of state shows that the chemical potential increases with the increasing baryon number density . this indicates that , in the present model , the repulsive force mediated by the u(1 ) gauge field is stronger than the attractive force mediated by the scalar degree included in x field . for studying the attractive force mediated by the scalar fields , we extract the density dependence of the effective nucleon mass using a walecka type model . our result shows that the effective mass decreases with increasing density . furthermore , the decreasing rate is more rapid for smaller value of @xmath60 . this is consistent with the fact that the percentage of the chiral invariant mass is larger for larger value of @xmath92 . in other word , the larger the percentage of the mass coming from the spontaneous @xmath0sb is , more rapidly the effective nucleon mass decreases with increasing baryon number density . we also studied the baryon number distribution in the holographic direction . our results show that the distribution is concentrated near the ir boundary for smaller @xmath89 . this indicates that the distribution becomes more important for larger density . in the present analysis , we made an analysis only at the mean field level . so a natural extension is to consider the fluctuations on the top of the mean field obtained here . it is also interesting to study the relation between the isospin chemical potential and the isospin density based on the approach developed in this paper , since the relation has a relevance to the symmetry energy . we leave these works to the future project . m.h . would like to thank useful discussions with youngman kim and chang - hwan lee . we are also grateful to yong - liang ma for helpful discussions . this work was supported in part by grant - in - aid for scientific research on innovative areas ( no . 2104 ) `` quest on new hadrons with variety of flavors '' from mext , and by the jsps grant - in - aid for scientific research ( s ) no . 22224003 , ( c ) no . br.h . would like to thank the nagoya university program for leading graduate schools `` leadership development program for space exploration and research '' for the financial support . in this appendix , we consider the parity transformation properties of the 5d fields . as in the 4-dimension , a parity transformation should flip the sign of normal three spatial coordinates . but the 5th coordinate @xmath93 , as it is defined in the range @xmath94 , does not participate in the parity transformation . @xmath95 using these conventions , we obtain the parity transformation of 5d fields as @xmath96 for the 5d spinors , their parity transformation properties are express as @xmath97 where @xmath98 and @xmath99 are arbitrary phases . c. e. detar and t. kunihiro , phys . rev . d * 39 * , 2805 ( 1989 ) . d. jido , m. oka and a. hosaka , prog . phys . * 106 * , 873 ( 2001 ) [ hep - ph/0110005 ] . c. sasaki and i. mishustin , phys . c * 82 * , 035204 ( 2010 ) [ arxiv:1005.4811 [ hep - ph ] ] . w. -g . paeng , h. k. lee , m. rho and c. sasaki , phys . d * 85 * , 054022 ( 2012 ) [ arxiv:1109.5431 [ hep - ph ] ] . j. erdmenger , n. evans , i. kirsch and e. threlfall , eur . phys . j. a * 35 * , 81 ( 2008 ) [ arxiv:0711.4467 [ hep - th ] ] . y. kim , i. j. shin and t. tsukioka , prog . part . nucl . phys . * 68 * , 55 ( 2013 ) [ arxiv:1205.4852 [ hep - ph ] ] . j. m. maldacena , adv . * 2 * , 231 ( 1998 ) [ int . j. theor . * 38 * , 1113 ( 1999 ) ] . k. -y . kim , s. -j . sin and i. zahed , hep - th/0608046 ; jhep * 0801 * , 002 ( 2008 ) [ arxiv:0708.1469 [ hep - th ] ] . n. horigome and y. tanii , jhep * 0701 * , 072 ( 2007 ) [ hep - th/0608198 ] . s. kobayashi , d. mateos , s. matsuura , r. c. myers and r. m. thomson , jhep * 0702 * , 016 ( 2007 ) [ hep - th/0611099 ] . y. kim , c. -h . lee and h. -u . yee , phys . d * 77 * , 085030 ( 2008 ) [ arxiv:0707.2637 [ hep - ph ] ] ; k - i . kim , s. h. lee and y. kim , in _ proceedings of workshop on hadron and nuclear physics , hnp09 , osaka , japan , 2009 _ ( world scientific , hackensack , 2010 ) . o. bergman , g. lifschytz and m. lippert , jhep * 0711 * , 056 ( 2007 ) [ arxiv:0708.0326 [ hep - th ] ] . y. seo and s. -j . sin , jhep * 0804 * , 010 ( 2008 ) [ arxiv:0802.0568 [ hep - th ] ] . s. nakamura , y. seo , s. -j . sin and k. p. yogendran , j. korean phys . soc . * 52 * , 1734 ( 2008 ) [ hep - th/0611021 ] ; prog . phys . * 120 * , 51 ( 2008 ) [ arxiv:0708.2818 [ hep - th ] ] . m. rozali , h. -h . shieh , m. van raamsdon and j. wu , jhep * 0801 * , 053 ( 2008 ) [ arxiv:0708.1322 [ hep - th ] ] . m. harada , s. nakamura and s. takemoto , phys . d * 86 * , 021901 ( 2012 ) [ arxiv:1112.2114 [ hep - th ] ] . d. k. hong , m. rho , h. -u . yee and p. yi , jhep * 0709 * , 063 ( 2007 ) [ arxiv:0705.2632 [ hep - th ] ] . t. sakai and s. sugimoto , prog . 113 * , 843 ( 2005 ) ; prog . phys . * 114 * , 1083 ( 2005 ) . a. dymarsky , d. melnikov and j. sonnenschein , jhep * 1106 * , 145 ( 2011 ) [ arxiv:1012.1616 [ hep - th ] ] . , , and , * * , ( ) , .
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we develope the holographic mean field approach in a bottom - up holographic qcd model including baryons and scalar mesons in addition to vector mesons and pions .
we study the effect of parity doubling structure of baryons at non - zero density to the equation of state between the chemical potential and the baryon number density .
we first show that we can adjust the amount of nucleon mass coming from the chiral symmetry breaking by changing the boundary value of the five - dimensional baryon fields .
then , introducing the mean field for the baryon fields , we calculate the equation of state between the baryon number density and its corresponding chemical potential .
then , comparing the predicted equation of state with the one obtained in a walecka type model , we extract the density dependence of the effective nucleon mass .
the result shows that the effective mass decreases with increasing density , and that the rate of decreasing is more rapid for larger percentage of the mass from the chiral symmetry breaking .
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pulsed high - frequency ( hf ) electromagnetic ( em ) waves from transmitters on the ground are regularly used for sounding the density profile and drift velocity of the overehead ionosphere [ _ hunsucker _ , 1991 ; _ reinisch et al . _ , 1995 , _ reinisch _ , 1996 ] . in 1971 , it was shown theoretically by _ perkins and kaw _ [ 1971 ] that if the injected hf radio beams are strong enough , weak - turbulence parametric instabilities in the ionospheric plasma of the type predicted by _ silin _ [ 1965 ] and _ dubois and goldman _ [ 1965 ] would be excited . ionospheric modification experiments by a high - power hf radio wave at platteville in colorado [ _ utlaut _ , 1970 ] , using ionosonde recordings and photometric measurements of artificial airglow , demonstrated the heating of electrons , the deformation in the traces on ionosonde records , the excitation of spread @xmath0 , etc . , after the hf transmitter was turned on . the triggering of weak - turbulence parametric instabilities in the ionosphere was first observed in 1970 in experiments on the interaction between powerful hf radio beams and the ionospheric plasma , conducted at arecibo , puerto rico , using a scatter radar diagnostic technique [ _ wong and taylor _ , 1971 ; _ carlson et al . _ , 1972 ] . a decade later it was found experimentally in troms that , under similar experimental conditions as in arecibo , strong , systematic , structured , wide - band secondary hf radiation escapes from the interaction region [ _ thid et al . _ , 1982 ] . this and other observations demonstrated that complex interactions , including weak and strong em turbulence , [ _ leyser _ , 2001 ; _ thid et al . _ , 2005 ] and harmonic generation [ _ derblom et al . _ , 1989 ; _ blagoveshchenskaya et al . _ , 1998 ] are excited in these experiments . numerical simulations have become an important tool to understand the complex behavior of plasma turbulence . examples include analytical and numerical studies of langmuir turbulence [ _ robinson _ , 1997 ] , and of upper - hybrid / lower - hybrid turbulence in magnetized plasmas [ _ goodman et al . _ , 1994 ; _ xi _ , 2004 ] . in this letter , we present a full - scale simulation study of the propagation of an hf em wave into the ionosphere , with ionospheric parameters typical for the high - latitude eiscat heating facility in troms , norway . to our knowledge , this is the first simulation involving realistic scale sizes of the ionosphere and the wavelength of the em waves . our results suggest that such simulations , which are possible with today s computers , will become a powerful tool to study hf - induced ionospheric turbulence and secondary radiation on a quantitative level for direct comparison with experimental data . we use the mks system ( si units ) in the mathematical expressions throughout the manuscript , unless otherwise stated . we assume a vertically stratified ion number density profile @xmath1 with a constant geomagnetic field @xmath2 directed obliquely to the density gradient . the em wave is injected vertically into the ionosphere , with spatial variations only in the @xmath3 direction . our simple one - dimensional model neglects the em field @xmath4 falloff ( @xmath5 is the distance from the transmitter ) , the fresnel pattern created obliquely to the @xmath3 direction by the incident and reflected wave , and the the influence on the radio wave propagation due to field aligned irregularities in the ionosphere . for the em wave , the maxwell equations give @xmath6 @xmath7 where the electron fluid velocity is obtained from the momentum equation @xmath8\ ] ] and the electron density is obtained from the poisson equation @xmath9 . here , @xmath10 is the unit vector in the @xmath3 direction , @xmath11 is the speed of light in vacuum , @xmath12 is the magnitude of the electron charge , @xmath13 is the vacuum permittivity , and @xmath14 is the electron mass . ms.,scaledwidth=48.0% ] the number density profile of the immobile ions , @xmath15 $ ] ( @xmath3 in kilometers ) is shown in the leftmost panel of fig . [ fig1 ] . instead of modeling a transmitting antenna via a time - dependent boundary condition at @xmath16 km , we assume that the em pulse has reached the altitude @xmath17 km when we start our simulation , and we give the pulse as an initial condition at time @xmath18 s. in the initial condition , we use a linearly polarized em pulse where the carrier wave has the wavelength @xmath19 ( wavenumber @xmath20 ) corresponding to a carrier frequency of @xmath21 ( @xmath22 ) . the em pulse is amplitude modulated in the form of a gaussian pulse with a maximum amplitude of @xmath23 v / m , with the @xmath24-component of the electric field set to @xmath25\sin(0.1047\times 10^{3 } z)$ ] ( @xmath3 in kilometers ) and the @xmath26 component of the magnetic field set to @xmath27 at @xmath18 . the other electric and magnetic field components are set to zero ; see fig . [ the spatial width of the pulse is approximately 30 km , corresponding to a temporal width of 0.1 milliseconds as the pulse propagates with the speed of light in the neutral atmosphere . it follows from eq . ( 1 ) that @xmath28 is time - independent ; hence we do not show @xmath28 in the figures . the geomagnetic field is set to @xmath29 tesla , corresponding to an electron cyclotron frequency of 1.4 mhz , directed downward and tilted in the @xmath30-plane with an angle of @xmath31 degrees ( @xmath32 rad ) to the @xmath3-axis , i.e. , @xmath33 . in our numerical simulation , we use @xmath34 spatial grid points to resolve the plasma for @xmath35 km . the spatial derivatives are approximated with centered second - order difference approximations , and the time - stepping is performed with a leap - frog scheme with a time step of @xmath36 s. the splitting of the wave is due to faraday rotation.,scaledwidth=48.0% ] ms . b ) a closeup of the region of the turning points of the r - x and l - o modes . we see that the wave - energy of the l - o mode is concentrated into one single half - wave envelop at @xmath37 km , while the turning point of the less localized r - x mode is at @xmath38 km.,scaledwidth=48.0% ] ms . b ) a closeup of the region of the turning points of the r - x and l - o modes . here , the l - o mode oscillations at @xmath37 km are radiating em waves with perpendicular ( to the @xmath3 axis ) electric field components.,scaledwidth=48.0% ] in the simulation , the em pulse propagates without changing shape through the neutral atmosphere , until it reaches the ionospheric layer . at time @xmath39 ms , shown in fig . [ fig2 ] , the em pulse has reached the lower part of the ionosphere . the initially linearly polarized em wave undergoes faraday rotation due to the different dispersion properties of the l - o and r - x modes ( we have adopted the notation `` l - o mode '' and `` r - x mode '' for the two high - frequency em modes , similarly as , e.g. , _ goertz and strangeway _ [ 1995 ] ) in the magnetized plasma , and the @xmath40 and @xmath41 components are excited . at @xmath42 ms , shown in fig . [ fig3 - 4 ] , the l - o and r - x mode pulses are in the vicinity of their respective turning points , the turning point of the l - o mode being at a higher altitude than that of the r - x mode ; see panel a ) of fig . [ fig3 - 4 ] . a closeup of this region , displayed in panel b ) , shows that the first maximum of the r - x mode is at @xmath38 km , and the one of the l - o mode is at @xmath37 km . the maximum amplitude of the r - x mode is @xmath43 v / m while that of the l - o mode is @xmath44 v / m ; the latter amplitude maximum is in agreement with those obtained by _ thid and lundborg _ , [ 1986 ] , for a similar set of parameters as used here . the electric field components of the l - o mode , which at this stage are concentrated into a pulse with a single maximum with a width of @xmath45 m , are primarily directed along the geomagnetic field lines , and hence only the @xmath46 and @xmath47 components are excited , while the magnetic field components of the l - o mode are very small . at @xmath48 ms , shown in panel a ) of fig . [ fig5 - 6 ] , both the r - x and l - o mode wave packets have widened in space , and the em wave has started turning back towards lower altitudes . in the closeup of the em wave in panel b ) of fig . [ fig5 - 6 ] , one sees that the l - o mode oscillations at @xmath37 km are now radiating em waves with significant magnetic field components . finally , shown in fig . [ fig7 ] at @xmath49 , the em pulse has returned to the initial location at @xmath17 km . due to the different reflection heights of the l - o and r - x modes , the leading ( lower altitude ) part of the pulse is primarily r - x mode polarized while its trailing ( higher altitude ) part is l - o mode polarized . in the center of the pulse , where we have a superposition of the r - x and l - o mode , the wave is almost linearly polarized with the electric field along the @xmath26 axis and the magnetic field along the @xmath24 axis . the direction of the electric and magnetic fields here depends on the relative phase between the r - x and l - o mode . ms.,scaledwidth=48.0% ] at @xmath50 km , near the turning point of the r - x mode , and b ) the amplitude of the electric field component @xmath46 at @xmath51 km , near the turning point of the l - o mode . c ) a snapshot of low - amplitude electrostatic waves of wavelength @xmath52 m ( wavenumber @xmath53 ) , observed at time @xmath54 ms , and d ) dispersion curves ( lower panel ) obtained from the appleton - hartree dispersion relation with parameters @xmath55 ( 5 mhz ) , @xmath56 ( 1.4 mhz ) and @xmath57 rad . we identify the high - frequency r - x and l - o modes , as well as the z - mode which extends to the electrostatic langmuir / upper hybrid branch for large wavenumbers ; the circles indicate the approximate locations on the dispersion curve for the electrostatic oscillations shown in panel c ) . for completeness we also show the low - frequency electron whistler branch in panel d).,scaledwidth=50.0% ] at the altitude @xmath50 km , and b ) of @xmath46 at the altitude @xmath51 km.,scaledwidth=48.0% ] in fig . [ fig8 - 9 ] , panel a ) , we have plotted the electric field component @xmath47 at @xmath50 km , near the turning point of the r - x mode and in panel b ) we have plotted the @xmath46 component at @xmath51 km , near the turning point of the l - o mode . we see that the maximum amplitude of @xmath47 reaches @xmath58 v / m at @xmath59 ms , and that of @xmath46 reaches @xmath60 v / m at @xmath61 ms . the electric field amplitude at @xmath50 km has two maxima , due to the l - o mode part of the pulse , which is reflected at the higher altitude @xmath51 km and passes twice over the altitude @xmath50 km . we also observe weakly damped oscillations of @xmath46 at @xmath51 km for times @xmath62 ms , which decrease exponentially in time between @xmath63 ms and @xmath64 ms as @xmath65 with @xmath66 s@xmath67 . we found from the numerical values that @xmath68 , where @xmath69 is the inverse ion density scale length at @xmath70 km , but we are not certain how general this result is . no detectable magnetic field fluctuations are associated with these weakly damped oscillations , and we interpret them as electrostatic waves that have been produced by mode conversion of the l - o mode . the amplitudes of the @xmath47 and @xmath40 components are also much weaker than that of the @xmath46 component for these oscillations . a closeup of these electrostatic oscillations at @xmath54 ms is displayed in panel c ) of fig . [ fig8 - 9 ] , where we see that they have a wavelength of approximately 33 m ( wavenumber @xmath71 ) . in panel d ) of fig . [ fig8 - 9 ] , we have plotted the frequency @xmath72 as a function of the wavenumber @xmath73 , where @xmath74 is obtained from the appleton - hartree dispersion relation [ _ stix _ , 1992 ] @xmath75 here @xmath76^{1/2}$ ] , @xmath77 ( @xmath78 ) is the electron plasma ( cyclotron ) frequency , and @xmath79 is the angle between the geomagnetic field and the wave vector @xmath80 , which in our case is directed along the @xmath3-axis , @xmath81 . we use @xmath82 ( corresponding to @xmath83 mhz ) , @xmath84 ( corresponding to @xmath85 mhz ) and @xmath57 rad . the location of the electrostatic waves whose wavelength is approximately 33 m and frequency 5 mhz is indicated with circles in the diagram ; they are on the same dispersion surface as the langmuir waves and the upper hybrid waves / slow z mode waves with propagation parallel and perpendicular to the geomagnetic field lines , respectively . the mode conversion of the l - o mode into electrostatic oscillations are relatively weak in our simulation of vertically incident em waves , and theory shows that the most efficient linear mode conversion of the l - o mode occurs at two angles of incidence in the magnetic meridian plane , given by , e.g. , eq . ( 17 ) in [ _ mjlhus _ , 1990 ] . the nonlinear effects at the turning point of the l - o and r - x modes are investigated in fig . [ fig10 ] which displays the frequency spectrum of the electric field component @xmath47 at the altitude @xmath86 km and of @xmath46 at the altitude @xmath51 km . the spectrum shows the large - amplitude pump wave at 5 mhz and the relatively weak second harmonics of the pump wave at 10 mhz at both altitudes ( the slight downshift is due to numerical errors produced by the difference approximations used in space and time ) . visible are also low - frequency oscillations ( zeroth harmonic ) due to the nonlinear down - shifting / mixing of the high - frequency wave field . in conclusion , we have presented a full - scale numerical study of the propagation of an em wave and its linear and nonlinear interactions with an ionospheric layer . we observe the reflection of the l - o and r - x modes at different altitudes , the mode conversion of the l - o mode into electrostatic langmuir / upper hybrid waves as well as nonlinear harmonic generation of the high - frequency waves . second harmonic generation have been observed in ionospheric heating experiments [ _ derblom et al . _ , 1989 ; _ blagoveshchenskaya et al . _ , 1998 ] and may be partially explained by the cold plasma model presented here . blagoveshchenskaya , n. f. , v. a. kornienko , m. t. rietveld , b. thid , a. brekke , i. v. moskvin , and s. nozdrachev ( 1998 ) , stimulated emissions around second harmonic of troms heater frequency observed by long - distance diagnostic hf tools . _ geophys . 25_(6 ) , 863876 . derblom , h. , b. thid , t. b. leyser , j. a. nordling , . hedberg , p. stubbe , h. kopka , and m. rietveld ( 1989 ) , troms heating experiments : stimulated emission at hf pump harmonic and subharmonic frequencies , _ j. geophys . 94_(a8 ) , 1011110120 . thid , b. , e. n. sergeev , s. m. grach , t. b. leyser , and t. d. carozzi ( 2005 ) , competition between langmuir and upper - hybrid turbulence in a high - frequency - pumped ionosphere , _ _ 95 _ , 255002 .
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the time evolution of a large - amplitude electromagnetic ( em ) wave injected vertically into the overhead ionosphere is studied numerically .
the em wave has a carrier frequency of 5 mhz and is modulated as a gaussian pulse with a width of approximately 0.1 milliseconds and a vacuum amplitude of 1.5 v / m at 50 km .
this is a fair representation of a modulated radio wave transmitted from a typical high - power hf broadcast station on the ground .
the pulse is propagated through the neutral atmosphere to the critical points of the ionosphere , where the l - o and r - x modes are reflected , and back to the neutral atmosphere .
we observe mode conversion of the l - o mode to electrostatic waves , as well as harmonic generation at the turning points of both the r - x and l - o modes , where their amplitudes rise to several times the original ones .
the study has relevance for ionospheric interaction experiments in combination with ground - based and satellite or rocket observations .
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chemical elements heavier than lithium are synthesized in stars . such `` metals '' are observed at times when the universe was only @xmath6% of its current age in the inter galactic medium ( igm ) as absorption lines in quasar spectra ( see ellison et al . 2000 , and references therein ) . hence , these heavy elements not only had to be synthesized but also released and distributed in the igm within the first billion years . only supernovae of sufficiently short lived massive stars are known to provide such an enrichment mechanism . this leads to the prediction that _ the first generation of cosmic structures formed massive stars ( although not necessarily only massive stars ) . _ in the past 30 years it has been argued that the first cosmological objects form globular clusters ( ) , super massive black holes ( ) , or even low mass stars ( ) . this disagreement of theoretical studies might at first seem surprising . however , the first objects form via the gravitational collapse of a thermally unstable reactive medium , inhibiting conclusive analytical calculations . the problem is particularly acute because the evolution of all other cosmological objects ( and in particular the larger galaxies that follow ) will depend on the evolution of the first stars . nevertheless , in comparison to present day star formation , the physics of the formation of the first star in the universe is rather simple . in particular : * the chemical and radiative of processes in the primordial gas are readily understood . * strong magnetic fields are not expected to exist at early times . * by definition no other stars exist to influence the environment through radiation , winds , supernovae , etc . * the emerging standard model for structure formation provides appropriate initial conditions . in previous work we have presented three dimensional cosmological simulations of the formation of the first objects in the universe ( , ) including first applications of adaptive mesh refinement ( amr ) cosmological hydrodynamical simulations to first structure formation ( , , abn hereafter ) . in these studies we achieved a dynamic range of up to @xmath7 and could follow in detail the formation of the first dense cooling region far within a pre galactic object that formed self consistently from linear density fluctuation in a cold dark matter cosmology . here we report results from simulations that extend our previous work by another 5 orders of magnitude in dynamic range . for the first time it is possible to bridge the wide range between cosmological and stellar scale . we employ an eulerian structured adaptive mesh refinement cosmological hydrodynamical code developed by bryan and norman ( , ) . the hydrodynamical equations are solved with the second order accurate piecewise parabolic method (; ) where a riemann solver ensures accurate shock capturing with a minimum of numerical viscosity . we use initial conditions appropriate for a spatially flat cold dark matter cosmology with 6% of the matter density contributed by baryons , zero cosmological constant , and a hubble constant of 50 km / s / mpc ( ) . the power spectrum of initial density fluctuations in the dark matter and the gas are taken from the computation by the publicly available boltzmann code cmbfast ( ) at redshift 100 ( assuming an harrison zeldovich scale invariant initial spectrum ) . we set up a three dimensional volume with 128 comoving kpc on a side and solve the cosmological hydrodynamics equations assuming periodic boundary conditions . this small volume is adequate for our purpose , because we are interested in the evolution of the first pre galactic object within which a star may be formed by a redshift of @xmath8 . first we identify the lagrangian volume of the first proto galactic halo with a mass of @xmath9 in a low resolution pure n body simulation . then we generate new initial conditions with four initial static grids that cover this langrangian region with progressively finer resolution . with a @xmath10 top grid and a refinement factor of 2 this specifies the initial conditions in the region of interest equivalent to a @xmath11 uni grid calculation . for the adopted cosmology this gives a mass resolution of @xmath12 for the dark matter ( dm , hereafter ) and @xmath13 for the gas . the small dm masses ensure that the cosmological jeans mass is resolved by at least ten thousand particles at all times . smaller scale structures in the dark matter will not be able to influence the baryons because of their shallow potential wells . the theoretical expectation holds , because the simulations of abn which had 8 times poorer dm resolution led to identical results on large scales as the simulation presented here . during the evolution , refined grids are introduced with twice the spatial resolution of the parent ( coarser ) grid . these child ( finer ) meshes are added whenever one of three refinement criteria are met . two langrangian criteria ensure that the grid is refined whenever the gas ( dm ) density exceeds 4.6 ( 9.2 ) its initial density . additionally , the local jeans length is always covered by at least 64 grid cells cells . ] ( 4 cells per jeans length would be sufficient , ) . we have also carried out the simulations with identical initial conditions but varying the refinement criteria . in one series of runs we varied the number of mesh points per jeans length . runs with 4 , 16 , and 64 zones per jeans length are indistinguishable in all mass weighted radial profiles of physical quantities . no change in the angular momentum profiles could be found , suggesting negligible numerical viscosity effects on angular momentum transport . a further refinement criterion that ensured the local cooling time scale to be longer than the local courant time also gave identical results . this latter test checked that any thermally unstable region was identified . the simulation follows the non equilibrium chemistry of the dominant nine species species ( h , h@xmath14 , h@xmath15 , e@xmath15 , he , he@xmath14 , he@xmath16 , h@xmath17 , and h@xmath18 ) in primordial gas . furthermore , the radiative losses from atomic and molecular line cooling , compton cooling and heating of free electrons by the cosmic background radiation are appropriately treated in the optically thin limit ( , ) . to extend our previous the studies to higher densities three essential modifications to the code were made . first we implemented the three body molecular hydrogen formation process in the chemical rate equations . for temperatures below 300 k we fit to the data of orel ( ) to get @xmath19 . above 300 k we then match it continuously to a powerlaw ( ) @xmath20 . secondly , we introduce a variable adiabatic index for the gas ( ) . the dissipative component ( baryons ) may collapse to much higher densities than the collisionless component ( dm ) . the discrete sampling of the dm potential by particles can then become inadequate and result in artificial heating of the baryons ( cooling for the dm ) once the gas density becomes much larger than the local dm density . to avoid this , we smooth the dm particles with a gaussian of width 0.05 for grids with cells smaller than this length . at this scale , the enclosed gas mass substantially exceeds the enclosed dm mass . the standard message passing library ( mpi ) was used to implement domain decomposition on the individual levels of the grid hierarchy as a parallelization strategy . the code was run in parallel on 16 processors of the sgi origin2000 supercomputer at the national center for supercomputing applications at the university of illinois at urbana champaign . we stop the simulation at a time when the molecular cooling lines reach an optical depth of ten at line center because our numerical method can not treat the difficult problem of time dependent radiative line transfer in multi dimensions . at this time the code utilizes above 5500 grids on 27 refinement levels with @xmath21 computational grid cells . an average grid therefore contains @xmath22 cells . [ colorplate ] our simulations ( fig . [ colorplate ] , fig . [ 5panel ] ) , identify at least four characterisic mass scales . from the outside going in , one observes infall and accretion onto the pre galactic halo with a total mass of @xmath23 , consistent with previous studies ( , , , abn , and for discussion and references ) . at a mass scale of about 4000 solar mass ( @xmath24 ) rapid cooling and infall is observed . this is accompanyed by the first of three valleys in the radial velocity distribution ( fig . [ 5panel]e ) . the temperature drops and the molecular hydrogen fraction increases . it is here , at number densities of @xmath25 , that the high redshift analog of a molecular cloud is formed . although the molecular mass fraction is not even 0.1% it is sufficient to cool the gas rapidly down to @xmath26 . the gas can not cool below this temperature because of the sharp decrease in the cooling rate below @xmath27 . at redshift 19 ( fig . [ 5panel ] ) , there are only two mass scales ; however , as time passes the central density grows and eventually passes @xmath28 , at which point the ro - vibrational levels of are populated at their equilibrium values and the cooling time becomes independent of density ( instead of inversely proportional to it ) . this reduced cooling efficiency leads to an increase in the temperature ( fig . [ 5panel]d ) . as the temperature rises , the cooling rate again increases ( it is 1000 times higher at 800 k than at 200 k ) , and the inflow velocities slowly climb . in order to better understand what happens next , we examine the stability of an isothermal gas sphere . the critical mass for gravitational collapse given an external pressure @xmath29 ( be mass hereafter ) is given by ebert ( ) and bonnor ( ) as : @xmath30 here @xmath29 is the external pressure and @xmath31 , @xmath32 , and @xmath33 the gravitational constant , the boltzmann constant and the sound speed , respectively . we can estimate this critical mass locally if we set the external pressure to be the local pressure to find @xmath34 where @xmath35 is the mean mass per particle in units of the proton mass . using an adiabatic index @xmath36 , we plot the ratio of the enclosed gas mass to this modified be mass in figure [ bemass ] . our modeling shows ( fig . [ bemass ] ) , that by the fourth considered output time , the central 100 exceeds the be mass at that radius , indicating unstable collapse . this is the third mass scale and corresponds to the second local minimum in the radial velocity curves ( fig . [ 5panel]e ) . the inflow velocity is @xmath37 is still subsonic . although this mass scale is unstable , it does not represent the smallest scale of collapse in our simulation . this is due to the increasing molecular hydrogen fraction . when the gas density becomes sufficiently large ( @xmath38 ) , three - body molecular hydrogen formation becomes important . this rapidly increases the molecular fraction ( fig . [ 5panel]c ) and hence the cooling rate . the increased cooling leads to lower temperatures and even stronger inflow and . at a mass scale of @xmath39 , not only is the gas nearly completely molecular , but the radial inflow has become supersonic ( fig . [ 5panel]e ) . when the mass fraction approaches unity , the increase in the cooling rate saturates , and the gas goes through a radiative shock . this marks the first appearance of the proto stellar accretion shock at a radius of about 20 astronomical units from its center . when the cooling time becomes independent of density the classical criterion for fragmentation @xmath40 ( ) can not be satisfied at high densities . however , in principal the medium may still be subject to thermal instability . the instability criterion is @xmath41 where @xmath42 denotes the cooling losses per second of a fluid parcel and @xmath43 and @xmath44 are the gas temperature and mass density , respectively . at densities above the critical densities of molecular hydrogen the cooling time is independent of density , i.e. @xmath45 where @xmath46 is the high density cooling function ( e.g. ) . fitting the cooling function with a power - law locally around a temperature @xmath47 so that @xmath48 one finds @xmath49 . hence , under these circumstances the medium is thermally stable if @xmath50 . because , @xmath51 for the densities and temperatures of interest , we conclude that the medium is thermally stable . the above analysis neglects the heating from contraction , but this only strengthens the conclusion . if heating balances cooling one can neglect the @xmath52 term in equation ( [ eq : ic ] ) and find the medium to be thermally stable for @xmath53 . [ bemass ] however , here we neglected the chemical processes . the detailed analysis for the case when chemical processes occur on the collapse time scale is well known ( ) . this can be applied to primordial star formation ( ) including the three body formation of molecular hydrogen ( ) which drives a chemo thermal instability . evaluating all the terms in this modified instability criterion ( , equation 36 ) one finds the simple result that for molecular mass fractions @xmath54 the medium is expected to be chemo thermally unstable . these large molecular fractions illustrate that the strong density dependence of the three body formation dominates the instability . examining the three dimensional temperature and density field we clearly see this chemo thermal instability at work . cooler regions have larger fractions . however , no corresponding large density inhomogeneities are found and fragmentation does not occur . this happens because of the short sound crossing times in the collapsing core . when the formation time scale becomes shorter than the cooling time the instability originates . however , as long as the sound crossing time is much shorter than the chemical and cooling time scales the cooler parts are efficiently mixed with the warmer material . this holds in our simulation until the final output where for the first time the formation time scale becomes shorter than the sound crossing time . however , at this point the proto stellar core is fully molecular and stable against the chemo thermal instability . consequently no large density contrasts are formed . because at these high densities the optical depth of the cooling radiation becomes larger than unity the instability will be suppressed even further . interestingly , rotational support does not halt the collapse . this is for two reaons . the first is shown in panel a of fig . [ angtrans ] , which plots the specific angular momentum against enclosed mass for the same seven output times discussed earlier . concentrating on the first output ( fig . [ angtrans ] ) , we see that the central gas begins the collapse with a specific angular momentum only @xmath55% as large as the mean value . this type of angular momentum profile is typical of halos produced by gravitational collapse ( e.g. ) , and means that the protostellar gas starts out without much angular momentum to lose . as a graphic example of this , consider the central one solar mass of the collapsing region . it has only an order of magnitude less angular momentum at densities @xmath56 than it had at @xmath57 although it collapsed by over a factor 100 in radius . the remaining output times ( fig . [ angtrans ] ) indicate that there is some angular momentum transport within the central @xmath58 ( since l plotted as a function of enclosed mass should stay constant as long as there is no shell crossing ) . in panel c , we divide @xmath42 by @xmath59 to get a typical rotational velocity and in panels b and d compare this velocity to the keplerian rotational velocity and the local sound speed , respectively . we find that the typical rotational speed is a factor two to three below that required for rotational support . furthermore , we see that this azimuthal speed never significantly exceeds the sound speed , although for most the mass below @xmath58 it is comparable in value . we interpret this as evidence that it is shock waves during the turbulent collapse that are responsible for much of the transported angular momentum . a collapsing turbulent medium is different from a disk in keplarian rotation . at any radius there will be both low and high angular momentum material , and pressure forces or shock waves can redistribute the angular momentum between fluid elements . lower angular momentum material will selectively sink inwards , displacing higher angular momentum gas . this hydrodynamic transport of angular momentum will be suppressed in situation where the collapse proceeds on the dynamical time rather on the longer cooling time as in the presented case . this difference in cooling time and the widely different initial conditions may explain why this mechanism has not been observed in simulations of present day star formation ( e.g. , and references therein ) . however , such situations may also arise in the late stages of the formation of present day stars and in scenarios for the formation of super massive black holes . to ensure that the angular momentum transport is not due to numerical shear viscosity ( ) we have carried out the resolution study discussed above . we have varied the effective spatial resolution by a factor 16 and found identical results . furthermore , we have run the adaptive mesh refinement code with two different implementations of the hydrodynamics solver . the resolution study and the results presented here were carried out with a direct piecewise parabolic method adopted for cosmology (; ) . we ran another simulation with the lower order zeus hydrodynamics ( ) and still found no relevant differences . these tests are not strict proof that the encountered angular momentum transport is not caused by numerical effects ; however , they are reassuring . the strength of magnetic fields generated around the epoch of recombination is minute . in contrast , phase transitions at the qantum chromo dynamic ( qcd ) and electro weak scales may form even dynamically important fields . while there is a plethora of such scenarios for primordial magnetic field generation in the early universe they are not considered to be an integral part of our standard picture of structure formation . this is because not even the order of these phase transitions is known ( ) , and references therein ) . unfortunately , strong primordial small - scale ( @xmath60 comoving ) magnetic fields are poorly constrained observationally ( ) . the critical magnetic field for support of a cloud ( ) allows a rough estimate up to which primordial magnetic field strengths we may expect our simulation results to hold . for this we also assume a flux frozen flow with no additional amplification of the magnetic field other than the contraction ( @xmath61 ) . for a comoving b field of @xmath62 on scales @xmath63 the critical field needed for support may be reached during the collapse possibly modifying the mass scales found in our purely hydrodynamic simulations . however , the ionized fraction drops rapidly during the collapse because of the absence of cosmic rays ionizations . consequently ambipolar diffusion should be much more effective in the formation of the first stars even if such strong primordial magnetic fields were present . previously we discussed the formation of the pre galactic object and the primordial `` molecular cloud '' that hosts the formation of the first star in the simulated patch of the universe ( ) . these simulations had a dynamic range of @xmath64 and identified a @xmath0 core within the primordial `` molecular cloud '' undergoing renewed gravitational collapse . the fate of this core was unclear because there was the potential caveat that three body formation could have caused fragmentation . indeed this further fragmentation had been suggest by analytic work ( ) and single zone models ( ) . the three dimensional simulations described here were designed to be able to test whether the three body process will lead to a break up of the core . _ no fragmentation due to three body formation is found . _ this is to a large part because of the slow quasi hydrostatic contraction found in abn which allows sub sonic damping of density perturbations and yields a smooth distribution at the time when three body formation becomes important . instead of fragmentation a single fully molecular proto star of @xmath65 is formed at the center of the @xmath66 core . however , even with extraordinary resolution , the _ final _ mass of the first star remains unclear . whether all the available cooled material of the surroundings will accrete onto the proto star or feedback from the forming star will limit the further accretion and hence its own growth is difficult to compute in detail . within @xmath67 about @xmath68 may be accreted assuming that angular momentum will not slow the collapse ( fig . [ acrete ] ) . the maximum of the accretion time of @xmath69 is at @xmath70 . however , stars larger than @xmath71 will explode within @xmath72 . therefore , it seems unlikely ( even in the absence of angular momentum ) that there would be sufficient time to accrete such large masses . a one solar mass proto star will evolve too slowly to halt substantial accretion . from the accretion time profile ( fig . [ acrete ] ) one may argue that a more realistic minimum mass limit of the first star should be @xmath73 because this amount would be accreted within a few thousand years . this is a very short time compared to expected proto stellar evolution times . however , some properties of the primordial gas may make it easier to halt the accretion . one possibility is the destruction of the cooling agent , molecular hydrogen , without which the acreting material may reach hydrostatic equilibrium . this may or may not be sufficient to halt the accretion . one may also imagine that the central material heats up to @xmath74 k , allowing lyman-@xmath75 cooling from neutral hydrogen . that cooling region may expand rapidly as the internal pressure drops because of infall , possibly allowing the envelope to accrete even without molecular hydrogen as cooling agent . additionally , radiation pressure from ionizing photons as well as atomic hydrogen lyman series photons may become significant and eventually reverse the flow . the mechanisms discussed by haehnelt ( 1993 ) on galactic scales will play an important role for the continued accretion onto the proto star . this is an interplay of many complex physical processes because one has a hot ionized strmgren sphere through which cool and dense material is trying to accrete . in such a situation one expects a raleigh taylor type instability that is modified via the geometry of the radiation field . at the final output time presented here there are @xmath76 hydrogen molecules in the entire protogalaxy . also the formation time scale is long because there are no dust grains and the free electrons ( needed as a catalyst ) have almost fully recombined . hence , as soon as the the first uv photons of lyman werner band frequencies are produced there will be a rapidly expanding photo dissociating region ( pdr ) inhibiting further cooling within it . this photo - dissociation will prevent further fragmentation at the molecular cloud scale . i.e. no other star can be formed within the same halo before the first star dies in a supernova . the latter , however , may have sufficient energy to unbind the entire gas content of the small pre galactic object it formed in ( ) . this may have interesting feedback consequences for the dispersal of metals , entropy and magnetic field into the intergalactic medium ( , ) . smoothed particle hydrodynamics ( sph , e.g. ) , used extensively in cosmological hydrodynamics , has been employed ( ) to follow the collapse of solid body rotating uniform spheres . the assumption of coherent rotation causes these clouds to collapse into a disk which developes filamentary structures which eventually fragment to form dense clumps of masses between @xmath77 and @xmath78 solar masses . it has been argued that these clumps will continue to accrete and merge and eventually form very massive stars . these sph simulation have unrealistic initial conditions and much less resolution then our calculations . however , they also show that many details of the collapse forming a primordial star are determined by the properties of the hydrogen molecule . we have also simulated different initial density fields for a lambda cdm cosmology . there we have focused on halos with different clustering environments . although we have not followed the collapse in these halos to proto - stellar densities , we have found no qualitative differences in the `` primordial molecular cloud '' formation process as discussed in abn . also other amr simulations ( ) give consistent results on scales larger than @xmath79 . in all cases a cooling flow forms the primordial molecular cloud at the center of the dark matter halo . we conclude that the molecular cloud formation process seems to be independent of the halo clustering properties and the adopted cdm type cosmology . also the mass scales for the core and the proto star are determined by the local bonnor ebert mass . consequently , we expect the key results discussed here to be insensitive to variations in cosmology or halo clustering . the picture arising from these numerical simulations has some very interesting implications . it is possible that all metal free stars are massive and form in isolation . their supernovae may provide the metals seen in even the lowest column density quasar absorption lines ( , and references therein ) . massive primordial stars offer a natural explanation for the absence of purely metal free low mass stars in the milky way . the consequences for the formation of galaxies may be even more profound in that the supernovae provide metals , entropy , and magnetic fields and may even alter the initial power spectrum of density fluctuations of the baryons . interestingly , it has been recently argued , from abundance patterns , that in low metallicity galactic halo stars seem to have been enriched by only one population of massive stars ( ) . these results , if confirmed , would represent strong support for the picture arising from our ab initio simulations of first structure formation . to end on a speculative note there is suggestive evidence that links gamma ray bursts to sites of massive star formation ( e.g. ) . it would be very fortunate if a significant fraction of the massive stars naturally formed in the simulations would cause gamma ray bursts ( e.g. ) . such high redshift bursts would open a remarkably bright window for the study of the otherwise dark ( faint ) ages . 1 . [ pd68 ] peebles , p. j. e. & dicke , r. h. 1968 , apj * 154 * , 891 2 . [ h69 ] hirasawa , t. 1969 , prog . . phys . * 42 * , 523 3 . [ pss83 ] palla , f. , salpeter , e.e . , stahler , s.w . 1983 , apj * 271 * , 632 4 . [ a95 ] abel , t. 1995 , thesis , university of regensburg . [ aanz ] abel , t. , anninos , p. , norman , m.l . , zhang , y. 1998 , apj * 508 * , 518 . [ abn99 ] abel , t. , bryan , g. l. , & norman , m. l. 1999 , in `` evolution of large scale structure : from recombination to garching '' , eds . banday , t. , sheth , r. k. and costa , l. n. 7 . [ abn00 ] abn : abel , t. , bryan , g.l . , norman , m.l . 2000 , apj * 540 * , 39 8 . [ cosmo ] friedmann models with a cosmological constant which currently seem to fit various observational test better differ from the standard cdm model considered here only slightly at the high redshifts modeled . [ bn97 ] bryan , g.l . , norman , m.l . 1997 , in _ computational astrophysics _ , eds . clarke and m. fall , asp conference # 123 10 . [ bn99 ] bryan , g.l . , norman , m.l . 1999 , in _ workshop on structured adaptive mesh refinement grid methods _ , i m a volumes in mathematics no . 117 , ed . n. chrisochoides , p. 165 [ wc84 ] woodward , p. r. , & colella , p. 1984 , j. comput 54 * , 115 12 . [ betal95 ] bryan , g.l . , cen , r. , norman , m.l . , ostriker , j.p . & stone , j.m . 1994 , apj * 428 * , 405 13 . [ sz96 ] seljak , u. , & zaldarriaga , m. 1996 , apj * 469 * , 437 14 . [ tkmh97 ] truelove , j. k. , klein , r. i. , mckee , c. f. , holliman , j. h. , howell , l. h. & greenough , j. a. 1997 , apjl * 489 * , l179 15 . [ aazn97 ] abel , t. , anninos , p. , zhang , y. , norman , m.l . 1997 , newa * 2 * , 181 16 . [ azan97 ] anninos , p. , zhang , y. , abel , t. , and norman , m.l . 1997 , newa * 2 * , 209 . [ orel ] orel , a.e . 1987 , j.chem.phys . * 87 * , 314 18 . [ on98 ] omukai , k. & nishi , r. 1998 , apj * 508 * , 141 19 . [ tegmark97 ] tegmark , m. , silk , j. , rees , m.j . , blanchard , a. , abel , t. , palla , f. 1997 , apj * 474 * , 1 . [ bcl99 ] bromm , v. , coppi , p. s. , & larson , r. b. 1999 , apjl * 527 * , l5 21 . [ ebert ] ebert , r. 1955 , zs . [ bonnor ] bonnor , w. b. 1956 , mnras * 116 * , 351 23 . [ field65 ] field , g. b. 1965 , apj * 142 * , 531 24 . [ gp98 ] galli , d. & palla , f. 1998 , a&a * 335 * , 403 25 . [ sy77 ] sabano , y. & yoshi , y. 1977 , pasj * 29 * , 207 26 . [ silk83 ] silk , j. 1983 , mnras * 205 * , 705 27 . [ qz88 ] quinn , p.j . & zurek , w.h . 1988 , apj * 331 * , 1 28 . [ bb93 ] burkert , a. & bodenheimer , p. 1993 , mnras * 264 * , 798 29 . [ nwb80 ] norman , m.l . , wilson , j.r . & barton , r.t . 1980 , apj * 239 * , 968 30 . [ sn92 ] stone , j.m . & norman , m.l . 1992 , apjs * 80 * , 791 31 . [ soj97 ] sigl , g. , olinto , a. v. & jedamzik , k. 1997 , phys.rev.d * 55 * , 4582 32 . [ bfs97 ] barrow , j. d. , ferreira , p. g. and silk , j. 1997 , physical review letters * 78 * , 3610 33 . [ ms76 ] mouschovias , t. ch . , spitzer , l. 1976 , apj * 210 * , 326 34 . [ haehnelt ] haehnelt , m.g . 1995 , mnras * 273 * , 249 35 . [ mf99 ] mac low , m. & ferrara , a. 1999 , apj * 513 * , 142 36 . [ f98 ] ferrara , a. 1998 , apjl * 499 * , l17 37 . [ cb00 ] cen , r. & bryan , g.l . 2000 , apjl * 546 * , l81 38 . [ m92 ] monaghan , j. j. 1992 , araa * 30 * , 543 39 . [ mba01 ] machacek , m.e . , bryan , g.l . , & abel , t 2001 , apj * 548 * , 509 40 . [ essp00 ] ellison , s. l. , songaila , a. , schaye , j. & pettini , m. 2000 , , 1175 41 . [ wq00 ] wasserburg , g. j. & qian , y. 2000 , apjl * 529 * , l21 42 . [ r99 ] reichart , d. e. 1999 , apjl * 521 * , l111 43 . [ cl00 ] ciardi , b. & loeb , a. 2000 , apj * 540 * , 687 44 . t.a . happily acknowledges stimulating and insightful discussions with martin rees and richard larson . glb was supported through hubble fellowship grant hf-0110401 - 98a from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . under nasa contract nas5 - 26555 .
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we describe results from a fully self consistent three dimensional hydrodynamical simulation of the formation of one of the first stars in the universe .
dark matter dominated pre - galactic objects form because of gravitational instability from small initidal density perturbations . as they assemble via hierarchical merging , primordial gas cools through ro - vibrational lines of hydrogen molecules and sinks to the center of the dark matter potential well .
the high redshift analog of a molecular cloud is formed .
when the dense , central parts of the cold gas cloud become self - gravitating , a dense core of @xmath0 undergoes rapid contraction . at densities
@xmath1 a @xmath2 proto - stellar core becomes fully molecular due to three body formation .
contrary to analytical expectations this process does not lead to renewed fragmentation and only one star is formed .
the calculation is stopped when optical depth effects become important , leaving the final mass of the fully formed star somewhat uncertain . at this stage
the protostar is acreting material very rapidly ( @xmath3 ) .
radiative feedback from the star will not only halt its growth but also inhibit the formation of other stars in the same pre galactic object ( at least until the first star ends its life , presumably as a supernova ) .
we conclude that at most one massive ( @xmath4 ) metal free star forms per pre galactic halo ,
consistent with recent abundance measurements of metal poor galactic halo stars .
# 1_[#1 ] _
# 110^#1 # 1n _ # 1 # 1k_#1 # 1@xmath5 # 1_#1 _ # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in
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the preceding paper @xcite discovered , among other findings , the independence of the sign of the polarization @xmath6 on the boundary conditions ( bcs ) for the one - dimensional ( 1d ) quantum well ( qw ) of the width @xmath7 placed into the uniform electric field @xmath0 that is directed perpendicular to its confining surfaces located at @xmath8 : the polarization @xmath9 of the ground state for any permutation of the dirichlet ( d ) , @xmath10 and neumann ( n ) , @xmath11 edge requirements imposed on the wavefunction @xmath12 is positive for all applied voltages while its excited - state counterparts @xmath13 , @xmath14 , for the small growing fields decrease from zero at @xmath15 to the negative values , pass through the minimum and only after this start to increase crossing zero at the @xmath16- and bc - dependent intensity @xmath17 . immediately , one wonders : for any kind of the particles , is it possible to observe the total statistically averaged polarization that is negative at the small electric forces ? analysis below answers this question together with the thermodynamic calculations of the corresponding energy @xmath18 and heat capacity @xmath1 . following the previous research @xcite , the qw with the particular distribution of the bcs will be denoted by the two characters , where the first ( second ) one corresponds to the edge condition at the left ( right ) interface . similar to the discussion of the spectrum @xmath19 and polarizations @xmath13 @xcite , all energies will be measured , if not specified otherwise , in units of @xmath20 , which is a ground - state energy of the dd qw , while the unit of the electric field will be @xmath21 , and that of the polarization - @xmath22 , with @xmath23 being the particle mass and @xmath24 denoting the absolute value of the electronic charge . in addition , heat capacity is expressed below in terms of boltzmann constant @xmath25 . discussion considers canonical as well as grand canonical ensembles . in this last case , the properties are calculated both for fermions and bosons . also , frequently we draw parallels with the 1d harmonic oscillator ( ho ) with the potential ( in regular units ) @xcite @xmath26 whose energies @xmath27 , upon application of the electric voltage , are @xmath28 for this configuration , the natural units that will be used below are : for the energy , @xmath29 ; for the length , @xmath30^{1/2}$ ] ; for the electric field , @xmath31 ; and for the polarization , @xmath32 . this type of the statistical ensemble assumes that the system under consideration is in the thermal equilibrium with the much larger bath characterized by the thermodynamic temperature @xmath2 . the fundamental quantity here is the partition function @xmath33 where the summation runs over all possible quantum states , and the parameter @xmath34 is ( in regular , unnormalized units ) @xmath35 . the probability @xmath36 of finding particle in the state @xmath16 depends on the temperature and the energy @xmath37 as @xmath38 as a result , the mean value @xmath39 of any physical quantity @xmath40 is calculated as @xmath41 for the @xmath3 particles in the system , this equation has to be multiplied by @xmath3 . applying these general results to the qw with the different bcs in the electric field @xmath0 , one derives the mean values of the energy @xmath42 and polarization @xmath5 [ canonicalmeanvalue2 ] @xmath43 where in the left - hand side we have explicitly underlined that they are functions of the temperature @xmath2 ( through the parameter @xmath34 ) and electric field [ through the corresponding dependence of @xmath19 and @xmath13 ] . equivalently , eq . can be written as : @xmath45 heat capacity at the constant volume @xmath1 is a work that has to be done to change the temperature of the system by one degree and , as a result of this , it is calculated as a derivative of the total energy with respect to the temperature @xmath2 : @xmath46 where regular , unnormalized units have been used . applying this generic definition to the canonical distribution from eq . , one gets fluctuation - dissipation theorem @xcite @xmath47 where , for convenience of the notation , the subscript @xmath48 has been dropped . energies @xmath37 and polarizations @xmath6 for the qw were calculated before @xcite while for the ho they are : [ ho_energypolarization ] @xmath49 note that , contrary to the hard - wall qw @xcite , for its ho counterpart the polarization is at any voltage a linear function of the field and is the same for all levels . accordingly , its mean value for the one particle is equal to @xmath0 too while the energy becomes : @xmath50 as a result , the electric field does _ not _ affect the ho canonical heat capacity , which reads @xcite : @xmath51 one can derive limiting cases of these dependencies : for the small temperatures ( @xmath52 ) : [ ho_canonicallimit1 ] @xmath53 for the large temperatures ( @xmath54 ) : [ ho_canonicallimit2 ] @xmath55 ( a ) heat capacity @xmath56 and ( b ) mean energy @xmath57 as a function of the normalized temperature @xmath58 for the canonical ensemble and pure dirichlet ( dotted line ) , neumann ( solid curve ) and nd ( dashed line ) qw at zero electric field . ] before discussing the electric field influence on the thermodynamic properties of the hard - wall qw , let us address first the voltage - free configuration . plugging in the well known expressions for the zero - field energies @xmath59 into eq . , one gets after some algebra : [ canonicalmeanenergy5 ] @xmath60 here , @xmath61 , @xmath62 , are theta functions @xcite . for small temperatures , @xmath52 , these equations degenerate to [ canonicalmeanenergy4 ] @xmath63 utilizing transformation properties of the theta functions @xcite [ thetatransformation1 ] @xmath64 one derives the energies in the opposite limit of the high temperatures : [ canonicalmeanenergy6 ] @xmath65 the corresponding heat capacities @xmath1 are calculated by applying the right - most part of eq . to the above dependencies ; in particular , one has for the cold " qw , @xmath52 : [ canonicalheatcapacity1 ] @xmath66 for the hot thermal bath : [ canonicalheatcapacity2 ] @xmath67 mean energy @xmath57 of the canonical ensemble in terms of the electric field @xmath68 and temperature @xmath58 for all permutations of the bcs . in each of the panels , the corresponding type of the edge requirements is denoted by the two characters . ] statistically averaged energies and corresponding heat capacities are shown in fig . [ electr0 ] . at the zero temperature , the total internal energy reduces to the ground - state energy and the heat capacity is zero , as it should be . as it follows from eqs . and , their avalanche growth from the @xmath69 values for the purely neumann qw takes place at the smaller temperatures as compared to the mixed bcs , which , in turn , is followed by the quantities for the dirichlet structure . this is explained by the growing difference between the two lowest energies for just consecutively mentioned bc configurations : the quantity @xmath70 at the zero field is the smallest ( largest ) for the neumann ( dirichlet ) structure : @xmath71 a remarkable feature of the heat capacity dependence is its nonmonotonic behaviour for the neumann qw : at @xmath72 ( @xmath73 ) it reaches a pronounced maximum @xmath74 that is followed by the minimum of @xmath75 located at @xmath76 ( @xmath77 ) . if the maximum is observed quite exactly by keeping only the first term in the parentheses of the right - hand side of eq . , the emergence and precision of the location and magnitude of the second extremum are described better by keeping more terms in the same expansion . physically , this nonmonotonicity of the heat capacity is attributed to the structure of the energy spectrum , see eqs . and ; namely , very small temperature promotes the particle mainly to the first excited level that is only one unit above the ground state , @xmath78 , with the contribution of the other levels being negligibly small due to the almost vanishing exponents in eq . or , equivalently , in eqs . and ; as a result , the heat capacity grows rapidly . for the larger temperatures , the occupations of the higher lying levels become essential ; however , the transitions to them are more difficult since the difference between , e.g. , second and first excited states @xmath79 is three times larger than that between the latter and the ground level . accordingly , the same speed of the heat capacity change can not be sustained what results in the observed maximum . for the other bcs , the ratio @xmath80 is smaller than for the neumann qw , as it follows from eq . : @xmath81 and @xmath82 ; as a result , for them no extrema are observed on the @xmath83 dependence at @xmath84 . mathematically , the drop of the nn specific heat is caused by the interplay between the counterbalancing terms @xmath85 and @xmath86 in eq . as the temperature grows . keeping only the first exponent in the parentheses of the right - hand side of this equation produces @xmath87 while the same procedure applied to the other bcs , see eqs . and , results in @xmath88 and @xmath89 , which are , respectively , three and two times smaller and lie beyond the range of the validity of these expansions . accordingly , for the latter two configurations , it is essential to keep other items in the corresponding series in order for them to be correct at the decreasing @xmath34 , and these extra exponents eliminate the resonance of the first - term approximation while for the neumann qw the ( negative ) second component simply improves the previous result . note that the ho leading term of the capacity expansion from eq . also results in @xmath90 ; however , the subsequent ( all positive ) items in the series wipe out the extremum . very broad and gentle asymmetric maximum is observed at @xmath91 for the dirichlet qw while for the mixed bc the heat capacity is a monotonically increasing function of the temperature , which , at quite large @xmath2 , rapidly approaches the asymptotic value of one half . on the contrary , the heat capacity of the symmetric qws reaches the same limit much slower , as eq . asserts and panel ( a ) of fig . [ electr0 ] exemplifies . note that the ho internal energy for the high temperatures is twice of that for the hard - wall qw : @xmath92 and @xmath93 in regular units , respectively . from point of view of classical equilibrium statistics that is applicable for @xmath94 , this difference is explained by the fact that in the former case the kinetic and potential parts of the motion make equal contributions of @xmath93 to the total energy @xcite while for the latter system it is the kinetic energy only that determines @xmath42 as the qw potential is zero . as a direct consequence of this , the qw heat capacities in the same limit are one half of their ho counterpart . applied electric field modifies the energy spectrum what , in turn , affects the thermodynamic properties of the wells . it was shown that the voltage increases the difference @xmath95 between the ground and first excited levels for any permutation of the bcs ( the only exception is the nd case at the small fields , see equations ( 50 ) in @xcite ) ; accordingly , the larger temperature is needed to push out the electron from its lowest state . this is reflected in figs . [ energycanonical ] and [ heatcapacitycanonical ] where the energy @xmath96 and heat capacity @xmath97 , respectively , are shown . it is seen that the @xmath98 range where the mean energy does not change appreciably from the ground - state value gets wider for the stronger intensities @xmath0 . the same is true for the heat capacity where the plateau with its almost zero value grows with the field . the increasing voltage wipes out the nn minimum of the heat capacity simultaneously moving the maximum to the higher temperatures and increasing its magnitude . for each of the mixed bcs , it also creates a maximum that was absent at @xmath15 . mentioned above dd extremum of the heat capacity gets narrower and its peak increases with the field growing . recalling again the language of the classical statistical mechanics @xcite , one qualitatively explains the larger heat capacities at the nonzero fields by the contribution of the electric potential ; namely , the thermally averaged value of the potential energy @xmath99 is : @xmath100 this classical expression is applicable to our quantum system for the large temperatures only : @xmath101 then , the potential contribution to the heat capacity reads : @xmath102 note that , contrary to the ho , the kinetic and potential contributions to the heat capacity in this case , generally , are not equal to each other . let us also mention once again that the electric field does _ not _ affect at all the ho heat capacity , see eq . , since it simply shifts all the levels by the same amount , according to eq . . the same as in fig . [ energycanonical ] but for the polarization @xmath103 ] fig . [ polarizationcanonical ] depicts statistically averaged polarizations @xmath104 in terms of @xmath0 and @xmath98 . growing temperature leads to the decrease of @xmath104 for all electric fields ; however , thermal energy is not strong enough to make the total dipole moment negative : for any bc the polarization stays positive . to understand the statistical properties better , it is instructive to consider the case of the low temperatures . for the small voltages , as a first approximation , we also accept undisturbed by the field energies from eq . . then , one has following dependencies : [ polarizationlimit1 ] @xmath105 these equations were derived under the assumption of @xmath106 , but the general property stating that the first - order temperature correction is determined by @xmath107 , holds for any electric intensities . for the small fields , this difference is negative @xcite what naturally explains the decrease of the total polarization with the temperature growing . in the opposite limit of the high voltages , the polarizations of the qw with the uniform bcs tend to the same level - independent value of one - half @xcite what requires larger temperatures in order to see the deviation of @xmath5 from its @xmath69 value . this is exemplified in fig . [ polarizationcanonical ] where the temperature - independent plateau at @xmath69 widens with the field growing . for the mixed edge requirements , this limiting quantity is supplemented by the term that is proportional to @xmath108 with its sign being determined by the orientation of the bcs @xcite ; so , for the dn case it is actually possible to observe the increase of the polarization with the temperature growing from zero . this feature is not shown in the corresponding panel of the figure since it takes place beyond the figure range @xmath109 . grand canonical distribution is used for the description of the quantum system that , in addition to the thermal balance with the external reservoir , is also in the chemical equilibrium with it . accordingly , the structure can exchange the energy as well as particles with the heat bath . so , the number of the quantum corpuscles @xmath3 in it can be changed . the fundamental role in this case is played by the chemical potential @xmath110 , which is defined from the condition @xmath111 where the upper sign corresponds to the fermi - dirac ( fd ) distribution while the lower one describes bose - einstein ( be ) particles . physically , the difference between these two statistics is in the fact that each quantum level can not be occupied by more than one fermion ( pauli exclusion principle ) while the arbitrary number of bosons can coexist in the same state . the distribution function now depends not only on the energies @xmath37 but also on the number of the particles in the system @xmath3 ; namely , for the physical quantity @xmath40 its grand canonical average value @xmath112 is @xmath113 applying the distribution from eq . for the calculation of the heat capacity , eq . , one finds that its grand canonical value @xmath114 is @xmath115 ^ 2}\,e^{(e_n-\mu)\beta},\ ] ] where the chemical potential , which , in the case of fermions , is also frequently called the fermi level , is calculated , as stated above , from eq . . physically , the value of @xmath110 corresponds to the energy that is needed for changing by one the number of the particles in the system : @xmath116 for calculating its partial derivative with respect to the temperature , one should consider eq . as a condition of zeroing of the implicit function @xmath117 of the chemical potential in terms of the variables @xmath34 and @xmath3 : @xmath118 the rule of differentiating implicit functions states @xcite : @xmath119 as a result , one finds : @xmath120 ^ 2}\,e^{(e_n-\mu)\beta}}{\sum_n\frac{1}{\left[e^{(e_n-\mu)\beta}\pm1\right]^2}\,e^{(e_n-\mu)\beta } } .\ ] ] the boson statistics is used for the particles with the integer spin such as photons or cooper pairs in superconductors while the fd distribution is applied for the system of the constituents with the half integer spins ; for example , the electron with its spin of @xmath121 has , for the same energy , two projections of its spin equal to @xmath122 . however , in our discussion below we will neglect this fact and will assume that the number of the fermions for each energy @xmath37 is not larger than one . [ fermigranddirichlet ] depicts the fd chemical potential for the pure dirichlet qw as a function of the electric field and temperature for several numbers @xmath3 . qualitatively , the same features are characteristic for other bc permutations too . there are several distinct regions of the fermi energy @xmath123 dependence on the temperature . from its @xmath124 value at @xmath69 it rapidly grows as @xmath125 until it reaches and stays exactly at the value of @xmath126 which is due to the interaction of the two corresponding levels that , at the zero temperature , were the highest occupied and lowest unoccupied states . the width of this @xmath2-independent plateau is determined by the number of the particles @xmath3 and electric field @xmath0 . for the still higher temperatures , the chemical potential for @xmath127 decreases while for the larger number of particles , @xmath128 , it grows with @xmath98 , reaches maximum and only after that decreases , passes zero at @xmath129 and continues to decline into the negative part of the spectrum . for the nonpositive chemical potentials , @xmath130 , eq . can be cast into the form @xmath131 for the zero field , @xmath15 , the energy spectrum from eqs . and simplifies this equation as follows : for the ho : [ numbern_3 ] @xmath132 putting here the chemical potential equal to zero , @xmath133 , leads to the calculation of @xmath129 . in known to us literature @xcite , there are no analytical expressions for these infinite series . however , for the very small @xmath34 , the @xmath134 terms in the above equations make the most significant contributions producing the following dependencies : for the ho : [ chemicalpotentiallimit3 ] @xmath135 superscripts @xmath136 and @xmath137 in eq . stand for any of the values of @xmath138 and/or @xmath3 . [ fermigranddirichlet ] manifests that , for the larger @xmath3 , these asymptotics are achieved at the higher temperatures . as a result , the grand canonical mean energy @xmath139 reads in the same limit : [ grandenergylimit1 ] @xmath140 what , by means of eq . , immediately leads to the associated heat capacities @xmath114 : [ grandcanonicalheatcapacity2 ] @xmath141 a comparison of these remarkable results with eqs . , and confirms the general property , which states that for the large temperatures there is no difference between canonical and grand canonical distributions @xcite . however , for the small @xmath2 these two statistics produce very different features . [ heatcapacitygranddirichlet ] shows the fd heat capacity of the pure dirichlet qw in terms of the temperature and electric field for the different @xmath3 corresponding to their counterparts from fig . [ fermigranddirichlet ] . it is seen that , for the larger number of the particles , the asymptotics from eq . is achieved at the higher @xmath2 . at the zero field , a prominent characteristic of the heat capacity dependence for the one particle ( top left panel of fig . [ heatcapacitygranddirichlet ] ) is a salient maximum @xmath142 observed at @xmath143 , i.e. , at the right edge of the plateau from eq . . accordingly , we attribute this extremum to the different behaviour of the chemical potential for @xmath127 and @xmath128 ; namely , as it was mentioned during discussion of fig . [ fermigranddirichlet ] , for one particle the fermi energy decreases after the flat part from eq . while for any other number @xmath3 it grows with @xmath2 . thus , their contributions to the heat capacity from eq . are opposite to each other what results in the resonance that is observed for the one particle only . even though the shape of this maximum is quite similar to its nn counterpart for the canonical ensemble , see sec . [ sec_canonical ] , its physical explanation is completely different . first , we point out that the very similar extrema are calculated also for the nd ( with @xmath144 and @xmath145 ) and pure neumann ( @xmath146 and @xmath147 ) qws too . the fact that the three @xmath148 are almost the same and the ratios of the three temperatures @xmath149 are practically equal to those of @xmath150 from eq . , undoubtedly proves that the origin of this effect is the bc independent one and that the interplay between the two lowest states plays a dominant role in it . to understand these resonances , let us recall that , for the very small temperatures , the properties of the fd well are determined only by the highest occupied level and its interaction with the nearest ( empty at @xmath69 ) above lying state , what is reflected in the extremely rapid approach by the chemical potential to the energy from eq . that is located exactly in the middle between them . for @xmath128 , a contribution from the lower lying members in this regime is negligibly small and can be safely neglected , while for the one - electron qw this addition is absent by definition . further growth of the temperature increases thermal energy but it is still too `` weak '' to compel the corpuscles , which at @xmath69 lied below the fermi energy , to contribute to the heat capacity . only at the right edge of the plateau , the thermodynamic quantum @xmath92 becomes strong enough and forces other particles to donate to @xmath123 and @xmath151 . therefore , for @xmath128 the heat capacity is a quite smoothly varying function of the temperature . however , for @xmath127 there are no such additional donors that aid to support the continuous growth of the heat capacity , which can not be sustained by the one particle only . as a result , the specific heat reaches maximum and drops . this qualitative physical reasoning can be corroborated by the simple quantitative mathematical analysis . fermi level from eq . defines the corresponding mean energy and heat capacity as : [ grandcanonicallimit2 ] @xmath152 on the other hand , for the chemical potential from eq . these quantities for one fermion , @xmath127 , become : [ grandcanonicallimit3 ] @xmath153,\quad\beta\gtrsim1,\end{aligned}\ ] ] where we take into consideration only the two lowest states . for the pure neumann qw without the field , this last expression takes an especially simple form : @xmath154 this function has a pronounced maximum of @xmath146 at @xmath155 . a perfect coincidence with the provided above exact results justifies a validity of the two - level approximation and proves that the electron transitions between them determine the specific heat resonance . it is also very instructive to contrast eq . with its canonical counterpart from eq . in sec . [ sec_canonical ] . the comparison shows that the magnitude of the grand canonical neumann extremum is almost two times larger and it is achieved at more than two times lower temperature . applied field @xmath0 smooths out and widens this maximum simultaneously increasing the heat capacity . similar to the canonical ensemble , this growth is explained by the contribution of the electrostatic potential . however , the electric influence is drastically decreased by the growing number of the particles in the qw ; for example , right - bottom panels exhibit the almost full independence of the fermi energy and heat capacity on the intensity @xmath0 already for @xmath156 . this is explained by the properties of the energy spectrum in the electric field when the higher lying states ( which , in the case of the fd distribution , determine the features of the system ) are less affected by the applied voltage @xcite . [ polarizationgranddirichlett0 ] demonstrates zero - temperature fd polarization for all possible bcs and several numbers @xmath3 . as the well accommodates more fermions , the total polarization becomes smoother function of the electric field . figure reveals that , independently of the edge demands , the magnitude of @xmath5 at @xmath157 grows linearly with the voltage and the slope of this almost straight line diminishes with @xmath3 . for any number of fermions , the total polarization remains positive at the arbitrary voltage . nonzero temperature leads to the dependencies that qualitatively are similar to the canonical patterns , fig . [ polarizationcanonical ] , and , because of this , the corresponding polarizations are not shown here . total polarization @xmath103 as a function of the electric field @xmath158 at zero temperature , @xmath69 , for different number of fermions @xmath3 that is depicted next to the corresponding curve . ] next , let us discuss bosonic structures . remarkable experimental observations of the be condensation in the vapors of rubidium @xcite and sodium @xcite spurred an avalanche of the research on the subject predicted almost ninety years ago @xcite , see , e. g. , reviews @xcite . theoretically , the main effort was devoted to the calculation of the properties of the be systems in the 3d isotropic or anisotropic harmonic traps @xcite and their existence / nonexistence in lower dimensions @xcite . however , other forms of the confining potentials @xcite , including the 3d box with the periodic @xcite or uniform @xcite bcs , were also discussed with the comparative analysis of their influence of the properties of the trap @xcite . from this point of view , an inclusion of the electric voltage and different bcs presents a generalization of the previous analysis . moreover , overwhelming majority of the research concentrated on the analysis of the be systems in the thermodynamic limit when the number of the particles and the volume containing them tend to infinity while the the density is kept constant . in this approximation , the infinite series above in this section can be safely replaced by the integrals @xcite . considering @xmath3 changing from one to the large values might help to understand the formation of the be processes with the the number of the particles growing . first , we state that eqs . - stay valid for the be statistics too since they were obtained as a result of retaining the first term in the series from eq . . in the opposite limit of the very small temperatures , it is elementary to derive : [ belimit1 ] @xmath159 ( a ) critical temperatures @xmath160 and ( b ) corresponding to them polarizations @xmath161 from eq . as functions of the applied electric field @xmath158 for different number @xmath162 of bosons that are depicted near the corresponding curves . solid ( dotted ) lines are for the pure dirichlet ( neumann ) bc while their dash ( dash - dotted ) counterparts denote dn ( nd ) geometry . in panel ( b ) , thin horizontal line is zero polarization . ] an important characteristic of the be system is its critical temperature @xmath4 . it corresponds to the situation when the chemical potential is equal to the energy of the lowest level , @xmath163 , and the number of the particles in this state @xmath164 is zero what leads to the implicit mathematical equation for finding @xmath165 @xmath166 physically , it is the largest temperature at which the be condensation still can be observed , and at the lower @xmath2 the fraction @xmath167 of the particles in the ground state will increase until at @xmath69 it becomes unity : @xmath168 fig . [ criticaltemppolariz](a ) shows dependencies of the critical parameter @xmath169 on the applied voltage for all possible bcs and several numbers @xmath3 . in accordance with the previous results @xcite , the temperature @xmath4 increases with @xmath3 . in the absence of the fields , the lowest ( highest ) temperature is observed for the pure neumann ( dirichlet ) qw what is a reflection of the corresponding spectrum from eq . and the energy difference between the affiliated states , see eq . . electric field leads to the modification of the mutual location of the levels on the energy axis ; in particular , at the small voltages , the two lowest states move closer to each other for the nd geometry while the difference @xmath170 grows with the field for all @xmath0 and any other bc configuration @xcite . as a result , the critical temperature for the former edge requirement decreases with the growing from zero field , passes through minimum and then unrestrictedly grows with the electric intensity while for all other bcs it is a continuously increasing function of @xmath0 . at the high voltages , the energy spectrum is determined mainly by the condition at the right wall @xcite what leads to almost the same critical temperature for , e. g. , the nn and dn wells . note that in this regime , contrary to the zero fields , the dirichlet requirement is more favorable to the formation of the be condensate as compared to the neumann interface . this is explained by the larger level separation for the latter geometry at the high voltages @xcite . as the ground - state polarization is positive for all fields and bcs @xcite , one can expect that at the onset of the be condensation the total statistically averaged dipole moment will , at the small voltages , be negative . this is exactly what is observed in panel ( b ) of fig . [ criticaltemppolariz ] that shows the polarizations @xmath171 corresponding to the critical temperatures @xmath169 from panel ( a ) . they were calculated from equation @xmath172 and , as stated above , the critical temperature @xmath169 was found from eq . . the characteristic features of the critical polarizations basically follow the properties of the first excited state : from zero they decrease with the growth of the field , reach minimum after which they increase . however , for the large voltages many upper lying states are occupied and contribute to the total dipole moment . as a result , the high - field @xmath171 for one boson is considerably smaller than @xmath173 . the absolute value of the negative polarization at the extremum grows with the number of the particles with the largest one , at the fixed @xmath3 , being observed for the pure neumann qw followed by its nd counterpart what is a replica of the similar behaviour for the first excited level @xcite . for the temperatures below @xmath4 , the nonzero occupation of the ground state contributes a positive term to the polarization what leads to the gradual disappearance of the negative region of the total dipole moment @xmath5 with the decreasing temperature until at @xmath69 it becomes @xmath174 , which is positive for any bc and arbitrary fields @xcite . bosonic heat capacity @xmath56 of the pure neumann qw as a function of the applied electric field @xmath68 and temperature @xmath58 for ( a ) @xmath175 , ( b ) @xmath176 and ( c ) @xmath177 . note that in the last two cases the temperature is scaled in units of the critical temperature @xmath178 while the corresponding @xmath179 axes measure specific heat per particle @xmath180 . panel ( d ) shows the chemical potential @xmath181 for @xmath177 with the corresponding heat capacity depicted in part ( c ) . ] as a final example , fig . [ boseneumann ] exhibits evolution of the heat capacity and chemical potential with the varying electric field and temperature for the pure neumann qw . it is seen that for the small number of bosons , say , @xmath182 in panel ( a ) , the applied voltage leads , at quite warm sample , to the increase of @xmath1 while at the small @xmath2 , the width of the temperature - independent zero - capacity plateau increases with the field . these features were discussed before for the canonical ensemble . increasing the number of the particles in the well leads to the suppression of the voltage dependence , as a transition from panel ( a ) to ( b ) with @xmath183 and ( c ) for @xmath184 vividly demonstrates . no any noticeable field dependence is seen there in the range @xmath109 . it is well known that for some potentials , such as , e.g. , the 3d isotropic harmonic trap @xcite , the heat capacity has a cusp - like peculiarity as it passes through the critical temperature while for the 1d quadratic potential it is a smooth function of @xmath2 @xcite . fig . [ boseneumann ] exemplifies that no any peculiarity is observed for the 1d hard - wall potential with neumann surfaces and arbitrary applied electric fields . our calculations confirm that the same is true for any other bcs . finally , panel ( d ) shows that the chemical potential @xmath110 is a monotonically decreasing function of both the electric field @xmath0 and temperature @xmath2 . it is seen that the growing temperature diminishes the voltage influence on the chemical potential . rigorous mathematical treatment of the qw with miscellaneous bcs under the applied voltage revealed a strong influence of the interplay between them on the thermodynamic properties of the structure . in particular , without the field the differences of the energy spectrum lead , for the canonical ensemble , to the conspicuous maximum followed by the minimum of the heat capacity @xmath1 on the temperature axis for the nn quantum box while for the other edge requirements no such adjacent extrema are observed . modification of the specific heat and statistically averaged polarization in the field is qualitatively explained by the influence of the associated electrostatic potential . numerical calculations , which predicted , for the flat potential with the arbitrary bc , a salient maximum of @xmath1 as a function of @xmath2 for one fermion and its absence for the larger @xmath3 , were corroborated by the two - level model that allows simple analytical treatment with its predictions perfectly coinciding with the exact results . from this , a clear physical explanation of this phenomenon follows that is based on the analysis of the associated fermi energy . it is predicted that the applied field , in general , favors the formation of the be condensate , and the differences and similarities of this process for the different bcs are discussed . the thermally averaged dipole moment is shown to take the negative values in some ranges of the fields and temperatures . it is also argued that for the larger number of either fermions or bosons in the qw , the influence of the electric field on the thermodynamic properties diminishes . dirichlet and neumann conditions are the limiting cases of the so called robin bc @xcite @xmath185 where @xmath186 is an inward unit normal to the surface , and the parameter @xmath187 has a dimension of length and is called the extrapolation distance . its variation allows a continuous transformation from the dirichlet ( @xmath188 ) to the neumann ( @xmath189 ) situation . without the field , especially intriguing are the properties of the qw at the small negative robin lengths , @xmath190 , when , in addition to the positive spectrum , two almost degenerate odd and even levels with the energies @xmath191 are created @xcite . analysis of the robin qw in the electric field might present an interesting extension of the present research . this project was supported by deanship of scientific research , college of science research center , king saud university . 100 o. olendski , ann . phys . 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thermodynamic properties of the one - dimensional ( 1d ) quantum well ( qw ) with miscellaneous permutations of the dirichlet ( d ) and neumann ( n ) boundary conditions ( bcs ) at its edges in the perpendicular to the surfaces electric field @xmath0 are calculated . for the canonical ensemble ,
analytical expressions involving theta functions are found for the mean energy and heat capacity @xmath1 for the box with no applied voltage . pronounced maximum accompanied by the adjacent minimum of the specific heat dependence on the temperature @xmath2 for the pure neumann qw and their absence for other bcs
are predicted and explained by the structure of the corresponding energy spectrum . applied field leads to the increase of the heat capacity and formation of the new or modification of the existing extrema what is qualitatively described by the influence of the associated electric potential .
a remarkable feature of the fermi grand canonical ensemble is , at any bc combination in zero fields , a salient maximum of @xmath1 observed on the @xmath2 axis for one particle and its absence for any other number @xmath3 of corpuscles .
qualitative and quantitative explanation of this phenomenon employs the analysis of the chemical potential and its temperature dependence for different @xmath3 .
it is proved that critical temperature @xmath4 of the bose - einstein ( be ) condensation increases with the applied voltage for any number of particles and for any bc permutation except the nd case at small intensities @xmath0 what is explained again by the modification by the field of the interrelated energies .
it is shown that even for the temperatures smaller than @xmath4 the total dipole moment @xmath5 may become negative for the quite moderate @xmath0 .
for either fermi or be system , the influence of the electric field on the heat capacity is shown to be suppressed with @xmath3 growing .
different asymptotic cases of , e.g. , the small and large temperatures and low and high voltages are derived analytically and explained physically .
parallels are drawn to the similar properties of the 1d harmonic oscillator , and similarities and differences between them are discussed .
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in the last years it has been largely developed the study of intrinsic submanifolds inside the heisenberg groups @xmath3 or more general carnot groups , endowed with their carnot - carathodory metric structure , also named sub - riemannian . by an intrinsic regular ( or intrinsic lipschitz ) hypersurfaces we mean a submanifold which has , in the intrinsic geometry of @xmath3 , the same role like a @xmath4 ( or lipschitz ) regular graph has in the euclidean geometry . intrinsic regular graphs had several applications within the theory of rectifiable sets and minimal surfaces in cc geometry , in theoretical computer science , geometry of banach spaces and mathematical models in neurosciences , see @xcite , @xcite and the references therein . we postpone complete definitions of @xmath3 to section [ s : hnrecalls ] . we only remind that the heisenberg group @xmath5 is the simplest example of carnot group , endowed with a left - invariant metric @xmath6 ( equivalent to its carnot - carathodory metric ) , not equivalent to the euclidean metric . @xmath3 is a ( connected , simply connected and stratified ) lie group and has a sufficiently rich compatible underlying structure , due to the existence of intrinsic families of left translations and dilations and depending on the horizontal vector fields @xmath7 . we call intrinsic any notion depending directly by the structure and geometry of @xmath3 . for a complete description of carnot groups @xcite are recommended . as we said , we will study intrinsic submanifolds in @xmath3 . an intrinsic regular hypersurface @xmath8 is locally defined as the non critical level set of an horizontal differentiable function , more precisely there exists locally a continuous function @xmath9 such that @xmath10 and the intrinsic gradient @xmath11 exists in the sense of distributions and it is continuous and non - vanishing on @xmath12 . intrinsic regular hypersurfaces can be locally represented as @xmath13-graph by a function @xmath14 , where @xmath15 , through an implicit function theorem ( see @xcite ) . in @xcite the parametrization @xmath16 has been characterized as weak solution of a system of non linear first order pdes @xmath17 , where @xmath18 and @xmath19 , ( see theorem [ miotheorem ] ) . by an intrinsic point of view , the operator @xmath20 acts as the intrinsic gradient of the function @xmath21 . in particular it can be proved that @xmath16 is a continuous distributional solution of the problem @xmath17 with @xmath22 if and only if @xmath16 induces an intrinsic regular graph , ( see @xcite ) . let us point out that an intrinsic regular graph can be very irregular from the euclidean point of view : indeed , there are examples of intrinsic regular graphs in @xmath23 which are fractal sets in the euclidean sense ( @xcite ) . the aim of our work is to characterize intrinsic lipschitz graphs in terms of the intrinsic distributional gradient . it is well - know that in the euclidean setting a lipschitz graph @xmath24 , with @xmath25 can be equivalently defined * by means of cones : there exists @xmath26 such that @xmath27 for each @xmath28 , where @xmath29 ; * in a metric way : there exists @xmath26 such that @xmath30 for every @xmath31 ; * by the distributional derivatives : there exist the distributional derivatives @xmath32 provided that @xmath33 is a regular connected open bounded set . intrinsic lipschitz graphs in @xmath3 have been introduced in @xcite , by means of a suitable notion of intrinsic cone in @xmath3 . as consequence , the metric definition ( see definition [ d : deflip ] ) is given with respect to the the graph quasidistance @xmath34 , ( see ) i.e the function @xmath35 is meant lipschitz in classical metric sense . this notion turns out to be the right one in the setting of the intrinsic rectifiability in @xmath3 . indeed , for instance , it was proved in @xcite that the notion of rectifiable set in terms of an intrinsic regular hypersurfaces is equivalent to the one in terms of intrinsic lipschitz graphs . we will denote by @xmath36 the class of all intrinsic lipschitz function @xmath37 and by @xmath38 the one of locally intrinsic lipschitz functions . notice that @xmath36 is not a vector space and that @xmath39 where @xmath40 and @xmath41 denote respectively the classes of euclidean lipschitz and @xmath42-hlder functions in @xmath33 . for a complete presentation of intrinsic lipschitz graphs @xcite are recommended . the first main result of this paper is the characterization of a parametrization @xmath37 of an intrinsic lipschitz graph as a continuous distributional solution of @xmath17 , where @xmath43 . [ t : firsttheorem ] let @xmath44 be an open set , @xmath45 be a continuous function and @xmath46 . @xmath47 if and only if there exists @xmath48 such that @xmath16 is a distributional solution of the system @xmath17 in @xmath33 . we stress that this is indeed different from proving a rademacher theorem , which is more related to a pointwise rather than distributional characterization for the derivative , see @xcite . nevertheless , we find that the density of the ( intrinsic ) distributional derivative is indeed given by the function one finds by rademacher theorem . we also stress that there are a priori different notions of _ continuous _ solutions @xmath49 to @xmath17 , which express the lagrangian and eulerian viewpoints . they will turn out to be equivalent descriptions of intrinsic lipschitz graphs , when the source @xmath1 belongs to @xmath50 . this is proved in section [ s : furthereq ] and it is summarized as follows . [ t : othertheorem ] let @xmath51 be a continuous function . the following conditions are equivalent 1 . [ item : distr ] @xmath16 is a distributional solution of the system @xmath17 with @xmath52 ; 2 . [ item : lagr ] @xmath16 is a broad solution of @xmath17 , i.e. there exists a borel function @xmath53 s.t . * @xmath54 @xmath55-a.e . @xmath56 ; * for every continuous vector field @xmath57 ( @xmath58 ) having an integral curve @xmath59 , @xmath60 is absolutely continuous and at points @xmath61 of differentiability it satisfies @xmath62 in the statement , @xmath63 denotes the set of measurable bounded functions from @xmath64 to @xmath65 , while @xmath50 denotes the equivalence classes of lebesgue measurable functions in @xmath63 which are identified when differing on a lebesgue negligible set . we will keep this notation throughout the paper : its relevance is remarked by examples [ ex:1 ] , [ ex:2 ] below . [ [ outline - of - the - proofs ] ] * outline of the proofs * + + + + + + + + + + + + + + + + + + + + + + + with the intention of focussing on the nonlinear field , we fix the attention on the case @xmath66 . the variables will be denoted by @xmath67 and @xmath68 , and the subscripts @xmath69_{t}$ ] , @xmath69_{z}$ ] will denote the distributional derivatives @xmath70 , @xmath71 in the euclidean sense w.r.t . these variables . given a continuous distributional solution @xmath72 of the pde @xmath73_{t}=w(z , t)\qquad { \rm for}\ ( z , t)\in { { \omega}},\ ] ] we first prove that it is lipschitz when restricted along any characteristic curve @xmath74 , where @xmath75 . the proof follows a previous argument by dafermos ( see lemma [ l : daflipschitz ] ) . by a construction based on the classical existence theory of odes with continuous coefficients , we can then define a change of variable @xmath76 which straightens characteristics . this change of variables does not enjoy @xmath77 or lipschitz regularity , it fails injectivity in an essential way , though it is continuous and we impose an important monotonicity property . this monotonicity , relying on the fact that we basically work in dimension @xmath78 , is the regularity property which allows us the change of variables . as we exemplify below , we indeed have an approach different from providing a regular lagrangian flow of ambrosio - di perna - lions s theory , and it is essentially two dimensional . after the change of variables , the pde is , roughly , linear , and we indeed find a family of odes for @xmath2 on the family of characteristics composing @xmath79 , with coefficients which now are not anymore continuous , but which are however bounded . by generalizing a lemma on odes already present in @xcite , we prove the @xmath42-hlder continuity of @xmath16 on the vertical direction ( @xmath80 constant ) , and a posteriori in the whole domain . this are the main ingredients for establishing that @xmath2 defines indeed a lipschitz graph : given two points , we connect them by a curve made first by a characteristic curve which joins the two vertical lines through the points , then by the remaining vertical segment . we manage this way to control the variation of @xmath2 between the two points with their graph distance @xmath81 , checking therefore the metric definition of intrinsic lipschitz graphs . the other implication of theorem [ t : firsttheorem ] is based on the possibility of suitably approximating an intrinsic lipschitz graph with intrinsic regular graphs . a geometric approximation is provided by @xcite . we also provide a more analytic , and weaker , approximation as a byproduct of the change of variable @xmath79 which straightens characteristics , by mollification ( see the proof of theorem [ t : conversedafermos ] ) . we stop now for a while in order to clarify the features of the statement in theorem [ t : othertheorem ] , and why it is so important to differentiate between @xmath63 and @xmath50 . lagrangian formulations are affected by altering the representative , as the following example stresses . [ ex:1 ] let @xmath82 and let @xmath83 be the functions defined as @xmath84 then it is easy to verify that @xmath16 is a continuous distributional solution of @xmath85 consider the specific characteristic curve @xmath86 . even if @xmath87 , the derivative of @xmath2 along this characteristic curve is not the right one : @xmath88 equation holds however on every characteristic curves provided we choose correctly an @xmath89-representative @xmath90 of the source @xmath1 : it is enough to consider @xmath91 notice that @xmath92 for @xmath93-a.e . @xmath94.@xmath95 ( -1.5,0 ) ( 5.5,0 ) node[above ] @xmath67 ; ( 2,0 ) ( 2,5 ) node[right ] @xmath80 ; ( -1 , -1 ) rectangle ( 5 , 5 ) ; in .25,.75, ... ,3.25 plot[domain=:4 ] ( 1/4*(-)^2 , ) ; in 0,.5, ... ,3.25 plot[domain=:4 ] ( 1/4*(-)^2 , ) ; in -3.25,-2.75, ... ,-0.25 plot[domain=0:4 ] ( 1/4*(-)^2 , ) ; in -3,-2.5, ... ,-0.25 plot[domain=0:4 ] ( 1/4*(-)^2 , ) ; in .25,.75, ... ,3.25 plot[domain=:4 ] ( -1/4*(-)^2,-+4 ) ; in 0,.5, ... ,3.25 plot[domain=:4 ] ( -1/4*(-)^2,-+4 ) ; in -3.25,-2.75, ... ,-0.25 plot[domain=0:4 ] ( -1/4*(-)^2,-+4 ) ; in -3,-2.5, ... ,-0.25 plot[domain=0:4 ] ( -1/4*(-)^2,-+4 ) ; ( -1.5,-2.5 ) ( 5.5,-2.5 ) node[above ] @xmath67 ; ( 2,-2.5 ) ( 2,-1 ) node[right ] @xmath96 ; plot[domain=0:1.25 ] ( ^2,-2.5 ) ; plot[domain=-1.25:0 ] ( ^2,--2.5 ) ; ( 2,0)(2,4 ) ; ( -1.5,0 ) ( 5.5,0 ) node[above ] @xmath67 ; ( 2,0 ) ( 2,5 ) node[right ] @xmath80 ; ( -1 , -1 ) rectangle ( 5 , 5 ) ; in 0,0.25, ... ,3.25 plot[domain=:4 ] ( 1/4*(-)^2 , ) ; in -3.25,-3, ... ,-0.25 plot[domain=0:4 ] ( 1/4*(-)^2 , ) ; in 0,0.25, ... ,3.25 plot[domain=:4 ] ( -1/4*(-)^2 , ) ; in -3.25,-3, ... ,-0.25 plot[domain=0:4 ] ( -1/4*(-)^2 , ) ; ( -1.5,-2.5 ) ( 5.5,-2.5 ) node[above ] @xmath67 ; ( 2,-2.5 ) ( 2,-1 ) node[right ] @xmath96 ; plot[domain=0:1.25 ] ( ^2,-2.5 ) ; plot[domain=-1.25:0 ] ( ^2,-2.5 ) ; ( -1.5,0 ) ( 5.5,0 ) node[above ] @xmath67 ; ( 2,0 ) ( 2,5 ) node[right ] @xmath80 ; ( -1 , -1 ) rectangle ( 5 , 5 ) ; in 0,0.25, ... ,3.25 plot[domain=:4 ] ( 1/4*(-)^2,-+4 ) ; in -3.25,-3, ... ,-0.25 plot[domain=0:4 ] ( 1/4*(-)^2,-+4 ) ; in 0,0.25, ... ,3.25 plot[domain=:4 ] ( -1/4*(-)^2,-+4 ) ; in -3.25,-3, ... ,-0.25 plot[domain=0:4 ] ( -1/4*(-)^2,-+4 ) ; ( -1.5,-2.5 ) ( 5.5,-2.5 ) node[above ] @xmath67 ; ( 2,-2.5 ) ( 2,-1 ) node[right ] @xmath96 ; plot[domain=0:1.25 ] ( ^2,--2.5 ) ; plot[domain=-1.25:0 ] ( ^2,--2.5 ) ; before outlining theorem [ t : othertheorem ] we exemplify other features mentioned above by similar examples . [ ex:2 ] let @xmath82 , and choose @xmath97 again @xmath98 one easily sees that characteristics do collapse in an essential way . considering instead @xmath99 characteristics do split in an unavidable way . therefore , while it is proved in @xcite that in example [ ex:1 ] one can choose for changing variable a flux which is better than a generic other_the regular lagrangian flow_we are not always in this case . this is the main reason why we refer in our first change of variables to a monotonicity property . we have now motivated the further study for the stronger statement of theorem [ t : othertheorem ] . in order to prove it , we consider also the weaker concept of _ lagrangian solution _ : the idea is that the reduction on characteristics is not required on _ any _ characteristic , but on a set of characteristics composing the change of variables @xmath79 that one has chosen . exhibiting a suitable set of characteristics for the change of variables @xmath79 is part of the proof . roughly , an @xmath100-representative @xmath101 for the source of the odes related to @xmath79 is provided by taking the @xmath80-derivative of @xmath102 , which is @xmath2 evaluated along the characteristics of the lagrangian parameterisation ; by construction , it coincides with the second @xmath68-derivative of @xmath103 . here we have hidden the fact that we need to come back from @xmath104 to @xmath105 , a change of variable that is not single valued , and not surjective since the second derivative was defined only almost everywhere . we overcome the difficulty choosing any value of the second derivative when present , and showing that it suffices . however , if one changes the set of characteristics in general one arrives to a different function @xmath106 . we have called _ broad solution _ a function which satisfies the reduction on every characteristic curve . in order to have this stronger characterization , we give a different argument borrowed from @xcite . we define a universal source term @xmath107 in an abstract way , by a selection theorem , at each point where there exists a characteristic curve with second derivative , without restricting anymore to a fixed family of characteristics providing a change of variables . after showing that this is well defined , we have provided a universal representative of the intrinsic gradient of @xmath2 . in cases as examples [ ex:1 ] , [ ex:2 ] it extends the one , defined only almost everywhere , provided by rademacher theorem . [ [ outline - of - the - paper ] ] * outline of the paper * + + + + + + + + + + + + + + + + + + + + + + the paper is organized as follows . in section [ s : hnrecalls ] we recall basic notions about the heinsenberg groups . in section [ s : pderecalls ] we fix instead notations relative to the pde , mainly specifying the different notions of solutions we will consider . one of them will involve a change of variables , for passing to the lagrangian formulation , which is mainly matter of section [ s : existencepartialpar ] and it is basically concerned with classical theory on odes . then appendix [ s : extpar ] also explains how to extend a partial change of variables of that kind to become surjective , and provides a counterexample to its local lipschitz regularity ; it is improved in @xcite showing that it is not in general of bounded variation . in section [ s : equivalence ] we prove the equivalence among the facts that either a continuous function @xmath2 describes a lipschitz graph or it is a distributional solution to the pde @xmath108 , @xmath109 . the further equivalencies are finally matter of section [ s : furthereq ] . with some simplification , we can illustrate the main connections by the following papillon . as mentioned above , there is also a connection with the existence of smooth approximations ( see the proof of theorem [ t : conversedafermos ] by mollification and @xcite with a more geometric procedure ) . @xmath110 & { \text{def.~\ref{d : lagrsol } } } \ar@{.}[dd ] & { \text{def.~\ref{d : broadsol } } } \ar@{.}[d ] \\ * + [ f]{\text{continuous distributional } } \ar@/^0pc/[rr]_{\text{th.\ref{t : univselection}+l.\ref{l : daflipschitz } } } \ar@/^1pc/[dr]_{\text{cor.\ref{c : lipordistrlagr } ( n=1 ) } } \ar@<1ex>[dd]^{\text{l.\ref{l : distrlip } } } & & * + [ f]{\text{continuous broad } } \ar@/^1pc/[dl]^{\text{l.\ref{l : globchi } } } \ar@/^1pc/@{.>}@<3ex>[dd]_{\text{clear } } \\ & * + [ f]{\text{continuous lagrangian } } \ar@/^1pc/[ul]^{\text{cor.\ref{t : conversedafermos } } } \ar@/^1pc/[ur]_{\text{th.\ref{t : univselection } } } \ar@{.>}[dr]^{\text{clear } } \\ * + [ f]{\text{intrinsic lipschitz } } \ar@{->}[ur]|{\text{cor.\ref{c : lipordistrlagr } ( n=1 ) } } \ar@/^1pc/[uu]^{\text{l.\ref{l : lipdistr } } } & & * + [ f]{\text{continuous broad * } } \ar@/^1pc/@{.>}[ul]^{\text{l.\ref{l : globchi } } } \\ { \text{def.~\ref{d : distribsol } } } \ar@{.}[u ] & & { \text{def.~\ref{d : broad*sol } } } \ar@{.}[u ] } \ ] ] * a remark about the identification of the source terms . * the intrinsic gradient is unique . suppose @xmath2 is both intrinsic lipschitz and a broad solution : then by definition at points of intrinsic differentiability the intrinsic gradient must coincide with @xmath107 . as a broad solution is also a lagrangian solution , at those points it must coincide also with @xmath111 . by rademacher theorem , these functions are therefore identified lebesgue almost everywhere , and by lemma [ l : lipdistr ] they are representative of the @xmath50-function which is the source term of the balance law . + we conclude remarking again that this identification lebesgue almost everywhere is not enough for fixing them , as both lagrangian solutions and broad solutions require the right representative of the source also on some of the points of non - intrinsic differentiability , not captured by the source term in the distributional formulation ( fig . [ fig:1 ] ) . by rademacher theorem , instead , we have that the source term in the broad formulation not only fixed dirctly the source term in the lagrangian formulation , but also in the distributional one . actually , it suffices the source of a lagrangian formulation in order to fix directly the one in the distributional one . _ acknowledgements . _ we warmly thank stefano bianchini and giovanni alberti for useful discussions and important suggestions , in particular on the subject of section [ s : furthereq ] . * definition : a noncommutative lie group . * we denote the points of @xmath112 by @xmath113 if @xmath114 , @xmath115 and @xmath116 , the group operation reads as @xmath117 the group identity is the origin 0 and one has @xmath118^{-1}=[-x ,- t]$ ] . in @xmath119 there is a natural one parameter group of non isotropic dilations @xmath120 $ ] , @xmath116 . the group @xmath121 can be endowed with the homogeneous norm @xmath122 and with the left - invariant and homogeneous distance @xmath123 the metric @xmath124 is equivalent to the standard carnot - carathodory distance . it follows that the hausdorff dimension of @xmath125 is @xmath126 , whereas its topological dimension is @xmath127 . the lie algebra @xmath128 of left invariant vector fields is ( linearly ) generated by @xmath129 and the only nonvanishing commutators are @xmath130 = t , \qquad j=1,\dots n.\ ] ] * horizontal fields and differential calculus . * we shall identify vector fields and associated first order differential operators ; thus the vector fields @xmath131 generate a vector bundle on @xmath3 , the so called _ horizontal _ vector bundle @xmath132 according to the notation of gromov ( see @xcite ) , that is a vector subbundle of @xmath133 , the tangent vector bundle of @xmath3 . since each fiber of @xmath132 can be canonically identified with a vector subspace of @xmath134 , each section @xmath135 of @xmath132 can be identified with a map @xmath136 . at each point @xmath137 the horizontal fiber is indicated as @xmath138 and each fiber can be endowed with the scalar product @xmath139 and the associated norm @xmath140 that make the vector fields @xmath131 orthonormal . a real valued function @xmath141 , defined on an open set @xmath142 , is said to be of class @xmath143 if @xmath144 and the distribution @xmath145 is represented by a continuous function . we shall say that @xmath146 is an _ @xmath147-regular hypersurface _ if for every @xmath148 there exist an open ball @xmath149 and a function @xmath150 such that * @xmath151 ; * @xmath152 . the _ horizontal normal _ to @xmath12 at @xmath153 is @xmath154 [ notation ] recalling that we denote a point of @xmath3 as @xmath155 , where @xmath156 , @xmath157 , it is convenient to introduce the following notations : * we will denote a point of @xmath158 as @xmath159 , where @xmath160 , @xmath157 . if @xmath161 , for given @xmath162 , we denote @xmath163 . when we will use this notation , we also denote a point @xmath164 as @xmath165 . * let @xmath166 , for given @xmath167 and @xmath157 we will denote @xmath168 * let @xmath166 , for given @xmath169 we will denote@xmath170 * let @xmath166 , for given @xmath167 we will denote @xmath171 * @xmath172 . a point @xmath173 will be identified with the point @xmath174 . * let @xmath175 , for given @xmath176 we will denote @xmath177 . a set @xmath8 is an @xmath13-graph if there is a function @xmath178 such that @xmath179 let us recall the following results proved in @xcite . [ dinitheorem ] let @xmath180 be an open set in @xmath3 , @xmath181 , and let @xmath182 be such that @xmath183 . let @xmath184 then there exist a connected open neighborhood @xmath185 of @xmath186 and a unique continuous function @xmath187 $ ] such that @xmath188 , where @xmath189 and @xmath190 is the map defined as @xmath191 and given explicitly by @xmath192 let @xmath193 , @xmath194 and define @xmath195 < r^2,\,|t - t^0|<r\right \}\ , . \end{array}\ ] ] when @xmath66 and @xmath196 let @xmath197 following @xcite we define the graph quasidistance @xmath81 on @xmath33 . we set @xmath198 , @xmath199 [ fidistanzadef2708 ] for @xmath200 we define @xmath201 following the notations [ notation ] we have explicitly if @xmath202 @xmath203 where @xmath204 . if @xmath66 and @xmath205 we have @xmath206 an intrinsic differentiable structure can be induced on @xmath207 by means of @xmath34 , see @xcite . we remind that a map @xmath208 is @xmath207-linear if it is a group homeomorphism and @xmath209 for all @xmath116 and @xmath210 . we remind then the notion of @xmath211-differentiablility . [ defiwfdiff ] let @xmath212 be a fixed continuous function , and let @xmath28 and @xmath213 be given . * we say that @xmath214 is @xmath211-differentiable at @xmath215 if there is a unique @xmath207-linear functional @xmath216 such that @xmath217 * we say that @xmath214 is uniformly @xmath211-differentiable at @xmath215 if there is a unique @xmath147-linear functional @xmath216 such that @xmath218 we will denote @xmath219 . if @xmath16 is uniformly @xmath211-differentiable at @xmath215 , then @xmath16 is @xmath211-differentiable at @xmath215 . in @xcite it has been proved that each @xmath147- regular graph @xmath220 admits an intrinsic gradient @xmath221 , in sense of distributions , which shares a lot of properties with the euclidean gradient . since @xmath222 , where the vector fields @xmath223 must be understood as restricted to @xmath207 , it is possible to define the differential operators given , in sense of distributions , by @xmath224 according with the notation introduced in [ notation ] , we also denote by @xmath225 the family of vector fields on @xmath226 defined by @xmath227 the following characterizations were proved in @xcite . the definitions of broad * and distributional solution of the system @xmath17 are recalled in section [ s : pderecalls ] . [ miotheorem ] let @xmath44 be an open set and let @xmath45 be a continuous function . then 1 . the set @xmath228 is an @xmath147-regular surface and @xmath229 for all @xmath148 , where @xmath230 is the horizontal normal to @xmath12 at @xmath153 . is equivalent to each one of the following conditions : 1 . there exists @xmath231 and a family @xmath232 such that , as @xmath233 , @xmath234 and @xmath235 in @xmath236 in sense of distributions . there exists @xmath231 such that @xmath2 is a _ broad * solution _ of the system @xmath17 . 3 . there exists @xmath231 such that @xmath16 is a distributional solution of @xmath17 . @xmath16 is uniformly @xmath211-differentiable at @xmath237 for all @xmath238 . it follows that , for given @xmath37 such that @xmath239 is @xmath147-regular , then at any point @xmath56 the differetial @xmath240 can be represented in terms of the intrinsic gradient @xmath241 . more precisely ( see @xcite ) @xmath242 [ d : gfderivative ] let @xmath37 be a continuous function and let @xmath56 be given . we say that @xmath16 admits @xmath243-derivative if there exists @xmath244 such that for each integral curve @xmath245 of @xmath243 with @xmath246 @xmath247 and @xmath248 . notice that if @xmath239 is an @xmath147-regular hypersurface then @xmath16 admits @xmath243-derivative at @xmath237 for every @xmath56 for @xmath162 ( see @xcite , @xcite ) . * introduction to the concern of this paper . * let us now introduce the concept of intrinsic lipschitz function and intrinsic lipschitz graph . [ d : deflip ] let @xmath249 . we say that @xmath16 is an intrinsic lipschitz continuous function in @xmath33 and write @xmath250 , if there is a constant @xmath26 such that latexmath:[\[\label{deflip } moreover we say that @xmath16 is a locally intrinsic lipschitz function in @xmath33 and we write @xmath252 if @xmath250 for every @xmath253 . we remark that when @xmath2 is intrinsic lipschitz , then there exists @xmath254 such that @xmath255 in particular , the graph distance @xmath256 is also equivalent to the carnot - carathodory distance restricted to the corresponding points on the graph of the lipschitz intrinsic hypersurface . this means that @xmath2 is lipschitz continuous also in the classical sense when evaluated on any fixed integral curve of the vector field @xmath257 , while it is @xmath42-hlder on the lines where @xmath67 is fixed . in @xcite is proved the following characterization for intrinsic lipschitz functions . [ t : tesipinamonti ] let @xmath175 be open and bounded , let @xmath37 . then the following are equivalent : 1 . @xmath252 2 . there exist @xmath258 and @xmath259 such that @xmath260 there exists @xmath261 such that 1 . @xmath262 uniformly converges to @xmath16 on the compact sets of @xmath33 ; 2 . @xmath264 @xmath265 ; 3 . @xmath266-a.e . @xmath238 . moreover if ( ii ) holds , then @xmath267 @xmath55-a.e . @xmath238 . let us finally recall the following rademacher type theorem , proved in @xcite . [ t : rademacher ] if @xmath268 then @xmath16 is @xmath211-differentiable for @xmath55-a.e @xmath56 . even when @xmath269 , where @xmath64 is an open subset of @xmath270 , the equation @xmath271_{t}=w(z , t)\qquad{\rm in\ } { { \omega}}\ ] ] allows in general for discontinuous solution . however , it is the case @xmath66 of the system @xmath272 where @xmath273 and this system , by theorem [ miotheorem ] , describes an @xmath147-regular surface @xmath228 which is an @xmath274-graph . since we want to study in the present paper intrinsic lipschitz graphs , then we do not require anymore the continuity of @xmath1 but we allow @xmath52 . notwithstanding that , the continuity of @xmath2 remains natural . there are a priori different notions of _ continuous _ solutions @xmath49 . we recall some of them in this section : distributional , lagrangian , broad , broad*. all of them will finally coincide . after giving in the present sections the definitions for all @xmath275 , we will focus in the next one the analysis on the non - linear equation in the case @xmath66 , which conveys the attention on the planar case . we will remind this reduction by adopting often the variables @xmath276 instead of @xmath277 . the generalization to other cases @xmath193 of most of the lemmas is straightforward , because the fields @xmath278 and @xmath279 are linear . it is not basically in lemma [ l : gfunztx ] , where we prefer taking advantage of the continuity of @xmath79 ; however , we have no reason to prove it in full generality . we recall that in general solutions are not smooth , even if we assume the continuity see e.g. example [ e : fillholesnogloblip ] below . the equation is then interpreted in a distributional way . [ d : distribsol ] a continuous function @xmath49 is a _ distributional solution _ to if for each @xmath280 @xmath281 and @xmath282 we consider now different versions for the lagrangian formulation of the pde . the first one somehow englobes a choice of trajectories for passing from lagrangian to eulerian variables , and imposes the evolution equation on these trajectories . [ d : lagrparam ] a family of _ partial lagrangian parameterisations _ associated to a continuous function @xmath49 and to the system is a family of couples @xmath283 ( @xmath58 ) with @xmath284 open sets and @xmath285 borel functions such that for each @xmath162 * the map @xmath286 , @xmath287 is valued in @xmath64 ; * for every @xmath288 , the function @xmath289 is nondecreasing ; * for every @xmath290 , for every @xmath291 , the function @xmath292 is absolutely continuous and @xmath293 we call it a family of _ ( full ) lagrangian parameterisations _ if @xmath294 is onto the section @xmath295 for all @xmath296 . we remark again that we emphasized in this definition the nonlinear pde of the system : a lagrangian parameterisaiton provides a covering of @xmath64 by characteristic lines for that equation . indeed , a covering by characteristic lines of the other equations is immediately given by an expression like @xmath297 moreover , the reduction along characteristics for the linear equations , and thus the equivalence between lagrangian and distributional solution , holds with less technicality . a family of ( partial ) parameterisations @xmath298 extends the family of ( partial ) parameterisations @xmath299 , we denote @xmath300 , if there exists a family of borel injective maps @xmath301 when @xmath302 and @xmath303 they are called equivalent . the notion of lagrangian parameterisation given above does not consist in a different formulation for the notion of regular lagrangian flow in the sense by ambrosio - di perna - lions ( see @xcite for an effective presentation ) . particles are really allowed both to split and to join , therefore in particular the compressibility condition here is not required , while instead we have a monotonicity property . when we need to distinguish letters , we denote with @xmath304 functions defined on @xmath64 but possibly related to a parameterisation , with @xmath305 functions defined on @xmath306 , and with @xmath307 functions defined on @xmath64 not related to specific parameterisations . notice that a full lagrangian parameterisation is continuous in the two variables for free : indeed , e.g. for @xmath66 , by monotonicity one has that for each @xmath61 @xmath308 by the surjectivity then equality must hold . considering the lipschitz continuity in the other variables one gains the joint continuity in @xmath276 . the same holds for @xmath309 provided that @xmath79 is continuous on the hyperplane @xmath310 , since the argument above gives the continuity only on the planes where @xmath311 is constant . we do not mind about continuity in @xmath311 . before giving the notion of lagrangian solution , we recall that a set @xmath312 is _ universally measurable _ if it is measurable w.r.t . every borel measure . universally measurable sets constitute a @xmath313-algebra , which includes analytic sets . a function @xmath314 is said universally measurable if it is measurable w.r.t . this @xmath313-algebra . in particular , it will be measurable w.r.t . any borel measure . + notice that borel counterimages of universally measurable sets are universally measurable . then the composition @xmath315 of any universally measurable function @xmath135 with a borel function @xmath316 is universally measurable . this composition would be nasty if @xmath317 were just lebesgue measurable . + since restrictions of borel functions on borel sets are borel , all the terms in the following definition are thus meaningful . [ d : lagrsol ] a continuous function @xmath51 is a _ lagrangian solution _ of if there exists a family of lagrangian parameterisations @xmath298 ( @xmath162 ) , associated to @xmath2 and , and a family of universally measurable functions @xmath318 ( @xmath162 ) , such that @xmath319 , @xmath320 , @xmath321 , it holds that * the function @xmath322 is absolutely continuous and @xmath323 where @xmath324 is given in ( l.1 ) of definition [ d : lagrparam ] . * if @xmath16 admits @xmath243-derivative at @xmath56 , then @xmath325 @xmath326 . we are going to prove in section [ ss : lagrdistr ] that @xmath2 above is a distributional solution to provided that the function @xmath327 is a pointwise representative of the source term @xmath328 . the fact that we do not require in the definition that @xmath329 and @xmath1 represent the same function is just a notational convenience for the proofs below . we give now the strongest notion of solution : the evolution equation is imposed on every trajectories . [ d : broadsol ] a continuous function @xmath49 is a _ broad solution _ of if there exists a universally measurable function @xmath330 such that * @xmath331 @xmath55-a.e . in @xmath33 ; * @xmath326 , for every integral curve @xmath332 of the vector field @xmath57 it holds that @xmath333 is absolutely continuous and @xmath334 * if @xmath16 admits @xmath243-derivative at @xmath56 , then @xmath335 @xmath336 . we also remind the intermediate notion of broad * solution introduced in @xcite . [ d : broad*sol ] a continuous function @xmath2 is a _ broad * solution _ of if for every @xmath28 there exists a map , that we will call exponential map , @xmath337\times{i_{{{\delta}}}(a_0)}\rightarrow{i_{\delta_1}(a_0 ) } \subset\omega\,,\ ] ] where @xmath338 such that , if @xmath339 for @xmath162 , then * @xmath340;\,{{\mathbb r}}^{2n})}\,,$ ] * @xmath341 * there exists a universally measurable function @xmath342 such that @xmath343 @xmath55-a.e . in @xmath33 , @xmath326 the map @xmath344 is absolutely continuous and @xmath345 * if @xmath16 admits @xmath243-derivative at @xmath56 , then @xmath346 @xmath336 . being a distributional solution to the pde , a lagrangian solution will be in particular a broad * solution with @xmath347 . viceversa , if the exponential map related to a broad solution satisfies the relative monotonicity property , then one can prove that the broad * solution is also a lagrangian solution , with the same @xmath111 . one can moreover derive a procedure for constructing a lagrangian parameterisation in lemma [ l : globchi ] , where the curves @xmath348 should be replaced by the ones of the exponential map . as mentioned , we can focus in this section on the case @xmath66 . we will give information about the generalization to the case @xmath193 in remark [ r : remarkdimnsez4 ] . given a continuous function @xmath2 on @xmath349 , we defined what we mean by ` lagrangian parameterisation ' . here we show that the definition is non - empty . it is part of the very classical theory on odes with continuous coefficients covering @xmath64 with integral curves of the continuous vector field @xmath350 , see for example the first chapters of @xcite . in the definition we of lagrangian parameterisation it is however essential that we cover the region with curves which are ordered and not with generic ones : we therefore briefly remind below one way of doing it , with also the aim being self - contained . firstly , in lemma [ l : solode ] we show , for familiarising with notations , that if one takes the integral curves through an @xmath61-section of @xmath64 which are minimal , in the sense that any other curve through that point lies on its right side , then we get a partial lagrangian parameterisation , as well as it happens for the maximal ones . extending it to a full one will be matter of the next section . a full lagrangian parameterisation basically amounts to an order preserving parameterisation , with a _ real _ valued parameter , of non - crossing curves through each point of the plane . we immediately give an example of a full one ( lemma [ l : globchi ] below ) . notice that the definition of lagrangian parameterisation concerns only classical theory on odes with continuous coefficients , as therefore lemmas [ l : solode ] , [ l : globchi ] , [ l : gfunztx ] , [ l : holderode ] , [ l : fillinode ] . [ l : solode ] let @xmath351 be a continuous function , @xmath352 . then there are domains @xmath353 , @xmath354 associated to the functions @xmath355 for which @xmath356 , @xmath357 are partial lagrangian parameterisaiton relative to @xmath2 . for every @xmath358 one could consider the minimal and maximal curves satisfying on @xmath359 the ode for characteristics and passing through that point : indeed the functions [ e : gammamm ] @xmath360 are well defined , lipschitz , and because of the continuity of @xmath2 they are still integral curves ( @xcite ) . denoting by @xmath361 the relative interior of a set , we define the domain @xmath362 where recall that @xmath363 . since we are assuming that @xmath64 contains the origin , @xmath364 is nonempty . the domain @xmath365 is analogous . being @xmath366 , @xmath367 lipschitz solutions to the ode with continuous coefficients , the functions @xmath368 in the statement are @xmath369 in the @xmath61 variable for every @xmath370 fixed . @xmath371 are jointly borel in @xmath372 by continuity in @xmath61 and monotonicity in @xmath373 , monotonicity that now we show . notice the semigroup property : for @xmath374 , for example for , @xmath375 this yields the monotonicity of @xmath376 for each @xmath61 fixed . indeed , if @xmath377 and @xmath378 at a certain @xmath379 , by the continuity of the curves there exists @xmath380 when @xmath381 . but then the curve @xmath382 is a good competitor for the definition of @xmath383 , which implies @xmath384 and therefore equality . for @xmath385 the argument is similar . in lemma [ l : fillinode ] we show how to make a partial parameterisation @xmath79 surjective : thus we cover @xmath64 by a family of characteristic curves which includes the ones of @xmath79 . in the following lemma , w.l.o.g . in a simpler setting , we provide instead a full lagrangian parameterisation , defined at once instead of extending a given one . [ l : globchi ] there exists a full lagrangian parameterisation associated to a continuous function @xmath386 which is also continuous on the closure @xmath387^{2}$ ] . step @xmath388 . let @xmath2 be a continuous function on @xmath387^{2}$ ] , we want to give a lagrangian parameterisation for his restriction to @xmath389 as we defined it on an open set . one can assume @xmath2 is compactly supported in @xmath390\times ( 0,1)$ ] . if not , one can extend it to a compactly supported function @xmath391 on @xmath387\times ( -1,2)$ ] : restricting the lagrangian parameterisation @xmath392 for @xmath391 , defined as described below , to the open set @xmath393 one will get a lagrangian parameterisation for @xmath2 . the assumption of @xmath394 compactly supported in @xmath395 implies that there are two characteristics , one starting from @xmath396 and one from @xmath395 , which satisfy @xmath397 . this means respectively @xmath398 and @xmath399 for each @xmath400 $ ] . step @xmath78 . after this simplification , we associate to each point @xmath401^{2}$ ] a curve @xmath402 which is minimal forward in @xmath61 , maximal backward : @xmath403 where @xmath404 , @xmath405 were defined in . notice that the curves @xmath406 are defined on the whole @xmath387 $ ] . let us denote by @xmath407\to[0,1],\ ( \bar s,\bar t)\in[0,1]^2\}.\ ] ] we will endow @xmath408 by the topology of uniform convergence on @xmath387 $ ] and the following total order relation @xmath409.\ ] ] let us denote by @xmath410 the closure of @xmath411)$ ] endowed with the topology of uniform convergence . we prove in the rest of this step the following claims : @xmath412 @xmath413 @xmath414 @xmath415 i.e. for each @xmath416 @xmath417 @xmath418 $ ] . since @xmath408 is a family of equi - lipschitz continuous and bounded functions of @xmath419)$ ] , then , from arzel - ascoli s theorem @xmath410 is compact . let @xmath420 and let us prove that @xmath421 or @xmath422 . by definition there are two sequences @xmath423 such that @xmath424 and @xmath425 uniformly on @xmath387 $ ] . assume @xmath426 , then there is @xmath427 $ ] such that @xmath428 for instance , let @xmath429 and let us prove that @xmath430.\ ] ] let @xmath431 , there exists @xmath432 such that @xmath433 because @xmath434 and @xmath435 are ordered , it follows that @xmath436 @xmath418 $ ] , @xmath437 . passing to the limit as @xmath438 in the previous inequality , we get . proof of . by contradiction , suppose that @xmath439 , with @xmath440 and @xmath441 non empty , closed sets in @xmath419)$ ] . it is well - know that , from and , for each subset @xmath442 there exists the least upper bound ( or supremum ) and greatest lower bound ( or infimum ) of @xmath443 ; we will denote respectively by @xmath444 , @xmath445 . thus let @xmath446 because @xmath447 and @xmath448 are closed , @xmath449 and @xmath450 . since @xmath451 , we have @xmath452 or @xmath453 . assume , for instance , that @xmath452 . then @xmath454 @xmath418 $ ] and @xmath455 for a suitable @xmath427 $ ] . let @xmath456.\ ] ] by definition , @xmath457 , but @xmath458 and therefore we have a contradiction since @xmath459 . let @xmath416 , then by definition there exists a sequence @xmath460 such that @xmath461 uniformly in @xmath387 $ ] . because @xmath462,\ \forall\ h,\ ] ] passing to the limit as @xmath438 in the previous identity we get . step @xmath463 . let us now consider the map @xmath464 defined by @xmath465 where @xmath466 is an enumeration of @xmath467 $ ] . notice that @xmath468 satisfies the following properties : @xmath469 @xmath470 @xmath471;\ ] ] @xmath472\to\mathcal c^*\ \text{continuous}.\ ] ] indeed and immediately follow by the definition of @xmath468 . equality follows by and ; is a consequence of , and . eventually let us consider the map @xmath473\times[0,2]\to[0,1]$ ] @xmath474\times[0,2].\ ] ] then , from , @xmath79 is continuous . moreover the map @xmath475 , defined by @xmath476 , turns out to be a full lagrangian parameterisation associated to @xmath477 . [ r : remarkdimnsez4]notice that the same construction works with more variables , considering analogous characteristic curves @xmath478having the same order relation at @xmath479 frozen and the relative parameterisation given by @xmath480 with @xmath481 defined as above . the continuity in @xmath311 however is not guaranteed . in this section we prove theorem [ t : firsttheorem ] , without dimensional restrictions . preliminary we highlight here two properties of continuous distributional solutions @xmath96 to the problem @xmath482 . in particular we need regularity results of the solution along the characteristics lines @xmath483 of the fields @xmath484 . in the case of the non linear field @xmath257 , the integral curves of @xmath485 exist by the continuity and boundedness of @xmath2 . [ l : daflipschitz ] let @xmath175 be an open set . a continuous distributional solution @xmath37 to @xmath17 is @xmath486-lipschitz along any characteristic line @xmath487\to { { \mathbb r}}$ ] satisfying @xmath488,\ \hat z_n\ { \rm fixed}.\ ] ] in the same way in dafermos @xcite we obtain for @xmath489 and @xmath311 fixed @xmath490 @xmath491 ^ 2\,ds\leq0\ ] ] and then @xmath492 dividing by @xmath493 and getting to the limit to @xmath494 we obtain @xmath495 the opposite inequality is obtained in a similar way integrating on the left of the characteristic . we obtain the same result of lemma [ l : daflipschitz ] for the linear fileds @xmath496 following the same proof . [ l : daflipschitzbis ] let @xmath175 be an open set . a continuous distributional solution @xmath37 to @xmath17 is @xmath497-lipschitz along any characteristic line @xmath498\to { { \omega}}$ ] satisfying @xmath499 we pass now to the hlder continuity in the other variable . [ l : holderode ] let @xmath500 be such that of all @xmath501 there are @xmath487\to { { \mathbb r}}$ ] satisfying @xmath502\\ \gamma(0)=\tau\end{cases}\ ] ] and that @xmath503 then we have @xmath504 let us denote @xmath505 let us observe that @xmath506 by contradiction , let us assume there exist @xmath507 , @xmath508 such that @xmath509 @xmath510 the idea of the proof is the following : the lipschitz condition in the hypothesis is an upper bound on the second derivative of the mentioned curves @xmath483 . therefore , if their first derivative wants to change it takes some time in @xmath61 . if we assume that at @xmath511 the first derivative differs at two points @xmath512 at least of the ratio , then the the relative curves @xmath513 starting from those points must meet soon . however , at the time they meet the first derivative did not have the time to change enough to become equal , providing an absurd . let us introduce curves @xmath514 such that for @xmath515 , @xmath516 @xmath517 @xmath518 we observe that , by our lipschitz assumption , @xmath519 is a function bounded by @xmath520 . therefore we can represent each @xmath521 for each @xmath522 $ ] as @xmath523 @xmath524\,.\ ] ] in particular by the second equality in , for @xmath525 , @xmath526 for @xmath527 $ ] . by we get @xmath528 or @xmath529 let us prove now that if holds then there exists @xmath530 such that @xmath531 let @xmath532 then @xmath533 , \quad { \gamma_1(s^*)\leq\gamma_2(s^*)}\,.\ ] ] indeed by and the definition of @xmath534 , @xmath535 . on the other hand by ( with @xmath536 ) , gives @xmath537 then follows . let @xmath538:\gamma_1(s)>\gamma_2(s)\}\ ] ] then by @xmath539 and it satisfies . if holds we can repeat the argument reversing the @xmath61-axis getting that there exist @xmath540 such that still holds . let us prove now that @xmath541 then a contradiction and the thesis will follow . indeed , for instance , let us assume . then by and the bound on @xmath542 @xmath543 @xmath544 @xmath545 @xmath546\,.\ ] ] therefore we get that @xmath547 and follows . we are now able to give the proof of theorem [ t : firsttheorem ] . we distinguish the two different implications in the following two lemmas . [ l : distrlip ] let @xmath175 be an open set . if @xmath16 is a distributional solution of @xmath17 , then @xmath252 . let @xmath548 for a sufficiently small @xmath549 . for @xmath202 let @xmath550 be the vector fields given by @xmath551 define @xmath552 observe that @xmath553 and @xmath554 are well defined . for @xmath66 , @xmath555 is not defined and we set @xmath556 and @xmath557 . we have to show that there exists @xmath26 such that @xmath558 @xmath559 , from lemma [ l : daflipschitz ] @xmath560 . by we can apply lemma [ l : holderode ] and obtain @xmath561(z_n'-z_n)-2\int_0^{z_n'-z_n}{{\phi}}(\exp(s{w^\phi})(b^\ast)\,ds \bigr|\vspace{0.2cm}\\ = : & \displaystyle{{d_\phi}}(b',b)^2+r_1(b',b)+r_2(b',b)+r_3(b',b ) . \end{array}\ ] ] for the case @xmath66 we arrive to with the same line ( it is sufficient to follow the same steps `` erasing '' the term @xmath562 ) . now we want to prove that for all @xmath563 there is a @xmath5640,{{\delta}}_0]$ ] such that , for @xmath5650,{{\delta}}_\epsilon[$ ] , @xmath566 for all @xmath567 and that there exist @xmath568 such that @xmath569 for all @xmath570 , these estimates are sufficient to conclude : in fact , choosing @xmath571 and using , , and , we get @xmath572 whence @xmath573 which is @xmath574 and then the thesis . by we obtain @xmath575 whence follows . observe that follows from @xmath576 if @xmath66 , and from @xmath577 if @xmath202 . finally , for @xmath578 $ ] we can define @xmath579s;\ ] ] we have @xmath580\,dr-\bigl [ { { \phi}}\bigl(\exp(s{w^\phi})(b^\ast)\bigr)-{{\phi}}(b^\ast)\bigr]s = o(s^2)\ ] ] because @xmath581 is lipschitz . therefore follows with @xmath582 . [ holder ] let @xmath583 be an open set . if @xmath584 is a distributional solution of @xmath17 in @xmath64 , then @xmath585 if @xmath252 , then @xmath16 is a distributional solution of @xmath17 in @xmath33 . by theorem [ t : tesipinamonti ] there exist @xmath258 , such that @xmath586 uniformly converges to @xmath16 on the compact sets of @xmath33 , @xmath587 @xmath588-a.e . @xmath56 for every @xmath265 and @xmath589 @xmath588-a.e . @xmath56 . therefore , denoting @xmath590 , we have for every @xmath265 and for every @xmath591 @xmath592 @xmath593 getting to the limit for @xmath594 we obtain @xmath595 @xmath596 i.e. @xmath16 is a distributional solution of the problem @xmath17 . in the previous section we established the equivalence between @xmath597 and @xmath598 we establish now in section [ s : distrbroad ] a characterization more related to the lagrangian formulation : we prove that one can reduce the pde @xmath599 along any integral line of the vector fields @xmath600 , @xmath601 , provided that one chooses suitably the @xmath602 representative @xmath107 of the distribution identified by @xmath1 . we first prove in section [ ss : distrtolagr ] , as an introduction , the weaker statement that one can reduce the pde to odes along a selected family of characteristic constituting a lagrangian parameterisation . as well , the converse holds : if the odes on characteristics are satisfied one has a distributional solution to the pde and the sources of the two formulations can be identified ; this , as recalled in lemma [ l : lipdistr ] , is known from @xcite and we prove it differently in section [ ss : lagrdistr ] below . the conclusion of this last section will be the following . the various notions of continuous solutions we have considered are equivalent . the present section is an introduction to the next one , which proves the stronger statement that distributional solutions are broad solutions . being technically simpler , this section provides a guideline for some ideas implemented next . one can in particular notice that the proof concerns only odes . in order to avoid technicalities we focus on @xmath603 and on @xmath604^{2}$ ] . we refer to section [ s : distrbroad ] for the stronger statement with the universal source term @xmath107 . the function @xmath111 below relative to the lagrangian formulation has not yet been related with the rhs of : their identification will come from theorem [ t : conversedafermos ] . [ l : gfunztx ] let @xmath2 be a continuous function . consider a lagrangian parameterisation @xmath605 and assume that @xmath606 is lipschitz in @xmath61 for all @xmath370 . then there exists a borel function @xmath607 such that for all @xmath370 @xmath608 [ c : lipordistrlagr ] if a function @xmath609 is * either an intrinsic lipschitz continuous function * or a continuous distributional solutions @xmath2 to the balance law then it is a lagrangian solution to the equation @xmath610 . the existence of a lagrangian parameterisation has been proved in lemma [ l : globchi ] . as recalled just after the definition [ d : deflip ] , an intrinsic lipschitz continuous function is continuous and lipschitz along characteristics . also continuous distributional solutions to the balance law are lipschitz continuous along characteristics by lemma [ l : daflipschitz ] , following @xcite ) . we have therefore the thesis directly by applying lemma [ l : gfunztx ] . we remind the notation @xmath611 second derivative in @xmath612 . by assumption @xmath613 is lipschitz in @xmath61 . being @xmath2 continuous , one can see that the subset @xmath614 of those @xmath276 where @xmath615 is twice @xmath61-differentiable is @xmath616 , and @xmath617 is a borel function on it . moreover , by rademacher s theorem the @xmath370-sections of @xmath618 have full measure and therefore by tonelli theorem also @xmath618 has full measure . however , the function @xmath617 is defined on @xmath619 , while we are looking for a function defined on @xmath64 . a preliminary comment . in order to check that @xmath620 lifts @xmath621 to a map @xmath111 a.e . defined on @xmath64 , which would provide our thesis , it would be natural to show that * @xmath622 is a lebesgue measurable subset of @xmath64 with full measure ; * @xmath623 is constant on the level set of @xmath620 intersected with @xmath618 . since @xmath624 is not lipschitz , we do not manage to prove at this point that @xmath625 has full measure . we instead assign a specific value , @xmath186 , to the function out of @xmath625 . this choice of the extension does not affect our claim . we notice moreover that the second point is true in that strong form , but we show that the points of the level set of @xmath620 corresponding to more values of @xmath626 are not relevant . analysis of @xmath622 and partial inverse of @xmath620 . the proof of the borel measurability of @xmath622 requires some technicality : we apply a theorem due to srivastava ( @xcite , theorem 5.9.2 ) deriving that there exists a borel restriction @xmath624 which is one - to - one to @xmath622 ; then theorem 4.12.4 in @xcite , due to lusin , would provide the thesis . in order to apply the first theorem , we partition @xmath619 into the level sets of @xmath620 , which are @xmath627 . we need also to observe that @xmath628 is borel for each open set @xmath629 . for simplicity , consider the case when @xmath624 is already a full parameterisation and thus it is continuous . every open set @xmath629 is @xmath313-compact : thus by continuity @xmath630 is @xmath313-compact , and finally @xmath628 is @xmath313-compact . therefore by srivastava s theorem there is a borel cross section @xmath12 for the partition : @xmath620 restricted to @xmath631 is borel , injective and onto @xmath622 . being a borel image by a one - to - one map , lusin s theorem asserts on one hand that that @xmath622 is borel , and moreover that this restriction has a borel inverse @xmath632 maybe there is a more elementary argument which allows to approximate @xmath618 with a @xmath313-compact subset @xmath633 whose @xmath370-section have full measure . then one could as well work with @xmath633 instead of @xmath618 avoiding measurability difficulties , even without investigating the size of its image . analysis of @xmath623 for @xmath634 . we can define @xmath111 as @xmath635 this map satisfies the claim by the next analysis of the set where @xmath636 is multivalued . analysis of multivalued points of @xmath637 . the set of points @xmath638 such that there exists @xmath639 satisfying @xmath640 and @xmath641 is borel : just see it as @xmath642 the value of @xmath111 on this set is not at all relevant : we show below that for any @xmath370 the intersection of @xmath643 with the characteristics curve @xmath644}$ ] is at most countable . + indeed , fix a curve @xmath645 , for simplicity fix @xmath646 . suppose @xmath647 intersects @xmath648 at @xmath536 with @xmath649 , @xmath650 fixed . by taylor s expansion , there exists a neighborhood @xmath651 where @xmath652 there can be at most @xmath653 such values @xmath654 of @xmath61 : otherwise by the last inequality two characteristics @xmath655 relative to two @xmath656 with @xmath657 would intersect at @xmath658 and cross each other . taking the union for a sequence @xmath659 , and then the union over @xmath650 , we get the thesis . below we show theorem [ t : univselection ] : there exists a borel function @xmath660 such that every curve @xmath648 satisfying the ode with continuous coefficient @xmath417 has time derivative lipschitz , and it has second derivative precisely @xmath661 for a.e . the remarkable fact is that @xmath660 could be defined independently of any set of characteristic curves . this is thus different , and stronger , from what we already proved , which is that there exists a lagrangian parameterisation @xmath79 and an associated function @xmath662 satisfying @xmath663 . the new point is indeed that @xmath107 is a universal representative for the source term . due to its nature this proof works for any @xmath275 with no further complication . the only difference is that @xmath64 will be a subset of @xmath226 instead of @xmath664 . we write it with @xmath66 only for notational convenience . in particular we generalize here the previous lemma [ l : gfunztx ] . since the argument is more intuitive , we mention first how to construct such a souslin function @xmath660 . we proceed then with the borel construction because it gives a better result . the first step is to define pointwise , but in a measurable way , a function @xmath660 such that @xmath67 is a lebesgue point for the second derivative of a curve @xmath665 with @xmath666 and satisfying the ode , whenever there exists one . therefore one applies von neuman s selection theorem ( section 5.5 in @xcite , from @xcite ) to the subset of @xmath667 defined by @xmath668 @xmath669 is borel . it has full measure projection on @xmath670 . components @xmath671 . the subset @xmath672 identified by the constraints @xmath673 , moreover , its projection on @xmath277 is all @xmath674 . by rademacher theorem , we have moreover seen in the same lemma also that the projection of @xmath669 on @xmath277 has full measure . component @xmath675 , discretization . in order to establish the existence ( and the value ) of the limit for the second derivative of @xmath483 at @xmath68 , it suffices to consider e.g. the sequence @xmath676 indeed , then for @xmath677 $ ] , for example at @xmath678 @xmath679 by construction however @xmath680 yielding that the existence of the limit along @xmath681 implies the existence of the limit for any @xmath682 . notice that this would not hold choosing a generic @xmath683 instead of @xmath684 . measurability of @xmath669 . the further constraint in @xmath483 can be written as @xmath685 therefore , we are considering the following subset of @xmath686 : @xmath687 since the set within brackets is closed , @xmath669 is borel . this allows to define an @xmath443-selection @xmath688 , which by construction is a measurable function assigning to every point @xmath277 an integral curve @xmath689 for the ode @xmath690 , whenever there exists a lipschitz one having @xmath68 as a lebesgue point for the right , left and total second derivative . as well , one can define the souslin function @xmath691 the importance of this selection is due to the following theorem . [ t : univselection ] let @xmath2 be a continuous function which is uniformly lipschitz along characteristics curves @xmath483 : @xmath692 . then curve @xmath483 satisfying the ode @xmath692 , one has @xmath693 [ c : distrbroad ] a continuous distributional solution @xmath2 to @xmath482 is also a broad solution . before proving the theorem , for the sake of completeness we show that one can define as well a borel fucntion , that we still denote as @xmath694 , for which theorem [ t : univselection ] still holds . this requires a bit more work than the previous argument , and it is conceptually a little more involved : we do not associate immediately to each point ( where it is possible ) an eligible curve and its second derivative , but something which must be close to it . we will find then with the proof of theorem [ t : univselection ] that we end up basically with the same selection . [ l : borelapprg ] for every @xmath695 , there is a borel function defined on the @xmath277-projection @xmath696 of @xmath669 @xmath697 such that @xmath698 of and that for @xmath699 sufficiently small @xmath700 we define the borel representative of @xmath1 as the function @xmath701 we apply arsenin - kunugui selection theorem ( @xcite , or th . 5.12.1 of @xcite ) to the set @xmath702 where @xmath686 was defined in and @xmath703 immediately below that , in the same proof . more precisely , @xmath686 is closed and @xmath704 is also closed : therefore each @xmath277-section of is @xmath313-compact . then the hypothesis of the theorem are satisfied : it assures that the projection of on the first factor @xmath705 is borel and there exists a borel section of defied on it , which is the function in our statement . notice that the domain fo this function , containing @xmath706 , has full lebesgue measure . we provide now the proof of theorem [ t : univselection ] with @xmath707 either the borel or the souslin one : we consider any characteristic @xmath708 for the balance law and we prove that for almost every @xmath68 its second derivative is precisely @xmath709 . we remind that @xmath710 is lipschitz ( lemma [ l : daflipschitz ] ) . step 1 , countable decomposition . the set of @xmath68 lebesgue point of @xmath711 with value different from @xmath712 can be reduced to @xmath713 if one is considering the souslin selection , clearly there is the simplification @xmath714 . step 2 , reduction argument . we prove that the set @xmath715 can not contain points @xmath716 with @xmath717 . then the thesis will follow : by the previous step the set of @xmath68 where the second derivative of @xmath718 exists and it is different from @xmath712 will be countable . therefore the second derivative of @xmath718 will be almost everywhere precisely @xmath719 . step 3 : analysis of the single sets . by contradiction , assume that contains two such points , for example @xmath720 , @xmath721 . by definition of the set of points we are considering , the two selected curves through @xmath722 , @xmath723 , @xmath724 must intersect in the time interval @xmath725 $ ] , say at time @xmath726 . since they satisfy the ode for characteristics , where they intersect they have the same derivative . being all of them lipschitz , we have then @xmath727 comparing the lhs and the rhs , one arrives to @xmath728 however , since the times @xmath729 belong by construction to the set one has @xmath730 reaching a contradiction . @xmath731 in this section we prove , without passing through the implicit function theorem , that if a continuous function @xmath2 satisfies the lagrangian formulation [ d : lagrsol ] of the balance law , then in the eulerian variables @xmath2 solves indeed the balance law in distributional sense . we instead rely on a mollification procedure in the legrangian variables . see also @xcite , where a different , pointwise approximation of the distributional solution is provided , starting from a broad * solution . this basically shows the converse of dafermos statement in @xcite . we notice that the fact that if a continuous @xmath732 solves the lagrangian formulation , then it is not difficult as we see in the next theorem proving that it solves a balance law in distributional sense . the identification of the lagrangian and distributional sources however is not trivial : we indeed use the converse implication of section [ s : distrbroad ] and rademacher theorem . we already motivated why focusing on the case @xmath66 , whereas the equation reduces to @xmath733_{t}=w(y , t).\ ] ] we give generalizations to the case @xmath193 in remark [ r : remarkdimnsection6 ] [ t : conversedafermos ] every lagrangian solution to is also a distributional solution , with source term given by the distribution identified by the function @xmath111 in the lagrangian formulation . [ c : broaddistrib ] any continuous broad solution @xmath2 to is also a distributional solution . let @xmath734 be a lagrangian parameterisation associated to @xmath2 , and @xmath735 such that @xmath736 we prove then that @xmath2 is a distributional solution of the balance equation @xmath737_{t}=w(y , t).\ ] ] smoothing of @xmath738 in the @xmath370-variable . consider a suitable convolution kernel @xmath739 and define the @xmath370-regularized function @xmath740 given by @xmath741 this function @xmath742 is smooth but possibly still not injective . however , by the monotonicity of @xmath79 one has that if @xmath743 then @xmath734 is constant for @xmath744 $ ] . as a consequence , @xmath745 is in turn constant for @xmath744 $ ] . the above observation allows to define consequently the approximation @xmath746 by @xmath747 notice that @xmath748 is well defined and continuous in the @xmath749-variables . the measurable subset of points where it is not differentiable is possibly non - empty as a consequence of the non - injectivity of @xmath742 , but it is at most countable on every @xmath61-section . @xmath748 is @xmath750 on the remaining full measure set . since both @xmath79 and @xmath2 are continuous , the above relations immediately imply the local uniform convergence of the regularized functions : @xmath751 convergence in the @xmath749-variables of @xmath748 . notice that @xmath752 is converging ( as a measure ) to @xmath753 , and that their variation is unifromly bounded . the above procedure not only defines correctly the continuous functions @xmath746 , but it allows to establish their convergence in @xmath754 : indeed @xmath755 the first factor in both the integrals is uniformly convergent to zero , while @xmath756 converges to the measure @xmath757 . the convergence of @xmath758 is then straightforward . approximation of the source . as @xmath748 is a smooth function a.e . , one can define an approximate source @xmath759 by @xmath760 the above relation , by the pointwise smoothness is immediately equivalent to @xmath761 since we started from a lagrangian parameterisation , the further regularity in the @xmath61 variable @xmath762 for the relative pointwise representative @xmath111 implies the relation @xmath763 in particular , the sources @xmath764 are uniformly bounded by the @xmath89 bound for @xmath111 . moreover , for each @xmath67 fixed @xmath765 converges in all @xmath766 to @xmath767 , and thus in @xmath768 ; the convergence is clearly uniform when @xmath111 is continuous . the lhs of equation passes to the weak limit by the @xmath754-convergence of @xmath769 to @xmath770 established above . the same holds as well for the rhs , since @xmath771 converge in @xmath772 to a function @xmath773 . thus @xmath774_{t}= \bar{\bar w}^{}(y , t).\ ] ] the function @xmath2 as we have seen is precisely the one in the lagrangian parameterisation . even if @xmath765 converges in all @xmath766 to @xmath767 , it is not instead trivial that above @xmath775 , lebesgue almost everywhere , holds . we now explain why it is so . we prove above in section [ s : distrbroad ] that if @xmath774_{t}= \bar{\bar w}^{}(y , t)\ ] ] holds then @xmath2 is also a broad solution : there is a point - wise representative @xmath776 of @xmath777 such that the broad formulation holds with @xmath778 . however , it is then also intrinsic lipschitz continuous with intrinsic gradient @xmath776 : by rademacher theorem the intrinsic gradient @xmath779 is uniquely defined point - wise almost everywhere , and it by lemma [ l : lipdistr ] it identifies the same distribution as @xmath780 . [ r : remarkdimnsection6]we finally remark that theorem [ t : conversedafermos ] works immediately also in higher dimensions , because for ( almost every ) @xmath781 the restriction to the plane @xmath782 of @xmath2 is still a lagrangian solution , with source term the restriction @xmath783 . for every test function @xmath784 we have then by fubini - tonelli theorem @xmath785 = \iint_{{{\mathbb r}}^{2(n-1 ) } } dv\int_{\omega_{v } } dy_{1}dt \left [ \varphi_{y_{1 } } \phi + \varphi_{t } \frac{\phi^{2}}{2 } \right]{\restriction_}{\omega_{v } } \\ & \stackrel{\text{th.~\ref{t : conversedafermos } } } { = } -\iint_{{{\mathbb r}}^{2(n-1 ) } } dv\int_{\omega_{v } } dy_{1}dt \left[\tilde w{\restriction_}{\omega_{v}}\right ] = -\iint_{\omega}\tilde w.\end{aligned}\ ] ] considering also the linear fields we gain the implication from to in theorem [ t : othertheorem ] , and more generally that a lagrangian solution is also a distributional solution . in the present section we deal with the issue of extending a partial lagrangian parameterisation to a ` full ' one . we construct a function @xmath734 satisfying the ode , both monotone and surjective in the @xmath370 variable , which extends a given one @xmath786 . this is the matter of lemma [ l : fillinode ] : we recall below how to extend a solution to an ode with continuous coefficients , whose existence is a classical result . the procedure can be first understood considering example [ e : fillholesnogloblip ] below , illustrated in figure [ fig : parameterisation ] . this deals with the simpler case of an @xmath61-independent @xmath2 , but it has all the ingredients of the general construction of lemma [ l : fillinode ] . moreover , example [ e : fillholesnogloblip ] provides a counterexample for the following fact : even if characteristics are @xmath787 in @xmath61 with lipschitz derivative , _ it is not possible in general to extend a partial , monotone lagrangian parameterisation to a full one which is locally lipschitz continuous_. the reduction of the balance law along characteristics , which is equation , has been inserted in the text ( lemma [ l : globchi ] and theorem [ t : univselection ] ) . here we just notice that it holds , with some @xmath788 pointwise defined in @xmath64 , for a particular lagrangian parameterisation @xmath789 . the argument is @xmath78-dimensional . let @xmath790 be a partial lagrangian parameterisation . focus e.g. the attention on @xmath791 $ ] and @xmath792 valued in @xmath387 $ ] and right continuous , the general case being similar . we fix also @xmath793 . we construct an extension @xmath794 by a recursive procedure . for convenience , the induction index is given by couples @xmath795 with @xmath796 and @xmath797 . the ordering is lexicographic , starting from the second variable : @xmath798 iff either @xmath799 or @xmath800 and @xmath801 . the starting point is @xmath802 defined for @xmath400 $ ] , @xmath803 $ ] . consider the dichotomous points @xmath804 , which go from @xmath805 to @xmath806 at step @xmath807 , associated to the indexes @xmath795 with @xmath796 and @xmath797 . induction step @xmath795 , @xmath808 : general description . assume you have been given @xmath79 defined on @xmath809\times t $ ] by a previous step . if at @xmath810 the map @xmath811 is not onto @xmath812 $ ] we construct an extension @xmath813 such that induction step @xmath795 , @xmath808 : change of parameter set . because of monotonicity the complementary of the image of @xmath819 is the at most countable union of disjoint intervals @xmath820 , which correspond to the discontinuities @xmath821 of this real valued map . at those parameters @xmath822 the two characteristics @xmath823 and @xmath824 bifurcate , and at time @xmath825 their opening is an interval @xmath826 : @xmath827 define consequently the strictly increasing map opening each of those parameters @xmath828 into an interval proportional ( with factor @xmath829 ) to the hole @xmath830 that the relative characteristics leave at @xmath825 : @xmath831 & & \\ j^{h , n } & : & \tau&&\mapsto & & \tau+{{\mathcal l}}^{1}(\cup_{\tau_{k}^{}\leq \tau}i_{k}^{})/2^{2n-1 } .&&\end{aligned}\ ] ] the inequality @xmath832 holds because we are assuming that @xmath79 is valued in @xmath387 $ ] , thus @xmath833 . the set @xmath834 will be the new space of parameters , and the injection @xmath816 will bring from the old set @xmath835 to the new one . induction step @xmath795 , @xmath808 : extension of the parameterisation . we just constructed a new parameter set @xmath834 and an immersion @xmath816 from the old one @xmath835 . in particular , the lagrangian parameterisation is fixed on @xmath836 , but in order to conclude the induction step we need to define the new lagrangian parameterisation on @xmath837 . we clearly need to respect also the monotonicity property , taking into account that part of the parameterisation is already fixed : the new characteristics that we are going to to define must not cross the old ones . for each @xmath838 , consider the @xmath787 maximal curve through @xmath839 defined at until it touches either @xmath840 or @xmath841 , in which case it goes on with the curve which has been touched . it is a little complicated to write formally , but it is just that . the times when this touching happens , if it ever happens , are @xmath842 with the notation that @xmath843 is empty if @xmath844 , a possible right continuous extension is then give by @xmath845 define finally on @xmath387\times t_{h , n}$ ] the lagrangian parameterisation which coincides with the previous one on the image of the old parameter set , and which is extended as described above elsewhere : @xmath846 for the second line , @xmath847 is the length of the interval @xmath848 , part of the ones added to the parameter set precisely at the @xmath795-th step . rescaled by @xmath849 , it gives the length of the segment @xmath850 : the point @xmath839 is where we start for defining the characteristic @xmath851 that we are inserting for extending the lagrangian parameterisation . notice that also the surjectivity property at @xmath810 is satisfied . conclusion . let us first look at how much the domain @xmath852 grows in the extension process . since @xmath853 couples of indices have second variable @xmath275 , then the total size of the intervals added by those couples alltogether is at most @xmath805 : thus , setting @xmath854 , @xmath855 , \quad t_{1}\subset [ 0,3/2 ] , \quad \dots\ , \quad t_{n}\subset [ 0,2 - 2^{-n } ] , \quad \dots\ .\ ] ] the maps @xmath856 are the lagrangian parameterisations @xmath861 and @xmath863 have different domains for the second components , the space of parameters @xmath834 , @xmath864 . however , as seen in the previous step there are strictly monotone injections from each of them to the interval @xmath865 given by @xmath866 and @xmath867 . being strictly monotone , they are invertible if we fill the graphs at discontinuity points : we can compare these compositions , with the same second component in @xmath865 , and we find for them @xmath868 the sequence of these compositions therefore converges uniformly . one verifies that the limit is a monotone lagrangian parameterisation @xmath79 which extends @xmath869 , with injection map @xmath860 . being surjective and monotone , each @xmath870 is continuous . by the continuity in @xmath61 we deduce then surjectivity also at the remaining times : indeed if by absurd we had @xmath871 , we could not have @xmath872 at @xmath873 arbitrarily close to @xmath654 . the following example introduces the extension of a lagrangian parameterisation . it shows moreover that it is not possible in general to get a full one which is lipschitz continuous , even though characteristics below are twice continuously differentiable . consider the very simple equation for @xmath105 in the rectangle @xmath874\times[0,1]$ ] @xmath875_{t}=w(z , t ) , \qquad \phi_{z}(z , t)=0.\ ] ] being @xmath2 dependent on one variable , we change notation and we write @xmath876 . + we consider the partial lagrangian parameterisation @xmath877 defined in lemma [ l : solode ] . this would need to specify the function @xmath2 : since the construction is involved , we specify it below and the reader can immediately visualise it in figure [ fig : parameterisation ] ( left side ) , where a family of integral curves @xmath878 is drown . step 1 : building block ( figure [ fig : parameterisation ] , right side ) . define first a smooth function @xmath648 , for @xmath400 $ ] , which increases continuously from @xmath186 to @xmath388 . let @xmath879 be even , strictly increasing in the first half interval from @xmath186 to its maximum . let @xmath880 vanish at @xmath881 and be positive in @xmath882 $ ] . for instance , consider @xmath883 step 2 : iteration step . we define now a first sequence of points @xmath884 and intermediate ones @xmath885 where @xmath886 and @xmath887 will vanish . the first ones are approximatively @xmath186 , @xmath888 , @xmath889 , @xmath890 , @xmath891 , @xmath892 , @xmath893 , , and each interval @xmath894 $ ] is divided into @xmath895 equal subintervals for determining the points @xmath896 . for @xmath897 , @xmath898 @xmath899 step 3 : iteration . now we define the functions @xmath886 , @xmath887 on each subinterval @xmath900 $ ] . as a preliminary half - step consider the rescaled smooth functions @xmath901 given by @xmath902,\ ] ] notice that each increases monotonically from @xmath903 to @xmath896 , and the first two derivatives vanish at the endpoints . in particular , we can associate to each @xmath904 $ ] a curve @xmath901 such that @xmath905 it will be unique out of the points @xmath906 , while at these junctions there will be two such curves , with however vanishing first two derivatives . observing figure [ fig : parameterisation ] , this curve @xmath901 is just the first segment of what will be part of @xmath877 @xmath907\ni s \mapsto \gamma_{h , i}(s ) \equiv \chi_{m}(s , z^{+}_{h+1,i}).\ ] ] define then , @xmath908 these functions are clearly continuous out of the nodes @xmath906 , and they vanish there , but the continuity at the limit point @xmath909 should be checked . it holds because @xmath910 and @xmath911 , which implies they vanish for @xmath912 . step 4 : non - lipschitz lagrangian parameterisation . having defined the function @xmath2 , we have already defined the partial lagrangian parameterisation @xmath877 of lemma [ l : solode ] . we now extend it to a surjective one , but there is no way of having it lipschitz continuous , as we compute now . different colors in figure [ fig : parameterisation ] show different steps of the extension process . minimal characteristics starting from @xmath913 do not cover almost all the interval at @xmath914 . adding those starting at @xmath914 , it remains to cover at @xmath915 open intervals of total length @xmath916 2 . including at the second step all those minimal characteristics which intersect the line @xmath915 , similarly , one does not cover the whole line @xmath917 : a length @xmath918 remains to cover at both those two values of @xmath80 . + i. at the subsequent @xmath919-th step , at each of the @xmath920 values of @xmath921 one needs covering a length @xmath922 . in the whole process , it must be covered a total length equal to @xmath923 any monotone , lagrangian parameterisation @xmath79 must map a disjoint family of real intervals @xmath924 with @xmath925 to the intervals @xmath926\}_{i}$ ] , respectively at the above valies @xmath927 . however , we have just computed that for all constants @xmath686 @xmath928 ) \\ > & c\sum_{h=0,\dots,2^{i-1}-1,\ i\in{{\mathbb n}}}| i_{h , i}| , \end{aligned}\ ] ] preventing any lipschitz regularity . indeed , the map @xmath929 can be lipschitz with some constant @xmath930 , but @xmath930 must blow up as @xmath931 . at other values of @xmath61 , this map is just @xmath932 . notice finally that the fact that the blow - up of the lipschitz constant is caused here by a behaviour approaching the boundary @xmath933 is incidental . indeed , one can extend the present construction also on @xmath934 for example reflecting the characteristics w.r.t . the axis @xmath935 .
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in this paper we provide a characterization of intrinsic lipschitz graphs in the sub - riemannian heisenberg groups in terms of their distributional gradients .
moreover , we prove the equivalence of different notions of continuous weak solutions to the equation @xmath0_{t}=w$ ] , where @xmath1 is a bounded measurable function depending on @xmath2 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the contact pattern among individuals in a population is an essential factor for the spread of infectious diseases . in deterministic models , the transmission is usually modelled using a contact rate function , which depends on the contact pattern among individuals and also on the probability of disease transmission . the contact function among individuals with different ages , for instance , may be modelled using a contact matrix @xcite or a continuous function @xcite . however , using network analysis methods , we can investigate more precisely the contact structure among individuals and analyze the effects of this structure on the spread of a disease . the degree distribution @xmath1 is the fraction of vertices in the network with degree @xmath2 . scale - free networks show a power - law degree distribution @xmath3 where @xmath4 is a scaling parameter . many real world networks @xcite are scale - free . in particular , a power - law distribution of the number of sexual partners for females and males was observed in a network of human sexual contacts @xcite . this finding is consistent with the preferential - attachment mechanism ( ` the rich get richer ' ) in sexual - contact networks and , as mentioned by liljeros et al . @xcite , may have epidemiological implications , because epidemics propagate faster in scale - free networks than in single - scale networks . epidemic models such as the susceptible infected ( si ) and susceptible infected susceptible ( sis ) models have been used , for instance , to model the transmission dynamics of sexually transmitted diseases @xcite and vector - borne diseases @xcite , respectively . many studies have been developed about the dissemination of diseases in scale - free networks @xcite and in small - world and randomly mixing networks @xcite . scale - free networks present a high degree of heterogeneity , with many vertices with a low number of contacts and a few vertices with a high number of contacts . in networks of human contacts or animal movements , for example , this heterogeneity may influence the potential risk of spread of acute ( e.g. influenza infections in human and animal networks , or foot - and - mouth disease in animal populations ) and chronic ( e.g. tuberculosis ) diseases . thus , simulating the spread of diseases on these networks may provide insights on how to prevent and control them . in a previous publication @xcite , we found that networks with the same degree distribution may show very different structural properties . for example , networks generated by the barabsi - albert ( ba ) method @xcite are more centralized and efficient than the networks generated by other methods @xcite . in this work , we studied the impact of different structural properties on the dynamics of epidemics in scale - free networks , where each vertex of the network represents an individual or even a set of individuals ( for instance , human communities or animal herds ) . we developed routines to simulate the spread of acute ( short infectious period ) and chronic ( long infectious period ) infectious diseases to investigate the disease prevalence ( proportion of infected vertices ) levels and how fast these levels would be reached in networks with the same degree distribution but different topological structure , using si and sis epidemic models . this paper is organized as follows . in section [ sec : hypothetical ] , we describe the scale - free networks generated . in section [ sec : model ] , we show how the simulations were carried out . the results of the simulations are analyzed in section [ sec : results ] . finally , in section [ sec : conclusions ] , we discuss our findings . we generated scale - free networks following the barabsi - albert ( ba ) algorithm @xcite , using the function barabasi.game(@xmath5 , @xmath6 , directed ) from the r package igraph @xcite , varying the number of vertices ( @xmath5 = @xmath7 , @xmath8 and @xmath9 ) , the number of edges of each vertex ( @xmath6 = 1 , 2 and 3 ) and the parameter that defines if the network is directed or not ( directed = true or false ) . for each combination of @xmath5 and @xmath6 , 10 networks were generated . then , in order to guarantee that all the generated networks would follow the same degree distribution and that the differences on the topological structure would derive from the way the vertices on the networks were assembled , we used the degree distribution from ba networks as input , to build the other networks following the method a ( ma ) @xcite , method b ( mb ) @xcite , molloy - reed ( mr ) @xcite and kalisky @xcite algorithms , all of which were implemented and described in detail in ref . as mentioned above , these different networks have distinct structural properties . in particular , the networks generated by mb are decentralized and with a larger number of components , a smaller giant component size , and a low efficiency when compared to the centralized and efficient ba networks that have all vertices in a single component . the other three models ( ma , mb and kalisky ) generate networks with intermediate characteristics between mb and ba models . the element @xmath10 of the adjacency matrix of the network , @xmath11 , is defined as @xmath12 if there is an edge between vertices @xmath13 and @xmath14 and as @xmath15 , otherwise . we also define the elements of the vector of infected vertices , @xmath16 . if vertex @xmath13 is infected , then @xmath17 , and , if it is not infected , @xmath18 . the result of the multiplication of the vector of infected vertices , @xmath16 , by the adjacency matrix , @xmath11 , is a vector , @xmath19 , whose element @xmath13 corresponds to the number of infected vertices that are connected to the vertex @xmath13 and may transmit the infection @xmath20 using matlab , the spread of the diseases with hypothetical parameters along the vertices of the network was simulated using the following algorithm : 1 . a proportion ( @xmath21 ) of the vertices is randomly chosen to begin the simulation infected . for our simulations , @xmath22 , since we are interested in the equilibrium state and this proportion guarantees that the disease would not disappear due to the lack of infected vertices at the beginning of the simulations . 2 . in the sis ( susceptible infected susceptible ) epidemic model , a susceptible vertex can get infected , returning , after the infectious period , to the susceptible state . for each time step : 1 . we calculate the probability ( @xmath23 ) of a susceptible vertex @xmath13 , that is connected to @xmath24 infected vertices , to get infected , using the following equation : @xmath25 where @xmath26 is the probability of disease spread . [ item2.b ] we determine which susceptible vertices were infected in this time step : if @xmath27uniform(0,1 ) @xmath28 , the susceptible vertex becomes infected . for each vertex infected , we generate the time ( @xmath29 ) that the vertex will be infected following a uniform distribution : @xmath30 uniform ( @xmath31 , @xmath32 ) , where @xmath31 and @xmath32 are , respectively , the minimum and the maximum time of the duration of the disease . 3 . decrease in 1 time step the duration of the disease on the vertices that were already infected , verifying if any of them returned to the susceptible state ; 4 . update the status of all vertices . for the si ( susceptible infected ) epidemic model , we chose @xmath29 in order to guarantee that an infected vertex remains infected until the end of the simulation . varying the values of the parameter @xmath29 , we simulated the behaviour of hypothetical acute and chronic diseases , using different values of @xmath26 , considering that once a vertex gets infected it would remain in this state during an average fixed time ( an approach that can be used when we lack more accurate information about the duration of the disease in a population ) or that there would be a variation in this period , representing more realistically the process of detection and treatment of individuals ( table [ tab : table1 ] shows the diseases simulated and the values of @xmath29 assumed ) . [ cols="^,^,^,^,^ " , ] we adopted a total time of simulation ( @xmath33 ) of 1000 arbitrary time steps . for each spreading model , we carried out 100 simulations for each network , calculating the prevalence of the simulated disease for each time step . then we calculated the average of the prevalence of these simulations on each network . after that we grouped the simulations by network algorithm . finally , we calculated the average prevalence of these network models . on the undirected networks , we observed that the disease spreads independently of the value of @xmath26 used , and that an increase in @xmath26 leads to an increase in the prevalence of the infection ( figure [ fig : figure1 ] ) . also , we observed that the prevalence tends to stabilize approximately in the same level despite the addition of vertices ( figure not shown ) . when we increase the number of edges of each vertex , there is an increase in the prevalence of the infection . a result that stands out is that , when @xmath34 , there is a great difference in the equilibrium level of the prevalence in each network . however , as we increase the value of @xmath6 , the networks tend to show closer values of equilibrium ( figure [ fig : figure3 ] ) . among the undirected networks , the networks generated using the mb algorithm presented the lowest values of prevalence in the spreading simulations ( figure [ fig : figure4 ] ) . on the directed networks , we observed that , despite the simulations of acute diseases , the disease spreads independently of the value of @xmath26 used and , as in the undirected networks , an increase in @xmath26 leads to an increase in the prevalence of the infection ( figure [ fig : figure5 ] ) . also , similarly to what was observed for undirected networks , the prevalence tends to stabilize approximately in the same level despite the addition of vertices ( figure not shown ) . when we increase the number of edges of each vertex , there is an increase in the prevalence of the infection ( figure [ fig : figure7 ] ) . a result that stands out is that , when @xmath35 , the prevalence in the kalisky networks tend to stabilize in a level a little bit higher than the other ones . among the networks , those generated using the ba algorithm presented the lowest values of prevalence in the spreading simulations ( figure [ fig : figure8 ] ) . to compare the numerical results of the simulation with a theoretical approach , it is possible to deduce , for an undirected scale - free network assembled following the ba algorithm , the equilibrium prevalence , given by @xcite @xmath36 . \label{eq:3}\ ] ] this expression applies to the ba undirected network with @xmath37 and a fixed infectious period of one time unit . for instance , for @xmath38 and @xmath39 , we obtain @xmath40 . in figure [ fig : figure9 ] , we observe that the equilibrium prevalence in the simulation reaches the value predicted by equation ( [ eq:3 ] ) . our approach , focusing on different networks with the same degree distribution , allows us to show how the topological features of a network may influence the dynamics on the network . analyzing the results of the spreading simulations , we have , as expected , that the variation in the number of vertices of the hypothetical networks had little influence in the prevalence of the diseases simulated , a result that is consistent with the characteristics of the scale - free complex networks as observed by pastor - satorras and vespignani @xcite . with respect to the effect of the increase of the probability of spreading on the prevalence in undirected networks ( figure [ fig : figure1 ] ) , we observed that the prevalence reaches a satured level . for undirected networks , if the probability of infection is high , there is a saturation of infection on the population for chronic diseases and therefore no new infections can occur . regarding the variation in the number of edges , the increase in the prevalence was also expected since it is known that the addition of edges increases the connectivity on the networks studied , allowing a disease to spread more easily . about the effect of considering the networks directed or undirected , we have that the diseases tend to stay in the undirected networks independently of the spreading model and the value of @xmath26 considered @xcite , while in the directed cases due to the limitations imposed by the direction of the movements , when @xmath34 , the acute diseases tend to disappear on some of the networks . also due to the direction of the links , in directed networks , the disease may not reach or may even disappear from parts of the network , explaining to some degree why the prevalence in directed networks ( figure [ fig : figure3 ] ) is smaller than in undirected networks ( figure [ fig : figure7 ] ) . in the si simulations , we could observe what would be the average maximum level of prevalence of a disease on a network and how fast this level would be reached . in the sis simulations , the oscillations on the stability of the prevalence observed result from the simultaneous recovery of a set of vertices . with a fixed time of infection , the set of vertices that simultaneously recover is greater than in the case of a variable time of infection , since in the latter , due to the variability of the disease duration , the vertices form smaller subsets that will recover in different moments of the simulation . a result that called attention is that , in the directed networks with @xmath34 , when we simulated the chronic diseases using a fixed time , the equilibrium levels achieved were lower than the ones achieved when we used a variable time . examining the results of the simulations on each network model , we have that among the undirected ones , the mb network has the lowest prevalence , with a plausible cause for this being how this network is composed , since there is a large number of vertices that are not connected to the most connected component of the network @xcite . among the directed networks , the ba network has the lowest prevalence observed , what is also probably due to the topology of this network , since it is composed of many vertices with outgoing links only and a few vertices with many incoming and few outgoing links , thus preventing the spread of a disease . using the methodology of networks , it is possible to analyze more clearly the effects that the heterogeneity in the connections between vertices have on the spread of infectious diseases , since we observed different prevalence levels in the networks generated with the same degree distribution but with different topological structures . moreover , considering that the increase in the number of edges led to an increase in the prevalence of the diseases on the networks , we have indications that the intensification of the interaction between vertices may promote the spread of diseases . so , as expected , in cases of sanitary emergency , the prevention of potentially infectious contacts may contribute to control a disease . this work was partially supported by fapesp and cnpq .
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the transmission dynamics of some infectious diseases is related to the contact structure between individuals in a network .
we used five algorithms to generate contact networks with different topological structure but with the same scale - free degree distribution .
we simulated the spread of acute and chronic infectious diseases on these networks , using si ( susceptible infected ) and sis ( susceptible infected susceptible ) epidemic models .
in the simulations , our objective was to observe the effects of the topological structure of the networks on the dynamics and prevalence of the simulated diseases .
we found that the dynamics of spread of an infectious disease on different networks with the same degree distribution may be considerably different .
* adv .
studies theor .
phys .
, vol . 7 , 2013 , no . 16 , 759 - 771 * * hikari ltd , www.m-hikari.com * * http://dx.doi.org/10.12988/astp.2013.3674 * * raul ossada , jos h. h. grisi - filho , * * fernando ferreira and marcos amaku * faculdade de medicina veterinria e zootecnia universidade de so paulo so paulo , sp , 05508 - 270 , brazil copyright @xmath0 2013 raul ossada et al .
this is an open access article distributed under the creative commons attribution license , which permits unrestricted use , distribution , and reproduction in any medium , provided the original work is properly cited .
* keywords : * scale - free network , power - law degree distribution , dynamics of infectious diseases
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the flexibility and bistability are the two key points in the next generation display technologies . the flexibility implies the display would be thin , lightweight , and ultimately , paper like , meaning it would be cheap enough to be disposable . the bistability implies the technology would be ecologically friendly . in the bistable display , the image does not need to be refreshed until rewritten and , therefore , a low level of power consumption is expected for still images . unfortunately , for motion pictures , the bistability has no advantages in power savings over nonbistable technologies such as liquid crystal displays ( lcds ) , plasma display panels ( pdps ) , and displays based on organic light emitting diodes commonly referred to as oleds . the display technologies based on particles are the most prominent candidates for a flexible and bistable displays . the earliest display based on particles dates back to 1970 s when ota@xcite filed for a patent . since then various particle - displays based on electrophoresis and electrowetting principles have emerged to form what is now referred to as `` e - paper technologies '' in the industry.@xcite in electrophoresis , particles are usually suspended in a fluid . because the speed at which particles move inversely vary with fluid density , particle - displays based on electrophoresis have slow response time , typically on the order of @xmath0@xcite this makes motion pictures unsuitable for electrophoretic particle - displays . the issue of slow response time in electrophoretic particle - displays , however , has been resolved with the unveil of quick response liquid powder display ( qr - lpd ) by hattori et al.@xcite the qr - lpd is distinguished from the rest of particle based displays in that it uses air as the particle carrying medium rather than fluid . because the particles in qr - lpd move in air , its response time is at @xmath1 which is even faster than lcds . the submillisecond response time makes qr - lpd the only candidate based on particle - display capable of handling motion pictures , and researches are being conducted for particle - displays with air as the particle carrying medium.@xcite common to all particle - displays , regardless of whether air or fluid is used as the particle carrying medium , is the lack of continuous grayscale . here , the terminology , `` continuous grayscale , '' is referred to as the number of grayscale levels required to produce desired number of colors . in principle , a display device with continuous grayscale can generate an infinite range of colors . that being clarified , a high grayscale range is essential to quality displays . without it , the images displayed on monitors would be dull.@xcite recently , it has been reported particle - display based on qr - lpd can generate up to 16 grayscale levels ( i.e. , 4 bits ) , which corresponds to the capability of generating 4096 colors.@xcite to the proponents of other competing flat panel display technologies , such as flexible lcds , pdps , or oleds , 4 bits of gray scale range could hardly be considered a technological milestone . however , considering only 4 grayscale levels were possible just a few years ago for particle - displays , 16 grayscale levels is a significant technological advancement for the e - paper technology.@xcite the most well known e - paper technology , e - ink , obtains different gray states through modulation of voltages supplied to the control electrodes . the same mechanism is employed by particle - displays based on qr - lpd for achieving different gray states.@xcite because the voltage is modulated at a value lower than the saturation voltage , which is the voltage required to display either all black or all white for a simple black - and - white display , the displayed image would have intensity somewhere between that of completely black and completely white . this approach to achieve different gray states , however , is done at the cost of losing bistability . since the pixel is being constantly modulated to sustain a gray state , the situation is equivalent to motion pictures and the power savings from bistability no longer applies to gray states . in spite of the lost bistability advantages over the other competing flexible display technologies , the speed at which particle can be modulated is limited by its finite inertia ( mass ) and this places a practical limit on the extent to which the grayscale levels of particle - display can be enhanced by the aforementioned method . most recently , chim@xcite demonstrated a 64 grayscale levels ( or 6bits ) for particle - display based on qr - lpd for his masters thesis project at delft university of technology . however , to generate 64 grayscale levels , the data driver , which is a serial to parallel shift register that shifts data words of 6 bits per clock cycle , is required . ( color online ) transflective display based on charged particles . ] in this work , a display architecture based on charged particles is presented . unlike the previous particle - display technologies , the proposed device has potential to generate continuous grayscale levels without sacrificing the benefit of bistability . also , the proposed particle - display technology , in principle , does not require complicated voltage modulation schemes to generate different gray states.@xcite the cross - sectional schematic of an optical shutter based on charged particles of same polarity is illustrated in fig . [ fig : transflective_0 ] . in the figure , the two electrodes , where each is labeled top and bottom electrodes , constitute the control electrodes . because the light must be transmitted through the control electrodes , the electrodes are chosen from optically transparent conductors . the chamber , wherein the particles reside , can be a vacuum , filled with noble gas , or filled with air . the metallic reflectors , which forms the lateral surface of the chamber , are electrically connected to one of the control electrodes . this make metallic reflectors not only to reflect light , but also function as to keep particles from aggregating to the lateral surface of chamber . the optically transparent passivation layer , which is treated on the inner surface of the chamber , functions to prevent charge transfer between particles and conductors . for positively charged particles inside the chamber , the hydrophobic treatment on the surface of the passivation layer , which is not explicitly shown in the figure , for example , prevents particles from sticking to the surface . ( color online ) transmission of light through a medium filled with total of @xmath2 suspended spherical particles . both of the compressed and the uncompressed cases are shown . ] the intensity of light transmitted through a medium filled with suspended spherical particles , illustrated in fig . [ fig : cylinder_0 ] , is given by @xmath3,\label{eq : transmitted_intensity_bh}\ ] ] where @xmath4 and @xmath5 are , respectively , the real and the imaginary part of the complex refractive index for the medium and the particles suspended in it ; @xmath6 is the initial intensity , @xmath7 is the wavelength of incidence light , @xmath8 is the height ( or thickness ) of the volume containing charged particles , and @xmath9 is the volume fraction of particles.@xcite in explicit form , the volume fraction of particles is @xmath10 where @xmath2 is the total number of particles in the optical chamber , @xmath11 is the volume of a single particle , @xmath12 is the height of compressed cylinder containing charged particles , and @xmath13 is the cross - sectional area of the cylindrical chamber illustrated in fig . [ fig : cylinder_0 ] . throughout this work , i shall refer to terms such as `` compressed state '' or `` compressed particle volume '' to denote the compression of the volume containing particles . insertion of @xmath9 into eq . ( [ eq : transmitted_intensity_bh ] ) gives @xmath14 where @xmath15 is just a constant . the complex dielectric constant , @xmath16 and the complex refractive index , @xmath17 are expressed in form as @xmath18 where @xmath19 and @xmath4 are the real parts , and @xmath20 and @xmath5 are the imaginary parts . for the complex refractive index , @xmath4 represents the real refractive index and @xmath5 is the extinction coefficient ( or the absorption coefficient ) . the two , @xmath21 and @xmath17 are related by the expression @xmath22 with the real and imaginary parts inserted for @xmath21 and @xmath23 from eq . ( [ eq : complex - optical - constants ] ) , i have @xmath24 the resulting expression can be rearranged to yield @xmath25 equation ( [ eq : complex - optical - constants - algebra ] ) can only be true if and only if the real and the imaginary parts are equal to zero independently . this requirement yields the two expressions connecting @xmath26 to the optical constants @xmath27 and , the two expressions are @xmath28 stoller et al.@xcite measured the complex dielectric constant , @xmath29 for the single gold nanoparticle . for the gold nanoparticles of diameters @xmath30 and @xmath31 assuming a spherical morphology , they observed a reasonably good correspondence existing between complex dielectric constants of the bulk gold and the gold nanoparticles for the wavelength range of roughly @xmath32 to @xmath33 their finding is of significant importance , as it allows the bulk gold dielectric constants , which is readily available , to be used for the gold nanoparticles , which is not so readily available . since the real and the imaginary parts of the complex dielectric constant are related to the optical constants @xmath34 thru eq . ( [ eq : e1-e2-complex - dielectric - constant ] ) , the @xmath4 and @xmath5 for the gold nanoparticles are readily available once @xmath19 and @xmath20 are known from the bulk counterpart . the vice versa is also true , of course . fortunately , johnson and christy measured the optical constants for the bulk copper , silver , and the bulk gold.@xcite justified by the findings of stoller et al.,@xcite for @xmath35 assuming the gold nanoparticle of radius @xmath36 the @xmath4 and @xmath5 measurements from the work of johnson and christy,@xcite @xmath37 can be utilized for the gold nanoparticles to plot eq . ( [ eq : transmitted_intensity ] ) for the transmitted intensity . for a spherical gold nanoparticle , the particle volume , @xmath38 is given by @xmath39 where @xmath40 is the particle radius . assuming the diameter of @xmath41 for the gold nanoparticle , @xmath11 becomes @xmath42 to plot eq . ( [ eq : transmitted_intensity ] ) , i shall assume the following values for the chamber parameters , ( the height @xmath8 and the cross - sectional area @xmath13 ) , and the particle number , @xmath43 @xmath44 schematic of curves illustrating the intensity , @xmath45 of transmitted light as a function of compression height , @xmath46 ] before going ahead with plotting eq . ( [ eq : transmitted_intensity ] ) , it is worthwhile to double check if the total number of particles , @xmath43 specified in eq . ( [ eq : han ] ) is reasonable . with chamber parameters as defined in eq . ( [ eq : han ] ) , the volume of the cylindrical chamber is @xmath47 suppose the charged spherical particles can be compressed and enclosed in a confined volume in such a way that neighboring particles actually touch each other . how many particles , under such restrictions , can be fitted inside the chamber defined by eq . ( [ eq : han ] ) ? the answer is @xmath48 and its expression is @xmath49 for the spherical particle of radius @xmath36 the @xmath48 is roughly @xmath50 for the charged particles , the condition of neighboring particles actually touching each other is not possible due to coulomb repulsion , unless the external compression force is infinite . nonetheless , the expression for @xmath51 defined in eq . ( [ eq : nmax_particle_num ] ) , provides the upper limit for @xmath2 inside the chamber . the physical optical shutter based on charged particles must be compressible in order to allow variations in transmission intensity , which imposes the condition , @xmath52 equivalently , the variability of transmission intensity thru compression requires @xmath53 and this further modifies the condition for @xmath2 as @xmath54 the inequality condition for @xmath43 defined in eq . ( [ eq : zero_n_nmax ] ) , can be understood from the illustration shown in fig . [ fig : transmission_intensity_schematic ] , where some of the possible curves for the transmission intensity , @xmath45 as a function of the compression height , @xmath55 are shown . the cases where @xmath56 and @xmath57 are represented by the curves corresponding to @xmath58 at @xmath59 and @xmath60 respectively . similarly , the the curve corresponding to @xmath58 at @xmath61 in the figure represents the case where @xmath62 for @xmath63 the particles in the chamber are already maximally compressed and , therefore , no light gets transmitted . on the other hand , for @xmath64 all light gets transmitted through the optical shutter , as the inside of the optical shutter is a void . array of metallic nanoparticles in grid formation . the parameter @xmath65 is the grid ( or particle - particle ) spacing and @xmath7 is the wavelength of the incidence light . for the charged particles inside the chamber , the parameter @xmath65 can only be thought of as the time averaged mean particle - particle distance between the nearest neighboring particles . nonetheless , the grid representation illustrated here serves the purpose of initiating the crude connection between the wavelength of incidence light , @xmath66 and the total particle number , @xmath67 ] for the case of any finite @xmath2 satisfying the condition defined in eq . ( [ eq : zero_n_nmax ] ) , the curve for the transmission intensity must necessarily lie in the region between the two extreme cases , @xmath56 and @xmath63 as schematically demonstrated in fig . [ fig : transmission_intensity_schematic ] . three such curves , corresponding to finite @xmath43 are illustrated in the figure : ( 1 ) the upper curve represented by @xmath68 ( 2 ) the curve represented by @xmath69 and ( 3 ) the lower curve represented by @xmath68 where @xmath70 is a constant and @xmath71 is a function of @xmath46 the two curves corresponding to @xmath72 only appreciably varies in transmission intensity with compression within the window of @xmath73 where @xmath74 such curves are associated with @xmath43 in which the @xmath2 is finite but close to either @xmath56 or @xmath62 because the transmission intensity only appreciably varies within @xmath75 for such choices of @xmath43 the gray states are difficult to achieve as it requires very precise control of the compression height . for example , assuming the particles can be compressed at the increment in @xmath76 the fact that @xmath77 makes it difficult to achieve gray states . to achieve gray states , the @xmath76 which defines the sensitivity of compression , must be much smaller than @xmath78 consequently , the transmission intensity curves for the aforementioned choices of @xmath2 are only good for displaying either completely bright or completely dark transmission states . the nanoparticles of radius @xmath40 are separated by @xmath79 which is the nearest surface to surface separation between two neighboring particles . the imaginary particles of radius @xmath80 are , however , in closed packed formation for a given nearest surface to surface separation of @xmath81 ] the optimal design for the presented optical shutter is achieved by choosing @xmath2 for the total particle number in the chamber in such way that the curve for the transmission intensity goes like @xmath69 where @xmath70 is a constant , in fig . [ fig : transmission_intensity_schematic ] . because the curve is linear for the transmission intensity , the window of range in which compression can be done is maximized , @xmath82 for this particular choice of @xmath43 the @xmath76 which is the increment at which particles can be compressed , is much less than @xmath83 i.e. , @xmath84 and the number of different gray states that can be achieved is given by @xmath85 in principle , the @xmath86 can be made finer and finer as desired . in reality , the fineness of @xmath86 is limited by the system design . nevertheless , with the right choice of @xmath43 the grayscale levels in number of @xmath87 as defined in eq . ( [ eq : ngray ] ) , is possible with the presented optical shutter based on charged particles . the question remains to be answered is this : how do we go about obtaining the right @xmath2 ? to answer this , i shall refer to the illustration shown in fig . [ fig : grid ] . due to the wave nature of light , the total particle number and the transmission intensity depend on the wavelength , @xmath66 of the incidence light . as a rudimentary assumption , the transmission loss of an electromagnetic wave passing through an array of metallic particles decreases for @xmath88 and increases for @xmath89 where @xmath90 which is schematically illustrated in fig . [ fig : grid ] . the case where @xmath88 represents the situation in which @xmath2 is very small , whereas the case where @xmath89 represents the situation in which @xmath91 provided the @xmath92 is not too large . it is , therefore , not too bad to impose the condition , @xmath93 for @xmath65 in estimating for the total number of particles in the chamber . for @xmath35 this condition for @xmath65 yields the value of @xmath94 to figure out exactly how man particles can be fitted inside the chamber under restriction @xmath93 the fig . [ fig : particle_num ] is referred to . since @xmath65 is the closest distance between surfaces of two nearest neighbor particles , i shall visualize an imaginary particle of radius @xmath95 @xmath96 assuming the imaginary particles are close - packed inside the chamber , the problem becomes identical to the previous case , which resulted in eq . ( [ eq : nmax_particle_num ] ) . with @xmath97 of eq . ( [ eq : r_imaginary ] ) inserted for @xmath40 in eq . ( [ eq : nmax_particle_num ] ) , the expression becomes @xmath98 where @xmath48 has been replaced by @xmath99 with @xmath100 defined in eq . ( [ eq : nmim_particle_num ] ) , the number of particles inside the chamber may be chosen according to the inequality , @xmath101 it can be easily verified that @xmath102 for @xmath103 which corresponds to the half wavelength of @xmath35 @xmath100 is roughly @xmath104 this value for @xmath100 is much larger than zero , but it is much smaller than @xmath51 which has the value @xmath105 from eq . ( [ eq : nmax_number ] ) . for the optical shutter involving charged particles of spherical morphology and the cylindrical chamber of specifications defined in eq . ( [ eq : han ] ) , the inequality condition for @xmath43 eq . ( [ eq : n_condition ] ) , becomes @xmath106 equation ( [ eq : transmitted_intensity ] ) has been computed using gold nanoparticles as the charged particles ( the gold nanoparticle was chosen only because its optical constant data , @xmath4 and @xmath107 were readily available ) . for the charged particles and the incidence light , the parameters defined in eq . ( [ eq : bulk - gold - n - k ] ) were used . the parameters defined in eq . ( [ eq : han ] ) were used for the chamber specification . using eq . ( [ eq : transmitted_intensity ] ) , the transmission intensity for different values of @xmath2 were considered and the results are shown in fig . [ fig : transmission_intensity ] . the @xmath108 curve is the case where particle number is relatively low in the chamber . as it can be observed , for low particle numbers in the chamber , the intensity does not vary well with compression except for small @xmath55 which is consistent with the curve represented by @xmath72 in fig . [ fig : transmission_intensity_schematic ] . contrarily , the @xmath109 curve corresponds to the case where too many particles are inside the chamber . although the transmission intensity varies linearly with compression , which is a good characteristic of an optical shutter , the output intensity is far too low even for the brightest state . for @xmath110@xmath111 the brightest state only transmits @xmath112 of the initial input intensity , i.e. , @xmath113 as expected , the @xmath114@xmath115 curve in fig . [ fig : transmission_intensity ] , which corresponds to the case where @xmath116 for @xmath117 most resembles the curve represented by @xmath118 in fig . [ fig : transmission_intensity_schematic ] . however , the brightest output intensity is only @xmath119 of the input intensity of the incidence light . compared to lcds , where only @xmath120 of the intensity of the incidence light from back light unit gets transmitted , the output intensity of @xmath119 is already @xmath30 times more efficient than lcds . ( color online ) transmission intensity as a function of particle number , @xmath43 and the compression height , @xmath46 the abbreviation `` mil '' in the figure denotes a million , i.e. , @xmath121 ] the same principle , which is inherent in eq . ( [ eq : transmitted_intensity ] ) , applies to the proposed optical shutter , fig . [ fig : transflective_0 ] . in the presented device , the chamber is filled with charged particles of same polarity and this can be identified with the medium filled with suspended particle in fig . [ fig : cylinder_0 ] . the on set of electric field inside the chamber , which is done by controlling the voltage over one of the electrodes , causes particles to be compressed in volume , as illustrated in fig . [ fig : transflective_1 ] . since the particles are assumed to be positively charged , they are compressed in the direction of electric field . eventually , the compression comes to a stop when particle - particle coulomb repulsion counterbalances the compression induced by the control electrodes . in principle , the level of compression for the particle volume can be varied continuously . because the intensity of transmitted light varies with compression , i.e. , eq . ( [ eq : transmitted_intensity ] ) , the display based on charged particles has the potential to generate continuous gray levels . also , since the control electrodes form a capacitor , provided there is no leakage ( or negligible ) current across the capacitor , the electric field inside the chamber can be sustained even when the device is disconnected from power . this opens up the possibility of a bistable mode for gray states as well . the presented device works as an optical shutter , provided the charged particles can be effectively prevented from piling up at the surface of dielectric walls . when a charged particle is brought close to the dielectric surface , the bound charges within the dielectric get redistributed in order to reduce the field originating from the charged particle placed near vicinity of dielectric surface . such is illustrated in fig . [ fig : bc1 ] . for the case where a positively charged particle is placed near the surface of a dielectric , the surface bound charges of opposite polarity get induced and distributed near the inner surface of dielectric . as a result , the positively charged external particle gets pulled to the dielectric surface and , eventually , sticking to the surface of dielectric , which process is illustrated in fig . [ fig : bc2 ] . if there are more than one positively charged particles placed at the vicinity of dielectric surface , the aforementioned process continues and , eventually , layers get formed , for example , the layers @xmath13 and @xmath122 in the figure . this process does not continue indefinitely , however , as the bound charges of opposite polarity within the dielectric get eventually shielded by the positively charged particles forming layers over the dielectric surface . because the particles in each layer are charged with same polarity , the coulomb repulsion keeps the two particles from touching each other . assuming the layer @xmath122 is sufficient to shield completely the negatively charged surface bound charges within the dielectric , the remaining positively charged particles in the region between two dielectric walls would have no other places to go , except to continuously bounce back and forth within the region , which region has been indicated by @xmath123 in fig . [ fig : bc2 ] . ( color online ) the on set of electric field compresses the filled volume for charged particles . ] the aforementioned description for an optical shutter , however , has a serious problem which is associated with the particle layers forming on the surface of dielectric walls . assuming the path of light propagation is along the horizontal axis , the light enters the device from the left and exits at the right or vice versa . in principle , the compression of charged particles in region , @xmath124 controls the intensity of transmitted light . the problem arises because the charged particles forming layers @xmath13 and @xmath122 on the surface of dielectric walls may no longer be optically transparent . as previously discussed , using the illustration in fig . [ fig : grid ] as an example , metallic particles in an array of grid formation severely reduces the intensity of transmitted light for grid spacing , @xmath79 much less than the wavelength , @xmath66 of the incidence light . i now discuss the ways to resolve this complication . ( color online ) charged particles near dielectric wall . ] ( color online ) charged particles in region @xmath123 are shielded from the negatively charged surface bound charges within the dielectric wall . ] ( color online ) transparent charged layer formed by isam or electron beam irradiation method effectively plays the role of positively charged particle layers at @xmath13 and @xmath125 ] it is well known that the surface of a typical @xmath126 glass becomes hydrophilic in an open air as a result of the silane group at the surface of glass combining with oxygen.@xcite this has the effect of making the surface of glass to be slightly negatively charged and any positively charged particles in vicinity would be attracted to the glass surface forming layers such as those , e.g. , @xmath13 and @xmath127 illustrated in fig . [ fig : bc2 ] . unless the attracted , positively charged , particles are optically transparent , the device would not be able to function as an optical shutter for no light would to pass through the device . necessarily , the role of layers @xmath13 and @xmath122 in fig . [ fig : bc2 ] must be played out by particles of same polarity as those residing in region @xmath124 but distinguished from those residing in region @xmath123 in that they are optically transparent . one way to achieve this is to chemically treat the surface of glass so as to make it hydrophobic . the hydrophobic treatment of glass electrically neutralizes the glass surface , thereby significantly reducing the oppositely charged particles from sticking to the glass surface . for large and weakly charged positive particles , simple electrical neutralization of glass surface by hydrophobic treatment is sufficient to keep particles from permanently sticking to the surface . in such system , charged particles are continuously bounced off of the wall and do not stick to the surface . however , for smaller and well charged positive particles , simple hydrophobic treatment of glass surface is not sufficient to prevent oppositely charged particles from sticking to the glass surface . for such cases , it is necessary to make the surface of glass net positively charged . this can be easily done by utilizing the technique known as `` ionically self assembled monolayer '' ( isam ) , in which technique charged particles , polymers , or monomers of preferred polarity are physically attached to the surface.@xcite alternatively , and more directly , the glass may be irradiated with electron beam to physically embed charged particles inside the glass medium.@xcite in this latter method , for example , net negative charges may be made to accumulate inside the glass . this has the effect of inducing positive charges on the surface of glass , which repels positive charged particles in vicinity of the glass surface , thereby preventing particles from sticking to the surface . in summary , the role of positively charged particle layers , @xmath13 and @xmath122 in fig . [ fig : bc2 ] , may be effectively get taken cared by the aforementioned isam or the electron beam irradiation methods ; and , using these alternative techniques , the glass surface repels charged particles inside the chamber and it is optically transparent , as illustrated in fig . [ fig : bc3 ] . to utilize charged particles in displays , a quantitative understanding of how design parameters , such as @xmath128 for the sub - pixel dimensions and @xmath129 for the charged particle , enter into the compression mechanism is required . here , @xmath130 is the particle radius ( assuming a spherical particle ) , @xmath131 is the particle mass density , and @xmath132 is the net charge which the particle holds . describing the compression mechanism in terms of the aforementioned design parameters is the task for the next section . the optical shutter presented in this proposal relies on the density of particles in chamber to control the intensity of transmitted light . an analogy can be made to the driving under misty weather . when the density of water vapor suspended in the atmosphere is heavy , one is obscured in his or her viewing distance , as less light reaches the eye . contrarily , the amount of light reaching the eye increases with a reduction in density of water vapor suspended in the atmosphere , thereby enabling the driver to see far distances . the particles in chamber of the proposed optical shutter ranges in diameter anywhere from a few nanometers to several microns , assuming a spherical morphology . this range for the particle size , although small macroscopically , is much too large to be considered for a treatment within quantum domain , where the quantum theory must be used for a description . therefore , the classical theory suffices for the description here . since the charged particles in the system , as a whole , behave like a classical gas , the description is carried out in the realm of statistical physics . to keep the topic presented here self - contained , i shall briefly summarize the kind of manipulations and approximations assumed in obtaining expressions which are considered crucial to the initial development of the analysis . the charged particles in chamber can be treated as classical particles obeying the maxwell - boltzmann statistics.@xcite the @xmath133 charged particle under influence of external forces , for example , gravitational and electric forces , assumes the energy @xmath134 where @xmath135 is the center of mass momentum for the @xmath133 charged particle and the term associated with it is the kinetic energy , @xmath136 is the interaction energy with external influences , and @xmath137 is the energy contribution arising only if the particle is not monatomic . in explicit form , @xmath136 can be expressed as @xmath138^{1/2}}+m_{i}gz_{i}\\ & + q_{i}ez_{i},\end{aligned}\ ] ] where @xmath2 is the number of particles in volume , the @xmath139 and @xmath140 denote respectively the net charges for the @xmath133 and @xmath141 particles , @xmath142 is the electric field magnitude , and the constants @xmath143 and @xmath144 in mks system of units . for a monatomic particle , the energy contributions from the internal rotation and vibration with respect to its center of mass vanishes , @xmath145 therefore , the @xmath133 monatomic charged particle under the influence of external forces assumes the energy @xmath146^{1/2}},\label{eq : ui}\end{aligned}\ ] ] where , for convenience , all particles in the system are assumed to be identically charged with same polarity , i.e. , @xmath147 the gravity and electric field have directions , and this information must be taken into account in eq . ( [ eq : ui ] ) . to do this , the parameter @xmath148 is first restricted to a domain @xmath149 with @xmath148 restricted to a domain defined by @xmath150 the gravitational potential energy of a particle , @xmath151 increases with positive @xmath152 and decreases with negative @xmath152 as @xmath148 increase . therefore , the direction of gravity in eq . ( [ eq : ui ] ) can be taken into account by @xmath153 the electric potential energy of a particle , @xmath154 increases with positive @xmath142 and decreases with negative @xmath142 as @xmath148 increase . for a positively charged particle , its electric potential energy increases as it moves against the direction of electric field and decreases as it moves in the direction of electric field . the direction of electric field inside the chamber can hence be taken into account in eq . ( [ eq : ui ] ) by @xmath155 the probability of finding the particle with its center of mass position in the ranges @xmath156 and @xmath157 can be expressed as @xmath158 where @xmath159 is the temperature in units of degree kelvin @xmath160 and @xmath161 is the boltzmann constant , @xmath162 with eq . ( [ eq : ui ] ) , @xmath163 becomes @xmath164^{1/2}}\right)\nonumber \\ & \times\exp\left[-\left(\frac{m_{i}g+qe}{k_{\textup{b}}t}\right)z_{i}\right]d^{3}\mathbf{r}_{i}.\label{eq : ps1}\end{aligned}\ ] ] the presence of repulsive coulomb interaction , @xmath165^{1/2}},\ ] ] makes eq . ( [ eq : ps1 ] ) difficult and this term must be approximated . the configuration depicted in fig . [ fig : cylinder ] is referred for the analysis . the @xmath133 particle is the top most particle and all other particles are ahead of it in the direction of compression . [ fig : cylinder ] ] as the @xmath133 particle gets compressed in the direction of electric field , it experiences net electric field given by @xmath166 where @xmath167 is the electric field inside the chamber generated by external electrodes and @xmath168 is the electric field produced by all other particles inside the chamber . because @xmath167 and @xmath168 are oppositely directed , the magnitude @xmath169 is given by @xmath170 to estimate @xmath171 fig . [ fig : cylinder ] is considered . the vectors @xmath172 @xmath173 and @xmath174 satisfy @xmath175 in cylindrical coordinates , @xmath176 @xmath177 and @xmath174 become @xmath178 where @xmath179 represents the cylindrical coordinates for particle labeled as @xmath180 in fig . [ fig : cylinder ] . with @xmath177 and @xmath174 thus defined , eq . ( [ eq : b_c_a ] ) becomes @xmath181 at @xmath182 the electric field contributed from particle labeled as @xmath180 is given by @xmath183}{\left[\rho_{2}^{2}+\left(\eta+\varepsilon - z_{2}\right)^{2}\right]^{3/2}}.\end{aligned}\ ] ] because both gravitational and electrical forces are assumed to depend on @xmath184 coordinate only , the @xmath185 and @xmath186 components of @xmath187 average to zero to become @xmath188^{3/2}}\mathbf{e}_{3}.\ ] ] all particles in the cylinder , not just the particle labeled as @xmath180 in fig . [ fig : cylinder ] , contributes to form @xmath189 hence , @xmath190^{3/2}}\mathbf{e}_{3},\label{eq : e_total_0}\ ] ] where @xmath2 is the number of charged particles in the chamber . for @xmath2 sufficiently large , the coordinates @xmath191 and @xmath192 can be replaced by @xmath193 and @xmath194 in the continuum limit , the summation symbol gets replaced by @xmath195 and the charge @xmath132 is replaced by @xmath196 where volume is that of cylinder illustrated in fig . [ fig : cylinder ] and @xmath197 is the total charge inside it . in the continuum limit then , where @xmath2 is assumed to be sufficiently large , eq . ( [ eq : e_total_0 ] ) can be approximated by @xmath198^{3/2}}\mathbf{e}_{3},\end{aligned}\ ] ] which result , integrating over the @xmath199 becomes @xmath200^{3/2}}\mathbf{e}_{3}.\label{eq : e_total_1}\end{aligned}\ ] ] with the change of variable , @xmath201 the @xmath202 integral in eq . ( [ eq : e_total_1 ] ) becomes @xmath203^{3/2 } } & \rightarrow\frac{\left(\eta+\varepsilon - z\right)}{2}\int\frac{dx}{x\sqrt{x}}\\ & = -\frac{\left(\eta+\varepsilon - z\right)}{\sqrt{x}}.\end{aligned}\ ] ] with @xmath204 reverted back to the original variable , the @xmath202 integral becomes @xmath205^{3/2 } } & = \left.-\frac{\left(\eta+\varepsilon - z\right)}{\sqrt{\rho^{2}+\left(\eta+\varepsilon - z\right)^{2}}}\right|_{0}^{\xi}\\ & = 1-\frac{\left(\eta+\varepsilon - z\right)}{\sqrt{\xi^{2}+\left(\eta+\varepsilon - z\right)^{2}}}.\end{aligned}\ ] ] insertion of the result into eq . ( [ eq : e_total_1 ] ) yields @xmath206 with the change of variable , @xmath207 the @xmath184 integral in eq . ( [ eq : e_total_2 ] ) becomes @xmath208 with @xmath209 reverted back to the original variable , the @xmath184 integral becomes @xmath210 insertion of the result into eq . ( [ eq : e_total_2 ] ) gives the @xmath211 @xmath212\mathbf{e}_{3}.\end{aligned}\ ] ] since the @xmath184 coordinate of the @xmath133 particle is given by @xmath213 the expression for @xmath168 may be rewritten in terms of @xmath148 as @xmath214\mathbf{e}_{3}.\end{aligned}\ ] ] the parameter @xmath215 has been introduced for a mathematical convenience to assure that the @xmath133 particle is the upper most particle residing at the top surface of the compressed volume . taking the limit @xmath216 the previous expression for @xmath168 becomes @xmath217 since @xmath148 is the @xmath184 coordinate for the @xmath133 particle , which is the particle residing at the top surface of the compressed volume in fig . [ fig : cylinder ] , the @xmath168 defined in eq . ( [ eq : e_repulsion ] ) represents the coulomb repulsion acting on the particle residing at the top surface of the compressed volume from all other ones within the compressed volume . insertion of eq . ( [ eq : e_repulsion ] ) into eq . ( [ eq : e_net ] ) gives @xmath218 the @xmath219 in current form is only an approximation because the expression for @xmath211 eq . ( [ eq : e_repulsion ] ) , is an approximation . the equality can be made by replacing @xmath220 where @xmath221 is the effective charge to be determined experimentally . with @xmath222 the expression for @xmath219 becomes @xmath223 where the subscript @xmath224 of @xmath225 has been dropped for convenience . what is the implication of @xmath219 ? the @xmath226 of eq . ( [ eq : ui ] ) , which is the energy term assumed by the @xmath133 charged particle under the influence of external forces , can be rearranged in form as @xmath227^{1/2}}\vphantom{\frac{\frac{\frac{1}{1}}{\frac{1}{1}}}{\frac{\frac{1}{1}}{\frac{1}{1}}}}\right\ } .\end{aligned}\ ] ] one notices that the term in the summation is the electric field contribution from the @xmath141 particle acting on the @xmath133 particle , @xmath228^{1/2}}.\end{aligned}\ ] ] furthermore , one finds @xmath229 and @xmath226 can be equivalently expressed as @xmath230 since @xmath231 eq . ( [ eq : e_net ] ) , the @xmath226 becomes @xmath232 insertion of eq . ( [ eq : ui_approximation ] ) into eq . ( [ eq : ps0 ] ) gives an alternate expression for probability density for finding particle with its center of mass position in the ranges @xmath156 and @xmath233 @xmath234d^{3}\mathbf{r}_{i}\nonumber \\ & \times\exp\left(-\frac{\mathbf{p}_{i}^{2}}{2m_{i}k_{\textup{b}}t}\right)d^{3}\mathbf{p}_{i},\label{eq : ps20}\end{aligned}\ ] ] which is different from the previous expression , eq . ( [ eq : ps1 ] ) , but now manageable . with eq . ( [ eq : e_net_final ] ) inserted for @xmath235 eq . ( [ eq : ps20 ] ) becomes @xmath236\exp\left(-\frac{\mathbf{p}_{i}^{2}}{2m_{i}k_{\textup{b}}t}\right)d^{3}\mathbf{r}_{i}d^{3}\mathbf{p}_{i},\label{eq : ps21}\end{aligned}\ ] ] equation ( [ eq : ps21 ] ) may be integrated over all possible @xmath204 and @xmath209 values lying within in the container and each components of the momentum may be integrated from @xmath237 to @xmath238 @xmath239dz_{i}\int_{x}\int_{y}dx_{i}dy_{i}.\label{eq : ps22}\end{aligned}\ ] ] the double integral over @xmath204 and @xmath209 gives slice area of the chamber at @xmath240 @xmath241 @xmath242 where @xmath243 is the radius of cylindrical chamber depicted in fig . [ fig : cylinder ] . the momentum integrals are obtained utilizing the well known integral formula@xcite @xmath244 where solutions are given by @xmath245 the momentum integrals become @xmath246 ] with eqs . ( [ eq : ps22a ] ) and ( [ eq : ps22b ] ) , eq . ( [ eq : ps22 ] ) becomes @xmath247dz_{i}.\end{aligned}\ ] ] with the following definitions , @xmath248 the previous expression for @xmath249 simplifies to @xmath250dz_{i},\label{eq : ps23}\end{aligned}\ ] ] where @xmath15 is the constant of proportionality to be determined from the normalization condition , @xmath251 it can be shown @xmath252dx_{i}\right\ } ^{-1},\label{eq : ccccc}\end{aligned}\ ] ] where @xmath253 is a dummy integration variable and it should not be confused with the coordinate @xmath204 of the cylinder . what can be said about @xmath221 defined in @xmath70 ? the effective coulomb repulsion from the remaining @xmath254 charged particles inside the chamber acting on the @xmath133 charged particle , see fig . [ fig : cylinder ] , is proportional to @xmath255 where @xmath221 must be determined empirically from measurements . in principle , @xmath221 takes into account the spatial configuration of the @xmath254 charged particles in the system because it effectively describes the system , which is illustrated in fig . [ fig : cylinder ] , in terms of the two body problem ( see fig . [ fig : cylinder_qeff ] ) . because the total charge in the imaginary cylinder must be conserved , it must be true that @xmath256 and , this implies the condition @xmath257 for describing the trend of volume compression involving charged particles , eq . ( [ eq : qeff_condition ] ) provides the way to estimate @xmath258 once @xmath221 is defined , eq . ( [ eq : ps23 ] ) may be plotted for the most probable height of the compressed volume , which volume contains the @xmath2 charged particles in the system . that being said , combining eqs . ( [ eq : ps23 ] ) and ( [ eq : ccccc ] ) , the probability density for the most probable height of the compressed volume containing @xmath2 charged particles becomes @xmath259dx_{i}\right\ } ^{-1}\exp\left[\alpha\left(\frac{1}{z_{i}}+\frac{1}{\xi}-\sqrt{\frac{1}{z_{i}^{2}}+\frac{1}{\xi^{2}}}\right)\right.\nonumber \\ & \left.-\beta z_{i}\vphantom{\sqrt{\frac{1}{z_{i}^{2}}}}\right]dz_{i},\label{eq : ps24}\end{aligned}\ ] ] where @xmath70 and @xmath260 are defined in eq . ( [ eq : alpha_beta ] ) . before plotting @xmath261 i shall explicitly define the charge @xmath262 particle mass @xmath263 and the electric field magnitude @xmath264 in nature , the charge is quantized and , therefore , it is convenient to express @xmath132 in terms of the charge number @xmath265 @xmath266 where @xmath267 is the fundamental charge unit and @xmath268 is the number of electrons removed from the particle . for the sake of simple analysis , the particles in the system are assumed to be spheres of identical radius . the mass of each particle would then be given by @xmath269 where @xmath130 is the radius of sphere and @xmath131 is the mass density . finally , the electric field generated in the chamber by control electrodes is @xmath270 where @xmath271 is the voltage applied to the top electrode ( the other electrode has been grounded ) . the approximation @xmath272 in electric field comes about because the effects of passivation layers inside the chamber have been neglected for simplicity . defined in eq . ( [ eq : ps24 ] ) . [ fig : cylinder2 ] ] having defined @xmath262 @xmath263 and @xmath273 the configuration illustrated in fig . [ fig : cylinder2 ] is referred to plot @xmath261 eq . ( [ eq : ps24 ] ) . the parameters for the configuration are given the following values : @xmath274 for the purpose of plotting @xmath275 defined in eq . ( [ eq : ps24 ] ) , i shall assume , see eq . ( [ eq : qeff_condition ] ) , @xmath276 where @xmath132 is defined in eq . ( [ eq : q - def ] ) . the voltages of @xmath277 @xmath278 @xmath279 @xmath280 and @xmath281 are considered for the top electrode ( the bottom electrode is grounded ) . with @xmath271 thus defined , the electric field generated inside chamber is given by eq . ( [ eq : e ] ) . the gravity of @xmath282 was assumed inside the chamber . the directions for the electric field @xmath283 and the gravitational force @xmath284 are determined from eqs . ( [ eq : g_direction ] ) and ( [ eq : e_direction ] ) . since both @xmath142 and @xmath152 are positive , according to the convention defined in eqs . ( [ eq : g_direction ] ) and ( [ eq : e_direction ] ) , the electric field @xmath283 and the gravitational force @xmath284 are both directed in @xmath285 which is the negative @xmath184 axis . the @xmath275 of eq . ( [ eq : ps24 ] ) is computed numerically utilizing simpson method for the integral.@xcite the simpson method routine was coded in fortran 90 . that being said , the results are summarized in figs . [ fig : probability ] and [ fig : probability_zoom ] , where the three smaller peaks of fig . [ fig : probability ] are magnified and replotted in fig . [ fig : probability_zoom ] . at @xmath277 that is , when there is no electric field inside the chamber other than the static fields from particles , the particles are distributed to occupy the entire volume of the chamber . this is indicated by the peak occurring at the physical height of the chamber , @xmath286 at @xmath287 an electric field of roughly @xmath288 is generated inside the chamber . because particles are positively charged , they are compressed in the direction of electric field , i.e. , the @xmath289 direction . this force , which induced particle volume compression , eventually gets counter balanced by the coulomb repulsion and the compression ceases . for the case where control electrode is held at @xmath287 the compression ceases at roughly @xmath290 ( see fig . [ fig : probability_zoom ] ) and this marks the most probable height of the compressed volume for the case . finally , with @xmath291 applied to the control electrode , the compressed volume state is reached where all particles are cluttered near the floor of the chamber , thereby resulting in very high particle density , as illustrated in fig . [ fig : probability ] . if charged particles are to be useful for any display applications , the charged particle system must be insensitive to gravitational effects , if not negligible . the effect of gravity on the most probable height for the compressed volume has been investigated by reversing the direction of gravity ( but , keeping all other conditions unchanged ) in fig . [ fig : probability_zoom ] . the case where @xmath292 was selected for comparison . the result is shown in fig . [ fig : probability_g_effect ] , where it shows that the most probable height for the compressed volume is only negligibly affected by the gravity . the influence of cylinder radius @xmath243 on the most probable height for the compressed volume has also been investigated . again , the case of @xmath292 was selected from fig . [ fig : probability_zoom ] for comparison by considering @xmath293 @xmath294 and @xmath295 all other conditions were kept unmodified . the result , fig . [ fig : probability_eta_effect ] , reveals a decrease in height for the most probable compressed volume with increasing @xmath243 as expected . ( color online ) most probable height of the compressed volume containing charged particles . ] ( color online ) the three small peaks in fig . [ fig : probability ] are magnified for detail . each peak represents the most probable height for the compressed volume , where there are @xmath2 charged nanoparticles inside the volume . ] ( color online ) gravity has negligible effect on most probable height of the compressed volume . ] ( color online ) the height of the compressed volume decreases with the increased radius of the cylindrical chamber . ] the intensity of light transmitted through a medium filled with charged particles goes like , eq . ( [ eq : transmitted_intensity ] ) , @xmath296 where both @xmath15 and @xmath12 have unit of meter , and @xmath12 is the height of the compressed volume containing charged particles . referring to fig . [ fig : probability_zoom ] , the compressed height @xmath12 corresponds to the @xmath184 axis where the probability curve is maximum . in principle , the compression height @xmath12 can be varied continuously by controlling the voltages applied to the control electrode . this implies the intensity of light output from the proposed optical shutter can be varied continuously , thereby generating continuous grayscale levels for the device . in reality , the number of grayscale levels that can be achieved in the presented optical shutter is given by eq . ( [ eq : ngray ] ) , @xmath297 where the fineness of @xmath86 is limited by the system design . this work is not the first kind to address potential applications with charged particles . szirmai@xcite experimented with alumina powders to study electrosuspension as early as 1990 s . in szirmais@xcite experiment , alumina powder of @xmath298 in diameter was placed inside an electrically insulating cylindrical vessel , which is similar in configuration with fig . [ fig : cylinder2 ] . the initial charging of alumina powder was done by a process of field emission , which can be achieved by applying high voltage to the control electrodes(field emission is the phenomenon in which electrons get emitted from the surface of host material , such as nanoparticles , due to the presence of high electric fields ) . szirmai,@xcite however , does not quantitatively address the compression states of volume containing charged particles in terms of the design parameters , as his motive was not in discussing possible applications of charged particles for displays . in his experiment , the control electrodes , in principle , could be supplied with whatever high voltages required by it to do the job ; therefore , the quantitative understanding of how design parameters , such as @xmath299 @xmath300 @xmath301 @xmath302 @xmath303 @xmath304 @xmath43 @xmath265 @xmath305 and @xmath306 enter into the picture of particle volume compression never was an issue . the competition has always been fierce and it will always remain so among different display manufacturers . in the near future , when paperlike displays become dominant , the most important deciding factor to who stays in and goes out of business would be determined by the power efficiency of their products . that being said , a low operation voltage for the control electrodes is crucial for all e - paper technologies and the proposed device based on charged particles is no exception . the light intensity out of each sub - pixel based on proposed charged particle display technology varies as illustrated in eq . ( [ eq : i_hc ] ) , where @xmath12 is the most probable compression height corresponding to the voltage difference of @xmath271 applied to the control electrodes , see fig . [ fig : probability ] . with the @xmath271 restricted to certain range , say @xmath307 one can not arbitrarily choose the other parameters which constitute the design parameters , i.e. , @xmath299 @xmath300 @xmath301 @xmath302 @xmath303 @xmath304 @xmath43 @xmath265 and @xmath308 for example , if too many electrons are removed from each of the aluminum particles , i.e. , @xmath265 the voltage of @xmath309 applied to one of the control electrodes ( the other grounded ) may not be sufficient enough to overcome the coulomb repulsion between particles and compress the particle volume to a level where dark state is reached , assuming @xmath310 defines the brightest state . on the other end , if too many charged particles are present in a chamber , i.e. , the particle number @xmath43 the brightest state achieved by setting @xmath310 for the control electrode may be too dark . the quantitative description of the height @xmath12 of the compressed particle volume in terms of the so called `` design parameters '' thru the expression @xmath261 eq . ( [ eq : ps24 ] ) , ables the design of particle based display with potential to generate continuous grayscale . the presented optical shutter based on charged particles portrays bistability at all states , including the gray states . this is possible because the two optically transparent electrodes act as a capacitor , which has the property of sustaining electric fields even when the device is removed of the power supply . to illustrated how the bistability is achieved for all states , including the gray states , the illustration shown in fig . [ fig : capacitor ] is considered . i shall begin with an isolated capacitor , in which the two electrodes of the capacitor are electrically neutral , resulting in zero electric field inside the region between the two electrodes . with the switch closed , the top electrode is quickly accumulated with a net positive charge , @xmath311 and the bottom electrode gets accumulated with a net negative charge , @xmath312 the potential difference between the two electrodes results in the creation of electric field inside the capacitor , as illustrated in stage 2 of fig . [ fig : capacitor ] . assuming the charged particles reside in the region between the two electrodes , the electric field generated inside the capacitor is responsible for the compression of volume containing charged particles . now , when the switch is opened , the net charge of @xmath313 remains in the top electrode and the net charge of @xmath314 remains in the bottom electrode , provided the capacitor is ideal , i.e. , free from the leakage of electrical current . therefore , for an ideal capacitor , the electric field is maintained forever inside the region between the two electrodes , thereby sustaining the gray states even when the device is removed of the power supply . ( color online ) the bistability is maintained by the electric field stored in the capacitor . ] in the real system , the role of switch is played by a semiconductor transistor , which is very far from being an ideal switch and it has finite leakage of current . this deficiency in semiconductor transistor makes the proposed device only semi - bistable , meaning the device must be refreshed regularly . this , however , is about to change with the current developments in micro electromechanical systems ( mems ) based switches , which literally has zero leakage current for the open state.@xcite initially , the mems based switch has been developed in an attempt to replace the dynamic random access memory ( dram ) architecture for the memory sector of business . however , as it lacks in switching speed , it will be a while before mems based switches can permanently replace the drams . as for its use as a switch in display technology , the mems based switches already show plenty of speed . combined with mems based switches , which has zero leakage current for the open switch mode , the proposed optical shutter based on charged particles opens up the possibility of realizing the bistability mode for all states , including the grayscale states . the pioneering work by szirmai,@xcite hattori et al.,@xcite and others have exposed the potential applications with charged particles . utilizing charged particles in display technologies , however , requires a quantitative understanding of how design parameters , such as @xmath299 @xmath300 @xmath301 @xmath302 @xmath303 @xmath304 @xmath43 @xmath265 @xmath305 and @xmath306 enter into particle volume compression . in this work , an expression for the compressed state , which incorporates the design parameters , has been presented . the result should find its role in the development of displays based on charged particles . the author acknowledges the support for this work provided by samsung electronics , ltd . w. chim , `` a flexible electronic paper with integrated display driver using single grain tft technology , '' msc . thesis , computer engineering , dept . of electrical engineering , delft university of technology ( 2009 ) .
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an optical shutter based on charged particles is presented .
the output light intensity of the proposed device has an intrinsic dependence on the interparticle spacing between charged particles , which can be controlled by varying voltages applied to the control electrodes .
the interparticle spacing between charged particles can be varied continuously and this opens up the possibility of particle based displays with continuous grayscale .
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a comprehensive understanding of superdeformed ( sd ) bands requires knowledge of the quantum numbers ( spin and parity ) and excitation energies of the levels in the second well . in particular , these quantities allow for stringent tests of configuration assignments and , more importantly , of the ability of theory to calculate shell - correction energies at large deformation . however , although more than 250 sd bands are known in the a=150 and 190 regions@xcite , only a few sd bands in @xmath2hg@xcite , @xmath2pb@xcite , @xmath3pb@xcite and in @xmath4dy@xcite , have the spins and excitation energies determined through one - step linking transitions . the yrast sd band in @xmath0hg was the first one to be discovered in the @xmath5 region @xcite . it is especially interesting to obtain the spins and excitation energies for an odd - a sd band which , combined with data on neighboring even - even nuclei , can give information on the relative pair correlation energies in normal - deformed ( nd ) and superdeformed ( sd ) states . so far , the main information on single - particle configurations has come from detailed analyses of the @xmath6 dynamic moments of inertia of the sd bands . with a knowledge of the level energies and the associated quantum numbers , calculations can be put to more extensive tests and information can be obtained on properties such as particle alignment . the work of vigezzi _ _ et al.__@xcite and recent improvements by refs . @xcite laid the theoretical foundation for treating the coupling of an isolated , cold sd state with a high - density of hot compound nd states , which forms the basis of the decay mechanism . the decay of sd bands happens suddenly , typically out of one to two sd states in the mass 190 region . one possible mechanism responsible for this sudden decay out of the sd band , proposed in ref . @xcite , is chaos - assisted tunneling . when the sd band decays , most of the strength is fragmented over numerous pathways , thus forming a quasi - continuum spectrum @xcite with sharp lines at high energy , which are due to direct decay to low - lying discrete nd levels . the decay spectrum from sd states is similar@xcite to the spectrum following the decay of neutron - capture states@xcite . in both cases , the decay originates from a discrete point in excitation energy and spin and proceeds to a multitude of final states . one way of determining the spins and excitation energies of sd bands is to analyze the quasi - continuum decay spectrum connecting sd and nd states . the technique to extract the quasi - continuum decay spectrum was pioneered on @xmath3hg in the work of henry _ _ et al.__@xcite . an improved method is described in detail by lauritsen _ _ et al.__@xcite and is the one used in this paper . the method has been successfully tested in the case of sd band 1 in @xmath2hg@xcite , where it determined the same spins and excitation energies as those known from several one - step @xmath1-ray transitions connecting the sd band to known nd states@xcite . the results from the quasi - continuum analysis are an important complement in cases where only one or two decay pathways are known . however , in most instances , one - step transitions are not observed , and it is then the only available option . in this work the quasi - continuum spectrum following the decay of the yrast sd band in @xmath0hg has been extracted . from this spectrum , we determine the excitation energy and spin of the sd band and also derive information on pairing in normal - deformed states . we also present two 1-step decay pathways , which directly connect the yrast sd band in @xmath0hg with known yrast levels in the nd level scheme . it will be shown that the results from the two methods agree very well and , thus , we can make a confident assignment of the spin and excitation energy of the yrast sd band in @xmath0hg . the experimental results are compared with theoretical calculations based on the hartree - fock - bogoliubov ( hfb ) theory with several skyrme interactions . we shall also extract and discuss information on pairing in sd states , by comparing the present results with those from the even - even hg nuclei . superdeformed states in @xmath0hg were populated using the @xmath7yb(@xmath8ne,5n)@xmath0hg reaction . the experiment was performed with the gammasphere array@xcite , which had 96 compton - suppressed ge detectors at the time of the experiment . the 120 mev @xmath8ne beam was provided by the 88 cyclotron at lawrence berkeley national laboratory . the 3.1 mg/@xmath9 @xmath7yb target had a 6.8 mg/@xmath9 @xmath10au backing to stop the recoiling nuclei . the decay - out @xmath1 rays were emitted after the recoils came to rest in the backing , so that the transitions will correspond to sharp lines rather than doppler broadened ones . a total of @xmath11 x @xmath12 triple- or higher - fold coincidence events were collected . the @xmath1-ray spectrum , obtained with pairwise coincidence gates on sd transitions , is shown in fig . [ fig : link ] . the lower panel presents the high - energy part of the spectrum , in particular , the two transitions at 2778 and 3310 kev , which will be shown to connect sd and known nd states@xcite . the stronger 2778-kev transition has an area of 6@xmath13 , while the 3310-kev transition has an area of 3@xmath13 , where @xmath13 is the statistical uncertainty . the intensities of the 2778 kev and 3310 kev lines are 0.8 % and 0.4 % , respectively , of the maximum sd intra - band intensity . figure [ fig : link - sd ] gives the coincidence spectra gated on a sd line and either the 2778-kev ( upper figure ) or 3310-kev ( lower figure ) transition . even though the statistics are low , the intensities of the nd lines suggest that the 2778-kev transition feeds the known nd 35/2@xmath14 yrast level at 3222 kev . the 3310-kev transition has been assigned to feed a 33/2@xmath14 known nd level at 2690 kev . the deduced decay scheme is shown in fig . [ fig : level ] . on the basis of the coincidence data , both one - step transitions place the deexciting sd level , i.e. the one fed by the 351 kev sd transition , at 6000 kev - see fig . [ fig : level ] . the angular distribution coefficient of the stronger one - step line ( 2778 kev ) , a@xmath15 0.57@xmath16 0.48 , is consistent with a @xmath17i = 0 dipole assignment , suggesting a 35/2 @xmath18 spin assignment for the level fed by the 351 kev sd transition . we rule out the possibility of it being a stretched e2 transition , because that would require m3 multipolarity for the 3310-kev one - step transition . the spin is consistent with a favored @xmath19 , @xmath20 configuration assignment , which is calculated to be yrast at high spin @xcite . the experimental data do not allow for a parity assignment . however , the @xmath20 configuration assignment requires the sd band to have negative parity , implying m1 multipolarity for the one - step transitions . this m1 assignment is discussed later . the partial level scheme in fig . [ fig : level ] shows the levels fed in the decay of the sd band . the energy of the 13/2@xmath21 state is set at zero to facilitate the comparison of the experimental and theoretical sd excitation energies and to circumvent the 22-kev uncertainity in its excitation energy @xcite . hence , the 3/2@xmath22 ground state , which is not populated by the sd band , has an energy of -128(22 ) kev in fig . [ fig : level ] . a method has been developed at argonne@xcite to isolate the quasi - continuum @xmath1-ray spectrum connecting the sd and nd states . to confirm the results from the one - step linking transitions , this method , which is described in detail in ref.@xcite , was followed here . first , the data were sorted with double coincidence gates on sd transitions to obtain clean spectra . only double gates which produce clean sd spectra , without significant contamination by nd transitions , were used . the background subtraction was done using the ful method@xcite . corrections were carried out for @xmath1-ray summing@xcite and for neutron interactions in the detectors@xcite . the spectra were unfolded @xcite to eliminate contributions from compton - scattered @xmath1 rays and corrected for the detector efficiency . the area of the spectrum was then normalized to multiplicity by requiring that the sum of the intensities of transitions feeding the ground state is 1 . the 390 kev line is a doublet composed of a sd transition and a transition feeding the nd ground state . the 390-kev sd component is taken to have multiplicity 1 , suggested by the plateau in the intensity of the sd transitions@xcite . the total normalized @xmath1 spectrum is shown on a logarithmic scale in fig . [ fig : sdnd ] , together with the equivalent spectrum , obtained by gating on two nd yrast lines . above 1 mev , the spectra are contracted to 32 kev / channel and below that to 1.33 kev / channel . there is clearly extra yield in the sd gated spectrum between 1 and 2.5 mev , which comes from the decay out of the sd band@xcite . the discrete peaks below 800 kev are subtracted from the continuum spectrum . they can be identified as transitions either along the yrast sd band or near the nd yrast line ( including previously unassigned transitions ) . the remaining continuum spectrum contains contributions from components ( of statistical , quadrupole and m1/e2 nature ) that feed the sd band , in addition to the sought - after decay - out spectrum . to extract the decay - out spectrum , the feeding components , starting with the one of statistical nature have to be subtracted . the feeding component of statistical nature can not be disentangled experimentally from the decay - out continuum . instead , it is obtained from a monte carlo simulation of the feeding of the sd band in @xmath0hg . the monte carlo code is described in refs . one of the inputs in the calculation of the statistical spectrum was the shape of the entry distribution . this was not measured for @xmath0hg in the experiment ; so the shape of the entry distribution for @xmath3hg from ref . @xcite was used . the distribution was shifted to have the right average entry spin and excitation energy ( 33.9(1.7 ) @xmath18 and 13.8(0.7 ) mev ) , as found from the analysis of quasi - continuum @xmath1 rays feeding all states in @xmath0hg . the average entry point for cascades feeding only the sd band was 44.1(2.2 ) @xmath18 and 17.3(0.9 ) mev , as given in table [ tab : comp ] . the feeding components of the spectrum are doppler shifted , since the speed of the recoiling compound nucleus is v / c = 0.019 ( for @xmath0hg nuclei formed halfway through the target ) . to take this into account , the spectra were transformed into the center - of - mass system . after the statistical feeding component was removed , the quadrupole and dipole feeding components could be separated based on the a@xmath23 coefficient of the angular distribution in the center - of - mass system - see fig . [ fig : a2 ] . the low - energy component ( @xmath24 kev ) is characterized by large negative @xmath25 coefficients , indicating m1/e2 nature ( as seen also in @xmath26hg@xcite ) . after extraction and subtraction of the quadrupole e2 component , the dipole m1/e2 feeding component and decay - out component remain . a sharp drop around 850 kev in the m1/e2 spectrum ( fig . [ fig : qc ] ) and the drop in the a@xmath23 coefficients in the same energy region indicates the presence of two components . the upper component is assigned to the decay - out of the sd band , following refs . @xcite . the different components of the continuum spectrum are presented in the upper part of fig . [ fig : qc ] . the energy and spin removed , on average , by the different @xmath1-ray components are listed in table [ tab : comp ] . for comparison , the different components of the spectrum feeding all states ( mostly of nd nature ) in @xmath0hg , from a similar analysis with two gates on yrast nd transitions ( 390.5 kev and 750.2 kev ) , is shown in the lower part of fig . [ fig : qc ] . there are two notable differences in the two spectra in fig . 6 . first , the quadrupole component feeding sd states has lower energy ( 0.70 _ vs. _ 0.77 mev ) , is narrower and has larger multiplicity ( 4.0 _ vs. _ 3.1 ) than that feeding nd states . the differences in this component , which arise from excited @xmath1 cascades , are attributed to the larger collectivity in the sd well . second , the top spectrum has an additional component , starting at 0.8 and extending to 3.3 mev , which arises from the decay - out quasicontinuum @xmath1 rays connecting sd and nd states . the decay - out spectrum connecting the sd and yrast nd states ( including statistical and discrete non - yrast transitions ) is given in fig . [ fig : doyr ] . from monte carlo simulations@xcite of the statistical decay , it is found that each quasi - continuum @xmath1 ray removes 0.5(1)@xmath18 of spin . the energy and spin removed , on the average , by the decay - out components are found by : @xmath27 and @xmath28 where @xmath29 and @xmath30 are the average energy and spin removed per @xmath1 ray ( for a given component @xmath31 ) and @xmath32 is the average multiplicity of this component . the total @xmath1-ray spectrum connecting sd and yrast nd states removes @xmath33 mev and @xmath34 . from the intensities of the nd yrast transitions in our sd gated spectrum , the average entry point into the nd yrast region is found to be 2.24(15 ) mev and 14.7(4 ) @xmath18 . the yrast transitions are taken from the nd level scheme of ref.@xcite . the energy and spin of the level fed by the 351 kev sd transition is , therefore , determined to be e@xmath35=5.7(5 ) mev and i@xmath35=17.8(8 ) @xmath18 . contributions to the uncertainty come from the calculated feeding statistical spectrum , the normalization to multiplicity , uncertainty in the spin removed by the quasi - continuum decay - out component and the multipolarities of the unknown lines . the errors are added in quadrature . in fig . [ fig : test ] , the experimental results from the 1-step linking transitions and from the quasi - continuum analysis are presented in a spin - energy diagram . the filled circles represent the yrast nd and sd levels , as given by the level scheme ( fig . [ fig : level ] ) , based on the one - step decay paths . the filled diamond denotes the sd level , which is fed by the 351 kev transition , and the open diamond the average entry point into the nd yrast band , as obtained from the quasi - continuum analysis . the box around the filled diamond shows the uncertainty in spin and energy from the quasi - continuum analysis . clearly , the results from the 1-step linking transitions and from the quasi - continuum analysis are in good agreement with each other . this gives confidence about the spin and energy assignments . with the spins now known , the kinematic moment of inertia @xmath36 can also be determined . figure [ j1j2 ] shows both @xmath36 and @xmath6 moments as a function of @xmath37 . the dynamical moment of inertia @xmath6 can be expressed in terms of the harris expansion@xcite : @xmath38 integration of eq . ( [ khoo1 ] ) gives @xmath39 and @xmath40 here @xmath37 is the rotational frequency , given by @xmath41 ; @xmath42 is the spin perpendicular to the symmetry axis and @xmath31 , the integration constant , represents the quasi - particle alignment . for @xmath0hg , fits of @xmath6 and @xmath42 _ vs. _ @xmath37 with eqs . ( [ khoo1 ] ) and ( [ khoo2 ] ) give @xmath43 . the behavior of @xmath36 of @xmath0hg is different from that of other a@xmath44190 sd bands , where the moments of inertia increase monotonically with @xmath37 . the u - shaped curve of @xmath36 arises from the @xmath31/@xmath37 term ( which causes the unusual rise at low @xmath37 ) and provides a characteristic signature for a band exhibiting finite alignment . knowledge of the spins allows configuration assignments to be made on a solid foundation . in the past , the assignments were largely based on the variation with rotational frequency @xmath37 of the dynamical moment of inertia @xmath6 . in only a handful of cases , spins were extracted using a fit method [ using eqs . ( [ khoo1 ] and [ khoo2 ] ) ] , with the assumption that @xmath31 = 0 ] . for sd band 1 in @xmath0hg , which exhibits particle alignment , this method can not be used and spins were proposed by carpenter @xmath45 @xcite , based on the entry spin into the nd yrast line after decay from the sd band . the present work firmly establishes the spins and confirms the assignment of ref . @xcite , thus validating the interpretation that sd band 1 in @xmath0hg is based on the n=7 @xmath20 [ 761]3/2 configuration . the alignment , @xmath46 , is an important ingredient in this assignment . together , this work and ref . @xcite add confidence about the single - particle orbitals calculated by theory . the woods - saxon potential gives this orbital as the yrast configuration at large deformation and at high spin . the neutron quasi - particle routhians for @xmath3hg @xcite suggest that this feature would also be given by hfb theory . however , details are not always correctly predicted , e.g. , at the lowest frequencies , there is a discrepancy of 2 @xmath18 in alignment between experiment and hfb theory for sd band 1 in @xmath0hg @xcite . altogether , mean - field theories provide good descriptions of the general features of sd bands in the mass 190 region , such as the rise of @xmath6 with @xmath37 ( due partly to the n=7 orbital ) and the convergence at high @xmath37 for most nuclei . this has been summarized in work by fallon _ et al . _ @xcite , which distills the main physics results . the assigned spin of the band is consistent with a favored @xmath19 , @xmath20 particle configuration assignment , which is calculated @xcite to be yrast at high spin . the experimental data do not allow for a parity assignment . however the @xmath20 assignment requires the sd band to have negative parity , implying m1 multipolarity for both of the one - step transitions . from neutron - capture data , it is known that , at e@xmath47 @xmath44 8 mev , the decay is dominated by e1 transitions@xcite . however , in @xmath0hg the one - step transitions have significantly lower energy , e@xmath47 @xmath44 3 mev . in fact , m1 transitions with similar energy have been observed to compete with e1 transitions in the decay out of the sd band in @xmath2pb@xcite . in addition , in neutron capture experiments on @xmath48dy targets , the m1 strength was found to be comparable to the e1 strength at @xmath49 3 mev @xcite . the scissor mode @xcite probably accounts for the enhanced m1 strength . the @xmath1 spectrum of fig . [ fig : doyr ] shows a region with depleted yield between 2.3 and 3.3 mev . following dssing _ et al . _ @xcite , this depletion can be explained by the reduction in level density in the interval from the nd yrast line up to the energy required to break the first pair of neutrons or protons . in ref . @xcite it is seen that the width of the depleted region in the @xmath1 spectrum is around 1.5 times the average pairing gap . the depleted region ( which is most clearly defined by the decay - out transitions with @xmath50 ) occurs between 2.3 and 3.3 mev , implying a pair gap of about 0.7 mev . for the non - rotating nucleus , @xmath51 ( or @xmath52 ) is approximately given by the five - point mass formula @xmath53 for a sequence of isotones ( or isotopes) see , for example , eq . ( 7 ) in ref . @xcite . around @xmath0hg , @xmath54 mev ( if one neglects mean - field contributions to @xmath53 , which are discussed in ref . although the information from the tail of the decay - out gamma - spectrum is quite uncertain , it yields a pairing gap similar to that given by @xmath53 . in table [ tab : theory ] , the experimental sd excitation energies are presented given for @xmath55hg . the excitation energies of the sd levels of @xmath3hg are given by two tentative decay - out pathways @xcite from the 10@xmath21 level , combined with limits imposed by the quasi - continuum analysis @xcite , giving @xmath56 mev . the sd bands are extrapolated to spin 2.9 and 0 @xmath18 , where the rotational frequencies are zero . ( for @xmath0hg , eq . ( [ khoo2 ] ) gives @xmath57 at @xmath37 = 0 , and @xmath58 . ) table [ tab : theory ] also presents the sd excitation energies from theoretical calculations based on the self - consistent hartree - fock - bogoliubov ( hfb ) approach with the effective skyrme interactions , skp , sly4 and skm@xmath59 , and the density - dependent zero - range interactions of ref . @xcite for the particle - particle ( pairing ) channel . the theoretical results were extrapolated in the same way as the experimental values in the case of @xmath0hg , while the theoretical value for the sd state in @xmath3hg was calculated directly for the ground state spin of 0 @xmath18@xcite . the calculations with the sly4 interaction show the best overall agreement with the experimental data . the excitation energies of sd states in odd - a and even - even nuclei give information on pairing in the sd well . in even - even nuclei pairing is stronger so the ground state has lower energy than that of the neighboring odd - even nuclei . this can be seen by comparing the nd and sd yrast bands of @xmath0hg and @xmath3hg after accounting for the difference in the mass excess of the two nuclei - see fig . [ fig : mass ] . the ground state of @xmath3hg is taken as a reference , i.e. set to zero . one sees that the 0@xmath21 state in @xmath3hg has a more negative mass excess than the 13/2@xmath21 state in @xmath0hg , implying an extra binding of the even - even nucleus by @xmath601.5 mev . the observed sd states are also shown in fig . [ fig : mass ] . here the energy of the sd `` ground '' state in @xmath3hg is lower than that in @xmath0hg by 0.8 mev . this smaller value ( compared to 1.5 mev for nd states ) , is consistent with reduced pairing in the sd well , as suggested before , e.g. ref . @xcite , from the increase of the @xmath6 moment of inertia with frequency . however , in addition to pairing , mean - field effects ( e.g. a change in the fermi energy and a polarization energy ) contribute to the binding energy @xcite . the convergence of the yrast lines of @xmath55hg around spin 10 and 25 @xmath18 , for nd and sd states , respectively , may be attributed to a reduction of pairing due to rotation . an alternative but equivalent way to present the differences in binding energies in fig . [ fig : mass ] is in terms of the neutron separation energies @xmath61 in the nd and sd wells . [ note that @xmath61 = mass excess ( @xmath0hg ) - mass excess(@xmath3hg ) + mass excess(neutron ) ] . table [ tab : theory ] compares the experimental and theoretical neutron separation energies s@xmath62 in the sd and nd wells . the s@xmath62 values and sd excitation energies from hfb calculations with the sly4 force are compared with experimental results in figure [ fig:1nsep ] . the experimental neutron separation energy in the sd well is found to be s@xmath63 mev , compared to s@xmath64 mev in the nd well ( to the 13/2@xmath21 state ) . the difference of 0.7 - 1.0 mev means that it is easier to remove a neutron from the sd well than from the nd well . as discussed above , part of the reduction in s@xmath62 in the sd well is due to a decrease in pairing , but other effects contribute as well . in order to gauge the reliability of extracting the pairing gap @xmath65 ( and the difference in the nd and sd wells ) from nuclear masses , we write the equations for 2- , 3- and 5-point mass differences in the form discussed by duguet @xmath45 @xcite . @xmath66 here n is the neutron number , @xmath65 the pairing gap , @xmath67 the polarization energy due to time - reversal symmetry breaking ( from the blocking of a single - particle level ) and @xmath68 the fermi level . these equations show that , in order to deduce @xmath65 from experimental masses , theoretical values for each well are also required for @xmath67 , as well as for @xmath68 or @xmath69 if @xmath70 or @xmath71 are employed . it would be best to use @xmath72 , since that requires a calculation of only @xmath73 . ( the value of @xmath73 is around @xmath16100 kev , but there is some uncertainty in its calculation @xcite . ) however , @xmath72 for sd states would need the sd excitation energies in 5 consecutive nuclides , @xmath74hg , and would require new experimental sd energies in @xmath75hg ( work on which is in progress@xcite ) , as well as in @xmath76hg . for the nd _ ground states _ , which have measured masses , @xmath77 yields @xmath78 excellent agreement of 1.1 mev is obtained using the experimental @xmath79 , together with a theoretical @xmath80 = -8.4 mev ( obtained with the sly4 force ) . this agreement provides validation of @xmath80 for hg nuclides around @xmath3hg , and is consistent with the reproduction of @xmath81 values ( within 0.2 mev ) for the nd ground states of nuclides @xcite in this region with the sly4 force . for the sd well , @xmath79 and @xmath82 ( @xmath83 mev ) give @xmath84 where the errors do not include the uncertainty in @xmath82 . this value of @xmath85 is a direct indicator of pair correlations in the sd well . there appears to a reduction of this value with respect to that in the nd well , but the uncertainties do not allow for a definitive conclusion . the spins and excitation energies of the yrast sd band in @xmath0hg have been determined from two single - step linking transitions and from the quasi - continuum spectrum that connects the sd and nd states . the results from the two methods are in good agreement , within the error bars , providing confidence about the spins and excitation energies of the yrast sd band . the sd level fed by the 351 kev sd transition has e@xmath86 = 6000 kev and i = 35/2 @xmath18 . excitation energies and spins provide a stringent test of orbital assignments . the spin is consistent with that expected for a @xmath20 orbital configuration , previously assigned to this sd band@xcite . this is the first time that the excitation energies and spins have been determined for a sd band in an odd - even nucleus in the mass a = 190 region . by comparing the results with those of neighboring even - even hg nuclei , we have obtained information on pairing in the sd states . the neutron separation energies in the nd and sd wells have been extracted by using data from @xmath55hg . the separation energy in the sd well is 0.7 - 1.0 mev smaller than in the nd well , due partly to a reduction in the pair gap @xmath65 with deformation and partly to an change in @xmath68 . we have compared the results with those from calculations based on hartree - fock - bogoliubov ( hfb ) theory with different skyrme interactions@xcite , and have found that the sly4 interaction , which yields 6.32 mev for the excitation energy of the i = 35/2 @xmath18 sd level , gives the best agreement . similarly , the same interaction gives the best reproduction of the neutron separation energies in the nd and sd wells . discussions with i. ragnarsson and a. afanasjev are gratefully acknowledged . this research is supported in part by the u.s . dept . of energy under contract nos . w-31 - 109-eng-38 and de - ac03 - 76sf00098 . s. siem acknowledges a nato grant through the research council of norway . + t. l. khoo , m.p . carpenter , t. lauritsen , d. ackermann , i. ahmad , d.j . blumenthal , s.m . fischer , r.v.f . janssens , d. nisius , e.f . moore , a. lopez - 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carlo simulations , the quasi - continuum decay - out and statistical feeding components have , respectively , @xmath87i@xmath88 = 0.5 and 0.66 @xmath18 per @xmath1 ray . the unknown lines are defined as non - yrast discrete transitions . [ cols="<,^,^,^,^,^ " , ] state , which is the termination of the sd band decay , has been set at zero since ( a ) it facilitates the comparison of experimental and theoretical sd excitation energies with respect to this state and ( b ) it circumvents the uncertainty in its energy ( 128@xmath1622 kev , given in ref . @xcite ) . ] spectra in @xmath0hg for decays going through the yrast sd band ( upper figure ) and for decays through all ( mostly nd ) states ( lower figure ) . the spectrum for feeding statistical transitions are from monte carlo simulations ; all other spectra are from experimental data . ]
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the excitation energies and spins of the levels in the yrast superdeformed band of @xmath0hg have been determined from two single - step @xmath1 transitions and the quasi - continuum spectrum connecting the superdeformed and normal - deformed states .
the results are compared with those from theoretical mean - field calculations with different interactions .
a discussion of pairing in superdeformed states is also included . 2
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experimental observations and robust theoretical arguments have established that our universe is essentially non - visible , the luminous matter scarcely accounting for one per cent of the critical density of a flat universe ( @xmath0 ) . the current picture describes an universe consisting of unknown species of dark energy ( @xmath1 ) and dark matter ( @xmath2 ) of which less than @xmath3 is of baryonic origin . most of the dark matter would then be made of non - baryonic particles filling the galactic halos , at least partially , according to a variety of models . weak interacting massive ( and neutral ) particles ( wimps ) are favourite candidates to such non - baryonic components . the lightest stable particles of supersymmetric theories , like the neutralino , describe a particular class of wimps . without entering into considerations about how about how large the baryonic dark component of the galactic halo could be , we take for granted that there is enough room for wimps in our halo , to try to detect them , either directly or through their by - products . discovering this form of dark matter is one of the big challenges in cosmology , astrophysics and particle physics . wimps can be looked for either directly or indirectly . the indirect detection of wimps proceeds currently through two main experimental lines : either by looking in cosmic rays experiments for positrons , antiprotons , or other antinuclei produced by the wimps annihilation in the halo ( caprice , bess , ams , glast , veritas , magic ... ) , or by searching in large underground detectors ( superkamiokande , sno , soudan , macro ) or underwater neutrino telescopes ( baikal , amanda , antares , nestor ) for upward - going muons produced by the energetic neutrinos emerging as final products of the wimps annihilation in celestial bodies ( sun , earth ... ) . this talk will deal with the direct detection of wimps . the direct detection of wimps relies in the measurement of the wimp elastic scattering off the target nuclei of a suitable detector . pervading the galactic halos , slow moving ( @xmath4 km / s ) , and heavy ( @xmath5 gev ) wimps could make a nucleus recoil with a small energy of a few kev ( @xmath6 kev ) , at a rate which depends of the type of wimp and interaction . only a fraction of the recoil , qt , is visible in the detector , depending on the type of detector and target and on the mechanism of energy deposition . the so - called quenching factor , q , is essentially unit in thermal detectors whereas for the nuclei used in conventional detectors it ranges from about 0.1 to 0.6 . for instance for a ge nucleus only about 1/4 of the recoil energy goes to ionization . on the other hand , the smallness of the neutralino - matter interaction cross - section makes the rates of the nuclear recoil looked for very small . the variety of models and parameters used to describe the astrophysics , particle physics and nuclear physics aspects of the process make the neutralino - nucleus interaction rate encompass several orders of magnitude , going from 10 to @xmath7 c / kg day , according to the susy model and parameters@xcite . in fact , it is not higher than @xmath8 c / kg day , for the neutralino parameters which provide the most favourable relic density ( @xmath9 ) . due to such small rate and small energy deposition produced by the wimp elastic scattering , the direct search for particle dark matter through their scattering by nuclear targets requires ultralow background detectors of a very low energy threshold . moreover , the ( almost ) exponentially decreasing shape of the predicted nuclear recoil spectrum mimics that of the low energy background registered by the detector . all these features together make the wimp detection a formidable experimental challenge . customarily , one compares the predicted event rate with the observed spectrum . if the former turns out to be larger than the measured one , the particle which would produce such event rate can be ruled out as a dark matter candidate . that is expressed as a contour line @xmath10(m ) in the plane of the wimp - nucleon elastic scattering cross section versus the wimp mass . that excludes , for each mass m , those particles with a cross - section above the contour line @xmath10(m ) . the level of background sets , consequently , the sensitivity of the experiment in eliminating candidates or in constraining their masses and cross sections . however , this mere comparison of the expected signal with the experimentally observed spectrum will not be able to detect the wimp , because such spectrum , even extremely small , could still be due to pure background sources . in other words , it would be difficult to be convinced that such signal is due to wimps and not to mere background sources . a convincing proof of the detection of wimps would need to find unique signatures in the data characteristic of them and not of the background . there exist some temporal or spatial asymmetries specific of the wimp interaction , which can not be faked by the background or by instrumental artifacts . they are due to the kinematics of the motion of the earth ( and of our detectors ) in the galactic halo . the only distinctive signature investigated up to now is a predicted annual modulation of the wimp signal rate due to the seasonal variation produced by the earth s motion with respect to the sun . such a seasonal affect has been found by the dama experiment at the @xmath11 level and has been associated to the existence of a wimp . the detectors used so far are : ionization detectors of ge ( igex , cosme , h / m , hdms ) and of si ( ucsb ) , scintillation crystals of nai ( zaragoza , dama , ukdmc , saclay , elegants ) , liquid or liquid - gas xenon detectors ( dama , ucla , ukdmc ) , calcium fluoride scintillators ( milan , osaka , roma ) , thermal detectors ( bolometers ) with saphire absorbers ( cresst , rosebud ) , with telurite absorbers , ( mibeta , cuoricino ) or with germanium absorbers ( rosebud ) as well as bolometers which also measure the ionization , like that of si ( cdms ) and of ge ( cdms , edelweiss ) . new detectors and techniques are entering the stage . worth to be mentioned are : scintillating bolometers of calcium tungstate which measure heat and high ( cresst and rosebud ) , and of bgo ( rosebud ) ; a tpc sensitive to the direction of the nuclear recoil ( drift ) ; devices which use superheated droplets ( simple and picasso ) , or those which use colloids of superconducting superheated grains ( orpheus ) . there exist also projects featuring a large amount of target nuclei in segmented detectors , both with ionization ge detectors ( genius , gedeon ) and cryogenic thermal devices ( cuore ) . table 4 gives an overview of the experiments on direct detection of wimps currently in operation or in preparation . a review of neutralino dark matter can be found in ref @xcite . wimp direct detection are reviewed , for instance in ref @xcite and @xcite . recent results are given in ref @xcite . the smallness and rarity of wimp signals dictate the experimental strategies for their detection : reduce first the background , by controlling the radiopurity of the detector , its components , the shielding and the environment . the best radiopurity has been obtained in conventional ge experiments ( igex , h / m ) . in the case of the nai scintillators ( dama , ukdm , anais ) , the achieved backgrounds are still one order of magnitude worse than those reached in ge , but the use of new radiopure powder in the crystals has lead to substantial improvements ( dama ) . the next step is to use discrimination mechanisms able to distinguish electron recoils ( tracers of the background ) from nuclear recoils ( originated by wimps or neutrons ) . two types of techniques have been applied for such purpose : a statistical pulse shape analysis ( psd ) based on the different timing behaviour of both types of pulses and a background rejection method based on the identification ( on an event - by - event basis ) of the nuclear recoils , by measuring at the same time two different mechanisms of energy deposition , like the ionization ( or scintillation ) and heat , capitalizing the fact that for a given deposited energy ( measured as phonons ) the recoiling nucleus ionizes less than the electrons . examples of psd are ukdmc , saclay , dama and anais , whereas the event by event discrimination has been successfully applied in cdms and edelweiss by measuring ionization and heat and now in cresst and rosebud by measuring light and heat . a promising discriminating technique is that used in the two - phase liquid - gas xenon detector with ionization plus scintillation , of the zeplin series of detectors . an electric field prevents recombination , the charge being drifted to create a second pulse in addition to the primary pulse . the amplitudes of both pulses are different for nuclear recoils and gammas allowing their discrimination . another technique is to discriminate gamma background from neutrons ( and so wimps ) using threshold detectors like neutron dosimeters which are blind to most of the low linear energy transfer ( let ) radiation ( e , @xmath12 , @xmath13 ) . detectors which use superheated droplets which vaporize into bubbles by the wimp ( or other high let particles ) energy deposition are simple and picasso . an ultimate discrimination will be the identification of the different kind of particles by the tracking they left in , say , a tpc , plus the identification of the wimp through the directional sensitivity of the device ( drift ) . obviously one should try to make detectors of very low energy threshold and high efficiency to see most of the signal spectrum , not just the tail . that is the case for the bolometer experiments which seen efficiently the energy delivered by the wimp ( quenching factor is unity ) , ( mibeta , cresst , rosebud , cuoricino , cdms and edelweiss ) . after the process of reducing and identifying the background is driven to its best , then one should search for distinctive signatures of the dark matter particles to prove that you are , indeed , seeing a wimp . identifying labels are derived from the earth motion through the galactic halo , which produces two assymetries distinctive of wimps : first , the earth orbital motion around the sun has a summer - winter variation which results in a small ( 5% ) annual modulation of the wimp interaction rates ; and second , the earth ( and solar system ) motion through the galactic centre produces a large ( @xmath14 ) directional assymetry forward - backward of the recoiling nucleus . the annual modulation signature has been already explored . pioneering searches for wimp annual modulation signals were carried out in canfranc ( nai-32 ) , kamioka ( elegants ) and gran sasso ( dama - xe ) . recently the dama experiment at gran sasso , using a set of nai scintillators reported in 1997 and 2000 an annual modulation effect of at the @xmath11 level interpreted as due to a wimp of about 60 gev of mass and scalar cross - section on protons of @xmath15 picobarns . as far as the forward - backward assymetry of the signal is concerned the drift project will search for it . information and current results of the experiments mentioned above can be found in references @xcite and @xcite . in particular , in the underground facility of the canfranc tunnel ( spain ) , various of the techniques mentioned above are currently employed in a search for wimps@xcite : ge ionization detectors , ( cosme and igex ) , sodium iodine scintillators ( nai-32 and anais ) , thermal detectors with al@xmath16o@xmath17 and ge absorbers ( rosebud - i ) as well as and with calcium tungstate and bgo scintillating bolometers ( rosebud - ii ) . to illustrate how these searches proceed we will present the case for ge detectors , and will describe its status , achievements and results . as one of the main achievements in ge experiments is their low radioactive background , essential ingredient in the search for dark matter , we will illustrate it with a saga of germanium experiments performed in canfranc and the succesive suppresion of the background obtained . such reduction has provided the lowest raw background rate obtained so far and , consequently , the stringest exclussion plots ever derived with ionization detectors . the high radiopurity and low background achieved in germanium detectors , their fair low energy threshold , their reasonable quenching factor ( about 25% ) ( nuclear recoil ionization efficiency relative to that of electrons of the same kinetic energy , or ionization yield ) and other nuclear merits make germanium a good option to search for wimps with detectors and techniques fully mastered . the first detectors applied to wimp direct searches ( as early as in 1987 ) were , in fact , ge diodes , as by - products of @xmath18-decay dedicated experiments . the exclusion plots @xmath10(m ) for spin - independent couplings obtained by former ge experiments [ pnnl / usc / zaragoza ( twin @xcite and cosme-1@xcite@xcite ) , ucsb @xcite , calt / neu / psi @xcite , h / m @xcite ] are still remarkable and have not been surpassed till recently by nai experiments ( dama)@xcite using statistical pulse shape discrimination ( psd ) or by cryogenic experiments measuring both heat and ionization like cdms@xcite and edelweiss@xcite . table 5 shows the germanium ionization detector experiments currently in operation . a ge detector of natural isotopic abundance ( cosme ) of the u. zaragoza / u. s. carolina / pnnl collaboration and another one ( rg - ii ) made of enriched @xmath19ge of the igex collaboration are being used in canfranc to search for wimps interacting coherently with the ge nuclei of the detectors . the cosme detector was fabricated at princeton gamma - tech , inc . in princeton , new jersey , using germanium of natural isotopic abundance . the refinement of newly - mined germanium ore to finished metal for this detector was expedited to minimize production of cosmogenic @xmath20ge . the detector is a p - type coaxial hyperpure natural germanium crystal with a mass of 254 g and an active mass of 234 g which has a long term resolution of 0.43 kev full width at half maximum ( fwhm ) at the 10.37 kev gallium x - ray . in its first canfranc installation ( at 675 m.w.e ) the detector was placed within a shielding of 10 cm of 2000 yr . old ( roman ) lead ( inner layer ) plus 20 cm of low activity lead ( about 70 yr old ) . a 3 mm thick pvc box sealed with silicone closed the lead shielding to purge the radon gas . the pvc box was covered by 1 mm of cadmium and 20 cm of paraffin and borated polyethylene . all the shielding and mounting was supported by 10 cm of vibrational and acoustic insulator sandwiched within two layers of 10 cm of wood mounted on a floor of concrete ( 20 cm ) . it was operated in the former canfranc underground facility at 675 m.w.e . in a small gallery of the canfranc railway tunnel in the spanish pyrenees closed to the traffic . in that set - up , the energy threshold was @xmath21 kev and the background at threshold was about 10 counts kev@xmath22 kg@xmath22 day@xmath22 . after about 500 days of data taking ( referred as cosme-1 experiment ) no signal originated by wimp appeared and , from the background spectrum , an exclusion plot in the cross - section versus wimp masses plane was derived , assuming wimp - matter spin - independent couplings . in the cosme-1 data , the energy range chosen to derive the exclusion plot was from 1.6 to 8 kev where the background was , approximately , 5 counts/(kev kg day ) . in spite of such modest figure , the results after 130 kg day of exposure improved the exclusion plots at low masses ( from 9 to 20 gev ) obtained with other ge experiments because of the low threshold energy of cosme-1 . the cosme detector has been reinstalled , in better background conditions in the new canfranc underground laboratory lsc ( at 2450 m.w.e . ) inside a marinelli beaker in roman lead ( cosme-2 ) in a common shielding ( described later ) together with the three 2.1 kg enriched germanium detectors of igex ( the international germanium experiment on double beta decay ) . in its new installation , cosme-2 has an energy threshold of @xmath23 kev , and an energy resolution of @xmath24 kev at 10 kev . the average background rate , in 311 days of exposure ( mt=72.8 kg day ) is 0.6 c/(kev kg day ) from 2 to 15 kev and 0.3 c/(kev kg day ) from 15 to 30 kev which is significantly better ( more than one order of magnitude ) than in cosme-1 . the cosme-2 spectrum is shown in fig . [ fig_1 ] together with that of cosme-1 , to show the remarkable background reduction obtained in the new set - up and shielding and the disappearance of the cosmogenically induced peaks in the 8 - 10 kev energy region due to the about ten years elapsed between the two experiments . the numerical data of cosme 2 are given in table [ tab - cos-9 ] . .low - energy data from the cosme-2 detector ( mt @xmath25 73 kg - d ) . [ cols="^,^,^,^,^,^ " , ] the exclusion plots resulting from the igex-2000 data are derived from the recorded spectrum fig . 2 in one - kev bins from 4 kev to 50 kev . the method followed in deriving the plot has been the same for all the detectors . as recommended by the particle data group , the predicted signal in an energy bin is required to be less than or equal to the ( 90% c.l . ) upper limit of the ( poisson ) recorded counts . the derivation of the interaction rate signal supposes that the wimps form an isotropic , isothermal , non - rotating halo of density @xmath26 gev/@xmath27 , have a maxwellian velocity distribution with @xmath28 km / s ( with an upper cut corresponding to an escape velocity of 650 km / s ) , and have a relative earth - halo velocity of @xmath29 km / s . the cross sections are normalized to the nucleon , assuming a dominant scalar interaction . the helm parameterization@xcite is used for the scalar nucleon form factor , and the recoil energy dependent ionization yield used is the same that in ref @xcite @xmath30 . the exclusion plots obtained from the cosme-2 spectrum ( fig . [ fig_1 ] and table [ tab - cos-9 ] ) and from cosme-1 ( fig . 1 ) have been calculated , as previously noted , in the same way as for igex . as shown in fig . 3 , the igex-2000 results@xcite ( thick - solid line ) exclude wimp - nucleon cross - sections above 1.3x10@xmath31 nb for masses corresponding to the 50 gev dama region . also shown is the combined germanium contour ( thin - dashed line ) , including the last heidelberg - moscow data@xcite ( recalculated from the original energy spectra with the same set of hypotheses and parameters ) , the dama experiment contour plot derived from pulse shape discriminated spectra@xcite , and the dama region corresponding to their reported annual modulation effect@xcite . the igex-2000 exclusion contour@xcite improves that of other germanium experiments for masses between 20 gev and 200 gev , which includes the mass region corresponding to the neutralino tentatively assigned to the dama modulation effect and results from using only raw data without background subtraction . the cosme-2@xcite@xcite exclusion contour also slightly improves the ge - combined plot for masses between 20 and 40 gev . then , a new set - up was constructed , with the purpose of further improving the background . in particular all the detectors except igex rg - ii were removed from the shielding , leaving rg - ii alone . a new , thorough cleaning of the inner layers of the bricks of lead was performed , and the free space left was stretched . in this new set - up , the detector rg - ii is surrounded by not less than 40 - 45 cm of lead of which 25 cm are archaeological . also the muon veto covers now more completely the ensemble , because three dewars have been removed . a substantied improvement is that of the neutron shielding which has been enlarged to up to 40 cm of thickners , consisting of polyethylene blocks and borated water tanks . = 7.5 cm the results obtained whit this new shielding are from a recent run and correspond to an exposure of mt=80 kg days@xcite . the spectrum obtained in this new set - up , and labelled igex-2001 , is shown in figure 4 compared with the previous igex-2000 spectrum@xcite . the numerical data are given in table 3 . = 7.5 cm the energy threshold of the detector is , as in previous runnings , 4 kev and the fwhm energy resolution at the 75 kev pb x - ray line is 800 ev . the background rate recorded was @xmath32 c / kev / kg / day between 410 kev , @xmath33 c / kev / kg / day between 1020 kev , and @xmath34 c / kev / kg / day between 2540 kev . as it can be seen , the background below 10 kev has been substantially reduced ( about a factor 50% ) with respect to that obtained in the previous set - up of igex-2000 , essentially due to the improved shielding ( both in lead and in polyethylene - water).this suggests that the neutrons could be an important component of the low energy background in igex@xcite . the new exclusion plot derived in this improved conditions igex-2001@xcite is shown in fig . [ dm - ig-5 ] ( thick solid line ) . it improves the igex-2000@xcite exclussion contourn ( thick dashed line ) as well as that of the other previous germanium ionization experiments ( and in particular that of the last result of heidelberg - moscow experiment@xcite now specifically depicted by the thick dotted line ) for a mass range from 20 gev zone to 200 gev , which encompass , as already said , that of the dama mass region . in particular , this new igex result excludes wimp - nucleon cross - sections above 7 @xmath35 pb for masses of 40 - 60 gev and enters the dama region . igex excludes the upper left part of this region . that is the first time that a direct search experiment without background discrimination , but with very low ( raw ) background , enters such region . also shown for comparison are the contour lines of the other experiments , cdms@xcite and edelweiss@xcite ( thin dashed line ) , which have entered that region by using bolometers which also measure ionization . the dama region ( closed line ) corresponding to the @xmath11 annual modulation effect reported by that experiment@xcite and the exclusion plot obtained by dama nai-0@xcite ( thin solid line ) by using statistical pulse shape discrimination are also shown . a remark is in order : for cdms two contour lines have been depicted according to a recent recommendation , the exclusion plot published in ref . @xcite ( thin dotted line ) and the cdms expected sensitivity contour ( thin dot - dashed line)@xcite . data collection is currently in progress and some strategies are being considered to further reduce the low energy background . another 50 % reduction from 4 kev to 10 kev ( which could be reasonably expected ) would allow to explore practically all the dama region in 1 kg y of exposure . in the case of reducing the background down to the flat level of 0.04 c / kg / kev / day ( currently achieved by igex for energies beyond 20 kev ) , the dama region would be widely surpassed . in figure [ dm - ig-6 ] we plot the exclusions obtained with a flat background of 0.1 c / kg / kev / day ( dot - dashed line ) and of 0.04 c / kg / kev / day ( solid line ) down to the current 4 kev threshold , for an exposure of 1 kg year . as can be seen , the complete dama region ( m=@xmath36 gev , @xmath37=(@xmath38)x10@xmath39 nb ) could be tested with a moderate improvement of the igex performances . table5 a new experimental project on wimp detection using larger masses of germanium of natural isotopic abundance ( gedeon , germanium detectors in one cryostat ) is planned@xcite . it will use the technology developed for the igex experiment and it would consist of a set of @xmath401 kg germanium crystals , of a total mass of about 28 kg , placed together in a compact structure inside one only cryostat . this approach could benefit from anticoincidences between crystals and a lower components / detector mass ratio to further reduce the background with respect to igex . the gedeon single cell is a cylindrical cryostat in electroformed copper ( dimensions 20 cm diameter @xmath41 32 cm long ) hosting 28 germanium crystals which share the same common copper cryostat ( 0.5 mm thick ) . the ge crystals , are arranged in four plates suspended from copper rods . the cell is embedded into a precision - machined hole made in a roman lead block providing a shield of 20 cm , and surrounded by another lead shielding 20 cm thick . a cosmic veto and a large neutron shield complete the shielding . the preliminary mc estimated background in the @xmath42 kev region ranges from @xmath43 to @xmath44 c / kev kg day , according to the level of radioimpurities included as input@xcite . the radiopurity assays have been carried out in the canfranc laboratory for the lead and copper components of the shielding . the background final goal of gedeon , below 100 kev , would be hopefully in the region of @xmath45 c / kev kg day and this value has been used to calculate anticipated @xmath10(m ) exclusion plots in the most favourable case . the expected threshold assumed has been @xmath46 kev and the energy resolution in the low energy region has been taken @xmath47 kev . the mc estimated background of the gedeon unit cell ( 28 crystals ) is given in fig . a detailed study is in progress to assess the physics potential of this device . the exclusion plot which could be expected with such proviso for 24 kg y of exposure is shown in the figure [ dm - ig-6 ] . moreover , following the calculations presented in@xcite , gedeon would be massive enough to search for the wimp annual modulation effect@xcite and explore positively an important part of the wimp parameter space including the dama region . a second phase of gedeon with four cryostats ( 112 detectors and a total mass of 92 kg of ge ) is also being considered . the data presented here regarding ge ionization detectors resulted from collaborative research with the members of the cosme and the igex collaborations , formed by c.e . aalseth , f.t . avignone , r.l . brodzinski , s. cebrin , e. garca , w.k . hensley , i.g . irastorza , i.v . kirpichnikov , a.a . klimenko , h.s . miley , j. morales , a. ortiz de solrzano , s.b . osetrov , v.s . pogosov , j. puimedn , j.h.reeves , m.l . sarsa , a.a . smolnikov , a.g . tamanyan , a.a . vasenko , s.i . vasiliev , j.a . villar members to whom i am deeply indebted . the results contained in this paper have already been published in the open literature . i wish to thank especially s. cebrin and i.g . irastorza for their invaluable collaboration in the making of the exclusion plots and to j. morales and i.g . irastorza for many useful discussions on igex - dm . i thank s.cebrian for allowing me the use of the mc results for gedeon and j.puimedon for the making of the radiopurity measurement . the present work was partially supported by the cicyt and mcyt ( spain ) under grant number aen99 - 1033 . a. morales , `` dark matter and its detection '' , summary talk given at the nupecc workshop on present and future of neutrino physics , frascati , nupecc report in highlights and opportunities in nuclear physics , ed . by j. vervier et al . , december 1997 . ( astro - ph/9810341 ) . avignone and a.morales , proc . conference on neutrino physics and astrophysics . helsinki , june 1996 , ed . k. enkvist et al . in world scientific pub . ( 1997 ) p. 413 ; a. morales `` selected projects in direct detection of dark matter '' , proc . neutrino telescopes workshop , venice , february 1999 . m. baldo - ceolin , p. 24 ; l. baudis and h.v . klapdor `` direct detection of non baryonic dark matter '' , astro - ph /0003434 and y. ramachers `` non baryonic dar matter searches '' ; xi rencontres de blois , june 1999 , astro - ph /9911260 . a. morales , `` direct detection of wimp dark matter '' ( astro - ph/9912554 ) . review talk at the taup 99 workshop , college de france , paris . b ( proc . suppl . ) 87 ( 2000 ) 477 and review talk at the taup 2001 workshop , laboratori nazionali del gran sasso , italy , sept . 2001 , to be published in nucl . b ( proc . suppl . ) 2002 . a. morales et al . [ igex collaboration ] , hep - ex/0110061 oct . 2001 and i.g . irastorza et al . ( igex coll . ) , talk given at the taup workshop sept . 2001 , ln gran sasso . to be published in nucl . b ( proc . suppl . ) 2002 .
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an overview of the main strategies followed in the search for non - baryonic particle dark matter in the form of wimps is given . to illustrate these searches the case for germanium ionization detectors
is selected .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the discovery of the cosmic acceleration @xcite and the search for dark energy responsible for its origin @xcite have stimulated the study of different field models driving the cosmological evolution . such a study usually is called the potential reconstruction @xcite , because the most typical examples of these models are those with a scalar field , whose potential should be found to provide a given dynamics of the universe . in the flat friedmann models with a single scalar field , the form of the potential and the time dependence of the scalar field are uniquely determined by the evolution of the hubble variable ( up to a shift of the scalar field ) . during last years the models with two scalar fields have also become very popular . this is connected with the fact that some observations point out that the relation between the pressure and the energy density could be less than -1 @xcite . such equation of state arises if the matter is represented by a scalar field with a negative kinetic term . this field is called `` phantom '' @xcite . moreover , according to some observations @xcite the universe undergoes a transition between normal and phantom phase . such an effect is dubbed `` phantom divide line crossing '' @xcite . in principle , the effect of phantom divide line crossing can be explained in the model with the only scalar field provided a special form of the potential and initial conditions is chosen @xcite or in models with a non - minimally coupled scalar field @xcite . however , the models with two scalar fields , one standard and one phantom , look more `` natural '' for the description of the phantom divide line crossing @xcite . in our preceding paper @xcite we have studied the procedure of reconstruction of the potential in two - field models . it was shown that there exists a huge variety of potentials and time dependences of the fields realizing the same cosmological evolution . some concrete examples were considered , corresponding to the evolution beginning with the standard big bang singularity and ending in the big rip singularity @xcite . one can ask oneself : what is the sense of studying different potentials and scalar field dynamics if they imply the same cosmological evolution ? the point is that the scalar and phantom field can interact with other fields and influence not only the global cosmological evolution but also other observable quantities . one of the possible effects of the presence of normal and phantom fields could be their influence on the dynamics of cosmic magnetic fields . the problem of the origin and of possible amplification of cosmic magnetic fields is widely discussed in the literature @xcite . in particular , the origin of such fields can be attributed to primordial quantum fluctuations @xcite and their further evolution can be influenced by hypothetic interaction with pseudoscalar fields breaking the conformal invariance of the electromagnetic field @xcite . in the present paper we consider the evolution of magnetic fields created as a result of quantum fluctuations , undergoing the inflationary period with unbroken conformal invariance and beginning the interaction with pseudoscalar or pseudophantom fields after exiting the inflation and entering the big bang expansion stage , which is a part of the bang - to - rip scenario described in the preceding paper @xcite . we shall use different field realizations of this scenario and shall see how the dynamics of the field with negative parity influences the dynamics of cosmic magnetic fields . to our knowledge the possible influence of the two - field dynamics , ( when one of two ( pseudo)scalar fields is a phantom one ) on the magnetic fields was not yet discussed in the literature . speaking of cosmic magentic fields we mean the large - scale galactic , intergalactic or super - cluster magnetic fields of order from @xmath0 to @xmath1 with correlation from 100 kpc to several mpc to the extent that they are originated from scalar and , possibly gauge field fluctuations after exiting the inflation . their seeds may well have @xmath2 or less ( see @xcite ) . the structure of the paper is as follows : in sec . 2 we recall the bang - to - rip scenario and describe some examples of different dynamics of scalar and phantom fields ; in sec . 3 we introduce the interaction of the fields ( phantom or normal ) with an electromagnetic field and write down the corresponding equations of motion ; in sec . 4 we describe the numerical simulations of the evolution of magnetic fields and present the results of these simulations ; sec . 5 is devoted to concluding remarks . we shall consider a spatially flat friedmann universe with the metric @xmath3 here the spatial distance element @xmath4 refers to the so called comoving coordinates . the physical distance is obtained by multiplying @xmath4 by the cosmological radius @xmath5 . we would like to consider the cosmological evolution characterized by the following time dependence of the hubble variable @xmath6 , where `` dot '' denotes the differentiation with respect to the cosmic time @xmath7 : @xmath8 this scenario could be called `` bang - to - rip '' @xcite because it is obvious that at small values of @xmath7 the universe expands according to power law : @xmath9 while at @xmath10 the hubble variable explodes and one encounters the typical big rip type singularity . ( the factor one third in ( [ btr ] ) was chosen for calculation simplicity ) . in our preceding paper @xcite ) we considered a class of two - field cosmological models , where one field was a standard scalar field @xmath11 , while the other was a phantom one @xmath12 . the energy density of the system of these two interacting fields is @xmath13 analyzing the friedmann equation and @xmath14 . in this system the planck mass @xmath15 , the planck length @xmath16 and the planck time @xmath17 are equal to @xmath18 . then when we need to make the transition to the `` normal '' , say , cgs units , we should simply express the planck units in terms of the cgs units . in all that follows we tacitly assume that all our units are normalized by the proper planck units . thus , the scalar field entering as an argument into the dimensionless expressions should be divided by the factor @xmath19 . ] @xmath20 we have shown that in contrast to models with one scalar field , there is huge variety of potentials @xmath21 realizing a given evolution . moreover , besides the freedom in the choice of the potential , one can choose different dynamics of the fields @xmath22 and @xmath23 realizing the given evolution . we have studied in @xcite some particular exactly solvable examples of forms of potentials and time dependences of fields @xmath11 and @xmath12 providing the evolution ( [ btr ] ) . here we shall present and apply some of them . consider the potential . ] @xmath24 and the fields @xmath25 @xmath26 ( here the expressions for the fields @xmath22 and @xmath23 should be multiplied by @xmath27 . for the relation between planck units and cgs ones see e.g. @xcite if we would like to substitute one of these two fields by the pseudoscalar field , conserving the correct parity of the potential , we can choose only the field @xmath11 because the potential @xmath28 is even with respect to @xmath11 , but not with respect to @xmath12 . in what follows we shall call the model with the potential ( [ v - i ] ) , the pseudoscalar field ( [ phi1 ] ) and the scalar phantom ( [ xi1 ] ) `` model @xmath29 '' . consider another potential @xmath30 with the fields @xmath31 @xmath32 this potential is even with respect to the field @xmath12 . hence our model @xmath33 is based on the potential ( [ v - ii ] ) , the pseudophantom field ( [ xi - new3 ] ) and the scalar field ( [ phi - new3 ] ) . they will be the fields with the negative parity which couple to the magnetic field . the action of an electromagnetic field interacting with a pseudoscalar or pseudophantom field @xmath11 is @xmath34 where @xmath35 is an interaction constant and the dual electromagnetic tensor @xmath36 is defined as @xmath37 where @xmath38 with the standard levi - civita symbol @xmath39},\ \epsilon_{0123 } = + 1 . \label{lc}\ ] ] variating the action ( [ action ] ) with respect to the field @xmath40 we obtain the field equations @xmath41 @xmath42 the klein - gordon equation for the pseudoscalar field is @xmath43 the klein - gordon equation for the pseudophantom field ( which is the one that couples with the magnetic field in the model @xmath33 ) differs from equation ( [ kg ] ) by change of sign in front of the kinetic term . in what follows we shall neglect the influence of magnetic fields on the cosmological evolution , i.e. we will discard the electromagnetic coupling in equation ( [ kg ] ) . if one wants to rewrite these formulae in terms of the three - dimensional quantities ( i.e. the electric and magnetic fields ) one can find the expression of the electromagnetic tensor in a generic curved background , starting from a locally flat reference frame in which it is well known the relation between electromagnetic fields and @xmath44 and using a coordinate transformation . it is easy to see that we have , for the metric ( [ friedmann ] ) : @xmath45 the field equations ( [ maxwell ] ) , ( [ maxwell1 ] ) and ( [ kg ] ) rewritten in terms of @xmath46 and @xmath47 become @xmath48 @xmath49 , \label{maxwell3}\ ] ] @xmath50 @xmath51 for a spatially homogeneous pseudoscalar field equations ( [ maxwell2 ] ) and ( [ maxwell3 ] ) look like @xmath52 @xmath53taking the curl of ( [ maxwell7 ] ) and substituting into it the value of @xmath46 from ( [ maxwell4 ] ) we obtain @xmath54 where @xmath55 stands for the three - dimensional euclidean laplacian operator . let us introduce @xmath56 and its fourier transform @xmath57 here the field @xmath47 is an observable magnetic field entering into the expression for the lorentz force . the field equation for @xmath58 is @xmath59 f(\vec{k},t)=0 , \label{max - four}\ ] ] where `` dot '' means the time derivative . this last equation can be further simplified : assuming @xmath60 and defining the functions @xmath61 one arrives to @xmath62f_{\pm}=0 , \label{max - four1}\ ] ] where we have omitted the arguments @xmath63 and @xmath7 . assuming that the electromagnetic field has a quantum origin ( as all the fields in the cosmology of the early universe @xcite ) the modes of this field are represented by harmonic oscillators . considering their vacuum fluctuations responsible for their birth we can neglect the small breakdown of the conformal symmetry and treat them as free . in conformal coordinates @xmath64 such that the friedmann metric has the form @xmath65 the electromagnetic potential @xmath66 with the gauge choice @xmath67 , @xmath68 satisfies the standard harmonic oscillator equation of motion @xmath69 hence the initial amplitude of the field @xmath66 behaves as @xmath70 , while the initial amplitude of the functions @xmath44 is @xmath71 . the evolution of the field @xmath44 during the inflationary period was described in @xcite , where it was shown that the growing solution at the end of inflation is amplified by some factor depending on the intensity of the interaction between the pseudoscalar field and magnetic field . here we are interested in the evolution of the magnetic field interacting with the pseudoscalar field after inflation , where our hypothetic bang - to - rip scenario takes place . more precisely , we would like to see how different types of scalar - pseudoscalar potentials and field dynamics providing the same cosmological evolution could be distinguished by their influence on the evolution of the magnetic field . we do not take into account the breaking of the conformal invariance during the inflationary stage and all the effects connected with this breakdown will be revealed only after the end of inflation and the beginning of the bang - to - rip evolution . this beginning is such that the value of the hubble parameter , characterizing this evolution is equal to that of the inflation , i.e. @xmath72 in turn , this implies that we begin evolution at the time moment of the order of @xmath73 . we shall consider both the components @xmath74 and @xmath75 and we shall dwell on the scenarios @xmath29 and @xmath33 described in the preceding section . anyway , our assumption regarding the initial conditions for equation ( [ max - four1 ] ) can be easily modified in order to account for the previous possible amplification of primordial magnetic fields as was discussed by @xcite . thus , all estimates for the numerical values of the magnetic fields in today s universe should be multiplied by some factor corresponding to the amplification of the magnetic field during the inflationary stage . hence , our results refer more to differences between various models of a post inflationary evolution than of the real present values of magnetic fields , whose amplification might be also combined effect of different mechanism @xcite . in this section we present the results of numerical simulations , for the two models @xmath29 and @xmath33 introduced in sec . 2 . in our models , the equation of motion for the modes @xmath76 ( [ max - four1 ] ) reads , where today - time is taken to be near the crossing of the phantom divide line , i.e. at @xmath77 . this implies in turn that at the beginning of the `` bang - to - rip '' evolution the cosmological radius is @xmath78 . ] : @xmath79f_{\pm}=0 , \label{edm}\ ] ] where @xmath80 stands for the scalar field @xmath11 in the model @xmath29 and for the phantom @xmath12 in the model @xmath33 , so that @xmath81 equation ( [ edm ] ) is solved for different values of the wave number @xmath63 and the coupling parameter @xmath35 . ( the parameter @xmath35 has the dimensionality inverse with respect to that of the scalar field ; the wave number @xmath63 has the dimensionality of inverse length ; the time @xmath82 ) . qualitatively we remark that in ( [ max - four1 ] ) the coupling term influence becomes negligible after some critical period . after that the magnetic fields in our different scenarios evolve as if the parameter @xmath35 in ( [ max - four1 ] ) had been put equal to zero . indeed , it can be easily seen that the interaction term vanishes with the growth of the cosmological radius @xmath83 . then the distinction between the two models is to be searched in the early time behavior of the field evolution . noting that in both our models @xmath29 and @xmath33 the time derivative @xmath84 is positive , by inspection of the linear term in equation ( [ max - four1 ] ) we expect the amplification to be mainly given for the mode @xmath75 provided the positive sign for @xmath35 is chosen ; so we will restrict our attention on @xmath75 . we can also argue that the relative strength of the last two terms in the left - hand side of ( [ edm ] ) is crucial for determining the behavior of the solution : when the coupling term prevails ( we remark that we are talking about @xmath85 so this term is _ negative _ in our models ) then we expect an amplification , while when the first term dominates we expect an oscillatory behavior . for future reference it is convenient to define @xmath86 which is just the ratio between the last two terms in the left - hand side of equation ( [ edm ] ) . indeed our numerical simulations confirm these predictions . let us consider the model @xmath29 with @xmath87 and @xmath88 , where @xmath16 is the planck length . such a value of the wave number @xmath63 corresponds to the wave length of @xmath89 at the present moment . we obtain an early - time amplification of about 2 orders of magnitude , with the subsequent oscillatory decay . notice that the parameter @xmath90 in this model at the beginning is very small : this corresponds to the dominance of the term proportional to @xmath84 and , hence , to the amplification of the field @xmath75 . at the time scale of the order of @xmath91 , where @xmath17 is the planck time , this regime turns to that with big values of @xmath90 where the influence of the term proportional to @xmath84 is negligible . for the same choice of the parameters @xmath35 and @xmath63 in the model @xmath33 the amplification is absent . for the model @xmath29 , with the parameter choice @xmath92 , @xmath88 . it can be easily seen that at a time scale of order @xmath91 the ratio becomes greater than 1 . ] ( given in planck units ) in model @xmath29 with the parameter choice @xmath92 , @xmath88 . the behavior , as said above in the text , consists in an amplification till a time of order @xmath91 , after which the oscillations begin . ] ( given in planck units ) in model @xmath29 with the parameter choice @xmath93 , @xmath88 . the behavior , consists in an amplification till a time of order @xmath94 , after which the oscillations begin . ] in figure [ mag1 ] we present the time dependence of the function @xmath90 for the model @xmath29 for the values of @xmath92 and @xmath88 chosen above . the figure [ mag2 ] manifests the amplification of the magnetic field in the model @xmath29 . naturally the effect of amplification of the magnetic field grows with the coupling constant @xmath35 and diminishes when the wave number @xmath63 increases . in figure [ mag3 ] we display the results for the case of @xmath95 , which is admittedly extreme and possibly non realistic , but good for illustrative purposes . here the amplification is more evident and extends for a longer time period . let us try to make some estimates of the cosmic magnetic fields in the universe today , using the correlation functions . the correlation function for the variable @xmath44 is defined as the quantum vacuum average @xmath96 and can be rewritten as @xmath97 integrating over the angles , we come to @xmath98 to estimate the integral ( [ corr2 ] ) we notice that the main contribution to it comes from the region where @xmath99 ( see e.g.@xcite ) and it is of order @xmath100 where @xmath101 . in this estimation the amplification factor is @xmath102 where the subscript @xmath103 is not present since we have taken the trace over polarizations . now we are in a position to give numerical values for the magnetic fields at different scales in the model @xmath29 for different values of the coupling parameter @xmath35 . these values ( see table [ tab ] ) correspond to three values of the coupling parameter @xmath35 ( 1,10 and 100 @xmath104 ) less than 1 make the effect of the coupling of the magnetic field with the pseudoscalar field negligible . ] and to two spatial scales @xmath105 determined by the values of the wave number @xmath63 . we do not impose some physical restrictions on the value of @xmath35 . it is easy to see that the increase of @xmath35 implies the growth of the value of the magnetic field @xmath106 . we also shall consider a rather large value of the wave number @xmath63 corresponding to the physical wavelength @xmath107 pc at the present moment when @xmath108 . while this scale looks too small for the description of large - scale cosmic magentic fields , we use it for illustrative purposes to underline the strong dependence of the amplification of magnetic field on the corresponing wavelength values . let us stress once again that we ignored the effects of amplification of the magnetic fields during inflation to focus on seizable effects during evolution . @lccc & @xmath109&@xmath110 & @xmath111 + & & & + @xmath112&@xmath113&@xmath114&@xmath115 + & & & + @xmath116&@xmath117&@xmath118&@xmath119 + [ tab ] finally notice that our quantum `` initial '' conditions correspond to physical magnetic fields which for presented values of @xmath63 are @xmath120 is equal to @xmath121 for @xmath122 and @xmath123 for @xmath124 . we have seen that the evolution of the cosmic magnetic fields interacting with a pseudoscalar ( pseudophantom ) field is quite sensitive to the concrete form of the dynamics of this field in two - field models where different scalar field dynamics and potentials realize the same cosmological evolution . we confirm the sensitivity of the evolution of the magnetic field with respect to its helicity given the sign of the coupling constant @xmath35 and that the @xmath80 is a monotonic function of time ( as it is really so in our models ) . we give also some numerical estimates of the actual magnetic fields up to the factor of amplification of such fields during the inflationary period . the toy model of the bang - to - rip evolution studied in this paper , can not be regarded as the only responsible for the amplification of cosmic magnetic fields implying their present observable values . it rather complements some other mechanisms acting before . however , the difference between cosmic magnetic fields arising in various models ( giving the same expansion law after the inflation ) is essential . it may provide a discriminating test for such models naturally , the influence of the interaction between a pseudoscalar ( phantom ) field and a cosmic magnetic field on the dynamics of the latter depends on the velocity of change with time of the former . the larger is the time derivative of the pseudoscalar field , the more intensive is the growth of the magnetic field ( remember that the evolution of the scalar field is monotonic ) . the results of numerical calculations illustrated in sec . 4 confirm these qualitative considerations . moreover , one can see that there exists a certain range of values of the wave number @xmath63 ( and hence of the corresponding wavelengths of the magnetic field ) where the effect is stronger . indeed , if @xmath63 is too small the interaction term is small as well and the evolution of the magnetic field is damped . on the other hand if the wave number @xmath63 is too large the interaction term is small compared to the oscillatory term , proportional as usual to @xmath125 and the evolution has practically oscillatory character . thus , the study of interplay between the dynamics of global scalar fields providing the cosmological evolution and the magnetic fields looks promising . this interplay has another interesting aspect . the pseudoscalar - 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we consider two different toy cosmological models based on two fields ( one normal scalar and one phantom ) realizing the same evolution of the bang - to - rip type .
one of the fields ( pseudoscalar ) interacts with the magnetic field breaking the conformal invariance of the latter .
the effects of the amplification of cosmic magnetic fields are studied and it is shown that the presence of such effects can discriminate between different cosmological models realizing the same global evolution of the universe . _ keywords _ : phantom dark energy , scalar fields , magnetic fields +
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in the arena of cosmology and stellar structure , the importance of the perfect fluid source can not be understated . the assumption of large - scale isotropy will demand , via the field equations , that the stress - energy tensor supporting the universe possess the algebraic structure of a perfect fluid . as well , cosmologies containing scalar fields have long been considered . the motivation being that the scalar field would represent some exotic component of matter , which could explain various puzzling phenomena presented by observational cosmology . more recently , viable alternatives to the standard big bang theory have been put forward , the most promising of which involve an inflationary era at some early time . most of these theories invoke a scalar field to play the role of the inflaton in the early universe . at late times , the scalar field may again play an important role as a `` dark energy '' field . it has been observed that the universe has recently entered an acceleration phase and some exotic dark energy must presently dominate @xcite , @xcite . it is well known that a scalar field provides a simple model to explain this acceleration . the scalar field may take many forms . aside from the traditional quintessence field , which arises from a lagrangian motivated by relativistic continuum mechanics , there is also the recently popular `` tachyonic '' scalar field from string theory , which possesses a born - infeld type action . both of these fields have been used extensively in cosmology as the inflaton , dark matter and dark - energy ( see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite @xcite and references therein ) . usual cosmological models are spherically symmetric . however , more general cosmological models , admitting a three - parameter group of motions ( bianchi type i - ix ) have also been studied . for the above reasons , we believe that it is important to study the combined system of einstein - scalar field - perfect fluid equations in detail . specifically , in section 2 we present the system in its full generality for both the quintessence field and tachyon field . we discuss the system of equations and the number of functions which can be apriori prescribed . this is dictated by the number of functions present as well as the number of equations and identities . the eigenvector and eigenvalue structure is briefly discussed as well as its relation to physical quantities such as energy density and pressure . the cauchy problem has long been studied in general relativity and is of great use due to the difficulty involved in obtaining exact solutions to physical problems within the theory ( see @xcite , @xcite , as well as @xcite where the gravitating complex scalar - maxwell field system has been studied ) . in solving einstein s equations , one often considers physically acceptable initial data ( for example : data from plasma physics for stellar collapse , a specific form for the scalar potential inspired by string theory or particle physics in cosmology , etc . ) the future evolution of this prescribed data is to be determined , ideally by analytically solving the einstein equations and matter field equations of motion . however , this is often not possible and either some simplifying assumptions must be made or else approximation methods must be utilized . it is useful therefore to numerically evolve the initial data of the full , unsimplified model to determine the behaviour of the system at later times and , in some cases , to aid in determining what form the exact solution should take . a thorough treatment of the system consisting of a massless self - gravitating scalar field in the context of gravitational collapse ( in the retarded time gauge ) may be found in @xcite . the cauchy problem in general relativity is non - trivial . initial data is not freely prescribable due to the existence of four constraining equations for the initial data . ( these arise due to the differential identities among the system of field equations . ) the constraint equations for both the quintessence plus perfect fluid and tachyon plus perfect fluid systems are derived in this paper along with the functions which one , in principle , may prescribe . in section 3 the cauchy problem for the einstein - perfect fluid - scalar field is properly posed using geodesic normal coordinates ( which are particularly well suited to cosmological studies ) . a scheme is presented which will evolve the this data from the initial @xmath0 hyper - surface analytically into the bulk ( @xmath1 ) . finally , we end with a simple example to illustrate the use of the scheme and its convergence properties . for an excellent review of the cauchy problem in general relativity , the reader is referred to @xcite . the system considered consists of gravity coupled to a perfect fluid and quintessence scalar field , @xmath2 . the fluid energy density , pressure and scalar potential are denoted by @xmath3 , @xmath4 and @xmath5 respectively . derivatives with respect to the scalar field are denoted by a prime . the quintessence lagrangian density has the form : @xmath6.\ ] ] the einstein and supplemental equations for this system are given by : @xmath7 \delta^{\mu}_{\;\nu } + \left(\rho + p \right ) u^{\mu}u_{\nu } + p\delta^{\mu}_{\;\nu } , \label{eq : tq } \\ & \mathcal{t}_{\nu}:=t^{\mu}_{\;\nu ; \mu}=0 , \label{eq : consq } \\ & \sigma:=\phi^{;\mu}_{\ ; ; \mu}-{v^{\prime}(\phi)}=0 , \label{eq : siq } \\ & \mathcal{u}:=u^{\alpha}u_{\alpha}+1=0 , \label{eq : uq } \\ & \mathcal{k}:=\left[(\rho + p)u^{\alpha}\right]_{;\alpha } -u^{\alpha}p_{,\alpha}=0 , \label{eq : kq } \\ & \mathcal{f}_{\nu}:=(\rho + p)u^{\alpha}u_{\nu;\alpha } + \left(\delta^{\alpha}_{\;\nu } + u^{\alpha}u_{\nu}\right)p_{,\alpha}=0 , \label{eq : fq } \\ & \mathcal{c}^{\nu}(g_{\alpha\beta } , g_{\alpha\beta,\gamma})=0 . \label{eq : cq}\end{aligned}\ ] ] ( the @xmath8 denote four possible coordinate conditions . ) the algebraic and differential identities are : @xmath9 in this system of equations the number of unknown functions is @xmath10 whereas , from ( [ eq : einstq]-[eq : cq ] ) , there exist twenty - five equations . however , there are nine identities ( [ eq : id1q]-[eq : id3q ] ) and therefore only sixteen independent equations yielding an underdetermined system . we can therefore prescribe two functions out of the eighteen . in case an equation of state is imposed , @xmath11^{2}+\left[\frac{\partial s(p,\rho)}{\partial p}\right]^{2}&>&0 , \nonumber\end{aligned}\ ] ] we can still prescribe one function . we can now explore eigenvalues of @xmath12 . assuming that @xmath13 we obtain from ( [ eq : tq]-[eq : uq ] ) and ( [ eq : eigassumpq ] ) that @xmath14u^{\alpha } , \label{eq : evec1q } \\ t^{\alpha}_{\;\beta}\phi^{;\beta}= & \left[p+ \frac{1}{2 } \phi^{;\beta}\phi_{,\beta } -{v(\phi)}\right]\phi^{;\alpha } + \left[\left(\rho + p \right)u_{\beta}\phi^{;\beta}\right ] u^{\alpha } , \label{eq : evec2q } \\ t^{\alpha}_{\;\beta}s^{\beta}=&\left[\phi_{,\beta}s^{\beta}\right]\phi^{;\alpha } -\left[\frac{1}{2 } \phi^{;\beta}\phi_{,\beta } + { v(\phi)}-p\right]s^{\alpha}. \label{eq : evec3q}\end{aligned}\ ] ] it is evident from the above equations that , in general , none of the vectors @xmath15 , @xmath16 , @xmath17 are eigenvectors . however , consider two special cases . in case - i , we take the scalar gradient perpendicular to the fluid velocity and @xmath17 : @xmath18 by ( [ eq : evec1q]-[eq : evec2q ] ) and ( [ eq : perpgradq ] ) , we get : @xmath19u^{\alpha } , \label{eq : class1vecsa } \\ t^{\alpha}_{\;\beta}\phi^{;\beta}= & \left[p + \frac{1}{2 } \phi^{;\beta}\phi_{,\beta } -{v(\phi)}\right]\phi^{;\alpha } , \label{eq : class1vecsb } \\ t^{\alpha}_{\;\beta}s^{\beta}=&-\left[-p + \frac{1}{2 } \phi^{;\beta}\phi_{,\beta } + { v(\phi)}\right]s^{\alpha}. \label{eq : class1vecsc}\end{aligned}\ ] ] in this case , @xmath15 , @xmath20 are eigenvectors and @xmath17 is a two - fold degenerate eigenvector . the corresponding eigenvalues in ( [ eq : class1vecsa]-[eq : class1vecsc ] ) represent the proper mass density and two principal pressures , respectively , of the matter . a positive value of the potential , @xmath5 , make a positive contribution to the energy density and negative contribution to the pressures . the structure here is algebraically similar to the _ anisotropic fluid_. in case - ii , assume that the scalar gradient is colinear with the fluid velocity : @xmath21 by equations ( [ eq : evec1q]-[eq : evec3q ] ) and ( [ eq : eigassumpq2 ] ) , we arrive at : @xmath22u^{\alpha } , \label{eq : class2vecsa } \\ t^{\alpha}_{\;\beta}\phi^{;\beta}= & -\left[\rho -\frac{1}{2 } \phi^{;\beta}\phi_{,\beta } + { v(\phi)}\right]\phi^{;\alpha } , \label{eq : class2vecsb } \\ t^{\alpha}_{\;\beta}s^{\beta}=&\left[p-\frac{1}{2 } \phi^{;\beta}\phi_{,\beta } -{v(\phi)}\right]s^{\alpha}. \label{eq : class2vecsc}\end{aligned}\ ] ] in this case , @xmath15 , @xmath20 and @xmath17 are all eigenvectors . the structure here is algebraically similar to a perfect fluid with proper mass density @xmath23 and pressure @xmath24 . the tachyon field lagrangian density is given by : @xmath25^{1/2},\ ] ] with @xmath26 representing the tachyon field . in the case of the tachyonic scalar field , the governing equations read : @xmath27 + \left(\rho+p\right ) u^{\mu}u_{\nu } + p\delta^{\mu}_{\;\nu } , \label{eq : tt } \\ & \mathcal{t}_{\nu}:=t^{\mu}_{\;\nu;\mu}=0 , \label{eq : const } \\ & \sigma:=\left[\frac{v({\phi})\ , { \phi}_{,\mu}}{{\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}}\right]_{;\mu } -v^{\prime}({\phi}){\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}=0 , \label{eq : sit } \\ & \mathcal{u}:=u^{\alpha}u_{\alpha}+1=0 , \label{eq : ut } \\ & \mathcal{k}=\left[(\rho + p)u^{\alpha}\right]_{;\alpha}-u^{\alpha } p_{,\alpha}= 0 , \label{eq : kt } \\ & \mathcal{f}_{\nu}=(\rho+p)u^{\alpha}u_{\nu;\alpha}+ \left(\delta^{\alpha}_{\;\nu } + u^{\alpha}u_{\nu } \right ) p_{,\alpha } = 0 , \label{eq : ft } \\ & \mathcal{c}^{\nu}(g_{\alpha\beta } , g_{\alpha\beta,\gamma})=0 . \label{eq : ct}\end{aligned}\ ] ] again the system is underdetermined by two which allows us to prescribe an equation of state and one quantity ( usually @xmath3 or @xmath4 ) . defining vectors as in ( [ eq : eigassumpq ] ) we find : @xmath28u^{\alpha } , \label{eq : evaleqta } \\ t^{\alpha}_{\;\beta}{\phi}^{;\beta}= & \left[p-\frac{v({\phi})}{{\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}}\right]{\phi}^{;\alpha } + \left[(\rho+p)u^{\beta } { \phi}_{,\beta}\right ] u^{\alpha } , \label{eq : evaleqtb}\\ t^{\alpha}_{\;\beta}s^{\beta}= & \frac{v({\phi})}{{\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa } } } } s^{\beta}{\phi}_{,\beta } { \phi}^{;\alpha } + \left[p - v({\phi}){\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}\right]s^{\alpha}. \label{eq : evaleqtc}\end{aligned}\ ] ] as with the quintessence field , none of the vectors @xmath15 , @xmath16 , @xmath17 are eigenvectors . in case the vectors are mutually orthogonal the following equations hold : @xmath29u^{\alpha},\label{eq : evaltaorth } \\ t^{\alpha}_{\;\beta}{\phi}^{;\beta}=&\left[p-\frac{v({\phi})}{{\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}}\right]{\phi}^{;\alpha } , \label{eq : evaltborth } \\ t^{\alpha}_{\;\beta}s^{\beta}=&\left[p - v({\phi}){\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}\right]s^{\alpha}. \label{eq : evaltcorth}\end{aligned}\ ] ] note that in this case , not only are @xmath15 , @xmath16 and @xmath17 eigenvectors , but the stress - energy tensor possesses the algebraic structure of an _ anisotropic fluid_. in case the fluid velocity and scalar gradient are colinear , the stress - energy tensor has following structure : @xmath30u^{\alpha } , \label{eq : evalta } \\ t^{\alpha}_{\;\beta}{\phi}^{;\beta}=&-\left[\rho+\frac{v({\phi})}{{\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}}\right]{\phi}^{;\alpha } , \label{eq : evaltb } \\ t^{\alpha}_{\;\beta}s^{\beta}=&\left[p - v({\phi}){\sqrt{1+{\phi}^{;\kappa}{\phi}_{,\kappa}}}\right]s^{\alpha } , \label{eq : evaltc}\end{aligned}\ ] ] which is similar to that of a _ here we present the cauchy problem for the self - gravitating system of a perfect fluid and scalar field . let us consider a contractible space - time domain ( see figure [ fig : cauchfig ] ) @xmath31 and is evolved to the hyper - surface domain @xmath32 . ] let a differentiable symmetric tensor field @xmath33 exist in @xmath34 of the space - time . we cite synge s lemma @xcite + * synge s lemma : * let @xmath33 be a symmetric , differentiable tensor field in the domain @xmath34 of space - time . then , the following two statements are mathematically equivalent : @xmath35 the above lemma can be applied to einstein s field equations . in that case , the following two statements are mathematically equaivalent : @xmath36 we shall investigate equations ( [ eq : einstlem ] ) involving a quintessence scalar field and perfect fluid in the following geodesic normal coordinates : @xmath37_{|t_{0}},\;\ ; \overline{g}_{ij,0,0}(\mathbf{x}):=\left[g^{\sharp}_{ij,0,0}(\mathbf{x},t)\right]_{|t_{0}}\end{aligned}\ ] ] to clarify , @xmath38 will subsequently be used to denote the _ bulk spatial metric _ and @xmath39 will be used to denote the _ spatial metric on the initial hyper - surface_. computing the ricci and einstein tensor components from ( [ eq : einstlem ] ) we obtain : @xmath40 , \label{eq : iconst2 } \\ g_{00}(x)= & \frac{1}{2 } r^{\sharp}(\mathbf{x},t ) + \frac{1}{8 } \left[g^{\sharp kl}g^{\sharp}_{kl,0}\right]^{2 } -\frac{1}{8 } g^{\sharp km}g^{\sharp ln}g^{\sharp}_{kl,0}g^{\sharp}_{mn,0}. \label{eq : iconst3}\end{aligned}\ ] ] here , sharp symbols ( @xmath41 ) on indices denotes covariant differentiation with respect to the three dimensional metric @xmath38 . the ricci and einstein tensor components can be expressed as ( from ( [ eq : cauchdefs ] ) and ( [ eq : iconst1 ] -[eq : iconst3 ] ) ) : @xmath42 , \label{eq : newiconst2 } \\ g_{00}(\mathbf{x},t_{0})= & \frac{1}{2}\overline{r}(\mathbf{x } ) + \frac{1}{8 } \left[\overline{g}^{kl}\overline{g}_{kl,0}\right]^{2 } -\frac{1}{8 } \overline{g}^{km}\overline{g}^{ln}\overline{g}_{kl,0}\overline{g}_{mn,0}. \label{eq : newiconst3}.\end{aligned}\ ] ] note that the _ extrinsic curvature components _ of the initial hyper - surface are given by @xmath43 and the above equations are also derivable from the gauss - codazzi equations @xcite , @xcite , @xcite . we shall now explore the field equations and subsequent evolution with quintessence scalar field and perfect fluid sources . using synge s lemma and consequent equations ( [ eq : einstlem ] ) and ( [ eq : iconst1 ] - [ eq : iconst3 ] ) we arrive at : @xmath44\right\}=0 , \label{eq : cauchqe } \\ & \sigma=\frac{1}{\sqrt{g^{\sharp } } } \left[\sqrt{g^{\sharp}}g^{\sharp kl}\phi _ { , k}\right]_{,l } -\phi_{,0,0 } -\left(\ln \sqrt{g^{\sharp}}\right)_{,0}\phi_{,0 } -{v^{\prime}(\phi)}=0 , \label{eq : cauchqs } \\ & u^{0}=u(u):=\sqrt{1+g^{\sharp mn}u_{m}u_{n}}\ , \geq 1 , \label{eq : cauchqu } \\ & \mathcal{k}=u(u)\rho_{,0 } + u^{l}\rho_{,l } + \frac{(\rho+p)}{\sqrt{g^{\sharp } } } \left\{\left ( \sqrt{g^{\sharp}}u^{k}\right)_{,k } + u(u)\left(\sqrt{g^{\sharp}}\right)_{,0 } + \sqrt{g^{\sharp}}u_{,0}\right\}=0 , \label{eq : cauchqk } \\ & \mathcal{f}_{0}=\left[1-\left(u(u)\right)^{2 } \right]p_{,0 } -u^{k}u(u)p_{,k } -\left(\rho+p\right ) \left\{u^{l}u_{,l } + \frac{1}{2}g^{\sharp bm}g^{\sharp}_{ma,0}u^{a}u_{b}\right\}-uu_{,0}=0 , \label{eq : cauchqf } \\ & \mathcal{f}_{j}=(\rho+p ) u(u ) u_{j,0 } -\frac{1}{2}(\rho + p)u g^{\sharp bm}g^{\sharp}_{mj,0}u_{b } + \left[p_{,j}+u^{b}u_{j}p_{,b } + uu_{j}p_{,0 } \right]=0 . \label{eq : cauchqf2}\end{aligned}\ ] ] moreover , on the initial hyper - surface ( @xmath45 ) , the initial data must satisfy : @xmath46 -8\pi \left[\phi_{,i}\phi_{,0}-(\rho + p)u u_{i}\right]_{|t_{0 } } , \label{eq : qinit1 } \\ \mathcal{e}_{00}(\mathbf{x},t_{0})=&\frac{1}{2 } \overline{r}(\mathbf{x } ) + \frac{1}{8 } \left[{\overline}{g}^{mn}(\mathbf{x } ) \overline{g}_{mn,0}\right]^{2 } -\frac{1}{8 } \overline{g}^{ma}\overline{g}^{nb } \overline{g}_{mn,0 } \overline{g}_{ab,0 } \nonumber \\ & -8\pi \left\{\frac{1}{2 } \left[(\phi_{,0})^{2 } + g^{\sharp ab}\phi_{,a}\phi_{,b}\right]+ v(\phi ) + ( \rho+p ) u^{2 } -p\right\}_{|t_{0 } } = 0 . \label{eq : qinit2}\end{aligned}\ ] ] the initial data is to be prescribed as : @xmath47 the given functions @xmath48 , @xmath49 , @xmath50 and @xmath51 are of class @xmath52 in @xmath53 , the functions @xmath54 are of class @xmath55 and the functions @xmath56 are of class @xmath57 . moreover , the consistency conditions ( [ eq : qinit1 ] - [ eq : qinit2 ] ) must be satisfied as : @xmath58 + 8\pi \left\ { \xi\chi_{,a } -\left[\mu+\eta\right]u\,w_{a}\right]=0 , \label{eq : qconsist1 } \\ & \frac{1}{2}\overline{r}+\frac{1}{8 } \left[\gamma^{mn } \psi_{mn}\right]^{2 } -\frac{1}{8 } \gamma^{ma } \gamma^{nb } \psi_{mn}\psi_{ab } \nonumber \\ & \;\;\ ; -8\pi \left[\frac{1}{2 } \left(\xi^{2 } + \overline{\gamma}^{ab } \chi_{,a}\chi_{,b } \right ) + v(\chi ) + \left(\mu+\eta\right ) u^{2 } -\eta \right]=0 . \label{eq : qconsist2}\end{aligned}\ ] ] ( here , barred quantities are derived from the prescribed metric @xmath59 ) . the system of three dimensional partial differential equations is underdetermined so that infinitely many solutions should exist locally . restricting the field equations to the initial hyper - surface , @xmath45 , we can rearrange the equations symbolically as : @xmath60p_{,0}\right\}_{|t_{0}}&= & \;\;\ ; " \;\;\;\;\;\;\;\ ; " \;\;\;\;\;\ ; " \;\;\;\;\;\ ; " \;\;\;\;\;\;\;\;\ ; " \;\;\;\;\;\;\;\;\ ; " \;\;\;\;\;\;\;\;\;\;,\nonumber \\ \left[\mu+\eta\right]u\,u_{a,0\;|t_{0}}&=&\;\;\ ; " \;\;\;\;\;\;\;\ ; " \;\;\;\;\;\ ; " \;\;\;\;\;\ ; " \;\;\;\;\;\;\;\;\ ; " \;\;\;\;\;\;\;\;\ ; " \;\;\;\;\;\;\;\;\;\ ; .\nonumber\end{aligned}\ ] ] thus , the higher derivatives of @xmath61 , @xmath2 , @xmath3 , @xmath4 , and @xmath62 at the initial hyper - surface are determined by allowable cauchy data and their spatial derivatives . assuming the functions @xmath63 , @xmath64 , @xmath65 , @xmath66 , and @xmath67 are _ real - analytic functions _ in @xmath68 , we can determine arbitrary higher order derivatives with respect to @xmath69 by differentiating the field equations in ( [ eq : cauchqe ] - [ eq : cauchqf2 ] ) and restricting subsequently those equations to the initial hyper - surface . thus , the power series : @xmath70 can be generated . by the real analyticity conditions , there exist a @xmath71 such that all the power series in ( [ eq : gevolve]-[eq : uevolve ] ) converge absolutely for all @xmath72 $ ] and converge uniformly for all @xmath73 $ ] for a sufficiently small @xmath74 . in the case of the tachyonic scalar field , the field and supplemental equations may be expressed as : @xmath75 + ( \rho+p)u_{i}u_{j } + \frac{1}{2}g^{\sharp}_{ij}(\rho -p ) \right\}=0 , \label{eq : etach } \\ & \sigma=\frac{1}{\sqrt{g^{\sharp } } } \left[\sqrt{g^{\sharp } } g^{\sharp ab}{\phi}_{,a } \right]_{,b } -{\phi}_{,0,0 } -\left(\ln \sqrt{g^{\sharp}}\right)_{,0 } { \phi}_{,0 } -\left[\ln |{v(\phi)}| \right]^{\prime } \nonumber \\ & \;\;\;\;\;\ ; -\left[1+{\phi}^{;\mu}{\phi}_{,\mu } \right]^{-1 } \left[{\phi}_{,i } { \phi}_{,j } g^{\sharp \,nj}g^{\sharp\ , bi } \left({\phi}_{,n;b^{\sharp } } + \frac{1}{2}g^{\sharp}_{bn,0}{\phi}_{,0}\right ) \left({\phi}_{,0}\right)^{2}{\phi}_{,0,0 } \right . \nonumber \\ & \;\;\;\;\;\ ; \left . -2{\phi}_{,0}{\phi}_{,j}g^{\sharp\,nj}{\phi}_{,n;0}\right ] = 0 , \label{eq : sigtach } \\ & u^{0}=:u:=\sqrt{1+g^{\sharp ab}u_{a}u_{b } } \geq 1 , \label{eq : utach } \\ & \mathcal{k}:=u\,\rho_{,0 } + u^{a } \rho_{,a } + \frac{(\rho+p)}{\sqrt{g^{\sharp } } } \left[\left(\sqrt{g^{\sharp}}u^{a}\right)_{,a}+ u\sqrt{g^{\sharp}}_{,0 } + \sqrt{g^{\sharp } } u_{,0 } \right]=0 , \label{eq : ktach } \\ & \mathcal{f}_{0}=\left[1-\left(u\right)^{2 } \right]p_{,0 } -u^{k}u(u)p_{,k } -\left(\rho+p\right ) \left\{u^{l}u_{,l } + \frac{1}{2}g^{\sharp bm}g^{\sharp}_{ma,0}u_{b}\right\}-uu_{,0}=0 , \label{eq : f0tach } \\ & \mathcal{f}_{j}=(\rho+p ) u(u ) u_{j,0 } -\frac{1}{2}(\rho + p)u g^{\sharp bm}g^{\sharp}_{mj,0}u_{b } + \left[p_{,j}+u^{b}u_{j}p_{,b } + uu_{j}p_{,0 } \right]=0 . \label{eq : f1tach}\end{aligned}\ ] ] as before , there exist constraints on the initial data on the @xmath45 hyper - surface : @xmath76 \nonumber \\ & - 8\pi \left\{{v(\phi)}\left[\frac{{\phi}_{,a}{\phi}_{,0}}{\sqrt{1 + g^{\sharp rs}{\phi}_{,r}{\phi}_{,s}-({\phi}_{,0})^{2}}}\right ] -(\rho + p)u u_{a } \right\}_{|t_{0}}=0 , \label{eq : tachcons1 } \\ \mathcal{e}_{00}(\mathbf{x},t_{0})= & \frac{1}{2 } \overline{r}+\frac{1}{8 } \left[\overline{g}^{mn } \overline{g}_{mn,0 } \right]^{2 } - \frac{1}{8 } \overline{g}^{ma } \overline{g}^{nb } \overline{g}_{mn,0 } \overline{g}_{ab,0 } \nonumber \\ & -8\pi \left\{{v(\phi)}\left[\frac{1+g^{\sharp ab}{\phi}_{,a } { \phi}_{,b}}{\sqrt{1+g^{\sharp mn } { \phi}_{,m}{\phi}_{,n } -({\phi}_{,0})^{2 } } } \right]+ ( \rho+p ) u^{2 } - p\right\}_{|t_{0 } } = 0 . \label{eq : tachcons2}\end{aligned}\ ] ] the initial data is prescribed as in the quintessence case ( [ eq : quintindat1])-([eq : quintindat5 ] ) with @xmath2 replaced by @xmath26 ( differentiability requirements remain the same ) . in this case , the consistency equations ( [ eq : tachcons1])-([eq : tachcons2 ] ) become : @xmath77 + 8\pi \left\{v(\chi ) \left[\frac{\xi\chi_{,a}}{\sqrt{1+\gamma^{rs}\chi_{,r}\chi_{,s } -\xi^{2 } } } \right ] -\left(\mu+ \eta\right)u\,w_{a } \right\}=0 , \label{eq : inittacha } \\ & -\frac{1}{2}\overline{r } -\frac{1}{8 } \left[\gamma^{mn } \psi_{mn}\right]^{2 } + \frac{1}{8 } \gamma^{ma } \gamma^{nb } \psi_{mn } \psi_{ab } \nonumber \\ & \;\;\ ; + 8\pi \left\{v(\chi ) \left[\frac{1+\gamma^{ab}\chi_{,a}\chi_{,b}}{\sqrt{1 + \gamma^{mn}\chi_{,m}\chi_{,n } -\xi^{2}}}\right ] + ( \mu+\eta ) u^{2 } -\eta\right\ } = 0 . \label{eq : inittachb}\end{aligned}\ ] ] at this point , the evolution is governed by the equations ( [ eq : gevolve ] ) - ( [ eq : uevolve ] ) as before . it is instructive to demonstrate how the above scheme works with an explicit example . we shall study the numerical evolution of a system whose analytic properties are known . comparison with a known solution will show whether or not the method works as well as provide a benchmark on its convergence properties . specifically , we consider the evolution of a constant scalar field in an otherwise empty , flat friedmann - lematre - robertson - walker ( flrw ) space - time . although extremely simple , this example is pedagogically useful as it serves well to elucidate the employment of the scheme without unnecessary complications which arise from more complex systems . as well , the constant field evolution is identical for both the quintessence and tachyonic scenario . we show how to extract various quantities for the cauchy evolution and compare with the known analytic result . since the source consists of only the constant scalar field , we can immediately set the following initial data : @xmath78 with this prescription , the equation pairs ( [ eq : qconsist1 ] - [ eq : qconsist2 ] ) and ( [ eq : inittacha ] - [ eq : inittachb ] ) both take the form : @xmath79 = 0 , \label{eq : exampconst1 } \\ & -\frac{1}{2}\overline{r } -\frac{1}{8}\left[\gamma^{mn}\psi_{mn}\right]^{2 } + \frac{1}{8 } \gamma^{ma } \gamma^{nb } \psi_{mn}\psi_{ab } + 8\pi v(\chi_{0 } ) = 0 . \label{eq : exampconst2}\end{aligned}\ ] ] it should be noted that in general , the consistency equations , although underdetermined , are difficult to solve even in vacuum and one must appeal to numerical techniques for solutions . as well , we consider the spatially flat flrw metric : @xmath80=a^{2}(t_{0})\left [ \begin { array}{ccc } 1&0&0\\ \noalign{\medskip } 0&r^{2}&0\\ \noalign{\medskip } 0&0&r^{2}\sin^{2}\theta \end { array } \right ] , \nonumber\ ] ] with @xmath81 the value of the scale factor on the initial data hyper - surface . we consider an inflationary scenario and prescribe : @xmath82_{,0|t = t_{0}}=&2e^{2t_{0}}. \nonumber\end{aligned}\ ] ] at this point , all the allowable initial data has been prescribed . the einstein - scalar field equations ( [ eq : etach]-[eq : sigtach ] ) dictate that @xmath83 at this point it is useful to reiterate that the above prescription is only valid _ if _ it satisfies the constraint equations ( [ eq : tachcons1 ] - [ eq : tachcons2 ] ) . in this example , the relevant values for the constraint equations are : @xmath84 it is a simple matter to check that these quantities indeed satisfy the constraint equations ( [ eq : exampconst1 ] ) and ( [ eq : exampconst2 ] ) . we show both the cauchy evolved and the analytic @xmath85 square of the scale factor in figure [ fig : evolved ] . as expected , there is excellent agreement at small values of @xmath86 . deviations at larger @xmath86 may be minimized by retaining more terms in the taylor series ( [ eq : gevolve ] ) as can be seen from the vatious dotted lines in the figure ( see figure caption ) . it may readily be verified that the other parameters to be evolved will display the proper evolution by a simple inspection of ( [ eq : gevolve ] - [ eq : uevolve ] ) . that is , in this simple example , all other parameters will retain their initial values as prescribed by ( [ eq : exinitdat ] ) . ( quadratic , cubic , quartic ) . the solid line represents the analytic result @xmath85 . ] both the quintessence and tachyonic scalar field , supplemented with a perfect fluid , were considered in the context of general relativity . the general mathematical properties of the system , including the eigenvalue structure were studied . it is seen that the system may behave either as a two component perfect fluid or an anisotopic fluid , the anisotropy being due to the properties of the scalar field . finally , in the geodesic coordinates , the cauchy problem as well as the initial constraint equations have been derived . assuming analyticity , the cauchy scheme presented here is convergent . the scheme is iterative and may easily be executed by computer .
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the system consisting of a self gravitating perfect fluid and scalar field is considered in detail .
the scalar fields considered are the quintessence and `` tachyonic '' forms which have important application in cosmology .
mathematical properties of the general system of equations are studied including the algebraic and differential identities as well as the eigenvalue structure .
the cauchy problem for both quintessence and the tachyon is presented .
we discuss the initial constraint equations which must be satisfied by the initial data .
a cauchy evolution scheme is presented in the form of a taylor series about the cauchy surface .
finally , a simple numerical example is provided to illustrate this scheme .
pacs numbers : 04.20.ex , 04.40.-b , 98.80jk + msc : 83f05 , 83c05 + key words : scalar fields , cauchy problem , relativity +
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three models which study the stochastic behaviour of the prices of commodities that take into account several aspects of possible influences on the prices were proposed by e schwartz @xcite in the late nineties . in the simplest model ( the so - called one - factor model ) schwartz assumed that the logarithm of the spot price followed a mean - reversion process of ornstein uhlenbeck type . the one - factor model is expressed by the following @xmath0 evolution equation@xmath1where @xmath2 measures the degree of mean reversion to the long - run mean log price , @xmath3 is the market price of risk , @xmath4 is the standard deviation of the return on the stock , @xmath5 is the stock price , @xmath6 is the drift rate of @xmath5 and @xmath7 is the time . @xmath8 is the current value of the futures contract which depends upon the parameters @xmath9 , _ i.e. _ , @xmath10 . generally @xmath11 , @xmath3 , @xmath4 and @xmath6 are assumed to be constants . in such a case the closed - form solution of equation ( [ 1fm.01 ] ) which satisfies the initial condition@xmath12was given in @xcite . it is @xmath13with @xmath14 . it has been shown that the closed - form solution ( [ 1fm.02 ] ) follows from the application of lie point symmetries . in particular it has been shown that equation ( [ 1fm.01 ] ) is of maximal symmetry , which means that it is invariant under the same group of invariance transformations ( of dimension @xmath15 ) as that of the black - scholes and the heat conduction equation @xcite . the detailed analysis for the lie symmetries of the three models , which were proposed by schwartz , and the generalisation to the @xmath16-factor model can be found in @xcite . other financial models which have been studied with the use of group invariants can be found in leach05a , leach06a , naicker , sinkala08a , sinkala08b , wafo , consta , lescot , dimas2 and references therein . solution ( [ 1fm.02 ] ) is that which arises from the application of the invariant functions of the lie symmetry vector @xmath17and also leaves the initial condition invariant . in a realistic world parameters are not constants , but vary in time and depend upon the stock price , that is , the parameters have time and space dependence @xcite , where as space we mean the stock price parameters as an analogue to physics . in this work we are interested in the case for which the parameters @xmath11 , @xmath3 , @xmath4 and @xmath6 are space dependent , _ ie _ , are functions of @xmath5 . we study the lie point symmetries of the space - dependent equation ( 1fm.01 ) . as we see in that case , when @xmath18 , there does not exist any lie point symmetry which satisfies the initial condition ( 1fm.01a ) . the lie symmetry analysis of the time - dependent black - scholes - merton equations was carried out recently in @xcite , it has been shown that the autonomous , and the nonautonomous black - scholes - merton equation are invariant under the same group of invariant transformations , and they are maximal symmetric . the plan of the paper is as follows . the lie point symmetries of differential equations are presented in section [ preliminaries ] . in addition we prove a theorem which relates the lie point symmetries of space - dependent linear evolution equations with the homothetic algebra of the underlying space which defines the laplace operator . in section [ space1 ] we use these results in order to study the existence of lie symmetries of for the space - dependent one - factor model ( 1fm.01 ) and we show that the space - dependent problem is not necessarily maximally symmetric . the generic symmetry vector and the constraint conditions are given and we prove a corollary in with the space - dependent linear evolution equation is always maximally symmetric when we demand that there exist at least one symmetry of the form ( [ 1fm.03 ] ) which satisfies the schwartz condition ( [ 1fm.01a ] ) . furthermore in section [ proof2 ] we consider the time - dependence problem and we show that the model is always maximally symmetric . finally in section [ con ] we discuss our results and we draw our conclusions . appendix[proof1 ] completes our analysis . below we give the basic definitions and properties of lie point symmetries for differential equations and also two theorems for linear evolution equations . by definition a lie point symmetry , @xmath19 of a differential equation @xmath20 where the @xmath21 are the independent variables , @xmath22 is the dependent variable and @xmath23 is the generator of a one - parameter point transformation under which the differential equation @xmath24 is invariant . let @xmath25 be a one - parameter point transformation of the independent and dependent variables with the generator of infinitesimal transformations being @xmath26 the differential equation @xmath24 can be seen as a geometric object on the jet space @xmath27 . therefore we say that @xmath24 is invariant under the one - parameter point transformation with generator , @xmath28 , if @xcite @xmath29 } } \theta = 0 . \label{go.11}\]]or equivalently@xmath29 } } \theta = \lambda \theta ~,~{mod}\theta = 0 , \label{go.12}\]]where @xmath30 } $ ] is the second prolongation of @xmath28 in the space @xmath31 . it is given by the formula @xmath32 } = x+\eta _ { i}\partial _ { u_{,i}}+\eta _ { ij}\partial _ { u_{,ij } } , \label{go.13}\]]where @xmath33 , @xmath34 and @xmath35 is the operator of total differentiation , _ ie _ , @xmath36@xcite . moreover , if condition ( [ go.11 ] ) is satisfied ( equivalently condition ( [ go.12 ] ) ) , the vector field @xmath28 is called a lie point symmetry of the differential equation @xmath24 . a geometric method which relates the lie and the noether point symmetries of a class of second - order differential equations has been proposed in jgp , ijgmmp . specifically , the point symmetries of second - order partial differential equations are related with the elements of the conformal algebra of the underlying space which defines the laplace operator . similarly , for the lie symmetries of the second - order partial differential equation,@xmath37where @xmath38 is the laplace operator , @xmath39 is a nondegenerate tensor ( we call it a metric tensor ) and @xmath40 , the following theorem arises . ] . [ theom1]the lie point symmetries of ( [ 1fm.04 ] ) are generated by the homothetic group of the metric tensor @xmath41 which defines the laplace operator @xmath42 . the general form of the lie symmetry vector is @xmath43where @xmath44 is the homothetic factor of @xmath45 , @xmath46 for the killing vector ( kv , @xmath47 for homothetic vector ( hv ) , @xmath48 and @xmath49 are solutions of ( [ 1fm.05 ] ) , @xmath50 is a kv / hv of @xmath51 and the following condition holds , namely,@xmath52note that @xmath53 . another important result for the @xmath0 linear evolution equation of the form of ( [ 1fm.04 ] ) is the following theorem which gives the dimension of the possible admitted algebra . [ theom2 ] the one - dimensional linear evolution equation can admits 0 , 1 , 3 and 5 lie point symmetries plus the homogenous and the infinity symmetries @xcite . however , as equation ( [ 1fm.04 ] ) is time independent , it admits always the autonomous symmetry @xmath54 . in the following we apply theorems [ theom1 ] and [ theom2 ] in order to study the lie symmetries of the space - dependent one - factor model the space - dependent one - factor model of commodity pricing is defined by the equation@xmath55 the parameters , @xmath56 , @xmath57 , @xmath58 and @xmath3 , depend upon the stock price , @xmath5 . in order to simplify equation ( [ 1fm.06 ] ) we perform the coordinate transformation @xmath59 , that is , equation ( 1fm.06 ) becomes@xmath60or @xmath61where @xmath42 is the laplace operator in the one - dimensional space with fundamental line element@xmath62and admits a two - dimensional homothetic algebra . the gradient kv is @xmath63 and the gradient hv is @xmath64with homothetic factor@xmath65 equation ( [ 1fm.08 ] ) is of the form of ( [ 1fm.04 ] ) where now @xmath66and @xmath67without performing any symmetry analysis we observe that , when @xmath68 , ( [ 1fm.08 ] ) is in the form of the heat conduction equation and it is maximally symmetric , _ ie _ , it admits @xmath15 symmetries . in the case for which @xmath69 from ( [ 1fm.10 ] ) we have that @xmath70where @xmath71 . however , this is only a particular case whereas new cases can arise from the symmetry analysis . let @xmath72 be the two hvs of the space ( [ 1fm.09 ] ) with homothetic factors @xmath44 . as ( [ 1fm.08 ] ) is autonomous and linear , it admits the lie symmetries @xmath73 , where @xmath74 is a solution of ( 1fm.08 ) , therefore from theorem [ theom1 ] we have that the possible additional lie symmetry vector is @xmath75for which the following conditions hold@xmath76@xmath77 we study two cases : a ) @xmath78 and b ) @xmath79 let @xmath78 . then ( [ 1fm.s05 ] ) is satisfied . hence from ( [ 1fm.s04 ] ) we have the system @xmath80where @xmath81 , _ ie _ , @xmath82 . this means that from any vector field @xmath83 we have only one symmetry . hence from theorem [ theom2 ] condition ( [ 1fm.s06 ] ) should hold for @xmath84 and @xmath85 in this case the space - dependent one - factor model admits @xmath86 lie point symmetries . consider that @xmath87 . from ( [ 1fm.s04 ] ) we have that@xmath88and then ( [ 1fm.s05 ] ) gives@xmath89 consider the case for which @xmath90 . recall that for the space ( [ 1fm.09 ] ) , @xmath91 that is , from ( [ 1fm.s07 ] ) we have the conditions@xmath92@xmath93where @xmath94 . we continue with the subcases : [ [ subcase - b1 ] ] subcase b1 + + + + + + + + + + let @xmath95 that is , @xmath96 . in this case the symmetry conditions are : @xmath97hence we have the following system @xmath98 if system ( [ 1fm.s11])-([1fm.s13 ] ) holds for @xmath84 or @xmath99 , then equation ( [ 1fm.08 ] ) admits @xmath86 @xmath100 lie symmetries and in the case for which conditions ( [ 1fm.s11])-([1fm.s13 ] ) hold , _ ie _ , admits @xmath15 lie symmetries which is the maximum for a @xmath0 evolution equation . [ [ subcase - b2 ] ] subcase b2 + + + + + + + + + + in the second subcase we consider that @xmath101 . hence , if b2.a ) @xmath102 , then from ( [ 1fm.s08 ] ) and ( [ 1fm.s09 ] ) it follows that@xmath103@xmath104@xmath105where from theorem [ theom2 ] these conditions must hold for @xmath84 and @xmath106 and equation ( [ 1fm.08 ] ) is maximally symmetric . b2.b ) let @xmath107 . then it follows that@xmath108@xmath109@xmath110these conditions hold for @xmath84 or @xmath99 . if these conditions hold for both @xmath84 and @xmath111 , then equation ( [ 1fm.08 ] ) is maximally symmetric . we collect the results in the following theorem . [ theom3]the autonomous @xmath0 linear equation ( [ 1fm.08 ] ) , apart from the symmetry of autonomy , the linear symmetry and the infinity symmetry , can admit : \a ) the two lie symmetries @xmath112 , where @xmath83 is a hv of the one - dimensional flat space with @xmath113 if and only if condition ( [ 1fm.s06 ] ) holds for @xmath84 and @xmath99 . b1 ) the two or four lie symmetries @xmath114 if conditions ( [ 1fm.s11])-([1fm.s13 ] ) hold for @xmath84 or @xmath115 and @xmath116 and @xmath111 , respectively , where @xmath117 and @xmath118 b2.a ) the four lie symmetries ( [ 1fm.s20a ] ) if conditions ( [ 1fm.s14])([1fm.s16 ] ) hold for @xmath84 and @xmath111 , where @xmath119 and @xmath120 is given by ( [ 1fm.s20 ] ) . b2.b ) the two or four lie symmetries ( [ 1fm.s20a ] ) if and only if conditions ( [ 1fm.s17])([1fm.s19 ] ) hold for @xmath84 or @xmath115 and @xmath84 and @xmath111 , respectively , where @xmath121 furthermore , we comment that theorem ( [ theom3 ] ) holds for all linear autonomous equations of the form of ( [ 1fm.04 ] ) . here we discuss the relation among the lie symmetries and the initial condition ( [ 1fm.01a ] ) . in the case of constant parameters , _ ie _ , in equation ( [ 1fm.01 ] ) the lie symmetry vector ( [ 1fm.03 ] ) is the linear combination among the linear symmetry @xmath122 and the symmetry which is generated by the kv of the underlying space , which is @xmath123for @xmath124 . however , for a general function , @xmath125 , in order for the symmetry which is generated by the kv @xmath126 to satisfy the initial condition @xmath127 or the initial condition has to change . consider now that @xmath128 and satisfies the conditions @xmath129then from theorem [ theom3 ] , b2.a , we have that @xmath130 is given by ( [ 1fm.11 ] ) and at the same time @xmath131 generates two lie point symmetries for equation ( [ 1fm.08 ] ) . the lie point symmetries are @xmath132@xmath133@xmath134plus the autonomous and trivial symmetries . the symmetry vector field @xmath135 is the kv of the one - dimensional space . therefore , if we wish the field @xmath136 to satisfy an initial condition such as @xmath137 , then it should be @xmath138 which gives @xmath139 . from this we can see that , when @xmath140 , we have the initial condition ( [ 1fm.01a ] ) . let @xmath11 , @xmath3 and @xmath6 be constants . hence from ( [ 1fm.11 ] ) we have that @xmath141where for @xmath142 we have @xmath143and the solution for position @xmath144@xmath145 let now @xmath146 and consider that the kv @xmath147 generates a lie point symmetry of equation ( 1fm.08 ) from case a of theorem [ theom3 ] . then from condition ( 1fm.s06 ) we have that@xmath148that is , @xmath149 however , in that case , equation ( [ 1fm.08 ] ) is maximally symmetric and admits @xmath150lie point symmetries . consider reduction with the lie symmetry @xmath151 which keeps invariant the initial condition@xmath152 the application of @xmath153 in ( [ 1fm.08 ] ) gives @xmath154@xmath155 as another application of theorem [ theom3 ] we select @xmath156 . then the kv @xmath126 is @xmath157 . let this generate a lie point symmetry for equation ( [ 1fm.08 ] ) from the case a of theorem [ theom3 ] , that is , conditions ( [ 1fm.s06 ] ) give@xmath158where now we can see that equation ( [ 1fm.08 ] ) is maximally symmetric and admits @xmath15 point symmetries . consider the lie symmetry @xmath159 , which leaves invariant the modified initial condition @xmath160 . the invariant solution which follows is @xmath161 we observe that , when @xmath126 generates a lie point symmetry for equation ( [ 1fm.08 ] ) , the functional form of @xmath162 , which includes @xmath163 and @xmath164 has a specific form , such that equation ( [ 1fm.08 ] ) is maximally symmetric and equivalent with the black - scholes and the heat equations . in general , for unknown function @xmath130 , from theorem [ theom1 ] we have the following corollary . [ cor]when the kv of the underlying space which defines the laplace operator in equation ( [ 1fm.08 ] ) generates a lie point symmetry , the functional form of @xmath162 is @xmath165and equation ( [ 1fm.08 ] ) is maximally symmetric . the symmetry vectors , among the autonomous , the homogeneous and the infinity symmetries , are:@xmath166@xmath167@xmath168for @xmath169 , @xmath170 , @xmath171@xmath172@xmath173for @xmath169 , @xmath174 and@xmath175@xmath176@xmath177for @xmath178 , @xmath179 , where @xmath180 and @xmath181 are the elements of the homothetic algebra of the underlying space . we note that corollary [ cor ] holds for all autonomous linear 1 + 1 evolution equations . in the following section we discuss the group invariants of the time - dependent problem . when the parameters @xmath182 of equation ( [ 1fm.01 ] ) depend upon time , the one - factor model can be written as @xmath183where@xmath184 without loss of generality we can select @xmath185 . by analysing the determining equations as provided by the sym package dimas05a , dimas06a , dimas08a we find that the general form of the lie symmetry vector is @xmath186 \partial _ { x } \nonumber \label{a3 } \\[1pt ] & & + \left [ f(t)+\frac{1}{4}\left ( 4xbq+x(1 - 2p)a^{\prime 2}qa^{\prime } -4x(b^{\prime } + ap^{\prime } ) \right . \right . \nonumber \\[1pt ] & & \left . + a^{\prime \prime } ) \right ) \right ] f\partial _ { f},\end{aligned}\]]where functions @xmath187 are given by the system of ordinary differential equations , @xmath188 & & + b^{\prime } -2pb^{\prime } -2f^{\prime } + ap^{\prime } -2app^{\prime } + aq^{\prime } -\frac{a^{\prime \prime } } { 2},\end{aligned}\]]@xmath189 & & + 3a^{\prime } p^{\prime } -aq^{\prime } -2bq^{\prime } + 2apq^{\prime } + 2b^{\prime \prime } + 2ap^{\prime \prime } \end{aligned}\]]and @xmath190 in addition to the infinite number of solution symmetries . consequently the algebra is @xmath191 so that it is related to the classical heat equation by means of a point transformation . in the following we discuss our results . in the models of financial mathematics the parameters of the models are assumed to be constants . however , in real problems these parameters can depend upon the stock prices and upon time . in this work we considered the one - factor model of schwartz and we studied the lie symmetries in the case for which the parameters of the problem are space - dependent . in terms of lie symmetries , the one - factor model it is maximally symmetric and it is equivalent with the heat equation , but in the case where the parameters are space dependent , that is not necessary true , and we show that the model can admit 1 , 3 or 5 lie point symmetries ( except the trivial ones ) . to perform this analysis we studied the lie symmetries of the autonomous linear evolution equation and we found that there exist a unique relation among the lie symmetries and the collineations of the underlying geometry , where as geometry we define the space of the second derivatives . however , for a specific relation among the parameters of the model the system is always maximally symmetric . in particular , that holds when @xmath130 is an arbitrary function and @xmath192where @xmath193 are constants . in that case , the correspoding symmetry ( 1fm.03 ) becomes @xmath194 . consider that @xmath195 , and ( con.01 ) holds . then the application of the lie symmetry @xmath196 in ( 1fm.08 ) gives the solution@xmath197where in the limit @xmath198 , solution ( [ con.02 ] ) becomes@xmath199which can compared with solution ( [ 1fm.02 ] ) . consider now that @xmath130 is periodic around the line @xmath200 . let @xmath201 that and ( [ con.01 ] ) holds . hence the solution of the space - dependent one - factor model ( [ 1fm.08 ] ) which follows from the lie symmetry @xmath196 is @xmath202which is a periodic function of the stock price @xmath203 . for @xmath204 the taylor expansion of the static solution ( [ con.04 ] ) around the point @xmath205 , is@xmath206 in figure [ fig3 ] we give the static evolution of the solutions , ( con.03 ) and ( [ con.04 ] ) , for various values of the constant @xmath207 . ) ( left figures ) and solution ( [ con.04 ] ) ( right . figure ) for various values of the constant @xmath208 . solid line is for @xmath209 , dash dash line is for @xmath209 , and the dash dot line is for @xmath210.,height=264 ] on the other hand , in section [ proof2 ] we studied the case for which the parameters of the one - factor model are time - dependent and we showed that the model is always maximally symmetric and equivalent with the heat equation , that is , the time - depedence does not change the admitted group invariants of the one - factor model ( [ 1fm.01 ] ) . a more general consideration will be to extend this analysis to the two - factor and three - factor models and also to study the cases for which the parameters are dependent upon the stock price and upon the time , _ ie _ , the parameters are space and time dependent . this work is in progress . finally we remark how useful are the methods which are applied in physics and especially in general relativity for the study of space - dependent problems in financial mathematics . the reason for this is that from the second derivatives a ( pseudo)riemannian manifold can be defined . this makes the use of the methods of general relativity and differential geometry essential . the research of ap was supported by fondecyt postdoctoral grant no . rmm thanks the national research foundation of the republic of south africa for the granting of a postdoctoral fellowship with grant number 93183 while this work was being undertaken . in @xcite it has been shown that for a second - order pde of the form , @xmath211the lie symmetries are generated by the conformal algebra of the tensor @xmath212 . specifically the lie symmetry conditions for equation ( [ eq.02 ] ) are @xmath213@xmath214@xmath215where@xmath216 by comparison of equations ( [ 1fm.04 ] ) and ( [ eq.02 ] ) we have that @xmath217 , _ ie _ , @xmath218 and@xmath219where @xmath220 . therefore the symmetry vector @xmath28 for equation ( [ 1fm.04 ] ) has the form @xmath221we continue with the solution of the symmetry conditions . when we replace @xmath222 in ( [ eq.05 ] ) , it follows that @xmath223 which means that @xmath224 , where @xmath225 is a ckv of the metric , @xmath51 , with conformal factor @xmath44,@xmath226 _ ie_,@xmath227 and @xmath228 furthermore , from ( eq.04 ) the following system follows ( recall that @xmath229 and @xmath230 ) @xmath231@xmath232moreover we observe that @xmath233 , where @xmath234 is a constant ; that is , @xmath50 is a kv / hv of @xmath51 . finally for the function , @xmath235 it holds that @xmath236 case i : let @xmath247 . then from ( [ eq.18 ] ) @xmath248 which means that @xmath249 . however , from ( eq.19 ) we have that @xmath250 which gives the linear symmetry @xmath251 . in that case from the form of @xmath252 the autonomous symmetry @xmath253 arises . sophocleuous c , leach pgl and andriopoulos k ( 2008 ) algebraic properties of evolution partial differential equations modelling prices of commodities _ mathematical methods in the applied sciences _ * 31 * 679 - 694 leach pgl , ohara jg & sinkala w ( 2006 ) symmetry - based solution of a model for a combination of a risky investment and a riskless investment _ journal of mathematical analysis and application _ * 334 * 368 - 381 sinkala w , leach pgl & ohara jg ( 2008 ) optimal system and group - invariant solutions of the cox - ingersoll - ross pricing equation _ mathematical methods in the applied sciences _ * 31 * 679 - 694 ( doi : 10.1002/maa.935 ) ibragimov nh & soh cw ( 1997 ) solution of the cauchy problem for the black - scholes equation using its symmetries , _ proceedings of the international conference on modern group analysis . mars , nordfjordeid , norway_. cimpoiasu r & constantinescu r , ( 2012 ) new symmetries and particular solutions for 2d black - scholes model , _ proceedings of the 7th mathematical physics meeting : summer school and conference on modern mathematical physics , belgrade , serbia _ tamizhmani km , krishnakumar k & leach pgl ( 2014 ) algebraic resolution of equations of the black scholes type with arbitrary time - dependent parameters , _ applied mathematics and computations _ * 247 * 115 - 124 paliathanasis a & tsamparlis m ( 2014 ) the geometric origin of lie point symmetries of the schrodinger and the klein - gordon equations , _ international journal of geometric methods in modern physics _ * 11 * 1450037 dimas s & tsoubelis d ( 2005 ) sym : a new symmetry - finding package for mathematica _ group analysis of differential equations _ ibragimov nh , sophocleous c & damianou pa edd ( university of cyprus , nicosia ) 64 - 70
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we consider the one - factor model of commodities for which the parameters of the model depend upon the stock price or on the time .
for that model we study the existence of group - invariant transformations . when the parameters are constant , the one - factor model is maximally symmetric .
that also holds for the time - dependent problem .
however , in the case for which the parameters depend upon the stock price ( space ) the one - factor model looses the group invariants . for specific functional forms of the parameters the model admits other possible lie algebras . in each case
we determine the conditions which the parameters should satisfy in order for the equation to admit lie point symmetries .
some applications are given and we show which should be the precise relation amongst the parameters of the model in order for the equation to be maximally symmetric . finally we discuss some modifications of the initial conditions in the case of the space - dependent model .
we do that by using geometric techniques .
* keywords : * lie point symmetries ; one - factor model ; prices of commodities * msc 2010 : * 22e60 ; 35q91
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globular clusters ( gcs ) are remarkable stellar systems where a variety of compact objects may form and evolve . numerous millisecond pulsars ( msps ) inhabit gcs as revealed by radio observations ( lyne 1996 ) , e.g. 11 msps in 47 tuc ( robinson et al . 1995 ) . in the x - ray band , the number of bright sources ( @xmath8 kev ) per unit stellar mass is two orders of magnitude larger for gcs than for the remainder of the galaxy ( clark 1975 ) . persistent x - ray sources in gcs divide into two luminosity groups : one with low luminosities ( @xmath9 erg s@xmath4 , @xmath10 kev ) , and the other with high luminosities ( @xmath11 erg s@xmath4 ) ( hertz and grindlay 1983 , hertz and wood 1985 , verbunt et al . the nature of the low luminosity , dim x - ray sources ( dxss ) has been controversial ( for reviews see bailyn 1995 , verbunt et al . 1994 , grindlay 1993b , hut et al . suggestions for their origin include cataclysmic variables , with mass accreting onto a white dwarf ( grindlay 1993a ) , low mass x - ray binaries ( lmxbs ) with accretion at quiescent levels onto a neutron star ( ns ) or black hole ( bh ) ( verbunt , van paradijs and elson 1984 ) , plerionic msp binaries ( tavani 1991 ) or chance superpositions of foreground / background objects ( margon and bolte 1987 ) . we have employed the batse instrument on board the compton gamma - ray observatory to search for transient and persistent hard x - ray emission ( @xmath12 kev ) . hard x - rays may be produced by gc compact objects in a number of ways . in a binary , hard x - ray emission can be powered by mass accretion onto the bh / ns primary , or in pulsars , from shock interactions in the relativistic pulsar winds in the binary . hard x - ray emission may also be produced by magnetospheric emission of isolated msps . transient emission has been observed previously from gcs . ngc 6440 , a gc containing dxss , has exhibited a high luminosity transient episode ( forman , jones and tananbaum 1976 ) . m15 may also have shown a transient event ( pye and mchardy 1983 ) . in the hard x - ray band , sigma has observed transient emission from the globular cluster terzan 2 . recent rosat hri observations indicate that the sigma source is most likely associated with the x - ray burster xb 1724 - 30 ( mereghetti et al . slx 1732 - 304 in the gc terzan 1 has also been detected as a hard x - ray source ( churazov et al . our search does not include terzan 2 , terzan 1 or ngc 6440 due to problems with nearby interfering sources . we have been able to observe m15 however . in section 2 , we describe hard x - ray emission mechanisms in detail . section 3 outlines the batse earth occultation flux measurement technique and the parameters of our search . in sections 4 and 5 we present the results of our search , providing upper limits on transient events and constraints on the recurrence times of transients from dxss . section 6 contains upper limits on persistent emission and constraints on the number of isolated and interacting msps . section 7 is a summary and conclusion . soft x - ray transients ( sxts ) have exhibited emission in excess of 10 kev ( e.g. white , kaluzienski and swank 1984 ) . these hard power law tails along with an anticorrelation of intensity and spectral hardness have been taken as indications of a bh primary ( tanaka 1989 , tanaka and lewin 1995 ) . originally , the discovery of x - ray sources in gcs was interpreted as a possible manifestation of massive accreting bhs ( e.g. , bahcall and ostriker 1975 ) . we know today that _ all _ bright x - ray sources in gcs show x - ray bursts , and therefore most likely contain neutron stars . though it is unlikely that clusters contain very massive black holes , they may harbor a population of stellar mass black holes kulkarni , hut and mcmillan ( 1993 ) estimate that there may be @xmath13 systems in galactic gcs in which a bh of mass @xmath13m@xmath14 has captured a companion . in a few of these systems the companion may have had time to evolve off the main sequence , producing an active x - ray binary . hard x - ray activity from such systems would be observable in our search . lmxbs that contain ns primaries and accrete with luminosities below some critical value also exhibit transient episodes of hard x - ray emission ( e.g. barret and vedrenne 1994 ) and an anticorrelation of spectral hardness and intensity , e.g. : 4u0614 + 091 ( barret and grindlay 1995a ) , 4u1608 - 522 ( mitsuda et al . 1989 ) . & ( kpc ) & ( erg s@xmath4 ) & + aql x-1 & 2.5 & @xmath15 & batse ( @xmath0 kev ) + cen x-4 & 1.2 & @xmath16 & signe 2mp ( @xmath17 kev ) + xb 1724 - 308 ( ter 2 ) & 14.0 & @xmath18 & sigma ( @xmath19 kev ) + ks1731 - 260 & 8.5 & @xmath18 & sigma + 4u1608 - 522 & 3.6 & @xmath20 & batse + a1742 - 294 & 8.5 & @xmath20 & sigma + table [ table : hardx ] lists the observed hard x - ray properties of several transient and persistent lmxbs . all of these sources have been observed as type - i x - ray bursters which indicate ns primaries . the hard x - ray outbursts generally exceed @xmath21 erg s@xmath4 , with emission extending above 100 kev . the time scales for the hard x - ray outbursts are typically @xmath22 days . such events would be easily detectable in our search . additional bursters are being detected as hard x - ray sources in a batse monitoring program ( barret et al . 1996 ) . persistent hard x - ray emission may also be produced by msps , and the fact that gcs contain large numbers of msps makes them ideal systems in which to search for such emission ( chen 1991 , tavani 1993b ) . recent theoretical work has shown that hard x - ray and gamma ray emission may be produced by msps either in isolation ( chen , middleditch and ruderman 1993 ) or in interacting binaries ( tavani 1993a ) . isolated msps could produce hard x - ray emission by the same mechanism operating in young crab - like pulsars . though the magnetic fields are smaller in recycled pulsars , the voltage drop at the light cylinder may be similar due to the smaller rotation periods ( ruderman and cheng 1988 ) . in binary systems , msps may emit hard x - rays by interaction with a companion ( tavani 1991 ) . this is in analogy to psr b1957 + 20 in which the pulsar is evaporating its companion ( e.g. kluzniak et al . 1988 ) . in such systems the pulsar is partly or fully buried in a gaseous envelope originating from its companion . the interaction of the msp wind with the surrounding material produces energetic emission by synchrotron radiation or inverse compton scattering of shock accelerated particles . emission , which can be explained by this model , has recently been observed in the @xmath23 kev range from the binary pulsar psr 1259 - 63 near periastron ( grove et al . 1995 , tavani et al . the batse instrument was designed to provide continuous coverage of the entire sky in the hard x - ray / gamma - ray band from @xmath24 kev @xmath25 2 mev ( harmon et al . 1992 ) , and has an optimal sensitivity in the range @xmath0 kev for a source with a typical photon index of 2 . source fluxes are measured by batse using the earth as an occulting object . as a source rises or sets , steps appears in the detector count rates due to attenuation by the earth s atmosphere . source fluxes are determined by fitting the amplitude of these steps . thus , depending on the geometry of the earth s rising and setting limbs , one can make two source measurements per gro orbit ( approximately 90 min . ) , and from these construct a light curve . fluxes in the resulting light curve , for a location in the sky with no sources , form a distribution about a mean flux of zero . both positive and negative fluxes are produced in the occultation edge fits , all with formal error estimates . the distribution of flux estimates has a width which is larger than expected for a poisson distribution of background rates . this is due to systematic effects , such as contributions in the occultation fits from nearby sources and spurious background fluctuations . the main limitation of the occultation technique is source confusion due to the large angular extent of the earth s occulting limbs . there are several diagnostics that can be employed to identify confused sources . one such check is to compare the measured relative rates in separate detectors to the rates expected given the batse detector responses ( pendleton et al . another technique is to compare the rates from the rising limb to those from the setting limb . for constant flux from the expected source , the rising and setting edges should give the same rates . the precession of the earth s limb can also be used to localize the emission , a technique that has been successfully employed to create images of sky regions for brighter , long duration events ( zhang et al . 1993 ) . we divide our search of 27 gcs into two parts , which we will refer to as search a and search b ( see figure [ fig : proj_full ] ) . search a is a lower sensitivity search of 23 gcs which are at larger distances and/or in regions of the sky prone to source confusion . search a includes gcs at distances @xmath26 kpc , gcs near the galactic center , gcs containing no known dxss , and gcs known to contain bright lmxbs . search b is a higher sensitivity search of four nearby gcs which are not subject to serious source confusion problems . we limit this search to gcs at @xmath26 kpc which also contain dxss near the cluster core . we also exclude clusters within 15@xmath27 of the galactic center , to avoid source confusion in this dense area . the gcs of search b are shown in figure [ fig : proj ] ( circled ) . in what follows , all cluster parameters have been taken from peterson ( 1993 ) . @xmath28 & @xmath29 & coverage & persistent + & ( kpc ) & ( cnt s@xmath4 ) & ( @xmath30 @xmath31s@xmath32 & ( erg s@xmath4 ) & ( % ) & x - ray + & & & [ @xmath33 & [ @xmath34 & & source(s ) + m4 & 2.0 & 3.5 & 2.8 & 0.9 & 80.0 & + ngc6544 & 2.5 & 4.0 & 3.2 & 1.7 & 77.1 & + ngc6656 & 3.0 & 2.6 & 2.1 & 1.6 & 82.4 & dim + ngc6838 & 3.9 & 4.0 & 3.2 & 4.0 & 80.5 & + ngc6539 & 4.0 & 4.4 & 3.5 & 4.7 & 70.9 & + ngc6366 & 4.0 & 6.5 & 5.2 & 6.9 & 74.9 & + ngc6760 & 4.2 & 6.3 & 5.0 & 7.2 & 38.7 & + ngc6254 & 4.3 & 3.3 & 2.6 & 4.0 & 74.3 & + ngc6809 & 4.8 & 4.5 & 3.6 & 6.8 & 72.3 & + ngc3201 & 5.0 & 7.0 & 5.6 & 11.6 & 77.6 & + ngc4372 & 5.2 & 4.0 & 3.2 & 7.2 & 69.8 & + ngc6218 & 5.6 & 3.8 & 3.0 & 7.8 & 74.5 & + ngc4833 & 5.8 & 2.9 & 2.3 & 6.4 & 69.0 & + ngc6626 & 5.9 & 4.0 & 3.2 & 9.2 & 79.4 & dim + ngc6541 & 6.6 & 3.3 & 2.6 & 9.3 & 73.3 & dim + ngc6712 & 6.8 & 4.4 & 3.5 & 13.4 & 72.9 & bright + ngc7099 & 7.4 & 3.0 & 2.4 & 10.9 & 88.4 & dim + ngc6341 & 7.5 & 3.5 & 2.8 & 13.0 & 85.7 & dim + ngc6624 & 8.1 & 3.5 & 2.8 & 15.2 & 84.2 & bright + ngc5272 & 10.1 & 2.4 & 1.9 & 16.0 & 88.0 & dim + m15 & 10.5 & 3.5 & 2.8 & 25.5 & 87.6 & bright + ngc1851 & 12.2 & 4.5 & 3.6 & 44.3 & 88.1 & bright + ngc1904 & 13.0 & 3.1 & 2.5 & 34.9 & 88.1 & dim + we have generated light curves for all the clusters extending from april 1991 to march 1995 ( @xmath11400 days ) . for search a , we have analyzed the rate history of each cluster for events with structure on a time scale of one or more days . generally in each light curve there are several events " apparent ; most of these can be attributed to interference from nearby bright sources . there are , however , a few spikes that can not readily be explained as interfering sources . none of these events are particularly outstanding in terms of amplitude or structure . for search b we have further analyzed such events to determine if they originate from the clusters , as described below . for the 23 clusters in search a , however , such an analysis is impractical . for search a , therefore , we take the largest of the outstanding features to set conservative upper limits to the minimum observable event amplitude ( table [ table : roughlimits ] ) . we initially find count rate upper limits ( @xmath35 in table [ table : roughlimits ] ) . to convert these limits to photon fluxes , we have multiplied by a constant conversion factor determined from other light curves for which we have properly deconvolved the instrument response with an assumed photon index of 2.0 to generate photon flux histories . this conversion is accurate to within approximately @xmath36 . from the photon rate , we calculate a luminosity upper limit by assuming power law spectra with index 2 . the effective time coverage for each cluster is shown in table [ table : roughlimits ] . coverage with batse is less than @xmath37 and varies as a function of sky position due to the geometry of the earth s occulting limbs and due to interfering sources . when target occultations occur within 10 seconds of the occultation of a bright interfering source , no edge fits are produced . this leads to data gaps , as in the case of ngc 6760 , which has highly non - uniform coverage due to the nearby source aql x-1 . note that we obtain a limit of @xmath38 erg s@xmath4 for ngc 5272 ( m3 ) which is known to contain a time variable supersoft x - ray source . note also that the cluster ngc 6440 has been excluded from our search due to nearby interfering sources . for search b , we have produced rate histories over the time period mentioned above and properly deconvolved the detector response to obtain photon light curves . we have searched these light curves for outburst events . candidate events are identified as features of duration one or more days which rise at least @xmath39 above the background flux distribution . with this criterion we have identified 40 possible candidate events . we analyzed each of these isolated events further to determine if they originate from the clusters . we check for nearby interfering sources and compare the relative rates in the detectors . this analysis eliminates all but 13 of the events as candidates . these 13 candidates consist of peak - like structures with durations of 1 to 8 days . the significance of these events , as measured from the photon flux error bars near the peaks , ranges from @xmath7 to @xmath40 . the maximum luminosities in these candidate events would range from @xmath41 erg s@xmath4 , and in one case @xmath42 erg s@xmath4 ( @xmath0 kev ) . these luminosities are similar to the expected transient peak luminosities ( c.f . table [ table : hardx ] ) and deserve further investigation . we further analyze candidate events by studying the correlation between the rising and setting rates within a given gro pointing period . figures [ fig : ngc6752cand ] and [ fig : ngc5139cand ] show example rate histories for candidate events in ngc 6752 and ngc 5139 . for each source , the rise and set rates clearly have different temporal behaviors . in each case nearly all the flux is from the fits in the rising limbs . formally we can calculate the correlation coefficient for the rising vs setting rates . as a test of this technique we have calculated the correlation coefficient for a two day data sample from the crab . the rise and set data for the crab are well correlated with a correlation coefficient of 0.77 and a corresponding probability of @xmath43 that the observed correlation could arise from a random sample . for the two events shown in figures [ fig : ngc6752cand ] and [ fig : ngc5139cand ] the correlation coefficients are -0.07 and -0.10 respectively , with a @xmath44 and @xmath45 probability of being random . we have calculated the correlation coefficient for the other 11 candidate events during the relevant time intervals . for each event the rises and sets either show very weak correlation or , in several cases , strong anticorrelation . the most significant correlation has a @xmath46 probability of being random . the lack of significant positive correlations imply that the events " observed in the light curves do not originate from the target sources but are due to interfering sources located elsewhere . as a check to this interpretation we have analyzed the event in ngc 6752 shown in figure [ fig : ngc6752cand ] in more depth . we have chosen to look further at this particular event because it is somewhat extended in time , lasting for approximately 5 days , and reaches moderate flux levels . we have produced a map in an approximately @xmath47 region centered on ngc 6752 during the time of the candidate event . the map is created by forming a grid of points separated by @xmath48 , and producing occultation histories for each grid point . the resulting rates for each point can be combined together to form a map . the map for the ngc 6752 event shows an emission enhancement well localized at @xmath49 ` northeast ' of the cluster during the first gro pointing period ( tjd @xmath50 ) . it is also clear from the limb geometry that such an event would be visible only on the setting rates , and indeed this is what is observed in the rate history for ngc 6752 . during the second pointing period ( tjd @xmath51 ) , the emission peak moves to @xmath52 ` southwest ' of the cluster , and is well localized only in the detector with the less optimal pointing . all of this leads us to conclude that the enhancement is not associated with the cluster . in the absence of any detected transient events , we are able to set a lower limit to the recurrence times of transient sources in the clusters of search b. for a given time interval in a gc light curve , there will be a minimum luminosity , @xmath53 , at which an event would be visible . @xmath53 is determined by the width of the flux distribution . we step through the cluster light curves and determine the total number of days that events of different @xmath53s would be observable . the result is a distribution of the number of days observable vs @xmath53 ( figure [ fig : cov_all ] ) . the @xmath53 distribution depends on the assumed duration of outburst events and the photon index used in the deconvolution . this dependence , however , is rather weak . the distribution depends more strongly on the time intervals into which the data are integrated . figure [ fig : cov_int ] shows the @xmath53 distribution for different data integration times . the minimum observable luminosities are shifted down at longer integration times due to an increased sensitivity , i.e. a smaller width in the distribution of fluxes . @xmath54 & @xmath55 & @xmath54 & @xmath55 + & ( kpc ) & & ( years ) & ( years ) & ( years ) & ( years ) + & & & @xmath56 erg s@xmath4 & & @xmath57 erg s@xmath4 & + 47 tuc & 4.6 & 5 & 2.0 & 3.4 & 3.5 & 5.8 + ngc 5139 & 5.2 & 2 & 0.6 & 0.4 & 2.9 & 1.9 + ngc 6397 & 2.2 & 5 & 3.6 & 6.0 & 3.6 & 6.0 + ngc 6752 & 4.1 & 3 & 3.2 & 3.2 & 3.6 & 3.6 + total & & & 9.4 & 13.0 & 13.6 & 17.4 + we determine limits to the recurrence times of outbursts from the @xmath53 distribution . for each @xmath53 there is an associated observable time , @xmath54 , for which an event exceeding @xmath53 would be detectable . @xmath54 is essentially the integral of the @xmath53 distribution . @xmath54 is related to the mean event recurrence time , @xmath58 , by @xmath59 , where @xmath60 is the probability of observing no events in time @xmath54 assuming a poisson distribution of events . we calculate a lower limit for @xmath58 using a 5% probability of detection . with multiple x - ray sources in each cluster , an event could originate from any of the sources . the minimum event recurrence time , @xmath58 , can be stated as a minimum recurrence time for sources , if we use the number of dxss in each cluster , @xmath61 . the minimum source recurrence time , @xmath55 , is given by @xmath62 ( table [ table : tau_limits ] ) . we also calculate a total recurrence time for all the sources as a class using the total @xmath54 for all clusters ( which gives the total @xmath58 ) and the total @xmath61 of 15 . figure [ fig : tau ] is plot of @xmath55 vs @xmath53 for the four clusters . recurrence times for sources in the clusters are constrained to lie above the lines . it is clear that with increasing event luminosity , @xmath54 increases and therefore @xmath55 increases , giving a tighter recurrence time constraint at higher luminosities . table [ table : tau_limits ] displays recurrence time lower limits for event luminosities of @xmath56 and @xmath57 erg s@xmath4 . the estimates of @xmath61 are taken from the literature : 47 tuc ( hasinger , johnston and verbunt 1994 ) , ngc 5139 ( @xmath63cen ) ( cool et al . 1995a ) , ngc 6397 ( cool et al . 1993 ) , and ngc 6752 ( grindlay 1993b ) . for 47 tuc we take only the dxss within the core . for the latter three clusters we have assumed , for consistency , that sources within 5 core radii of each cluster center are associated with the gc and others note that changing @xmath61 simply scales the numbers in table [ table : tau_limits ] and the curves in figure [ fig : tau ] . these recurrence time limits can be compared to the several recurrent transients that have been discovered with all - sky x - ray surveys . one such transient is aql x-1 , observed over a 7 year period with the vela 5b satellite . it has an outburst recurrence time measured at 1.2 years ( priedhorsky and terrell 1984 ) . another example is cen x-4 from which hard x - ray outbursts were observed in 1969 and 1979 ( kaluzienski , holt and swank 1980 , bouchacourt et al . 1984 ) , indicating that its recurrence time is approximately 10 years or less , putting it at the outer edge of our limit . both aql x-1 and cen x-4 have exhibited hard x - ray emission during outburst in excess of @xmath15 erg s@xmath4 ( @xmath0 kev ) ( c.f . table [ table : hardx ] ) . a number of other sources have also displayed recurrent outbursts : 1608 - 522 , 1630 - 472 , 1730 - 335 ( van paradijs and verbunt 1984 , chen , shrader and livio 1996 ) . these systems have recurrence times of approximately 0.5 to a few years . our recurrence limits indicate that transient sources with outburst luminosities greater than approximately @xmath15 erg s@xmath4 ( @xmath0 kev ) and recurrence times less than about 2 to 6 years can not constitute the population of dim gc sources in the four sampled clusters . the recurrence time lower limit for all sources is @xmath5 years , if one is willing to consider them as a class . this suggests that the dim sources in these clusters are not quiescent lmxbs of a type similar to aql x-1 or cen x-4 . some caution should be taken with this interpretation . the outburst recurrence times are determined well only for a few systems which may not be typical . also lmxbs in gcs may have systematically different properties than the field binaries in which the hard x - ray emission and recurrent outbursts have been observed . indeed some bright lmxbs in gcs have unusual properties , for example very short orbital periods ( bailyn 1996 ) . one might argue that a substantial number of x - ray outbursts from aql x-1-like lmxbs might be detectable only below 20 kev . if this were the case , such events would be not detectable by batse . however , the recently monitored behavior of aql x-1 shows clearly that batse can usually detect hard x - ray emission at times when the optical counterpart of aql x-1 is excited . these events are most likely associated with major x - ray outbursts ( harmon et al . @xmath64 & @xmath65 & @xmath66 + & ( @xmath30 @xmath31s@xmath4 ) & ( erg s@xmath4 ) & & ( @xmath30 @xmath31s@xmath4 ) & + & [ @xmath67 & [ @xmath68 & & [ @xmath69 & + 47 tuc & 3.3 & 5.8 & 19 & 0.71 & 46 + ngc 5139 & 4.1 & 9.2 & 31 & 0.62 & 66 + ngc 6397 & 4.2 & 1.7 & 6 & 3.10 & 14 + ngc 6752 & 3.6 & 5.0 & 17 & 0.85 & 42 + we have searched for longer time scale persistent emission using the flux histories for the four gcs in search b. previous upper limits have been obtained only for 47 tuc . observations with sigma spanning seven days set a limit at @xmath70 erg s@xmath4 ( @xmath71 kev ) ( barret et al . 1993 ) and an observation by the balloon - borne exite instrument found an upper limit of @xmath72 erg s@xmath4 ( @xmath73 kev ) ( grindlay et al . upper limits have also been obtained for 47 tuc with comptel , osse and egret in their respective bands ( oflaherty et al . 1995 ) . our batse search offers improved sensitivity as a result of the very long integration time . we have detected no emission for the four clusters . the upper limits are summarized in table [ table : persist_limits ] . we have obtained these upper limits by formal error calculations integrating over large time windows . as discussed in section 3 , such a procedure yields error values systematically larger than the actual flux distribution . we have studied this effect for the four gcs of search b and determined a correction factor of 1.6 based on fitted widths of flux distributions . we have used the factor of 1.6 correction to the formal error estimates for the upper limit values of table [ table : persist_limits ] . the clusters of search b were chosen to limit the systematic effects of interfering sources , a selection which permits very sensitive flux limits . we note that the sensitivities may depend on the assumed spectral shape used in deconvolution . we have employed power law spectra with photon index 2.0 . using our upper limits , we constrain the number of isolated high energy emitting msps in the clusters . the luminosity for a typical non - interacting msp can be calculated . chen ( 1991 ) does this using the outer - gap emission model of cheng , ho and ruderman ( 1986 ) and the observed period distribution for msps . the resulting luminosity from a cluster is a constant times the number of isolated msps , assuming isotropic emission . for a cluster with one msp , this would yield a luminosity of approximately @xmath74 erg s@xmath4 . combining this luminosity estimate with our luminosity upper limit , we find the maximum number of isolated msps , @xmath64 , in each of the clusters ( table [ table : persist_limits ] ) . the estimate of @xmath64 depends on the period distribution of msps . in the luminosity calculation , chen ( 1991 ) , uses the period distribution of all the 28 then - known msps , fit as a falling power law . this distribution is poorly constrained due to selection effects and small number statistics and may vary from cluster to cluster . this adds additional uncertainty to our values of @xmath64 . the flux from an interacting binary msp has been estimated by tavani ( 1993b ) . using this model we calculate the estimated flux from each cluster for a single interacting msp system , @xmath65 ( table [ table : persist_limits ] ) . combined with our flux upper limits , the flux estimate yields an upper limit on the number of interacting msps , @xmath66 . the constraints on @xmath66 are weak . this calculation uses an average msp spin - down power of @xmath75 erg s@xmath4 ( taylor et al . msp spin - down powers are in the range @xmath76-@xmath21 erg s@xmath4 but are still not well constrained . we have monitored 27 galactic globular clusters ( gcs ) with batse during a period of approximately four years each . we have detected no distinct hard x - ray outburst episodes and no persistent emission . the lack of detected events has several interesting implications . for the gcs with dim x - ray sources ( dxss ) , we find a lower limit for the outburst recurrence time from dxss of @xmath77 years . this limit excludes the existence of ` aql x-1-like ' objects ( i.e. , persistent x - ray sources subject to major x - ray outbursts with a time scale of @xmath78 year ) , since @xmath55 is comparable to or greater than this outburst recurrence time scale . this suggests that the dxss in these clusters are not lmxbs similar to aql x-1 . the lack of strong outburst events in our search means we also have no evidence of accreting bhs in gcs . if active bh binaries exist in gcs , they would be clearly detectable with hard x - ray luminosities of order of @xmath79 erg s@xmath4 . we have calculated upper limits on persistent hard x - ray emission from gcs . the limiting luminosity is @xmath80 erg s@xmath4 for the closest clusters . the limit for 47 tuc is somewhat more sensitive than previous measurements . a model - dependent but reasonable estimate of the expected hard x - ray magnetospheric emission from isolated msps ( chen 1991 ) implies that the number of isolated msps emitting hard x - rays in 47 tuc is less than 19 . taking the observed 11 msps in 47 tuc , and a beaming factor of 2 for short period pulsars , this upper limit is comparable to actual number of msps in 47 tuc . our results are therefore a constrain the magnetospheric model of msp hard x - ray emission . our upper limits on persistent hard x - ray emission also provide a mild constraint on the number of interacting pulsars in binaries . the observed efficiency ( a few percent ) of conversion of pulsar spindown luminosity into hard x - ray emission in the case of the periastron passage of the be star / pulsar system psr b1259 - 63 ( tavani et al . 1996 , grove et al . 1995 ) together with the observed msp spindown average luminosity of @xmath75 erg s@xmath4 implies a limit to the gc population interacting pulsar binaries . we find limits of @xmath81 interacting msps in the clusters studied . we would like to acknowledge the batse instrument team for their support . we thank jonathan grindlay , didier barret and andrew chen for helpful comments . we thank mark stollberg for assistance with data analysis . this work is columbia astrophysics laboratory contribution number 596 . this work is supported in by nasa grants nag 5 - 2235 and ngt 8 - 52806 . bahcall , j. and ostriker , j. 1975 , nature , 256 , 23 bailyn , c. 1996 , in the origins , evolution , and destinies of binary stars in clusters , eds . milone and j.c . mermilliod , ( asp conf . ser . ) , in press bailyn , c. 1995 , araa , 33 , 133 barret , d. and grindlay , j. 1995 , , 440 , 841 barret , d. et al . 1996 , 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we have monitored a sample of 27 nearby globular clusters in the hard x - ray band ( @xmath0 kev ) for @xmath11400 days using the batse instrument on board the compton gamma - ray observatory .
globular clusters may contain a large number of compact objects ( e.g. , pulsars or x - ray binaries containing neutron stars ) which can produce hard x - ray emission .
our search provides a sensitive ( @xmath150 mcrab ) monitor for hard x - ray transient events on time scales of @xmath2 day and a means for observing persistent hard x - ray emission .
we have discovered no transient events from any of the clusters and no persistent emission .
our observations include a sensitive search of four nearby clusters containing dim x - ray sources : 47 tucanae , ngc 5139 , ngc 6397 , and ngc 6752 .
the non - detection in these clusters implies a lower limit for the recurrence time of transients of 2 to 6 years for events with luminosities @xmath3 erg s@xmath4 ( @xmath0 kev ) and
@xmath5 years if the sources in these clusters are taken collectively .
this suggests that the dim x - ray sources in these clusters are not transients similar to aql x-1 .
we also place upper limits on the persistent emission in the range @xmath6 erg s@xmath4 ( @xmath7 , @xmath0 kev ) for these four clusters . for 47 tuc
the upper limit is more sensitive than previous measurements by a factor of 3 .
we find a model dependent upper limit of 19 isolated millisecond pulsars ( msps ) producing gamma - rays in 47 tuc , compared to the 11 observed radio msps in this cluster .
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multi - output ( mimo ) systems can be designed to provide two types of gains : transmit diversity gain and spatial multiplexing gain@xcite . the full - rate full - diversity space - time block codes ( stbc ) in @xcite can achieve both for 2@xmath12 mimo systems . recently , a fast - decodable full - rate stbc is proposed by s. sezginer , h. sari and e. biglieri @xcite@xcite : @xmath2 \end{split}\ ] ] where @xmath3 with @xmath4 are information symbols , @xmath5 are design coefficients and @xmath6 denotes the complex conjugate . due to its code structure in ( [ bostbc ] ) , @xmath7 has additional zero entries appearing in the upper - triangular matrix after qr decomposition of the equivalent channel matrix , thus making it fast - decodable @xcite@xcite . it is shown in @xcite that the code structure ( [ bostbc ] ) after optimizing for non - vanishing determinant can be rewritten with a single design coefficient . base on this knowledge , in this paper we adopt a simplified version of the code structure ( [ bostbc ] ) by setting @xmath8 to obtain : @xmath9 \end{split}\ ] ] where @xmath10 and @xmath11 is the design coefficient with @xmath12 . our objective is to analytically optimize the design coefficient @xmath13 in ( [ rate2coderot ] ) to enable the full - rate stbc to achieve non - vanishing determinants . in particular , we will consider the influence of different signal constellation topologies , including rectangular quadrature amplitude modulation ( qam ) , amplitude - phase shift keying ( apsk ) and phase shift keying ( psk ) , on the existence of non - vanishing determinants . the rest of this paper is organized as follows . in section [ secdiversity ] , the methods to optimize the design coefficient in ( [ rate2coderot ] ) for both integer - coordinate and non - integer - coordinate signal constellations are described . comparisons of the code in ( [ rate2coderot ] ) with other full - rate codes are shown in section [ secsimulation ] . this paper is concluded in section [ secconclusion ] . in what follows , bold lower case and upper case letters denote vectors and matrices ( sets ) , respectively ; @xmath14 and @xmath15 denote the real and the complex number fields , respectively ; @xmath16 and @xmath17 stand for the real and imaginary parts of a complex element vector and matrix , respectively ; @xmath18^h$ ] denotes the complex conjugate transpose of a matrix ; @xmath19 denotes the determinant of a square matrix . following @xcite , the diversity gain of @xmath20 in ( [ rate2coderot ] ) is denoted as @xmath21 @xcite where the difference matrix @xmath22 , @xmath23 and @xmath24 are stbc matrices based on different information symbols . a full - rank @xmath25 guarantees that @xmath26 is full - rank , and the code @xmath20 in ( [ rate2coderot ] ) will achieve full diversity . when @xmath25 is full rank , the coding gain can be defined as @xmath27\\ & = \underset{\delta \textbf{x}}{\min}\left(\left| det\left(\delta\textbf{x}\right)\right|^2 \right ) \end{split}\ ] ] where @xmath28 and @xmath29 are the difference symbols of @xmath30 . @xmath31 can be split into two parts : @xmath32 where @xmath33 @xmath34\\ & ( 1-j ) . \end{split}\ ] ] note that @xmath35 is dependent on the design coefficient @xmath13 , while @xmath36 is decided by the difference symbols only . since @xmath36 is in the form of @xmath37 multiplied by a real number ( determined by specific values of @xmath38 ) , if @xmath39 is plotted on the @xmath40-axis and @xmath41 is plotted on the @xmath42-axis , @xmath36 lies discretely on the line @xmath43 , as shown in fig . [ smc2x2_d1d21 ] and fig . [ smc2x2_d1d22 ] . since @xmath44=\underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min}(|d_1-d_2|^2)$ ] , @xmath45 is the necessary and sufficient condition for full diversity , and this can be achieved by influencing @xmath35 using the design coefficient @xmath13 . let @xmath46 where @xmath47 and @xmath48 , we have @xmath49+\\ & \left[\left(| \delta s_3|^2+| \delta s_4|^2\right)v-\left(| \delta s_1|^2+|\delta s_2|^2\right)u\right]j . \end{split}\ ] ] the coding gain can be analyzed in two different cases as shown below : in this case , @xmath50(1-j).\ ] ] similar to @xmath36 , @xmath35 lies on the line @xmath43 if @xmath51 is plotted on the @xmath40-axis and @xmath52 is plotted on the @xmath42-axis ( as illustrated in fig [ smc2x2_d1d21 ] ) . the discrete loci of @xmath35 on the line @xmath43 depend not only on @xmath38 , but also on @xmath53 and @xmath54 ( the design coefficients ) . let @xmath55 and @xmath56 , from ( [ d_1 ] ) and ( [ d_2 ] ) we get @xmath57 to achieve full diversity gain ( @xmath45 ) , @xmath53 and @xmath54 must be chosen to achieve @xmath58 . note that @xmath59 $ ] due to @xmath48 . hence , in this case the * case i * coding gain is @xmath60\\ = & \underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min}\left(|d_1-d_2|^2 \right)\\ = & \underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min}\left(2| \tilde d_1-\tilde d_2|^2\right ) . \end{split}\ ] ] , @xmath36 and @xmath61 of * case i * illustrated in real - imaginary axis graph.,width=259 ] , @xmath36 and @xmath61 of * case ii * illustrated in real - imaginary axis graph.,width=288 ] in this case , we have @xmath62 if @xmath63 , then @xmath64 and @xmath35 will never lie on the line @xmath43 , as shown in fig . [ smc2x2_d1d22 ] . since we have earlier shown that @xmath36 always lies on the line @xmath43 , it implies that @xmath45 , hence full diversity is always achieved by * case ii*. as shown in fig . 2 , the euclidean distance between @xmath35 and @xmath36 can be lower bounded by the perpendicular distance between @xmath35 and the line @xmath43 where @xmath36 lies on . [ smc2x2_d1d22 ] . hence , the * case ii * coding gain is lower bounded as @xmath65\\ = & \underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min}\left(|d_1-d_2|^2 \right)\\ \geq&\underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min}\left [ \emph{\emph{d}}^2(\emph{\emph{point } } : d_1,~\emph{\emph{line } } : x+y=0)\right ] \\=&\underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min}\left[(| \delta s_3|^2+| \delta s_4|^2-| \delta s_1|^2-| \delta s_2|^2)^2\right.\\ & \left.(u+v)^2/2\right ] \end{split}\ ] ] where @xmath66 denotes the perpendicular distance from a point to a line . next , we will find the @xmath53 and @xmath54 that satisfy the above full diversity conditions and maximize the coding gain @xmath44 $ ] for the above two cases . to make the optimization process tractable , we will first find the @xmath53 and @xmath54 that maximize the * case i * coding gain ( [ rate2det2 ] ) . then we show that the * case ii * coding gain ( [ rate2det1 ] ) substituted with the @xmath53 and @xmath54 obtained are larger than the maximized ( [ rate2det2 ] ) . hence the * case i * coding gain ( [ rate2det2 ] ) dominates the performance of the code @xmath20 , and the @xmath53 and @xmath54 obtained by maximizing ( [ rate2det2 ] ) will be the global optimum design coefficients . when integer - coordinate signal constellations ( such as rectangular qam ) are applied , the difference symbols also have integer coordinates @xcite@xcite , where the minimum euclidean distance in the signal constellation is fixed at 1 . [ lemma3_conclusion ] _ when integer - coordinate signal constellations are applied , the coding gain ( [ rate2det2 ] ) of the code @xmath20 in * case i * is upper bounded by @xmath67 , and the maximum value can be achieved if and only if @xmath68 . _ in appendix [ proof_theorem1 ] . the following theorem establishes the non - vanishing determinant of @xmath69 with integer - coordinate signal constellations . [ theorem1 ] _ when integer - coordinate signal constellations are applied , the optimum design coefficient @xmath46 to achieve full diversity and maximum non - vanishing coding gain for the code @xmath20 in ( [ rate2coderot ] ) is given by : _ [ rate2det_con11 ] @xmath70 let us first consider * case i*. ( [ rate2det_con11 ] ) can be obtained by combining @xmath68 from lemma [ lemma3_conclusion ] and @xmath48 ( by definition ) . next , for * case ii * , since @xmath71 and integer - coordinate signal constellations are applied , we have @xmath72 . substituting the @xmath53 and @xmath54 in ( [ rate2det_con11 ] ) to ( [ rate2det1 ] ) , the coding gain becomes @xmath73\\ \geq&\underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min } \left[(| \delta s_3|^2+| \delta s_4|^2-| \delta s_1|^2-| \delta s_2|^2)^2\right.\\ & \left.(u+v)^2/2\right]\\ = & 7/8 . \end{split}\ ] ] comparing the * case ii * coding gain expressed in ( [ redet2 ] ) with the * case i * coding gain expressed in lemma [ lemma3_conclusion ] , the * case i * coding gain is lower , hence it is the overall coding gain , and theorem [ theorem1 ] is proved . _ remark _ : the method of proof in this paper , specifically * case i * and * case ii * , are presented in a different way than the proof provided in @xcite . interestingly , however , the optimized design coefficients in both papers are found to be the same . apsk ( amplitude - phase shift keying ) is a high - order modulation scheme commonly used in siso ( single - input single - output ) communications . conventional apsk topology resembles multi - ring psk , or circular qam , as illustrated in fig . [ fig_8apsk ] and fig . [ fig_16apsk ] . compared with rectangular qam , apsk has advantages such as lower constellation peak - to - average - power ratio ( papr ) and robustness against nonlinear distortion in siso communications @xcite . moreover , apsk may lead to larger minimum euclidean distance per unit average power for certain constellation dimension such as 8-apsk @xcite . hence apsk has been adopted by the dvb - s2 standard @xcite . in order for the apsk constellation with arbitrary constellation dimension to achieve non - vanishing coding gain with the code @xmath20 in ( [ rate2coderot ] ) , we may deduce from theorem [ theorem1 ] that : \(1 ) the apsk constellation points should lie on square grids and ring radius @xmath0 ; \(2 ) the design coefficient in ( [ rate2det_con11 ] ) should be adopted for the code @xmath20 . two examples of the proposed apsk constellations are shown in fig . [ fig_8apskm ] and fig . [ fig_16apskm ] . with minimum euclidean distance fixed at 1 , they lead to a non - vanishing coding gain of @xmath67 for the code @xmath20 ( same proof as theorem [ theorem1 ] ) . when non - integer - coordinate signal constellations such as @xmath74-ary phase shift keying ( @xmath74-psk ) are applied , the difference symbols @xmath75 and @xmath76 do not have integer coordinates . this leads to a vanishing determinant for the code @xmath20 in ( [ rate2coderot ] ) even when the minimum euclidean distance is fixed at 1 . the proof is straightforward , hence omitted . although the code @xmath20 in ( [ rate2coderot ] ) with @xmath74-psk constellations has vanishing determinant , the code can still be analytically optimized for a specific constellation size based on the mathematical framework presented earlier . the optimization methodology is described below : step 2 consider * case i * : @xmath77 , whose coding gain expression is shown in ( [ rate2det2 ] ) . given a signal constellation , find out all the values of @xmath78 ; for each value of @xmath78 , find out all the values of @xmath79 . since @xmath80 is a function of @xmath81 , the expression of @xmath82 as a function of @xmath81 can be evaluated . based on these expressions of @xmath82 and @xmath83 $ ] , obtain the maximum value of @xmath84 , and the corresponding @xmath81 . combining the @xmath81 obtained with @xmath48 , we can obtain the corresponding @xmath53 , @xmath54 and the maximized coding gain ; next , consider * case ii * : @xmath85 . substitute the @xmath53 and @xmath54 obtained in step 1 into ( [ rate2det1 ] ) to obtain the * case ii * coding gain . if the * case ii * coding gain is higher than that of * case i * , then the latter is the overall coding gain by definition , and we conclude that the code @xmath20 in ( [ rate2coderot ] ) with design coefficients @xmath46 obtained in * case i * achieves full diversity gain and maximum coding gain . for psk and conventional apsk , this is found to be always true . applying the optimization steps described above to the code @xmath20 in ( [ rate2coderot ] ) with 8-psk constellation , the optimum design coefficients shown in ( [ rate2det_con21 ] ) and the maximum coding gain of @xmath86 are obtained . [ rate2det_con21 ] @xmath87 similarly , the optimized design coefficients and coding gains for the conventional 8-apsk shown in fig . [ fig_8apsk ] and the conventional 16-apsk shown in fig . [ fig_16apsk ] can be found , and are listed in table [ table_apsk ] . they will be used later in fig . [ in the simulations , we assume that the rayleigh fading channel is quasi - static in the sense that the channel coefficients do not change within a codeword , and the channel state information ( csi ) is perfectly known at the receiver . firstly , we show the ml bit error rate ( ber ) performances of the proposed code @xmath20 in ( [ rate2coderot ] ) with the optimized coefficient ( [ rate2det_con11 ] ) , the golden @xcite , pga @xcite , mtd and mcc @xcite codes for 2@xmath12 mimo systems with 4-qam and 16-qam in fig . [ fig_qr21 ] . the ssb code @xcite is equivalent to the proposed code @xmath20 . the results show that the proposed code @xmath20 in ( [ rate2coderot ] ) with design coefficients ( [ rate2det_con11 ] ) has ber performance slightly worse than golden code @xcite , comparable with pga code @xcite , and better than mtd and mcc codes @xcite . on the other hand , as the proposed code structure ( [ rate2coderot ] ) is fast - decodable , it has computational complexity order @xmath88 @xcite , same as the codes in @xcite@xcite . since the computational complexity order of golden code @xcite is @xmath89 , the small performance loss of the proposed code compared to golden code can be viewed as a small penalty to be paid for the complexity reduction . 2 mimo systems with 4-qam and 16-qam constellations.,width=355 ] the ber performance of the code @xmath20 in ( [ rate2coderot ] ) is next compared with other full - rate codes @xcite@xcite@xcite with 8-psk in fig . [ fig_qr22 ] . here the design coefficients in ( [ rate2det_con21 ] ) are adopted for the code * x * in ( [ rate2coderot ] ) , while the optimum design coefficients for the other codes are taken from their respective publications . from the simulation results , we can see that the code * x * in ( [ rate2coderot ] ) achieves a larger ber slope when the snr is high . this is because the other codes , including golden code , were optimized for qam , not psk . 2 mimo systems with 8-psk constellation.,width=367 ] the coding gains of the full - rate stbcs with qam and psk constellations are tabulated in table [ table_codinggain ] with the average power of information symbols normalized to 1 . in all cases , they concur with the ber observations made in fig . [ fig_qr21 ] and [ fig_qr22 ] . 0.8 mm [ cols="^,^,^,^",options="header " , ] comparisons of the properties and performance of the code @xmath20 in ( [ rate2coderot ] ) , when used with the conventional apsk topology versus the proposed apsk topology shown in fig . [ fig_cqam ] , are presented in table [ table_apsk ] and fig . [ 8apsk ] , respectively . note that fig . [ fig_8apsk ] is the best known conventional 8-apsk ( in siso sense ) , while fig . [ fig_16apsk ] is the 16-apsk adopted by the dvb - s2 standard @xcite . in the ber simulations , the corresponding optimum code design coefficients @xmath13s from table [ table_apsk ] are applied . note from table [ table_apsk ] that the proposed apsk does not need to change its design coefficient @xmath13 for different constellation dimensions as it achieves non - vanishing determinant , but this is not true for the conventional apsk . interestingly , table [ table_apsk ] shows that although the proposed apsk shown in fig . [ fig_8apskm ] and fig . [ fig_16apskm ] have smaller minimum euclidean distance ( hence lower papr for the proposed 8-apsk ) , they achieve higher coding gain than the conventional apsk . this is because the coding gains do not depend linearly nor solely on the minimum euclidean distance , as shown in ( [ rate2det2 ] ) and ( [ rate2det1 ] ) . [ 8apsk ] shows that the code @xmath20 in ( [ rate2coderot ] ) with the proposed 8-apsk has much better performance than the conventional 8-apsk , while the proposed 16-apsk has similar performance as the conventional 16-apsk at high snr . [ 8apsk ] also testifies that the code design coefficients shown in table [ table_apsk ] for the conventional 8/16 apsk achieve full diversity . 0.8 mm ' '' '' [ 0pt]apsk & euclidean dis . & coefficient@xmath90 : @xmath13 & [ 0pt]coding gain + conventional & & & + 8-apsk @xcite & [ 0pt]0.9194 & [ 0pt]@xmath91 & [ 0pt]0.0230 + proposed & & & + 8-apsk & [ 0pt]0.8165 & [ 0pt]@xmath92 & [ 0pt]0.2222 + conventional & & & + 16-apsk @xcite & [ 0pt]0.5848 & [ 0pt]@xmath93 & [ 0pt]0.0004 + proposed & & & + 16-apsk & [ 0pt]0.5 & [ 0pt]@xmath92 & [ 0pt]0.03125 + in ( [ rate2coderot ] ) in 2@xmath12 mimo systems with the conventional and proposed 8/16-apsk constellations shown in fig . [ fig_cqam].,width=355 ] [ proof_theorem1 ] let us first introduce lemma [ lemma1_conclusion ] and lemma [ lemma2_conclusion ] which will be used later to prove lemma [ lemma2_conclusion ] and lemma [ lemma3_conclusion ] , respectively . in the following , @xmath97 denotes that @xmath98 divides @xmath99 , and @xmath100 denotes that @xmath98 can not divide @xmath99 . as the value of @xmath109 is equal to 0 or 1 for any integer @xmath99 and @xmath110 , one of the following two equations must hold @xmath111 @xmath112 in other words , @xmath113 , @xmath114 , @xmath115 , @xmath116 must hold at the same time , or @xmath117 , @xmath118 , @xmath119 , @xmath120 must hold at the same time . : since @xmath121 , we have @xmath122 . as the value of @xmath123 is equal to 1 for any odd integer @xmath99 , it follows from @xmath122 that @xmath124 are even integers , i.e. , @xmath125 , @xmath126 , @xmath127 and @xmath128 . then , @xmath129 where @xmath130 , @xmath131 , @xmath132 and @xmath133 are integers . applying the conclusions in case @xmath134 , we have @xmath135 , @xmath136 , @xmath137 , @xmath138 at the same time , or @xmath139 , @xmath140 , @xmath141 , @xmath142 at the same time . since @xmath102 , @xmath144 , @xmath145 , @xmath146 and @xmath147 , we have @xmath148 where @xmath149 , @xmath150 , @xmath151 and @xmath152 are integers . following the conclusions in case @xmath153 , it can be shown that @xmath149 , @xmath150 , @xmath151 and @xmath152 are even integers , i.e. , @xmath154 , @xmath155 , @xmath156 and @xmath157 . then , @xmath158 where @xmath159 , @xmath160 , @xmath161 and @xmath162 are integers . applying the conclusions in case @xmath134 , we have @xmath163 , @xmath164 , @xmath165 , @xmath166 at the same time , or @xmath167 , @xmath168 , @xmath169 , @xmath170 at the same time . hence , @xmath171 , @xmath172 , @xmath173 , @xmath174 at the same time , or @xmath175 , @xmath176 , @xmath177 , @xmath178 at the same time . \2 ) when @xmath194 , @xmath195 at the same time , then @xmath197 and @xmath198 where @xmath199 and @xmath200 are odd integers . thus , @xmath201\emph{\emph { mod } } 2^k=2^{k-1}[(m_1+m_2)\emph{\emph { mod } } 2]=0 $ ] , i.e. , @xmath196 . in * case i * , @xmath203 . for integer - coordinate signal constellations , the difference symbols can be denoted as @xmath204 where @xmath179 are integers and will not be zeros at the same time , i.e. , @xmath205 . hence , we have let @xmath207 be expressed as @xmath208 where @xmath104 is a non - negative integer and @xmath209 is an odd integer . following lemma [ lemma2_conclusion ] , it can be shown that @xmath210 , i.e. , @xmath211 . hence , @xmath212 where @xmath213 is an integer and we have @xmath214 since @xmath213 is an integer decided by @xmath79 and @xmath209 is an odd integer , we have @xmath215 . the equality holds if and only if @xmath216 . since the coding gain of @xmath20 in * case i * is @xmath44=\underset{\delta s_1\emph{\emph { to } } \delta s_4}{\min}(2|\tilde d_1-\tilde d_2|^2)$ ] , it is easy to see that @xmath44\leq1/2 $ ] and the equality holds if and only if @xmath216 . j. paredes , a. b. gershman , and m. g. alkhanari , a new full - rate full - diversity space - time block code with nonvanishing determinants and simplified maximum - likelihood decoding , _ ieee trans . signal process . 2461 - 2469 , june 2008 . v. tarokh , n. seshadri , and a. r. calderbank , space - time codes for high data rate wireless communication : performance criterion and code construction , _ ieee trans . inf . theory _ , 744 - 765 , mar . 1998 . etsi tr 102 376 v1.1.1 ( 2005 - 02 ) , digital video broadcasting ( dvb)-user guidelines for the second generation system for broadcasting , interactive services , news gathering and other broadband satellite applications ( dvb - s2 ) , 2005 .
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full - rate stbc ( space - time block codes ) with non - vanishing determinants achieve the optimal diversity - multiplexing tradeoff but incur high decoding complexity . to permit fast decoding , sezginer , sari and biglieri proposed an stbc structure with special qr decomposition characteristics . in this paper
, we adopt a simplified form of this fast - decodable code structure and present a new way to optimize the code analytically .
we show that the signal constellation topology ( such as qam , apsk , or psk ) has a critical impact on the existence of non - vanishing determinants of the full - rate stbc . in particular
, we show for the first time that , in order for apsk - stbc to achieve non - vanishing determinant , an apsk constellation topology with constellation points lying on square grid and ring radius @xmath0 needs to be used . for signal constellations with vanishing determinants
, we present a methodology to analytically optimize the full - rate stbc at specific constellation dimension .
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ultracold neutral atoms in an optical lattice with single - atom and single - site resolution constitute an ideal physical system to investigate strongly correlated quantum phases @xcite which , in turn , has interesting applications in quantum optics @xcite , quantum simulation @xcite and quantum information processing @xcite , among others . the first approaches towards single - site addressing considered the use of lattices with relatively large site separations @xcite . however , to have access to the regime of strongly correlated systems , typical lattice spacings well below @xmath2 m are needed since the tunneling rate has to be comparable to the on site interactions . in this case , the diffraction limit imposes strong restrictions on the addressability of individual lattice sites . to overcome this limitation different techniques have been investigated . for instance , spatially dependent electric and magnetic fields have been used to induce position dependent energy shifts on the atom @xcite , allowing for site - selective addressability . alternatively , a scanning electron microscopy system to remove atoms from individual sites with a focused electron beam @xcite has been reported . however , in this case , atoms need to be reloaded into the emptied sites after each detection event . more recently , high resolution fluorescence imaging techniques , that make use of an optical system with high numerical aperture , have been implemented to perform _ in situ _ single - atom and single - site imaging for strongly correlated systems @xcite . in this context , a single - site addressing ( ssa ) scheme based on focused laser beams inducing position - dependent energy shifts of hyperfine states has been theoretically @xcite and experimentally reported @xcite . in the experiment , an intense addressing beam is tightly focused by means of a high resolution optical system . this beam produces spatial dependent light shifts bringing the addressed atom into resonance with a chirped microwave pulse and eventually inducing a spin - flip between two different hyperfine levels of the atom . on the other hand , during past years , several proposals based on the interaction of spatially dependent fields , _ e.g. _ , standing waves , with three - level atoms in a @xmath0-type configuration have been considered , not only for single - site addressing in optical lattices but , more generally , for sub - wavelength resolution and localization @xcite . in the first approaches @xcite , a spatially modulated dark state is created by means of either electromagnetically induced transparency ( eit ) or coherent population trapping ( cpt ) @xcite , which allows for a tight localization of the atomic population in one of the ground states , around the position of the nodes of the spatially dependent field . more recently , it has been shown that the resolution achieved with those cpt or eit based techniques can be surpassed using stimulated raman adiabatic passage ( stirap ) @xcite processes , by means of the so - called sub - wavelength localization via adiabatic passage ( slap ) technique @xcite . the slap technique relies on a position - dependent stirap of atoms between the two ground states of a three - level @xmath0 atomic system , and has additional advantages compared with cpt or eit techniques such as ( i ) robustness against parameter variations , ( ii ) coherence of the transfer process , that allows for its implementation also in bose - einstein condensates @xcite , and ( iii ) the absence of photon induced recoil . in this work , we apply the slap technique @xcite to ultracold atoms in an optical lattice , where single - site addressing ( ssa ) requires one to overcome the diffraction limit . in order to address only a single site , we use here stokes and pump pulses with gaussian shaped spatial distributions , with the pump presenting a node centered at the lattice site that we want to address . assuming that all the atoms in the optical lattice are initially in the same internal ground state and applying the standard stirap counterintuitive temporal sequence for the light pulses @xcite , we will demonstrate that it is possible to adiabatically transfer all the atoms , except the one at the node of the pump field , to an auxiliary ground state . we will show that this process is performed with higher efficiency and yields better spatial resolution than the cpt based techniques @xcite . also , we will demonstrate that our addressing technique requires shorter times than in the adiabatic spin - flip technique discussed in ref . @xcite , and that larger addressing resolutions can be achieved using similar focusing of the addressing fields . moreover , our technique has the additional advantage that it can be applied between two degenerated ground - state levels . the paper is organized as follows . in sec . [ sec : physicalmodel ] we introduce the physical system under consideration . in sec . [ sec : singlesiteaddressing ] , we present a protocol to achieve ssa , and we derive analytical expressions for the spatial resolution and addressing efficiency of our technique . in addition , a comparison with cpt based techniques is provided . next , in sec . [ sec : numerical simulations ] , we perform a numerical investigation of the proposed technique for a single - occupancy optical lattice loaded with @xmath1rb atoms by integrating the corresponding atomic density - matrix equations . finally , in sec . [ sec : conclusions ] , we summarize the results and present the conclusions . figure [ f : fig1ab](a ) illustrates the physical system under consideration . it consists of a sample of ultracold neutral atoms loaded into a two - dimensional ( 2d ) square optical lattice with spatial period @xmath3 , placed in the plane ( @xmath4 ) and illuminated by a pump and a stokes laser pulse with rabi frequencies @xmath5 and @xmath6 , respectively , propagating in the @xmath7 direction with a selectable time delay . the spatial profile of the pump pulse has a node coinciding with a particular lattice site , our target site , at which the stokes pulse is also centered . with the spatial profiles of the pulses having revolution symmetry around the propagation axis [ dashed line in fig . [ f : fig1ab](a ) ] , in the following we consider only the transverse spatial dimension @xmath8 , without loss of generality . in our model , we assume the system to be in the mott insulator regime with only one atom per lattice site . each atom is considered to have only three relevant energy levels in a @xmath0-type configuration , defined by the interaction with the light pulses as shown in fig . [ f : fig1ab](b ) . here , @xmath9 ( @xmath10 ) is the spontaneous transition rate from the excited state @xmath11 to the ground state @xmath12 ( @xmath13 ) and @xmath14 ( @xmath15 ) is the detuning of the pump ( stokes ) field . we assume that all atoms are initially in state @xmath12 . and @xmath6 , propagate in the @xmath7 direction and interact with the atoms of a single - occupancy optical lattice located in the ( @xmath4 ) plane . ( b ) scheme of the @xmath0-type three - level atoms , initially in state @xmath12 , that interact with pump and stokes pulses . excited level @xmath11 has spontaneous transition rate @xmath9 ( @xmath10 ) to level @xmath12 ( @xmath13 ) and @xmath14 ( @xmath15 ) is the detuning of the pump ( stokes ) field.,title="fig : " ] our approach to achieve single site addressing is based on the slap technique @xcite , where , depending on their position , the atoms are transferred between two internal ground states by means of the stirap technique @xcite . stirap consists in adiabatically following one of the energy eigenstates of the @xmath0 system , the so - called dark state , which under the two - photon resonance condition , _ i.e. _ , @xmath16 , has the form @xmath17 where @xmath18 . starting with all the population in @xmath12 , it is possible to coherently transfer the atomic population to state @xmath13 changing adiabatically @xmath19 from @xmath20 to @xmath21 by means of a convenient time sequence of the fields . this time sequence corresponds to apply first the stokes pulse and , with a certain temporal overlap , the pump pulse . since the process involves one of the eigenstates of the system , the population transfer is robust under fluctuations of the parameter values if these are adiabatically changed and the system does not evolve near degenerate energy eigenvalues . in the slap technique , the pump field has a spatial structure with nodes yielding state - selective localization at those positions where the adiabatic passage process does not occur , _ i.e. _ , those atoms placed at the nodes of the pump field remain in @xmath12 , while those interacting with both fields , pump and stokes , are transferred to @xmath13 . for our purposes , we use the slap technique with a pump field having a single node at the position of the target site . therefore , at the end of the slap process the population of all atoms illuminated is transferred from @xmath12 to @xmath13 except for the one at the node of the pump field . the spatial and temporal profiles for pump and stokes rabi frequencies are given by @xmath22 where @xmath23 and @xmath24 are the peak rabi frequencies , @xmath25 and @xmath26 are the centers of the temporal gaussian profiles , @xmath27 and @xmath28 are the spatial widths of the node in the pump and of the stokes field , respectively , and @xmath29 is the temporal width . there exist several methods to create the required pump intensity profile with a central node : e.g. ( i ) re - imaging of a gaussian beam with a dark central spot created by a circular absorption mask , using ( ii ) a laguerre - gaussian laser beam @xcite or ( iii ) a `` bottle beam '' created by the interferometric overlap of two gaussian beams with differing waists @xcite , or ( iv ) a flexible intensity pattern generated by spatial light modulators and subsequent imaging @xcite . in our model , we assume that the spatial wavefunctions of the individual atoms placed at the different sites , centered at @xmath30 ( being @xmath31 the site index ) , correspond to the ground state of the trapping potential , which in first approximation can be considered harmonic . therefore , the full atomic distribution in the lattice is given , initially , by @xmath32},\end{aligned}\ ] ] where @xmath33 is the width of the initial atomic distribution at an individual site , @xmath34 is the mass of the trapped atom and @xmath35 is the harmonic trapping frequency . we assume that the addressed site is @xmath36 and their nearest neighbors @xmath37 are at a distance @xmath38 where @xmath39 is the wavelength of the fields that create the optical lattice . in order to characterize our single - site addressing technique we consider that , once the slap technique has been applied , the final atomic population distribution in @xmath12 , @xmath40 , is given by @xmath41 where @xmath42 is the probability distribution that an atom remains in state @xmath12 after the slap process . using the slap technique , the addressing resolution that one can obtain is related to the global adiabaticity condition @xcite at each spatial position @xmath8 , @xmath43 ^ 2\geq\left(\frac{a}{t}\right)^2 , \notag \\\end{aligned}\ ] ] where @xmath44 and @xmath45 is a dimensionless constant that , for optimal gaussian temporal profiles and overlapping times , takes values around 10 @xcite . in eq . ( [ eq : adiabatic condition ] ) , the equality gives a spatial threshold @xmath46 above which the adiabaticity condition is fulfilled . assuming that the full width at half maximum ( fwhm ) of @xmath42 is @xmath47 and expanding eq . ( [ eq : adiabatic condition ] ) up to first order in @xmath8 one obtains @xmath48 where @xmath49 and @xmath50 . equation ( [ eq : fwhm prob slap ] ) gives the width of the addressing region , and it tends to zero as @xmath51 increases . moreover , since @xmath52 must be real valued , we find that the inequality @xmath53 must be fulfilled . in this paper we will consider that @xmath24 is fixed , so @xmath54 can be varied through @xmath55 , and @xmath28 . two conditions should be satisfied for our ssa technique to work . first , the population of the atom in the addressed site must remain in state @xmath12 after the action of the fields and , second , the rest of the atoms of the lattice have to be transferred to level @xmath13 . therefore , taking into account the overlap between @xmath42 and @xmath56 in eq . ( [ eq : final distribution ] ) , it is clear that the fwhm of the probability distribution @xmath42 should satisfy @xmath57 where @xmath58 , and @xmath59 is the position of the nearest - neighboring site . using eq . ( [ eq : fwhm prob slap ] ) , it is easy to see that these conditions fix the range for @xmath60 to obtain ssa using the slap technique : @xmath61 where @xmath62 ^ 2+r'}},\end{aligned}\ ] ] with @xmath63 and @xmath64 . note that the upper limit for eq . ( [ eq : ssa range ] ) is more restrictive than eq . ( [ eq : math limit ] ) . in order to have a quantitative description of the ssa performance , let us introduce the ssa efficiency as @xmath65 where @xmath66 corresponds to the probability of finding the atom at the addressed site @xmath67 in state @xmath12 , while @xmath68 corresponds to the probability that the atom in the neighbor site @xmath69 has been transferred to a different internal state . we define @xmath70 with @xmath71 and @xmath72 , whereas @xmath40 and @xmath56 have been defined in eqs . ( [ eq : final distribution ] ) and ( [ eq : wavefunction ] ) , respectively . using eqs . ( [ eq : wavefunction ] ) and ( [ eq : final distribution ] ) , the explicit forms for eqs . ( [ eq : partialefficiency ] ) are @xmath73}. \label{eq : explicitefficiencyx1}\end{aligned}\ ] ] from these expressions , it can be seen that , for @xmath74 , the limits given by eq . ( [ eq : deltax ssa range ] ) , _ i.e. _ , @xmath75 and @xmath76 , correspond to ssa efficiencies of @xmath770.70 and @xmath770.94 , respectively . an alternative technique to perform atomic localization based on spatial dependent dark states is the coherent population trapping ( cpt ) technique @xcite . in the cpt @xcite technique the dark state is populated after several cycles of coherent excitation followed by spontaneous emission from @xmath11 to the ground states . note that , while cpt relies on spontaneous emission , the slap technique is fully coherent . moreover , the latter provides higher resolution , as shown in ref . @xcite , and does not suffer from recoil since the localized atoms have not interacted with light . in what follows , we compare the range of parameters necessary to perform ssa considering both slap and cpt techniques . note that we focus our comparative analysis in the cpt technique , although similar results are obtained considering the method proposed in ref . @xcite , where the spatial dependent dark state , @xmath78 , is created via the stirap technique by switching off the fields before completing the transfer process . in order to compare both techniques , we define the final population distribution in @xmath12 using cpt as @xmath79 in an analogous way as it has been done for the slap technique in eq . ( [ eq : final distribution ] ) . then the fwhm of the corresponding probability function @xmath80 is obtained by imposing that @xmath81 and @xmath82 in eqs . ( [ eq : fieldp ] ) and ( [ eq : fields ] ) : @xmath83 fixing the desired @xmath84 and using eqs . ( [ eq : fwhm prob slap ] ) and ( [ eq : fwhm prob cpt ] ) for slap and cpt , respectively , the constraints for the relevant parameters for each technique can be obtained . for simplicity , we consider that the stokes pulse parameters @xmath28 and @xmath24 are fixed , and only the node width @xmath27 and pump peak rabi frequency @xmath23 can be varied . note that , for the slap case , we have to fix also @xmath45 and @xmath85 . needed to perform ssa as a function of @xmath27 , using slap ( solid line ) and cpt ( dashed line ) techniques . the addressing probability distribution widths are taken as @xmath86 in both cases . ( b ) @xmath87 ( solid line ) and @xmath88 ( dashed line ) as a function of @xmath51 . the fwhm of the atomic distribution , @xmath89 corresponds to the horizontal dotted line and we have taken @xmath90 . the parameters used in both ( a ) and ( b ) are @xmath91 , @xmath92 , and @xmath93 . , title="fig : " ] taking @xmath86 , half of the site separation , the required values for @xmath51 and @xmath27 are plotted in fig . [ f : fig2](a ) . to simultaneously illuminate a large number of sites , we use a large stokes beam waist of @xmath91 , where @xmath94 is the wavelength of both pump and stokes fields @xcite . the solid ( dashed ) line corresponds to the slap ( cpt ) case with the parameter values @xmath92 and @xmath93 @xcite . as @xmath27 decreases , in both slap and cpt cases , lower values of @xmath51 are needed to reach the fixed resolution @xmath84 , since the narrower the node of the pump field , the narrower the probability distribution of atoms remaining in @xmath12 . in addition , for any given width of the node of the pump field @xmath27 , the required values of @xmath51 are lower in the slap case than in the cpt case . it is important to realize that eqs . ( [ eq : fwhm prob slap ] ) and ( [ eq : fwhm prob cpt ] ) show the possibility to obtain , for certain parameter values , widths of the probability distribution , @xmath87 or @xmath88 , smaller than @xmath95 . in particular , using the slap technique this can be achieved with moderate @xmath51 values . this is shown in fig . [ f : fig2](b ) , where @xmath87 ( solid line ) and @xmath88 ( dashed line ) are represented as a function of @xmath51 for @xmath96 and the rest of the parameters as in fig . [ f : fig2](a ) . the fwhm of the atomic distribution , @xmath89 , is depicted with a horizontal dotted line to indicate the values where @xmath97 . as we stated in the discussion of eqs . ( [ eq : explicitefficiencyx0 ] ) and ( [ eq : explicitefficiencyx1 ] ) , this limit corresponds to a ssa efficiency of @xmath98 . this regime of parameters is interesting because it shows that the slap technique could be used for applications in site - selective imaging with a resolution down to the width of the atomic distribution at each site . in this section , by numerically integrating the corresponding density - matrix equations , we study the implementation of the slap - based ssa technique for @xmath0-type three - level @xmath1rb atoms in a single - occupancy optical lattice . numerical calculations using the cpt technique are also presented for comparison . the wavelength of the lasers that create the optical lattice , red detuned with respect to the @xmath99 line of @xmath1rb , is @xmath100 nm . the potential depth of the optical lattice is chosen as @xmath101 , where @xmath102 is the recoil energy . this corresponds to a harmonic trapping frequency of @xmath103 khz @xcite . therefore , the fwhm of the atom distribution at each site due to the confining potential is @xmath104 nm . pump and stokes fields with @xmath105 nm are coupled to @xmath106 and @xmath107 , respectively , where @xmath108 , @xmath109 and @xmath110 are hyperfine energy levels of the @xmath99 line of @xmath1rb . the excited state @xmath11 has a spontaneous transition rate @xmath111 mhz ( @xmath112 mhz ) to state @xmath12 ( @xmath13 ) , and we assume no spin decoherence during the interaction time . we consider equal temporal pulse widths of @xmath113s with a temporal delay @xmath114 , in such a way that the total ssa process time is @xmath115 . the stokes pulse has a maximum rabi frequency @xmath116 mhz , while the maximum rabi frequency of the pump is varied through the parameter @xmath51 , since @xmath117 . concerning the spatial profiles of the fields , we assume a wide stokes profile , @xmath118 , and a narrow node for the pump , @xmath90 . in state @xmath12 ( a ) , the probability to transfer it from @xmath12 to another state ( b ) , and the efficiency @xmath119 as a function of @xmath51 ( c ) , for the ssa with slap ( circles ) and cpt ( crosses ) techniques . analytical curves for slap ( solid line ) and cpt ( dashed lines ) , computed from eqs . ( [ eq : totalefficiency ] ) and ( [ eq : partialefficiency ] ) , are added in ( c ) for comparison ( see text for the rest of the parameters).,title="fig : " ] as it has been discussed in the previous section , to properly perform ssa , the population of all the atoms in the lattice , except the one in the addressed site , must be transferred from @xmath12 to @xmath13 with high probability . in what follows , those requirements for the realization of the ssa are studied by numerically evaluating the ssa efficiency . the signatures of ssa are shown in fig . [ f : efficienciesnum ] , where the numerically evaluated efficiency and probabilities defined in eqs . ( [ eq : totalefficiency ] ) and ( [ eq : partialefficiency ] ) are plotted as a function of @xmath51 for both the slap ( circles ) and cpt ( crosses ) techniques . figures [ f : efficienciesnum](a ) and [ f : efficienciesnum](b ) show the probabilities @xmath120 and @xmath121 , respectively . for large values of @xmath51 the probability of finding the atom at the addressed site in @xmath12 is higher with the cpt than with the slap technique [ see fig . [ f : efficienciesnum](a ) ] , while for small values of @xmath51 the probability of removing the atom from the neighboring site [ see fig . [ f : efficienciesnum](b ) ] is higher in the slap case . this is explained by the fact that @xmath122 for a given value of @xmath51 , as shown in fig . [ f : fig2](b ) . the ssa efficiency results obtained by the numerical evaluation of eqs . ( [ eq : totalefficiency ] ) and ( [ eq : partialefficiency ] ) are shown in fig . [ f : efficienciesnum](c ) , together with the corresponding analytical curves ( solid and dashed lines ) obtained using eqs . ( [ eq : explicitefficiencyx0 ] ) and ( [ eq : explicitefficiencyx1 ] ) , added for comparison . a good agreement is found between numerical and analytical results . from fig . [ f : efficienciesnum](c ) it is clear that the slap technique is more efficient for lower values of the intensity ratio of the addressing fields ( @xmath123 ) than the cpt technique . certainly , this is advantageous for the experimental implementation with limited laser power available . . ( b ) final population distribution remaining in @xmath12 using slap ( circles ) and cpt ( crosses ) single - site addressing techniques for @xmath124 . the initial atomic distribution in @xmath12 , @xmath56 , is shown as a solid line ( see text for the parameters values ) . , title="fig : " ] an example of the final population distribution after performing ssa with the slap ( circles ) and the cpt ( crosses ) techniques is plotted in fig . [ f : rubidiumpeaks](b ) for the particular case of @xmath124 and with the rest of the parameters as in fig . [ f : efficienciesnum ] . the initial population distribution in @xmath12 , @xmath56 , at the addressed site ( @xmath125 ) , and the two next neighbors @xmath126 nm is shown as a solid line . note that , in the slap case ( circles ) , the population of state @xmath12 around @xmath67 remains almost the same after the addressing process , while in the first neighbor sites it is practically zero . on the other hand , for the cpt case ( crosses ) , the population in the addressed site remains also nearly unchanged , but it exhibits a significant amount of population in the neighbor sites . this is in full agreement with the discussion following figs . [ f : efficienciesnum](a ) and [ f : efficienciesnum](b ) . for this example , the total efficiencies found are @xmath127 and @xmath128 according to the corresponding values shown in fig . [ f : efficienciesnum](c ) . in addition , as shown in fig . [ f : rubidiumpeaks](a ) , the width of the pump node required to perform the ssa method is much larger than the addressed region @xmath87 . in particular , for the case shown in fig . [ f : rubidiumpeaks ] , @xmath129 and @xmath130 , thus obtaining addressing resolution beyond the diffraction limit . finally , we carry out a comparison between our proposal and the experiment reported in ref . @xcite , where a focused laser beam induces position - dependent light shifts , allowing one to perform a spin flip by means of a resonant microwave pulse at the addressed site . since the microwave field involved in the experiment has a rabi frequency of khz , the total spin - flip time is in the order of ms . in contrast , as our proposal makes use of only optical fields , the addressing time is three orders of magnitude below ( @xmath131s ) . specifically , to achieve similar values of the addressing resolution in both techniques , @xmath132 nm , we have obtained an addressing time of @xmath7740@xmath133s . this decrease of the total addressing time needed implies a reduction of the effects caused by spontaneous scattering of photons , which are a limitation for the light shifts - based proposals @xcite . these effects could be strongly reduced in our case . also , we have compared the resolution obtained in both techniques . in ref . @xcite , they use an addressing beam with an intensity fwhm of approximately 600 nm , and obtain a spin - flip probability distribution with fwhm=330 nm . in our technique , using @xmath134 and a width of the pump node @xmath135 nm , which corresponds to the width of their addressing beam , a very similar fwhm of the addressing probability distribution is obtained : @xmath136 nm . note that this value can be reduced by increasing the ratio between the pump and the stokes intensities , _ e.g. _ , @xmath137 implies @xmath138 nm and for @xmath139 we obtain @xmath140 nm . in this work we have discussed a proposal to perform single - site addressing ( ssa ) in an optical lattice by using the slap technique . with respect to other dark state based techniques such as cpt , this method is fully coherent , robust against variations of the parameter values , and we have found that it yields higher efficiencies for smaller values of the intensity ratio between the pump and the stokes fields . moreover , the addressed atom does not interact with the fields , minimizing all possible decoherent processes . on the other hand , with respect to the recent experiment on ssa using adiabatic spin flips @xcite , the present proposal allows one to use two degenerate ground levels , takes shorter times to perform the addressing process , and provides similar or even larger addressing resolutions for similar focusing of the addressing fields . the proposed method provides an achievable addressing resolution that can be pushed well below the diffraction limit of the addressing light field and of the optical setup used for addressing or detection of atoms at closely spaced lattice sites . this relaxes the requirements on the optical setup or extends the achievable spatial resolution to lattice spacings smaller than accessible to date . through analytical considerations , we have derived the range of parameters for which ssa is properly achieved . moreover , we have obtained expressions to estimate the resolution and the efficiency of the slap - based addressing method , and we have compared them with the analogous expressions obtained using the cpt technique . next , by integrating the density - matrix equations with realistic parameter values for state - of - the - art optical lattices loaded with @xmath1rb atoms , we have checked the validity of the analytical approach . we thank albert benseny for fruitful discussions and comments . we acknowledge financial support from the spanish ministry of science and innovation under contracts no . fis2008 - 02425 , no . fis2011 - 23719 , and no . csd2006 - 00019 ( consolider project quantum optical information technologies ) , from the catalan government under contract no . sgr2009 - 00347 , from the german research foundation dfg ( contract no . bi 647/3 - 1 ) , and from the german academic exchange service daad ( contract no . 0804149 ) . 99 m. endres , m. cheneau , t. fukuhara , c. weitenberg , p. schau , c. gross , l. mazza , m. c. banuls , l. pollet , i. bloch and s. kuh , science * 334 * , 200 ( 2011 ) . n. s. ginsberg , s. r. garner , and l. v. hau , nature * 445 * , 623 ( 2007 ) . i. bloch , j. dalibard , and w. zwerger , rev . phys . * 80 * , 885 ( 2008 ) . g. k. brennen , c. m. caves , p. s. jessen , and i. h. deutsch , phys . lett . * 82 * , 1060 ( 1999 ) . r. raussendorf and h. j. briegel , phys . rev . lett . * 86 * , 5188 ( 2001 ) . d. jaksch and p. zoller , ann . 315 , * 52 * ( 2005 ) . r. scheunemann , f. s. cataliotti , t. w. hnsch , and m. weitz , phys . a * 62 * , 051801(r ) ( 2000 ) ; 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we propose a single - site addressing implementation based on the sub - wavelength localization via adiabatic passage ( slap ) technique .
we consider a sample of ultracold neutral atoms loaded into a two - dimensional optical lattice with one atom per site .
each atom is modeled by a three - level @xmath0 system in interaction with a pump and a stokes laser pulse . using a pump field with a node in its spatial profile ,
the atoms at all sites are transferred from one ground state of the system to the other via stimulated raman adiabatic passage , except the one at the position of the node that remains in the initial ground state .
this technique allows for the preparation , manipulation , and detection of atoms with a spatial resolution better than the diffraction limit , which either relaxes the requirements on the optical setup used or extends the achievable spatial resolution to lattice spacings smaller than accessible to date . in comparison to techniques based on coherent population trapping
, slap gives a higher addressing resolution and has additional advantages such as robustness against parameter variations , coherence of the transfer process , and the absence of photon induced recoil .
additionally , the advantages of our proposal with respect to adiabatic spin - flip techniques are highlighted .
analytic expressions for the achievable addressing resolution and efficiency are derived and compared to numerical simulations for @xmath1rb atoms in state - of - the - art optical lattices .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
non - equilibrium growth and aging phenomena are of great interest due to their wide applications across various scientific fields of study , including many body statistical systems , condensed matter systems , biological systems and so on @xcite-@xcite . they are complex physical systems , and details of microscopic dynamics are widely unknown . thus it is best to describe these systems with a small number of variables , their underlying symmetries and corresponding universality classes , which have been focus of nonequilibrium critical phenomena . one particular interesting class is described by the kardar - parisi - zhang ( kpz ) equation @xcite@xcite . recently , this class is realized in a clean experimental setup @xcite@xcite , and their exponents for one spatial dimension @xmath11 is confirmed : the roughness @xmath12 , the growth @xmath13 and the dynamical @xmath14 exponents . along with the experimental developments , there have been also theoretical developments in the context of aging . the kpz class reveals also simple aging in the two - time response functions @xcite@xcite . in these works , it is shown that the autoresponse function of the class is well described by the logarithmic ( log ) and logarithmic squared ( log@xmath15 ) extensions of the scaling function with local scale invariance for @xmath11 . in a recent paper @xcite based on @xcite@xcite , we have considered log extensions of the two - time correlation and response functions of the scalar operators with the conformal schrdinger and aging symmetries for the spatial dimension @xmath16 and the dynamical exponent @xmath17 , in the context of gauge / gravity duality @xcite@xcite . the power - law and log parts are determined by the scaling dimensions of the dual field theory operators , the eigenvalue of the internal coordinate and the aging parameter , which are explained below . interestingly , our two - time response functions show several qualitatively different behaviors : growth , aging ( power - law decaying ) or both behaviors for the entire range of our scaling time , depending on the parameters in our theory @xcite . we further have made connections to the phenomenological field theory model @xcite in detail . the two - time response functions and their log corrections of our holographic model @xcite are completely fixed by a few parameters and are valid for @xmath18 , while the field theory model @xcite has log@xmath15 extensions and is valid for @xmath19 . closing the gap between these two models from the holographic side is one of the main motivation of this work . in this paper , we generalize our analysis @xcite in two different directions : ( 1 ) by applying to general dynamical exponent @xmath1 ( not conformal ) as well as to any number of spatial dimensions @xmath0 and ( 2 ) by including log@xmath15 corrections in two - time response functions . in [ sec : ntlft ] , we first analytically compute the correlation and response functions for general @xmath1 and @xmath0 along with their log and log@xmath15 extensions . then we add the aging generalizations of our non - conformal results in [ sec : aging ] . we try to make contact with kpz class in [ sec : kpz ] , and conclude in [ sec : conclusion ] . logarithmic conformal field theory(lcft ) is a conformal field theory(cft ) which contains correlation functions with logarithmic divergences . they typically appear when two primary operators with the same conformal dimensions are indecomposable and form a jordan cell . the natural candidates for the bulk fields , in the holographic dual descriptions of lcft , of the pair of two primary operators forming jordan cell are given by a pair of fields with the same spin and a special coupling . after integrating out one of them , it becomes the model with higher derivative terms . the ads dual construction of the lcft was first considered in @xcite@xcite@xcite using a higher derivative scalar field on ads background . recently , higher derivative gravity models on ads geometry with dual lcft have got much attention , starting in three dimensional gravity models @xcite-@xcite . in four and higher dimensional ads geometry , the gravity models with curvature - squared terms typically contain massless and ghostlike massive spin two fields . when the couplings of the curvature - squared terms are tuned , the massless and massive modes become degenerate and turn into the massless and logarithmic modes @xcite . this , so called , critical gravity has the boundary dual lcft which contains stress - energy tensor operator and its logarithmic counterpart . more recently , studies on generalizations of lcft in the context of ads / cft correspondence @xcite@xcite , in particular the correlation functions of a pair of scalar operators , have been made in two different directions . one is on the non - relativistic lcft . in @xcite the dual lcft to the scalar field in the lifshitz background has been investigated . the study on the dual lcft to the scalar field in the schrdinger and aging background was made in @xcite . the other is on the lcft with @xmath20 divergences . correlation functions with log@xmath15 corrections have been investigated in several works . in the context of the gravity modes of tricritical point , they are interpreted as rank-3 logarithmic conformal field theories ( lcft ) with log@xmath15 boundary conditions @xcite . explicit action for the rank-3 lcft is considered in @xcite . see also a recent review on these developments in @xcite . in this section we would like to accomplish two different things motivated by these developments along with those explained in the introduction . first , we compute correlation and response functions for ads in light - cone ( alcf ) and schrdinger type gravity theories , which are dual to some non - relativistic field theories with galilean invariance for general dynamical exponent @xmath1 and @xmath0 . second , we generalize our correlation and response functions with log@xmath15 as well as log contributions for @xmath1 and @xmath0 . the log correction has been investigated in @xcite in the context of schrdinger geometry and aging geometry for @xmath16 and @xmath17 as mentioned in the introduction . we first consider the alcf @xcite@xcite . due to several technical differences , we present our computations in some detail , building up correlation functions [ sec : alcfcorr ] , their log extensions [ sec : alcfcorrlog ] and log@xmath15 extension [ sec : alcfcorrlog2 ] . for schrdinger backgrounds @xcite@xcite , we comment crucial differences in [ sec : schrlog ] . and then we present the correlation functions for @xmath17 in [ sec : ztwo ] and for @xmath21 in [ sec : zthreehalf ] . we summarize our results in [ sec : ntlftsummary ] . let us turn to the ads in light - cone ( alcf ) with galilean symmetry studied in @xcite@xcite@xcite . the case for general @xmath22 is also considered in @xcite for zero temperature and in @xcite for finite temperature . the metric is given by @xmath23 which is invariant under the space - time translations @xmath24 , galilean boost @xmath25 , @xmath26 scale transformation @xmath27 , @xmath28 and translation along the @xmath29 coordinate , which represents the dual particle number or rest mass . , @xmath30 it turns out that this does not provide a closed algebra with other symmetry generators for @xmath31 . ] the geometry satisfies vacuum einstein equations with a negative cosmological constant . the finite temperature generalizations of the alcf for @xmath17 and @xmath16 have been considered in @xcite@xcite@xcite . we compute correlation functions of the geometry ( [ alcfmetric ] ) by coupling a probe scalar . @xmath32 where @xmath33 is a coupling constant , and @xmath34 . we use @xmath35 for our boundary cutoff . the field equation of @xmath36 for general @xmath1 and @xmath0 is @xmath37 note that we treat @xmath38 coordinate special and replace all the @xmath39 as @xmath40 . this is in accord with the fact that the coordinate @xmath38 plays a distinguished role in the geometric realizations of schrdinger and galilean symmetries @xcite@xcite@xcite . the general solution is given by @xmath41 where @xmath42 , @xmath43 represent bessel functions . we choose @xmath44 over @xmath45 due to its well defined properties deep in the bulk . we follow @xcite@xcite to compute correlation functions by introducing a cutoff @xmath35 near the boundary and normalizing @xmath46 , which fixes @xmath47 . we compute an on - shell action to find @xmath48 & = \int d^{d+1 } x \frac{l^{d+3}}{u^{d+3 } } ~\phi^ * ( u , t , \vec x ) ~\frac{u^2}{l^2}\partial_u \phi ( u , t , \vec x ) \big |_{u_b } \;. \end{aligned}\ ] ] using @xmath49 , @xmath50 , @xmath51 and @xmath52 the onshell action can be rewritten as @xmath53 \int [ x , \vec k ' , \vec k ] \quad \phi_0^ * ( k ' ) \mathcal f_1 ( u , k',k ) \phi_0 ( k ) \big |_{u_b } \ ; , \label{wholeeq}\end{aligned}\ ] ] where @xmath54 \int [ x , \vec k ' , \vec k ] = \int d t ~\frac{d \omega'}{2 \pi } \frac{d \omega}{2 \pi } e^{-i ( \omega ' -\omega ) t } \int d^d x \int \frac{d^d k'}{(2\pi)^d } \frac{d^d k}{(2\pi)^d } e^{i(\vec k ' - \vec k ) \cdot \vec{x } } $ ] . @xmath55 is given by @xmath56 this function appears again when we construct the log and log@xmath15 extensions below . for general @xmath0 we find @xmath57 one can evaluate the @xmath58-dependent part of @xmath55 at the boundary , @xmath59 , by expanding it for small @xmath35 . we obtain the following non - trivial contribution @xmath60 note that the function @xmath55 is only a function of @xmath61 when it is evaluated at the boundary . by inverse fourier transform of ( [ f1function ] ) for an imaginary parameter @xmath62 , we get the following coordinate space correlation functions for a dual field theory operator @xmath36 @xmath63 these are our correlation functions evaluated for general dynamical exponent @xmath1 and for the number of spatial dimensions @xmath0 , which are direct generalizations of a previous result for @xmath16 and @xmath17 @xcite . note that the result is valid for the field theories with galilean boost without conformal symmetry . in particular , the parameter @xmath2 carries scaling dimensions @xmath64 = z-2 $ ] , and the exponent is actually dimensionless . this is consistent with the scaling properties written in ( [ alcfscaling ] ) . the result ( [ f1correlator ] ) is independent of @xmath1 , while depending on the number of dimensions @xmath0 . this is expected because the alcf metric ( [ alcfmetric ] ) is independent of @xmath1 , which is a special feature for the alcf . this is not true for schrdinger background as we see below . motivated by the recent interests on lcft from the holographic point of view @xcite@xcite , we consider two scalar fields @xmath36 and @xmath65 in the background ( [ alcfmetric ] ) @xmath66 where @xmath67 represents a cutoff near the boundary . we take @xmath68 and @xmath69 . the field equations for @xmath36 and @xmath65 of the action ( [ schractionlcft ] ) become @xmath70 where @xmath71 \ ; , \end{aligned}\ ] ] which is a differential operator for alcf . following @xcite@xcite , we construct bulk to boundary green s functions @xmath72 @xmath73\ ; , \nonumber \\ & \tilde \phi ( u , x ) = \int \frac{d^{d+1 } k'}{(2\pi ) ^{d+1 } } e^{-i k ' \cdot x } \left [ g_{21 } ( u , k ' ) j(k')+ g_{22 } ( u , k ' ) \tilde j(k')\right]\;. \end{aligned}\ ] ] we have @xmath74 , which follows from the structure of the equations of motion given in ( [ diffeqschr ] ) and the action ( [ schractionlcft ] ) . the green s functions satisfy @xmath75 where @xmath76 and @xmath77 . solutions of @xmath78 and @xmath79 are given by @xmath80 where @xmath81 . the normalization constants @xmath82 can be determined by requiring that @xmath83 @xcite@xcite . there exists another green s function @xmath84 due to a coupling between @xmath36 and @xmath65 in the action ( [ schractionlcft ] ) , which satisfies @xmath85 to evaluate @xmath84 , we use the same methods used in @xcite@xcite . using @xmath86 = 2 \nu \ ; , \end{aligned}\ ] ] and the fact that @xmath87 , we get @xmath88 thus we have an explicit form . @xmath89 after plugging the bulk equation of motion into the action ( [ schractionlcft ] ) , the boundary action becomes of the form @xmath90 \int [ x , \vec k ' , \vec k ] ~~\tilde j ( k ' ) \big [ \mathcal f_1 ( u_b , k ' , k ) j ( k ) + \mathcal f_2 ( u_b , k ' , k ) \tilde j ( k ) \big ] \;. \end{aligned}\ ] ] for alcf , the system has the time translation invariance , thus the time integral is trivially evaluated to give a delta function . the @xmath91 s are given by @xmath92 we note that @xmath55 leads the same result as ( [ f1function ] ) and ( [ f1correlator ] ) . let us evaluate @xmath93 , which is @xmath94 derivative of @xmath55 given in ( [ g12expressionlcft ] ) . the result is @xmath95 \right ) \;. \end{aligned}\ ] ] these are our correlation and response functions with log extensions for general @xmath1 and @xmath0 . this is a direct generalization of the previous result for @xmath16 and @xmath17 @xcite . note that the result is valid for the field theories with galilean boost without conformal symmetry similar to the result ( [ f1correlator ] ) . again it is independent of @xmath1 . motivated by the recent interests on tricritical log gravity @xcite , we consider three scalar fields @xmath96 , @xmath97 and @xmath98 in the background ( [ alcfmetric ] ) with the following action @xmath99 + \frac{\phi_3 \phi_2}{2l^2 } \right ] \ ; , \end{aligned}\ ] ] where we take @xmath100 for @xmath101 . this action is previously considered in @xcite in a different context . the field equations for @xmath36 s of the action ( [ schractionlcft ] ) become @xmath102 where @xmath103 is given in ( [ diffopalcf ] ) . we construct the bulk to boundary green s function @xmath72 in terms of @xmath104 as @xmath105 we choose @xmath106 , which is in accord with the structure of the equations of motion given in ( [ diffeqschrtlcft ] ) . the green s functions satisfy @xmath107 where @xmath108 and @xmath77 . the green s functions @xmath109 are @xmath110 where @xmath111 with the same normalization constant given in ( [ onshact1 ] ) . there exist other green s functions @xmath112 for the action ( [ schractiontlcft ] ) , which satisfies @xmath113 in particular , we have @xmath114 to evaluate them , we generalize the methods used in @xcite@xcite to the tricritical case . using again @xmath115 = 2 \nu $ ] , and the fact that @xmath116 , we get @xmath117 thus @xmath118 note that the last expression has second order derivative of @xmath94 , which leads log@xmath15 contributions . after plugging the bulk equation of motion into the action ( [ schractiontlcft ] ) , the boundary action becomes of the form @xmath119 \int[\vec x , \vec k ' , \vec k ] ~~ j_i ( k'_\mu ) \mathcal f_{ij } ( u_b , k'_\mu , k_\mu ) j_j ( k_\mu ) \;. \end{aligned}\ ] ] the system has time translation invariance , thus the time integral is trivially evaluated to give delta function . the @xmath91 s are given by @xmath120 note that the first terms in @xmath121 and @xmath122 are @xmath123 when evaluated at @xmath59 . we also notice that @xmath124 and @xmath125 are identical to ( [ f1function ] ) and ( [ f2function ] ) , and thus the corresponding correlation functions ( [ f1correlator ] ) and ( [ f2correlator ] ) , respectively . now we are ready to evaluate @xmath126 . using ( [ f3function ] ) and ( [ f1correlator ] ) , we get @xmath127 - \nu^2 \ln [ \frac{\mathcal m u_b^2}{2(t_2 - t_1)}]^2 \right ] \ ; , \nonumber \end{aligned}\ ] ] where @xmath128 . this is our main result in this section , response functions with log and log@xmath15 extensions , which is valid for general @xmath1 and @xmath0 . note also that this result is valid for the systems without non - relativistic conformal invariance . we notice that various coefficients in the square bracket are completely determined once @xmath94 is fixed . we first establish the schrdinger type solutions with galilean symmetry with @xmath129 following @xcite , see also @xcite@xcite . finite temperature generalizations for general @xmath1 is considered in @xcite , while those for @xmath17 are considered in @xcite@xcite@xcite@xcite@xcite . the metric at zero temperature is given by @xmath130 which is invariant under the space - time translations @xmath24 , galilean boost @xmath25 , scale transformation @xmath131 and translation along the @xmath29 coordinate . their explicit forms are given in ( [ galileantr1 ] ) and ( [ scaletr1 ] ) . there exists additional special conformal transformation for @xmath17 , which has been focus of previous investigations . there have been more general class of gravity backgrounds with so - called hyperscaling violation . these backgrounds are described by @xmath132 $ ] considered in @xcite , where @xmath133 is a hyperscaling violation exponent . @xmath133 is first introduced in @xcite based on @xcite . this hyperscaling violation might be also interesting in the general context of aging and growth phenomena . the associated matter fields are a gauge field , a scalar and the non - trivial coupling between them . the geometry ( [ schrmetric ] ) is not a solution of vacuum einstein equations . thus we require to support it with some matter fields . one particular example is the ground state of an abelian higgs model in its broken phase @xcite @xmath134 where @xmath135 , @xmath136 , and @xmath0 is the number of spatial dimensions . it is not hard to find a different matter system that supports the metric @xcite . @xmath137 where @xmath138 . we are interested in constructing correlation and response functions using three different actions , ( [ scalaractionalcf ] ) , ( [ schractionlcft ] ) and ( [ schractiontlcft ] ) , as in the previous section [ sec : alcf ] . here we briefly show that the procedure is the same as before . thus we can compute the logarithmic ( squared ) extensions by taking a simple @xmath94 derivatives of the correlation function obtained from the action ( [ scalaractionalcf ] ) . we start by considering correlation functions of the geometry ( [ schrmetric ] ) by coupling a probe scalar with the same action as ( [ scalaractionalcf ] ) . the field equation for @xmath36 becomes @xmath140 note @xmath141 , which is one of the main differences between the schrdinger background ( [ schrmetric ] ) and alcf ( [ alcfmetric ] ) . again , we treat @xmath38 coordinate special and replace all the @xmath39 as @xmath40 . with this differential equation , one can compute the correlation function @xmath55 . for general @xmath1 , analytic solutions are not available . the resulting correlation function for @xmath17 is already computed in @xcite@xcite@xcite , while that of @xmath21 is computed below in [ sec : zthreehalf ] . previously , several special cases also have been computed in @xcite . to compute the corresponding log and log@xmath15 extensions , we consider a schrdinger differential operator @xmath142 \ ; , \end{aligned}\ ] ] where @xmath143 and @xmath144 for @xmath129 . for the special case @xmath17 , we have @xmath145 from ( [ diffopschr ] ) . with the differential operator @xmath146 , we can still use the relation @xmath147 = 2 \nu \ ; , \end{aligned}\ ] ] to compute the correlation ( response ) functions with the logarithmic extensions . for that purpose , we use the equations ( [ bulktoboundarygreensfunctionlcft ] ) - ( [ f1function2 ] ) with appropriate @xmath148 and @xmath149 . we also get the response functions with the log@xmath15 extension using ( [ bulktoboundarygreensfunctiontlcft ] ) - ( [ lastfstlcft ] ) with appropriate @xmath150 and @xmath151 . the upshot is that the logarithmic extensions can be computed by taking one or two derivatives of the correlation functions available . let us compute these correlation and response functions for @xmath17 and @xmath21 in turn . we comment for the conformal case @xmath17 here . the differential equation ( [ blukscalareqschr ] ) simplifies to @xmath152 this is similar to that of alcf given in ( [ blukscalareqalcf ] ) , the only difference is the presence of the parameter @xmath153 , which modifies @xmath94 as @xmath154 . this observation leads us that we can compute correlation and response functions with logarithmic extensions as in [ sec : alcf ] . these are ( [ f1correlator ] ) , ( [ f2correlator ] ) and ( [ f3correlator ] ) with modified @xmath94 . the case @xmath19 is our main interest for the application to kpz universality class . for the time being , we work on general spatial dimensions @xmath0 . the corresponding solution is @xmath155 \ ; , \end{aligned}\ ] ] where @xmath156 , @xmath157 , @xmath144 and @xmath143 . @xmath158 and @xmath159 represent the confluent hypergeometric function and the generalized laguerre polynomial . we choose @xmath158 for our regular solution . the momentum space correlation function can be evaluated as the ratio between the normalizable and non - normalizable contributions at the boundary expansion of the solution ( [ schr32sol ] ) , which is given by @xcite @xmath160 } { \gamma [ 2 \nu ] } \frac{\gamma \left[\frac{1 + 2\nu}{2}+\frac{m^2}{2 q}\right ] } { \gamma \left[\frac{1 - 2\nu}{2}+\frac{m^2}{2 q}\right ] } \ ; , \end{aligned}\ ] ] where we only keep momentum dependent parts . one can restore the @xmath35 dependence using scaling arguments . for the general case , fourier transforming back analytically to the coordinate space is difficult . thus we would like to consider some special cases . * @xmath161 : the momentum space correlator has the same form as aging in alcf @xmath162 } { \gamma [ 2 \nu ] } \frac{\gamma \left[\frac{1 + 2\nu}{2}\right]}{\gamma \left[\frac{1 - 2\nu}{2}\right ] } \;. \end{aligned}\ ] ] for imaginary parameter @xmath62 , we get @xmath163 this result is for @xmath164 . the dependence on time and space is identical to the result of the aging in alcf . + we are interested in log and log@xmath15 extensions . for this purpose , we consider @xmath165 where @xmath166 collectively denotes the other @xmath94 dependent parts . we also use the same actions ( [ schractionlcft ] ) and ( [ schractiontlcft ] ) to get the correlation functions with the logarithmic extension @xmath167 \right ) \langle \phi ( x_{2 } ) \phi ( x_{1 } ) \rangle _ { \mathcal f_1}^{m_0 } \ ; , \end{aligned}\ ] ] and the log@xmath15 extension @xmath168 + a_2 \ln [ \frac{\mathcal m u_b^2}{(t_2 - t_1)}]^2 \right ] \langle \phi _ 2 \phi_1 \rangle _ { \mathcal f_1}^{m _ 0 } \ ; , \end{aligned}\ ] ] where @xmath169+\nu h''[\nu ] } { h[\nu ] } $ ] , @xmath170 + 2 \nu h'[\nu ] } { h[\nu ] } $ ] and @xmath171 . * @xmath172 : we use the asymptotic expansion form from 5.11 of @xcite @xmath173 } { \gamma [ z+b ] } \sim z^{a - b } \sum_{k=0}^{\infty } \frac{g_k ( a , b)}{z^k } \ ; , \quad g_{k}(a , b)=\binom{a - b}{k } b^{{(a - b+1)}}_{{k } } ( a ) \;,\end{aligned}\ ] ] where @xmath174 are binomial coefficients and @xmath175 s are generalized bernoulli polynomials . for our case , @xmath176 . + the momentum space correlation function is @xmath177 } { \gamma [ 2 \nu ] } \left ( \frac{m^2}{2 q } \right)^{2\nu } \sum_{n=0}^{\infty } g_{2n } \left(\frac{q}{m^2 } \right)^{2n } \;,\end{aligned}\ ] ] the first term is independent of momenta , which we ignore . for the rest of the terms , the inverse fourier transform of @xmath178 gives us @xmath179 , which vanishes for integer @xmath180 . thus the coordinate correlation function identically vanishes except the case @xmath181 . thus we get , using @xmath182 @xmath183 } \frac{\mathcal m^{\frac{d}{2}-2\nu}}{\pi 2^{1 - 2\nu } } \cdot \frac{\theta ( x_2^+ ) \theta(t_2 - t_1 ) } { ( t_2 - t_1)^{\frac{d+2}{2}+2\nu } } \cdot \exp \left ( - { \frac{\mathcal m ( \vec{x}_2 - \vec{x}_1)^2 } { 2 ( t_2 - t_1)}}\right ) \;. \end{aligned}\ ] ] + we are interested in the response functions with log and log@xmath15 extensions , we consider @xmath184 where @xmath185 collectively denotes the other @xmath94 dependent parts . we also use the same actions ( [ schractionlcft ] ) and ( [ schractiontlcft ] ) to get the correlation functions with the logarithmic extension @xmath186 \right ) \langle \phi^ * \phi \rangle _ { \mathcal f_1}^{m_\infty } \ ; , \end{aligned}\ ] ] and the log@xmath15 extension @xmath187 + \tilde a_2 \ln [ \frac{\mathcal m u_b^2}{(t_2 - t_1)}]^2 \right ] \langle \phi^ * \phi \rangle _ { \mathcal f_1}^{m_\infty } \ ; , \end{aligned}\ ] ] where @xmath188+\nu \tilde h''[\nu ] } { \tilde h[\nu ] } $ ] , @xmath189 + 4 \nu \tilde h'[\nu ] } { \tilde h[\nu ] } $ ] and @xmath190 . these two extreme cases , @xmath191 and @xmath192 , signal that the parameter @xmath2 can bring some quantitatively different behaviors of the correlation and response functions because of the different power in time dependent denominators @xmath193 and @xmath194 in ( [ schrcorrf1 ] ) and ( [ schrcorrlargemf1 ] ) , respectively . in this section we summarize [ sec : ntlft ] by considering the two - time correlation and response functions with logarithmic extensions . from the various results of alcf and schrdinger backgrounds , equations ( [ f3correlator ] ) , ( [ schrcorrf3 ] ) and ( [ schrcorrlargemf3 ] ) , we observe that the correlation functions with(out ) log extensions show qualitatively similar properties . some typical two - time correlation and response functions can be obtained by putting @xmath195 in equation ( [ f3correlator ] ) . @xmath196 + a_2 \ln [ \frac{\mathcal m_b}{t_2 - t_1}]^2 \right ] \ ; , \end{aligned}\ ] ] where @xmath197 , and the coefficients @xmath198 we note that these coefficients , @xmath199 and @xmath200 , are determined once @xmath94 is fixed . @xmath201 is invariant under the time translation transformation , and so @xmath202 . the so - called `` waiting time '' @xmath203 does not have a physical meaning . thus @xmath204 is completely fixed as a function of @xmath205 , once @xmath0 , @xmath94 and @xmath206 are given . physically , this time translation invariant two - time response functions describe either constant growth or constant aging ( decaying ) phenomena . further physical significances are considered in detail in [ sec : kpz ] . equipped with the generalization of our correlation and response functions for general @xmath1 and @xmath0 , non - relativistic and ( non-)conformal geometries , we would like to add yet another ingredient to them : aging , one of the simplest time - dependent physical phenomena . typically aging is realized when the system is rapidly brought out of equilibrium . for this simple time - dependent phenomena , time translational invariance is broken . there are two important time scales : ( 1 ) _ waiting time _ which marks the time scale when the system is perturbed after it is put out of equilibrium and ( 2 ) _ response time _ which marks when the perturbation is measured . typical properties of aging are described by the two - time response functions in terms of these waiting time and response time , and are power law decay , broken time translation invariance and dynamical scaling between the time and spatial coordinates . these are shown in holographic model in @xcite as well as various field theoretical models , see _ e.g. _ @xcite@xcite@xcite . in the context of anti - de sitter space / conformal field theory correspondence ( ads / cft ) @xcite@xcite and its extension to schrdinger geometries @xcite@xcite@xcite@xcite , the geometric realizations of aging have been put forward in @xcite@xcite by generalizing the background with explicit time dependent terms . these terms are generated by a singular time dependent coordinate transformation , which itself has significant physical meaning in the context of holography @xcite . furthermore , there exists a time boundary at @xmath207 and physical boundary conditions are explicitly imposed : ( 1 ) by complexifying time in @xcite or ( 2 ) by introducing some decay modes of the bulk scalar field along the ` internal ' spectator direction @xmath38 , which is not explicitly visible from the dual field theory in @xcite . we prefer the option ( 2 ) in this paper as in @xcite , where the resulting two - time correlation functions show a dissipative behavior and exhibit the three characteristic features of the aging system mentioned above . thus the time translation symmetry is broken globally , and the aging symmetry is realized as conformal schrdinger symmetry modulo time translation symmetry @xcite@xcite . their finite temperature properties with asymptotic aging invariance are also investigated in @xcite . see also a recent review @xcite . in this section we would like to generalize this aging construction to the case with general dynamical exponent @xmath1 and for general dimensions @xmath0 . the generalization of the singular coordinate transformation and the corresponding aging geometries are constructed in [ sec : agingtransform ] . in [ sec : agingalcf ] , we construct the two point correlation and response functions for alcf in the context of @xcite@xcite , while similarly in [ sec : agingback ] for schrdinger background in the context of @xcite@xcite . their log and log@xmath15 extensions are explained in [ sec : aginglogextension ] . physical properties of aging is explored in holography by using a _ singular _ coordinate transformation @xmath208 which is first introduced in @xcite , specifically for @xmath17 case . it is important to impose physical boundary conditions on the time boundaries in addition to the spatial boundaries . the simplest possibility in this context has been explored in @xcite we would like to extend this singular transformation for general @xmath1 in a direct manner . @xmath209 note that for general @xmath1 , the coordinate @xmath38 has non - trivial dimensions , @xmath210 = 2-z $ ] , under the scaling transformation . one immediate consequence is a nontrivial scaling dimension of our parameter @xmath64 = z-2 $ ] . this is already observed in the exponent of the correlation and response functions in previous section . now for the aging extension , we observe that the parameter @xmath3 also has a definite scaling dimension @xmath211 = 2-z$ ] . these two parameters conspire to provide us a rather simple and elegant generalization to the aging correlation and response functions for general @xmath1 . the background metric extended to the aging is correspondingly modified to @xmath212 where @xmath213 corresponds to aging in alcf . there exists a slight change in metric compared to ( [ alcfmetric ] ) or ( [ schrmetric ] ) : the coefficient of the term @xmath214 has a factor of @xmath1 instead of @xmath215 . one can check that the matter contents without the singular transformation would solve the corresponding einstein equation . these cases can be considered as _ locally galilean_. to compute the correlation functions of the probe scalar fields in the background geometry ( [ agingmetric ] ) with general @xmath1 and @xmath0 , we consider the action given in ( [ scalaractionalcf ] ) . the field equation for @xmath36 becomes @xmath216 \phi \\ & = \frac{\partial^2 \phi}{\partial u^2 } + \frac{z i m\alpha -d-1}{u } \frac{\partial \phi}{\partial u } -\left [ \frac { 4 m^2 l^2 + 2(d+2 ) z i m \alpha + z^2 m^2 \alpha^2 } { 4 u^2 } + \frac{\gamma m^2 } { u^{2z-2 } } + \vec k^2 \right ] \phi \;. \nonumber \end{aligned}\ ] ] note that here we treat @xmath38 coordinate special and replace all @xmath39 as @xmath40 , because this coordinate plays a distinguished role in galilean and corresponding aging holography @xcite@xcite . to find the solution of the equation ( [ blukscalareqagingalcf ] ) , we use the fourier decomposition as @xmath217 where @xmath218 is the momentum vector for the corresponding coordinates @xmath219 . @xmath220 is introduced for the calculation of the correlation functions and is determined by the boundary condition with the normalization @xmath221 . and @xmath222 is the kernel of integral transformation that convert @xmath223 to @xmath224 , which is necessary for our time dependent setup @xcite . with this fourier mode , the differential equation ( [ blukscalareqagingalcf ] ) decomposes into time dependent part and radial coordinate dependent one . the time dependent equation and solution read @xmath225 the radial dependent equation is given by @xmath226 where @xmath227 . from this point we can not carry on the analysis for both the aging in alcf , @xmath228 , and aging background @xmath141 simultaneously . thus , we first consider the correlation functions of the scalar operator in aging alcf . for @xmath228 , an analytic solution of the equation ( [ feq ] ) is available as @xmath229 where @xmath230 and @xmath231 are bessel functions with @xmath232 and @xmath233 . note the overall @xmath3 dependent factor , which is a non - trivial feature of our model . we also consider the boundary condition along time direction near the boundary . the solution behaves as @xmath234 along with the time dependent factor @xmath235 in ( [ agingtimesol ] ) , whose inverse fourier transform is given by @xmath236 this wave function converges for @xmath237 if @xmath238 , and for @xmath239 due to the exponential factor if @xmath240 . in particular , this condition allows the parameter @xmath241 to be negative @xmath242 especially for the case @xmath243 . similar result for @xmath17 and @xmath16 is already considered in @xcite . note that we only consider the imaginary @xmath244 . we follow @xcite to compute the correlation functions by introducing a cutoff @xmath35 near the boundary and normalizing @xmath46 , which fixes @xmath245 . the on - shell action is given by @xmath246 & = \int d^{d+1 } x \frac{l^{d+3}}{u^{d+3 } } ~\phi^ * ( u , t , \vec{x } ) ~\left(\frac{u^2}{l^2}\partial_u + i m \frac{z \alpha u}{2 l^2 } \right ) \phi ( u , t , \vec{x } ) \big |_{u_b } \;. \end{aligned}\ ] ] this can be recast using @xmath247 as @xmath248 where @xmath249 represents the existence of a physical boundary in the time direction , @xmath250 , and @xmath91 is @xmath251 note that the spatial integration along @xmath252 can be done trivially to give a delta function @xmath253 . one can bring @xmath254 factors in @xmath255 and @xmath256 together to cancel each other . this removes the second part in @xmath257 . from this point it is straight forward to check that @xmath91 is given by ( [ f1function ] ) at the boundary . further details can be found in @xcite . for imaginary parameter @xmath62 , we get the same correlation function as @xmath258 this is one of our main results . the aging correlation functions for general dynamical exponent @xmath1 and @xmath0 have a direct relation with those of schrdinger as @xmath259 the overall time dependent factor @xmath260 comes from the inverse fourier transform of the time part , which has been evaluated in great detail @xcite . thus the result is independent of @xmath1 , while depending on the number of dimensions @xmath0 . the corresponding extensions with log and log@xmath15 are considered below in [ sec : aginglogextension ] . for aging backgrounds , we have nontrivial @xmath1 dependence , and we need to treat them separately . fortunately , an analytic solution is available for @xmath21 @xmath261 + c_2 l_a^{2\nu } ( 2 q u ) \right ) \ ; , \end{aligned}\ ] ] where @xmath262 , @xmath263 and @xmath143 . @xmath158 and @xmath159 represent the confluent hypergeometric function and the generalized laguerre polynomial . we choose @xmath158 for our regular solution . the momentum space correlation function turns out to be the same as ( [ schrmomentumcorr ] ) as explained there . for the rest , we follow similarly [ sec : agingalcf ] to get the aging correlation and response functions . finally , we arrive general conclusion @xmath264 the overall time dependent factor @xmath265 comes from the inverse fourier transform of the time part , which has been evaluated in great detail @xcite . as we mentioned in [ sec : ntlftsummary ] , the aging in alcf and aging background have similar properties as far as the correlation and response functions are concerned . thus we present the logarithmic extension of the aging correlation functions using ( [ ficorrelatoraginglc ] ) . in the previous sections , [ sec : agingalcf ] and [ sec : agingback ] , we establish the fact that the correlation functions have the overall time dependent factor @xmath265 from the time dependent part of the momentum correlation function . in section [ sec : ntlft ] , on the other hand , we developed the algorithm to generate the logarithmic extensions using @xmath94 derivatives from the fact @xmath266 = 2 \nu$ ] . these two generalizations are independent of each other . thus we safely generate the logarithmic extensions of the aging correlation functions by differentiating the aging correlation functions in terms of @xmath94 . @xmath267\ ; , \label{generalformulalogaging2 } \\ & \langle \phi ( x_{2 } ) \phi ( x_{1 } ) \rangle_{aging}^{\mathcal f_3 } = \left ( \frac{t_2}{t_1 } \right)^{-\frac{\alpha \mathcal m}{2 } } \frac{1}{4\nu } \frac{\partial}{\partial \nu } \left [ \frac{1}{2\nu } \frac{\partial}{\partial \nu } \left [ \langle \phi ( x_{2 } ) \phi ( x_{1 } ) \rangle_{schr}^{\mathcal f_1 } \right ] \right ] \;. \label{generalformulalogaging3 } \end{aligned}\ ] ] the results @xmath268 are given in equations ( [ f1correlator ] ) , ( [ schrcorrf1 ] ) and ( [ schrcorrlargemf1 ] ) . their specific forms are @xmath269 \right ) \langle \phi ( x_{2 } ) \phi ( x_{1 } ) \rangle_{aging}^{\mathcal f_1 } \ ; , \\ \langle \phi_3 ( x_{2 } ) \phi_3 ( x_{1 } ) \rangle^{\mathcal f_3}_{aging } \label{f3correlatoraging } & = - \frac{1}{8 \nu^4 } \left [ a_0 + a_1 \ln [ \mathcal m_{\delta t } ] + a_2 \ln [ \mathcal m_{\delta t}]^2 \right ] \langle \phi ( x_{2 } ) \phi ( x_{1 } ) \rangle_{aging}^{\mathcal f_1 } \ ; , \end{aligned}\ ] ] where @xmath270 these are main results of our aging response functions . physical significances related to them are discussed in the following section . in this section we would like to seek a connection to kpz universality class , its growth , aging or both phenomena at the same time . our investigation is concentrated on the generalizations of two - time response functions for general dynamical exponent @xmath1 and for general spatial dimensions @xmath0 , along with their generalizations with the log and log@xmath15 contributions . previously , we observed that our two - time response functions reveal several qualitatively different behaviors , such as growth , aging or both in our holographic setup @xcite . in a particular case , @xmath17 and @xmath16 , both growing and aging behaviors have been observed for a parameter range @xmath271 @xcite . this was motivated by a recent progress on field theory side @xcite@xcite along with some clear experimental realization of the kpz class in one spatial dimension @xcite@xcite . here we obtain additional properties of the two - time response functions as well as to extend our results for general @xmath1 and @xmath0 . before presenting the details , we comment their general behaviors . * due to the simple broken time translation invariance of our system , signified by the parameter @xmath241 , our two - time response function reveals a power - law scaling behavior at early time region , which is distinct from another power - law scaling at late time region . the turning point between the two time regions , @xmath272 , is marked by the waiting time @xmath203 . if @xmath273 , there exists either only growth or aging behavior . * the initial power scaling behaviors , growth or aging , are crucially related to the parameter @xmath274 , especially the combination @xmath275 , which is the scaling dimensions of the dual field theory operators we consider . the late time scaling behavior is further modified by @xmath276 , which is aging parameter , in addition to the scaling dimensions . * the power - law part of the two - time response functions show the growth and aging behaviors , while the log and log@xmath15 corrections provides further modifications that would match detailed data by tuning available parameters . we consider a typical correlation and response functions for general @xmath1 and @xmath0 , extending previous results for @xmath17 and @xmath16 @xcite @xmath277 with a waiting time , @xmath203 , a scaling time , @xmath278 and two other free parameters @xmath279 and @xmath241 , which satisfies the condition ( [ conditiononalpham ] ) coming from the time boundary @xmath280 . the response function ( [ twotimeresponsetlcfty ] ) is the general form for the aging alcf for all the cases considered in [ sec : agingalcf ] . this is also valid for the alcf in [ sec : alcf ] without the condition ( [ conditiononalpham ] ) if we set @xmath281 . . the parameters @xmath0 and @xmath282 do not change the qualitative behaviors , while the parameters @xmath94 , actually @xmath274 , and @xmath241 are important for the early time and late time power law scaling . , title="fig:",scaledwidth=47.0% ] . the parameters @xmath0 and @xmath282 do not change the qualitative behaviors , while the parameters @xmath94 , actually @xmath274 , and @xmath241 are important for the early time and late time power law scaling . , title="fig:",scaledwidth=47.0% ] let us comment for the case with positive @xmath94 . this case has only aging properties if the parameters @xmath94 and @xmath241 are not too large . for @xmath283 and @xmath284 , which is allowed by the time boundary condition ( [ conditiononalpham ] ) , the bending point , around @xmath285 in the figure [ fig : positivenus ] , sits deep down and the second leg of the plot becomes horizontal . as we increase either @xmath94 or @xmath241 , the bending point goes up . this is depicted in the figure [ fig : positivenus ] . for @xmath286 or a particular value of @xmath274 , we can get a straight line , which is identical to the time independent case . if one is interested in growth phenomena , it is more interesting to consider @xmath287 . due to the form of the response function ( [ twotimeresponsetlcfty ] ) , the part @xmath288 determines the properties at early time @xmath289 . for @xmath290 , actually @xmath291 determines the slope at early time . for @xmath292 , the slope of the first leg is negative , while that is positive for @xmath293 . this can be verified directly in the figure [ fig : negativenus ] . . each panel has a fixed value of @xmath274 , which determines the slope of the first leg at early time , while the sign of @xmath241 determines the relative slope of the second leg at late time , compared to the first leg . left : the case saturated with the bf bound @xmath294 and @xmath295 . three plots are for @xmath296 , which are blue straight , black dotted and red dashed lines , respectively . similarly for the middle with @xmath297 and right plots with @xmath298 with the same values of @xmath296 . , title="fig:",scaledwidth=49.0% ] . each panel has a fixed value of @xmath274 , which determines the slope of the first leg at early time , while the sign of @xmath241 determines the relative slope of the second leg at late time , compared to the first leg . left : the case saturated with the bf bound @xmath294 and @xmath295 . three plots are for @xmath296 , which are blue straight , black dotted and red dashed lines , respectively . similarly for the middle with @xmath297 and right plots with @xmath298 with the same values of @xmath296 . , title="fig:",scaledwidth=49.0% ] + . each panel has a fixed value of @xmath274 , which determines the slope of the first leg at early time , while the sign of @xmath241 determines the relative slope of the second leg at late time , compared to the first leg . left : the case saturated with the bf bound @xmath294 and @xmath295 . three plots are for @xmath296 , which are blue straight , black dotted and red dashed lines , respectively . similarly for the middle with @xmath297 and right plots with @xmath298 with the same values of @xmath296 . , title="fig:",scaledwidth=49.0% ] on the other hand , the late time behavior is determined by a factor @xmath299 . if @xmath300 , the slope does not change . the relative slope of the second leg is determined by the sign of @xmath241 . this is verified in the figure [ fig : negativenus ] . the real slope of the second leg is governed by the sign of @xmath301 . in particular , early time growth and late time aging happens for @xmath302 the second condition is similar to our time boundary condition ( [ conditiononalpham ] ) , but not identical . for growth phenomena , the roughness of interfaces is quantified by their inter - facial width , @xmath303 ^ 2 \rangle_l } $ ] , defined as the standard deviation of the interface height @xmath304 over a length scale @xmath305 at time @xmath306 @xcite . an equivalent way to describe the roughness is the height - difference correlation function @xmath307 ^ 2 \rangle $ ] . @xmath308 and @xmath309 denote the average over a segment of length @xmath310 and all over the interface and ensembles , respectively . both @xmath311 and @xmath312 are common quantities for characterizing the roughness , for which the so - called family - vicsek scaling @xcite is expected to hold . the dynamical scaling property is @xmath313 with two characteristic exponents : the roughness exponent @xmath314 and the growth exponent @xmath315 . the dynamical exponent is given by @xmath316 , and the cross over length scale is @xmath317 . for an infinite system , the correlation function behaves as @xmath318 at some late time region @xmath319 . from the two - time correlation function in equation ( [ twotimeresponsetlcfty ] ) , we can get the growth exponent @xmath320 where @xmath321 . note that the parameters satisfy the condition ( [ conditiononalpham ] ) from the time boundary conditions . we notice that our system size is infinite , and thus it is not simple matter to obtain the corresponding roughness exponent . the dynamical exponent is not fixed in alcf , even though the differential equation has @xmath1 dependence , which can be checked in ( [ blukscalareqagingalcf ] ) . for kpz universality class , there is a nontrivial scaling relation between the roughness exponent @xmath322 and the dynamical exponent @xmath1 , chapter 6 in @xcite @xmath323 while this relation is remained to be checked in our holographic model , we assume it is valid to make contact with some field theoretical models . using the relation @xmath324 , we get @xmath325 let us examine these critical exponents against the known case for @xmath11 @xmath326 these can be reproduced with the condition @xmath327 which can be matched for @xmath328 and @xmath329 for negative @xmath94 . we choose @xmath329 for the simple growth behavior . there are two independent critical exponents . one particular interesting exponent is the so called growth exponent @xmath330 , where @xmath321 . we consider a dual field theory operator with @xmath331 , @xmath332 . then by expanding for small @xmath333 , we get @xmath334 using again @xmath335 and the relation @xmath324 , we get @xmath336 if we further restrict our attention to the case @xmath337 for considering only the growth phenomena , we have the following dependence on the number of spatial dimensions @xmath338 where @xmath339 . for @xmath340 , these exponents match ( [ kpzexponents ] ) for @xmath11 . the corresponding roughness exponent is depicted as blue line in the left panel of the figure [ fig : roughnessexponents ] , which is referred as `` alcf '' . to compare with other growth models @xcite , we also depicted the roughness exponents of the kim - kosterlitz model @xmath341 @xcite as well as wolf - kertsz model @xmath342 @xcite . of alcf for @xmath343 and @xmath344 that matches kpz exponents for @xmath11 given in ( [ kpzexponents ] ) . kk represents @xmath341 from kim - kosterlitz @xcite , while wk @xmath342 from wolf - kertsz @xcite . right panel : plot for @xmath322 of alcf@xmath345 for @xmath343 and @xmath346 that matches ( [ kkexponents ] ) only when @xmath347 . , title="fig:",scaledwidth=48.0% ] of alcf for @xmath343 and @xmath344 that matches kpz exponents for @xmath11 given in ( [ kpzexponents ] ) . kk represents @xmath341 from kim - kosterlitz @xcite , while wk @xmath342 from wolf - kertsz @xcite . right panel : plot for @xmath322 of alcf@xmath345 for @xmath343 and @xmath346 that matches ( [ kkexponents ] ) only when @xmath347 . , title="fig:",scaledwidth=48.0% ] for @xmath348 , we get @xmath349 our results ( [ kkexponents ] ) are only valid for @xmath347 , which is referred as `` alcf@xmath345 '' in the right panel of the figure [ fig : roughnessexponents ] . these exponents have been conjectured for growth in a restricted solid - on - solid model by kim - kosterlitz @xcite@xcite . we have shown in previous sections that the response functions reveal growth and aging behaviors without log or log@xmath15 corrections . the log and log@xmath15 corrections have been considered to match further details at early time region @xcite@xcite . in this section , we would like to investigate some more details related to those corrections based on previous results @xcite . our two - time response functions with log correction are given by @xmath350 \bigg\ } \ ; , \end{aligned}\ ] ] where @xmath203 , @xmath351 , @xmath94 and @xmath241 are free parameters , while the coefficients are given by @xmath352 } \;.\end{aligned}\ ] ] we note that the coefficients are completely fixed by two fixed parameters @xmath353 note that similar result for @xmath17 and @xmath16 has been available in @xcite . the detailed comparisons between ( [ twotimeresponsetlcftylog ] ) and the phenomenological field theory model @xcite@xcite were investigated . we noted that the terms proportional to @xmath354 and @xmath355 are not considered in @xcite@xcite , which do not modify qualitative features of the response functions . for this case , the analysis done in @xcite is still valid . we obtain the log@xmath15 extension of the response function using holographic approach @xmath356 \ln [ 1 - \frac{1}{y } ] + \tilde r_2 \ln [ 1 - \frac{1}{y}]^2 \bigg\ } \ ; , \end{aligned}\ ] ] where @xmath357}{\tilde a_0 + \tilde a_1 \ln [ \mathcal m_b ] + \tilde a_2 \ln [ \mathcal m_b]^2 } \ ; , \\ & \tilde r_2 = \frac{\tilde a_2}{\tilde a_0 + \tilde a_1 \ln [ \mathcal m_b ] + \tilde a_2 \ln [ \mathcal m_b]^2 } \ ; , \\ & \tilde a_0 = 1 + \nu \psi - \nu^2 ( \psi^2- \psi ' ) \ ; , \\ & \tilde a_1 =( 2\nu^2 \psi - \nu ) \ ; , \qquad \tilde a_2 = - \nu^2 \;.\end{aligned}\ ] ] these coefficients are also completely fixed by two fixed parameters given in ( [ mbpsiparameters ] ) . compared to @xmath358 , a new parameter @xmath359 determines the behaviors of response functions related to the log and log@xmath15 contributions . as we explicitly check in the figure [ fig : lognegativenus ] , the qualitative behavior of the response functions does not change with the log and log@xmath15 contributions for reasonably small @xmath359 . the growth exponent and aging properties are determined by the two parameters @xmath274 and @xmath241 , which define our theory . with blue straight , @xmath360 with red dashed and @xmath361 with black dot - dashed lines for @xmath243 , @xmath11 , @xmath362 and @xmath363 . for the response functions with log corrections , we need one more input @xmath359 , which we took @xmath364 for these plots . the smaller the value of @xmath359 , the smaller the differences between @xmath365 and its log extensions . , title="fig:",scaledwidth=47.0% ] with blue straight , @xmath360 with red dashed and @xmath361 with black dot - dashed lines for @xmath243 , @xmath11 , @xmath362 and @xmath363 . for the response functions with log corrections , we need one more input @xmath359 , which we took @xmath364 for these plots . the smaller the value of @xmath359 , the smaller the differences between @xmath365 and its log extensions . , title="fig:",scaledwidth=47.0% ] with black dot - dashed and @xmath366 with brown straight lines for @xmath243 , @xmath11 , @xmath362 and @xmath363 . left : @xmath364 , right : @xmath367 . we check that there is no qualitative changes due to the unwanted terms explained in ( [ twotimeresponsetlcftylog2wanted ] ) . , title="fig:",scaledwidth=47.0% ] with black dot - dashed and @xmath366 with brown straight lines for @xmath243 , @xmath11 , @xmath362 and @xmath363 . left : @xmath364 , right : @xmath367 . we check that there is no qualitative changes due to the unwanted terms explained in ( [ twotimeresponsetlcftylog2wanted ] ) . , title="fig:",scaledwidth=47.0% ] we would like to compare our results ( [ twotimeresponsetlcftylog2 ] ) with the following equation , which is equation ( 10 ) of @xcite ( or equation ( 4.3 ) of @xcite ) , obtained from the phenomenological field theory model . @xmath368-\frac{1}{2 } f_{0}\tilde \xi'^2\ln^2 [ 1-\frac{1}{y}]-g_{21,0}\xi'\ln[y-1]+\frac{1}{2}f_{0}\xi'^2\ln^2 [ y-1]\right ) \ ; , \nonumber \end{aligned}\ ] ] where the parameters @xmath369 come from the logarithmic extension in the field theory side . from the phenomenological input @xcite `` the parenthesis becomes essentially constant for sufficiently large @xmath370 , '' the condition @xmath371 is imposed to remove the last two terms . the other parameters in ( [ ftresponsefunction ] ) are determined to match available data . we can identify the exponents @xmath372 by comparing the equation to our result @xcite @xmath373 by including the log@xmath15 corrections , we provide the necessary terms @xmath374 ^ 2 $ ] as well as @xmath375 $ ] . the relative coefficients between them are fixed by ( [ log2coefficients ] ) . on the other hand , there exist also several unwanted terms inside the parenthesis of ( [ twotimeresponsetlcftylog2 ] ) . first , we note new terms @xmath376 and @xmath377 $ ] due to log@xmath15 corrections , in addition to @xmath378 coming from the log correction . all these terms might spoil the desired properties of the phenomenological response function ( [ ftresponsefunction ] ) . to examine the effects coming from these unwanted terms , we plot the response function with only wanted terms as @xmath379 + \tilde r_2 \ln [ 1 - \frac{1}{y}]^2 \bigg\ } \;. \end{aligned}\ ] ] we explicitly check in the figure [ fig : logunwantedterms ] that the full response functions ( [ twotimeresponsetlcftylog2 ] ) have a qualitatively similar behavior compared to those ( [ twotimeresponsetlcftylog2wanted ] ) with only wanted terms in the field theory approach . while the dynamical exponent @xmath1 for the alcf is not fixed in obtaining the correlation and response functions , those of the schrdinger backgrounds crucially depend on @xmath1 . in fact , obtaining analytic solutions of the differential equation ( [ blukscalareqschr ] ) is a highly non - trivial task . fortunately , we are able to get response functions for @xmath164 with some approximations as @xmath380 where @xmath203 , @xmath351 , @xmath94 and @xmath241 are free parameters . this is valid for the aging background ( [ agingschrcorrf1 ] ) as well as the schrdinger background , @xmath281 , given by ( [ schrcorrf1 ] ) with @xmath381 for @xmath191 and by ( [ schrcorrlargemf1 ] ) with @xmath382 for @xmath383 . from the two - time response functions in equation ( [ twotimeresponsetlcfty ] ) , we can get the growth exponent @xmath384 where @xmath321 . now the dynamical exponent is fixed as @xmath164 for the aging background . using the relation @xmath324 , we get @xmath385 the critical exponents for the kpz universality class given in ( [ kpzexponents ] ) can be reproduced with the condition @xmath386 which can be matched for @xmath387 and @xmath388 for negative @xmath94 . the value of @xmath274 for @xmath382 becomes negative , yet is allowed as we can see from the expression of @xmath94 . we have extended our geometric realizations of aging symmetry in several different ways based on previous works for @xmath17 and @xmath16 @xcite@xcite . first , we generalize our correlation and response functions to the non - conformal setup with general dynamical exponent @xmath1 and for arbitrary spatial dimensions @xmath0 . they have galilean symmetries with time translation symmetry , which are summarized in the equations ( [ f1correlator ] ) , ( [ schrcorrf1 ] ) and ( [ schrcorrlargemf1 ] ) . for convenience , we reproduce equation ( [ f1correlator ] ) here @xmath389 which is valid for general @xmath1 and @xmath0 . second , these are extended with log and log@xmath15 corrections with appropriate bulk actions . practically , these corrections can be computed using simple properties of the differential operators ( [ diffopalcf ] ) , ( [ diffopschr ] ) and their commutation relations ( [ commutationrelation ] ) . the results are listed in ( [ f2correlator ] ) , ( [ f3correlator ] ) for alcf , ( [ schrcorrf2 ] ) , ( [ schrcorrf3 ] ) for schrdinger backgrounds . all these response functions have time translation invariance . third , on top of these extensions , we also compute response functions with aging symmetry , by breaking the global time translation invariance using a singular coordinate transformation ( [ coordinatechangez ] ) . we check the general relation between the aging response functions and those of schrdinger backgrounds holds ( [ generalformulalogaging1 ] ) @xmath390 this generalization is independent of the logarithmic extensions . with these results , we investigate our two - time response functions for general @xmath1 and especially for arbitrary number of spatial dimensions @xmath0 with log and log@xmath15 extensions ( [ twotimeresponsetlcftylog2 ] ) @xmath391 where @xmath392 represent various contributions from the log and log@xmath15 extensions . from the systematic analysis , we have found that our two - time response functions reveal a power - law scaling behavior at early time region , which is distinct from another power - law scaling at late time region . this can be explicitly checked in the figure [ fig : negativenus ] . the early time power scaling behaviors are governed by the scaling dimensions of the dual field theory operators . in particular , their growth and aging is determined by the sign of the parameter @xmath274 . the late time behaviors are modified by @xmath276 , the aging parameter . if @xmath286 , the initial behaviors persist without change , which is expected due to its time translation invariance . the turning point between these two time regions is marked by the waiting time @xmath203 . the log and log@xmath15 corrections provide further modifications that would match detailed data by turning available parameters . let us conclude with some observations and future directions toward holographic realizations of kpz universality class . our generalizations of the holographic response functions to general @xmath1 and @xmath0 open up some possibilities to have contact with the higher dimensional growth and aging phenomena . we have done the first attempt to do so in [ sec : alcfcriticalexponent ] . we make some contacts with kim - kosterlitz model @xcite at higher spatial dimensions with an assumptions ( [ azrelationkpz ] ) for some particular dual scalar operators . although it is not perfect , we consider this as a promising sign for the future developments along the line . we mention two pressing questions we would like to answer in a near future . our holographic model is an infinite system , and thus obtaining the roughness exponent `` @xmath322 '' is rather challenging . progresses on this point will provide a big step toward realizing holographic kpz class . assumption ( [ azrelationkpz ] ) is well understood in the field theoretical models @xcite . there galilean invariance was a crucial ingredient , which is also important in our holographic model . verifying this relation would be an important future challenge we thank to e. kiritsis , y. s. myung , v. niarchos for discussions and valuable comments on the higher dimensional kpz class and the logarithmic extensions of cft . sh is supported in part by the national research foundation of korea(nrf ) grant funded by the korea government(mest ) with the grant number 2012046278 . sh and jj are supported by the national research foundation of korea ( nrf ) grant funded by the korea government(mest ) through the center for quantum spacetime(cquest ) of sogang university with grant number 2005 - 0049409 . jj is supported in part by the national research foundation of korea(nrf ) grant funded by the korea government(mest ) with the grant number 2010 - 0008359 . bsk is grateful to the members of the crete center for theoretical physics , especially e. kiritsis , for his warm hospitality during his visit 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we analytically compute correlation and response functions of scalar operators for the systems with galilean and corresponding aging symmetries for general spatial dimensions @xmath0 and dynamical exponent @xmath1 , along with their logarithmic and logarithmic squared extensions , using the gauge / gravity duality .
these non - conformal extensions of the aging geometry are marked by two dimensionful parameters , eigenvalue @xmath2 of an internal coordinate and aging parameter @xmath3 .
we further perform systematic investigations on two - time response functions for general @xmath0 and @xmath1 , and identify the _ growth exponent _ as a function of the scaling dimensions @xmath4 of the dual field theory operators and aging parameter @xmath3 in our theory .
the initial growth exponent is only controlled by @xmath4 , while its late time behavior by @xmath3 as well as @xmath4 .
these behaviors are separated by a time scale order of the waiting time .
we attempt to make contact our results with some field theoretical growth models , such as kim - kosterlitz model at higher number of spatial dimensions @xmath0 .
cquest-2013 - 0593 + taup-2963/13 seungjoon hyun@xmath5 , jaehoon jeong@xmath6 and bom soo kim@xmath7 @xmath8_department of physics , college of science , yonsei university , seoul 120 - 749 , korea _ + @xmath9_center for quantum spacetime , sogang university , seoul 121 - 742 , korea _
+ @xmath10_raymond and beverly sackler school of physics and astronomy , _ +
_ tel aviv university , 69978 , tel aviv , israel _
+ [email protected] , [email protected] , [email protected]
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the interchain , interplane transport of electrons in low dimensional ( quasi 1d , 2d ) materials attracts much attention @xcite in view of striking differences between longitudinal and transverse transport mechanisms revealing a general problematics of strongly correlated electronic systems . beyond the low field ( linear ) conduction , the tunnelling current - voltage j - u characteristics @xmath0 , @xmath1 are of particular importance . the interest has been renewed thanks to recently developed @xcite design of intrinsic tunnelling devices where electronic transitions between weakly coupled chains or planes take place in the bulk of the unperturbed material . the first feature one expects to see at any tunnelling experiment in gapful conductors is the regime of free electrons when the current onset corresponds to the voltage @xmath2 of the gap in the spectrum of electrons . but contrarily to usual systems , like semi- or even superconductors , there is also a possibility for tunnelling within the subgap region @xmath3 . it is related to the pseudogap ( pg ) phenomenon known for strongly correlated electrons in general , well pronounces in quasi 1d systems and particularly in cases where the gap is opened by a spontaneous symmetry breaking ( see @xcite and refs . therein ) . the pg is originated by a difference , sometimes qualitative , between three forms of electronic states : a ) short living excitations which are close to free electrons , b ) dressed stationary excitations of the correlated systems , and c ) added particles which modify the ground state itself @xcite . for our typical examples of electrons on a flexible lattice , the modification results in self - trapped states ( b ) like single particle @xmath4 polarons with energies @xmath5 below the single electron ( a ) activation energy @xmath6 ; then the new gap @xmath7 will be observed as a true threshold with the pg in between . there may be also contributions of two - particle @xmath8 states ( c ) - bipolarons , which energy gain per electron is larger than for polarons @xmath9 . while the cases ( a , b ) are common for low symmetry and discrete symmetry cases , for ( c ) there is a further drastic effect of a spontaneous symmetry breaking like the case of the polyacethylene @xmath10 or of some doubly commensurate cdws . now the bipolarons are decoupled into particles with a nontrivial topology , solitons or kinks , changing the sign of the order parameter of the dimerization . the situation is further intricate in systems with a continuous gs degeneracy like incommensurate charge density waves ( icdw ) or wigner crystals . here even the self - trapping of a single electron is allowed to lead to topologically nontrivial states , the amplitude solitons ass . in the same class we find a more common case of acoustic polarons in a 1d semiconductor @xcite . properties of systems with different types of the gs degeneracy , and required theoretical approaches , are quite different . here we shall concentrate on systems with a discrete , precisely double , degeneracy which also include most basic elements of non - degenerate systems . theoretically , the tunnelling in cdws was studied in details for regimes of free electrons @xcite when the current onset corresponds to the voltage @xmath2 of the gap in the spectrum of electrons . we shall consider the tunnelling in the pg regime . we shall follow the method @xcite developed for studies of single particle spectral density @xmath11 in applications to pes and arpes intensities . we refer to this publication for details in techniques and literature . a word of notations . in the following we shall invoke many quantities with the dimension of energy ( or frequency , since we shall keep @xmath12 ) which will be classified according to different characters ( with indices ) . @xmath13 and @xmath14 will be the external voltage difference for tunnelling and the external frequency for pes or optics . @xmath15 will always stay for electronic eigenvalue in a given potential , negative values will be addressed explicitly as @xmath16 . @xmath17 will be branches of a total energy ( of deformations together with electronic energies ) supporting eigenstates @xmath18 which may be filled with occupation numbers @xmath19 . @xmath20 will be total energies of stationary states , that is @xmath21 . @xmath22 will be the frequency of a collective mode ( phonons specifically to cdws ) which interaction with electrons is responsible for their self - trapping . the collective deformation @xmath23 will also be measured as the potential energy experienced by electrons . we shall keep the electron charge @xmath24 hence potentials will be measured as energies and the interchain current @xmath25 will have the dimension _ number - of - particles / unit - time / unit - length_. the indices @xmath26 will number coupled chains ; indices @xmath27 will number moments @xmath28 of time for virtual processes . a possibility of tunnelling or of other excitations within the gap in spectra of free electrons @xmath29 is related to a more general phenomenon of the pseudogap pg . for electrons , the pg signifies the remnants of the spectral density @xmath30 , or the integrated one @xmath31 , at @xmath32 where @xmath33 is the absolute boundary of the spectrum . @xmath33 is the energy of a fully dressed state of one electron interacting with other degrees of freedom . ( there may be totally external modes like deformations or polarizations for usual polarons , external modes essentially modified by the bath of electrons like in cdws , internal collective modes of electronic system itself like in sdws . ) most commonly , the self - trapped state of one electron is known as the `` polaron '' while more complex objects , solitons , can appear for systems with continuously degenerate gss ( see @xcite for a review ) . the functions @xmath34 and @xmath30 are measured directly in pes and arpes experiments ( these abbreviations stay for the integrated photo - emission spectroscopy and for the angle ( that is momentum ) resolved one ) . as such they have been studied theoretically for the pg region by the present authors @xcite and we refer to these publications for a more comprehensive discussion and for the literature review . the one electron spectra can be accessed also in traditional external tunnelling experiments : junctions or stm . for the last case , and practically for macroscopic point junctions , only the integrated @xmath34 is measured . elements of the full dependence @xmath30 become necessary to describe strongly anisotropic materials ( layered quasi-2d or chain quasi-1d ones ) where the coherent tunnelling is realized in internal junctions of `` mesa '' type devices @xcite . here the tunnelling goes between adjacent layers within the single crystal of the same material , hence the momentum is preserved . in a simpler version , the internal subgap tunnelling takes place from free electrons of some metallic bands or pockets to polaronic states within gapful spectra which probably takes place in @xmath35 @xcite . otherwise it measures actually the joint spectral density for creation a particle - hole excitation at adjacent chains ( the interchain exciton ) . in this respect it will be instructive to compare the coherent tunnelling and the subgap optical absorption oa ( see a short excursion and references in @xcite , iii.e ) . a less expected version of the internal tunnelling is a possibility for bi - electronic transfers ( tunnelling of bipolarons or of kink - antikink pairs ) which usually is attributed only to superconductors . we shall see that these processes extend the pg further down to even lower voltages . in any case , the tunnelling current @xmath0 is given by the transition rate of electrons between two subsystems @xmath36 kept at the potential difference @xmath37 . for a weak coupling @xmath38 , the electron tunnelling from @xmath39 to @xmath40 is given by the convolution of spectral densities @xmath41 if the momentum is conserved , or of their integrals @xmath42 for the incoherent tunnelling . ( everywhere we assume @xmath43 . ) recall that for free electrons with a spectrum @xmath44 we have @xmath45 while @xmath34 becomes the dos @xmath46 , e.g. for @xmath47 @xmath48 near the bottom of the free band where the electron effective mass is @xmath49 . consider briefly the case where one of reservoirs , say @xmath50 , is composed by free electrons with a known dos @xmath51 . one of applications of a tunnelling between the free spectrum and the pg may be the case of several families of conjugated polymers ( polypyrolle , polythiophene ) where origins of filled , @xmath52 or empty , @xmath53 bands below and above the gap are essentially different . then the polaronic effect , hence the pg , may be pronounced only for one type of particles : electrons or holes . the same concerns 1d systems made of semiconducting wires where both effective masses and deformation potentials for electrons and holes are usually very different . then , for the incoherent tunnelling , @xmath54 gives directly either the tunnelling current @xmath55 ( if @xmath56 has a sharp peak at the fermi level , which is typical for using junctions with superconductors ) or the tunnelling differential conductance @xmath57 ( if @xmath58 at the fermi surface ) . below we shall be mostly interested in systems with the charge conjugated symmetry ( or qualitatively equivalent ones ) ; the examples are carbon nanotubes , symmetric conjugated polymers like polyphenylenes , polyanilines and polymers where the gap is formed ( partly at least ) by the spontaneous symmetry breaking : the polyacethylenes ( @xcite ) . numerical details will be presented for the last rich case . in all these cases the pg will exist near both rimes @xmath59 of the free excitation gap @xmath60 . recall now some known results for @xmath34 within the pg @xcite . it has the form @xmath61 where the action @xmath62 is proportional to the big parameter of our adiabatic approximation : @xmath63 . @xmath64 is determined by an optimal fluctuation localized in space and time ( an instanton ) which supports the necessary split - off local level @xmath65 . in principle , the prefactor @xmath66 also depends on @xmath67 and may show power law dependencies near extremals @xmath68 . but within constraints of the adiabatic approximation @xmath69 the dependence @xmath70 is negligible in comparison with the one of @xmath64 . the characteristic value of @xmath71 may be important for estimates of the overall magnitude of observable effects . thus for the single particle integrated intensity @xmath72 and @xmath73 for the momentum resolved intensity ; here @xmath49 is the effective electron mass @xmath74 . appendix [ app:1 ] contains derivation of the prefactor specifically for the tunnelling processes . in limiting cases we have @xcite \1 . near the entry to the pg , just below the free edge @xmath75 : @xmath76 . \label{nh}\ ] ] \2 . near the low end of the pg , just above the true spectral boundary @xmath33 : @xmath77 . \label{nl}\]]the total dependence @xmath34 and the values of numerical constants in the above limiting laws , can be determined approximately @xcite with the help of the instanton techniques simplified by the zero dimensional reduction ( the anzats of an effective particle which we shall recall and extend below ) . the resulting curve is plotted at the figure [ fig : s4pol ] . moreover , the regime ( 1 . ) can be mapped exactly @xcite upon the problem of a particle in a quenched random uncorrelated potential which here is created by instantaneous quantum fluctuations of the media . the known exact solution @xcite provides the reference value of the coefficient in the exponent of ( [ nh ] ) , from which our approximate value differs only by @xmath78 @xcite . recall for comparison the usual regime @xmath79 of the allowed tunnelling which is dominated by free electronic states . the current of the coherent tunnelling between chains @xmath36 is given as @xmath80with @xmath81 and @xmath82 being the bloch functions . it is instructive to compare the interchain tunnelling probability with the on chain interband optical absorption oa when the matrix element of density @xmath83 changes to the one of the momentum : @xmath84 . in both cases the @xmath85 pair is created and the same spectral densities are involved . the difference is in matrix elements : the oa takes place between states of opposite parity while the tunnelling requires for the same parity . the on - chain oa between edges @xmath86 of the free gap is known to be allowed since the parity of states near @xmath86 is opposite , hence @xmath87 is finite and the oa intensity as a function of frequency @xmath88 rises as @xmath89 . but for the same reason , the tunnelling matrix element between identical chains is prohibited at @xmath90 and the tunnelling will show only a weak edge onset @xmath91 . nevertheless , in many cases of gaps opened due to spontaneous dimerization , the neighboring chains tend to order in antiphase . now the shift by half a period permute states with @xmath92 then the parity of states near opposite rims @xmath86 at neighboring chains is equal , the tunnelling becomes allowed and the usual singularity is restored : @xmath93 . going down into the pg @xmath94 , the above analysis applies to the on - chain optics but changes drastically for the interchain tunnelling . the tunnelling will be studied in details below , here we shall only mention in advance an effect of spatial incoherence of optimal quantum fluctuations at different chains which removes completely the constraints of orthogonality . the case of the on - chain oa can be analyzed briefly already here . the oa is given by the convolution of two fast decaying functions of the energy @xmath95 \left\vert \lambda ( \omega_{1},\omega_{1}-\omega)\right\vert^{2}d\omega_{1}\sim \nonumber\\ \lambda^{2}(\omega /2,-\omega /2)\left(s^{\prime \prime}\right)^{-1/2 } \exp \left [ -2s(\omega /2)\right]= \nonumber\\ \left(s^{\prime \prime}(\omega /2)\right)^{-1/2 } i^{2}(\omega /2);\ s^{\prime \prime}=\frac{d^{2}s}{d\omega^{2}}. \label{ioa}\end{aligned}\]]here we have used that for the convex function @xmath64 , as given by ( [ nh],[nl ] ) , the minimum of the expression @xmath96 lies at the middle @xmath97 . at this point the electron levels @xmath65 and @xmath98 are placed symmetrically , wave functions have opposite parity , hence @xmath99 is finite . this is the case of typical peierls insulators . but for systems where the basis wave functions of valent and conductive bands have the same parity ( the dipole oa is not allowed ) , @xmath100 and we have to consider in ( [ ioa ] ) the deviations from the symmetry condition . now @xmath101 and the saddle point integration in ( [ ioa ] ) gives another factor of @xmath102 which is small as @xmath103 . we arrive at the answer similar to ( [ ioa ] ) but with the small prefactor @xmath104 . until now we did not consider the dependencies on the momentum @xmath105 . in the full range of @xmath67 and @xmath105 , the spectral function @xmath106 has a rich structure which can be tested in the arpes experiments . in observable quantities , the momentum dependence appears twice : via the matrix element @xmath107 and via the action @xmath108 . the analysis is simplified for the regime 2 : the low polaron boundary @xmath109 . here the action dependence on @xmath88 and @xmath105 comes through the single variable @xmath110 where @xmath111 , is a heavy mass of the polaron center motion . this kinetic energy contribution can be neglected in compare to the matrix element dependence on @xmath105 which confines @xmath112 within the characteristic momenta distribution @xmath113 of the wave function @xmath114 of the self - trapped electronic state localized over the scale @xmath115 : beyond @xmath116 , the function @xmath107 falls off exponentially . ( at this scale , the recoil kinetic energy @xmath117 is small in compare to the energy width @xmath118 . then the final integration over @xmath105 affects only @xmath107 and gives a constant factor @xmath119 . ) altogether we find for tunnelling just the law ( [ nl ] ) with @xmath120 . in the regime 1 . , near the free edge , the states are shallow @xmath121 and extended @xmath122 . the effective mass @xmath123 for the center of motion becomes light , energy dependent @xmath124 but the characteristic energy scale of the form factor @xmath125 is still small in comparison with the characteristic energy width @xmath126 of ( [ nh ] ) . so again we integrate separately the factor @xmath127 to obtain an additional prefactor @xmath128 for the tunnelling law ( [ nh ] ) with @xmath120 . recall that for the arpes with independent variations of @xmath88 and @xmath105 , their interference may lead to rather unexpected and potentially observable phenomena ( @xcite , section iii.d ) . one of them is the `` quasi spectrum '' : the intensity maximum over the line @xmath129 within the pg @xmath130 ( @xcite , section iii.d , case b1 , eq.49 ) . another effect is the emergence of instantons at high @xmath105 within the domain of free electron region @xmath131 leading to the enhanced intensity within the band @xmath132 ( @xcite , section iii.d , case b3 , eq.51 ) . we shall follow the adiabatic method of earlier publications @xcite assuming a smallness of collective frequencies @xmath133 in compare with the electronic gap : @xmath134 . now , electrons are moving in a slowly varying potential @xmath135 , so that at any instance @xmath136 their energies @xmath137 and wave functions @xmath138 are defined from a stationary schrodinger equation @xmath139 ( eq . ( [ dirac ] ) below will give an example ) . the hamiltonian @xmath140 depends on the instantaneous configuration @xmath135 so that @xmath141 and @xmath142 depend on time only parametrically . exponentially small probabilities which we are studying here are determined by steepest descent paths in the joint space @xmath143 $ ] of configurations and the time , that is by a proximity of the saddle point of the action @xmath144 . it is commonly believed , in analogy with the usual wkb , that the saddle point , the extremum of @xmath144 over @xmath145 and @xmath136 , lie at the imaginary axis of @xmath136 so that , as usual , we shall assume @xmath146 and correspondingly @xmath147 since now on . consider the system of two weakly coupled chains @xmath26 which are put at the electric potential difference @xmath37 . the system is described by the total action @xmath148where @xmath149 $ ] are single chain actions and the term @xmath150 describes the interchain hybridization of electronic sates . @xmath151 are operators of electronic states.the average transverse current is given by the functional integral @xmath152~ it_{\perp } ( \hat{\psi}_{a}^{\dag}(x , t)\hat{\psi}_{b}(x , t)- \hat{\psi}_{b}^{\dag}(x , t)\hat{\psi}_{a}(x , t))\exp [ -s_{ab } ] } { \int dx d[\delta_{j}(x , t)]\exp [ -s_{ab}]}. \label{j}\end{aligned}\ ] ] we consider first the processes originated by the transfer of one electron between the chains . they appear already in the first order of expansion of the exponent in ( [ s - total ] ) in powers of @xmath153 , which contribution to the current ( [ j ] ) can be written as @xmath154 \int d(x - y)\int d(\tau_{1}-\tau_{2 } ) \\ \nonumber [ \psi_{a}^{\ast}(x,\tau_{1})\psi_{b}(x,\tau_{1 } ) \psi_{a}(y,\tau_{2})\psi_{b}^{\ast}(y,\tau_{2 } ) \exp(-s(\tau_{1}-\tau_{2},\delta_{j}(x , t ) ) ) , \label{j1}\end{aligned}\]]where the normalizing factor @xmath155 is the denominator in ( [ j ] ) taken at @xmath156 . here the time dependent action @xmath157 describes ( in imaginary time ) the process of transferring one particle from the doubly occupied level @xmath158 of the chains @xmath39 to the unoccupied level @xmath159 of the chain @xmath40 at the time @xmath160 and the inverse process at the time @xmath161 . we have @xmath162+\int_{\tau_{1}}^{\tau_{2}}dt[(l_{a}(-1)+l_{b}(1)-u ] \nonumber\\ & = & \int_{-\infty}^{\infty}dt[l_{a}(0)+l_{b}(0)]+ \int_{\tau_{1}}^{\tau_{2}}dt[(e_{b}+e_{a}-u ] , \label{s(1,2)}\end{aligned}\]]where @xmath163,\nu)$ ] are lagrangians of the @xmath164-th chain with the number of electrons changed by @xmath165 . they are given as a sum of the kinetic term and the potential @xmath166 : @xmath167~;\ v_{\nu}=v_{0}+|\nu |e . \label{l(nu)}\ ] ] here the potential term @xmath166 contains the energy of deformations and the sum over electron energies in filled states @xmath168 which include both the vacuum states and the split off ones : @xmath169= \int dx\frac{\delta^{2}}{2g^{2}}+\sum_{e_{\alpha}<e_{f}}e_{\alpha}[\delta ( x , t)]-w_{gs } \label{v(nu)}\ ] ] ( here @xmath170 is the coupling constant ) . @xmath166 is counted with respect to the gs energy @xmath171 so that in the non perturbed @xmath172 state @xmath173 ( the particle , electron for @xmath174 or hole for @xmath175 , added instantaneously to the non deformed gs is placed at the lowest allowed energy , the gap rim @xmath75 ) . the exact extremal ( saddle point ) trajectory is defined by equations @xmath176 actually the explicit calculation of the action requires for approximations . we shall follow a way @xcite of the zero dimensional reduction which reduces the whole manyfold of functions @xmath177 to a particular class @xmath178\rightarrow s[e_{j}(t),x_j(t ) ] \label{reduction}\ ] ] of a given function @xmath179 of @xmath180 ( relative to a time dependent center of mass coordinate @xmath181 ) . @xmath182 is parameterized by a conveniently chosen ( see @xcite for examples ) parameter for which a universal and economic choice is the eigenvalue @xmath137 . the requirement for the manyfold @xmath183 is that it supports a pair of eigenvalues @xmath18 split off inside the gap @xmath184 which span the whole necessary interval . the last simplification is to assume , in the spirit of all approaches of optimal fluctuations @xcite , that the potential supports one and only one pair of localized eigenstates @xmath185 . explicit formulas for the peierls case are given in the appendix [ app:2 ] . recall that for the oa problem we deal with one chain characterized by one pair of functions @xmath141 and @xmath186 . but for the interchain tunnelling , the functions @xmath137 at chains @xmath26 are not obliged to be identical and also the wells may be centered around different points @xmath181 . within such a parametrization the variational equation in ( [ dd , d1,d2 ] ) yields the equation of motion for @xmath141 @xmath187 where @xmath188 are the hamiltonians which must be constants within each interval of integration in ( [ s(1,2 ) ] ) . apparently , at the outer intervals @xmath189,@xmath190 @xmath191 to provide the return to the gs with @xmath192 at @xmath193 . at the inner interval @xmath194 @xmath195 to preserve the continuity of velocities @xmath196 at @xmath197 . since the values @xmath198 are determined uniquely by the equation of motion at the outer intervals , then @xmath198 coincide for both @xmath26 , hence @xmath199 and the functions @xmath137 become identical at any time @xmath200 . ( still , the shapes are allowed to be shifted by different centers @xmath181 : @xmath201 ) . finally the extremal conditions ( [ dd , d1,d2 ] ) with respect to impact times @xmath202 in ( [ s(1,2 ) ] ) yield @xmath203 the action is finite @xmath204 , hence the transition probability is not zero , only for a closed trajectory , that is at presence of a turning point ( as examples , see figures [ fig : u12],[fig : u14],[fig : u18],[fig : u16+c ] in the appendix [ app:2 ] ) . there must be a minimal value of @xmath205 where @xmath206 hence @xmath207 and @xmath208 . the last condition requires for @xmath209 that is for @xmath210 which determines the threshold voltage at twice the polaron energy . we arrive at the effective one chain problem with the doubled effective action . the extremal tunnelling action is @xmath211 which is twice the exponent appearing in the spectral density @xmath212 with limiting laws ( [ nh],[nl ] ) . the full expression is @xmath213 we obtain a final expression for the current after integration over @xmath177 around the extremal taking into account the zero modes related with translations of the instanton centers positions @xmath181 . ( details of calculations are given in the appendix [ app:1 ] ) @xmath214 , \label{pre0}\]]where @xmath215 is the fourier transforms of the wave functions @xmath114 , the time @xmath216 is defined as @xmath217 . the mean fluctuational displacement @xmath218 of the center of mass between the impact moments is given as @xmath219 where @xmath220 is the translational mass : @xmath221 note that the prefactor in eq . ( [ pre0 ] ) , which is the matrix element between orthogonal states @xmath222 and @xmath223 , is always nonzero due to the integration over zero modes @xmath181 ( in contrast to results for the rigid lattice where it obeys the selection rules ) ; see more in the appendix [ app:1 ] . comparing with the pes intensity @xmath34 calculated in @xcite we see that , up to pre - exponential factors , the tunnelling current is proportional to the square of the pes intensity @xmath212 : @xmath224 . e.g. near the threshold @xmath225 we can write @xmath226 \exp \left [ c_{2}\frac{(u-2w_{1})}{g\omega_{0}}\log \frac{2c_{3}\delta_{0}}{(u-2w_{1})}\right ] . \label{j1low}\ ] ] the coefficients @xmath227 can be found numerically from ( [ s(u ) ] ) as ( for the peierls model ) @xmath228 , @xmath229 , @xmath230 . ( these values differ from the corresponding ones in @xcite because of different normalizations of frequency @xmath133 in compare to @xmath231 ) . it is known that the joint self - trapping of two electrons allows to further gain the energy resulting in stable states different from independent polarons . in general nondegenerate systems this is the bipolaron , confined within the length scale twice smaller than that of the polaron , the energy gain of the bound state @xmath232 is four times that of the polaron and the total energy gain of the bipolaron @xmath233 is also four times that of two polarons . ( certainly these results neglect the energy loss due to the coulomb repulsion which may become critical for the stability of a shallow bipolaron . ) the same time , the total energy of one bipolaron @xmath234 is larger than the energy of one polaron @xmath235 and even than the free electron energy @xmath75 . this is why bipolarons can not be seen as thermal excitations while they are favored in case of doping . the information on their existence comes from the ground state of doped systems where bipolarons are recognized by their spinless character and special optical features ( see @xcite for experimental examples on conducting polymers and @xcite for relevant theoretical models ) . an important advantage of tunnelling experiments is a possibility to see bipolarons directly , at voltages @xmath37 below the two - polaron threshold @xmath236 that is within the true single particle gap . this possibility comes from the fact that , for bipolarons as particles with the double charge @xmath237 , the voltage gain by transferring from one chain to another is @xmath238 , hence the threshold will be at @xmath239 . the probability of the bi - electron tunnelling is small as it appears only in the higher order @xmath240 in interchain coupling . but it can be seen as extending below the one - electron threshold where no other excitations can contribute to the tunnelling current . the bi - electronic contribution to the current can be written , by expanding ( [ s - total ] ) and ( [ j ] ) , as @xmath241d[\delta_b]\int \prod_{i=1}^{3}dy_{i}d\tau_{i}\exp \left(-s(\tau_{i})\right ) \nonumber\\ & & \left [ \psi_{a}^{\ast}(x,\tau)\psi_{b}(x,\tau)\psi_{a}^{\ast}(y_{1},\tau_{1 } ) \psi_{b}(y_{1},\tau_{1})\psi_{b}^{\ast}(y_{b},\tau_{2 } ) \psi_{a}(y_{2},\tau_{2})\psi_{b}^{\ast}(y_{3},\tau_{3})\psi_{a}(y_{3},\tau_{3 } ) -\{2\leftrightarrow 3\}\right ] , \nonumber\end{aligned}\]]which generalizes expressions ( 7 ) and ( 8) for the one electron tunnelling . here + \nonumber\\ & & \left\{\int_{\tau}^{\tau_{1}}+\int_{\tau_{2}}^{\tau_{3}}\right\ } dt[(l_{a}(1)+l_{b}(1)-u]+\int_{\tau_{1}}^{\tau_{2}}dt[l_{a}(2)+l_{b}(2)-2u ] . \label{2e}\end{aligned}\ ] ] within our model ( [ l(nu)],[v(nu ) ] ) the potentials @xmath243 are additive in energy @xmath65 , then the action can be simplified as @xmath244 the extremal solution is defined , as above , by equations of the type ( [ dd , d1,d2 ] ) but with four impact times @xmath28 instead of two . ( actually , in view of the time reversion symmetry , the number of boundary conditions is twice smaller . ) a similar analysis of the extremal solution shows that optimal fluctuations @xmath177 are identical in shape , up to shifts of their centra : @xmath181 , @xmath245 . hence the energies are identical @xmath246 , and also the resonance conditions @xmath247 take place at the impact moments @xmath28 . moreover , the simple hierarchy of our model @xmath248 shows that all branches @xmath17 cross at the same point @xmath249 ( see figures [ fig : u12],[fig : u14],[fig : u18 ] below ) . then the evolution @xmath141 switches directly from the branch @xmath250 to the branch @xmath8 and back , without following the intermediate branch @xmath4 . it means that the intervals @xmath251 and @xmath252 of one - electron transfers @xmath4 are confined to zero : @xmath253 . in other words , only processes of simultaneous tunnelling of pairs of particles are left . notice that this picture changes in more general models , particularly taking into account important coulomb interactions . they add , to the energy branch of a shallow bipolaron , the energy @xmath254 where @xmath255 is the dielectric susceptibility of the media in the interchain direction , @xmath256 is the localization length of @xmath257 , such that @xmath258 . now the intermediate intervals @xmath259,@xmath260 appear where the evolution follows the @xmath4 branches , see figure [ fig : u16+c ] . with increasing coulomb interactions this single particle interval becomes more pronounced and the bipolaronic threshold is shifted towards the one of two independent polarons . in any case , the extremum solution for the action ( [ 2e ] ) is achieved on the instanton trajectory given be the equation @xmath261 the extremal action is @xmath262 this action is finite if the turning point @xmath263 does exist , that is if @xmath264 . notice that , neglecting coulomb interactions , the energy @xmath166 is determined only by the total number @xmath265 of electrons and holes . then the energy of the bipolaron ( both @xmath266 and @xmath267 are either empty or doubly occupied ) and the energy of the exciton ( both @xmath268 and @xmath269 are singly occupied ) are the same . then the trajectory of the bi - electronic tunnelling becomes the same as the one for the case of optical absorption @xcite , only the action is doubled @xmath270 . up to the pre - exponential factor we have @xmath271^{2},\]]where @xmath272 is the optical absorption probability for one chain . for common systems with a nondegenerate ground state , the dependence @xmath273 resembles qualitatively the law [ j1low ] for the one - electron contribution , with a similar behavior near the threshold @xmath274 . the situation changes for a doubly degenerate ground state where the bipolaron dissolves into a diverging pair of solitons ( dimerization kinks ) . thus for the peierls model the evaluation of ( [ s2(u ) ] ) gives , similar to the oa law of @xcite , near the two particle threshold @xmath275 with @xmath276 . the overall dependence for the @xmath277 is shown at the figure [ fig : s4sol ] . here we see explicitly that in the order @xmath240 the threshold voltage @xmath278 is smaller than @xmath279 obtained in the order @xmath280 . therefore this is the main contribution to the current in the region @xmath281 . figure [ fig : s4sol ] shows that the dependence of @xmath282 near the low @xmath37 onset is much sharper than that of @xmath0 at the figure [ fig : s4pol ] near the polaronic onset which corresponds to the higher singularity in the limiting formula ( [ j2min ] ) in compare to ( [ j1low ] ) . in quasi 1d systems with a gapful electronic spectrum , the interchain tunnelling ( as well as pes or oa ) can be used to test virtual electronic states within the pseudogap . due to the interaction of electrons with a low frequency mode , phonons in our examples , the tunnelling is allowed in the subgap region @xmath283 which forms the pseudogap . the one electron processes lead to universal results similar both for systems with the build - in gap and for those where the gap is due to the spontaneous breaking of a discrete symmetry . the pg is entered with the law ( [ nh ] ) and continues down to the threshold @xmath284 , approached with the law ( [ nl ] ) . this threshold corresponds to the interchain transfer of fully dressed particles : polarons with the energies @xmath33 . but in tunnelling the pg is stretched even further down thanks to processes of a simultaneous tunnelling of two electrons . it terminates at the lower threshold @xmath285 or @xmath286 , @xmath287 . here @xmath288 is the energy of the bipolaron - a bound state of two electrons selftrapped together . in degenerate systems the bipolaron dissolves into unbound solitons , hence the threshold at @xmath289 with a more pronounced dependence of the tunnelling rate ( [ j2min ] ) as well of the oa . numerical results are presented at figures [ fig : s4pol],[fig : s4sol ] . there is an important difference between subgap processes and the usual overgap transitions at @xmath290 of free electrons in a rigid system . it comes , beyond intensities , from different character of matrix elements . actually within the pg region there are no particular selection rules since the wave functions of virtual electronic states split off within the gap are localized having a broad distribution of momenta . then the pg absorption is allowed independent on the interchain ordering . contrarily , the regular tunnelling across the free gap shows an expected dos singularity @xmath291 for the out of phase interchain order while for the in - phase order the threshold is smooth @xmath292 . this difference may be important to choose an experimental system adequate for studies of pgs . the smearing of the free edge singularity is a natural criterium for existence of the pg below it @xcite . but the total absence of this strong feature in systems with forbidden overgap transitions can allow for a better resolution of the whole pg region , down to the absolute threshold . probably a very smooth manifestation of gaps in usual tunnelling experiments on cdws @xcite , while the gaps show up clearly through activation laws , is related to this smooth crossover from the overgap to the subgap region . ( notice that the existing experiments refer mostly to icdws which , with their continuous degeneracy of the gs , must be studied specially which is beyond the scope of this article . ) finally we shall discuss relations with other theoretical approaches . most theories of tunnelling , see @xcite , keep the following assumptions : i. they refer to the overgap region where interactions or fluctuations are not important and usually are not taken into account . they refer to the incoherent tunnelling , local in space , which is a usual circumstance of traditional experiments . the pg in tunnelling was considered by monz et al in @xcite in the framework of the approach @xcite . this method became popular recently in theories of the pg thanks to its easy implementation : it is sufficient to average results for a rigid system over a certain distribution of the gap values . apparently this is the way to describe an average over a set of measurements performed on similar systems with various values of the gap , e.g. manipulating with the temperature , the pressure or a composition . but actually , as we could see above , the pg is formed by fluctuations localized both in space and time , the instantons , with localization parameters depend on the energy deficit being tested . there is an intermediate approach applied @xcite to a complex of the pg phenomenon from optics to conductivity and susceptibility . it treats fluctuations as an instantaneous disorder due to quantum zero point fluctuations of the gap . indeed , this picture can be well applied , as it was done already in @xcite , but only to dynamical processes and only in the upper pg region , just below the free gap @xmath293 , which leads to the law ( [ nh ] ) . but deeper within the pg , the fluctuations are not instantaneous : they require for an increasingly longer time and become self - consistent with the measured electronic state leading to another law and to appearance of the lower threshold . generalizations and deeper analysis of the model of the instantaneous disorder lead to interesting theoretical studies @xcite , but their applicability is very limited unless the variable time scale is realized as we have demonstrated in this and preceding articles . our approach can be compared to the work @xcite on the fluctuational creation of pairs of phase solitons in a 1d commensurate cdw under the longitudinal electric field . but in our case me deal , in effect , with the interchain tunnelling of pairs of solitons under the transverse field ; also the solitons have a more complex character of a multielectronic origin . in conclusion , the presented and earlier @xcite studies recall for the necessity of realizing the variable time scale of subgap processes both in theory and in diverse interpretations of different groups of experiments ( dynamic , kinetic , thermodynamic ) which address excitations with very different life times . s. m. acknowledges the hospitality of the laboratoire de physique thorique et des modle statistiques , orsay and the support of the cnrs via the ens - landau foundation . we consider the system of weakly coupled dimerised chains . each chain is described by a usual electron - phonon hamiltonian ( peierls , ssh models ) . electron levels @xmath65 and wave functions @xmath294 are determined by equations @xmath295\mathbf{\psi}=e\mathbf{\psi } , \label{dirac}\ ] ] where @xmath296 are the pauli matrices , @xmath297 , @xmath298 are the components of electron wave functions near fermi points @xmath299 , and the real function @xmath182 is the amplitude of the alternating dimerization potential . the ground state of each chain is the peierls dielectric with the gap @xmath300 . the electron spectrum has the form @xmath301 ( in the following we shall put the fermi velocity @xmath302 , @xmath303 and , as everywhere , the plank constant @xmath304 ) . the excited states are solitons ( kinks ) , polarons and bi - solitons ( kink - antikink pairs ) which are characterized by electron levels localized deeply within the gap ( see the review @xcite ) . the one parametric family of configurations @xmath305 supporting the single split - off pair of levels @xmath18 can be written as @xmath306 evolving from a shallow potential well at @xmath307 through the stationary configuration for a polaron @xmath4 to the pair of diverging kinks at @xmath308 as shown at the figure [ fig : shapes ] . the potentials @xmath166 ( for the @xmath65 level filling @xmath165 ) as functions of @xmath65 are given as @xmath309 the translational mass can be found as @xmath310.\ ] ] consider the matrix element between levels @xmath18 in the peierls state . the wave function has two components @xmath311 according to @xmath312 . explicit expressions for split - off states are @xmath313 . the equation for the bound eigenstate ( [ dirac ] ) shows the following symmetry : @xmath314 , @xmath315 , @xmath316 . then @xmath317 , @xmath318 , with @xmath319 , which demonstrates explicitly the orthogonality of @xmath257 and @xmath320 . the matrix element in eq . ( [ pre0 ] ) becomes @xmath321 . at @xmath90 , @xmath322 , hence for identical chains the transition at the free gap @xmath323 is forbidden which removes the singularity at the gap threshold in a rigid system . but the true threshold at @xmath236 for the subgap absorption or tunnelling are not subjected to this selection rule since the wave functions of localized states associated to the optimal fluctuation are distributed over the momentum region @xmath324 . figure [ fig : shapes ] shows exact shapes @xmath183 of the equilibrium polaron ( upper thick line ) and of a well formed @xmath325 pair of solitons ( lower thick line ) . thin lines show exact shapes of optimal fluctuations necessary to create these states by tunnelling . notice the much less pronounced shapes for optimal fluctuations in compare to the final states which facilitates the tunnelling . of the equilibrium polaron , @xmath326 ( upper thick line ) and of a nearly formed ( @xmath327 ) pare of solitons ( lower thick line ) . thin lines show exact shapes of optimal fluctuations necessary to create these states by tunnelling.,height=207 ] figure [ fig : v1 ] plots the total energy @xmath328 of the single particle branch as a function of the associated energy of the bound state . @xmath329 , @xmath330 corresponds to the particle added to the unperturbed ground state , at the bottom of the continuous spectrum . @xmath331 is the mid - gap state reached for the limit of two divergent solitons when the total energy approaches the maximal value @xmath332 . in between , at @xmath333 , @xmath334 , the minimum corresponds to the stationary polaronic state . the short thin vertical line between plots @xmath65 and @xmath328 points to the configuration ( upper thin curve at the figure [ fig : shapes ] ) of the fluctuation necessary for tunnelling to the polaron ( the minimum of @xmath328 , upper thick curve at the figure [ fig : shapes ] ) . as a function of the energy @xmath65 of the associated bound state.,height=207 ] next three figures plot the total energies @xmath335 for different branches as a function of the energy @xmath65 of the associated bound state ( all in units of @xmath75 ) . branches are distinguished by their ordering at @xmath329 . figure [ fig : u12 ] corresponds to the potential @xmath336 which is below the bi - electronic threshold ; no branch is crossing @xmath337 axis , hence no final action is allowed and the current is zero . for different tunnelling branches @xmath335 as a function of the energy @xmath65 of the associated bound state . this figure corresponds to @xmath336 which is below the bi - electronic threshold.,height=207 ] figure [ fig : u14 ] corresponds to the potential @xmath338 which is between the bi - electronic threshold @xmath339 and the polaronic one @xmath340 ; the bi - electronic branch crosses the axis @xmath337 at the point @xmath263 , the action is finite , hence a nonzero tunnelling of two electrons is allowed . as a function of the energy @xmath65 of the associated bound state this figure corresponds to @xmath338 which is between the thresholds for tunnelling of bipolarons and polarons.,height=207 ] figure [ fig : u18 ] corresponds to the potential @xmath341 , above the bi - electronic threshold @xmath342 , exactly at the polaronic one @xmath340 . now two parallel processes of one- and two- electron tunnelling are allowed . as a function of the energy @xmath65 of the associated bound state ( over a selected interval ) . this figure corresponds to @xmath341 which is just at the polaronic threshold , above the bipolaronic one.,height=207 ] the figure [ fig : u16+c ] corresponds to the potential @xmath343 between the bi - electronic threshold @xmath342 , and the polaronic one @xmath340 . contrary to the figure [ fig : u14 ] , the coulomb interaction is taken into account which lifts the degeneracy of the earlier crossing point of three branches . the one electron term @xmath4 does not cross @xmath337 axis yet , but it passes below two other terms in a vicinity of their crossing . now the optimal bi - electronic tunnelling takes place via a sequence of two single electronic processes confined in time . as a function of the energy @xmath65 of the associated bound state . this figure corresponds to @xmath343 which is between the thresholds for tunnelling of bipolarons and polarons . contrarily to the figure [ fig : u14 ] , the coulomb interaction is taken into account which lifts the crossing degeneracy.,height=207 ] we need to perform the integration over @xmath177 around the extremal taking into account the zero modes related to translations of positions @xmath181 of the instanton centers . the path integration over the gapless mode @xmath186 is important , particularly for the matrix element : the overlap of wave functions evolves following @xmath344 while their localization follows the evolution of @xmath141 . we shall work within the zero dimensional reduction of eq.([reduction ] ) . @xmath346d[x_j]j_{x_j}j_{e_j}\exp(-s)\nonumber\\ \psi_{a}^{\ast}(x - x_{a}(\tau_1),e(\tau_1))\psi_{a}(y - x_{a}(\tau_2),e(\tau_2 ) ) \psi_{b}(x - x_{b}(\tau_1),-e(\tau_1))\psi_{b}^{\ast}(y - x_{b}(\tau_2),-e(\tau_2 ) ) , \label{jjj}\end{aligned}\ ] ] where @xmath347 , @xmath348 are the jacobians of the transformation ( [ reduction ] ) . ( @xmath349 is the number of points for the intermediate discretization of the time axis . ) we integrate over the zero mode @xmath186 and take into account fluctuations of the instanton shape due to variations of the parameter @xmath350 . the action in ( [ reduction ] ) has the form @xmath351=\sum_{j = a , b}dt\left(m(e_{j}(t))\dot{x_{j}}^{2}/2+f(e_{j})\dot{e}_{j}^{2}/2 + { v}_u(e_j)\right)\ ] ] with @xmath352 from ( [ s2(u ) ] ) . the integration over @xmath353 is carried out exactly after the transformation @xmath354 using the known expression here @xmath363 , and the same for @xmath364 , are functions of energies in these points which finally become @xmath365 . using fourier transforms , we rewrite the product of wave functions as @xmath366 integration over @xmath367 gives @xmath368 , @xmath369 and integration over @xmath370 gives @xmath371 . after integration over @xmath372 we arrive at the result ( [ pre0 ] ) . the factor @xmath373 in ( [ pre0 ] ) after integration over @xmath374 $ ] which was performed using again the equation ( [ dashen ] ) . i. latyshev , a. a. sinchenko , l. n. bulaevskii , v.n . pavlenko , p. monceau , jetp letters , * 75 , * 93 ( 2002 ) ; yu.i . latyshev , l. n. bulaevskii , t. kawae , a. ayari , and p. monceau , j. phys . iv france , * 12 * , pr9 , 109 ( 2002 ) . we view the pg as a partial filling of an expected clear spectral gap , which is adequate to cdw physics . another view , typical in the field of hihg - tc superconductors , is that the pg is a suppressiion of the expected metalic dos . these too approaches might be convergent . artemenko , a.f.volkov , sov . : jetp * 60 * , 395 ( 1984 ) ; k.m . munz , w.wonneberger , z.phys . b**79 * * , 15 ( 1990 ) ; a.m. gabovich , a.i . voitenko , phys . b * 52 * , 7437 ( 1995 ) ; k. sano eur . phys . j. b * 25 * , 417 ( 2002 ) _ and rfs . theirin_. s. brazovskii , n. kirova , `` electron selflocalization and superstructures in quasi one - dimensional dielectrics '' in _ soviet scientific reviews _ , i. m. khalatnikov ed . ( harwood ac . publ . , ny , 1984 ) , vol . * 5 * , p. 99 . latyshev , p.monceau , a. sinchenko , l. bulaevskii , s. brazovskii , t. kawae , t. yamashita , _ in _ proceedings of the international workshop on strongly correlated electrons in new materials ( scenm02 ) , j. physics a : * 36 * , 9323 ( 2003 ) . b. halperin , m. lax , phys . rev . * 148 * , a722 ( 1966 ) ; j. zittarts , j.s . langer , phys . rev . * 148 * , a741 ( 1966 ) ; i.m . lifshits , s.a . gredeskul and l.a . `` introduction to the theory of disordered systems '' , _ ( wiley , new york , 1988 ) . lifshitz , yu . m. kagan , sov . jetp * 35*,206 ( 1972 ) ; s.v . iordanskii , a. m. finkelshtein , sov . jetp * 35 * , 215 ( 1972 ) ; s. v. iordanskii and e. i. rashba , sov . : jetp * 47 * , 975 ( 1978 ) . epstein , a.g . macdiarmid , journal of molecular electronics * 4 * , 161(1988 ) ; z. vardeny , e. ehrenfreund , and o. brafman , m. nowak , h. schaffer , a. j. heeger , and f. wudl , phys . lett . * 56 * , 671 ( 1986 ) . s. brazovskii , n. kirova , jetp letters * 33 * , 4 ( 1981 ) _ and _ chemica scripta * 17 * , 171 ( 1981 ) ; d. k. campbell and a. r. bishop , phys . b 24 , r4859 ( 1981 ) ; k. fesser , a.r . bishop , d.k . campbell , phys . b , * 27 * , 4804 ( 1983 ) ; s. brazovskii , n. kirova , s. matveenko , sov . phys . : jetp * 59 * , 434 ( 1984 ) ; s. matveenko , sov . : jetp * 60 * , 1026 ( 1984 ) .
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we suggest a theory of internal coherent tunnelling in the pseudogap region where the applied voltage is below the free electron gap . we consider quasi 1d systems where the gap is originated by a lattice dimerization ( peierls or ssh effect ) like in polyacethylene , as well as low symmetry 1d semiconductors .
results may be applied to several types of conjugated polymers , to semiconducting nanotubes and to quantum wires of semiconductors .
the approach may be generalized to tunnelling in strongly correlated systems showing the pseudogap effect , like the family of high tc materials in the undoped limit .
we demonstrate the evolution of tunnelling current - voltage characteristics from smearing the free electron gap down to threshold for tunnelling of polarons and further down to the region of bi - electronic tunnelling via bipolarons or kink pairs . the interchain tunnelling is described in a parallel comparison with the on chain optical absorption , also within the subgap region .
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with the discovery of neutrino masses and mixing in neutrino oscillation experiments , leptogenesis @xcite has become the most attractive model of baryogenesis to explain the observed matter - antimatter asymmetry of the universe . this can be expressed for example in terms of the baryon - to - photon number ratio and is very well measured by cmb observations @xcite to be [ etabobs ] _ b^cmb = ( 6.2 0.15)10 ^ -10 . leptogenesis originates from the see - saw mechanism @xcite that is based on a simple extension of the standard model where right - handed ( rh ) neutrinos with a majorana mass matrix @xmath15 and yukawa couplings @xmath16 to leptons and higgs are added . within @xmath0 models , three rh neutrinos @xmath17 ( @xmath18 ) are nicely predicted and for this reason they are traditionally regarded as the most appealing theoretical framework to embed the seesaw mechanism . however , within the simplest set of assumptions inspired by @xmath0 models @xcite , barring strong fine - tuned degeneracies in the rh neutrino mass spectrum and using the experimental information from neutrino oscillation experiments , the traditional @xmath19-dominated leptogenesis scenario predicts an asymmetry that falls many orders of magnitudes below the observed one @xcite . this is because , within @xmath19-dominated leptogenesis , where the spectrum of rh neutrinos is hierarchical and the asymmetry is produced from the decays of the lightest ones , successful leptogenesis implies a stringent lower bound on their mass @xcite , @xmath20 . on the other hand , @xmath0 grand - unified theories typically yield , in their simplest version and for the measured values of the neutrino mixing parameters , a hierarchical spectrum with the rh neutrino masses proportional to the squares of the up - quark masses , leading to @xmath21 and therefore to a final asymmetry much below the observed one . however , it has been shown @xcite that , when the production from the next - to - lightest rh neutrinos @xcite and lepton flavour effects @xcite are simultaneously taken into account @xcite , the final asymmetry can be generated by the decays of the next - to - lightest rh neutrinos and allowed regions in the low energy neutrino parameter space open up . in this paper we proceed with the analysis of @xcite and present the resulting constraints on all low energy neutrino parameters . the paper is organized as follows . in section 2 we discuss the current experimental status on low energy neutrino parameters , we set up the notation and describe the general procedure to calculate the the asymmetry and find the constraints . in section 3 we first consider the case already studied in @xcite , when the dirac basis and the charged lepton basis coincide and then , in section 4 , we allow for a misalignment between the two bases not larger than that one described by the ckm matrix in the quark sector . finally , in section 5 we present a global scan in the space of parameters where all possible cases between the case of no misalignment and the case of a misalignment at the level of the ckm matrix are taken into account . we also discuss two scenarios , one at small @xmath22 and one at large @xmath22 , and show how , within @xmath0-inspired models , minimal leptogenesis could be tested in future low energy neutrino experiments . notice that our discussion is made within a non - supersymmetric framework . recently a study of @xmath0-inspired models within a supersymmetric framework has also enlightened interesting potential connections with lepton flavour violating decays and dark matter @xcite . an analysis of leptogenesis within left - right symmetric models , where a type ii seesaw contribution to the neutrino mass matrix is also present , has been performed in @xcite . within these models , the minimal type i scenario considered here represents a particular case recovered under specific conditions . after spontaneous symmetry breaking , a dirac mass term @xmath23 , is generated by the vacuum expectation value ( vev ) @xmath24 gev of the higgs boson . in the see - saw limit , @xmath25 , the spectrum of neutrino mass eigenstates splits in two sets : three very heavy neutrinos , @xmath26 and @xmath27 respectively with masses @xmath28 almost coinciding with the eigenvalues of @xmath15 , and three light neutrinos with masses @xmath29 , the eigenvalues of the light neutrino mass matrix given by the see - saw formula @xcite , m_= - m_d1d_mm_d^t , that we wrote in a basis where the majorana mass matrix is diagonal defining @xmath30 . the symmetric light neutrino mass matrix @xmath31 is diagonalized by a unitary matrix @xmath32 , @xmath33 with @xmath34 , that , in the basis where the charged lepton mass matrix is diagonal , can be identified with the lepton mixing matrix . neutrino oscillation experiments measure two neutrino mass - squared differences . for no one has @xmath35 and @xmath36 . the two heavier neutrino masses can therefore be expressed in terms of the lightest neutrino mass @xmath22 as [ m2m3nor ] m_2 = , m_3 = , where we defined @xmath37 and @xmath38 @xcite . recently , a conservative upper bound on the sum of neutrino masses , @xmath39 , has been obtained by the wmap collaboration @xcite combining wmap 7 years data plus baryon acoustic oscillations observations and the latest hst measurement of @xmath40 . considering that it falls in the quasi - degenerate regime , it straightforwardly translates into [ upperbound ] m_1 < 0.19ev ( 95% cl ) . we will adopt the following parametrization for the matrix @xmath32 in terms of the mixing angles , the dirac phase @xmath41 and the majorana phases @xmath42 and @xmath43 @xcite @xmath44 and the following @xmath45 ranges for the three mixing angles @xcite [ twosigma ] _ 12= ( 31.3 ^ -36.3^ ) , _ 23= ( 38.5 ^ -52.5^ ) , _ 13= ( 0 ^ -11.5^ ) . in the case of io the expression of @xmath46 in terms of @xmath22 becomes [ m2m3inv ] m_2 = , while the expression for @xmath47 does not change . with the adopted convention for the light neutrino masses , @xmath48 , the case of io corresponds to relabel the column of the leptonic mixing matrix performing a column cyclic permutation , explicitly @xmath49 the predicted baryon - to - photon ratio @xmath50 is related to the value of the final @xmath51 asymmetry @xmath52 by @xcite [ etab ] _ b 0.9610 ^ -2 n_b - l^f , where @xmath53 is the @xmath54 number in a co - moving volume that contains on average one rh neutrino @xmath55 in thermal ultra - relativistic equilibrium abundance ( @xmath56 ) . the dirac mass matrix can be diagonalized by a bi - unitary transformation m_d = v_l^d_m_du_r , where @xmath57 . the matrix @xmath58 can be obtained from @xmath59 , @xmath32 and @xmath60 , considering that it provides a takagi factorization @xcite of @xcite m^-1 d^-1_m_dv_lud_mu^tv_l^td^-1_m_d , explicitly [ takagi ] m^-1 = u_rd_m^-1u_r^t . for non degenerate @xmath61 , the matrix @xmath58 can be determined noticing that it diagonalizes @xmath62 , i.e. m^-1(m^-1)^ = u_rd_m^-2u_r^ . this relation determines @xmath58 unless a diagonal unitary transformation , since any @xmath63 is also a solution . however , given a @xmath64 , one can fix @xmath65 from the eq . ( [ takagi ] ) , [ dphi ] d _ = and in doing so @xmath58 is unambiguously determined . inspired by @xmath0 relations , we can parameterize the eigenvalues of @xmath66 in terms of the up quark masses as [ so(10 ) ] _1= _ 1m_u , _2= _ 2 m_c , _3= _ 3m_t . within @xmath0 models one can expect @xmath67 and we will refer to this case . the reader is invited to read ref . @xcite for a more comprehensive discussion about these @xmath0-inspired relations . notice however that our results will be valid for a much broader range of values , since , quite importantly , it turns out that they are independent of @xmath68 and @xmath69 provided @xmath70 and @xmath71 . with the parametrization eq . ( [ so(10 ) ] ) and barring very special choices of parameters where the rh neutrino masses can become degenerate @xcite , and @xmath72 . this is clearly a conservative condition , since the asymmetry gets enhanced when @xmath73 or @xmath74 . however , in this way , we only neglect very special points in the parameter space yielding @xmath75 and @xmath76 . we will comment again later on this point . ] the rh neutrino mass spectrum is hierarchical and of the form ( for generic expressions in terms of the low energy parameters , see ref . @xcite ) [ alpha ] m_1:m_2:m_3=(_1m_u)^2:(_2m_c)^2:(_3m_t)^2 . as we said , the values of @xmath77 and @xmath78 are actually irrelevant for the determination of the final asymmetry ( unless @xmath77 is unrealistically large to push @xmath79 from @xmath80 above the lower bound @xmath81 gev to achieve successful @xmath19 leptogenesis ) . on the other hand , the value of @xmath82 is relevant to set the scale of the mass @xmath83 ( valid for @xmath84 ) of the next - to - lightest rh neutrino mass , but it does not alter other quantities crucial for thermal leptogenesis , such as the amount of wash - out from the lightest rh neutrinos . defining the flavoured @xmath85 asymmetries as _ 2-_2-_2 _ 2+_2 , these can be calculated using @xcite [ eps2a ] _ 2 \ { im+ } , where ( x)=x and @xmath86 is the decay rate of the rh neutrino @xmath1 into the flavor @xmath87 with couplings given by the yukawa s matrix @xmath16 . we will assume an initial vanishing @xmath1-abundance instead of an initial thermal abundance as in @xcite . in this way , a comparison of the results in the two analyses gives a useful information about the dependence of the final asymmetry on the initial @xmath1 abundance when successful leptogenesis is imposed . let us now define the flavored decay parameters as k_i = = , where @xmath88 is the hubble rate , m_= 1.0810 ^ -3ev , @xmath89 is the number of the effective relativistic degrees of freedom and @xmath90 is the planck mass . the total decay parameters are then just simply given by @xmath91 . it is also convenient to introduce the quantities @xmath92 . from the decay parameters one can then calculate the efficiency factors that are the second needed ingredient , together with the @xmath85 asymmetries , for the calculation of the final asymmetry . these can be well approximated by the following analytical expression @xcite at the peak for @xmath93 . for @xmath94 , the difference is below @xmath95 . ] ( k_2,k_2 ) = _ -^f(k_2,k_2)+ _ + ^f(k_2,k_2 ) , where the negative and the positive contributions are respectively approximately given by [ k- ] _ -^f(k_2,k_2)-2p_2 ^ 0 e^-3k_2 8 ( e^p_2 ^ 02n_n_2(z_eq ) - 1 ) , @xmath96 and [ k+ ] _ + ^f(k_2,k_2 ) ( 1-e^-k_2z_b(k_2)n_n_2(z_eq)2 ) , where z_b(k_2 ) 2 + 4k_2 ^ 0.13e^-2.5k_2=o(110 ) . the @xmath0-inspired conditions @xmath97 , yield a rh neutrino mass spectrum with @xmath98 , though , as we already noticed , this spectrum is obtained for a broader range of @xmath99 values . in this situation , the asymmetry is dominantly produced from @xmath1 decays at @xmath100 in a two flavour regime , i.e. when final lepton states can be described as an incoherent mixture of a tauon component and of coherent superposition of a an electron and a muon component . therefore , at the freeze - out of the @xmath1 wash - out processes , the produced asymmetry can be calculated as the sum of two contributions , [ nbmltm2 ] n_b - l^t~m_2 _ 2(k_2,k_2)+ _ 2e+(k_2,k_2e+ ) , where @xmath101 stands for @xmath102 and @xmath103 . more precisely , notice that each flavour contribution to the asymmetry is produced in an interval of temperatures between @xmath104 $ ] and @xmath105 $ ] , with @xmath106 . at @xmath107 the coherence of the @xmath108 quantum states breaks down and a three flavour regime holds , with the lepton quantum states given by an incoherent mixture of @xmath109 , @xmath110 and @xmath111 flavours . the asymmetry has then to be calculated at the @xmath19 wash - out stage as a sum of three flavoured contributions . the assumption of an initial vanishing @xmath1-abundance allows to neglect the phantom terms in the muon and in the electron components @xcite so that the final asymmetry can be calculated using the expression n_b - l^f _ 2e+(k_2 e+ ) e^-38k_1 e+ p^0_2p^0_2e+_2e+(k_2 e+ ) e^-38k_1 + _ 2 ( k_2 ) e^-38k_1 . notice that successful leptogenesis relies on points in the parameter space where one out of the three @xmath112 . from this point of view the constraints on low energy neutrino experiments that we will obtain should be quite stable against effects that could enhance the asymmetry such as a resonant enhancement for special points where @xmath113 . such effects are however still able to relax the lower bound on @xmath114 and on the @xmath115 , since the @xmath116 s do not depend on @xmath114 . we start from the case @xmath117 that has been studied already in @xcite deriving constraints in the plane @xmath118 for no . here we show constraints on all low energy neutrino parameters , including the case of io . let us first discuss the case of no . in fig . 1 we plotted the final asymmetry @xmath50 for the same three sets of values of the involved parameters as in the fig . 4 of ref . @xcite , where these three choices were corresponding to three different kinds of solutions for successful leptogenesis . + + + + this time the third solution ( right panel ) , is suppressed and successful leptogenesis is not attained . in @xcite , this was the only solution corresponding to a final asymmetry dominantly in the muon flavour instead than in the tauon flavour ( as for the first two ) . the suppression that we find now is explained partly because we are adopting a correct determination of the phases in the @xmath58 matrix ( cf . ( [ dphi ] ) ) and partly because we are now assuming an initial vanishing @xmath1-abundance instead than an initial thermal one . we will see however that , allowing for @xmath119 , this kind of solution will again yield successful leptogenesis in some allowed regions of the parameter space , characterized in particular by large values of @xmath120 . the solution in the central panel is also partly suppressed and successful leptogenesis is not attained . however , in a parameter scan , we find that this kind of solution can still give successful leptogenesis for slightly different values of the parameters than those indicated in the figure caption . in this case the difference with respect to the results in @xcite is explained just in terms of the different assumption on the initial abundance . this dependence on the initial conditions is due to the fact that @xmath121 , i.e. the solution falls in the weak wash - out regime at the production . finally , the first solution ( left panel ) is fully unchanged . it therefore exhibits a full independence of the initial conditions and this is in agreement with the fact that it respects all the necessary conditions for the independence on the initial conditions found in @xcite . notice that these conditions also enforce an efficient wash - out of a possible pre - existing asymmetry . a scan in the space of parameters confirms that these three solutions obtained for special sets of values are actually representative of the three general kinds of solutions that come out and , therefore , the drawn conclusions apply in general . in the panels of figure 2 we show the results of such a scan that highlight the allowed regions in the parameter space projected on different two - parameter planes . + + the scatter plots have been obtained scanning the three mixing angles @xmath122 and @xmath6 over the @xmath123 ranges eqs.([twosigma ] ) , the three phases @xmath124 over the ranges @xmath125 $ ] and the absolute neutrino mass scale for @xmath126 . these ranges also coincide with those shown in the plots , except for @xmath22 where the plots are for @xmath127 simply because no allowed solutions have been found for lower values . the shown results have been obtained in two steps . a first scan of @xmath128 points has yielded a first determination of the allowed regions . with a second scan of additional @xmath129 points , restricted to the excluded regions , we have then more robustly and sharply determined the contours of the allowed regions . notice that these regions have no statistical significance and the random values of the parameters have been generated uniformly . in the top left panel the three rh neutrino masses are plotted versus @xmath22 . we have also plotted the lower bound on the reheating temperature calculated as [ trhmin ] t_rh^min . this calculation relies on the fact that in the case @xmath117 the solutions , as we commented , always fall in a tauon @xmath1-dominated scenario . it can be seen that the lowest bound is given by @xmath130 that in a supersymmetric version , if unchanged , would be marginally reconcilable with the upper bound from the gravitino problem @xcite . this is another reason to extend our investigation to cases with @xmath119 in next sections . in the top central panel we have then plotted the allowed region in the @xmath118 plane that can be compared with an analogous figure in @xcite . here , however , we show only those points that respect the condition @xmath131 but for 2 different values of @xmath132 ( in @xcite we were only showing points for @xmath133 ) . the ( red ) star represents a point found for a minimum value @xmath134 . this point basically roughly indicates where the maximum of the asymmetry occurs in the parameter space for a fixed value of @xmath135 . we will continue to use this convention ( yellow circles for @xmath133 , green squares for @xmath136 and red stars for minimum found @xmath135 value ) throughout the next figures . the structure of the allowed region in the @xmath118 plane can be understood as follows . since @xmath137 and @xmath138 , we immediately deduce that a large lepton asymmetry in the tau flavor may be produced only for sufficiently large values of @xmath22 . this is rather easy to understand . if @xmath22 tends to zero , we go into the so - called decoupling limit , @xmath139 . as the @xmath85 asymmetry needs ( at least ) two heavy states to be generated at the one - loop level , and disregarding the contribution from the @xmath19 , @xmath140 must vanish . the wash - out parameter @xmath141 is @xmath142 @xcite and therefore the final baryon asymmetry may be estimated to be _ b510 ^ -3_2 5 ( ) 10 ^ -10 , which requires m_1()^210 ^ -3ev , for no . this estimate holds if the wash - out from the interaction with @xmath19 is negligible , _ i.e. _ @xmath143 . of course , the smaller is @xmath22 , the smaller @xmath144 needs to be . for @xmath145 ev , the only possibility is that @xmath144 is significantly below unity . extending the analysis of ref . @xcite , one finds [ pp ] s_13(-2 ) > 0.04 . to get the feeling of the figures involved , we may set @xmath146 and find that the wash - out mediated by the @xmath19 s vanishes for an experimentally allowed value of the mixing between the first and the third generation of lh neutrinos , @xmath147 in agreement with our numerical results . if @xmath22 is larger than @xmath148 ev , then @xmath149 is allowed and @xmath6 can be taken to be vanishing . notice also that the lower bound eq . ( [ pp ] ) on @xmath6 increases with @xmath150 . this nicely reproduces the linear dependence emerging from the numerical results in the _ left column middle panel _ for the plane @xmath151 and that is described , roughly for @xmath133 and more accurately for @xmath136 , by [ linear ] _ 2344^+4(_13 - 7^ ) , represented with a dashed line in the panel . in the _ top right panel _ we show the allowed region in the plane @xmath152 . the @xmath85 non conserving terms in neutrino oscillation probabilities can be expressed in terms of the _ jarlskog invariant _ @xmath153 given by @xcite j_cp & = & im[u_3u_e 2u_2^u_e3^ ] + & = & c_12s_12c_23s_23c^2_13s_13 , such that p___-p_|_|_= 4j_cp_k > js_;kj(m^2_kjl2e ) , where @xmath154 . in the bottom left panel we show the allowed points in the plane @xmath155 . it can be noticed that a non zero value of @xmath153 is not crucial . looking at the bottom - central panel , it is interesting to notice that the allowed regions for the majorana phases are centered approximately around @xmath156 and @xmath157 . these play a role in the determination of the _ effective majorana mass _ of @xmath158 in @xmath159 decays that is given by m_ee & = & |_im_iu_ei^2 | + & = & | m_1c_12 ^ 2c_13 ^ 2e^2i+m_2s_12 ^ 2c_13 ^ 2 + m_3s^2_13e^2i(- ) | . in the bottom - right panel one can see how there is a precise relation between @xmath160 and @xmath22 , given approximately by @xmath12 . it can be also noticed that there is quite a strict lower bound @xmath161 . lowest values @xmath162 are the most favoured ones in this case . though current planned experiments will not be able to test the full allowed range , it is still interesting that they will test it partially , tightening the constraints on the other parameters as well . we have also made an interesting exercise . we determined the constraints without making use of any experimental information on the mixing angles and letting them just simply variate between @xmath163 and @xmath164 . the results are shown in fig . [ thijarb ] . first , notice that the lower bound on @xmath22 relaxes of a few orders of magnitude ( see left panel ) . then notice quite interestingly that small values @xmath165 are well allowed for @xmath166 but values @xmath167 would have been very marginally consistent . therefore , the current bound @xmath165 seems to match quite well with successful @xmath0-inspired leptogenesis on the other hand , values @xmath168 would have been more optimal for @xmath165 than the current experimental large atmospheric values ( see the central panel in the figure ) . however , they are still allowed thanks to the observed range of values of the solar neutrino mixing angle ( see the right panel ) . for the solar neutrino mixing angle there is no real favourite range of values for @xmath165 . let us now discuss the results for io . it has been shown @xcite that in grand unified models with conventional type i seesaw mechanism one can always find , for any no model satisfying the low energy neutrino experimental constraints , a corresponding io model . therefore , though they exhibit some unattractive features that quite strongly disfavour them ( e.g. instability under radiative corrections ) , io models within grand unified theories are not unequivocally excluded . it is therefore legitimate to check whether the requirement of successful leptogenesis can somehow provide some completely independent information . we repeated the same scan performed in the case of no and the results are shown in figure [ constrio ] , the analogous of the figure [ constrno ] for the no case . one can see that io is only very marginally allowed . for @xmath133 , there is only a small region at large values of @xmath169 . extending the analysis in ref . @xcite , this is explained by the fact that the wash - out parameter @xmath144 turns out to be k_110 ^ 3evm_atm , while in the no case @xmath144 was proportional to @xmath170 . this constrains @xmath171 to be as large as possible , thus ruling out small values of @xmath22 . it is interesting to notice that in this case the allowed values for @xmath7 lie in the second octant and correspond to the largest ones compatible with the current experimental limits . the allowed values of the effective neutrino mass fall in a narrow range , @xmath172 . + + therefore , io will be in any case fully tested from cosmology and @xmath159 experiments during next years . we will see that this conclusion will hold also allowing @xmath119 . as usual , in the plots the red star corresponds to the minimum value of @xmath135 for which we have found a solution , @xmath173 . the corresponding set of values indicates approximately where the asymmetry has a maximum for a fixed @xmath135 value . for this choice of values , in figure [ io ] , we show the plots of the rh neutrino masses , of the asymmetry @xmath50 , of the @xmath85 asymmetries @xmath174 , @xmath175 , of @xmath176 and of @xmath177 versus @xmath22 . + one can see how the heaviest rh neutrino mass @xmath178 decreases with @xmath22 much faster and at @xmath179 one has @xmath180 . therefore , as one can see from the central top panel , the @xmath85 asymmetries are this time strongly suppressed at @xmath181 . on the other hand , in the range @xmath182 the @xmath85 asymmetries are large enough that successful leptogenesis is still possible . notice , that this kind of solutions are a sort of modification of the solution obtained at large @xmath22 values for no , simply shifted at somehow larger values . the asymmetry is therefore strongly depending on the initial conditions ( @xmath183 ) . the first kind of solution , at small @xmath22 values , is completely absent . therefore , though io is strongly disfavoured , it is not completely ruled out , a conclusion somehow very similar to that one obtained from completely independent arguments @xcite . in this case , however , leptogenesis provides quite a precise quantitative test . we can conclude this section saying that these results confirm and complete those shown in @xcite . in particular it is confirmed that there are viable solutions corresponding to the different points shown in the figures falling in the currently experimentally allowed ranges of the parameters , . the model is therefore not ruled out . a further step is now to understand whether the model is predictive , excluding regions of the parameter space that future experiments can test . from the figures , as we have discussed , it is clear that assuming @xmath117 such excluded regions exist and therefore one obtains interesting constraints . however , it is important to go beyond the simple condition @xmath117 in order to test the stability of the constraints for variations of @xmath59 . this is the main objective of the next sections . we now study how the constraints change when a misalignment between the physical basis where @xmath66 is diagonal and the flavour basis , where the charged lepton mass matrix is diagonal , is considered , corresponding to @xmath119 . since @xmath59 is unitary , we can parameterize it similarly to the leptonic mixing matrix introducing three mixing angles , one dirac - like phase and two majorana - like phases , @xmath185 where we defined @xmath186 and @xmath187 . therefore , we have now six additional parameters that give much more freedom . we will not explore the full parameter space but , in the spirit of @xmath0-inspired models , we will allow only small mixing angles @xmath188 at the level of the mixing angles in the ckm matrix . as a first definite example we repeat the analysis performed for the case @xmath117 for a definite case where the @xmath189 are exactly equal to the mixing angles in the ckm matrix and therefore we set @xmath190 , @xmath191 , @xmath192 , where the latter is the measured value of the cabibbo angle . for no the results are shown in figure 6 . + + there is a first result to highlight : @xmath135 values as low as @xmath193 are now allowed . this is an interesting result in connection with the study of realistic @xmath0 models . at the same time this result also implies slightly lower values of @xmath114 and consequentially of the minimum value of @xmath194 that can be now as low as @xmath195 , as it can be noticed in the top - left panel in fig . 6 . in this case we have more generally calculated the minimum reheat temperature as [ trhmin2 ] t_rh^min , considering that in the case the asymmetry at the production can be either tauon dominated or @xmath108 dominated . this is because the third kind of solution that was highly suppressed in the case @xmath117 , the right panel in fig . 1 , becomes now viable and is @xmath108 dominated , as we will discuss soon in more detail . notice that if we compare the allowed points for @xmath136 with those found for @xmath117 , the constraints on the low energy neutrino parameters are now less stringent . in particular an allowed region for values @xmath196 is also found for very small values of @xmath6 . indeed , in the case @xmath117 , and for small values of @xmath22 , the suppression of the wash out value @xmath144 imposed a lower bound on @xmath6 . by choosing @xmath197 introduces the possibility of getting vanishing @xmath144 even for zero @xmath6 angles . extending the analysis of ref . @xcite , one finds indeed that one configuration where @xmath144 is smaller than unity is attained if @xmath198 ( mod @xmath199 ) and @xmath200\,[m_{\rm sol}^4/(m_2 ^ 3\ , m_3)]\sim -(5/12)(m_{\rm sol}/m_{\rm atm})\sim -10^{-1}$ ] . this implies @xmath201 ( mod @xmath199 ) , as confirmed by our numerical results . including in the analysis the atmospheric neutrino mixing angle , as one can see from the panel with the constraints in the @xmath151 plane , only values @xmath202 for @xmath165 are allowed for @xmath203 . notice that , for @xmath204 , the allowed region in @xmath205 only marginally overlaps , at small values of @xmath7 , with the region for the case @xmath117 . this means that a measurement of these three quantities can distinguish between the two cases , @xmath117 and @xmath197 , and not all combinations of these three quantities seem to be possible . we will be back on this point in the next section . in figure 7 we plotted the relevant quantities for three particular choices of the parameters , as indicated in the figure caption , corresponding to the three kinds of solutions found for @xmath197 . + + + + these three sets of values correspond to the three kinds of solutions that are found in the scan plots . the first two sets , corresponding to the left and central panels , give a tauon dominated asymmetry , while the third set , corresponding to the right panels , yields a muon dominated asymmetry . notice that these three kinds of solutions are the same three kinds , with slight modifications , found for the case @xmath117 . however , one can see that this time the third kind of solution , where the final asymmetry is muon dominated , also yields successful leptogenesis . the major difference that explains this result , is that for @xmath197 the flavoured @xmath85 asymmetries @xmath175 are not as hierarchical as in the case @xmath117 , as it can be clearly seen in the three panels showing the @xmath85 asymmetries in figure 7 . we have also repeated , as for the case @xmath117 , the exercise to leave the mixing angles completely free , without imposing any experimental constraint finding the results shown in fig . [ thijarb2 ] . one can see that in this case the points found when the current experimental constraints are imposed ( the green points ) fall in more marginally allowed regions , also for @xmath6 . this might suggest that @xmath117 seems to be a more attractive case than @xmath197 . finally , we also present in figure 9 the constraints obtained for io . even though there is again a remarkable suppression of the allowed regions compared to no , they are somehow less restrictive than for @xmath117 . in particular now a broader range of values for @xmath22 is allowed and @xmath7 can be as low as @xmath206 for @xmath204 . + + this is also confirmed by the fact that lowest allowed value is now @xmath207 , much lower than in the case @xmath117 ( it was @xmath208 . however , it is still fair to say that the io case is only marginally allowed and certainly disfavoured compared to the no case . the two specific cases that we discussed , @xmath117 and @xmath197 , suggest an interesting sensitivity of @xmath0-inspired leptogenesis to slight deviations of @xmath59 from the identity . this sensitivity was absent in the results found in @xmath19-dominated leptogenesis @xcite . in this way it seems that one could even gain some information on @xmath59 from low energy neutrino experiments . however , there is a potentially dangerous aspect of such a sensitivity : if for a slight variation of @xmath59 the entire space of low energy neutrino parameters becomes accessible , then any chance to test @xmath0-inspired leptogenesis is lost . on the other hand , from a comparison of the results obtained for the two definite cases , @xmath117 and @xmath197 , one can understand that this does not happen . one can still suspect that for a continuous variation of the parameters in @xmath59 , such that @xmath59 changes from @xmath117 to @xmath197 , new solutions appear so that any point in the space of the low energy neutrino parameters can be obtained for a proper choice of @xmath59 . in this section we study this issue . we perform a global continuous scan of the parameters for @xmath59 between @xmath117 and @xmath197 . obviously a precise limit @xmath197 for such a global scan is somehow arbitrary . it should be therefore taken as a working assumption defining @xmath0-inspired leptogenesis , even more than the condition @xmath97 that , as we stressed many times , should not be regarded as a very restrictive assumption . clearly within well defined realistic @xmath0 models , more specific conditions on @xmath59 should be obtained . in any case one expects that if the @xmath59 satisfies the condition @xmath209 , then the allowed values for the low energy parameters should fall in the allowed regions for @xmath0-inspired leptogenesis . therefore , in this section we present the constraints on the low energy neutrino parameters for a continuous variation of the values of the mixing angles @xmath188 in the range @xmath210 ( i.e. for @xmath209 ) . more explicitly the shown scatter plots are obtained for the low energy neutrino parameters scanned over exactly the same ranges as for the case @xmath117 . the three angles in @xmath59 are scanned over the ranges @xmath211 , @xmath212 , @xmath213 , while the three phases are scanned over @xmath125 $ ] . in order to determine the allowed regions , we have followed the same strategy as in the case @xmath117 , with a similar total number of scanned points , @xmath214 . the results for no are shown in figure 10 . one can see how the allowed regions are approximately given by a super - position of those found for @xmath117 and @xmath197 plus all intermediate solutions . the result is that now the correlations among the parameters found in the two special cases seem to disappear . there are however still interesting non trivial constraints . what clearly survives is that the allowed points still cluster within two distinguished ranges of values for @xmath22 , one range at small values , @xmath215 , and one range at high values , @xmath216 , a distinction that is sharp for @xmath217 ( green squares ) while it is softer for @xmath133 ( yellow circles ) . + + + at the same time one can see that a global scan actually shows a slight correlation between @xmath22 and @xmath6 in the low @xmath22 range while the interesting linear dependence between @xmath6 and @xmath7 found for @xmath117 seems now to be lost . however , it should be considered that these plots are projections on two - parameters planes of an allowed region in a seven - parameter space . therefore , only a full multi - parameters analysis would be able to unreveal correlations involving more than two parameters . nevertheless , thanks to the distinct analysis that we carried out for the two special cases @xmath117 and @xmath197 , one can catch sight of an interesting correlation among @xmath22 , @xmath218 , @xmath6 and @xmath160 . to this extent , this time we have also plotted the constraints in the plane @xmath219 , showing how the lower bound on @xmath160 increases with @xmath6 . in order to find out whether the linear dependence between @xmath6 and @xmath7 found for @xmath117 ( cf . ( [ linear ] ) ) still holds for a global scan , we show in fig . 11 the same constraints as in fig . 10 imposing the condition @xmath5 , since the linear dependence was found in that range of values . we only show the constraints on the relevant parameters , therefore only those in the plane @xmath151 , in the plane @xmath220 and in the plane @xmath219 . this time we could also easily find points for @xmath221 ( red stars ) , showing again how allowing for a @xmath119 the allowed regions get larger . one can see that the quite clear linear dependence eq . ( [ linear ] ) between @xmath6 and @xmath7 holding for @xmath117 , now turns more , for the red star points at @xmath221 , into an allowed region below the dashed line showed in the figure and corresponding approximately to [ th13th23 ] _ 23 49^+ 0.65(_13 - 5^ ) . this result should be also understood in terms of the condition @xmath222 ( cf . ( [ pp ] ) ) when a very small @xmath59 is allowed clearly yielding a dispersion around the linear dependence eq . ( [ linear ] ) . notice that inside this region there are still sort of sub - regions that seem to be excluded . we can summarize these results saying that , at low values of @xmath5 , there is an interesting testable constraints in the plane @xmath151 given by the relation eq . ( [ th13th23 ] ) . in particular experiments that are already taking data such as the nuclear reactor experiment double chooz @xcite and the long baseline experiment t2k @xcite have the capability of a @xmath223 discovery of values @xmath224 . our results seem to suggest that if such high @xmath6 values will not be found , then a restricted range of values for @xmath7 is predicted . for example , if @xmath225 then @xmath226 , and if @xmath227 then @xmath202 . such a constraint on @xmath7 should be also tested during next years with quite a good accuracy by the t2k experiment @xcite . these constraints in the plane @xmath151 should be considered at this level indicative , and should also consider that they are quite sensitive to the value of @xmath135 . notice that at the same time , cosmological observations and/or neutrinoless double beta decay experiments should also be able to test the condition @xmath228 . it should be therefore appreciated that this scenario will be tested during next years . it is also interesting to notice ( see right panel in fig . 11 ) that there is a linear dependence between @xmath160 and @xmath6 as well . in particular , for @xmath229 , at large values @xmath230 one has @xmath231 and even for @xmath224 one has @xmath232 . these values for @xmath160 are below the sensitivity of future planned experiments ( @xmath233 ) such as exo @xcite . however , at least , @xmath160 can not be arbitrary small but has a lower bound that , for sufficiently large @xmath6 values , is @xmath234 times below the currently planned reachable experimental sensitivity , a very small value but maybe not completely hopeless . within the two - parameter analysis we are presenting , we can not draw sharper predictions but is seems quite plausible that from a more involved multi - parameter analysis precise correlations could emerge , maybe also involving the solar neutrino angle @xmath218 . in this respect the central panel in figure 11 suggests that the solar mixing angle could indeed play also a role and that maybe sharper predictions in the 3 parameter space @xmath235 exist . we can also study how the allowed regions would reduce requiring large values @xmath236 . the results are shown in figure 12 . one can see that in this case one obtains very clear constraints that will allow to test this scenario during next years in a quite unambiguous way . first of all from the fig . 10 , thanks to the very precise values of the majorana phases , one can notice that there is a very clear relation between @xmath22 and @xmath160 . second , one can see from the left panel of fig . 12 how there is an upper bound @xmath13 for @xmath229 . for values of @xmath237 , one has even @xmath238 . it should be said however that at these large @xmath22 values , one typically obtains a final asymmetry that depends on the initial conditions . since we are assuming vanishing initial @xmath1 abundance and vanishing initial asymmetry , these constraints should be regarded as the most stringent ones , but likely also the best motivated ones . finally , we repeated the global scan for io as well and the results are shown in fig . 13 . + + + one can see how the allowed regions somehow merge those found for the two extreme cases @xmath117 and @xmath197 . there is therefore nothing really new . io is quite strongly constrained and it will be fully tested in next years . in particular we can notice again how there is a clear lower bound on @xmath7 rather than an upper bound as in no . more particularly , one can notice that the allowed region in the plane @xmath152 is approximately described by [ m1th23io ] _ 2343^+12^(0.2ev / m_1 ) ( the dashed line in the upper right panel ) . it is then quite interesting that @xmath0 inspired leptogenesis is able to distinguish no and io even at @xmath11 , when the same values of @xmath160 and of @xmath239 ( the quantity tested by cosmological observations ) are found both for io and for no . from this point of view @xmath0 inspired leptogenesis provides a way to solve this ambiguity . we have derived constraints on the low energy neutrino parameters from @xmath0-inspired leptogenesis . our investigation shows that even minimal leptogenesis , based on a type i seesaw mechanism and assuming a thermal production of the rh neutrinos and with a traditional high mass scale rh neutrino spectrum , can be testable within a well motivated framework , where the see - saw parameter space is restricted by the @xmath0-inspired conditions . the role played by the @xmath1 decays is crucial in this respect , not only in re - opening the viability of these models . the presence in the @xmath1-dominated regime of a double stage , a production stage and a lightest rh neutrino wash - out stage , seems to introduce , as shown simultaneously both by the numerical and by the analytical results , a strong direct dependence on neutrino mixing angles as well , in addition to the dependence on the absolute neutrino mass scale , already found in usual @xmath19-dominated leptogenesis @xcite . in the significant case of no with low @xmath22 values , the neutrinoless double beta decay effective mass seems to be too small to be measured but not arbitrary small and in any case future experimental results can be anyway useful to restrict the allowed regions for the other parameters and sharpening the predictions . the results for @xmath119 seems also to be sensitive to @xmath59 itself and they therefore suggest that there is an opportunity to gain information on it , an interesting point within studies of specific @xmath0 models . it is quite interesting that there is an allowed region in the parameter space that allows large values of @xmath6 testable with on - going reactor neutrino experiments and that for these large values the models favours either large or small @xmath7 values depending whether @xmath5 or @xmath240 . in the small @xmath22 range it is also interesting that the constraints are completely independent of any assumption on the initial conditions , a point that maybe makes this option more attractive . it is actually quite interesting that this conclusion is also supported by completely independent and general considerations based on the possibility to reproduce , without a particularly fine tuned @xmath58 matrix , the observed atmospheric to solar neutrino mass ratio , @xmath241 , starting from hierarchical neutrino yukawa couplings . it is found @xcite that this experimental observation is far more natural if the lightest neutrino presents a much stronger hierarchy than the the two heavy ones , as it occurs in the region that we have found at small @xmath22 . it should be also stressed again , that since our results are independent of @xmath78 and @xmath77 , as far as @xmath242 and @xmath243 , they hold even for a yukawa couplings hierarchy milder than in the case of up quark masses . this can help to make even more natural to reproduce the result @xmath241 without a fine tuned @xmath58 . a more precise measurement of @xmath218 could also play a relevant role in testing these models , a point that should be addressed by a more involved multi - parameter analysis . a future accurate determination of the neutrino mixing angles will be therefore crucial to test @xmath0-inspired leptogenesis and could even yield some interesting information on the matrix @xmath59 . in conclusion , it seems that @xmath0-inspired leptogenesis provides an interesting well justified example that gives some hopes about the possibility of testing minimal leptogenesis even only with low energy neutrino experiments . it will be then quite interesting in next years to compare the experimental results with the constraints and the predictions from @xmath0-inspired models that we discussed . we wish to thank s. blanchet , m. chun chen , f. feruglio , carlo giunti , s. king , r. mohapatra , m. schmidt for useful discussions . p.d.b . acknowledges financial support from the next institute and sepnet . for a recent review , see s. davidson , e. nardi and y. nir , arxiv:0802.2962 [ hep - ph ] . see also , g. f. giudice , a. notari , m. raidal , a. riotto and a. strumia nucl . b * 685 * , 89 ( 2004 ) ; w. buchmuller , p. di bari and m. plumacher , annals phys . * 315 * , 305 ( 2005 ) . e. komatsu _ et al . _ , arxiv:1001.4538 [ unknown ] . p. minkowski , phys . b * 67 * , 421 ( 1977 ) ; m. gell - mann , p. ramond and r. slansky , _ proceedings of the supergravity stony brook workshop _ , new york 1979 , eds . p. van nieuwenhuizen and d. freedman ; t. yanagida , _ proceedings of the workshop on unified theories and baryon number in the universe _ , tsukuba , japan 1979 , ed.s a. sawada and a. sugamoto ; r. n. mohapatra , g. senjanovic , phys . . lett . * 44 * , 912 ( 1980 ) . a. abada , s. davidson , f. x. josse - michaux , m. losada and a. riotto , jcap * 0604 * , 004 ( 2006 ) ; e. nardi , y. nir , e. roulet and j. racker , jhep * 0601 * , 164 ( 2006 ) ; a. abada , s. davidson , a. ibarra , f. x. josse - michaux , m. losada and a. riotto , jhep * 0609 * , 010 ( 2006 ) . s. blanchet , d. marfatia and a. mustafayev , jhep * 1011 * ( 2010 ) 038 [ arxiv:1006.2857 [ hep - ph ] ] . their analysis is restricted to normal hierarchical models with @xmath244 . though we find agreement on some general features , for example on the existence of a lower bound on @xmath22 , in some cases our results seem to indicate different conclusions , like for example on the lower bound on the reheating temperature and on many features of the low energy neutrino constraints when a comparison is possible , for example we do not find a lower bound on @xmath6 when we allow @xmath119 . however , it should be said that a precise understanding of the different results is made difficult because of the different framework , non supersymmetric in our case and supersymmetric in @xcite , and because the exact assumptions on @xmath59 and the procedure for the calculation of @xmath58 and of the rh neutrino mass spectrum are not specified in @xcite . s. antusch , s. f. king and a. riotto , jcap * 0611 * ( 2006 ) 011 . s. antusch , p. di bari , d. a. jones and s. f. king , arxiv:1003.5132 . e. bertuzzo , p. di bari and l. marzola , arxiv:1007.1641 [ hep - ph ] . m. y. khlopov and a. d. linde , phys . b * 138 * ( 1984 ) 265 . j. r. ellis , j. e. kim and d. v. nanopoulos , phys . b * 145 * ( 1984 ) 181 ; k. kohri , t. moroi and a. yotsuyanagi , phys . d * 73 * ( 2006 ) 123511 .
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we extend the results of a previous analysis of ours showing that , when both heavy and light flavour effects are taken into account , successful minimal ( type i + thermal ) leptogenesis with @xmath0-inspired relations is possible . barring fine tuned choices of the parameters , these relations enforce a hierarchical rh neutrino mass spectrum that results into a final asymmetry dominantly produced by the next - to - lightest rh neutrino decays ( @xmath1 dominated leptogenesis ) .
we present the constraints on the whole set of low energy neutrino parameters . allowing a small misalignment between the dirac basis and the charged lepton basis as in the quark sector
, the allowed regions enlarge and the lower bound on the reheating temperature gets relaxed to values as low as @xmath2 .
it is confirmed that for normal ordering ( no ) there are two allowed ranges of values for the lightest neutrino mass : @xmath3 and @xmath4 . for @xmath5 the allowed region in the plane
@xmath6-@xmath7 is approximately given by @xmath8 , while the neutrinoless double beta decay effective neutrino mass falls in the range @xmath9 for @xmath10 .
for @xmath11 , one has quite sharply @xmath12 and an upper bound @xmath13 .
these constraints will be tested by low energy neutrino experiments during next years .
we also find that inverted ordering ( io ) , though quite strongly constrained , is not completely ruled out .
in particular , we find approximately @xmath14 , that will be fully tested by future experiments .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
mechanical systems subject to rolling kinetic constraints are one of the most studied argument of classical mechanics , especially for its wideness of applicability in several branches of mechanical sciences : contact mechanics , tribology , wear , robotics , ball bearing theory and control theory applied to moving engines and vehicles are only some of the important fields where the results about pure rolling constraint can be fruitfully used . it is well known that , when a mechanical system moves in contact with an assigned rough surface , the effective fulfilment of the kinetic conditions determined by the rolling without sliding requirement of the system on the surface depends on the behavior , with respect to the considered law of friction , of the reaction forces acting on the system in the contact points . for example , the roll of a disk on a rough straight line , considering the coulomb s law of friction , can happen only if the contact force lie inside the friction cone ( see example 1 below ) . however , even in the simplest case of a mechanical system formed by a single rigid body , in the case of multiple contact points between the rigid body and the rough surface , it could be an hard task to obtain sufficient information about the contact reactions in order to determine if the laws of friction are satisfied or not during the motion . in fact the most common methods to determine information about the reactions , starting from the simple application of linear and angular momenta equations ( see e.g. @xcite ) to most refined techniques such as lagrangian multipliers in lagrangian mechanics ( see e.g. @xcite ) or deep analyses of the contact between the system and the surface ( see e.g. @xcite ) , have a global character . then these methods , for their very nature , can determine only a reactive force system equivalent to the real one but , in the general case , these methods can not determine the single reactive forces in the contact points . the problem becomes even more complicated in case of multibody system , due to the presence of the internal reactions in the link between the parts of the system . in this paper we consider the motion of a mechanical system having two or more distinct contact points with one or more assigned rough surfaces , and we determine necessary conditions for which in all the contact points the pure rolling kinetic constraint can hold . we also analyze the sufficiency of these conditions by generalizing to this case a well known and usually accepted assumption on the behavior of pure rolling constraint . moreover , we briefly discuss the possible behaviors of the system when the necessary conditions are not fulfilled . the procedure to determine if the rolling condition can be fulfilled can be applied both to systems formed by a single rigid body and to multibody systems . it is essentially based on the application of linear and angular momenta equations to the ( parts forming the ) mechanical system , and therefore it gives an underdetermined system in the unknown single contact reactions . nevertheless , we show that the lack of complete knowledge of the single contact reactions is not an obstacle to determine the feasibility of the rolling conditions . it is however important to remark that , although the procedure has a very simple and unassailable theoretic foundation , its effective application to general systems could present insurmountable difficulties . this is essentially due to the fact that the general procedure explicitly requires the knowledge of the motion law of the system , and in the general case the explicit time dependent expression of the motion can not be obtained because of complications determined by the geometry of the system itself and/or by the integrability of the equations of motion . nevertheless there are several significative cases where the procedure can be explicitly performed . in the paper , we illustrate three examples with rising complication : the well known case of a disk falling in contact with an inclined plane ( that is presented only to point out some key points of the general procedure ) ; the case of a system formed by a non coupled pair of disks connected with a bar and moving on the horizontal plane ; the case of a heavy sphere falling in contact with a guide having the form of a v groove non symmetric with respect to the vertical axis and inclined with respect to the horizontal . the main content of this paper can be approached starting from a very standard background knowledge , essentially focused to the linear and angular equations of motion for a mechanical system , the so called cardinal equations , and the basic theory of pure rolling conditions and kinetic constraints . on the other hand , the list of possible references involving theory and application of pure rolling constraint is almost endless . therefore we chose to cite only a very limited list of references sufficient to make the paper self consistent : the classical book of levi civita and amaldi @xcite and the book of goldstein @xcite for the cardinal equations and the basic concepts about pure rolling conditions ; the book of neimark and fufaev @xcite and the paper of massa and pagani @xcite for the behavior of systems subject to kinetic constraints . the interested reader can find in the wide but not exhaustive lists of references of @xcite as a useful starting point to delve in the expanse of the material related to this argument . the paper is divided in four sections . section 1 contains a very brief preliminary description of the well known analysis of the rolling condition for a disk in contact with an inclined plane . this remind is motivated by some useful affinities with the general procedure for generic systems . section 2 contains the discussion of the general case , and the determination of the necessary conditions for pure rolling conditions simultaneously hold . section 3 presents the example of the system formed by the non coupled disks and the example of the heavy sphere falling in the v groove . section 4 is devoted to open problems , remarks and conclusions . an homogeneous disk of mass @xmath0 and radius @xmath1 moves in the vertical plane being in contact with a rough guide inclined with slope angle @xmath2 . [ disco ] considering the system subject to the coulomb s law of friction , with obvious notation clarified by fig . [ disco ] , the feasibility of pure rolling condition of the disk can be determined with the following procedure : * we determine the relative velocity @xmath3 of the contact point @xmath4 of the disk at the instant @xmath5 with respect to the inclined plane as function of the initial data of the motion . the pure rolling condition requires of course that @xmath6 . if so * we assume that the disk rolls without sliding on the inclined plane . then the system has a single degree of freedom ( for example the coordinate @xmath7 of @xmath4 along the inclined plane ) and we can determine the equation of motion @xmath8 * we determine the corresponding reaction @xmath9 as a ( in this case constant ) function of time @xmath10 * we test the coulomb s law of friction condition @xmath11 where @xmath12 are the parallel and orthogonal component of @xmath13 with respect to the inclined plane ; * we assume that , if and until the coulomb s condition is verified , the disk moves rolling on the plane and that if and when the coulomb s condition is not verified , the disk changes its dynamic evolution beginning to slide on the plane ( until the first time @xmath14 such that @xmath15 ) . some remarks are in order to focus the possibility to generalize the procedure to more complicated systems . step 2 consists in the determination of the reaction acting on the disk as function of time . the utmost simplicity of the specific problem can hide the fact that in a more general situation the information about the reaction sufficient to analyze the rolling condition could require an explicit determination of the motion of the system as function of time . step 3 tests the compatibility of the reaction evaluated in step 2 with the coulomb s law of friction assumed as the constitutive characterization of the rough surface in contact with the disk . of course the feasibility of the rolling condition can be tested with any other significative constitutive law . in step 4 we assume that , roughly speaking , if the disk can roll then it does . this is of course an arbitrary assumption , but the hypothesis is well confirmed by experimental results . in the next section , we will confirm this assumption in the more general situation of generic system . to conclude the section , let us note that , in this very simple case , both the behaviors of the disk when the coulomb s friction condition is or is not verified are determinable . in the general case , when the constitutive law is not verified , the behavior of the system turns out to be not so straight to determine , although some reasonable assumptions can be done . we will go back on this arguments in section 4 . in this section , following a line of though similar to the one applied in the previous section , we discuss the possibility that a mechanical system @xmath16 having two points @xmath17 in contact with a fixed surface @xmath18 moves such that in both the contact points the rolling conditions can subsist respecting the coulomb s law of friction . the arguments of the discussion can be easily extended to cases with more ( but a finite number ) than two contact points and possibly to different friction constitutive laws . the discussion is based on the fact that , along the motion , the reactive forces acting on the system must validate the linear and angular momenta equations @xmath19 where @xmath20 is the total mass of the system , @xmath21 is the center of mass of the system and @xmath22 are respectively the sum of the active and reactive forces and active and reactive momenta acting on the whole system . in this specific situation we have that : @xmath23 it is however well known @xcite that eqs . ( [ sistema_generale ] ) are not sufficient to determine the motion of the mechanical system and the single reactions @xmath24 along the motion , since the system @xmath25 is by its very nature under determined . in fact the projection of the angular momenta equation of ( [ sistema_sottodeterminato ] ) in the direction of @xmath26 is a pure equation of motion of the system where no reactions appear . then ( [ sistema_sottodeterminato ] ) can give no more than 5 relations on the components of @xmath27 and @xmath28 . unfortunately , due to the roughness of the contacts , no preliminary conditions can be imposed on the components of the reactions , so that , even when the motion of the mechanical system is known , ( [ sistema_sottodeterminato ] ) is a linear system with @xmath29 unknowns that is not of maximum rank . nevertheless the parametric solution of the system ( [ determinazione_reazioni ] ) , and an assumption parallelizing the one of step 4 of the case of section 1 , give us the possibility of determining if the rolling conditions in @xmath30 and @xmath31 are or not verified . the procedure to test the feasibility of pure rolling condition of the disk can be then based on the following steps : * we test if the initial relative velocities of the contact points @xmath17 with respect to the surface are null or not . if they are null * we suppose that the system rolls without sliding in both the contact points . this assumption fixes the dynamics ( for example the number of degrees of freedom ... ) of the system and consequently allows the determination of the motion of the system ; * we write the linear and angular momenta equations for the whole system , for example in the form : @xmath32 since the motion of the system is known , both the right hand sides of the equations , together with the position vectors @xmath33 and @xmath34 , are known as function of time . therefore eqs . ( [ sistema_reazioni ] ) turn out to be a time dependent under determined linear system in the six scalar unknowns given by the components of the vectors @xmath35 ; * we solve the linear under determined system ( [ sistema_reazioni ] ) , obtaining the expression of the reaction @xmath27 and @xmath28 as function of time and parameters @xmath36 , where of course the integer @xmath37 is related to the rank of ( [ sistema_reazioni ] ) . then we can determine the tangent and orthogonal components @xmath38 of the reactions with respect to the surface @xmath18 as functions of @xmath39 . the pure rolling conditions then can subsist in both the contact points only in the time interval @xmath40 $ ] such that for every @xmath41 $ ] there exists at least one admissible @xmath37uple @xmath42 such that the system @xmath43 holds ; * we assume that , if for every @xmath41 $ ] there exists at least one admissible @xmath37uple @xmath42 such that ( [ condizioni_cm ] ) are verified , the system moves rolling without sliding in both points @xmath30 and @xmath31 during the time interval @xmath40 $ ] . it is clear that the general procedure described above parallelizes as possible and generalizes the one of the disk on the inclined plane . the most significant differences consist in the explicit determination of the motion of the system ( since otherwise eqs . ( [ sistema_reazioni ] ) could not admit a simple parametric solution for the reactions @xmath44 and @xmath28 ) and in the fact that , when the coulomb conditions ( [ condizioni_cm ] ) are not verified , being understood that the system does not roll in both contact points , the determination of the behavior of the system could require a more subtle analysis . we will go back on these arguments in section 4 . we also remark that not all the @xmath37uple @xmath36 could be admissible in the discussion of the inequalities ( [ condizioni_cm ] ) . for example , if the system is leaned on the surface , we have to restrict our attention to the @xmath37uple such that @xmath45 ( where @xmath46 is the unit normal vector to the surface @xmath18 in the point @xmath47 and orientated toward the side of the system ) since otherwise the system detaches from the surface . [ ruote ] a mechanical system is formed by two equal disks of mass @xmath0 and radius @xmath1 and a rod , of mass @xmath20 and length @xmath48 . the rod is constrained to remain orthogonal to the two planes of the disks with its endpoints coinciding with the two centers of the disks ( see fig . [ ruote ] ) so that the disks remain vertical . the whole system is leaned on a rough horizontal plane . the system has then @xmath49 degrees of freedom : the coordinates @xmath50 of the center of mass @xmath21 of the rod , the angle @xmath51 formed by the plane of the disks with the @xmath52 plane and the two rotation angles @xmath53 of the disks . the rolling conditions in the contact points @xmath30 and @xmath31 are equivalently expressed by : @xmath54 tedious but straightforward computations ( see @xcite ) give the equations of motion of the system @xmath55 if we suppose assigned the almost generic initial data @xmath56 with the only condition @xmath57 , the motion of the system is given by @xmath58 + x_0 \\ \\ { y}(t ) & = & - \dfrac12 l \dfrac{\dot{\vph_1}_0 + \dot{\vph_2}_0}{\dot{\vph_1}_0 - \dot{\vph_2}_0}\left[\cos\left(\dfrac{r}{l } \left(\dot{\vph_1}_0 - \dot{\vph_2}_0\right)t + \vth_0 \right ) - \cos \vth_0 \right ] + y_0 \\ \\ \vth ( t ) & = & \dfrac{r}{l } ( \dot{\vph_1}_0 - \dot{\vph_2}_0)t \\ \\ { \vph_1}(t ) & = & \dot{\vph_1}_0 t \\ \\ { \vph_2}(t ) & = & \dot{\vph_2}_0 t \end{array } \right.\end{aligned}\ ] ] the linear and angular momenta equations for the system can be written as @xmath59 taking into account the motion of the system ( [ moto_es1 ] ) and introducing the orthonormal base @xmath60 with @xmath61 , with obvious notation we obtain @xmath62 \\ \\ \phi_{2_z } & = & \dfrac12 \left[(2m+m)g - ( 3m+m ) \dfrac{r^3}{l^2}(\dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) \right ] \\ \\ \phi_{1_v } & = & 0 \\ \\ \phi_{2_v } & = & 0 \\ \\ \phi_{1_u } + \phi_{2_u } & = & - \dfrac12 ( 2m+m ) \dfrac{r^2}{l } ( \dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) \end{array } \right.\end{aligned}\ ] ] note that , if the system leans on the horizontal plane , we must add the requirement @xmath63 since otherwise one between @xmath64 and @xmath65 becomes negative ( and this is not acceptable , because the system lifts from the horizontal plane , and the initial assumptions of five degrees of freedom is violated ) . if ( [ condizioni_esistenza_es1 ] ) is fulfilled , then we can chose for example @xmath66 and we find the reactions @xmath24 as functions of @xmath67 : coulomb conditions ( [ condizioni_cm ] ) then takes the form : @xmath68 \\ \\ \left| \dfrac12 ( 2m+m ) \dfrac{r^2}{l } ( \dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) + \lambda \right| \le \mu_2 \ , \dfrac12 \left[(2m+m)g - ( 3m+m ) \dfrac{r^3}{l^2}(\dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) \right ] \end{array } \right.\end{aligned}\ ] ] in conclusion , the pure rolling of the disks can subsist if and only if ( [ condizioni_esistenza_es1 ] ) holds and there is a @xmath67 such that @xmath69 , \right . \\ \qquad\qquad\left . - \dfrac12 ( 2m+m ) \dfrac{r^2}{l } ( \dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) \right . \\ \qquad\qquad\qquad\qquad\left . - \dfrac12 \mu_2 \left[(2m+m)g - ( 3m+m ) \dfrac{r^3}{l^2}(\dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) \right ] \right\ } \\ \\ \qquad\qquad\qquad\qquad\qquad\qquad \le \lambda \le \\ \\ \qquad\qquad \min \left\ { \dfrac12 \mu_1 \left[(2m+m)g - ( 3m+m ) \dfrac{r^3}{l^2}(\dot{\vph_2}^2_0 - \dot{\vph_1}^2_0 ) \right ] , \right . \qquad\qquad \qquad\qquad \dfrac12 \mu_2 \left[(2m+m)g - ( 3m+m ) \dfrac{r^3}{l^2}(\dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) \right ] \right . \\ \qquad\qquad . - \dfrac12 ( 2m+m ) \dfrac{r^2}{l } ( \dot{\vph_1}^2_0 - \dot{\vph_2}^2_0 ) \right\}. \end{array}\end{aligned}\ ] ] [ sfera ] a mechanical system is formed by a sphere of mass @xmath0 and radius @xmath1 leaned in an inclined v groove whose walls are described by the equations @xmath70 we introduce an orthonormal base @xmath71 where @xmath72 are orthogonal to @xmath73 respectively and @xmath74 . the center @xmath75 of the sphere is then determined by the vector @xmath76 , where @xmath7 is the distance of @xmath75 from a fixed plane orthogonal to @xmath77 ( see fig . [ sfera ] ) . the rolling conditions in the contact points @xmath78 determines the angular velocity of the sphere in the form @xmath79 and the system has one degree of freedom : the coordinate @xmath7 . the linear and angular momenta equations for the system can be written as @xmath80 the projection of the angular momenta equation in the direction of @xmath81 gives the equation of motion of the sphere , that is @xmath82 . this relation suffices to obtain from ( [ sistema_generale_es2 ] ) the under determined system of the reactions : if we decompose the reactions along the basis introduced above @xmath83 the system takes the form @xmath84 to analyze the parametric solution of the system we chose @xmath85 . in this case , and once again supposing the sphere leaned on the groove , we must require the condition @xmath86 since otherwise @xmath87 and the sphere comes off the groove . conditions ( [ condizioni_cm ] ) take in this case the form @xmath88 with @xmath89 . a straightforward minimum computation for the functions on the left - hand side of ( [ condizioni_cm_es2 ] ) shows then that the system can roll on both the contact points if and only if @xmath90 the procedure described in sec . 2 in the case of two contact points can be generalized to ( multibody ) systems with three or more contact points ( think for example of a steering tricycle formed by three vertical disks connected with three rods leaned on the horizontal plane ) . of course , an increase of the number of contact points implies in general an increase of the technical difficulties in practical applications . this is principally due to the fact that step 1 of the general procedure is not a straightforward passage . the effective knowledge of the motion of the system can be achieved only in some particular cases . insurmountable technical difficulties can arise both for geometrical reasons ( think of a convex rigid body moving in contact with a surface , both having generic shapes with the only requirement that the contact between rigid body and groove happens in two points . for a more detailed discussion on the argument , see , e.g. @xcite ) , and/or for computational reasons ( even when the equations of motion of the system are explicitly obtained , it could be hard to integrate them to obtain the motion of the system ) . nevertheless note that , as pointed out by the examples in sec . 3 , not for all the systems the explicit integration of the equations of motion is required . a second remark is that the general procedure gives necessary conditions such that the pure rolling subsists in all contact points ( conditions that become sufficient if we take into account step 4 of the procedure ) but it does not give any information on the behavior of the system if the pure rolling is not possible even in a single contact point . in fact , analogously to what happens in the simple case of ex . 1 , in the instant when ( [ condizioni_cm ] ) stops to hold , the dynamics of the system ( for example , the number of degrees of freedom ) changes abruptly . to clarify this fact , suppose that , at the instant @xmath91 of the study of the system of ex . 2 , a sudden variation of the friction coefficient @xmath92 in the point @xmath31 ( an oil spot on the plane ? ) causes the invalidity of the second relation of ( [ condizioni_cm_es1 ] ) , while the first relation still holds . of course , even if we chose the assumption of step 4 of the procedure as a fixed point of our argument , we can not suppose that the system continues to roll in @xmath30 ( and begins to slide in @xmath31 ) since the beginning of sliding in @xmath31 can affect the pure rolling behavior of the system in the point @xmath30 . we must perform a new analysis of the behavior of the system , possibly supposing the system rolling in @xmath30 and sliding in @xmath31 , we must determine ( if possible ) the new equations of motion of the system ( with the additional difficulties of different friction laws in the point @xmath30 and @xmath31 and possibly increased number of degrees of freedom ) , the motion of the system , the new ( parametric ) system of reactions acting on the system and then we can test the coulomb condition in the point @xmath30 .
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we illustrate a theoretical procedure determining necessary conditions for which simultaneous pure rolling kinetic constraints acting on a mechanical system can be fulfilled .
we also analyze the sufficiency of these conditions by generalizing to this case a well known and usually accepted assumption on the behavior of pure rolling constraint .
we present in detail the application of the procedure to some significative mechanical systems .
0.5truecm * pacs : * 45.40.-f , 45.50.-j * 2000 mathematical subject classification : * 70e18 70f25 70e60 * keywords : * rolling constraint linear and angular momenta equations
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models of particle physics including weak scale supersymmetry ( susy ) are amongst the most promising candidates@xcite for new physics at the tev scale . of this class of models , the minimal supergravity ( msugra ) model stands out as providing one of the most economic explanations for the diversity of soft supersymmetry breaking terms in the susy lagrangian@xcite . in this model@xcite , supersymmetry is communicated from a hidden sector ( whose dynamics leads to the breaking of supersymmetry ) to the observable sector ( consisting of the fields of the minimal supersymmetric standard model or mssm ) via gravitational interactions . with the assumption of canonical kinetic terms for scalars in the lagrangian , this leads to a universal mass @xmath13 for all scalar particles at some high scale @xmath16 , usually taken to be @xmath17 . at @xmath17 , gaugino masses and trilinear terms are assumed to unify at @xmath14 and @xmath18 , respectively . these parameters , along with the bilinear soft term @xmath19 , provide boundary conditions for the renormalization group evolution of the various soft terms from @xmath17 to @xmath20 . requiring in addition radiative electroweak symmetry breaking leaves a rather small parameter set @xmath21 from which the entire susy particle mass spectrum and mixing parameters may be derived . the flavor changing neutral current decay of the bottom quark @xmath4 is well known to be particularly sensitive to new physics effects . new weak scale particles ( _ e.g. _ , a chargino @xmath22 and top squark @xmath23 ) could give loop contributions which would be comparable to the standard model ( sm ) @xmath24 loop amplitude . measurements from the cleo experiment@xcite restrict the inclusive @xmath25 branching ratio to be @xmath26 , where @xmath27 at 95% cl . many analyses have been performed@xcite which compare theoretical predictions of susy models to experimental results . in a previous report@xcite , predictions of the @xmath4 decay rate were made as functions of the msugra model parameter space . in this study , a number of qcd improvements were incorporated into the calculation which reduced the inherent uncertainty of the @xmath4 decay rate predictions due to the qcd scale choice from @xmath28 down to @xmath29 . susy contributions to the @xmath4 decay amplitude included @xmath5 , @xmath30 and @xmath31 loops . results were presented for @xmath32 and @xmath33 , and for both signs of @xmath34 . for @xmath10 , large regions of parameter space were excluded , especially for @xmath35 . for @xmath12 , all the parameter space scanned was allowed by cleo data : in fact , for some ranges of parameters , the model predicts values of @xmath36 close to the central value measured by the cleo collaboration . recently , sparticle mass spectra and sparticle decay branching ratios in the msugra model have been reanalysed for large values of the parameter @xmath0@xcite . in the msugra model , the range of @xmath0 is typically @xmath37 , where the lower limit depends somewhat on the precise value of @xmath38 . for @xmath8 , @xmath39 and @xmath40 yukawa couplings become non - negligible and can affect the sparticle mass spectrum and decay branching fractions . the upper and lower limits on @xmath0 are set by a combination of requiring a valid solution to radiative electroweak symmetry breaking , and requiring perturbativity of third generation yukawa couplings between the scales @xmath20 and @xmath17 . some optimization of scale choice at which the one - loop effective potential is minimized was found to be needed in ref . @xcite in order to gain stable sparticle and higgs boson mass contributions . this scale optimization effectively includes some portion of two - loop corrections to the effective potential@xcite . it was shown that the mass of the pseudoscalar higgs boson @xmath41 , and the related masses of @xmath42 and @xmath43 , suffer a sharp decrease as @xmath0 increases . in addition , the masses of the lighter tau slepton @xmath44 and bottom squark @xmath45 also decrease , although less radically . naively , one might expect corresponding increases in the loop contributions to @xmath4 decay involving @xmath45 and @xmath43 . indeed , borzumati has shown in ref . @xcite that as @xmath46 decreases , the charged higgs contribution to @xmath4 decay does increase . however , for large values of @xmath0 , the chargino loop contributions increase even more dramatically , and dominate the decay amplitude . she further notes that at intermediate to large @xmath0 values , there is a non - negligible contribution from @xmath6 loops . in this paper , we re - examine constraints on the msugra model from @xmath4 decay at large @xmath0 . in doing so , we incorporate several improvements over previous analyses . * we present our analysis using updated msugra mass predictions for large @xmath0 , using a renormalization group improved one - loop effective potential with optimized scale choice @xmath47 . we use an updated value of top mass @xmath48 gev . * we include in this analysis contributions from @xmath49 and @xmath50 loops . these contributions require knowledge of the full squark mixing matrices , and hence an improved calculation of renormalization group evolution of soft susy breaking parameters . * as in ref . @xcite , we include the dominant next - to - leading order ( nlo ) virtual and bremsstrahlung corrections to the operators mediating @xmath4 decay at scale @xmath51 . in addition , we include nlo rg evolution of wilson coefficients between scales @xmath52 and @xmath53 . we also include appropriate renormalization group evolution of wilson coefficients at high scales @xmath54 for @xmath5 , @xmath30 and @xmath31 loops following the procedure of anlauf@xcite . the corresponding rg evolution of wilson coefficients for @xmath49 and @xmath50 loops is not yet available . * we compare our results to recent calculations at large @xmath0 of the neutralino relic density and direct dark matter detection rates for the msugra model . in sec . ii of this paper , we present some details of our calculations , especially those regarding the inclusion of @xmath49 and @xmath50 loops . in sec . iii , we present qcd improved results for the @xmath4 branching fraction in msugra parameter space for large @xmath0 . in this section , we also make comparisons with cosmological and collider search expectations . in sec . iv , we relax some of the assumptions of the msugra framework to see whether @xmath49 loops can become large or even dominant . this question is important when considering the model dependence of our results . the calculation of the width for @xmath4 decay proceeds by calculating the loop interaction for @xmath4 within a given model framework , _ e.g. _ , msugra , at some high mass scale @xmath55 , and then matching to an effective theory hamiltonian given by @xmath56 where the @xmath57 are wilson coefficients evaluated at scale @xmath16 , and the @xmath58 are a complete set of operators relevant for the process @xmath59 , given , for example , in ref . all orders approximate qcd corrections are included via renormalization group resummation of leading logs ( ll ) which arise due to a disparity between the scale at which new physics enters the @xmath60 loop corrections ( usually taken to be @xmath55 ) , and the scale at which the @xmath60 decay rate is evaluated ( @xmath51 ) . resummation then occurs when we solve the renormalization group equations ( rge s ) for the wilson coefficients @xmath61 where @xmath62 is the @xmath63 anomalous dimension matrix ( adm ) , and @xmath64 the matrix elements of the operators @xmath58 are finally calculated at a scale @xmath51 and multiplied by the appropriately evolved wilson coefficients to gain the final decay amplitude . the dominant uncertainty in this leading - log theoretical calculation arises from an uncertainty in the scale choice @xmath16 at which effective theory decay matrix elements are evaluated . varying @xmath16 between @xmath65 to @xmath66 leads to a theoretical uncertainty of @xmath67% . recently , next - to - leading order qcd corrections have been completed for @xmath4 decay . these include _ i _ ) complete virtual corrections@xcite to the relevant operators @xmath68 and @xmath69 which , when combined with bremsstrahlung corrections@xcite results in cancellation of associated soft and collinear singularities ; _ ii _ ) calculation of @xmath70 contributions to the adm elements @xmath71 for @xmath72 ( by ciuchini _ et al._@xcite ) , for @xmath73 by misiak and mnz@xcite , and for @xmath74 by chetyrkin , misiak and mnz@xcite . in addition , if two significantly different masses contribute to the loop amplitude , then there can already exist significant corrections to the wilson coefficients at scale @xmath52 . in this case , the procedure is to create a tower of effective theories with which to correctly implement the rg running between the multiple scales involved in the problem . the relevant operator bases , wilson coefficients and rge s are given by cho and grinstein@xcite for the sm and by anlauf@xcite for the mssm . the latter analysis includes contributions from just the @xmath5 , @xmath75 and @xmath76 loops ( which are the most important ones ) . we include the above set of qcd improvements ( with the exception of @xmath74 , which has been shown to be small@xcite ) into our calculations of the @xmath4 decay rate for the msugra model . the contributions to @xmath77 and @xmath78 from @xmath6 and @xmath7 loops ( susy contributions to @xmath79 are suppressed by additional factors of @xmath80 ) have been presented in ref . @xcite , although some defining conventions must be matched between ref . @xcite and ref . @xcite and ref . the only complication is that the squark mixing matrix @xmath81 which enters the couplings must be derived . to accomplish this , we incorporate the following procedure into our program for renormalization group running . * we first calculate the values of all running fermion masses in the sm at the mass scale @xmath82 . from these , we derive the corresponding yukawa couplings @xmath83 , @xmath84 and @xmath85 for each generation , and construct the corresponding yukawa matrices @xmath86 , @xmath87 and @xmath88 , where @xmath89 runs over the 3 generations . we choose a basis that yields flavor diagonal matrices for @xmath87 and @xmath88 , whereas the ckm mixing matrix creates a non - diagonal matrix @xmath86@xcite . * the three yukawa matrices are evolved within the mssm from @xmath90 up to @xmath91 and the values are stored . we use 1-loop rges except for the evolution of gauge couplings . * at @xmath91 , the matrices @xmath92 , @xmath93 and @xmath94 are constructed ( assuming @xmath95 ) . the squark and slepton mass squared matrices @xmath96 are also constructed , where @xmath97 and @xmath98 . these matrices are assumed diagonal at @xmath91 with entries @xmath99 . * the @xmath100 and @xmath96 matrices are evolved along with the rest of the gauge / yukawa couplings and soft susy breaking terms between @xmath82 and @xmath17 iteratively via runge - kutta method until a stable solution is found . the entire solution requires the simultaneous solution of 134 coupled rge s ( with some slight redundancy ) . we use 1-loop rges except for the evolution of gauge couplings . * at @xmath90 , the @xmath101 @xmath102-squark mass squared matrix is constructed . numerical diagonalization of this matrix yields the squark mass mixing matrix @xmath81 which is needed for computation of the @xmath6 and @xmath7 loop contributions . at this point , the wilson coefficients @xmath77 and @xmath78 can be calculated and evolved to @xmath51 as described above , so that the @xmath4 decay rate can be calculated @xcite . as an example , we show in fig . [ nfig1 ] the calculated contributions to the wilson coefficient @xmath77 versus @xmath0 for the msugra point @xmath103 gev , @xmath104 and @xmath12 . in frame _ a _ ) , we show contributions from @xmath105 loops , as well as from @xmath5 and @xmath75 . the @xmath5 contribution is of course constant , while the @xmath75 contribution is of the same sign , and increasing slightly in magnitude . the various contributions from chargino loops increase roughly linearly with @xmath0 at a much faster rate , and thus form the dominant components of the @xmath4 decay amplitude . in the case shown , there are several large negative as well as positive contributions , so that significant cancellations take place . the sum of all chargino loop contributions is shown by the dotted curve . in frame _ b _ ) we show the contributions to @xmath77 from different @xmath6 loops . these contributions vary with @xmath0 as well and are comparable to corresponding contributions from chargino loops . the sum of all gluino loop contributions is shown by the dotted curve ; in this case , however , the cancellation amongst the various loop contributions is nearly complete . in frame _ c _ ) , we show the individual and summed contributions from @xmath7 loops . these also increase with @xmath0 but , as expected , are tiny compared to the @xmath105 and @xmath6 loop contributions . the sum is again shown by the dotted curve . here , the cancellations are not as complete as in the gluino loop case due to the higgsino interactions of the neutralinos which increase with @xmath0 . our first numerical results for @xmath11 decay are shown in fig . [ nfig2 ] , where we plot the branching fraction versus @xmath0 for the msugra point @xmath103 gev , @xmath104 and for _ a _ ) @xmath10 and _ b _ ) @xmath12 . the sm value , after qcd corrections , is @xmath106 , where the error comes from varying the scale choice @xmath107@xcite . the sm result is denoted by the dot - dashed line , and of course does not vary with @xmath0 . if we include in addition the contribution from the @xmath75 loop , then we obtain the dotted curves , which always increase the value of @xmath11 . for this parameter space point , including the @xmath75 loop always places the value of @xmath11 above the cleo 95% cl excluded region of @xmath108 . if we include the full contribution of supersymmetric particles to the computation of @xmath11 , then we arrive at the solid curves in fig . [ nfig2 ] . for the @xmath10 case in frame _ a _ ) , the susy loops increase the branching fraction , which increases with @xmath0 , so that the cleo restriction on @xmath11 severely constrains the msugra model for large @xmath0 . for the frame _ b _ ) case with @xmath12 , the susy loop contributions generally act to decrease the branching fraction , so that much of the parameter space is allowed for moderate values of @xmath0 . ultimately , as @xmath0 increases , the decrease in @xmath11 becomes so severe that the msugra model becomes in conflict with the cleo lower 95% cl bound that @xmath109 , so that for this particular msugra point , all values of @xmath110 are excluded for the particular choice of msugra parameters . in figure [ nfig3 ] , we show the main result of this paper : the contours of constant @xmath11 in the @xmath111 parameter plane for large @xmath112 , for @xmath104 and for _ a _ ) @xmath10 and _ b _ ) @xmath12 . the contours are evaluated at a renormalization scale choice @xmath113 . the region marked by th is disallowed by theoretical considerations : either electroweak symmetry is not properly broken ( the large @xmath13 , small @xmath14 region ) or the lightest neutralino @xmath114 is not the lightest susy particle ( lsp ) . for small @xmath13 , the light tau slepton @xmath44becomes so light that in the th region , @xmath115 . the region denoted by ex is excluded by lep2 constraints which require that the light chargino mass @xmath116 gev . in frame _ a _ ) , we see the value of @xmath11 is large throughout the entire parameter space plane . the region with small values of @xmath13 and @xmath14 which is most favored by fine - tuning considerations@xcite is in the most severe violation of the cleo constraint . the region below the dotted contour is in violation of the cleo 95% cl bound for _ all _ choices of renormalization scale @xmath107 . thus , susy models allowed by cleo for @xmath112 and @xmath10 would be required to have @xmath117 gev ( at @xmath118 gev ) and @xmath119 gev ( for @xmath120 gev ) . in frame _ b _ ) for @xmath12 , we see that the values of @xmath11 are uniformly _ below _ the sm value , and so usually in better agreement with the cleo measured value of @xmath121 ( where errors have been combined in quadrature ) . in fact , we note that the region with @xmath122 gev agrees with the cleo central value for @xmath11 ! this region corresponds to parameter space points with @xmath123 gev , and @xmath124 gev . in this frame , the entire plane shown , except the region below the dotted contour , is _ allowed _ by the cleo constraint . the region below the dotted contour falls below the cleo 95% cl value of @xmath109 for all values of scale choice @xmath107 . this is again the region most favored by fine - tuning . in this plane , @xmath125 gev and @xmath126 gev . up to this point , we have only shown results for a constant value of @xmath104 . in figure [ nfig4 ] , we show contours of constant @xmath11 in the @xmath127 plane for @xmath128 gev , for @xmath112 and _ a _ ) @xmath10 and _ b _ ) @xmath12 . for frame _ a _ ) , we see that the branching fraction can change by typically a factor of 2 over the parameter range shown , with most of the variation occuring for changes in @xmath13 , instead of with @xmath18 . the entire plane shown in frame _ a _ ) is excluded by the cleo bound . in frame _ b _ ) , for @xmath12 , the branching fraction can change by up to a factor of @xmath129 over the plane shown , again with most of the variation coming due to changes in @xmath13 . in this case , the region to the left of the dotted contour is excluded by the cleo bound for all choices of renormalization parameter @xmath130 . an important constraint on the msugra model comes from implications for the relic density of dark matter in the universe . the idea here is that in the very early universe , the lsp ( the lightest neutralino ) was a constituent of the matter and radiation assumed to be in thermal equilibrium at some very high temperature . as the universe expanded and cooled , the lsp s could no longer be produced , although they could still annihilate with one another . upon further expansion , the neutralino flux dropped to such low levels that further annihilations would rarely occur , and a relic abundance of neutralinos was locked in . these relic lsp s could make up the bulk of dark matter in the universe today . the neutralino relic density is calculable as a function of msugra model parameter space@xcite . the relic density is usually parametrized in terms of @xmath131 , where @xmath132 , @xmath133 is the relic density , @xmath134 is the critical closure density of the universe ( @xmath135 ) and @xmath136 km / sec / mpc is the scaled hubble constant with @xmath137 . a value of @xmath138 implies a universe with age less than 10 billion years , in conflict with the ages of the oldest stars . if @xmath139 , then the relic density of neutralinos can not even account for the dark matter required by galactic rotation curves . some popular cosmological models that account for the cobe cosmic microwave background measurements as well as structure formation in the universe actually prefer a mixed dark matter ( mdm ) universe@xcite , with a matter density ratio of 0.3/0.6/0.1 for a hot dark matter / cold dark matter / baryonic matter mix . in this case , values of @xmath140 are preferred . in fig . [ nfig5 ] , we show contours of constant relic density @xmath131 . the region to the right of the solid @xmath141 contour is excluded by the age of the universe constraint , while the region below the dotted contour has @xmath139 . the region betwen the dashed - dotted contours is favored by a mdm universe . the region excluded by cleo data is below the solid contour labelled @xmath4 . in frame _ a _ ) , we see that combining the two constraints allows only a small patch of allowed parameter space with @xmath142 gev and @xmath143 gev . over almost all of this region , the relic density @xmath144 . in frame _ b _ ) , the @xmath4 excluded region hardly intersects with the mdm region , so that large regions of parameter space are favorable for cosmology as well as for cleo constraints ! in fig . [ nfig5 ] we also plot one contour for expected rates for direct detection of neutralino dark matter via cryogenic dark matter detectors@xcite . the calculations have been performed for neutralino scattering from a @xmath145 detector . current experiments are sensitive to detection rates of 1 - 10 events / day / kg of detector . the goal of such experiments is to achieve a sensitivity of @xmath146 events / kg / day by about the year 2000 . towards this end , we show the 0.01 event / kg / day contour in the @xmath111 plane ; below the contour the event rates exceed the 0.01/kg / day benchmark . in frame _ a _ ) , we see that the region accessible to direct neutralino detection coincides with the region with very large @xmath11 rates well beyond the cleo 95% cl limit . however , in frame _ b _ ) , for @xmath12 , there exists a significant region with large direct detection rates , which is allowed by the cleo constraint , and is also in the favorable cosmological region ! the lep2 @xmath147 collider is expected to reach a peak cm energy of @xmath148 gev , which should allow sm higgs bosons of mass @xmath149 gev to be explored . for the @xmath112 value shown in fig . [ nfig5 ] , the light higgs scalar @xmath150 gev over the entire @xmath111 plane shown . hence , for these values of @xmath0 , we would expect no higgs signals to be seen at lep2 . the reach of lep2 via @xmath151 and @xmath152 searches is shown by the dashed contour just above the region marked ex . this contour is defined by requiring @xmath153 gev and @xmath154 gev . for both cases of @xmath10 and @xmath12 shown in fig . [ nfig5 ] , the lep2 sparticle reach falls below both the @xmath11 excluded region , and below the @xmath155 contours . if we accept the msugra model literally , then the prediction is that lep2 should see no evidence for either a higgs or susy if @xmath156 . the fermilab tevatron @xmath15 collider is expected to operate at @xmath157 tev in run 2 , and to amass @xmath158 fb@xmath159 of integrated luminosity by use of the main injector ( mi ) . ultimately , experiments hope to acquire @xmath67fb@xmath159 of integrated luminosity under the tev33 program . recently completed calculations of the reach of the tevatron mi for msugra at large @xmath156 show a maximal reach in @xmath14 to @xmath160 gev in the @xmath161jets channel@xcite . a similar reach has been calculated for tev33 , and finds points with @xmath162 accessible . comparing these regions to fig . [ nfig5 ] shows that , like lep2 , the reach of tevatron mi and tev33 are below both the @xmath11 excluded contour and the @xmath155 contour , making discovery of susy particles highly unlikely for msugra if @xmath0 is large . over much of the parameter space plane in frame _ b _ ) of fig . [ nfig5 ] , however , @xmath163 gev , which ( optimistically ) corresponds to the maximal reach for @xmath164 at tev33 . hence , if msugra is correct and @xmath165 , then tev33 experiments may see a hint of the higgs boson in their data sample . of course , the entire large @xmath0 parameter space shown should be easily visible in at least the jets@xmath166 channel at the cern lhc , even with modest integrated luminosity . it is well known that within the msugra framework the chargino loop gives the dominant susy contribution to the amplitude for the decay @xmath59 . this can also be seen from fig . [ nfig1 ] where we see that while the contributions from the gluino loops are individually comparable ( or even larger ! ) than those from chargino loops , these cancel out almost completely leaving only a small residual contribution . in contrast , while there is indeed considerable cancellation amongst the various chargino contributions , there is nonetheless a sizeable residue that remains . we may understand the large cancellations among the gluino contributions in analogy with the familiar gim cancellation in the sm : indeed such a cancellation would be exact if squarks were precisely degenerate ( _ i.e. _ , the squark mass matrix is proportional to the unit matrix ) because we can then , by a unitary transformation , align the squark and quark mass matrices , so that the gluino - squark - quark vertex is exactly flavour diagonal . within the msugra framework with universal soft breaking squark mass matrices at the unification scale , squarks are indeed ( approximately ) degenerate , and gluino loop contributions to the flavour violating @xmath59 decay are suppressed . the gim - like cancellation that we have described above does not occur when yukawa couplings enter the calculation as occurs , for instance , via the higgsino components of chargino and neutralino loops . in this case , as can be seen from fig . [ nfig1 ] , the cancellation is indeed incomplete ( particularly for the chargino where the large top quark yukawa coupling enters ) . these considerations lead us to examine whether the breaking of the degeneracy of the soft susy - breaking squark masses at the unification scale so strongly upsets the delicate cancellations that it results in large gluino contributions to the amplitude for @xmath59 decay . of course , by allowing soft - breaking mass squared matrices with arbitrary off - diagonal entries , it should be possible to get very large flavour violating gluino interactions . the issue that we address , however is whether large gluino contributions are possible even if we choose these soft squark matrices to be diagonal at the unification scale . as we will soon see , the physics of this ansatz is basis - dependent . to parametrize the breaking of the squark degeneracy we begin by noting that we may always choose a quark basis so that _ either _ the down or the up type yukawa couplings are diagonal at the weak scale . we will call these the @xmath102- and @xmath167- cases , respectively . next , these couplings are evolved to @xmath17 , where both up and down yukawa matrices have off - diagonal components . in the @xmath102-case , the down type yukawa matrices get off - diagonal contributions just from the rge , while the up type yukawas start off off - diagonal right at the weak scale ; in the @xmath167-case , the situation is reversed . up to now , the choice to work in the @xmath167 or @xmath102 cases is purely a matter of convention , and indeed in previous sections we have used the @xmath102 case . this is , however , no longer the case if we further assume that the soft breaking squark mass squared matrix is diagonal ( but not a multiple of the identity ) at the gut scale . this is because the transformation that takes us from the @xmath102-case to the @xmath167-case does not leave the squark mass squared matrix diagonal ( except in the case when this matrix is @xmath168 ) . the @xmath167- and @xmath102- cases are thus physically distinct . we have , therefore , studied these two cases separately . to keep things simple , we split only the @xmath39-squarks ( @xmath169 splits with @xmath170 , of course ) keeping the others degenerate at @xmath13 . the splitting is given by a single parameter @xmath171 where @xmath53 and @xmath13 are soft squark masses at the gut scale . thus @xmath172 corresponds to the msugra case . we further consider three possibilities where ( _ i _ ) just @xmath173 , ( _ ii _ ) just @xmath174 and ( _ iii _ ) both @xmath173 and @xmath174 masses are split from those of other squarks . the results of our calculation of the gluino contribution to @xmath77 where the squark degeneracy is broken as described above is shown in fig . [ nfig6 ] for the @xmath102-case labelled @xmath102-diagonal , and for the @xmath167-case , labelled @xmath167-diagonal . in each frame , we have three curves labelled @xmath175 , @xmath176 and @xmath177 for the cases where just left , just right , and both left and right sbottom soft masses are different from @xmath13 . in our calculation , we have chosen @xmath178 gev , @xmath128 gev , @xmath179 , @xmath104 and @xmath180 . we choose a large value of @xmath13 so that the squark masses are not dominated by @xmath181 ( in which case splitting due to non - universal soft mass term would be unimportant ) . also , for the large value of @xmath0 the bottom yukawa coupling is significant . the following features are worth noting . * for @xmath172 the value of @xmath182 is the same for the up and down cases for reasons that we have already explained . * for non - degenerate squarks , gluino loop effects are significantly larger in the @xmath167-case . this may be understood if we recall that in the @xmath102-case , the mixing of down type yukawas at @xmath17 arises _ only _ due to rge . * non - degeneracy effects when just the right squarks are split show only small variation with the non - degeneracy parameter @xmath183 because flavour mixing in the right squark sector is suppressed . * somewhat surprising is the fact that despite the large degree of non - degeneracy , the gluino contribution to @xmath77 , which increases by up to an order of magnitude relative to that in the msugra case , never becomes really large . for our choice of parameters , this may be partly due to the fact that gluinos and squarks are significantly heavier than @xmath184 and @xmath185 . this is not the complete reason though . in our computation we find that the three main contributions ( from the two sbottoms and the @xmath186 ) cancel one another leaving a remainder that is typically smaller than 10 - 15% of the largest contribution . while this cancellation is much less complete than in the msugra case , we are unable to give a simple argument for why such a cancellation occurs . finally , we comment on the neutralino contributions for the case of non - degenerate squarks . we have already noted that cancellations among various chargino contributions are incomplete because of the effect of large higgsino couplings to the @xmath187 system . for large values of @xmath0 , we may expect a similar effect for neutralinos . we may further guess that this effect is largest in the @xmath167-case where flavor mixing does not originate solely in the rge . we have checked that for values of parameters in fig . [ nfig6 ] above a small value of @xmath188 , the neutralino contribution to @xmath77 is indeed enhanced by a factor of 6 - 7 above its msugra value , and further , that this enhancement is largely due to incompleteness in the cancellation between various contributions . for very large values of @xmath183 , @xmath77 is about 1.5 times its msugra value , but opposite in sign . we thus conclude that while the neutralino contribution is somewhat sensitive to the splitting of squark masses , it never appears to become very dominant . to summarize the results of this section , we see that with our assumptions , gluino contributions to the amplitude for the @xmath189 decay never dominate the susy contribution . this contribution may nonetheless be non - negligible even if gluinos and squarks are well beyond the reach of the tevatron ( and its proposed upgrades ) as seen in fig . [ nfig6 ] . we emphasize though that our conclusion is special to models where all the flavour violation in the gluino - squark - quark vertex at the gut scale comes from non - diagonal yukawa interactions . larger contributions from gluino loops may be possible in other models . _ note added : _ after completion of this manuscript , a related paper by blazek and raby appeared on the topic of @xmath4 constraints on @xmath190 susy models at large @xmath0@xcite . since ref . @xcite adopts a particular @xmath190 framework and does not include the radiative electroweak symmetry breaking constraint , comparison of results between the two papers is not straightforward . also , we have not included @xmath191 mixing included in ref . @xcite in our evaluation of the chargino loop . we are grateful to m. drees and t. ter veldhuis for helpful conversations and comments . we thank the aspen center for physics for hospitality while a portion of this work was completed . this research was supported in part by the u. s. department of energy under grant number de - fg-05 - 87er40319 . e.g. _ , h. haber in _ woodlands superworld _ , hep - ph/9308209 ( 1993 ) . e.g. _ , m. drees and s. martin in _ electroweak symmetry breaking and new physics at the tev scale _ , edited by t. barklow , s. dawson , h . haber and j. seigrist , ( world scientific ) 1995 ; see also j. amundson _ _ in _ new directions for high energy physics _ , edited by d. g. cassel , l. trindle gennari and r. h. siemann , ( stanford linear accelerator center , 1996 ) , hep - ph/9609374 . a. chamseddine , r. arnowitt and p. nath , phys . lett . * 49 * , 970 ( 1982 ) ; r. barbieri , s. ferrara and c. savoy , phys . lett . * b119 * , 343 ( 1982 ) ; l.j . hall , j. lykken and s. weinberg , phys . 2359 ( 1983 ) . m. s. alam _ et al . _ , ( cleo collaboration ) , phys . rev . lett . * 74 * , 2885 ( 1995 ) . note that recently the aleph collaboration ( at the international europhysics conference on high energy physics , jerusalem , israel , aug . 19 - 26 , 1997 ) has made a preliminary announcement of a measurement of @xmath192 . we do not include this preliminary result in our analysis , and caution the reader that some of our conclusions may change if the @xmath193 decay rate turns out to be closer to this aleph value . s. bertolini , f. borzumati , a. masiero and g. ridolfi , nucl . b*353 * , 591 ( 1991 ) . r. barbieri and g. f. giudice , phys . b*309 * , 86 ( 1993 ) ; 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m. carena , m. quiros , c.e.m . wagner , nucl . * b461 * , 407 ( 1996 ) ; h. haber , r. hempfling and a. hoang , z. phys . * c*75 , 539 ( 1997 ) . see f. borzumati , ref . h. anlauf , nucl . b*430 * , 245 ( 1994 ) . e.g. _ , b. grinstein , m. j. savage and m. wise , nucl . b*319 * , 271 ( 1989 ) ; a. ali , in _ 20th international nathiagali summer college on physics and contemporary needs _ , bhurban , pakistan , 1995 , hep - ph/9606324 . c. greub , t. hurth and d. wyler , phys . b*380 * , 385 ( 1996 ) and phys . rev . d*54 * , 3350 ( 1996 ) . a. ali and c. greub , z. phys . c*49 * , 431 ( 1991 ) , phys . b*259 * , 182 ( 1991 ) , * 287 * , 191 ( 1992 ) and * 361 * , 146 ( 1995 ) ; z. phys . c*60 * , 433 ( 1993 ) ; n. pott , phys . * , 938 ( 1996 ) ; m. ciuchini , e. franco , g. martinelli and l. reina , nucl . b*415 * , 403 ( 1994 ) . m. misiak and m. mnz , phys . b*344 * , 308 ( 1995 ) . k. g. chetyrkin , m. misiak and m. mnz , phys . b400 * , 206 ( 1997 ) . p. cho and b. grinstein , nucl . b*365 * , 279 ( 1991 ) . h. arason _ et al . _ , rev . d*46 * , 3945 ( 1992 ) . @xcite , a. buras , a. kwiatkowski and n. pott , hep - ph/9707482 ( 1997 ) and c. grueb and t. hurth , hep - ph9708214 ( 1997 ) quote a somewhat higher sm value of @xmath11 than is presented here . the above works include a 3% increase due to non - perturbative effects calculated by m. b. voloshin , phys . lett . * b*397 , 275 ( 1997 ) plus the corrected value of @xmath194 in their calculations , which conspire to increase the sm decay rate by about 10% . these effects would similarly increase our estimate of the susy contributions . g. anderson and d. castao , phys . b347 * , 300 ( 1995 ) and phys . rev . d*52 * , 1693 ( 1995 ) ; see also k. l. chan , u. chattopadhyay and p. nath , hep - ph/9710473 ( 1997 ) . for a recent review , see g. jungman , m. kamionkowski and k. griest , phys . rep . * 267 * , 195 ( 1996 ) . see also m. drees and m. nojiri , phys . rev . * d47 * , 376 ( 1993 ) . the result shown here are from h. baer and m. brhlik , phys . * d53 * , 597 ( 1996 ) . further references are included in these reports . e.g. _ , j. primack , astro - ph/9707285 ( 1997 ) . h. baer and m. brhlik , phys . d , in press , hep - ph/9706509 . h. baer , c. h. chen , m. drees , f. paige and x. tata , manuscript in preparation . t. blazek and s. raby , hep - ph/9712257 ( 1997 ) .
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in the minimal supergravity model ( msugra ) , as the parameter @xmath0 increases , the charged higgs boson and light bottom squark masses decrease , which can potentially increase contributions from @xmath1 , @xmath2 and @xmath3 loops in the decay @xmath4 .
we update a previous qcd improved @xmath4 decay calculation to include in addition the effects of gluino and neutralino loops .
we find that in the msugra model , loops involving charginos also increase , and dominate over @xmath5 , @xmath1 , @xmath6 and @xmath7 contributions for @xmath8 .
we find for large values of @xmath9 that most of the parameter space of the msugra model for @xmath10 is ruled out due to too large a value of branching ratio @xmath11 . for @xmath12 and large @xmath0 ,
most of parameter space is allowed , although the regions with the least fine - tuning ( low @xmath13 and @xmath14 ) are ruled out due to too _ low _ a value of @xmath11 .
we compare the constraints from @xmath4 to constraints from the neutralino relic density , and to expectations for sparticle discovery at lep2 and the fermilab tevatron @xmath15 colliders .
finally , we show that non - universal gut scale soft breaking squark mass terms can enhance gluino loop contributions to @xmath4 decay rate even if these are diagonal .
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throughout this article , we will let @xmath8 , @xmath9 , and @xmath10 denote the set of positive integers , the set of nonnegative integers , and the set of prime numbers , respectively . the lowercase letter @xmath11 will always denote a prime number , and @xmath12 will denote the exponent of @xmath11 in the prime factorization of a positive integer @xmath13 . furthermore , for any nonzero complex number @xmath14 , we let @xmath15 denote the principal argument of @xmath14 with the convention that @xmath16 . for any complex number @xmath0 , the divisor function @xmath17 is the arithmetic function defined by @xmath2 for all @xmath3 . the function @xmath5 is a multiplicative arithmetic function that satisfies @xmath18 for all primes @xmath11 and positive integers @xmath19 . of course , if @xmath20 , then we may write @xmath21 . divisor functions are some of the most important functions in number theory ; their appearances in various identities and applications are so numerous that we will not even attempt to list them . however , divisor functions other than @xmath22 , and @xmath23 are rarely studied . recently , the author @xcite has studied the ranges of the functions @xmath5 for real @xmath0 and has shown that there exists a constant @xmath24 such that if @xmath25 , then the range of the function @xmath26 is dense in the interval @xmath27 if and only if @xmath28 . for any complex @xmath0 , we will let @xmath29 be the range of the function @xmath5 . in this article , we will study the basic topological properties of the sets @xmath6 for various complex numbers @xmath0 . more specifically , we will direct the bulk of our attention toward answering the following questions : 1 . for which complex @xmath0 is @xmath6 bounded ? for which complex @xmath0 does @xmath6 have isolated points ? 3 . what can we tell about the closure @xmath30 of the set @xmath6 for given values of @xmath0 ? in particular , what are the values of @xmath0 for which @xmath6 is dense in @xmath7 ? we begin with a number of useful lemmas . some of these lemmas not only aid in the proofs of later theorems , but also provide some basic yet interesting information that serves as a nice introduction to the sets @xmath6 . henceforth , @xmath0 will denote a complex number with real part @xmath31 and imaginary part @xmath32 . [ lem1.1 ] for any @xmath33 , @xmath34 . we have @xmath35 lemma [ lem1.1 ] tells us that @xmath36 is simply the reflection of the set @xmath6 about the real axis . in many situations , this simple but useful lemma allows us to restrict our attention to complex numbers @xmath0 in the upper half plane and then use symmetry to deduce similar results for values of @xmath0 in the lower half - plane . [ lem1.2 ] we have @xmath37 if and only if @xmath38 and @xmath39 for some prime @xmath11 and some rational @xmath40 that is not an even integer . first , suppose @xmath38 and @xmath41 , where @xmath11 is a prime and @xmath40 is a rational number that is not an even integer . as @xmath40 is not an even integer , @xmath42 . we may write @xmath43 for some nonzero integers @xmath44 and @xmath45 with @xmath46 . then @xmath47 so @xmath37 . conversely , suppose @xmath37 . then there exists some @xmath3 with @xmath48 . clearly @xmath49 , so we may let @xmath50 be the canonical prime factorization of @xmath13 . then @xmath51 , so @xmath52 for some @xmath53 . let @xmath54 and @xmath55 . we know that @xmath20 because , otherwise , we would have @xmath56 . therefore , @xmath57 , so @xmath58 . now , @xmath59 so we must have @xmath38 and @xmath60 for some integer @xmath61 . letting @xmath62 , we see that @xmath32 has the desired form . finally , @xmath40 is not an even integer because @xmath63 . [ lem1.3 ] suppose @xmath38 and @xmath64 . let @xmath65 , and let @xmath66 be the circle @xmath67 . then @xmath68 is a dense subset of @xmath66 . by lemma [ lem1.1 ] , it suffices to prove our claim in the case @xmath69 . furthermore , because @xmath70 for all primes @xmath11 , it suffices to show that the set @xmath71 is a dence subset of the circle @xmath72 . we know that every point in @xmath73 lies on the circle @xmath74 because @xmath75 for all primes @xmath11 . now , choose some @xmath76 and some @xmath77 . we may write @xmath78 for some @xmath79 $ ] . we wish to show that there exists a prime @xmath11 such that @xmath80 for some integer @xmath81 . equivalently , we need to show that there exists a prime @xmath11 and a positive integer @xmath13 such that @xmath82 . setting @xmath83 , @xmath84 , and @xmath85 , we may rewrite these inequalities as @xmath86 . it follows from the well - known fact that @xmath87 that such a prime @xmath11 is guaranteed to exist for sufficiently large @xmath13 ( here , we let @xmath88 denote the @xmath89 prime number ) . [ lem1.4 ] if @xmath90 , then @xmath91 for all @xmath3 . suppose @xmath90 . for any prime @xmath11 and positive integer @xmath19 we have @xmath92 therefore , for any @xmath3 , @xmath93\left[\prod_{\substack{p\vert n \\ p\geq 2^{1/a}}}\left\lvert\sigma_c\left(p^{\nu_p(n)}\right)\right\rvert\right]\ ] ] @xmath94\left[\prod_{\substack{p\vert n \\ p\geq 2^{1/a}}}(p^a-1)\right]\geq\prod_{p<2^{1/a}}(p^a-1).\ ] ] in the third question that we posed above , we asked if we could find the values of @xmath0 for which @xmath6 is dense in @xmath7 . lemma [ lem1.4 ] gives us an immediate partial answer to this question . if @xmath90 , then @xmath6 can not be dense in @xmath7 because there is a neighborhood of @xmath95 of radius @xmath96 that contains no elements of @xmath6 . we will see in theorem [ thm2.2 ] that , in some sense , @xmath6 is very far from being dense when @xmath90 . the following lemma simply transforms an estimate due to rosser and shoenfeld into a slightly weaker inequality which is more easily applicable to our needs . [ lem1.5 ] if @xmath97 , then @xmath98}\left(1-\frac{1}{p}\right)<\frac{\log y}{\log x}+\frac{2}{\log^2y}.\ ] ] rosser and shoenfeld s estimate ( * ? ? ? * theorem 7 ) states that if @xmath99 , then @xmath100 where @xmath101 is the euler - mascheroni constant . therefore , if @xmath97 , then @xmath98}\left(1-\frac{1}{p}\right)<\left(\frac{e^{-\gamma}}{\log x}\left(1+\frac{1}{2\log^2x}\right)\right)\left(\frac{e^{-\gamma}}{\log y}\left(1-\frac{1}{2\log^2y}\right)\right)^{-1}\ ] ] @xmath102 @xmath103 [ lem1.6 ] suppose @xmath104 and @xmath69 . fix some @xmath105 . for each nonnegative integer @xmath61 , let @xmath106}$ ] . let @xmath107 . if @xmath108 , then @xmath109 and @xmath110 in addition , @xmath111 suppose @xmath112 . then @xmath113 , so @xmath114 . by the definition of @xmath115 , @xmath116 $ ] for some nonnegative integer @xmath61 . therefore , @xmath117 . the inequality @xmath118 follows from the law of cosines because @xmath119 @xmath120 we now wish to prove that @xmath121 . this sum makes sense ( the order of the summands is immaterial ) because all summands are positive by the preceding inequality . for sufficiently large @xmath61 , we may use lemma [ lem1.5 ] to write @xmath122 @xmath123 @xmath124 also , if @xmath125 , then @xmath126 hence , for sufficiently large @xmath61 , we have @xmath127 @xmath128 the desired result then follows from the fact that @xmath129 [ lem1.7 ] suppose @xmath130 and @xmath69 . fix some @xmath105 . for each nonnegative integer @xmath61 , let @xmath131}$ ] . let @xmath132 . if @xmath133 , then @xmath134 and either @xmath135 or @xmath136 . in addition , @xmath137 the proof is quite similar to that of lemma [ lem1.6 ] . suppose @xmath138 . then @xmath113 , so @xmath114 . by the definition of @xmath139 , @xmath140 $ ] for some positive integer @xmath61 . therefore , either @xmath135 or @xmath136 . the law of cosines yields @xmath141 we now show that @xmath142 first , we need to check that the order of the summands in this summation does not matter . for all @xmath143 , we have @xmath144 because @xmath145 and @xmath146 . therefore , @xmath147 for all @xmath143 . in fact , if @xmath11 is sufficiently large , then @xmath148 as all summands are negative , their order does not matter . using lemma [ lem1.5 ] , we see that if @xmath61 is sufficiently large , then @xmath149 @xmath150 hence , for sufficiently large @xmath61 , we have @xmath151 @xmath152 we now obtain the desired result from the fact that @xmath153 we omit the proof of the following lemma because it is very similar to ( and quite a bit easier than ) the proofs of lemmas [ lem1.6 ] and [ lem1.7 ] . we remark that , for the proof of the following lemma , it is more convenient to use theorem 5 in @xcite than it is to use theorem 7 . [ lem1.8 ] suppose @xmath69 . for each nonnegative integer @xmath61 , let @xmath154 $ ] , and let @xmath155 . if @xmath156 , then @xmath157 also , @xmath158 . it turns out that questions @xmath159 and @xmath160 posed in the introduction are not too difficult to handle , so we will give complete answers to them in this section . [ thm2.1 ] the set @xmath6 is bounded if and only if @xmath161 . if @xmath161 , then @xmath6 is bounded because @xmath162 for all @xmath33 . now , suppose @xmath104 . by lemma [ lem1.1 ] , we see that it suffices to prove the result for @xmath163 . if @xmath164 , then @xmath165 if @xmath69 , then the proof follows from lemma [ lem1.6 ] because @xmath166 where @xmath115 is defined as in the lemma . note that we have used the fact that @xmath5 is multiplicative . [ thm2.2 ] if @xmath167 and @xmath168 , then @xmath6 has no isolated points . if @xmath90 or @xmath169 , then every point of @xmath6 is an isolated point of @xmath6 . first , suppose @xmath145 , and let @xmath170 . note that @xmath171 by lemma [ lem1.2 ] . we may write @xmath172 for some @xmath3 . choose some @xmath77 . to show that @xmath173 is not an isolated point of @xmath6 , we simply need to exhibit a positive integer @xmath174 such that @xmath175 . as @xmath145 , we may choose some prime @xmath176 such that @xmath177 . let @xmath178 . as @xmath40 is relatively prime to @xmath13 , we have @xmath179 @xmath180 this also shows that @xmath181 because @xmath182 . we now handle the case @xmath38 , @xmath168 . in this case , @xmath183 . we wish to show that @xmath6 has no isolated points . in fact , we will prove the much stronger assertion that @xmath6 is dense in @xmath7 . by lemma [ lem1.1 ] , we see that it suffices to prove this claim when @xmath69 . fix some @xmath184 and some @xmath185 $ ] . choose @xmath186 . we wish to exhibit a positive integer @xmath174 such that @xmath187 and @xmath188 for some integer @xmath81 . this will show that @xmath189 is either in @xmath6 or is a limit point of @xmath6 . because we may choose @xmath189 to be any nonzero complex number , this will prove the assertion that @xmath6 is dense in @xmath7 . by lemma [ lem1.3 ] , it is possible to find distinct primes @xmath190 such that @xmath191 for each @xmath192 . note that @xmath193 , so @xmath194 for each @xmath192 . for each @xmath192 , we know that @xmath195 lies on the circle @xmath67 and that @xmath196 , so it is easy to verify that @xmath197 . therefore , one may easily verify that @xmath198 let @xmath199 be a positive integer such that @xmath200 now , if @xmath11 is any prime such that @xmath201 , then it is easy to see from the law of cosines ( just as in the proof of lemma [ lem1.6 ] ) that @xmath202 because @xmath203 . lemma [ lem1.3 ] tells us that it is possible to choose distinct primes @xmath204 such that @xmath205 and @xmath206 for all @xmath207 . let @xmath208 , and let @xmath209 . using the fact that @xmath210 , we have @xmath211 therefore , @xmath212 . this implies that there exists a complex number @xmath173 such that @xmath213 , @xmath214 , and @xmath215 . by lemma [ lem1.3 ] , it is possible to choose distinct primes @xmath216 and @xmath217 such that @xmath218 , @xmath219 , @xmath220 , @xmath221 , @xmath222 , and @xmath223 . essentially , we have just chosen @xmath216 and @xmath216 so that @xmath224 is sufficiently close to @xmath173 and @xmath225 is sufficiently close to @xmath226 . if we let @xmath227 , then @xmath228 also , there exists some integer @xmath81 such that @xmath229 @xmath230 @xmath231 @xmath232 this complete the proof of the fact that @xmath6 is dense in @xmath7 when @xmath38 and @xmath64 . now , assume @xmath90 . choose some @xmath233 , and let @xmath45 be a positive integer such that @xmath234 . suppose @xmath235 for some prime @xmath236 and some positive integer @xmath237 . let us write @xmath238 , where @xmath239 and @xmath240 are integers , @xmath241 and @xmath242 . note that @xmath243 and @xmath244 by lemma [ lem1.4 ] . for the sake of brevity , let @xmath245 . then @xmath246 if we fix @xmath236 , we see that @xmath237 must be bounded above . similarly , if we fix @xmath237 , we see that @xmath236 must be bounded above . consequently , there are only finitely many prime powers @xmath247 that can divide @xmath45 . this implies that there are only finitely many positive integers @xmath45 such that @xmath248 . it follows that any disk in the complex plane contains finitely many points of @xmath6 , so every point of @xmath6 is an isolated point of @xmath6 . the final case we have to consider is when @xmath169 . this is easy because @xmath249 is simply the number of divisors of @xmath13 . we see that @xmath250 , so every point in @xmath251 is an isolated point of @xmath251 . the third question posed in the introduction proves to be more difficult than the first two . for one thing , the first part of the question is fairly open - ended . what exactly would we like to know about the sets @xmath30 ? in order to ask more specific and interesting questions , we wish to gain a bit of basic information . first of all , if @xmath90 or if @xmath169 , then there is not much use in inquiring about the set @xmath30 because this set is the same as @xmath6 by theorem [ thm2.2 ] . if @xmath164 ( so that @xmath0 is real ) , then @xmath6 is a subset of @xmath252 . in that case , there are many fascinating questions we may ask . for example , as mentioned in the introduction , the author has classified those real @xmath0 for which @xmath30 is a single closed interval @xcite . here , however , we will not pay too close attention to the sets @xmath30 for real @xmath0 . for the moment , let us streamline our attention toward the sets @xmath30 when @xmath253 and @xmath64 . by theorem [ thm2.1 ] , these sets are not bounded . at the same time , theorem [ thm2.2 ] tells us that @xmath6 has no isolated points for such values of @xmath0 , so we might hope to obtain more interesting sets than those that arise when @xmath90 . in fact , if we recall the second part of the third question posed in the introduction and decide to embark on a quest to find those complex @xmath0 for which @xmath254 , then we need only consider the case @xmath253 , @xmath64 . to begin this quest , we define a set @xmath255 $ ] for each complex @xmath0 by @xmath256 . the set @xmath257 is simply the set of arguments of nonzero points in @xmath6 . clearly , if @xmath254 , then @xmath258 $ ] . in fact , the truth of the converse of this assertion ( in the case @xmath253 ) will allow us to deduce our main result . first , we need the following lemma . [ lem3.1 ] if @xmath130 and @xmath64 , then @xmath258 $ ] . choose some @xmath259 . we will find a positive integer @xmath13 such that @xmath260 for some integer @xmath81 , and this will prove the claim . consider the sets @xmath261 and @xmath262 that were defined in lemma [ lem1.8 ] , and let @xmath263 , where @xmath264 . pick some positive integer @xmath61 . note that @xmath265 . because lemma [ lem1.8 ] tells us that @xmath266 , it is not difficult to see from a bit of geometry that @xmath267 using the fact that @xmath268 along with the inequality @xmath269 , which holds for all real @xmath270 , we have @xmath271 it then follows from lemma [ lem1.8 ] that @xmath272 . because @xmath145 , @xmath273 . let @xmath274 be a positive integer such that @xmath275 for all @xmath276 . as @xmath277 , there exists some integer @xmath278 such that @xmath279 . setting @xmath280 , we have @xmath281 for some integer @xmath81 , from which we obtain the desired inequalities @xmath260 . [ thm3.1 ] the set @xmath6 is dense in @xmath7 if and only if @xmath253 and @xmath64 . we have seen that @xmath253 and @xmath64 if @xmath254 , so we now wish to prove the converse . if @xmath38 and @xmath64 , then we saw in the proof of theorem [ thm2.2 ] that @xmath6 is dense in @xmath7 . therefore , let us assume that @xmath130 . with the help of lemma [ lem1.1 ] , we may also assume that @xmath69 . fix @xmath0 ( with @xmath282 ) , and choose some @xmath283 and some @xmath184 . let @xmath13 be a positive integer such that @xmath284 . we wish to show that @xmath189 is a limit point of @xmath6 . if we can accomplish this goal , then we will know that any arbitrary nonzero complex number @xmath14 is a limit point of @xmath6 . indeed , lemma [ lem3.1 ] allows us to choose @xmath285 arbitrarily close to @xmath15 , and we may set @xmath286 . it will then follow that @xmath95 is also a limit point of @xmath6 so that @xmath254 . if @xmath287 , then we are done by theorem [ thm2.2 ] . therefore , let us assume that @xmath288 . in particular , @xmath289 because @xmath290 . choose some @xmath291 with @xmath292 . if @xmath293 , let @xmath294 ; if @xmath295 , let @xmath296 . we will produce a positive integer @xmath174 such that @xmath297 and @xmath298 for some integer @xmath81 . as @xmath299 and @xmath300 are fixed , we may make @xmath301 as small as we wish by initially choosing sufficiently small values of @xmath302 and @xmath303 . therefore , the construction of such an integer @xmath174 will prove that @xmath189 is a limit point of @xmath6 ( technically , we must insist that @xmath304 , but this follows from our assumption that @xmath288 ) . let us define @xmath305 , @xmath306 , @xmath115 , and @xmath139 as in lemmas [ lem1.6 ] and [ lem1.7 ] . we will also write @xmath307 and @xmath308 , where @xmath309 and @xmath310 . because @xmath145 , we have @xmath311 and @xmath312 for all sufficiently large integers @xmath61 . let us fix some positive integer @xmath274 large enough so that for all integers @xmath313 , we have @xmath311 , @xmath312 , @xmath314 , @xmath315 , @xmath316 , and @xmath317 . suppose @xmath293 . because @xmath311 for all @xmath313 and @xmath318 by lemma [ lem1.6 ] , there must exist some integer @xmath278 such that @xmath319 this yields the inequalities @xmath320 let @xmath321 . because @xmath322 is relatively prime to @xmath13 for all @xmath313 , we have @xmath323 therefore , @xmath324 , so @xmath325 as desired . we now prove that @xmath298 for some integer @xmath81 . we have @xmath326 for some integer @xmath81 . therefore , @xmath327 let us fix some @xmath328 . because @xmath329 , we know from lemma [ lem1.6 ] that @xmath330 . also , @xmath331 . using basic trigonometry , we have @xmath332 if we write @xmath333 , then lemma [ lem1.6 ] tells us that @xmath334 . recall that we chose @xmath274 large enough to ensure that @xmath335 . also , @xmath336 because we chose @xmath337 . we have @xmath338 , so @xmath339 @xmath340 this shows that @xmath341 from these inequalities , we get @xmath342 @xmath343 @xmath344 we now use the fact that @xmath345 to conclude that @xmath346 this completes the proof of the case @xmath293 . let us now assume @xmath295 . the proof of this case is quite similar to the previous case . because @xmath312 for all @xmath313 and @xmath347 by lemma [ lem1.7 ] , there must exist some integer @xmath348 such that @xmath349 this yields the inequalities @xmath350 in this case , we will let @xmath351 . because @xmath352 is relatively prime to @xmath13 for all @xmath313 , we have @xmath353 therefore , @xmath354 , so @xmath297 once again . we have @xmath355 for some integer @xmath81 . therefore , @xmath356 let us fix some @xmath357 . because @xmath358 , we know from lemma [ lem1.7 ] that @xmath359 or @xmath360 . also , @xmath361 . trigonometry tells us that @xmath362 recalling that we chose @xmath274 large enough to ensure that @xmath363 , we see that @xmath364 . in addition , because @xmath365 , we have @xmath366 . if we write @xmath367 , then lemma [ lem1.7 ] tells us that @xmath368 . we also see that @xmath369 , so @xmath370 . using the inequality @xmath371 , which holds for all @xmath372 , we have @xmath373 thus , @xmath374 . combining these inequalities , we get @xmath375 @xmath376 @xmath377 using the fact that @xmath378 , we have @xmath379 hence , we conclude that @xmath380 this completes the proof of the case @xmath295 . recall the section of the proof of theorem [ thm2.2 ] in which we proved that @xmath254 whenever @xmath38 and @xmath64 . we proved this fact by describing the construction of an integer @xmath174 and showing that we could make @xmath381 arbitrarily close to any predetermined complex number . the observant reader may have noticed that the integer @xmath174 that we constructed was squarefree . furthermore , we constructed @xmath174 using primes that could have been arbitrarily large . in other words , there was never an upper bound on the sizes of the required primes . the same observations are true of the proofs of lemma [ lem3.1 ] and theorem [ thm3.1 ] . in particular , we chose @xmath13 in the proof of theorem [ thm3.1 ] to be an arbitrary element of @xmath257 , and we could have chosen @xmath13 to be a squarefree number with large prime divisors . therefore , we obtain the following corollary . [ cor3.1 ] let @xmath382 be a positive integer , and let @xmath383 be the set of squarefree integers whose prime factors are all greater than @xmath382 . suppose @xmath253 and @xmath64 . let @xmath384 . then @xmath385 is dense in @xmath7 . we now focus on the sets @xmath6 that arise when @xmath161 . we saw in theorem [ thm2.1 ] that these sets are bounded , so it is natural to inquire about the shapes of their closures . we will prove two theorems in order to provide a taste of the questions that one might wish to ask about these sets . the first theorem is very straightforward , but leads to an interesting question . [ thm3.2 ] if @xmath161 , then @xmath386 . as @xmath387 , we have @xmath388 @xmath389 now , @xmath390 , and the result follows . it is clearly not difficult to extract explicit bounds for @xmath391 from the proof of theorem [ thm3.2 ] , but we are more concerned with the following obvious corollary . [ cor3.2 ] if @xmath161 , then @xmath392 . suppose @xmath161 . since @xmath30 is lebesgue measurable , we may define @xmath393 to be the area of @xmath30 . provided @xmath394 , we may let @xmath395 be the centroid of @xmath30 ( where @xmath396 represents a differential area element in the complex plane ) . corollary [ cor3.2 ] endorses @xmath397 as a potential candidate for @xmath398 , although it certainly does not provide a proof . unfortunately , a rigorous determination of the value of @xmath398 seems to require knowledge of the shape of @xmath30 because of the apparent necessity of calculating the integrals involved in the definitions of @xmath393 and @xmath398 . the limit in corollary [ cor3.2 ] does not appear to be too useful for these purposes because its value depends on the ordering of @xmath8 . that is , if we let @xmath399 be some enumeration of the positive integers , then it could very well be the case that @xmath400 ( or that this limit does not exist ) . hence , for now , we will let @xmath398 be . it seems natural to ask for the values of @xmath0 with @xmath161 for which the sets @xmath30 are connected . the following theorem will show that if @xmath31 is sufficiently negative ( meaning negative and sufficiently large in absolute value ) , then @xmath30 is separated . in fact , the theorem states that for any positive integer @xmath174 , if @xmath31 is sufficiently negative , then @xmath30 is a disjoint union of at least @xmath174 closed sets . this is somewhat unintuitive since , for each @xmath3 , @xmath401 as @xmath402 . in other words , the sets @xmath30 shrink " while becoming more separated " as @xmath402 . [ thm3.3 ] for each positive integer @xmath174 , there exists a real number @xmath403 such that if @xmath404 , then @xmath30 is a union of at least @xmath174 disjoint closed sets . we will assume that @xmath161 throughout this proof . note that we may set @xmath405 . we will let @xmath406 denote the image under @xmath5 of a set @xmath407 of positive integers . for each @xmath408 , let @xmath409 denote the @xmath410 prime number . let @xmath411 be the set of positive integers that are not divisible by any of the first @xmath412 primes , and let @xmath413 denote the set of positive integers @xmath13 such that the smallest prime divisor of @xmath13 is @xmath409 . in symbols , @xmath414 , and @xmath415 , where we convene to let @xmath416 . fix some positive integer @xmath61 . we will show that there exists a real number @xmath417 and a function @xmath418 such that if @xmath419 , then @xmath420 for all @xmath421 and @xmath422 . this will show that @xmath423 is disjoint from @xmath424 whenever @xmath419 . in particular , it will follow that @xmath423 is disjoint from @xmath425 for all integers @xmath426 whenever @xmath419 . setting @xmath427 , we will see that if @xmath428 , then @xmath429 are @xmath430 disjoint closed sets whose union is @xmath30 . choose some @xmath421 and @xmath422 . note that we may write @xmath431 , where @xmath81 is a positive integer and @xmath432 . if we let @xmath433 for all @xmath434 , then we have @xmath435 similarly , @xmath436 . observe that @xmath437 , so there exists some number @xmath438 such that @xmath439 for all @xmath419 . let us define @xmath418 by @xmath440 if @xmath419 , then @xmath441 @xmath442 @xmath443 @xmath444 so @xmath445 . therefore , @xmath446 @xmath447 @xmath448 @xmath449 @xmath450 for all @xmath419 , and this is what we sought to prove . the expressions in the proof of theorem [ thm3.3 ] get a bit messy , and we probably could have simplified the argument by using more careless estimates . however , we organized the proof in order to give an upper bound for the values of @xmath451 . that is , we showed that we may let @xmath438 be any number such that holds for all @xmath419 . in particular , we have the following corollary . [ cor3.3 ] if @xmath452 , then @xmath30 is separated . preserve the notation from the proof of theorem [ thm3.3 ] . note that @xmath453 therefore , when @xmath454 , becomes @xmath455 we may rewrite this inequality as @xmath456 noting that @xmath457 and dividing each side of this last inequality by @xmath458 yields @xmath459 therefore , we simply need to show that holds for all @xmath460 . if @xmath461 , then @xmath462 , so @xmath463 @xmath464 hence , holds for @xmath461 . we now show that holds for @xmath465 . the reader who wishes to evade the banalities of the following fairly computational argument may wish to simply plot the values of @xmath466 and @xmath467 to see that appears to hold for these values of @xmath31 . let @xmath468 and @xmath469 so that our goal is to prove that @xmath470 for all @xmath471 . in order to reduce the number of necessary computations , we will partition @xmath472 into the two intervals @xmath473 $ ] and @xmath474 . for all @xmath471 , we have @xmath475 @xmath476 now , if @xmath477 , then @xmath478 numerical calculations show that if @xmath479 then @xmath480 . because @xmath481 is continuous , we see that @xmath482 for all @xmath477 . if @xmath483 , then @xmath484 for all @xmath485 numerical calculations show that @xmath486 . because @xmath481 is continuous , we see that @xmath487 for all @xmath483 . = 8.5 cm + = 8.29 cm we have obtained a decent understanding of the ranges @xmath6 of the functions @xmath5 for values of @xmath0 with real part @xmath488 , but many problems concerning the sets @xmath6 that arise when @xmath161 remain open . for example , it would be quite interesting to determine the values of @xmath0 for which @xmath30 is connected . corollary [ cor3.3 ] provides an initial step in this direction , but there is certainly much work that remains to be done . for example , it seems as though the bound @xmath460 in that corollary is quite weak since the estimates used to derive it are far from optimal . recall that we derived that bound by showing that if @xmath460 , then @xmath30 is the disjoint union of the closed sets @xmath489 and @xmath490 defined in the proof of theorem [ thm3.3 ] . we remark , however , that it might be more useful to look at the value of @xmath0 for which @xmath491 is disjoint from @xmath492 . indeed , if we confine @xmath0 to the real axis and decrease @xmath0 from @xmath493 to @xmath494 , we will see that @xmath30 first separates into the disjoint union of the two connected sets @xmath491 and @xmath492 . more formally , one may use the methods described in @xcite to show that there exist constants @xmath24 and @xmath495 such that if @xmath0 is real and @xmath496 , then @xmath30 is the disjoint union of the two connected sets @xmath491 and @xmath492 ( this phenomenon essentially occurs because the largest value of @xmath497 occurs when @xmath498 ) . nonetheless , a full determination of those complex @xmath0 for which @xmath30 is connected seems to require knowledge about the specific shapes of the sets @xmath30 . using mathematica to plot points of some of these sets allows one to see the emergence of sets that appear to have certain fractal - like properties ( see figure 1 ) . however , as we can only plot finitely many points , it is difficult to predict the shapes of the full sets @xmath6 and their closures . we mention once again the open problem of determining the values of @xmath393 and @xmath398 defined after corollary [ cor3.2 ] above . observe the point @xmath397 plotted in the top image of figure 1 ( corresponding to @xmath499 ) . upon visual inspection , this point does not appear to be the centroid @xmath398 of @xmath30 . this is likely due to the fact that points of @xmath6 clustered in the lower part of the image are packed more densely than those in the middle part . finally , we remark that , while investigating the topics discussed in section 3 , the author found that it would be useful to have good upper and lower bounds for @xmath257 . thus , we state the derivation of such bounds as an additional potential topic for future research .
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for any complex number @xmath0 , let @xmath1 denote the divisor function defined by @xmath2 for all @xmath3 , and define @xmath4 to be the range of @xmath5 .
we study the basic topological properties of the sets @xmath6 .
in particular , we determine the complex numbers @xmath0 for which @xmath6 is bounded and determine the isolated points of the sets @xmath6 . in the third section , we find those values of @xmath0 for which @xmath6 is dense in @xmath7 .
we also prove some results and pose several open problems about the closures of the sets @xmath6 when these sets are bounded .
.2 in 2010 _ mathematics subject classification _ : primary 11b05 ; secondary 11a25 . _
keywords : divisor function ; complex number ; range ; dense ; isolated point ; closure . _
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You are an expert at summarizing long articles. Proceed to summarize the following text:
discovering pulsars typically involves identifying periodic signals in observational data , then reducing each of them into a set of diagnostic values and graphical representations referred to as a _ candidate_. a modern all - sky pulsar survey such as the high time resolution universe ( htru ; * ? ? ? * ) produces several million such candidates , the overwhelming majority of which are either the result of human - made radio - frequency interference ( rfi ) , or due to various forms of noise . the selection of promising candidates to be observed again for confirmation remains up to this day heavily dependent on human inspection , a very time - consuming process becoming increasingly unmanageable as surveys continue to evolve into ever larger scale operations over time . next generation instruments such as the square kilometre array ( ska ) , can be expected to find 20,000 pulsars @xcite , but not before an estimated 200 million candidates are properly classified , if we are to conservatively assume that the fraction of pulsars to be found among them ( one in ten thousand at most ) is comparable to current surveys @xcite . this implies that , among other challenges , the problem of automated candidate selection must be decisively solved . supervised machine learning ( ml ) classifiers offer great promise in this area , and were first introduced into the field by @xcite . they are general purpose methods that can be used to classify instances of multi - class data , by operating on a well - chosen set of their numerical properties called _ features_. they first build an internal model of the underlying statistical distributions of these features for each data class through the process of _ training_. in the context of supervised learning , this requires a labelled data set carefully prepared by a human expert , or _ training set_. once learned , that internal representation enables the classifier to subsequently label previously unseen data . supervised ml algorithms are well suited to classification problems where no reliable and simple rules are available to perform the task . in this work we use such a class of algorithms , namely artificial neural networks ( ann ) , and attempt to exceed the performance of previous automated candidate selection tools , aiming in particular to correctly label pulsars with a 100% success rate . to achieve that goal , we used a larger training data set that included 1196 pulsar observations from 542 distinct pulsars , wrote a custom ann implementation for increased control over its training process and designed new features to describe the nature of a pulsar candidate . this paper first outlines the problem of candidate selection by visual inspection , and offers a review of existing methods to either reduce the workload or automate the process . in section 3 , we present a detailed introduction to ann . section 4 details the features we use to capture candidate characteristics , and the rationale for their design . in section 5 , our ann implementation is evaluated on a set of candidates from the intermediate galactic latitude area of the htru survey ( htru - medlat ) . its efficiency when used on new data is shown in section 6 . we conclude with a discussion of the reasons of spinn s success , its current limitations , possible future improvements , and how future pulsar surveys should be run with ml classifiers in mind , if human intervention in classification is to be eventually reduced to a bare minimum . identifying new pulsar signals in observational radio data can be done either via single pulse searches @xcite , or periodicity searches which we briefly summarize here . the first computational step , the so - called de - dispersion or dm search , consists in correcting for the dispersive properties of the interstellar medium , which induce a delay in the observed arrival time of pulses that is both dependent on the observational frequency and the a priori unknown free - electron density integrated along the line of sight , a parameter called _ dispersion measure _ ( dm ) . discovering pulsars in binary systems may also require the application of methods to compensate for the effect of orbital motion of the radio source , as its change of velocity along the line of sight causes its apparent pulse period to vary over the course of an observation , as a result of the doppler effect . one of these methods is time domain resampling @xcite , also referred to as acceleration search , working under the assumption that orbital motion during an observation sufficiently shorter than the orbital period of the source is well approximated by a constant acceleration . a thorough processing of the radio data therefore involves a grid search in both dm and acceleration , and for each _ trial _ [ postulated ( dm , acceleration ) pair ] , the time series is transformed accordingly , and periodic signals are identified using a fast fourier transform ( fft ) . finally , the transformed time series can be folded " modulo the period of every significant periodic signal found , coherently stacking and summing the train of pulses of a potential pulsar . the folding process returns the final product of a pulsar survey : candidates with their set of diagnostic information , described shortly after . more detailed information about modern pulsar searching methods can be found in the standard references @xcite . fig.[fig : diagnosticplot ] shows the diagnostic information for a known pulsar that exhibits all the typical characteristics . the plots in the left - hand column describe , from top to bottom , the pulse in different bands of observed frequencies ( sub - bands plot ) , the evolution of the pulse during the observation ( sub - integrations plot ) , and the folded profile " which is the pulse averaged across all the observed frequencies for the entire observation . a pulsar is expected to emit in a broad range of wavelengths , with its signal remaining visible for most of the observation with a stable pulse shape . most pulsars also display a folded profile made of a single narrow peak , although wide and/or multi - peak profiles are not uncommon . the right - hand column of plots contains from top to bottom : the period - dm plane , which represents the evolution of the signal - to - noise ratio ( s / n ) of the signal as it is folded with slightly different values of period and dm . darker colors denote a brighter signal . below , the dm - s / n and acceleration - s / n curves summarize the results of earlier dm and acceleration trials , before the time series was folded , associating the s / n of the candidate in the fourier domain with dm and acceleration trial values . these plots are used to determine that the signal of a prospective pulsar is associated with well - defined and unique values of acceleration , period and dm . for the latter , an unambiguously non - zero value strongly indicates an extra - terrestrial origin for the source . visual inspection of every candidate produced by a modern survey is no longer a reasonable option . as an example , our latest processing of the htru intermediate latitude survey @xcite returned 4.3 million candidates , which , at a very optimistic rate of one candidate per second , would require approximately 1200 person hours to classify . that proposition can be made even less economically interesting in the case of re - processing of data analyzed one or more times before , in which the number of expected new discoveries is much lower . the repetitive nature of the work also leads to errors during long inspection sessions . as a consequence , techniques to reduce the required amount of human intervention have been used for more than a decade . graphical selection tools such as reaper @xcite and jreaper @xcite enable the user to project up to several thousand candidates at once in scatter plots , representing one of their features versus another , leading to rejection en masse of candidates not exhibiting the desired properties , for example excessively faint candidates , or ones found too close to narrow frequency bands polluted by rfi . scoring algorithms such as peace @xcite have also been developed , combining 6 numerical candidate quality factors into one formula that produces a subjective ranking where pulsars are expected to be found close to the top . machine learning ( ml ) solutions have also been proposed , first by @xcite , who used an artificial neural network to classify outputs from the pmps survey , operating on 8 to 12 numerical features extracted from candidate diagnostic information . @xcite applied the same technique on the htru survey , extending the number of features to 22 . more recently , @xcite combined a variety of ml algorithms that perform pattern recognition directly on candidate plots such as those shown in fig . [ fig : diagnosticplot ] , instead of first attempting to reduce them into features . in this work , we use an approach most similar to @xcite . supervised learning is the task of inferring a real - valued function from a set of labelled data points called _ training examples_. a training example consists of a pair @xmath0 where @xmath1 is the input or feature vector , and @xmath2 the _ target value _ or desired output ( typically @xmath3 ) chosen by human experts or gathered from experimental measurements . supervised learning can be used to solve regression problems , where the goal is to predict a continuous variable from a set of inputs , and classification problems , where one tries to assign a discrete class label to new unlabelled data points . in the context of binary classification , the two possible class labels are encoded in the target value , which may be set for example to 1 for members of the positive " class , and to 0 for members of the negative " class . a wide range of supervised learning algorithms is available , including artificial neural networks . one of the valuable features of anns that motivated us to choose them is that they naturally produce a real - valued continuous output , the _ activation value_. while it can be easily converted to a binary class label by applying a threshold , the activation value also represents a level of confidence in the class label obtained . in the context of pulsar candidate classification , this can be used as a way to prioritize inspection and confirmation of candidates as we will see later . in this section we only present an introduction to anns geared more specifically towards their use as binary classifiers . for a more advanced and general overview see e.g. @xcite or @xcite . inputs . the bias term @xmath4 can be seen as a weight operating on an extra constant input equal to one . the activation function @xmath5 usually chosen is the logistic sigmoid or similar.,scaledwidth=45.0% ] an artificial neuron is a computational model ( see fig . [ fig : artificialneuron ] ) inspired by its biological counterpart , which constitutes the basic building block of a network . it is parametrized by a vector of _ weights _ @xmath6 of pre - determined dimension , a scalar _ bias term _ @xmath4 , and an _ activation function _ @xmath5 . for a given feature vector @xmath7 , it outputs an _ activation value _ @xmath8 given by @xmath9 common choices for the activation function are sigmoid shaped non - linear functions such as the hyperbolic tangent or the logistic sigmoid function , the expression of the latter being @xmath10 which takes values between 0 and 1 ( see fig . [ fig : sigmoidfunction ] ) . a very useful geometric interpretation is to visualize a single artificial neuron as defining a separating hyperplane in feature space ( fig . [ fig : neuronhyperplanes ] , top panel ) , with a normal vector @xmath6 defining its orientation , and the bias term @xmath4 defining its altitude at the origin . note that the norm of the weights vector @xmath11 is a meaningful parameter on its own , despite not having any effect on the orientation of the hyperplane : it defines its sharpness " . unsurprisingly , an individual neuron performs poorly on non linearly separable data , but any number of them can be connected into layered networks capable of carving boundaries of arbitrarily high complexity in feature space ( fig . [ fig : neuronhyperplanes ] , bottom panel ) training is the process of finding an adequate set of weights for the given classification problem . this is posed as an optimization problem , where a _ cost function _ or _ loss function _ , which measures the discrepancy between target values and actual outputs of the network on the training set , must be minimized with respect to the weights and biases of the whole network . a common choice of cost function is the mean squared error @xmath12 where @xmath13 and @xmath14 are respectively the weights and bias term of the @xmath15-th neuron in layer number @xmath16 , @xmath17 and @xmath18 are respectively the target value and activation value for training example number @xmath19 , and @xmath20 the total number of training examples . the cost function is minimized using gradient descent , starting from a random initialization and going through iterations where the following three steps are performed in succession 1 . compute the activation values of the network on the data set , and the differences with the target values . 2 . compute the derivative of the cost function with respect to the network parameters ( weights and biases ) , using the _ backpropagation algorithm _ ( see below ) . 3 . correct every network parameter @xmath21 with the following update rule , where @xmath22 is the _ learning rate _ : @xmath23 the backpropagation algorithm ( see e.g. * ? ? ? * for a description and proof ) is a very computationally efficient way to compute the gradient of the cost function with respect to the weights and biases of the network , that historically made the training of large networks tractable . the training process only yields the best network weights and biases to properly label the training set , which does not necessarily imply optimal classification performance on unseen data . the model learned may capture not only legitimate patterns in the data , but also fit irregularities specific to the training set ( due for example to its limited size ) , a situation referred to as _ overfitting_. regularization consists in limiting the complexity of a model to improve its ability to generalize to new data . one such method that we used is _ l2 weight decay _ , where a penalty term is introduced into the neural network cost function usually written as @xmath24 with @xmath25 being the _ weight decay _ parameter . the net effect is to prevent the weights of the network from growing excessively large during training , therefore simplifying the decision boundary shape in feature space . the optimal value of @xmath25 , along with the optimal number of neurons to use , is found through grid search and cross - validation , described in section 5 . features are the properties of an unlabelled data instance upon which a ml algorithm decides to which class it is most likely to belong . the main part of the present work consisted in reducing pulsar candidates into maximally relevant features , i.e. that take values as different as possible for pulsars and non - pulsars , and ensure that these features capture a wide range of information and domain knowledge of a human classifier . to ensure maximum classification performance , particularly with respect to the identification of faint pulsars , we obeyed the following set of guidelines : 1 . reduce selection effects against faint or more exotic pulsars , especially msps or the ones with large duty cycles , which have been the most difficult to identify in the past @xcite . as an example , the number of dm trials or acceleration trials above a certain signal - to - noise threshold were found to introduce a strong and unjustified bias against short period candidates , regardless of their brightness . that feature was used in the past @xcite but not in our own work . 2 . ensure complete robustness to noisy data . as a result , no curve fitting to folded profiles or dm search graphs was attempted , as the results are very difficult to exploit properly in the low s / n regime , in which we are most interested . 3 . limit the number of features to a set of very relevant ones , as the very limited sample of pulsar observations is unlikely to sufficiently cover all the degrees of freedom of a large feature space . an excessive number of features induces a reduction in classification performance . this is a facet of the curse of dimensionality " problem in machine learning known as the _ hughes effect _ @xcite . this also implies avoiding the use of correlated features , as any extra feature must capture additional information . 1 . * signal - to - noise ratio of the folded profile ( log - scale ) . * s / n is a measure of signal significance , which can be defined in various ways . we use the definition ( see e.g. * ? ? ? * ) given by equation . for a given contiguous pulse window @xmath26 , @xmath27 where @xmath28 is the amplitude of the @xmath29-th bin of the folded profile , @xmath30 is the width of the pulse region @xmath26 measured in bins , @xmath31 and @xmath32 are respectively the mean value and the standard deviation of the folded profile in the off - pulse region . the position and width of the pulse are determined by an exhaustive search that maximizes s / n . once determined , the indices of the bins corresponding to pulse and baseline regions are retained in memory for further data processing . we also compute the equivalent width of the profile @xmath33 for further processing , defined by @xmath34 which , in other words , would be the width of a top - hat pulse window that would have the same area and peak height as the original pulse . + since the values of s / n can span a wide range across candidates , we use its logarithm as a feature . past a certain level , an increase of s / n does not make a candidate any more significant to a human expert on its own . all other things equal , two candidates with extremely significant signal - to - noise ratios of 50 and 500 can be considered equally likely to be legitimate pulsars . * intrinsic equivalent duty cycle of the pulse profile . * the duty cycle of a pulsar is the ratio of its pulse width @xmath30 expressed in seconds to its spin period . most folded profiles of pulsars show a narrow pulse with a duty cycle typically below 5% , while a significant amount of terrestrial signals reach much higher values up to 50% . this can often be due to the significant amount of phase drift exhibited by artificial sources during an observation , leading to apparent smearing of their folded profiles . that being said , some pulsars , especially among the millisecond population , can have legitimately large duty cycles , which can be further increased by dispersive smearing . to avoid penalizing such objects , and further increase the usefulness of the duty cycle feature for classification , we remove the effect of dispersive smearing by defining the _ intrinsic equivalent duty cycle _ of a candidate as @xmath35 where @xmath36 is the period of the candidate , @xmath33 its equivalent width defined in expressed in units of time , and @xmath37 the dispersive smearing time across a frequency channel . a first - order approximation of @xmath37 is given by @xmath38 where @xmath39 is the width of an observational frequency channel , @xmath40 the centre observation frequency , and dm the dispersion measure of the candidate . negative intrinsic equivalent duty cycle values are possible , if the dispersive smearing time exceeds the equivalent width . strongly negative values of @xmath41 are not expected for a genuine astronomical signal , and this constitutes an extra selection pattern that can be learned by a ml algorithm . * ratio between barycentric period and dispersion measure ( log - scale ) . * as far as the htru data is concerned , the most pulsar - like rfi candidates ( bright and persistent in time ) tend to appear at periods longer than 1 second , and at dispersion measures close to zero . a way to combine these two selection criteria into one is by considering the ratio between period and dm . as we did for s / n , since the values of this ratio span a large range across the pulsar and rfi population , we actually use the logarithm of the ratio between period and dm as a feature . as shown in fig.[fig : iewvspdm ] , the combination of @xmath42 and intrinsic equivalent duty cycle offers a powerful selection tool that is _ fully independent from s / n _ , splitting clearly the data into three distinct clusters : pulsars , noise candidates ( faint ) , and rfi ( bright ) . note that the usefulness of this feature is dependent on the rfi landscape at the place and even time of observation , and its portability to other surveys is unknown . * validity of optimized dispersion measure . * pulsars can have a wide variety of dispersion measures while their rfi counterparts usually exhibit dm values very close to zero , but no other truly selective pattern based solely on dm can be found . therefore we define the _ validity of dispersion measure _ as @xmath43 the purpose of this feature is to ensure that the classifier learns to very strongly reject candidates with a dm below a certain threshold , below which no pulsars are ever found . we used @xmath44 for htru - medlat data . persistence of signal through the time domain . * a genuine pulsar is expected to be consistently visible during most of an observation , and this provides a selection criterion against impulsive man - made signals . refining an interesting idea proposed by @xcite , we attribute a score " to every sub - integration of the candidate , based on its s / n ( see equation [ eq : snr ] ) measured with respect to the pulse window and baseline region defined by the folded profile . note that negative s / n values are possible if signal is found outside the expected window , a common property of rfi . the scoring function ( see fig . [ fig : scoringfunction ] ) is defined as @xmath45 where @xmath46 is the signal - to - noise ratio of the candidate in a sub - integration , measured as described above , and @xmath4 the benchmark signal - to - noise ratio which is a user - defined parameter . the average of the scores obtained through all sub - integrations constitutes the persistence of the candidate through the time domain . note that @xmath4 should be chosen low enough to filter out signals visible only for a small fraction of the observation , but not so much as to excessively penalize the class of nulling " pulsars @xcite that can become invisible for a part of the observation . we found a sensible , albeit arbitrary choice to be + @xmath47 + with @xmath48 being an estimation of the overall signal - to - noise ratio of the faintest pulsars still clearly visible to the trained eye , and @xmath49 the total number of sub - integrations . we set @xmath50 , also in accordance with the fact that no pulsar discovered with a s / n below 9.5 was ever confirmed over the course of the htru - medlat survey . * root - mean - square distance between the folded profile and the sub - integrations . * the persistence through time " feature is insufficient on its own to capture the information that rfi signals tend to show some amount of drift in phase or even shape changes during an observation , enough to easily betray their non - astronomical nature even to a moderately well - trained human eye . to alleviate this problem , we define a measure of the variability of the pulse shape though the observation . to compute it , we first normalize the folded profile to values between 0 and 1 , and also normalize individually every sub - integration in the same fashion . let @xmath28 be the value of the @xmath29-th bin of the folded pulse profile , and @xmath51 be the value of the @xmath29-th bin of the @xmath15-th sub - integration . let @xmath26 be once more the set of bin indexes that constitute the pulse window in the folded profile , and @xmath30 the pulse window width . we then simply define the root mean square distance between the folded profile of the candidate and each of its sub - integrations as : @xmath52 this feature helps characterize persistent rfi in the medium to high s / n regime . + as a final word , a persistence of signal through the frequency domain " similarly defined as its time domain counterpart was tried and initially believed to provide a very useful selection criterion to separate broadband pulsar signals from all the others . it was eventually removed from the feature set , as its addition proved slightly detrimental to classification performance on htru medlat data . we attributed this effect to the relative absence of candidates that arise from narrowband radio frequency interference , but it might be useful for other surveys facing different rfi populations . computing persistence features as defined earlier relies on comparing the brightness of the signal vs. a benchmark , which is constant in the time domain . for surveys with large fractional bandwiths , correcting for the average spectral index of pulsars when computing that benchmark in the frequency domain could be beneficial . training examples were gathered from the outputs of a new processing of htru medlat data , performed as a test run of the new high - performance gpu - based peasoup pipeline @xcite , with acceleration searching enabled . dms from 0 to 400 @xmath53 , and accelerations from @xmath54 to @xmath55 @xmath56 were searched , yielding in excess of a thousand candidates in each of the 95,725 beams . signal periods simultaneously found in three beams or more of the same pointing were ignored , and the resulting 50 brightest candidates in every beam were folded , with the exception that any signal with a fourier - domain s / n in excess of 9 was automatically folded as well . this processing strategy returned 4.34 million folded candidates . each of them was individually matched against the atnf pulsar catalogue @xcite to label all known pulsars ( and their harmonics ) that were found by the pipeline . a total of 1196 observations of 542 distinct known pulsars were identified after being carefully reviewed by eye to confirm their nature . they constituted the positive class of the ann training set , to which we added 90,000 non - pulsar observations picked at random as a negative class , to obtain a varied and representative sample of the population of spurious folded candidates . we assumed that none of these were undiscovered pulsars : in past htru medlat processings , approximately 100 new pulsars were found in 10 million folded candidates , a discovery rate of 1:100,000 . in our re - processing set of candidates , it is expected to be significantly lower . the training set contains pulsars with varied spin periods , duty cycles , and signal - to - noise ratios . among them , 77 have periods shorter than 50 ms and 46 have duty cycles larger than 20% . a total of 78 observations of pulsars have folded signal - to - noise ratios below 10 , down to a minimum of 7.3 . for the sake of comparison , no pulsar discovered with a folded s / n below 9.5 was ever confirmed over the entire course of the htru survey . low s / n pulsars are difficult to distinguish from noise fluctuations even by eye , and the limited observation time available imposes a conservative s / n selection threshold ( a consensual value is 10 ) on the folded candidates to confirm . despite these limitations , fainter pulsar observations were kept in the training set to ensure maximum sensitivity . to obtain fine control over the training process , a custom ann implementation was written . we followed some practical recommendations detailed at length in @xcite , which we enumerate here . 1 . feature scaling was performed before training , ensuring that every individual feature has zero mean and unit standard deviation over the entire training set . the hyperbolic tangent activation function was used instead of the logistic sigmoid , yielding activation values ranging from @xmath57 to @xmath58 . the ann was trained using mini - batches " , whereby during each training epoch the network is presented only with a small , changing subset of the training set . this is not only a much faster process than standard batch " training , but also its noisy nature can help to avoid local minima of the network cost function . to overcome the large class imbalance of the training set , we oversampled the pulsars to obtain a 4:1 ratio of non - pulsars to pulsars , so that pulsar candidates were seen " much more often by the ann during training , while preserving the variety of non - pulsars . obviously , the goal of an automated classifier is to identify the largest possible fraction of pulsars while returning a minimal amount of mislabelled noise or rfi ( false positives ) . the two natural performance metrics for this problem , measured on a test sample of labelled data , are therefore : @xmath59 @xmath60 the true positives are the positive examples correctly labelled as such , and false negatives are the positive examples incorrectly labelled as negatives . recall is therefore the fraction of positives properly labelled . likewise , the false positive rate is the fraction of negatives mislabelled as positives . other common metrics such as _ accuracy _ or _ f - score _ depend explictly on class imbalance ( the ratio of positive to negative examples in the test data ) , and as such are not suitable if we are to compare classifier performances across different test samples or even different pulsar surveys . since pulsars remain arguably rare objects up to this day , missing any of them carries a heavy cost , and emphasis must be put on maximising recall above all else . the amount of visual inspection required to select folded candidates for confirmation is proportional to the false positive rate of the classifier , for which low values are desirable . we performed a 5-fold cross - validation procedure , to choose an optimal network architecture and weight decay , and evaluate classification performance . this consists in randomly partitioning all of the labelled data into 5 equally sized subsets , each of them being successively held out as a test set , while only the remaining four are used to train the ann upon . this ensures that performance is always evaluated on data unseen during training , and that over the five iterations ( folds " ) of this procedure every candidate ends up in the test set exactly once , at which point its ann activation value or score " ( that takes continuous values between -1 and + 1 ) was recorded . we repeated the entire process 20 times , with different random data partitions , obtaining a representative average score for every candidate from our labelled sample that does not depend on a specific training / test set split . the resulting list of scores allowed to determine , for every score decision threshold between @xmath57 and @xmath58 , what were the associated recalls and false positive rates , ie . how many pulsars were below that decision threshold , and how many rfi or noise candidates were above . when later deploying the ann on new data , this gives an estimate of how far down the score ladder candidates should be inspected by eye , where the tradeoff between recall and false positive rate is left to the user s discretion . furthermore , examining consistently low scoring pulsars gave insight into which ones the ann was biased against , which we will discuss later . finally , by repeating this whole procedure with various network architectures and weight decay values , we were able to settle on an optimal ann configuration . it was chosen so as to minimize the number of non - pulsars scoring better than the worst - scoring pulsar , that is minimize the false positive rate at 100% recall . a simple two - layered 8:1 network ( 8 hidden units , one output unit ) was found to yield the best results , with performance progressively degrading with more units . [ fig : xvresults ] shows the distribution of scores obtained by all candidates in our labelled sample during cross - validation , and illustrates the recall / false positive rate tradeoff . table [ table : performancesummary ] summarizes expected classification performance for various score decision thresholds . the score distribution of pulsars shows a long tail where no more than a dozen low - scoring pulsars are responsible for the major part of the false positive rate . their close inspection reveals that they always share at least two of the following characteristics , making them similar to noise or rfi candidates with respect to the feature space we used : large duty cycles in excess of 20% , low s / n ( below 9 ) , high value of @xmath61 . a few other low - scoring pulsar observations were found to exhibit abnormally low persistence through time , being rendered invisible during a part of the observation by short bursts of rfi . .classification performance during cross - validation , summarizing some key values from fig . [ fig : xvresults ] . [ cols="<,>,>",options="header " , ] [ table : discoveries ] a fully trained ann was deployed on all 4.34 million candidates returned by the processing previously described in section 5.1 , a process that takes only 400 cpu - hours , despite being severely limited in speed by a large amount of small file i / o operations . using 64 cpus on swinburne university s gstar cluster , this can be done overnight . candidates were sorted by decreasing score , and known pulsars and their harmonics were removed from the list . in light of the cross - validation results ( see fig . [ fig : xvresults ] and table [ table : performancesummary ] ) , one can reasonably expect to find all potential discoveries above a score threshold of @xmath62 , which left approximately 27000 candidates to review . [ subsec : visualinspectionreport ] so far all 2400 candidates that scored above @xmath63 returned by spinn have been inspected , a process that will continue as observation time to confirm possible discoveries becomes available . table [ table : discoveries ] summarizes the attributes of the most promising candidates found among them . it shows that spinn is very sensitive to pulsar - like signals down to s / n = 8 , and that it can also highly rank broad pulses ( duty cycles in excess of 20% ) and potential millisecond pulsars , which is a known blind spot of some previous ml solutions @xcite . it should be noted that a considerable amount of the reviewed candidates were rfi signals with very specific periods . [ fig : logperioddistribution ] presents the distribution in @xmath64 of all previously unknown 7094 candidates that obtained a positive score . one percent of the bins account for 50% of these candidates . one could either postpone or skip the inspection of heavily polluted period intervals , or adjust _ a posteriori _ all ann scores via bayesian inference , using the period distribution of high scoring candidates and that of known pulsars @xcite . we intend to implement such a scheme in the future . the candidates of table [ table : discoveries ] with a s / n above 9.5 have been reobserved at the parkes observatory and four were confirmed as new pulsars , three of which have spin periods shorter than 5 ms . [ fig : discoveries ] shows their candidate plots exactly as they were evaluated by spinn . the details of these four new discoveries will be discussed in a future paper @xcite once their long - term coherent timing solutions have been obtained from currently ongoing observations . we have described spinn , an automated pulsar candidate classifier designed with maximum recall in mind . being essentially an artificial neural network that produces a real - valued and continuous output instead of a binary class label , it can also produce a subjective ranking of candidates that can be used to prioritize visual inspection . spinn was cross - validated on a data set containing all known pulsars found by the peasoup pipeline in a re - processing of htru - medlat and 90,000 non - pulsar candidates chosen at random . its expected recall and false positive rates were evaluated ( see table [ table : performancesummary ] ) and it was found to be capable of reducing the survey s outputs by a factor of approximately 150 while identifying all potential pulsars . reduction factors of several thousand can be achieved at the cost of postponing a small fraction of new discoveries , an interesting prospect for future surveys . spinn was deployed on all candidates produced during re - processing and four new pulsars were discovered in the 2,400 candidates it ranked most highly ( less than 0.06% of the survey s output ) . three of them are millisecond pulsars , one of which was found with a signal - to - noise ratio below 10 . while spinn seems to be a significant step forward compared to previous solutions , the performance of automated candidate classifiers depends very significantly on the properties of the data they are evaluated upon . the amount of bright spurious candidates to be sifted through will be affected by the rfi landscape at the observation site , and rfi mitigation techniques used during early stages of observational data processing . the available quantity and variety of known pulsar candidates to train ml algorithms upon also plays a role , and the absence of even a handful of pulsars difficult to detect even by visual inspection in test data can skew the results heavily . [ fig : xvresults ] illustrates this fact , as the removal of ten well - chosen " pulsars from our training data could have unjustifiably reduced the reported false positive rate by an order of magnitude , while obviously reducing spinn s sensitivity . comparisons between automated solutions are therefore limited at best , unless they are made on a common data set . to address this issue , we have made publicly available the set of candidates on which spinn was cross - validated ( see appendix [ sec : trainingdata ] ) . this will allow other authors of classifiers to evaluate their own solutions , and hopefully stimulate interest in the pulsar candidate classification problem , even from machine learning enthusiasts not necessarily acquainted with astronomy . a summary of reported performance of existing automated candidate classifiers is provided in table [ table : performancecomparison ] with all the previous caveats in mind . lllllr classifier & type & recall & false positive rate & test data origin & comments + @xcite & ann & 85% & 1% & htru@xmath65 medlat & + & & 93% & 1% & & 12:2 network + & & 92% & 0.5% & & 8:2 network + & & 100% & 3.7% & & + & & 95% & 0.34% & & + & & 68% & 0.17% & & + & & 92% & 1% & palfa@xmath66 & + & & 100% & 3.8% & gbncc@xmath67 & trained on palfa data + & & 100%@xmath68 & 1.1%@xmath68 & gbncc@xmath67 & trained on palfa data + & & 68% & 0.16% & gbncc@xmath67 & trained on palfa data + & & 100% & 0.64% & & + & & 99% & 0.11% & & + & & 95% & 0.01% & & + + + + + + [ table : performancecomparison ] ml algorithms operate on numerical features that carry no label or context , and unlike human classifiers , can not rely on any domain knowledge associated with these features . therefore , the statistical distributions of these features for different classes ( pulsars and non - pulsars ) should overlap as little as possible , so that these classes are more easily separated in feature space ( see fig . [ fig : neuronhyperplanes ] for a visual interpretation on a toy example ) . different features were tried and discarded in the context of this work and the best subset of them , reported in section [ sec : features ] , selected through cross validation with maximum recall in mind . also , while this may appear counter - intuitive , ml algorithms do not always perform better with more features . for a given amount of training data , larger feature spaces will be more sparsely sampled , and the data distributions more difficult to infer accurately . the decision boundaries separating these distributions must be of appropriate complexity , which , in the case of artificial neural networks , increases with the number of layers and neurons in the network . we attribute part of spinn s success to its relatively simple internal model of a pulsar candidate . the low number of features and neurons used are in accordance with the limited amount of pulsar observations available . this can be quantitatively supported by the fact that the pics ( without score adjustment ) and peace classifiers show almost identical performance on candidates taken from gbncc data , as shown in table [ table : performancecomparison ] . pics makes use of a committee of ml classifiers , including deep neural networks containing in excess of eight thousand neurons , while peace relies on a linear function of six numerical features . in this case , the large increase in model complexity did not translate into a significant improvement of classification performance . spinn s limitations are closely related to the features it relies upon . the @xmath69 feature has provided a very simple and efficient selection criterion against artificial signals on htru - medlat data partly because a large majority of rfi appears at periods in excess of one second ( fig . [ fig : logperioddistribution ] ) , a rule not guaranteed to hold true in other surveys . cross - validation also indicated that spinn carries a bias against pulsar signals showing a large duty cycle , and scores them even lower if they also exhibit a high @xmath69 value , as any pulsar signal with these two properties becomes difficult to distinguish from spurious ones in spinn s feature space ( fig . [ fig : iewvspdm ] ) . characterization of rfi is therefore largely incomplete , which was confirmed by the visual inspection of high scoring candidates ( see section [ subsec : visualinspectionreport ] and table [ table : discoveries ] ) . with the known pulsars and their harmonics ignored , the remaining shortlist was dominated by artificial signals that had pulsar - like properties with respect to spinn s feature space , yet their nature was often recognizable to the eye . some signals were found to drift erratically over time , or to emit only in a set of narrow frequency bands . while spinn has been successful in finding new pulsars , its false positive rate would have to be further reduced by about two orders of magnitude for it to be a match for the human eye in terms of accuracy , and while this is an ambitious goal , it leaves the problem of automated pulsar candidate classification largely open . we have previously suggested that spinn s success comes from its low - complexity model of a pulsar candidate , relying on a lower number of features and neurons than what has been previously used , a consequence of the limited number of known pulsars observable from any given site on earth . this would have two main implications for any present or future ml solutions . firstly , collecting as many pulsar observations as possible for training should be a top priority . observing known sources multiple times could be a viable option , purposely not pointing exactly at them to simulate a blind all - sky search , or purposely processing only a part of such an observation to reduce the s / n of the output candidate . artificially generating credible pulsar candidates is another possibility previously proposed several times @xcite . this would come with the challenges associated with realistically simulating the varying properties of pulsar signals and all forms of noise or interference that affect them . secondly , additional efforts should be undertaken to improve the quality of the folded data if we are to increasingly rely on ml for pulsar candidate selection . dealing with candidates folded with wrong periods , dispersion measures or accelerations vastly increases the number of degrees of freedom of the classification problem by perturbating the candidate plots in various ways . while such mistakes can be easily spotted by eye with domain knowledge and the wealth of information available in candidate plots , ml algorithms would require more features to achieve the same , and enough training data to properly sample the space of possible processing errors . instead , extra features that can be afforded would find better use in improving rfi characterization , which will be the main focus of our future efforts to further increase classification performance . the parkes observatory is part of the australia telescope which is funded by the commonwealth of australia for operation as a national facility managed by csiro . this work was supported by the australian research council centre for excellence for all - sky astrophysics ( caastro ) , through project number ce110001020 . the re - processing and scoring of the htru medlat survey made extensive use of the gpu supercomputer for theoretical astrophysics research ( gstar ) hosted at swinburne university , and funded by a grant obtained via astronomy australia limited ( aal ) . we thank the anonymous referee for their helpful remarks . the entire training data set that was used to cross - validate spinn in this work is available at http://astronomy.swin.edu.au/~vmorello/
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we describe spinn ( straightforward pulsar identification using neural networks ) , a high - performance machine learning solution developed to process increasingly large data outputs from pulsar surveys .
spinn has been cross - validated on candidates from the southern high time resolution universe ( htru ) survey and shown to identify every known pulsar found in the survey data while maintaining a false positive rate of 0.64% .
furthermore , it ranks 99% of pulsars among the top 0.11% of candidates , and 95% among the top 0.01% . in conjunction with the peasoup pipeline @xcite
, it has already discovered four new pulsars in a re - processing of the intermediate galactic latitude area of htru , three of which have spin periods shorter than 5 milliseconds .
spinn s ability to reduce the amount of candidates to visually inspect by up to four orders of magnitude makes it a very promising tool for future large - scale pulsar surveys . in an effort to provide a common testing ground for pulsar candidate selection tools and
stimulate interest in their development , we also make publicly available the set of candidates on which spinn was cross - validated .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
lagrangian formulations of general relativity abound in literature , but rather than differing in the choice of the lagrangian density they differentiate one another by the variable upon which the variation of the action has to be taken . the most famous examples are pheraps the metric and palatini variational approaches . the first one , sometimes called second order variation , considers the metric tensor as the only dynamical field variable while the second one , also known as first order variation , assumes the connection to be an independent field . though both approaches reproduce the einstein field equations in vacuum , the latter gives rise to another theory whenever the independent connection is allowed to appear inside the matter action . in this case the theoretical framework goes under the name of metric - affine variational principle and it is known not to recover general relativity in the presence of spinor fields , for example . the scope of the present paper is to summarize the basic features of a new variational approach recently proposed@xcite . together with the metric , two completely independent connections , with no _ a priori _ assumpions , are considered as dynamical variables entering the gravitational action for general relativity . the variation has thus to be taken independently with respect to the metric and both the connections . as we are going to show , applying this approach to a suitable generalization of the einstein - hilbert action leads to general relativity even when the matter action is allowed to depend on one of the connections . because of the presence of two connections and in analogy with bimetric thoeries of gravity , this approach will be called _ biconnection variational principle_. denote with @xmath0 and @xmath1 two independent connections , both containing torsion and non - metricity . in order to produce a new gravitational lagrangian within our framework , we consider a generalization of the riemann tensor obtained by symmetrizing the anticommutation of the covariant derivatives over the two connections . for any vector @xmath2 we have then @xmath3v^\alpha \ , , \label{001}\end{gathered}\ ] ] where @xmath4 , @xmath5 and @xmath6 , @xmath7 are the torsion tensors and covariant derivatives of @xmath8 and @xmath9 respectively . the biconnection riemann tensor @xmath10 has been defined as @xmath11 notice that whenever the two connections coincide this reduces to the usual ( metric - affine ) riemann tensor and ( [ 001 ] ) becomes the standard formula for the commutation of covariants derivatives , which in general includes a torsion term . the action we propose is thus @xmath12 where @xmath13 in analogy with the ricci tensor and the curvature scalar . when @xmath14 this reduces to the palatini ( or metric - affine ) action . to ( [ 002 ] ) we want to add a suitable matter action , where we have to make a choice upon the connection used to build covariant derivatives . in both metric and palatini approaches this connection is assumed to be the levi - civita connection , while in metric - affine theories this is taken to be the independent connection which prevents the dynamical equivalence with general relativity . since we will make the rather physical assumptions that all the matter fields couple to the same connection , in our case we have three choices : the levi - civita connection , @xmath8 or @xmath9 . the first choice would reduce the theory to palatini general relativity and thus will not be considered . because of the exchange symmetry @xmath15 of action ( [ 002 ] ) , it does not matter which one among this two connections enters the matter action . we choose @xmath8 . the total action of the theory is thus @xmath16 where @xmath17 denotes all matter fields collectively . variation with respect to @xmath9 eventually yields the condition has to be considered here because of a projective symmetry of ( [ 002 ] ) . see ref . for full details . ] @xmath18 which means that @xmath8 reduces to be the levi - civita connection . the role of @xmath9 is thus similar to the one played by a lagrange multiplier field . the variation with respect to @xmath8 leads , after one takes into account condition ( [ 003 ] ) , to @xmath19}_\sigma\right ) \ , , \label{004}\ ] ] where @xmath20 and @xmath21 is the variation of the matter action with respect to @xmath8 , the so - called _ hypermomentum_. finally , variation of action ( [ 005 ] ) with respect to the metric tensor produces , again after having taken into account ( [ 003 ] ) , the field equations @xmath22 where @xmath23 is the standard covariant derivative of the levi - civita connection , @xmath24 is the einstein tensor and @xmath25 is the variation of the matter action with respect to the metric . note that this differs from the usual energy - momentum tensor @xmath26 of general relativity inasmuch as it lacks of the variation of the connection with respect to the metric . using now equations ( [ 004 ] ) and some algebra , we can rewrite field equations ( [ 006 ] ) as @xmath27 the divergence in the right hand side of ( [ 007 ] ) represents exactly the missing variation of the connection with respect to the metric in the tensor @xmath28@xcite . we thus have that ( [ 007 ] ) simply reduces to @xmath29 which are nothing but the _ einstein field equations_. to conclude , the biconnection variational approach for general relativity we have presented here , is physically equivalent to the more standard metric formulation of the theory . it allows the connection employed to minimally couple matter fields to be completely independent of the metric . in this respect , the biconnection approach is similar to the metric - affine one but , in contrast with the latter one , it always permits to reduce the equations of motion to the einstein field equations and thus do not alter the phenomenology of gravitation . applications of the biconnection formulation of general relativity can be relevant in quantum gravity theories , such as loop quantum gravity , which assume a metric - affine classical framework .
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a recently proposed variational approach for general relativity where , in addition to the metric tensor , two independent affine connections enter the action as dynamical variables , is revised .
field equations always reduce to the einstein field equations for any dependence of the matter action upon an independent connection .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
astronomical images obtained form the ground suffer from serious degradation of resolution , because the light passes through a turbulent medium ( the atmosphere ) @xcite before reaching the detector . a number of methods was developed to alleviate this phenomenon @xcite and multiple special - case solutions were implemented as well ( e.g. @xcite ) , but none of them is able to provide a perfect correction and restore the diffraction - limited ( dl ) image . therefore , the best observatories , in terms of angular resolution , are the spaceborne ones , since they are limited only by the dl . recently , there is an increasing interest in the attempts to overcome the dl boundary . devices which can give such possibility are called the quantum telescopes ( qt ) . first qts will probably work in the uv , optical and ir bands , mainly because of the speed , maturity and reliability of the detectors . latest progress in adaptive optics ( ao ) , especially so called extreme ao , makes the use of qts realistic also in ground - based observatories @xcite . in this letter the general idea of quantum telescopes is considered . in particular we refer to the setup proposed by @xcite , since , to our knowledge , it is the only existing detailed description of a qt . in fig . [ fig : qtscheme ] we propose an upgraded version of this setup . according to @xcite , each photon coming from the extended source triggers a signal by qnd detection and gets cloned @xcite . the coincidence detector controlled by the trigger is turned on for a short period @xcite and registers the clones . after that it is quickly turned off , so that it receives only a small fraction of spontaneous emission from the cloning medium and virtually no clones from other photons . if the source is too bright and emits too many photons per unit of time , a gray filter should be installed . as a result , a set of clones is produced and registered . the centroid position of the clones cloud is used to add 1 adu . ] at the corresponding position of the high resolution output image . the exposure time has to be much longer than in classic telescopes ( ct ) , since ( a ) in most cases a narrowband filter has to be applied and ( b ) the qnd detection efficiency is much below 100% @xcite . below we present the results of our detailed simulations of the qt system , discuss the feasibility of building such a device and predict its expected performance . to our knowledge , this is the first paper describing the detailed simulations of a qt of any design , as this is a preliminary concept . in our simulations as input images we used parts of real images obtained by the hubble space telescope ( hst ) . such images are optimal for the purpose of qt testing because this observatory is working at its dl . we cropped and decimated them to 200@xmath0200 pixels to speed up the computations . for the comparison , we also simulated the process of digital image formation in the case of a ct ( for a review on high angular resolution imaging see @xcite ) using the same images ( fig . [ fig : obscuredpupil]e ) . we assumed a real telescope for which the pupil is obscured by a secondary mirror and its truss ( spider ) @xcite . we assumed a similar size of a secondary mirror and truss as it is installed on the hst ( fig . [ fig : obscuredpupil]a ) . in the case of a ct , we sampled the counts ( photons ) from the original image ( fig . [ fig : obscuredpupil]d ) and distributed them according to the airy diffraction pattern decimated to 61x61 pixels ( fig . [ fig : obscuredpupil]b ) . for the simulation of a qt , each photon from the reference image was converted to @xmath1 cloned photons ( fig . [ fig : obscuredpupil]c ) . for @xmath1 we assumed the poissonian distribution . the clones were arranged using the gaussian profile ( see @xcite fig . 2 therein for justification ; we assumed @xmath2 = 10 ) centered around the photon s position . to include the effects of spontaneous emission , on the top of it we added counts distributed equally within the coincidence detector plane and governed by the poissonian distribution . the mean noise level was tuned to achieve a given signal - to - noise ratio ( snr ; see exemplary simulated exposure in fig . [ fig : drawing]b ) . the assessment of the snr was based on the comparison of the number of cloned photons with the number of counts originating from the spontaneous emission within the circular aperture of 3@xmath2 radius ( i.e. within the aperture , for which nearly all the clones are received ) . it follows the approach of snr derivation presented in @xcite . in the next stage of simulations of the qt image formation , we computed the centroid of clones employing the matched filtering approach . as the cloned photons exhibit gaussian spatial distribution @xcite , in our calculations the image registered on the coincidence detector was first convolved with the gaussian ( @xmath2 = 10 ) and then the centroid was obtained from the position of the maximum value of such a filtered image ( fig . [ fig : drawing]c ) . @xmath3 = @xmath4(@xmath5,@xmath6 ) ) generate @xmath7 generate @xmath1 clones in the @xmath7 around ( @xmath5,@xmath6 ) compute @xmath8 paste @xmath8 to @xmath9(@xmath5,@xmath6 ) we ran the simulations for different numbers of clones , reaching also very high numbers ( up to the expected value of ~10k , see @xcite for justification ) . the mean level of the poissonian noise of spontaneous emission was set so that snr was : 3/1 , 2/1 , 1 , 1/2 , 1/3 , 1/4 , 1/5 , 1/6 , 1/7 , 1/8 , 1/9 , 1/10 , 1/11 , 1/12 , 1/13 and 1/14 . such a selection of snr includesa value of 1/7.3 which was assessed for qt in @xcite . the quality of the simulated qt outcomes was assessed by two indicators : peak signal - to - noise ratio ( psnr ) and mean centroid error ( rms value ) . for 16-bit pixel representation the psnr measure is defined as follows : @xmath10 @xmath11 where @xmath12 and @xmath13 denotes the intensity of pixel at ( @xmath14 ) in respectively qt simulated image and reference high - fidelity image . the psnr and the mean centroid errors are depicted in fig . [ fig : surfs ] . the place where both dependencies meet each other was retrieved and presented in fig . [ fig : betterworse ] . it shows the minimal requirements for the clones number and the snr level , which should be satisfied to achieve the resolution enhancement in qt . this curve can be treated as a first guidance for selection of qt parameters . as seen from figs . [ fig : surfs ] and [ fig : betterworse ] , assuming snr @xmath15 7.3 , in average @xmath1657 clones per 1 detected photon are necessary to produce an image sharper than ct . for @xmath1710k clones the image is virtually ideally restored , even for the lowest values of snrs . the noticeable improvement of the outcome with higher clones count is related to the generally better estimation of the centroid when using the mf approach . for very small number of clones , the photons distribution is dominated by isolated photon - detection events and the mf output is strongly dependent on actual arrangement of registered clones . in contrast , for higher numbers of clones , the gaussian becomes more uniform and therefore , the mf provides much more reliable estimations even if the snr remains the same . in fig . [ fig : outputimages ] we show exemplary resulting qt images for various snrs and clones counts . the improvement in the image quality is easily noticeable as both parameters increase . below we summarize and discuss the feasibility of the qts as emerging from the most recent literature . a constant progress in the qnd increases the qt feasibility . there remains a possibility that some other kind of discriminator of spontaneous emission will be found . possibly qnd does not require to hold the photon in a resonant cavity for several hundreds microseconds , as it was claimed before @xcite . according to @xcite , in the case of the cavity - free qnd the photons momentum change is also negligible . however , this setup is suspected to produce false detections ( < 5% ) and can not be miniaturized . authors are presently working on a version easier to miniaturize ( keyu xia priv . com . ) . we see no strong limitations for the size of the device if it could be placed e.g. at the ` cassegrain ` or ` coud ` plane optically conjugated to the pupil plane of the optical system near the focus ( fig . [ fig : focus ] ) . another technological issue concerning the qt is the spontaneous emission from the process of cloning ( for details see @xcite ) . as we showed , in our model , for a sufficiently large number of clones this problem could be overcome . however , in the general case of the qt some more sophisticated methods might be needed ( e.g. different geometrical setup or filters ) . given the recent progress in cloning , it might also become possible to clone photons from a wider wavelength span and at a greater amount @xcite . in conclusion , the general idea of the quantum telescopes seems feasible and in the near future it might be possible to construct a technology demonstrator . in this paper we presented the updated schematic `` toy model '' of a qt system and the first quantitative results of simulations of such an imager , aiming at predicting the conditions which would guarantee its optimal performance . we found that it is generally more important to provide more clones than to reduce the noise background . given the predicted snr = 7.3 , a satisfactory results are obtained from less than 60 clones . using 10k clones , the signal is almost perfectly restored for any noise level . we encourage the interest and discussion on the idea of the qt , since even a small increase in the resolution might lead to a major breakthrough in astronomical imaging . * funding . * marian smoluchowski research consortium matter energy future from know . polish national science center grant number umo-2012/07/b / st9/04425 . polish national science center , grant no . umo-2013/11/n / st6/03051 . poig.02.03.01 - 24 - 099/13 grant : geconii - upper silesian center for computational science and engineering . v. m. tikhomirov , `` dissipation of energy in isotropic turbulence '' , _ selected works of a. n. kolmogorov _ ( springer science + business media , 1993 ) http://dx.doi.org/10.1007/978-94-011-3030-1_47 [ online version ] beckers , j. m. , `` adaptive optics for astronomy principles , performance , and applications '' , annual review of astronomy and astrophysics * 31 , * 13 - 62 ( 1993 ) http://adsabs.harvard.edu/abs/1993ara%26a..31...13b [ online version ] tonry , j. ; burke , barry e. ; schechter , paul l , `` the orthogonal transfer ccd '' , publications of the astronomical society of the pacific * 109 , * 1154 - 1164 ( 1997 ) http://adsabs.harvard.edu/abs/1997pasp..109.1154t [ online version ] roggemann , michael c. ; welsh , byron m. ; fugate , robert q. , `` improving the resolution of ground - based telescopes '' , reviews of modern physics . * * 69,**issue 2 , 437 - 505 ( 1997 ) http://adsabs.harvard.edu/abs/1997rvmp...69..437r [ online version ] hecquet , j. ; coupinot , g. , `` a gain in resolution by the superposition of selected recentered short exposures '' , journal of optics * 16 , * 21 - 26 ( 1985 ) http://adsabs.harvard.edu/abs/1985jopt...16...21h [ online version ] buie , m. w. and grundy , w. m. and young , e. f. and young , l. a. and stern , s. a. , `` pluto and charon with the hubble space telescope . ii . resolving changes on pluto s surface and a map for charon '' , the astronomical journal , * 139 , * , 1128 - 1143 ( 2010 ) http://adsabs.harvard.edu//abs/2010aj....139.1128b [ online version ] v. korkiakoski , c. vrinaud , `` extreme adaptive optics simulations for the european elt '' , frontiers in optics , optics & photonics technical digest ( 2009 ) https://www.osapublishing.org/abstract.cfm?uri=aopt-2009-aotud1 [ online version ] kellerer , a. , `` beating the diffraction limit in astronomy via quantum cloning '' , astronomy & astrophysics , * 561 , * i d . ( 2014 ) http://adsabs.harvard.edu/abs/2014a%26a...561a.118k [ online version ] xia , keyu ; johnsson , mattias ; knight , peter l. ; twamley , jason , `` cavity - free scheme for nondestructive detection of a single optical photon '' , phys . 116 , 023601 ( 2016 ) http://journals.aps.org/prl/abstract/10.1103/physrevlett.116.023601 [ online version ] mutchler , m. ; beckwith , s. v. w. ; bond , h. ; christian , c. ; frattare , l. ; hamilton , f. ; hamilton , m. ; levay , z. ; noll , k. ; royle , t. , `` hubble space telescope multi - color acs mosaic of m51 , the whirlpool galaxy '' , bulletin of the american astronomical society * 37 , * p.452 ( 2005 ) http://adsabs.harvard.edu/abs/2005aas...206.1307m [ online version ] guerlin , c. ; bernu , j. ; delglise , s. ; sayrin , c. ; gleyzes , s. ; kuhr , s. ; brune , m. ; raimond , j.- . ; haroche , s. , `` progressive field - state collapse and quantum non - demolition photon counting '' , nature * 448 , * 889 - 893 ( 2007 ) http://adsabs.harvard.edu/abs/2007natur.448..889g [ online version ] lamas - linares , a. ; simon , c. ; howell , j. c. ; bouwmeester , d. , `` experimental quantum cloning of single photons '' , science * 296 , * 712 - 714 ( 2002 ) http://adsabs.harvard.edu/cgi-bin/bib_query?arxiv:quant-ph/0205149 [ online version ] barbieri , m. ; ferreyrol , f. ; blandino , r. ; tualle - brouri , r. ; grangier , ph . , `` nondeterministic noiseless amplification of optical signals : a review of recent experiments '' , laser physics letters * 8 , * issue 6 , article i d . 417 ( 2011 ) http://adsabs.harvard.edu/abs/2011laphl...8..417b [ online version ] ralph , t. c. and lund , a. p. , `` nondeterministic noiseless linear amplification of quantum systems '' , american institute of physics conference series * 1110 , * p. 155 - 160 ( 2009 ) http://adsabs.harvard.edu/abs/2009aipc.1110..155r [ online version ] s. e. derenzo , woon - seng choong and w. w. moses , `` fundamental limits of scintillation detector timing precision '' , physics in medicine and biology * 59 , * 3261 - 3286 ( 2014 ) http://dx.doi.org/10.1088/0031-9155/59/13/3261 [ online version ] p. grangier , j. a. levenson and jean - philippe poizat , `` quantum non - demolition measurements in optics '' , nature * 396 , * 537 - 542 ( 1998 ) http://www.nature.com/nature/journal/v396/n6711/abs/396537a0.html [ online version ] harvey , j. e. and ftaclas , c. , `` diffraction effects of telescope secondary mirror spiders on various image - quality criteria '' , applied optics . * 34 , * 6337 - 6349 ( 1995 ) http://adsabs.harvard.edu/abs/1995apopt..34.6337h [ online version ] kellerer , a. , `` beating the diffraction limit in astronomy via quantum cloning ( corrigendum ) '' , astronomy & astrophysics , * 582 , * i d . c3 ( 2015 ) http://www.aanda.org/articles/aa/abs/2015/10/aa22665e-13/aa22665e-13.html [ online version ] c. w. helstrom , d. w. fry , l. costrell and k. kandiah , `` statistical theory of signal detection '' , international series of monographs in electronics and instrumentation , vol . 9 , ( pergamon press , oxford ; new york ) , 2nd edition ( 1968 ) http://www.sciencedirect.com/science/book/9780080132655 [ online version ] woodward , p.m. , `` probability and information theory with applications to radar '' , ( norwood , ma : artech house ) , 2nd edition ( 1980 ) http://www.sciencedirect.com/science/book/9780080110066 [ online version ] de martini , f. ; sciarrino , f. , `` investigation on the quantum - to - classical transition by optical parametric amplification : generation and detection of multiphoton quantum superposition '' , optics communications * 337 , * 4452 ( 2015 ) http://www.sciencedirect.com/science/article/pii/s0030401814007640 [ online version ] nogues , g. ; rauschenbeutel , a. ; osnaghi , s. ; brune , m. ; raimond , j. m. ; haroche , s , `` seeing a single photon without destroying it '' , nature * 400 , * 239 - 242 ( 1999 ) http://adsabs.harvard.edu/abs/1999natur.400..239n [ online version ]
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quantum telescope is a recent idea aimed at beating the diffraction limit of spaceborne telescopes and possibly also other distant target imaging systems .
there is no agreement yet on the best setup of such devices , but some configurations have been already proposed .
+ in this letter we characterize the predicted performance of quantum telescopes and their possible limitations .
our extensive simulations confirm that the presented model of such instruments is feasible and the device can provide considerable gains in the angular resolution of imaging in the uv , optical and infrared bands . we argue that it is generally possible to construct and manufacture such instruments using the latest or soon to be available technology .
we refer to the latest literature to discuss the feasibility of the proposed qt system design .
copyright : the optical society ( osa publishing ) 2016 . published in optics letters vol .
41 no . 6 ( 2016 ) +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in march of 2010 , the _ fermi_-lat collaboration announced a new gev transient in the galactic plane , fgl j2102 + 4542 , that was identified as a nova outburst in the symbiotic binary v407 cygni ( hereafter , v407 cyg ) . at least seven gev transients located near the galactic plane have been discovered by egret , _ fermi_-lat and agile . only two have been identified at other wavelengths : v407 cyg , which is the first nova to be detected at gev energies , and j0109 + 6134 , which was likely a background blazar @xcite . the physical nature of the other five sources is unknown @xcite , and some of these gev transients may represent a new class of gamma - ray emitting objects . the _ fermi_-lat collaboration reported variable gamma - ray emission in the 0.110 gev band from fgl j2102 + 4542 during 2010 march 1026 ( mjd 5526555281 ) @xcite . its flux in gamma rays , binned on a day - to - day basis , peaked 2010 march 1314 with a flux of @xmath2 above 100 mev @xcite . the gev gamma - ray activity lasted approximately two weeks . the initial report of gev emission triggered very energetic radiation imaging telescope array system ( veritas ) observations of the object at very high energy ( vhe ; e @xmath0 100gev ) as part of an ongoing campaign to observe transients detected by _ fermi_-lat . using multi - wavelength data , it was determined that the new transient was most likely associated with v407 cyg , a binary system consisting of a mira - type pulsating red giant and a white dwarf companion @xcite . a nova outburst from v407 cyg was detected in the optical waveband on 2010 march 10 @xcite with a magnitude of @xmath3 , while pre - outburst magnitudes from the previous two years of monitoring ranged between magnitude 9 and 12 @xcite . v407 cyg has been optically monitored for decades and has experienced previous outbursts , but the system had never been observed to be as bright as during the nova ( e.g. , * ? ? ? * ; * ? ? ? the onset of the optical outburst corresponds to the first significant detection of the source by the _ fermi_-lat on 2010 march 10 . novae in red giant / white dwarf systems have been known to produce expanding shocks that can result in x - ray emission ( e.g. , the recurrent nova rs oph , * ? ? ? * ; * ? ? ? * ) , and indeed , x - ray emission from v407 cyg was detected after the nova @xcite . based on the observed x - ray emission from the 2006 nova outburst of rs oph , before the launch of _ fermi_-lat , @xcite suggested that particles could be accelerated in novae up to tev energies , but gamma - ray emission from a nova had never previously been detected . here , we discuss the veritas observations of v407 cyg and their implications for gamma - ray emission from the nova . we also describe an improved event reconstruction technique for stereo observations by imaging atmospheric cherenkov telescopes ( iacts ) made at large zenith angles ( lza ) . veritas is a ground - based vhe gamma - ray observatory located at the fred lawrence whipple observatory in southern arizona . it consists of four iacts sensitive from approximately 100 gev to above 30 tev . each veritas telescope has a 12 m tessellated reflector with a total area of 110m@xmath4 . each camera s focal plane contains 499 closed - packed circular photomultiplier tubes , giving a total field - of - view of @xmath5 . gamma - rays incident onto the upper atmosphere induce a particle cascade , called an air shower , in which some charged particles have sufficient speed to emit cherenkov light . the direction and energy of the original gamma ray can be reconstructed from images of the cherenkov light recorded by the telescopes . when observing at small zenith angles ( @xmath6 ) , the array has an energy resolution of @xmath7 at 1 tev and an angular resolution of better than @xmath8 at 1 tev @xcite . for observations at lza , the energy and angular resolution are degraded and the energy threshold is increased . veritas observed v407 cyg for several nights after the announcement of the _ fermi_-lat detection , during days 916 of the outburst ( 2010 march 1926 ) . the zenith angle of these observations ranged between @xmath9 and @xmath10 . the veritas telescopes are regularly operated in a mode called wobble mode , during which the location of the object to be observed is offset from the center of the field of view ( fov ) by 0.5@xmath11 , allowing for simultaneous source and background measurements @xcite . the offset direction cycles between north , south , east and west for sequential observing segments to reduce systematic effects . after filtering the data for contamination due to poor weather or instrumental problems , 304 minutes of live time remained from the original 335 minutes of observations , see table 1 . to test the improved reconstruction technique discussed in section [ sec : two reconstruction methods ] , veritas observations of the crab nebula were also analyzed . we selected 203 minutes of good time intervals from 17 data segments taken on the crab nebula during 2010 march 1216 ( mjd 5526755271 ) with similar zenith angles ranging from @xmath12 to @xmath13 . all data were analyzed using the standard analysis package for veritas data @xcite . the raw data were calibrated and cleaned , and quality selection criteria based on the number of photomultiplier tubes contained in the images and the position of the image in the camera were applied . the shape and orientation of the gamma - ray images were parametrized by their principal moments @xcite . in order to produce gamma - ray images of the sky , it is necessary to reconstruct the putative source location for each shower in the camera plane ( hereafter arrival direction " ) . when imaging showers with multiple iacts , the arrival direction of a shower is usually found using simple geometric arguments . the major axes of the images produced by a shower in each iact camera intersect near the location of the arrival direction . the shower arrival direction is calculated by minimizing the perpendicular distance to each image s semi - major axis , weighted by the size of each image . this method , here called the standard method , is effective at small zenith angles . however , at lza , the major axes of the air shower images from an individual gamma - ray event are generally close to parallel . thus , the uncertainty of the intersecting point increases , resulting in a loss of angular resolution . due to this effect , a reconstruction technique that does not depend on the intersection of the axes is desirable for lza observations . the displacement method is a direction reconstruction algorithm that is useful for lza observations @xcite . in very general terms , it consists of calculating the arrival direction using the shape and brightness of a given air shower image . more specifically , it relies on the determination of the _ disp _ parameter , defined as the angular distance from the image centroid to the arrival direction . this method was used by several experiments in the past @xcite , with varying ways of calculating _ disp_. the basis of the displacement method is the relationship of the _ disp _ parameter to other image parameters @xcite . the implementation of the algorithm in veritas is as follows : we estimate _ disp _ as a function of three other image parameters , _ size _ , _ length _ and _ width _ @xcite , using monte carlo simulated gamma - ray showers . the method results in two different arrival directions , one on each side of a telescope image along the semi - major axis , also known as head - tail ambiguity @xcite . this ambiguity is eliminated by choosing the cluster of arrival directions closest to one another , one coming from each image . finally , the arrival direction is estimated independently for each telescope image and an average weighted by _ size _ is taken . this method proves to be more powerful than the standard method @xcite when reconstructing events with zenith angle larger than @xmath9 . quantitatively , an improvement of @xmath1430% in detection significance for a source having 1% of the strength of the crab nebula has been observed . the cosmic ray background rate for iacts is typically more than @xmath15 times the gamma ray rate , so it is important to reduce this background while retaining as many gamma - ray events as possible . by exploiting the differences in the development of gamma ray and cosmic ray induced showers , the background due to cosmic rays can be reduced significantly , while still retaining a high fraction of gamma - ray like events . the background reduction is performed by placing standard selection criteria , optimized using monte carlo simulations and real data from the crab nebula on the shower image parameters . the selection criteria for the size of the telescope images , the mean scaled width and mean scaled length parameters @xcite , the height of maximum cherenkov emission and the angular distance from the anticipated source location to the reconstructed arrival direction of each shower ( @xmath16 ) are given in table [ cuts ] . to perform a background subtraction of the surviving cosmic ray events , an estimation of these background counts is made using the reflected - region background model @xcite . events within an angular distance @xmath16 of the anticipated source location are considered on events . background measurements ( off events ) are taken from regions of the same size and at the same angular distance from the center of the fov . for this analysis , a minimum of eight background regions was used . the excess number of events from the anticipated source location is found by subtracting the number of off events ( scaled by the relative exposure , @xmath17 ) from the on events . statistical significances are calculated using a modified version of eqn . 17 of @xcite to allow for varying number of off - source regions due to nearby star 60 cygni @xcite . more details about veritas , the calibration procedure and the analysis techniques can be found in @xcite . analysis of the veritas data did not show a significant detection at the location of v407 cyg . the results from both event reconstruction methods were used to calculate upper limits on the flux from v407 cyg with the method described by @xcite and the assumption of a gaussian - distributed background . the upper limits for v407 cyg are calculated at the decorrelation energies of 1.8 tev for the standard method and 1.6 tev for the displacement method and assume that any emission takes the form of a power law with a photon index of -2.5 . the decorrelation energy is the energy at which the dependence of the upper limit calculation on the assumed photon index is minimized . this energy is found by performing multiple upper limit calculations , with different spectral indices , and determining the region where the resulting upper limit functions intersect . the energy threshold for the observations of v407 cygni with veritas , defined as the maximum of the product of the assumed spectral shape and the effective area , is 1.2 tev for both methods . the analysis results for v407 cyg are presented in table [ results ] and figure [ fig : mapfermi ] . in addition , results from observations of the crab nebula at similar zenith angles are presented in table [ results ] and figure [ fig : mapcrab ] . the efficiency of the displacement method for event reconstruction can be observed in the increase in both gamma - ray rate and significance for the crab measurements . the increased sensitivity also results in a reduction of the upper limit for v407 cyg compared to the standard method . the gev detection of v407 cyg provides evidence for previously unobserved gamma - ray emission from novae in white dwarf / red giant systems . expanding shock waves have been known to accelerate particles to high energies , and gamma rays are observed from supernova remnants . the discovery by the _ fermi_-lat team , however , suggests that the same phenomenon occurs in some novae , adding a new class of gamma - ray emitting objects . the lack of a significant detection in the vhe band suggests that either particles were not accelerated to sufficient energies to produce vhe photons during the v407 cyg outburst or that vhe photons were produced , but then absorbed . the key to creating gamma rays is the acceleration of sufficiently energetic charged particles . in the case of v407 cyg , the expanding matter from the nova collides with the stellar wind from the red giant and causes a shock , which accelerates the particles near the shock to relativistic energies . a rough estimation for the maximum energy attainable by first order fermi acceleration of a particle at a shock can be found , following the discussion of @xcite . if @xmath18 is the magnetic flux density where a shock proceeds and the shock travels with velocity @xmath19 , the maximum energy of a particle with charge @xmath20 is @xmath21 , where @xmath22 is the time allowed for particle acceleration . this means that the highest attainable energy is proportional to the magnetic field in which the nova travels , the square of shock speed and the time for acceleration . the mean magnetic field in the shock can be estimated as @xmath23^\frac{1}{2}$ ] , where @xmath24 is the density of gas molecules with respect to the distance from the center of the red giant and t is the wind temperature @xcite . we assume a wind temperature of t=700k , corresponding to the temperature of the dust envelope measured by @xcite . @xcite were able to directly measure the temperature of the red giant wind in a similar symbiotic system , eg and , and found that it can reach @xmath148000 k near the red giant . using this wind temperature would increase the estimate for maximum particle energy by a factor of three . @xcite carried out detailed hydrodynamic simulations of the v407 cyg nova with various gas distribution models and could accurately reproduce the x - ray light curve of v407 cygni . their model for the distribution of gas that best reproduced the light curve included what they call a circumbinary density enhancement , " a region of density exceeding the typical @xmath25 profile of the stellar wind in the binary system , and had a binary separation of 15.5 au @xcite . for the temporal profile of the nova shock velocity , we used the equation that @xcite found from fits to the broad components of the h@xmath17 spectra they measured beginning at day 2.3 after the outburst ( 13 march 2010 ) and thereafter : @xmath26 . for the velocity between day 0 and day 2 of the outburst , we can assume two cases that bound the possible velocity profiles : i ) the nova shell experienced free expansion at a constant velocity before day 2 , with the assumption that the mass collected by the nova shell during this period was small ( free expansion model ) or ii ) extrapolate the above equation for the velocity to times before day 2 ( extrapolation model ) . we then find that at the start of veritas observations of v407 cyg ( day 9 of the outburst ) , @xmath27 tev for the free expansion model and @xmath28 tev for the extrapolation model . this suggests that particles could have been accelerated to tev energies by the time of the veritas observations . to check the importance of absorption , we calculated the opacity along the photon path for gamma rays generated at the shock front . electron - positron pair production via photon - photon collision is the dominant interaction @xcite . we modeled the red giant spectrum as a blackbody with a temperature of 2500 k and a radius of 500 @xmath29 , found the photon density as a function of position and energy following @xcite , and used the cross section for the photon - photon collision from @xcite . we found that the opacity for tev photons only becomes significant when the tev emission region is located directly behind the red giant with the system viewed edge on . though this case can not be ruled out , it is statistically unlikely . in addition , if the suggested orbital parameters of @xcite are accurate , the system is unlikely to have been in such an orientation at the time of the nova . the upper limits placed by veritas can put some restrictions on the gamma - ray emission mechanism in v407 cyg . two physical models of gamma - ray production at the shock - front have been suggested @xcite . in the hadronic model , gamma rays are produced in the decay of @xmath30 particles generated by collisions of high energy protons accelerated in the shock . in the leptonic model , gamma rays are produced via inverse - compton scattering of infrared photons emitted from the red giant on high energy electrons accelerated in the shock . the electron threshold energy for production of a gamma - ray photon via inverse - compton scattering off the red giant photons can be estimated as : @xmath31 , where @xmath32 is the electron threshold energy , @xmath33 is the gamma - ray energy and @xmath34 is the energy of the red giant photons . the electron threshold energy for a 1 tev gamma ray scattering off 0.6 ev photons at the peak of the red giant spectrum is 1.1 tev . though the above calculation indicates that particles could reach tev energies if continuously accelerated for the full nine days from the initial outburst to the start of the veritas observations , the inverse - compton cooling time would be significantly less than a day ( the time estimated by @xcite for 5 gev electrons ) , meaning electrons that are accelerated in the first few days of the outburst would not likely retain sufficient energy to produce vhe photons by the time of the veritas observations . vhe emission near the time of the veritas observations would therefore require freshly accelerated particles , however , recently accelerated particles would likely not have enough time to reach tev energies . if electrons did reach tev energies , they would be approaching the klein - nishina regime , where the cross section for inverse - compton scattering would be reduced , resulting in a longer cooling time . however , electrons with tev energies would be well above the exponential cutoff , @xmath35 gev , of the electron spectrum in the best - fit leptonic model of @xcite . these two factors imply that the veritas upper limits place no new constraints on leptonic models . for hadronic models , the _ fermi_-lat data alone provide relatively poor constraints on the extension of the proton spectrum to high energies . in particular , the cutoff energy @xmath36 is not well bounded from above if the spectral index is steep . the veritas data can be used to improve the constraints on the hadronic model parameters . to do so , we re - fit the hadronic model used by @xcite to the _ fermi_-lat points with the addition of the veritas upper limit . figure [ fig : spectrum ] shows the _ fermi_-lat data @xcite and the veritas flux upper limit compared to the best fitting hadronic model . the gamma - ray spectrum is calculated via the method of @xcite , assuming a cosmic proton spectrum of the following form : @xmath37 ( protons gev@xmath38 ) , where ( @xmath39 are kinetic energy , cut - off energy , and mass of the proton and @xmath40 is the power law index ) . figure [ fig : revisedmap ] shows a confidence region map for the parameters of the hadronic model using both the _ fermi_-lat and veritas data . the gamma - ray spectrum was modeled as described above and fitted to the _ fermi_-lat data by varying @xmath40 and @xmath41 . the spectrum was then compared to the veritas upper limit , and a @xmath42 value for the veritas data point alone was calculated . specifically , we calculated the model flux in the veritas energy band and compared this to the flux upper limit determined via the displacement method . this @xmath42 value was then added to the @xmath42 calculated for the _ fermi_-lat data . the confidence levels were then calculated for the two parameters of interest , @xmath40 and @xmath41 . as can be seen from figure [ fig : revisedmap ] , the veritas observations place greater restrictions on the model proton spectral index for high cutoff energies . the 90% confidence limits are @xmath43 5 tev ( comparable to @xmath44 tev , calculated above for the extrapolation model ) and @xmath45 0.01 tev ( much lower than @xmath46 ) . it is possible that the peak energy of the particles produced by the shock could be reduced if the magnetic field is weaker than estimated above . @xcite argue for a larger binary separation of 2025 au , based on the presence of lithium burning in the mira , and evidence that the white dwarf in the system is massive . using this larger separation distance would lead to weaker magnetic fields . the limits placed by the veritas observations are near the threshold for the observations , so it is also possible that simply not enough particles were accelerated to high enough energies to produce a significant detection by veritas . _ acknowledgments . _ we would like to thank pierre jean of the _ fermi_-lat team for providing the contour data from the _ fermi_-lat results and useful discussion . this research is supported by grants from the u.s . department of energy office of science , the u.s . national science foundation and the smithsonian institution , by nserc in canada , by science foundation ireland ( sfi 10/rfp / ast2748 ) and by stfc in the u.k . we acknowledge the excellent work of the technical support staff at the fred lawrence whipple observatory and at the collaborating institutions in the construction and operation of the instrument . abdo , a. a. , et al . , 2010 , science , 329 , 817 acciari , v. a. , et al . 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we report on very high energy ( e @xmath0 100 gev ) gamma - ray observations of v407 cygni , a symbiotic binary that underwent a nova outburst producing 0.110 gev gamma rays during 2010 march 1026 .
observations were made with the very energetic radiation imaging telescope array system during 2010 march 1926 at relatively large zenith angles , due to the position of v407 cyg .
an improved reconstruction technique for large zenith angle observations is presented and used to analyze the data .
we do not detect v407 cygni and place a differential upper limit on the flux at 1.6 tev of @xmath1 ( at the 95% confidence level ) .
when considered jointly with data from _ fermi_-lat , this result places limits on the acceleration of very high energy particles in the nova .
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a programme has been started , using the recently commissioned nagoya - south african 1.4 m infrared survey facility ( irsf ) at saao sutherland , to study the stellar populations , evolution and structures of local group galaxies . one aim of this programme is to detect long period variables ( miras and other types ) in these systems and to derive their infrared light curves . the programme will necessarily take several years to complete . in the present communication we discuss the light that initial observations of the dwarf spheroidal galaxy , leo i , throw on the agb star population of that galaxy . the irsf is a 1.4-m telescope constructed and operated in terms of an agreement between saao and the graduate school of science and school of science , nagoya university , to carry out specialized surveys of the southern sky in the infrared . the telescope is equipped with a 3-channel camera , sirius , constructed jointly by nagoya university and the national astronomical observatory of japan ( nagashima et al . 1999 ) , that allows _ j , h _ and _ k@xmath5 _ images to be obtained simultaneously . the field of view is 7.8 arcmin square with a scale of 0.45 arcsec / pixel . images centred on leo i ( referred to hereafter as field a ) were obtained at two epochs , 2001 - 01 - 16 and 2001 - 12 - 19 , and processed by means of the standard irsf pipeline ( nakajima , private communication ) . a single image comprises 10 dithered 30-s exposures . three such sets of frames were combined to give an effective 900-s exposure in each of _ j , h _ and @xmath6 at both epochs . at this stage , the effective field of view is reduced to 7.2 arcmin square . standard stars from persson et al . ( 1998 ) were observed on each night and the results presented here are in the natural system of the sirius camera , but with the zero point of the persson et al . standards . at the first epoch , we obtained a supplementary set of images of an adjacent field ( field b ) centred 7 arcmin to the east of field a. the two fields overlap by only about 20 arcsec . photometry was carried out on the images with the aid of dophot ( schechter , mateo & saha 1993 ) used in fixed - position mode . since the seeing was much better at the first epoch ( 1.6 arcsec as opposed to 2.6 arcsec at the second epoch ) , the @xmath7 image obtained then was used as a template to measure a complete sample of stars to a limiting magnitude of about @xmath8 . the data are plotted in figs . 1 ( @xmath1 vs @xmath9 ) and 2 ( @xmath10 vs @xmath11 ) . in the past , @xmath12 , derived from burstein and heiles ( 1984 ) has generally been adopted for this galaxy ( e.g. lee et al . the results of schlegel et al . ( 1998 ) suggest that a larger value ( @xmath13 ) is appropriate . in neither case will this lead to significant reddening at jhk and we have neglected it . the stars lying to the blue of the main concentration of stars in fig . 1(a ) are shown as crosses there and are similarly marked in fig . they are likely to be foreground field stars . this view is strengthened by the results for the adjacent field b where the stars in the almost vertical sequence are almost certainly field dwarfs . two points ( filled squares ) at @xmath141.5 in fig . 1(a ) and one in fig . 1(b ) are likely , from their colours , to be due to background galaxies . indeed , close inspection of our images shows evidence for extended emission associated with two of them , one of which is clearly a galaxy on publicly available hst images . apart from the field stars discussed above and the four very red objects discussed in the next section , all the stars in field a lie on a sequence in fig . objects identified as carbon stars by azzopardi , lequeux & westerlund ( 1986 = alw ) or by demers & battinelli ( 2002 = db ) are indicated by star symbols . photometry was obtained for 21 known or suspected carbon stars in leo i , which account for all the stars in the alw and db lists except for the following : db 4 and 8 which are seen on the edges of our frames but were not measured ; db 13 and alw 4 and 6 which are outside our fields . + using the bolometric corrections for carbon stars as a function of @xmath15 given by frogel , persson and cohen ( 1980 ) and a distance modulus of 22.2 for leo i based on the rgb tip ( lee et al . 1993 ) one finds that the carbon star sequence runs from @xmath16 at @xmath17 to @xmath18 at @xmath19 . however , as can be seen from work on galactic carbon stars ( whitelock 2000 ) , the stars at the redder end of this sequence may well be mira variables and can not be taken as defining the upper limit of the sequence without more observations . all the stars of this sequence are agb stars . the rgb tip is expected to be fainter than @xmath20 for any reasonable metallicities or ages ( see for instance castellani et al . 1992 , salaris & cassisi 1998 ) . the present results show clearly how the blue - green grism results of alw miss the brighter carbon stars and would therefore lead to an underestimate of the brightness of the agb tip . a similar underestimate of the agb tip is present in _ vi _ work ( e.g. lee et al . 1993 , fig 4d ) . all but one of the brightest , reddest objects constituting the top of the agb sequence appear in the db list , and it is interesting to note that the obscured objects discussed below would , when dereddened , extend this sequence to even brighter k@xmath5 magnitudes . + at the lower ( bluer ) end of the agb sequence in fig . 1(a ) ( which is of course determined by our adopted magnitude cut off ) there is a group of objects without spectral classification . they lie mainly to the blue of the known carbon stars in fig 2(a ) . it would be interesting to know whether these are o or c rich objects . a few of them may be foreground stars . + fig 1(a ) contains an object , without spectral classification , near the top of the agb sequence with @xmath21 . in view of its position in figs 1(a ) and 2(a ) it seems likely that it is also a carbon star . the fact that it was not found in the survey of db may mean that it is a variable and was below the magnitude limit of db at the time of their observations . the star s colour and luminosity are similar to those expected for carbon miras . four very red objects are conspicuous in figs 1(a ) and 2(a ) ( field a ) while there is an even redder one , though it is rather fainter in k@xmath5 , in field b ( figs 1(b ) and 2(b ) ) . the positions and photometry of these five objects at jd 2451962.52 are listed in table 1 . their locations in the colour - magnitude and two - colour diagrams are consistent with their being bright agb stars obscured by circumstellar dust ( see e.g. whitelock 2000 ) . in view of the fact that the top magnitude or more of the agb sequence just discussed is heavily , and perhaps entirely , populated by carbon stars , it seems very likely that these obscured stars are also carbon stars . as noted by nikolaev & weinberg ( 2000 ) several other types of object have colours that would put them in this part of the two - colour diagram , for example , oh / ir stars and protostars . for the reasons outlined above , it seems more likely they are carbon- rather than oxygen stars , though the discussion on ages below is not significantly affected if the latter is the case . the presence of protostars of this brightness in a dwarf spheroidal galaxy would be quite remarkable . these five stars are the most extreme examples yet found of this type of obscured object in dwarf spheroidal galaxies . the nearest comparable objects , though they are somewhat bluer(@xmath22 ) , are the two carbon miras that have been found in the sagittarius dwarf spheroidal ( whitelock et al . 1999 ) . .photometry and positions ( equinox 2000 ) of 5 very red objects in the field of leo i [ cols="^ , < , < , < , < , < , < " , ] obscured stars are denoted by their letters in table 1 , other stars by their db numbers ( c10 etc . ) or their alw designation . a full discussion of the nature of the obscured stars must obviously be deferred till further variability studies have been made . neverthless it is of interest to draw some preliminary conclusions . + the results we have obtained are strikingly similar to those of nishida et al . ( 2000 ) ( see also tanab et al . 1997 , 1999 ) who found an obscured carbon mira in each of the intermediate age clusters , ngc 419 , ngc 1783 and ngc 1978 , in the magellanic clouds . these variables have values of @xmath15 ( saao system ) between 3.75 and 4.76 . thus , in colour , the obscured leo i stars lie between the sagittarius dwarf spheroidal miras , mentioned above , and the lmc cluster miras . in the magellanic cloud clusters the tip of the unobscured agb is at @xmath23 and @xmath24 ( using data from frogel et al . 1990 and distance moduli of 18.6 and 19.0 for the lmc and smc ) . in leo i the corresponding values are 8.3 and 5.1 which however remain uncertain pending full variability studies . + in the magellanic cloud clusters the miras have a mean @xmath25 of 8.0 and @xmath18 or slightly brighter . the corresponding figures for the obscured stars in leo i are quite uncertain both because three , and possibly all , are variable and also because the estimation of bolometric corrections for such stars from @xmath26 colours is rather uncertain . the three obscured stars found to be variable have @xmath27 and @xmath28 whilst all five obscured stars yield @xmath29 and @xmath30 . to derive these results for leo i we have used a relation of @xmath31 to bolometric correction for carbon miras derived by whitelock ( to be published ) which includes the use of iso data . thus the tip of the unobscured agb in @xmath25 and @xmath32 is very similar in leo i to that in the three magellanic cloud clusters . the obscured agb may be fainter in @xmath32 than the cluster miras but that is not certain . + it is of some interest to note that if the intrinsic ( underlying ) colours of the five obscured stars were @xmath33 , they would all move to a position near the top of the agb in @xmath1 , when dereddened using the reddening law ( @xmath34 ) found for the circumstellar envelope of the galactic carbon mira r for ( feast et al . 1984 ) . + the magellanic cloud clusters are estimated to have ages in the range 1.6 to 2.0 gyr , metallicities , [ fe / h ] @xmath35 , and turn - off masses of @xmath36 . in view of the above discussion it seems likely that the leo i obscured stars are in the same age and mass range . it is therefore interesting to note that gallart et al ( 1999 ) suggest that major star formation in leo i stopped about 2 gyr ago and that a metallicity of that population as high as [ fe / h ] @xmath35 is possible . the results thus suggest that the obscured stars , and the most luminous unobscured agb stars , at least , belong to this youngest major stellar component of leo i. + the very red star ( star e ) in field b is especially interesting . it is @xmath37 arcmin from the centre of leo i , much further out than any of the other stars we discuss . even the c stars outside field a , mentioned in section 2.2 , are within @xmath38 arcmin of the centre . there is hi in a wide area around leo i and apparently associated with it ( blitz & robishaw 2000 ) . star counts ( irwin & hatzidimitriou 1995 ) show that leo i has a tidal radius of 13 arcmin , which extends well beyond the distance of star e from the centre . however the stellar density at these distances is very low compared with that in the main body of the galaxy . furthermore , in dwarf spheroidals the younger populations are generally more concentrated to the centre than the older ones ( e.g. harbeck et al . thus if we have correctly interpreted this star as similar to the obscured agb stars in field a , and hence relatively young , its position at such a large distance from the centre is remarkable . this is particularly so when it is recalled that such objects are short lived and are therefore tracers of much larger populations . since the star has only been observed at one epoch we can not comment on its possible variability . we have obtained jhk@xmath5 photometry of a complete sample of stars to k@xmath5=16 in a 7.2 arcmin square field centred on leo i and in an adjacent field . this sample includes all 21 known carbon stars falling in the imaged area . our results show that the top one magnitude or more of the agb in @xmath1 is populated entirely or almost entirely by carbon stars . these stars form a sequence in the @xmath39 diagram and several of them are variable . in addition there are five very red stars , at least three of them variable , which from their magnitudes and colours are deduced to be agb tip stars obscured by dust shells . they are strong candidates for mira variability . these stars , and at least the brightest unobscured agb stars , probably belong to the youngest and most metal rich of the significant stellar populations in leo i. comparison with carbon stars in magellanic cloud clusters suggests ages of about 2 gyr for these stars in agreement with the age of the youngest major population in this galaxy as derived in other ways . surprisingly , in view of the fact that younger populations are generally more centrally concentrated than others in dwarf spheroidals , one of the obscured stars lies about 8 arcmin from the centre of the galaxy , compared with a tidal radius of 13 arcmin . we are grateful to noriyuki matsunaga for help at the telescope . we also thank yasushi nakajima for providing the reduction pipeline and for information on the photometric system . the irsf project was initiated and supported by nagoya university and the national astronomical observatory of japan , and we thank professor sato , professor nagata and all others involved in the project . azzopardi m. , lequeux j. , westerlund b.e . , 1986 , a&a , 161 , 232 (= alw ) blitz l. , robishaw t. , 2000 , apj , 541 , 675 . burstein d. , heiles , c. , 1984 , apj sup . 54 , 33 castellani v. , chieffi a. , straniero o. , 1992 , apj sup , 78 , 517 demers s. , battinelli p. , 2002 , aj , 123 , 238 (= db ) feast m.w . , whitelock p.a . , catchpole r.m . , overbeek m.d . , 1984 , mnras , 211 , 331 frogel j.a . , mould j. , blanco v.m . , 1990 , apj , 352 , 96 frogel j.a . , persson s.e . , cohen j.g . , 1980 , apj , 239 , 495 gallart c. et al . 1999 , apj , 514 , 665 guarnieri m.d . , renzini a. , ortolani s. , 1997 , apj , 477 , l21 harbeck d. , grebel e.k . , holtzman j. , guhathakurta p. , brandner w. , geisler d. , sarajedini a. , dolphin a. , hurley - keller d. , mateo m. , 2001 , aj , 172 , 3092 held e.v . , clementini g. , rizzi l. , momany y. , saviane i. , di fabrizio l. , 2001 , apj , 562 , l39 hernandez x. , gilmore g. , valls - gabaud d. , 2000 , mnras , 317 , 831 irwin m. , hatzidimitriou d. , 1995 , mnras , 277 , 1354 lee g.l . , freedman w. , mateo m. , thompson i. , roth m. , ruiz m - t . , 1993 , aj , 106 , 1420 nagashima c. et al . , 1999 , in `` star formation 1999 '' , ed . nakamoto t.,(nobeyama radio observatory ) . nikolaev s. , weinberg , m.d . , 2000 , apj , 542 , 804 nishida s. , tanab t. , nakada y. , matsumoto s. , sekiguchi k. , glass i.s . , 2000 , mnras , 313 , 136 persson s.e . , murphy d.c . , krzeminski w. , roth m. , rieke m.j . , 1998 , aj , 116 , 2475 . salaris m. , cassisi s. , 1998 , mnras , 298 , 166 schecter p.l . , mateo m. , saha a. , 1993 , pasp , 105 , 1342 schlegel d. , finkbeiner d.p . , davis m. , 1998 , apj , 500 , 525 tanab t. , nishida s. , matsumoto s. , onaka t. , nakada y. , soyano t. , ono k. , sekiguchi t. , glass i.s . , 1997 , nature , 385 , 509 tanab t. , nishida s. , nakada y. , onaka t. , glass i.s . , sauvage m. , 1999 in `` asymptotic giant branch stars '' , ( iau symp . 191 ) , ed . le bertre t. , asp , p.573 whitelock p. , 2000 , in `` the carbon star phenomenon '' , ( iau symp . 177 ) , ed . wing r.f . , kluwer , dordrecht , p.179 whitelock p. , menzies j. , irwin m. , feast m. , 1999 , in `` the stellar content of local group galaxies '' , ( iau symp . 192 ) , eds . whitelock p. , cannon r. , asp , p.136
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the first results of a study of the dwarf spheroidal galaxy , leo i , using the new nagoya - south african infrared survey facility ( irsf ) are presented .
@xmath0 observations show that most , if not all , of at least the top magnitude of the agb in @xmath1 is populated by carbon stars .
in addition there are five very red objects which are believed to be dust enshrouded agb stars .
one of these is , remarkably , well outside the main body of the galaxy .
three of these obscured stars and five known carbon stars show variability in observations 11 months apart .
one of the obscured stars has @xmath2 making it highly likely that it , at least , is a mira variable .
the tip of the agb is at @xmath3 , but further variability studies are necessary to obtain a definitive value .
comparison with carbon stars , both miras and non - miras , in magellanic cloud clusters and taking into account other evidence on the ages and metallicities of leo i populations suggests that these obscured stars belong to the youngest significant population of leo i and have ages of @xmath4 gyr .
galaxies : dwarf - galaxies : stellar content - stars : agb and post - agb - stars : variable : other - local group
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the goal of this paper is to derive an effective string theory of vortices beginning with a field theory containing classical vortex solutions . the abelian higgs model is an example of such a theory . nielsen and olesen @xcite showed that this model has classical magnetic vortex solutions . these vortices are tubes of magnetic flux with constant energy per unit length . the motivation for this work came from the dual superconductor picture of confinement @xcite . in this picture , a dual meissner effect confines electric color flux ( @xmath0 flux ) to narrow tubes connecting quark antiquark pairs . calculations with explicit models of this type @xcite have been compared both with experimental data and with monte carlo simulations of qcd @xcite . to a good approximation , aside from a color factor , the dual abelian higgs model , coupling dual potentials to a scalar higgs field carrying magnetic charge , can be used to describe the results of these calculations . however , these calculations neglect the effect of fluctuations in the shape of the flux tube on the @xmath1 interactions . we show in this paper that taking account of those fluctuations leads to an effective string theory of long distance qcd . well before the introduction of the idea of dual superconductivity , string models @xcite had been used to understand the origin of regge trajectories , and they have continued to be used to describe other features of hadron physics , such as the spectrum of hybrid mesons . in the dual superconductor picture , a string arises because the dual potentials couple to a quark antiquark pair via a dirac string whose ends are a source and sink of electric color flux . the effect of the string is to create a flux tube ( or abrikosov nielsen olesen vortex @xcite ) connecting the quark antiquark pair . as the pair moves , this flux tube sweeps out a space time surface on which the dual higgs field must vanish . this condition determines the location of the qcd string in the dual superconductor picture . the effort to obtain an effective string theory for abrikosov nielsen olesen vortices has a long history , independent of any connection to qcd . nambu @xcite attached quarks to the ends of superconducting vortices , and found an expression for the classical action of the resulting ano vortex in the singular london limit of infinite higgs mass . he introduced a cutoff to render this action finite , and showed that it was proportional to the area of the worldsheet ( the nambu goto action ) . frster @xcite took into consideration the curvature of the worldsheet . he showed that in the strong coupling limit , with the ratio of vector and scalar masses held fixed , the effects of curvature were unimportant , and the classical action for the vortex reduced to the nambu goto action . this limit can be regarded as the long distance limit , since only zero mass excitations are left in the theory . equivalently , since the flux tube radius vanishes in this limit , all physical distances , measured in units of the flux tube radius , are becoming large . all degrees of freedom except the transverse oscillations of the vortex are frozen out . gervais and sakita @xcite first considered the quantum theory of the vortices of the abelian higgs model in the same long distance limit . they used the results of frster to define collective coordinates for the vortices , by means of which they constructed an effective vortex action . they also obtained a formal expression for the feynman path integral of the abelian higgs model as an integration over vortex sheets . however , they were not able to write this expression as an integral over the physical degrees of freedom of the vortices . lscher , symanzik , and weisz @xcite considered the leading semiclassical corrections to the classical nambu goto action due to transverse string fluctuations , and showed how to regulate the resulting divergences . they showed that for a string of length @xmath2 with fixed ends , the leading semiclassical contribution to the heavy quark potential is @xmath3 . in a second paper , lscher @xcite showed that this result was unaffected by the addition of other terms to the effective string action . polchinski and strominger @xcite discussed the relation of the abelian higgs model to fundamental string theory , regarding the theory of ano vortices as an effective string theory . they explained how existing string quantization methods were inappropriate for quantizing the vortices . to compensate for the anomalies @xcite in these quantization methods , they introduced an additional term , the `` polchinski strominger term , '' into the effective vortex action . akhmedov , chernodub , polikarpov , and zubkov @xcite studied the quantum theory of ano vortices in the london limit , in particular , they studied the transformation from field degrees of freedom to vortex degrees of freedom . they showed that the jacobian of this transformation contained the `` polchinski strominger term '' as a factor . although they , like gervais and sakita , did not obtain a complete expression for the path integral , this paper provided an important stimulus to our own work . in the current paper , we simplify and extend work done in an earlier paper @xcite . we begin with the path integral representation of a field theory having vortex solutions . it is an effective field theory describing phenomena at distances greater than the flux tube radius . we end up with an effective string theory of vortices in a form suitable for explicit calculations . we apply this theory to calculate the energy @xmath4 and angular momentum @xmath5 of the fluctuations of a string bounded by the curve generated by the worldlines of a quark antiquark pair separated by a fixed distance and rotating with fixed angular velocity . this gives the contribution of string fluctuations to the regge trajectory @xmath6 , which we compare with the experimental @xmath7 and @xmath8 trajectories . in section [ partition section ] , we rewrite the path integral over field configurations of the abelian higgs model containing vortices as an integral over surfaces on which the higgs field vanishes . this introduces a jacobian due to the change from field variables to string variables ( surfaces ) . this jacobian is the key to determining the action of the effective string theory , and to defining the integral over all surfaces . we next use the formalism described in section [ partition section ] to obtain an effective theory of ano vortices . in section [ jacobian factor ] , we show how the jacobian divides into a field part and a string part . the two parts of the jacobian play different roles in the effective theory . in section [ effective action section ] , we define an expression for the action of the effective string theory . all the dependence on the abelian higgs model is contained in the string action . we also obtain an expression for the path integral over vortices . in section [ coordinate fix ] , we show how to express the integral over surfaces as an integral over the two physical degrees of freedom of the vortex , and obtain the final form of the effective string theory . in the remaining sections we compute the leading semiclassical contribution to regge trajectories due to the fluctuations of the string . we obtain an expression for the contribution of string fluctuations to the effective action in section [ small f section ] , and in sections [ regularization section ] and [ w_2 section ] describe how to regularize this expression , making use of the results of lscher , symanzik , and weisz @xcite . in section [ regge effective potential section ] we calculate the contribution of string fluctuations to the effective action for a straight , rotating string , and in section [ regge plot section ] obtain the resulting corrections to regge trajectories . in this section we consider the abelian higgs model coupled via a dirac string to a moving quark antiquark pair . we transform the path integral over field configurations containing vortices to an integral over the surfaces @xmath9 determining the location of the vortices . we denote the ( dual ) potentials by @xmath10 and the complex ( monopole ) higgs field by @xmath11 . the dual coupling constant is @xmath12 , where @xmath13 is the yang mills coupling constant ( @xmath14 ) . the worldlines of the quark and antiquark trajectories form the closed loop @xmath15 ( see fig . [ loop figure ] ) . = 1.8 in = 1.8 in the moving quark antiquark pair couples to the dual potentials @xmath10 via a dirac string tensor @xmath16 , which is nonvanishing along some line @xmath17 connecting the @xmath1 pair . as the pair moves , the line @xmath17 sweeps out a worldsheet @xmath18 parameterized by coordinates @xmath19 , @xmath20 . the field @xmath11 vanishes on this worldsheet , @xmath21 the corresponding dirac string tensor @xmath16 is given by @xmath22 the action @xmath23 of a field configuration which has a vortex on the sheet @xmath18 is @xmath24 \ , , \label{firstaction}\ ] ] where the field strength @xmath25 is given by @xmath26 the higgs mechanism gives the vector particle ( dual gluon ) a mass @xmath27 and the scalar particle a mass @xmath28 , where @xmath29 is the vacuum expectation value of the higgs field . we have introduced the color factor @xmath30 in ( [ g mu nu def ] ) because we are interested in using @xmath23 as a model for long distance qcd . we consider @xmath23 to be an effective action describing distances greater than the flux tube radius @xmath31 . the long distance @xmath1 interaction is determined by the wilson loop @xmath32 $ ] , @xmath33 = \int \scrd\phi^ * \scrd\phi \scrd c^{\mu } e^{i(s[\phi , c ] + s_{gf } ) } \ , , \label{originalpartition}\ ] ] where @xmath34 is a gauge fixing term . the functional integrals are cut off at the momentum scale @xmath35 . the action ( [ firstaction ] ) describes a field theory having classical vortex solutions . the functional integral ( [ originalpartition ] ) goes over all field configurations containing a vortex bounded by @xmath15 . previous calculations @xcite of @xmath32 $ ] were carried out in the classical approximation ( corresponding to a flat vortex sheet @xmath9 ) , and showed that the landau ginzburg parameter @xmath36 is approximately equal to @xmath37 . this corresponds to a superconductor on the border between type i and type ii . in this situation , both particles have the same mass @xmath38 , the string tension is @xmath39 , and the flux tube radius is @xmath40 . to take into account the fluctuations of these vortices , we must evaluate @xmath32 $ ] beyond the classical approximation . we carry out the functional integration ( [ originalpartition ] ) in two steps : ( 1 ) we fix the location of a vortex sheet @xmath9 , and integrate only over field configurations for which @xmath41 vanishes on @xmath9 . ( 2 ) we integrate over all possible vortex sheets . to implement this procedure , we introduce into the functional integral ( [ originalpartition ] ) the factor one , written in the form @xmath42 \int \scrd \tilde x^\mu \delta\left[\re\phi(\tilde x^{\mu}(\xi))\right ] \delta\left[\im\phi(\tilde x^{\mu}(\xi))\right ] \ , . \label{inserttildex}\ ] ] the integration @xmath43 is over the four functions @xmath18 . the functions @xmath18 are a particular parameterization of the worldsheet @xmath9 . the expression ( [ inserttildex ] ) implies that the string worldsheet @xmath9 , determined by the @xmath44 functions , is the surface of the zeros of the field @xmath11 . the factor @xmath45 $ ] is a jacobian , and is defined by eq . ( [ inserttildex ] ) . inserting ( [ inserttildex ] ) into ( [ originalpartition ] ) puts the wilson loop in the form @xmath33 = \int \scrd\phi^ * \scrd\phi \scrd c^{\mu } e^{i(s[\phi , c ] + s_{gf } ) } j[\phi ] \int \scrd \tilde x^\mu \delta\left[\re\phi(\tilde x^{\mu}(\xi))\right ] \delta\left[\im\phi(\tilde x^{\mu}(\xi))\right ] \ , . \label{beforeswitch}\ ] ] we then reverse the order of the field integration and the string integration over surfaces @xmath18 , @xmath33 = \int \scrd \tilde x^\mu \int \scrd\phi^ * \scrd\phi \scrd c^{\mu } j[\phi ] \delta\left[\re\phi(\tilde x^{\mu}(\xi))\right ] \delta\left[\im\phi(\tilde x^{\mu}(\xi))\right ] e^{i(s[\phi , c ] + s_{gf } ) } \ , . \label{afterswitch}\ ] ] in eq . ( [ beforeswitch ] ) , the @xmath44 functions fix @xmath9 to lie on the surface of the zeros of a given field @xmath11 , while in eq . ( [ afterswitch ] ) , they restrict the field @xmath11 to vanish on a given surface @xmath9 . the integral over @xmath11 in eq . ( [ afterswitch ] ) is therefore restricted to functions @xmath11 which vanish on @xmath9 , in contrast to the integral over @xmath11 in eq . ( [ beforeswitch ] ) , in which @xmath11 can be any function . to evaluate @xmath32 $ ] we divide @xmath45 $ ] into two parts . the jacobian @xmath45 $ ] in eq . ( [ afterswitch ] ) is evaluated for field configurations @xmath11 which vanish on a particular surface @xmath9 . we make this explicit by writing ( [ inserttildex ] ) as @xmath46^{-1 } = \int \scrd \tilde y^\mu \delta\left[\re\phi(\tilde y^{\mu}(\tau))\right ] \delta\left[\im\phi(\tilde y^{\mu}(\tau))\right ] \ , , \label{jdef}\ ] ] where @xmath47 is some other string worldsheet , distinct from @xmath9 . the evaluation of the jacobian is the essential new ingredient in deriving @xmath32 $ ] . the @xmath44 functions in ( [ jdef ] ) select surfaces @xmath48 which lie in a neighborhood of the surface @xmath18 of the zeros of @xmath11 . we separate @xmath48 into components lying on the surface @xmath18 and components lying along vectors @xmath49 normal to @xmath18 at the point @xmath50 : @xmath51 the point @xmath52 is the point on the surface @xmath18 lying closest to @xmath48 , and the magnitude of @xmath53 is the distance from @xmath48 to @xmath52 ( see fig . [ normal figure ] ) . we evaluate the jacobian ( [ jdef ] ) by making the change of variables @xmath54 defined by ( [ localcoords ] ) . although the @xmath44 functions in ( [ jdef ] ) force @xmath55 to vanish , the integrations over @xmath55 give a contribution to the jacobian . furthermore , this contribution depends on the field variable @xmath11 in a neighborhood of the surface . the integration over the reparameterizations @xmath56 of the surface @xmath18 , on the other hand , depends upon the surface , but not on the fields . the change of variables ( [ transformation ] ) leads to a factorization of the jacobian into a field contribution , and into a contribution depending only on the intrinsic properties of the worldsheet @xmath57 . we now exhibit the factorization of the jacobian . under the transformation ( [ transformation ] ) , the integral over @xmath47 becomes @xmath58 \scrd y_\perp^a \scrd\xi \nonumber \\ & = & \det_\tau\left[\sqrt { -\frac{1}{2 } \left ( \epsilon^{ab } \frac{\partial \tilde x^\mu}{\partial\xi^a } \frac{\partial \tilde x^\nu}{\partial\xi^b } \right)^2 \frac{1}{2 } \left ( \epsilon^{ab } n_\alpha^a n_\beta^b \right)^2}\right ] \scrd y_\perp^a \scrd\xi \nonumber \\ & = & \det_\tau\left[\sqrt{-g(\xi)}\big|_{\xi=\xi(\tau)}\right ] \scrd y_\perp^a \scrd\xi \ , , \label{y jacobian}\end{aligned}\ ] ] where @xmath59 is the square root of the determinant of the induced metric @xmath60 evaluated on the worldsheet @xmath9 . appendix [ notation appendix ] gives a summary of our notation , and of the relations used to obtain ( [ y jacobian ] ) . the functional determinant in ( [ y jacobian ] ) is the product of its argument evaluated at all points @xmath61 on the sheet , in the same way that the integration over @xmath62 is a product of integrals at all points @xmath61 . making the change of coordinates ( [ localcoords ] ) , ( [ transformation ] ) in the jacobian ( [ jdef ] ) gives @xmath63^{-1 } & = & \int \scrd \xi \scrd y_{\perp}^a \det_\tau[\sqrt{-g } ] \delta\left[\re\phi\left(\tilde x^{\mu}(\xi(\tau ) ) + y_{\perp}^a(\xi(\tau ) ) n_a^{\mu}(\xi(\tau))\right)\right ] \nonumber \\ & & \times \delta\left[\im\phi\left(\tilde x^{\mu}(\xi(\tau ) ) + y_{\perp}^a(\xi(\tau ) ) n_a^{\mu}(\xi(\tau))\right)\right ] \ , . \label{j partial split}\end{aligned}\ ] ] eq . ( [ j partial split ] ) has the form : @xmath64^{-1 } = \int \scrd \xi(\tau ) \det_{\tau}\left[\sqrt{-g}\right ] j_{\perp}[\phi , \tilde x^{\mu}(\xi(\tau))]^{-1 } \ , , \label{newjacob}\ ] ] where @xmath65^{-1 } & = & \int \scrd y_{\perp}^a \delta\left[\re\phi\left(\tilde x^{\mu}(\xi(\tau ) ) + y_{\perp}^a(\xi(\tau ) ) n_a^{\mu}(\xi(\tau))\right)\right ] \nonumber \\ & & \times \delta\left[\im\phi\left(\tilde x^{\mu}(\xi ( \tau ) ) + y_{\perp}^a(\xi(\tau ) ) n_a^{\mu}(\xi(\tau))\right)\right]\end{aligned}\ ] ] contains all the dependence on @xmath11 . since @xmath66 is independent of the parameterization @xmath56 , the jacobian factors into two parts : @xmath64^{-1 } = j_{\parallel}[\tilde x]^{-1 } j_{\perp}[\phi , \tilde x]^{-1 } \ , , \label{jfactor}\ ] ] where @xmath67^{-1 } = \int \scrd \xi \det_{\tau}\left[\sqrt{-g}\right ] \ , . \label{jparallel}\ ] ] the string part @xmath68 of the jacobian arises from the parameterization degrees of freedom . in the next section , we show that @xmath66 is the faddeev popov determinant for the @xmath44 functions in ( [ afterswitch ] ) . this allows us to define the action of the effective string theory . in the following section , we will use @xmath68 to fix the reparameterization degrees of freedom . inserting the factorized form ( [ jfactor ] ) of @xmath45 $ ] into the expression ( [ afterswitch ] ) for @xmath32 $ ] gives the wilson loop the form @xmath33 = \int \scrd \tilde x^\mu j_\parallel[\tilde x ] e^{is_{{\rm eff } } } \ , , \label{stringrep}\ ] ] where the action @xmath69 of the effective string theory is given by @xmath70 } = \int \scrd\phi^ * \scrd\phi \scrd c^\mu j_\perp[\phi ] \delta\left[\re\phi(\tilde x^\mu(\xi))\right ] \delta\left[\im\phi(\tilde x^\mu(\xi))\right ] e^{i(s + s_{gf } ) } \ , . \label{seff}\ ] ] the string action ( [ seff ] ) was obtained previously by gervais and sakita @xcite . the novel feature of our result is the string integration measure of the wilson loop ( [ stringrep ] ) . the string action depends upon the field part @xmath66 of the jacobian , @xmath71^{-1 } = \int \scrd y_{\perp}^a \delta\left[\re\phi\left(\tilde x^{\mu } + y_{\perp}^a n_{\mu a } \right)\right ] \delta\left[\im\phi\left(\tilde x^{\mu } + y_{\perp}^a n_{\mu a } \right)\right ] \ , . \label{j perp of y}\ ] ] the @xmath44 functions force @xmath72 to be zero , so we can expand their arguments in a power series in @xmath72 , @xmath73 the zeroth order term in ( [ phi expand in y ] ) vanishes because @xmath74 is the surface of the zeros of @xmath11 . the integration ( [ j perp of y ] ) over @xmath55 gives the result @xmath71^{-1 } = \det^{-1}_{\xi}\left [ \epsilon^{ab } n_a^{\mu } n_b^{\nu } \left(\partial_{\mu}\re\phi\right ) \left(\partial_{\nu}\im\phi\right ) \big|_{x^{\mu } = \tilde x^{\mu } } \right ] \ , . \label{j_perp expressed}\ ] ] the jacobian @xmath66 is a faddeev popov determinant , which we discuss in appendix [ j_perp fp appendix ] . ( [ seff ] ) gives the action @xmath75 of the effective string theory as an integral over field configurations which have a vortex fixed at @xmath9 . since the vortex theory ( [ originalpartition ] ) is an effective long distance theory , the path integral ( [ originalpartition ] ) for @xmath32 $ ] , written in terms of the fields of the abelian higgs model , is cut off at a scale @xmath76 which is on the order of the mass @xmath77 of the dual gluon . furthermore , the integration ( [ stringrep ] ) over @xmath9 includes all the long distance fluctuations of the theory . therefore , the path integral ( [ seff ] ) contains neither short distance nor long distance fluctuations , and is determined by minimizing the field action @xmath78 $ ] for a fixed position of the vortex sheet : @xmath79 = s[\tilde x^\mu , \phi^{{\rm class } } , c_\mu^{{\rm class } } ] \ , , \kern 0.5 in \phi^{{\rm class}}(\tilde x^\mu ) = 0 \ , . \label{classical string}\ ] ] the fields @xmath80 and @xmath81 are the solutions of the classical equations of motion , subject to the boundary condition @xmath82 . the action @xmath69 depends both on the distance @xmath2 between the quarks , and the radius of curvature @xmath83 of the vortex sheet bounded by @xmath15 . in the long distance limit , when the length of the string @xmath2 and its radius @xmath83 are large compared to the thickness of the flux tube @xmath31 , the string action ( [ classical string ] ) becomes the nambu goto action @xmath84 , @xmath85 where @xmath86 is the classical string tension , determined from the solution of the nielsen olesen equations for a straight , infinitely long string . it is convenient to separate the action ( [ classical string ] ) into its perturbative and nonperturbative parts : @xmath79 = s[\tilde x^\mu , \phi^{{\rm class } } , c_\mu^{{\rm class } } ] = s^{{\rm maxwell}}[\tilde x^\mu ] + s^{{\rm np}}[\tilde x^\mu ] \ , , \label{break abelian action}\ ] ] where @xmath87 is the action obtained by setting @xmath88 in eq . ( [ firstaction ] ) . the value of @xmath87 depends only upon the boundary @xmath15 , and is the usual electromagnetic interaction between charged particles : @xmath89 = \frac{4}{3 } \frac{e^2}{2 } \oint dx^\mu \oint d{x'}^\mu { \cal d}_{\mu\nu}(x^\mu - { x'}^\mu ) \ , , \label{maxwell action}\ ] ] where @xmath90 is the photon propagator . to calculate the wilson loop @xmath32 $ ] from the effective string theory ( [ stringrep ] ) , we must also examine @xmath69 at smaller values of @xmath2 and @xmath83 , on the order of the string thickness @xmath31 . we first consider the dependence of @xmath69 on @xmath2 for a flat string , where @xmath91 . in this case , the curve @xmath15 is a rectangle of length @xmath92 in the time direction , and width @xmath2 in the space direction . in the large @xmath92 limit , the action @xmath69 reduces to the product of @xmath92 and the potential @xmath93 previously used to fit the energy levels of heavy quark systems . evaluation of ( [ break abelian action ] ) for a flat sheet gives a corresponding decomposition of @xmath93 , @xmath94 for small @xmath2 , @xmath95 while eq . ( [ ng action def ] ) gives the large @xmath2 behavior @xmath96 recent numerical studies @xcite of the classical equations of motion for a flat sheet have shown that for a superconductor on the i ii border , the long distance behavior ( [ v class long ] ) of @xmath97 persists to small values of @xmath2 , even to values less than the string thickness @xmath31 . therefore , for a superconductor on the i ii border , @xmath93 is , to a good approximation , equal to the cornell potential @xcite : @xmath98 in other words , for a flat sheet , @xmath99 thus , for short straight strings the nambu goto action remains a good approximation to the nonperturbative part of the classical action for a superconductor on the type i ii border . next , consider the nonperturbative contribution to the classical action for a long bent string . ( the maxwell action has the value ( [ maxwell action ] ) independent of the shape of the vortex . ) the leading correction to the nambu goto action when the string is bent is the curvature term : @xmath100 where @xmath101 is the extrinsic curvature . @xmath102 the magnitude of @xmath101 is of the order of @xmath103 , so that @xmath104 . the calculation of the `` rigidity '' @xmath105 determining the size of @xmath106 has been considered by a number of authors @xcite , but the value of @xmath105 for a superconductor on the i ii border was never calculated . we conjecture that the value of @xmath105 is small , because de vega and schaposnik @xcite have shown that the components of the stress tensor perpendicular to the axis of a straight nielsen olesen flux tube vanish for a superconductor on the border between type i and type ii . in other words , there are no `` bonds '' perpendicular to the field lines of a straight flux tube of infinite extent . when the flux tube is bent slightly , there are no perpendicular bonds to be stretched or compressed , and the change in the energy is just the string tension multiplied by the change in length . that is , the curvature term , which in a sense represents the attraction or repulsion between neighboring parts of the string , should vanish . a more formal argument can be made by regarding the borderline superconductor as the long distance limit of a theory where the forces between vortices become weak . polyakov @xcite has shown , using renormalization group methods , that @xmath105 also vanishes in this limit . similar heuristic arguments give a reason for the above mentioned result that the nambu goto action is a good approximation for short , straight strings on the i ii border . the bending of the field lines as the quark antiquark separation becomes smaller causes no additional changes in the energy . we therefore take the action of the effective string theory to be equal to the sum of the maxwell action ( [ maxwell action ] ) and the nambu goto action ( [ ng action def ] ) : @xmath79 = s^{{\rm maxwell}}[\gamma ] - \sigma \int d^2\xi \sqrt{-g } \ , . \label{given action}\ ] ] we use eq . ( [ given action ] ) for the full range of string lengths @xmath2 and radii of curvature @xmath83 greater than the inverse of the mass @xmath77 of the dual gluon , which is the cutoff of the effective string theory ( [ stringrep ] ) . ( [ given action ] ) for @xmath107 $ ] is the generalization of ( [ v class ] ) to a general sheet . the first term , @xmath108 $ ] , is just a boundary term , independent of the fluctuating string , and we take take @xmath109 for the calculations carried out in the rest of this paper . in the next section we show how to carry out the integration over @xmath9 in ( [ stringrep ] ) by separating the degrees of freedom of the worldsheet @xmath9 into two physical degrees of freedom and two reparameterization degrees of freedom . this treatment makes no use of ( [ given action ] ) , and is applicable to any effective string theory of vortices . we next show how to evaluate the integral over @xmath18 in eq . ( [ stringrep ] ) , @xmath33 = \int \scrd \tilde x^\mu j_\parallel[\tilde x^\mu ] e^{is_{{\rm eff } } } \ , . \label{vortex partition}\ ] ] the integration over @xmath18 is the product of an integral over string worldsheets and an integral over reparameterizations of the coordinates of the string . the jacobian @xmath68 is the inverse of the integration ( [ jparallel ] ) over reparameterization degrees of freedom . in this section , we fix the parameterization of the string , and show that @xmath68 cancels the integration over reparameterizations . any surface @xmath9 has only two physical degrees of freedom . the other two degrees of freedom represent the invariance of the surface under coordinate reparameterizations . we fix the coordinate reparameterization symmetry by choosing a particular `` representation '' @xmath110 of the surface , which depends on two functions @xmath111 , @xmath112 , @xmath113 \ , . \label{x_p def}\ ] ] a particular example of a representation @xmath110 is obtained by expanding in transverse fluctuations @xmath114 about a fixed sheet @xmath115 , @xmath116 = \bar x_p^\mu(\xi ) + x_\perp^a(\xi ) \bar n^\mu_a(\xi ) \ , . \label{normal rep}\ ] ] the vectors @xmath117 are orthogonal to the surface @xmath115 . in this example , @xmath118 and @xmath119 are the transverse coordinates @xmath114 . any physical surface can be expressed in terms of @xmath110 by a suitable choice of @xmath118 and @xmath119 . in particular , the worldsheet @xmath18 appearing in the integral ( [ vortex partition ] ) can be written in terms of a reparameterization @xmath120 of the representation @xmath110 , @xmath121 \ , . \label{x tilde of x_p}\ ] ] the four degrees of freedom in @xmath18 are replaced by two physical degrees of freedom @xmath111 , @xmath112 and two reparameterization degrees of freedom @xmath120 . we can write the integral over @xmath18 in ( [ vortex partition ] ) in terms of integrals over @xmath122 and @xmath120 , @xmath123 \scrd f^1 \scrd f^2 \scrd\tilde\xi \,.\ ] ] noting that the derivative of @xmath9 with respect to @xmath120 is , @xmath124 \bigg|_{\xi=\tilde\xi } \,,\ ] ] we can write @xmath125 \scrd f^1 \scrd f^2 \scrd \tilde\xi \nonumber \\ & = & \det\left [ \tilde t_{\mu\nu } \sqrt{-g^p } \frac{\partial x_p^\mu}{\partial f^1 } \frac{\partial x_p^\nu } { \partial f^2 } \bigg|_{\xi = \tilde\xi } \right ] \scrd f^1 \scrd f^2 \scrd \tilde\xi \ , , \label{measure shift}\end{aligned}\ ] ] where @xmath126 is the antisymmetric tensor normal to the worldsheet and , @xmath127 is the induced metric of @xmath128 . the metric @xmath129 is related to the metric @xmath130 of @xmath18 , @xmath131 the induced metric @xmath130 of the original worldsheet @xmath18 does not appear in eq . ( [ measure shift ] ) because the determinant is independent of @xmath132 . only the induced metric @xmath129 of the worldsheet @xmath128 enters into the determinant . with the parameterization ( [ x tilde of x_p ] ) of @xmath9 , the path integral ( [ vortex partition ] ) takes the form @xmath33 = \int \scrd\tilde\xi \scrd f^1 \scrd f^2 \det\left [ \tilde t^{\mu\nu } \frac{\partial x_p^\mu } { \partial f^1 } \frac{\partial x_p^\nu } { \partial f^2 } \right ] \det[\sqrt{-g^p } ] j_\parallel e^{is_{{\rm eff } } } \ , . \label{w subbed}\ ] ] the action @xmath69 is parameterization independent , so it is independent of @xmath120 . the same is true for @xmath68 . furthermore , @xmath133 is parameterization independent , so that the product @xmath134\ ] ] is independent of @xmath120 . therefore , this product , along with @xmath68 and @xmath135 , can be brought outside the @xmath132 integral in ( [ w subbed ] ) . the path integral then takes the form @xmath33 = \int \scrd f^1 \scrd f^2 \det\left [ \tilde t^{\mu\nu } \frac{\partial x_p^\mu } { \partial f^1 } \frac{\partial x_p^\nu } { \partial f^2 } \right ] j_\parallel e^{is_{{\rm eff } } } \int \scrd \tilde\xi \det[\sqrt{-g^p } ] \ , . \label{before cancel j parallel}\ ] ] the remaining integral over reparameterizations @xmath132 is equal to @xmath136 , defined by ( [ jparallel ] ) , and is canceled by the explicit factor of @xmath68 appearing in ( [ before cancel j parallel ] ) . this means we do not need to evaluate @xmath68 , and can avoid the complications inherent in evaluating the integral over reparameterizations of the string coordinates . the anomalies produced in string theory by evaluating this integral are not present , so we do not have a polchinski strominger term in the theory . ( [ before cancel j parallel ] ) gives the final result for the wilson loop @xmath33 = \int \scrd f^1 \scrd f^2 \det\left [ \tilde t_{\mu\nu } \frac{\partial x_p^\mu}{\partial f^1 } \frac{\partial x_p^\nu } { \partial f^2 } \right ] e^{is_{{\rm eff } } } \ , , \label{param measure}\ ] ] as an integration over two function @xmath111 and @xmath112 , the physical degrees of freedom of the string . the path integral ( [ param measure ] ) is invariant under reparameterizations of the string , and describes a two dimensional field theory with two degrees of freedom , the two transverse oscillations of a two dimensional sheet . the integration ( [ param measure ] ) goes over the normal fluctuations of the string worldsheet . the components of @xmath118 and @xmath119 along the sheet are nonphysical . the determinant in ( [ param measure ] ) is a normalization factor for @xmath118 and @xmath119 . this can be seen by applying the identity @xmath137 to the determinant , @xmath138 = \det\left [ \epsilon^{ab } \left ( n_{\mu a } \frac{\partial x_p^\mu } { \partial f^1 } \right ) \left ( n_{\nu b } \frac{\partial x_p^\nu } { \partial f^2 } \right ) \right ] \,.\ ] ] the factors of @xmath139 determine the amount of the fluctuation @xmath140 which is in a direction normal to the sheet . ( [ param measure ] ) is the string representation of any field theory containing classical vortex solutions . the expression ( [ seff ] ) for @xmath69 had been obtained previously by gervais and sakita @xcite ; we are unaware of any previous derivation of the string representation ( [ param measure ] ) for the path integral . we now show how it provides a method for explicit calculations . in this section we carry out the semiclassical expansion of @xmath32 $ ] about a classical solution of the effective string theory , and find the leading contribution of string fluctuations to the effective action @xmath141 $ ] . the ends of the string follow the path @xmath15 fixed by the prescribed trajectory of the quarks , and the fluctuations of the string are cutoff at the momentum scale @xmath77 of the inverse string radius . as explained in section [ effective action section ] , we take the action @xmath69 of the effective string theory to be the nambu goto action @xmath142 and ( [ param measure ] ) becomes @xmath33 = \int \scrd f^1(\xi ) \scrd f^2(\xi ) \det\left [ \tilde t^{\mu\nu } \frac{\partial x_p^\mu}{\partial f^1 } \frac{\partial x_p^\nu}{\partial f^2 } \right ] e^{-i\sigma \int d^2\xi \sqrt{-g } } \ , . \label{w with action}\ ] ] we expand ( [ w with action ] ) in small fluctuations @xmath140 of @xmath143 $ ] around a fixed sheet @xmath144 , subject to the condition that the boundary of @xmath115 lies on the curve @xmath15 , @xmath145 where @xmath146 is the position of the string worldsheet when @xmath147 . expanding @xmath59 to quadratic order in small @xmath118 and @xmath119 , we obtain @xmath148 & = & \int \scrd f^i(\xi ) \det\left [ \frac{1}{2 } \tilde t^{\mu\nu } \epsilon^{ij } \frac{\partial x_p^\mu}{\partial f^i } \frac{\partial x_p^\nu}{\partial f^j } \bigg|_{f^i=0 } \right ] \nonumber \\ & & \times \exp\left\ { -i\sigma \int d^2\xi \sqrt{-\bar g } \left [ 1 + \bar g^{ab } \frac{\partial x_p^\mu}{\partial\xi^a } \bigg|_{f^i=0 } \frac{\partial}{\partial\xi^b } \left ( \frac{\partial x_p^\mu } { \partial f^i } \bigg|_{f^i=0 } f^i \right ) + \frac{1}{2 } f^i g^{-1}_{ij } f^j \right ] \right\ } \ , , \label{semiclassical string partition}\end{aligned}\ ] ] where @xmath149 is @xmath150 the metric of the fixed worldsheet @xmath115 , and @xmath151 we choose @xmath115 to be the surface which minimizes the action . then @xmath115 satisfies the `` classical equation of motion '' @xmath152 where the covariant laplacian is @xmath153 using the fact that the covariant derivative of the metric is zero , we show in appendix [ notation appendix ] that @xmath154 the vectors @xmath155 and @xmath156 form a complete basis , so ( [ classical string equation ] ) and ( [ covariant identity ] ) imply @xmath157 evaluating the @xmath140 integral in eq . ( [ semiclassical string partition ] ) gives @xmath33 = e^{-i\sigma \int d^2\xi \sqrt{-\bar g } } \det\left [ \frac{1}{2 } \tilde t_{\mu\nu } \epsilon^{ij } \frac{\partial x_p^\mu}{\partial f^i } \frac{\partial x_p^\nu}{\partial f^j } \bigg|_{f^i=0 } \right ] \det^{-1/2}\left[g^{-1}_{ij}\right ] \ , . \label{z of g}\ ] ] the inverse propagator @xmath158 ( [ g^-1_ij def ] ) can be shown to be @xmath159 \bar n_{\nu b } \frac{\partial x_p^\nu}{\partial f^j } \bigg|_{f^i=0 } \ , , \label{g inverse def}\ ] ] where @xmath160 is the extrinsic curvature tensor of the sheet @xmath115 . a derivation of ( [ g inverse def ] ) is given in appendix a of @xcite . the @xmath161 are vectors normal to the worldsheet @xmath115 . ( [ g inverse def ] ) gives @xmath162 = \det^{-1/2}\left [ -\nabla^2 \delta_{ab } - \bar\scrk^a_{ab } \bar\scrk^{bab } \right ] \det^{-1}\left[\frac{1}{2 } \epsilon^{ab } \bar n_{\mu a } \bar n_{\nu b } \epsilon^{ij } \frac{\partial x_p^\mu}{\partial f^i } \frac{\partial x_p^\nu}{\partial f^j } \bigg|_{f^i=0 } \right ] \ , . \label{g det}\ ] ] from the identity ( [ dual normal ] ) , @xmath163 we see that the first determinant in ( [ z of g ] ) and the second determinant in ( [ g det ] ) cancel . the determinant appearing in ( [ param measure ] ) produces exactly the correct normalization for the green s function . the functional integral ( [ z of g ] ) becomes @xmath33 = e^{-i\sigma \int d^2 \xi \sqrt{-\bar g } } \det^{-1/2}\left [ -\nabla^2 \delta_{ab } - \bar\scrk^a_{ab } \bar\scrk^{bab } \right ] \ , . \label{last z}\ ] ] we note that ( [ last z ] ) is independent of the factors of @xmath164 which appeared in the inverse propagator ( [ g inverse def ] ) . these factors are the projections of the fluctuations @xmath140 normal to the string worldsheet . for small @xmath140 , the worldsheet @xmath110 is @xmath165 the perturbation of the worldsheet in the direction @xmath166 is @xmath167 the factors of @xmath168 in ( [ g inverse def ] ) project out the part of the fluctuations @xmath140 perpendicular to the worldsheet @xmath115 . only normal fluctuations contribute to @xmath32 $ ] , since fluctuations along the worldsheet are equivalent to a reparameterization of the sheet coordinates . the effective action obtained from ( [ last z ] ) is @xmath169 = s_{cl } + s_{{\rm fluc } } \ , . \label{effective action def}\ ] ] the first term in ( [ effective action def ] ) is the nambu goto action evaluated at the `` classical '' worldsheet @xmath144 , @xmath170 the semiclassical correction @xmath171 due to the transverse string fluctuations is @xmath172 \ , . \label{w unreg}\ ] ] to summarize , we have integrated out the string fluctuations , and reduced the problem to the evaluation of the determinant in ( [ last z ] ) . this is a quantum mechanical scattering problem in the background of the solution of the classical equation ( [ class string simple ] ) , with appropriate boundary conditions . in the next section , we describe how to evaluate this determinant . the argument of the logarithm in ( [ w unreg ] ) is the inverse propagator for fluctuations on the string . this inverse propagator can also be obtained by direct variation of the nambu goto action with respect to any transverse coordinates @xmath114 , @xmath173 up to an overall normalization factor . in fact , the correction ( [ w unreg ] ) to the effective action has already been studied by lscher , symanzik , and weisz ( lsw ) @xcite in the case of a straight string with fixed ends . we describe their results , which we will use in evaluating ( [ w unreg ] ) . lsw used pauli villars regularization to obtain a regulated form @xmath174 of the trace in ( [ w unreg ] ) , @xmath175 the @xmath176 are the masses of the regulators , and the @xmath177 are suitably chosen coefficients . the laplacian in ( [ regulated action ] ) has been wick rotated from minkowski to euclidean space . the regulated quantity @xmath174 is separated into a divergent part @xmath178 and a finite part @xmath179 , @xmath180 lsw evaluated the divergent part @xmath181 , and obtained terms which are quadratically , linearly , and logarithmically divergent in the cutoffs @xmath176 . the quadratic term is a renormalization of the string tension , the linear term is a renormalization of the quark masses , and the logarithmically divergent term is proportional to the integral over all space of the scalar curvature @xmath182 of the string worldsheet . lsw also obtained a formal expression for the finite part @xmath179 . they evaluated this expression only for the case of a straight string of length @xmath2 with fixed ends , and calculated a correction @xmath183 to the static potential : @xmath184 we are interested in calculating @xmath171 for rotating quarks , so we must evaluate @xmath174 for a more general surface . we break ( [ w unreg ] ) into two parts : @xmath185 + \frac{i}{2 } \tr\ln\left [ \frac{-\nabla^2 \delta_{ab } - \bar \scrk^a_{ab } \bar \scrk^{bab}}{-\nabla^2 } \right ] \ , . \label{separated effective action}\ ] ] we will evaluate the first term in ( [ separated effective action ] ) by generalizing the calculation of lsw . we will calculate the second term directly . the first term in ( [ separated effective action ] ) , @xmath186 \ , , \label{w_1}\ ] ] involves the laplacian in the curved background of the classical solution @xmath115 . in the flat case studied by lsw , the laplacian is equal to @xmath187 , @xmath188 the coordinate @xmath189 is the time in the lab frame , and @xmath190 is a radial coordinate which takes the values @xmath191 and @xmath192 at the two ends of the string . the length of the string is @xmath193 . to calculate @xmath194 we extend the calculation of lsw to more general coordinate systems . we make a coordinate transformation @xmath195 to conformal coordinates , where the transformed metric @xmath196 , @xmath197 . this transformation puts the laplacian in a form similar to the flat sheet laplacian ( [ flat sheet laplacian ] ) , and allows us to evaluate ( [ w_1 ] ) by extending the calculation of lsw . to see how this works , we express @xmath198 as a functional integral , @xmath199 the transformation to conformal coordinates @xmath200 gives @xmath201 which is of the form of @xmath174 treated by lsw . we will need @xmath194 in the limit of large @xmath92 , and hence are only interested in strings whose metric is time independent . to determine which metrics are time independent , we must choose a coordinate system . we choose coordinates @xmath190 and @xmath189 , where @xmath189 is the time in the lab frame , and @xmath190 is orthogonal to @xmath189 ( @xmath202 ) . this guarantees that @xmath189 is the physical time . from now on we consider only metrics @xmath130 which are independent of @xmath189 . in appendix [ luscher appendix ] , we show that @xmath203 where @xmath204 contains quark mass and string tension renormalizations . the finite part of @xmath194 is @xmath205 where @xmath206 the results ( [ effective r proper ] ) and ( [ r_p def text ] ) are valid for any orthogonal coordinate system with a time independent metric . we show in appendix [ luscher appendix ] that @xmath207 is equal to the classical energy of the string divided by the string tension @xmath86 . we call @xmath207 the `` proper length '' of the string . for a flat metric , where @xmath208 , the proper length @xmath207 of the string reduces to the distance @xmath2 between its endpoints . in the previous section , we evaluated the finite part of @xmath194 for a sheet with a time independent metric using the results of lsw . in this section we evaluate the second term in ( [ separated effective action ] ) , @xmath209 \ , . \label{w_2 def}\ ] ] the trace in ( [ w_2 def ] ) is over functions of @xmath190 and @xmath189 . we first make the coordinate transformation @xmath210 , @xmath211 the coordinate @xmath212 runs from @xmath213 to @xmath214 , @xmath215 and @xmath216 . in appendix [ luscher appendix ] , we show that the metric in the system @xmath217 is conformal ( @xmath218 , @xmath219 ) . in this coordinate system , the inverse propagator for string fluctuations is @xmath220 the string has infinite extent in time , and the curvature @xmath221 is independent of @xmath189 , so we can take the fourier transform with respect to the time coordinate . we express the trace in ( [ w_2 def ] ) over functions of @xmath189 and @xmath190 as an integral over a frequency @xmath222 and a trace over functions of a single variable @xmath212 , @xmath223 \ , . \label{w_2 of nu}\ ] ] in going from ( [ w_2 def ] ) to ( [ w_2 of nu ] ) , we have also carried out the wick rotation @xmath224 . the integration over @xmath222 gives @xmath225 \ , . \label{w_2 given}\ ] ] eq . ( [ w_2 given ] ) expresses @xmath226 as the trace of the difference of two operators . the first has the form of a hamiltonian for a relativistic particle in the local potential @xmath227 . the second operator has the form of a free hamiltonian . the square roots enter because we are working with relativistic degrees of freedom . the terms @xmath194 and @xmath226 are proportional to the time @xmath92 , but are otherwise time independent . we define the `` effective lagrangian '' of the string to be the effective action divided by the time @xmath92 . the sum of ( [ effective r proper ] ) and ( [ w_2 given ] ) gives the effective lagrangian @xmath228 determining the contribution of the string fluctuations to @xmath32 $ ] , @xmath229 in the next section , we will evaluate @xmath228 for a string of length @xmath2 rotating with angular velocity @xmath8 . in appendix [ log cutoff appendix ] , we show that , for a general sheet , @xmath226 is logarithmically divergent . we show that its divergent part is given by @xmath230 where @xmath182 is the scalar curvature , @xmath231 this result agrees , in the large time limit , with the logarithmically divergent term in the cutoff dependent part of the effective string action ( [ luscher reg ] ) found by lsw . we now evaluate the effective lagrangian of a string with boundary @xmath15 generated by a quark antiquark pair separated at fixed distances @xmath232 and @xmath192 from the origin , and rotating with angular velocity @xmath8 . this lagrangian has two parts , the classical string lagrangian and the contribution ( [ rotating potential ] ) of string fluctuations . we evaluate the classical string lagrangian first . the solution to the classical equations of motion ( [ class string simple ] ) yields the classical , straight rotating string , @xmath233 the coordinate @xmath190 is chosen so that the velocity of the string is zero when @xmath234 . the coordinate @xmath190 runs from @xmath191 to @xmath192 , and @xmath189 runs from @xmath235 to @xmath236 . the vectors @xmath237 and @xmath238 are two orthogonal unit vectors in the plane of rotation , and @xmath239 is a unit vector in the time direction . the classical lagrangian @xmath240 obtained from ( [ s_ng def ] ) is @xmath241 next , we calculate the contribution @xmath228 ( [ rotating potential ] ) due to string fluctuations . the metric of the sheet ( [ x bar def ] ) is @xmath242 this metric is independent of @xmath189 , and @xmath243 . we make the transformation ( [ coord x def ] ) from coordinates @xmath190 and @xmath189 to coordinates @xmath212 and @xmath189 , and find @xmath244 the coordinate @xmath212 runs from @xmath213 to @xmath214 , where @xmath245 and the proper length @xmath207 of the string is @xmath246 using this coordinate system , we evaluate @xmath228 in appendix [ eigenfunctions and sum ] for the case of equal quark masses , @xmath247 : @xmath248 - \frac{v^2 \gamma}{\pi r } f\left(v\right ) \ , , \label{v string}\ ] ] where @xmath249 and the function @xmath250 is @xmath251 \,.\ ] ] eq . ( [ v string ] ) becomes the lscher term in the zero velocity limit . we are interested in the large @xmath2 limit , where the quark velocity is close to the speed of light . for @xmath252 close to one , ( [ v string ] ) becomes @xmath253 + \frac{7}{6r } + o\left(\frac{\ln\gamma}{\gamma r}\right ) \ , . \label{l fluc calculated}\ ] ] furthermore , for the semiclassical expansion to be valid , the theory must be weakly coupled . that is , @xmath228 must be less than @xmath240 ( [ l classical string ] ) . for large @xmath2 , @xmath254 the semiclassical expansion is valid , since , as we will see , @xmath2 grows like @xmath255 in the @xmath256 limit . in this case , the long distance limit where the effective theory is applicable is automatically the region of weak coupling . we calculate classical regge trajectories for equal mass quarks by adding a quark mass term to the string lagrangian @xmath240 , @xmath257 we have used eq . ( [ l classical string ] ) with @xmath247 . the quark velocity is @xmath258 , and @xmath259 is the quark boost factor . the lagrangian ( [ l_cl def ] ) is a function of @xmath2 and @xmath8 , @xmath260 the angular momentum of the meson is obtained by varying the lagrangian with respect to the angular velocity , @xmath261 the meson energy is given by the hamiltonian , @xmath262 the classical equation of motion @xmath263 for the quarks determines @xmath2 as a function of @xmath8 , @xmath264 eq . ( [ boundary condition ] ) shows that @xmath2 is proportional to @xmath255 for large @xmath265 . expanding ( [ j equation ] ) and ( [ e equation ] ) in the large @xmath2 limit , where the quark velocity @xmath252 goes to one , yields the result : @xmath266 the first term in ( [ classical regge eqn ] ) is the classical formula for the slope of a regge trajectory . the second term is the leading correction for nonzero classical quark mass , where @xmath267 . we now calculate the correction to the energy obtained by considering @xmath228 a small perturbation to the classical lagrangian @xmath268 . the lagrangian , @xmath269 depends on only one degree of freedom , the rotation angle @xmath270 , through its time derivative @xmath271 , to first order in @xmath228 , the correction to the energy is minus the correction to the lagrangian @xcite , @xmath272 \bigg|_{\omega = \omega(j ) } \ , , \label{e of j}\ ] ] where @xmath8 is given as a function of @xmath5 through the classical relation ( [ j equation ] ) . the correction ( [ e of j ] ) to the energy of the meson gives a correction to the slope ( [ classical regge eqn ] ) of a regge trajectory , @xmath273 using ( [ classical regge eqn ] ) for @xmath274 and ( [ l fluc calculated ] ) for @xmath228 , we obtain @xmath275 - \frac{4}{3\pi^2\sigma } \gamma^{-3 } + \frac{7}{6 \pi \sigma r e } + o\left(\gamma^{-5 } , \frac{1}{re\gamma}\right ) \ , . \label{regge corrected}\ ] ] we write @xmath2 and @xmath265 as functions of @xmath4 using the definition ( [ e equation ] ) of @xmath4 and the classical equation of motion ( [ boundary condition ] ) . because @xmath2 and @xmath265 only appear in the small correction terms in the result ( [ regge corrected ] ) , we only need their leading order dependence on @xmath4 , @xmath276 substituting ( [ r , gamma of e ] ) in ( [ regge corrected ] ) gives @xmath277 - \frac{4}{3\sigma } \sqrt{\frac{m^3 e}{\pi } } + \frac{7}{12 } + o\left ( e^{-1/2 } \right ) \ , . \label{j result}\ ] ] the leading term is the classical regge formula . the next term is the leading correction due to string fluctuations . the third term is a nonzero quark mass correction . the fourth term is another correction due to string fluctuations . ( [ j result ] ) gives a meson regge trajectory @xmath6 . [ cols=">,^ " , ] we used values @xmath278 and @xmath279 obtained from the cornell fits of heavy quark potentials @xcite . this gives @xmath280 . the only other parameter is the quark mass @xmath281 . in figure [ regge figure ] , we plot @xmath5 versus the square of the energy ( [ e of j ] ) for quark masses of 30 mev , 100 mev , and 300 mev . for comparison , we also plot the classical formula @xmath282 . the points @xcite plotted on the graph are the @xmath283 , @xmath284 , @xmath285 , @xmath286 , @xmath287 , and @xmath288 mesons . x s are the @xmath289 , @xmath119 , @xmath290 , @xmath291 , and @xmath292 . we have added one to the value of the angular momentum @xmath5 in figure [ regge figure ] to account for the contribution of the spin of the quarks . we have chosen a range of values for the quark masses in fig . [ regge figure ] in order to give a qualitative picture of the dependence of the regge trajectory on the quark mass . since ( [ j result ] ) does not include the contribution of quark fluctuations to the regge trajectory , this formula is incomplete . we are now in the process of including the quark degrees of freedom in the functional integral ( [ param measure ] ) . the boundary @xmath15 of the sheet @xmath9 becomes dynamical , and couples to the string fluctuations . it is clear that a calculation of the contribution of these degrees of freedom is essential to understanding why the classical formula for regge trajectories works so well . the primary results of the paper are ( [ param measure ] ) and ( [ j result ] ) . we have expressed the path integral @xmath32 $ ] ( [ originalpartition ] ) of a renormalizable quantum field theory having classical vortex solutions as the path integral formulation of an effective string theory of vortices ( [ param measure ] ) . this theory describes the two transverse fluctuations of the vortex at scales larger than the inverse mass of the lightest particle in the field theory . our method is applicable to any field theory containing vortex solutions . using the string representation of @xmath32 $ ] , we carried out a semiclassical expansion of the effective action @xmath141 $ ] about a classical solution of the effective string theory . we calculated the contribution of these string fluctuations explicitly for the case where the worldline @xmath15 is generated by the trajectory of a quark antiquark pair separated by a distance @xmath2 . we are now calculating the contribution to the effective action @xmath141 $ ] due to the quantum fluctuations of the boundary . we would like to thank n. brambilla for very helpful conversations . this work was supported in part by the u. s. department of energy grant de - fg03 - 96er40956 . we describe the string worldsheet by the function @xmath18 of the coordinates @xmath50 . the physics of the vortex should be independent of the coordinate system we choose , so we require the theory to be invariant under a reparameterization of the coordinates , @xmath293 . the tangent vectors to the vortex worldsheet are defined by taking derivatives of @xmath18 , @xmath294 where @xmath295 is a partial derivative with respect to one of the vortex coordinates . the induced metric on the worldsheet @xmath18 is @xmath296 it is also convenient to define the square root of the determinant of the metric , @xmath297 we use the @xmath298 to define an antisymmetric tensor which describes the orientation of the string worldsheet , @xmath299 this quantity was defined by polyakov @xcite . it is the projection of the antisymmetric tensor @xmath300 into the space of four dimensional tensors . this tensor defines the orientation of the two dimensional vortex worldsheet in four space . the quantity ( [ t mu nu def ] ) is also independent of the coordinate parameterization of the worldsheet @xmath18 . we now describe the curvature of the vortex worldsheet . we do this by taking covariant derivatives of the tangent vectors . the covariant derivative of @xmath298 is @xmath301 where the @xmath302 are christoffel symbols , @xmath303 the covariant derivatives of the tangent vectors are orthogonal to the worldsheet , @xmath304 this identity is derived using the definition ( [ gamma def ] ) of the christoffel symbols , the definition ( [ metric appendix def ] ) of the metric , and the relationship between derivatives of different @xmath298 , @xmath305 the covariant derivatives of the tangent vectors are normal to the string worldsheet . we therefore define a basis of normal vectors @xmath306 , which satisfy the conditions @xmath307 the @xmath306 are an orthonormal basis for the vectors normal to the worldsheet . ( [ no tangent covariant ] ) implies that @xmath308 for some tensor @xmath101 . the tensor @xmath101 is called the extrinsic curvature tensor of the string worldsheet . with the definition ( [ curvature def ] ) , the curvature tensor @xmath101 is @xmath309 it is symmetric in the indices @xmath31 and @xmath310 due to the relationship ( [ tangent vector derivatives ] ) between derivatives of tangent vectors . the extrinsic curvature of the string worldsheet can also be described using derivatives of the normal vectors . the orthogonality of the @xmath311 and the @xmath306 implies @xmath312 therefore , the derivatives of the normal vectors can be written as @xmath313 the tensor @xmath314 is called the torsion , and it describes the twisting of the basis of normal vectors as we move along the worldsheet . the torsion depends on our choice of the @xmath306 , so we will choose them so that the torsion is zero . this is done by requiring that the @xmath306 satisfy the differential equation @xmath315 the equation ( [ torsion free ] ) is equivalent to the statement @xmath316 . it is consistent with the conditions ( [ n def ] ) which define the normal vectors . as long as the normal vectors have an orthonormal basis at one point , eq . ( [ torsion free ] ) guarantees they will be orthonormal in a neighborhood of that point . therefore , it is always possible to find a local , orthonormal , torsion free basis for the normal vectors . there is one additional property of the normal vectors we will use . the antisymmetric combination of the normal vectors is ( with proper ordering ) equal to the dual of the worldsheet orientation tensor @xmath317 ( [ t mu nu def ] ) , @xmath318 where @xmath319 the relationship ( [ dual normal ] ) can be understood by noting that any antisymmetric tensor is of the form @xmath320 the tensor @xmath321 is orthogonal to the @xmath298 , so it must be proportional to @xmath322 . squaring both of these tensors gives @xmath323 therefore , @xmath321 and @xmath322 are equal up to an overall sign , which is fixed by choosing an appropriate ordering for the normal vectors . the jacobian @xmath66 ( [ j_perp expressed ] ) is a faddeev popov determinant , because fixing the position of the string in the field integrals is analogous to fixing a gauge in a gauge theory . in the string action , we fix the degrees of freedom which generate the transformation @xmath324 which displaces the vortex . the jacobian @xmath66 is analogous to the faddeev popov determinant in a gauge theory . in a gauge theory , where the @xmath44 function fixes the symmetry generated by the transformation @xmath325 the faddeev popov determinant appears as a normalization for the @xmath44 function @xcite , @xmath326 \delta_{fp } e^{-s } \ , , \label{gauge part}\ ] ] where @xmath327 \,.\ ] ] the wilson loop ( [ gauge part ] ) is analogous to our equation ( [ seff ] ) for the effective action . the determinant @xmath328 is analogous to @xmath66 . in the gauge theory , the faddeev popov method is used to remove nonphysical degrees of freedom from the problem . the @xmath44 function is inserted in the path integral eqnlessrefgauge part to fix the fields in some particular gauge . this creates an integral over all gauges which appears as a normalization factor , and is removed . the @xmath44 function in eq . ( [ seff ] ) , on the other hand , fixes the position of the vortex sheet , which is a physical degree of freedom . we want to evaluate the term , @xmath329 \,,\ ] ] in the effective action for a general string worldsheet . we work in coordinates @xmath190 and @xmath189 , such that @xmath189 is the time in the lab frame , and @xmath190 is orthogonal to @xmath189 ( @xmath202 ) . in these coordinates , the functional integral ( [ w_1 int ] ) for @xmath198 takes the form @xmath330 \right\}\,.\ ] ] we consider the case where the metric is independent of @xmath189 , and we make the coordinate transformation @xmath210 defined by @xmath331 the coordinate @xmath212 runs from @xmath213 to @xmath214 , @xmath332 in these coordinates , the length of the string is @xmath216 , the proper length of the string , @xmath333 in the coordinate system @xmath217 , the metric is conformal , @xmath334 and @xmath335 \right\ } \ , . \label{w_1 of x and t}\ ] ] we evaluate ( [ w_1 of x and t ] ) in a manner analogous to our treatment of @xmath226 in section [ w_2 section ] . we fourier transform in both space and time , introducing variables @xmath222 and @xmath336 . this transformation puts the action in ( [ w_1 of x and t ] ) in a diagonal form . doing the @xmath337 and @xmath338 integrals gives @xmath339 \ , , \label{w_1 sum}\ ] ] where we have wick rotated @xmath224 to avoid the poles at @xmath340 . the length @xmath207 is just the classical string energy @xmath341 divided by the string tension @xmath86 , since @xmath341 is @xmath342 the quantity @xmath343 is the length of the string measured in local co - moving coordinates , which are at rest with respect to the string . this is different from the string length @xmath2 in the laboratory frame . we will regulate @xmath194 using the results of lsw . their result for @xmath174 is the following , @xmath344 where @xmath345 is the number of dimensions , @xmath346 is the area of the string worldsheet , @xmath347 is the length of its boundary , and @xmath182 is the scalar curvature of the sheet . the @xmath176 are regulator masses , and the @xmath177 are appropriate coefficients . the final term , @xmath179 , is finite in the limit where the @xmath348 . lsw evaluate the finite term @xmath179 only for a straight string of length @xmath2 with fixed ends . in this case , the area of the sheet @xmath349 , the length of the boundary @xmath350 , and the curvature of the sheet is zero . they then obtained the explicit contribution to the heavy quark potential : @xmath351 the first term in ( [ luscher reg ] ) renormalizes the string tension . the second renormalizes the quark mass . the third is the well known lscher term in the heavy quark potential . since the extrinsic curvature vanishes for a flat sheet , we can identify the result ( [ luscher reg ] ) with our expression ( [ w_1 sum ] ) for @xmath194 , with @xmath207 replaced by @xmath2 : @xmath352 = \frac{1}{2\pi } r \sum_j \epsilon_j { \cal m}_j^2 \ln { \cal m}_j^2 + \sum_j \epsilon_j { \cal m}_j - \frac{\pi}{12 r } \ , . \label{flat eigenvalue sum}\ ] ] eq . ( [ flat eigenvalue sum ] ) tells us how to regulate @xmath194 . replacing @xmath2 by @xmath207 in ( [ flat eigenvalue sum ] ) gives the regulated form of @xmath194 : @xmath353 the first term in ( [ modified luscher ] ) is still a string tension renormalization , since both the string tension contribution to the energy ( [ e sigma ] ) and the first term in eq . ( [ modified luscher ] ) are proportional to @xmath207 , the second term in eq . ( [ modified luscher ] ) is , as before , a renormalization of the quark mass . the finite part of the contribution of @xmath194 to the action is @xmath354 this is the result stated in section [ regularization section ] . the result ( [ luscher r_p appendix ] ) is the lscher term , with the distance @xmath2 between the quarks replaced by the proper length @xmath207 of the string . our result is ( [ r_p appendix ] ) , the derivation of @xmath207 . in this appendix , we show that the divergent part of @xmath226 for a general sheet is proportional to the integral of the scalar curvature @xmath182 . this agrees with the logarithmic divergence ( [ luscher reg ] ) derived by lsw by other means . to obtain the divergent part of @xmath226 , we carry out a fourier transform with respect to the variable @xmath212 . for functions defined on the interval @xmath355 , the @xmath44 function can be expressed as a sum of sines , @xmath356 where @xmath336 . the fourier transform of an operator of the form @xmath357 can then be written @xmath358 using the formula ( [ fourier transform ] ) to evaluate ( [ w_2 given ] ) gives @xmath359 \ , , \label{fourier transformed}\end{aligned}\ ] ] the trace is over indices @xmath360 which run from 1 to 2 , and indices @xmath361 which run from 1 to @xmath236 . the trace is cutoff at @xmath362 , the mass of the vector particle in the original field theory . we expand ( [ fourier transformed ] ) for large @xmath363 and obtain the cutoff dependent part of @xmath226 , @xmath364 the term @xmath365 is equal to minus the scalar curvature @xmath182 , @xmath366 since the equation of motion ( [ class string simple ] ) implies @xmath367 . the cutoff dependent part of @xmath226 is therefore @xmath368 eq . ( [ w_2 divergence ] ) agrees with the result of lsw for the leading semiclassical logarithmic divergence . we want to evaluate @xmath226 , @xmath369 for the fluctuations about a straight string of length @xmath2 rotating with angular velocity @xmath8 . to evaluate this , we must determine the value of the extrinsic curvature @xmath101 . the definition of the extrinsic curvature is @xmath102 the string @xmath370 is @xmath371 the @xmath372 are a basis of orthonormal unit vectors in minkowski space . the @xmath306 are a basis for the vectors normal to @xmath370 . we choose the basis @xmath373 with this choice for the @xmath306 , @xmath374 is zero , because the @xmath375 component of @xmath370 is zero . the only nonzero component of @xmath376 is @xmath377 now that we know what @xmath101 is , we can evaluate ( [ v trace ] ) . inserting ( [ nonzero k ] ) into ( [ v trace ] ) gives @xmath378 the traces in ( [ final rotating potential ] ) are defined as sums over the eigenvalues of the given operators . replacing the traces with explicit sums gives @xmath379 where @xmath380 the eigenvalues @xmath381 are determined by the eigenfunction equation @xmath382 with the boundary conditions @xmath383 . the difference between the traces in eq . ( [ final rotating potential ] ) is logarithmically dependent on the cutoff @xmath76 ( the mass of the dual gluon ) . ( [ eigenfunction equation ] ) has the form of the schredinger equation , with the potential @xmath384 . this potential is an analytic continuation of the potential @xmath385 , whose eigenfunctions can be expressed terms of hypergeometric functions @xcite . using this result , we find the eigenfunctions @xmath386 the eigenvalues @xmath381 are @xmath387 where @xmath388 satisfies the transcendental equation @xmath389 and @xmath390 . there is no @xmath391 eigenvalue , despite the fact that @xmath392 satisfies ( [ alpha def ] ) , because the corresponding eigenvalue @xmath393 makes @xmath394 zero everywhere . we will carry out the sum ( [ sum to do ] ) , @xmath395 by converting it to a contour integral . we will find a function @xmath396 which has zeros whenever @xmath397 . we will find another function @xmath398 which has zeros whenever @xmath399 . we will then define a function @xmath400 , @xmath401 the function @xmath400 has poles of residue @xmath402 when @xmath403 and poles of residue @xmath404 when @xmath399 . we then rewrite the sum ( [ w_2 sum before contour ] ) as a contour integral , @xmath405 the contour of the integral ( [ w_2 contour abstract ] ) lies along the imaginary axis , and on a semicircle at @xmath406 with the real part of @xmath407 positive . to write @xmath226 as the contour integral ( [ w_2 contour abstract ] ) , we need to find the functions @xmath396 and @xmath398 . the function @xmath398 is @xmath408 which is zero for @xmath399 . we find @xmath396 by recalling that the eigenfunctions ( [ eigenfunctions ] ) vanish at @xmath409 . therefore , @xmath410 has zeros at @xmath411 for @xmath412 odd , and @xmath413 has zeros at @xmath411 for @xmath412 even . thus , @xmath414 the factor @xmath415 removes nonphysical zeros which appear because @xmath416 . these zeros correspond to the @xmath391 `` eigenfunction '' which is zero everywhere for @xmath393 . the function @xmath400 is @xmath417 \ , . \label{f int found}\ ] ] inserting ( [ f int found ] ) in ( [ w_2 contour abstract ] ) and integrating by parts gives @xmath418 \ , . \label{contour defined}\ ] ] now , instead of having poles at @xmath411 and @xmath419 , the integrand has branch points at these points . the branch cuts run from @xmath420 to @xmath421 for all @xmath412 along the real axis . there is one branch cut for each value of @xmath412 . since the contour either includes both @xmath420 and @xmath421 or excludes both these points , none of these branch cuts cross the contour of integration . there are no branch points at @xmath422 , since both the numerator and denominator vanish there . none of the branch cuts crosses the contour , so the contour is still closed , and the integration by parts does not produce a boundary term . the contour of the integral ( [ contour defined ] ) lies on the imaginary axis and a semicircle passing through positive real infinity . we rewrite ( [ contour defined ] ) as two integrals over the different pieces of the contour . for large values of the cutoff @xmath76 , the action @xmath226 is @xmath423 \nonumber \\ & & - t \frac{1}{4\pi } \int_{-\pi/2}^{\pi/2 } d\theta \lambda e^{i\theta } \left[-2\frac{\omega}{\lambda } e^{-i\theta } \tan\left(\frac{r_p}{2 } \omega\right ) \cot\left(r_p \lambda e^{i\theta}\right ) + o(\lambda^{-2 } ) \right ] \,.\end{aligned}\ ] ] for large @xmath76 , @xmath424 is proportional to the sign of @xmath270 , so the @xmath270 integral vanishes . the @xmath425 integral is symmetric under @xmath426 . changing variables to @xmath427 gives @xmath428 \ , . \label{s integral}\ ] ] the numerator of in ( [ s integral ] ) is approximately @xmath429 for large values of @xmath430 . we use this fact to extract the divergent part of ( [ s integral ] ) , getting @xmath431 - t \frac{v^2 \gamma}{\pi r } f\left ( v \right ) \ , . \label{s_2 lambda cutoff}\ ] ] we have replaced @xmath207 with its definition @xmath432 @xmath433 is the quark boost factor . the function @xmath250 contains the rest of the integral , @xmath434 \,.\ ] ] for @xmath256 , the asymptotic value of @xmath250 is @xmath435 the cutoff @xmath76 used in ( [ s_2 lambda cutoff ] ) is the cutoff in the @xmath212 coordinate . we must express @xmath76 in terms of the cutoff @xmath77 for the @xmath190 coordinate , which measures physical distance . the cutoffs @xmath76 and @xmath77 are related by the equation @xmath436 , or @xmath437 inserting ( [ lambda of m ] ) into ( [ s_2 lambda cutoff ] ) gives @xmath438 - t \frac{v^2 \gamma}{\pi r } f\left ( v \right ) \,.\ ] ] using the classical equation of motion ( [ boundary condition ] ) then gives @xmath439 - t \frac{v^2 \gamma}{\pi r } f\left ( v \right ) \ , . \label{s_2 final}\ ] ]
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starting from a field theory containing classical vortex solutions , we obtain an effective string theory of these vortices as a path integral over the two transverse degrees of freedom of the string .
we carry out a semiclassical expansion of this effective theory , and use it to obtain corrections to regge trajectories due to string fluctuations .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
cosmic baryogenesis stands as one of the unresolved problems of particle cosmology . most models address baryogenesis after the inflationary epoch . recently the authors of ref . @xcite demonstrated that the baryon asymmetry can be generated during inflation from gravity waves . in this model the lepton number was generated by a quantum expectation value of the chern - simons density from ultra - violet ( uv ) , birefringent gravitational waves during the inflationary epoch . in a subsequent paper the authors showed that this model can be embedded in string theory , in a model independent manner , through the green - schwarz mechanism @xcite . in the stringy embedding there was a huge enhancement of the lepton asymmetry due to a hierarchy in the fundamental string scale and the four dimensional planck scale . other investigators have searched for parity violation in the cosmic microwave background ( cmb ) which ultimately leads to a birefringence in gravitational waves @xcite . in this note , we shall study the super - horizon power spectrum and tensor to scalar ratio of scalar and tensor birefringent perturbations produced during inflation . specifically , we study the spectrum of super - horizon gravity waves whose uv counterparts were responsible for leptogenesis . is it possible to see a signature of the leptogenesis mechanism in the super - horizon power spectrum ? to address this question we shall derive the tensor to scalar ratio and show that it contains a direct signature of the leptogenesis mechanism which occurred in the uv . furthermore we show that the scalar to tensor ratio contains the string scale in a model independent way and is in an observable window to this physics . the paper is organized as follows . in section ii we derive the equations for the gravitational waves in the presence of the chern - simons term . in section iii we provide the exact solutions at various scales and derive the power spectrum as well as the corrected tensor to scalar ratio . in section iv we relate this modified tensor to scalar ratio to the stringy embedding of gravitational leptogenesis and we conclude with some open issues concerning a consistent quantization and further directions . the starting point of inflationary leptogenesis is the einstein - hilbert action coupled to the gravitational chern - simons term , which is necessarily present in string theory . this last term can be written as @xmath3 where @xmath4 , @xmath5 being the planck mass . we proceed to linearize the einstein - hilbert action with the chern - simons term in a friedmann - lematre - robertson - walker ( flrw ) background in the presence of tensor perturbations ( _ i.e . _ in presence of gravitational waves ) . the corresponding metric tensor takes the form ( assuming that the space - like sections are flat ) @xmath6 \ , , \end{aligned}\ ] ] with @xmath7 , being a transverse and traceless tensor , i.e. @xmath8 , @xmath9 and @xmath10 , the flrw scale factor , being a function of the conformal time @xmath11 . due to the symmetries of the flrw metric , the inflaton field @xmath12 in eq . ( [ csaction ] ) is also a function of the conformal time only . expanding the action up to second order in the gravitational waves tensor @xmath7 ( which is necessary in order to obtain first order equations of motion ) , after lengthy but straightforward calculations , one obtains the following expression @xmath13 -f'\epsilon ^{ijk } \biggl[\left(h^q{}_i\right)'\left(\partial _ jh_{kq}\right ) ' -\left(\partial ^rh^q{}_i\right)\partial _ j\partial _ rh_{kq}\biggr]\biggr\}\ , , \end{aligned}\ ] ] where a prime stands for a derivative with respect to conformal time and @xmath14 , @xmath15 being the totally antisymmetric tensor . in the above expression , one recognizes the standard ( _ i.e . _ einstein - hilbert ) expression of the perturbed action ( first term between squared brackets ) while the term proportional to @xmath16 represents the correction coming from the chern - simons contribution . varying this action with respect to the gravitational waves tensor , one obtains the first order equation of motion which reads @xmath17 & = & 0 \ , , \end{aligned}\ ] ] next , following ref . @xcite , we define the tensor @xmath18 by the following equation @xmath19 and , then , the equation of motion takes the form @xmath20=0 \ , , \ ] ] where we have defined @xmath21 . this equation is similar to eqs . ( 11 ) and ( 12 ) of ref . @xcite , except that we have written the equation in terms of the conformal time rather than in terms of the cosmic time . the next step consists in going to the fourier space . for this purpose , we write the metric tensor as @xmath22 in the above expression , @xmath23 is the linear polarization tensor ( @xmath24 corresponds to @xmath25 ) . concretely , if the wave - vector is written in polar coordinates as @xmath26 , then two vectors perpendicular to @xmath27 are given by @xmath28 for the first vector and @xmath29 for the second vector but only if @xmath30 . if @xmath31 , _ i.e. _ if the wave - vector points to the bottom , the expression of the second vector should in fact read @xmath32 . it is also interesting to mention how these quantities transform under the change @xmath33 . it is easy to see that this corresponds to the transformation @xmath34 . then , we have @xmath35 and @xmath36 . finally , the polarization tensor can be written as @xmath37 due to the properties of the vectors @xmath38 and @xmath39 established above , it is easy to check that @xmath40 and @xmath41 . using these properties and the fact that @xmath7 is real , @xmath42 , one can also establish that @xmath43 the next step consists in defining two other states of polarization , the so - called right and left polarization states . the corresponding polarization tensors are given by @xmath44 from the above expressions , using the properties of the linear polarization tensors , one can show that @xmath45 these expressions are of course valid only if the polarization tensors are evaluated for the same wave - number . we also have @xmath46 with @xmath47 . then , using the expression of the vectors @xmath48 and @xmath49 , it is easy to show that @xmath50 where @xmath51 and @xmath52 and where the upper sign refers to @xmath30 while the lower one refers to @xmath31 . we are now in a position where one can re - write the gravitational waves tensor in terms of the left and right polarization states . this gives @xmath53 where we have introduced the definitions @xmath54 let us notice that it is straightforward to demonstrate that @xmath55 and @xmath56 . the next step consists in introducing the new expansion of the gravitational waves tensor given by eq . ( [ gwpolalr ] ) into the equation of motion and in using eq . ( [ eigenpola ] ) to arrive at a new form of the equation of motion . one obtains @xmath57 finally , the last step consists in introducing the quantity @xmath58 defined by @xmath59 and the new amplitude @xmath60 defined by @xmath61 . then , the equation of motion for @xmath62 has the traditional form of the equation of motion for a parametric oscillator , namely @xmath63 the effective potential @xmath64 depends on time , on polarization ( birefringence ) but also on the wave - number which is an important difference with respect to the standard case where the effective potential depends on conformal time only . this equation has been derived for the first time in ref . @xcite ; see eq . ( 15 ) of that paper . however , in ref . @xcite , it is also assumed that the effective potential takes the form @xmath65 where @xmath66 is a constant . in particular , one notices that , with this ansatz , the scale dependence of the effective potential has disappeared . this permits to find simple solutions in terms of bessel functions . however , we will see that , in the present context , the effective potential is different and more complicated . let us now calculate the effective potential explicitly . using the formulas established previously , one obtains that the exact expression of the potential can be written as @xmath67 ^ 2}{\left[1-\lambda ^sk\left(f'/a^2\right)\right]^2}\ , .\end{aligned}\ ] ] to go further , we need to postulate the function @xmath68 . following ref . @xcite , we choose @xmath69 where @xmath70 is the reduced planck mass and @xmath2 a number that we discuss in more details in the last section and that can be related to the string scale . with this definition @xmath71 is dimensionless , as it should , if the scale factor has the dimension of a length ( which is our convention ) . in terms of the slow - roll parameters @xmath72 , @xmath73 and @xmath74 ( a dot means a derivative with respect to cosmic time ) , we have at leading order in the slow - roll parameters , see also ref . @xcite @xmath75 from this expression , one deduces that ( at leading order in the slow - roll parameters ) @xmath76 because @xmath77 . from that expression , one arrives at the two following formulas which are useful in order to calculate the effective potential @xmath78 and @xmath79 inserting the above equations into the formula giving the potential , namely eq . ( [ poteff ] ) , one obtains @xmath80^{-1 } \nonumber \\ & & -\frac{k^2}{4}\frac{{\cal n}^2}{256\pi ^4}\left(\frac{h_{_{\rm inf}}}{m_{_{\rm pl}}}\right)^4(2\epsilon ) \left[1-\lambda ^s\frac{{\cal n}}{16\pi ^2 } \left(\frac{h_{_{\rm inf}}}{m_{_{\rm pl}}}\right)^2 \sqrt{2\epsilon } \times ( k\eta ) \right]^{-2}\ , , \end{aligned}\ ] ] where we have ignored sub - dominant term in the slow - roll parameters . it is important to notice that , in order to obtain the above equation , we have never expanded a term like @xmath81 in the slow - roll parameters . we notice the presence of @xmath82 in the numerator of the second term . this is in full agreement with ref . @xcite where it has been noticed that a term like @xmath83 in the effective potential necessarily implies a new characteristic scale . here , the characterized scale defined in ref . @xcite could be written as ( at this level , the two situations are not yet totally equivalent because the above effective potential is not exactly similar to the potential studied in ref . this will be the case below . ) now let @xmath84 where we have defined @xmath85 by the following relation . ( see eq . ( 13 ) in ref . @xcite ) : @xmath86 in the present context , somehow , the characteristic scale @xmath87 `` depends on the scale '' ( _ i.e . _ on @xmath82 ) . however , we see that the large - scale limit , as defined in ref . @xcite _ i.e. _ @xmath88 , corresponds in the present context to the condition @xmath89 . the only way to satisfy this condition is to have a large @xmath2 which could compensate the smallness of @xmath90 and of the slow - roll parameter . for convenience , we now introduce the variable @xmath91 defined by @xmath92 . then , the equation of motion takes the form @xmath93\mu = 0\ , , \ ] ] with @xmath94 the functions @xmath95 and @xmath96 are represented in fig . [ pot ] . from this figure , the different behavior of the two states of polarization is apparent . the l mode ( dashed line ) undergoes a `` kick '' at @xmath97 where the effective potential blows up . at the same point the potential of the r mode is perfectly regular ( solid line ) . therefore , we expect the r mode function to propagate smoothly through @xmath97 while the behavior of the l mode function can be more problematic . we now turn to this question in more details . let now us study the equation of motion for the left mode in the vicinity of @xmath98 . it is easy to check that a very good approximation of the potential is @xmath99 in fact the approximation is good even far form @xmath98 provided @xmath100 since , on small scales , _ i.e. _ in the limit @xmath101 , we have @xmath102 . in the limit , the solution can be written as @xmath103 where @xmath104 and @xmath105 are two constants that are fixed by the choice of the initial conditions . usually , one requires that , on sub - hubble scales @xmath106 this prescription completely fixes the coefficients @xmath104 and @xmath105 which read @xmath107 in the above equation , @xmath108 is the planck length and @xmath109 is some initial time at the beginning of inflation . the knowledge of this time is not important since it will drop out from the final result . with the potential given by eq . ( [ approxpotdiv ] ) , the equation of motion can be solved exactly . indeed , if we define @xmath110 then the equation of motion takes the form @xmath111\mu _ { \mathbf k}^{_{\rm l}}=0\ , .\ ] ] this is the well - known whittaker equation , see eq . ( 9.220.1 ) of ref . the corresponding solution , correctly normalized , see eqs . ( [ cidiv ] ) , reads @xmath112\ , , \ ] ] where @xmath113 is the whittaker function . let us now study how the mode function behaves when @xmath114 . the whittaker function can be expressed in terms of the confluent hypergeometric function , see eq . ( 13.1.33 ) of ref . one obtains @xmath115\ , , \end{aligned}\ ] ] where @xmath116 is the above - mentioned confluent hypergeometric function . using eq . ( 13.5.9 ) of ref . @xcite which says that , when @xmath117 , @xmath118/\gamma ( a)$ ] , where @xmath119 is the euler s integral of the second kind and where @xmath120 , see ref . @xcite , one deduces that @xmath121 but what really matters is not the intermediate variable @xmath122 but in fact the amplitude of the gravitational waves itself given by @xmath123 , see eq . ( [ defz ] ) . since @xmath124 , one obtains @xmath125 the conclusion is that the amplitude of the mode @xmath126 blows up at the time corresponding to @xmath97 , that is to say at the time @xmath127 defined by @xmath128 at this point the linear theory of cosmological perturbations breaks down and becomes non linear . an important feature of @xmath129 is that it is scale dependent . this means that the physical wavelength of the fourier modes @xmath130 , at time @xmath131 , are all equal to the same physical length . explicitly , one has @xmath132 somehow , this is reminiscent of one of the possible formulations of the trans - planckian problem of inflation @xcite where it is postulated that a mode of comoving wavenumber @xmath82 is `` created '' when its physical wavelength equals a given new fundamental scale in the theory ( the idea being to test the robustness of the inflationary predictions to short distance modifications of the theory ; therefore , it is typical in this context to consider that the new scale is the planck length , see refs . @xcite for more details ) . it is then easy to show that the `` time of creation '' is inversely proportional to @xmath82 as it is the case for @xmath129 , see in particular the fifth paper in ref . @xcite . as a consequence , we see from eq . ( [ lengththeta ] ) that @xmath85 defines in fact a new scale the value of which depends on the inflation scale but also on the string scale since we will see that the string scale is hidden into the number @xmath2 which participates to the definition of @xmath85 , see eq . ( [ deftheta ] ) . a possible way out to the question of the divergence would be to push the problem in the trans - planckian regime . from the above equation , this means that the parameter @xmath133 should satisfy @xmath134 therefore , this boils down to a quite stringent constraint on @xmath133 , typically @xmath135 . unfortunately , we will see in the following that , for such small values of @xmath85 , the modifications on large scales are not observable . if one wants to consider larger values of @xmath85 , it seems that much more refined ( _ i.e . _ non - linear ) calculations are necessary . this calculation are obviously beyond the scope of the present paper which is just exploratory . let us conclude this subsection by stressing the fact that , a priori , trans - planckian effects do not play a deep role in the chern - simons theory under considerations in this work . here , it is merely a technical trick which allows to avoid the non - linear regime and to adopt the common assumption that the fourier modes emerge from the trans - planckian region in the vacuum state . but clearly , if @xmath136 , then the non - linear calculation is in principle feasible without any trans - planckian considerations . let us now study what happens on very large scales , _ i.e. _ in the limit where @xmath91 vanishes . in the situation , the effective potentials can be very well approximated by the following equations @xmath137 where we remind that @xmath51 and @xmath52 . birefringence enters this equation via the term proportional to @xmath138 which changes its sign according to the polarization state under considerations . the term proportional to @xmath139 is the standard slow - roll term . the corresponding equation of motion takes the form @xmath140\mu _ { \mathbf k}^s=0\ , .\ ] ] once again , we have to deal with a whittaker equation . in fact this equation ( and the corresponding power spectrum ) has been studied in details in ref . @xcite , see eq . ( 8) , and the corresponding power spectrum has been derived in that reference , see eq . therefore , in the present paper we can use the results obtained in ref . @xcite and follow the procedure utilized in that reference . let us introduce the new definitions @xmath141 in the following , we will consider that @xmath142 ( see the discussion at the end of the previous subsection ) and , as a consequence , will simply approximate @xmath143 by @xmath144 . with the new definitions taken into account , the equation of motion takes the form @xmath145\mu _ { \mathbf k}^s=0\ , , \ ] ] which is again the whittaker equation . the situation is exactly similar to the one studied around eq . ( 12 ) of ref . the exact general solution to this equation is given in terms of whittaker functions @xmath146 where @xmath147 and @xmath148 are two constants fixed by the choice of the initial conditions . as discussed in the preceding subsection , we will fix the initial conditions in the region @xmath149 which is free of divergences . in this regime , the only natural choice that we have is to postulate a plane wave . this is equivalent to postulating that the non - linear phenomena occurring around the divergence of the effective potential , provided they happen in the trans - planckian region , will not affect the standard choice of the initial conditions in the region @xmath150 . somehow , this is the same assumption that is made in the standard inflationary scenario . indeed , despite the fact that the modes of astrophysical interest today originate from the trans - planckian region @xcite , the vacuum is assumed to be the correct initial state . let us stress , however , that a possible weakness of the above comparison is that , in the case of the trans - planckian problem of inflation @xcite , one can show that the final result can be robust to changes in the short distance physics @xcite ( under some conditions , _ i.e. _ adiabatic evolution of the fourier modes ) . in the present context , however , it is more difficult to imagine that the non - linearities will not affect the initial conditions . on the other hand , in the absence of a second - order calculations and as a first approach to the problem , this seems to be quite reasonable . as shown in ref . @xcite , see eqs . ( 14 ) , this choice amounts to @xmath151 where we have used the fact that , see eq . ( 9.227 ) of ref . @xcite , @xmath152 . the sign of the argument in the exponential depends on the polarization state considered , as expected . we conclude that the solution to the mode equation on very large scales is now known explicitly . usually , the power spectrum is given by the two - point correlation function calculated in the vacuum state . another way to calculate the same quantity is to view it as a classical spatial average . since a fully consistent quantum formulation of the present theory is not yet available , we adopt the second point of view . therefore , the two - point correlation function can be written as @xmath153 with @xmath154 is the total volume . using the properties of the polarization tensor , straightforward calculations show that @xmath155 from which we deduce the power spectrum @xmath156 let us notice that , usually , the power spectrum is proportional to the factor @xmath157 . here we do nt have the factor @xmath158 because we consider the two states of polarization separately ( _ i.e . _ usually , these two states are summed and produce the factor @xmath158 ) . a priori , using the solution obtained in the previous subsection , we can calculate the spectrum exactly in terms of the whittaker function . but only the spectrum on large scales is needed and in this regime one has ( for details , see ref . @xcite ) @xmath159 let us notice that we have neglected the factor @xmath160 because @xmath161 is proportional to @xmath162 and hence negligible on large scales . the above expression is similar to eq . ( 15 ) of ref . @xcite . at this stage , the only thing which remains to be done is to expand the above expression at first order in the slow - roll parameter . after lengthy but straightforward calculations , one obtains the following result @xmath163 \ , , \end{aligned}\ ] ] with , @xmath164 at this point some remarks are in order . as required one can check that , when @xmath165 , the standard inflationary result is recovered . this is the case because @xmath166 and @xmath167 . as already mentioned , a factor @xmath168 is left because the ( now identical ) contribution from the two states of polarization should be added . the function @xmath169 describes the dominant modification in the amplitude of the power spectrum ( the contribution originating from the function @xmath170 is clearly sub - dominant since it is proportional to the slow - roll parameter @xmath171 ) . for small values of @xmath85 we have @xmath172 where @xmath173 and @xmath174 . therefore , the amplitude of the right polarization state is reduced while the one of the left polarization state is enhanced . however , for small values of @xmath85 , the effect is clearly not very important . another conclusion that can be obtained from the above spectrum is that , at leading order in the slow roll parameter , the spectral index remains unmodified . indeed , one has @xmath175 for each polarization state . finally , let us now compute how the ratio @xmath0 is modified . the scalar power spectrum is not modified ( see also ref . @xcite ) and reads @xcite @xmath176\ , .\ ] ] therefore , we conclude that the consistency check of inflation , at leading order in the slow - roll parameters , can now be written as @xmath177 \\ & \simeq & 16\epsilon \times \left[1+\left(\frac{\pi ^2}{384}-\frac{1}{256}\right)\theta ^2\right]\ , .\end{aligned}\ ] ] unfortunately , the linear corrections in @xmath85 cancels out and we are left with a correction which is quadratic in @xmath85 . another way to express the above result is to calculate the ratio of @xmath0 with the chern - simon modification taken into account to @xmath0 obtained in the standard case . one gets @xmath178 it is clear from this expression that the modification is not observable at all since we have seen before that , typically , @xmath179 in order for the calculations presented here to be consistent ( _ i.e . _ for the divergence of the effective potential to be in the trans - planckian region ) . we have evaluated the super - hubble power spectrum and the tensor to scalar ratio for birefringent gravitational waves produced during inflation . the power spectrum exhibits two interesting regimes , linear and non - linear . the non - linear regime occurs when @xmath180 because the effective potential controlling the evolution of the linear perturbations blows up . at this point , the linear theory of cosmological perturbations that we used is no longer valid . this divergence occurs for all modes ( _ i.e . _ for all comoving wavenumber @xmath82 ) but at different times . in this present investigation we only considered the linear regime since at the present moment we were not able to perform a rigorous analysis of the non - linear phenonema we found corrections which survive to second order in @xmath181 . therefore , in this regime the tensor to scalar ratio gets corrected by @xmath181 but this effect is very small . if @xmath135 , one can push the non - linear regime ( _ i.e . _ the divergence in the effective potential ) into the trans - planckian region where , anyway , other effects ( for instance , non perturbative stringy effects ) are likely to become important . somehow , this corresponds to the standard situation where the evolution from the planck scale to the super - horizon scales is under control and where the perturbations are assumed to emerge from the trans - planckian regime in the vacuum state , thus ignoring the modifications of the initial conditions that the trans - planckian physics could cause ( in the very same way that we have ignored the effect of the divergence in the potential , provided it is in the trans - planckian region ) . however , it is important to keep in mind that this is mostly a technical trick which allows us to work with the linear theory . at a deeper level , the trans - planckian effects are not expected to play a more important role than in the standard situation . in particular , if the divergence is not in the trans - planckian regime , only the non - linear theory of cosmological perturbations is necessary in order to calculate the modified @xmath0 irrespectively of any trans - planckian effects . it is interesting to note that the linear regime ( where @xmath182 ) is compatible with the stringy embedding of inflationary baryogenesis ( seb ) @xcite . in this context , the value of @xmath181 enhances and gives the resonant frequency associated with the observed baryon asymmetry . as already mentioned before , this value is completely fixed by the string scale and coupling in a model independent fashion . explicitly , the value of the number @xmath2 which appears in the definition of @xmath85 , see eq . ( [ deftheta ] ) , is given by @xmath183 where @xmath184 is the ten - dimensional fundamental scale and @xmath185 is the string coupling . therefore , we established a direct link between stringy quantities and cmb anisotropies . explicitly , eq . ( [ cccheck ] ) can be re - written as @xmath186 in a recent paper the authors of ref . @xcite found that for reasonable values of string coupling ( _ i.e . _ weak ) and the string scale set to @xmath187 , both @xmath85 can be as small as @xmath188 and the observed baryon asymmetry can be generated . of course the stringy embedding admits much larger values of @xmath181 putting our analysis into the non - linear regime . if @xmath189 , a non - linear calculation is mandatory and one can hope to obtain a significative modification of the ratio @xmath0 , maybe observable by future high accuracy cmb experiments . clearly , the present situation is not very satisfactory since the regime for which sizable effects are expected turns out to be very complicated from the technical point of view . furthermore , we suspect that the scale associated to the divergence of the effective potential corresponds to a resonant production of lepton number . we wish to report on this issue in a future paper . m. lemoine , m. lubo , j. martin and j. p. uzan , phys . d * 65 * , 023510 ( 2002 ) , hep - th/0109128 ; j. c. niemeyer , phys . d * 63 * , 123502 ( 2001 ) , astro - ph/0005533 ; a. kempf , phys . d * 63 * , 083514 ( 2001 ) , astro - ph/0009209 ; r. easther , b. r. greene , w. h. kinney and g. shiu , phys . d * 64 * , 103502 ( 2001 ) , hep - th/0104102 ; j. martin and r. h. brandenberger , phys . d * 68 * , 063513 ( 2003 ) , hep - th/0305161 . j. martin and c. ringeval , phys . d * 69 * , 083515 ( 2004 ) , astro - ph/0310382 ; j. martin and c. ringeval , phys . d * 69 * , astro - ph/0402609 ; j. martin and c. ringeval , astro - ph/0405249 .
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in this work we show that the gravitational chern - simons term , aside from being a key ingredient in inflationary baryogenesis , modifies super - horizon gravitational waves produced during inflation .
we compute the super - hubble gravitational power spectrum in the slow - roll approximation and show that its overall amplitude is modified while its spectral index remains unchanged ( at leading order in the slow - roll parameters ) .
then , we calculate the correction to the tensor to scalar ratio , @xmath0 . we find a correction of @xmath0 which is dependent on @xmath1 ( more precisely quadratic in @xmath2 ) , the parameter characterizing the amplitude of the chern - simons terms . in a stringy embedding of the leptogenesis mechanism , @xmath1 is the ratio between the planck scale and the fundamental string scale .
thus , in principle , we provide a direct probe of leptogenesis due to stringy dynamics in the cosmic microwave background ( cmb ) .
however , we demonstrate that the corresponding correction of @xmath0 is in fact very small and not observable in the regime where our calculations are valid . to obtain a sizable effect
, we argue that a non - linear calculation is necessary .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
among the heavy quarkona , the @xmath15 meson being the unique meson with two different heavy quarks in the standard model has aroused great interest since its discovery by the cdf collaboration @xcite . although the direct " hadronic production of the @xmath15 meson has been systematically studied in refs . @xcite , as a compensation to understand the production mechanism and the @xmath15 meson properties , it would be helpful to study its production indirectly " through @xmath20(@xmath21)-quark , @xmath22-boson , and @xmath23-boson decays , as too many directly produced @xmath15 events shall be cut off by the trigging condition @xcite . a systematical study on the indirect production of @xmath15 mesons through @xmath20(@xmath21)-quark , @xmath22-boson , and @xmath23-boson decay can be found in refs . @xcite , refs . @xcite , and refs . @xcite , respectively . meanwhile , it has been found that the higher excited states like @xmath24 and @xmath25 wave states can provide sizable contributions in the @xmath15 meson s indirect production through @xmath26-boson decay in ref . @xcite ; one should take these higher fock states contributions into account so as to make a better estimation with other indirect production mechanisms . therefore , to present a systematic study on higher fock states indirect production of @xmath15 meson indirect production through top quark decays under the nonrelativistic quantum chromodynamics framework ( nrqcd ) @xcite is one of the purposes of the present paper . with the lhc running at the center - of - mass energy @xmath13 tev and luminosity raising up to @xmath12 , about @xmath27 @xmath20-quark or @xmath21-quark events per year will be produced @xcite . this makes the lhc much better than tevatron , since more @xmath28-quark rare decays can be adopted for precise studies . a systematic study on the production of the @xmath29 or @xmath15 meson and its excited states via @xmath20-quark or @xmath21-quark decays , e.g. @xmath30\rangle + cw^{+}$ ] or @xmath31\rangle + \bar{c}w^{-}$ ] , can be found in refs . @xcite , where @xmath32 wave states . their results show that large number of heavy quarkonium events through top quark decays can be found at lhc(slhc , dlhc , and tlhc , etc . @xcite ) , so these channels shall be helpful for studying heavy quarkonium properties . in addition to the production of the two color - singlet @xmath33 wave states @xmath34 and @xmath35 , naive nrqcd scaling rule shows that the production of the four color - singlet @xmath36-wave states @xmath37 and @xmath38 ( with @xmath39 ; @xmath40 ) shall also give sizable contributions in @xmath41\rangle+w^{+}q$ ] , where @xmath8 stands for @xmath9 or @xmath10 quark accordingly . these higher excited @xmath42\rangle$ ] quarkonium states , may directly or indirectly ( in a cascade way ) decay to its ground state with almost 100% possibility via electromagnetic or hadronic interactions . so , it would be interesting to study higher fock states contributions so as to make a more sound estimation of the production of the heavy quarkonium through top quark decays , and hence to be a useful reference for experimental studies . in the framework of an effective theory of the nrqcd , a doubly heavy meson system is considered as an expansion of various fock states . the relative importance among those infinite ingredients is evaluated by the velocity scaling rule . in evaluating the production and decay amplitude of the heavy quarkonium , each factor can be separated into a short - distance factor and a long - distance coefficient . the short - distance factor can be computed using perturbative quantum chromodynamics ( pqcd ) , and the long - distance factor is associated with quarkonia structure , which is expressed in terms of nonperturbative matrix elements @xmath43 . the matrix elements @xmath44 can be expressed in terms of the meson s nonrelativistic wave function , or its derivatives , evaluated at the origin under the color - singlet model @xcite , and the wave function is identified with the schr@xmath0dinger wave function calculated in potential models for heavy quarkonium . as a result , the rigorously calculated nonrelativistic wave function at the origin is very important in studying heavy quarkonium decay and production . because of the emergence of massive fermion lines in @xmath41\rangle+w^{+}q$ ] , the analytical expression for the squared amplitude becomes too complex and lengthy . for such complicated processes , one important way out is to deal with it directly at the amplitude level . for this purpose , the improved trace technology " suggested and developed by refs . @xcite shows that the hard scattering amplitude can also be expressed by the dot products of the concerned particle momenta like that of the squared amplitudes . in the present paper , we shall adopt improved trace technology to derive the analytical expression for all the mentioned fock states , and to be a useful reference , we shall transform its form to be as compact as possible by fully applying symmetries and relations among them . this paper is organized as follows : in sec . ii , we show our calculation techniques for the mentioned top quark semi exclusive decays to heavy quarkonium . in order to calculate the production of the excited heavy quarkonium via top quark decays , we present five qcd - motivated potential models for heavy @xmath45\rangle$ ] quarkonium in sec.iii . in sec.iv , we calculate and tabulate all the values of the schr@xmath0dinger radial wave functions , its first nonvanishing derivative , and its second nonvanishing derivative at zero quark - antiquark separation . furthermore , we also present numerical results and make some discussions on the properties of the heavy quarkonium production through top quark decays . the final section is reserved for a summary . we shall deal with some typical top quark semiexclusive processes for the heavy quarkonium production , i.e. , @xmath46\rangle(q_3 ) + w^{+}(q_2 ) + q(q_1)$ ] , where @xmath47 ( @xmath48 ) are the momenta of the corresponding particles . according to the nrqcd factorization formula @xcite , its total decay widths @xmath49 can be factorized as @xmath50\rangle+ q+w^{+ } ) \langle{\cal o}^h(n ) \rangle,\ ] ] where the nonperturbative matrix element @xmath51 describes the hadronization of a @xmath52 pair into the observable quark state @xmath53 and is proportional to the transition probability of the perturbative state @xmath52 into the bound state @xmath42\rangle$ ] . as for the color - singlet components , the matrix elements can be directly related to the schr@xmath0dinger wave functions at the origin for the @xmath33 wave states , the first derivative of the wave functions at the origin for the @xmath36 wave states , or the second derivative of the wave functions at the origin for the @xmath54 wave states @xcite , which can be computed via potential nrqcd ( pnrqcd ) @xcite , lattice qcd @xcite and/or the potential models @xcite . the short - distance decay widths are given by @xmath55\rangle+q+w^{+})= \frac{1}{2 q^0_{0 } } \overline{\sum } |m|^{2 } d\phi_3,\ ] ] where @xmath56 means that we need to average over the spin and color states of the initial particle and to sum over the color and spin of all the final particles . in the top quark rest frame , the three - particle phase space can be written as @xmath57 we have done a calculation to simplify the @xmath58 phase space with massive quark / antiqark in the final state in refs . @xcite . to shorten the paper , we shall not present it here and the interested reader may turn to these three references for the detailed technology . with the help of the formulas listed in refs . @xcite , one can not only derive the whole decay widths but also obtain the corresponding differential decay widths that are helpful for experimental studies , such as @xmath59 , @xmath60 , @xmath61 , and @xmath62 , where @xmath63 , @xmath64 , @xmath65 is the angle between @xmath66 and @xmath67 , and @xmath68 is the angle between @xmath66 and @xmath69 . in particular , the partial decay widths over @xmath70 and @xmath71 can be expressed as @xmath72 where @xmath73 is the mass of the top quark . the color - singlet nonperturbative matrix element @xmath44 can be related to the schr@xmath0dinger wave function @xmath74 at the origin or the first derivative of the wave function @xmath75 at the origin for @xmath33- and @xmath36-wave quarkonium states . for convenience , we have adopted the convention of refs . @xcite for the nonperturbative matrix element . @xmath76\rangle}(0)|^2,\nonumber\\ \langle{\cal o}^h(np ) \rangle & \simeq & |\psi^\prime_{\mid(b\bar{q})[np]\rangle}(0)|^2.\end{aligned}\ ] ] since the spin - splitting effects are small , we will not distinguish the difference between the wave function parameters for the spin - singlet and spin - triplet states at the same @xmath77th level . and then our task is to deal with the hard - scattering amplitude for specified processes @xmath78\rangle + cw^{+},~~t\rightarrow |(b\bar{b})[n]\rangle + bw^{+}.\end{aligned}\ ] ] for convenience , we shorten the two processes as @xmath79\rangle(q_3 ) + w^{+}(q_2)+q(q_1)$ ] , where @xmath8 stands for @xmath9 or @xmath10 quark accordingly . the feynman diagrams of the process are presented in fig . [ feyn1 ] , where the intermediate gluon should be hard enough to produce a @xmath80 pair or @xmath81 pair , so the amplitude is pqcd calculable . \rangle(q_3 ) + w^{+}(q_2)+ q(q_1)$ ] , where @xmath8 stands for the @xmath9 and @xmath10 quark in the left and right panels , respectively . @xmath42\rangle$ ] quarkonium stands for a heavy quarkonium fock state.,scaledwidth=45.0% ] these amplitudes can be generally expressed as @xmath82 where @xmath83 stands for the number of feynman diagrams , @xmath84 and @xmath85 are spin states , and @xmath86 and @xmath87 are color indices for the outing @xmath8 quark and the initial top quark , respectively . the overall factor @xmath88 , here @xmath89 is the cabibbo - kobayashi - maskawa ( ckm ) matrix element . @xmath90s in the formulas are listed in ref . @xcite . as mentioned above , we adopt the improved trace technology to simplify the amplitudes @xmath91 at the amplitude level . in a difference from the helicity amplitude approach @xcite , only the coefficients of the basic lorentz structures are numerical at the amplitude level . however , by using the improved trace technology in refs . @xcite , one can sequentially obtain the squared amplitudes , and the numerical efficiency can also be greatly improved . the standard procedures of the improved trace technology for @xmath79\rangle(q_3 ) + w^{+}(q_2)+q(q_1)$ ] have been presented in ref . nonperturbative matrix elements @xmath44 can be related to the wave function at the origin @xcite . in the rest frame of the @xmath45\rangle$ ] quarkonium , it is convenient to separate the schr@xmath0dinger wave function into radial and angular pieces as @xmath92 where @xmath77 is the principal quantum number , and @xmath93 and @xmath83 are the orbital angular momentum quantum number and its projection . @xmath94 and @xmath95 are the radial wave function and the spherical harmonic function accordingly . further on , the value of the radial wave function , its first nonvanishing derivative or its second nonvanishing derivative at the origin can be obtained as in @xcite @xmath96 where @xmath97 , @xmath98 , and @xmath99 correspond to the radial wave functions @xmath100\rangle}(0)$ ] , @xmath101\rangle}(0)$ ] , and @xmath102\rangle}(0)$ ] at the origin . the wave function @xmath103\rangle}(0)$ ] , the first derivative of the wave function @xmath104\rangle}(0)$ ] , and the second derivative of the wave function @xmath105\rangle}(0)$ ] at the origin are related to the radial wave function @xmath100\rangle}(0)$ ] , the first derivative of the radial wave function @xmath101\rangle}(0)$ ] , and the second derivative of the radial wave function @xmath102\rangle}(0)$ ] at the origin , accordingly . @xmath106\rangle}(0)&=&\sqrt{{1}/{4\pi}}r_{|(q\bar{q}')[ns]\rangle}(0),\nonumber\\ \psi'_{|(q\bar{q'})[np]\rangle}(0)&=&\sqrt{{3}/{4\pi}}r'_{|(q\bar{q}')[np]\rangle}(0),\nonumber\\ \psi{''}_{|(q\bar{q'})[nd]\rangle}(0)&=&\sqrt{{5}/{16\pi}}r{''}_{|(q\bar{q}')[nd]\rangle}(0).\end{aligned}\ ] ] next , we will give a brief introduction to the five qcd - motivated potentials that give reasonable accounts of the @xmath107\rangle$ ] , @xmath108\rangle$ ] , and @xmath3\rangle$ ] quarkonium . \(1 ) buchm@xmath109ller and tye have given the qcd - motivated potential ( b.t . potential ) with two - loop correction @xcite as @xmath110 and in which @xmath112 , with @xmath113 , is the regge slope . @xmath114 , @xmath115 , where here @xmath116 is the number of active flavor quarks . @xmath117 stands for the scale parameters , and @xmath118 is the modified minimal subtraction scheme . the parameter @xmath119 can be expressed in terms of the string constant @xmath120 . and @xmath121\sin(\frac{q}{\lambda}~r),\ ] ] with @xmath122},\ ] ] where @xmath123 , and @xmath124 is the euler constant . @xmath125 is a physical quantity and therefore independent of the choice of gauge and the subtraction scheme . for small values of @xmath126 , it has the form of @xmath127 for large @xmath126 , perturbative qcd implies @xmath128 here @xmath129 is the transfer momentum in the rest frame of the @xmath107\rangle$ ] , @xmath108\rangle$ ] , and @xmath3\rangle$ ] quarkonium . \(2 ) the qcd - motivated potential with one - loop correction is given by john l. richardson ( j. potential ) @xcite as @xmath130,\end{aligned}\ ] ] where @xmath131.\ ] ] \(3 ) the qcd - motivated potential with two - loop correction is given by k. igi and s. ono ( i.o . potential ) @xcite as @xmath132,\end{aligned}\ ] ] with @xmath133,\ ] ] in which @xmath134 , @xmath135 , where @xmath136 has the same meaning of the first potential model , and @xmath137 or @xmath138 is the mass of the heavy quark @xmath8 or @xmath139 , accordingly . \(4 ) the qcd - motivated potential with two loop correction is given by yu - qi chen and yu - ping kuang ( c.k . potential ) as @xcite @xmath140,\end{aligned}\ ] ] with @xmath141 and @xmath142}{\lambda_{\overline{ms}}~ r},\ ] ] where @xmath143 . \(5 ) the qcd - motivated coulomb - plus - linear potential ( cor . potential ) @xcite has the form of @xmath144 for calculating the wave function at the origin of the five potentials @xcite , we adopt the scale parameters @xmath117 as @xmath145=0.386 gev , @xmath146=0.332 gev , @xmath147=0.231 gev , @xmath148=0.0938 gev @xcite . the quark mass is adopted as the values of the constituent quark mass of the @xmath149\rangle$ ] quarkonium derived in refs . the quantities @xmath150\rangle}(0)|^2 $ ] , @xmath151\rangle}(0)|^2 $ ] , and @xmath152\rangle}(0)|^2 $ ] are presented in tables [ tabrpa ] , [ tabrpb ] , and [ tabrpc ] for the five potential models . during the following calculation , we adopt the values of wave functions at the origin under the b.t . potential as the central values for calculations of the decay widths of @xmath153\rangle + qw^{+}$ ] , [ @xmath136=3 is for @xmath108\rangle$ ] quarkonium and @xmath136=4 for @xmath3\rangle$ ] quarkonium ] , since it is noted that the b.t . model potential has the correction of two - loop short - distance behavior in pqcd @xcite . the results for the other four potential models , i.e. , the j. model @xcite , the i.o . model @xcite , the c.k . model @xcite , and the cor . model @xcite , will be adopted as an error analysis . .radial wave functions at the origin and related quantities for @xmath107\rangle$ ] quarkonium . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] [ tabrpf ] in the present paper , we only calculate and discuss the decay widths of @xmath24 and @xmath25 wave of @xmath108\rangle$ ] and @xmath3\rangle$ ] quarkonium via the top - quark decays under the five potential models . yet we believe that the values of the wave functions at the origin of @xmath24 , @xmath25 , and @xmath154 wave of @xmath155 , @xmath156 , @xmath157 in tables [ tabrpa ] , [ tabrpb ] , and [ tabrpc ] under the five potential models are helpful for both theoretical and experimental study . in the present paper , we have calculated the values of the schr@xmath0dinger radial wave function at the origin of @xmath107\rangle$ ] , @xmath108\rangle$ ] , and @xmath3\rangle$ ] quarkonium for the five potential models , and made a detailed study on the higher excited heavy quarkonium production through top quark semiexclusive decays , i.e. , @xmath158\rangle + cw^{+}$ ] and @xmath159\rangle + bw^{+}$ ] , within the nrqcd framework . results for @xmath42\rangle$ ] quarkonium fock states , i.e. , @xmath160\rangle$ ] and @xmath161\rangle$ ] , and @xmath162\rangle$ ] and @xmath163\rangle$ ] ( @xmath164 ) have been presented . and to provide the analytical expressions as simply as possible , we have adopted the ` improved trace technology ' developed in refs . @xcite to derive lorentz- invariant expressions for top quark decay processes at the amplitude level . such a calculation technology shall be very helpful for dealing with processes with massive spinors . numerical results show that higher @xmath24 and @xmath25 wave states in addition to the ground @xmath165 wave states can also provide sizable contributions to heavy quarkonium production through top quark decays , so one needs to take the higher @xmath24 and @xmath25 wave states into consideration for a sound estimation . if all the excited states decay to the ground state @xmath166\rangle$ ] with @xmath167 efficiency , we can obtain the total decay width for @xmath168 quarkonium production through top quark decays as shown by eqs . ( [ bct ] ) and ( [ bbt1 ] ) . at the lhc , due to its high collision energy and high luminosity , sizable heavy quarkonium events can be produced through top quark decays , i.e. , @xmath169 @xmath170 quarkonium events and @xmath171 @xmath172 bottomonium events per year can be obtained . therefore we need to take these higher excited states into consideration for a sound estimation . ( cdf collaboration ) , phys . d * 58 * , 112004 ( 1998 ) ; a. abulencia _ ( cdf collaboration ) , phys . . lett . * 96 * , 082002 ( 2006 ) ; a. abulencia _ et al . _ , cdf collaboration , phys . 97 * , 012002 ( 2006 ) . c.h . chang and y.q . chen , phys . d * 48 * , 4086 ( 1993 ) ; c.h . chang , y.q . chen , g.p . han , and h.t . jiang , phys . lett.364 b , 78 ( 1995 ) ; c.h . chang and x.g . j. c * 38 * , 267 ( 2004 ) ; r.m . thurman - keup , a.v . kotwal , m. tecchio , and a.b . wagner , rev . phys * 73 * , 267 ( 2001 ) . chang , c. driouich , p. eerola , and x.g . wu , comput . . commun . * 159 * , 192 ( 2004 ) ; c.h . chang , j.x . wang , and x.g . wu , comput . . commun . * 174 * , 241 ( 2006 ) ; * 175 * , 624 ( 2006 ) ; x.y . wang and x.g . wu , comput . . commun . * 183 * , 442 ( 2006 ) . chang , nucl . b * 172 * , 425 ( 1980 ) ; r. baier and r. rueckl , phys . * 102 * b , 364 ( 1981 ) ; e.l . berger and d.jones , phys . d * 23 * , 1521 ( 1981 ) ; h. krasemann , z. phys . c * 1 * , 189 ( 1979 ) ; g. guberina , j. kuhn , r. peccei , and r. rueckl , nucl . b * 174 * , 317 ( 1980 ) .
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the value of the quarkonium wave function at the origin is an important quantity for studying many physical problems concerning a heavy quarkonium .
this is because it is widely used to evaluate the production and decay amplitudes of the heavy quarkonium within the effective field theory framework , e.g. , the nonrelativistic qcd ( nrqcd ) . in this paper , the values of the schr@xmath0dinger radial wave function or its first nonvanishing derivative at zero quark - antiquark separation , i.e. , @xmath1\rangle$ ] , @xmath2\rangle$ ] , and @xmath3\rangle$ ] quarkonium ,
have been tabulated under five potential models with new parameters for the heavy quarkonium .
moreover , the production of the lower - level fock states @xmath4\rangle$ ] and @xmath5\rangle$ ] , together with the higher excited fock states @xmath6\rangle$ ] and @xmath7\rangle$ ] ( @xmath8 stands for the @xmath9 or @xmath10 quark ; @xmath11 ) through top quark decays has been studied with the new values of heavy quarkonium wave functions at the origin under the framework of nrqcd . at the lhc with the luminosity @xmath12 and the center - of - mass energy @xmath13 tev
, sizable heavy quarkonium events can be produced through top quark decays , i.e. , @xmath14 @xmath15 and @xmath16 , and @xmath17 @xmath18 and @xmath19 events per year can be obtained according to our calculation . + * pacs numbers : * 12.38.bx , 14.65.ha , 14.40.nd , 14.40.pq
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we will discover a large class of positive finite presentations of the @xmath0-braid group . roughly speaking , a presentation in this class has a set of generators determined by a connected graph with @xmath0 vertices immersed in a plane in such a way that each pair of edges intersects at most once . our class of presentations includes all known positive finite presentations such as the artin presentation@xcite , the band - generator presentation in @xcite and all of sergiescu s presentations in @xcite . our first objective is to find a ( minimal ) collection of positive relations among generators given by a graph as above so that it becomes a presentation of the @xmath0-braid groups . the semigroup of all positive words plays a crucial role in the solutions to the word problem and the conjugacy problem for the @xmath0-braid group given by garside@xcite , thurston@xcite , and elrifi - morton@xcite . their solutions are based on the property that the semigroup embeds in the whole group . birman - ko - lee s recent work@xcite also requires this embedding property in order to give a fast solution to the word problem in the band - generator presentation . our second objective is to prove that with a few exception all presentations in our class have the embedding property . the artin and the band - generator presentations are the only nontrivial exceptions . consequently these two among all presentations are natural and interesting for further study . the @xmath0-braid group @xmath1 is the group of isotopy classes of orientation - preserving automorphisms of an @xmath0 punctured plane @xmath2 that fix the set of punctures and the outside of a disk containing all punctures . it is customary to place punctures on the @xmath3-axis and equally spaced . ignoring the punctures , such an automorphism is a homeomorphism of @xmath2 permuting the punctures and so is isotopic to the identity map of @xmath2 . a geometric braid is determined by taking the trace of the punctures under this isotopy . when the punctures are not in the customary location , there is a choice of homeomorphisms of @xmath4 that send the punctures to their customary location and a geometric braid is uniquely determined up to conjugation in this case . since the symmetric group over @xmath0 letters is naturally a quotient of @xmath1 , an automorphism that exchanges two punctures is a candidate for a generator of @xmath1 and a typical automorphism of this type can be easily depicted as an arc connecting the two punctures so the automorphism exchanges two punctures clockwise or counterclockwise inside an annulus lying along the arc . see figure [ f : string ] for an example . this type of braids will be called the _ half twist along the given arc_. therefore by regarding the punctures as vertices and a collection of these arcs as edges , we have a graph @xmath5 immersed in @xmath2 satisfying the following properties : 1 . there are no loops ; 2 . edges have no self - intersections ; 3 . two distinct edges intersect either at interior points transversely or at common end points there is at least a vertex inside a pseudo digon " , that is , a region cobounded by two subarcs from two edges so that corners of the region are either two interior intersection points or one vertex and one interior intersection point . if the number of intersections among edges is minimized , for example , by making all edges geodesics in the hyperbolic structure of the punctured plane , the condition ( 4 ) holds automatically . two immersed graphs are regarded as equivalent if one graph can be transformed to the other by an orientation - preserving homeomorphism of @xmath4 . thus we may assume that vertices lie on the customary locations if necessary . throughout this article , graphs mean immersed graphs in @xmath6 satisfying the condition discussed . if graphs are embedded in @xmath4 , we call them _ planar_. the set of edges in a graph @xmath5 with @xmath0 vertices corresponds to a set of elements in @xmath1 as in figure [ f : string ] . two equivalent graphs determine two sets of braids that differ by an inner automorphism of @xmath1 . if the set corresponding to edges generates @xmath1 , we say that @xmath5 _ generates _ @xmath1 and we are interested in graphs that generate @xmath1 . since an edge of @xmath5 corresponds to a transposition in the symmetric group , a graph that generates @xmath1 must be connected in order to permute any two vertices . but there are connected graphs that do not generate @xmath1 . the problem of deciding when an immersed graph generates the braid group is considered elsewhere@xcite . we now discuss how a braid word corresponds to an edge of a graph with @xmath0 vertices in the customary location . the artin generator @xmath7(or @xmath8 ) for @xmath9 corresponds to the clockwise(counterclockwise , respectively ) half twist along the straight edge joining the @xmath10-th and the @xmath11-st vertices . given an edge of a graph , we simplify the edge via a sequence of half twists @xmath7 or @xmath8 for @xmath9 until it becomes a straight edge corresponding to , say , @xmath12 . the required sequence can be expressed as a word @xmath13 in artin generators in which a half twist applied later is written on the left . then the edge that we started with corresponds to the braid word @xmath14 . the word @xmath13 is not uniquely expressed but @xmath13 as an element of @xmath1 is well - defined . the inverse of this correspondence is similar . given a conjugate @xmath14 , apply a sequence of half twists determined from right to left by @xmath15 to the straight edge @xmath12 . consequently each edge of a graph with @xmath0 vertices is uniquely represented by an element in @xmath1 that can be written as a conjugate of an artin generator . the following lemma summarizes this discussion . a braid @xmath16 in @xmath1 is the half twist along an arc joining two punctures in the plane if and only if @xmath16 can be written as a conjugate of an artin generator or its inverse . from now on as long as no confusion arises , we will not distinguish three concepts , namely an edge of a graph , the half twist along the edge , and a conjugate word expressing the edge . since any two artin generators are conjugate each other , any half twist along an arc corresponds to a conjugate of a fixed artin generator . the following is immediate from the above lemma . [ conjugate ] a graph @xmath5 of @xmath0 vertices generates @xmath1 if and only if each artin generator can be expressed as @xmath17 for some edge @xmath18 and some word @xmath13 on edges of @xmath5 . in the view of the above corollary , it is important to know how the conjugate of an edge by another edge in a graph looks like . let @xmath18 and @xmath16 be words in @xmath1 that express edges of a graph . if the edges @xmath18 and @xmath16 do not intersect each other and are not adjacent , they commute and so @xmath19 . if the edges @xmath18 and @xmath16 do not intersect each other and are adjacent , the three edges @xmath18 , @xmath16 , and @xmath20 form a triangle counterclockwise and the three edges @xmath18 , @xmath16 , and @xmath21 form a triangle clockwise as in figure [ f : triangle ] . we note that there should be no other vertex inside the triangles and this can be achieved by drawing thin triangles inside a sufficiently small neighborhood of the union of two edges @xmath18 and @xmath16 . when the edges @xmath18 and @xmath16 intersect at an interior point , one can still describe the edges @xmath22 and @xmath23 but we do not need them in this article . give a set @xmath24 of generators of the @xmath0-braid group @xmath1 , a _ positive word _ in @xmath24 is a product of positive powers of generators in @xmath24 . a presentation of a group is _ finite _ if there are finitely many generators and relations . a presentation is _ positive _ if all defining relations are equations of positive words in generators . we now introduce the graphs that corresponds to some of known positive finite presentations of the braid groups . the set of artin generators @xmath25 of @xmath1 forms the graph in figure [ f : artin ] and a minimal set of defining relations is given by @xmath26 the graphs in figure [ f : artin ] will be called the _ artin _ graph . sergiescu @xcite showed how a finite presentation of @xmath1 can be obtained from any planar connected graph with @xmath0 vertices . he described a sufficient set of positive relations that depend only on the geometry of a given planar graph . we will give a minimal set of positive relations as a corollary of the main theorem in 3 . let @xmath0 be the number of vertices in @xmath5 . by choosing a fixed @xmath0 points in @xmath27 and a homeomorphism of @xmath27 that sends vertices of @xmath5 to the @xmath0 fixed points , the group @xmath28 introduced in @xcite is identified with the @xmath0-braid group @xmath1 . these presentations of @xmath1 will be called _ sergiescu s presentations . _ the relations in a sergiescu s presentation are highly redundant but they are useful when we need to find a relation locally given by a planar graph . recently a new presentation of @xmath1 called the band - generator presentation has been developed by birman - ko - lee@xcite . this presentation has @xmath29 generators @xmath30 for @xmath31 corresponds to the graph in figure [ f : band ] and defining relations : @xmath32 the artin generator and the band - generators are related as @xmath33 and @xmath34 the set of band - generators are depicted as the graph in figure [ f : band ] that will be called the _ inner complete _ graph . the relations @xmath35 will be called a _ triangular _ relation . a triangular relation is derived whenever a new generator is introduced by means of a conjugation of an edge by an adjacent edge . triangular relations serve as building blocks of positive relations in the braid groups . after an immersed graph is turned into a planar graph by regarding all interior intersections as vertices , a region bounded by a closed edge - path is called a _ pseudo face _ if the edge - path contains at least one vertex that is an interior intersection of two edges as in figure [ f : pseudo ] . a graph @xmath5 is said to be _ linearly spanned _ if it is connected and there is no vertex in any pseudo face of @xmath5 . a connected subgraph of a linearly spanned graph is clearly linearly spanned . two edges of a linearly spanned graph intersect each other at most once since any pseudo digon cobounded by two edges can not exist in a linearly spanned graph . all planar graphs and all subgraphs of the inner - complete graph are linearly spanned . for example the graph on the left in figure [ f : linear ] is neither planar nor a subgraph of the inner - complete graph but it is linearly spanned , and the graph on the right is not linearly spanned the following lemma justifies the terminology `` linearly spanned '' . a linearly spanned graph that is a tree is equivalent to a subgraph of the inner - complete graph . choose a point @xmath3 far away from the graph . since the graph is a tree and no vertices are surrounded by edges , there are @xmath0 arcs that join @xmath3 to each vertex and are disjoint each other and are disjoint from the graph as in figure [ f : tree ] . choose a new horizontal axis disjoint from the graph and move each vertex along each arc by a homeomorphism so that it lies on the new axis as figure [ f : tree ] . then the result is a subgraph of the inner complete graph . a linearly spanned graph @xmath5 with @xmath0 vertices generates @xmath1 . a maximal tree of @xmath5 is equivalent to a connected subgraph of the inner - complete graph from which we can obtain the inner complete graph by adding missing edges that are conjugates of an existing edge by an adjacent edge . [ tree ] let @xmath36 be a linearly spanned tree with @xmath0 vertices . then @xmath36 generates @xmath1 with @xmath37 positive relations described as in the proof below . induction on @xmath0 . it is trivial when @xmath38 . suppose a linear spanned tree @xmath39 with @xmath40 vertices generates @xmath41 with @xmath42 positive relations . we add a new vertex and a new edge @xmath18 to @xmath39 . for each edge @xmath16 of @xmath39 , we will have a new positive relation so that @xmath43 new positive relations will be added . in the following we use tietze transformations @xcite that add and delete a generator(s ) denoted by @xmath44 or @xmath45 to utilize triangular relations . 1 . if @xmath18 and @xmath16 have no intersection , add the positive relation @xmath46 2 . for each set of edges @xmath47 simultaneously adjacent to @xmath18 at a vertex of valency @xmath48 as in figure [ f : induct1](a ) , add @xmath49 positive relations @xmath50 3 . if @xmath18 and @xmath16 intersect and form a pseudo face as in figure [ f : induct1](b ) , add the positive relation @xmath51 that is derived from @xmath52 where @xmath53 4 . if @xmath18 and @xmath16 intersect and form a pseudo face as in figure [ f : induct2](a ) , add the positive relation @xmath54 that is derived from @xmath55 where @xmath53 5 . if @xmath18 and @xmath16 intersect and form a pseudo face as in figure [ f : induct2](b ) , add the positive relation @xmath56 that is derived from @xmath57 where @xmath58 the following is the main theorem of this section and the known presentation mentioned in the previous section can be obtained from this . [ circuit ] let @xmath5 be a linearly spanned graph with @xmath0 vertices . then @xmath5 generates @xmath1 with @xmath59 positive relations described as in the proof below where @xmath5 has @xmath60 edges . choose a spanning tree @xmath36 of @xmath5 . then a linearly spanned tree @xmath36 generates @xmath1 with @xmath37 positive relations as in lemma [ tree ] . when each edge @xmath18 in @xmath61 are added , a circuit is formed and the circuit give a new positive relation as follows : 1 . if @xmath18 forms a circuit with no intersection with other edges as in figure [ f : cutopen ] , add the positive relation @xmath62 where we regard that the circuit bounds a polygonal disk by cutting open all edges inside the circuit as in figure [ f : cutopen ] . 2 . if @xmath18 forms a circuit with some intersections with other edges as in figure [ f : face ] , add the positive relation @xmath63 two positive words @xmath64 , @xmath65 in a positive presentation will be said to be _ positively equivalent _ if they are identically equal or they can be transformed into each other through a sequence of positive words such that each word of the sequence is obtained from the preceding one by a single direct application of the defining relations . and we will write @xmath66 if @xmath64 and @xmath65 are positively equivalent . given a positive finite presentation @xmath67 of @xmath1 , let @xmath68 be the free semigroup generated by @xmath69 modulo @xmath70 . if any two equivalent positive words @xmath64 and @xmath65 are positively equivalent , then we say that the semigroup @xmath68 _ embeds _ in @xmath1 or the presentation @xmath67 has the _ embedding property_. a set @xmath69 of generators is said to have the _ embedding property _ if a presentation @xmath67 has the embedding property for some finite set @xmath70 of positive relations . the _ embedding problem _ of a graph that generates @xmath1 with positive relations is to decide whether the set of generators given by the graph has the embedding property . thus a graph does not have the embedding property if and only if no finite set of positive relations over the set of generators given by the graph form a presentation with the embedding property . garside@xcite showed that the artin presentation has the embedding property and birman - ko - lee@xcite showed that the band - generator presentation has the embedding property . we will show that linearly spanned graphs with more than 3 vertices do not have the embedding property except these two presentations . there are 4 possible linearly spanned graphs with 3 vertices as in figure [ f:3vertex ] . the first two graphs give the artin and the band - generator presentations of @xmath71 . the last two graphs have multiple edges . we do not know whether these graphs have the embedding property . in particular the following presentation with 4 defining relations : @xmath72 may have the embedding property . but all known techniques as in @xcite fail to apply to this example . we will try to avoid these unknown exceptions in the following discussion by allowing no multiple edges . [ circle ] let @xmath73 be a connected full subgraph of a linearly spanned graph @xmath5 . if a graph @xmath5 has the embedding property and there is a circle @xmath74 such that @xmath74 contains @xmath73 inside and all vertices in @xmath75 lie outside , then @xmath73 also has the embedding property . let @xmath69 and @xmath76 be the set of generators given by @xmath5 and @xmath73 , respectively and let @xmath70 be a finite set of positive relations on @xmath69 such that @xmath67 has the embedding property . let @xmath77 be the set of relations on @xmath76 which is a full " subset of @xmath70 in the sense that @xmath77 contains all relations in @xmath70 written on @xmath76 . we will prove by contradiction that @xmath78 has the embedding property . suppose that there exists a pair of positive words @xmath79 on @xmath76 such that @xmath80 are equivalent in the braid group but are not positively equivalent under @xmath77 . since @xmath81 has the embedding property , @xmath66 under @xmath70 . thus there is a sequence of positive words @xmath82 over @xmath69 such that @xmath83 under @xmath70 and each positive equivalence is obtained by one direct application of relations in @xmath70 . the sequence must contain at least one positive word , say @xmath84 , that is not written solely on @xmath76 , otherwise @xmath66 under @xmath77 . in the view of lemma [ tree ] and theorem [ circuit ] , it is impossible that @xmath70 contains any relation @xmath85 such that @xmath86 is a positive word over @xmath76 and @xmath13 is a positive word over the edges that are not incident to any vertex of @xmath73 . furthermore @xmath73 is a full subgraph of @xmath5 . thus @xmath84 must contain an edge that joins a vertex @xmath87 in @xmath73 to a vertex @xmath88 not in @xmath73 . let @xmath16 be such an occurrence that comes last in @xmath84 . since @xmath73 is connected , the vertex @xmath87 is joined to another vertex @xmath89 in @xmath73 by an edge @xmath18 . as an automorphism of the punctured plane , @xmath64 does not change the circle @xmath74 because the edges for @xmath64 never touch @xmath74 . on the other hand we will show @xmath84 must change @xmath74 and this is a contradiction because @xmath64 and @xmath84 are isotopic as automorphisms of the punctured disk . the automorphism @xmath84 is the composition of the counterclockwise half twists along edges in @xmath84 . then the counterclockwise half twist along @xmath16 creates an intersection @xmath3 of @xmath74 with the edge @xmath18 . the intersection @xmath3 can disappear only via a clockwise half twist along an edge incident at either @xmath89 or @xmath87 as in figure [ f : circle ] . but all of half twists in @xmath84 are counterclockwise because @xmath84 is a positive word over edges , an intersection between two adjacent edges always creates a pseudo face that must contain a vertex and so there is no intersection between adjacent edges in a linearly spanned graph . suppose @xmath5 has more than one intersection . figure [ f : oneinter ] shows a typical situation with one intersection and another intersection @xmath3 . the edge @xmath18 must join vertices @xmath90 and @xmath91 , otherwise @xmath3 is an intersection between two adjacent edges . one can easily check that all possibilities of completing @xmath18 create a pseudo face containing at least a vertex . [ length ] let @xmath5 be a linearly spanned graph and @xmath69 be the set of generators given by edges of @xmath5 . suppose that for each @xmath92 , there are positive words @xmath93 and @xmath94 over @xmath69 for @xmath95 in @xmath69 such that let @xmath101 be any positive finite presentation of the braid group . the hypothesis ( iii ) implies that any shorter positive relation than @xmath85 itself can not make @xmath13 positively equivalent to @xmath86 . choose a large @xmath102 such that @xmath85 is longer than any relation in @xmath70 . then @xmath13 is not positively equivalent to @xmath86 over @xmath70 . the following theorem completely determines when a linearly spanned graph with 4 vertices and no multiple edges has the embedding property . in the proof given below , @xmath103 or @xmath104 will denote a positive word over the generators @xmath105 . in order to show that all other linearly spanned graphs with 4 vertices do not have the embedding property , we appeal to lemma [ length ] . to check the hypothesis ( iii ) of the lemma , we use the fact that the band - generator presentation has the embedding property so that the positive equivalence in the presentation is the same as the equivalence in the braid group . we also utilize the left and right cancellation theorem and the left and right canonical forms in the band - generator presentation in @xcite . planar graphs with 4 vertices can be divided further into three types : graphs containing neither a triangle nor a rectangle , graphs containing at least a rectangle , and graphs containing at least a triangle but no rectangle , where a triangle ( or a rectangle ) is non - degenerate , that is , must have 3 ( or 4 , respectively ) vertices and must contain no other vertices inside . [ [ i - graphs - containing - a - vertex - adjacent - to - all - of - remaining - three - vertices ] ] * ( i ) graphs containing a vertex adjacent to all of remaining three vertices * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we may have two possible graphs as in figure [ f : star1 ] up to equivalence . for each of these two graphs , let @xmath106 and @xmath107 . we will show that @xmath13 and @xmath86 satisfy lemma [ length ] . add the edges @xmath108 so that these edges together with @xmath109 form an inner - complete graph . then @xmath110 and so @xmath85 . suppose @xmath111 for some positive word @xmath112 in the band - generator presentation . then @xmath113 by the left cancellation in the band - generator presentation . but @xmath114 may only start with @xmath44 or @xmath115 which commute . thus @xmath112 can not be written over @xmath109 . similarly @xmath116 . thus @xmath13 satisfies ( iii ) of lemma [ length ] . due to the triangular relation @xmath117 , @xmath13 can not be equivalent to a positive word over @xmath118 that contains @xmath119 since @xmath13 is not positively equivalent to a positive word in the band - generator presentation that contains the subword @xmath120 . thus @xmath121 also satisfy the hypothesis of lemma [ length ] over @xmath118 . [ [ ii - graphs - containing - no - vertex - adjacent - to - all - of - remaining - three - vertices ] ] * ( ii ) graphs containing no vertex adjacent to all of remaining three vertices * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + suppose @xmath126 for some positive word @xmath112 in the band - generator presentation . then @xmath127 by the left cancellation in the band - generator presentation . thus @xmath112 can not be written over @xmath128 . similarly @xmath129 . thus @xmath13 satisfies ( iii ) of lemma [ length ] . due to the triangular relation @xmath130 , @xmath13 can not be equivalent to a positive word over @xmath131 that contains @xmath132 since @xmath13 is not positively equivalent to a positive word in the band - generator presentation that contains the subword @xmath133 . thus @xmath121 also satisfy the hypothesis of lemma [ length ] over @xmath131 . let @xmath137 . . the right canonical form of @xmath139 is @xmath140 . so @xmath139 may finish with @xmath141 over the band - generators and so @xmath44 or @xmath45 must appear in @xmath112 . . in this case triangles must share at least 1 edge since we consider only 4 vertices . however if two triangles share 2 edges then it must contain a multiple edge . thus we have two possible graphs as in figure [ f : trian1 ] . for these two graphs , let @xmath143 and @xmath144 . the left canonical form of @xmath146 is @xmath147 and so it can not start with @xmath119 over the band - generators . thus @xmath148 . the right canonical form of @xmath149 is @xmath150 and so it can not finish with @xmath151 . thus @xmath152 . due to the triangular relation @xmath153 , @xmath13 can not be equivalent to a positive word over @xmath131 that contains @xmath154 since @xmath13 is not positively equivalent to a positive word in the band - generator presentation that contains the subword @xmath133 . thus @xmath121 also satisfy the hypothesis of lemma [ length ] over @xmath131 . the edges @xmath151 and @xmath115 that intersect each other are transformed to the diagonals of a rectangle . then 4 more edges are needed to make an inner - complete graph . we have four distinct types of graphs , depending on how many edges are missing from the inner - complete graph . let @xmath160 . then @xmath161 . to remove @xmath162 , @xmath163 must start with @xmath164 . but the left canonical form of @xmath163 is @xmath165 and so it may start only with @xmath166 . thus @xmath167 . similarly @xmath168 . the left canonical form of @xmath174 is @xmath175 and so it can not start with @xmath162 over the band - generators . thus @xmath176 . the right canonical form of @xmath177 is @xmath178 and so it can not finish with @xmath119 . thus @xmath179 . the left canonical form of @xmath185 is itself and so it can not start with @xmath162 over the band - generators . thus @xmath186 . let @xmath187 then @xmath188 and @xmath189 commute , so @xmath162 can not be removed . thus @xmath190 . we have already discussed about linearly spanned graphs with 3 vertices . let @xmath5 be a linearly spanned graph with more than 3 vertices that has the embedding property . first we choose a connect full subgraph @xmath73 with 4 vertices from @xmath5 so that there is a separating circle satisfying the hypothesis of theorem [ circle ] . choose 4 vertices that form a connected subtree in a spanning tree of @xmath5 . take the full subgraph with these 4 vertices . if there is no other vertices in faces of this full subgraph , then this is a desired full subgraph . if there is other vertices in the faces of this full subgraph and none of them is adjacent to the chosen 4 vertices , then there is an edge - path starting at a vertex @xmath87 on a face and ending at one of the 4 vertices such that the edge - path intersect the full subgraph of the 4 vertices and so @xmath87 is contained in a pseudo face . since @xmath5 is linearly spanned , this can not happen . thus at least one of vertices on faces , say @xmath88 , is adjacent to one of the 4 vertices . then we have the less number of unwanted vertices contained in faces of a new connected full subgraph that is obtained by replacing one of the 4 vertices by @xmath88 . by repeating this process , we eventually obtain a connect full subgraph @xmath73 with 4 vertices @xmath191 that can be separated by a circle from other vertices of @xmath5 . theorem [ circle ] and theorem [ 4vertex ] say that @xmath73 must be either the artin graph or the inner - complete graph with 4 vertices . choose a vertex @xmath192 in @xmath75 that is adjacent to one of @xmath191 . if @xmath73 is an inner - complete graph , then each full subgraph with the 4 vertices consisted of @xmath192 and any three vertices from @xmath191 must be an inner - complete graph by theorem [ circle ] and theorem [ 4vertex ] . thus the full subgraph with the 5 vertices @xmath193 is an inner - complete graph . by repeating this process , @xmath5 itself eventually becomes an inner - complete graph . we now suppose that @xmath73 is the artin graph with 4 vertices @xmath191 where @xmath194 are of valency 1 and the other vertices are of valency 2 . choose a vertex @xmath192 in @xmath75 that is adjacent to any one of @xmath191 . by theorem [ circle ] and theorem [ 4vertex ] , @xmath192 can be adjacent to either one or both of vertices @xmath194 . if @xmath192 is adjacent to both vertices , we stop the process . if @xmath192 is adjacent to one vertex , say @xmath195 , then @xmath196 form an artin graph and we repeat this process . eventually we see that @xmath5 must contain one of the following two types of graphs as a full subgraph separated by a circle as in theorem [ circle ] unless @xmath5 itself is an artin graph . the proof will be completed when we show that both types of graphs do not have the embedding property . the graph of this type is depicted in figure [ f : kcycle ] . let @xmath197 and @xmath198 . then one can show that @xmath121 satisfy the hypothesis of lemma [ length ] by a similar but longer argument as for figure [ f : cycle ] . thus @xmath5 does not have the embedding property we now show that @xmath203 suppose that @xmath204 then @xmath205 where @xmath206 . due to the triangular relation @xmath207 , the positive word @xmath208 must end with @xmath209 in order for @xmath210 to disappear . we will show that @xmath208 can not end with @xmath209 by an induction on @xmath211 . clearly @xmath212 can not end with @xmath164 in the band - generator presentation . it is easy to check that @xmath214 ends with @xmath209 if and only if @xmath215 ends with @xmath209 if and only if @xmath216 ends with @xmath217 in the band - generator presentation . by the induction hypothesis , @xmath216 can not end with @xmath217 . similarly we have that @xmath218 consequently @xmath219 satisfy the hypothesis of lemma [ length ] . mm e. artin , _ theori der zpfe _ , hamburg * 4 * ( 1925 ) , 4772 . j. s. birman , _ braid groups and mapping class group _ , annals of math . studies , no . * 82 * , princeton university press , 1974 . j. s. birman , k. h. ko and s. j. lee , _ a new approach to the word and conjugacy problem in the braid groups _ , adv . math . * 139 * ( 1998 ) 322353 a. h. clifford and g. b. preston , _ algebraic theory of semi - groups _ survey 7 ( 1961 ) . e. a. elrifai and h. r. morton , _ algorithms for positive braids _ , quart . j. math . oxford * 20 * ( 1994 ) 479497 d. b. a. epstein ( with cannon , holt , levy , patterson and thurston ) , _ word processing in groups _ , jones and bartlett , boston , mass . ( 1992 ) , isbn 0 - 86720 - 244 - 0 f. a. garside , _ the braid group and other groups _ , quart . j. math . oxford * 20 * ( 1969 ) 235254 . j. w. han and k. h. ko , conjugate presentations of the braid groups , preprint . e. s. kang , k. h. ko and s. j. lee , _ band - generator presentation for the 4-braid group _ , topology appl . * 78 * ( 1997 ) 3960 j. h. remmers , _ on the geometry of semigroup presentations _ , advance math . * 36 * , ( 1980 ) , 283296 . v. sergiescu , _ graphes planaires et prsentations des groupes de tresses _ , math . z. * 214 * ( 1993 ) 477490 . h. tietze , _ ber die topologischen invarianten mehrdimensionalen mannigfaltigkeiten _ monatsh . f. math . u. physik * 19 * ( 1908 )
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a large class of positive finite presentations of the braid groups is found and studied .
it is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the topic of control over a communication channel has been extensively studied in the past decade , with issues such as the minimum data rate for stabilization @xcite and optimal quadratic closed - loop performance @xcite being the main focus . other issues of interest concern effects of channel - induced packet drops and/or time - varying delays on closed - loop performance . the majority of papers concerned with control over networks regards the mechanism of information loss in the network as probabilistic but not strategic . in contrast , in the problem of control over an _ adversarial _ channel , the communication link is controlled by a rogue jammer whose intention is to mount a cyber attack on the system by actively jamming the communication link . its objectives are to impose on the controller a control law which can not be expected under regular operating conditions in a packet - dropping network . if the controller is unaware of the jammer s actions and continues to follow a control policy designed for a regular network , the system performance is likely to be inferior . it is this situation that is considered as a scenario of a successful cyber attack by the jammer . a natural way to describe the problem of control over an adversarial channel is to employ a game - theoretic formulation . originally proposed in @xcite , this idea has been followed upon in a number of recent papers including @xcite . a zero - sum dynamic game between a controller performing a finite horizon linear - quadratic control task and a jammer , proposed in @xcite , specifically accounted for the jammer s strategic intentions and limited actuation capabilities , but was otherwise agnostic regarding the type of channel involved . a startling conclusion of @xcite was that in order to maximally disrupt the control task , the jammer must act in a markedly different way than a legitimate , non - malicious , packet - dropping channel . once this deterministic behavior is observed by the controller , it can establish with certainty that an attack has taken place . in @xcite , we introduced a different model , which , while capturing the same fundamental aspects of the problem as in @xcite , modified the jammer s action space so that each jammer decision corresponded to a choice of channel rather than to passing / blocking transmission . the corresponding one - step zero - sum game was shown to have a unique saddle point in the space of mixed jammer strategies . in turn , the controller s best response to the jammer s optimal randomized strategy was to act as if it was operating over a packet - dropping channel whose statistical characteristics were controlled by the jammer . since under normal circumstances the controller can not be aware of these characteristics , and can not implement such a best response strategy , we regard the zero - sum game in @xcite as an example of a successful cyber attack . in this paper , we show that such a situation is not specific to the zero - sum game considered in @xcite . we introduce a class of zero - sum stochastic games that generalize the model introduced in @xcite . for these games we obtain necessary and sufficient conditions which guarantee the existence of optimal jammer s strategies whose nature suggests that the jammer must select its actions randomly , in order to make a maximum impact on the control performance . our conditions are quite general , they apply to nonlinear systems and draw on standard convexity / coercivity properties of payoff functions . furthermore , we specialize these conditions to the linear - quadratic control problem over a packet - dropping link considered in @xcite and show that our conditions allow for an express characterization of a set of plant s initial states for which optimal randomized jammer strategies exist ( this is in contrast to @xcite where a complete analysis of the state space had to be performed to determine such regions ) . we also compute an optimal controller response to those strategies , which turns out to be nonlinear . our analysis is restricted to one - step zero - sum games . although such a formulation is admittedly simple , due to the general nature of the game under consideration , it can be thought of as reflecting a more general situation where one is dealing with a one - step hamilton - jacobi - bellman - isaacs min - max problem associated with a multi - step optimal control problem . also , even a one - step formulation provides a rich insight into a possible scenario of cyber attacks on controller networks . we believe that such an insight can be valuable as was the case , e.g. , in early studies of adversarial channels and multi - agent decision problems involving incomplete information . we present our model in section [ sec_mod ] . the problem formulation , its assumptions and preliminary results are given in section [ sec_form ] . the main result of the paper that gives a necessary and sufficient condition for the game under consideration to have a nontrivial saddle point is presented in section [ main ] . next , in section [ examples ] we demonstrate an application of this result to the linear - quadratic static problem which is an extension of the problem in @xcite . in this problem , the jammer is offered an additional reward for undertaking actions concealing its presence . conclusions are given in section [ conclusions ] . the general model description is an extension of that in @xcite . we consider a situation where a strategic jammer is attacking the link in the feedback loop between a controller and a plant . the plant is a general discrete - time system described by @xmath0 with a given initial condition ; @xmath1 is the state , @xmath2 is a scalar input , @xmath3 is an @xmath4-valued function defined on @xmath5 . the plant input @xmath2 and the control signal @xmath6 are related by the equation @xmath7 which describes the transmission of information from the controller to the plant over a packet - dropping communication link . here , @xmath8 is a discrete random variable taking value in @xmath9 , that describes the transmission state of the link . the value of @xmath8 depends on actions of the jammer and the state of the communication link , as explained below . the communication link consists of a finite set @xmath10 of channels ( with @xmath11 ) out of which the jammer can draw with certain probability a channel to replace the currently active channel so as to optimally disrupt the control task . each channel can be either in passing or blocking state , and the transmission states of _ all _ channels randomly change _ after _ one of them is selected as a replacement . hence , each channel @xmath12 represents a binary channel with the state space @xmath9 , as pictured in figure [ fig_2 ] . to describe the probability model of channel transitions , let @xmath13 denote the vectors of transmission states of _ all _ channels before and after the jammer has selected one of them to replace the current one , respectively , with the @xmath14th component @xmath15 , @xmath16 denoting the corresponding transmission state of channel @xmath17 . the probability of channel @xmath17 to become `` passing '' after the replacement is selected , given its and all other channels previous transmission states , is then @xmath18 the jammer strategy is to choose a probability distribution over @xmath10 , indicating which channel it desires to switch to . we denote this distribution by a vector @xmath19 in the unit simplex @xmath20 of @xmath21 . that is , the jammer s strategy is to influence the selection of a channel linking the controller to the plant . let the index of the selected channel be @xmath22 , then @xmath22 is a discrete random variable taking values in @xmath23 , distributed in accordance with the vector @xmath19 . the latter depends on the information set available to the jammer which includes the current state of the plant @xmath24 , the index of the channel occupying the link @xmath25 , and the vector of transmission states of all channels @xmath26 which jammer observes before the link switches from channel @xmath27 to a new channel : @xmath28 in addition , if the control input @xmath6 is available to the jammer , the vector @xmath19 may depend on @xmath6 as well . after the jammer has made its decision , the random variable @xmath22 is realized and the link switches to the channel @xmath29 . after that , the transmission state of all channels including @xmath29 changes , according to ( [ q_j ] ) . thus , the jammer can not predict the transmission state of the channels in @xmath10 when selecting the probability vector @xmath19 . in accordance with this channel switching mechanism , the transmission state of the link between the controller and the plant is determined by the binary random variable @xmath30 , i.e. , @xmath31 , which takes value 1 with probability @xmath32 and value 0 with probability @xmath33 . clearly , @xmath8 and @xmath22 are statistically dependent . all random variables considered in this paper will be adapted to the joint conditional probability distribution of @xmath22 and @xmath8 , given @xmath24 , @xmath25 and @xmath26 . the expectation with respect to this conditional probability distribution is denoted @xmath34 $ ] . 1.5ex is a binary channel . shown here is channel @xmath17 . [ fig_2 ] ] we now introduce a general two - player stochastic one - step zero - sum game as follows . in this game , we assume that the initial state of the plant @xmath24 , the initial vector of transmission states @xmath26 and the channel that initially occupies the link @xmath27 are known to both the jammer and controller . let @xmath35 be a scalar function of @xmath36 . this function will determine the payoff of the game played by the controller and the jammer . the standing assumptions regarding this function are summarized below : [ sigma.a1 ] for all @xmath12 , @xmath37 . [ sigma.a3 ] for each @xmath12 and @xmath38 , the functions @xmath39 and @xmath40 are coercive . [ u ] under assumptions [ sigma.a1 ] and [ sigma.a3 ] , for every @xmath41 there exists a compact set @xmath42 with the properties : a. for all @xmath43 , @xmath44 } & & \nonumber \\ & & = \inf_{u\in u(x)}\max_{j } \mathbb{e}[\sigma(x^+,u , f_s)|s = j ) ] . \label{max.property}\end{aligned}\ ] ] b. @xmath45=\inf_{u\in u(x)}\mathbb{e}[\sigma(x^+,u , f_s)].\ ] ] the proof is omitted for the sake of brevity . it proceeds by first proving that the coercivity of the functions involved ensures that the infima on the left hand side of ( [ max.property ] ) and ( [ infu.1 ] ) exist . in particular , @xmath46 , @xmath47 , @xmath48 $ ] . next we show that a suitably defined set @xmath49 , with a sufficiently large @xmath50 can be chosen as @xmath51 . we now define the stochastic zero - sum min - max game of interest for the plant ( [ plant ] ) . in this game , the controller is a minimizing player who selects the control input @xmath52 based on @xmath24 , @xmath25 and @xmath26 . also , the jammer is the maximizing player who chooses a probability distribution vector @xmath53 for the ` channel selection ' random variable @xmath22 , as the function of @xmath24 , @xmath25 , @xmath26 and possibly @xmath6 , as in ( [ p_j ] ) . the controller s best action is determined by computing @xmath54 . \label{inf_sup } \end{aligned}\ ] ] while the jammer s best action is obtained by computing @xmath55 . \label{sup_inf } \end{aligned}\ ] ] our goal is to show that @xmath56 , i.e. , that the corresponding zero - sum game has a value . lemma [ u ] allows to reduce the minimization over @xmath57 in ( [ inf_sup ] ) and ( [ sup_inf ] ) to minimization over a compact set @xmath51 . indeed , the cost function of the inner maximization problem ( [ inf_sup ] ) is linear in @xmath19 , therefore using claim ( i ) of lemma [ u ] leads to the conclusion that @xmath58 \nonumber \\ & = & \inf_{u\in u(x)}\max_j \mathbb{e } \left[\sigma(x^+,u , f_j)\right ] \label{inf_u_max } \\ & = & \inf_{u\in u(x)}\max_{p \in \mathcal{s}_{n-1 } } \mathbb{e } \left[\sigma(x^+,u , f_s)\right ] . \label{inf_u_sup}\end{aligned}\ ] ] also , it follows from claim ( ii ) of lemma [ u ] that for every @xmath19 , the inner minimization problem in ( [ sup_inf ] ) can be carried out over @xmath51 . thus , @xmath59 . \label{sup_inf_u}\end{aligned}\ ] ] we make an additional assumption about the set @xmath51 . [ sigma.a4 ] the set @xmath51 is connected . under this assumption , the set @xmath51 is a closed bounded interval , hence it is a convex set . of course , this can be guaranteed when @xmath60 is chosen so that each @xmath61 is convex . [ bo.t4 ] under assumptions [ sigma.a1 ] , [ sigma.a3 ] , and [ sigma.a4 ] , the value of the game ( [ inf_sup ] ) exists , i.e. , @xmath62 furthermore , the game has a ( possibly non - unique ) saddle point . it is not unreasonable to assume that in the game ( [ inf_sup ] ) the jammer , who observes the controller action @xmath6 , can rank all the channels according to the contribution they make towards the payoff and order them accordingly . it can do so by comparing the conditional expected cost values @xmath63 $ ] . [ sigma.a2 ] for any two channels @xmath64 , @xmath65 , if @xmath66 then @xmath67\ge \mathbb{e}[\sigma(x^+,u , f_ss)|s = k ] \\ \nonumber \forall u\in u(x ) . \end{aligned}\ ] ] assumption [ sigma.a2 ] generalizes the situation considered in @xcite where all channels were ranked according to the probability of becoming passing , @xmath68 . in section [ examples ] we will show that such a natural ranking leads to satisfaction of assumption [ sigma.a2 ] . according to this assumption , the jammer who seeks a higher value of payoff should favour channels with lower numbers , since a larger reward is associated with these channels . in contrast , the controller actions should be directed towards forcing the jammer into utilizing channels with higher numbers . also , the channel @xmath27 is excluded from this ranking . this is done to allow the jammer to consider contributions to payoff other than those based on blocking / passing . these considerations may either discourage the jammer from switching , or conversely encourage it to undertake a denial - of - service attack . such decisions can be influenced by a number of factors that are not related to channel properties . the cost of channel switching is one reason as to why the jammer may decide not to change the channel . under another scenario , the jammer may be offered a reward for remaining stealthy , and may choose this reward over disrupting the control loop . for instance , when the controller monitors the link , an anomaly in the channel transition probabilities could signal the attack . in this case , rewarding the jammer for not defaulting to the most blocking channel unless it is absolutely necessary will provide it with an incentive for not revealing itself . in yet another class of problems , the jammer s decision could be based on the knowledge that the system is prepared to tolerate service disruptions as long as the cost of such disruptions is below the cost of rectifying them . we defer detailed analyses of these situations to section [ examples ] . it should be stressed that jammer decisions in each of these scenarios will depend on the plant state @xmath24 , the channel @xmath27 and the channel ranking ( the latter may require knowledge of @xmath6 ) . using the channel ranking introduced in assumption [ sigma.a2 ] , the value and saddle points of the game ( [ inf_sup ] ) can be characterized by solving a game over a reduced jammer strategy space . this reduced game focuses on two channels , namely the channel that currently occupies the link and the channel that delivers the highest payoff to the jammer when it seeks to block communications between the controller and the plant . the latter channel is indexed as channel @xmath69 , by assumption [ sigma.a2 ] . let us introduce the reduced jammer action vector @xmath70 , @xmath71 , @xmath72 . also , consider payoffs associated with selecting channel @xmath69 and keeping the current channel @xmath27 : @xmath73 , \\ \tilde h_2(u)&= & \mathbb{e}\left[\sigma(x^+,u , f_s)|s = j^-\right ] , \end{aligned}\ ] ] and define @xmath74 . consider the following ` reduced ' two - player game with upper value @xmath75 and lower value @xmath76 [ only.two ] suppose assumptions [ sigma.a1]-[sigma.a2 ] are satisfied . then @xmath77 furthermore , the zero - sum game ( [ inf_sup.red ] ) has a ( possibly non - unique ) saddle point . also , if @xmath78 is such a saddle point , then @xmath79 is a saddle point of the game ( [ inf_sup ] ) , where @xmath80 lemma [ only.two ] allows the jammer to constrain its actions to the set @xmath81 . among these actions there are two trivial actions : choose the most blocking channel ( channel @xmath69 in our notation ) by using @xmath82 and @xmath83 , or stay put by allocating @xmath84 and @xmath85 , so that the controller continues communicating with the plant over channel @xmath27 . however , the question arises as to whether there exist _ optimal mixed strategies _ in @xmath86 for the jammer to undertake , i.e. , optimal policy vectors @xmath19 such that @xmath87 , @xmath88 . the first main result of this paper provides necessary and sufficient conditions for the existence of nontrivial saddle points . these conditions characterize the controller - jammer games in which the jammer randomizes its choice of optimal strategies . as we will see , this will force the controller to respond in a non - obvious manner in order to remain optimal , which ultimately represents a signature of an attack on the communication link . [ saddle.point ] suppose @xmath89 , @xmath90 are strictly convex functions of @xmath6 for all @xmath24 . for every @xmath24 , the zero - sum game ( [ inf_sup.red ] ) admits a nontrivial saddle point @xmath78 if and only if there exists @xmath91 such that @xmath92 and one of the following conditions hold : either @xmath93 or @xmath94 in this section , we specialize theorem [ saddle.point ] to the controller - jammer game where the plant is linear , @xmath95 and the performance cost is quadratic . in this game the jammer is rewarded for remaining stealthy . we show that in this game , there is a region in the plant state space where the jammer s optimal policy is to randomize its channel selection . furthermore , an optimal control response to this optimal jammer action is nonlinear . consider a controller - jammer game for the plant ( [ plant.lin ] ) with the quadratic payoff @xmath96 here , @xmath97 is the kronecker symbol , and @xmath98 is the constant ` reward for stealthiness ' which the jammer receives if the channel does not change as a result of its action . as explained earlier , the rationale here is to reward the jammer for keeping the current channel in the link when excessive switching may reveal its presence , or may drain its resources . this controller - jammer game was analyzed in @xcite ( for a one - dimensional plant ) , where the region in the state - space was found where the game has a unique saddle point corresponding to a jammer s nontrivial strategy . such a region was found by computing the game value directly , which required a quite tedious analysis . here , we revisit this result of @xcite from a more general perspective , using conditions of theorem [ saddle.point ] . the corresponding function @xmath60 in this case is @xmath99 clearly , the function @xmath60 defined in ( [ lq.cost ] ) satisfies assumptions [ sigma.a1 ] and [ sigma.a3 ] . also in this case , the functions @xmath61 have the form @xmath100 \nonumber \\ & = & \gamma_j(x)+u^2+r_jq_ju(u+2\beta(x ) ) , \label{hj.lau3}\end{aligned}\ ] ] where @xmath101 @xmath102 , and @xmath103 for all @xmath14 . also , the available channels are assumed to be ordered according to their probability to become passing , that is , @xmath104 [ u.lq ] under condition ( [ qq ] ) , the set @xmath105 verifies properties ( i ) and ( ii ) stated in lemma [ u ] , and also satisfies assumptions [ sigma.a4 ] and [ sigma.a2 ] . under the above assumptions , the payoff functions @xmath89 and @xmath90 for the reduced zero - sum game become @xmath106 with these definitions , condition ( [ f = f ] ) reduces to the equation @xmath107 which admits real solutions if @xmath108 . also , condition ( [ df.df<0 ] ) reduces to the condition @xmath109 let @xmath110 the analysis of conditions ( [ f = f.lq ] ) , ( [ df.df<0.lq ] ) shows that only one of the solutions of equation ( [ f = f.lq ] ) , @xmath111 satisfies ( [ df.df<0.lq ] ) provided @xmath112 condition ( [ z.lau3a ] ) describes the region in the state space in which the jammer s optimal policy is to choose randomly between the channel @xmath27 currently in use and the most blocking channel @xmath69 . observe that in the case where the plant ( [ plant.lin ] ) is scalar and @xmath113 , we recover the exactly same condition as that obtained in @xcite by direct computation . that is , theorem [ saddle.point ] confirms the existence of the nontrivial optimal jammer s strategy for this region . we refer the reader to @xcite for the exact value of the optimal vector @xmath114 ; the calculation for the multidimensional plant ( [ plant.lin ] ) follows the same lines , and is omitted for the sake of brevity . we also point out that the optimal controller s policy ( [ u * ] ) is nonlinear . hence , any linear feedback policy that controller may employ assuming that its signals are transmitted over a _ bona fide _ packet dropping channel will lead to an inferior control performance . we interpret this situation as a signature of a successful dos attack by the jammer . in this paper we have analyzed a class of control problems over adversarial channels , in which the jammer actively attempts to disrupt communications between the controller and the plant . we have posed the problem as a static game , and have given necessary and sufficient conditions for such a game to have a nontrivial saddle point . the significance of these conditions is to allow a characterization of a set of plant s initial states for which a dos attack can be mounted that requires a nontrivial controller s response . for instance , in the linear quadratic problem analyzed in the paper the optimal control law is nonlinear . this gives the jammer an advantage over any linear control policy in those problems . the jammer achieves this outcome by randomizing its choice of a packet - dropping channel rather than operating packet dropping facility directly . on the other hand , the part of the state space where the jammer randomizes is determined by the jammer s cost of switching ( reward for not switching ) and transition probabilities of the current and the most blocking channels . if these parameters can be predicted / estimated by the controller , it has a chance of mitigating the attack by either eliminating those regions , or steering the plant so that it avoids visiting those regions . future work will be directed to further understanding conditions for dos attacks , with the aim to consider dynamic / multi - step control problems . another interesting question is whether associating a distinct payoff with one of the channels is necessary for the jammer to resort to randomization . e. garone , b. sinopoli , and a. casavola , lqg control over lossy tcp - like networks with probabilistic packet acknowledgements , international journal of systems , control and communications , vol . 55 - 81 , 2010 . a. gupta , c. langbort , and t. baar , optimal control in the presence of an intelligent jammer with limited actions , in proc . of 49th ieee conference on decision and control ( cdc ) , 1096 - 1101 , december 2010 . t. baar and y - w . wuh , a complete characterization of minimax and maximin encoder - decoder policies for communication channels with incomplete statistical description , ieee transactions on information theory , vol . 31 , no . 4 , 1985
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we revisit a one - step control problem over an adversarial packet - dropping link .
the link is modeled as a set of binary channels controlled by a strategic jammer whose intention is to wage a ` denial of service ' attack on the plant by choosing a most damaging channel - switching strategy .
the paper introduces a class of zero - sum games between the jammer and controller as a scenario for such attack , and derives necessary and sufficient conditions for these games to have a nontrivial saddle - point equilibrium . at this equilibrium ,
the jammer s optimal policy is to randomize in a region of the plant s state space , thus requiring the controller to undertake a nontrivial response which is different from what one would expect in a standard stochastic control problem over a packet dropping channel .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the need for the efficient use of the scarce spectrum in wireless applications has led to significant interest in the analysis of cognitive radio systems . one possible scheme for the operation of the cognitive radio network is to allow the secondary users to transmit concurrently on the same frequency band with the primary users as long as the resulting interference power at the primary receivers is kept below the interference temperature limit @xcite . note that interference to the primary users is caused due to the broadcast nature of wireless transmissions , which allows the signals to be received by all users within the communication range . note further that this broadcast nature also makes wireless communications vulnerable to eavesdropping . the problem of secure transmission in the presence of an eavesdropper was first studied from an information - theoretic perspective in @xcite where wyner considered a wiretap channel model . in @xcite , the secrecy capacity is defined as the maximum achievable rate from the transmitter to the legitimate receiver , which can be attained while keeping the eavesdropper completely ignorant of the transmitted messages . later , wyner s result was extended to the gaussian channel in @xcite . recently , motivated by the importance of security in wireless applications , information - theoretic security has been investigated in fading multi - antenna and multiuser channels . for instance , cooperative relaying under secrecy constraints was studied in @xcite@xcite . in @xcite , for amplify and forwad relaying scheme , not having analytical solutions for the optimal beamforming design under both total and individual power constraints , an iterative algorithm is proposed to numerically obtain the optimal beamforming structure and maximize the secrecy rates . although cognitive radio networks are also susceptible to eavesdropping , the combination of cognitive radio channels and information - theoretic security has received little attention . very recently , pei _ et al . _ in @xcite studied secure communication over multiple input , single output ( miso ) cognitive radio channels . in this work , finding the secrecy - capacity - achieving transmit covariance matrix under joint transmit and interference power constraints is formulated as a quasiconvex optimization problem . in this paper , we investigate the collaborative relay beamforming under secrecy constraints in the cognitive radio network . we first characterize the secrecy rate of the amplify - and - forward ( af ) cognitive relay channel . then , we formulate the beamforming optimization as a quasiconvex optimization problem which can be solved through convex semidefinite programming ( sdp ) . furthermore , we propose two sub - optimal null space beamforming schemes to reduce the computational complexity . we consider a cognitive relay channel with a secondary user source @xmath0 , a primary user @xmath1 , a secondary user destination @xmath2 , an eavesdropper @xmath3 , and @xmath4 relays @xmath5 , as depicted in figure [ fig : channel ] . we assume that there is no direct link between @xmath0 and @xmath2 , @xmath0 and @xmath1 , and @xmath0 and @xmath3 . we also assume that relays work synchronously to perform beamforming by multiplying the signals to be transmitted with complex weights @xmath6 . we denote the channel fading coefficient between @xmath0 and @xmath7 by @xmath8 , the fading coefficient between @xmath7 and @xmath2 by @xmath9 , @xmath7 and @xmath1 by @xmath10 and the fading coefficient between @xmath7 and @xmath3 by @xmath11 . in this model , the source @xmath0 tries to transmit confidential messages to @xmath2 with the help of the relays on the same band as the primary user s while keeping the interference on the primary user below some predefined interference temperature limit and keeping the eavesdropper @xmath3 ignorant of the information . it s obvious that our channel is a two - hop relay network . in the first hop , the source @xmath0 transmits @xmath12 to relays with power @xmath13=p_s$ ] . the received signal at the @xmath14 relay @xmath7 is given by @xmath15 where @xmath16 is the background noise that has a gaussian distribution with zero mean and variance of @xmath17 . in the af scenario , the received signal at @xmath7 is directly multiplied by @xmath18 without decoding , and forwarded to @xmath2 . the relay output can be written as @xmath19 the scaling factor , @xmath20 is used to ensure @xmath21=|w_m|^2 $ ] . there are two kinds of power constraints for relays . first one is a total relay power constraint in the following form : @xmath22 where @xmath23^t$ ] and @xmath24 is the maximum total power . @xmath25 and @xmath26 denote the transpose and conjugate transpose , respectively , of a matrix or vector . in a multiuser network such as the relay system we study in this paper , it is practically more relevant to consider individual power constraints as wireless nodes generally operate under such limitations . motivated by this , we can impose @xmath27 or equivalently @xmath28 where @xmath29 denotes the element - wise norm - square operation and @xmath30 is a column vector that contains the components @xmath31 . @xmath32 is the maximum power for the @xmath14 relay node . the received signals at the destination @xmath2 and eavesdropper @xmath3 are the superposition of the messages sent by the relays . these received signals are expressed , respectively , as @xmath33 where @xmath34 and @xmath35 are the gaussian background noise components with zero mean and variance @xmath36 , at @xmath2 and @xmath3 , respectively . it is easy to compute the received snr at @xmath2 and @xmath3 as @xmath37 where @xmath38 denotes the mutual information . the interference at the primary user is latexmath:[\ ] ] where superscript @xmath43 denotes conjugate operation . then , the received snr at the destination and eavesdropper , and the interference on primary user can be written , respectively , as @xmath44 with these notations , we can write the objective function of the optimization problem ( i.e. , the term inside the logarithm in ( [ srate ] ) ) as @xmath45 if we denote @xmath46 , @xmath47 , define @xmath48 , and employ the semidefinite relaxation approach , we can express the beamforming optimization problem as @xmath49 the optimization problem here is similar to that in @xcite . the only difference is that we have an additional constraint due to the interference limitation . thus , we can use the same optimization framework . the optimal beamforming solution that maximizes the secrecy rate in the cognitive relay channel can be obtained by using semidefinite programming with a two dimensional search for both total and individual power constraints . for simulation , one can use the well - developed interior point method based package sedumi @xcite , which produces a feasibility certificate if the problem is feasible , and its popular interface yalmip @xcite . it is important to note that we should have the optimal @xmath50 to be of rank - one to determine the beamforming vector . while proving analytically the existence of a rank - one solution for the above optimization problem seems to be a difficult task , we would like to emphasize that the solutions are rank - one in our simulations . thus , our numerical result are tight . also , even in the case we encounter a solution with rank higher than one , the gaussian randomization technique is practically proven to be effective in finding a feasible , rank - one approximate solution of the original problem . details can be found in @xcite . obtaining the optimal solution requires significant computation . to simplify the analysis , we propose suboptimal null space beamforming techniques in this section . we choose @xmath51 to lie in the null space of @xmath52 . with this assumption , we eliminate @xmath3 s capability of eavesdropping on @xmath2 . mathematically , this is equivalent to @xmath53 , which means @xmath51 is in the null space of @xmath54 . we can write @xmath55 , where @xmath56 denotes the projection matrix onto the null space of @xmath54 . specifically , the columns of @xmath56 are orthonormal vectors which form the basis of the null space of @xmath54 . in our case , @xmath56 is an @xmath57 matrix . the total power constraint becomes @xmath58 . the individual power constraint becomes @xmath59 under the above null space beamforming assumption , @xmath60 is zero . hence , we only need to maximize @xmath61 to get the highest achievable secrecy rate . @xmath61 is now expressed as @xmath62 the interference on the primary user can be written as @xmath63 defining @xmath64 , we can express the optimization problem as @xmath65 this problem can be easily solved by semidefinite programming with bisection search @xcite . in this section , we choose @xmath51 to lie in the null space of @xmath52 and @xmath66 . mathematically , this is equivalent to requiring @xmath67 , and @xmath68 . we can write @xmath69 , where @xmath70 denotes the projection matrix onto the null space of @xmath54 and @xmath71 . specifically , the columns of @xmath70 are orthonormal vectors which form the basis of the null space . in our case , @xmath70 is an @xmath72 matrix . the total power constraint becomes @xmath73 . the individual power constraint becomes @xmath74 . with this beamforming strategy , we again have @xmath75 . moreover , the interference on the primary user is now reduced to @xmath76 which is the sum of the forwarded additive noise components present at the relays . now , the optimization problem becomes @xmath77 again , this problem can be solved through semidefinite programming . with the following assumptions , we can also obtain a closed - form characterization of the beamforming structure . since the interference experienced by the primary user consists of the forwarded noise components , we can assume that the interference constraint @xmath78 is inactive unless @xmath41 is very small . with this assumption , we can drop this constraint . if we further assume that the relays operate under the total power constraint expressed as @xmath79 , we can get the following closed - form solution : @xmath80 where @xmath81 is the largest generalized eigenvalue of the matrix pair @xmath82 . and positive definite matrix @xmath83 , @xmath84 is referred to as a generalized eigenvalue eigenvector pair of @xmath82 if @xmath84 satisfy @xmath85 @xcite . ] hence , the maximum secrecy rate is achieved by the beamforming vector @xmath86 where @xmath87 is the eigenvector that corresponds to @xmath88 and @xmath89 is chosen to ensure @xmath90 . the discussion in section [ sec : op ] can be easily extended to the case of more than one primary user in the network . each primary user will introduce an interference constraint @xmath91 which can be straightforwardly included into ( [ optimal ] ) . the beamforming optimization is still a semidefinite programming problem . on the other hand , the results in section [ sec : op ] can not be easily extended to the multiple - eavesdropper scenario . in this case , the secrecy rate for af relaying is @xmath92 , where the maximization is over the rates achieved over the links between the relays and different eavesdroppers . hence , we have to consider the eavesdropper with the strongest channel . in this scenario , the objective function can not be expressed in the form given in ( [ srate ] ) and the optimization framework provided in section [ sec : op ] does not directly apply to the multi - eavesdropper model . however , the null space beamforming schemes discussed in section [ sec : null ] can be extended to the case of multiple primary users and eavesdroppers under the condition that the number of relay nodes is greater than the number of eavesdroppers or the total number of eavesdroppers and primary users depending on which null space beamforming is used . the reason for this condition is to make sure the projection matrix @xmath93 exists . note that the null space of @xmath94 channels in general has the dimension @xmath95 where @xmath4 is the number of relays . we assume that @xmath96 , @xmath97 are complex , circularly symmetric gaussian random variables with zero mean and variances @xmath98 , @xmath99 , @xmath100 and @xmath101 respectively . in this section , each figure is plotted for fixed realizations of the gaussian channel coefficients . hence , the secrecy rates in the plots are instantaneous secrecy rates . in fig . [ fig:1 ] , we plot the optimal secrecy rates for the amplify - and - forward collaborative relay beamforming system under both individual and total power constraints . we also provide , for comparison , the secrecy rates attained by using the suboptimal beamforming schemes . the fixed parameters are @xmath102 , @xmath103 , and @xmath104 . since af secrecy rates depend on both the source and relay powers , the rate curves are plotted as a function of @xmath105 . we assume that the relays have equal powers in the case in which individual power constraints are imposed , i.e. , @xmath106 . it is immediately seen from the figure that the suboptimal null space beamforming achievable rates under both total and individual power constraints are very close to the corresponding optimal ones . especially , they are nearly identical in the high snr regime , which suggests that null space beamforming is optimal at high snrs . thus , null space beamforming schemes are good alternatives as they are obtained with much less computational burden . moreover , we interestingly observe that imposing individual relay power constraints leads to small losses in the secrecy rates . in fig . [ fig:11 ] , we change the parameters to @xmath107 , @xmath108 and @xmath104 . in this case , channels between the relays and the eavesdropper and between the relays and the primary - user are on average stronger than the channels between the relays and the destination . we note that beamforming schemes can still attain good performance and we observe similar trends as before . in fig . [ fig:2 ] , we plot the optimal secrecy rate and the secrecy rates of the two suboptimal null space beamforming schemes ( under both total and individual power constraints ) as a function of the interference temperature limit @xmath41 . we assume that @xmath109 . it is observed that the secrecy rate achieved by beamforming in the null space of both the eavesdropper s and primary user s channels ( bnep ) is almost insensitive to different interference temperature limits when @xmath110 since it always forces the signal interference to be zero regardless of the value of @xmath41 . it is further observed that beamforming in the null space of the eavesdropper s channel ( bne ) always achieves near optimal performance regardless the value of @xmath41 under both total and individual power constraints . in this paper , collaborative relay beamforming in cognitive radio networks is studied under secrecy constraints . optimal beamforming designs that maximize secrecy rates are investigated under both total and individual relay power constraints . we have formulated the problem as a semidefinite programming problem and provided an optimization framework . in addition , we have proposed two sub - optimal null space beamforming schemes to simplify the computation . finally , we have provided numerical results to illustrate the performances of different beamforming schemes . a. wyner `` the wire - tap channel , '' _ bell . syst tech . j _ , vol.54 , no.8 , pp.1355 - 1387 , jan 1975 . i. csiszar and j. korner `` broadcast channels with confidential messages , '' _ ieee trans . inform . theory _ , vol.it-24 , no.3 , pp.339 - 348 , may 1978 . v. nassab , s. shahbazpanahi , a. grami , and z .- q . luo , `` distributed beamforming for relay networks based on second order statistics of the channel state information , '' _ ieee trans . on signal proc . 56 , no 9 , pp . 4306 - 4316 , g. zheng , k. k. wong , a. paulraj , and b. ottersten , `` robust collaborative - relay beamforming , '' _ ieee trans . on signal proc . 57 , no . 8 , aug . 2009 z - q luo , wing - kin ma , a.m .- c . so , yinyu ye , shuzhong zhang `` semidefinite relaxation of quadratic optimization problems '' _ ieee signal proc . magn . 3 , may 2010 j. lofberg , `` yalmip : a toolbox for modeling and optimization in matlab , '' _ proc . the cacsd conf . _ , taipei , taiwan , 2004 . s. boyd and l. vandenberghe , convex optimization . cambridge , u.k . : cambridge univ . press , 2004 .
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in this paper , a cognitive relay channel is considered , and amplify - and - forward ( af ) relay beamforming designs in the presence of an eavesdropper and a primary user are studied .
our objective is to optimize the performance of the cognitive relay beamforming system while limiting the interference in the direction of the primary receiver and keeping the transmitted signal secret from the eavesdropper .
we show that under both total and individual power constraints , the problem becomes a quasiconvex optimization problem which can be solved by interior point methods .
we also propose two sub - optimal null space beamforming schemes which are obtained in a more computationally efficient way .
_ index terms : _ amplify - and - forward relaying , cognitive radio , physical - layer security , relay beamforming .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
atmospheric hazes , present in a range of solar system and extrasolar planetary atmospheres , play an important role in physical and chemical processes occurring in the atmosphere and for terrestrial planets , on the surface . haze particles affect the radiative balance of an atmosphere , may serve as condensation nuclei for clouds and rain , play a role in fluvial and aeolian processes @xcite , and affect the elemental budget of an atmosphere and surface . the effect on haze particles on the temperature structure of an atmosphere has implications for the habitability of a planet and an organic haze , such as seen in the atmosphere of saturn s moon titan , may potentially serve as the building blocks of life @xcite . while haze formation in n@xmath0/ch@xmath1 atmospheres has been extensively studied for decades in the laboratory through the production of titan aerosol analogues or `` tholins '' ( see @xcite ) , the effect of other atmospheric constituents on the formation of haze in planetary atmospheres has not been well studied . carbon monoxide ( co ) is particularly interesting because it might serve as a source of oxygen for incorporation into photochemical aerosols affecting both their radiative and chemical properties and it is found in numerous atmospheres throughout the universe . it is present in the hazy , reducing n@xmath0/ch@xmath1 atmospheres of titan @xcite , pluto @xcite , and triton @xcite and in the h@xmath0 dominated atmospheres of the giant planets @xcite . recently co and ch@xmath1 , in addition to handful of other molecules , have also been detected in the atmospheres of extrasolar planets ( see e.g. @xcite ) and haze layers have also been invoked to explain relatively featureless spectra of a number of exoplanets ( see e.g. @xcite ) . it is therefore important to understand the effect of co on the formation of planetary atmospheric hazes . a few previous titan atmosphere simulation experiments have included co in their initial gas mixtures . @xcite and @xcite focused on the effects of co on the production of gas phase products . @xcite focused on both gas and solid phase composition and observed the formation of ketones and carbonyls in their solid phase products . @xcite reported the detection of amino acids and nucleotide bases in the solid products . however , none of these investigations reported the effect of co on the size and number of haze particles produced , which are important parameters for the radiative effects of haze particles and for the total organic inventory found in haze particles . we present here an experimental investigation of the effect of co on the formation of planetary atmospheric hazes including measurements of the size and number density of haze particles produced using a spark discharge source or uv photons to irradiate a range of mixtures of ch@xmath1 , co , and n@xmath0 . figure [ fig : experiment ] shows a schematic of our experimental setup . previous uv and spark experiments were performed using a similar setup by @xcite and @xcite , respectively . we introduced co ( 99.999% airgas ) in volume mixing ratios ranging from 50 ppm ( the abundance in titan s atmosphere @xcite ) to 5% and ch@xmath1 ( 99.99% airgas ) in volume mixing ratios of 0.1% and 2% ( see table 1 ) into a stainless steel mixing chamber , then filled the mixing chamber to 600 psi with n@xmath0 ( 99.999% airgas ) . we allow the gases to mix for a minimum of 8 hours before running the experiment . the reactant gases continuously flowed through a cold trap , to remove trace impurities in the gases , before flowing through a glass reaction cell . we maintain a flow rate of 100 standard cubic centimeters per minute ( sccm ) using a mass flow controller ( mykrolis fc-2900 ) . the glass cold trap consists of two lines ; one line is immersed in a slurry of 200 proof ethanol and liquid nitrogen , while the other line bypasses the cold trap entirely . for this work , the bypass line remained closed . the slurry remained at a temperature of @xmath2 - 115 @xmath3c . the temperature of the gas line into the production cell was also monitored and was found to be unaffected by the use of the cold bath . we maintain the pressure in the reaction cell between 620 and 640 torr ( atmospheric pressure in boulder , co ) at room temperature . we expose the reactant gases to one of two energy sources , spark discharge from a tesla coil or fuv photons , which initiate chemistry leading to particle formation . the experimental setup is the same for both energy sources until the gases reach the reaction cells . a tesla coil ( electro technic products ) is connected to the spark reaction cell , while the uv reaction cell is connected to a deuterium lamp with a mgf@xmath0 window ( hamamatsu l1835 ) . aerosol particle formation results from gas phase chemistry initiated by energy from the tesla coil or the deuterium lamp . photons play a dominant role in the dissociation and ionization of chemical species that eventually lead to the formation of aerosols in titan s atmosphere @xcite . it is therefore of paramount importance to investigate aerosol formation from photochemistry . the deuterium lamp we used for these experiments is a continuum source that produces photons from 115 - 400 nm ( with major peaks near 121 and 160 nm ) . although these photons are not sufficiently energetic to directly dissociate n@xmath0 and co , @xcite and @xcite demonstrated that nitrogen is participating in the chemistry in our reaction cell . work is ongoing in our laboratory to understand the mechanism(s ) responsible for the observed nitrogen incorporation . it seems likely that co is dissociated through an analogous mechanism due to the similarity of their bonds . we use the electrical discharge because it is known to dissociate the triple bonds of co and n@xmath0 and is therefore an analog of the relatively energetic environment of upper atmospheres . however , we acknowledge that the resulting energy density is higher than the energy available in most planetary atmospheres to initiate chemistry . we use a tesla coil that can operate at a range of voltages . as described in @xcite , we set the tesla coil to minimize the energy density while still producing sufficient aerosol using 2% ch@xmath1 in n@xmath0 for our analytical techniques and used that setting for every experiment . the flow exits the reaction cell and flows into a scanning mobility particle sizer ( smps ) , which measures the distribution of particle sizes . the smps has three parts : an electrostatic classifier ( tsi 3080 ) , a differential mobility analyzer ( dma , tsi 3081 ) , and a condensation particle counter ( cpc , tsi 3775 ) . the polydisperse aerosol first enters the dma , where an electric field is applied to the flow of particles , which are then size selected based on their electrical mobility against the drag force provided by the sheath flow . sheath flows of either 3 l / min or 10 l / min were used depending on the range of particle sizes produced in the experiment ( covering @xmath4 ranging from 14.5 to 673 nm or 7.4 to 289 nm , respectively ) . once size - selected , the particles enter the cpc where the number of particles is measured by light scattering . in this manner , we measure the number of particles as a function of their mobility diameter ( @xmath4 ) . our standard flow rate of 100 sccm is determined by requirements of other instruments ( see e.g. @xcite ) and is used here for consistency ; however , the smps requires a higher flow rate . we therefore add an additional flow of n@xmath0 after the particles exit the reaction chamber bringing the total flow rate to 260 sccm . the dilution caused by the additional flow of n@xmath0 is accounted for during data analysis . our _ in situ _ analysis technique prevents the particles from being exposed to earth s atmosphere and does not require sample collection , which could alter the observed particle sizes and number densities . however , real time analysis requires higher production rates . additionally , the smps requires pressures at or near atmospheric pressure for operation . for those two reasons , we ran the experiments presented here at 620 - 640 torr ( boulder , co , atmospheric pressure , altitude @xmath21600 m ) . this pressure is higher than the surface pressure on pluto and triton and in titan s thermosphere where the chemical processes that result in formation of aerosol begin . here we are interested only in comparing differences resulting from the addition of co and from the choice of energy source at our standard experimental pressure . our previous titan simulation experiments have used 0.1% ch@xmath1 for uv experiments @xcite and 2% ch@xmath1 for spark experiments @xcite . while 2% ch@xmath1 is analogous to titan s atmosphere , the 0.1% ch@xmath1 is determined by experimental production constraints . aerosol formation in our setup from the fuv lamp peaks near 0.1% ch@xmath1 due to optical depth in the reaction cell @xcite and previous work has determined that the aerosol composition does not vary strongly with ch@xmath1 concentration @xcite . both for comparison purposes and to extend the range of planetary atmospheres where our results may shed light on aerosol formation , experiments were run for a range of co concentrations using 0.1% and 2% ch@xmath1 . in titan s atmosphere the ch@xmath1 concentration has almost certainly varied over time , and since the co abundance is tied both to ch@xmath1 chemistry and the plumes of enceladus @xcite , the co abundance has almost certainly varied over time as well . measurement of absolute mixing ratios of co and ch@xmath1 in exoplanet atmospheres is still quite difficult ( see e.g. the discussion in @xcite ) and further emphasizes the need to explore a range of co and ch@xmath1 concentrations . the smps measurements of particle size as a function of initial co concentration are shown in panel a of figure [ fig : diameter ] . for the spark experiments , the addition of 50 ppm of co results in a decrease in particle diameter compared to experiments performed with no co , while even the addition of 50 ppm co results in an increase in particle size for the uv experiments . in general , the particle diameter increases as a function of increasing co starting at 50 ppm for both spark and uv energy sources regardless of methane concentration . remarkably , the addition of 5% co results in the formation of particles with diameters 2 - 3 times larger than the experiments that did not include co ; particle distributions emphasizing this point are shown in figure [ fig : dist ] . while the strong influence of co on particle diameter could potentially be attributed to the increase of carbon in the system , this behavior is not observed as a function of increasing ch@xmath1 concentration in n@xmath0/ch@xmath1 in the same experiment @xcite . for uv experiments , particle size decreases as a function of increasing ch@xmath1 from 0.01% to 10% ch@xmath1 , while for the spark experiments the particle diameter increased until a peak at 2% ch@xmath1 and then decreased . the number density of particles , shown in panel b of figure [ fig : diameter ] , exhibits behavior very similar to that of particle diameter ; the number density increases as a function of increasing co concentration for both energy sources at both ch@xmath1 concentrations from 50 ppm co to 5% co. as we observed with the n@xmath0/ch@xmath5 only experiments @xcite , the uv experiments always produce more particles than the spark discharge experiments . for number density , the addition of 50 ppm co results in decreases compared to no co for both spark experiments ( 0.1% ch@xmath1 and 2% ch@xmath1 ) and a slight decrease for the 0.1% ch@xmath1 uv experiment . however , the addition of 50 ppm co to the 2% ch@xmath1 uv experiment results in an increase in number density . this indicates that the presence of co in titan s atmosphere can not be ignored in titan atmosphere simulation experiments . the aerosol mass loading calculations are performed assuming that the density of the particle is 1 g/@xmath6 . an extensive discussion of the assumptions made in analysis of smps measurements , as well experimental determinations of tholin particle density for n@xmath0/ch@xmath1 experiments can be found in @xcite . a variation of particle density is observed in both spark and uv experiments in the absence of co. density calculations require additional measurements not obtained for this work . however , the aerosol mass loading calculations are presented here , despite the assumption of density , so that the results may be compared to other works . since both particle size and number density increase , aerosol mass loading also increases as a function of increasing co concentration . the increase is most pronounced for the spark and 2% ch@xmath1 uv experiments . for the 2% ch@xmath1 uv experiment , an increase of more than 2 orders of magnitude is observed in the mass loading with a concentration of 5% co compared to the case where no co is used . taken together , the measurements of particle diameter and number density demonstrate that the addition of co to n@xmath0/ch@xmath1 experiments results in more , larger particles . co is therefore affecting both the formation and growth of aerosol particles in our experiments , even at relatively low concentrations . further work , particularly on the composition of these particles , is necessary to fully understand the chemical mechanisms by which co is affecting particle formation and growth . however , we have a few possible explanations for the observed behavior . first , previous tholin works have suggested that the presence of h@xmath0 and h can decrease particle production . this explanation is often invoked to explain results of multiple plasma experiments which demonstrate that aerosol production first increases with increasing ch@xmath1 concentration , reaches a peak , and then decreases . the decrease is attributed to an increase in the production of h@xmath0 and h at high ch@xmath1 concentrations @xcite . the addition of h@xmath0 has also been shown to reduce aerosol formation in early earth simulation experiments that photolyzed ch@xmath1 , co@xmath0 , and n@xmath0 @xcite . perhaps oxygen , produced from co in our experiments , is reacting with h@xmath0 and h and removing them from the system . this could then lead to an aerosol formation by reducing one of the limiting factors . second , as mentioned earlier , the presence of co in the gas mixture does increase the total amount of carbon atoms present in the system . based on our work looking at aerosol production as a function of ch@xmath1 concentration , we know that simply increasing the amount of carbon in the system does not result in the formation of more aerosol for spark or uv experiments . however , the molecule that carries the carbon atom play a significant role . in the uv experiments , the decrease in aerosol production as a function of increasing ch@xmath1 in the gas mixture has been attributed to the increase in optical depth at the wavelengths produced by our fuv lamp @xcite . however , co does not absorb these wavelengths and therefore may serve as source of carbon in the experiment without increasing the optical depth in the cell ; thus more photons are available to drive aerosol chemistry . for spark experiments , increasing co in the system increases the amount of carbon available without increasing the amount of hydrogen present in the system , which may also result in an increase of aerosol formation . third , if the degree of oxygen participation in the chemistry is increasing as a function of increasing co then the vapor pressures of the molecules produced may be lower , on average , than the molecules produced from n@xmath0/ch@xmath1 mixtures . this oxygen incorporation may shift the partitioning of gas and solid phase species toward the solid phase , which would result in the formation of more aerosol . the actual chemistry occurring in the reaction cell may be a combination of these three ideas or some other possibility . future measurement of the aerosol and gas phase composition will provide insight into the partitioning of gas and solid phase species and allow us to assess the degree that oxygen is participating in the chemistry occurring in our experiment and the possible effect of co on hydrogen chemistry . isotopic labeling experiments will help determine the degree to which carbon in the aerosol originates from co or ch@xmath1 . we obtained _ in situ _ particle size and number density measurements for tholins produced using co concentrations from 50 ppm to 5% and ch@xmath1 concentrations of 0.1% and 2% and two different energy sources , spark discharge and uv as summarized in table 1 . for both energy sources and both ch@xmath1 concentrations investigated , the particle size , number density and aerosol mass loading all increase as a function of increasing co concentration above 50 ppm co. the inclusion of co has a dramatic effect on aerosol production , increasing the aerosol mass loading by orders of magnitude over the range of co mixing ratios investigated . the fact that both the particle size and number density increase indicates that inclusion of co increases both particle formation and growth . intriguingly , the behavior as a function of co mixing ratio is quite similar for both energy sources , in contrast to the behavior observed with only n@xmath0/ch@xmath1 gas mixtures where the production rate trends differ greatly with ch@xmath1 concentrations based on energy source @xcite . products of co destruction may be decreasing the presence of h@xmath0 and h in the reaction cell , species which are believed to inhibit aerosol formation , thus resulting in an increase in particle formation and growth . co may effect aerosol formation by serving as an additional source of carbon , without affecting the optical depth at fuv wavelengths or introducing more hydrogen into the system . the increase in aerosol formation may also result from a shift in the partitioning between gas and solid phase species due to changes in vapor pressures of the molecules produced . however , further work is necessary to understand the effects of co on the gas phase and particle phase composition before the role of co in aerosol formation can be fully understood . , r. , irwin , p. g. j. , teanby , n. a. , lellouch , e. , bzard , b. , vinatier , s. , nixon , c. a. , fletcher , l. , howett , c. , calcutt , s. b. , bowles , n. e. , flasar , f. m. , & taylor , f. w. 2007 , icarus , 186 , 354 , l. a. , tomasko , m. g. , archinal , b. a. , becker , t. l. , bushroe , m. w. , cook , d. a. , doose , l. r. , galuszka , d. m. , hare , t. m. , howington - kraus , e. , karkoschka , e. , kirk , r. l. , lunine , j. i. , mcfarlane , e. a. , redding , b. l. , rizk , b. , rosiek , m. r. , see , c. , & smith , p. h. 2007 , planet . space sci . , 55 , 2015
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organic haze plays a key role in many planetary processes ranging from influencing the radiation budget of an atmosphere to serving as a source of prebiotic molecules on the surface .
numerous experiments have investigated the aerosols produced by exposing mixtures of n@xmath0/ch@xmath1 to a variety of energy sources .
however , many n@xmath0/ch@xmath1 atmospheres in both our solar system and extrasolar planetary systems also contain co. we have conducted a series of atmosphere simulation experiments to investigate the effect of co on formation and particle size of planetary haze analogues for a range of co mixing ratios using two different energy sources , spark discharge and uv .
we find that co strongly affects both number density and particle size of the aerosols produced in our experiments and indicates that co may play an important , previously unexplored , role in aerosol chemistry in planetary atmospheres .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the possibility of muon colliders was introduced by skrinsky et al.@xcite , neuffer@xcite , and others . more recently , several workshops and collaboration meetings have greatly increased the level of discussion@xcite,@xcite . a detailed feasibility study@xcite was presented at snowmass 96 . hadron collider energies are limited by their size , and technical constraints on bending magnetic fields . lepton ( or ) colliders , because they undergo simple , single - particle interactions , can reach higher energy final states than an equivalent hadron machine . however , extension of @xmath1 colliders to multi - tev energies is severely performance - constrained by beamstrahlung , the luminosity @xmath2 of a lepton collider can be written : [ lumeq ] 1 4e n_2 r_o p_beam _ y n_collisions where @xmath3 is the average vertical ( assumed smaller ) beam spot size , @xmath4 is the beam energy , @xmath5 is the total beam power , @xmath6 is the electromagnetic constant , @xmath7 is the classical radius , and @xmath8 is the number of photons emitted by one bunch as it passes through the opposite one . if this number is too large then the beamstrahlung background of electron pairs and other products becomes unacceptable . as the energy rises , the luminosity , for the same event rate , must rise as the square of the energy . for an electron collider , @xmath9 , and , for a fixed background , we have the severe requirement : e^3 in a muon collider there are two significant changes : 1 ) the classical radius @xmath7 is now that for the muon and is 200 times smaller ; and 2 ) the number of collisions a bunch can make @xmath10 is no longer 1 , but is now related to the average bending field in the muon collider ring , for 6 t , it is 900 . in addition , with muons , synchrotron radiation is negligible , and the collider is circular . in practice this means that it can be much smaller than a linear electron machine . the linacs for the 0.5 tev nlc will be 20 km long . the ring for a muon collider of the same energy would be only about 1.2 km circumference . there are , of course , technical difficulties in making sufficient muons , cooling and accelerating them before they decay and dealing with the decay products in the collider ring . despite these difficulties , it appears possible that high energy muon colliders might have luminosities comparable to or , at energies of several tev , even higher than those in colliders . the basic parameters of a 4 tev collider are shown schematically in fig.[schematic ] and given in tb.[sum ] together with those for a 0.5 tev demonstration machine based on the ags as an injector . it is assumed that a demonstration version based on upgrades of the fermilab machines would also be possible . .parameters of collider rings [ cols="<,<,^,^ " , ] the proton driver is a high - intensity ( four bunches of @xmath11 protons per pulse ) 30 gev proton synchrotron , operating at a repetition rate of 15 hz . two of the bunches are used to make @xmath12 s and two to make @xmath13 s . prior to targeting the bunches are compressed to an rms length of 1 ns . for a demonstration machine using the ags@xcite , two bunches of @xmath14 at a repetition rate of 2.5 hz at @xmath15gev could be used . predictions of nuclear monte - carlo programs@xcite@xcite@xcite suggest that @xmath16 production is maximized by the use of heavy target materials , and that the production is peaked at a relatively low pion energy ( @xmath17mev ) , substantially independent of the initial proton energy . cooling requirements dictate that the target be liquid : liquid lead and gallium are under consideration . in order to avoid shock damage to a container , the liquid could be in the form of a jet . pions are captured from the target by a high - field ( @xmath18 t , 15 cm aperture ) hybrid magnet : superconducting on the outside , and a water cooled bitter solenoid on the inside . a preliminary design@xcite has a bitter magnet with an inside coil diameter of 24 cm ( space is allowed for a 4 cm heavy metal shield inside the coil ) and an outside diameter of 60 cm ; it provides half ( 10 t ) of the total field , and would consume approximately 8 mw . the superconducting magnet has a set of three coils , all with inside diameters of 70 cm and is designed to give 10 t at the target and provide the required tapered field to match into the decay channel . the decay channel consists of a periodic superconducting solenoidal ( @xmath19 t and radius @xmath20 cm ) . a linac is introduced along the decay channel , with frequencies and phases chosen to deaccelerate the fast particles and accelerate the slow ones ; i.e. to phase rotate the muon bunch . [ evsctpol2 ] shows the energy vs ct at the end of the decay channel . the selected muons have a mean energy 150 mev , rms bunch length @xmath21 m , and rms momentum spread @xmath18% ( @xmath22% , @xmath23 ) . the number of muons per initial proton in this selected bunch is @xmath24 0.3 . if nothing is done then the average muon polarization is about 0.19 . if higher polarization is desired , some selection of muons from forward pion decays @xmath25 is required . this can be done by momentum selecting the muons at the end of the decay and phase rotation channel . a snake@xcite is used to generate the required dispersion . varying the selected minimum momentum of the muons yields polarization as a function of luminosity loss as shown in fig.[polvscut ] . dilutions introduced in the cooling have been calculated@xcite and are included . a siberian snake is also required in the final collider ring . for the required collider luminosity , the phase - space volume must be greatly reduced ; and this must be done within the @xmath26 lifetime . cooling by synchrotron radiation , conventional stochastic cooling and conventional electron cooling are all too slow . optical stochastic cooling@xcite , electron cooling in a plasma discharge@xcite and cooling in a crystal lattice@xcite are being studied , but appear very difficult . ionization cooling@xcite of muons seems relatively straightforward . in ionization cooling , the beam loses both transverse and longitudinal momentum as it passes through a material medium . subsequently , the longitudinal momentum can be restored by coherent reacceleration , leaving a net loss of transverse momentum . the equation for transverse cooling ( with energies in gev ) is : @xmath27 where @xmath28 is the normalized emittance , @xmath29 is the betatron function at the absorber , @xmath30 is the energy loss , and @xmath31 is the radiation length of the material . the first term in this equation is the coherent cooling term , and the second is the heating due to multiple scattering . this heating term is minimized if @xmath29 is small ( strong - focusing ) and @xmath31 is large ( a low - z absorber ) . energy spread is reduced by placing a transverse variation in absorber density or thickness at a location where position is energy dependent , i.e. where there is dispersion . the use of such wedges can reduce energy spread , but it simultaneously increases transverse emittance in the direction of the dispersion . it thus allows the exchange of emittance between the longitudinal and transverse directions . the cooling is obtained in a series of cooling stages . in general , each stage consists of three components with matching sections between them : 1 . a fofo lattice consisting of spaced axial solenoids with alternating field directions and lithium hydride absorbers placed at the centers of the spaces between them , where the @xmath29 s are minimum . a lattice consisting of more widely separated alternating solenoids , and bending magnets between them to generate dispersion . at the location of maximum dispersion , wedges of lithium hydride are introduced to interchange longitudinal and transverse emittance . 3 . a linac to restore the energy lost in the absorbers . in a few of the later stages , current carrying lithium rods replace item ( 1 ) above . in this case the rod serves simultaneously to maintain the low @xmath29 , and attenuate the beam momenta . similar lithium rods , with surface fields of @xmath32 t , were developed at novosibirsk and have been used as focusing elements at fnal and cern@xcite . the emittances , transverse and longitudinal , as a function of stage number , are shown in fig.[cooling ] . in the first 10 stages , relatively strong wedges are used to rapidly reduce the longitudinal emittance , while the transverse emittance is reduced relatively slowly . the object is to reduce the bunch length , thus allowing the use of higher frequency and higher gradient rf in the reacceleration linacs . in the next 7 stages , the emittances are reduced close to their asymptotic limits . in the last 3 stages , using lithium rods , there are no wedges and the energy is allowed to fall to about 15 mev . transverse cooling continues , and the momentum spread is allowed to rise . the total length of the system is 750 m , and the total acceleration used is 5 gev . the fraction of muons remaining at the end of the cooling system is calculated to be @xmath33% . following cooling and initial bunch compression the beams must be rapidly accelerated to full energy ( 2 tev , or 250 gev ) . a sequence of recirculating accelerators ( similar to that used at cebaf)could be used but would be relatively expensive . a more economical solution would be to use fast pulsed magnets in synchrotrons with rf systems consisting of significant lengths of superconducting linac . for the final acceleration to 2 tev in the high energy machine , the power consumed by a ring using only pulsed magnets would be excessive and alternating pulsed and superconducting magnets@xcite are used instead . after acceleration , the @xmath12 and @xmath13 bunches are injected into a separate storage ring . the highest possible average bending field is desirable to maximize the number of revolutions before decay , and thus maximize the luminosity . collisions occur in one , or perhaps two , very low-@xmath34 interaction areas . the magnet design is complicated by the fact that the @xmath26 s decay within the rings ( @xmath35 ) , producing electrons whose mean energy is approximately 0.35 that of the muons . these electrons travel toward the inside of the ring dipoles , radiating a fraction of their energy as synchrotron radiation towards the outside of the ring , and depositing the rest on the inside . the total average power deposited , in the ring , in the 4 tev machine is 13 mw . the beam must thus be surrounded by a @xmath24 6 cm thick warm shield@xcite , which is located inside a large aperture conventional superconducting magnet . the quadrupoles can use warm iron poles placed as close to the beam as practical , with coils either superconducting or warm , as dictated by cost considerations . in order to maintain a bunch with rms length 3 mm , without excessive rf , an isochronous lattice , of the dispersion wave type@xcite is used . for the 3 mm beta at the intersection point , the maximum beta s in both x and y are of the order of 400 km ( 14 km in the 0.5 tev machine ) . local chromatic correction is essential . two lattices have been generated@xcite@xcite , one of which@xcite , after the application of octupole and decapole correctors , has been shown to have an adequate calculated dynamic aperture . studies of the resistive wall impedance instabilities indicate that the required muon bunches would be unstable if uncorrected . in any case , the rf requirements to maintain such bunches would be excessive . bns@xcite damping , applied by rf quadrupoles@xcite , is one possible solution , but needs more careful study . monte carlo study@xcite,@xcite indicated that the background , though serious , should not be impossible . further reductions are expected as the shielding is optimized , and it should be possible to design detectors that are less sensitive to the neutrons and photons present . there would also be a background from the presence of a halo of near full energy muons in the circulating beam . the beam will need careful preparation before injection into the collider , and a collimation system will have to be designed to be located on the opposite side of the ring from the detector . there is a small background from incoherent ( i.e. @xmath36 ) pair production in the 4 tev collider case . the cross section is estimated to be @xmath37 , which would give rise to a background of @xmath38 electron pairs per bunch crossing . approximately @xmath39 of these , will be trapped inside the tungsten nose cone , but those with energy between 30 and @xmath40mev will enter the detector region . * considerable progress has been made on a scenario for a 2 + 2 tev , high luminosity collider . much work remains to be done , but no obvious show stopper has yet been found . * the two areas that could present serious problems are : 1 ) unforeseen losses during the 25 stages of cooling ( a 3% loss per stage would be very serious ) ; and 2 ) the excessive detector background from muon beam halo . * many technical components require development : a large high field solenoid for capture , low frequency rf linacs , multi - beam pulsed and/or rotating magnets for acceleration , warm bore shielding inside high field dipoles for the collider , muon collimators and background shields , etc . but : * none of the required components may be described as _ exotic _ , and their specifications are not far beyond what has been demonstrated . * if the components can be developed and the problems overcome , then a muon - muon collider could be a useful complement to colliders , and , at higher energies could be a viable alternative . this research was supported by the u.s . department of energy under contract no . de - aco2 - 76-ch00016 and de - ac03 - 76sf00515 . 99 e. a. perevedentsev and a. n. skrinsky , proc . 12th int . conf . on high energy accelerators , f. t. cole and r. donaldson , eds . , ( 1983 ) 485 ; _ early concepts for @xmath41 colliders and high energy @xmath26 storage rings _ , _ physics potential & development of @xmath41 colliders . 2@xmath42 workshop _ , sausalito , d. cline , aip press , woodbury , new york , ( 1995 ) . d. neuffer , ieee trans . * ns-28 * , ( 1981 ) 2034 . , napa ca , nucl inst . and meth . , * a350 * ( 1994 ) ; proceedings of the muon collider workshop , february 22 , 1993 , los alamos national laboratory report la- ur-93 - 866 ( 1993 ) and _ physics potential & development of @xmath41 colliders 2@xmath42 workshop _ , sausalito , ca , ed . d. cline , aip press , woodbury , new york , ( 1995 ) . 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report # 61983 ( 1995 ) . n. v. mokhov , _ the mars code system user s guide _ , version 13(95 ) , fermilab - fn-628 ( 1995 ) . j. ranft , dpmjet code system ( 1995 ) . r. weggel , private communication ; physics today , pp . 21 - 22 , dec . f. chen , _ introduction to plasma physics _ , plenum , new york , pp . 23 - 26 ( 9174 ) ; t. tajima , _ computational plasma physics : with applications to fusion and astrophysics _ , addison - wesley publishing co. , new york , pp . 281 - 282 ( 1989 ) . b. norum and r. rossmanith , _ polarized beams in a muon collider _ , in physics potential & development of @xmath41 colliders , proc . , 3rd int . conf . , san francisco , dec . 1995 , elsevier , in press . a. a. mikhailichenko and m. s. zolotorev , phys . lett . * 71 * , ( 1993 ) 4146 ; m. s. zolotorev and a. a. zholents , slac - pub-6476 ( 1994 ) . a. hershcovitch , brookhaven national report ags / ad / tech . note no . 413 ( 1995 ) . z. huang , p. chen and r. ruth , slac - pub-6745 , _ proc . workshop on advanced accelerator concepts _ , lake geneva , wi , june ( 1994 ) ; p. sandler , a. bogacz and d. cline , _ muon cooling and acceleration experiment using muon sources at triumf _ , _ physics potential & development of @xmath41 colliders 2@xmath42 workshop _ , sausalito , ca , ed . d. cline , aip press , woodbury , new york , pp . 146 a. n. skrinsky and v.v . parkhomchuk , sov . j. of nucl . physics * 12 * , ( 1981 ) 3 ; d. neuffer , particle accelerators , * 14 * , ( 1983 ) 75 ; d. neuffer , proc . 12th int . conf . on high energy accelerators , f. t. cole and r. donaldson , eds . , 481 ( 1983 ) ; d. neuffer , in advanced accelerator concepts , aip conf . 156 , 201 ( 1987 ) . g. silvestrov , proceedings of the muon collider workshop , february 22 , 1993 , los alamos national laboratory report la - ur-93 - 866 ( 1993 ) ; b. bayanov , j. petrov , g. silvestrov , j. maclachlan , and g. nicholls , nucl . inst . and meth . * 190 * , ( 1981 ) 9 ; colin d. johnson , hyperfine 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( 1995 ) , to be published . a. garren , et al . , _ design of the muon collider lattice : present status _ , in physics potential & development of @xmath41 colliders , proc . , 3rd int . , san francisco , dec . 1995 , elsevier , in press . k. oide , private communication . v. balakin , a. novokhatski and v. smirnov , proc . 12_th _ int . conf . on high energy accel . , batavia , il , 1983 , ed . f.t . cole , batavia : fermi natl . accel . lab . ( 1983 ) , p. 119 . a. chao , _ physics of collective beam instabilities in high energy accelerators _ , john wiley & sons , inc , new york ( 1993 ) . g. w. foster and n. v. mokhov , _ backgrounds and detector performance at 2 + 2 tev @xmath41 collider _ , _ physics potential & development of @xmath41 colliders 2@xmath42 workshop _ , sausalito , ca , ed . d. cline , aip press , woodbury , new york , pp . 178 ( 1995 ) . presentation at the unpublished .
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muon colliders have unique technical and physics advantages and disadvantages when compared with both hadron and electron machines .
they should be regarded as complementary .
parameters are given of 4 tev high luminosity collider , and of a 0.5 tev lower luminosity demonstration machine .
we discuss the various systems in such muon colliders .
@xmath0 ) = = 10000
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You are an expert at summarizing long articles. Proceed to summarize the following text:
let @xmath2 be a finite group . factor equivalence of finitely generated @xmath3-free @xmath4$]-modules is an equivalence relation that is a weakening of local isomorphism . it has been used e.g. in @xcite among many other works to derive restrictions on the galois module structure of rings of integers of number fields and of their units in terms of other arithmetic invariants . more recently , a set of rational numbers has been attached to any finitely generated @xmath4$]-module , called regulator constants @xcite , with the property that if two modules are locally isomorphic , then they have the same regulator constants . these invariants have been used in @xcite and in @xcite to investigate the galois module structure of integral units of number fields , of higher @xmath0-groups of rings of integers , and of mordell - weil groups of elliptic curves over number fields . it is quite natural to ask whether there is a connection between the two approaches to galois modules and whether the results of one can be interpreted in terms of the other . it turns out that there is indeed a strong connection , which we shall investigate here . we will begin in the next section by recalling the definitions of factorisability , of factor equivalence , and of regulator constants . we will then establish some purely algebraic results that link factor equivalence and regulator constants . in [ sec : results ] we will revisit the relevant results of @xcite on galois module structures and will use the link established in [ sec : conn ] to compare them to each other . finally , in [ sec : k ] we will use the results of [ sec : conn ] to prove a factorisability result on @xmath0-groups of rings of integers that is a direct analogue of ( * ? ? ? * theorem 5.2 ) . throughout the paper , whenever there will be mention of a group @xmath2 , we will always assume it to be finite . all @xmath4$]-modules will be assumed to be finitely generated and all representations will be finite - dimensional . this work is partially funded by a research fellowship from the royal commission for the exhibition of 1851 . part of this work was done , while i was a member of the mathematics department at postech , korea . it is a great pleasure to thank both institutions for financial support , and to thank postech for a friendly and supportive working environment . i am also grateful to an anonymous referee for carefully reading the manuscript and for helpful suggestions . we will begin by recalling the definition of factorisability and of factor equivalence , and by discussing slight reformulations . this concept first appears in @xcite and plays a prominent rle e.g. in the works of frhlich . [ def : factorisability ] let @xmath2 be a group ( always assumed to be finite ) , and let @xmath5 be an abelian group , written multiplicatively . a function @xmath6 on the set of subgroups @xmath7 of @xmath2 with values in @xmath5 is _ factorisable _ if there exists an injection of abelian groups @xmath8 and a function @xmath9 on the irreducible characters of @xmath2 with values in @xmath10 , with the property that @xmath11\rangle}\ ] ] for all @xmath12 , where @xmath13 denotes the set of irreducible characters of @xmath2 , and @xmath14 denotes the usual inner product of characters . the definition one often sees in connection with galois module structures is a special case of this : @xmath5 is usually taken to be the multiplicative group of fractional ideals of the ring of integers @xmath15 of some number field @xmath16 , and @xmath10 is required to be the ideal group of @xmath17 for some finite galois extension @xmath18 with galois group @xmath2 , with @xmath19 being the natural map @xmath20 . let us introduce convenient representation theoretic language to concisely rephrase the above definition . the _ burnside ring _ @xmath21 of a group @xmath2 is the free abelian group on isomorphism classes @xmath22 $ ] of finite @xmath2-sets , modulo the subgroup generated by elements of the form @xmath23+[t ] - [ s\sqcup t],\ ] ] and with multiplication defined by @xmath23\cdot [ t ] = [ s\times t].\ ] ] the _ representation ring _ @xmath24 of a group @xmath2 over the field @xmath25 is the free abelian group on isomorphism classes @xmath26 $ ] of finite dimensional @xmath25-representations of @xmath2 , modulo the subgroup generated by elements of the form @xmath27+[\tau ] - [ \rho\oplus \tau],\ ] ] and with multiplication defined by @xmath27\cdot [ \tau ] = [ \rho\otimes \tau].\ ] ] in the case that @xmath28 , which will be the main case of interest , we will omit the subscript and simply refer to the representation ring @xmath29 of @xmath2 . there is a natural map @xmath30 that sends a @xmath2-set @xmath5 to the permutation representation @xmath31 $ ] . denote its kernel by @xmath32 . by artin s induction theorem , this map always has a finite cokernel @xmath33 of exponent dividing @xmath34 . moreover , @xmath33 is known to be trivial in many special cases , e.g. if @xmath2 is nilpotent , or a symmetric group . the cokernel @xmath33 is important when strengthenings of the notion of factorisability are considered , such as @xmath35-factorisability , but will not be important for us . it follows immediately from definition [ def : factorisability ] and from standard representation theory that for @xmath36 to be factorisable , it has to be constant on conjugacy classes of subgroups . there is a bijection between conjugacy classes of subgroups of @xmath2 and isomorphism classes of transitive @xmath2-sets , which assigns to @xmath12 the set of cosets @xmath37 with left @xmath2-action by multiplication , and to a @xmath2-set @xmath1 the conjugacy class of any point stabiliser @xmath38 , @xmath39 . an arbitrary @xmath2-set is a disjoint union of transitive @xmath2-sets , and so an element of @xmath21 can be identified with a formal @xmath3-linear combination of conjugacy classes of subgroups of @xmath2 . so if @xmath36 is a factorisable function , then it can be thought of as a function on conjugacy classes of subgroups of @xmath2 , equivalently on transitive @xmath2-sets , and then extended linearly to yield a group homomorphism @xmath40 . [ prop : factorisability ] let @xmath41 be a group homomorphism , where @xmath5 is an abelian group . the following are equivalent : 1 . @xmath36 is factorisable in the sense of definition [ def : factorisability ] . 2 . there exists an injection @xmath42 of abelian groups such that the composition @xmath43 factors through the natural map @xmath44 . 3 . there exists an injection @xmath45 such that @xmath46 factors through the natural map @xmath30 , i.e. there is a homomorphism @xmath47 that makes the following diagram ( whose first row is exact ) commute : @xmath48 & k(g ) \ar[r ] & b(g ) \ar[r ] \ar[d]_f & r(g)\ar[d]^{g ' } \ar[r ] & c(g)\ar[r ] & 0\\ & & x \ar@{^{(}->}[r]_{\iota ' } & y ' . & & } \ ] ] 4 . the homomorphism @xmath36 is trivial on @xmath49 . the condition ( 2 ) is just a reformulation of ( 1 ) . suppose that condition ( 2 ) is satisfied , and let us deduce ( 3 ) . let @xmath50 be the map @xmath51 whose existence is postulated by ( 2 ) . define @xmath52 to be the subgroup of @xmath10 generated by @xmath53 and by @xmath54 , define @xmath55 to be the restriction of @xmath50 to @xmath29 , followed by the inclusion @xmath56 , and @xmath57 to be @xmath19 , followed by the inclusion @xmath58 . then @xmath52 , @xmath57 , @xmath55 satisfy ( 3 ) . a brief diagram chase shows that ( 3 ) implies ( 4 ) : since @xmath57 , is an injection , @xmath59 . so for the diagram in ( 3 ) to commute , we must have @xmath60 . incidentally , exactly the same proof shows also that ( 2 ) implies ( 4 ) . finally , the implication ( 4 ) @xmath61(2),(3 ) follows from two standard facts about abelian groups : * any abelian group can be embedded into a divisible abelian group , * and any homomorphism from a subgroup @xmath62 of an abelian group @xmath63 to a divisible group @xmath64 extends to a homomorphism from @xmath63 to @xmath64 . since @xmath36 is trivial on @xmath32 , it induces a homomorphism from @xmath65 , which is canonically identified with a subgroup of @xmath66 . now , embed @xmath5 into a divisible group @xmath10 , and extend @xmath67 to a homomorphism @xmath68 . 1 . it follows from the last part of the proof that if @xmath5 is divisible , then @xmath52 can be taken to be equal to @xmath5 in proposition [ prop : factorisability ] . also , if @xmath33 is trivial , then @xmath69 , and again @xmath52 can be taken to be equal to @xmath5 . if @xmath5 is the group of fractional ideals of a number field @xmath16 , and if @xmath36 vanishes on @xmath32 , then @xmath52 can always be taken to be the group of fractional ideals of a suitable galois extension @xmath18 , so this is not an additional restriction . indeed , a sufficient condition on @xmath52 is that elements of @xmath65 that are @xmath70-divisible in @xmath29 are mapped under @xmath36 to elements of @xmath5 that become @xmath70-divisible in @xmath52 . so if @xmath5 is the group of fractional ideals of a number field @xmath16 , this condition translates into relative ramification indices of some integral ideals of @xmath0 being divisible by some integers , and some elements of @xmath16 having certain @xmath70-th roots in @xmath0 . in @xcite , the word `` representation - theoretic '' has been used in place of `` factorisable '' . let @xmath2 be a group , and let @xmath71 , @xmath72 be two @xmath3-free @xmath4$]-modules such that there is an isomorphism of @xmath73$]-modules @xmath74 . fix an embedding @xmath75 of @xmath2-modules with finite cokernel . then @xmath71 and @xmath72 are said to be _ factor equivalent _ , written @xmath76 , if the function @xmath77 $ ] is factorisable . the notion of factor equivalence is independent of the choice of the embedding @xmath78 , and defines an equivalence relation on the set of @xmath3-free @xmath4$]-modules . if @xmath79 for some prime @xmath80 , then @xmath78 can be chosen to have a cokernel of order coprime to @xmath80 . indeed , @xmath79 if and only if @xmath81 ( @xcite , see also @xcite ) , and an isomorphism @xmath82 gives rise to an embedding @xmath78 with cokernel of order coprime to @xmath80 , by composing it with multiplication by an integer to clear denominators . it follows that two modules that are locally isomorphic at all primes @xmath80 are factor equivalent . the above definition is the one usually appearing in the literature , but it will be convenient for us to follow @xcite in defining factor equivalence for @xmath4$]-modules that are not necessarily @xmath3-free : let @xmath2 be a group , and let @xmath71 , @xmath72 be two @xmath4$]-modules such that there is an isomorphism of @xmath73$]-modules @xmath83 . fix a map @xmath84 of @xmath2-modules with finite kernel and cokernel . then @xmath71 and @xmath72 are said to be _ factor equivalent _ if the function @xmath85\cdot|\ker(i)^h|^{-1}$ ] is factorisable . again , this notion is independent of the choice of the map @xmath78 , and defines an equivalence relation on the set of @xmath4$]-modules that weakens the relation of lying in the same genus ( where @xmath71 and @xmath72 are said to lie in the same genus if @xmath79 for all primes @xmath80 ) . we continue to denote by @xmath2 an arbitrary ( finite ) group . we also continue to use the identification between conjugacy classes of subgroups of @xmath2 and isomorphism classes of transitive @xmath2-sets . under this identification , a general element of @xmath21 will be written as @xmath86 with the sum running over mutually non - conjugate subgroups , and with @xmath87 . an element of @xmath32 is such a linear combination with the property that the virtual permutation representation @xmath88^{\oplus n_h}$ ] is 0 . alternatively , more down to earth , if we write @xmath89 as @xmath90 with all @xmath91 , @xmath92 non - negative , then @xmath89 is in @xmath32 if and only if the permutation representations @xmath93^{\oplus n_i}$ ] and @xmath94^{\oplus n_j'}$ ] are isomorphic . an element @xmath95 of @xmath32 is called a _ brauer relation_. the following invariants of @xmath4$]-modules were introduced in @xcite and used e.g. in @xcite to investigate galois module structures , as we shall review in the next section : let @xmath2 be a group and @xmath71 a @xmath4$]module . let @xmath96 be a bilinear @xmath2invariant pairing that is non degenerate on @xmath97 . let @xmath98 be a brauer relation . the regulator constant of @xmath71 with respect to @xmath89 is defined by @xmath99 here and elsewhere , the abbreviation @xmath100 refers to the @xmath3-torsion subgroup . this is independent of the choice of pairing ( * ? ? ? * theorem 2.17 ) . as a consequence , @xmath101 is always a rational number , since the pairing can always be chosen to be @xmath102-valued . it is also immediate that @xmath103 , so given a @xmath4$]-module , it suffices to compute the regulator constants with respect to a basis of @xmath32 . in other words , this construction assigns to each @xmath4$]-module essentially a finite set of rational numbers , one for each element of a fixed basis of @xmath32 . one can show that if @xmath71 , @xmath72 are two @xmath4$]-modules such that @xmath104 , then for all @xmath105 the @xmath80-parts of @xmath106 and @xmath107 are the same . so , like factor equivalence , regulator constants provide invariants of a @xmath4$]-module that , taken together , are coarser than the genus . let @xmath71 , @xmath72 be two @xmath4$]-modules with the property that @xmath74 , let @xmath84 be a map of @xmath2-modules with finite kernel and cokernel . fix a @xmath108-valued bilinear pairing @xmath14 on @xmath72 that is non - degenerate on @xmath109 . the following immediate observation is crucial for linking regulator constants with the notion of factorisability : @xmath110 ^ 2\cdot \det\left(\langle\cdot,\cdot\rangle\big|n/\tors\right)\\ & = & \frac{[n : i(m)]^2}{|\ker i|^2}\cdot\frac{|m_{\tors}|^2}{|n_{\tors}|^2}\cdot \det\left(\langle\cdot,\cdot\rangle\big|n/\tors\right).\end{aligned}\ ] ] we deduce [ lem ] let @xmath71 , @xmath72 be two @xmath4$]-modules such that @xmath74 , let @xmath95 be a brauer relation . then @xmath111}{|\ker(i|_m^h)|}\cdot\frac{|m_{\tors}^h|}{|n_{\tors}^h|}\right)^{2n_h}\cdot\cc_\theta(n)\ ] ] for any map @xmath84 of @xmath2-modules with finite kernel and cokernel . by combining this with proposition [ prop : factorisability ] , we obtain [ cor ] two @xmath4$]-modules @xmath71 and @xmath72 with the property that @xmath112 are factor equivalent if and only if @xmath113 for all brauer relations @xmath114 . in particular , if @xmath71 and @xmath72 are @xmath3-free and satisfy @xmath112 , then they are factor equivalent if and only if @xmath115 for all @xmath105 . we shall now show by way of several examples how lemma [ lem ] and corollary [ cor ] link known results on galois module structures with each other . throughout this section , let @xmath18 be a finite galois extension of number fields with galois group @xmath2 . the ring of integers @xmath17 , and its unit group @xmath116 are both @xmath4$]-modules . more generally , if @xmath1 is any @xmath2-stable set of places of @xmath0 that contains the archimedean places , then the group of @xmath1-units @xmath117 of @xmath0 is a @xmath4$]-module . it is a long standing and fascinating problem to determine the @xmath2-module structure of these groups , e.g. by comparing it to other well - known @xmath2-modules or by linking it to other arithmetic invariants . a starting point is the observation that @xmath118^{\oplus [ k:\q]}$ ] as @xmath73$]-modules . also , by dirichlet s unit theorem , @xmath119 , where @xmath120\rightarrow \z\right),\ ] ] with the map being the augmentation map that sends each @xmath121 to 1 . it is therefore natural to compare the galois module @xmath17 to @xmath4^{\oplus [ k:\q]}$ ] and @xmath117 to @xmath122 . it had been known since e. noether that @xmath17 lies in the same genus as @xmath4^{\oplus [ k:\q]}$ ] if and only if @xmath18 is at most tamely ramified . the following is therefore particularly interesting in the wildly ramified case : we always have that @xmath17 is factor equivalent to @xmath4^{\oplus[k:\q]}$ ] . we will now give a very short proof of this result in terms of regulator constants . first , note that by corollary [ cor ] the statement is equivalent to the claim that for any @xmath105 , @xmath123^{\oplus[k:\q]})$ ] . since regulator constants are multiplicative in direct sums of modules ( ( * ? ? ? * corollary 2.18 ) ) , and since @xmath124)=1 $ ] for all @xmath105 ( ( * ? ? ? * example 2.19 ) ) , we have reduced the proof of the theorem to showing that @xmath125 for all @xmath105 . if we choose the pairing on @xmath17 defined by @xmath126 with the sum running over all embeddings @xmath127 , then the determinants on @xmath128 , @xmath12 , appearing in the definition of regulator constants are nothing but the absolute discriminants @xmath129 . the fact that these vanish in brauer relations follows immediately from the conductor - discriminant formula . as we have mentioned above , it is natural to compare @xmath117 with @xmath122 , since they span isomorphic @xmath73$]-modules . for @xmath12 , let @xmath130 denote the set of places of @xmath131 below those in @xmath1 , and let @xmath132 denote the @xmath1-class number of @xmath131 . [ thm : mult ] fix an embedding @xmath133 of @xmath2-modules with finite cokernel . for @xmath134 , let @xmath135 be its residue field degree in @xmath136 , define @xmath137 then the function @xmath138\frac{n(h)}{h_s(k^h)l(h)}\ ] ] is factorisable . as in the additive case , we want to understand and to reprove this theorem in terms of regulator constants . more specifically , we will show it to be equivalent to [ thm : propmult ] for @xmath139 , let @xmath140 be the decomposition group of a prime @xmath141 above @xmath142 ( well - defined up to conjugacy ) . for any brauer relation @xmath143 , we have @xmath144 ) } \prod_h \left(\frac{w(k^h)}{h_s(k^h)}\right)^{2n_h},\ ] ] where @xmath145 denotes the number of roots of unity in @xmath131 , i.e. the size of the torsion subgroup of @xmath146 . note that since @xmath122 is torsion free and @xmath147 is injective , proposition [ prop : factorisability ] and lemma [ lem ] imply that theorem [ thm : mult ] is equivalent to the following statement : for any brauer relation @xmath148 , @xmath149 the equivalence of theorems [ thm : mult ] and [ thm : propmult ] will therefore be established if we show that @xmath150 ) } \prod_h\left(\frac{l(h)}{n(h)}\right)^{2n_h}.\ ] ] this is just a linear algebra computation that we will not carry out in full detail , since it is a combination of the computations of @xcite and @xcite . indeed , it is shown in @xcite that under the embedding @xmath151\hookrightarrow \z[s ] , \;\;\;\fp\mapsto \sum_{\fq\in s,\fq | \fp}f_\fp\fq\end{aligned}\ ] ] we have @xmath152 = \frac{n(h)}{l(h)}$ ] . so , instead of computing @xmath153 for a suitable choice of pairing @xmath14 on @xmath122 , we may compute @xmath154 where @xmath155 is identified with a submodule of @xmath122 as in ( [ eq : embedding ] ) . to do that , we note that for any @xmath12 , @xmath155 is generated by @xmath156 , @xmath157 for any fixed @xmath158 , and that there is a natural @xmath2-invariant non - degenerate pairing on @xmath122 that makes the canonical basis of @xmath159 $ ] orthonormal . it is now a straightforward computation , which has essentially been carried out in @xcite , to show that the quantity ( [ eq : regconst ] ) is equal to @xmath160)},\ ] ] as required . as another illustration of the connection we have established , we will give an easy proof of an analogue of ( * ? ? ? * theorem 5.2 ) for higher @xmath0-groups of rings of integers . the main ingredient will be the compatibility of lichtenbaum s conjecture on leading coefficients of dedekind zeta functions at negative integers with artin formalism , as proved in @xcite . let @xmath161 be an integer . let @xmath162 , respectively @xmath163 denote the set of real embeddings , respectively of representatives from each pair of complex conjugate embeddings of a number field @xmath35 , and denote their cardinalities by @xmath164 , respectively @xmath165 . denote @xmath166 by @xmath167 . it is shown in @xcite that the ranks of the higher @xmath0-groups or rings of integers are as follows : @xmath168 let @xmath18 be a finite galois extension with galois group @xmath2 , and let @xmath169 denote the set of real places of @xmath16 that become complex in @xmath0 . for @xmath170 , let @xmath171 denote the non - trivial one - dimensional @xmath102-representation of the decomposition group @xmath140 , which has order 2 . by artin s induction theorem , a rational representation of a finite group is determined by the dimensions of the fixed subrepresentations under all subgroups of @xmath2 . it therefore follows that we have the following isomorphisms of galois modules : @xmath172\nonumber\\ & \cong & \bigoplus_{\fp\in s_\infty(k)}\q[g / d_\fp]\;\;\;\text{if $ n$ is odd , and}\label{eq : kqodd}\\ k_{2n-1}(\co_k)\otimes \q & \cong & \bigoplus_{\fp \in s_r(k / k)}\ind_{g / d_\fp}\epsilon_{\fp}\oplus \bigoplus_{\fp\in s_2(k)}\q[g]\nonumber\\ & \cong & \bigoplus_{\fp \in s_r(k / k)}\q[g]\big/\q[g / d_\fp ] \oplus \bigoplus_{\fp\in s_2(k)}\q[g]\;\;\;\text{if $ n$ is even.}\label{eq : kqeven}\end{aligned}\ ] ] we are thus led to compare , using the machine of factorisability , the galois module structure of @xmath173 with @xmath174 $ ] when @xmath70 is odd , and with @xmath175 % = \bigoplus_{\fp \in s_r(k / k ) } \z[g]\big/\z[g / d_\fp]\oplus \bigoplus_{\fp\in s_2(k)}\z[g]\ ] ] when @xmath70 is even . here and elsewhere , we write @xmath176 interchangeably for the rational representation and for the unique ( up to isomorphism ) @xmath3-free @xmath177$]-module inside it . let @xmath18 be a finite galois extension of number fields with galois group @xmath2 , let @xmath161 be an integer . then the function @xmath178}{|k_{2n-2}(\co_{k^h})|}\ ] ] is factorisable at all odd primes , where @xmath179\cong \bigoplus_{\fp\in s_\infty(k)}\z[g / d_\fp]\;\;\;\;\text{if $ n$ is odd , and}\\ m & = & \bigoplus_{\fp \in s_r(k / k)}\ind_{g / d_\fp}\left(\epsilon_\fp\right ) \oplus \bigoplus_{\fp\in s_2(k)}\z[g ] \;\;\;\text{if $ n$ is even},\end{aligned}\ ] ] and where @xmath180 is any inclusion of @xmath2-modules . proposition [ prop : factorisability ] and lemma [ lem ] imply that the assertion of the theorem is equivalent to the claim that for any brauer relation @xmath148 , @xmath181^{2n_h}}{|k_{2n-2}(\co_{k^h})|^{2n_h}}\\ & = _ { 2 ' } & \frac{\cc_{\theta}(m)}{\cc_{\theta}(k_{2n-1}(\co))}\cdot \prod_h \left(\frac{|k_{2n-1}(\co_{k})^h_{\tors}|}{|k_{2n-2}(\co_{k^h})|}\right)^{2n_h } , \end{aligned}\ ] ] where @xmath182 means that the two sides have the same @xmath80-adic valuation for all odd primes @xmath80 . now , for any odd prime @xmath80 and any subgroup @xmath12 , we have @xmath183 this is a consequence of the quillen lichtenbaum conjecture ( see e.g. ( * ? ? ? * proposition 2.9 and the discussion preceding it ) ) , which is known to follow from the bloch kato conjecture , which in turn is now a theorem of rost , voevodsky , and weibel @xcite . moreover , it follows from @xcite ( see ( * ? ? ? * equation ( 2.6 ) ) ) that @xmath184 putting this together , we see that the assertion of the theorem is equivalent to the claim that @xmath185 for all brauer relations @xmath89 . but @xmath186 ( not just up to powers of 2 ) by ( * ? ? ? * corollary 2.18 and proposition 2.45 ( 2 ) ) , and because cyclic groups have no non - trivial brauer relations .
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we compare two approaches to the study of galois module structures : on the one hand factor equivalence , a technique that has been used by frhlich and others to investigate the galois module structure of rings of integers of number fields and of their unit groups , and on the other hand regulator constants , a set of invariants attached to integral group representations by dokchitser and dokchitser , and used by the author , among others , to study galois module structures .
we show that the two approaches are in fact closely related , and interpret results arising from these two approaches in terms of each other .
we then use this comparison to derive a factorisability result on higher @xmath0-groups of rings of integers , which is a direct analogue of a theorem of de smit on @xmath1-units .
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low - dimensional time - discrete maps are among the most important models for exploring different aspects of chaos . these systems display a very rich dynamical behavior but are still very amenable to straightforward computer simulations . even more , in some cases rigorous analytical solutions are possible . after it was realized that diffusion processes can be generated by microscopic deterministic chaos in the equations of motion , time - discrete maps became useful tools in deterministic transport theory . the analysis of these simple models required to suitably combine nonequilibrium statistical mechanics with dynamical systems theory leading to a more profound understanding of transport in nonequilibrium situations @xcite . however , time - discrete maps provide not only a suitable starting point for studying normal diffusion but also for investigating the anomalous case @xcite . moreover , there are certain classes of more realistic models which share specific properties of maps such as being low - dimensional and exhibiting certain periodicities . indeed , theoretical investigations of chaotic billiards subject to external fields @xcite , of periodic lorentz gases @xcite , and of pendulum - like differential equations @xcite showed that many properties of deterministic transport in maps carry over to these more complex chaotic dynamical systems . in this framework , recently a new feature of deterministic diffusion was discovered . for simple one - dimensional hyperbolic maps it was shown that the diffusion coefficient is typically a fractal function of control parameters @xcite . subsequently an analogous behavior was detected for other transport coefficients @xcite , and in more complicated models @xcite . however , up to now the fractality of transport coefficients could be assessed for hyperbolic systems only , whereas , to our knowledge , the fractal nature of classical transport coefficients in the broad class of nonhyperbolic systems was not discussed . on the other hand , studying nonhyperbolic dynamics appears to be more relevant in order to connect fractal transport coefficients to some known experiments . here we think particularly of dissipative systems driven by periodic forces such as josephson junctions in the presence of microwave radiation @xcite , superionic conductors @xcite , and systems exhibiting charge - density waves @xcite in which certain features of deterministic diffusion were already observed experimentally . for these systems the equations of motion are typically of the form of some nonlinear pendulum equation . in the limiting case of strong dissipation they can be reduced to nonhyperbolic one - dimensional time - discrete maps sharing certain symmetries @xcite . the so - called climbing sine map is a well - known example of this class of maps @xcite . in this paper we pursue a detailed analysis of the diffusive and dynamical properties of the climbing sine map . particularly , we show that the nonhyperbolicity of this map does not destroy the fractal characteristics of deterministic diffusive transport as they were found in hyperbolic systems . on the contrary , fractal structures appear for normal diffusive parameters as well as for anomalous diffusive regions . we argue that higher - order memory effects are crucial to understand the origin of these fractal hierarchies in this nonhyperbolic system . by using a green - kubo formula for diffusion , the dynamical correlations are recovered in terms of fractal takagi - like functions . these functions appear as solutions of a generalized integro - differential de rham - type equation . we furthermore show that the distribution of periodic windows exhibiting anomalous diffusion forms devil s staircase like structures as a function of the parameter and that the complementary sets of chaotic dynamics have a positive measure in parameter space that increases by increasing the parameter value . our paper is organized as follows . in sec . ii we introduce the model . in sec . iii we explore the coarse functional form of the parameter - dependent diffusion coefficient and discuss it in relation to previous results on hyperbolic maps . in sec . iv our analysis is refined revealing complex scenarios of anomalous diffusion , which are explained in terms of correlated random walk approximations . in sec . v generalized fractal takagi functions are constructed for the climbing sine map and the connection to the diffusion coefficient is worked out . periodic windows exhibiting anomalous diffusion are studied in detail in sec . we then draw conclusions and discuss our results in the final section . the one - dimensional climbing sine map is defined as . the dashed line indicates the orbit of a moving particle starting from the initial position @xmath0.,title="fig:",scaledwidth=50.0% ] + @xmath1 where @xmath2 is a control parameter and @xmath3 is the position of a point particle at discrete time @xmath4 . obviously , @xmath5 possesses translation and reflection symmetry , @xmath6 the periodicity of the map naturally splits the phase space into different boxes , @xmath7 $ ] , @xmath8 , as shown in fig.[fig1 ] . ( [ map ] ) as restricted to one box , i.e. , on a circle , we call the reduced map , @xmath9 the probability @xmath10 to find a particle at a position between @xmath11 and @xmath12 at time @xmath4 then evolves according to the continuity equation for the probability density @xmath13 , which is the frobenius - perron equation @xcite @xmath14 the stationary solution of this equation is called the invariant density , which we denote by @xmath15 . due to its nonhyperbolicity , the climbing sine map possesses a rich dynamics consisting of chaotic diffusive motion , ballistic dynamics , and localized orbits . under parameter variation these different types of dynamics are highly intertwined resulting in complicated scenarios related to the appearance of periodic windows @xcite . in order to study diffusion we will be interested in parameters that are greater than @xmath16 for which the extrema of the map exceed the boundaries of each box for the first time indicating the onset of diffusive motion . in this section we explore the relationship between nonlinear maps like the climbing sine map and simple piecewise linear maps for which , in contrast to the climbing sine map , the diffusion coefficient can be calculated exactly . it is well - known that in special cases such different types of maps are linked to each other via the concept of conjugacy . indeed , we show that maps which are approximately conjugate to each other exhibit a very similar oscillatory behavior in the diffusion coefficient on coarse scales . our argument refers to some existing methods for calculating the diffusion coefficient of piecewise linear maps , which we briefly review . we then describe how we numerically calculated the complete parameter dependence of the diffusion coefficient for the climbing sine map and discuss a first result . one speaks of normal deterministic diffusion if the mean square displacement of an ensemble of moving particles grows linearly in time . the diffusion coefficient is then given by the einstein relation @xmath17 where the brackets denote an ensemble average over the moving particles . there exist various efficient numerical as well as , for some system parameters , analytical methods to exactly compute diffusion coefficients for piecewise linear hyperbolic maps , such as transition matrix methods based on markov partitions @xcite , cycle expansion methods@xcite , and more recently a very powerful method related to kneading sequences @xcite . we first restrict our analysis of diffusion in the climbing sine map to parameters for which there are simple markov partitions . for one - dimensional maps , a partition is a markov partition if and only if parts of the partition get mapped again onto parts of the partition , or onto unions of parts of the partition , see ref . @xcite and further references therein . an example of a markov partition consisting of five parts is shown in the inset of fig.[d_mark ] . in case of the climbing sine map markov partitions can be constructed simply by forward iteration of one of the critical points @xmath18 defined by the condition that @xmath19 in the reduced map . if higher iterations of this point fall onto a periodic orbit a markov partition exits . indeed , if a markov partition is known , for piecewise linear maps the diffusion coefficient can often be calculated analytically via calculating the invariant measure of the map or via computing the second largest eigenvalue of the frobenius - perron operator written in form of a transition matrix . one can now identify an infinite series of parameter values corresponding to a certain type of markov partition @xcite . for parameter values which belong to such a markov partition series the corresponding invariant densities @xmath20 have a very similar functional form . note that , in case of nonlinear maps , singularities in the invariant density exactly correspond to the iteration of the critical point @xmath18 @xcite . an example of @xmath20 for one series of parameter values ( marked as filled circles ) is shown in the inset of fig . [ d_mark ] . by using respective series of markov partitions piecewise linear maps can be related to nonlinear maps . for this purpose let us consider , along with the climbing sine map , ( i ) the piecewise linear zig - zag map @xcite @xmath21 with @xmath22 , @xmath23 and @xmath24 , and ( ii ) the nonlinear cubic map @xcite @xmath25 the definitions of both maps are given on the unit interval . in order to compare the diffusion coefficient of these different maps , the parameters @xmath26 were chosen such that the maps all display the same height @xmath27 , defined as the distance between the first iteration of the leftmost critical point @xmath28 and the zero bound in the first box @xmath29 $ ] . thus , @xmath30 corresponds to the onset of diffusion for all three maps . for the two simple markov partition series @xmath31 and @xmath32 , corresponding to integer and half - integer values of @xmath27 , respectively , the diffusion coefficient of the zig - zag map can be calculated analytically . for integer values of @xmath27 the result reads @xcite @xmath33 for half - integer values of @xmath27 the diffusion coefficient is easily calculated analytically , e.g. , by transition matrix methods @xcite , to @xmath34 in case of the climbing sine map and of the cubic map the diffusion coefficient was obtained from computer simulations by evaluating the mean square displacement eq . ( [ d_a ] ) for the same series of markov partitions . results are shown in fig . [ d_mark ] . for the climbing sine map some more markov partition series points ( alltogether five different series ) were included . for all three maps there is a very analogous oscillatory behavior of the parameter - dependent diffusion coefficient . these oscillations can be explained in terms of the changes of the microscopic dynamics under parameter variation , that is , whenever there is a local maximum there is an onset of strong backscattering in the dynamics yielding a local decrease of the diffusion coefficient in the parameter , and vice versa at local minima @xcite . however , the five markov partition series for the climbing sine diffusion coefficient already indicate that there are more irregularities on finer scales . for piecewise linear maps , the origin of these irregularities was identified to be the topological instability of the dynamics under parameter variation @xcite . that is , a small deviation of the parameter changes the markov partition and the corresponding invariant density which , in turn , is reflected in a change of the value of the diffusion coefficient . note that the dependence of the diffusion coefficient for a single markov partition series appears to be a monotonously increasing function of the parameter @xcite . nevertheless , computing @xmath35 for more and more markov partitions series will reveal more and more irregularities in @xmath35 thus forming a fractal structure @xcite . since the climbing sine map shares the same topological features as piecewise linear maps in terms of these series of markov partitions , one may wonder whether it is not possible to straightforwardly calculate the diffusion coefficient for nonlinear maps from the one of piecewise linear maps by using the concept of conjugacy @xcite , see also the definition in appendix a. in fact , it was stated by grossmann and thomae @xcite that the diffusion coefficient is invariant under conjugacy , however , without giving a proof . in appendix a such a proof is provided . unfortunately , conjugacies are explicitly known only in very specific cases and for maps acting on the unit interval @xcite . as soon as the map extrema exceed the unit interval , which is reminiscent of the onset of diffusive behavior , only some approximate , piecewise conjugacies could be constructed in a straightforward way , see ref . @xcite for an example . we now apply this reasoning along the lines of conjugacy in order to understand the similarities between the diffusion coefficient of the three maps as displayed in fig . 2 . the functional form of the cubic map can be obtained from a taylor series expansion of @xmath36 by keeping terms up to third order thus representing a low - order approximation of the climbing sine map . this seems to be reflected in the fact that at any odd integer parameter value of @xmath27 the climbing sine map has an invariant density whose functional form is very close to the one of the cubic map at parameter value @xmath30 , @xmath37 . hence , one may expect that both diffusion coefficients are possibly trivially related to each other , however , note the increasing deviations between the respective results at larger @xmath27 . for @xmath30 , the cubic map and the piecewise linear zig - zag map are now in turn conjugate to each other @xcite . however , for @xmath38 we are not aware of the existence of any exact conjugacy between zig - zag and cubic map . still , along the lines of ref . @xcite one can at least approximately relate both maps to each other via using piecewise conjugacies . this explains why the zig - zag map and the climbing sine map display qualitatively the very same oscillatory behavior in the diffusion coefficient , somewhat linked by diffusion in the cubic map . in summary , by using markov partitions and by arguing with the concept of conjugacy we have shown that the structure of the diffusion coefficient for the nonlinear climbing sine map has much in common with the one of respective piecewise linear maps , in the sense of displaying a non - trivial oscillatory parameter dependence . however , to use conjugacies in order to exactly calculate the diffusion coefficient for nonlinear maps does not appear to be straightforward @xcite , hence in the following we restrict ourselves to alternative methods as discussed in the next subsection . ) ( squares ) , the climbing sine map ( circles ) , and the cubic map eq . ( [ cub_map ] ) ( triangles ) . the values for the zig - zag map represent analytical results , see eqs.([cub_d1],[cub_d2 ] ) , the remaining values are from computer simulations . the lines are guides for the eyes . the inset shows an example of a markov partition for the climbing sine map on the unit interval and the corresponding invariant density.,title="fig:",scaledwidth=70.0% ] + in order to obtain the full parameter dependence for the diffusion coefficient of the climbing sine map we numerically evaluated the green - kubo formula for diffusion in maps @xcite reading @xmath39 here the angular brackets denote an average over the invariant density of the reduced map , @xmath40 . the jump velocity @xmath41 is defined by @xmath42-[x_n]\equiv [ m_a(x_n ) ] \ : , \label{jumps_vel}\ ] ] where the square brackets denote the largest integer less than the argument . points.,title="fig:",scaledwidth=70.0% ] + the sum @xmath43 gives the integer value of the displacement of a particle after @xmath4 time steps that started at some initial position @xmath11 , and we call it jump velocity function . ( [ gk ] ) defines a time - dependent diffusion coefficient which , in case of normal diffusion , converges to @xmath44 in our simulations for fig . [ d_5box ] we truncated @xmath45 after @xmath46 time steps . the invariant density was obtained by solving the continuity equation for @xmath47 eq . ( [ fp_eq ] ) with the histogram method of ref . note that both the integer displacement and the density are coupled via eq . ( [ gk ] ) . results for @xmath35 are shown in fig . [ d_5box ] for a large range of parameters demonstrating a highly non - trivial behavior of the diffusion coefficient . the large - scale oscillations as predicted from the simple markov partition series ( see fig.[d_mark ] ) are still clearly seen , however , on top of this there exist further oscillations on finer scales . these are regions of anomalous diffusion that manifest themselves in form of abrupt divergences , @xmath48 , or by a vanishing diffusion coefficient , @xmath49 . in this section we study the parameter - dependent diffusion coefficient in more detail . based on the green - kubo formula we derive a systematic hierarchy of approximations for the diffusion coefficient and show how they can be used to understand the complex behavior of this curve in more detail . we are first interested in understanding the coarse functional form of the parameter - dependent diffusion coefficient in the limit of very small and very large parameter values . for this purpose we use simple random walk approximations that are based on the assumption of a complete loss of memory between the single jumps . such an analysis was already performed for hyperbolic piecewise linear maps @xcite . here we apply the same reasoning to the nonlinear case of the climbing sine map . we start in the limit of very small parameter values , i.e. , near the onset of diffusion . here we assume that particles make either a step of length one to the left or to the right , or just remain in the box . the transition probability is then given by integrating over the respective invariant density in the escape region . putting all this information into eq . ( 5 ) yields @xcite @xmath50 where @xmath51 . making the additional approximation that @xmath52 we get @xmath53 the other limiting case concerns values of @xmath54 . here the precise value of the width of the escape region is much less important than the precise value of the step length which is very large , hence by again assuming that @xmath52 eq . ( 5 ) can be approximated to @xcite @xmath55 , which give an indication of the exact functional form . the magnification ( inset in normal scale ) of a small region around @xmath56 shows more irregularities on a finer scale pointing towards a fractal structure of the diffusion coefficient.,title="fig:",scaledwidth=70.0% ] + these two asymptotic random walk approximations are shown in fig.[crisis ] as bold lines . one can clearly see that there is a dynamical crossover between the different functional forms of these two asymptotic regimes . this crossover was first observed in piecewise linear maps and appears to be typical for diffusive systems exhibiting some spatial periodicity @xcite . it was lateron also verified for diffusion in the periodic lorentz gas @xcite . the coarse functional form of the random walk approximations should be compared to a certain series of higher - order approximations based on the green - kubo formula eq . ( [ gk ] ) , which is also shown in the figure . these refined approximations are getting closer and closer to the exact functional form , as explained below , and thus give a good indication for these exact values . in the previous subsection we have outlined two simple random walk approximations for diffusion that do not include any memory effects . however , one can do better by systematically evaluating the single terms as contained in the series expansion of the green - kubo formula eqs.([gk],[vel_func ] ) . for a simple piecewise linear map and for the periodic lorentz gas this was done in ref . @xcite and provided a simple approach to understand the origin of complex structures in the diffusion coefficient on fine scales . the basic idea of this approach is as follows : the green - kubo formula eq.([gk ] ) splits the dynamics into an inter - box dynamics , in terms of integer jumps , and into an intra - box dynamics , as represented by the invariant density . we first approximate the invariant density in eq . ( [ gk ] ) to @xmath57 irrespective of the fact that it is typically a very complicated function of @xmath11 and @xmath26 @xcite . the resulting approximate diffusion coefficient we label with a superscript in eq . ( [ gk ] ) , @xmath58 . the term for @xmath59 obviously excludes any higher - order correlations and was already worked out in form of the simple random walk approximation eqs . ( [ rw])-([rw_2 ] ) . the generalization @xmath60 , which systematically includes more and more dynamical correlations , may consequently be denoted as _ correlated random walk approximation _ @xcite . we now use this expansion to analyze the parameter dependence of the diffusion coefficient of the climbing sine map in terms of such higher - order correlations . , see eq . ( [ gk ] ) with uniform invariant density @xmath61 , for @xmath62 . note the quick convergence for normal diffusive parameters . the dashed lines define periodic windows , which are the same as in fig.[dif ] . the insets ( b ) and ( c ) contain two magnifications of ( a ) in the region close to the onset of diffusion for @xmath63 only . they show self - similar behavior on smaller and smaller scales.,title="fig:",scaledwidth=70.0% ] + fig . [ crisis ] depicts results for @xmath64 at @xmath65 over a large range of parameters , whereas fig . [ frw ] ( a ) presents a respective detailed analysis for the region close to the onset of diffusion , i.e. , for parameters @xmath66 $ ] , showing results for @xmath64 at @xmath67 . the series of approximations in fig.[crisis ] clearly reveals finer and finer sequences of oscillations that eventually converge to a fractal structure , as is particularly shown in the inset of this figure . however , here the order of the expansion is not large enough to identify parameter regions of anomalous diffusion . these regions can be better seen in fig . [ frw ] , where three different cases of parameter regions can be distinguished : ( i ) regions with quick convergence of this approximation corresponding to normal diffusion ( ii ) divergence of @xmath58 corresponding to ballistic motion , in agreement with @xmath68 , and ( iii ) localized dynamics where @xmath58 alternates in @xmath4 between two solutions , with @xmath69 for the exact diffusion coefficient . this oscillation points to the dynamical origin of localization in terms of certain period - two orbits . that these approximate solutions are non - zero is due to the fact that the invariant density was set equal to one . the dashed lines in fig.[frw ] indicate the largest regions of anomalous diffusion . the approximate diffusion coefficient @xmath63 of this figure is compared to the `` numerically exact '' one in fig . here `` numerically exact '' we wish to be understood in the sense that no further ad hoc - approximations are involved , i.e. , we evaluated the green - kubo formula according to the numerical method described in sec . iii . by truncating it after @xmath70 time steps . this comparison shows that in case of normal diffusion our approximation nicely reproduces the irregularities in the non - approximated diffusion coefficient . like the inset of fig . [ crisis ] , the magnifications in fig.[frw ] give clear evidence for a self - similar structure of the diffusion coefficient . these results thus show that dealing with correlated jumps only yields a qualitative and to quite some extent even quantitative understanding of the regions of normal and anomalous diffusion in the climbing sine map . the impact of specific features of the microscopic dynamics on the diffusion coefficient is nicely elucidated by comparing the bifurcation diagram of the reduced climbing sine map eq . ( [ reduced_map ] ) with the numerically exact diffusion coefficient , see fig . [ dif ] . as one can see in the upper panel of fig . [ dif ] , the bifurcation diagram consists of ( infinitely ) many periodic windows . whenever there is a window the dynamics of eq . ( [ map ] ) is either ballistic or localized @xcite . fig.[dif ] demonstrates the strong impact of this bifurcation scenario on the diffusion coefficient . for localized dynamics , orbits are confined within some finite interval in phase space implying subdiffusive behavior for which the diffusion coefficient vanishes , whereas for ballistic motion particles propagate superdiffusively with a diverging diffusion coefficient being proportional to @xmath71 . only for normal diffusion @xmath35 is nonzero , finite and the limit in eq . ( [ dlim ] ) exists . at the boundaries of each periodic window the diffusion coefficient is related to intermittent - like transient behavior eventually resulting in normal diffusion with @xmath72 @xcite . in comparison with the correlated random walk approximation @xmath63 ( dots ) . the dashed vertical lines connect regions of anomalous diffusion , @xmath68 or @xmath73 , with ballistic and localized windows , respectively , of the bifurcation diagram.,title="fig:",scaledwidth=70.0% ] + in this section we further analyze the dynamical origin of the different structures in the parameter - dependent diffusion coefficient by constructing objects called fractal generalized takagi functions . these functions somewhat resemble usual takagi functions but , as will be shown , they fulfill a more complicated type of functional recursion relations than standard de rham - type equations . interestingly , takagi functions were known to mathematicians since about a hundred years @xcite , however , in the field of chaotic transport they were appreciated by physicists only very recently @xcite . we first show how to construct fractal generalized takagi functions and study their properties with respect to the three different types of dynamics in the climbing sine map . we then relate these objects directly to the diffusion coefficient . from the definition of the time - dependent jump velocity function there follows the recursion relation @xcite @xmath74 since the time - dependent jump velocity function @xmath75 is getting extremely complicated after some time steps , we introduce the more well - behaved function @xmath76 integration of eq . ( [ rec ] ) yields the recursive functional equation @xmath77 with the integral term @xmath78 ( upper curve ) and @xmath79 ( lower curve ) . ( b ) , ( c ) : an example of nonconverging iterations of the generalized fractal takagi functions for the climbing sine map with parameters corresponding to ( b ) ballistic dynamics at @xmath80 and to ( c ) localized dynamics at @xmath81 , both for the time steps @xmath82 . note the divergence of the iterations in ( b ) and the alternation between two states in ( c).,title="fig:",scaledwidth=55.0% ] + where @xmath83 is the second derivative of the inverse function of @xmath84 . the function @xmath85 is given by @xmath86 where @xmath87 is defined to be constant on each subinterval where the jump velocity @xmath41 has a given value . this constant is fixed by the condition for @xmath88 to be continuous on the unit interval supplemented by @xmath89 the generalized takagi function is now defined in the long - time limit of eq.([takeq ] ) , @xmath90 for piecewise linear hyperbolic maps @xmath91 in eq . ( [ takeq2 ] ) simply disappears , and the derivative in front of the second term of eq . ( [ takeq ] ) reduces to the local slope of the map thus recovering ordinary de rham - type equations @xcite . it should be noted that for smooth nonlinear maps like the climbing sine map the reduced map @xmath84 is generally not invertible . in order to define a local inverse of @xmath84 , we split the unit interval into subintervals on which this function is piecewise invertible . thus eq . ( [ takeq ] ) should be understood as a series of equations where each part is defined for a respective piecewise invertible part of @xmath84 . the detailed derivation of eq . ( [ takeq ] ) is given in appendix b. it is not known to us how to directly solve this generalized de rham - type integro - differential equation for the climbing sine map , however , solutions can alternatively be constructed from eqs . ( [ takagi ] ) , ( [ t_a ] ) , ( [ t_0 ] ) starting from simulations . results are shown in fig.[takfig ] . for normal diffusive parameters the limit of eq.([takfun ] ) exists and the respective curves are fractal on the whole unit interval somewhat resembling standard takagi functions @xcite . however , in case of periodic windows @xmath92 either diverges due to ballistic flights , or it oscillates indicating localization . interestingly , in these functions the corresponding attracting sets appear in form of smooth , non - fractal regions on fine scales as marked by the dashed lines in fig . [ takfig ] . the diffusion coefficient can now be formulated in terms of these fractal functions by carrying out the integrations contained in eq . ( [ gk ] ) . for simplicity we restrict ourselves to the parameter region of @xmath93 $ ] in which the respective solution reads @xmath94 - d_0^{\rho}(a ) , \label{dtak}\ ] ] where @xmath95 , is defined by @xmath96:=1 $ ] , and @xmath97 . our previous approximation @xmath58 with @xmath98 is recovered from this equation in form of @xmath99 - d_0 ^ 1(a)\;. \label{dtak2}\ ] ] hence , eqs . ( [ dtak ] ) , ( [ dtak2 ] ) explicitly relate the generalized fractal takagi functions shown in fig . [ takfig ] to the fractals of fig.[dif ] . one of the most important problems regarding periodic windows in maps remains the question of their total measure . much understanding has been achieved for one - dimensional unimodal maps @xcite . based on the sharkovskii theorem about the ordering of periodic orbits @xcite , metropolis , stein and stein organized periodic windows in universal symbolic sequences ( u - sequences ) such that the sequence of next order is uniquely determined by the previous one @xcite . later jacobson came up with the proof that chaotic parameter values in one - dimensional unimodal maps with a single maximum do have positive measure @xcite . related numerical studies were made by farmer @xcite . furthermore , it was shown that periodic windows in such a map form so - called fat fractal cantor - like sets with positive measure . however , for diffusive maps on a line , apart from the preliminary studies of refs.@xcite , nothing appears to be known . on the other hand , as was exemplified in sec . iv there is an intimate relation between the irregular structures of the diffusion coefficient and the occurrence of periodic windows . hence , in this section we investigate the periodic windows for the climbing sine map in full detail . due to the spatial extension of our model a new type of periodic motion , which is not present in unimodal maps , exists , which is that particles move on average in one direction . ballistic solutions _ are born through tangent bifurcations , further undergo a feigenbaum - type scenario and die at crises points @xcite . _ localized solutions _ occurr at even periods only and start with tangent bifurcations followed by a symmetry breaking at slope - type bifurcation points @xcite . in this case the bifurcation scenario is much more complex . obviously , periodic windows are related to certain periodic orbits , thus there are infinitely many of them , and they are believed to be dense in the parameter space @xcite . of all period @xmath100-windows ( lines with symbols ) for the first three parameter intervals ( from top to bottom ) as defined in the text . the dotted lines represent exponential fits ; for the parameters see table [ fit ] . the measures corresponding to windows with localized orbits are shown as pluses . lower panel : the partial sum @xmath101 for all periodic windows at a certain period @xmath100 . the dashed curves represent approximations as calculated from eq . ( [ sp ] ) , the straight lines are their limiting values at @xmath102.,title="fig:",scaledwidth=60.0% ] + of all period @xmath100-windows ( lines with symbols ) for the first three parameter intervals ( from top to bottom ) as defined in the text . the dotted lines represent exponential fits ; for the parameters see table [ fit ] . the measures corresponding to windows with localized orbits are shown as pluses . lower panel : the partial sum @xmath101 for all periodic windows at a certain period @xmath100 . the dashed curves represent approximations as calculated from eq . ( [ sp ] ) , the straight lines are their limiting values at @xmath102.,title="fig:",scaledwidth=60.0% ] + by dividing the parameter line into subsets labeled by the integer value of the map maximum on the unit interval , @xmath103 = j , \ ; j \in\mathbb{z}$ ] , we computed all windows of a certain period @xmath100 in a certain subset . the numerical procedure which was used for these computations is outlined in appendix c. let @xmath104 denote the total measure of all period @xmath100-windows in a subset @xmath105 and let @xmath101 be the partial sum of @xmath104 defined by @xmath106 . in fig . [ fig5 ] @xmath107 is plotted as a function of the period for the three first subsets @xmath108 . the measures corresponding to windows with localized orbits are shown in fig . [ fig5 ] as pluses . is it clear that they make the major contribution to the total measure for even periods hence explaining the origin of the pronounced oscillatory behavior of @xmath109 . .fit parameters for the exponential decrease of the measure at even and odd periods for the first three subsets of the map control parameter labeled by @xmath105 . [ fit ] [ cols="<,<,<,<,<",options="header " , ] is the integrated number of period six - windows . the inset shows a blowup of the initial region.,title="fig:",scaledwidth=70.0% ] + the main message of table [ mes ] is that the measure of periodic windows is different for a different subset @xmath105 and obviously decreases by increasing the control parameter . correspondingly , the measure of chaotic solutions increases as the parameter of the system is getting larger . in order to make this more quantitative we consider the dependence of the measure of period @xmath100-windows as a function of the box index @xmath105 . first we check period one - windows . since in each box there exists only one window of this period , we are able to go to up to @xmath110 . in the upper panel of fig . [ fig6 ] the logarithm of @xmath111 is plotted against @xmath112 . we find that the slope of this function almost exactly equal to @xmath113 . the behavior of @xmath114 for different @xmath100 is shown in the lower panel of fig . the slope of the line for period two is also @xmath113 , for period three and four it is @xmath115 , and for period five it is @xmath116 . [ fig6 ] shows that even and odd periods decrease with respectively different laws , where the decay rate appears to be precisely given by the periodicity of the windows according to @xmath117 for even periods and @xmath118 for odd periods . , where @xmath119 is the total measure of windows that are smaller than @xmath120 . results are plotted for the first three subsets @xmath121 of the map control parameter , from top to bottom . in all cases , the slope of the solid lines is approximately @xmath122 . the dashed lines in each graph represent the corresponding coarse - grained window distribution functions . in the inset of fig . ( c ) the measure of any single period five - window in the subset @xmath123 is shown with respect to an integer label that accounts for the ordering according to the map control parameter.,title="fig:",scaledwidth=50.0% ] + , where @xmath119 is the total measure of windows that are smaller than @xmath120 . results are plotted for the first three subsets @xmath121 of the map control parameter , from top to bottom . in all cases , the slope of the solid lines is approximately @xmath122 . the dashed lines in each graph represent the corresponding coarse - grained window distribution functions . in the inset of fig . ( c ) the measure of any single period five - window in the subset @xmath123 is shown with respect to an integer label that accounts for the ordering according to the map control parameter.,title="fig:",scaledwidth=50.0% ] + , where @xmath119 is the total measure of windows that are smaller than @xmath120 . results are plotted for the first three subsets @xmath121 of the map control parameter , from top to bottom . in all cases , the slope of the solid lines is approximately @xmath122 . the dashed lines in each graph represent the corresponding coarse - grained window distribution functions . in the inset of fig . ( c ) the measure of any single period five - window in the subset @xmath123 is shown with respect to an integer label that accounts for the ordering according to the map control parameter.,title="fig:",scaledwidth=50.0% ] + in order to analyze the structure of the regions of anomalous diffusion in the parameter space , we sum up the number of period six - windows as a function of the parameter , that is , the total number is increased by one for any parameter value at which a new period six - window appears . this sum forms a devil s staircase like structure in parameter space indicating an underlying cantor set like distribution for the corresponding anomalous diffusive region , see fig . since the lebesque measure of periodic windows is positive , this set must be a fat fractal @xcite . its self - similar structure can quantitatively be assessed by computing the so - called fatness exponent . following @xcite , let @xmath124 be the total measure of all periodic windows whose width is greater than or equal to @xmath120 . define the coarse - grained measure as @xmath125 , where @xmath126 is the total measure of a box related to the control parameter . for quadratic maps on the interval , it was conjectured and confirmed numerically that @xmath119 asymptotically scales as a power law in the limit of @xmath127 , @xmath128 where @xmath129 is the measure of chaotic parameters . @xmath130 was called the fatness exponent . for quadratic maps it was found to be @xmath131 . since our map belongs to the same universality class as considered in farmer s case , namely the map has a single quadratic maximum , one may expect that @xmath130 will have the same value . for the climbing sine map a double - logarithmic plot of @xmath132 is shown in fig . [ fig8 ] for the first three boxes , @xmath108 . in all cases the bold lines have the slope @xmath122 with errors of @xmath133 , respectively , which seems to be in agreement with farmer s conjecture about the universality of @xmath130 . however , apart from this coarse linear behavior one can see an interesting oscillatory behavior in @xmath134 with respect to @xmath120 . this fine structure can be explained with respect to the histogram distribution functions @xmath135 of the window sizes @xmath136 , which are plotted in fig . [ fig8 ] in form of dashed lines . somewhat surprisingly , the periodic windows are not distributed uniformly or smoothly with respect to their size but form certain peaks , in which preferably windows of certain periods are grouped together . this non - uniformity is clearly reflected in the oscillations of @xmath137 . moreover , we find that the size distributions of periodic windows have a fine structure that appears to resemble a fractal function . some evidence for this property is given in the inset of fig . [ fig8 ] ( c ) , which shows the size of every window of period five in the subset @xmath123 as a function of its appearance with respect to the map control parameter @xmath26 , i.e. , not the parameter itself is plotted but just an integer running index @xmath138 is given instead . particularly the height of the peaks is important , and one can clearly see a complicated hierarchy of different peaks which are reminiscent of the fine structure in the corresponding distribution function shown in fig . [ fig8 ] ( c ) . in this paper we have performed a detailed analysis of the parameter dependence of the diffusion coefficient in a nonhyperbolic dynamical system . the climbing sine map has been chosen as a paradigmatic example of such a system . we have shown that , on a coarse scale , there are certain analogies between the parameter - dependent diffusion coefficient of this map and the ones in simple hyperbolic piecewise linear maps , such as the existence of an oscillatory structure , and the existence of asymptotic functional forms as derived from simple random walk models . however , in contrast to hyperbolic maps showing normal diffusion only , in the nonhyperbolic climbing sine map fractal structures appear for both normal and anomalous diffusive regions of the diffusion coefficient . an understanding of the origin of these fractal structures was given in terms of dynamical correlations starting from the green - kubo formula for diffusion . we furthermore related these irregularities in the diffusion coefficient more microscopically to different characteristics in corresponding fractal generalized takagi functions . for this purpose we derived a new functional recursion relation that defines these fractal forms and generalizes ordinary de rham - type equations . our analysis was completed by extensive numerical studies of the periodic windows of the climbing sine map showing that both the periodic and the chaotic parameter regions have positive measures in the parameter space . however , these measures are themselves parameter - dependent , and by increasing the parameter we found that the chaotic regions occupy larger and larger measures . we finally provided evidence that these different sets form fat fractals on the parameter axis . in conclusion , we wish to remark that the climbing sine map is of the same functional form as the respective nonlinear equation in the two - dimensional standard map , which is considered to be a standard model for many physical hamiltonian dynamical systems . indeed , both models are motivated by the driven nonlinear pendulum , both are strongly nonhyperbolic , and though the standard map is area - preserving it too exhibits a highly irregular parameter - dependent diffusion coefficient . understanding the origin of these irregularities was the subject of intensive research @xcite , however , so far the complexity of this system did not enable to reveal its possibly fractal nature . a suitably adapted version of our approach to nonhyperbolic diffusive dynamics as presented in this paper may enable to make some progress in this direction . another interesting problem is to possibly further exploit the concept of conjugacy between nonlinear and piecewise linear maps , as explained in sec.iii , in order to exactly calculate diffusion coefficients for nonlinear maps . a very promising approach in this direction was presented in ref.@xcite . based on these techniques we are planning to perform a spectral analysis of the frobenius - perron operator governing the probability density of the diffusive climbing sine map . combining such an analysis with the takagi function approach outlined here may lead to a general theory of nonhyperbolic transport . it would furthermore be important to check out the applicability of periodic orbit theory for computing the parameter - dependent diffusion coefficient of the climbing sine map , which may provide an alternative method @xcite . another promising direction of future research concerns establishing crosslinks between our work and the realm of strange kinetics and stochastic modeling as described in refs.@xcite , e.g. , by trying to apply continuous time random walk techniques to more complicated chaotic models exhibiting fractal diffusion coefficients such as the climbing sine map . we finally emphasize the importance to look for possibly fractal transport coefficients in experiments . a very promising candidate appears to be the phase dynamics in squid s , which was very recently analyzed theoretically @xcite and studied experimentally @xcite . in this appendix we give a proof of the statement of grossmann and thomae @xcite that two diffusive maps which are conjugate to each other have the same diffusion coefficient . two diffusive maps @xmath139 and @xmath140 are called conjugate @xcite if there exists a map @xmath141 such that @xmath142 . let us assume in the following that the conjugation function @xmath143 is sufficiently smooth . let the invariant densities of the corresponding reduced ( @xmath144 ) maps be @xmath145 for @xmath146 and @xmath147 for @xmath148 ; then it is , according to conservation of probability , @xmath149 . the diffusion coefficients of the maps @xmath150 and @xmath151 we denote by @xmath152 and @xmath153 , respectively . without loss of generality let us furthermore assume that the maxima of both maps are in the interval @xmath154 $ ] . we now start with the green - kubo formula written in _ correlated random walk _ terms as @xmath155 ^ 2 \tilde{\rho } ( x ) dx + \int\limits^1_0 \left [ f(x ) \right ] \cdot b(x ) \ ; \tilde{\rho } ( x ) dx , \label{ap1}\ ] ] where @xmath156 + ... + \left [ f\left ( \ { f \left ( \{ ... \left ( \ { f(x ) \ } \right) ... \ } \right ) \ } \right)\right ] + ... \ ; , \label{ap2}\ ] ] or shortly @xmath157 where @xmath158 ^ 2 \ ; \tilde{\rho } ( x ) \ ; dx , \label{ap3}\ ] ] @xmath159 \ ; \left[f\{f(x)\}\right ] \ ; \tilde{\rho } ( x ) \ ; dx , \label{ap4}\ ] ] and so on . focusing on the first term , one can rewrite this expression using the symmetry of the map to @xmath158 ^ 2 \ ; \tilde{\rho}(x ) \ ; dx = \int\limits^{x2}_{x1 } \tilde{\rho}(x ) \ ; dx , \label{ap5}\ ] ] where @xmath160 defines an escape region . for the conjugate map @xmath151 the respective term reads @xmath161 ^ 2 \ ; \rho(y ) \ ; dy = \int\limits^{y2}_{y1 } \rho(y ) \ ; dy , \label{ap6}\ ] ] where @xmath162 is the corresponding escape region for @xmath163 . note that the escape regions @xmath164 and @xmath165 are not the same , however , it is straightforward to show that @xmath166 , that is , the topology of both maps is conserved such that the two escape regions are mapped onto each other under conjugacy . taking into account the conservation of probability mentioned before one immediately gets @xmath167 all other terms @xmath168 and @xmath169 have the form @xmath170 where @xmath171 is a constant , @xmath172 is the respective escape region and @xmath173 is the corresponding invariant measure . thus , the same argument can be applied to show that @xmath174 , ( @xmath175 ) . combining all results we arrive at we start with the recursion relation for the jump velocity function eq.([rec ] ) , @xmath177 by recalling the definition of the generalized takagi function eq.([takagi ] ) , @xmath178 or differently @xmath179 . piecewise invertible branches are labeled by the integer numbers @xmath180.,title="fig:",scaledwidth=60.0% ] + we have to integrate eq . ( [ appb_1 ] ) , @xmath181 by using of eq . ( [ appb_2 ] ) we get @xmath182 , \label{appb_4}\ ] ] where @xmath183 . \label{appb_5}\ ] ] without loss of generality let us assume that the maximum of the map is in the interval @xmath154 $ ] . depending on @xmath11 the integral in eq . ( [ appb_5 ] ) can be decomposed into @xmath184 ; \label{appb_6}\ ] ] @xmath185 ; \end{aligned}\ ] ] @xmath186 ; \ ; ... \ ; i_6(x ) = ... , \ ; x \in ( x_5,x_6];\ ] ] @xmath187 . \end{aligned}\ ] ] each integral in eq . ( [ appb_6 ] ) now contains only one piecewise invertible branch of the reduced map @xmath188 as shown in fig.[app_l ] . here , the piecewise invertible branches of the reduced map are labeled by integers , and the corresponding branches of the inverse function @xmath189 have the same indices @xmath180 . since all integrals in eq.([appb_6 ] ) have the same form ( only the inverse parts of the reduced map are different ) , we restrict ourselves to the integral @xmath190 making the change of variables @xmath191 and using the definition of the generalized takagi function eq . ( [ appb_2_a ] ) we get @xmath192 using integration by parts we arrive at @xmath193 now recall that according to eq . ( [ appb_2 ] ) it is @xmath194 , where the @xmath195 define the boundaries of the piecewise invertible parts of @xmath84 , see fig . [ app_l ] , and that @xmath196 . thus , by formally defining the inverse function @xmath197 as consisting of all branches @xmath198 , we can finally write eq . ( [ appb_4 ] ) in the form @xmath199 with the integral term @xmath200 the parameter values @xmath201 which correspond to the tangent bifurcations of the @xmath100-periodic windows were found by solving the two coupled transcendental equations @xmath202 where @xmath203 denotes the @xmath100-times iterated reduced map . this corresponds to the situation where @xmath203 touches the bisector . somewhat after a tangent bifurcation one will unavoidably find a situation where a critical point @xmath204 , which corresponds to an extremum of @xmath203 , crosses the bisector . when this critical point is exactly located on the diagonal , the reduced map or its higher iterations have a fixed point and there exists a specific markov partition on the interval @xcite . the periodic orbit generated by the corresponding parameter value @xmath205 is superstable , @xmath206 by further increasing the parameter value up to @xmath207 a crisis takes place , and this again corresponds to the existence of a certain markov partition . based on this scenario , the full numerical procedure which was used for calculating the measure of periodic windows is as follows : the values of @xmath205 corresponding to superstable solutions were first calculated by a combination of bisection with the newton method . the parameters for the tangent bifurcations could then usually be found by the modified two - dimensional newton method @xcite . however , the highly discontinuous nature of @xmath203 made its implementation very inefficient . instead , starting in the vicinity of each @xmath205 we again combined the one - dimensional newton and bisection methods . this ensured that no windows were missed . finally , the parameter values corresponding to crisis points @xmath207 , which are also defined by markov partitions , can be found by solving respective equations that are formally analogous to eq . ( [ sstable ] ) .
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the nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter .
we show that on a suitable coarse scale this map generates an oscillating parameter - dependent diffusion coefficient , similarly to hyperbolic maps , whose asymptotic functional form can be understood in terms of simple random walk approximations . on finer scales
we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter . by using a green - kubo formula for diffusion the origin of these different regions
is systematically traced back to strong dynamical correlations .
starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized takagi functions obeying generalized de rham - type functional recursion relations .
we finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive , and that for normal diffusion it increases by increasing the parameter value . deterministic diffusion , nonhyperbolic maps , fractal diffusion coefficient , anomalous diffusion , periodic windows 05.45.df , 05.10.-a , 05.45.ac , 05.60.-k
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the final stages of the evolution of low and intermediate mass stars involve complex and poorly understood physical phenomena that lead to the formation of axially symmetric bipolar planetary nebulae ( pne ) . the origin of these spectacular structures is generally associated to interactions between the material expelled from the evolved star and an orbiting companion ( lagagec & chesneau @xcite ; soker & livio @xcite ) . the presence of a circumstellar disk is also invoked to collimate mass loss and foster the appearance of the axial symmetry , but such disks have only been observed in a few evolved stars . l@xmath0 pup ( hd56096 ) is one of the nearest ( @xmath3 , van leeuwen @xcite ) and brightest ( @xmath4 , @xmath5 ) asymptotic giant branch ( agb ) star that exhibits mira - like pulsations with a period of 141days ( bedding et al . this star is particularly interesting as kervella et al . ( @xcite ) discovered a circumstellar disk around l@xmath0pup using adaptive optics in the near - infrared . this discovery was recently confirmed from polarimetric imaging at visible wavelengths by kervella et al . ( @xcite ) , that also revealed the presence of a secondary source . we present these two sets of observations in sect . [ observations ] , and we discuss the spectral energy distribution and disk geometry in sect . kervella et al . ( @xcite ) reported observations of l@xmath0 pup using the vlt / naco ( rousset et al . @xcite ) adaptive optics system . using a `` lucky imaging '' approach , they obtained diffraction limited images in a series of narrow - band filters ranging in wavelength from 1 to 4@xmath2 m . these images show the presence of a dust band in front of l@xmath0 pup . the band exhibits a high opacity in the @xmath6 band ( @xmath7 m ) and becomes translucent in the @xmath8 and @xmath9 bands ( @xmath10 m ) where the light scattering becomes less efficient ( fig . [ naco ] ) . in the @xmath11 band , the thermal emission from the inner rim of the dust disk was also detected , as well as a large loop extending to more than 10au from the star . new observations by lykou et al . ( @xcite ) using the sparse aperture masking mode of naco also showed the presence of a disk - like structure , and ohnaka et al . ( @xcite ) confirmed the presence of an elongated central emission in the east - west direction . the spectro - polarimetric high - contrast exoplanet research ( sphere , beuzit et al . @xcite ) is a high performance adaptive optics ( fusco et al . @xcite ) recently installed at the very large telescope . kervella et al . ( @xcite ) observed l@xmath0 pup using sphere equipped with the imaging polarimeter zimpol ( roelfsema et al . this instrument provides diffraction limited imaging at visible wavelengths , down to @xmath12 nm . the observed intensity image and the map of the degree of linear polarization in the @xmath13 band ( @xmath14 nm ) are presented in fig . [ zimpol ] , together with a nomenclature of the observed features . the right panel shows the observed features ( field of view 0.6 , kervella et al . @xcite).,title="fig:",width=151 ] pup . the right panel shows the observed features ( field of view 0.6 , kervella et al . @xcite).,title="fig:",width=151 ] pup . the right panel shows the observed features ( field of view 0.6 , kervella et al . @xcite).,title="fig:",width=151 ] the intensity image shows a bright unresolved source at a separation of 0.032 from the agb star ( labeled b in fig . [ zimpol ] , right panel ) . although the nature of this source is unknown , its brightness and color are consistent with a late k giant star . several spiral structures are also observed , as well as two thin plumes extending perpendicularly to the disk plane . the map of the degree of linear polarization @xmath15 shows a maximum at a radius of 6au from the central star . kervella et al . ( @xcite ) interpret this maximum as 90@xmath16 scattering occurring at the inner rim of the dust disk . the radius is consistent with the observed extension of the thermal emission in the @xmath9 band ( fig . [ naco ] , central panel ) . the radiative transfer modeling ( using the radmc-3d code ; dullemond @xcite ) presented by kervella et al . ( @xcite , fig . [ sedmodel ] , right panel ) shows that the circumstellar disk is seen almost edge on , with an inclination of @xmath17 on the line of sight . this model reproduces satisfactorily the observed spectral energy distribution ( fig . [ sedmodel ] , left ) , that exhibits a remarkably flat flux density between 1 and @xmath18 m . the total dust mass in the radmc-3d disk model is @xmath19 m@xmath20 . assuming a gas - to - dust ratio of 100 , this translates to a total mass including gas of @xmath21 to @xmath22 m@xmath20 . pup s disk ( right , taken from kervella et al . @xcite).,title="fig:",height=158 ] pup s disk ( right , taken from kervella et al . the environment of l@xmath0 pup hosts the two major ingredients suspected to play a role in the shaping of pne : a close - in companion ( whose physical nature is still largely enigmatic ) and a circumstellar dust disk . thanks to its proximity and the availability of the sphere adaptive optics providing 20mas angular resolution in the visible , it is now possible to monitor in real time the dynamical evolution of the different components of the immediate environment of l@xmath0 pup ( companion , spirals , plumes , ... ) . considering their linear scales , we expect a significant evolution over timescales of only a few years . l@xmath0 pup thus exhibits a strong potential to improve our understanding of the mass loss of low and intermediate mass stars and the formation of bipolar pne . _ j .- ph . berger : _ if you have such a massive companion you should have strong tidal truncation and a cavity . it is surprising not to see any sign of cavity in the spectral energy distribution ( sed ) . + _ answer : _ the sed of l@xmath0 pup is atypical due to its flatness over a major part of the infrared domain ( @xmath23 m ) . the fact that the disk is seen almost edge - on affects the flux contribution from the inner rim of the disk through absorption and reddening , as shown by the radmc-3d model . this may explain that the signature of the cavity is not immediately observable in the sed . _ j. groh : _ is this object representative of its class , or is it a peculiar object ? would it be possible to observe a larger sample to investigate this phenomenology ? + _ answer : _ the agb star itself appears to be close in physical properties to the mira star r vir , whose pulsation period is 145.5 days ( see e.g. eisner et al . more generally , l@xmath0 pup a could well be classified as a low amplitude mira ( kervella et al . the presence of circumstellar disk has been proposed around the mira v cvn by neilson et al . ( @xcite ) based on polarimetry , but this is a rare example of such a detection around a mira . the difficulty to extend the sample is the angular scale of the disks : l@xmath0 pup is very nearby , so the angular scale is resolvable using sphere , but for more distant stars , optical or millimeter interferometry would be the only options . _ a. lobel : _ when you overplot the @xmath18 m image showing the extended loop , it appears to be projected on the northern cone . could it tell about the direction of the orbital motion on the sky ? + _ answer : _ there is unfortunately an ambiguity whether the loop is located on the nearby or far surface of the northern cone . the difficulty is that the thermal emission is isotropic , while the scattering is preferentially in the forward direction . we therefore very likely observe in fig . [ zimpol ] the nearby side of the cones , but we can not be sure for the thermal emission of the loop . _ t. ueta : _ 1 ) bright spots a and b in the zimpol image could still be just brightly lit surfaces of the top of the disk . polarization vector maps should help pinpoint the location of the central bright source . does the centrosymmetric structure of vector maps point to the location of a ? + 2 ) the distribution of scattering matter can be better probed by the polarized flux @xmath24 rather than the degree of polarization @xmath15 . the disk edge might show differently in @xmath24 than in @xmath15 , or do they look the same ? + _ answer : _ 1 ) the nature of the secondary source is indeed uncertain . but its flux contribution is considerable in the visible ( @xmath25% of the primary ) , pointing at a stellar origin . regarding the polarization vector , the difficulty is that the secondary is located so close to the primary ( 32 mas ) , that its measurement is highly uncertain . we see a slight deviation from the classical single central source vector map ( purely orthoradial ) , but not at a sufficiently significant level . + 2 ) the degree of polarization @xmath15 has the advantage to be a simple proxy for the scattering angle @xmath26 ( in the single scattering approximation ) , and therefore well suited to probe the overall geometry of the disk . the polarized flux @xmath24 is sensitive to both the scattering angle @xmath26 and the scattering dust density @xmath27 , therefore relatively more complex to model . in l@xmath0 pup , the @xmath24 map is generally similar to the total intensity map , except close to the primary star where the polarization is low ( forward scattering ) . the secondary source exhibits a relatively strong @xmath24 contribution , suggesting that it contains a significant fraction of scattered light from the primary . but its overall degree of polarization @xmath15 is low , indicating that it is not purely scattering . the presence of an accreting disk around a companion star is a possibility that we plan to investigate . _ j. milli : _ is the angle of polarization deduced from zimpol observations consistent with the morphological model ? + _ answer : _ for most of the field of view apart from the very central part , the polarization vector is orthoradial , consistent with a central source illuminating the dust envelope ( as in the radmc-3d model ) . but very close to the star ( within @xmath28mas ) , the determination of the polarization angle is uncertain ( see also the answer to dr . ueta s question 1 ) . _ j. hoffman : _ does your radiative transfer model predict your observed polarization results ? + _ answer : _ we have encountered some difficulties with the numerical computation of the polarization with the radmc-3d code . but we indeed plan to compare the observed polarization maps to the predictions of the model . 99 bedding , t. r. , zijlstra , a. a. , jones , a. , et al . 2002 , mnras , 337 , 79 beuzit , j .- l . , feldt , m. , dohlen , k. , et al . 2008 , spie conference series , 7014 , 18 dullemond , c. p. 2012 , radmc-3d : a multi - purpose radiative transfer tool , astrophysics source code library eisner , j. a. , graham , j. r. , akeson , r. l. , et al . 2007 , apj , 654 , l77 fusco , t. , sauvage , j .- f . , petit , c. , et al . 2014 , spie conference series , 9148 , 1 kervella , p. , montargs , m. , ridgway , s. t. , et al . 2014 , a&a , 564 , a88 kervella , p. , montargs , m. , lagadec , e. , et al . 2015 , a&a , 578 , a77 lagadec , e. & chesneau , o. 2014 , arxiv:1410.3692 lykou , f. , klotz , d. , paladini , c. , et al . 2015 , a&a , 576 , a46 neilson , h. r. , ignace , r. , & henson , g. d. 2014 , iau symposium , vol . 301 , 463 ohnaka , k. , schertl , d. , hofmann , k. h. , & weigelt , g. 2015 , a&a , in press , arxiv:1507.06668 roelfsema , r. , bazzon , a. , schmid , h. m. , et al . 2014 , spie conference series , 9147 , 3 rousset , g. , lacombe , f. , puget , p. , et al . 2003 , spie conference series , 4839 , 140 soker , n. & livio , m. 1989 , apj , 339 , 268 van leeuwen , f. 2007 , a&a , 474 , 653
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adaptive optics observations in the infrared ( vlt / naco , kervella et al . 2014 ) and
visible ( vlt / sphere , kervella et al . 2015 ) domains revealed that the nearby agb star l@xmath0 pup ( @xmath1pc ) is surrounded by a dust disk seen almost edge - on .
thermal emission from a large dust `` loop '' is detected at 4@xmath2 m up to more than 10au from the star .
we also detect a secondary source at a separation of 32mas , whose nature is uncertain .
l@xmath0 pup is currently a relatively `` young '' agb star , so we may witness the formation of a planetary nebula .
the mechanism that breaks the spherical symmetry of mass loss is currently uncertain , but we propose that the dust disk and companion are key elements in the shaping of the bipolar structure .
l@xmath0 pup emerges as an important system to test this hypothesis .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
let @xmath8 be a field of characteristic @xmath9 . let @xmath10 be a reducive algebraic group over @xmath11 , @xmath12 a borel subgroup and @xmath13 the weyl group and its simple reflections . denote by @xmath14 the bounded derived category of sheaves of @xmath8-vector spaces on @xmath1 constructible along @xmath15-orbits . in @xmath14 there exist the intersection cohomology sheaves @xmath16 . these sheaves are a certain extension of the constant sheaf in degree @xmath17 on the bruhat cell @xmath18 to its closure . let @xmath19 be the hecke algebra of @xmath20 over @xmath21 $ ] normalised so as to satisfy @xmath22 and let @xmath23 be the kazhdan - lusztig basis of @xmath19 . it satisfies @xmath24 h_x$ ] . given a finite dimensional graded vector space @xmath25 let @xmath26 be its poincar polynomial . the character of a sheaf @xmath27 is the element of @xmath19 given by @xmath28 if @xmath8 is of characteristic zero , a theorem of kazhdan and lusztig @xcite says that @xmath29 . thus the poincar polynomials of the stalks of the intersection cohomology sheaves are given by kazhdan - lusztig polynomials . it then follows that the same is true in almost all characteristics , however for any given characteristic almost nothing is known . it is a difficult question to determine over which fields one has @xmath29 and , if not , what these characters are . it has been known since the original papers of kazhdan and lusztig ( @xcite and @xcite ) that in non - simply laced cases the intersection cohomology complexes may have a different character in characteristic 2 . ( this happens , for example , in the only non - smooth schubert variety in the flag variety of @xmath30 . ) in 2002 braden discovered examples of schubert varieties in simply laced types @xmath31 and @xmath32 where the character of the interesection cohomology sheaf in characteristic 2 is different to all other characteristics ( see the appendix ) . in this article we define combinatorially a certain subset @xmath33 of _ separated elements _ and show : [ thm - sep ] suppose that @xmath34 , then @xmath29 for any field @xmath8 . the determination of the characters of @xmath16 is closely related to the decomposition theorem in positive characteristic . given a simple reflection @xmath35 let @xmath36 be the corresponding standard minimal parabolic subgroup and consider the quotient map @xmath37 if @xmath8 is of characteristic zero , the decomposition theorem implies that @xmath38 is a direct sum of shifts of intersection cohomology sheaves . this need not be true if @xmath8 is of positive characteristic . given @xmath39 and @xmath35 let @xmath40 be all the parameters of kazhdan - lusztig basis elements that appear in the product @xmath41 . then we have : [ thm - dec ] suppose that @xmath42 and @xmath43 lie in @xmath44 . then the decomposition theorem holds for @xmath45 . whilst being of considerable intrinsic interest , these questions are also important in representation theory . assume that @xmath8 is algebraically closed , that @xmath46 is greater than the coxeter number of @xmath0 and let @xmath47 be a split semi - simple and simply connected algebraic group over @xmath8 whose root system is dual to that of @xmath10 , and let @xmath48 be a borel subgroup and maximal torus in @xmath47 . fix positive roots @xmath49 so that the roots corresponding to @xmath15 are those lying in @xmath50 . to each weight @xmath51 one may associate a standard module @xmath52 which is non - zero if and only if @xmath53 is dominant , in which case it contains a unique simple submodule @xmath54 . a conjecture of lusztig @xcite expresses the characters of the simple modules @xmath54 in terms of the ( known ) characters of standard modules . a particular case of the conjecture is the following : let @xmath55 denote the half - sum of the positive roots , and @xmath56 the steinberg weight , then it is conjectured that , for all @xmath57 , @xmath58 = h_{x , y}(1)\ ] ] where @xmath59 $ ] is the kazhdan - lusztig polynomial indexed by @xmath57 . a theorem of soergel @xcite says that ( [ eq : lusztig ] ) is equivalent to the semi - simplicity of @xmath60 for all @xmath61 and @xmath35 . of course , in order to apply theorems [ thm - sep ] and [ thm - dec ] it is necessary to know the set @xmath44 . the essential ingredient in the calculation of @xmath44 is the @xmath0-graph of the coxeter system @xmath20 . unfortunately , even in simple situations the @xmath0-graph can be very complicated and no general description is known . however , using fokko du cloux s program _ coxeter _ @xcite it is possible to use a computer to determine the set @xmath44 for low rank weyl groups . the simplest situation is when @xmath62 . this only occurs in type a in low rank : [ thm : a6 ] if @xmath10 is of type @xmath2 for @xmath63 then @xmath62 . hence , in all characteristics the intersection cohomology complexes have characters given by kazhdan - lusztig basis elements and the decomposition theorem holds for @xmath64 for all @xmath35 and @xmath61 . it also follows that ( [ eq : lusztig ] ) holds for @xmath65 with @xmath5 if @xmath8 has characteristic at least @xmath66 . using results of braden in the appendix , we are also able to extend this theorem to cover @xmath47 of type @xmath32 . in other types and type @xmath2 for @xmath67 our techniques are not as effective . in most examples that we have computed @xmath44 is not the entire weyl group . however , we are able to confirm the characters and decomposition theorem for many intersection cohomology complexes in ranks @xmath68 . it also seems that the elements @xmath69 for which our methods fail will provide an interesting source of future research . indeed in the appendix braden shows that , both in type @xmath32 and @xmath31 , if @xmath8 is of characteristic 2 , then for minimal elements in @xmath70 one has @xmath71 . these two examples , combined with the case of dihedral groups , leads one to suspect a close relationship between @xmath70 and those intersection cohomology complexes for which one has @xmath72 for some field of coefficients @xmath8 . let us briefly mention that , in @xcite braden and macpherson give an algorithm for the calculation of the stalks of the intersection cohomology complexes , using only the fixed points and one - dimensional orbits of a maximal torus on the flag variety . ( this data is encoded in the so - called `` moment graph '' of the flag variety . ) the forthcoming paper @xcite of fiebig and the author extends this result , showing that the moment graph of the flag variety can be used to calculate the characters of parity sheaves , and hence determine those intersection cohomology complexes for which @xmath71 . thus the results of this paper could ( at least in principle ) be deduced from the moment graph . in fact , the computations of torsion in the appendix translate easily into the moment graph language and give a proof that the moment graph sheaves corresponding to certain bott - samelson varieties do not split as expected unless @xmath73 is invertible in the coefficient ring lastly , in this article we only consider field coefficients . however versions of the intersection cohomology complex exist with coefficients in @xmath74 and the statement that @xmath75 for a field of characteristic @xmath76 is equivalent to the absence of @xmath76-torsion in the stalks and costalks of an intersection cohomology complex over @xmath74 . the reader is referred to @xcite for more details . the structure of the paper is as follows . in section [ sec - hecke ] we review the hecke algebra and kazhdan - lusztig basis in more detail and recall the @xmath0-graph associated to @xmath20 . in section [ sec - sheaves ] we discuss the _ parity sheaves _ , introduced in @xcite , which are our main theoretical tool . in section [ sec - sep ] we define the subset @xmath33 and prove theorems [ thm - sep ] and [ thm - dec ] . in section [ sec - comp ] we discuss the calculation of the sets @xmath44 via computer and give some examples of the sets @xmath44 for low rank weyl groups . i would like to thank tom braden for useful correspondence , pointing out errors in previous versions , and contributing the appendix . i would also like to thank daniel juteau and carl mautner for comments on a previous version of this paper . in this section we recall the hecke algebra and kazhdan - lusztig basis in slightly more detail . let @xmath20 be a coxeter system with bruhat order @xmath77 and length function @xmath78 . given @xmath79 we define the _ left _ and _ right descent set _ to be @xmath80 recall that the hecke algebra is the free @xmath81$]-module with multiplication given by @xmath82 the elements @xmath83 are invertible and there is an involution @xmath84 on @xmath19 which sends @xmath83 to @xmath85 and @xmath86 to @xmath87 . we will call elements fixed by this involution _ self - dual_. there exists a basis @xmath88 of @xmath19 called the _ kazhdan - lusztig basis _ which is uniquely determined by requiring : 1 . the @xmath89 are self - dual ; 2 . @xmath90 where @xmath91 and @xmath92 $ ] for @xmath93 . the polynomials @xmath94 are ( up to a renormalisation ) the _ kazhdan - lusztig polynomials_. one may check , for example , that @xmath95 . the action of @xmath96 for @xmath35 on the kazhdan - lusztig basis has a particularly simple form . we denote by @xmath97 the coefficient of @xmath87 in @xmath94 . then : @xmath98 ( a similar formula describes multiplication by @xmath96 on the left ) . thus all the information about the action of @xmath96 on the left and right on the kazhdan - lusztig basis may be encoded in a labelled graph , known as the _ @xmath0-graph_. the vertices correspond to the elements of @xmath0 and are labelled with the left and right descent sets . there is a directed edge between @xmath99 and @xmath100 if @xmath101 , in which case the edge is labelled with the value of @xmath102 . for more details on the kazhdan - lusztig basis and @xmath0-graphs the reader is referred to @xcite , @xcite or @xcite . in this section we briefly recall some basic properties of `` parity sheaves '' introduced in @xcite and motivated by @xcite . these are our main technical tool . we recall briefly the setting of @xcite . we fix a field of coefficients @xmath8 . all spaces will be complex algebraic @xmath103-varieties , for @xmath103 a complex linear algebraic group . given an @xmath103-space @xmath104 , we write @xmath105 for the bounded derived category of constructible @xmath8-sheaves on @xmath104 and @xmath106 for the bounded @xmath103-equivariant derived category of constructible sheaves of @xmath8-vector spaces on @xmath104 ( see @xcite ) . by abuse of language , we call objects in @xmath107 complexes . we denote by @xmath108 the forgetful functor ( see @xcite ) . if @xmath103 has finitely many orbits on @xmath104 then the image of the forgetful functor is contained in @xmath109 , the full subcategory of @xmath105 consisting of sheaves whose cohomology is locally constant along @xmath103-orbits . the endomorphism ring of any indecomposable object in @xmath107 is local , and hence the krull - schmidt theorem holds in @xmath107 . all maps will be equivariant morphisms of complex algebraic varieties . given a map @xmath110 we have functors @xmath111 , @xmath112 from @xmath107 to @xmath113 and @xmath114 , @xmath115 from @xmath113 to @xmath107 . similar functors exist between @xmath105 and @xmath116 . on the categories @xmath117 and @xmath116 we have the verdier duality functor , which we denote by @xmath118 . we have isomorphisms of functors @xmath119 and @xmath120 . all functors @xmath121 commute with the forgetful functor . now let @xmath10 denote a reductive complex algebraic group and @xmath122 a borel subgroup and maximal torus . let @xmath0 denote the weyl group and @xmath123 the set of simple reflections . throughout @xmath124 , where @xmath125 is either @xmath15 or a minimal standard parabolic subgroup @xmath36 corresponding to @xmath35 ( i.e. @xmath126 ) . we regard @xmath104 as a @xmath15-variety . each @xmath15-orbit is isomorphic to an affine space and the strata are classified by @xmath0 if @xmath127 and @xmath128 if @xmath129 . given @xmath39 ( resp . @xmath130 ) we denote by @xmath131 ( resp . @xmath132 ) the stratum @xmath18 ( resp . @xmath133 ) , by @xmath134 ( resp . @xmath135 ) its inclusion and by @xmath136 ( resp . @xmath137 ) in the @xmath15-equivariant constant sheaf on @xmath131 ( resp . @xmath138 ) . for brevity , @xmath139 . a complex @xmath140 is _ @xmath141-even _ if @xmath142 ( resp . @xmath143 ) is isomorphic to a direct sum of constant sheaves concentrated in even degrees , for all strata @xmath144 ( resp . @xmath145 ) . a complex is _ even _ if it is both @xmath146- and @xmath147-even . a complex @xmath148 is _ ( @xmath141- ) odd _ if @xmath149 $ ] is ( @xmath141- ) even . a complex @xmath140 is ( ? ) -_parity _ if we have an isomorphism @xmath150 with @xmath151 ( ? ) -even and @xmath152 ( ? ) -odd . note that direct sums and summands of ( ? ) -parity sheaves are ( ? ) -parity . an indecomposable parity complex is called a _ parity sheaf_. the following theorem shows that one may classify parity sheaves on the flag variety in a similar way to intersection cohomology complexes : [ thm - clas ] there exists ( up to isomorphism ) a unique parity sheaf @xmath153 with support contained in @xmath154 and @xmath155 $ ] . each @xmath156 is self - dual and any indecomposable parity complex is isomorphic to @xmath157 $ ] for some @xmath39 and @xmath158 . given a @xmath146-parity complex @xmath159 and @xmath39 we may write @xmath160 for some finite dimensional graded @xmath8-vector space @xmath161 . we define the character of @xmath162 in the hecke algebra to be : @xmath163 given a @xmath146-parity complex @xmath162 it is easily seen that @xmath164 agrees with the character of @xmath165 as defined in the introduction . a similar character map exists for @xmath166 for a simple reflection @xmath35 . let @xmath167 denote the left ideal @xmath168 in @xmath19 . then @xmath167 is free with basis @xmath169 for @xmath170 , where @xmath171 denotes the subset of elements @xmath39 such that @xmath172 . given a @xmath146-parity complex @xmath173 and @xmath170 we can write @xmath174 for some graded @xmath8-vector space @xmath175 . we define the character of @xmath162 to be : @xmath176 for any @xmath35 we have obvious maps given by inclusion and multiplication : @xmath177^ { \cdot \h_s } & \ar@(dl , dr)[l]^{\operatorname{inc } } \he_s}\ ] ] the quotient map @xmath178 induces functors : @xmath179^ { \pi_{s * } } & \ar@(dl , dr)[l]^{\pi_s^ * } d^b_{b}(g/{p_s})}\ ] ] the following lemma is well - known ( see @xcite , lemme 2.6 ) : [ lem - parchar ] 1 . if @xmath159 is @xmath146-parity , then so is @xmath180 and @xmath181 2 . if @xmath182 is @xmath146-parity , then so is @xmath183 and @xmath184 ) = \operatorname{inc}(\operatorname{ch}(\se)).\ ] ] 3 . if @xmath159 or @xmath185 is @xmath146-parity then @xmath186 ) \cong v^{-1 } \operatorname{ch}(\se).\ ] ] we first show the character relations . statement ( 3 ) is a straightforward consequence of the definitions and ( 2 ) follows from the definitions and the fact that @xmath187 for @xmath39 . it remains to show ( 1 ) . we prove ( 1 ) by induction on the number of @xmath39 for which @xmath188 . if this number is one , then ( by definition of @xmath146-parity ) @xmath162 is necessarily a direct sum of shifts of @xmath189 , for some @xmath79 . we may assume that @xmath190 $ ] . let us write @xmath191 for the image of @xmath42 in @xmath192 . if @xmath193 then @xmath194 restricts to an isomorphism @xmath195 . hence @xmath196.\ ] ] if @xmath197 then the restriction of @xmath194 to @xmath131 induces a ( trivial ) @xmath198-bundle over @xmath132 , hence @xmath199.\ ] ] a simple calculation in the hecke algebra then shows that in both cases @xmath200 as claimed . we now turn to the general case . we may assume without loss of generality that @xmath162 is @xmath146-even . choose @xmath39 so that @xmath131 is open in the support of @xmath162 and let @xmath201 denote the inclusion . then @xmath202 and we have a distinguished triangle of @xmath146-even sheaves @xmath203}{\to}\ ] ] by induction @xmath204 applied to the first or third terms is @xmath146-parity and ( 1 ) holds . it follows that the same is true of @xmath162 because @xmath205 and @xmath206 it remains to see that @xmath207 and @xmath208 preserve the classes of parity sheaves . however this follows immediately because @xmath209 ( as @xmath210 is proper ) and @xmath211 ) \mathbb{d } \cong \mathbb{d } ( \pi^*[1])$ ] ( because @xmath210 is a smooth fibration with fibres of complex dimension 1 ) . consider @xmath10 as a @xmath212-space via @xmath213 . as the second copy of @xmath15-acts freely on @xmath10 , the quotient equivalence ( @xcite ) yields an equivalence of triangulated categories : @xmath214 consider the inversion map @xmath215 . then this is @xmath216-equivariant with respect to the swap map @xmath217 . this induces an equivalence @xmath218 consider the functor @xmath219 . then @xmath220 commutes with @xmath118 . it is easy to see @xmath220 preserves parity complex and that , for a parity complex @xmath159 , @xmath221 where @xmath222 is the anti - involution @xmath223 . define endofunctors on @xmath224 by @xmath225 \text { and } \ver_s ( - ) : = \iota^ * \pi_s^ * { \pi_s } _ * \iota^*(-)[1].\ ] ] then the functors @xmath226 and @xmath227 preserve parity sheaves ; the shift is chosen so that @xmath228 and @xmath229 . by and the above lemma , @xmath230 for parity sheaves @xmath159 . the first result about the characters of parity sheaves is the following : [ cor - dual ] for all @xmath39 , @xmath231 is self - dual . we proceed via induction on @xmath232 with the base case being trivial . let us fix @xmath42 and choose @xmath35 with @xmath233 . by theorem [ thm - clas ] we may write @xmath234 where @xmath235 ^{\oplus m_{x,\eta}}.\ ] ] the verdier self - duality of @xmath236 and each @xmath237 for @xmath238 together with krull - schmidt implies @xmath239 by induction , the @xmath240 for @xmath238 are self - dual . hence both @xmath241 and @xmath242 are self - dual . thus so is @xmath243 . let @xmath35 be a simple reflection . the next proposition relates parity sheaves on @xmath244 to those on @xmath1 : [ cor : ps ] let @xmath39 be such that @xmath197 and denote by @xmath191 the image of @xmath42 in @xmath128 . we have isomorphisms @xmath245 \cong \se(w)\ ] ] and @xmath246 \oplus \se({\overline{w}})[1].\ ] ] as @xmath156 is a direct summand of @xmath247 $ ] we have @xmath248 it follows ( by considering @xmath249 ) that @xmath250 \oplus \se({\overline{w}})[-1 ] \oplus \sg\ ] ] for some parity complex @xmath251 . we may also decompose @xmath245 \cong \se({w } ) \oplus \sg^{\prime}.\ ] ] hence @xmath252 \cong \se({\overline{w}})[1 ] \oplus \se({\overline{w}})[-1 ] \oplus \sg \oplus { \pi_s } _ * \sg^{\prime}.\ ] ] however , by lemma [ lem - parchar ] , @xmath253 ) = ( v + v^{-1 } ) \operatorname{ch } ( \se(\overline{w}))\ ] ] and so @xmath254 . hence @xmath251 and @xmath255 are zero . in @xmath14 there exist the intersection cohomology sheaves @xmath16 which may be defined as @xmath256 $ ] ( see @xcite ) . the intersection cohomology complexes admit equivariant lifts @xmath257 which are uniquely determined ( up to isomorphism ) by requiring @xmath258 ( see @xcite ) . we will need the following in the next section : [ prop : ic ] for @xmath39 , @xmath259 if and only if @xmath260 . hence if @xmath261 then @xmath262 . recall that @xmath16 is the unique complex in @xmath14 satisfying the four conditions : 1 . @xmath16 is verdier self - dual ; 2 . @xmath263 for @xmath264 ; 3 . @xmath265 $ ] ; 4 . @xmath266 is concentrated in degrees @xmath267 for @xmath238 . note that 1 ) , 2 ) and 3 ) are always satisfied by @xmath268 . now , if @xmath269 then ( 4 ) is also satisfied and so @xmath270 . on the other hand , if @xmath271 then @xmath243 is a self - dual and positive combination of kazhdan - lusztig basis elements containing more than one term ( again by proposition [ cor - dual ] ) . it is straightforward to see that if this is the case then @xmath268 can not satisfy ( 4 ) . let @xmath272 be a set of representatives of isomorphism classes of indecomposable parity sheaves on @xmath1 . we would like to investigate when their characters are equal to the kazhdan - lusztig basis . the set @xmath273 yields a self - dual basis of @xmath274 with certain positivity properties which are shared by the kazhdan - lusztig basis . in this section we investigate to what extent these properties already determine the basis . as motivation , let us consider some examples : 1 . _ ( simplistic ) _ let @xmath61 and suppose that @xmath275 , @xmath276 and @xmath277 . we know that @xmath278 is a direct summand of @xmath279 $ ] with self - dual character . hence @xmath280 and so @xmath281 by the uniqueness of the kazhdan - lusztig basis . ( more realistic ) _ fix @xmath61 and suppose that we can show that @xmath282 for all @xmath283 . suppose further that the only kazhdan - lusztig basis element that appears with non - zero coefficient in all expressions @xmath284 with @xmath285 is @xmath286 . then , using the fact that @xmath237 occurs as a direct summand in @xmath287 $ ] for all @xmath283 , it follows that @xmath288 . in the above examples it is essential to know how the elements @xmath96 for @xmath35 act on the kazhdan - lusztig basis this is precisely the information provided by the @xmath0-graph discussed in section [ sec - hecke ] . we start with some definitions . given an element @xmath289 we define the _ kl - support _ to be the set : @xmath290 we say that @xmath291 is _ kl - supported in degree 0 _ if , in addition , all @xmath292 . given @xmath293 we may write the difference @xmath294 in the standard basis as @xmath295 if all @xmath296 $ ] we write @xmath297 . note that if @xmath162 is a direct summand of a parity complex @xmath251 then @xmath298 . [ support - lemma ] suppose @xmath162 is a direct summand of a parity complex @xmath251 whose character is kl - supported in degree 0 . then the character of @xmath162 is also kl - supported in degree zero and @xmath299 because @xmath241 is kl - supported in degree 0 , it is isomorphic to a direct sum of indecomposable parity sheaves @xmath156 without shifts . moreover , each @xmath156 that occurs is kl - supported in degree zero . hence @xmath162 is also isomorphic to a direct sum of such indecomposable parity sheaves and the lemma follows . a more conceptual proof of the above lemma is provided by the following ( which will not be needed below ) . the character of a self - dual parity complex @xmath162 is kl - supported in degree zero if and only if @xmath162 is perverse . now if @xmath251 is as in the lemma it is perverse and splits into a direct sum @xmath300 ^{\oplus m_{x,\eta}}.\ ] ] however we must have @xmath301 if @xmath302 as each direct summand must be perverse . hence any direct summand is self - dual and perverse . given a subset @xmath303 we denote by @xmath304 we now define a function @xmath305 from some subset @xmath306 ( to be defined below ) to the power set of @xmath0 . this function and its domain are defined inductively as follows : 1 . suppose we have defined @xmath308 on all @xmath309 . then @xmath99 belongs to @xmath310 if there exists @xmath311 or @xmath312 such that either @xmath313 in this case we define @xmath314 to be the set : @xmath315 [ ex - dih ] let @xmath0 be a dihedral group : @xmath316 if @xmath317 ( resp . @xmath318 ) is not the longest element then : @xmath319 for the longest element @xmath320 one has @xmath321 . if @xmath308 is defined on @xmath61 and @xmath322 we say that @xmath99 is _ separated_. the set of all separated elements will be denoted by @xmath44 . if @xmath0 is a dihedral group @xmath323 then it follows from above that the separated elements are @xmath324 . in particular , @xmath325 and @xmath326 are the only rank two weyl groups in which @xmath62 . one may show ( for example using the results of @xcite ) that if @xmath10 is a kac - moody group having weyl group @xmath0 a dihedral group @xmath323 , then the elements @xmath327 are precisely those for which @xmath29 in all characteristics . theorem [ thm - sep ] in the introduction is an immediate consequence ( using proposition [ prop : ic ] ) of the following : [ prop - sup ] if @xmath308 is defined on @xmath61 then @xmath240 is kl - supported in degree 0 and @xmath328 in particular , for all @xmath34 we have @xmath277 . clearly @xmath329 and so we may assume by induction that @xmath330 for all @xmath331 , where @xmath61 is some element on which @xmath308 is defined . without loss of generality we may assume , by the inductive definition of @xmath308 above , that there exists some @xmath311 so that @xmath332 . hence @xmath333 is kl - supported in degree zero . we may now apply lemma [ support - lemma ] to conclude that @xmath240 is kl - supported in degree zero and that @xmath334 however , as @xmath240 is kl - supported in degree zero and is a direct summand of all @xmath335 for all @xmath336 we conclude that @xmath337 an identical argument applies on the right . the proposition then follows by intersecting these conditions . the following remains from the introduction : under the assumptions of the theorem , @xmath259 and @xmath338 splits as a direct sum of parity sheaves , all of which are isomorphic to equivariant intersection cohomology complexes . we are then done by the above proposition , combined with proposition [ prop : ic ] and proposition [ cor : ps ] . in this section we give some examples of the sets @xmath33 for low rank weyl groups . as is clear from the definition of @xmath308 , the only piece of information needed to calculate the @xmath44 and @xmath308 is the @xmath0-graph of @xmath20 . however , no general description of the @xmath0-graph is known ( for descriptions of some subgraphs see @xcite and @xcite and for a description of the computational aspects of the problem see @xcite and @xcite ) . thus , in order to calculate @xmath308 and @xmath44 we have to restrict ourselves to examples . this involves two steps : 1 . calculation of the @xmath0-graph of @xmath20 , and 2 . calculation of the function @xmath308 using the @xmath0-graph . step 1 ) is computationally quite difficult , especially when the weyl group is large . luckily there exists the program _ coxeter _ written by fokko du cloux @xcite , which calculates the @xmath0-graph very efficiently . step 2 ) is then relatively straightforward . a crude implementation in magma ( whose routines for handling coxeter groups proved very useful ) as well as the @xmath0-graphs obtained from _ coxeter _ are available at : http://people.maths.ox.ac.uk/williamsong/torsion/ this site also contains a complete description of the sets @xmath44 and @xmath308 for all weyl groups of ranks less than 6 . we will now describe examples of the sets @xmath44 . the values of @xmath308 on values not in @xmath44 may be found on the above web page . here thus all intersection cohomology complexes with coefficients of any characteristic have the same characters as in characteristic zero and the decomposition theorem is always true . this is the statement of theorem [ thm : a6 ] . let @xmath340 with coxeter generators @xmath341 with @xmath342 corresponding to the simple transpositions @xmath343 . in @xmath0 , 38 of the 40 320 elements do not belong to @xmath44 . the elements which do not lie in @xmath44 break up naturally into five groups , which we now describe . consider the following elements of @xmath0 : @xmath344{perm7a } \end{array } , \quad w_2 = \begin{array}{c } 67823451 \\ \includegraphics[width=2cm]{w2 } \end{array } , \quad w_3 = \begin{array}{c } 84567123 \\ \reflectbox{\includegraphics[width=2cm]{w2 } } \end{array } \\ w_4 = \begin{array}{c } 62845173 \\ \includegraphics[width=2cm]{perm4a } \end{array } \ ; \text { and } \ ; w_5 = \begin{array}{c } 84627351 \\ \includegraphics[width=2cm]{perm1a } \end{array}. \end{array}\ ] ] the first group consists of @xmath345 the second group consists of @xmath346 ( note that @xmath347 is maximal in @xmath348 ) . the third group @xmath349 is obtained from @xmath348 by inversion ( or by applying the automorphism @xmath350 ) . it contains @xmath351 as a maximal element . the fourth group consists of @xmath352 the fifth group consists of @xmath353 it would be interesting to investigate the intersection cohomology complexes corresponding to the minimal elements in @xmath354 for @xmath355 directly . note that the set @xmath356 has already arisen in kazhdan - lusztig combinatorics ; these are the so - called `` hexagon permutations '' of billey and warrington ( see @xcite , the name refers to a characteristic hexagon shape appearing in their heap representation ) . we say that a permutation @xmath357 contains the pattern of a permutation @xmath358 if there is a collection of indices @xmath359 so that @xmath360 are in the same relative order as @xmath361 . otherwise , we say that @xmath42 avoids the pattern @xmath362 . for a more general notion of pattern avoidance which works for general coxeter groups , . a geometric interpretation of pattern avoidance is given in @xcite . the significance of hexagon permutations is explained by the following result : let @xmath363 be a reduced word for an element @xmath357 . then @xmath364 is a small map if and only if the permutation @xmath42 avoids the pattern @xmath365 and the four hexagon permutations . the bott - samelson resolutions of the hexagon permutations are semi - small , but not small . in the appendix , braden treats the hexagon permutations in detail and shows that @xmath271 if @xmath8 is of characteristic 2 . consider @xmath367 with generators @xmath368 & t\ar@2{-}[r ] & v}.\ ] ] this is the first group on which @xmath308 is not defined everywhere ; it is not defined on the element @xmath369 . this means that there may be a characteristic in which the parity sheaf corresponding to @xmath369 is not perverse . of the 48 elements of @xmath0 , 20 do not lie in @xmath44 . they are : @xmath370 . in @xmath371 , which contains 384 elements , @xmath308 is not defined on 8 elements , and 180 elements do not lie in @xmath44 . in @xmath372 , which contains 3840 elements , @xmath308 is not defined on 26 elements and 1696 elements do not lie in @xmath44 . we label our generators @xmath373 , @xmath374 , @xmath375 and @xmath86 of @xmath0 as follows : @xmath376 \\ s \ar@{-}[r ] & t\\ & & v\ar@{-}[ul]}\ ] ] here , @xmath308 is defined on all of @xmath0 and 7 elements do not belong to @xmath44 . let @xmath377 be the automorphism of @xmath0 mapping @xmath378 . the elements not in @xmath44 are @xmath379 , @xmath380 and @xmath381 as well as @xmath382 , @xmath383 , @xmath384 and @xmath385 . in the appendix , braden discusses the case of @xmath347 in more detail . in fact , in this example one may extend the techniques of section [ sec - sep ] to deduce combinatorially that one always has @xmath386 clearly it is enough to see this for @xmath387 . as @xmath388 is separated , @xmath389 occurs as a direct summand of @xmath390 . but as @xmath391 is separated and @xmath392 we conclude that @xmath393 . in @xmath394 , @xmath308 is not defined on one element , and 176 of the 1920 elements in @xmath0 do not lie in @xmath44 . in @xmath395 , which contains 23040 elements , @xmath308 is not defined on 33 elements and 3259 elements of @xmath0 do not lie in @xmath44 . in @xmath396 , @xmath308 is not defined on 23 elements and 621 of the 1152 do not lie in @xmath44 . in this case we have already calculated @xmath44 in example [ ex - dih ] . here we obtain nothing new . if @xmath398 then @xmath399 . however these schubert varieties are smooth , and so the @xmath400 in any characteristic if @xmath327 . let us briefly describe how the above algorithm may be taken further with some of the geometric input contained in the appendix . for example , if @xmath0 is of type @xmath31 then , with notation as in [ subsec : a7 ] , the calculations in the appendix allow one to conclude that @xmath401 if @xmath402 . then the above algorithm may be used to deduce that @xmath269 for all hexagon permutations @xmath42 . similarly in @xmath32 if one knows , with notation as in [ subsec : d4 ] , that @xmath403 then it follows that @xmath404 , @xmath405 and @xmath406 all have characters given by kazhdan - lusztig basis elements . by the calculations of the appendix this occurs if and only if @xmath402 . it follows that lusztig s conjecture is true around the steinberg weight in type @xmath32 . one can also turn these arguments around to deduce that one has @xmath271 ( and hence @xmath71 ) for families of elements , once one knows that it occurs once . for example , if @xmath0 is of type @xmath32 and one knows that @xmath407 ( as is the case in characteristic 2 ) one can not have @xmath408 for @xmath409 . indeed , assume that @xmath410 . then , as @xmath411 is separated @xmath412 occurs as a direct summand of @xmath413 . but @xmath414 and so the calculations of [ subsec : d4 ] show that @xmath415 . similar arguments apply in the other cases . similarly one may show in type @xmath31 that if @xmath416 for one hexagon permutation then this is true for all of them . a geometric explanation for this is given in remark [ rem : otherhex ] of the appendix . _ _ by tom braden _ _ let @xmath10 be a semisimple complex group , and fix a choice of a borel subgroup @xmath15 , maximal torus @xmath417 , and opposite borel subgroup @xmath418 . let @xmath419 be the corresponding sets of roots and positive roots , chosen so that the weights of @xmath420 acting on @xmath421 are @xmath422 . let @xmath423 denote the weyl group and @xmath123 the set of simple reflections . let @xmath424 be the length function , and @xmath77 the bruhat - chevalley order on @xmath0 . consider the flag variety @xmath425 ; @xmath10 acts on @xmath104 by left multiplication . the set of @xmath417-fixed points is in bijection with @xmath0 by @xmath426 , where @xmath427 is any lift of @xmath42 to @xmath10 . using this bijection , we abuse notation and refer to points in @xmath428 and elements of @xmath0 by the same symbols . the flag variety @xmath104 has two decompositions by bruhat cells and dual bruhat cells @xmath429 , where @xmath430 and @xmath431 . the notation @xmath432 is chosen to indicate that it should be thought of as a normal slice at @xmath42 to the stratification @xmath433 . it is also easy to describe the one - dimensional @xmath417-orbits in @xmath104 . the closure of a one - dimensional orbit is a closed irreducible @xmath417-invariant curve ; we will refer to such curves as `` @xmath417-curves '' for short . for any positive root @xmath434 ( simple or not ) , let @xmath435 denote the corresponding reflection . [ flag 1d orbits ] for any @xmath434 and any @xmath39 , there is a unique @xmath417-curve @xmath436 which contains @xmath42 and @xmath437 , and all @xmath417-curves are of this form . the weight of the action of @xmath417 on the tangent space @xmath438 is @xmath439 . since @xmath440 , the above formula for the tangent weight can also be given , up to sign , by saying that if @xmath436 is the @xmath417-curve joining @xmath42 and @xmath441 for some @xmath442 , then the @xmath417-weight of @xmath443 is @xmath444 . the sign can then be specified by noting that the weight is in @xmath445 if and only if @xmath446 . let @xmath447 be a sequence of simple roots , not necessarily distinct , and let @xmath448 be the corresponding sequence of simple reflections : @xmath449 . put @xmath450 . then @xmath363 is a reduced word for @xmath42 if @xmath451 . for each simple reflection @xmath341 , let @xmath452 be the corresponding minimal parabolic containing @xmath15 , whose lie algebra is @xmath453 . the bott - samelson variety @xmath454 associated to @xmath363 is defined to be the quotient @xmath455 where @xmath456 acts on @xmath457 on the right by @xmath458 let @xmath459 $ ] denote the point of @xmath460 corresponding to @xmath461 . we wish to describe the set @xmath467 of @xmath417-fixed points . let @xmath468 . for each @xmath469 , define @xmath470 and @xmath471 by @xmath472 , \ ; { \mathbf{w}}^{{\varepsilon}}= s_1^{{{\varepsilon}}(1)}\cdots s_l^{{{\varepsilon}}(l)}.\ ] ] for any @xmath473 and any @xmath462 , define @xmath474 = ( { { \varepsilon}}(1),\dots,{{\varepsilon}}(k),0\dots,0 ) \in d_l$ ] . as usual , we will refer to elements of @xmath0 and points of @xmath428 by the same symbols . let @xmath482 . let @xmath483 denote the longest element in @xmath0 . then there are five @xmath417-fixed points in @xmath484 , namely @xmath485 , where @xmath486 let us denote these five elements of @xmath394 by @xmath487 . there exists a @xmath417-curve containing @xmath488 and @xmath489 if and only if @xmath490 . the tangent weights of the curves joining @xmath488 to @xmath491 for @xmath492 are @xmath493 , @xmath494 , @xmath480 , @xmath495 , and @xmath481 . the composition of @xmath496 with the projection @xmath497 restricts to a birational map @xmath498 which identifies @xmath484 with the blow - up of @xmath499 at two points . the exceptional fibers are the @xmath417-curves joining @xmath500 to @xmath501 and @xmath502 to @xmath503 . the one - dimensional @xmath417-orbits of @xmath504 are more difficult to classify than the fixed points . unlike the flag variety @xmath1 , bott - samelson varieties generally have infinitely many @xmath417-curves . we will describe a collection of @xmath417-curves which span the tangent space at each fixed point , but there are in general many other @xmath417-curves . denote the standard basis of @xmath476 by @xmath505 , where @xmath506 is the kronecker @xmath507-function . for any @xmath473 and @xmath508 , we have a @xmath417-curve joining @xmath485 and @xmath509 , namely @xmath510\mid x'_j = x_j , j\ne i\},\ ] ] where @xmath511 $ ] . this curve projects under @xmath210 to the @xmath417-curve in @xmath1 which joins @xmath512 and @xmath513 , and so the tangent weight of this curve at @xmath485 is @xmath514}(\alpha_i)$ ] . note that @xmath417-curves which project down to fixed points , such as the ones in example [ sl3 example ] , are not of this type besides their definition as @xmath15-orbits , the bruhat cells @xmath433 in the flag variety @xmath104 can can also be described as bialynicki - birula cells for the action of a dominant cocharacter @xmath515 . for any @xmath39 , we have @xmath516 the bott - samelson variety @xmath504 will not in general have finitely many @xmath15-orbits , but we can still consider its biaynicki - birula cells ( for the same cocharacter @xmath517 ) . given @xmath473 , we define @xmath518 [ bb cells ] the dimension of the bialynicki - birula cells is given by @xmath519}(\mu_k ) \in -\phi^+\}\\ = \#\{1 \le k \le l \mid \ell({\mathbf{w}}^{{{\varepsilon}}[k ] } ) > \ell({\mathbf{w}}^{{{\varepsilon}}[k]}s_k)\}. \end{split}\ ] ] the cell @xmath520 fibers linearly over @xmath131 , @xmath521 , with fiber of dimension @xmath522}(\mu_k ) \in -\phi^+\}\\ = \#\{1 \le k \le l \mid \ell({\mathbf{w}}^{{{\varepsilon}}[k-1 ] } ) > \ell({\mathbf{w}}^{{{\varepsilon}}[k-1]}s_k)\}. \end{split}\ ] ] these cells give a paving by affines of the fiber @xmath523 . fix a word @xmath363 as above , and let @xmath524 , an object in @xmath525 . we want to understand when does the decomposition theorem hold for @xmath526 , i.e. when does it split as a direct sum of ( shifted ) intersection cohomology sheaves . put @xmath527 . we will be interested in the natural homomorphism @xmath528 of cohomology groups . because @xmath210 is proper , this is the same as the map obtained by applying hypercohomology to the adjunction morphism @xmath529 , where @xmath530 is the inclusion . on the other hand @xmath531 is an isomorphism if @xmath532 and is zero if @xmath533 , so if the decomposition theorem holds for the projection @xmath210 with @xmath534 coefficients , then for any @xmath535 the cokernel of @xmath536 will have no torsion . when the resolution @xmath210 is semi - small , this map was previously considered by @xcite , in the guise of an intersection form on the top borel - moore homology of @xmath537 with @xmath534 coefficients . in the semi - small case @xmath536 is an isomorphism modulo torsion , and it is an isomorphism if and only if the decomposition theorem holds . when @xmath210 is not semi - small we do not know to what extent the lack of torsion in @xmath538 implies the splitting of @xmath526 . because there is a cocharacter of @xmath417 which contracts @xmath541 onto @xmath362 , we have an isomorphism between @xmath542 and @xmath543 . using the properness of @xmath210 , this is the same as the cohomology of the fiber @xmath537 . since this fiber has a paving by affines , its cohomology is torsion free . the torsion freeness of the domain of the map @xmath539 follows from lefschetz duality and the following lemma . let @xmath545 , @xmath546 be the unipotent parts of @xmath15 , @xmath418 . let @xmath547 . then @xmath548 is a @xmath417-invariant open neighborhood of @xmath362 , and the map @xmath549 , @xmath550 is an isomorphism . this map identifies @xmath551 with the cell @xmath552 and @xmath553 with the slice @xmath541 . since @xmath554 is @xmath555-equivariant and @xmath556 , it follows that @xmath557 . since @xmath558 is an open subvariety of a smooth variety , @xmath544 must be smooth . if the fiber @xmath559 is smooth , then we can replace the pair @xmath560 by @xmath561 , where @xmath545 is the total space of the normal bundle @xmath562 to @xmath310 in @xmath544 , and @xmath563 is the complement of the the zero section in @xmath545 . in this case , by the thom isomorphism theorem we have @xmath564 , @xmath565 , and the map @xmath536 can be identified with multiplication by the euler class @xmath566 . the spaces involved are all @xmath417-invariant , so we can also consider the corresponding map in equivariant cohomology : @xmath567 the same arguments as the non - equivariant case , together with standard arguments about equivariant formality and localization , prove the following . we let @xmath572 , so @xmath0 is the symmetric group on the set @xmath573 . taking the torus @xmath417 to be the diagonal matrices , the lattice @xmath574 of characters is naturally identified with @xmath575 let @xmath576 be the @xmath577th standard basis vector of @xmath575 . the roots of @xmath10 are then the vectors @xmath578 , for @xmath579 , @xmath580 . choose the borel subgroup @xmath15 to be the lower triangular matrices . with this choice , the positive roots are @xmath581 with @xmath582 , and the simple roots are @xmath583 , @xmath584 . the simple reflection @xmath585 corresponding to @xmath586 is the transposition of @xmath577 and @xmath587 . we now fix @xmath588 where @xmath589 is the shortest of the `` hexagon permutations '' introduced in [ subsec : a7 ] . the one - line notation of @xmath42 is @xmath590 . a reduced word @xmath363 for @xmath42 can be given by the sequence of simple roots @xmath591 reduced words for the other three hexagon permutations are obtained from this by appending @xmath592 at the beginning or the end , or both . to prove this , we look more closely at the fiber @xmath597 . we will compute the map @xmath598 by working with @xmath417-equivariant cohomology and localization . first we describe the @xmath417-fixed points in @xmath310 . @xmath310 has two irreducible components . the first , call it @xmath609 , is isomorphic to @xmath610 , where @xmath611 is isomorphic to @xmath612 blown up at two points . the @xmath417-fixed points in @xmath609 are the ones given by ( a ) above . let @xmath617 be the ( non - reduced ) word corresponding to the sequence of simple roots @xmath618 it is a subword of @xmath363 ; specifically , the roots @xmath480 , @xmath592 and @xmath619 have been omitted . there is an embedding @xmath620 which fills in the coordinates for the missing roots with the identity element of @xmath621 . the simple reflections @xmath622 that appear in @xmath617 generate the weyl group of the group @xmath623 , embedded into @xmath624 as block diagonal matrices acting on the middle two factors in the decomposition @xmath625 . it follows that we have an isomorphism @xmath626 , where the factors are the bott - samelson varieties for @xmath627 and @xmath628 , respectively . both @xmath629 and @xmath630 are isomorphic to the bott - samelson variety in example [ sl3 example ] , and it is easy to see that @xmath631 is a product of two copies of the fiber from that example . to see that @xmath632 is an irreducible component of @xmath310 , note that a computation with theorem [ bb cells ] shows that the paving by affines of @xmath310 given by intersecting with the biaynicki - birula cells has only one cell of dimension four , namely @xmath633 , and all other cells are of smaller dimension . so the closure of this cell must be a component of @xmath310 , and it is the only four - dimensional component , so it is equal to @xmath609 . to understand the other component , note that there is only one cell in @xmath634 of dimension three , namely @xmath635 , and all other cells are of smaller dimension . if we let @xmath636 be the smallest subword of @xmath363 containing the nonzero entries of @xmath637 , and @xmath638 , then the @xmath417-fixed points of @xmath639 are the ones given in ( b ) in the statement of the theorem . it is easy to check that @xmath640 , @xmath641 , and @xmath642 each map these fixed points onto a pair of fixed points in @xmath104 , and so they each map @xmath613 onto a @xmath417-curve . the map @xmath643 gives the required isomorphism of @xmath613 with @xmath614 . the fiber @xmath310 is not smooth , but there is only one component of dimension four , so we can reduce the computation of @xmath598 to the smooth case : it is isomorphic to the restriction map @xmath644 , where @xmath545 is the total space of the normal bundle @xmath562 to @xmath609 in @xmath645 . as remarked earlier , @xmath646 can be identified with multiplication by the euler class @xmath647 , so the image of @xmath598 is spanned by @xmath566 . we will compute this class by computing the equivariant euler class @xmath648 and then finding its image in ordinary cohomology . to do this , we split the normal bundle @xmath562 into line bundles . we have seen in the proof of proposition [ y components ] that @xmath649 ; we claim that this intersection is transverse . since @xmath650 and @xmath210 is @xmath15-equivariant , @xmath210 is a submersion over the cell @xmath552 , and the required transversality follows from the transversality of @xmath541 and @xmath552 in @xmath104 . let @xmath651 be the subwords of @xmath363 of length @xmath652 which contain all of the simple reflections in @xmath617 , together with one of the remaining simple reflections @xmath653 . if @xmath654 is the restriction to @xmath609 of the normal bundle to @xmath655 in @xmath656 , then we have @xmath657 . to compute the equivariant euler classes @xmath658 , we compute their restrictions to the fixed point set @xmath659 . the restriction @xmath660 is just the @xmath417-weight of the tangent space to the unique @xmath417-curve containing @xmath661 , contained in @xmath656 and not contained in @xmath655 . this curve is the curve joining @xmath661 and @xmath662 , where @xmath663 , @xmath664 , @xmath665 , @xmath666 . using propositions [ flag 1d orbits ] and [ bs fixed points ] , we can compute that its @xmath417-weight is the sum of the entries under @xmath667 and @xmath668 in the following tables : the equivariant class @xmath671 induces the same class in @xmath672 as @xmath673 where we abuse notation and write a weight @xmath674 instead of its pullback under the map @xmath675 . after a little computation one sees that this class restricts to zero at every point of @xmath676 except @xmath677 and @xmath678 , where it has the same restriction as @xmath679 , the equivariant euler class of the tangent bundle to @xmath609 . ( to compute the localization of @xmath679 to the fixed points , use the identification of the weights of @xmath417-curves in example [ sl3 example ] . note that the labeling of the fixed points @xmath680 in that example corresponds to the labeling of the fixed points @xmath681 and @xmath682 of @xmath629 and @xmath630 . ) then the atiyah - bott localization formula shows that @xmath566 is twice a generator of @xmath672 , as desired . [ rem : otherhex ] the other hexagon permutations can be shown to have @xmath73-torsion by a similar computation ; we give the main points . let @xmath683 and @xmath684 for @xmath685 . the fiber @xmath686 is still four - dimensional , but now the union of the components of maximal dimension is isomorphic to @xmath687 , where @xmath688 and @xmath689 , the union taken so that @xmath690 is identified with a @xmath417-curve in @xmath611 with trivial normal bundle . the excess intersection formula then implies that matrix of @xmath598 is diagonal under the natural bases given by the components of @xmath691 ( in other words , the components are orthogonal under the intersection form ) . the normal bundle to the component @xmath610 is the same as before , so we have @xmath692 . let @xmath693 . we follow the notation of section [ subsec : d4 ] : the simple reflections in @xmath0 are @xmath694 where @xmath695 all commute with each other . let @xmath363 be the word @xmath696 , put @xmath697 , and let @xmath698 . the @xmath417-fixed points in @xmath559 are @xmath699 the fiber @xmath310 is the transverse intersection of @xmath700 and @xmath544 . it is @xmath417-equivariantly isomorphic to @xmath614 , where the @xmath417-weights on the three factors are @xmath701 , @xmath702 , and @xmath703 , respectively . since @xmath310 is smooth , we have @xmath704 , and the map @xmath705 can be identified with multiplication by @xmath706 on @xmath707 , where @xmath708 is the normal bundle to @xmath310 in @xmath544 . as in the previous example we compute this by computing the localization of the equivariant class @xmath709 to the fixed points @xmath616 . we have @xmath710 so @xmath711 , where @xmath712 are the pullbacks of a generating class of @xmath713 by the three projection maps . multiplication by this class from @xmath714 to @xmath715 is given by the matrix @xmath716\ ] ] with respect to the natural monomial basis in @xmath717 , @xmath718 , @xmath719 . this matrix has determinant @xmath720 , so @xmath721 has @xmath73-torsion . a. belinson , j. bernstein , and p. deligne . faisceaux pervers . in _ analyse et topologie sur les espaces singuliers , i ( luminy , 1981 ) _ , volume 100 of _ astrisque _ , pages 5171 . france , paris , 1982 . f. du cloux . coxeter . a computer program available from http://math.univ - lyon1.fr/~ducloux / coxeter / coxeter3/english / coxeter3_% e.html[math.univ-lyon1.fr/ ducloux / coxeter / coxeter3/english / coxeter3_e.html ] . d. kazhdan and g. lusztig . schubert varieties and poincar duality . in _ geometry of the laplace operator ( proc . pure math . hawaii , honolulu , hawaii , 1979 ) _ , proc . pure math . , xxxvi , pages 185203 . soc . , providence , r.i . , 1980 . a. lascoux and m .- polynmes de kazhdan & lusztig pour les grassmanniennes . in _ young tableaux and schur functors in algebra and geometry ( toru , 1980 ) _ , volume 87 of _ astrisque _ , pages 249266 . france , paris , 1981 . g. lusztig . some problems in the representation theory of finite chevalley groups . in _ the santa cruz conference on finite groups ( univ . california , santa cruz , calif . , 1979 ) _ , volume 37 of _ proc . pure math . _ , pages 313317 . soc . , providence , r.i .
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we present a combinatorial procedure , based on the @xmath0-graph of the coxeter group , which shows that the graded dimension of the stalks of intersection cohomology complexes of certain schubert varieties is independent of the characteristic of the coefficient field .
our procedure exploits the existence and uniqueness of parity sheaves . in particular
we are able to show that the characters of all intersection cohomology complexes with coefficients in a field on the flag variety @xmath1 of type @xmath2 for @xmath3 are given by kazhdan - lusztig basis elements . by results of soergel
, this implies a part of lusztig s conjecture for @xmath4 with @xmath5 .
we also give examples where our techniques fail . in the appendix by tom
braden examples are given of intersection cohomology complexes on the flag varities of @xmath6 and @xmath7 whose stalks have different graded dimension in characteristic 2 .
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the physical picture of our theory for the forced dissociation of receptor - ligand bonds is very similar with the small ligand binding to heme proteins @xcite : there is a energy surface for dissociation which dependents on both the reaction coordinate for the dissociation and the conformational coordinate @xmath7 of the complex , while the later is perpendicular to the former ; for each conformation @xmath7 there is a different dissociation rate constant which obeys the bell rate model , while the distribution of @xmath7 could be modulated by the force component along x - direction ; higher temperature or larger diffusivity ( low viscosities ) allows @xmath7 variation within the complex to take place , which results in a variation of the energy barrier of the bond with time . there are two types of experimental setups to measure forced dissociation of receptor - ligand complexes . first we consider constant force mode @xcite . a diffusion equation in the presence of a coordinate dependent reaction is given by @xcite @xmath8 where @xmath9 is probability density for finding a value @xmath7 at time @xmath10 , and @xmath11 is the diffusion constant . the motion is under influence of a force modulating potential @xmath12 , where @xmath13 is intrinsic potential in the absence of any force , and a coordinate - dependent bell rate . in the present work [ bellmodel ] depends on @xmath7 through the intrinsic rate @xmath14 , and the distance @xmath15 is assumed to be a constant for simplicity . here @xmath16 and @xmath17 are respective projections of external force @xmath4 along the reaction and conformational diffusion coordinates : @xmath18 and @xmath19 is the angle between @xmath4 and the reaction coordinate . we are not ready to study general potentials here . instead , we focus on specific @xmath13s , which make @xmath20 to be @xmath21 where @xmath22 and @xmath23 are two constants with length and force dimensions . for example for a harmonic potential @xmath24 with a spring constant @xmath25 in which we are interested , it gives @xmath26 and @xmath27 defining a new coordinate variable @xmath28 , we can rewrite eq . [ origindiffusionequation ] with the specific potentials into @xmath29 where @xmath30 . compared to the original work by agmon and hopfield @xcite , our problem for the constant force case is almost same except the reaction rate now is a function of the force . hence , all results obtained previously could be inherited with minor modifications . considering the requirement of extension of eq . [ origindiffusionequation ] to dynamic force in the following , we present the essential definitions and calculations . substituting @xmath31 into eq . [ newdiffusionequation ] , one can convert the diffusion - reaction equation into schr@xmath32dinger - like presentation @xcite . @xmath33 where @xmath34 is the normalization constant of the density function at @xmath35 , and the effective " potential @xmath36+k_f(y).\nonumber\end{aligned}\ ] ] we define @xmath37 for it is independent of the force @xmath4 . . [ schodingerequation ] can be solved by eigenvalue technique @xcite . at larger @xmath11 in which we are interested here , only the smallest eigenvalue @xmath38 mainly contributes to the eigenvalue expansion which is obtained by perturbation approach @xcite : if the eigenfunctions and eigenvalues of the unperturbed " schr@xmath32dinger operator @xmath39 in the absence of @xmath40 have been known , @xmath41 and @xmath42 is adequately small , the first eigenfunction @xmath43 and eigenvalue @xmath38 of the operator @xmath44 then are respectively given by @xmath45 and @xmath46 considering that the system is in equilibrium at the initial time , _ i.e. _ , no reactions at the beginning , the first eigenvalue @xmath47 must vanish . on the other hand , because @xmath48 and the square of @xmath49 is just the equilibrium boltzmann distribution @xmath50 with the potential @xmath51 , we rewritten the first correction of @xmath38 as @xmath52.\nonumber\end{aligned}\ ] ] substituting the above formulaes into eq . [ transform ] , the probability density function then is approximated to @xmath53\phi_0(f)\end{aligned}\ ] ] the quantity measured in the constant force experiments is the mean lifetime of the bond @xmath54 , @xmath55 where the survival probability @xmath56 related to the probability density function is given by @xmath57.\end{aligned}\ ] ] in addition to the constant force mode , force could be time - dependent , _ e.g. _ , force increasing with a constant loading rate in biomembrane force probe ( bfp ) experiment @xcite . in principle the scenario would be more complicated than that for the constant force mode . we assume that the force is loaded slowly compared to diffusion - reaction process . we then make use an adiabatic approximation analogous to what is done in quantum mechanics . the correction of this assumption would be tested by the agreement between theoretical calculation and experimental data . we still use eq . [ origindiffusionequation ] to describe bond dissociations with the dynamic force , therefore we obtain the almost same eqs . [ forcedependentpotential]-[forcedependentquantumpotential ] except that the force therein is replaced by a time - dependent function @xmath58 . we immediately have @xcite @xmath59\phi_0(f_t),\end{aligned}\ ] ] where the berry phase " @xmath60 and @xmath61 is the first eigenfunction of the time - dependent sch@xmath32dinger operator @xmath62 because the eigenvalues and eigenfunctions of the above operator can not be solved analytically for general @xmath63 , we also apply the perturbation approach . hence , we obtain @xmath61 and @xmath64 by replacing @xmath42 in eqs . [ eigenfunctionexpansion ] and [ eigenvalueexpansion ] with @xmath63 . the berry phase then is approximated to @xmath65 finally , the survival probability for the dynamic force is given by @xmath66\nonumber\\\end{aligned}\ ] ] different from the constant force mode , data of the dynamic force experiments is typically presented in terms of the force histogram , which corresponds to the probability density of the dissociation forces @xmath67 @xmath68 particularly , when the force is a linear function of time @xmath69 , where @xmath70 is the loading rate , and zero or nonzero of @xmath71 respectively corresponds to the steady- or jump - ramp force mode in the dynamic force experiment @xcite , we have @xmath72\times\\ & & \exp\left[-\frac{1}{r}\int_{f_0}^f\left(\lambda^{(1)}_0(f')+ \lambda^{(2)}_0(f')+ b(f')\right)df'\right].\nonumber\end{aligned}\ ] ] we consider a bounded diffusion in the harmonic potential eq . [ harmonicpotential ] . then @xmath73 reduces to a harmonic oscillator operator with @xmath74 its eigenvalues and eigenfunctions are @xmath75 and @xmath76 respectively , where @xmath77 and @xmath78 is the hermite polynormials @xcite . given that the intrinsic dissociation rate satisfies the arrenhenius form @xmath79,\end{aligned}\ ] ] where the height of the energy barrier along the reaction coordinate @xmath80 is a function of the conformational coordinate @xmath7 . according to the form of barrier , we first analyze two simple and meaningful cases . + * bell - like forced dissociations*. the simplest function of the energy barrier might be linear with respect to @xmath7 , @xmath81 where @xmath82 is the height at position @xmath83 , and the slope @xmath84 ( its dimension force ) for the perturbation requirement in solving eq . [ schodingerequation ] . according to eqs . [ eigenvalueexpansion ] and [ berryphase ] , we easily get @xmath85\times\nonumber\\ & & \exp\left[\beta\left(\xi^\ddag f_\parallel-\frac{k_g}{k_x}f_\perp\right)\right],\nonumber\\ \lambda^{(2)}_0(f)&=&\frac{-k_0 ^ 2}{\beta dk_x}\exp{\left[-2\beta\delta g^\ddag_0+\frac{\beta k_g^2}{k_x}\right]}\times\\ & & \exp{\left[2\beta\left(\xi^\ddag f_\parallel-\frac { k_g}{k_x}f_\perp\right)\right]}\sum_{n=1}^{\infty}\frac{1}{nn!}\left(\frac{\beta k_g^2}{k_x}\right)^n,\nonumber\end{aligned}\ ] ] and @xmath86}\times\\ & & \exp{\left[2\beta\left(\xi^\ddag { f_t}_\parallel-\frac { k_g}{k_x}{f_t}_\perp\right)\right ] } \sum_{n=1}^{\infty}\frac{1}{n^2n!}\left(\frac{\beta k_g^2}{k_x}\right)^n,\nonumber\end{aligned}\ ] ] where @xmath87 . for large @xmath11 or @xmath25 ( or very small @xmath6 ) , the second correctness and the berry phase tend to zero . under these limitations the first eigenvalue of eq . [ origindiffusionequation ] is approximated to be @xmath88 \exp[\beta d^\ddag f].\end{aligned}\ ] ] here we define a new distance @xmath89 where @xmath90 whose dimension is distance . we see that the presence of the complex conformational coordinate could modify the original bell model in novel ways : ( i ) @xmath91 , eq . [ correctedbellform ] is indistinguishable from the origin bell model , although the projection distance @xmath92 from the bound state to the energy barrier may be increased or decreased in terms of the orientation of the applied force . in particular , if the force is antiparallel to @xmath7 , _ i.e. _ , @xmath93 , we get a bell - like rate expression with a distance " @xmath94 ; ( ii ) @xmath95 , the force does not affect dissociations of the bonds , which have been named ideal " bonds by dembo @xcite ; ( iii ) @xmath96 , the force slows down dissociations of the bonds . it is catch " bonds in which we are interested . in contrast to the catch behavior suggested by dembo @xcite , the rate decays exponentially with respect to the force instead of the square of the force . given the linear function eq . [ linearheightfunction ] and @xmath97 , increasing of the force only stabilizes the bonds by dragging the system to the higher energy barriers ( catch ) , whereas the other force component @xmath98 destabilizes the complex by lowering the energy barriers ( slip ) . therefore the sign of the distance @xmath99 in fact reflects a competition of the two contrast effects of the same force . + * dembo - like forced dissociations*. another function of the energy barrier is a harmonic with a spring constant @xmath100 @xmath101 where @xmath102 is the barrier height at position @xmath103 . because for any form of the barrier height , the dependence of @xmath104 and @xmath105 on @xmath11 is the same from eq . [ unpertubatedeigenvaluesharmonic ] , we only consider the large @xmath11 limitation in the following . hence we have @xmath106\times\nonumber\\ \exp\left[-\frac{\beta k_g(f_\perp - k_x(x_1-x_0))^2}{2k_x(k_x+k_g)}\right]\end{aligned}\ ] ] given @xmath107 and @xmath108 , we find that there is a interesting transition from slip to catch bond when the force increase over a threshold @xmath109 ; otherwise only catch bond presents . we note that the latter is very similar to the result proposed by dembo @xcite even their physical origins are completely different : both of them exponentially dependent on the square of the force . + * comparison with the experiments . * . @xmath110 and @xmath111 are the value of the linear functions in eq . [ bendfunction ] at position @xmath83 , while @xmath112 is the intersection of the functions at @xmath113 . the bold solid line is the minimum model which is used to fit the experiment . the shaded area represents the equilibrium distribution of the conformational coordinate under the potential @xmath51 ( eq . [ equilibriumdistribution ] ) . ] in the constant force rupture experiment of the psgl-1@xmath0p - selectin complex , the dissociation rate as the inverse mean lifetime of the complex first decreased and then increased when the applied force increased beyond a force threshold @xcite . we now can easily understand this counterintuitive transition according to the previous discussion about the bell - like dissociation rate : the dissociation effect of @xmath98 regains its dominance when the force is beyond the threshold . although in principle we can construct various barrier height functions which result into catch - slip transitions , the most simplest form may be a composition of two linear functions @xmath114 where we require that the distances defined in eq . [ newdistance ] with @xmath115 and @xmath116 are respectively minus and positive . for convenience , their absolute values are correspondingly denoted by @xmath117 and @xmath118 . define two intrinsic " dissociation constants @xmath119,\nonumber\\ k^s_0=k_0\exp[-\beta\delta g^\ddag_s(x_0)].\end{aligned}\ ] ] fig . [ figure1 ] shows the characteristics of the function . we then have @xmath120\exp\left[-\beta d^\ddag_c f\right]\nonumber\\ & & \times{\rm erfc}\left[-\left(\delta+\frac{k_c}{k_x}\right)\sqrt{\frac{\beta k_x}{2}}+f\sqrt{\frac{\beta}{2k_x}}\sin\theta \right ] \nonumber\\ & & + \text { } \frac{k_0^s}{2}\exp\left[\frac{\beta k_s^2}{2k_x}\right]\exp\left[\beta d^\ddag_s f\right]\\ & & \times{\rm erfc}\left[\left(\delta+\frac{k_s}{k_x}\right)\sqrt{\frac{\beta k_x}{2}}-f\sqrt{\frac{\beta}{2k_x}}\sin\theta \right]\nonumber\end{aligned}\ ] ] where @xmath121 , and the complementary error function @xmath122 before fixing numerical values of the parameters in eq . [ reactionratebendlandscape ] , we first simply analyze the main properties of @xmath38 given @xmath123 : ( i ) in the absence of force , due to @xmath124 and @xmath125 , we have @xmath126 which is the same with that obtained by agmon and hopfield @xcite ; ( ii ) if force is nonzero and smaller , @xmath127 which means that the bond is catch ; and finally ( iii ) , when the force is sufficiently large , eq . [ reactionratebendlandscape ] reduces to @xmath128 it is the ordinary slip bond . there are total eight independent parameters presenting in eq . [ reactionratebendlandscape ] : @xmath129 , and @xmath130 . it is not necessary to determine all of them , which is also impossible only through fitting to the experimental data @xcite . for example , the latter two parameters are lumped into @xmath131 , while and @xmath15 always presents with @xmath132 together . what we really concern with is the coefficients of the force and the factors before the error functions in eq . [ reactionratebendlandscape ] . they can be obtained by least square fit . guided by the properties eqs [ smallforcelimit ] and [ largeforcelimit ] , the fitting process in fact is simple . even so , we are still able to fix all parameters from the fitting results if we study a minimum model in which the slop @xmath116 is zero and @xmath133 . here the particular value of the angle is actually of no particular significance and it is only as a reference . we immediately have : @xmath134 , @xmath135 , @xmath136 , @xmath137 ; the other interesting parameters see tab . 1 , where the values are independent of the angle . .comparison of the parameters of the present theory , and the two - pathway and one energy - well model presented by thomas et al . @xcite on the constant force ( cf ) and the dynamic force ( df ) modes . the parameters for the slip behavior of the p - selectin are also listed as a reference @xcite . [ cols="^,^,^,^,^,^",options="header " , ] [ table ] substituting these values into eq . [ reactionratebendlandscape ] and according to eq . [ averagelifetime ] , we calculate the mean lifetime of the psgl-1@xmath0p - selectin complex with respect to different constant force in fig . [ figure2 ] : the agreement between theory and the experimental data is quite good . with the same parameters . ] more challenging experiments to our theory are the force steady- and jump - ramp modes @xcite . under large @xmath11 limitation , eq . [ ruputureforcedistribution ] reduces to @xmath138 . \label{approxruputureforcedistribution}\end{aligned}\ ] ] we see that the mean lifetime can be extracted from the above equation by setting @xmath139 , _ i.e. _ , @xmath140 . we calculate the dissociation force distributions of the steady- and jump - ramp modes at three loading rates to compare with the bfp experiments performed by evans _ et al . _ here we are not ready to fit the experiments afresh ; instead we directly apply the parameters obtained from the constant force mode to current case . because the bfp experimental data is for _ dimeric _ ligand psgl-1 , whereas our parameters are from monomeric ligand spsgl-1 . therefore it is necessary to map our predictions for the single bond spsgl-1@xmath0p - selectin to the double bonds . a natural assumption is that the two bonds share the same force and fail randomly . the same assumption has been used in previous works @xcite . hence , the probability density of the dissociation force for the double bond psgl-1@xmath0p - selectin complex is related to the single case by @xmath141 fig . [ figure3 ] presents the final result . we see that the theoretical prediction agrees to the data very well . hence we conclude that the adiabatic approximation proposed at the beginning is reasonable . under the different loading rates predicted by our theory ( solid curves ) for the psgl-1@xmath0p - selectin complex . the symbols are from the force steady- and jump - ramp experimental data @xcite . the apparent deviations between the theory and the data in the last column may be from the invalidation of the assumption of two equivalent bonds at higher loading rates . ] the previous works @xcite have claimed that they could not fit the experimental data from the constant force experiment using atomic force microscopy ( afm ) and the force jump - ramp experiment using bfp with the same parameters , _ e.g. _ , see tab . the authors simply contributed it to the different equipment and biological constructs though the experiments studied the same complexes @xcite . our calculations however show that the mechanical parameters defined by us have almost the same values . in addition , the tendencies of our density functions for the first two panels of the second array in fig . [ figure3 ] are closer to the data than that predicted by the two - pathways models @xcite . the density functions and the force histograms in the experiments reach the maximum and minimum at two distinct forces , which are named @xmath142 and @xmath143 in the following , respectively . this observation could be understood by setting the derivative of eq . [ ruputureforcedistribution ] with respect to @xmath4 equal to zero , @xmath144 we immediately see that the values of @xmath142 and @xmath145 must be larger than the catch - slip transition force @xmath146 for the left term in eq . [ extremacondition ] is negative as the bond is catch . indeed the experimental observations show that the force values at the minimum histograms are around a certain values even the loading rates change 10-fold . ( see figs . 2 and 4 in ref . the above equation could have no solutions when the loading rate is smaller than a critical rate @xmath147 , which can be obtained by simultaneously solving eq . [ extremacondition ] and its first derivative . we estimate @xmath148 pn / s using the current parameters , while the force @xmath149 pn ( about 26 pn in the double bond cases ) . if @xmath150 , then the density function is monotonous and decreasing function . therefore , the most probable force at the bond dissociation is zero . the most interesting characteristics of eq . [ extremacondition ] are the dependence of the maximum and minimum forces on the loading rate . in particular the latter is an important index in dynamic force spectroscopy ( dfs ) theory since it corresponds to the most possible dissociation force @xcite . when the loading rate is sufficiently large , and correspondingly @xmath143 is larger , the approximation of eq . [ reactionratebendlandscape ] at large force eq . [ largeforcelimit ] implies that @xmath151}\propto \ln r.\end{aligned}\ ] ] the experimental measurement supported this prediction ; see fig . 3a in ref . @xcite . on the other hand , due to that @xmath142 is very close to @xmath146 at the larger loading rate , employing the taylor s expanding approach we have @xmath152,\end{aligned}\ ] ] where @xmath153 . substituting it into eq . [ extremacondition ] , we get @xmath154 it means that @xmath142 tends to @xmath146 very fast . different from @xmath143 , the loading rate dependence of @xmath142 is an intrinsic property of the catch - slip bond ; a unique requirement is that the dissociation rate @xmath38 has a minimum at the transition force @xmath146 . therefore @xmath142s observed in experiment performed by evans _ et al . _ @xcite are almost the catch - slip transition force observed in the constant force rupture experiment performed by marshall _ _ @xcite . indeed , the force values of the minimum force histograms for the former are about 26 pn , while the transition force for the latter ( _ dimeric _ psgl-1@xmath0p - selectin ) is also about 26 pn . we know that the dissociation forces distribution of a simple slip bond only has a maximum at a certain force value that depends on loading rate @xcite . therefor the catch - slip bond can easily be distinguished from the slip case by the presence of a minimum on the density function of the dissociation forces at a nonvanished force . because the above analysis is independent of the initial force @xmath71 , in order to track the catch behaviors in the force jump - ramp experiments , @xmath71 should be chosen to be smaller than @xmath146 . compared to the chemical kinetic schemes , our theory should be more attractive on the following aspects . first of all , we suggest that the counterintuitive catch - slip transition is a typical example of the rate processes with dynamic disorder . because this concept has been broadly and deeply studied from theory and experiment during the past two decades , extensive experience and knowledge could be used for reference . for example , we suggest that a new receptor - ligand forced dissociation experiment could be performed over a large range of temperatures and solvent viscosities . according to eq . [ origindiffusionequation ] , if the viscosity is so higher that @xmath155 , we could predict @xmath156.\end{aligned}\ ] ] we know that such a dissociation reaction is a typical example of the rate processes with _ static _ disorder @xcite . in addition that the survival probability of the bond converts into multiple exponential decay at a single force from the single exponential decay at the large @xmath11 limitation ( see eq . [ survivalprobability ] ) , the mean lifetime is @xmath157 which means that the catch - slip bond changes into slip bond only . then our theory gives a intuitively obvious physical explanation of catch bonds in an apparent expression ( eq . [ correctedbellform ] ) : they could arise from a competition of the two components of applied external force along the dissociation reaction coordinate and the molecular conformational coordinate ; the former accelerates the dissociation by lowering the height of the energy barrier , while the latter stabilizes the complex by dragging the system to the higher barrier height . finally , the time - dependence of the forced dissociation rates could be induced by either global conformational changes of the complex or local conformational changes at the interface between the receptor and ligand ; no separated bound states and pathways are needed in the current theory . therefore it is possible that one can not find new stable complex structures through experiments or detailed molecular dynamics ( md ) simulations . even there are many advantages in the present theory . we can not definitely distinguish which theory or model is the most reasonable and more close real situations with existing experimental data . moreover , except the coarse - grain physical picture our theory does not reveal the detailed structural information of the catch behavior of the ligand - receptor complexes , while biologists might be more interested in it . we could correspond the increasing height of the energy barrier with respect to the conformational coordinate to the hook structure @xcite or more affinity bound states @xcite , however we believe that further single - molecule experiments including micromanipulation experiments and fluorescence spectroscopy , more crystal structure data and detailed md simulations from the atomic interactions are essential to elucidate the real molecular mechanism of the catch bonds . + + fl thanks prof . mian long and dr . fei ye for their helpful discussion about the work .
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recently experiments showed that some adhesive receptor - ligand complexes increase their lifetimes when they are stretched by mechanical force , while the force increase beyond some thresholds their lifetimes decrease .
several specific chemical kinetic models have been developed to explain the intriguing transitions from the catch - bonds " to the slip - bonds " . in this work
we suggest that the counterintuitive forced dissociation of the complexes is a typical rate process with dynamic disorder . an uniform one - dimension force modulating agmon - hopfield model is used to quantitatively describe the transitions observed in the single bond p - selctin glycoprotein ligand 1(psgl-1)@xmath0p - selectin forced dissociation experiments , which were respectively carried out on the constant force [ marshall , _
et al . _ , ( 2003 ) nature * 423 * , 190 - 193 ] and the force steady- or jump - ramp [ evans _ et al . _ , ( 2004 ) proc .
natl .
acad .
sci .
usa * 98 * , 11281 - 11286 ] modes .
our calculation shows that the novel catch - slip bond transition arises from a competition of the two components of external applied force along the dissociation reaction coordinate and the complex conformational coordinate : the former accelerates the dissociation by lowering the height of the energy barrier between the bound and free states ( slip ) , while the later stabilizes the complex by dragging the system to the higher barrier height ( catch ) .
adhesive receptor - ligand complexes with unique kinetic and mechanical properties paly key roles in cell aggregation , adhesion and other life s functions in cells .
a well studied example is the receptors in selectin family which comprises e- , l- and p - selectin interacting and forming bonds " with their glycoprotein ligands .
these bonds are primarily responsible for the tethering and rolling of leukocytes on inflamed endothelium under shear stress @xcite . in particular , in the past two years great experimental efforts @xcite have been devoted to study the surprising kinetic and mechanical behaviors of the bonds between l- and p - selectin and p - selectin glycoprotein ligand 1 ( psgl-1 ) at the single molecule level : the lifetimes of these bonds first increase with initial application of small force , which are termed catch " bonds , and subsequently decrease , which are termed slip " bonds when the force increases beyond some thresholds .
the most important biological meaning of this discovery is that the catch - slip transitions of the psgl-1@xmath0l- and @xmath0p - selectin bonds may provide a direct experimental evidence at the single - molecule level to account for the shear threshold effect @xcite , in which the number of rolling leukocytes first increases and then decreases while monotonically increasing shear stress . on the theoretical side , it is a challenge to give a reasonable physical theory or model to explain the counterintuitive bond transitions .
bell @xcite firstly suggested that the force induced dissociation rate of adhesive receptor - ligand complex could be described by , @xmath1 , \label{bellmodel}\end{aligned}\ ] ] where @xmath2 is the intrinsic dissociation rate constant in the absence of force , @xmath3 is the distance from the bound state to the energy barrier , @xmath4 is a projection of external applied force along the dissociation coordinate , @xmath5 the boltzmann s constant , and @xmath6 is absolute temperature .
the validity of the model has been demonstrated in experiments @xcite . although later at least four models have been put forward to explain and understand various receptor - ligand forced dissociation experiments @xcite , they can not predict catch bonds because force in these models only lowers height of the energy barrier while shortening lifetimes of the bonds .
an exception is the hookean spring model proposed by dembo many years ago @xcite , in which a catch bond was raised in mathematics .
compared to the exponential decay of the lifetimes of slip - bonds with respect to force in experiments @xcite , the model claimed that the lifetimes decrease exponentially with the square of force @xcite .
in addition , the hookean spring model can not account for the catch - slip bond transitions in self - consistent term .
prompted by the intriguing experimental observations , three chemical kinetic models have been developed . evans _
et al . _
presented a two pathways , two bound states model with rapid equilibrium assumption between the two states .
they suggested that the catch - slip bond transitions take place due to applied force switching the pathways from the one with slower dissociation rate to the fast one @xcite .
although this insightful viewpoint well described force jump - ramp experiments , there are two apparent flaws in physics .
first if the forced dissociation experiments were performed at very low temperatures or higher solvent viscosities , force would be independent of the bond dissociations since the force in the two pathways and two bound states model only acts on the inner bound states , while these states would be frozen " under this circumstances .
the other is that force does not accelerate the dissociation processes further when the force is sufficiently large for the fast dissociation rate is a constant .
the next model given by barsegov and thirumalai @xcite with same kinetic scheme seems to improve the two flaws in which the dissociation rates of the two pathways were allowed to be force - dependent with the bell formulas .
unfortunately , so many independent reaction constants with arbitrary dependencies on the force parameters ( total seven parameters ) and the final dissociation rate depending on them in a complicated way make the physical explanations and the determination of the parameters difficult to track . very recently , a competitive two pathways and one bound state model was proposed by thomas _
et al . _
@xcite .
this model is distinct from the others because there is a catch pathway therein , which was thought to arise from a backward unbinding pathway .
but the model is not intuitively obvious just like the authors pointed out . as one type of noncovalent bonds , interactions of adhesive receptors and their ligands
are weaker .
moreover , the interfaces between them have been reported to be broad and shallow , such as the crystal structure of the psgl-1@xmath0p - selectin bond revealed @xcite .
therefore it is plausible that the energy barriers for the bonds are fluctuating with time due to either global conformational changes or local conformational changes at the interfaces .
association / dissociation reactions with fluctuating energy barrier have been deeply studied by statistical physicists in terms of rate processes with _ dynamic disorder _
@xcite during the past two decades .
a prototype is the ligand rebinding in myoglobin where the rate constant depends on a protein coordinate @xcite .
hence it is of interest to determine whether the fluctuation of energy barrier responses to catch - slip bond transitions .
such studies should be meaningful since in the bell s initial work and the other models developed later , the intrinsic rate constants @xmath2 were deterministic and time - independent .
it is possible to derive unexpected results from the relaxation of this restriction .
stimulated by the two considerations , in the present work we propose that the intrinsic dissociation rate in the bell model is controlled by a conformational coordinate of receptor - ligand complex , while the coordinate is fluctuating as a brownian motion in a bound harmonic potential ; applied force not only lowers the height of the energy barrier as described in eq . [ bellmodel ] but also modulates the distribution of the conformational coordinate .
in addition to well predicting the experimental data , our theory may also provide a new physical mechanism for the dissociation rates suggested by bell @xcite and dembo @xcite early .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in january 2016 , a crocheter on the social networking site ravelry @xcite posted images of a coin purse she had made as a christmas present , and started a discussion thread asking readers of a particular forum if they could help her identify the name of its 3d shape ( hereafter termed _ the shape _ ) . the purse , shown in figure [ purse ] , had been constructed following a pattern @xcite by making a flat oval , and attaching a zipper to its edge in such a way that when the zipper closed , the purse was not folded over exactly in half onto itself ( flat ) , but sat up , enclosing an intriguingly - shaped volume . the crocheter had read the german version @xcite of an ian stewart column on the sphericon @xcite , and believed that the shape might be sort - of - but - not - quite a sphericon . however , the paper templates for making one s own sphericons that she had seen @xcite have curved edges only , which her oval did not . in fact _ oval _ is rather loose a term , and broadly means _ egg - like_. the purse in question had been constructed from a _ stadium _ , the 2d shape made from a rectangle and two semi - circles , as shown in figure [ stad ] . over the next few weeks , various ravelry members , including the author of this paper , joined in on - line discussion and brain - storming about the shape . more than 500 of ravelry s ( admittedly 6 million ) members read at least part of the thread ( before , as threads do , it wandered off to shapes more generally ) . patterns for objects with similar construction were pointed out @xcite . as well as the sphericon family , d - forms @xcite were mentioned . so , was the shape one of these , and if so , which was it ? this seemingly simple question , prompted by a hand - made christmas gift , has triggered this survey and investigation of sphericons and d - forms . the main conclulsions of this investigation are to be found in section [ concl ] . the sphericon has been described as ` [ a ] solid ... not as widely known as it should be ' @xcite , and conversations with mathematical colleagues , and presentations about this work , have shown this to be so . thus , an introduction to the history , properties and generalisations of sphericons , and of d - forms , make up the next two sections of the paper . the classification of the shape is then clear . the paper concludes with some more general results , summarised in table [ classify ] , and includes templates for the surfaces required to make several particular generalised sphericons by d - forming , thereby stitching ( literally ) a connection between the sphericon family and d - forms . \(a ) at ( 0,0 ) ; ( b ) at ( 0,2 * 1.5 cm ) ; ( c ) at ( 4.71cm,0 ) ; ( d ) at ( 4.71cm,2 * 1.5 cm ) ; ( a ) node[above ] @xmath1 ( c ) ; ( c ) arc [ start angle=-90 , end angle=90 , radius=1.5 cm ] ( b ) arc [ start angle=90 , end angle=270 , radius=1.5 cm ] ; ( a ) node[left ] @xmath2 ( b ) ; ( c ) node[right , pos=0.75 ] @xmath3 ( d ) ; at ( @xmath4 ) ; the sphericon , at least as a mathematical object , came to public attention in stewart s mathematical recreations column @xcite in 1999 . colin roberts , a joiner who had enjoyed geometry at school , sent stewart his creation , after seeing him on television . the column was written shortly after this . roberts had first made a sphericon from mahogany some thirty years earlier in 1967 @xcite . independently , in the early 1980s , an israeli designer david hirsch submitted a patent application for ` meander motion ' toys constructed on the same principles @xcite . roberts motivation , however , lay in producing a one - surfaced object , with no hole ( unlike the well - known mbius strip ) . the sphericon can be constructed from a pair of identical cones of apex angle 90 degrees joined at their bases ( a _ bicone _ ) . cut the bicone in half through the two apices so as to reveal its square cross - section , twist one half through 90 degrees , and then ` glue ' the square faces together again , as shown in figure [ cut ] . especially when made of grained wood , it is a beautiful object . it has a single , sinuous surface @xcite , and two disjoint curved edges , set at right angles ; see figure [ sphericon ] . the surface has the property that if one traces a finger along the middle of it , one comes back to the starting point @xcite . ( rather inaccurately from the point of view of mathematical terminology , this has been called a ` continuous ' surface @xcite ; we will avoid that term . ) the two curved edges each end in vertices ( four in total ) . the sphericon rolls in an amusing manner , what stewart terms ` a controlled wiggle ' @xcite , the meander motion of hirsch s toys @xcite . a _ sphere _ will roll in a straight line , and a _ cone _ rolls in a circle . the sphericon changes direction as the parts of its surface come into contact with the surface it is rolling on , but on average the motion is in a line . hence the name _ sphericon_. sphericons are one of the intriguing solids that can be assembled , and their rolling behaviour predicted , at momath in new york @xcite . the sphericon featured next in the monthly feature column of the american mathematical society @xcite . the sphericon was used as an interesting examplar of how edges and apices have to be treated carefully , in order to calculate the total gaussian curvature of the figure to be the expected @xmath5 . phillips introduced the _ hexasphericon _ and found connections between such higher order sphericons ( to be introduced below ) and his work on mazes @xcite . some particular closed space curves were found to lie on the sphericon surface @xcite , and an explicit hybrid function form ( in cartesian coordinates ) is given . perhaps not surprisingly , given its origins , wood - turners began experimenting with the sphericon . a special join , called a paper join , enables the cutting - open process to be done without damage . the bicone can be replaced by any solid of revolution with appropriate cross - section symmetry . some makers introduced different profiles ( producing what they termed _ femispheres _ ) or non - convex cross - sections ( the _ streptohedrons _ ) @xcite . other makers @xcite generated some of the convex geometric objects that are classified as the series of @xmath6-icons @xcite . sphericons have been made of ceramic , glass , plastic ( 3d printing ) and as metal sculptures , as well as of wood . think of the bicone as the solid of revolution of a square about its diagonal ( as in the image ( a ) of figure [ cut ] ) . now replace the square by any other @xmath6-vertex regular convex polygon , and use one of the symmetry axes to generate a solid of revolution @xcite . when @xmath6 is odd , this axis will run through one vertex and one straight edge of the polygon . when @xmath6 is even , the polygonal cross - section can be rotated about an axis through two opposite vertices or through the midpoints of two opposing edges . for @xmath7 , different ` @xmath6-icons ' can be produced by more than one rotation ( in the cut - rotate - reglue process ) , due to the rotational symmetries of the cross - section ; these may be completely different or may come in right- and left - hand pairs . a full classification of the family of solids requires one to specify @xmath8 where @xmath9 is the rotation ( in radians ) between cutting and regluing , termed the twist @xcite . two special cases will now be considered : * the dual sphericon * rotating a square about an axis through the midpoints of opposite edges results in a cylinder of equal height and diameter . applying the cutting and rotating process , an object with _ two surfaces _ and _ one closed edge _ is produced , as illustrated in figure [ dual ] . this object is dual to the original sphericon in a particular sense ; the sphericon has _ two disjoint edges _ ending in vertices and _ one surface _ that can be traced as described above . springett @xcite described this object , which wood - turners term _ side - cut _ , as being ` rather unexciting ' . in the cutting process , four semicircles are produced , which are then joined smoothly to the sides of the cylinder . the paper template for this object consists of not one stadium , but two . the dual sphericon is a possible two - sided die . indeed , such dice and various @xmath10-icons are available from various 3d printing artists . \(a ) at ( 60pt,7.5pt ) ; ( b ) at ( 108.5pt,40pt ) ; ( c1 ) at ( 64pt,63.75pt ) ; ( c2 ) at ( 50pt,63.25pt ) ; ( d ) at ( 9pt,100pt ) ; ( e ) at ( 65pt,143pt ) ; ( f ) at ( 108.5pt,130pt ) ; ( g ) at ( 65pt,110pt ) ; ( h ) at ( 9pt,50pt ) ; ( f ) ( b ) ; ( d ) ( h ) ; ( a ) .. controls ( @xmath11 ) and ( @xmath12 ) .. ( b ) ; ( b ) .. controls ( @xmath13 ) and ( @xmath14 ) .. ( c1 ) .. controls ( @xmath15 ) and ( @xmath16 ) .. ( c2 ) .. controls ( @xmath17 ) and ( @xmath18 ) .. ( d ) ; ( d ) .. controls ( @xmath19 ) and ( @xmath20 ) .. ( e ) .. controls ( @xmath21 ) and ( @xmath22 ) .. ( f ) .. controls ( @xmath23 ) and ( @xmath24 ) .. ( g ) .. controls ( @xmath25 ) and ( @xmath26 ) .. ( h ) .. controls ( @xmath27 ) and ( @xmath28 ) .. ( a ) ; * the trisphericon * a _ single _ cone may be created as the solid of revolution of an equilateral triangle , the @xmath29 case . it can be cut to produce a triangular cross - section . when one half is rotated through 120 degrees and the form reglued , a solid with one curved surface , one -shaped edge and 2 vertices is produced ; see figure [ trisph ] . in the sense described above , this object is self - dual . following a similar argument to stewart @xcite we can establish what the paper template for this object is ( figure [ temp ] ) . this object has also been called the ` kleines sphericon ' @xcite , the ` conicon ' @xcite or the ` equilaticon ' @xcite . while ` one - sided dice ' based on the mbius band are available to purchase , trisphericons are even better candidates for this title , as they have the desirable feature of having no hole . \(a ) at ( 0,0 ) ; ( b ) at ( 0,2 * 1.5 cm ) ; ( c ) at ( 2 * 1.5cm,2 * 1.5 cm ) ; ( d ) at ( 2 * 1.5cm,4 * 1.5 cm ) ; ( a ) node[left ] @xmath2 ( b ) ; ( b ) node[above ] @xmath2 ( c ) ; ( c ) node[right ] @xmath2 ( d ) ; ( a ) ( c ) ; ( b ) ( d ) ; ( d ) arc [ start angle=90 , end angle=180 , radius=2 * 1.5 cm ] arc [ start angle=90 , end angle=270 , radius=1.5 cm ] arc [ start angle=270 , end angle=360 , radius=2 * 1.5 cm ] arc [ start angle=-90 , end angle=90 , radius=1.5 cm ] ; ( @xmath30 ) + + ( 0.2 * 1.5cm,0 ) + + ( 0,0.2 * 1.5 cm ) ; ( @xmath31 ) + + ( 0,0.2 * 1.5 cm ) + + ( 0.2 * 1.5cm,0 ) ; more generally , the @xmath6-icons can always be considered as formed by cutting , twisting and reglueing a solid made of @xmath32 sections of cones ( possibly including a cone with apex at infinity , a cylinder ) . the @xmath6-icons for odd @xmath6 are self - dual and the @xmath6-icons for even @xmath6 have dual forms , generated by the two different symmetry axes @xcite . when it is pertinent to specify the twist @xmath9 used in construction , the notation @xmath0-icon will be used . stadia may be present in the dual @xmath6-icons , as we can see from the four semicircles created in the cutting process . new features arise as @xmath6 increases , such as surfaces enclosing other surfaces @xcite think of a velodrome , with the sloped surface around the centre . it is worthwhile , at this point , if you have resisted the temptation so far , to make a sphericon from paper or light card using templates such as those of pppe @xcite . ( indeed , the crocheter mentioned in the introduction had done exactly this with her family , which is why she was familiar with them . ) of course , you will not have a solid sphericon ; you will have the surface and will manipulate and tape it to enclose a volume shaped like the solids . the d - form entered mathematics in a similar way to the sphericon but this time from design . they are 3d objects formed when the boundaries of two 2d shapes of equal perimeter length are joined . the british designer and inventor tony wills says they came to him in a dream , as a way of expanding his working vocabulary of forms @xcite . wills has used d - forms in street architecture ( benches , bins , bollards ) @xcite . the constituent 2d shapes need to be made of a material that does not stretch or shear ; a typical cushion is not a d - form ! two circles can not produce a ( non - trivial ) d - form , but two identical offset ellipses do , as seen in figure [ ms ] . the cross - section of the object produced using two ellipses with their major axes set at right angles looks like the letter capital . ( d is for dream , for the shape of this cross - section and also for developable surfaces @xcite . ) as the shapes are joined , they inform each other as to how their joined boundary should curve in space . wills observed that choosing different points on the boundaries of the 2d shapes to start the seam between them changes the resulting d - form @xcite . he investigated joining the boundary of a circle to that of a square ( i.e. not a smooth boundary ) producing an object he termed a squaricle , and also used non - convex shapes . baseball construction @xcite is an example of d - forming that long pre - dates wills ; two flat pieces of leather have been designed ( by trial and error initially ) to be joined in such a way that their seam lies on a sphere . the initial 3d form is not spherical the leather is subsequently stretched over a hard spherical core . d - forms are piece - wise _ developable surfaces _ , in that they can be cut open and each piece flattened , without creasing , to be planar @xcite , although they have been described thus far as being formed by the reverse process . developable surfaces have the advantage for designers , artists and builders that they can be constructed from rolled sheet metal ; even ships and airplanes are made this way @xcite . d - forms have been observed to have much in common with various sculptural works @xcite . predicting in advance what the d - form will look like , and hence what 2d shapes to use , is not easy , though computational methods ( based on meshes ) have been developed @xcite . a recent paper @xcite describes software used ( in the education of architecture students ) to match the perimeter length of different planar shapes , scaling them to be isoperimetric , in order that they can be used to create d - forms . as with the sphericon , making a paper model , right now , will help you understand how d - forms evolve in front of you and how beautiful they can be . you can use a template @xcite or you can easily cut your own using a double layer of paper . d - forms were introduced to the mathematical world by wills and sharp before their publications @xcite . questions @xcite regarding their creasing , and whether or not a d - form is always the convex hull of a space curve , were posed in 2001 and addressed in 2007 , with the boundaries of the 2d shapes restricted to being convex @xcite . demaine and orourke also introduced the concept of a _ pita - form _ a pita - form is defined to be the 3d shape obtained by glueing ( or zipping ) the boundary of a _ single _ convex 2d shape to itself . demaine returned to the open questions surrounding d - forms with a different co - author in 2010 @xcite . they established that both d - forms and pita - forms are the convex hull of their seams , that d - forms are always crease - free away from the seam , but that a pita - form may have at most one crease , between the two endpoints of its seam . ( of course , ` crease ' has to be defined a bit more carefully ( in terms of differentiability properties ) than by simply looking at a paper model , as this initially led to the proposition that pita - forms also might never have creases @xcite . ) also introduced were _ seam forms _ , a natural generalization to more flat ( convex ) pieces than two . ( it may be pertinent to mention one more concept introduced by demaine and collaborators , that of _ zipper unfolding _ of a polyhedron @xcite , to prevent confusion . in this case , the surface being zipped is deliberately folded , marking it into the polygons that constitute the faces of the polyhedron . ) as mentioned in the introduction , both sphericons and d - forms were suggested as describing the shape ( figure [ purse ] ) , formed from a crocheted stadium , lined so that there is some rigidity to the surface and no shearing or stretching , before the zipper is added . the astute reader will by now have realised that , simply , it is a pita - form , made from a stadium . this author has experimented with making various pita - forms from crocheted stadia , varying the ratio between the rectangle s height and the radius of the semi - circles , and with different starting points for the seam ( figure [ pita ] ) . tips for making your own from fibre are given in appendix a. but does the shape belong on the sphericon family tree also ? as should have become clear from section [ solid ] , any @xmath6-icon s surface is subject to symmetry , side length and angle constraints prescribed by its construction from a regular 3d object . what if one steps back from these origins , and hence from these constraints ? could not the stadium ( figure [ stad ] ) be regarded as the outcome of relaxing conditions which led to figure [ temp ] ? just as the two shapes in figure [ hex ] are hexagons , one regular and one irregular , the shape , with its one surface and one -shaped seam between two distinct endpoints , could be termed an ` irregular trisphericon ' , to coin a phrase . iin 0, ... ,5 ( i ) at ( i60:1.5 ) ; iin 0, ... ,5 let 1=int(mod(i+1,6 ) ) in ( i ) ( 1 ) ; ( 0,0 ) + + ( 35:1.5 ) + + ( 65:1 ) + + ( 120:1 ) + + ( 200:2 ) + + ( 250:0.75 ) cycle ; to put this around the other way , the [ regular ] trisphericon is a pita - form , formed from a particular 2d shape ( figure [ temp ] ) which is _ not _ convex . moving on further from the original question about the shape that motivated this investigation , what of other relationships between the sphericon family , and pita- and d - forms ? a kinship is foreshadowed in the works of the turkish artist ilhan koman . consider the piece called _ rolling lady _ which dates from 1983 @xcite . it consists of two bicones ( four developable surfaces ) interlocked at right angles in such a way as to roll . at its elegant heart lie two perpendicular circles , just as two perpendicular edges , circle sectors , are a striking feature of the sphericon . ( in this article , the connection of the sphericon to the rollers and wobblers of @xcite , and to the oloid , has not been explored , being somewhat tangential to the identification question under consideration . ) let s look at the shape that did not much interest springett , the dual sphericon ( figure [ dual ] ) . again , the astute reader has probably pre - empted the statement that this is precisely a d - form obtained from two stadia , with @xmath33 , and attached to one another symmetrically . this inspired the crocheted piece _ d - based _ shown in figure [ ball ] . d - forms made using stadia seem not to have been mentioned or visualised prior to 2007 @xcite . authors moved directly from dismissing the circle to trying out the ellipse . perhaps this springs from the fact that the cone is probably the most well - known developable surface @xcite , so that conic sections seemed a natural constituent for d - forms ( despite the fact that there is no formula for the perimeter of an ellipse , and matching perimeter length is the name of the game in making d - forms @xcite ) . of course , a stadium has a perimeter that can be calculated as precisely as one might wish : @xmath34 . two identical stadia can always be joined to form a d - form , but this will only be a dual sphericon if the midpoints of the semicircles on one stadium are joined to the midpoints of the straight sides of the other , and if @xmath35 . when @xmath36 , and the attachment is midpoint - to - midpoint , the elongated object is stable on one of its four semi - circular faces . such a d - form makes an excellent four - sided die . in fact , it is arguably nicer than the one based on the tetrahedron , which has a point , not a face , in the air ! when @xmath37 , and the attachment is symmetric , the object is stable only when it rests on the centre - line of one or other of its two curved surfaces . if tilted slightly from this position , it will rock back . hollow mock - ups of these two d - forms can be seen in figure [ stable ] . a perfectly constructed dual sphericon has stability properties that transition between these two cases . tilting the dual sphericon when it is sitting on the centre - line of one of its two curved surfaces does not raise or lower the centre of gravity , so that this is a position of neutral equilibrium ; it rolls , until it sits flat on one of its two semicircular faces . it is interesting to classify which @xmath0-icons , over and above the trisphericon and the dual sphericon , are d - forms and pita - forms ( in the sense of permitting the convexity condition on the constituent surfaces to be relaxed ) . the information needed to perform this classification is available , though using unconventional mathematical nomenclature , and at least one error , in early studies at un - maintained web sites @xcite . it is collected and corrected in table [ classify ] . in interpreting this table , the values of @xmath38 are restricted to @xmath39 for @xmath6 odd , and to @xmath40 for @xmath6 even , limiting the forms to those which are unique ( up to chirality ) . [ classify ] an @xmath0-icon will be a pita - form if it has one surface and one edge that runs between two distinct endpoints ; it will be a @xmath41-form if it has two surfaces and one closed edge . from this we can conclude : * when @xmath6 is odd , and co - prime to @xmath38 , the @xmath0-icon is a generalised pita - form . ( this includes the trisphericon . the first values for which this does not apply are @xmath42 . ) * when @xmath6 is even , and the @xmath0-icon has been generated using the midpoint - to - midpoint symmetry axis , it will be a d - form , if @xmath43 and @xmath38 are co - prime . ( this includes the dual sphericon . it also , for example , includes all 5 possible @xmath44 forms . ) by way of illustration , figure [ hexasphericon ] shows a wooden version of the dual hexasphericon ( @xmath45 , @xmath46 ) ; the two semicircular ends that are visible are from the same surface . when a regular hexagon ( side length @xmath3 ) is rotated to form a solid of revolution ( about the midpoints of a pair of opposite straight edges ) , its vertices trace out three circles . two of these have radius @xmath3 , and the other has radius @xmath2 , the distance between opposite vertices in a regular hexagon , by elementary geometry ( see figure [ hex ] ) . each of these circles is cut in half , relocated in space and then joined to half of another circle , when the cut - rotate - reglue process is carried out . apart from the circular ends , the other parts of the dual hexasphericon s surface are pieces of cones , and each is formed ( developed ) from a quarter of an annulus . thus each of the two surfaces that form the dual hexasphericon by d - forming is of the shape shown in figure [ hextemp ] . ( 0,0 ) arc [ start angle=0 , end angle = 90 , radius=2 ] ; ( 0,0 ) arc [ start angle=0 , end angle=-180 , radius=0.5 ] ; ( -1,0 ) arc [ start angle = 0 , end angle = 90 , radius = 1 ] ; ( -2,1 ) arc [ start angle = -90 , end angle = -180 , radius = 2 ] ; ( -2,2 ) arc [ start angle = -90 , end angle = -180 , radius = 1 ] ; ( -3,3 ) arc [ start angle = 0 , end angle = 180 , radius = 0.5 ] ; ( -1.5,0.5 ) ( -1.29,0.707 ) ; ( 0,0 ) ( -1,0 ) ; ( -3,3 ) ( -4,3 ) ; ( -2,1 ) ( -2,2 ) ; for the octasphericon ( @xmath47 , @xmath46 ) a similar geometric argument shows that each of its constituent surfaces is made up of a rectangle , two semi - circles and two sectors ( opening angle @xmath48 ) from an annulus . the inner arc of the annulus sector has the same length as the semi - circles ( @xmath49 ) , and the length of the outer arc and of the rectangle side is @xmath50 . the template is shown in figure [ dualoct ] . ( 3.8,0 ) ; ( 0,1 ) ( 3.8,1 ) ; ( 3.8,1 ) arc [ start angle=90 , end angle = -37.3 , radius = 1.71 ] ; ( 3.8,0 ) arc [ start angle=90 , end angle = -37.3 , radius = 0.71 ] ; ( 0,0 ) arc [ start angle = -90 , end angle = -217.3 , radius = 1.71 ] ; ( 0,1 ) arc [ start angle = -90 , end angle= -217.3 , radius = 0.71 ] ; ( -0.56 , 2.14 ) arc [ start angle = -37.3 , end angle = 142.7 , radius = 0.5 ] ; ( 4.36 , -1.14 ) arc [ start angle = -217.3 , end angle = -37.3 , radius = 0.5 ] ; for @xmath51 , there are two possible @xmath52-icons . their two constituent surfaces are each bounded by 2 semicircles , 2 arcs of length @xmath49 and @xmath53 , and four of length @xmath54 . the order in which they connect to each other is different for @xmath46 and @xmath55 . in the static representation of words and diagrams , it is not easy to appreciate the features of these forms . as well as making some of their own , readers are encouraged to enjoy some remarkable on - line animations of them ( for numerous higher values of @xmath6 and @xmath38 ) @xcite . as far as can be judged from the published literature , these instructions for creating the ` piece - wise circular ' surfaces of the dual @xmath0-icons by d - forming have not been explicitly stated previously , though presentations on sphericons and d - forms have been made at the same conference @xcite ! this may be because of the very different geography of their birthplaces , in the realm of solids and of surfaces respectively . it seems to have taken crochet fibre to link them . i wish to acknowledge the contributors to the thread _ how would you call this shape ? _ in the woolly thoughts forum of ravelry , in particular annagret , bevbh , irishlacenet , madeleines , soyloquesoy and woolhelmina , for their inspiration of this piece , and for pooling their collective wisdom about fibre arts and mathematics . thanks also to jane pitkethly for creating four of the line drawings and to the fibre artists who permitted photos of their work to be used . particular thanks to the estate of the late steve mathias for giving permission for photos of his wood - turning to be used , and to craig kaplan for figure 3 . the perceptive comments of editor and associate editor have also improved this paper significantly , and i thank them . 12 , ravelry website ; available at http://www.ravelry.com/about . t. akgn , a. koman and e. akleman , _ developable sculptural forms of ilhan koman _ , in _ bridges london : mathematics , music , art , architecture , culture _ , london , uk , 48 august 2006 , the bridges organization , pp . 343 - 350 . available at http://archive.bridgesmathart.org/2006/bridges2006-343.html . l.c . bardos , _ d - forms _ website ; available at http://www.cutoutfoldup.com/1602-d-forms.php . cherry berry crochet , _ coins purse or sewing necessaire crochet pattern _ ; available at + http://www.ravelry.com/patterns/library/coins-purse-or-sewing-necessaire-crochet-pattern . c. belden , _ sphericon _ ; + available at http://www.hypersurf.com/ charlie2/turning / streptohedrons / sphericon / sphericon.html p. bourke , _ dforms _ website ; available at http://paulbourke.net / geometry / dform/. f. brown , _ zippy strip _ ; available at http://www.ravelry.com/patterns/library/zippy-strip . e.d . demaine , m.l . demaine , a. lubiw , a. sharlo and j.l . shallit , _ zipper unfoldings of polyhedral complexes _ in _ proceedings of the 22nd canadian conference on computational geometry ( cccg 2010 ) _ , winnipeg , manitoba , canada , 9 - 11 august 2010 , pp . 219 - 222 ; + images available at http://gallery.bridgesmathart.org/exhibitions/2011-bridges-conference/alubiw e.d . demaine and j. orourke , _ geometric folding algorithms _ , cambridge university press , cambridge , 2007 . demaine and g. n. price , _ generalized d - forms have no spurious creases _ , discrete comput geom 43 ( 2010 ) , pp . 179 - 186 . c. engelhardt , and c. ucke , _ zwei - scheiben - roller _ , mathematisch - naturwissenschaftlicher unterricht 48 ( 1995 ) , pp . 259 - 263 . m. fessenden , _ please play with your math : new musuem opens in new york city _ , scientific american , december 19 , 2012 . available at : http://blogs.scientificamerican.com / observations / please - play - with - your - math - new - museum - opens - in - new - york - city/. d. henderson and d. taimina , _ crocheting the hyperbolic plane _ , the mathematical intelligencer 23 ( 2001 ) , pp . . gnen , e. akleman and v. srinivasan , _ modeling d - forms _ , in _ bridges donostia : mathematics , music , art , architecture , culture _ , san sebastian , spain , 24 27 july 2007 , the bridges organization , pp . 209 - 216 . available at http://archive.bridgesmathart.org/2007/bridges2007-209.html . d.h . hirsch , _ patent no . 59720 : a device for generating a meander motion _ ( 1980 ) ; available at https://commons.wikimedia.org/wiki/file:patent59720-claims.pdf and + https://commons.wikimedia.org/wiki/file:patent59720-drawings.pdf . r. kaufman , _ the @xmath6-icon study _ website ; available at http://www.interocitors.com/polyhedra/ c. kimberling and p. j. c. moses , _ closed space curves made from circles on polyhedra _ , journal for geometry and graphics 15 ( 2011 ) , pp . i. koman , _ rolling lady _ ( 1983 ) . image available at http://www.koman.org/work/work_1983rollinglady.html r.r . orduo , n . winard , s. bierwagen , d . shell , n. kegar , a. borhani and e. akleman , _ a mathematical approach to obtain isoperimetric shapes for d - form construction _ , in _ proceedings of bridges 2016 : mathematics , music , art , architecture , education , culture _ , jyvskyl , finland , 913 august 2016 , the bridges organization , pp . 277 - 284 . available at http://archive.bridgesmathart.org/2016/bridges2016-277.pdf h.m . osinga and b. krauskopf , _ crocheting the lorentz manifold _ , the mathematical intelligencer 26 , ( 2004 ) , pp . a. phillips , _ the differential geometry of the sphericon _ , feature column of the american mathematical society , october 1999 . available at http://www.ams.org/samplings/feature-column/fcarc-sphericon1 a. phillips , _ meander mazes on polysphericons _ , in _ the visual mind ii _ , m. emmer ( ed . ) , mit press , cambridge , ma , 2005 and http://www.ams.org/samplings/feature-column/fcarc-octo-cretan . c. pppe and i. stewart , _ der kegel mit dem dreh _ , spektrum der wissenschaft 7 ( 2000 ) , pp . 114 - 116 . + available at http://www.spektrum.de/magazin/der-kegel-mit-dem-dreh/826587 + and http://www.wissenschaft-online.de/spektrum/themen/sphericon-text.html h. pottmann and j. wallner , _ computational line geometry _ , springer - verlag , berlin , 2001 . p. roberts , _ the sphericon _ ; + available at http://web.archive.org/web/20141219072120/http://www.pjroberts.com/sphericon/ j. sharp , _ d - forms and developable surfaces _ , in _ renaissance banff : mathematics , music , art , culture _ , banff , alberta , canada , 31 july3 august 2005 , the bridges organization , pp . available at http://archive.bridgesmathart.org/2005/bridges2005-121.html . j. sharp , _ d - forms : surprising new 3-d forms from flat curved shapes _ , tarquin publications , hertfordshire , uk , 2009 . d. springett , _ streptohedrons ( twisted polygons ) _ , in _ bridges london : mathematics , music , art , architecture , culture _ , london , uk , 48 august 2006 , the bridges organization , pp . 285292 . available at http://archive.bridgesmathart.org/2006/bridges2006-285.html . d. springett , _ woodturning full circle _ , sussex , uk , guild of master craftsmen publications , 2008 . website ; available at http://www.mathias.org / steve / sphericons/. i. stewart , _ cone with a twist _ 281(1999 ) , pp.116 - 117 . r.b . thompson , _ designing a baseball cover _ , coll . j. 29 ( 1998 ) , pp.48 - 61 . weisstein , _ sphericon _ , mathworld a wolfram web resource . + available at http://mathworld.wolfram.com/sphericon.html . t. wills , _ d - forms : 3d forms from two 2d sheets _ , in _ bridges london : mathematics , music , art , architecture , culture _ , london , uk , 48 august 2006 , the bridges organization , pp . 503 - 510 . t. wills , _ d - form street furniture _ website ; available at http://www.wills - watson.co.uk / portfolio - item / d - form - street - furniture/. to crochet a stadium is not as daunting a task as making the lorentz manifold @xcite or even a hyperbolic plane @xcite . to make a dual sphericon , you will need to crochet two identical pieces . to make a pita - form , you need only one . you will need yarn , a crochet hook , a sewing needle and scissors ( only to cut threads , not to cut the surface ! ) . any ( free or purchased ) non - lacy pattern for an ` oval ' will suffice . this could be a pattern for a rug or table mat . ` oval ' is used in craft forums ( and daily conversation ) for just about anything which is not symmetric enough to be a circle and not angular enough to be a rectangle , and even for 3d forms , so check to see that it is actually going to produce a stadium ( figure [ stad ] ) . if you are familiar with both knitting and crochet , you will realise that to produce a stadium , crochet is easiest . this is related to the reason that crochet is more natural for producing hyperbolic surfaces @xcite ; because one works with a stitch at a time on the hook , and can control stitch height , different types of shapes can be made in crochet from those made with knitting . i have chosen to make my models so far using plarn ( plastic yarn , made by recycling shopping bags ) which produces quite a stiff fabric even for treble ( uk ) stitches . using acrylic yarn and a smaller - than - usual hook to produce a dense fabric in double ( uk ) crochet would also work @xcite . if you decide to stuff your models , use just enough stuffing that they do not collapse , but not so much as to distort them . my models are quite small , but larger models will probably need stuffing . another method to get around the inherent stretchiness and floppiness of most knitted or crocheted fabric , and the shaping disadvantages of knitting , is to use felted knitted fabric , cut to shape , as seen in figure [ ms ] . i closed my models by overstitching , pulling only tight enough to close the seam , again to avoid distortion . for the dual sphericon , begin by joining the midpoint of a semicircle on one stadium to the midpoint of the straight side of the other , and then join stitch by stitch , until the two pieces are joined . for the pita - form , you can pick any starting point ( apart from the exact midpoints ) for your seam . it s fun to make several , using different starting points . to control the size of the rectangular part of your stadium , vary the length of the starting chain . the diameter of the semi - circular parts depends on the stitch height ( double or treble ) and the number of rounds completed ; you will need to make a gauge swatch if you want stadia of precise measurements . it might be fun to use @xmath56 interesting pita - forms are obtained for @xmath58 such as those created using the pattern _ zippy strip _ @xcite . in this case , the stadium continues to wrap around itself , with the seam forming a helix in space ( rather than just an ) . again , the starting point of the seam will determine the final outcome . some makers describe a flattish purse ( figure [ zippy ] ) , while others obtain something more ` pyramidal ' . in making one myself , i observed that remarkable feature of d - forms , that they continue to develop right up until they are closed . this development is captured in figure [ contort ] .
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sphericons and d - forms are 3d objects created and described by artists , which have separately received attention in the mathematical literature in the last 15 or so years .
the attempt to classify a seamed , crocheted form geometrically led to the observation , which appears not to have been previously made explicit , that these objects are related .
general results concerning @xmath0-icons and seam- , d- and pita - forms are given .
instructions to crochet such forms are provided in the appendix .
sphericon , d - form , pita - form , developable surface , crochet 51m04 ; 97m80
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we consider the basic multiple - access communication problem in a distributed setting . in gallager s survey paper @xcite , it is pointed out that the multiple - access problem has been studied from a number of different perspectives , each having its own advantages and shortcomings . in the data networking community , a well - known distributed multiple access scheme is aloha @xcite . in aloha , it is assumed that a collision " happens whenever more than one user transmit simultaneously . those packets involved in a collision are discarded and retransmitted according to some retransmission probability . the collision channel model , however , does not accurately describe the underlying physical multiple - access channel . it is well known that there exist coding techniques which can decode multiple users messages when simultaneous transmissions occur . indeed , more sophisticated models such as signal capture @xcite , spread aloha @xcite , and multi - packet reception @xcite have been developed to enhance aloha . even these improved schemes , however , are not optimal from the viewpoint of information theory . in the information theory literature , the capacity regions of various multiple - access channel ( mac ) models have been characterized ( see @xcite ) . rate splitting multiple - access techniques ( or generalized time - sharing ) are presented in @xcite to achieve every point in the gaussian or the discrete memoryless mac capacity region using only single - user codes . these schemes , however , require a pre - defined decoding order , which makes distributed implementation difficult . finally , in the spread spectrum community , cdma techniques are adopted . here , users are decoded regarding all other users signals as interference . this , however , is not optimal from the information theoretic viewpoint . to address some of the shortcomings mentioned above , medard _ et al . _ @xcite use information - theoretic techniques to analyze different notions of capacity for time - slotted aloha systems . a coding / decoding scheme which combines rate splitting and superposition coding is constructed . this scheme allows some bits to be reliably received even when collision occurs , and more bits to be reliably received in the absence of collisions . shamai @xcite proposes a similar scheme to apply a broadcast strategy to multiple - access channel under static fading where the fading coefficients are not available to the transmitters or the receiver . to implement the scheme in @xcite , however , a pre - defined decoding order is required , as in @xcite . in @xcite , cheng proposes a distributed scheme called `` stripping cdma '' for the @xmath1 out of @xmath2 gaussian mac . here , no pre - defined decoding order is required . it is shown in @xcite that stripping cdma is asymptotically optimal , although the optimal operating parameters are not specified . in this paper , we investigate distributed multiple - access schemes based on the idea of rate splitting for both the @xmath0-user additive white gaussian noise mac and the @xmath0-user discrete memoryless mac . we characterize the optimal operating parameters as well as the asymptotic optimality of these schemes from the viewpoint of information theory . assume that every user has an infinite backlog of bits to send , and that every user knows the total number of users @xmath0 . we propose a distributed scheme , called _ distributed rate splitting _ ( drs ) , to achieve the optimal communication rates allowed by information theory . in this scheme , each real user creates a number of virtual users via a power / rate splitting mechanism in the @xmath0-user gaussian channel or via a random switching mechanism in the @xmath0-user discrete memoryless channel . at the receiver , all virtual users are successively decoded . a possible advantage of the drs scheme is that it can be implemented with lower complexity when compared with multiple - access schemes such as joint coding and less coordination when compared with time - sharing and rate splitting . in sections [ sec : gau ] and [ sec : dmc ] , we focus first on symmetric situations where the channel capacity regions are symmetric and every real user creates the same number of virtual users . in this case , the drs scheme entails the following . each user @xmath3 creates @xmath1 virtual users indexed by @xmath4 , @xmath5 . the virtual user class @xmath6 consists of users @xmath7 ( i.e. we have altogether @xmath1 virtual user classes and there are @xmath0 virtual users in each class ) . in the @xmath0-user gaussian mac , virtual users are created via a power / rate splitting mechanism . the signal transmitted by a real user is the superposition of all its virtual users signals . the receiver receives the sum of the virtual users signals plus noise . all virtual users are then successively decoded in increasing order of their class . that is , all virtual users in class @xmath6 , @xmath8 , are decoded before any virtual user in @xmath9 , where @xmath10 , is decoded . in contrast to @xcite , the optimal operating parameters , such as power and rate , are explicitly specified for any finite @xmath1 . in the @xmath0-user discrete memoryless mac , virtual users with the same input distribution as the real users are created , and the transmitted signal of a real user is determined by a random switch . the receiver successively decodes all virtual users in increasing order of their class given the side information of already decoded virtual users . the optimal switch is found for any finite @xmath1 for the @xmath11-user case . finally , it is shown that for both channel models , the rate tuple achieved by the drs scheme converges to the maximum equal rate point allowed by the information - theoretic bound as the number of virtual users per real user tends to infinity . next , in section [ sec : variation ] , we consider more general situations where the capacity regions can be asymmetric and real users may generate different numbers of virtual users . for the case of asymmetric capacity regions , new operating parameters are specified for any finite number of virtual users per real user . we show that the drs scheme still can achieve a point on the dominant face as the number of virtual users per real user tends to infinity . for the case of unequal number of virtual users per real user , we present a variation of drs which supports differential user rate requirements in a distributed manner . in this new scheme , each user @xmath3 , independently from other users , generates @xmath12 virtual users according to its own rate requirement . all virtual users are then decoded reliably at the receiver . furthermore , as each real user generates more virtual users , the rate tuple achieved under this variation of drs converges the maximum equal rate point on the dominant face . we first examine a gaussian mac with a symmetric capacity region . later in section [ sec : variation ] , we consider the asymmetric case . consider an @xmath0-user gaussian mac where each transmitter has transmission power @xmath13 and the receiver has noise variance @xmath14 . the capacity region @xmath15 is the set of @xmath16 satisfying to denote @xmath17 throughout . ] @xmath18 where @xmath19 is the cardinality of the set @xmath20 . the _ dominant face _ @xmath21 is the subset of rate tuples which gives equality in for @xmath22 . for this symmetric setting , it is easy to see that the maximum common rate that every user can achieve is achievable if for any @xmath23 , there exists an @xmath24 multiple - access code with overall error probability @xmath25 , where @xmath26 is the block length . ] is @xmath27 . it is well - known that rate tuples on the dominant face other than the vertices can not be achieved via standard successive decoding @xcite . note that the optimal rate tuple @xmath28 , called the _ maximum equal rate point _ , is such a point . for the two - user gaussian mac , the maximum equal rate point is shown in fig . [ fig : twogaussian ] . currently , three methods are known to achieve general points on the dominant face : joint encoding / decoding , time - sharing , and rate - splitting . joint encoding / decoding is not practical because of its high complexity @xcite . in time - sharing , all @xmath0 users need to coordinate their transmissions . therefore , some communication overhead is required . the rate - splitting method in @xcite achieves every point in @xmath29 via a generalized successive decoding scheme . for the two - user case , user @xmath30 creates two virtual users , say @xmath31 and @xmath32 , by splitting its power @xmath13 into @xmath33 and @xmath34 and setting @xmath35 , @xmath36 . user @xmath11 does not split its power and sets its rate to @xmath37 . the decoding order is @xmath38 . in order to achieve the maximum equal rate point , we solve @xmath39 , yielding @xmath40 . thus , both time - sharing and rate splitting require some coordination among users . in this paper , we focus on _ distributed _ multiple - access communication schemes . in particular , we introduce the _ distributed rate splitting _ ( drs ) scheme . the drs scheme offers the possibility of multiple - access communication with lower complexity when compared with joint coding , and communication with less coordination when compared with the time - sharing or rate splitting method . moreover , we show that drs can achieve the maximum equal rate point of the mac capacity region asymptotically . we now formally present the drs scheme . in this scheme , each user creates @xmath1 virtual users by splitting its power @xmath13 into @xmath41 , where @xmath42 is the power allocated to the @xmath43th virtual user and @xmath44 . each user then assigns transmission rate @xmath45 to virtual user @xmath43 . note that the proposed drs scheme is symmetric , i.e. all @xmath0 users split their powers and set their rates in the same way . the signal transmitted by a user is the superposition of its virtual users signals . as defined in section [ sec : intro ] , virtual user class @xmath6 consists of all virtual users indexed by @xmath43 . the receiver receives the sum of all virtual users signals plus noise . all virtual users are then successively decoded in increasing order of their class . to illustrate the drs scheme , consider the case @xmath46 . each real user splits its power @xmath13 into @xmath33 and @xmath34 . notice there are two major differences between our scheme and the traditional rate splitting scheme in @xcite . first , in our scheme , all real users split in the same way , whereas there is at least one user who does not split in the traditional rate splitting scheme . second , virtual users in the same class , ( i.e. with the same index @xmath43 ) , are allocated the same rate in our scheme , whereas all virtual users have different rates according to the pre - defined decoding order in the traditional rate splitting scheme . is not optimal . later in this section , we demonstrate the asymptotic optimality of drs by taking @xmath1 to infinity . ] these differences are illustrated in fig . [ fig : difference ] . since we assume the receiver uses successive decoding method , some virtual user must be decoded first . without loss of generality we assume one of the @xmath33 virtual users is decoded first . for the case @xmath46 , we show that there is a unique way for a real user to split its power in order to maximize its total throughput . for @xmath47 and for a fixed @xmath33 , each real user s throughput is maximized by setting @xmath48 and @xmath49 . _ proof : _ the @xmath33 virtual user who is decoded first must have @xmath50 ( i.e. the virtual user regards all other virtual users as interference ) in order to be decoded successfully . due to symmetry , all other @xmath33 virtual users must have the same @xmath51 . then the problem of maximizing each real user s throughput reduces to @xmath52 , subject to ( i ) @xmath53 , ( ii ) one of the @xmath33 virtual users is decoded first and ( iii ) @xmath54 note that @xmath55 is maximized when the interference plus noise faced by all the @xmath56 virtual users is minimized , and the only way to minimize the interference plus noise faced by all the @xmath56 virtual users is to decode all the @xmath33 virtual users before decoding any @xmath56 virtual user . therefore , the minimum interference plus noise faced by any @xmath56 virtual user is @xmath57 . hence , the maximum rate associated with a @xmath56 virtual user is @xmath58 . @xmath59 + using the drs scheme with @xmath47 , each user can strictly increase its throughput relative to the case where users do not split their powers and decode against each other as noise . this is easily verified by observing that for any @xmath60 , @xmath61 now consider the case where each user creates more than two virtual users @xmath62 . here , we show that each user s throughput increases further . [ le : morevuser ] given a drs scheme with @xmath1 virtual users per real user , where @xmath63 are the virtual users powers , it is possible to strictly increase the throughput via an @xmath64 virtual user system with powers @xmath65 , where @xmath66 . _ proof : _ suppose that every user splits its power into @xmath1 virtual users : @xmath67 subject to @xmath44 , where @xmath1 is an arbitrary integer and @xmath42 is the power of @xmath43th virtual user . since virtual user @xmath1 is decoded last , following the reasoning in the proof of lemma @xmath30 , we have @xmath68 . we now split the virtual user with power @xmath69 into two new virtual users with powers @xmath70 and @xmath71 , where @xmath72 . we set @xmath73 now each real user has @xmath74 virtual users . notice that we do not change the power and decoding order of any of the other virtual users ( i.e. virtual users @xmath75 ) . from a real user s view point , the virtual user with @xmath76 is decoded second to last _ among all virtual users generated by this real user _ and the virtual user with @xmath77 is decoded last . thus , all virtual users can be decoded and from , @xmath78 . therefore , every real user with @xmath1 virtual users can strictly increase its throughput by splitting its power among @xmath74 virtual users . @xmath59 + before we examine the asymptotic behavior of drs , we solve the problem of how to split a user s power _ optimally _ among a fixed number of virtual users . the main difficulty here is that the objective function is not concave . in order to find the optimal splitting method , we prove the following lemma . [ lemma : maximization ] consider the following optimization problem : @xmath79 subject to @xmath42+@xmath80 = @xmath81 and @xmath82 , where @xmath83 , @xmath0 and @xmath81 are positive constants and @xmath84 . the unique solution to is also the unique solution to @xmath85 , where @xmath86 . _ proof : _ substitute @xmath87 into the objective function , we have @xmath88setting @xmath89 subject to @xmath90 , the unique solution is @xmath91 . thus , @xmath92 is the unique stationary point of @xmath93 . we can also verify that @xmath94 and @xmath95 . so @xmath96 is the unique solution to our maximization problem . we can directly solve @xmath97 subject to @xmath98 and @xmath82 . the unique solution is also @xmath96 . @xmath59 + we now present the optimal splitting method . theorem [ thm : nece ] states a necessary condition for the optimal splitting method , and theorem [ thm : unique ] implies there is a unique optimal splitting method . in corollary [ cor : optimal ] , we formally present the optimal splitting method and the required power levels . [ thm : nece ] let each real user split its power into @xmath1 virtual users . let @xmath42 be the power allocated to the @xmath43th virtual user and @xmath99 @xmath100 . if @xmath101 maximizes @xmath102 and satisfies @xmath103 , @xmath104 for @xmath105 , then @xmath106 , for all @xmath43 . that is , the optimal power split must lead to equal transmission rates for all virtual users . _ proof : _ we use a perturbation argument . suppose @xmath107 maximizes @xmath108 and satisfies @xmath109 , @xmath110 and the resulting @xmath111 is not the same for all @xmath43 . then we can find a pair of virtual users @xmath112 , where virtual user @xmath43 and @xmath113 are decoded at the @xmath43th and @xmath114th places respectively , and @xmath115 . without loss of generality , let us consider the case where @xmath116 @xmath117 . by the definition of @xmath45 , we have @xmath118 we can verify that if we change @xmath119 to @xmath120 and @xmath121 to @xmath122 , where @xmath123 is a small positive number , then the first term of decreases and the second term of increases . let @xmath124 be the solution to @xmath125 ( the existence of @xmath126 can be demonstrated ) . let @xmath127 and @xmath128 . notice that @xmath129 . since the maximization considered in lemma [ lemma : maximization ] has a unique solution , @xmath130 . this contradicts our assumption that @xmath107 maximizes + @xmath108 . therefore , the theorem follows . @xmath59 + by theorem [ thm : nece ] , if @xmath131 maximizes @xmath132 and satisfies @xmath133 , @xmath134 for @xmath105 , then we must have @xmath135 where @xmath136 . therefore , if we show that @xmath137 for all @xmath138 , has a unique solution , then there is _ at most _ one feasible solution to the maximization problem . [ thm : unique ] the set of equations : @xmath139 , subject to @xmath140 and @xmath141 @xmath142 , has a unique solution . _ proof : _ see appendix i. [ cor : optimal ] if a real user splits its power @xmath13 into @xmath1 virtual users , then the unique way to maximize this user s throughput is to set @xmath143 $ ] for @xmath144 . _ proof : _ since the constraint region @xmath145 is a simplex and @xmath108 is continuous , there exists at least one solution . we denote one solution by @xmath146 . by the necessary condition stated in theorem [ thm : nece ] , @xmath146 must satisfy @xmath147 . moreover , by the uniqueness property stated in theorem [ thm : unique ] , @xmath146 is the unique solution to @xmath148 subject to @xmath149 next , we plug @xmath42 into the expression of @xmath150 . let @xmath151 , we are able to verify that @xmath152 , which is independent of @xmath43 . hence , the corollary follows . @xmath59 + we now examine the asymptotic behavior of the drs scheme . we first demonstrate the interesting fact that the rate tuple converges to the maximum equal rate point for a general power split as long as all virtual users powers go to @xmath153 as @xmath154 . this implies a convergence result for the optimal power split . we then analyze the rate of convergence under the optimal power split . [ thm : genesplit ] given any power split @xmath155 , a sufficient condition for @xmath156 is @xmath157 as @xmath158 . _ @xmath159where @xmath160 , @xmath161 . the equality in is justified as follows . note that , @xmath162 . hence , for sufficiently small positive @xmath33 , @xmath163 . now , we examine the error term @xmath164 \right| \nonumber \\ & \leq & \sum_{k=1}^l \left| \log \left ( 1 + \frac{p_k}{mp - m\sum_{j < k}p_j - p_k + n } \right ) \right . \nonumber \\ & & \hspace{1 cm } \left . - \left(\frac{p_k}{mp - m\sum_{j < k}p_j - p_k + n } \right ) \right| \nonumber \\ & \leq & \sum_{k=1}^l \left(\frac{p_k}{mp - m\sum_{j < k}p_j - p_k + n } \right)^2 \label{eq : small } \\ & \leq & \sum_{k=1}^l \left(\frac{1}{n}\right)^2 p_k^2 \label{inq : powerconv}\\ & \leq & \left(\frac{1}{n}\right)^2 \max_{k } p_k \sum_{k=1}^l p_k \\ & = & p \left(\frac{1}{n}\right)^2 \max_{k } p_k\end{aligned}\ ] ] where inequality in holds because @xmath165 when @xmath166 , and the inequality in follows from the fact that @xmath167 . since @xmath168 as @xmath154 , the error term goes to zero in the limit . this justifies the equality in . using the capacity bound , we also have @xmath169 . therefore , @xmath170 @xmath59 + note that our optimal power split satisfies the sufficient condition in theorem [ thm : genesplit ] . therefore , its convergence is assured . [ cor : conv ] if each real user adopts the optimal splitting method specified in corollary [ cor : optimal ] , then @xmath171 next , we examine the rate of convergence to the maximum equal rate point under the optimal power split . define the error term @xmath172 \equiv r^ * - \sum_{k=1}^l r_k$ ] , we analyze how fast this error term tends to @xmath153 as @xmath154 . we prove the following : = \omega\left(g\left[n\right]\right)$ ] if there exist positive constants @xmath173 and @xmath174 such that @xmath175 \geq c_1 g\left[n\right ] $ ] for all @xmath176 , and @xmath177 = o(g[n])$ ] if there are positive constants @xmath178 and @xmath174 , such that @xmath179 \leq c_2 g[n]$ ] for all @xmath180 . finally , @xmath177 = \theta \left(g[n]\right)$ ] if @xmath177 = \omega \left(g[n]\right)$ ] and @xmath177 = o \left(g[n]\right)$ ] . ] [ thm : speed ] @xmath181 = \theta \left(\frac{1}{l}\right)$ ] . _ proof : _ let @xmath182 . @xmath183 \nonumber \\ & = & \lim_{l \rightarrow \infty } l \left ( \frac{1}{2m}\log \left(a\right ) - \frac{l}{2 } \log \left ( \frac{m}{m-1 + a^{-\frac{1}{l}}}\right)\right ) \nonumber \\ & = & \lim_{y \rightarrow 0 } \frac{\frac{y\log\left(a\right)}{2m}-\frac{1}{2 } \log \left(\frac{m}{m-1+a^{-y}}\right)}{y^2}\\ & = & \lim_{y \rightarrow 0 } \frac{\frac{\log\left(a\right)}{2m}-\frac{1}{2}\frac{a^{-y}\log\left(a\right)}{m-1+a^{-y}}}{2y } \label{eq : l1}\\ & = & \frac{\left(m-1\right)\left(\log\left(a\right)\right)^2}{4m^2}.\label{eq : l2}\end{aligned}\ ] ] note that equalities in and can be verified by lhospital s rule . consequently , given any @xmath23 , there exists a positive integer @xmath174 such that for all @xmath184 , we have @xmath185 - \frac{\left(m-1\right)\left(\log\left(a\right)\right)^2}{4m^2}\right| < \varepsilon$ ] . this implies @xmath186 < \frac{\left(m-1\right)\left(\log\left(a\right)\right)^2}{4m^2 } + \varepsilon$ ] . therefore , we can choose a small enough @xmath123 such that @xmath187 . let @xmath188 , we have @xmath189 \leq c_2 $ ] . this implies that there exists a positive integer @xmath174 and for all @xmath184 , we have @xmath190 \leq \frac{c_2}{l}$ ] . @xmath59 + finally , we note that all virtual users in one virtual user class can be decoded _ in parallel_. thus , the decoding delay of drs is proportional to the number of virtual users @xmath1 and independent of the number of real users . since @xmath1 is controlled by the designer , drs offers a tradeoff between the throughput of a real user and the decoding delay . in fig . [ fig : simulation ] , we present some numerical simulations illustrating the tradeoff between the number of virtual users and the throughput for both the high and low snr regimes . a system with @xmath191 real users is used in the simulations . an @xmath0-user discrete memoryless mac is defined in terms of @xmath0 discrete input alphabets @xmath192 , @xmath193 , an output alphabet @xmath194 and a stochastic matrix @xmath195 with entries @xmath196 . for any product input distribution @xmath197 , let the achievable region @xmath198 $ ] be the set of @xmath199 satisfying @xmath200 where @xmath201 and @xmath202 . the capacity region of the asynchronous mac is @xcite @xcite @xmath203.\ ] ] we fix the input product distribution @xmath204 and focus on achieving the desired operating point in @xmath205 $ ] . in this section , we consider only discrete memoryless channels satisfying the following symmetry condition : @xmath206 @xmath207 such that @xmath208 . later , in section [ sec : variation ] , we consider the more general asymmetric case . we further assume that for @xmath209 , if @xmath210 , then @xmath211 . under our symmetric setting , the maximum common rate that every user can achieve is @xmath212 . in the gaussian mac , virtual users are created via a power / rate splitting mechanism . for the discrete memoryless mac , we adopt the random switching mechanism of @xcite where virtual users with the same input distribution as the real users are created and the transmitted signal of a real user is determined by a random switch . we first consider the two - user discrete memoryless mac @xmath213 , and illustrate the random switching mechanism . the optimal random switches and the asymptotic behavior of drs under the optimal switching are presented . we then examine the @xmath0-user case @xmath214 , and present a sufficient condition for the random switching mechanism to converge to the information theoretic upper bound . finally , we investigate the rate of convergence for a simple suboptimal random switch . consider a two - user mac , @xmath215 . for a fixed input product distribution @xmath216 , the achievable region is given by : @xmath217 under our symmetry assumption ( cf ) , we have @xmath218 , @xmath219 , @xmath220 , and the optimal rate tuple is @xmath221 let us consider the random switching mechanism for this channel . we first consider the case where each real user generates two virtual users . later , we consider the case where the number of virtual users per real user goes to infinity . we split by means of two switches , as shown in fig . [ fig : switch ] . each switch has two inputs , @xmath222 and @xmath223 and one output @xmath224 . switch @xmath3 is controlled by a random variable @xmath225 with @xmath226 . the output is given by @xmath227 if @xmath228 , and @xmath229 if @xmath230 . the switching random variables @xmath231 are independent of the channel inputs . we also assume that @xmath231 are available at the receiver . in practice , one would generate @xmath232 and @xmath233 at the transmitters and at the receiver , e.g. by means of a pseudorandom sequence generator . assign to the channel inputs @xmath234 , @xmath235 , @xmath236 and @xmath237 the probability mass function @xmath238 . notice that @xmath239 and @xmath240 are independent and each has the same probability mass function as the random variable @xmath241 for @xmath242 . in successive decoding for the discrete memoryless mac , the signals of decoded virtual users are used as side information to aid the decoding process of subsequent virtual users . the first constituent decoder observes the output @xmath243 and tries to decode @xmath244 and @xmath236 . the second constituent decoder is informed of the decision about @xmath245 made by the previous constituent decoder and tries to decode @xmath235 and @xmath237 . without loss of generality , let us focus on real user 1 . @xmath246where the second equality follows from the independence between @xmath234 and @xmath232 , and the last equality follows from the fact that when @xmath247 , @xmath234 is independently of the output @xmath248 and @xmath233 . similarly , we have @xmath249.\end{aligned}\ ] ] it can be verified that in the two - user discrete memoryless mac , both real users throughput can be strictly increased by splitting their inputs via a random switch , relative to the case where they do not split , ( i.e. @xmath250 for @xmath251 ) . next , we show that by generating more virtual users , the throughput of each real user increases further . for @xmath252 , consider a distributed rate splitting scheme with @xmath1 virtual users per real user . the random switch for user @xmath3 is controlled by @xmath253 , where @xmath254 for @xmath144 . it is possible to strictly increase the throughput via an @xmath64 virtual user system by splitting the @xmath1th virtual user into two virtual users . _ proof : _ without loss of generality , we consider user @xmath30 . for the @xmath43th virtual user , we have @xmath255 \nonumber \\ & = & \lambda_k \left [ \left(\sum_{j < k } \lambda_j \right)i \left(x_1;y , x_2 \right ) \right . \nonumber \\ & & \hspace{2 cm } + \left . \left ( 1 - \sum_{j < k } \lambda_j \right ) i\left(x_1;y \right ) \right ] , \label{eq : lelast}\end{aligned}\]]where equality in is due to the independence between @xmath256 and @xmath232 , and equality in follows from the fact that when @xmath257 , @xmath256 is independent of the output @xmath248 and all the other random variables . finally , equality in holds because when @xmath258 , one of the random variables @xmath259 is the switch output , and when @xmath260 , none of them is the switch output . therefore , @xmath261.\ ] ] now let us split @xmath262 into @xmath263 and @xmath264 by using a switch controlled by a binary random variable @xmath265 with @xmath266 . we have @xmath267 where @xmath268 . hence , @xmath269 \\ & = & \alpha \overline{\alpha } \lambda_l^2 \left[i \left(x_1;y , x_2 \right ) - i \left(x_1;y \right ) \right ] \\ & \geq & 0\end{aligned}\ ] ] with strict inequality if @xmath270 . @xmath59 + before we examine the asymptotic behavior of the drs scheme for @xmath252 , we first solve the problem of how to find the optimal switches for a fixed number of virtual users per real user . for @xmath271 , if a real user has @xmath1 virtual users , then the optimal random variable to control the switch for user @xmath3 is @xmath272 with @xmath273 for @xmath144 and @xmath242 . _ proof : _ we use a perturbation argument . suppose the random variables @xmath274 with @xmath275 maximize @xmath276 , where @xmath277 and @xmath278 . moreover , suppose there exists @xmath279 such that @xmath280 . let @xmath279 be the first element which is not equal to @xmath281 . we consider the pair @xmath282 . we have @xmath283 @xmath284.\end{gathered}\ ] ] therefore , @xmath285i(x_1;y , x_2 ) \\ & & + \left[\left(\lambda_k + \lambda_{k+1}\right)\frac{l - k+1}{l } - \lambda_k \lambda_{k+1 } \right ] i(x_1;y).\end{aligned}\ ] ] first , we consider the case where @xmath286 . we let @xmath287 and @xmath288 for @xmath289 . we have @xmath290i(x_1;y , x_2 ) \\ + \left [ \left(\lambda_k + \lambda_{k+1 } \right)\frac{l - k+1}{l } - \left(\lambda_k-\varepsilon \right ) \left(\lambda_{k+1}+\varepsilon \right ) \right ] i(x_1;y ) . \end{gathered}\ ] ] thus , @xmath291 + @xmath292 - @xmath293 + @xmath294 = @xmath295 + @xmath296 @xmath123 -@xmath297 @xmath298-@xmath299 notice that the second term of the r.h.s . expression is positive and there exists @xmath123 such that @xmath300 + @xmath301 @xmath123-@xmath302 . this , however , contradicts our assumption that the random variables @xmath274 with @xmath303 , maximize @xmath276 . we obtain similar contradictions for the case where @xmath304 . @xmath59 + we now examine the asymptotic behavior of the drs scheme in the two - user discrete memoryless mac . [ thm : twoconv ] for @xmath252 , if both real users adopt @xmath305 for @xmath8 and @xmath306 , then @xmath307 for @xmath308 . moreover , if we define the error term @xmath172 \equiv r^ * - \sum_{k=1}^l r_{x_{ik}}$ ] , then @xmath172 = \theta ( \frac{1}{l } ) $ ] for @xmath306 . _ proof : _ without loss of generality , we consider real user @xmath30 . since @xmath309 we have @xmath310 \\ & = & \lim_{l \rightarrow \infty}\left[\frac{l+1}{2l}i(x_1;y ) + \frac{l-1}{2l}i(x_1;yx_2)\right ] \\ & = & \frac{1}{2 } \left[i(x_1;y ) + i(x_1;y , x_2 ) \right ] \\ & = & \frac{1}{2}i(x_1,x_2;y),\end{aligned}\]]where the last equality follows from the fact that @xmath219 in the symmetric setting . next , we examine the rate of convergence . we have the error term @xmath172 = \frac{1}{2}i(x_1;y ) + \frac{1}{2}i(x_1;y , x_2 ) - ( \frac{l+1}{2l}i(x_1;y ) + \frac{l-1}{2l}i(x_1;y , x_2))$ ] . thus , @xmath311 = \frac{1}{2l}\left ( i(x_1;y , x_2 ) - i(x_1;y ) \right),\ ] ] which implies @xmath312 = \frac{1}{2}\left ( i(x_1;y , x_2 ) - i(x_1;y ) \right ) \triangleq \frac{c}{2}.\ ] ] for any @xmath313 , there exists an @xmath174 such that for all @xmath184 , @xmath314 - \frac{c}{2}|<\varepsilon$ ] . hence , @xmath315 < \frac{c}{2}+\varepsilon $ ] . we can choose @xmath123 small enough such that @xmath316 . this implies @xmath172 = \theta ( \frac{1}{l } ) $ ] . @xmath59 + there are @xmath0 real users and each real user creates @xmath1 virtual users . we have @xmath0 switches @xmath317 to do the splitting with probabilities @xmath254 for @xmath318 and @xmath319 . we assume the receiver also knows @xmath317 . this may require common randomness to exist between all transmitters and the receiver . due to symmetry , we focus on one user , say user @xmath30 . for @xmath144 , we set @xmath320 , \label{eq : mlast}\end{aligned}\]]where @xmath321 and @xmath322 . equality in is due to the independence between @xmath256 and @xmath232 . equality in holds because when @xmath257 , @xmath256 is independent of the output @xmath248 and all the other random variables . the first term in follows from @xmath323 = @xmath324 . the second summation term in follows from the fact that the probability of @xmath3 switching random variables among @xmath325 having values less than @xmath43 is @xmath326 . it can be verified that real user @xmath30 with @xmath1 virtual users can strictly increase its total throughput via an @xmath74 virtual user system . in order to maximize the total throughput of real user 1 for fixed @xmath1 , we need to find the optimal @xmath327 to maximize @xmath328 . this is a non - convex optimization problem and appears to be difficult . we are able to verify that for the general @xmath0-user case ( unlike the two - user case ) , random switches with a uniform distribution are in general suboptimal . nevertheless , it is possible to generalize the asymptotic result of theorem [ thm : twoconv ] . we first demonstrate the fact that the convergence result holds for a general switch controlled by @xmath253 , where @xmath329 for @xmath8 , as long as @xmath330 as @xmath154 . we then analyze the rate of convergence for a particular suboptimal switch , the uniform switch . [ thm : mconv ] for a general random switch controlled by @xmath253 , where @xmath331 for @xmath144 , a sufficient condition for @xmath332 is @xmath330 as @xmath154 for @xmath318 . _ proof : _ without loss of generality , let us examine real user @xmath30 . @xmath333 \\ & = & \lim_{l \rightarrow \infty}\sum_{i=0}^{m-1}c_i i_i,\end{aligned}\]]where @xmath334 for @xmath335 . it is sufficient to prove @xmath336 for all @xmath3 . for @xmath335 , @xmath337where @xmath160 , @xmath338 and @xmath339 is the beta function . the term @xmath340 can be shown to converge to @xmath341 as @xmath154 by a similar argument . @xmath59 + next , we analyze a particular suboptimal switch , the uniform switch . since the uniform switch satisfies the sufficient condition in theorem [ thm : mconv ] , the convergence result holds . the next lemma presents its rate of convergence . consider an @xmath0-user discrete memoryless mac . let each real user have @xmath1 virtual users and each switch be controlled by an i.i.d . random variable @xmath342 with @xmath343 , @xmath8 . define the error term @xmath172 \equiv \frac{1}{m}i(x_1,x_2, ... ,x_m;y ) - \sum_{k=1}^l r_{x_{ik } } $ ] . then @xmath172 = o ( \frac{1}{l } ) $ ] for all @xmath3 . _ proof : _ in the uniform switch setting , @xmath343 for @xmath318 and @xmath144 . without loss of generality , we examine the total throughput of real user @xmath30 . @xmath344,\end{gathered}\]]where @xmath345 for @xmath346 . we denote @xmath347 , where @xmath348 is defined in the proof of theorem [ thm : mconv ] for @xmath349 , and @xmath350for @xmath335 . therefore , the error term can be calculated as follows , @xmath351 & = & \left| \frac{1}{m } i(x_1,x_2, ... \sum_{i=0}^{m-1}c_i i_i \right|\\ & = & \left| \sum_{i=0}^{m-1 } \left(\frac{1}{m } - c_i \right ) i_i \right| \\ & \leq & \sum_{i=0}^{m-1 } \left| \frac{1}{m } - c_i \right| i_i \\ & \leq & m \left[\max_i \left(\left| \frac{1}{m } - c_i \right| i_i \right ) \right].\end{aligned}\ ] ] note that @xmath352 is maximized at @xmath353 . for @xmath335 , it can be verified that @xmath354 @xmath355 multiplying both sides of the above two inequalities by @xmath356 , we have @xmath357 therefore , @xmath358 \frac{1}{l } \\ & \equiv & \alpha \frac{1}{l},\end{aligned}\ ] ] so @xmath172 \leq m \alpha \frac{1}{l}$ ] , where @xmath359 . for the term @xmath340 , a similar argument can be used to show that @xmath360 . therefore , @xmath172 = o \left ( \frac{1}{l } \right)$ ] . in section [ sec : gau ] and section [ sec : dmc ] , we imposed two symmetry constraints . the first is that the capacity region for the gaussian mac and the achievable rate region for the discrete memoryless mac are symmetric . the second is that users generate the same number of virtual users . in this section , we describe two variations of drs . the first variation is presented in section [ sec : unequalpower ] , where we relax the symmetric region constraint . in this case , we show that as the number of virtual user per real user tends to infinity , the rate tuple achieved under drs approaches a point on the dominant face . the second variation is presented in section [ sec : differentvuser ] , where each real user may generate a different number of virtual users . the main advantage of this variation is that it can accommodate different user rate requirements in a distributed fashion . in this section , we consider the case where real users in a gaussian mac may have different transmission powers ( i.e. the capacity region may not be symmetric ) . we assume that user @xmath3 has transmission power @xmath361 and the power vector @xmath362 is known to all users . we also assume that all real users split their powers into @xmath1 virtual users according to the common power splitting rule defined by the vector @xmath363 , where @xmath364 @xmath365 and @xmath366 . the power vector for the virtual users generated by user @xmath3 is @xmath367 for @xmath368 . for any real user with @xmath1 virtual users , the unique way to maximize this user s throughput is to set + + @xmath369 $ ] + + for @xmath144 . moreover , if all real users adopt this power allocation rule , then @xmath370for @xmath318 . _ proof : _ by replacing @xmath42 by @xmath371 for @xmath144 , we can use arguments similar to those in section [ sec : gau ] to prove the following : 1 . given a drs scheme with @xmath1 virtual users per real user , it is possible to strictly increase the throughput via an @xmath64 virtual user system . 2 . under the optimal power split , all virtual users generated by real user @xmath3 must have the same rate for @xmath372 . ( virtual users generated by different real users may have different rates . ) 3 . for any real user with @xmath1 virtual users , the unique way to maximize this user s throughput is to set + + @xmath373 $ ] + + for @xmath144 . if all real users adopt this power allocation rule , then @xmath374for @xmath318 @xmath59 + we illustrate this achievable point on the dominant face for a two - user gaussian mac in fig . [ fig : asy_reg ] . in section [ sec : dmc ] , we considered the symmetric setting ( cf ): @xmath375 @xmath207 such that @xmath208 . in this section , we relax this constraint and consider an asymmetric achievable region . we require only that for @xmath376 , if @xmath377 , then @xmath378 . the @xmath0 switches @xmath379 have probabilities @xmath380 for @xmath318 and @xmath381 . consider a general random switch controlled by @xmath253 , where @xmath380 , @xmath381 . define @xmath382 . if @xmath383 as @xmath154 , then @xmath384 for @xmath318 . _ proof : _ we can replace @xmath385 by @xmath386 and use arguments similar to those for theorem [ thm : mconv ] to prove the above lemma . @xmath59 + in this section , we retain the assumption that every user has the same transmission power @xmath13 , but we do not require all real users to create the same number of virtual users . that is , user @xmath3 and user @xmath387 create @xmath12 and @xmath388 virtual users independently , where @xmath12 may not be equal to @xmath388 . the signal transmitted by a real user is the superposition of all its virtual users signals . we also assume in this section that user @xmath3 transmits the number @xmath12 in a header message to the receiver . the receiver receives the sum of @xmath389 signals plus noise . we now describe a protocol which allows each user to split its power and set its rates independently , and allows the receiver to decode all virtual users one by one via a generalized successive decoding mechanism . recall that for the gaussian mac , successive decoding works as follows . users are decoded one after another regarding all other users that have not been decoded as interference , and the signals of decoded users are subtracted from the overall received signal . _ protocol @xmath30 _ : for user @xmath3 , the power split and rate allocation rule are defined as follows : for @xmath390 , @xmath391,\\ r_{ik } & = \frac{1}{2 } \log \left(1 + \frac{p_{ik}}{m\left(p - \sum_{j < k } p_{ij } \right ) - p_{ik } + n } \right ) . \ ] ] note that the power split and rate allocation rule in _ protocol @xmath30 _ are the same as that discussed in section [ sec : gau ] . the generalized successive decoding algorithm is given by the following pseudo - program . note that after a virtual user is decoded , its signal is subtracted from the overall received signal . * decode * virtual users @xmath392 in any order or in parallel . * set * @xmath393 * while * ( some virtual users are not decoded ) , find the minimal element in @xmath394 , say the @xmath3th entry ; decode the subsequent virtual user of user @xmath3 ; update the @xmath3th entry of @xmath394 : @xmath395 , where @xmath396 is the power of the virtual user being decoded in the previous step ; * end * [ le : adrs ] if all @xmath0 users adopt the power split and the rate allocation rule described in _ protocol @xmath30 _ , then for any @xmath397 , @xmath398 , the decoder can decode all virtual users one by one following the decoding algorithm . _ proof : _ by the rate allocation rule , @xmath399 for @xmath400 . thus , each of them can tolerate the maximum amount of interference plus noise , @xmath401 . it is then easy to see that virtual users @xmath402 can be decoded reliably in any order or in parallel . now , we set @xmath403 . in the first run of the * while * loop , if @xmath404 is the minimum entry in @xmath405 , the receiver decodes virtual user @xmath406 . by the rate allocation rule , @xmath407 this implies that the maximum amount of interference plus noise that virtual user @xmath406 can tolerate is @xmath408 . however , the real amount of interference plus noise it faces is @xmath409 , which is smaller than or equal to what it can tolerate because @xmath410 . therefore , virtual user @xmath406 can be decoded reliably at the receiver . suppose the decoding process succeeds in the @xmath411th run of the * while * loop . now , @xmath412 . let us consider the @xmath413th run of the * while * loop . suppose the @xmath3th entry , @xmath414 , is the minimum entry in @xmath394 . the receiver decodes the subsequent virtual user of user @xmath3 , denoted by @xmath415 . by the rate allocation rule , @xmath416 this implies the maximum amount of interference plus noise that virtual user @xmath415 can tolerate is @xmath417 = @xmath418 . however , the real amount of interference plus noise it faces is @xmath419 , which is smaller than or equal to what virtual user @xmath415 can tolerate because @xmath420 . therefore , virtual user @xmath415 can be decoded reliably . hence , the lemma follows by induction . @xmath59 + to illustrate the decoding algorithm , let us carefully examine a three - user example shown in fig . [ fig : examgau ] . the shaded regions correspond to the virtual users @xmath421 , which are decoded in any order . suppose we decode `` @xmath422 '' first . by the rate allocation rule , @xmath423 , which means the maximum amount interference plus noise that virtual user `` @xmath422 '' can tolerate is @xmath424 . this is exactly the amount of interference plus noise it faces . therefore , `` @xmath422 '' can be decoded reliably and we can subtract the signal of virtual user `` @xmath422 '' from the overall received signal . similarly , @xmath425 can be decoded reliably and subtracted from the overall received signal . now @xmath426 . the subsequent decoding order is illustrated by the numbers in fig . [ fig : examgau ] . in the first run of the * while * loop , since @xmath427 is the minimum in @xmath394 , the receiver decodes virtual user `` @xmath428 '' . by the rate allocation rule , @xmath429 , which implies the maximum amount interference plus noise it can tolerate is @xmath430 . however , the real interference plus noise it faces is @xmath431 , which is smaller than what it can tolerate since @xmath432 . so virtual user `` @xmath428 '' can be decoded reliably and subtracted from the received signal . by searching for the minimum entry in @xmath394 in each run , we always decode a virtual user that can tolerate more interference than what it really faces . this assures the validity of our decoding algorithm . all of the other virtual users can be decoded in a similar fashion . by lemma [ le : morevuser ] and lemma [ le : adrs ] , user @xmath3 can choose any @xmath397 , independently from other users , and have all virtual users decoded reliably at the receiver . therefore , user @xmath3 can choose @xmath12 according to its own service requirement . for example , if user @xmath3 wants to send low rate voice communication packets , it can set @xmath433 , which corresponds to the basic cdma scheme . if user @xmath3 wants to send high rate stream video , it can set @xmath12 equal to a large value in order to get higher throughput at the expense of higher coding complexity . thus , this variation of drs provides an explicit way for end users to trade off throughput and coding complexity , making differential rate requirements achievable in a distributed manner . finally , corollary [ cor : conv ] and theorem [ thm : speed ] demonstrate the asymptotic optimality of this scheme and its rate of convergence . in this section , we describe a variation of the drs scheme for the discrete memoryless mac which supports differential rate requirements to end users in a distributed manner . in this scheme , we adopt the uniform switch , but we do not require every user to have the same number of virtual users . we split by means of @xmath0 independent switches . without loss of generality , let us consider user @xmath3 . if user @xmath3 has @xmath12 virtual users , then switch @xmath3 has @xmath12 inputs , @xmath434 for @xmath390 , and one output @xmath224 . switch @xmath3 is controlled by a uniform random variable @xmath435 with @xmath436 for @xmath437 . the output is given by : @xmath438 , if @xmath439 . we now describe the protocol for the discrete memoryless mac which allows user @xmath3 to choose @xmath12 independently . we show the asymptotic optimality of this variation of drs under the protocol . _ protocol 2 _ : for user @xmath3 with @xmath12 virtual users , the switch @xmath3 is controlled by a uniform random variable @xmath253 where @xmath436 for @xmath437 . the rate allocation rule is defined as follows : @xmath440.\end{gathered}\ ] ] the decoding algorithm is given by the following pseudo - program . note that after a virtual user is decoded , its signal is used as side information to aid the decoding process of subsequent virtual users . * decode * virtual users @xmath392 in any order or in parallel . * set * @xmath441 * while * ( some virtual users are not decoded ) , find the minimal element in @xmath442 , say the @xmath3th entry ; decode the subsequent virtual user of user @xmath3 ; update the @xmath3th entry of @xmath442 : @xmath443 ; * end * if all @xmath0 users adopt the rate allocation rule in _ protocol 2 _ , then for any @xmath397 @xmath444 , the decoder can decode all virtual users one by one following the decoding algorithm . _ proof _ : virtual users @xmath392 can be decoded in any order if and only if @xmath445 for all @xmath3 . this is true because under our rate allocation rule , @xmath446 . we set @xmath447 in the first run of the * while * loop , if @xmath448 is the minimum entry of @xmath442 , the receiver decodes virtual user @xmath406 . by the rate allocation rule @xmath449 \label{eq : i2}\end{aligned}\ ] ] virtual user @xmath406 can be decoded reliably if @xmath450 this mutual information can be simplified in the same way as described in equations - . recall the definition @xmath382 . @xmath451 we can verify that the @xmath452 term in is less than or equal to the corresponding @xmath452 term in equation . this follows from @xmath453since @xmath454 for all @xmath455 . similarly , it can be verified that the second term in is less than or equal to the corresponding term in equation . therefore , virtual user @xmath406 can be decoded reliably . suppose the decoding process succeeds in the @xmath411th run of the * while * loop . now , @xmath456 . let us consider the @xmath413th run of the * while * loop . suppose the @xmath3th entry , @xmath457 , is the minimum entry in @xmath442 . the receiver decodes the subsequent virtual user of user @xmath3 , denoted by @xmath458 . by the rate allocation rule @xmath459.\end{aligned}\ ] ] again , we can simplify the mutual information @xmath460 and show that it is great than or equal to @xmath461 , which implies that virtual user @xmath458 can be decoded reliably . hence , the lemma follows by induction . @xmath59 + let us illustrate the decoding algorithm by the following example . we consider a two - user discrete memoryless mac where user @xmath30 creates @xmath11 virtual users and user @xmath11 creates @xmath462 virtual users . random switch @xmath30 is controlled by @xmath232 where @xmath463 , and random switch @xmath11 is controlled by @xmath233 where @xmath464 for @xmath465 . by the rate allocation rule , the virtual users rates can be simplified as follows @xmath466\end{aligned}\ ] ] @xmath467\\ r_{x_{23 } } & = & \frac{1}{3}\left[\frac{1}{3 } i(x_1;y ) + \frac{2}{3}i(x_1;y , x_2 ) \right].\end{aligned}\ ] ] we first decode @xmath468 in any order . suppose we decode `` @xmath469 '' first . virtual user `` @xmath469 '' can be decoded reliably if @xmath470 . the condition holds because @xmath471 virtual user `` @xmath422 '' can be decoded similarly . now , we set @xmath472 . in the first run of the * while * loop , the receiver decodes `` @xmath473 '' since @xmath474 . let us calculate the mutual information between @xmath237 and @xmath248 given @xmath475 and previously decoded @xmath476 . @xmath477 the third equality is due to our symmetric assumption . therefore , virtual user `` @xmath473 '' can be decoded . by searching for the minimum entry in @xmath442 in each run , we always decode a virtual user whose rate is smaller than or equal to the corresponding mutual information . this guarantees the correctness of our decoding algorithm . virtual users @xmath478 can be decoded reliably at the receiver in a similar fashion . the asymptotic optimality of this scheme in the discrete memoryless mac can be demonstrated by theorem [ thm : mconv ] . in this paper , we take an information - theoretic approach to the problem of distributed multiple - access communication . we present a distributed rate splitting scheme whereby each real user creates a number of virtual users and all virtual users are successively decoded at the receiver . one possible advantage of distributed rate splitting is that it can be implemented with lower complexity when compared with joint coding schemes , and less coordination among users when compared with either time - sharing or rate splitting . for the symmetric @xmath0-user gaussian mac , each real user creates the same number of virtual users via a power / rate splitting mechanism . the transmitted signal of a real user is the superposition of all its virtual users signals . for the symmetric @xmath0-user discrete memoryless mac , each real user creates the same number of virtual users via a random switching mechanism , and the transmitted signal of a real user is determined by a random switch . all virtual users are successively decoded at the receiver . it is shown that drs can achieve the maximum equal rate point for both channel models as the number of virtual users per real user tends to infinity . finally , we present two variations of the drs scheme . for the case of asymmetric capacity regions , we show that a point on the dominant face can be achieved asymptotically . for the case of an unequal number of virtual users , we show that different user rates requirements can be accommodated independently in a distributed manner . _ proof : _ we use induction on the number of virtual users . for @xmath46 , the original problem reduces to : @xmath479 subject to @xmath480 and @xmath481 . the unique solution is @xmath482 for @xmath483 , suppose @xmath484 uniquely solves @xmath485 for any @xmath486 , subject to @xmath487 and @xmath488 . let us consider the @xmath387 virtual users case . given any tuple @xmath489 such that @xmath490 , we can fix @xmath80 , so @xmath491 . we now solve @xmath492 subject to @xmath493@xmath42 = @xmath494 , @xmath42 @xmath495 for @xmath43=@xmath496 for fixed @xmath80 , by the induction hypothesis , we have a unique solution @xmath497@xmath498 , ... , @xmath499@xmath500 which solves equation ( [ eq : hminusone ] ) . let @xmath501 @xmath502 . we are able to verify that @xmath503 is a strictly decreasing function of @xmath80 , and @xmath504 = @xmath505 is a strictly increasing function of @xmath80 . moreover , the function values at boundary points satisfy @xmath506 and @xmath507 . so there exists a unique @xmath508 such that @xmath509 . hence , we conclude @xmath510 is the unique solution to the @xmath387 virtual users case . the theorem follows by induction . edmund yeh gratefully acknowledges useful discussions with prof . shlomo shamai on this topic . m. medard , j. huang , a. goldsmith , s. meyn , and t. coleman , `` capacity of time - slotted aloha packetized multiple - access systems over the awgn channel , '' _ ieee trans . wireless commun . . 3 , pp . 486499 , mar . 2004 . r. cheng , `` stripping cdma an asymptotically optimal coding scheme for l - out - of - k white gaussian channels , '' in _ proc . ieee global telecommunications conference _ , sorrento , italy , nov . 1996 , pp . 142146 . jian cao jian cao received the b.s . degree in electrical and electronics engineering with first class honor from the hong kong university of science and technology , hong kong , in 2002 , the m.s . degree in electrical engineering from yale university , new haven , in 2003 , and is currently working toward the ph.d . degree in electrical engineering at yale . his research interests include information theory and queueing theory . edmund yeh edmund yeh received his b.s . in electrical engineering with distinction from stanford university in 1994 , m.phil in engineering from the university of cambridge in 1995 , and ph.d . in electrical engineering and computer science from mit in 2001 . since july 2001 , he has been on the faculty at yale university , new haven , connecticut , where he is currently an associate professor of electrical engineering and computer science . yeh is a recipient of the army research office ( aro ) young investigator program ( yip ) award ( 2003 ) , the winston churchill scholarship ( 1994 ) , the national science foundation and office of naval research fellowships ( 1994 ) for graduate study , the frederick e. terman award from stanford university ( 1994 ) and the barry m. goldwater scholarship from the united states congress ( 1993 ) . yeh is a member of phi beta kappa , tau beta pi , and ieee . he has been visiting faculty at mit , princeton university , university of california at berkeley , and swiss federal institute of technology , lausanne .
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we consider the multiple - access communication problem in a distributed setting for both the additive white gaussian noise channel and the discrete memoryless channel .
we propose a scheme called distributed rate splitting to achieve the optimal rates allowed by information theory in a distributed manner . in this scheme ,
each real user creates a number of virtual users via a power / rate splitting mechanism in the @xmath0-user gaussian channel or via a random switching mechanism in the @xmath0-user discrete memoryless channel . at the receiver ,
all virtual users are successively decoded .
compared with other multiple - access techniques , distributed rate splitting can be implemented with lower complexity and less coordination .
furthermore , in a symmetric setting , we show that the rate tuple achieved by this scheme converges to the maximum equal rate point allowed by the information - theoretic bound as the number of virtual users per real user tends to infinity . when the capacity regions are asymmetric , we show that a point on the dominant face can be achieved asymptotically . finally , when there is an unequal number of virtual users per real user , we show that differential user rate requirements can be accommodated in a distributed fashion . multiple access , rate splitting , successive decoding , stripping , interference cancellation , aloha .
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the color of a galaxy is determined primarily by its star formation history ( sfh ) and the amount of dust attenuation present , with significant additional contributions from metallicity and dust geometry . an empirical relation between sfh , dust attenuation , and color is therefore a useful constraint on models of galaxy formation , and can aid in the interpretation of high redshift galaxy observations where measurements are difficult . here we investigate such an empirical relation , using a representative sample of galaxies observed from the ultraviolet to the infrared by _ galex _ , sdss , and _ spitzer_. the long wavelength coverage allows us to construct simple but robust measures of dust attenuation that are relatively free of a dependence on sfh . similarly , the sdss spectroscopy allows us to measure sfh diagnostics we use a 4000 break measure that are largely free of a dependence on dust attenuation . earlier studies of the effect of dust on galaxy colors have focused on the ultraviolet ( uv , @xmath6 ) colors of galaxies . a primary reason for this is that the intrinsic , underlying uv color ( before attenuation ) is relatively insensitive to the sfh of the galaxy , when compared to the effects of dust attenuation . also , studies of the highest redshift galaxies are often restricted to the restframe uv so that estimates of dust attenuation must be made using uv colors . @xcite derived the effective attenuation properties of dust in star - bursting galaxies by comparing the total dust absorption measured from the ratio of dust emission to uv emission to the change in uv color of a sample of starforming galaxies ( the so - called irx-@xmath5 relation ) . this locally derived relation has been used extensively to determine the dust attenuation in galaxies at higher redshift using their uv colors ( e.g. @xcite ) . however , more recent studies @xcite have shown that there is significant scatter in this relation , especially when less rapidly starforming galaxies are included . @xcite show , using stellar population synthesis models with significant recent bursts , that both sfh and dust are expected to affect the uv color of normal galaxies since at a constant value of dust attenuation galaxies with an older stellar population should appear redder in the uv because of their redder intrinsic uv spectra . they thus propose that a measure of sfh specifically the 4000 break can be used to explain the scatter in the irx-@xmath5 relation . we have constructed a sample of galaxies observed from the uv to the mid / far - infrared . our primary data set is the sample of local ( @xmath2 ) galaxies observed spectroscopically by sdss and analyzed by @xcite and @xcite ( hereafter sdss / mpa galaxies ) . the uv data is taken from pipeline processed _ galex _ observations of the lockman hole , with exposure times of @xmath7 ks . the pipeline - produced _ galex _ catalogs are searched for objects within 3 `` of the sdss / mpa galaxy locations , and the nearest object is taken as the match to the sdss galaxy . uv flux measurements are made in elliptical kron apertures . in the optical we use sdss petrosian magnitudes . the infrared data is provided by the swire _ spitzer _ observations of the lockman hole @xcite . we have performed aperture photometry in the swire team processed 3.6 through 7.8 irac images and 24 mips images at the location of each of the sdss / mpa galaxies , using a 7 '' radius aperture ( 12 " at 24 ) . the fluxes are then aperture corrected to total magnitudes . systematic errors in ir flux due to calibration uncertainty , aperture corrections , and the resolved nature of many of the sources amount to @xmath8 . the resulting uv through 3.6 magnitudes are @xmath9-corrected to @xmath10 ( e.g. @xmath11 , @xmath12 , etc . ) using the method of @xcite . at longer wavelengths dust emission becomes more important than stellar emission , and we use a different method to ` @xmath9-correct ' the data : we choose the best fitting redshifted @xcite model ir sed , on the basis of the observed 8 to 24 flux ratio . this sed is then normalized using the measured 24 flux , and the integrated far - infrared ( 8 - 1000 ) dust luminosities ( @xmath13 ) are derived . note that the different @xcite seds have @xmath14 ratios that are different by a factor of up to five . we have checked that our results would not change significantly if we use the model seds of @xcite ( see @xcite for a detailed discussion of predicting ir luminosities from _ spitzer _ data ) . our final selection of galaxies consists of those sdss / mpa galaxies with detections in the far - uv ( @xmath15 , @xmath16 ) and near - uv ( @xmath17 , @xmath18 ) through 24 bands . this is a total of 467 of the 645 sdss / mpa galaxies within the @xmath19 deg@xmath20 of sdss/_galex_/_spitzer _ overlap . these galaxies have a stellar mass range of @xmath21 @xmath22 and a sfr range of @xmath23 @xmath22@xmath24 , as determined by @xcite and @xcite from the optical spectra and photometry . to simplify the present analysis we do not consider upper limits or selection effects except to note that we are biased against galaxies with very low @xmath25 or @xmath13 . nevertheless , we recover a significant number of galaxies which appear ` old ' and elliptical but have low levels of uv and ir emission . one of the primary motivations for compiling the sample of galaxies described above was to construct a robust and model independent measure of dust attenuation , the so - called infrared excess ( irx , see @xcite for a discussion of the relation of infrared excess to uv attenuation ) . we adopt the definition @xmath26 where @xmath13 is the 8 - 1000 dust luminosity as determined above and @xmath27 is the luminosity in the @xmath28 band ( @xmath29 ) . a second available dust indicator is the h@xmath30to h@xmath31decrement ( see @xcite and references therein ) , measured from the sdss spectra . this dust measure is only well defined for those galaxies with strong emission lines , and can not be easily compared to global galaxy measures due to spectroscopic aperture effects . we do not consider it in the present study . we take the strength of the 4000 break in galaxies as an indicator of sfh @xcite . this has been measured from the deredshifted sdss spectra by @xcite using the ratio of the flux in two narrow bands ( @xmath32 ) centered at 4050 and 3900 . this narrowband color , d@xmath33 , is less sensitive to reddening by dust than broadband colors . it is not , however , completely insensitive to dust effects @xcite . note that d@xmath33 is only measured within the 3 " sdss aperture , which can cause an overestimate of the integrated d@xmath33 for galaxies with moderate bulge to disk ratios . we do not consider in detail the relation between d@xmath33 and more physical measures of galaxy age or sfh ( e.g. the ratio of current to past averaged star formation rate or the specific star formation rate ) , which is metallicity dependent and requires population synthesis modeling . models of galaxy spectra have suggested that the uv color of galaxies can be decomposed into contributions from dust and sfh , though the effect of dust is the dominant contribution @xcite . in figure [ fig : bigfig ] we show the relation between irx and color for different ranges of d@xmath33 . we find a marginal dependence of the irx vs. @xmath34 relation ( analogous to the irx-@xmath5 relation ) on d@xmath33 . only the very oldest galaxies ( red points ) have systematically redder @xmath34 for the same value of irx , but also have larger errors . for galaxies with d@xmath33@xmath35 there appears to be little dependence of the scatter on galaxy sfh . this is similar to the result of @xcite , who use the @xmath36k color as a measure of sfh ( but see below for the effect of dust on the @xmath36nir color ) . this result can be reconciled with the models of @xcite in several ways . first , some stars contributing to the dust heating ( and therefore the ir emission ) may not be contributing to the uv emission or , conversely , some stars contributing to the uv emission are not contributing to the ir emission ( i.e. a decoupling of ir and uv emission ) . this may occur for various reasons including star - dust geometry ( e.g. @xcite ) and contributions to dust heating from older stellar populations . related to this , d@xmath33 may not be measuring the relevant timescale for changes in the uv spectrum . the sfh diagnostic that is likely to be most relevant to changes in the uv spectral shape is the ratio of current sfr to the sfr averaged over the last 100 myr @xcite , which is not probed well by d@xmath33 . second , weak agn may affect the uv colors . third , the entire range of @xmath34 color is @xmath37 mag , compared to a median error of @xmath38 mag . the scatter due to measurement error may obscure a trend in the scatter with sfh . the results of @xcite were based on a library of 95000 model spectra , which may not be well represented by our sample . these and other possibilities will be investigated in future work . for @xmath39 color it is easy to see the effect of sfh that was predicted by @xcite for the @xmath34 color . at a given low d@xmath33 ( i.e. younger mean stellar age , purple points ) the relation between dust and color is clear , and has low scatter . this relation is closely related to the effective attenuation curve of the dust in these galaxies @xcite . for intermediate d@xmath33 ( i.e. intermediate mean stellar age , orange or green points ) the relation between dust and color persists with low scatter , but the entire relation is shifted to redder color . this is presumably because of the redder intrinsic spectrum of an older stellar population , on top of which the effect of dust attenuation on color persists relatively unchanged . for @xmath40 the ratio of the median error to the range in color is significantly smaller than for @xmath34 , making trends with color easier to see . at the largest d@xmath33 the scatter in the irx - color relation increases this may be due to several causes . first , there are larger errors in irx ( and color ) for these galaxies , which have relatively little star formation and are systematically dimmer in @xmath15 and at 24 . second , the effects of metallicity on d@xmath33 become more pronounced at high d@xmath33 . third , some uv emission may be due to evolved populations ( e.g. bhb stars ) . finally , these galaxies may well host agn that affect the ir and/or uv emission , changing irx . in figure [ fig : model ] we present another projection of the dust - sfh - color relation . here we show d@xmath33 as a function of @xmath39 color for different ranges of irx . irx appears to be well determined using just @xmath39 and d@xmath33 . the relation between d@xmath33 and @xmath40 shifts to redder color for galaxies with more dust , while the slope of the relation remains nearly constant . galaxies in the lower right of this plot are predominantly dusty star - forming galaxies . galaxies in the lower left are typically blue star - forming galaxies or dwarfs / irregulars . galaxies in the upper right of figure [ fig : model ] are red - sequence galaxies composed primarily of old stars , with very low levels of uv and ir emission . we have conducted quantitative parametric fits to the relation between dust attenuation , sfh , and the @xmath41 color . considering a simple effective attenuation law we are motivated to consider polynomial fits to the color ( @xmath42 ) , d@xmath33 , and an expression of irx that is linearly related to color @xmath43 where @xmath44 is the bolometric correction to the uv and the bolometric correction to the ir has been made in [ sec : data ] ( see @xcite for a discussion of the relation between irx and the true uv attenuation @xmath45 ) . we treat the color and d@xmath33 as independent variables ( since the errors are much smaller than for irx ) . we assume either a linear or quadratic form for the relation between color and d@xmath33 . the results are given in table [ table : fit ] . examination of the residuals shows that a cross term ( @xmath46d@xmath33@xmath47 ) is required , and fits including such a term are also given in table [ table : fit ] . the fit from the 3rd row of table [ table : fit ] is overplotted in figure [ fig : model ] , and the residuals in irx as a function of color and d@xmath33 are shown in figure [ fig : fit ] . figure [ fig : fit ] shows that the fit is poorer for redder colors and larger d@xmath33 . this may be due to the effects listed in [ sec : irx_color ] for the @xmath41 color , as well as the effect of aperture on d@xmath33 ( [ sec : sfh ] ) . at bluer colors ( @xmath41@xmath48 ) and smaller d@xmath33 the residuals are lower , @xmath49 rms in @xmath50 . table [ table : fit ] also includes fits using the @xmath51 color since this is more easily measured for a large sample of galex observed sdss galaxies ( e.g. @xcite ) . the behavior is similar to the @xmath41 color though the residuals are larger . cccccccc + 1.31 & -3.46 & & 0.84 & & 0.57 & 0.33 + 1.21 & -1.59 & -2.73 & 0.83 & & 0.55 & 0.31 + 1.08 & -1.80 & & 1.03 & -0.72 & 0.51 & 0.31 + 1.07 & -2.39 & 1.59 & 1.10 & -0.94 & 0.50 & 0.32 + + 1.60 & -3.27 & & 0.87 & & 0.75 & 0.43 + 1.43 & -0.58 & -4.12 & 0.89 & & 0.70 & 0.39 + 1.27 & -1.56 & & 1.35 & -1.24 & 0.65 & 0.38 + 1.25 & -2.91 & 3.30 & 1.56 & -1.82 & 0.64 & 0.39 we have used a simple treatment of attenuation to motivate our fits that does not self consistently consider , e.g. , the heating of dust by old stars . also , d@xmath33 is subject to aperture effects that may affect the fits . it is thus difficult to interpret the fit coefficients as physical parameters . the fits in table [ table : fit ] are referenced to d@xmath33@xmath52 , @xmath53 because this region is typical of ` blue sequence ' galaxies . the constant a term thus gives the typical @xmath50 for such galaxies . the b term ( with contributions from the c and e terms when present ) gives the trend in @xmath50 with d@xmath33 , and is negative since galaxies that are intrinsically redder have less attenuation for a given color . similarly , d gives the trend in @xmath50 with color , and is positive as a redder galaxy at a given d@xmath33 has more attenuation . while we recover only a weak sfh dependence for the short wavelength - baseline uv color , we have shown that long wavelength - baseline colors specifically @xmath39 can be decomposed into contributions from dust and sfh with low scatter . this is the dust - sfh - color relation . such a decomposition is possible due to the use of a relatively dust - insensitive ( though perhaps metallicity dependent ) sfh indicator in combination with a robust measure of the dust attenuation in galaxies . at large d@xmath33 the relation between dust , sfh , and color is more scattered , suggesting that an additional parameter may be necessary to explain the color of such galaxies , or that our sfh and/or dust indicators become less reliable here . deep _ spitzer _ irac data , when combined with optical data , is ideally suited to measuring the @xmath39 color for galaxies at @xmath54 , allowing similar analyses for galaxies at much earlier epochs ( e.g. @xcite and @xcite ) . the mpa / jhu collaboration for sdss studies has very generously made their catalogs publicly available . the publicly available _ spitzer _ data obtained and reduced by the swire team have been essential to this work . we gratefully acknowledge nasa s support for construction , operation , and science analysis for the _ galex _ mission , developed in cooperation with the centre national detudes spatiale of france and the korean ministry of science and technology . bdj was supported by nasa gsrp grant nng05go43h .
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we combine data from sdss and the _ galex _ and _ spitzer _ observatories to create a sample of galaxies observed homogeneously from the uv to the far - ir .
this sample , consisting of @xmath0460 galaxies observed spectroscopically by sdss provides a multiwavelength ( 0.15 - 24 @xmath1 m ) view of obscured and unobscured star formation in nearby ( @xmath2 ) galaxies with sfrs ranging from 0.01 to 100 m@xmath3 yr@xmath4 .
we calculate a robust dust measure from the infrared to uv ratio ( irx ) and explore the influence of star formation history ( sfh ) on the dust - uv color relation ( i.e. the irx-@xmath5 relation ) .
we find that the uv colors of galaxies are only weakly dependent on their sfh as measured by the 4000 break .
however , we find that the contributions of dust and sfh are distinguishable when colors at widely separated wavelengths ( e.g. 0.23 - 3.6 ) are introduced .
we show this explicitly by recasting the irx-@xmath5 relation as a more general irx - sfh - color relation , which we examine in different projections .
we also determine simple fits to this relation .
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the ` knee ' , a rather sharp steepening in the primary cosmic ray ( cr ) energy spectrum at about 3 pev , was inferred from the observation of a similar feature in the measured size spectrum of extensive air showers by kulikov and khristiansen ( 1958 ) . the knee is commonly asserted to be due to an increasing failure of ` galactic containment ' of the cr generated by sources within the galaxy , the containment being caused by the magnetic fields in the interstellar medium ( ism ) , however , it is the firmly - held view of the present authors that the knee is too sharp for this explanation and we have advanced what we claim to be a more realistic model . this is our ` single source ( ss ) model ' ( see erlykin and wolfendale , 1997 , 2001b for recent details ) which comprises cosmic ray acceleration up to the knee energy by supernova remnants , the knee itself being due to the truncation that occurs at 3 pev for oxygen nuclei from a single , recent , nearby snr . the other main accelerated nucleus at these energies is iron and its termination occurs at about 12 pev where , it is claimed by us that there is a small second knee ( when the spectrum is plotted as @xmath6 vs. @xmath7 , the knees appear as small peaks ) . the remainder of the cr spectrum ( at least to some 10@xmath8 gev , or so ) is presumed due to ` super'-snr and other sources and their spatial distribution is such as to give a comparatively smooth spectrum in the pev region . erlykin and wolfendale ( to be referred to henceforth as ew ) have examined a variety of other cosmic ray data and concluded that there is either support for the model or that the data are neutral . very recently , low energy gamma ray data have also been studied ( ew , 2002 ) and the well - known ` gamma ray excess ' in the inner galaxy , and deficit in the outer galaxy , have been explained in terms of propagation differences dependent on the conditions in the ism from which the gamma rays come . the results relate to galaxy - wide properties and , although the snr acceleration hypothesis has been invoked , there is no significant information about the single source . it is at higher gamma ray energies where potential problems exist ( e.g. drury et al . , 1994 ) . most recently , bhadra ( 2002 ) has argued that the single source should be visible in tev gamma rays , and it is not . this is the topic to be addressed here . we use the results found in a very recent paper ( ew , 2003a to be referred to as i ) , where we made predictions of the fluxes and the angular distribution of gamma ray intensity from snr of different ages and at different distances from the sun . the threshold energies were taken as 0.1 gev and 1 tev . a critical feature of the bhadra estimate was the ` normalisation ' of the snr ` conditions ' so as to give the cr energy density created by the single source at earth . we regard this as a legitimate procedure and we follow this path , although other features of our model differ considerably from those adopted by bhadra . our calculations are thus not simply a ` re - run of the bhadra calculations with different values for the parameters ' but , rather , for what is certainly a more appropriate model of snr acceleration and ( less certainly , perhaps ) a significantly different model of cosmic ray propagation . bhadra s model is rather straightforward , in principle , at least : particles are accelerated by the snr shock to give a differential spectrum @xmath9 , with @xmath10 and these particles interact with the ambient ( or swept - up ) gas of density @xmath11 , with @xmath12 @xmath13 . at this stage it is necessary to make critical remarks , however . implicit in the bhadra calculations is the assumption that the snr shock accelerates the cr instantaneously , at ` @xmath14 ' . although this can be used to give a viable mathematical model , such a situation is certainly not appropriate to a real snr where the acceleration occurs over an extended period : @xmath15 years in our model , and little different in other snr acceleration models . nevertheless , we continue to describe the bhadra calculations . it is assumed that the particles diffuse from the source a distance @xmath16 from the sun in a normal , gaussian fashion with diffusion coefficient @xmath17 @xmath18s@xmath19 . the parameters are chosen to give the required cr energy density at the earth created by the single source . for instance , if the source is at the distance of 300 pc and it is 10@xmath20 years old the total energy transferred from sn to cr is required to be 1.9@xmath2110@xmath22 erg . the table summarizes the most important parameters . . values of the parameters adopted by bhadra ( 2002 ) in comparison with those in the present work ( denoted ew ) . the remnant is taken to be at 300 pc from the sun . @xmath23 is the cosmic ray energy input from the snr ; @xmath24 is the diffusion coefficient for normal , gaussian diffusion , @xmath7 is in gev ; @xmath11 is the density of the interstellar medium ( ism ) in _ h_-atoms @xmath13 ; @xmath25 is the exponent of the differential proton spectrum ; @xmath26 and @xmath27 are the predicted gamma ray fluxes . [ cols="<,^,^ " , ] the expected minimum gamma ray flux above 0.1 gev rises with the age of the sn from @xmath28 at 10@xmath29 years to @xmath30 at 10@xmath31 years and from @xmath32 to @xmath33 above 1 tev , respectively . for the real candidates for the single source discussed in ew , ( 1997 ) ( eg . loop i , clayton snr ) the expected fluxes are substantially higher . comparing these fluxes with the diffuse gamma ray background bhadra found that for the present gamma ray telescopes it should have been possible to observe the single source . since there has been no claim for an observation bhadra concluded that the single source can not be such a snr . turning to our remarks on the validity of bhadra s model , in addition to the basic problem with the assumption about the instantaneous acceleration there are two further reasons why we can not allow this conclusion to stand and , in fact , bhadra made the neeeded reservations , viz . _ the detection could be crucial , depending on the angular size of the object _ and _ unless the source is in a lower density environment_. these are the points addressed in the following sections . we require , first , the likely whereabouts of the single source such that it can give the particle spectrum needed in the knee region . we have calculated cosmic ray energy spectra originating from snr of different ages and distances . the model of acceleration was described in ew ( 2001a ) and briefly in i. the propagation of accelerated cosmic rays through the ism was calculated using two alternative assumptions about the mode of propagation from the source : ` anomalous ' diffusion , viz . making allowance for the fractal - like nature of the ism , and normal , gaussian , diffusion . following the work of lagutin et al.(2001a , b ) we distinguish these two modes by a parameter @xmath34 determined by the fractal nature of the medium ; @xmath34 is equal to 1 or 2 for anomalous or normal diffusion , respectively . the difference between the two modes can be seen most clearly in the shape of the lateral distribution function for the cosmic ray intensity : @xmath35 for @xmath36 and @xmath37 for @xmath38 , with @xmath39 , @xmath16 being the distance from the radius @xmath40 pc where the particles start to diffuse and @xmath41 being the diffusion radius which is defined as @xmath42 , i.e. , there is a different time dependence for the two modes . kpc for the vertical scale of the galactic halo and @xmath44 and @xmath45 are the diffusion time and time against escape , respectively , for the protons . details of the propagation model are given in ew ( 2002 ) and in i. the difference in the two lateral distributions is quite dramatic . thus , for @xmath46 2 , 4 , 6 and 8 , the ratio of the intensity for anomalous diffusion to that for normal diffusion changes as 0.11 , 0.19 , 5.9 and 2.1@xmath2110@xmath29 , respectively . the long tail for anomalous diffusion - the occasional considerable ` penetration ' ( levy flights ) - can have important implications . an example is that for the secondary to primary ratio ; to our knowledge the implications have not been worked out . we have determined the proton energy spectra expected for the two values of @xmath34 and these are given in figures 1 and 2 . also indicated in the figures is the spectrum ` required ' by the ss model . we have argued that the ` needed ' particles are oxygen nuclei for the first ` peak ' at 3 pev and iron nuclei for the second ` peak ' at @xmath47 pev , and the requirement has been converted to rigidity before plotting , in figures 1 and 2 . an alternative association of the ` peaks ' with helium and oxygen changes the indicated ss spectrum no more than by 17% . ( ew , 2003b ) . dotted lines above and below the ` ss ' curve indicate its uncertainty limits ; the least uncertainty and therefore the most important constraint of the ss model is in the knee region . the limits come from the fit to the experimental data on the primary energy spectrum measured by means of the cherenkov light emitted by extensive air showers ( ew , 2001b ) . these data determine the magnitude of the uncertainty in the range of about a decade below the knee . at lower energies , - @xmath48 gev - the upper limit is determined by the uncertainty of the direct measurements of the primary cr energy spectrum ( biermann and wiebel - sooth , 1999 ) . although the shape of the ss spectrum has been adopted from the theoretical model of berezhko et al . ( 1996 ) its lower experimental limit at low energies is completely uncertain because the contribution of the single source to the total cr intensity at these energies is negligibly small and even consistent with zero . however , the actual value of the uncertainty at tev energies is not important for this analysis because we have made a quantitative comparison of the calculated proton spectra with the ss spectrum only in the most important range of energies , viz . that covering a decade below the knee . the energy range of @xmath49 used for this comparison is indicated in the figures by _ ` min ' _ and _ ` max'_. ( ew , 2002 ) ( rigidity and energy are , of course , the same for protons ) . the energy interval used for the comparison of proton spectra with the ss model is marked by _ min _ and _ max_. the result of the comparison in terms of the reduced @xmath50 , i.e. @xmath50 divided by the number of degrees of freedom , @xmath51 , is shown inside the figures.,width=604,height=529 ] [ fig : bhad1 ] .,width=604,height=529 ] [ fig : bhad2 ] some comments are needed about the figures . it is evident that @xmath52 always gives a bad fit ( figure 2 ) whereas @xmath53 can give a reasonable one , at least for energies above 10@xmath20 gev , up to the cut - off at 4@xmath54 gev , for a range of age ( t ) and distance ( r ) values . has a formally acceptable value for @xmath38 at t@xmath55 kyear the slopes of the proton spectra are too steep to give the sharp knee and we do not consider it as a good contender . ] calculations made for a wide range of t and r allow us to estimate the range over which there is satisfactory agreement between calculations and the ss model ; it is @xmath56 kyear and @xmath57 pc for the adopted set of input parameters . it is relevant to point out that larger distances , too , would give a good fit to the spectral _ shape_. the necessary upward movement in intensity could be effected by increasing the fraction of the shock energy going into cr . as was pointed out in i , berezhko et al.,(1996 ) used much bigger values than the 10% used here - their highest being 80% . specifically , for 100 kyear an increase in cr yield by a factor 8 would allow the source to be at @xmath58 pc . the shape and the absolute intensity of the cr energy spectrum give the most stringent constraints on the age and distance . the t - r region of snr which could give an acceptable spectrum ( for our standard ` 10% ' ) is shown in figure 3 by the 95% confidence level contour . following bhadra s approach we have also used such an integrated characteristic of the spectrum as the energy density contained in it . again , we used for the comparison just the last decade of the spectrum below the knee , because intensities at low energies are so poorly determined by the single source model . the energy density contained in the spectrum of our single source between @xmath59 gev and @xmath60 gev is 1.84@xmath61 . comparison of this value , allowing for uncertainty , with those expected for snr of different ages and distances gives an acceptable t - r region indicated by the dashed line in figure 3 . it overlaps with the region deduced from the comparison of the spectral shapes . this proves the consistency of these results , although we must admit that the two methods are not completely independent , because consistent values for the spectra should inevitably give consistent energy densities contained in them . in any case this analysis indicates that our single source should be located at about 300 - 350 pc from the sun and should be about 90 - 100 kyears old . we know of no objection to such parameters ( see 4 for its likely location ) . s@xmath19 . , width=566,height=264 ] [ fig : bhad3 ] fluxes of gamma rays from snr of different ages and at different distances from the sun have been calculated in i. in figure 3 we show contours of the fluxes calculated for the interactions of protons , accelerated by the snr , which propagate through the ism by way of anomalous diffusion ( @xmath36 ) and collide with nuclei of the ism , the density of the ism for these calculations has been taken to be our standard @xmath62 . it is seen that the expected fluxes are about 3@xmath63 for @xmath64 gev and about 2@xmath65 for @xmath66 tev . it is necessary now to study the implications of the fact that the cosmic rays from our single sourse are not only protons , but that the composition is mixed . at the same rigidity it consists of 21% p , 48% o , 13% ` heavy ' nuclei ( @xmath67 ) and 18% fe . because nuclei of the same rigidity are more efficient in the production of gamma quanta ( see i ) , the expected fluxes should be increased by a factor of 10.5 . this gives expected fluxes of @xmath68 for @xmath64 gev and @xmath69 for @xmath66 tev . it is necessary to point out that the cr energy injection is higher by the same amount . since this is a first order effect , a detailed examination of the problem is required . as pointed out in i , there are several aspects that need attention , specifically : + ( i ) the effect of the pre - snr winds from the progenitor star in excavating a ` hole ' into which the snr shock expands , this is essentially the origin of the hism ( note that for vey young snr , however , the progenitor star s wind may have _ enhanced _ the density ) ; + ( ii ) the general ism density in the region where the progenitor star was situated and + ( iii ) the pressure , or otherwise , of clumpy gas ( often molecular ) in the vicinity of the snr . in the present case , it is almost certain that the progenitor star - at 300 - 350 pc from the sun - was in the hot interstellar medium , where the density is often quoted as @xmath70 @xmath13 ( berezhko et al . , 1996 ; cox and reynolds , 1987 , for example quote a density of @xmath71@xmath13 ) . indeed , since all our model calculations assume this to be the case , a low density target material is a prerequisite . the assumption of the hism comes from two factors : + ( i ) the absolute maximum particle energy needed to explain the knee ( @xmath72 gev for oxygen nuclei ) appears only for this density , in the berezhko et al . model , and + ( ii ) the hism is eminently reasonable for a nearby source . factor ( ii ) can be examined in more detail . insofar as the sun is located on the edge of a spiral arm , in one hemisphere ( the south ) , at least , the gas density will be low . frish ( 1997 ) has considered the situation in detail . beyond the very local region , where there is the ` local fluff ' ( of extent @xmath733 pc ) in certain directions the column density of atomic ( and molecular ) hydrogen is very low . the ` local bubble ' ( loop i ) is quickly reached , this bubble being caused by several sn over the past myear . here , one expects the hism , with its density @xmath74 @xmath13 . concerning the interarm region , frish quotes an undisturbed part as having a density of @xmath75 @xmath13 , i.e. even smaller . however , much of the local bubble is here and the density will be higher because of material brought in from elsewhere . interestingly , frish ( 1981 ) suggested , earlier , that the sun is embedded in one of the super - bubble shells associated with the formation of the scorpius - centaurus association . another possibility for the formation of loop i ( but very similar to that given above is that it was caused by activity in the ` upper centaurus lupus ' subgroup some 14 - 15 myear ago ( de geus , 1991 ) ) . a distinction must be made between a ` recent ' sn exploding in loop i , i.e. in the low density hism caused by previous snr and stellar activity , and the low level gamma ray flux from loop i as a whole . this latter was considered in i and it was argued there that the mean density overall , allowing for molecular clouds inside the loop , and the piled - up gas in the edges of the loop , is @xmath76 0.1 @xmath13 . there is no conflict with our @xmath77 @xmath13 if the single source is well into the interior . with @xmath78 , the predicted flux is reduced to @xmath79 @xmath80s@xmath19 for @xmath810.1 gev and @xmath82@xmath80s@xmath19 for @xmath83 1 tev . if , surprisingly , the source is not in the hism in the local bubble , but is isolated , then as was discussed in i , @xmath84 and the predicted fluxes are @xmath85 for @xmath64 gev and @xmath86 for @xmath66 tev . the predicted fluxes are shown in figure 4 , for the various possibilities of _ n _ and _ z_. gev and @xmath87 tev for a variety of scenarios . the source would be at @xmath88 pc from us and have an age of @xmath89 kyear , i.e. it would be only 10 - 20 kyear since the particles were released by the remnant . the dotted lines marked ` needed ' are the minimum fluxes required for the source to have been detected by the arrays in use up to now . as an indication of the future we also give the estimated minimum detectable flux for @xmath66 tev for a source of 20@xmath1 radius , from the work of aharonian et al.,(1997 ) . these authors give results for 1000 hours of observation of a point source and one of 1@xmath1 extent ; our estimate arises from an extrapolation , based on results described in i. the work reported relates to the then proposed iact ( 100 gev - class imaging atmospheric telescope array ) , denoted ` hess'.,width=566,height=340 ] [ fig : bhad4 ] the sources must be detected against a background due to the galactic diffuse emission arising from cr - ism nucleus interactions ( and , in the case of tev gamma rays , protons interacting in the atmosphere ) . this background can be allowed for in a straightforward manner for ` point ' sources , where a subtraction can be made of the signal nearby to the source , but for extended sources the problem is much more severe . the angular radius of a snr , which is 90 - 100 kyear old , seen from a distance of 300 - 350 pc is about 20@xmath1 ( see also figure 5 in i ) . for such a large size the determination of the background by a linear interpolation of the intensity between the adjacent regions is not at all accurate . estimates of the background based on the known column density of the target gas , although more appropriate , are again not sufficiently accurate because the intensity of the initiating cosmic rays along the line of sight can not be assumed to be strictly constant . estimates of the limiting fluxes were made from the available data and they are given in i. for a source of radius 20@xmath1 the average limits are @xmath90 for @xmath64 gev and @xmath91 for @xmath66 tev . these limiting sensitivities , which relate to observations made so far , are indicated in figure 4 . the lower energy limit comes from an extrapolation of the egret results ( hartman et al , 1999 ) , which refer to small angular sizes , to 20@xmath1 . that for the upper energy band comes largely from the tibet and hegra arrays ( amenomori et al . , 2001 and lampeitl et al . , 2001 , respectively ) . it is instructive to examine these sensitivities in terms of the background fluxes . for @xmath64 gev , the intensity at low latitudes towards the galactic anticenter is about @xmath92 and for our 20@xmath1-radius source ( @xmath93 = 0.4 sr ) the flux would be @xmath94 . the plotted limit of @xmath95 is thus 8% of the background . for the direction to the galactic center , where the background is higher by a factor of 3 , this percentage is reduced to ( 2 - 3)% . when allowance is made for other sources of uncertainty our adopted limit seems reasonable . for @xmath66 tev , the corresponding background in the galactic anticenter direction has been estimated to be @xmath96 ( porter and protheroe , 1999 ) , viz a flux of @xmath97 over a 20@xmath1-radius source . our adopted limit of @xmath98 is thus the same : @xmath738% of the background . in the direction of the galactic center the background is expected to be twice as large and the fraction of the limiting flux falls to @xmath734% . this figure , too , is reasonable in view of misidentified proton contributions and a variety of technical problems associated with making absolute measurements at different zenith angles . the conclusions are indicated in figure 4 . only if the single source is in a ` high ' density region ( @xmath99 ) and the primaries are ` heavy ' nuclei ( z = 10 ) ( with consequent high cr energy injection ) will it be possible to detect the single source at high energies . such a situation is not impossible but there would be problems for our single source model with such a high ambient density , specifically that it would not be possible with the model in use to reach the required 3 pev energy at the knee . insofar as we consider it very likely that the source is in the hism and is extended , the chance of detecting it with contemporary instruments is considered , by us , to be very low . there are , however , hopes for the future . it is germane to consider in which direction future , improved , gamma ray detectors should be pointed in order to see it . the best that can be done here is to suggest the general direction of loop i. however , since this structure occupies about 25% of the sky the ` advice ' is not very helpful . the authors are grateful to the royal society and the university of durham for financial support . paula chadwick , of the hess collaboration , is thanked for providing useful information . two unknown referees are also thanked for useful comments and suggestions . aharonian , f.a . et al . , 1997 , _ astroparticle physics _ , * 6 * , 369 + amenomori , m. et al . , 2001 , _ _ ( hamburg ) , * 6 * , 2344 + berezhko , e.g. , elshin , v.k . and ksenofontov , l.t . , 1996 , _ , * 82 * , 1 + bhadra , a. , 2002 , _ j. phys . g : nucl . part . _ , * 28 * , 397 + biermann p.l . and wiebel - sooth b. , 1999 , _ astronomy & astrophysics - interstellar matter , galaxy , universe _ , landolt - brnstein , springer verlag , berlin / heidelberg , * 3 * , 37 + cox d.p . and reynolds r.j . , 1987 , _ annual rev . astrophys . _ * 25 * , 303 + de geus , e.j . , 1991 , _ _ , * 262 * , 258 + drury , l.oc . , aharonian , f.a . and vlk , 1994 , _ astron . _ , * 287 * , 959 + erlykin , a.d . and wolfendale , a.w . , 1997 , _ j. phys . g : nucl . part . _ , * 23 * , 979 + erlykin , a.d . and wolfendale , a.w . , 2001a , _ j. phys . g : nucl . part . _ , * 27 * , 941 + erlykin , a.d . and wolfendale , a.w . , 2001b , _ j. phys . _ , * 27 * , 1005 + erlykin , a.d . and wolfendale , a.w . , 2002 , _ j. phys . g : nucl . part . _ , * 28 * , 2329 + erlykin , a.d . and wolfendale , a.w . , 2003a ( submitted to j. phys . g ) + erlykin , a.d . and wolfendale , a.w . , 2003b , _ nucl . b ( proc . suppl . ) _ ( in press ) + frisch , p.c . , 1981 , _ nature _ , * 293 * , 377 + frisch , p.c . , 1997 , astro - ph/9705231 + hartman , r.c . et al . , 1999 , _ _ , * 123 * , 179 + kulikov , g.v . and khristiansen , g.b . , 1958 , _ j. exp _ , * 35 * , 635 + lagutin , a.a . 2001a , _ nucl . b ( proc . suppl . ) _ , * 97 * , 267 + lagutin , a.a . et al . 2001b , _ proc . _ ( hamburg ) , * 5 * , 1900 + lampeitl , h. et al . , 2001 , _ 27th int . cosm . ray conf . _ ( hamburg ) , * 6 * , 2348 + porter t.a . and protheroe r.j . , 1999 , _ 26th int . _ , salt lake city , * 4*. 306 +
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some six years ago , we ( erlykin and wolfendale , 1997 ) proposed the ` single source model ' in which a local , recent supernova remnant ( snr ) was responsible for the ` knee ' in the cosmic ray ( cr ) energy spectrum at @xmath0 pev . stimulated by the paper by bhadra ( 2002 ) , which drew attention to a possible gamma ray signature of this local remnant , we now study the situation for the local source and we conclude that , in contrast to bhadra s conclusion , the non - observation of this remnant is understandable - at least using our snr model .
it is due to the fact that this snr , being local , develops in the local hot interstellar medium ( hism ) with its low density of gas and also being nearby it will be an extended source occupying up to 40@xmath1 of the sky and thus indistinguishable from the background . plus 2 mm minus 2 mm 23.0 cm 17.0 cm -1.0 in -42pt * high - energy cosmic gamma rays + from the ` single source ' * ' '' '' a.d . erlykin@xmath2 and a.w .
wolfendale@xmath3 @xmath4 lebedev physical institute , leninsky prospekt , moscow , russia + @xmath5 department of physics , university of durham , dh1 3le , uk +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
mannheim @xcite ( henceforth m6 ) has developed a cosmological model ( mm ) based on conformal invariance . this model has a number of attractive features , but has not so far been accepted as a viable candidate for the correct theory of gravitation , in large part because it predicts , apparently unambiguously , a negative value for the effective gravitational constant , @xmath0 . from this follows a thermal history markedly different from the usual one , and consequent difficulties in addressing questions such as primordial nucleosynthesis @xcite . in this paper we point out a mathematical problem in the formulation of the model , and present a new approach that yields a positive @xmath0 , so that the model closely resembles the conventional one . the mm involves a scalar field , @xmath1 , in a friedmann - robertson - walker ( frw ) background metric . it is this field @xmath1 that we will be concentrating on in this paper . the treatment of both @xmath1 and the gravitational field in m6 is classical . important subsequent work has been done on the quantization of the fourth - order equations for the gravitational field that result from the conformal lagrangian @xcite . the scalar field , @xmath1 , however , has always been treated classically , a procedure that we will question in this paper . there are several things that are required of @xmath1 in a viable model : @xmath2 an equation of motion that is conformally invariant . an energy - momentum tensor whose 00 component ( hamiltonian ) has an energy spectrum that is bounded below . a constant vacuum expectation value , @xmath3 , derived from the equation of motion . the @xmath4 in the action generates the effective gravitational constant , @xmath0 . a @xmath5 term in the action that gives the correct sign for the cosmological constant . let us see whether these criteria are met in the mm . using ( as mannheim does ) a metric signature @xmath6 , and neglecting the coupling to the fermion field , the terms in mannheim s lagrangian involving @xmath1 are ( m6 ( 61 ) ) : @xmath7 this results in an equation of motion ( m6 ( 63 ) ) : @xmath8 this equation , as required , is conformally invariant . @xmath1 is assumed to acquire a non - zero vacuum expectation value , @xmath3 , found by setting the derivative term in ( [ eq : s ] ) equal to zero . this gives @xmath9 as in @xcite ( 13 ) , with @xmath10 . further development of the model ( m6 , section 10 ) shows that , in order to meet observational criteria at the present time , @xmath11 and @xmath12 are both negative , so that @xmath4 is real and positive . the effective gravitational constant , @xmath0 , in the mm turns out to be negative ( m6 ( 224 ) ) : @xmath13 choosing units with @xmath14 . the stress tensor in the mm model is obtained by taking the variation of the action with respect to the metric , in the usual way ( m6 ( 64 ) ; a similar expression is given in @xcite ) . retaining just the terms involving @xmath1 we get @xmath15 the lagrangian ( [ eq : l ] ) is analogous to the minkowski space lagrangian for a @xmath16 field theory ( @xcite ( 7.2.14 ) , with @xmath17 ) : @xmath18 the @xmath19 term in ( [ eq : l ] ) has the `` right '' sign for an @xmath20 term . but with @xmath21 , the @xmath22 term has the `` wrong '' sign , and , in a conventional treatment , will lead to a spectrum that is not bounded below . a theory of this sort has to be treated by methods appropriate to non - hermitean hamiltonians @xcite . at the level of quantum mechanics , `` wrong sign '' @xmath16 theory is well developed . nevertheless , even in minkowski space , constructing a corresponding quantum field theory is a difficult problem , still incompletely understood @xcite . conformal cosmology will remain flawed until we can make progress in understanding the scalar field . a conformally invariant theory with a single scalar field has a unique action , m6 ( 61 ) , provided we use the familiar techniques appropriate for hermitean lagrangians . once we recognize that our theory involves a non - hermitean lagrangian , however , a new approach is suggested , that we introduce in this section . we begin by working in minkowski space , but retain @xmath23 and @xmath24 in formulae to simplify the transition to a frw space . a @xmath16 theory with a `` wrong sign '' @xmath16 term is non - hermitean but is nevertheless @xmath25 symmetric , and can be treated by the methods outlined in @xcite . the distinctive feature of this approach is the use of the @xmath26 norm in place of the usual dirac norm ; for a quantized field , @xmath27 , we write this norm as @xmath28 here @xmath29 and @xmath30 represent the usual parity and time - reversal operations , while @xmath31 represents a special operation designed to ensure the norm is real and positive definite and the theory is unitary . the @xmath31 operator has to be specifically calculated for each hamiltonian . our cosmological model is written in terms of classical fields ( expectation values , @xmath32 ) , which we take to be real . we assume @xmath33 can then be expressed as @xmath34_{\forall \rho : z^{\rho } = -y^{\rho } } \label{eq : cptdef}\end{aligned}\ ] ] ( compare @xcite ( 78 ) ) , so that @xmath35_{\forall \rho : z^{\rho } = -y^{\rho } } s ( x ) \label{eq : normex2}\end{aligned}\ ] ] in an expression of this sort , @xmath36 and @xmath37 describe the field @xmath1 at the same physical point , but use different coordinate systems to refer to that point . take the complex conjugate of ( [ eq : normex2 ] ) , and let @xmath38 and @xmath39 : @xmath40_{\forall \rho : z^{\rho } = x^{\rho } } \left [ s^ { * } ( u ) \right]_{\forall \rho : u^{\rho } = -y^{\rho } } \nonumber \\ & = & \int { \rmd}^4 x \int { \rmd}^4 y \ , c^ { * } ( x^{\mu } - y^{\mu } ) s^ { * } ( -y ) s(x)\end{aligned}\ ] ] showing that @xmath41 must be real . we define our action by @xmath42^{\cal cpt } \frac{\partial s ( x)}{\partial x^{\nu } } \right . \nonumber \\ & & \left . { } + \frac{\sigma_m}{2 } m^2 s^{\cal cpt}(x ) s(x ) + \sigma_{\mu } { \mu}^2 \left [ s^{\cal cpt}(x ) s(x ) \right]^2 \right\ } \label{eq : imink } \end{aligned}\ ] ] where @xmath43 , @xmath44 and @xmath45 are simply `` sign factors '' , each of which can be equal to @xmath46 . we will determine their actual values as we proceed . the energy - momentum tensor is obtained in the usual way by varying the action with respect to the metric : @xmath47^{\cal cpt } \frac{\partial s ( x)}{\partial x^{\beta } } \right . \nonumber \\ & & \left . { } + \frac{\sigma_m}{2 } m^2 s^{\cal cpt } ( x ) s(x ) + \sigma_{\mu } { \mu}^2 \left [ s^{\cal cpt } ( x ) s(x ) \right]^2 \right\ } \nonumber \\ & & \left . { } - ( -g ) ^{1/2 } \frac{\sigma_k}{2 } g^{\mu \alpha } g^{\nu \beta } \left [ \frac{\partial s ( x)}{\partial x^{\alpha } } \right]^{\cal cpt } \frac{\partial s ( x)}{\partial x^{\beta } } \right ) \nonumber \\ & = & g^{\mu \nu } \left\ { \frac{\sigma_k}{2 } g^{\alpha \beta } \left [ \frac{\partial s ( x)}{\partial x^{\alpha } } \right]^{\cal cpt } \frac{\partial s ( x)}{\partial x^{\beta } } \right . \nonumber \\ & & \left . { } + \frac{\sigma_m}{2 } m^2 s^{\cal cpt } ( x ) s(x ) + \sigma_{\mu } { \mu}^2 \left [ s^{\cal cpt } ( x ) s(x ) \right]^2 \right\ } \nonumber \\ & & { } - \sigma_k g^{\mu \alpha } g^{\nu \beta } \left [ \frac{\partial s ( x)}{\partial x^{\alpha } } \right]^{\cal cpt } \frac{\partial s ( x)}{\partial x^{\beta}}\end{aligned}\ ] ] setting @xmath48 , the hamiltonian is given by @xmath49^{\cal cpt } \frac{\partial s(x)}{\partial x^{\beta } } \right . + \frac{\sigma_m}{2 } m^2 \left [ s(x ) \right]^{\cal cpt } s(x ) \nonumber \\ & & \left . { } + \sigma_{\mu } { \mu}^2 \left\ { \left [ s(x ) \right]^{\cal cpt } s(x ) \right\}^2 \right ) - \sigma_k \left [ \frac{\partial s(x)}{\partial x^{0 } } \right]^{\cal cpt } \frac{\partial s(x)}{\partial x^{0 } } \nonumber \\ & = & - \frac{\sigma_k } { 2 } \left [ \frac{\partial s(x)}{\partial x^{0 } } \right]^{\cal cpt } \frac{\partial s(x)}{\partial x^{0 } } - \frac{\sigma_k } { 2 } \left [ \nabla s(x ) \right]^{\cal cpt } \nabla s(x ) \nonumber \\ & & { } - \frac{\sigma_m}{2 } m^2 \left [ s(x ) \right]^{\cal cpt } s(x ) - \sigma_{\mu } { \mu}^2 \left\ { \left [ s(x ) \right]^{\cal cpt } s(x ) \right\}^2 \label{eq : ham}\end{aligned}\ ] ] all four terms in ( [ eq : ham ] ) include a ( positive definite ) @xmath26 norm . by analogy with ordinary @xmath16 theory , the first two terms and the last must be positive if we are to have an energy spectrum that is bounded below . we therefore set @xmath50 . the sign of the @xmath44 term will be determined by the requirement of conformal invariance when we go to frw space . the equation of motion for @xmath1 is obtained in the usual way , by requiring the action to be stationary under small variations , @xmath51 . the appearance of @xmath52 in the action is unusual , and the variation requires some care ; details are given in the appendix . the equation of motion is given in ( [ eqapp : motion ] ) . writing @xmath50 , this becomes @xmath53 s(x ) = 0 \label{eq : motion}\ ] ] comparing this with mannheim s equation of motion , ( [ eq : s ] ) , we infer that we go over to frw space by setting @xmath54 . since @xmath55 , we must set @xmath56 . to maintain mannheim s notation as far as possible , we will here define @xmath57 , giving the equation of motion @xmath58 s(x ) = 0 \label{eq : motion2}\ ] ] following mannheim , we assume that @xmath1 develops a constant vacuum expectation value , calculated by setting the derivative term in ( [ eq : motion2 ] ) equal to zero . the result is the same as ( [ eq : svev ] ) , @xmath59 , where , as before , @xmath55 and @xmath60 . we can now use ( [ eq : imink ] ) to write the action in frw space : @xmath61^{\cal cpt}_{;\mu } \left[s ( x ) \right]_{;\nu } \right . \nonumber \\ & & \left . { } - \frac{r^{\sigma}_{\;\;\sigma}}{12 } s^{\cal cpt}(x ) s(x ) + \lambda_{\rm m } \left [ s^{\cal cpt}(x ) s(x ) \right]^2 \right\ } \label{eq : ifrw } \end{aligned}\ ] ] from this we can infer the energy - momentum tensor analogous to ( [ eq : t_mann ] ) . for a cosmological model comparable to mannheim s only the last two terms are of interest , which are @xmath62 + g^{\mu \nu } \lambda_{m } \left [ s^{\cal cpt}(x ) s(x ) \right]^2 \label{eq : t_mann2}\ ] ] replacing @xmath32 by its vacuum expectation value , @xmath3 , we get @xmath63 the important point is that the signs of both these terms are reversed in comparison to ( [ eq : t_mann ] ) . mannheim points out ( m6 , section 10 ) that because the frw space is conformally flat , the cosmological equation of motion ( cem ) reduces to @xmath64 @xmath65 is just the sum of ( [ eq : t_mann3 ] ) and @xmath66 , the contribution from ordinary matter ( fermion fields , electromagnetic radiation , etc . ) . so our cem , analogous to m6 ( 222 ) , becomes @xmath67 or , as in m6 ( 223 ) @xmath68 with @xmath69 . the @xmath4 term defines the effective gravitational constant in the theory . we find @xmath70 analogous to m6 ( 224 ) , but wth a positive sign . of particular interest , since it permits an analytic solution , is a model containing radiation only ( m6 , section 10.2 ) . a corresponding solution exists for the present model also ; let us see how it differs . the equation analogous to m6 ( 226 ) is ( with @xmath71 ) : @xmath72 @xmath73 @xmath74 since @xmath75 for the open geometry of the mm , all three terms in ( [ eq : omega ] ) are positive , in contrast to m6 , where @xmath76 is negative . write the solution as in m6 ( 230 ) , with @xmath77 : @xmath78 @xmath79 the first term on the right of ( [ eq : r ] ) is negative , not positive as in the mm . this means that @xmath80 must start from zero , as in the standard cosmology , not from some finite value , as in the mm . this is illustrated in figure [ fig : ccpgfigure1a_1b ] . both curves were drawn using the formula ( [ eq : r ] ) , but in the upper graph @xmath81 , while in the lower graph @xmath82 . the origin of @xmath83 is conventionally shifted in the lower graph so that @xmath84 is at point b , where @xmath85 . two curves drawn using ( [ eq : r ] ) , but with different values of parameters . ( a ) @xmath86 , @xmath87 , @xmath88 . @xmath80 starts from a non - zero value at @xmath84 ( point a ) . ( b ) @xmath86 , @xmath87 , @xmath89 . part of the curve now lies below the horizontal axis , and is non - physical . the origin of @xmath83 is conventionally shifted to b , where @xmath85 . ] the present model has two features that are not present in conventional cosmology based on einstein s equations . first , the model includes a scalar field that is essentially massless . the non - zero vacuum expectation value of this field is essential , but we have ignored any excitations . this may be permissible because the field couples very weakly to normal matter , and is difficult to observe , or it may undergo a spontaneous transition that renders it massive . second , mannheim uses ( [ eq : motion3 ] ) for his basic cosmological equation , rather than the more complete one that results from the conformal action , m6 ( 188 ) : @xmath90 where @xmath91 is the weyl tensor , defined in m6 ( 107 ) , ( 108 ) and ( 185 ) . mannheim justifies the neglect of @xmath91 by observing that a frw metric is conformally flat , and in such a space @xmath92 . this is true , but if we are to include perturbations to the metric ( as , for example , in the study of anisotropies of the cmb ) then this neglected term may become important . we have pointed out two flaws in mannheim s conformal cosmological model . @xmath2 the model predicts , apparently unambiguously , a negative value for the effective gravitational constant , @xmath0 . the model involves a scalar field , @xmath36 , that satisfies a conformally invariant equation of motion and develops a vacuum expectation value , @xmath3 . the values of the parameters that are needed to satisfy observations lead to a `` wrong sign @xmath22 theory '' , with a hamiltonian that has a spectrum that is unbounded below . we have attempted to apply the techniques appropriate for such hamiltonians to this cosmological problem , restricting ourselves to the classical limit of field equations that are still imperfectly understood . in this limit , using assumptions that appear reasonable , we find both a positive value for @xmath0 and a positive definite spectrum for the hamiltonian . our derivation depends on one simple observation , the change of sign as we go from ( [ eq : deltaika ] ) to ( [ eq : deltaikb ] ) . the derivation presented here will remain conjectural until progress is made in two main directions : @xmath2 the techniques that have been successfully applied to @xmath16 quantum mechanics will have to be developed to cover the corresponding quantum field theory ; this seems not to have been achieved at this time @xcite . in particular , the form ( [ eq : cptdef ] ) for @xmath33 must be shown to be appropriate at the classical level . the various manipulations we have employed are suitable for minkowski space , but more detailed investigations are needed to show whether they can legitimately be extended to a frw space . if , on the other hand , we can accept the present model as viable , without first filling in these important gaps in our understanding , then we have to face the difficult question : how can we conclusively distinguish this model from the conventional one ? the department of physics at washington university , in particular carl bender , have given invaluable support to this retired colleague . the action is defined in ( [ eq : imink ] ) . we will start with the simplest term , @xmath93_{\forall \rho : z^{\rho } = -y^{\rho } } s(x ) \label{eq : ima}\end{aligned}\ ] ] let @xmath32 vary by a small amount @xmath94 . then @xmath95 will vary by @xmath96 $ ] . the variation of @xmath97 will be the sum of two terms , @xmath98 and @xmath99 : @xmath100_{\forall \rho : z^{\rho } = -y^{\rho } } \delta s(x ) \label{eq : deltaim1}\ ] ] @xmath101_{\forall \rho : z^{\rho } = -y^{\rho } } s(x ) \label{eq : deltaim2}\ ] ] take the complex conjugate of ( [ eq : deltaim2 ] ) , and let @xmath38 , @xmath39 : @xmath102 so that @xmath103 \nonumber \\ \fl & = & \int { \rmd}^4 x \int { \rmd}^4 y \ , ( -g)^{1/2 } \sigma_m m^2 c(x^{\mu } - y^{\mu } ) { \rm re } \left\{\left [ s^{*}(z ) \right]_{\forall \rho : z^{\rho } = -y^{\rho } } \delta s(x ) \right\ } \label{eq : deltaimb}\end{aligned}\ ] ] the @xmath104 term in the action can be treated in the same way , to give @xmath105 { \rm re } \left\ { \left [ s^ { * } ( z ) \right]_{\forall \rho : z^{\rho } = - y^{\rho } } \delta s ( x ) \right\}\end{aligned}\ ] ] calculating the variation of the kinetic term starts just as with conventional lagrangians . we imagine a variation @xmath106 , and convert this to a variation of @xmath36 by an integration by parts , discarding a surface term . recalling that we are working in minkowski space , where the metric tensor is constant , we get @xmath107^{\cal cpt } \delta s ( x)\ ] ] using ( [ eq : cptdef ] ) we write this as @xmath108_{\forall \rho : z^{\rho } = -y^{\rho } } \delta s ( x ) \\ \fl & = & \int { \rmd}^4 x \int { \rmd}^4 y \ , ( -g)^{1/2 } \frac{\sigma_k } { 2 } g^{\mu \nu } \left [ \frac { \partial } { \partial y^{\nu } } c(x^{\mu } - y^{\mu } ) \right ] \nonumber \\ \fl & & \times \ , \left [ \frac{\partial s^ { * } ( z)}{\partial z^{\mu } } \right]_{\forall \rho : z^{\rho } = -y^{\rho } } \delta s ( x)\end{aligned}\ ] ] now integrate by parts and discard a surface term : @xmath109_{\forall \rho : z^{\rho } = -y^{\rho } } \delta s ( x ) \label{eq : deltaika } \\ \fl & = & \int { \rmd}^4 x \int { \rmd}^4 y \ , ( -g)^{1/2 } \frac{\sigma_k } { 2 } g^{\mu \nu } c(x^{\mu } - y^{\mu } ) \left [ \frac{\partial^2 s^ { * } ( z)}{\partial z^{\mu } \partial z^{\nu } } \right]_{\forall \rho : z^{\rho } = -y^{\rho } } \delta s ( x ) \label{eq : deltaikb } \end{aligned}\ ] ] the passage from ( [ eq : deltaika ] ) to ( [ eq : deltaikb ] ) is best illustrated by an example . suppose @xmath110 where @xmath111 is some fixed vector . then @xmath112_{\forall \rho : y^{\rho } = - x^{\rho } } = n ( -1)^{n } \left ( k_{\sigma } x^{\sigma } \right)^{n-1 } k_{\mu } k_{\nu } \label{eqapp : difex}\ ] ] and @xmath113_{\forall \rho : y^{\rho } = -x^{\rho } } & = & \left [ n \left ( k_{\sigma } y^{\sigma } \right)^{n-1 } k_{\mu } k_{\nu } \right]_{\forall \rho : y^{\rho } = - x^{\rho } } \nonumber \\ & = & n ( -1)^{n-1 } \left ( k_{\sigma } x^{\sigma } \right)^{n-1 } k_{\mu } k_{\nu } \label{eqapp : difex2}\end{aligned}\ ] ] note the change in sign from ( [ eqapp : difex ] ) to ( [ eqapp : difex2 ] ) ; it implies that the variation of the kinetic term results in @xmath114_{\forall \rho : z^{\rho } = -y^{\rho } } \delta s ( x ) \right\ } \label{eq : deltaikc}\ ] ] now use @xmath115 for arbitrary ( possibly complex ) variations @xmath116 to get the equation of motion @xmath117 s(x ) = 0 \label{eqapp : motion}\ ] ]
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the conformal cosmological model presented by mannheim predicts a negative value for the effective gravitational constant , @xmath0 .
it also involves a scalar field , @xmath1 , which is treated classically . in this paper
we point out that a classical treatment of @xmath1 is inappropriate , because the hamiltonian is non - hermitean , and the theory must be developed in the way pioneered by bender and others .
when this is done , we arrive at a hamiltonian with an energy spectrum that is bounded below , and also a @xmath0 that is positive .
the resulting theory closely resembles the conventional cosmology based on einstein relativity .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in the accordance with the conventional regge - gribov approach , the one - pomeron contribution to the differential cross section of shdid can be expressed at @xmath25 in the form : @xmath26 where @xmath27 and @xmath28 are cms total interaction energy , total cross section , transverse momentum transferred , and invariant masses of final diffractively excited states respectively , @xmath29 is three - pomeron vertex and @xmath30 is the pomeron trajectory ; the parameter @xmath31 is to be chosen to single out diffraction processes from other ones [ 7 ] . since the mean slope of the pomeron trajectory is the only dimensional parameter which can be responsible for the decrease of the function @xmath29 as @xmath32 is increased , the domain where @xmath29 is expected to be nearly constant is estimated as @xmath33 where @xmath34 is an effective mean value of the derivative @xmath35 there which is reasonably evaluated to be @xmath36 . it is why this domain is expected to be remarkably large , from @xmath37 to @xmath38 or even larger ( it has been observed long ago by comparison of the elastic and single inelastic diffraction differential cross sections that @xmath39 at @xmath40 [ 7 ] , wherefrom , in particular , a rather slow @xmath41-dependence of double inelastic diffraction differential cross section at @xmath42 follows ) . the double inelastic diffraction is the only type of hadron interaction which is expected to exhibit such slow transverse momentum dependence . at still larger values of squared 4-momentum transferred pomeron is expected to be dissolved to its constituents [ 6 ] that begin to interact independently , so that the `` normal '' qcd regime @xmath43 is to be approached gradually . in what follows the logarithmic dependence on @xmath41 and rather ambiguous but definitely slow decrease of @xmath44 in the right - hand side of eq.(1 ) are accounted on the average as @xmath45 . the rough estimate of screening corrections to the one - pomeron shdid scattering amplitude @xmath46 associated with diagrams depicted in fig.3 shows that @xmath47 , @xmath48 being the corrected amplitude . it is reasonable to adopt @xmath49 and enhance the above correction ( i.e.,to multiply the denominator in eq.(2 ) ) by the phenomenologically approved ( for forward elastic scattering amplitude ) factor about 1.5 , accounting the shadowing by the inelastic intermediate states . then the corrected shdid amplitude is expected to be @xmath50 and the corresponding differential cross section is @xmath51 after integration of eq.(1 ) over the region @xmath52 one obtains the total cross section of shdid @xmath53 if one chooses a reasonable values @xmath54 , @xmath55 and the experimental value of @xmath56 , @xmath57 , then the fraction of shdid is expected to be @xmath58 and 0.10 at @xmath59 and @xmath60 respectively . it can be several times less or larger , since the above estimate is rather rough , but its smooth logarithmic threshold - like energy increase is independent of the choice of parameters . it seems reasonable to expect that hadronization of diffractively excited final states produced by shdid is dominated by mechanism of string rupture as shown in fig.4 , string been formed between scattered colored hadron constituent ( quark , diquark or gluon ) and remnant of the same hadron . any alternative string configuration would be unfavorable since it implies formation of some strings of a very high energy ( it is worthy to mention that diffractively produced state associated with target particle was always out of the game in cosmic ray experiments under discussion because it is never seen within the area of observation ; it is why the projectile inelastic diffraction only is thought of throughout the paper ) . at the same time , transferred momentum @xmath61 is insufficiently large for the fragmentation mechanism of hadronization to prevail . let us consider the above string in its own cms and adopt that secondary particle rapidity and transverse momentum distributions in pomeron - proton interaction is similar to that in real hadron one at cms energy @xmath62 ( as to the rapidity distribution , it is supported by the well known result of ua4 collaboration [ 8 ] ) . since what is observed is nothing else , than transverse plane projection of the picture which is resulted from its rupture , it becomes obvious that the typical ratio of a secondary transverse momentum projection normal to reaction plane ( i.e. , to the plane of draft ) to `` transverse momentum string length '' ( i.e. to ls relative transverse momentum of leading particles oppositely directed in string cms ) is about @xmath63 where @xmath64 is mean transverse momentum of secondaries in hadron interactions , and mean leading particle energy is experimentally proved to be about half of incident particle one . at @xmath65 this ratio is about 0.13 . the only point what remains to be discussed to compare the above consideration to the experimental data is an obvious estimate of the role of atmospheric cascade . since the atmosphere thickness above the altitude where the calorimeter is mounted corresponds to about 3.5 nuclear mean free paths , the probability of at least one shdid collision is about @xmath66 at @xmath67 . if it does happen , then the subsequent soft collisions can not , most probably , blur essentially the target plane picture it initiates , especially for energy distinguished cores . it is why the additional assumption suggested by experimenters [ 2 ] seems to be not necessary , that alignment is caused by some peculiarities of the lowest nuclear collision above the chamber only . at the same time , the threshold - like dependence of alignment on core energies is associated , may be , with the violating role of nuclear cascade . thus , the main puzzling experimental features of alignment phenomenon , namely , the fraction of alignment events about ( 20 - 40)% and the ratio of mean value of normal to reaction plane projection of core transverse momentum to maximal value of core relative transverse momenta ( @xmath68 ) ( string `` half - thickness '' to its `` length '' in transverse momentum space ) are compatible qualitatively with the above theoretical consideration ( 30% and 0.13 respectively ) , if one adopts that each core is originated ( due to electromagnetic cascade ) from a hadron created along with string rupture . the threshold - like dependence of shdid cross section on interaction energy can elucidate why the phenomenon has not been noticed at lower energies ( especially , accounting a poor statistics and other ambiguities of cosmic ray experiments ) . however , this point as well as some other features of the phenomenon , such as its threshold - like dependence on core energies , core energy distribution , their energy sequence along the alignment line , etc . , needs both the enrichment of statistics and mc simulation of cascade and shdid collisions themselves ( especially , accounting that hadrons of different masses can be produced at the end of string and along its length ) which are in progress . unfortunately , it is rather questionable , whether an attempt to observe the alignment phenomenon will be undertaken in accelerator experiments soon . 99 pamir collaboration , `` analysis of structure of halo in families with energy @xmath69 '' , proceedings 5th international symposium on very high energy cosmic ray interactions , lodz , 1988 , v. contributed papers , p. 9 . amineva , g.f . fedorova et al . , `` alignment of increased background region in gamma - hadron superfamilies '' , proceedings 6th international symposium on very high energy cosmic ray interactions , tarbe , 1990 v. contributed papers , p. 264 ; i.p . ivanenko , v.k . kopenkin , a.k . managadze and i.v . rakobolskaya , pisma jetf , 1992 , v. 56 , p. 192 . f. halzen and d.a . morris , phys . , 1990 , v. d42 , p. 1435 . ua1 collaboration , g. arnison et al . lett . , 1985 , v.158b , p. 494 a. managadze , private communication . a. mironov and i. royzen , sov . fiz . , 1988 , v. 47 , p. 1125 ; v. 48 , p. 194 . a. alberi and g. goggi , phys . rep . 1984 , v. 74 , no . ua4 collaboration , d. bernard et al . lett . , 1986 , v. 166b , p.459 fig.1 . the example of target plane picture with energy distinguished cores for event with alignment , @xmath70 ; figures stand for energy in tev ( already multiplied by factor 3 for hadrons);andor stand for electromagnetic halo and hadrons of high energy respectively . other particles of the family are marked as(@xmath1 -quanta ) and ( hadrons ) . one - pomeron exchange approximation to shdid . wavy lines refer to pomeron exchange , @xmath71 and @xmath28 are invariant masses of diffractively excited states , q is 4-momentum transferred , @xmath72 is triple - pomeron vertex function .
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an explanation of the puzzling alignment effect observed in cosmic ray experiments is suggested * theoretical approach to alignment phenomenon * + * i. royzen * + p.n .
lebedev physical institute of russian academy of sciences 53 leninsky prospect , 117924 moscow , russia + few years ago the observation has been made [ 1 ] in cosmic ray experiments that the alignment of the main energy fluxes along a straight line in target ( transverse ) plane exceeds significantly the background level .
more precisely , at superhigh energies of initial particle ( @xmath0 ) the secondary particle superfamilies detected by deep lead x - ray emulsion chamber appeared to be situated almost along straight line in target plane ( fig.1 ) .
the coplanar scattering of such a type was so surprising that an attempt has been made to revise the result but instead they were confirmed with much better confidence level [ 2 ] .
the analysis of the alignment effect for 74 high energy @xmath1-families induced by hadrons above and within the chamber has been carried out .
their energies energies are selected to be @xmath2 ( hadron energies being restored , accounting that the energy of induced @xmath1-family is about @xmath3 of the hadron energy it is originated from ) .
this analysis suggested that superfamily production happened predominantly rather low above the chamber ( at the altitude @xmath4 , since it seemed that nuclear - electromagnetic cascade development would blur alignment , if several interactions contributed ) .
it confirmed a coplanar scattering and scaling - like fragmentation spectrum of energy distinguished cores .
the alignment parameter @xmath5 , @xmath6 is used as the alignment criterion where @xmath7 stands for a number of centers of highest energy and @xmath8 is the angle between the two - dimensional vectors @xmath9 and @xmath10 in target plane , an event being recognized to have alignment , if @xmath11 .
actually , events with @xmath12 were chosen only because of too high statistical background for @xmath13 and rather poor statistics for @xmath14 .
the threshold - like behavior of the effect has been observed : no alignment at @xmath1-family energies @xmath15 , then its gradual increase within energy range @xmath16 to manifest itself finally in ( 20 - 40)% of total number of events .
14 events with @xmath17 have been observed , exhibiting most striking alignment structure ( @xmath18 ) .
core transverse momentum @xmath19 was estimated by rough relation @xmath20 , @xmath21 being the distance of a spot from the interaction axis .
the mean ratio of value of maximal relative core transverse momentum to its normal to the alignment line projection ( in target plane ) @xmath22 is @xmath23 .
no other peculiarities of alignment events compared to `` usual '' cascade have been noticed .
the first attempt of theoretical consideration of the above alignment phenomenon has been made by f. halzen and d.a .
morris [ 3 ] , whose approach was based on the assumption that semihard gluon jets is a feature of all events at energies above @xmath24 .
it was shown that within this approach the cosmic ray observations were associated probably with the jet alignment in three - jet events observed already in the collider experiment [ 4 ] .
i would like to suggest an alternative treatment which makes it possible to understand many features of cosmic ray alignment observations quite naturally , including the threshold - like energy behavior and fraction in extensive atmospheric showers as well as the typical projections of core transverse momenta to the alignment line and normal to it , and allowing for events with more , than four cores aligned , that have been extracted recently from cosmic ray data [ 5 ] .
the main point of the approach under consideration is that the alignment events are assumed to be associated with semihard double inelastic diffraction ( shdid ) of hadrons [ 6 ] .
let us trace them step - by - step .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
quantum simulators , as first proposed by feynman @xcite , which are devices built to evolve according to a postulated quantum hamiltonian and thus `` compute '' its properties , are one of the hot ideas which may provide a breakthrough in many - body physics . while one must be aware of possible difficulties ( see , e.g. , @xcite ) , impressive progress has been achieved in recent years in different systems employing cold atoms and molecules , nuclear magnetic resonance , superconducting qubits , and ions . the latter are extremely well controlled and already it has been demonstrated that , indeed , quantum spin systems may be simulated with cold - ion setups @xcite . quantum spin systems on a lattice constitute some of the most relevant cases for quantum simulations as there are many instances where standard numerical techniques to compute their dynamics or even their static behavior fail , especially for two- or three - dimensional systems . chronologically the first proposition to use trapped ions to simulate lattice spin models came from porras and cirac @xcite who derived the effective spin - hamiltonian for the system , @xmath1-\mu\sum\limits_{i } s_{i}^{z } , \label{hamspin}\end{aligned}\ ] ] where @xmath2 is the chemical potential ( which in this case acts as an external magnetic field ) , @xmath3 is the interaction strength , and @xmath4 are the spin operators at site @xmath5 . all the long - ranged interactions fall off with a @xmath6 dipolar decay , which for the ions is due to the fact that they are both induced by the same mechanism , namely lattice vibrations mediated by the coulomb force @xcite . hamiltonian is a dipolar xxz model where the ratio of hopping to dipolar repulsion can be scaled by @xmath7 @xcite . another route to simulate quantum magnetism of dipolar systems , using the rotational structure of the ultracold polar molecules , has been proposed in refs . @xcite . in this case , the xy ( or tunneling ) term is restricted to nearest neighbors ( nn ) and only the zz ( or interaction ) term is dipolar . precisely this long - range ( lr ) dipolar character makes this system interesting and challenging . dipolar interactions introduce new physics to conventional short - range ( sr ) systems . for example , for soft - core bosons , the system with long - range interactions and short - range tunneling , can apart from mott - insulating ( mi ) , superfluid ( sf ) , and crystal phases host a haldane - insulating phase @xcite . this is characterized by antiferromagnetic ( afm ) order between empty sites and sites with double occupancy , with an arbitrarily long string of sites with unit occupancy in between . another case where dipolar interactions play a crucial role is the appearance of the celebrated supersolid . the extended bose - hubbard model ( i.e. , with nn interactions ) with soft - core on - site interactions shows stable supersolidity in one - dimensional ( 1d ) @xcite and square lattices @xcite , where in the 1d case a continuous transition to the supersolid phase occurs , in contrast to the first - order phase transition in two - dimensional ( 2d ) lattices . long - ranged dipolar interactions on the other hand allow for the appearance of supersolidity even in the hard - core limit @xcite . in this limit , dipolar interactions give rise to a large number of metastable states @xcite ( for a review see @xcite ) . by tuning the direction of the dipoles , incompressible regions like devil s staircase structures have been predicted in ref . @xcite . while being interesting , long - range interactions make computer simulations of such systems quite difficult . on the other hand , since ions in optical lattices may be extremely well controlled , they form an ideal medium for a quantum simulator . in our previous paper @xcite , we considered both the mean - field phase diagram for the system ( [ hamspin ] ) as well as a 1d chain using different quasi - exact techniques , such as density matrix renormalization group ( dmrg ) , and exact diagonalization for small systems . here , we would like to concentrate on frustration effects on a 2d triangular lattice using a quantum monte carlo ( qmc ) approach . for the system consisting of spin - half particles ( as , e.g. , originating from two internal ion states in the porras cirac @xcite proposal ) the spins can be mapped onto a system of hard - core bosons , using the holstein primakoff transformation , @xmath8 , @xmath9 , and @xmath10 , where @xmath11 @xmath12 are the creation ( annihilation ) operators of hard - core bosons and @xmath13 is the number operator at site @xmath5 . these bosons obey the normal bosonic commutation relationships , @xmath14=1 $ ] , but are constrained to only single filling , @xmath15 . this is the limit when the on - site repulsion term @xmath16 goes to infinity in the standard bose - hubbard model . in this representation , a spin up particle is represented by a filled site and a spin down particle by an empty site . for convenience , we will mostly use the language of hard - core bosons in this paper . before presenting the results for the 2d triangular lattice , let us briefly review the behavior of the system for a 1d chain of bosons as obtained in ref . in order to form some intuition about possible physical effects due to the lr interaction and tunneling , we now discuss briefly the ground - state phase diagram of the hamiltonian in 1d . for hard - core bosons , which is the topic of this article , the one - dimensional version of the hamiltonian has been thoroughly investigated in the past @xcite ( see also @xcite for the special cases @xmath17 and @xmath18 ) . for reference , we reproduce in fig . [ fig : phasediagram1d ] the phase diagrams for the system with dipolar interaction and nn tunneling , as well as the system with both the interaction and the tunneling terms dipolar . phase diagram of the 1d system with dipolar interactions and * ( a ) * nearest - neighbor tunneling , * ( b ) * dipolar tunneling . the phases are labeled according to density matrix renormalization results , while the actual data shown comes from the infinite time evolving block decimation method ( with interactions truncated at next - to - nearest - neighbors ) , both from ref . @xcite . along the black line , there exists a devil s staircase of crystal states . at finite tunneling , for ( a ) , these spread into conventional insulating states , while for ( b ) , they form quasi - supersolids . the light blue lines near the center of the plots sketch the crystal lobes at @xmath19-filling . their cusp - like structure is typical for 1d systems . in 2d , they are expected to be rounded off , similar to mott - lobes of the bose hubbard model @xcite . for nearest - neighbor tunneling , ( a ) , the superfluid ( sf ) phases can be mapped into one another , while for nearest - neighbor dipolar tunneling , ( b ) , they are distinct on the ferromagnetic ( fm , @xmath20 ) and the antiferromagnetic side ( afm , @xmath21 ) . note also that in ( b ) frustration leads to an asymmetry between @xmath20 and @xmath21 . ] at zero tunneling , the ground states are periodic crystals where to minimize the dipolar interaction energy occupied sites are as far apart as possible for a given filling factor @xcite . for finite 1d systems and very small tunneling such a situation persists as exemplified in @xcite . for infinite chains in 1d , every fractional filling factor @xmath22 is a stable ground state for a portion of @xmath2 parameter space . the extent in @xmath2 decreases with @xmath23 , since at large distances the dipolar repulsion is weak and thus can not efficiently stabilize crystals with a large period . this succession of crystal states is termed the devil s staircase . this name derives from its surprising mathematical properties , challenging naive intuitions about continuity and measure : since all rational fillings are present , it is a continuous function ; moreover , its derivative vanishes almost everywhere ( i.e. , it is non - zero only on a set of measure zero ) and still it is not a constant , but covers a finite range . at finite tunneling , the crystals spread into lobes similar to the mott lobes of the bose hubbard model . if the tunneling is only over nns , these mott lobes are not sensitive to the sign of the tunneling and they form standard insulating states with diagonal long - range order ( lro ) and off - diagonal short - range order ( fig . [ fig : phasediagram1d]a ) . for long - range dipolar tunneling , the extent of the lobes is asymmetric under sign change @xmath24 : frustration effects stabilize the crystal states for @xmath20 , while the ferromagnetic ( fm ) tunneling for @xmath21 destabilizes them ( fig . [ fig : phasediagram1d]b ) . moreover , the crystal states acquire off - diagonal correlations which follow the algebraic decay of the dipolar interactions @xcite . this coexistence of diagonal and off - diagonal ( quasi-)lro turns the crystal states into ( quasi-)supersolids . the occurrence of such phases for hard - core bosons in 1d is truly exceptional , since the systems where it appears consist typically of soft - core bosons @xcite , or two - dimensional lattices @xcite . furthermore , this 1d quasi - supersolid defies luttinger - liquid theory , which typically describes 1d systems very well , even in the presence of dipolar interactions @xcite . where luttinger - liquid theory applies , diagonal and off - diagonal correlations decay algebraically with exponents which are the inverse of one another . therefore , if the diagonal correlations show lro , the corresponding exponent is effectively @xmath25 , and the exponent for the off - diagonal correlations is infinite , describing an exponential decay . in our case , this exponent remains finite in the quasi - supersolid phase and the above relationship clearly does not hold . at even stronger tunneling strengths , the crystal melts and the system is in a sf phase . the lr tunneling and interactions influence the correlation functions at large distances and therefore also modify the character of this phase @xcite . these results show that in this system the dipolar interactions considerably modify the quantum - mechanical phase diagram . in higher dimensions , we can expect the influence of long - range interactions to be even stronger , which makes extending these studies to a two - dimensional lattice highly relevant . for example , one can expect that if quasi - supersolids appear already in 1d the long - range tunneling has a profound effect on the stability of two - dimensional supersolids , which appear in triangular lattices at the transition between crystal and superfluid phases @xcite . also , the frustration effects already observed in the 1d system should be much more pronounced in the triangular lattice , simply due to the increased number of interactions , our method of choice , quantum monte carlo , fails due to the sign problem , invoked by frustration . see section [ cha : results ] for details . ] . further , such an analysis is especially relevant at finite temperature . in fact , a recent work scanned the phase diagram of hamiltonian along the line @xmath26 in a square lattice @xcite . there , the authors found that above the superfluid on the fm side ( i.e. , @xmath20 ) , the continuous u(1 ) symmetry of the off - diagonal correlations remains broken even at finite temperatures . thus , the long - range nature of the tunneling leads to a phase which defies the mermin wagner theorem @xcite . for these reasons , we can expect intriguing effects of the long - range tunneling for the two dimensional triangular lattice , to which we turn now . to analyze the system of dipolar bosons on the two dimensional triangular lattice , we employ quantum monte carlo ( qmc ) simulations at a finite but low temperature . specifically , we study rhombic lattices with periodic boundary conditions and @xmath27 sites , with @xmath28 . we will in particular thoroughly investigate the wigner crystal at @xmath19-filling . the triangular lattice is frustrated even with only nn interactions . when any longer ranged interactions are added , this only increases the frustration , which also makes qmc calculations more difficult . also , in frustrated systems @xcite and systems with long - range interactions @xcite typically many metastable states appear , making finding the ground state a somewhat difficult task . to avoid complications with the sign problem , caused by negative probabilities in qmc codes , we will take only negative values of @xmath7 under consideration . note that this sign problem appears only for the long - ranged xy interactions , while the zz ( ising - like ) interactions are despite frustration sign - problem free . in this study , we compare the hamiltonian with both long - range interactions and hoppings ( lr lr ) , with long - ranged dipolar interactions but nn hopping ( lr sr ; relevant for polar molecules @xcite ) , as well as with hopping and interactions truncated at nns ( sr sr ; this is the nn xxz model , relevant to magnetic materials with planar anisotropy in their couplings ) , this amounts to including interactions up to distances of the fifth - nearest neighbor . ] . each of these systems will display different crystal , superfluid , and supersolid regions . comparing these cases will give valuable insight into the influence of the long - range terms . all the calculations were performed using the worm algorithm of the open source alps ( algorithms and libraries for physics simulations ) project @xcite . this algorithm , first created by n. prokofev , works by sampling world lines in the path integral representation of the partition function in the grand canonical ensemble @xcite . the first calculation , which creates the motivation for the rest of the paper , is to look at the case of vanishing tunneling and temperature for each system ( lr lr , lr sr , and sr sr ) . here , similar to the 1d devil s staircase , at vanishing temperature a series of insulating crystal states is expected to cover the entire range of @xmath29 . since we are interested in finite temperature results , we set @xmath30 which should still be low enough to reflect the characteristics of the ground - state phase diagram and look for plateaus in the density . we distinguish short- and long - ranged zz interactions . from fig . [ devil ] , left panel , we can see that the only plateau ( besides the completely filled system ) that appears is at @xmath31 ( corresponding either to @xmath19 boson filling or in spin terms , a lattice with @xmath19 of the spins oriented up and @xmath32 oriented down ) for both short- and long - ranged interactions . scaling the system size from @xmath33 to @xmath34 causes no change for the short - ranged interactions , and minimal change for the long - ranged ones . the key difference is in the size and position of the short - ranged and the long - ranged plateaus . for short - ranged interactions this plateau is larger and centered around @xmath35 , while the long - ranged interactions have a smaller plateau centered around @xmath36 . the finite width of these plateaus suggests that a @xmath19-filling wigner crystal persists also for some finite @xmath7 . the right panels of fig . [ devil ] show how the plateau shrinks with increasing temperature , as well for sr interactions ( top right panel ) as for lr interactions ( bottom right panel ) . in the latter case , in fact , by @xmath37 the plateau has completely disappeared . we can also notice that at @xmath38 there are signs of some of the other plateaus , most noticably the @xmath39-filling plateau . the rest of the paper will focus on the @xmath19-filling crystal lobes and their properties . and @xmath30 , where the density shows a single plateau for @xmath19-filling . for sr interactions ( solid blue line ) , different curves for @xmath40coincide , and for lr interactions ( solid red : @xmath33 , dashed red : @xmath41 and @xmath42 ) the size dependence is small . the panels on the right are at fixed @xmath34 and @xmath17 for different temperatures @xmath43 ( from dotted to dashed to solid ) . both for sr ( top right ) and lr interactions ( bottom right ) , the @xmath19-filling plateau shrinks as t increases . , scaledwidth=100.0% ] we now introduce a finite tunneling by choosing @xmath20 in our hamiltonian , and study the properties around the @xmath19-filling crystal . we calculate the density and the superfluid fraction . the superfluidity is measured using the winding numbers calculated from the movement of the worms in the qmc code . in order to get this value the system must have periodic boundary conditions so that the world lines can properly `` wind '' around the system . the superfluid fraction is @xmath44 where @xmath45 is the winding number fluctuation of the world lines and @xmath46 is the inverse temperature . figure [ lobes ] shows the results for the boson density of a @xmath33 triangular lattice at @xmath30 . for all types of interactions we see that as @xmath7 increases in absolute value the @xmath47 plateau shrinks , because larger @xmath48 increases the ratio of hopping to dipolar interactions . this introduces more kinetic energy and the crystal melts into a superfluid . it can be seen for the sr sr system that the boson density lobe extends to @xmath49 , while for the lr sr interactions it ends at @xmath50 , and finally for lr lr interactions the lobe is smaller still , only going out to @xmath51 . the behavior is explained by the fact that the increased amount of interactions cause a quicker melting of the lobe . lobes , evidenced by particle density ( top row ) and the superfluid fraction ( bottom row ) that arise under varying the ratio , @xmath7 , and the chemical potential , @xmath29 ( data for @xmath33 ) . the left column corresponds to the sr sr system , the middle one shows the lr sr system and the right column depicts the lobes for the lr lr system . long - ranged dipolar interactions decrease the lobes in @xmath29 due to the appearance of devil s staircase like features , and long - range tunneling decreases the extent of the lobe in @xmath7.,scaledwidth=100.0% ] a second major observation is the shift in the position of the lobes . while the short - ranged lobe exists approximately for @xmath52 , the long - ranged lobes lie generally on the interval @xmath53 . for a system like this at @xmath54 , one expects the @xmath47 lobe to exist on the range @xmath55 with a mirrored image of the @xmath56 lobe at @xmath57 . the existence of the two identical lobes is explained by particle - hole symmetry @xcite . the lobes are separated with a kind of mixed solid in between ( with coexistence of @xmath32- and @xmath19-filling regions ) the existence of this region may be caused by several phenomena . it could either be an effect of the finite temperature as in ref . @xcite , or due to the existence of many metastable states caused by a devil s staircase like behavior , similar to what was observed in ref . @xcite . again referring to the @xmath54 phase diagram , we expect that there is a region of supersolidity that extends in between the lobes and goes all the way to their base at @xmath58 @xcite . in our system this region should exist near the tip of the lobe but not extend all the way to the base due to the finite temperature and the resulting mixed solid . looking at fig . [ lobes ] , it is obvious indeed that , if a supersolid region exists , it can only be near the tip of the lobe because the superfluidity is zero a significant way up the lobe . judging by the increased separation of the long - ranged lobes , we can assume that the supersolid region for these systems should increase in size to fill the region in between . to search for the supersolid phase , we now compare the superfluidity with the static structure factor . the structure factor is defined as the fourier transform of the density - density correlations , @xmath59 here , we focus on the wave vector @xmath60 , which corresponds to the @xmath61 order parameter that is associated with @xmath32- and @xmath19-filling crystals on the triangular lattice . for the case of the @xmath19-filling lobe that we are interested in , it will show plateaus over the same range of @xmath2 as the density , but additionally gives insight into the arrangement of the bosons on the lattice . this makes it a useful quantity in searching for supersolid regions . in fact , a supersolid exists when both the structure factor and the superfluid fraction have non - zero values . the physical mechanism behind the supersolid phenomenon is based upon the appearance of extra holes ( particles ) . the underlying crystal structure has @xmath61 order on a triangular sublattice of the physical lattice . the extra holes ( particles ) are free to move around on the rest of the lattice as superfluid objects . in this way , the system retains a crystal structure , while it acquires at the same time the long - range coherence of a superfluid . due to the hole ( particle ) doping , it forms in sections away from commensurate filling , in this case in between the @xmath32- and @xmath19-filling lobes . for sr sr ( left top ) , lr sr ( left bottom ) and lr lr ( right ) for a @xmath62 , and @xmath42 lattice ( lines become thicker and darker with increasing system size ) . for all cases , the structure factor ( solid blue ) is finite at small @xmath7 and the superfluid fraction ( dashed black ) at large @xmath7 . at the system sizes studied , there appears a supersolid region at intermediate @xmath7 where both structure factor and superfluid fraction are finite . in the lr lr system there is a reversal of finite size effects . in this case the superfluid fraction for larger systems becomes higher instead of lower.,scaledwidth=100.0% ] taking `` slices '' out of the crystal lobes we now check where there is a supersolid region and where the system transitions directly from crystal to superfluid . also we study the nature of these transitions to see if they are of first or second order . the most logical region to look for supersolids is directly in between the @xmath32- and @xmath19-filling lobes , at @xmath63 . figure [ horizss ] shows the behavior of the sr sr , the lr sr , and the lr lr system at this cut for several system sizes . structure factor and superfluidity reveal , for all the systems , three different regions . in each case , the system starts at @xmath17 in a solid phase where the superfluid fraction is zero but the structure factor is finite . it transitions smoothly into a supersolid region where both superfluidity and structure factor are non - zero . finally , the structure factor smoothly drops away and leaves just a non - zero superfluid fraction , making the final phase a superfluid . in each system , the size of the supersolid region is different . in the sr sr case , the supersolid region begins to appear at @xmath64 for a @xmath33 lattice . as the size grows to @xmath34 , the region has shifted to @xmath65 with the superfluid curves becoming sharper . the increased system size also reduces the value of the structure factor a little . from @xcite , we know that at even larger sizes ( but @xmath54 ) the supersolid will continue to exist in this type of system . for the lr sr system the supersolid region appears at a similar point and also shifts with system - size increase . the structure factor on the other hand has a significant decrease for larger system sizes . it is difficult to tell if at greater sizes the existence of the supersolid will persist . the final graph shows the lr lr system . in this system , superfluidity appears even before @xmath29 reaches @xmath66 . in this system , the superfluid fraction is much greater than in the previous two because of the long - ranged tunneling . this means that at small system sizes the supersolid region is much more prominent relative to the crystal lobe . the structure factor diminishes with system size almost exactly as in the lr - sr case except that the transition is at a different value of @xmath7 , and near @xmath63 it drops to slightly lower values . due to this strong decrease , for any situation with long - range interactions we can not clearly state whether the supersolid region survives at larger system sizes . for sr sr ( left top ) , lr sr ( left bottom ) , lr lr ( right ) for a @xmath62 , and @xmath42 lattice ( lines become thicker and darker with increasing system size ) . solid blue : structure factor . dashed black : superfluid fraction . at this value of @xmath7 , for short - range tunneling , the superfluid fraction disappears rapidly with increasing system size , while for long - range tunneling it even increases . at @xmath67 , possibly a supersolid may survive in large lattices . again in the lr lr system there is a reversal of finite size effects . the superfluid fraction for larger systems becomes higher instead of lower . , scaledwidth=100.0% ] next we take vertical cuts at a value of @xmath68 , since this is a reasonable place for a supersolid to exist for the lr - lr system ( @xmath69 of the tip of the lobe ) . we compare all the systems at this cut , and study the behavior of the superfluid fraction and the structure factor , plotted in fig . [ vertcut15 ] . for sr sr interactions , no supersolid region appears . on one side of the lobe there is a sharp phase transition directly from the crystal to the superfluid phase , while on the other side there is a slower change from one solid form to another ( @xmath70 ) . the finite - size scaling in the figure shows that as the size increases the transitions of the structure factor become even sharper , although they stay continuous due to the finite temperature . the values of the superfluid fraction decrease as the system sizes increases and essentially disappear at @xmath34 . in the lr sr system , a hint of the supersolid phase begins to appear on either side of the lobe . it is a bit more evident on the side where @xmath29 is small ( as is to be expected from references such as @xcite ) , but it also arises on the opposite side . this is contrary to a system of only short - ranged interactions where this supersolid region appears only on one side of the lobe and not both . at larger sizes , also in the lr sr system the transitions become sharper and the superfluidity gets smaller . the final and most interesting cut is taken out of the lr lr lobe . in this system , we see a smooth transition from crystal to supersolid at @xmath71 and at @xmath72 for @xmath33 . for larger systems the transition at @xmath71 occurs at the same spot but becomes sharper , making the supersolid region disappear . at @xmath72 the transition shifts to a higher value of @xmath29 , making the @xmath19-filling plateau smaller . it also becomes less smooth but the supersolid region remains longer than for the sr sr case . in the lr lr system , the superfluid fraction for larger systems has the opposite effect than for the previous cases , it becomes higher instead of lower . and lr sr ( right ) at @xmath73 . solid blue : structure factor . dashed black : superfluid fraction . lines become thicker and darker as system size goes up . in the sr sr system , a supersolid at small @xmath29 persists at large systems , while at @xmath74 the transition from crystal to superfluid becomes a direct first - order transition . for the lr sr system , the curves change slowly at both sides of the crystal lobe , leading to persisting supersolids . , title="fig:",scaledwidth=49.0% ] and lr sr ( right ) at @xmath73 . solid blue : structure factor . dashed black : superfluid fraction . lines become thicker and darker as system size goes up . in the sr sr system , a supersolid at small @xmath29 persists at large systems , while at @xmath74 the transition from crystal to superfluid becomes a direct first - order transition . for the lr sr system , the curves change slowly at both sides of the crystal lobe , leading to persisting supersolids . , title="fig:",scaledwidth=49.0% ] a perhaps fairer comparison is to look at a cut through a region where we are sure the supersolid exists for all three systems . therefore , in fig . [ vertss ] we look at two more cuts that are now taken closer to the tips of the sr sr and lr sr lobes . as with the lr - lr system ( last panel of fig . [ vertcut15 ] ) , these lie at around @xmath75 of the tip of the lobe . in the sr sr lobe , the cut is taken at @xmath76 . here we see a similar behavior for the structure factor as we did in the @xmath77 cut of the lobe , but this time the superfluid fraction plays a much more important role . on the one side , @xmath71 , both the superfluid fraction as well as the structure factor have sharp transitions that become even sharper at larger sizes . in fact , at @xmath54 these transitions have been shown to be of first order , and the system goes directly from crystal to superfluid . due to the finite temperature , they are continuous in our case . on the other side , where @xmath78 , there appears a second order phase transition into a supersolid region that spans all the way to @xmath79 . finally , we take a cut at @xmath73 of the lr sr lobe . the behavior of this system seems to be quite different . the first thing to notice is that the transitions on either side of the lobe are of second order . the other , and more important , observation is that now it appears that this system has supersolid behavior on both sides of the lobe : in addition to the expected supersolid at smaller @xmath29 , a region at @xmath29 above the crystal lobe appears where both structure factor and superfluid fraction are finite . if we recall fig . [ vertcut15 ] , right panel , the lr lr system showed that at @xmath77 as @xmath80 increased the supersolid region disappeared from the upper side of the lobe . in the case of the lr sr system for @xmath81 the increase in the system size does not get rid of this supersolid phase . , middle column yields lr sr at @xmath73 , right column is for lr lr at @xmath68 . for all three systems , the three distinct quantum phases crystal , supersolid , and superfluid survive over some temperature range before they melt . , scaledwidth=100.0% ] as a final calculation , we take a look at the important role that the temperature plays in both the melting of the crystal as well as the supersolid region . in this section , we will use the same cuts as in the previous section ( @xmath82 for sr sr , @xmath73 for lr sr and @xmath68 for lr - lr ) so that each system will posses all the possible phases : crystal , superfluid , and supersolid . each cut is investigated for @xmath83 at a system size of @xmath33 . first , we analyze the structure factor to study how the @xmath31 crystal melts with an increase in temperature ( second row of fig . [ ssmelt ] ) . for the sr sr interactions , even at a temperature of @xmath84 there still exists a bump in the structure factor which indicates that the crystal has not completely melted yet , while for both of the long - ranged lobes the crystal melts by @xmath85 . interestingly , the sr sr crystal and the lr lr crystal are approximately the same size at @xmath38 , but by @xmath86 one has melted while the other still exists . that means that the system with short - ranged interactions holds its crystal structure better at higher temperatures than does our system with all long - ranged interactions . looking at the lr sr lobe , we see that its crystal at this cut starts off smaller , yet it melts at about the same temperature as the one for lr lr interactions . it seems that the dipolar repulsion helps stabilize the crystal structure over a larger temperature range , while the long - ranged hopping destroys the crystal more quickly because of the extra kinetic energy . maybe more importantly , we now study the melting of the supersolid for these same cuts . figure [ ssmelt ] shows the structure factor , superfluidity , and supersolidity as a function of temperature for each of the different systems . since the supersolid is defined by having both non - zero structure factor and non - zero superfluid fraction , by combining the graphs we are able to see where these regions exist and also how they melt with increased temperature ( the bottom row of fig . [ ssmelt ] shows a product of structure factor and superfluid fraction , which remains finite only where the two coexist ) . a common feature of all the graphs are the spikes on either side of the plateaus . these are regions where a phase transition occurs but does not necessarily imply that a supersolid region exists . most likely , these features appear due to the finite size of the system and the resulting smooth transitions of superfluid fraction and structure factor . at larger sizes , the transitions would be much sharper at these points , the regions where a finite structure factor and superfluid fraction coexist would shrink , and the spikes would diminish . from the previous section , we can assume that for the sr sr system they would disappear completely at the upper transition from the crystal lobe while for the other two systems there would still exist a small supersolid region . returning to the main focus , the small-@xmath2 region , we see that in each case a supersolid region appears that extends from the left side of the plateau all the way to @xmath63 . in every system , this supersolid region also exists for a finite range of temperatures . for sr sr interactions , it gradually decreases but still extends all the way out past @xmath86 . for the lr sr interactions , the supersolid region again slowly melts but now disappears at @xmath87 , just below the spot where the crystal melted . the supersolid region for the lr lr system appears to have the largest magnitude of the three systems , but then rapidly melts at @xmath85 . , lr sr at @xmath81 and lr lr at @xmath68 ( left to right ) . all cuts are taken at @xmath63 . the structure factor ( solid blue ) attains similar values for all three systems . the superfluid fraction ( dashed black ) is largest in the lr lr system and melts fastest in the lr sr one.,title="fig:",scaledwidth=32.0% ] , lr sr at @xmath81 and lr lr at @xmath68 ( left to right ) . all cuts are taken at @xmath63 . the structure factor ( solid blue ) attains similar values for all three systems . the superfluid fraction ( dashed black ) is largest in the lr lr system and melts fastest in the lr sr one.,title="fig:",scaledwidth=32.0% ] , lr sr at @xmath81 and lr lr at @xmath68 ( left to right ) . all cuts are taken at @xmath63 . the structure factor ( solid blue ) attains similar values for all three systems . the superfluid fraction ( dashed black ) is largest in the lr lr system and melts fastest in the lr sr one.,title="fig:",scaledwidth=32.0% ] in order to compare these transitions more quantitatively , we take a cut along @xmath63 for each system , shown in fig . [ mumelt ] . all three systems show a relatively similar and steady value for the structure factor . hence , the values of the superfluid fraction are going to determine the existence of the supersolid regions . the first plot shows the sr sr system at the @xmath82 cut , and we can see that the superfluid fraction stays non - zero all the way out to @xmath88 . the lr sr system has a very similar behavior at the @xmath73 cut , but in this case the supersolid is nearly completely melted by @xmath88 . the final plot is the lr lr system at @xmath68 , which behaves slightly differently . the most important difference is that the starting value of the superfluid fraction is higher than in the first two plots . this should therefore make the supersolid region more pronounced . but even though the superfluid fraction has the highest value for this system , it decays more quickly and reaches values similar to the sr sr system at @xmath89 . in this paper , we have presented a quantum monte carlo study of dipolar spin models that describe various systems of ultracold atoms , molecules , and ions . we have presented predictions concerning the phase diagram of the considered systems at zero and finite temperatures , and described the appearance and some properties of the superfluid , supersolid , and crystalline phases . while the results are not surprising and resemble earlier obtained results for similar systems in 1d and 2d , the main advantage of our study is that it is directly relevant to the current experiments : * the results for the xxz model with short - range tunneling apply for ultracold gases of polar bosonic molecules in the limit of hard - core bosons . note that the earlier works @xcite have concentrated on the appearance of the supersolid phase and devil s staircase of crystalline phases in the square lattice @xcite , and supersolid and emulsion phases in the triangular lattice @xcite . here we focus on the hard - core - spin limit , and compare it and stress differences with other models , such as the ones with long - range tunneling , i.e. , long - range xx interactions . * the results for the xxz models with long - range tunneling apply for systems of trapped ions in triangular lattices of microtraps . these results are novel , since so far such models have been only studied using various techniques in 1d , and using the mean - field approach in 2d . while the first experimental demonstrations of such models were restricted to a few ions ( see for instance @xcite ) , many experimental groups are working on an extension of such ionic quantum simulators to systems of many ions in microtraps @xcite . in fact , very recently the nist group has engineered 2d ising interactions in a trapped - ion quantum simulator with hundreds of spins @xcite . although in this experiment the quantum regime has not yet been achieved , it clearly opens the way toward quantum simulators of spin models with long - range interactions . we expect that in the near future the result of our present theoretical study will become directly relevant for experiments . this work was supported by the international phd projects programme of the foundation for polish science within the european regional development fund of the european union , agreement no . mpd/2009/6 . we acknowledge financial support from spanish government grants toqata ( fis2008 - 01236 ) and consolider ingenio qoit ( cds2006 - 0019 ) , eu ip aqute , eu strep namequam , erc advanced grant quagatua , catalunyacaixa , alexander von humboldt foundation and hamburg theory award . m.m . and j.z . thank lluis torner , susana horvth and all icfo personnel for hospitality . j.z . acknowledges support from polish national center for science project no . dec-2012/04/a / st2/00088 . 16 b. bauer , l. d. carr , h. g. evertz , a. feiguin , j. freire , s. fuchs , l. gamper , j. gukelberger , e. gull , s. guertler , a. hehn , r. igarashi , s. v. isakov , d. koop , p. n. ma , p. mates , h. matsuo , o. parcollet , g. pawowski , j. d. picon , l. pollet , e. santos , v. w. scarola , u. schollwck , c. silva , b. surer , s. todo , s. trebst , m. troyer , m. l. wall , p. werner and s. wessel , _ j. stat _ p05001 ( 2011 ) .
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we use a quantum monte carlo method to investigate various classes of 2d spin models with long - range interactions at low temperatures . in particular , we study a dipolar xxz model with @xmath0 symmetry that appears as a hard - core boson limit of an extended hubbard model describing polarized dipolar atoms or molecules in an optical lattice .
tunneling , in such a model , is short - range , whereas density - density couplings decay with distance following a cubic power law .
we investigate also an xxz model with long - range couplings of all three spin components - such a model describes a system of ultracold ions in a lattice of microtraps .
we describe an approximate phase diagram for such systems at zero and at finite temperature , and compare their properties .
in particular , we compare the extent of crystalline , superfluid , and supersolid phases .
our predictions apply directly to current experiments with mesoscopic numbers of polar molecules and trapped ions .
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the abundance evolution of galaxy structures is a major prediction of cosmological models ( e.g. oukbir @xmath11 blanchard 1992 , 1997 , romer et al . 2001 ) . the more distant the structure , the strongest the constraint . until now , however , we are limited to the study of the most massive distant structures . this is because these structures are the easiest to detect at high redshift using for example x - ray selected samples ( e.g. borgani et al . 1999 , burke et al . 1997 , ebeling et al . 2000 , gioia et al . 1990 , romer et al . 2000 , vikhlinin et al . these distant massive structures are the progenitors of the most massive z@xmath120 clusters . the hierarchical models predict , however , that nearby intermediate mass clusters have formed from low mass high redshift systems as groups of galaxies . the new generation of deep galaxy surveys ( vlt and keck surveys ) will be able to detect such high redshift low mass structures . this will add an element to the large scale structure formation models . we present in this paper a new efficient method to detect and study such structures in very large galaxy redshift samples ( section 2 ) . we applied this method to the ssrs2 ( da costa et al . 1998 ) galaxy redshift catalog in order to quantify our detection efficiency at low redshift . we have created a catalog of groups from this survey and we have studied the properties of these structures ( section 3 ) , comparing our results to literature studies ( carlberg et al . 2001 , girardi et al . 2000 , merchan et al . we conclude in section 4 . the structure detection we used is an hybrid method derived from the method used for the enacs sample of galaxy clusters ( e.g. katgert et al . 1996 ) and from classical friend - of - friend detection methods ( e.g. huchra @xmath11 geller 1982 ) . the friend - of - friend algorithm is applied to a preprocessed catalog of elementary galaxy associations and not to individual galaxies . in this way we reduce the computation time by a large factor ( see section 3.3 ) . the procedure consists of four steps : * we apply a running window ( with an overlap of the half of the window size ) to the original catalog of galaxies to divide the total area covered by the survey in several elementary lines of sight . the angular size @xmath13 of the running window is chosen as a function of the kind of structures we are trying to detect ( @xmath14 has to be close to the typical size of the structures ) and as a function of the redshift of these structures . * we search for compact galaxy redshift associations in each of these elementary beams by using a technique similar to katgert et al . this technique is basically detecting gaps ( fixed redshift separations between successive galaxies ) greater than a given value ( noted `` gap analysis '' in the following ) . this value is given by the typical maximal velocity dispersion of the structures we try to detect . for groups , we searched galaxies separated by more than 600 km / s in each elementary beam . if two galaxies are separated by more than this value , these galaxies belong to different structures . an elementary structure has more than 2 galaxies . * we apply a friend - of - friend algorithm to merge the elementary structures into larger real structures . two elementary structures were assumed to be connected if they were closer than @xmath14 . this value is an approximation of the maximal diameter of a group in real space . this allows to generate a preliminary catalog of interesting regions with potential structures inside . * finally , we re - apply the gap method for each of these regions in order to detect real structures in the catalog . the ssrs2 catalog is a sample of 5369 galaxies , covering a region of 1.70 sr of the southern sky ( @xmath15 and @xmath16 for the southern galactic cap and @xmath17 and @xmath18 for the northern galactic cap ) . this catalog is 99% complete down to m@xmath19 . the m@xmath20 magnitude is close to a b magnitude ( see da costa et al . the precision of the individual galaxy redshift measurements is @xmath1240 km / s . detailed information on this catalog can be found in da costa et al . we limited our analysis to the 2000@xmath21cz@xmath2120000 km / s range . to detect groups , we used @xmath22 . this value is an approximation of the maximal diameter of a group and gives the most similar results compared to the ssrs2 group analysis by merchan et al . we kept only the structures with more than 4 galaxies and less than 40 galaxies ( to be consistent with merchan et al . structures with less than or with 3 galaxies are possibly close superposition effects and not real dynamical structures ( ramella et al . structures with more than 40 galaxies are likely to be rich clusters or diffuse elongated filaments of galaxies . the detected structures are described in table 1 and 2 . the mean number of galaxies per structure is @xmath23 and 90% of the structures have less than 11 galaxies . the largest group has 32 galaxies . the richest groups are the closer ones , in agreement with the detection of a uniform richness class of groups . this is because , due to apparent limiting magnitude , the closest groups have more galaxies brighter than the ssrs2 catalog magnitude limit . the groups are more or less circular : they have the same angular extension in @xmath24 ( right ascension ) and @xmath25 ( declination ) . we have @xmath26arcmin ( computed with 80 groups ) . this means that we are not detecting systematically elongated structures such as poor filaments . the mean extension of the detected groups ( computed as the dispersion of the coordinates of the galaxies of a given group ) is @xmath27 @xmath28 . we have no variation of the group size as a function of the redshift , but the richest groups are also the larger ones . the groups with more than 10 galaxies have a size in the range [ 0.45;0.8 ] @xmath28 . the groups with less than 10 galaxies have a size in the range [ 0.05;0.65 ] @xmath28 . finally , the mean virial radius r@xmath29 of the groups is @xmath30 ( in the range [ 0.06;0.87 ] @xmath28 , see table 1 and 2 ) . we computed velocity dispersions using a robust estimator ( beers et al . 1990 ) . this gives a more realistic estimate of the velocity dispersion for groups with more than 10 galaxies , but still a low number , compared to classical estimators . for groups with less than 10 galaxies , there is no significant difference between this robust estimator and the classical estimator of the velocity dispersion ( lax 1985 ) . the mean velocity dispersion of the groups is @xmath31km / s . none of them has a velocity dispersion greater than 500 km / s ( see fig 1 ) . these values are consistent with several other computations for groups ( e.g. carlberg et al . 2001 with a median velocity dispersion of 200 km / s ) . this means that we detected no rich clusters . the groups with the highest velocity dispersions are also the richest ( see fig 2 ) but we have no significant relation between the group mean redshift and velocity dispersion . this is , one more time , in favor of a uniform detection of groups as a function of redshift . in order to check how many spurious groups we are detecting ( groups with a large velocity dispersion , but a low number of galaxies : chance alignments ) , we plotted the velocity dispersion as a function of the number of galaxies inside the groups ( fig 2 ) . only 10% of the groups have less than 5 galaxies and a velocity dispersion greater than 300 km / s . this can be assumed to be the contamination rate of our catalog . combining the virial radius and the velocity dispersion , we computed the mass using the standard virial mass estimator ( m @xmath32 virial radius @xmath33 ( velocity dispersion)@xmath34 ) ( e.g. ramella et al . 1997 ) . about 85% of the sample has a mass in the range [ @xmath35;@xmath36 solar masses and 97.5% has a mass in the range [ @xmath37;@xmath36 solar masses . none of the groups has a mass greater than @xmath38 solar masses . the mean mass is ( @xmath39)@xmath40 solar masses . all the masses computed by merchan et al . ( 2000 ) are in the range [ @xmath35;@xmath36 solar masses . this means that we have slightly lower masses in the sample ( for 15% of the sample ) . this is probably due to the fact that we are using robust estimators to compute velocity dispersions . such estimators are less biased toward high values for small samples of galaxies . the velocity dispersions and masses are given in table 1 and 2 . in order to estimate the mass uncertainty , we used galaxy groups with more than 10 galaxies . for these groups , a reliable error can be computed for velocity dispersion using 100 bootstrap resamplings . this is obviously a lower limit of the velocity dispersion uncertainty as this uncertainty is probably larger for smaller groups . the 1-@xmath8 uncertainty is : @xmath41 . the mass uncertainty can be written as : @xmath42 assuming that the velocity dispersion uncertainty is the dominant factor , we have the 1-@xmath8 error on the mass : @xmath43 . this large error raises the question of the validity of virial mass estimates for groups of galaxies as these structures are probably not all in equilibrium ( e.g. dos santos @xmath11 mamon 1999 ) . we compared our catalog with the one of merchan et al . ( 2000 ) that used the same galaxy sample . fig 3 shows the group redshift distribution of the two catalogs . we are detecting similar numbers of groups up to cz@xmath128000 km / s . the mean difference per bin is 5% ( with redshift bins of 1250 km / s width ) . merchan et al . are detecting more groups at higher redshift . however , their catalog begins to be incomplete above 5000 km / s . this means that we have the same detection completeness level , in agreement with press @xmath11 schechter ( 1974 ) predictions ( merchan et al . 2000 ) . finally , we estimated the 2-point correlation function for the sample . using a power law approximation of the form @xmath44 , we have @xmath45 and @xmath46 . this is consistent at the 1@xmath8 level with the values of merchan et al . ( 2000 ) : @xmath47 and @xmath48 , girardi et al . ( 2000 ) : @xmath49 and @xmath50 or carlberg et al . ( 2001 ) : @xmath51 and @xmath52 ( fixed ) . we detect a positive signal up to @xmath1290 @xmath28 ( similar to the values of merchan et al . these estimates are all in good agreement and give confidence in the detection method . the only significant difference is a lower group detection rate at the faint end of the galaxy catalog . however , this is not affecting the completeness level of our group sample as we detect the same number of groups up to cz@xmath128000 km / s ( assuming that the merchan et al . completeness limit is correct ) . the redshift completeness limit close to 5000 km / s is directly related to the magnitude limit of the catalog : m@xmath53 . it gives an absolute magnitude of -18 + 5logh . this is the magnitude limit for the galaxies lying inside groups before the completeness limit . we can extrapolate this redshift completeness limit assuming the typical depth of the new generation of redshift surveys using 10 meter class telescopes ( keck and vlt ) . for example , the deep part of the virmos survey should provide a redshift survey 80% complete down to b@xmath1226 , and the shallow part of this survey a 50% complete redshift catalog down to b@xmath1224 . assuming a k - correction proportional to the redshift , it should provide a catalog of groups complete up to [email protected] for the deep catalog and [email protected] for the shallow catalog . this is enough to sample nearly all classes of galaxy structures up to [email protected] . this will allow to put strong constraints on cosmological models ( e.g. romer et al . 2001 ) as well as sampling the internal structure evolution of groups ( and more massive clusters ) up to these redshifts . the time required to treat these future samples with our method is very short . we used the simulations of steve hatton ( private communication ) in order to estimate these times . these simulations cover 1 deg@xmath34 and reproduce an @xmath55=0.3 and @xmath56=0.7 universe , including several structures as clusters or groups . we used a single 6 running window for simplicity and we selected randomly different sub - samples of galaxies in this catalog . the computation times are given in fig 4 . we see that a catalog of 150000 galaxies , comparable to the spectroscopic catalogs which will be produced by the vlt and keck surveys , is completely analyzed in about 2 minutes ( using an es40 compaq , processor eb67 at 600 mhz ) . we computed the luminosity function of the group galaxies of the detected sample . the galaxy catalog is complete down to m@xmath57 with groups up to 8000 km / s . limiting ourselves to groups with velocity lower than 6000 km / s ( this velocity limit is close to the completeness limit of the group sample ) , we have a galaxy catalog complete down to m@xmath20=-18.4 + 5logh . using the estimate of zabludoff @xmath11 mulchaey ( 2000 ) , this corresponds to about m@xmath58 + 1.5 . we fitted a schechter function on the magnitude distribution down to m@xmath20=-18.4 + 5logh and for groups with cz@xmath596000 km / s using a maximum likelihood technique ( e.g. lobo et al . 1997 ) . the absolute magnitudes were k - corrected ( but this is a minor correction due to the low redshift of the sample : see table 1 and 2 ) using the k - correction -3.5@xmath60z ( rauzy et al . we also corrected for galactic extinction using the work by schlegel et al . the correction we applied for the magnitudes was -4.325@xmath33e(b - v ) . the mean correction was 16% of the luminosity ( but up to 60% for the worse cases : see table 1 and 2 ) . the best fit parameters of a schechter function are m@xmath58=([email protected])+5logh and @[email protected] ( estimates with 1-@xmath8 error ) . this is consistent at the 1-@xmath8 level with the estimates of zabludoff @xmath11 mulchaey ( 2000 ) : m@xmath61=([email protected])+5logh ( assuming m@[email protected] ) and @[email protected] for groups in 2800@xmath21cz@xmath217700 km / s . these values are also similar at the 1-@xmath8 level with the estimates of rauzy et al . ( 1998 ) for rich clusters of galaxies . in order to check the robustness of our estimates , we split the sample in two parts : the northern and the southern galactic cap . the southern cap gives m@xmath58=([email protected])+5logh and @[email protected] and the northern cap gives m@xmath58=([email protected])+5logh and @[email protected] . despite the larger uncertainty , these values are still consistent , with , however , a mild tendency to have more faint galaxies in the southern galactic cap . we computed the total luminosity of each group summing up all the individual magnitudes ( see table 1 and 2 ) . we used m@xmath64=5.53 ( approximation of m@xmath65 ) . we corrected these values for incompleteness due to galaxy catalog magnitude limit by using the ratio between the luminosity function integrated from m@xmath66=-23 + 5logh . to -10 + 5logh and integrated between the faintest magnitude of each group and m@xmath66=-10 + 5logh . this is because we assumed that the faintest galaxies were not fainter than -10 + 5logh . this limit has , however , a moderate influence on this correction factor . we used the luminosity function computed in the previous section with m@xmath58=([email protected])+5logh and @[email protected] . the mean correction was 135% of the raw group luminosity . the total luminosities and the individual completeness corrections are given in table 1 and 2 . combining estimates of the mass and of the luminosity , we computed the mass to light ratio ( see table 1 and 2 ) . the mean value is 550 hm@xmath7/l@xmath67 and the median is 250 hm@xmath7/l@xmath7 . we compared these estimates with those of carlberg et al . ( 2001 ) . in the r band , they computed median values in the range [ 150;250 ] hm@xmath7/l@xmath7 . assuming m@xmath68-m@xmath69=-1.17 ( allen 1973 ) and m@xmath70-m@xmath71=-1.5 for galaxies in nearby structures ( e.g. katgert et al . 1998 ) , it corresponds to median values in the range [ 204;340 ] hm@xmath7/l@xmath67 in the ssrs2 magnitude passband ( multiplication factor of 10@xmath72 ) . this is in good agreement with our estimates . we found a relation between the mass to light ratio and the velocity dispersion @xmath73 for our sample ( see fig . 5 and table 3 ) . these relations are qualitatively similar to the relation we found between rich cluster velocity dispersion and cluster mass to light ratio ( adami et al . 1998 ) : @xmath74 . the slope is , however , significantly steeper for groups than for clusters . the slope of these relations is also similar to that of carlberg et al . ( 2001 ) which is close to 3.5 . the more massive the group ( or the cluster ) , the higher the mass to light ratio , and therefore , the larger the amount of dark matter inside the group ( or cluster ) . another explanation would be a very efficient stripping of the gas of the galaxies in rich groups ( as opposed to very low mass groups ) . this would be due to a denser intra group medium or a more efficient tidal stripping due to the higher number of galaxies ( e.g. dos santos @xmath11 mamon 1999 ) in rich groups . it would lower the star formation rate in galaxies , inducing a higher mass to light ratio for the richest groups ( see also carlberg et al . the dispersion of the relation is also smaller for richer groups . this is probably due to a better computation of the velocity dispersion due to the larger number of available galaxies . schaeffer et al . ( 1993 ) or adami et al . ( 1998 ) have shown , using optical data , that clusters of galaxies populate a well defined area in the [ total luminosity : l , characteristic radius of a beta profile : r@xmath75 , velocity dispersion : @xmath8 ] space . adami et al . ( 1998 ) found the following relation : l@xmath77 r@xmath78@xmath79@xmath80 with a dispersion of the relation of 5@xmath81 using the virial radius instead of the characteristic radius of a beta profile would give : l@xmath77 r@xmath82@xmath79@xmath83 this relation called the fundamental plane is also detected for elliptical galaxies , but with different coefficients ( see adami et al . 1998 for a comparison ) . this is interpreted as the different relaxation state of elliptical galaxies and clusters of galaxies . we searched for the same kind of relation in the groups we detected . using all the groups , we found no relation between l , r@xmath84 and @xmath8 . using only groups with more than 8 galaxies , we found : l@xmath77 r@xmath85@xmath79@xmath86 with a dispersion of 58@xmath81 the coefficients are consistent with those of clusters of galaxies , but the uncertainties are very large , and the intrinsic dispersion of the relation is 10 times larger than for clusters . we conclude that a mean relation between l , r@xmath29 and @xmath8 probably exists for groups , but this relation is much less well defined than for rich clusters . moreover , poor groups ( less than 7 galaxies ) probably have too large a velocity dispersion uncertainty to allow any detection of this relation . crrccrccccrrr n & @xmath87 & @xmath88 & z & obs . lum . & e(b - v ) & ext . & comp . & tot . mass & m / l & @xmath89 & rvir + 5 & 3.2488 & -24.3504 & 0.0255 & 5.33e+10 & 0.02220 & 1.09 & 3.20 & 1.86e+11 & 1.90e+12 & 10.2 & 82 & 0.13 + 4 & 3.2868 & -7.4360 & 0.0181 & 4.54e+10 & 0.03389 & 1.14 & 1.94 & 1.01e+11 & 4.91e+13 & 486.9 & 286 & 0.27 + 4 & 5.5908 & -4.3517 & 0.0138 & 1.13e+10 & 0.03145 & 1.13 & 1.60 & 2.04e+11 & 2.13e+14 & 1043.7 & 401 & 0.6 + 5 & 6.8999 & -9.0373 & 0.0192 & 2.99e+10 & 0.03553 & 1.15 & 2.19 & 7.56e+10 & 1.44e+14 & 1905.7 & 359 & 0.51 + 4 & 7.0446 & -37.3208 & 0.0245 & 3.91e+10 & 0.02375 & 1.10 & 3.13 & 1.34e+11 & 4.04e+12 & 30.1 & 86 & 0.25 + 4 & 7.1985 & -22.9825 & 0.0268 & 7.17e+10 & 0.01305 & 1.05 & 3.06 & 2.31e+11 & 4.44e+12 & 19.2 & 113 & 0.16 + 5 & 7.3508 & -10.9007 & 0.0122 & 5.79e+10 & 0.03334 & 1.14 & 2.15 & 1.42e+11 & 2.45e+13 & 172.4 & 189 & 0.31 + 5 & 8.0602 & -3.3412 & 0.0196 & 3.70e+10 & 0.03581 & 1.15 & 2.23 & 9.51e+10 & 1.98e+13 & 208.3 & 215 & 0.2 + 4 & 8.4047 & -28.5031 & 0.0234 & 3.47e+10 & 0.02017 & 1.08 & 2.62 & 9.86e+10 & 2.45e+13 & 248.4 & 149 & 0.5 + 4 & 8.8372 & -23.7145 & 0.0128 & 1.90e+10 & 0.01679 & 1.07 & 1.80 & 3.65e+10 & 3.89e+12 & 106.5 & 99 & 0.18 + 7 & 10.6675 & -9.3056 & 0.0197 & 4.22e+10 & 0.03604 & 1.15 & 1.89 & 9.20e+10 & 6.88e+13 & 747.4 & 281 & 0.4 + 4 & 12.7327 & -13.4234 & 0.0376 & 9.85e+10 & 0.02390 & 1.10 & 7.94 & 8.60e+11 & 2.06e+13 & 24.0 & 286 & 0.12 + 4 & 14.2872 & -9.1834 & 0.0150 & 2.88e+10 & 0.08095 & 1.38 & 2.07 & 8.22e+10 & 1.35e+13 & 164.2 & 113 & 0.48 + 9 & 18.0011 & -32.3378 & 0.0192 & 1.25e+11 & 0.02365 & 1.10 & 2.18 & 3.00e+11 & 5.37e+13 & 179.0 & 346 & 0.2 + 4 & 18.0096 & -6.6967 & 0.0204 & 2.76e+10 & 0.11385 & 1.57 & 2.37 & 1.03e+11 & 1.10e+13 & 107.0 & 285 & 0.06 + 14 & 20.7478 & -35.3655 & 0.0193 & 1.23e+11 & 0.01880 & 1.08 & 2.17 & 2.88e+11 & 5.47e+13 & 190.1 & 212 & 0.56 + 5 & 20.7672 & -38.5622 & 0.0205 & 4.90e+10 & 0.01960 & 1.08 & 2.18 & 1.15e+11 & 7.60e+12 & 65.8 & 159 & 0.14 + 4 & 25.2701 & -34.4858 & 0.0128 & 1.14e+10 & 0.01997 & 1.08 & 1.57 & 1.93e+10 & 4.52e+12 & 233.7 & 190 & 0.06 + 5 & 25.3051 & -4.2930 & 0.0180 & 3.59e+10 & 0.02266 & 1.09 & 2.05 & 8.05e+10 & 1.92e+13 & 238.4 & 157 & 0.36 + 5 & 26.2856 & -13.8575 & 0.0181 & 2.74e+10 & 0.01640 & 1.07 & 2.08 & 6.09e+10 & 1.94e+13 & 318.4 & 175 & 0.29 + 4 & 28.2577 & -4.2047 & 0.0164 & 2.91e+10 & 0.02386 & 1.10 & 1.85 & 5.92e+10 & 1.25e+14 & 2110.9 & 331 & 0.52 + 4 & 28.7924 & -9.3172 & 0.0173 & 1.91e+10 & 0.02394 & 1.10 & 1.95 & 4.10e+10 & 1.08e+14 & 2635.4 & 265 & 0.7 + 4 & 29.5184 & -25.5656 & 0.0144 & 3.66e+10 & 0.01053 & 1.04 & 1.98 & 7.56e+10 & 2.34e+13 & 309.5 & 256 & 0.16 + 5 & 30.2063 & -32.1871 & 0.0183 & 4.08e+10 & 0.01977 & 1.08 & 2.33 & 1.03e+11 & 1.17e+14 & 1138.0 & 288 & 0.64 + 4 & 30.6148 & -29.3072 & 0.0169 & 3.96e+10 & 0.01656 & 1.07 & 2.15 & 9.10e+10 & 1.34e+14 & 1471.9 & 289 & 0.73 + 4 & 31.4930 & -6.8843 & 0.0169 & 1.60e+10 & 0.01961 & 1.08 & 1.83 & 3.16e+10 & 3.71e+13 & 1172.2 & 271 & 0.23 + 4 & 31.5281 & -23.1068 & 0.0178 & 5.10e+10 & 0.01674 & 1.07 & 2.12 & 1.16e+11 & 9.35e+13 & 808.6 & 316 & 0.43 + 4 & 34.8054 & -38.3532 & 0.0167 & 2.95e+10 & 0.02016 & 1.08 & 1.83 & 5.85e+10 & 4.24e+12 & 72.4 & 55 & 0.64 + 7 & 36.4329 & -11.4888 & 0.0153 & 3.37e+10 & 0.02209 & 1.09 & 1.75 & 6.44e+10 & 6.41e+12 & 99.6 & 181 & 0.09 + 9 & 39.0671 & -13.1909 & 0.0150 & 5.33e+10 & 0.03001 & 1.13 & 1.68 & 1.01e+11 & 4.99e+13 & 494.9 & 243 & 0.39 + 9 & 40.8864 & -25.7116 & 0.0229 & 1.15e+11 & 0.01903 & 1.08 & 3.60 & 4.45e+11 & 8.34e+13 & 187.4 & 219 & 0.79 + 8 & 41.4914 & -31.8348 & 0.0221 & 1.13e+11 & 0.02305 & 1.10 & 2.55 & 3.17e+11 & 1.66e+14 & 523.7 & 393 & 0.49 + 5 & 42.5306 & -8.9778 & 0.0178 & 2.59e+10 & 0.02907 & 1.12 & 2.04 & 5.93e+10 & 9.15e+13 & 1542.9 & 274 & 0.56 + 14 & 46.6818 & -10.4633 & 0.0155 & 8.66e+10 & 0.07630 & 1.36 & 1.67 & 1.96e+11 & 1.17e+14 & 597.3 & 422 & 0.3 + 5 & 47.4664 & -25.3934 & 0.0212 & 2.99e+10 & 0.01680 & 1.07 & 2.33 & 7.45e+10 & 3.07e+12 & 41.2 & 113 & 0.11 + 4 & 48.1498 & -7.5486 & 0.0177 & 3.10e+10 & 0.06842 & 1.31 & 2.15 & 8.76e+10 & 7.86e+13 & 896.8 & 203 & 0.87 + 10 & 49.0948 & -4.7276 & 0.0079 & 2.40e+10 & 0.06228 & 1.28 & 1.26 & 3.87e+10 & 8.08e+13 & 2087.3 & 387 & 0.25 + 4 & 52.7168 & -4.8493 & 0.0198 & 3.00e+10 & 0.04634 & 1.20 & 2.24 & 8.10e+10 & 8.13e+13 & 1004.3 & 235 & 0.07 + 4 & 53.8719 & -18.4927 & 0.0142 & 2.26e+10 & 0.06259 & 1.28 & 2.06 & 5.97e+10 & 1.94e+12 & 32.5 & 67 & 0.2 + 11 & 55.2961 & -4.5489 & 0.0138 & 1.01e+11 & 0.06435 & 1.29 & 1.58 & 2.07e+11 & 3.45e+13 & 166.7 & 216 & 0.34 + 5 & 149.4540 & -2.7081 & 0.0210 & 5.70e+10 & 0.04201 & 1.18 & 2.13 & 1.43e+11 & 2.49e+13 & 173.6 & 308 & 0.12 + 7 & 150.3229 & -6.2040 & 0.0164 & 4.93e+10 & 0.03796 & 1.16 & 1.78 & 1.02e+11 & 3.36e+13 & 329.0 & 249 & 0.25 + 5 & 151.6062 & -2.4202 & 0.0202 & 3.65e+10 & 0.04465 & 1.19 & 2.37 & 1.03e+11 & 2.66e+12 & 25.8 & 58 & 0.36 + 5 & 155.7194 & -2.3807 & 0.0186 & 5.69e+10 & 0.04830 & 1.21 & 1.93 & 1.33e+11 & 4.88e+13 & 366.3 & 218 & 0.47 + 6 & 160.6563 & -10.2180 & 0.0077 & 1.29e+10 & 0.00910 & 1.04 & 1.38 & 1.77e+10 & 6.25e+13 & 3522.1 & 245 & 0.48 + 4 & 162.5583 & -12.2600 & 0.0154 & 1.79e+10 & 0.03559 & 1.15 & 1.86 & 3.84e+10 & 2.77e+13 & 721.7 & 248 & 0.21 + 4 & 164.2253 & -9.5503 & 0.0270 & 6.25e+10 & 0.03783 & 1.16 & 3.15 & 2.29e+11 & 2.44e+13 & 106.6 & 191 & 0.31 + 17 & 165.7276 & -14.5042 & 0.0144 & 1.09e+11 & 0.00944 & 1.04 & 1.52 & 1.66e+11 & 2.97e+14 & 1787.6 & 475 & 0.6 + 5 & 168.0586 & -13.6485 & 0.0171 & 3.23e+10 & 0.07506 & 1.35 & 1.93 & 8.42e+10 & 1.25e+14 & 1484.7 & 351 & 0.46 + 4 & 170.6570 & -13.4858 & 0.0173 & 5.77e+10 & 0.03418 & 1.15 & 1.85 & 1.22e+11 & 7.72e+13 & 630.9 & 244 & 0.59 + [ ] crrccrccccrrr n & @xmath87 & @xmath88 & z & raw lum . & e(b - v ) & ext . & comp . & tot . lum . & mass & m / l & @xmath89 & rvir + 4 & 171.4233 & -11.2592 & 0.0174 & 2.83e+10 & 0.04699 & 1.21 & 1.99 & 6.79e+10 & 3.97e+13 & 585.1 & 165 & 0.66 + 4 & 175.3333 & -11.9775 & 0.0174 & 4.43e+10 & 0.03384 & 1.14 & 1.97 & 9.98e+10 & 1.35e+14 & 1352.6 & 395 & 0.39 + 6 & 179.4168 & -20.3306 & 0.0222 & 4.46e+10 & 0.04524 & 1.20 & 2.31 & 1.23e+11 & 4.77e+13 & 386.6 & 272 & 0.29 + 4 & 184.8120 & -12.4777 & 0.0144 & 2.48e+10 & 0.03033 & 1.13 & 1.67 & 4.14e+10 & 1.98e+13 & 477.9 & 127 & 0.56 + 4 & 190.5120 & -20.5779 & 0.0224 & 5.86e+10 & 0.05480 & 1.24 & 2.45 & 1.79e+11 & 8.54e+13 & 478.2 & 313 & 0.4 + 4 & 191.9568 & -4.7924 & 0.0148 & 4.57e+10 & 0.01745 & 1.07 & 1.93 & 8.82e+10 & 1.28e+14 & 1450.5 & 326 & 0.54 + 5 & 192.5408 & -26.9663 & 0.0115 & 2.64e+10 & 0.07663 & 1.36 & 2.16 & 5.69e+10 & 2.71e+13 & 476.1 & 166 & 0.45 + 26 & 192.5900 & -13.3802 & 0.0148 & 2.04e+11 & 0.05606 & 1.25 & 1.47 & 3.01e+11 & 3.63e+14 & 1207.8 & 466 & 0.76 + 4 & 193.6383 & -20.2686 & 0.0229 & 4.89e+10 & 0.06569 & 1.30 & 2.66 & 1.69e+11 & 5.99e+12 & 35.5 & 223 & 0.06 + 4 & 195.6448 & -4.6870 & 0.0103 & 1.72e+10 & 0.01346 & 1.06 & 1.55 & 2.66e+10 & 2.04e+12 & 76.7 & 79 & 0.15 + 9 & 199.7299 & -16.4609 & 0.0229 & 9.04e+10 & 0.08047 & 1.38 & 2.41 & 3.00e+11 & 9.28e+13 & 309.1 & 291 & 0.5 + 5 & 200.9558 & -11.9613 & 0.0223 & 6.17e+10 & 0.03190 & 1.14 & 2.31 & 1.62e+11 & 3.86e+13 & 238.4 & 238 & 0.31 + 17 & 201.6564 & -20.3991 & 0.0183 & 1.10e+11 & 0.08540 & 1.41 & 1.88 & 2.06e+11 & 5.93e+13 & 287.5 & 242 & 0.46 + 4 & 201.9665 & -1.8782 & 0.0136 & 2.51e+10 & 0.03450 & 1.15 & 1.69 & 4.25e+10 & 4.04e+13 & 951.6 & 262 & 0.27 + 4 & 201.9803 & -21.4628 & 0.0243 & 2.69e+10 & 0.09882 & 1.48 & 2.52 & 1.00e+11 & 2.59e+12 & 25.8 & 93 & 0.14 + 11 & 202.8497 & -24.7016 & 0.0159 & 7.93e+10 & 0.06650 & 1.30 & 1.74 & 1.38e+11 & 1.39e+14 & 1007.2 & 278 & 0.82 + 4 & 229.1948 & -13.2268 & 0.0073 & 1.25e+10 & 0.03372 & 1.14 & 1.43 & 1.78e+10 & 1.56e+12 & 87.6 & 104 & 0.07 + 4 & 318.7207 & -23.0257 & 0.0269 & 6.63e+10 & 0.05581 & 1.25 & 3.3 & 2.73e+11 & 1.06e+13 & 38.8 & 228 & 0.09 + 32 & 329.3656 & -33.4588 & 0.0087 & 1.01e+11 & 0.03357 & 1.14 & 1.26 & 1.46e+11 & 2.54e+13 & 174.4 & 166 & 0.42 + 6 & 331.6321 & -27.8423 & 0.0238 & 1.49e+11 & 0.02030 & 1.08 & 2.64 & 4.28e+11 & 8.80e+12 & 20.6 & 209 & 0.09 + 5 & 332.2561 & -27.0691 & 0.0086 & 6.75e+09 & 0.02354 & 1.10 & 1.32 & 9.78e+09 & 5.11e+11 & 52.3 & 30 & 0.26 + 4 & 332.4736 & -22.8598 & 0.0178 & 4.14e+10 & 0.02640 & 1.11 & 2.5 & 1.15e+11 & 2.63e+12 & 22.9 & 52 & 0.44 + 4 & 332.6806 & -30.0202 & 0.0139 & 4.61e+10 & 0.01484 & 1.06 & 2.39 & 1.17e+11 & 6.94e+13 & 593.3 & 262 & 0.46 + 4 & 332.9872 & -27.7711 & 0.0178 & 5.87e+10 & 0.01950 & 1.08 & 2.74 & 1.74e+11 & 1.01e+13 & 58.1 & 122 & 0.31 + 7 & 333.7648 & -21.2985 & 0.0087 & 1.61e+10 & 0.02895 & 1.12 & 1.33 & 2.40e+10 & 5.09e+11 & 21.2 & 59 & 0.07 + 4 & 335.6633 & -31.4416 & 0.0285 & 6.64e+10 & 0.01282 & 1.05 & 3.9 & 2.72e+11 & 6.78e+13 & 248.9 & 295 & 0.36 + 16 & 341.0783 & -22.2565 & 0.0107 & 9.25e+10 & 0.02412 & 1.10 & 1.38 & 1.41e+11 & 2.88e+13 & 204.9 & 130 & 0.78 + 4 & 343.0948 & -34.2753 & 0.0291 & 6.90e+10 & 0.01265 & 1.05 & 3.86 & 2.80e+11 & 1.57e+12 & 5.6 & 93 & 0.08 + 14 & 356.4653 & -28.3663 & 0.0282 & 2.60e+11 & 0.01613 & 1.07 & 3.79 & 1.05e+12 & 3.56e+13 & 33.9 & 315 & 0.16 + 4 & 357.4948 & -29.5501 & 0.0291 & 5.24e+10 & 0.01603 & 1.07 & 3.97 & 2.22e+11 & 1.62e+13 & 73.1 & 143 & 0.36 + [ ] rrrr selection & slope & constant & dispersion + all ssrs2 groups & [email protected] & 294@xmath90570 & 8@xmath81 + ssrs2 groups with & [email protected] & 462@xmath90431 & 6@xmath81 + more than 7 galaxies & & & + ssrs2 groups with & [email protected] & 579@xmath90470 & 4@xmath81 + more than 8 galaxies & & & + cluster sample & [email protected] & 12@xmath90236 & - + [ ] we demonstrated that we are able to detect similar populations of groups with our method compared to classical friend - of - friend algorithms . the groups detected with our method have the same physical properties than those detected by merchan et al . ( 2000 ) : nearly same mass distribution ( most of our groups are in the range [ @xmath35;@xmath36 solar masses ) , same abundances up to cz@xmath128000 km / s ( and , therefore , same completeness limit : @xmath125000 km / s and abundances in agreement with press @xmath11 schechter models ) , and same 2-point correlation function ( modeled by a power law : @xmath91 with @xmath1 and @xmath2 ) . this redshift completeness limit close to 5000 km / s is directly related to the magnitude limit of the catalog : m@xmath53 . extrapolating these limits assuming the typical depth of the new generation of redshift surveys using 10 meter class telescopes ( keck and vlt ) leads to a redshift completeness limit of [email protected] for the groups in these samples . we found for the ssrs2 catalog a similar luminosity function for nearby group galaxies to that of zabludoff et al . ( 2000 ) : m@xmath58=([email protected])+5logh and @[email protected] . we computed a similar mass to light ratio compared to carlberg et al . ( 2001 ) ( median value of 250 hm@xmath7/l@xmath7 in the b ssrs2 magnitude passband ) and we deduced a similar relation between the mass to light ratio and velocity dispersion ( @xmath92 ) . this relation is qualitatively similar to that detected for rich clusters of galaxies ( adami et al . 1998 ) , but with a significantly steeper slope . the more massive the group ( or the cluster ) , the higher the mass to light ratio , and therefore , the larger the amount of dark matter inside group ( and cluster ) . another explanation is a significant stripping of the gas of the galaxies in rich groups as opposed to poorer groups . finally , we detected a fundamental plane for these groups ( @xmath93 for groups with more than 8 galaxies ) but much less narrow and barely significant compared to clusters of galaxies . we conclude that a mean relation between l , r@xmath29 and @xmath8 probably exists for groups , but this relation is much less well defined compared to clusters .
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we present an automated method to detect populations of groups in galaxy redshift catalogs .
this method uses both analysis of the redshift distribution along lines of sight in fixed cells to detect elementary structures and a friend - of - friend algorithm to merge these elementary structures into physical structures .
we apply this method to the ssrs2 galaxy redshift catalog .
the groups detected with our method are similar to group catalogs detected with pure friend - of - friend algorithms .
they have similar mass distribution , similar abundance versus redshift , similar 2-point correlation function ( modeled by a power law : @xmath0 with @xmath1 and @xmath2 ) and the same redshift completeness limit , close to 5000 km / s .
if instead of ssrs2 , we use catalogs of the new generation ( deep redshift surveys obtained with 10 meters class telescopes ) , it would lead to a completeness limit of [email protected] .
we model the luminosity function for nearby galaxy groups by a schechter function with parameters m@xmath4=([email protected])+5logh and @[email protected] to compute the mass to light ratio .
the median value of the mass to light ratio is 360 hm@xmath7/l@xmath7 ( in the ssrs2 band , close to a b band magnitude ) and we deduce a relation between mass to light ratio and velocity dispersion @xmath8 ( @xmath9 ) .
the more massive the group , the higher the mass to light ratio , and therefore , the larger the amount of dark matter inside the group .
another explanation is a significant stripping of the gas of the galaxies in massive groups as opposed to low mass groups .
this extends to groups of galaxies the mild tendency already detected for rich clusters of galaxies .
finally , we detect a barely significant fundamental plane for these groups ( @xmath10 for groups with more than 8 galaxies ) but much less narrow than for clusters of galaxies .
galaxies : clusters : general ; cosmology : large - scale structure of universe
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the idea of a macroscopic state demonstrating quantum mechanical behaviour was introduced by schrdinger in 1935 @xcite . his famous thought experiment considered how a macroscopic entity ( in this case a domestic cat ) could evolve into a superposition of two distinct physical states ( alive and dead ) when entangled with a microscopic system that obeyed the laws of quantum mechanics . this possibility of a macroscopic system being simultaneously in two distinct physical states was initially considered to be a flaw in quantum mechanical theory @xcite . however , experiments have shown the predictions of quantum mechanics to be correct as superpositions of macroscopically distinct physical states have been produced in a variety of systems . these are often referred to as ` schrdinger cat states ' . for example , a group of six beryllium ions has been put into a superposition of two hyperfine states @xcite , and persistent currents ( of a few @xmath0a ) of opposing circulation in squids have been detected @xcite . bose - einstein condensates ( becs ) are attractive candidates for generating macroscopic superpositions due to the large number of atoms that share a single quantum state . this may provide a useful system in which to further test the validity or boundaries of the assumption of macroscopic realism . a macroscopic superposition has yet to be demonstrated in a bose - einstein condensate , although there exist numerous proposals for generating either a superposition of relative phase or number states @xcite . in the following we consider a particular kind of superposition state of a single component bec in a double well , that is , a superposition of the two states where the entire condensate is localised in one of the wells ( sometimes referred to as a ` noon ' state ) . if realisable , this kind of superposition promises to be useful in quantum information applications and precision interferometry , due to the measurement uncertainty scaling inversely with the number of particles ( the so - called heisenberg limit ) . a major difficulty in realising macroscopic quantum superpositions is decoherence , which occurs when interactions with the surrounding environment cause the pure superposition state to decay into a statistical mixture . however , this paper is not concerned with avoiding decoherence in the realisation of macroscopic superpositions . instead , we concentrate on measurements aimed at distinguishing a coherent superposition from a statistical mixture . we find even in a decoherence - free environment , demonstrating a superposition presents several practical challenges . once realised , measurements of the purity of the state could be useful in studying , for example , rates of decoherence . this paper begins with an introduction of the two - mode description of a double - well condensate in section [ sec2 ] . in section [ sec3 ] we consider measurements aimed at distinguishing between a coherent superposition and a statistical mixture . we focus on quadrature - based measurements , analogous to those used in quantum optics , that can be realized with a ramsey - type interference experiment . the coherence of the noon state is evident in parity measurements of the number distribution after the two modes are interfered a difficult measurement with standard atom counting techniques . we analyse the effects of atomic interactions and discuss atom loss during the interference procedure . finally , we show in section [ sec4 ] that a mesoscopic superposition of 20 atoms could be generated in a reasonable time frame if a feshbach resonance can be used to tune the atomic interactions , before concluding in section [ sec5 ] . the hamiltonian for a condensate in an external trapping potential , @xmath1 , is @xmath2 } , \label{hamiltonian}\ ] ] where @xmath3 is the field operator for the condensate , and the non - linear interaction parameter is @xmath4 , ( @xmath5 is the s - wave scattering length describing two - body collisions within the condensate , @xmath6 is the atomic mass ) . we consider the case where the external potential provides a double well confinement for the condensate . double well potentials can be generated by an optical lattice with an additional harmonic confinement to reduce the number of occupied lattice sites to two @xcite . they can also be realised on chips , where suitably arranged current carrying wires create a magnetic confinement for the condensate atoms @xcite . when a double well potential is considered , the above hamiltonian can be simplified by making use of the two - mode approximation . this means we consider each atom to be in some linear superposition of being in the left well and being in the right well . we consider the zero temperature case , where all atoms in the system are condensed . if the ground state energies of the condensate in the two single ( and separate ) wells are sufficiently separated from the energies of the condensate in all other excited single particle states , transitions to or from the two modes of interest and these higher lying states can be neglected . this is required for the two - mode description to be valid . in the two - mode approximation , the field operator is expanded as @xmath7 where @xmath8 and @xmath9 are discrete bose annihilation operators for the left and right well respectively , and @xmath10 are the ground state spatial wave functions of the condensate in the left and right wells . substituting this into equation ( [ hamiltonian ] ) , we find an effective hamiltonian @xmath11 where we have neglected the spatial overlap of the left and right well densities . the single well bound state energies , @xmath12 , are @xmath13 @xmath14 , the tunnel coupling , is @xmath15 and the effective non - linear interaction terms are @xmath16 for the remainder of this paper we assume a symmetric potential , where @xmath17 and @xmath18 . by ignoring all possibility of atom loss and decoherence , we can efficiently represent the @xmath19-body wave function using the basis @xmath20 , representing states with @xmath21 atoms in the left well , and @xmath22 atoms in the right well . any wave function can be written as a superposition of these number states , i.e. @xmath23 where @xmath24 . in this representation , the expectation value of the number of atoms in the left well is @xmath25 and the variance in the number difference is @xmath26 ^ 2 .\ ] ] for any initial state , @xmath27 , these coefficients have a time - dependence given by @xmath28 taking the derivative with respect to time , we find @xcite @xmath29 inserting the hamiltonian , ( [ hamiltonian ] ) , into this expression gives the equations of motion for the number state coefficients @xmath30 c_n(t ) + \hbar u_r \left[n^2-n \right ] c_n(t ) \nonumber \\ & - & \hbar\kappa \sqrt{\left(n - n+1 \right ) n } ~c_{n+1}(t ) -\hbar\kappa \sqrt{\left(n+1 \right ) \left(n - n \right ) } ~c_{n-1}(t ) . \label{cnbydt}\end{aligned}\ ] ] the ground states and dynamics of a condensate in the double well can then be found by solving these equations numerically . [ fig : gndstates ] shows the ground state probability distributions for the atom number in the left well ( found using imaginary time propagation @xcite ) for three different regimes - negligible , intermediate , and strongly attractive interactions . the variance in the number difference for these are 12 , 293 , and 396 respectively ( a perfect noon state would have a variance of 400 ) . when non - linear interactions are weak compared to the tunneling , the ground state number distribution is essentially binomial ( fig . [ fig : gndstates ] ( a ) ) . for large n this can be approximated as a poissonian distribution and the ground state would be a coherent state ( which has equal mean and variance of atom number in each well ) . for intermediate attractive interactions , ground states exist consisting of two well separated peaks in the number state distribution . these can be considered as superpositions of distinct physical states , and we refer to them as ` mesoscopic ' superpositions @xcite . the ground state in the limit of infinitely strong attractive interactions is a macroscopic superposition of the entire condensate localized in each well . this can be understood by realizing that it is energetically preferable for the condensate to be localized in one well , but that given the symmetry of the double well potential these two localized states are degenerate . thus in the ground state it is equally likely that the condensate will be found entirely in one well as in the other ( hence the two peaks in the probability distribution ) . this state is the macroscopic superposition that we are interested in . [ fig : gndstates ] ( c ) shows that with a ratio of non - linear interaction strength to tunnelling rate @xmath31 , the ground state of the double well condensate is close to , but not quite , an ideal superposition . we return to the problem of creating such a state in sec . [ sec4 ] . is the atom number in the left well . interaction strengths are ( a ) @xmath32 , ( b ) @xmath33 , and ( c ) @xmath34.,title="fig : " ] is the atom number in the left well . interaction strengths are ( a ) @xmath32 , ( b ) @xmath33 , and ( c ) @xmath34.,title="fig : " ] is the atom number in the left well . interaction strengths are ( a ) @xmath32 , ( b ) @xmath33 , and ( c ) @xmath34.,title="fig : " ] we now consider the problem of demonstrating that a state is a macroscopic quantum superposition , as opposed to a statistical mixture . this will allow tests of macroscopic realism , may be an important tool for studying decoherence , and is a prerequisite to building a practical interferometer using noon states . the obvious measurement to make on such a system is the atom number in each well . for the noon state , we have equal probability of finding @xmath19 atoms in the left well and none in the right , or vice - versa . however , such a measurement can not distinguish the coherent noon state @xmath35 from the statistic mixture with density operator @xmath36 there exist complimentary measurements that distinguish between a coherent superposition and a statistical mixture . one method of finding such an measurement is motivated by the expansion of the density operator of the statistical mixture as the average of all coherent phases @xmath37 , @xmath38 we see that the determination of @xmath37 constitutes proof of coherence as @xmath37 is undefined for the mixed state . consider an arbitrary operator @xmath39 ; for the mixed state in ( [ mixed_state ] ) , the expectation value of @xmath39 is the average of the two pure , separable states , @xmath40 on the other hand , for the pure noon state @xmath41 , the expectation value is @xmath42 the additional interference terms clearly display a dependence on the value of @xmath37 . therefore , an appropriate observable @xmath39 has non - zero @xmath43 . an example of an operator that achieves this is @xmath44 , which coherently transfers @xmath19 atoms from one well to the other or vice - versa . unfortunately , there is no clear way of directly measuring this observable in an experimental setting . we now consider a quadrature - based method for distinguishing the entangled noon state from the statistical mixture . in this paper , we define the quadrature operator as @xmath45 the measurement of this observable can be achieved using simple linear interference and number measurements . such a procedure is analogous to quantum optics experiments using a 50 - 50 beam splitter to interfere two photonic modes before intensity measurement , and allows access to phase information . unlike common quantum optics experiments , both modes contain a similar number of atoms and neither mode can be interpreted as a local oscillator . this explains the difference between the above definition and the standard quadrature arising from homodyne measurement , proportional to @xmath46 . both definitions have been employed in theoretical discussions of becs in the past @xcite . to realize these quadrature measurements , we propose a ramsey - type experiment ( see ref . @xcite for a description of the ramsey technique ) . similar experiments have been proposed for double well condensates ( for example , to detect a weak force @xcite ) . the first step , after creating the superposition state , is to set the tunneling rate between the wells @xmath14 and interaction strength @xmath47 to zero and let the system evolve for some time , @xmath48 , during which an energy imbalance exists between the wells ( i.e. @xmath49 is non - zero ) . the quadrature angle is set by @xmath50 . the second stage is to restore the symmetry of the wells ( i.e. set @xmath51 to zero ) , and switch on tunneling for a time of @xmath52 ( this is analogous to a beam splitting operation ) . after this we make a measurement of the atom number in each well . the difference in atom number is exactly proportional to @xmath53 . we simulate the entire procedure using the equations of motion of the number state coefficients ( [ cnbydt ] ) . it is then straightforward to numerically calculate the distribution of measurement outcomes , as given by @xmath54 , and thus any moment ( e.g. mean , variance , etc ) of the the quadrature @xmath53 . the next step is to extract information about the off - diagonal terms of the density matrix by measuring the interference terms in ( [ expectation_pure ] ) . it is straightforward to see that @xmath55 for @xmath56 . in the special case of @xmath57 , interference is observed by a sinusoidal dependence on the value of @xmath58 , as seen in fig . [ ramsey1atomuzero ] ( a ) ( where @xmath37 is the phase angle used in ( 13 ) ) . in fig . [ ramsey1atomuzero ] ( b ) we plot the same quantity for @xmath59 , and it is unsurprising that inteference is lacking for this case . however , higher moments of the quadrature measurements do contain information that can distinguish a pure noon state from a classical mixture . for @xmath19 atoms , the @xmath19th moment @xmath60 contains exactly one subterm equal to @xmath44 , while all lower moments lack such a term . in fig . [ ramsey1atomuzero ] ( c ) , we see interference fringes in the quadrature variance for the case @xmath59 . the frequency of these fringes is doubled compared to the case @xmath57 . as a function of the accumulated phase shift for an ideal superposition containing 1 atom . ( b ) the same quantity for an ideal superposition containing 2 atoms displays no interference fringes . ( c ) for the two atom superposition state , the variance ( @xmath61 ) is sensitive to the phase @xmath62 accumulated during the ramsey simulation.,title="fig : " ] as a function of the accumulated phase shift for an ideal superposition containing 1 atom . ( b ) the same quantity for an ideal superposition containing 2 atoms displays no interference fringes . ( c ) for the two atom superposition state , the variance ( @xmath61 ) is sensitive to the phase @xmath62 accumulated during the ramsey simulation.,title="fig : " ] as a function of the accumulated phase shift for an ideal superposition containing 1 atom . ( b ) the same quantity for an ideal superposition containing 2 atoms displays no interference fringes . ( c ) for the two atom superposition state , the variance ( @xmath61 ) is sensitive to the phase @xmath62 accumulated during the ramsey simulation.,title="fig : " ] in general the @xmath19 atom noon state will display fringes with frequency @xmath19 times greater than the first - order coherence measured by @xmath63 allows . specifically , the @xmath19th quadrature moment contains terms proportional to @xmath64 . it is this scaling that makes noon states of interest for precision interferometry a noon state with known @xmath37 could potentially be used to measure @xmath65 with accuracy proportional to @xmath66 ( the so - called heisenberg limit ) , compared with the @xmath67 scaling typical when using ` classical ' inteferemetric techniques . such scaling has been observed in single - photon experiments @xcite , but to - date neither with atoms nor noon states . the quadrature moments are intrinsically linked with the number distribution @xmath54 after the ramsey interference procedure , and so it follows that the off - diagonal terms in eq . ( [ expectation_pure ] ) are directly visible in this distribution . we have plotted the output in fig . [ ramseycninterference ] ( a ) for @xmath68 , and we observe a pattern where each second @xmath69 is zero . this interference pattern is sensitive to the accumulated relative phase , so that for certain values of @xmath58 the interference pattern is absent and the number state coefficients are given by a binomial distribution . on the other hand , if the initial state were a statistical mixture a binomial distribution would be expected for all values of the relative phase . after the ramsey procedure is simulated , with accumulated phase @xmath70 . initial state has 20 atoms and @xmath71 . non - linear strength @xmath72 is ( a ) zero , ( b ) @xmath73 , and ( c ) @xmath74.,title="fig : " ] after the ramsey procedure is simulated , with accumulated phase @xmath70 . initial state has 20 atoms and @xmath71 . non - linear strength @xmath72 is ( a ) zero , ( b ) @xmath73 , and ( c ) @xmath74.,title="fig : " ] after the ramsey procedure is simulated , with accumulated phase @xmath70 . initial state has 20 atoms and @xmath71 . non - linear strength @xmath72 is ( a ) zero , ( b ) @xmath73 , and ( c ) @xmath74.,title="fig : " ] the interference can be seen most clearly in the parity , defined as @xmath75 which is the difference in probability that an even or odd number of atoms in measured in one of the modes at the end of the ramsey experiment . the expectation value of the parity is an oscillatory function of the relative phase between the wells , again with a frequency proportional to the total atom number @xmath19 , as seen in fig . [ parity ] ( a , b ) . measurements of the parity were considered in detail in ref . @xcite , where the authors discuss methods for obtaining heisenberg - limited phase resolution . line lies under the @xmath32 line.),title="fig : " ] line lies under the @xmath32 line.),title="fig : " ] accurate measurement of the @xmath19th quadrature moment , or equivalently the atom parity after the ramsey experiment , is experimentally very difficult . measuring the interference pattern in the @xmath76 coefficients would require very accurate atom counting , as the relative phase value giving maximum fringe visibility for an even total atom number gives minimum fringe visibility if the total atom number is odd , and vice versa . indeed , it is apparent that the loss of a single atom would completely destroy the coherence of the noon state . therefore , the required counting efficiency @xmath77 is such that @xmath78 , which would be challenging for large values of @xmath19 . an alternative method of determining the parity has been suggested in ref . this method does not require extremely accurate atom counting ; however it does involve a third condensate mode and precise control of the nonlinearity . there are additional complications that may make this procedure difficult , even for moderate values of @xmath19 . we now consider the effect of finite non - linear interactions during the ramsey interference procedure . with an ideal superposition as the initial state , the non - linear interactions degrade the visibility of the interference pattern in the @xmath76 coefficients , as shown in fig . [ ramseycninterference ] ( b , c ) . the oscillations in the expectation value of parity are also reduced by the presence of non - linear interactions ( see fig . [ parity ] ) . for a given value of @xmath47 , the visibility of the parity oscillations decreases as the number of atoms in the superposition increase . however , we note that the phase of the interference fringes is unaffected by the non - linearity , which is important for possible interferometric applications . finally , when the initial state is an ideal noon state , we observed that including non - linear interactions can result in small fringes in the mean quadrature value even for @xmath56 . the maximum amplitude of these induced fringes is much less than the total atom number . these fringes are most noticeable for small atom numbers , as they have amplitudes of only one or two atoms regardless of the total atom number . there have been a number of proposals for generating superpositions of various kinds involving bose - einstein condensates . many of these consider superpositions of two - component condensates . in this case , methods involving adiabatic manipulation @xcite , and dynamical evolution ( making use of the interplay between nonlinear interactions and tunneling , with a specific initial relative phase ) @xcite , have been suggested to generate superpositions . superpositions of phase states have also been considered @xcite . we are interested in superpositions of number states in a single component condensate in a double well potential . one proposal for generating superpositions of this kind involves using a feshbach resonance to produce a sudden change in the interaction strength , initiating a dynamical evolution where a macroscopic superposition emerges periodically @xcite . in ref . @xcite , it was claimed that the adiabatic method was not feasible due to the infinitely long evolution time required because of the near degeneracy of the ground and first excited states in the strongly attractive regime . we find that with realistic parameters a superposition of two well - separated wave packets can be generated on a time scale of seconds using a smooth change in interaction strength . although this evolution is not necessarily adiabatic it can generate a ` mesoscopic ' superposition state . to estimate some suitable parameters for our effective hamiltonian we have considered a small condensate of rubidium atoms . around a magnetic field strength of 155 g there is a feshbach resonance for @xmath79rb atoms , allowing the s - wave scattering length to be tuned from at least 2000 @xmath80 to @xmath81 ( @xmath80 being the bohr radius ) . assuming a cigar - shaped condensate confined by trapping frequencies of 1 khz in both tight directions and 100 hz in the longitudinal direction , and using a gaussian approximation for the condensate wavefunction , we estimate that the interaction parameter , @xmath47 , given by equation ( [ interactioneqn ] ) could range from approximately 30 s@xmath82 to @xmath83 s@xmath82 . we look at simulations which are a few seconds in length ( note that condensate lifetimes of greater than 10 seconds are experimentally achievable @xcite ) . [ linchangeu ] shows the wave function as the interaction strength is changed linearly from 1 s@xmath82 to @xmath83 s@xmath82 over a timespan of 0.5 and 4 seconds . the initial state is the ground state at u=1 s@xmath82 , and @xmath84 s@xmath82 . clearly the slower change in interaction strength results , as per the adiabatic theorem , in a final state that is closer to the ideal macroscopic superposition . this can be seen in fig . [ fidelity ] , which plots the fidelity of the evolving wave function , i.e. the overlap , @xmath85 , of the ideal superposition state with the evolving wave function . the variance in the number difference between the wells of the final states in fig . [ linchangeu ] are approximately 283 and 371 , compared with 400 for an ideal superposition state containing 20 atoms . once the interaction strength is held constant at @xmath83 s@xmath82 the probability distributions @xmath54 do not change significantly with time . the parameters could be optimized further to obtain better superposition states . in general , better states are obtained when the final ratio @xmath72 is large and negative . on the other hand , the timescale for the evolution to be adiabatic is inversely proportional to both @xmath47 and @xmath14 . the requirement of physically separated wells implies a relatively small value for @xmath14 , while the timespan of the experiment is limited by atomic loss . we have not attempted to optimize the process of generating a superposition state within these bounds , but rather concentrate on measurements aimed at distinguishing even an imperfect superposition from a statistical mixture . to @xmath83 s@xmath82 in ( a ) 0.5 , and ( b ) 4 seconds . initial wavefunction is the ground state at u=1 s@xmath82 and @xmath84 s@xmath82.,title="fig : " ] to @xmath83 s@xmath82 in ( a ) 0.5 , and ( b ) 4 seconds . initial wavefunction is the ground state at u=1 s@xmath82 and @xmath84 s@xmath82.,title="fig : " ] s@xmath82 and @xmath84 compared to the ideal noon state . a slower ramp time results in a final wave function closer to that of an ideal superposition state . ] we now perform the ramsey procedure followed by parity measurements in order to detect coherence in the non - ideal superposition states generated above . [ ramseynonideal ] and [ paritynonideal ] show typical fringes in the number difference and parity for an initial non - ideal superposition state of a 20 atom condensate . in the parity , we see clear high frequency components in the interference fringes that correspond to the off - diagonal coherence between states of large atom number difference . however , as the state is not perfect , other frequency components are present , resulting in beating and a low frequency envelope . in the appendix , we formalise the relationship between the parity frequency components and coherence and show that @xmath86 for some real numbers @xmath87 . the component with angular frequency @xmath6 corresponds to coherence between elements separated by @xmath6 atoms in fock space . in fact , the highest frequency component with angular frequency @xmath19 is proportional only to @xmath88 , and is the only component observed in the results for an ideal superposition ( fig . [ parity ] ) . the presence of the same high frequency component in fig . [ paritynonideal ] ( a ) is an unambiguous demonstration that the generated state contains coherence and is _ not _ a statistical mixture . with the introduction of non - linear interactions during the ramsey interference , the amplitude and frequency of the parity fringes become less regular and are no longer periodic over @xmath58 modulo @xmath89 . it would not be easy to use the results when @xmath47 is large as an indicator of the presence of a superposition due to their irregularity , and the fact that high frequency components can not be said to indicate noon - type coherence . s@xmath82 , and ( c ) @xmath90 s@xmath82 . initial state was generated from the ground state at u=1 s@xmath82 , @xmath84 s@xmath82 , and linearly changing the interaction strength to @xmath91 s@xmath82 over 4 seconds.,title="fig : " ] s@xmath82 , and ( c ) @xmath90 s@xmath82 . initial state was generated from the ground state at u=1 s@xmath82 , @xmath84 s@xmath82 , and linearly changing the interaction strength to @xmath91 s@xmath82 over 4 seconds.,title="fig : " ] s@xmath82 , and ( c ) @xmath90 s@xmath82 . initial state was generated from the ground state at u=1 s@xmath82 , @xmath84 s@xmath82 , and linearly changing the interaction strength to @xmath91 s@xmath82 over 4 seconds.,title="fig : " ] s@xmath92 , and ( c ) @xmath90 s@xmath92 . initial state was generated from the ground state at u=1 s@xmath92 , @xmath84 s@xmath92 , and linearly changing the interaction strength to @xmath91 s@xmath92 over 4 seconds . , title="fig : " ] s@xmath92 , and ( c ) @xmath90 s@xmath92 . initial state was generated from the ground state at u=1 s@xmath92 , @xmath84 s@xmath92 , and linearly changing the interaction strength to @xmath91 s@xmath92 over 4 seconds . , title="fig : " ] s@xmath92 , and ( c ) @xmath90 s@xmath92 . initial state was generated from the ground state at u=1 s@xmath92 , @xmath84 s@xmath92 , and linearly changing the interaction strength to @xmath91 s@xmath92 over 4 seconds . , title="fig : " ] our simulations indicate that a quantum superposition of a small bose - einstein condensate may be generated in a reasonably short time by a smooth change in the atomic interaction strength from repulsive to attractive . however , we find that verifying that a superposition has been generated may not be straightforward , even in the ideal case of a perfect ` noon ' state superposition and no decoherence . we have considered a ramsey - type interference experiment as a method to measure the quadrature operator and parity thereof of a double well wave function . for a noon superposition , only the highest - order quadrature moment ( and not the mean ) is sensitive to the accumulated relative phase and is therefore useful in distinguishing between a statistical mixture and a coherent superposition . parity measurements after a ramsey - type experiment on an ideal superposition state show a dependence on the relative phase that could in principle be used to verify the presence of the superposition . a statistical mixture would not be expected to show any dependence on the relative phase if parity measurements were performed . the presence of non - linear interactions during the ramsey experiment degrades the amplitude of the parity oscillations . measurements of a non - ideal superposition state also show a high - frequency dependence on the relative phase a smoking gun indication of noon - type coherence . however , we observed that the imperfect state displays additional frequencies resulting in a beating pattern . for some choices of phase angle , the visibility is close to one while the period decreases to @xmath93 , allowing it to remain useful for heisenberg - limited phase measurements . similarly as for the ideal state , non - linear interactions degrade the amplitude and regularity of the parity oscillations , however for small non - linearity , the high frequency component could still be used to verify that a mesoscopic superposition has been generated . there are many factors that may make these measurements difficult in practice . for example , phase diffusion due to non - linear interactions dramatically reduces the visibility of the parity oscillations . other effects not considered here would also reduce the fringe visibility , such as 3-body loss , and other sources of decoherence . highly accurate atom counting is required to observe the interference patterns in the @xmath76 coefficients , and the associated parity oscillations as a function of accumulated phase , unless an alternative measurement scheme could be realised . the authors wish to acknowledge support from the australian research council . for a state with precisely @xmath19 atoms , the ( discrete ) quadrature operator can take @xmath94 values . therefore , it is possible to express any linear combination of the measurement probabilities ( i.e. the @xmath95 after the ramsey interference ) as a linear sum of the first @xmath19 quadrature moments . specifically , the parity can be expressed this way , i.e. @xmath96 where @xmath97 are real numbers . expanding the power results in @xmath98 where @xmath99 represents the symmetric ordering of @xmath39 . the operator @xmath100 transfers @xmath101 atoms from the right well to the left , and therefore we can say @xmath102 where @xmath103 are appropriately chosen constants . putting everything together , we finally arrive at eq . ( [ frequencies ] ) , where the the constants @xmath87 can be obtained from the above .
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the availability of bose - einstein condensates as mesoscopic or macroscopic quantum objects has aroused new interest in the possiblity of making and detecting coherent superpositions involving many atoms . in this article
we show that it may be possible to generate such a superposition state in a reasonably short time using feshbach resonances to tune the inter - atomic interactions in a double - well condensate .
we also consider the important problem of distinguishing whether a coherent superposition or a statistical mixture is generated by a given experimental procedure .
we find that unambiguously distinguishing even a perfect ` noon ' state from a statistical mixture using standard detection methods will present experimental difficulties .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
diffusion in stationary states may be encountered either in equilibrium , where no macroscopic mass or energy fluxes are present in a system of many diffusing particles , or away from equilibrium , where diffusion is often driven by a density gradient between two open segments of the surface that encloses the space in which particles diffuse . in equilibrium states , one is interested in the _ self - diffusion _ coefficient @xmath3 , as given by the mean - square displacement ( msd ) of a tagged particle . this quantity , also called tracer diffusion coefficient , can be measured using e.g. neutron scattering , nmr or direct video imaging in the case of colloidal particles . in gradient - driven non - equilibrium steady states , there is a particle flux between the boundaries which is proportional to the density gradient . this factor of proportionality is the so - called transport or collective diffusion coefficient @xmath4 . often these two diffusion coefficients can not be measured simultaneously under concrete experimental conditions and the question arises whether one can infer knowledge about the other diffusion coefficient , given one of them . generally , in dense systems these diffusion coefficients depend in a complicated fashion on the interaction between the diffusing particles . in the case of diffusion in microporous media , e.g. in zeolites , however , the mean free path of the particles is of the order of the pore diameter or even larger . then diffusion is dominated by the interaction of particles with the pore walls rather than by direct interaction between particles . in this dilute so - called knudsen regime neither @xmath3 nor @xmath4 depend on the particle density anymore , but are just given by the low - density limits of these two quantities . one then expects @xmath3 and @xmath4 to be equal . this assumption is a fundamental input into the interpretation of many experimental data , see e.g. @xcite for an overview of diffusion in condensed matter systems . not long ago this basic tenet has been challenged by monte - carlo simulation of knudsen diffusion in pores with fractal pore walls @xcite . the authors of these ( and further ) studies concluded that self - diffusion depends on the surface roughness of a pore , while transport diffusion is independent of it . in other words , the authors of @xcite argue that even in the low density limit , where the gas particle are independent of each other and interact only with the pore walls , @xmath5 , with a dependence of @xmath3 on the details of the pore walls that @xmath4 does not exhibit . this counterintuitive numerical finding was quickly questioned on physical grounds and contradicted by further simulations @xcite which give approximate equality of the two diffusion coefficients . these controversial results gave rise to a prolonged debate which finally led to the consensus that indeed both diffusion coefficients should agree for the knudsen case @xcite . it has remained open though whether these diffusion coefficients are generally exactly equal or only approximately to a degree depending on the details of the specific setting . a physical argument put forward in @xcite suggests general equality . to see this one imagines the following _ gedankenexperiment_. imagine one colours in a equilibrium setting of many non - interacting particles some of these particles without changing their properties . at some distance from this colouring region the colour is removed . then these coloured particles experience a density gradient just as `` normal '' particles in an open system with the same pore walls would . since the walls are essentially the same and the properties of coloured and uncoloured particles are the same , the statistical properties of the ensemble of trajectories remain unchanged . hence one expects any pore roughness to have the same effect on diffusion , irrespective of whether one consider transport diffusion or self - diffusion . notice , however , that this microscopic argument , while intuitively appealing , is far from rigorous . first , the precise conditions under which the independence of the diffusion coefficients on the pore surface is supposed to be valid , is not specified . this is more than a technical issue since one may easily construct surface properties leading to non - diffusive behaviour ( cf . second , there is no obvious microscopic interpretation or unique microscopic definition of the transport diffusion coefficient for arbitrary surface structures . @xmath4 is a genuinely macroscopic quantity and a proof of equality between @xmath4 and @xmath3 ( which is naturally microscopically defined through the asymptotic long - time behaviour of the msd ) requires some further work and new ideas . one needs to establish that on large scales the knudsen process converges to brownian motion ( which then also gives @xmath3 ) . moreover , in order to compare @xmath4 and @xmath3 one needs a precise macroscopic definition of @xmath4 which is independent of microscopic properties of the system . the first part of this programme is carried out in @xcite . there we proved the quenched invariance principle for the horizontal projection of the particle s position using the method of considering the environment viewed from the particle . this method is useful in a number of models related to markov processes in a random environment , cf . e.g. @xcite . the aim of this paper is to solve the second problem of defining @xmath4 and proving equality with @xmath3 . as in @xcite we consider a random tube to model pore roughness . in contrast to @xcite , we now have to consider tubes of finite extension along the tube contour and introduce open segments at the ends of the tube . doing this rigorously then clarifies some of the salient assumptions underlying the equality of @xmath4 and @xmath3 . naturally , since we are in the dilute gas limit , there is no dependence on the particle density in either of the two diffusion constants . this obvious point has not been controversial and will not be stressed below . we note that we define @xmath4 through stationary transport in an open system since this is accessible experimentally as well as numerically in monte carlo simulation . indeed , in the literature that gave rise to the controversy that we address here , this way of defining @xmath4 is used , albeit in a non - rigorous fashion . sticking to this experimentally motivated setting we shall give below a precise definition that can be used to prove rigorously that under rather generic circumstances @xmath6 , which means that both diffusion constants depend on the pore surface in the same way . as pointed out above , this equality is expected from independence of the particles and the invariance principle for the process and its time - reversed . however , we could not find a general result applying here , and moreover , as it turns out , the proof is not entirely trivial . there are some technical difficulties to overcome because the quenched invariance principle of definition [ def_invariance_principle ] below is not very `` strong '' ( there is no uniformity assumption on the speed of convergence as a function of the initial conditions ) and the jumps of the embedded discrete - time billiard are not uniformly bounded . let us mention here that it is generally difficult to obtain stronger results in the above sense , since the corrector technique , generally used in the proof of quenched central limit theorems for reversible markov processes in random environment , is still not sufficiently well understood . to further illuminate the contents of our results we point out that in a bulk system the equality of the self - diffusion coefficient and the transport diffusion coefficient for the spread of _ equilibrium _ density fluctuations in an infinite system may be taken for granted in the case of particles that have no mutual interaction . hence another way of stating the main conclusion of our work is the assertion that the transport diffusion coefficient as defined here in a stationary _ far - from - equilibrium _ setting coincides with the usual equilibrium transport diffusion coefficient . we also address finite - size effects coming from the fact that we are dealing with diffusion in a finite , open geometry . this causes deviations from bulk results for first - passage - time properties if a tagged particle starts its motion close to one boundary . in particular , we compute the permeation time and the milne extrapolation length that characterizes the survival time of a particle injected at a boundary . as a final introductory remark , it is worth noting that the case of knudsen gas with the cosine reflection law ( which is the model considered in this paper ) is particularly easy to analyse because the stationary state can be written in an explicit form , cf.theorem [ t_stat_measure ] . as explained below , this is related to the following facts : ( i ) there is no interaction between particles , ( ii ) for random billiard ( i.e. , a motion of only one particle in a closed domain ) with the cosine reflection law the stationary measure is quite explicit , as shown in @xcite . similar questions are much more complicated when the explicit form of the stationary state is not known . this is the general situation for non - equilibrium steady states . we refer to e.g. the model of @xcite ( a chain of coupled oscillators ) where one resorts to a bound on the entropy production . this paper is organized in the following way . in section [ s_defin ] we define the infinite random tube , and then introduce the process we call random billiard . in section [ s_transport ] , we then consider a gas of independent particles with absorption / injection in a finite piece of the random tube , and we formulate our results on the stationary measure for that gas and on the transport diffusion coefficient . in section [ s_perm ] , we go on to formulate first passage time results that concern exit from and crossing of the finite tube by a tagged particle . the remaining part of the paper is devoted to the proof of our results . in section [ s_pr_prelim ] we mainly use the reversibility of the process to obtain several technical facts used later . in section [ s_pr_steady ] we prove the result on the stationary measure of the knudsen gas in the finite tube . section [ s_pr_trans ] contains the proofs of the results related to the transport diffusion coefficient , and in section [ s_pr_perm ] we prove the results related to the crossing of the finite tube . naively the transport diffusion coefficient in tube direction @xmath7 may be defined through the diffusion equation for the probability density @xmath8 , where a possible @xmath7-dependence may originate from a spatial inhomogeneity of the tube . denote by @xmath9 the particle current in the system ; assuming stationarity with a probability density @xmath10 one has @xmath11 . with fixed external densities @xmath12 at @xmath13 and @xmath14 at @xmath15 one finds by integration @xmath16 with density gradient @xmath17 and @xmath18 . by measuring the current and the boundary densities one can thus obtain the transport diffusion coefficient without having to determine the local quantity @xmath19 . this result , however , implies knowledge of the local coarse - grained boundary densities @xmath20 to be able to make any comparison with @xmath3 . in a real experimental setting as well as for a given microscopic model these boundary densities @xmath20 are difficult to obtain . in particular , there is no well - defined prescription where precisely on a microscopic scale these boundary quantities should be measured . we circumvent the problem of computing these quantities from microscopic considerations by considering the total number of particles in the tube rather than local properties of the boundary region of the tube . together with proving a large - scale linear density profile in a stationary open random tube , one may then infer the macroscopic density gradient , see the definition ( [ defdtrans ] ) below . thus one obtains a macroscopic definition of the transport diffusion coefficient which is independent of microscopic details of the model . in order to fix ideas in a mathematically rigorous form we first recall some notations from @xcite . let us formally define the random tube in @xmath21 , @xmath22 . in this paper , @xmath23 will always stand for the linear subspace of @xmath21 which is perpendicular to the first coordinate vector @xmath24 , we use the notation @xmath25 for the euclidean norm in @xmath21 or @xmath23 . for @xmath26 let @xmath27 be the open @xmath28-neighborhood of @xmath29 . define @xmath30 to be the unit sphere in @xmath21 . let @xmath31 be the half - sphere looking in the direction @xmath32 . for @xmath33 , sometimes it will be convenient to write @xmath34 , being @xmath35 the first coordinate of @xmath7 and @xmath36 ; then , @xmath37 , and we write @xmath38 , being @xmath39 the projector on @xmath23 . fix some positive constant @xmath40 , and define @xmath41 let @xmath42 be an open connected domain in @xmath23 or @xmath21 . we denote by @xmath43 the boundary of @xmath42 and by @xmath44 the closure of @xmath42 . the random tube is viewed as a stationary and ergodic process @xmath45 , where @xmath46 is a subset of @xmath47 ; cf . @xcite for a more detailed definition . we denote by @xmath48 the law of this process ; sometimes we will use the shorthand notation @xmath49 for the expectation with respect to @xmath48 . with a slight abuse of notation , we denote also by @xmath50 the random tube itself , where the billiard lives . intuitively , @xmath46 is the `` slice '' obtained by crossing @xmath51 with the hyperplane @xmath52 . we will assume that the domain @xmath51 is defined in such a way that it is an open subset of @xmath21 , and that it is connected . we write also @xmath53 for the closure of @xmath51 . in order to define the random billiard correctly , following @xcite , throughout this paper we suppose that @xmath48-almost surely @xmath54 is a @xmath55-dimensional surface satisfying the lipschitz condition . this means that for any @xmath56 there exist @xmath57 , an affine isometry @xmath58 , a function @xmath59 such that * @xmath60 satisfies lipschitz condition , i.e. , there exists a constant @xmath61 such that @xmath62 for all @xmath63 ; * @xmath64 , @xmath65 , and @xmath66 roughly speaking , lipschitz condition implies that any boundary point can be `` touched '' by a piece of a cone which lies fully inside the tube . this in its turn ensures that the ( discrete - time ) process can not remain in a small neighborhood of some boundary point for very long time ; in section 2.2 of @xcite one can find an example of a non - lipschitz domain where the random billiard behaves in an unusual way . we keep the usual notation @xmath67 for the @xmath55-dimensional lebesgue measure on @xmath47 ( usually restricted to @xmath46 for some @xmath35 ) or haar measure on @xmath68 . we write @xmath69 for the @xmath70-dimensional lebesgue measure in case @xmath71 , and haar measure in case @xmath72 . also , we denote by @xmath73 the @xmath55-dimensional hausdorff measure on @xmath54 ; since the boundary is lipschitz , one obtains that @xmath73 is locally finite ( cf . the proof of lemma 3.1 in @xcite ) . we assume additionally that the boundary of @xmath48-a.e . @xmath51 is @xmath73-a.e . _ continuously _ differentiable , and we denote by @xmath74 the set of boundary points where @xmath54 is continuously differentiable . to avoid complications when cutting a ( large ) finite piece of the infinite random tube , we assume that there exists a constant @xmath75 such that for @xmath48-almost all environments @xmath51 we have the following : for any @xmath76 with @xmath77 there exists a path connecting @xmath78 that lies fully inside @xmath51 and has length at most @xmath75 . for all @xmath79 , let us define the normal vector @xmath80 pointing inside the domain @xmath51 . we say that @xmath81 is _ seen from _ @xmath82 if there exists @xmath83 and @xmath84 such that @xmath85 for all @xmath86 and @xmath87 . clearly , if @xmath88 is seen from @xmath7 then @xmath7 is seen from @xmath88 , and we write `` @xmath89 '' when this occurs . next , we construct the knudsen random walk ( krw ) @xmath90 , which is a discrete time markov process on @xmath54 , cf . section 2.2 of @xcite . it is defined through its transition density @xmath91 : for @xmath92 @xmath93 where @xmath94 is the normalizing constant , and @xmath95 stands for the indicator function . this means that , being @xmath96 the quenched ( i.e. , with fixed @xmath51 ) probability and expectation , for any @xmath79 and any measurable @xmath97 we have @xmath98 = { \int\limits}_b k(x , y)\ , d{\nu^\omega}(y).\ ] ] we also refer to the knudsen random walk as the random walk with cosine reflection law , since it is elementary to obtain from that the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector . [ rem_not_to_infty ] in fact , in the general setting of @xcite , for unbounded domains , one has to consider the following possibility : at some moment the particle chooses the outgoing direction in such a way that , moving in this direction , it never hits the boundary of the domain again , thus going directly to the infinity . however , it is straightforward to see that , since @xmath99 , in our situation @xmath100-a.s . this can not happen . it is immediate to obtain from that @xmath101 is symmetric ( that is , @xmath102 for all @xmath92 ) ; for both the discrete- and continuous - time processes this leads to some nice reversibility properties , exploited in @xcite . clearly , @xmath91 depends on @xmath51 as well , but we usually do not indicate this in the notations in order to keep them simple . also , let us denote by @xmath103 the @xmath104-step transition density ; clearly , one obtains that @xmath105 is symmetric too for any @xmath106 . now , we define the knudsen stochastic billiard ( ksb ) @xmath107 , which is the main object of study in this paper . first , we do that for the process starting on the boundary @xmath54 from the point @xmath108 . let @xmath109 be the trajectory of the random walk , and define @xmath110 then , for @xmath111 , define @xmath112 in proposition 2.1 of @xcite it was shown that , provided that the boundary satisfies the lipschitz condition , we have @xmath113 @xmath100-a.s . , and so @xmath114 is well - defined for all @xmath115 . the quantity @xmath114 stands for the position of the particle at time @xmath116 ; since it is not a markov process by itself , we define also the cdlg version of the motion direction at time @xmath116 : @xmath117 observe that @xmath118 . recall also another notation from @xcite : for @xmath119 , @xmath120 , define ( with the convention @xmath121 ) @xmath122 so that @xmath123 is the next point where the particle hits the boundary when starting at the location @xmath7 with the direction @xmath124 . of course , we can define also the stochastic billiard starting from the interior of @xmath51 by specifying its initial position @xmath125 and initial direction @xmath126 : the particle starts at the position @xmath125 and moves in the direction @xmath126 with unit speed until hitting the boundary at the point @xmath127 ; then , the previous construction is applied , being @xmath127 the starting boundary point . we denote by @xmath128 the ( quenched ) law of ksb in the tube @xmath51 starting from @xmath7 with the initial direction @xmath124 . consider the rescaled projected trajectory @xmath129 of ksb . [ def_invariance_principle ] we say that the quenched invariance principle holds for the knudsen stochastic billiard in the infinite random tube if there exists a positive constant @xmath130 such that , for @xmath48-almost all @xmath51 , for any initial conditions @xmath131 such that @xmath132 , the rescaled trajectory @xmath133 weakly converges to the brownian motion as @xmath134 . also , for some of our results we will have to make more assumptions on the geometry of the random tube . consider the following * condition t. * * there exists a positive constant @xmath135 and a continuous function @xmath136 such that @xmath137 * in the case @xmath138 , we assume that there exist @xmath139 such that for all @xmath140 with @xmath141 there exists @xmath142 such that @xmath143 . * in the case @xmath144 , we assume that @xmath145 [ rem_almost_all ] from the fact that @xmath99 and @xmath73-almost all points of @xmath54 belong to @xmath146 , it is straightforward to obtain that for lebesgue@xmath147haar - almost all @xmath148 we have @xmath149 ( see lemma 3.2 ( i ) of @xcite ) . [ rem_clt ] in the paper @xcite we prove that , if the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle is finite ( which is true for @xmath138 , but not always for @xmath144 ) , then under certain additional conditions ( related to condition t of the present paper ) , the quenched invariance principle holds for the knudsen stochastic billiard in the infinite random tube , cf . theorem 2.2 , propositions 2.1 and 2.2 of @xcite . let us comment more on the above condition t : * in @xcite , instead of the `` uniform dblin condition '' ( ii ) , we assumed a more explicit ( although a bit more technical ) condition p , which implies that ( ii ) holds ( see lemma 3.6 of @xcite ) . in fact , in the proof of the quenched invariance principle the technical condition of @xcite is used only through the fact that it implies the uniform dblin condition . * the assumption we made for @xmath144 may seem to be too restrictive . however , is it only a bit more restrictive that the assumption that the random tube does not contain an infinite straight cylinder . as it was shown in proposition 2.2 of @xcite , if the random tube contains an infinite straight cylinder , then the averaged second moment of the projected jump length is infinite in dimension @xmath150 , and so the ( quenched ) invariance principle can not be valid . now , let us introduce the notations specific to this paper . consider a positive number @xmath151 ( which is typically supposed to be large ) ; denote by @xmath152 the part of the random tube @xmath51 which lies between @xmath153 and @xmath151 : @xmath154\}.\ ] ] denote also @xmath155 so that @xmath156 ( see figure [ f_piece_of_tube ] ) . ] observe that @xmath152 can , in fact , consist of several separate pieces , namely , one big piece between @xmath153 and @xmath151 , and possibly several small pieces near the left and the right ends ( we suppose that @xmath157 , so that there could not be two or more big pieces ) . it can be easily seen that those small pieces have no influence on the definition of the transport diffusion coefficient ; for notational convention , we still allow @xmath152 to be as described above . then , we consider a gas of independent particles in @xmath152 , described as follows . there is usual reflection on @xmath158 ; any particle which hits @xmath159 , disappears . in addition , for a given @xmath160 , new particles are injected in @xmath161 with intensity @xmath162 per unit surface area . every newly injected particle chooses the initial direction at random according to the cosine law . in other words , the injection in @xmath161 is given by an independent poisson process in @xmath163 with intensity @xmath164 . [ rem_cosine_chosen ] the choice of the cosine law for the injection of new particles is justified by theorem 2.9 of @xcite : for the ksb in a finite domain , the long - run empirical law of intersection with a @xmath55-dimensional manifold is cosine . one may think of the following situation : the random tube is connected from its left side @xmath161 to a very large reservoir containing the knudsen gas in the stationary regime ; then , the particles cross @xmath161 with approximately cosine law ( at least on the time scale when the density of the particles in the big reservoir remains unaffected by the outflow through the tube ) . in section [ s_pr_steady ] ( proof of theorem [ t_stat_measure ] ) we use this kind of argument to obtain a rigorous characterization of the steady state of this gas . we now consider this gas in the stationary regime . let @xmath165}:=[a , b]\times\xi$ ] , and let @xmath166 be the mean number of particles in @xmath167}$ ] , in a fixed environment @xmath51 . in theorem [ t_dens_grad ] below we shall see that there exists a constant @xmath168 such that @xmath169 which means that , after coarse - graining , the particle density profile is asymptotically linear . the above quantity @xmath168 is called the ( rescaled ) density gradient . we define also the current @xmath170 as the mean number of particles absorbed in @xmath171 per unit of time , and let the rescaled current be defined as @xmath172 then , consistently with the discussion in the beginning of this section , the _ transport diffusion coefficient _ @xmath4 is defined by @xmath173 now , suppose that the quenched invariance principle with constant @xmath174 holds for the stochastic billiard . our goal is to prove that @xmath4 is equal to the _ self - diffusion coefficient _ @xmath175 . to this end , we prove the following two results . first , we prove that the coarse - grained density profile is indeed linear : [ t_dens_grad ] suppose that the quenched invariance principle holds . then , for any @xmath176 there exists @xmath177 such that @xmath48-a.s . @xmath178 then , we calculate the limiting current : [ t_current ] suppose that the quenched invariance principle holds with constant @xmath174 , and assume also that condition t holds . then , we have @xmath48-a.s . @xmath179 some remarks are in place that illustrate the significance of the above theorems . theorem [ t_dens_grad ] means that @xmath180 , and using also theorem [ t_current ] , we obtain that @xmath6 . at the same time it becomes clear that such a statement can be true only asymptotically since in a finite open tube one has to expect finite size corrections of the mean particle number . these corrections may , in fact , depend strongly on the microscopic shape of the tube near the open boundaries . this implies that in experiments on real spatially inhomogeneous systems some care has to be taken as to what is measured as macroscopic density gradient . notice that with theorem [ t_current ] we also prove fick s law for diffusive transport of matter in the random knudsen stochastic billiard . since the velocity of the particles does not change at collisions with the tube walls , mass transport is proportional to energy transport . in this interpretation theorem [ t_current ] implies fourier s law for heat conduction , see e.g. @xcite for recent work on other processes . for a function @xmath181 and @xmath182 , denote @xmath183 as mentioned in the introduction , in the proof of theorems [ t_dens_grad ] and [ t_current ] we use the explicit form of the steady state for the knudsen gas in the random tube with injection from one side . let us formulate the following theorem : [ t_stat_measure ] * for the knudsen gas with absorption / injection in @xmath184 ( as before , with intensity @xmath162 per unit surface area ) the unique stationary state is poisson point process in @xmath185 with intensity @xmath186 . * for the gas with injection in @xmath161 only , the unique stationary distribution of the particle configuration is given by a poisson point process in @xmath187 with intensity measure @xmath188 \,d\alpha\ , du\ , dh.\ ] ] also , in both cases , for any initial configuration the process converges to the stationary state described above . of course , the above result is not quite unexpected . it is well known that independent systems have poisson invariant distributions ( with the single particle invariant measure for poisson intensity ) , let us mention e.g. @xcite ( section viii.5 ) and @xcite . still , we decided to include the proof of this theorem because ( as far as we know ) , it does not directly follow from any of the existing results available in the literature . let us introduce some more notations for the finite random tube . we denote by @xmath189 the set of points of @xmath190 , from where the particle can reach @xmath171 by a path which stays within @xmath152 and set @xmath191 ( see figure [ f_permeation ] ) , and let @xmath192 be the corresponding finite tube . since we are going to study now how long a tagged particle stays inside the tube and how it crosses ( i.e. , goes to the right boundary without going back to the left boundary ) , the idea is to inject it in a place from where it can actually do it . our interest is then in certain first - passage properties , in particular , the total life time of the particle inside @xmath193 ( i.e. , the time until the particle first exits @xmath193 ) and the permeation time which the particle needs to first exit @xmath193 at the end of the tube segment `` opposite '' to that where it was injected , i.e. , after crossing the tube . , and the event @xmath194 ( a trajectory crossing the tube is shown ) ] so , suppose that one particle is injected ( uniformly ) at random at @xmath195 into the tube @xmath193 ( that is , the starting location has the uniform distribution in @xmath195 , and the direction is chosen according to the cosine law ) , and let us denote by @xmath194 the event that it crosses the tube without going back to @xmath195 , i.e. , @xmath196 ( here , @xmath197 and @xmath198 are , respectively , entrance and hitting times for the discrete - time process , see and for the precise definitions ) . also , define @xmath199 to be the total lifetime of the particle , i.e. , if @xmath114 is the location of the particle at time @xmath116 , then @xmath200 . first , we calculate the asymptotic behaviour of the quenched and annealed ( averaged ) expectation of @xmath199 : [ t_exp_perm ] suppose that the quenched invariance principle holds with constant @xmath174 . we have @xmath201 observe that condition t ( i ) implies that @xmath202 is bounded away from @xmath153 , and so @xmath203 . at this point we remind the reader that here and in the next theorem the expected `` times '' are actually expected lengths of flight , related through the corresponding times through the trivial generic relation _ _ length__@xmath204__velocity__@xmath147_time_. in our knudsen gas we always assume unit velocity @xmath205 so that times can be identified with the appropriate lengths . to elucidate the physical significance of theorem [ t_exp_perm ] we observe that for usual brownian motion the expected lifetime @xmath206 of particle in an interval @xmath207 $ ] is given by @xmath208 , where @xmath209 is the starting position and @xmath210 is the diffusion coefficient . so , in particular , for a particle starting at the boundary @xmath211 ( or at @xmath212 ) the expected life time is @xmath153 . however , in a microscopic model of diffusion in a finite open system , this result can not be expected to be generally valid because of a positive probability that a particle which starts at @xmath213 would escape through the other boundary at @xmath214 . often it is found empirically that the expected life time can be approximated by @xmath215 with an effective shifted coordinate @xmath216 and effective interval length @xmath217 . the empirical shift length @xmath218 is known as milne extrapolation length @xcite , for a recent application to diffusion in carbon nanotubes see @xcite . from the definition ( [ milne ] ) one can see that the life time of a particle starting at the origin @xmath213 allows for the computation of the milne extrapolation length through the asymptotic relation @xmath219 provided the diffusion coefficient @xmath210 is known . in a physical system the milne extrapolation length depends on molecular details of the gas such as type of molecule or temperature , but in a knudsen gas also on the tube surface . in our model the properties of the gas are encoded in the unit velocity @xmath205 of the particles . observe now that the quantity @xmath220 corresponds to @xmath221 in our setting . hence , by identifying @xmath222 and using @xmath223 , theorem [ t_exp_perm ] furnishes us with the dependence of the milne extrapolation length on the tube properties through @xmath224 interestingly , @xmath218 depends only on very few generic properties of the random tube . the next result relies on theorem [ t_current ] , so we need to assume a stronger condition on the geometry of the tube . [ t_perm ] let us suppose that the quenched invariance principle is valid with @xmath174 , and assume that condition t holds . for the asymptotics of the probability of crossing , we have @xmath225 = \frac{\gamma_d|{{\mathbb s}}^{d-1}|{\hat\sigma}^2{\big\langle |\omega_0| \big\rangle_{\!{}_{{\mathbb p } } } } } { 2|{\tilde\omega}_0| } \qquad \text{${{\mathbb p}}$-a.s . , } \label{q_perm_prob}\\ \lim_{h\to\infty } h{\big\langle { { \mathtt p}_\omega}[{{\mathfrak c}}_h ] \big\rangle_{\!{}_{{\mathbb p } } } } = \frac{1}{2 } \gamma_d|{{\mathbb s}}^{d-1}|{\hat\sigma}^2{\big\langle |\omega_0| \big\rangle_{\!{}_{{\mathbb p } } } } { \big\langle |{\tilde\omega}_0|^{-1 } \big\rangle_{\!{}_{{\mathbb p}}}}. \label{a_perm_prob}\end{aligned}\ ] ] for the quenched behaviour of the conditional expectations , we have , @xmath48-a.s . @xmath226 and for the annealed ones @xmath227 as one sees from theorems [ t_exp_perm ] and [ t_perm ] , all our annealed results in fact say that one can interchange the limit as @xmath228 with integration with respect to @xmath48 . we still decided to include these results ( even though they are technically not difficult ) because , in models related to random environment , it is frequent that the annealed behaviour differs substantially from the quenched behaviour . one may find it interesting to observe that , by and @xmath229 to obtain another interesting consequence of our results , let us suppose now that @xmath48-a.s . the random tube is such that we have @xmath230 . observe that , by jensen s inequality , it holds that @xmath231 ( and the inequality is strict if the distribution of @xmath232 is nondegenerate ) , so `` roughness '' of the tube makes the quantities @xmath233 and @xmath234 increase . in other words , these quantities as well as the milne correlation length are minimized on the tubes with constant section ( which , by the way , do not have to be necessarily `` straight cylinders '' ! ) . the remaining part of the paper is devoted to the proofs of our results , and , as mentioned in the introduction , it is organized in the following way . in section [ s_pr_prelim ] we obtain several auxiliary results related to hitting of sets by the random billiard . in section [ s_pr_steady ] we obtain the explicit form of the stationary measure of the knudsen gas in the finite tube @xmath152 by using the corresponding result from @xcite about the stationary distribution of one particle in a finite domain . then , in section [ s_pr_trans ] , we apply the results of sections [ s_pr_prelim ] and [ s_pr_steady ] to obtain the explicit form of the transport diffusion coefficient . finally , in section [ s_pr_perm ] we use little s theorem to prove the results related to the crossing time of the random tube . we need first to prove several auxiliary facts for random billiard in arbitrary finite domains . as in @xcite , let @xmath235 be a bounded domain with lipschitz and a.e . continuously differentiable boundary . we keep the notation @xmath100 to denote the law of our processes , and we still use @xmath73 to denote the @xmath55-dimensional hausdorff measure on the boundary @xmath236 . consider a markov chain @xmath237 on @xmath236 , which has a transition density @xmath238 with the property @xmath239 for all @xmath240 . observe that the knudsen random walk @xmath241 has the above property , but we need to formulate the next results in a slightly more general framework , since we shall need to apply them to some other processes built upon @xmath241 . let us introduce the notations @xmath242 for the entrance and the hitting time of @xmath243 . also , for measurable @xmath244 such that @xmath245 we shall write @xmath246 = \frac{1}{{\nu^\omega}(b)}{\int\limits}_b{{\mathtt p}_\omega}^x[\cdot]\,d{\nu^\omega}(x),\ ] ] so that @xmath247 is the law for the process starting from the uniform distribution on @xmath248 . taking advantage of the reversibility of the process @xmath241 , we prove the following [ l_hitting_revers ] consider two arbitrary measurable sets @xmath249 such that @xmath250 . * suppose that @xmath251 . for any @xmath252 , we have @xmath253 } \nonumber\\ & = \frac{1}{{\nu^\omega}(b){{\mathtt p}_\omega}^b[\tau(f)<\tau^+(b ) ] } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[\tau(b)<\tau^+(f)]\ , d{\nu^\omega}(y ) \nonumber\\ & = \frac{1}{{\nu^\omega}(f){{\mathtt p}_\omega}^f[\tau(b)<\tau^+(f ) ] } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[\tau(b)<\tau^+(f)]\ , d{\nu^\omega}(y ) . \label{hitting_rev1}\end{aligned}\ ] ] * suppose that @xmath254 . for any @xmath255 , we have @xmath256 \,d{\nu^\omega}(x ) } \nonumber\\ & = { \int\limits}_{b '' } { { \mathtt p}_\omega}^x[\xi_{\tau^+(b)}\in b ' , \tau^+(b)<\tau(f ) ] \,d{\nu^\omega}(x ) . \label{hitting_rev2}\end{aligned}\ ] ] one immediately obtains the following consequence of lemma [ l_hitting_revers ] ( ii ) : [ c_trans_dens ] for any @xmath249 such that @xmath257 and @xmath254 , we have the following . * for @xmath258 , let us define the conditional ( on the event @xmath259 ) transition density @xmath260 : @xmath261 = { \int\limits}_{b '' } { \bar k}_{b , f}(x , y)\,d{\nu^\omega}(y).\ ] ] then , we have @xmath262 { \bar k}_{b , f}(x , y ) = { { \mathtt p}_\omega}^y[\tau^+(b)<\tau(f ) ] { \bar k}_{b , f}(y , x),\ ] ] that is , the random walk conditioned to return to @xmath248 without hitting @xmath263 is reversible with the reversible measure @xmath264 defined by @xmath265.\ ] ] * in particular ( take @xmath266 in the previous part ) the random walk observed at the moments of successive visits to @xmath248 is reversible with the reversible measure @xmath73 . _ proof of lemma [ l_hitting_revers ] . _ abbreviate for the moment @xmath267 . first , write using the fact that @xmath268 is symmetric @xmath269 & = \sum_{n=1}^\infty { { \mathtt p}_\omega}^b[\tau(f)=n,\tau^+(b)>n]\\ & = \sum_{n=1}^\infty { \int\limits}_b \frac{d{\nu^\omega}(x_0)}{{\nu^\omega}(b ) } { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_1 ) \ldots d{\nu^\omega}(x_{n-1 } ) \\ & \qquad\qquad\times { \int\limits}_f d{\nu^\omega}(x_n ) { \bar k}(x_0,x_1)\ldots { \bar k}(x_{n-1},x_n)\\ & = \frac{{\nu^\omega}(f)}{{\nu^\omega}(b)}\sum_{n=1}^\infty { \int\limits}_f \frac{d{\nu^\omega}(x_n)}{{\nu^\omega}(f ) } { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_{n-1 } ) \ldots d{\nu^\omega}(x_1 ) \\ & \qquad\qquad\times { \int\limits}_b d{\nu^\omega}(x_0 ) { \bar k}(x_n , x_{n-1})\ldots { \bar k}(x_1,x_0)\\ & = \frac{{\nu^\omega}(f)}{{\nu^\omega}(b ) } \sum_{n=1}^\infty { { \mathtt p}_\omega}^f[\tau(b)=n,\tau^+(f)>n]\\ & = \frac{{\nu^\omega}(f)}{{\nu^\omega}(b ) } { { \mathtt p}_\omega}^f[\tau(b)<\tau^+(f)].\end{aligned}\ ] ] then , similarly @xmath253 } \\ & = \frac{1}{{{\mathtt p}_\omega}^b[\tau(f)<\tau^+(b ) ] } \sum_{n=1}^\infty { { \mathtt p}_\omega}^b[\tau(f)=n,\xi_{\tau(f)}\in f ' , \tau^+(b)>n]\\ & = \frac{{\nu^\omega}(b)}{{\nu^\omega}(f){{\mathtt p}_\omega}^f[\tau(b)<\tau^+(f ) ] } \sum_{n=1}^\infty { \int\limits}_b \frac{d{\nu^\omega}(x_0)}{{\nu^\omega}(b ) } { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_1 ) \ldots d{\nu^\omega}(x_{n-1 } ) \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\times { \int\limits}_{f ' } d{\nu^\omega}(x_n ) { \bar k}(x_0,x_1)\ldots { \bar k}(x_{n-1},x_n)\\ & = \frac{1}{{\nu^\omega}(f){{\mathtt p}_\omega}^f[\tau(b)<\tau^+(f ) ] } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[\tau(b)<\tau^+(f)]\ , d{\nu^\omega}(y),\end{aligned}\ ] ] so is proved . let us prove . analogously to the previous computation , we write @xmath270 \,d{\nu^\omega}(x)}\\ & = { \int\limits}_{b'}d{\nu^\omega}(x ) \sum_{n=1}^\infty { { \mathtt p}_\omega}^x[\xi_{\tau^+(b)}\in b '' , \tau^+(b)=n,\tau(f)>n]\\ & = \sum_{n=1}^\infty { \int\limits}_{b ' } d{\nu^\omega}(x_0 ) { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_1 ) \ldots d{\nu^\omega}(x_{n-1 } ) \\ & \qquad\qquad\times { \int\limits}_{b '' } d{\nu^\omega}(x_n ) { \bar k}(x_0,x_1)\ldots { \bar k}(x_{n-1},x_n)\\ & = \sum_{n=1}^\infty { \int\limits}_{b '' } d{\nu^\omega}(x_n ) { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_{n-1 } ) \ldots d{\nu^\omega}(x_1 ) \\ & \qquad\qquad\times { \int\limits}_{b ' } d{\nu^\omega}(x_0 ) { \bar k}(x_n , x_{n-1})\ldots { \bar k}(x_1,x_0)\\ & = { \int\limits}_{b '' } { { \mathtt p}_\omega}^x[\xi_{\tau^+(b)}\in b ' , \tau^+(b)<\tau(f ) ] \,d{\nu^\omega}(x),\end{aligned}\ ] ] and is proved . this concludes the proof of lemma [ l_hitting_revers ] . next , we recall the dirichlet s principle : [ p_dirichlet ] consider @xmath271 with @xmath272 and @xmath254 , and denote @xmath273 $ ] ( so that , in particular , @xmath274 for all @xmath275 and @xmath276 for all @xmath277 ) . define @xmath278 , h(x)=0 \text { for all } x\in b , h(x)=1 \text { for all } x\in f\}.\ ] ] then @xmath279 = { { \mathcal e}}({\hat h},{\hat h } ) = \min_{h\in { { \mathcal h}}}{{\mathcal e}}(h , h),\ ] ] where @xmath280 _ proof . _ for the proof , we refer to the discrete case , e.g. proposition 3.8 in @xcite , and observe that the proof applies to the space - continuous case , using that , on general spaces , harmonicity in the analytic sense and in the probabilistic sense are equivalent notions by @xcite . indeed , minimizers @xmath32 of the dirichlet form are harmonic in the analytic sense , i.e. , there are in the kernel of the form ( see ( 2.10 ) in @xcite ) , though the left - hand side of ( 24 ) is the value of @xmath281 when @xmath32 is harmonic in the probabilistic sense , i.e. , the expectation of the process at some exit time ( see theorem 2.7 in @xcite ) with the appropriate boundary conditions . now , we go back to the knudsen random walk in the random tube @xmath51 . recall that @xmath105 stands for the the @xmath104-step transition density of krw , and that we have @xmath282 for all @xmath78 . let us define for an arbitrary @xmath283 @xmath284 in case @xmath42 is an interval , say , @xmath285 , we write @xmath286 instead of @xmath287 . there is the following apriory bound on the size of the jump of the random billiard : there exists a constant @xmath288 , depending only on @xmath289 and the dimension , such that for @xmath48-almost all @xmath51 @xmath290 \leq { \tilde\gamma}_1 u^{-(d-1)},\ ] ] for all @xmath56 , @xmath291 , see formula ( 54 ) of @xcite . moreover , using , for any @xmath292 it is straightforward to obtain that , for some @xmath293 @xmath294 \leq { \tilde\gamma}^{(n)}_1 u^{-(d-1)},\ ] ] for all @xmath56 , @xmath291 ( also , without restriction of generality , we can assume that @xmath295 is nondecreasing in @xmath104 ) . now , with the help of the above formula we prove the following result : [ l_separated ] for any @xmath106 there exists @xmath296 such that for all @xmath291 and @xmath182 we have @xmath297 _ proof . _ abbreviate @xmath298 . the main idea is the following : if at some step the knudsen random walk jumped from some point of @xmath299 to @xmath300 , it must cross @xmath301 , so the probability of such a jump is the same as the probability of the jump to @xmath301 in the semi - infinite tube with the boundary @xmath302 . so , we obtain @xmath303}d{\nu^\omega}(x ) { \int\limits}_{{\tilde f}^\omega[a+u,\infty)}d{\nu^\omega}(y ) k^n(x , y)}\qquad\\ & = { \int\limits}_{{\tilde f}^\omega(-\infty , a ] } { { \mathtt p}_\omega}^x[\xi_n\in { \tilde f}^\omega[a+u,\infty)]\,d{\nu^\omega}(x)\\ & \leq { \int\limits}_{{\tilde f}^\omega(-\infty , a ] } { { \mathtt p}_\omega}^x\big[\bigcup_{k=1}^n\{\xi_k\cdot{\mathbf{e}}\geq a+u , \xi_j\cdot{\mathbf{e } } < a+u \text { for all } j < k\}\big]\ , d{\nu^\omega}(x)\\ & \leq { \int\limits}_{{\tilde f}^\omega(-\infty , a]}d{\nu^\omega}(x_0 ) \sum_{k=1}^n { \int\limits}_{({\tilde f}^\omega(-\infty , a+u))^{k-1 } } d{\nu^\omega}(x_0)\ldots d{\nu^\omega}(x_{k-1})\\ & \qquad\qquad\qquad\qquad\qquad\qquad { \int\limits}_{{\tilde f}^\omega[a+u,\infty)}d{\nu^\omega}(x_k ) k(x_0,x_1)\ldots k(x_{k-1},x_k)\\ & \leq { \int\limits}_{{\tilde f}^\omega(-\infty , a]}d{\nu^\omega}(x ) { \int\limits}_v d{\nu^\omega}(y ) \big(k(x , y)+k^2(x , y)+\cdots+k^n(x , y)\big).\end{aligned}\ ] ] by symmetry of @xmath91 , we have for any @xmath177 @xmath304\ , d{\nu^\omega}(y),\ ] ] so lemma [ l_separated ] now follows from . let us consider a sequence of i.i.d . random variables @xmath305 with uniform distribution on @xmath306 ( where @xmath307 is from condition t ( ii ) ) , independent of everything . also , let us define @xmath308 . then , it is straightforward to obtain that , for any @xmath56 and @xmath309 , we have @xmath310 \geq n^{-1}r_1{\nu^\omega}(b)\ ] ] for some @xmath311 . let @xmath312 be the transition density of the process @xmath313 . observe that this process is still reversible with the reversible measure @xmath73 , so that @xmath314 for all @xmath92 . similarly to @xcite , let us define @xmath315 and @xmath316 we suppose that @xmath317 and @xmath318 are defined as in but with @xmath241 instead of @xmath237 , and let @xmath319 and @xmath320 be the corresponding quantities for the process @xmath321 . [ l_escape_upper ] suppose that @xmath322 with @xmath323 . moreover , assume that @xmath324 for all @xmath275 , and @xmath325 for all @xmath326 ( of course , the same result is valid if we assume that @xmath324 for all @xmath277 , and @xmath325 for all @xmath327 ) . then , there exist positive constants @xmath328 , @xmath329 , such that @xmath330 \leq { \tilde\gamma}_3u^{-(d-1 ) } + \frac{1}{u^2}{\int\limits}_{{\tilde f}^\omega[a , a+u]}b(x)\ , d{\nu^\omega}(x),\ ] ] and @xmath331 \leq { \tilde\gamma}_4u^{-(d-1 ) } + \frac{1}{u^2}{\int\limits}_{{\tilde f}^\omega[a , a+u]}{\hat b}(x)\ , d{\nu^\omega}(x).\ ] ] moreover , and are valid also in the finite tube @xmath152 ( in this case we assume that @xmath332 and @xmath333 ) . we now work in finite tube @xmath152 . let us use the abbreviations @xmath339 , and @xmath340 . observe that , by condition t ( i ) , we have that for some @xmath341 @xmath342 for all @xmath104 and for @xmath48-a.a . @xmath51 . to distinguish between the seconds moments of the projected jump length in finite and infinite tubes , we modify our notations in the following way . for @xmath343 , let @xmath344 and @xmath345 be the quantities defined as in and , but in the finite tube @xmath152 . let us use the notations @xmath346 and @xmath347 for the corresponding quantities in the infinite tube . now , we need an estimate on the integrals appearing in the right - hand sides of and , for the case of the finite tube : [ l_int_b ] suppose that @xmath348 and assume that @xmath138 and condition t holds . then , we have @xmath349}b_h(x)\ , d{\nu^\omega}(x ) < \infty \qquad { { \mathbb p}}\text{-a.s.},\ ] ] and the same is valid with @xmath350 on the place of @xmath351 . _ let us recall some notations from @xcite . define @xmath352 define the probability measure @xmath353 on @xmath354 by @xmath355 where @xmath356 is the @xmath357-dimensional hausdorff measure on the boundary of @xmath190 , @xmath358 is the scalar product of the normal vectors pointing inside the section and inside the tube ( see section 2 of @xcite for details ) , and @xmath359 is the normalizing constant . in lemma 3.1 of @xcite it is shown that @xmath353 is the invariant law of the environment seen from the walker , that is @xmath360 \big\rangle_{\!{}_{{\mathbb q}}}}={\big\langle f \big\rangle_{\!{}_{{\mathbb q}}}}.\ ] ] using also that @xmath361 and , it is straightforward to obtain that @xmath362 implies @xmath363 . so , using the notations of @xcite , by the ergodic theorem we obtain @xmath364}b_\infty(x)\ , d{\nu^\omega}(x ) & = \frac{1}{h } { \int\limits}_0^h d\alpha { \int\limits}_{\xi}d{\mu^\omega}_\alpha(v)\kappa^{-1}_{\alpha , v } b_\infty(\theta_\alpha\omega , v)\nonumber\\ & \to { \big\langle b_\infty \big\rangle_{\!{}_{{\mathbb q } } } } \qquad \text{as $ h\to\infty$ } , \label{conv_b}\end{aligned}\ ] ] a.s . and in @xmath365 , and the same with @xmath366 on the place of @xmath367 . then , follows from the fact that , for all @xmath151 , @xmath368 for all @xmath369 . now , with @xmath366 instead of @xmath367 , the previous inequality is not necessarily valid . so , to prove for @xmath350 instead of @xmath351 , consider @xmath56 such that @xmath370 $ ] , and write ( note that for all @xmath343 we have @xmath371 ) @xmath372\\ & \leq c_1 h^{-(d-3)}\end{aligned}\ ] ] ( recall that @xmath138 ) , and then we obtain for @xmath350 as well . next , we obtain a lower bound for certain escape probabilities : [ l_dirichlet_lower ] suppose that @xmath373 , and @xmath374 . also , assume that @xmath138 and condition t holds . then , there exist positive constants @xmath375 , @xmath376 , such that @xmath377 \geq \frac{{\tilde\gamma}_7}{n - m},\ ] ] and @xmath378 \geq \frac{{\tilde\gamma}_8}{h}.\ ] ] _ proof . _ let @xmath379 be the dirichlet form corresponding to @xmath380 ( cf . ) . first , let us prove . as in proposition [ p_dirichlet ] , we use the notation @xmath381 $ ] ; observe that @xmath274 for all @xmath382 and @xmath383 for all @xmath384 ( and hence for all @xmath385 ) . using this fact together with and cauchy - schwarz inequality , we write ( abbreviating @xmath386 ) @xmath387}\\ & = { \hat{{\mathcal e}}}({\hat h},{\hat h})\\ & \geq \sum_{j=0}^u { \int\limits}_{u_{m+j}}d{\nu^\omega}(x_j ) { \int\limits}_{u_{m+j+1}}d{\nu^\omega}(x_{j+1 } ) { \hat k}(x_j , x_{j+1 } ) ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & = \big(\prod_{j=0}^{u+1}{\nu^\omega}(u_{m+j})\big)^{-1 } { \int\limits}_{u_m}d{\nu^\omega}(x_0)\ldots { \int\limits}_{u_{m+u+1}}d{\nu^\omega}(x_{u+1})\\ & \qquad\qquad\qquad \sum_{j=0}^{u}{\nu^\omega}(u_{m+j}){\nu^\omega}(u_{m+j+1 } ) { \hat k}(x_j , x_{j+1 } ) ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & \geq n^{-1}r_1{\tilde\gamma}_5 ^ 2 \big(\prod_{j=0}^{u+1}{\nu^\omega}(u_{m+j})\big)^{-1 } { \int\limits}_{u_m}d{\nu^\omega}(x_0)\ldots\\ & \qquad\qquad\qquad \ldots { \int\limits}_{u_{m+u+1}}d{\nu^\omega}(x_{u+1})\sum_{j=0}^{u } ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & \geq \frac{n^{-1}r_1{\tilde\gamma}_5 ^ 2}{u+1 } \big(\prod_{j=0}^{u+1}{\nu^\omega}(u_{m+j})\big)^{-1 } { \int\limits}_{u_m}d{\nu^\omega}(x_0)\ldots{\int\limits}_{u_{m+u+1}}d{\nu^\omega}(x_{u+1})\\ & = \frac{n^{-1}r_1{\tilde\gamma}_5 ^ 2}{n - m+1},\end{aligned}\ ] ] and this proves . by denoting @xmath388 $ ] and writing @xmath389}\\ & \geq \sum_{j=1}^{n+1 } { \int\limits}_{u_j}d{\nu^\omega}(x_j ) { \int\limits}_{u_{j+1}}d{\nu^\omega}(x_{j+1 } ) { \hat k}(x_j , x_{j+1 } ) ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & \quad + { \int\limits}_{{{\hat d}_\ell}}d{\nu^\omega}(x_0 ) { \int\limits}_{u_1}d{\nu^\omega}(x_1 ) { \hat k}(x_0,x_1 ) ( { \hat h}(x_0)-{\hat h}(x_1))^2\end{aligned}\ ] ] in exactly the same way one can show . this concludes the proof of lemma [ l_dirichlet_lower ] . next , we need ( pointwise ) estimates on the probabilities of exiting the tube at the left boundary : [ l_gamblers_ruin ] assume condition t and @xmath138 . suppose also that @xmath390 , and @xmath391 $ ] . then , there exists @xmath392 such that for all @xmath382 we have @xmath393 \leq \frac{{\tilde\gamma}_9(n - m+1)}{h}.\ ] ] _ proof . _ from now on , we assume for technical reasons that @xmath394 ( in any case , otherwise the upper bound @xmath395 is good enough for us ) . first , by lemmas [ l_escape_upper ] and [ l_int_b ] , we obtain that @xmath396 \leq \frac{c_1}{h}.\ ] ] next , lemma [ l_dirichlet_lower ] implies that @xmath397 \geq \frac{c_2}{n - m+1}.\ ] ] also , from it is clear that for any @xmath382 we have @xmath398 \geq { { \mathtt p}_\omega}^x[{\hat\xi}_1\in u_m ] \geq c_3\ ] ] for some @xmath399 . now , denote @xmath400 , @xmath401 to be the successive times when the set @xmath402 is visited . by corollary [ c_trans_dens ] ( i ) and , we obtain that , conditional on not hitting @xmath403 , the process of successive returns to @xmath402 is reversible with the reversible density @xmath404 , such that for all @xmath382 @xmath405 for some positive constants @xmath406 . using also and , we obtain that there are constants @xmath407 such that for any @xmath70 @xmath408 & \leq \frac{c_6}{h},\\ { { \mathtt p}_\omega}^{u_m}[{\hat\tau}(v_n)<{\hat\tau}^+(u_m ) \mid { \hat\tau}({{\hat d}_\ell}\cup v_n)>\sigma_k ] & \geq \frac{c_7}{n - m+1}.\end{aligned}\ ] ] so , we can write @xmath409 & = \sum_{k=1}^\infty { { \mathtt p}_\omega}^{u_m}\big[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n ) \mid { \hat\tau}({{\hat d}_\ell}\cup v_n)\in(\sigma_{k-1},\sigma_k]\big ] \nonumber\\ & \qquad\qquad \times { { \mathtt p}_\omega}^{u_m}\big[{\hat\tau}({{\hat d}_\ell}\cup v_n ) \in(\sigma_{k-1},\sigma_k]\big ] \nonumber\\ & \leq \sum_{k=1}^\infty \frac{c_6/h}{c_7/(n - m+1 ) } { { \mathtt p}_\omega}^{u_m}\big[{\hat\tau}({{\hat d}_\ell}\cup v_n ) \in(\sigma_{k-1},\sigma_k]\big ] \nonumber\\ & = \frac{c_6c_7^{-1}(n - m+1)}{h}. \label{bound_set}\end{aligned}\ ] ] now , the `` pointwise '' version of is substantially more difficult to prove . consider a sequence of i.i.d . random variables @xmath410 with @xmath411 = n^{-1}r_1{\tilde\gamma}_5\ ] ] ( recall and ) . then , one can couple the random sequences @xmath412 with @xmath413 in such a way that when the event @xmath414 occurs , @xmath415 has the stationary distribution on @xmath416}$ ] . we denote by @xmath417 and @xmath418 the probability and expectation with fixed @xmath51 and @xmath419 , and let @xmath420 be the expectation with respect to @xmath419 . one can formally define @xmath417 in the following way . for any @xmath421 , define the transition density @xmath422 by @xmath423 let @xmath424 be the distribution on @xmath54 with the density @xmath422 , and let @xmath425 be the uniform distribution on @xmath426 . then , given @xmath427 , the law of @xmath415 under @xmath417 is given by @xmath428 also , let us define @xmath429 . now , observe that @xmath430 = [ { \hat\xi}_{j-1}\cdot{\mathbf{e } } ] \text { on } \{j={\hat\kappa}\}\ ] ] and , for @xmath431 such that @xmath432 , @xmath433 \mid { \hat\kappa}=j \big ) } \nonumber\\ & = e^{\zeta } \big({{\mathtt p}_{\omega,\zeta}}^x\big[|({\hat\xi}_i- { \hat\xi}_{i-1})\cdot{\mathbf{e}}| \geq u \big ] \mid \zeta_i=0 \big ) \nonumber\\ & \leq \frac{1}{p^{\zeta}[\zeta_i=0 ] } { { \mathtt p}_\omega}^x\big[|({\hat\xi}_i- { \hat\xi}_{i-1})\cdot{\mathbf{e}}| \geq u \big ] \nonumber\\ & \leq c_8h^{-(d-1 ) } , \label{<j}\end{aligned}\ ] ] recall . then , write using and @xmath434 e^{\zeta}\big({{\mathtt p}_{\omega,\zeta}}^{u_0}\big [ \max_{\ell\leq{\hat\kappa } } |({\hat\xi}_\ell-{\hat\xi}_0)\cdot{\mathbf{e}}| \geq s\big]\mid { \hat\kappa}=j\big ) \nonumber\\ & \leq \sum_{j=1}^\infty p^{\zeta}[{\hat\kappa}=j ] e^{\zeta}\big({{\mathtt p}_{\omega,\zeta}}^x\big[\text{there exists } i\leq j \text { such that } \nonumber\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad \geq s / j\big]\mid { \hat\kappa}=j\big ) \nonumber\\ & \leq \sum_{j=1}^\infty p^{\zeta}[{\hat\kappa}=j ] jc_9\big(\frac{s}{j}\big)^{-(d-1 ) } \nonumber\\ & = c_9 s^{-(d-1 ) } \sum_{j=1}^\infty j^d p^{\zeta}[{\hat\kappa}=j ] \nonumber\\ & = c_{10 } s^{-(d-1)}. \label{oc_kappa_1}\end{aligned}\ ] ] now , using , we have for an arbitrary @xmath382 @xmath435 } \nonumber\\ & = e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n ) ] \nonumber\\ & \leq e^{\zeta } { { \mathtt p}_{\omega,\zeta}}^x\big[\max_{j\leq { \hat\kappa } } { \hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n)\big ] \nonumber\\ & \qquad + e^{\zeta } { { \mathtt p}_{\omega,\zeta}}^x\big[\max_{j\leq { \hat\kappa } } \nonumber\\ & \leq e^{\zeta } { { \mathtt p}_{\omega,\zeta}}^x\big[\max_{j\leq { \hat\kappa } } { \hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n)\big ] + c_{11}h^{-(d-1)}. \label{i+ii}\end{aligned}\ ] ] let us deal with the first term in . we have , taking advantage of and ( recall that @xmath138 ) @xmath436}\\ & \leq \sum_{\ell\geq h/16 } e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x\big[[{\hat\xi}_{{\hat\kappa}}]=\ell\big ] { { \mathtt p}_\omega}^{u_\ell}[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n ) ] \\ & \leq \frac{c_{12}(n - m+1)}{h}\sum_{\ell\geq m } e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x\big[[{\hat\xi}_{{\hat\kappa}}]=\ell\big]\\ & \quad + \sum_{\frac{h}{16}\leq\ell < m } \frac{c_{12}\big((n - m+1)+(m-\ell)\big)}{h } e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x\big[[{\hat\xi}_{{\hat\kappa}}]=\ell\big]\\ & \leq \frac{c_{13}(n - m+1)}{h},\end{aligned}\ ] ] and this concludes the proof of lemma [ l_gamblers_ruin ] . next , we prove a result which shows that it is unlikely that a particle crosses the tube @xmath152 `` too quickly '' . suppose that one particle is injected ( uniformly ) at random at @xmath161 into the tube @xmath152 , and we still denote by @xmath194 the event that it crosses the tube without going back to @xmath161 , i.e. , @xmath437 ( one can see that there is no conflict with the notation of section [ s_perm ] ) . also , recall that @xmath199 stands for the total lifetime of the particle as defined in section [ s_perm ] , i.e. , if @xmath114 is the location of the particle at time @xmath116 , then @xmath438 . [ l_cross ] for any @xmath439 there exists ( large enough ) @xmath177 with the following property : there exists large enough @xmath440 such that for all @xmath441 @xmath442 \leq \frac{{\varepsilon}}{h}.\ ] ] _ proof . _ for @xmath443 , we say that @xmath56 is @xmath444-good if @xmath445 \geq 1-{\varepsilon}_1.\ ] ] let @xmath446 be a large positive parameter to be specified later ; for @xmath447 denote @xmath448 ; denote also @xmath449 now , consider first the case @xmath138 . from now on we suppose that @xmath177 is sufficiently large to assure the following : @xmath450 \geq 1-\frac{{\varepsilon}_1}{2},\ ] ] where @xmath451 is the standard brownian motion and @xmath452 is the corresponding probability measure . in this case , if the invariance principle holds , then for any fixed @xmath453 every @xmath7 is @xmath444-good for all large enough @xmath151 . using the monotone convergence theorem , it is straightforward to obtain that for fixed @xmath454 there exists large enough @xmath455 such that for all @xmath441 @xmath456 > \frac{3}{4}.\ ] ] then , by the ergodic theorem , there exists large enough @xmath455 such that for all @xmath441 there exists @xmath457 such that @xmath458 , and @xmath459 . now , let us consider also the event @xmath460 ( that is , with respect to the process @xmath461 , the particle enters @xmath462 before coming back to @xmath161 ) . then , write @xmath463 & \leq { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h , { { \mathcal t}}_h\leq m^{-1}h^2 ] + { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h^c,{{\mathfrak c}}_h ] \nonumber\\ & = { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h]{{\mathtt p}_\omega}^{{{\hat d}_\ell}}[{{\mathcal t}}_h\leq m^{-1}h^2\mid { \hat{{\mathfrak c}}}_h ] \nonumber\\ & \quad + { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{{\mathfrak c}}_h]{{\mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h^c\mid{{\mathfrak c}}_h ] . \label{cross_decompose}\end{aligned}\ ] ] now , by lemmas [ l_escape_upper ] and [ l_int_b ] , we can write for some @xmath464 @xmath465,{{\mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h]\ } \leq \frac{c_1}{h}.\ ] ] then , from we obtain that @xmath466 \leq \sup_{x\in{{\hat d}_r } } { { \mathtt p}_\omega}^x\big[\max_{j\leq n}|\xi_j\cdot{\mathbf{e}}-h|\geq h/4\big ] \leq c_2 h^{-(d-1)}\ ] ] for some @xmath467 . so , to complete the proof of , it remains to prove that the term @xmath468 $ ] in is small . to do this , let us recall that , by lemma [ l_hitting_revers ] ( i ) , for any @xmath469 , we have @xmath470 = \big({\nu^\omega}({{\hat d}_\ell}){{\mathtt p}_\omega}^{{{\hat d}_\ell}}[\hat{{\mathfrak c}}_h]\big)^{-1 } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}^+(v_{ln_0})]\,d{\nu^\omega}(y).\ ] ] by lemma [ l_dirichlet_lower ] , we have that for some @xmath399 @xmath471\big)^{-1 } \leq c_3 h.\ ] ] for @xmath472 denote @xmath473 . using lemma [ l_gamblers_ruin ] , we can write for any @xmath474 @xmath475 & = { \int\limits}_{\partial\omega}{\hat k}(y , z ) { { \mathtt p}_\omega}^z[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_{ln_0})]\,d{\nu^\omega}(z ) \nonumber\\ & \leq { \int\limits}_{{{\hat d}_\ell}}{\hat k}(y , z)\,d{\nu^\omega}(z ) \nonumber\\ & \qquad + \sum_{j=1}^{ln_0 } \frac{{\tilde\gamma}_9(ln_0-j+1)}{h } { \int\limits}_{u_j}{\hat k}(y , z)\,d{\nu^\omega}(z)\nonumber\\ & \leq \frac{{\tilde\gamma}_9}{h}\sum_{j=1}^{ln_0 } { \int\limits}_{s_j}{\hat k}(y , z)\,d{\nu^\omega}(z ) . \label{upper_py}\end{aligned}\ ] ] so , by , in the case @xmath138 , we obtain from that for some positive constant @xmath476 @xmath477 \leq \frac{c_4}{h}\ ] ] and , by , , and the construction of @xmath478 we obtain that @xmath479 \leq c_3c_4{\varepsilon}_2.\ ] ] next , integrating over @xmath480 , we obtain from lemma [ l_separated ] that @xmath481\,d{\nu^\omega}(y ) & \leq \frac{{\tilde\gamma}_9}{h}\sum_{j=1}^{ln_0 } { \int\limits}_{v_{n_0}\setminus i_{n_0}}d{\nu^\omega}(y ) { \int\limits}_{s_j}d{\nu^\omega}(z){\hat k}(y , z)\\ & \leq \frac{c_5{\tilde\gamma}_9}{h } \sum_{j=1}^{ln_0 } ( ln_0+l - j)^{-(d-1)}\\ & \leq \frac{c_6}{h } l^{-(d-2)}.\end{aligned}\ ] ] again using , , we obtain that @xmath482 \leq c_3c_6 l^{-(d-2)}.\ ] ] so , and imply that for any @xmath483 there exists large enough @xmath214 such that for all large enough @xmath151 we have @xmath484 \geq 1-{\varepsilon}_3.\ ] ] but then , since all @xmath485 are @xmath444-good , from we obtain that @xmath486 \leq 1-(1-{\varepsilon}_1)(1-{\varepsilon}_3).\ ] ] using , , and in , we conclude the proof of in the case @xmath138 . let us prove the lemma in the case @xmath144 . take @xmath487 note that @xmath488 , so lemma [ l_escape_upper ] implies that @xmath489\leq c_7h^{-1}$ ] for some @xmath490 . by condition t ( iii ) we obtain that @xmath491 = 0,\ ] ] and , since for any @xmath492 , @xmath493 we have @xmath494 , we then obtain @xmath495 \geq 1-{\varepsilon}_4\ ] ] for a small @xmath496 . the proof of in the case @xmath144 then follows in the same way . in this section we prove the theorem that characterizes the stationary regime for the knudsen gas in a finite tube . _ proof of theorem [ t_stat_measure ] . _ in order to prove item ( i ) , we consider the process with absorbing / injection boundaries in _ both _ @xmath161 and @xmath171 ( that is , the injection is given by two independent poisson processes in @xmath497 and @xmath498 with intensities @xmath499 in both cases ) . fix a sequence of positive numbers @xmath500 such that @xmath501 for all @xmath70 . for each @xmath70 , consider a domain @xmath502 with the following properties * @xmath503 , @xmath504 , @xmath505 ; * @xmath506 ; * any segment @xmath507 with @xmath508 , @xmath509 has length at least @xmath510 ] ( one may construct such a domain e.g. as shown on figure [ f_reservoir ] ) . now , let us consider @xmath511 independent particles in @xmath502 . by theorem 2.4 of @xcite , the unique invariant measure of this system is product of uniform measures in location and direction . we are going to compare this process ( observed only on @xmath152 ) with the process with absorbing / injection boundaries in both @xmath161 and @xmath171 ( naturally , we assume that the injection is with the cosine law and with the same intensity mentioned in theorem [ t_stat_measure ] . let @xmath512 be the expectation for the above process in @xmath502 with @xmath511 particles , with respect to the invariant measure . also , we denote by @xmath513 the expectation with respect to the process with absorbing / injection boundaries in @xmath514 at time @xmath116 , with the initial configuration chosen from the poisson point process in @xmath185 with intensity @xmath186 . let @xmath515 be a function on @xmath187 , taking values on the interval @xmath516 $ ] . for a configuration @xmath517 in @xmath187 ( which means that we have @xmath518 particles with positions @xmath519 and vector speeds @xmath520 ) , write @xmath521 denote also by @xmath522 the mean value of @xmath515 on @xmath187 . clearly , we have @xmath523 also , it is straightforward to obtain that @xmath524 since , as @xmath525 , the binomial distribution with parameters @xmath511 and @xmath526 converges to the poisson distribution with parameter @xmath527 , for any @xmath515 we have @xmath528 now , let us fix @xmath529 and prove that for any @xmath439 @xmath530 for all large enough @xmath70 . for this , denote by @xmath531 the total number of particles which entered @xmath152 through the right boundary @xmath171 up to time @xmath529 . for the process with absorption / injection , an elementary calculation shows that @xmath531 has poisson distribution with parameter @xmath532 . let us suppose without restriction of generality that @xmath533 and denote @xmath534 \text { such that } x+vt\in{{\hat d}_r}\};\ ] ] observe that @xmath535 . now , a particle starting in @xmath536 with the direction @xmath124 will cross @xmath171 by time @xmath529 iff @xmath537 . so , it is straightforward to obtain that , for the process in @xmath502 , the random variable @xmath531 has the binomial distribution with parameters @xmath511 and @xmath538 as @xmath525 . then , conditioned on @xmath539 , for both processes the @xmath104 entering particles to @xmath171 ( seen as a point process on @xmath540 $ ] ) are independent , each having density @xmath541 . observe that the same considerations apply also to the particles which enter through @xmath161 . to obtain , we use now the following coupling argument . first of all , as we already know , the initial configurations restricted to @xmath152 for both processes can be successfully coupled with probability that converges to @xmath395 as @xmath525 . then , by the argument we just presented , the same applies for the process of particles entering through @xmath514 . this shows that , with large probability , both processes can be successfully coupled . now , combining with and using the fact that a point process is uniquely determined by its characteristic functional ( cf . e.g. section 5.5 of @xcite ) , we obtain that the poisson point process in @xmath185 with intensity @xmath186 is invariant for the knudsen gas with absorption / injection in @xmath184 . as for the convergence to the stationary state and the uniqueness , this follows from an easy coupling argument . indeed , consider one process starting from the invariant measure defined above , and another process starting from an arbitrary ( fixed ) configuration . the initial particles are independent , but the newly injected particles are the same for both processes . then , since any fixed particle will eventually disappear , the coupling time is a.s.finite , and so the system converges to the unique stationary state . ( using theorem 2.1 of @xcite , with some more work one can show that , for _ fixed _ tube , this convergence is exponentially fast ; however , we do not need this kind of result in the present paper . ) this concludes the proof of the part ( i ) . let us prove the part ( ii ) . still considering the process with absorption and injection in @xmath184 , suppose that the particles entering through @xmath171 are coloured red , and the particles entering through @xmath161 are coloured green . so , we need to compute the stationary measure for green particles . using the ( quasi ) reversibility of knudsen stochastic billiard ( see theorem 2.5 of @xcite ) , we obtain that , given that there is a particle in @xmath34 with the vector speed @xmath32 , the probability that it is green equals @xmath542.\ ] ] using also the part ( i ) , we obtain that , for the gas with injection only in @xmath161 , the stationary measure is that of poisson point process with intensity @xmath543 \,d\alpha\ , du\ , dh.\ ] ] note also that convergence and uniqueness follow from the same coupling argument as in part ( i ) . this concludes the proof of theorem [ t_stat_measure ] . let us observe also that theorem [ t_stat_measure ] allows us to characterize the stationary measure for knudsen gas where the injection takes place from both sides , but with different intensities ( which are constant on @xmath161 and @xmath171 ) . we have [ cor_different ] consider now knudsen gas with injection from both sides , with respective intensities @xmath162 and @xmath544 on @xmath161 and @xmath171 ( without restriction of generality , let us suppose that @xmath545 ) . then , a poisson point process with intensity measure @xmath546\big ) \,d\alpha\ , du\ , dh\ ] ] is the steady state of the knudsen gas . indeed , one may imagine that particles of type @xmath395 are injected from both sides with intensity @xmath544 and particles of type @xmath150 are injected only from the left with intensity @xmath547 , and use theorem [ t_stat_measure ] . for integers @xmath548 define @xmath549 let @xmath550 be a brownian motion with diffusion constant @xmath174 , starting from the origin ; we define ( being @xmath551 the expectation with respect to the probability measure on the space where the brownian motion is defined ) @xmath552 to be the probabilities of the corresponding events for this brownian motion . fix an integer @xmath177 . for @xmath553 and @xmath453 define @xmath554 intuitively , @xmath555 is the scaling factor one needs to use in order to assure that the rescaled ( and projected on @xmath556 ) trajectory of the knudsen stochastic billiard stays sufficiently close to the brownian motion . by the portmanteau theorem , observe that , if the knudsen stochastic billiard starting from @xmath557 satisfies the quenched invariance principle , this means that for any @xmath453 it holds that @xmath558 . since , for @xmath48-almost every @xmath51 , the invariance principle holds for a.a . starting points @xmath557 , we have @xmath559 by the monotone convergence theorem , we obtain that for all @xmath560 there exists @xmath561 such that @xmath562 so , using the ergodic theorem , we obtain for almost all @xmath51 and all @xmath151 large enough @xmath563 then , by theorem [ t_stat_measure ] , we can write @xmath564.\ ] ] now , let us prove that the rescaled density gradient is given by @xmath565 . _ proof of theorem [ t_dens_grad ] . _ fix an arbitrary @xmath176 and suppose that @xmath177 is a ( large ) integer . consider the quantity @xmath566 defined by , and suppose that @xmath567 is large enough to assure that ( recall ) @xmath568 abbreviate @xmath569 and consider any integer @xmath570 $ ] . suppose that @xmath571 , @xmath572 are such that @xmath573 $ ] , and @xmath574 . then , since @xmath575 , from we obtain that @xmath576 & \leq { { \mathtt p}_\omega}^{z , h}[\wp_{-(m - j)}({\hat z}^{(\phi)})<\wp_j({\hat z}^{(\phi ) } ) ] \nonumber\\ & = { { \mathtt e}_\omega}^{z , h } \big(1-r_{m - j , j}({\hat z}^{(\phi)})\big)\nonumber\\ & \leq \frac{j}{m } + m^{-2}\nonumber\\ & \leq \frac{j+1}{m},\label{upper_b_gambler}\end{aligned}\ ] ] and @xmath576 & \geq { { \mathtt p}_\omega}^{z , h}[\wp_{-(m - j+1)}({\hat z}^{(\phi)})<\wp_{j-1}({\hat z}^{(\phi ) } ) ] \nonumber\\ & = { { \mathtt e}_\omega}^{z , h } \big(1-r_{m - j+1,j-1}({\hat z}^{(\phi)})\big ) \nonumber\\ & \geq \frac{j-1}{m } - m^{-2}\nonumber\\ & \geq \frac{j-2}{m}. \label{lower_b_gambler}\end{aligned}\ ] ] also , by the ergodic theorem , we can choose @xmath151 large enough so that for all @xmath577 @xmath578 } \big| -{\big\langle |\omega_0| \big\rangle_{\!{}_{{\mathbb p}}}}\big| = \bigg|\frac{m}{h}{\int\limits}_{\frac{(j-1)h}{m}}^{\frac{jh}{m } } - { \big\langle |\omega_0| \big\rangle_{\!{}_{{\mathbb p}}}}\bigg| \leq m^{-1}.\ ] ] so , by , , , , @xmath579 analogously , using instead of , we obtain @xmath580 then , we obtain from and , and so the proof of theorem [ t_dens_grad ] is concluded . at this point , let us formulate an additional result which will be used in section [ s_pr_perm ] . [ p_1/6 ] define @xmath581\nonumber\\ & \qquad\qquad\qquad\qquad\qquad \times { { \mathtt p}_\omega}^{(\alpha , u),h}[\wp_{h-\alpha}(x\cdot{\mathbf{e } } ) < \wp_{-\alpha}(x\cdot{\mathbf{e } } ) ] , \label{def_m*}\end{aligned}\ ] ] and suppose that the quenched invariance principle holds . then , for any @xmath176 there exists @xmath177 such that @xmath48-a.s . @xmath582 _ proof . _ the proof is quite analogous to the proof of theorem [ t_dens_grad ] . now , we calculate the limiting rescaled current . _ proof of theorem [ t_current ] . _ first , we obtain an upper and a lower bounds for @xmath583 , where @xmath584 . by e.g. the formula * 1*.2.0.2 of @xcite , we have @xmath585 = { \int\limits}_0^t\frac{|a|}{\sqrt{2\pi}{\hat\sigma}s^{3/2 } } \exp\big(-\frac{a^2}{2{\hat\sigma}^2s}\big)\ , ds.\ ] ] so , for @xmath586 @xmath587 - p[\wp_{-j}(b^{({\hat\sigma } ) } ) < \wp_i(b^{({\hat\sigma})})]\nonumber\\ & \geq -m^{-2/5 } + { \int\limits}_0^m\frac{i}{\sqrt{2\pi}{\hat\sigma}s^{3/2 } } \exp\big(-\frac{i^2}{2{\hat\sigma}^2s}\big)\ , ds . \label{gtil_lower}\end{aligned}\ ] ] also , for any @xmath588 , @xmath589\nonumber\\ & = { \int\limits}_0^m\frac{i}{\sqrt{2\pi}{\hat\sigma}s^{3/2 } } \exp\big(-\frac{i^2}{2{\hat\sigma}^2s}\big)\ , ds . \label{gtil_upper}\end{aligned}\ ] ] in particular , for @xmath590 , we obtain after some elementary computations that there exists a positive constant @xmath591 such that @xmath592 next , we employ the same strategy as in the proof of theorem [ t_dens_grad ] . fix a large @xmath177 , and suppose that @xmath593 is such that holds . now , let @xmath594 be the expected number of particles that were absorbed in @xmath161 up to time @xmath595 , in the stationary regime . clearly , we have then @xmath596 . so , one can write @xmath597\nonumber\\ & \qquad { } \times { { \mathtt p}_\omega}^{(\alpha , u),h}\big[\wp_{h-\alpha } ( x\cdot{\mathbf{e}})\leq \frac{h^2}{m } , \wp_{h-\alpha}(x\cdot{\mathbf{e } } ) < \wp_{-\alpha } ( x\cdot{\mathbf{e}})\big]\nonumber\\ & + { { \mathtt e}_\omega}{\widetilde w}_{h , m},\label{calc_current}\end{aligned}\ ] ] where @xmath598 is the mean number of particles that were injected in @xmath161 , successfully crossed the tube , and then hit @xmath171 before time @xmath595 . to obtain the corresponding upper bound , fix an arbitrary @xmath439 and suppose that @xmath177 is large enough so that of lemma [ l_cross ] holds for those @xmath610 . the term @xmath611 of can be estimated in the following way : @xmath612 \leq c_4 \frac{h^2}{m}\times\frac{{\varepsilon}}{h},\ ] ] so @xmath613 . then , analogously to , using also , we have for some @xmath614 @xmath615 now , observe that @xmath616 with this observation , theorem [ t_current ] follows from and . observe that , since the particles are independent , the knudsen gas in the finite tube @xmath193 can be regarded as a @xmath617 queueing system ; moreover , using e.g.theorem 2.1 of @xcite it is straightforward to obtain that the service time ( which is the lifetime of a newly injected particle ) is a random variable with exponential tail . then , let us recall the following basic identity of queuing theory ( known as little s theorem ) : [ prop_little ] suppose that @xmath618 is the arrival rate , @xmath619 is the mean number of customers in the system , and @xmath620 is the mean time a customer spends in the system , then @xmath621 . _ _ see e.g. section 5.2 of @xcite . to understand intuitively why this fact holds true , one may reason in the following way : by large time @xmath116 , the total time of all the customers in the system would be ( approximately ) @xmath622 on one hand , and @xmath623 on the other hand . _ proof of theorem [ t_exp_perm ] . _ this result almost immediately follows from theorem [ t_dens_grad ] by using proposition [ prop_little ] . first , for the gas of independent particles the arrival rate is @xmath624 recall that the particles are injected in @xmath195 only . then , from theorem [ t_dens_grad ] it is straightforward to obtain that for the mean number of particles @xmath625 in the system , we have @xmath626 then , proposition [ prop_little ] implies . to prove the corresponding annealed result , note that @xmath627 by theorem [ t_stat_measure ] ( ii ) . so , applying the bounded convergence theorem , we obtain . _ proof of theorem [ t_perm ] . _ first , observe that in the stationary regime the particles leave the system at the right boundary with rate @xmath628 , and this should be equal to the entrance rate @xmath629 $ ] of the particles which cross the tube , with @xmath618 from . so , follows from theorem [ t_current ] . to prove , observe that , by using lemma [ l_escape_upper ] with @xmath630 and @xmath631 , we obtain that for some positive constants @xmath632 which do not depend on @xmath51 @xmath633 \leq c_1 + \frac{c_2}{h } { \int\limits}_{{\tilde f}^\omega(0,h)}b(x)\ , d{\nu^\omega}(x).\ ] ] by , the collection of random variables @xmath634,h>1)$ ] is uniformly integrable , and this implies . in order to prove , denote by @xmath635 the mean number of particles in the stationary regime that _ will exit _ at @xmath171 . observe that , by theorem [ t_stat_measure ] ( ii ) and proposition [ p_1/6 ] , @xmath636 so , using and proposition [ prop_little ] , we obtain . the relations and follow from and . now , observe that and immediately follow from , , and , so now it remains only to prove . let @xmath637 , @xmath638 be the moments of successive visits to @xmath195 for the process in the finite tube . by corollary [ c_trans_dens ] , @xmath639 is uniformly distributed in @xmath195 for all @xmath70 , and so we can write @xmath640 \leq k{{\mathtt p}_\omega}[{{\mathfrak c}}_h].\ ] ] then , using , lemma [ l_dirichlet_lower ] , and the fact that the random variables @xmath641 are independent of everything , we obtain @xmath642\\ & \leq \sum_{k=1}^\infty { { \mathtt p}_\omega}[{\hat\tau}({{\hat d}_r})<{\hat\tau}({{\tilde d}_\ell } ) , \sigma_{k-1}<z_1+\cdots+z_{{\hat\tau}({{\hat d}_r})}<\sigma_k]\\ & \leq { { \mathtt p}_\omega}[z_1+\cdots+z_j\neq \sigma_\ell \text { for all $ \ell < k$ and all~$j$ } \mid \tau({{\hat d}_r})<\sigma_k]\\ & \qquad \times { { \mathtt p}_\omega}[\tau({{\hat d}_r})<\sigma_k]\\ & \leq { { \mathtt p}_\omega}[{{\mathfrak c}}_h ] \sum_{k=1}^\infty k(1-n^{-1})^{\lceil\frac{k-1}{n}\rceil},\end{aligned}\ ] ] and this implies that @xmath643\geq c_4/h$ ] for some @xmath644 not depending on @xmath51 . since @xmath645 , one obtains from the bounded convergence theorem . we thank takashi kumagai for pointing us reference @xcite . the work of f.c . was partially supported by cnrs ( umr 7599 `` probabilits et modles alatoires '' ) and anr polintbio . s.p.was partially supported by cnpq ( 300886/20080 ) . thanks dfg ( priority programme spp 1155 ) for financial support . the work of m.v . was partially supported by cnpq ( 304561/20061 ) . s.p . and m.v . also thank fapesp ( 2009/523798 ) , cnpq ( 471925/20063 , 472431/20099 ) , and capes / daad ( probral ) for financial support . f. comets , s. popov , g.m . schtz , m. vachkovskaia ( 2010 ) quenched invariance principle for knudsen stochastic billiard in random tube . to appear in : _ ann . cooper ( 1981 ) _ introduction to queueing theory _ ( 2nd ed . ) . north holland . s. zschiegner , s. russ , r. valiullin , m .- o . coppens , a .- dammers , a. bunde , j. krger ( 2008 ) normal and anomalous diffusion of non - interacting particles in linear nanopores . . phys . j. _ * 161 * ( 109 ) .
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we consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate ( the tube axis ) . inside the random tube , which is stationary and ergodic ,
non - interacting particles move straight with constant speed . upon hitting the tube walls ,
they are reflected randomly , according to the cosine law : the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector .
steady state transport is studied by introducing an open tube segment as follows : we cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis .
particles which leave this piece through the segment boundaries disappear from the system . through stationary injection of particles at one boundary of the segment a steady state with non - vanishing stationary particle current
is maintained .
we prove ( i ) that in the thermodynamic limit of an infinite open piece the coarse - grained density profile inside the segment is linear , and ( ii ) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube .
thus we prove fick s law and equality of transport diffusion and self - diffusion coefficients for quite generic rough ( random ) tubes .
we also study some properties of the crossing time and compute the milne extrapolation length in dependence on the shape of the random tube . +
* keywords : * cosine law , knudsen random walk , random medium , self - diffusion coefficient , transport diffusion coefficient , random walk in random environment + * ams 2000 subject classifications : * 60k37 .
secondary : 37d50 , 60j25 @xmath0universit paris 7 , ufr de mathmatiques , case 7012 , 2 , place jussieu , f75251 paris cedex 05 , france + e - mail : ` [email protected] ` , url : ` http://www.proba.jussieu.fr/\simcomets ` @xmath1department of statistics , institute of mathematics , statistics and scientific computation , university of campinas unicamp , rua srgio buarque de holanda 651 , cep 13083859 , campinas sp , brazil + e - mails : ` [email protected] ` , ` [email protected] ` + urls : ` http://www.ime.unicamp.br/\simpopov ` , ` http://www.ime.unicamp.br/\simmarinav ` @xmath2forschungszentrum jlich gmbh , institut fr festkrperforschung , d52425 jlich , deutschland + e - mail : ` [email protected] ` , + url : ` http://www.fz-juelich.de/iff/staff/schuetz_g/ `
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the existence of feeble magnetic fields of several microgauss in our galaxies @xcite , as well as of gigagauss in intense laser - plasma interaction experiments @xcite and of billions of gauss in compact astrophysical objects @xcite ( e.g. super dense white dwarfs , neutron stars / magnetars , degenerate stars , supernovae ) is well known . the generation mechanisms for seed magnetic fields in cosmic / astrophysical environments are still debated , while the spontaneous generation of magnetic fields in laser - produced plasmas is attributed to the biermann battery @xcite ( also referred to as the baroclinic vector containing non - parallel electron density and electron temperature gradients ) and to the return electron current from the solid target . computer simulations of laser - fusion plasmas have shown evidence of localized anisotropic electron heating by resonant absorption , which in turn can drive a weibel - like instability resulting in megagauss magnetic fields @xcite . there have also been observations of the weibel instability in high intensity laser - solid interaction experiments @xcite . furthermore , a purely growing weibel instability @xcite , arising from the electron temperature anisotropy ( a bi - maxwellian electron distribution function ) is also capable of generating magnetic fields and associated shocks @xcite . however , plasmas in the next generation intense laser - solid density plasma experiments @xcite would be very dense . here the equilibrium electron distribution function may assume the form of a deformed fermi - dirac distribution due to the electron heating by intense laser beams . it then turn out that in such dense fermi plasmas , quantum mechanical effects ( e.g. the electron tunneling and wave - packet spreading ) would play a significant role @xcite . the importance of quantum mechanical effects at nanometer scales has been recognized in the context of quantum diodes @xcite and ultra - small semiconductor devices @xcite . also , recently there have been several developments on fermionic quantum plasmas , involving the addition of a dynamical spin force @xcite , turbulence or coherent structures in degenerate fermi systems @xcite , as well as the coupling between nonlinear langmuir waves and electron holes in quantum plasmas @xcite . the quantum weibel or filamentational instability for non - degenerate systems has been treated in @xcite . in this work , we present an investigation of linear and nonlinear aspects of a novel instability that is driven by equilibrium fermi - dirac electron temperature anisotropic distribution function in a nonrelativistic dense fermi plasma . specifically , we show that the free energy stored in electron temperature anisotropy is coupled to purely growing electromagnetic modes . first , we take the wigner - maxwell system @xcite with an anisotropic fermi - dirac distribution for the analysis of the linearly growing electromagnetic perturbations as a function of the physical parameters . second , we use a fully kinetic simulation to assess the saturation level of the magnetic fields as a function of the growth rate . the treatment is restricted to transverse waves , since the latter are associated with the largest weibel instability growth rates . the nonlinear saturation of the weibel instability for classical , non - degenerate plasmas has been considered elsewhere @xcite . it is well known @xcite that a dense fermi plasma with isotropic equilibrium distributions does not admit any purely growing linear modes . this can be verified , for instance , from the expression for the imaginary part of the transverse dielectric function , as derived by lindhard @xcite , for a fully degenerate non - relativistic fermi plasma . it can be proven ( see eq . ( 30 ) of @xcite ) that the only exception would be for extremely small wavelengths , so that @xmath0 , where @xmath1 is the wave number and @xmath2 the characteristic fermi wave number of the system . however , in this situation the wave would be super - luminal . on the other hand , in a classical vlasov - maxwell plasma containing anisotropic electron distribution function , we have a purely growing weibel instability @xcite , via which dc magnetic fields are created . the electron temperature anisotropy arises due to the heating of the plasma by laser beams @xcite , where there is a signature of the weibel instability as well . in the next generation intense laser - solid density plasma experiments , it is likely that the electrons would be degenerate and that electron temperature anisotropy may develop due to an anisotropic electron heating by intense laser beams via resonant absorption , similar to the classical laser plasma case @xcite . in a dense laser created plasma , quantum effects must play an important role in the context of the weibel instability . in order to keep the closest analogy with the distribution function in phase space for the classical plasma , we shall use the wigner - maxwell formalism for a dense quantum plasma @xcite . here the distribution of the electrons is described by the wigner pseudo - distribution function @xcite , which is related to the fermi - dirac distribution widely used in the random phase approximation @xcite . proceeding with the time evolution equation for the wigner function ( or quantum vlasov equation @xcite ) , we shall derive a modified dispersion relation accounting for a wave - particle duality and an anisotropic wigner distribution function that is appropriate for the fermi plasma . the results are consistent with those of the random phase approximation , in that they reproduce the well - known transverse density linear response function for a fully degenerate fermi plasma @xcite . consider linear transverse waves in a dense quantum plasma composed of the electrons and immobile ions , with @xmath3 , where @xmath4 is the wave vector and @xmath5 is the wave electric field . following the standard procedure , one then obtains the general dispersion relation @xcite for the transverse waves of the wigner - maxwell system @xmath6 where @xmath7 is the frequency , @xmath8 is the speed of light in vacuum , @xmath9 is the planck constant divided by @xmath10 , @xmath11 the rest electron mass , @xmath12 the unperturbed plasma number density , @xmath13 the electron plasma frequency , @xmath14 is the velocity vector , and @xmath15 is the equilibrium wigner function associated to fermi systems . for spin @xmath16 particles , the equilibrium pseudo distribution function is in the form of a fermi - dirac function . here we allow for velocity anisotropy and express @xmath17 + 1 } \,,\ ] ] where @xmath18 is the chemical potential , @xmath19 the boltzmann constant , and the normalization constant is @xmath20 here @xmath21 is a polylogarithm function @xcite . also , @xmath22 $ ] , where @xmath23 and @xmath24 are related to velocity dispersion in the direction perpendicular and parallel to @xmath25 axis , respectively . in the special case when @xmath26 , the usual fermi - dirac equilibrium is recovered . the chemical potential is obtained by solving the normalization condition ( [ e3 ] ) , yielding , in particular , @xmath27 in the limit of zero temperature , where @xmath28 is the fermi energy . also , the fermi - dirac distribution @xmath29 , where @xmath4 is the appropriated wave vector in momentum space , is related to the equilibrium wigner function ( [ e2 ] ) by @xmath30 , with the factor @xmath31 coming from spin @xcite . however , these previous works refer to the cases where there is no temperature anisotropy . notice that it has been suggested @xcite that in laser plasmas the weibel instability is responsible for further increase of @xmath24 with time . inserting ( [ e2 ] ) into ( [ e1 ] ) and integrating over the perpendicular velocity components , we obtain @xmath32 where @xmath33\bigr\ } \nonumber \\ & - & { \rm li}_{2}\bigl\{-\exp\bigl[-\bigl(\nu - \frac{h}{2}\bigl)^2 + \beta\mu\bigl]\bigr\}\biggr ) \ , . \nonumber \end{aligned}\ ] ] in ( [ e5 ] ) , @xmath34 is the dilogarithm function @xcite , @xmath35 is a characteristic parameter representing the quantum diffraction effect , @xmath36 , and @xmath37 , with @xmath38 . in the simultaneous limit of a small quantum diffraction effect ( @xmath39 ) and a dilute system ( @xmath40 ) , it can be shown that @xmath41 , where @xmath42 is the standard plasma dispersion function @xcite . it is important to notice that either ( [ e1 ] ) or ( [ e4 ] ) reproduces the transverse dielectric function calculated from the random phase approximation for a fully degenerate quantum plasma @xcite , in the case of an isotropic system . the simple way to verify this equivalence is to put @xmath43 in ( [ e1 ] ) and then take the limit of zero temperature , so that @xmath44 for @xmath45 , and @xmath46 otherwise , where @xmath47 is the fermi velocity . however , to the best of our knowledge , there is no corresponding calculation for an anisotropic fermi equilibrium , as necessary in laser - solid interaction experiments with an anisotropic electron heating due to resonant absorption . also notice that in this letter we are mainly interested in the real part of the transverse response function , since we are looking for purely growing instabilities ( @xmath48 ) , so that the contribution from the poles at ( [ e4 ] ) is not relevant . ( @xmath49 ) and @xmath50 , relevant for the next generation inertially compressed material in intense laser - solid density plasma interaction experiments . the temperature anisotropies are @xmath51 ( dashed line ) , @xmath52 ( solid line ) and @xmath53 ( dotted line ) , yielding , respectively , @xmath54 , @xmath55 and @xmath56 . , width=321 ] ( @xmath49 ) . here the temperature anisotropy is @xmath52 . we used @xmath57 ( dashed line ) , @xmath50 ( solid line ) and @xmath58 ( dotted line ) , yielding @xmath59 , @xmath60 and @xmath61 , respectively.,width=321 ] we next solve our new dispersion relation ( [ e4 ] ) for a set of parameters that are representative of the next generation laser - solid density plasma interaction experiments . the normalization condition ( [ e3 ] ) can also be written as @xmath62=(4/3\sqrt{\pi})(\beta{\cal e}_f)^{3/2}$ ] , which is formally the same relation holding for isotropic fermi - dirac equilibria @xcite . for a given value on the product @xmath63 and the density , this relation yields the value @xmath64 , from which the temperatures @xmath65 and @xmath66 can be calculated , if we know @xmath67 . consider only purely growing modes . from the definition ( [ e5 ] ) , one can show that @xmath68 when @xmath69 for a finite wavenumber @xmath1 . from ( [ e4 ] ) we then obtain the maximum wavenumber for instability as @xmath70 . when @xmath71 , the range of unstable wavenumbers shrinks to zero . in figs . 1 and 2 , we have used the electron number density @xmath72 , which can be obtained in laser - driven compression schemes . the growth rate for different values on @xmath67 is displayed in fig . we see that the maximum unstable wavenumber is @xmath70 , as predicted , and that the maximum growth rate occurs at @xmath73 . figure 1 also reveals that the maximum growth rate of the instability is almost linearly proportional to @xmath74 . in fig . 2 , we have varied the product @xmath63 , which is a measure of the degeneracy of the quantum plasma . we see that for @xmath63 larger than @xmath75 , the instability reaches a limiting value , which is independent of the temperature , while thermal effects start to play an important role for @xmath63 of the order unity . ( top panel ) and @xmath76 ( bottom panel ) as a function of space and time , for @xmath50 and @xmath77 . the magnetic field has been normalized by @xmath78 . we see a nonlinear saturation of the magnetic field components at an amplitude of @xmath79.,width=321 ] , over the simulation box ( top panel ) , and the logarithm of the magnetic field maximum ( bottom panel ) as a function of time , for @xmath52 and @xmath50 . the magnetic field has been normalized by @xmath78 . from the logarithmic slope of the magnetic field in the linear regime we find @xmath80 . , width=321 ] from several numerical solutions of the linear dispersion relation , we have been able to deduce an approximate scaling law for the instability as @xmath81 , where the constant is approximately @xmath82 . using that @xmath83 , we have @xmath84 for the maximum growth rate of the weibel instability in a degenerate fermi plasma . this scaling law , where the growth rate depends on the fermi energy and the temperature anisotropy , should be compared to that of a classical plasma @xcite , where the growth rate depends on the thermal energy and the temperature anisotropy . for a maxwellian plasma , it has been found @xcite that the weibel instability saturates nonlinearly once the magnetic bounce frequency @xmath85 has increased to a value comparable to the linear growth rate . in order to assess the nonlinear behavior of the weibel instability for a degenerate plasma , we have carried out a kinetic simulation of the wigner - maxwell system . we have assumed that the quantum diffraction effect is small , so that the simulation of the wigner equation can be approximated by simulations of the vlasov equation by means of an electromagnetic vlasov code @xcite . as an initial condition for the simulation , we used the distribution function ( 2 ) . in order to give a seed for any instability , the plasma density was perturbed with low - frequency fluctuations ( random numbers ) . the results are displayed in figs . 3 and 4 , for the parameters @xmath50 and @xmath52 , corresponding to the solid lines in figs . 1 and 2 . figure 3 shows the magnetic field components as a function of space and time . we see that the magnetic field initially grows , and saturates to steady state magnetic field fluctuations with an amplitude of @xmath86 . the maximum amplitude of the magnetic field over the simulation box as a function of time is shown in fig . 4 , where we see that the magnetic field saturates at @xmath87 , while the linear growth rate of the most unstable mode is @xmath88 . similar to the classical maxwellian plasma case @xcite , we can thus estimate the magnetic field ( in tesla ) as @xmath89 for a degenerate fermi plasma . for our parameters parameters relevant for intense laser - solid interaction experiments , we will thus have magnetic fields of the order @xmath90 ( one gigagauss ) . in conclusion , we have demonstrated the existence of the weibel instability for a wigner - maxwell dense quantum plasma , taking into account an anisotropic fermi - dirac equilibrium distribution function and the quantum diffraction effect . numerically solving the dispersion relation for transverse waves , we found the dependence of the growth rate on the fermi energy and the temperature anisotropy . the nonlinear saturation level of the magnetic field was found by means of kinetic simulations , which show a linear dependence between the growth rate and the saturated magnetic field . the present results may account for intense magnetic fields in dense quantum plasmas , such as those in the next generation of intense laser - solid density plasma interaction experiments . 99 l. w. wildrow , rev . phys . * 74 * , 775 ( 2002 ) ; 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we present an investigation for the generation of intense magnetic fields in dense plasmas with an anisotropic electron fermi - dirac distribution . for this purpose , we use a new linear dispersion relation for transverse waves in the wigner - maxwell dense quantum plasma system .
numerical analysis of the dispersion relation reveals the scaling of the growth rate as a function of the fermi energy and the temperature anisotropy .
the nonlinear saturation level of the magnetic fields is found through fully kinetic simulations , which indicates that the final amplitudes of the magnetic fields are proportional to the linear growth rate of the instability .
the present results are important for understanding the origin of intense magnetic fields in dense fermionic plasmas , such as those in the next generation intense laser - solid density plasma experiments .
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for almost four decades , since the discovery of quasars , mounting observational evidence has accumulated that black holes indeed exist in nature . recent observations ( wilms et al . 2001 ) of the steep emissivity of seyfert 1 galaxy mcg6 - 30 - 15 , indicating strong photon emission at radii near the event horizon ; and observations of the lack of evidence of the expected ion `` dusty '' torus of m87 ( perlman et al . 2001 ) , have prompted astrophysicists to suggest a new energy source . however , it is hardly a new energy source to relativists , i.e. , those who study einstein s theory of general relativity . they knew for sometime , at least theoretically , what black holes were capable of doing ( williams 1991 , 1995 ) . williams ( 1995 , 2003 , 2002 ) shows , through theoretical and numerical ( monte carlo ) _ n - particle _ calculations of penrose ( 1969 ) processes , occurring at radii inside the ergosphere of a rotating black hole near the event horizon : including the `` plunging '' regimes ( bardeen , press , & teukolsky 1972 ; williams 1995 ; krolik 1999 ) , that the black hole can yield escaping particles with energies up to @xmath3 gev . these particles escape in the form of collimated , symmetrical and asymmetrical jets about the polar axis , confirming the existence of intrinsically collimated vortical jets , found theoretical by de felice & calvani ( 1972 ) ; de felice & curir ( 1992 ) ; de felice & carlotto ( 1997 ) ; de felice & zanotti ( 2000 ) : from geometrical studies of particle trajectories in a kerr ( 1963 ) metric . the kerr metric in general describes the spacetime separation of events in the gravitational field of a rotating compact massive object . in light of the above observational surprises , particularly the steep emissivity of x - rays producing the broad fe k@xmath4 emission line at @xmath5 kev in mcg6 - 30 - 15 ( wilms et al . 2001 ) and similar agns , it appears that gravity has triumphed over proposed forms of electromagnetic energy extraction from a black hole , as will be described in this paper . this should be of no surprise near the event horizon , where the gravitational forces are so strong that electromagnetic radiation itself becomes trapped . overall , energy extraction from black holes and the production of their associated jets have been the most poorly understood phenomena of today . it is clear that gravitational accretion and magnetic fields play a role , but how ? has been the mystery . we observe these jets in quasars and microquasars due to supermassive and stellar size black holes , respectively . therefore , we know that any effective model must have the commonality to explain jets in both systems . at present there are two popular trains of thought associated with energy extraction and the production of jets in black holes : one is that the jets are inherent properties of geodesic trajectories in the kerr metric of a rotating black hole , and thus , can be described by einstein s general theory of relativity ; and the other is that the accretion disk and its magnetic field through magnetohydrodynamics ( mhd ) are producing the jets . perhaps it could be a combination of the two , with gravity controlling the flow near the event horizon ( williams 2004 ) , and mhd controlling the flow at distances farther away . the observations of the jet of m87 suggest this may be the case ( junor , biretta , & livio 1999 ; perlman et al . 2001 ) . there are some proposed mhd model calculations using a general relativistic accretion disk that involve having the magnetic field lines of the disk `` anchor '' to conductive ionized particles of the disk , inside the ergospheric region , extracting rotational energy from a kerr black hole , by way of a poynting flux of electromagnetic energy , out to infinity . such models have been proposed to explain recent observations of possible direct evidence for the extraction of energy from a rotating black hole ( wilms et al . 2001 ) . in this paper , however , i point out problems with such models that make these models highly improbable to be at work , i.e. , extracting the energy needed to be consistent with general observations of sources powered by black holes . it is agreed by the author that some form of the penrose mechanism is employed , but it is argued below that electromagnetic energy extraction is not an effective way to use this mechanism . associated problems with such models are described in detail in the appendix . in a classical paper by bardeen et al . ( 1972 ) , astrophysical implausible penrose processes are discussed concerning the breakup of subrelativistic objects in the ergosphere . however , i point out that the `` penrose - williams '' mechanism , described by williams ( 1995 ) , involves relativistic scattering processes : such processes can be very efficient ( piran , shaham , & katz 1975 ; williams 1995 ) , and do not fall under the category of being implausible due to hydrodynamical constraints ( bardeen et al . 1972 ) , since the incident and target particles in the collisions are already relativistic , having speeds @xmath6 . the penrose mechanism as described here ( williams 1995 ) has a `` one - on - one '' consistent relationship with accretion disk particles . for example , particles from the accretion disk can populate the high energy gravitationally blueshifted trapped orbits ( or so - called plunging orbits ) at @xmath7 , the marginal stable orbit ( bardeen et al . particles in these now populated orbits can undergo penrose processes with lower soft x - ray energy infalling accretion disk photons : penrose compton scattering ( pcs ) produces copious distributions of high energy x - rays and soft @xmath8-rays , and penrose pair production ( ppp ) ( @xmath0 ) produces copious distributions of relativistic electron - positron ( @xmath9 ) pairs , with up to @xmath10% of the particles escaping along vortical orbits ( [ sec : collimation ] ) that circle the polar axis of the kbh many times , as spacetime itself is dragged around because of gravity ( williams 2004 ) . the particles escape to infinity along well defined four - momentum trajectories , with some intersecting the disk ( i.e. , returning to be reprocessed and/or escaping to infinity ) . this scenario is particularly consistent with recent observations of mcg6 - 30 - 15 ( [ sec : mcg ] ) , and other black hole sources ( [ sec : observation ] ) . importantly , in these penrose processes we do not need the magnetic field of the accretion disk to `` communicate '' between the accretion disk and the black hole . therefore , there is no need for the blandford and znajek ( bz ) ( 1977 ) proposed type models ( and their many associated problems ) in the direct role of energy extraction from a spinning black hole . however , their presence appears to be need once particles are on escaping orbits , serving the same effects they do in the jets of protostars , i.e. , appearing to have a dominant role on a large scale , within the weak field limit , at distances outside the strong effects of general relativity . as for producing the observed synchrotron radiation , indicating the present of a magnetic field near the core region , this radiation could very well be produced by the intrinsically self - induced magnetic field due to the dynamo - like action of the escaping penrose produced @xmath9 pairs , escaping on vortical , coil - like trajectories concentric the polar axis , in the form of a swirling `` current '' plasma . this , therefore , adds more to the unimportance of the accretion disk magnetic field near the event horizon . moreover , although suggested to be evidence of rotational magnetic energy extraction from the seyfert 1 galaxy mcg6 - 30 - 15 ( wilms et al . 2001 ) , it appears , as we shall see in this paper , that it is gravitational energy momentum being extracted , in the form of a relativistic particle flux via penrose processes , as described by williams ( 1995 ) , and _ not _ the poynting flux of electromagnetic energy suggested : produced by magnetic field lines torquing the black hole or plunging accretion disk material , as described by the bz - type models . in the penrose - williams mechanism , the steep emissivity profile [ @xmath11 of x - ray photons observed ( wilms et al . 2001 ) , requiring a x - ray source that is both powerful and very centrally concentrated ( which can not be explained by standard accretion disk models ) , is consistent with energy being extracted by penrose compton scattering processes , occurring at radii between the marginally bound and marginally stable orbits , @xmath12 and @xmath13 , respectively ( williams 1995 ) . this black hole source mcg6 - 30 - 15 will be discussed further in [ sec : mcg ] . nevertheless , once these penrose processes have occurred and particles are on escaping trajectories , they can then interact , say with the expected large scale structure disk magnetic field , at some effective radius @xmath14 where this field becomes important in jet collimation , probably similar to a relative radius existing for the collimated bi - polar jets of protostars , which , as mentioned above , appear to be undergoing some type of bz effect the direct effect is still somewhat unclear . it appears that the magnetic field of the accretion disk serves to aid in collimating into jets gravitational binding energy release due to gravitational accretion , in both protostars and agns ( or microquasars ) ; however , in the latter it appears that the jets are superimposed with collimated particles from penrose processes . so , overall , in this paper , an analysis of the penrose mechanism is presented to describe gravitational - particle interactions close to the event horizon at radii @xmath15 and down to the _ photon orbit _ , @xmath16 , for a canonical kbh with @xmath17 ( thorne 1974 ) , where @xmath18 is the angular momentum per unit mass parameter . in this fully general relativistic description , polar jets of relativistic particles of photons and @xmath9 pairs are produced and collimated by gravity alone , without the necessity of the external magnetic field of the accretion disk . this theoretical and numerical model of penrose processes can apply to any size black hole , and suggests a complete theory for the extraction of energy momentum from a rotating black hole . in [ sec : model ] a summary of the general formalism of the model is presented . in [ sec : results ] , results of theoretical and numerical calculated luminosities and energies are presented , along with discussion of the escaping particles space momentum trajectories : featuring asymmetrical polar distributions and vortical orbits . also in [ sec : results ] , agreement with observations of specific sources are presented . finally , in [ sec : conclusions ] a summary and conclusions are presented . the primary model ( williams 1995 ) consists of a supermassive @xmath19 rotating kerr black hole plus particles from an assumed relativistic bistable thin disk / ion corona [ or torus , i.e. , advection - dominated accretion flow ( adaf ) ] , two - temperature [ separate temperatures for protons ( @xmath20 k ) and electrons ( @xmath21 k ) ] accretion ( williams 2003 ; novikov & thorne 1973 ; eardley & lightman 1975 ; eilek 1980 ; eilek & kafatos 1983 ) . the bistable accretion disk can exist either in the thin disk phase and/or the ion corona ( adaf ) phase , or oscillate between the two ( see williams 2003 and references therein ) in various degrees which could be responsible for the observed variability . the penrose effect as employed here can operated in either phase . the penrose mechanism is used to extract rotational energy momentum by scattering processes inside the ergosphere ( @xmath22 , in the equatorial plane for @xmath17 ) . see williams ( 1995 ) for a detailed description of the model . the `` quasi - penrose '' ( williams 1991 , 1995 ) processes investigated are ( a ) penrose compton scattering ( pcs ) of equatorial low energy radially infalling photons by equatorially confined ( @xmath23 ) and nonequatorially confined ( @xmath24 ) orbiting target electrons , at radii between the marginally bound ( @xmath25 ) and marginally stable ( @xmath26 ) orbits , where @xmath27 is the so - called carter constant ( carter 1968 ) , referred to as the @xmath28 value ( williams 1995 ) ; ( b ) ppp ( @xmath29 ) at @xmath12 ; and ( c ) ppp ( @xmath0 ) by equatorial low energy radially infalling photons and high energy gravitationally blueshifted ( by factor @xmath30 ) nonequatorially confined @xmath8-rays at the photon orbit ( @xmath16 ) , where @xmath31 is the `` blueshift '' factor given by the @xmath32 component of the kerr metric ( see williams 1995 ) . although in the scattering plane the incident angle of the infalling photon relative to the target particle is expected in general to be at least between @xmath33 ( due to the bending of light and/or inertial frame dragging ) , maximum energy is extracted in the process when the incident angle is @xmath34 , as it is for radially infalling photons ( @xmath35 , where @xmath36 is the azimuthal coordinate angular momentum ) . note , the target particles are initially in bound ( marginally stable or unstable ) trapped orbits , trapped in the sense of possibly having no other way of escaping save these penrose processes ( bardeen et al . 1972 ; williams 1995 ) . note also that , as the nonequatorially confined target particle , whose orbital trajectory is derived by williams ( 1991 , 1995 ; see also williams 2002 ) , passes through the equatorial plane , in its bound circular orbit at constant radius , the @xmath28 value , a constant of motion as measured by an observer at infinity ( carter 1968 ; williams 1995 ) , equals @xmath37 , where @xmath38 is the polar coordinate momentum of the particle . setting @xmath39 , in the carter constant expression for the orbital @xmath28 value , gives the maximum and minimum latitudinal angles of the trajectories about the equatorial plane for wilkins ( 1972 ) `` spherical - like '' nonequatorially confined orbits ( see williams 2003 ) . these unstable , bound or marginally bound orbits ( equatorial , nonequatorial ) of the target particles are assumed to be populated by accretion disk instability processes and prior penrose processes . such particles must satisfy conditions to have a turning point at the scattering radius ( note , a bound stable orbit is considered to have a `` perpetual '' turning point ) . these conditions depend on the orbital conserved parameters of the particle : @xmath40 , the energy , and @xmath36 , or @xmath28 . in williams ( 1995 , 2002 ) such conditions are discussed in detail ; see also the possible scenario discussion in [ sec : asymmetry ] of this present manuscript . in addition , the `` instability phase , '' during which the target particle orbits are presumed to be populated , could very well be related to the timescale of the prominent observed variabilities of the source . radial infalling equatorially confined incident photons are assumed , not only for maximum energy extraction but for the simplicity of their geodesics as well , since it appears that an infalling equatorially confined photon will not acquire gravitationally blueshifted orbital energy momentum as measured by an observer at infinity , only frame dragging blueshifted energy ( eq . [ 2.8@xmath41 of williams 1995 ) . this is because the @xmath28 value of such photons is zero ( see eq . [ 2.27 ] of williams 1995 ) . the incoming photons , however , need not be confined to the equatorial plane . in these calculations if equatorially confined infalling photons were not desired , @xmath42 of the initial photon would not be set equal zero . that is , the model calculation is set up such that one can change the initial energy - momentum four vector components ( or four - momenta ) of the incident and target particles to accommodate any 3-space dimensional geometrical disk configuration . moreover , in an adaf ( including the ion corona ) , during the infall of particles , through the ergosphere , some of the particles are expected to become trapped in nonequatorial `` spherical - like orbits '' ( wilkins 1972 ) : such orbits would past through the equatorial plane : here is where the scattering takes place in these calculations . note , the target photons at the photon orbit can only exist in nonequatorially confined orbits ( williams 1995 ) ; this is also pointed out by bardeen ( 1973 ) . monte carlo _ n - particle _ computer simulations of up to @xmath43 scattering events of infalling accretion disk photons ( normalized to a power - law distribution ) are executed for each computed penrose produced luminosity spectrum ( williams 2003 ) . energy and momentum ( i.e. , four - momentum ) spectra of escaping particles ( @xmath8-rays , @xmath9 pairs ) , as measured by an observer at infinity , are obtained per each 2000 scattering events per monochromatic infalling photon distribution . the following constituents are used ( williams 1995 ) : ( 1 ) general relativity is used [ the kerr metric spacetime geometry yields equatorially and nonequatorially confined spherical - like ( wilkins 1972 ) particle orbits and escape conditions , conserved energy and angular momentum parameters , and transformations from the boyer - lindquist coordinate frame ( blf ) to the local nonrotating frame ( lnrf ) ] . note , blf is the observer at infinity ( boyer & lindquist 1967 ) ; lnrf is the local minkowski ( flat ) spacetime . ( 2 ) special relativity is used [ in the lnrf , physical processes ( i.e. , the scatterings ) are done ; lorentz transformations between inertial frames are performed ; and lorentz invariant laws are applied ] . ( 3 ) cross sections are used [ application of the monte carlo method to the cross sections , in the electron rest frame for pcs , in the proton rest frame for ppp(@xmath29 ) , and in the center of momentum frame for ppp(@xmath0 ) , give the distributions of scattering angles and final energies ] . in general , energies attained using the proposed accretion disk model are the following ( williams 1995 ) : i. _ pcs_.for the input photon energy range @xmath44 kev to 0.15 mev , the corresponding output energy range is @xmath45 3 kev to 7 mev . the input photon range covers the range of photons in a thin disk ( @xmath46 kev ) , thin disk / ion corona ( @xmath46 kev , @xmath47 kev ) , and adaf ( @xmath47 kev ) for a @xmath19 kbh ( williams 2003 , 1995 ) . the input luminosity spectra are based on observations , consistent with a power - law distribution in the x - ray , and accretion disk theory . typical output luminosity spectral distributions from pcs are displayed in figure 1@xmath18 , the curves passing through numbers @xmath48 ( as will be described below in the discussion of the model produced luminosity ) . ( @xmath29).there are no escaping pairs for radially infalling equatorially confined @xmath8-rays ( @xmath4940 mev ) and no energy boost : implying that the assumption : negligible recoil energy given to the proton , made in the conventional cross section , and perhaps the geometry of the scattering must be modified . it had been predicted ( leiter & kafatos 1978 ) that pairs with energies ( @xmath491 gev ) could escape . see williams ( 1995 ) for further details of this ppp process . ( @xmath0).an input photon energy range @xmath45 3.5 kev to 200 mev yields output ( @xmath9 ) energy range @xmath45 1 mev to 10 gev ( for a proton maxwell - boltzmann distribution ) , and higher up to @xmath45 54 gev ( for a proton power - law distribution , with input photon energy @xmath50 gev ) , where maxwell - boltzman and power - law distributions are for the accretion disk protons : undergoing nuclear proton - proton scatterings , which yield neutral pion decays @xmath51 ( eilek 1980 ; eilek & kafatos 1983 ; mahadevan , narayan , & krolik 1997 ) that can possibly populate the photon orbit . below , i refer to such decay produced photons and subsequent @xmath9 pair production ( from such photons ) , which can occur in adafs , as eilek s particles ( eilek 1980 ) . specific disk model correlations are the following [ see williams ( 1995 , 2003 ) for further details ] : \1 . without instabilities [ implying the classical thin disk ( novikov & thorne 1973 ) ] : * pcs can convert infalling ( incident ) soft x - rays [email protected] kev to moderate x - rays , escaping with energies in the range @xmath45 3@xmath52262 kev . the upper and lower bounds on the energy of the outgoing photons are set by the initial four - momentum conditions of the target orbiting electron ( with @xmath53 mev at @xmath12 ) and the incident photon ( with @xmath54 kev ) undergoing pcs . these initial four - momenta are consistent in general with the following : theoretical accretion disk models , the threshold energy values for the scattering process to occur , and what brings about the most `` efficient '' energy extraction process [ see williams ( 1995 ) for details defining the efficiencies ] . these initial momenta are substituted into appropriate theoretical analytically derived model equations ( describing the penrose scattering process in the ergosphere of a kerr black hole ; see williams 1995 ) , and the equations are computed . the output energy range presented above gives the lowest and highest energy values obtained by the escaping penrose compton scattered ( pcs ) photons , for the given input energy range of the incident photons . * inwardly directed pcs photons that have an appropriate turning point ( see williams 2002 ) can serve as seed @xmath8-rays for the ppp ( @xmath0 ) at the photon orbit . specifically , the pcs photons that satisfy conditions to have a turning point , acquire gravitationally blueshifted energies as high as @xmath55 mev . * ppp ( @xmath0 ) can convert infalling soft x - ray photons to relativistic @xmath9 pairs , escaping with energies in the range @xmath45 2@xmath526 mev , i.e. , infalling photons can pair produce at the photon orbit with photons populated by prior pcs . \2 . with instabilities ( implying the thin disk / ion corona or adaf ) : * pcs can convert infalling x - rays [email protected] mev to escaping photon energies in the range @xmath45 0.4@xmath527 mev . the differences in the calculations of the above case in item 1 and the present case of item 2 are the energies of the infalling incident photons and the target electrons ( @xmath56 mev ) , including nonequatorially confined targets , with @xmath57 mev , for @xmath58 , corresponding to adafs with @xmath59 , respectively , after being gravitationally blueshifted at @xmath12 by factor @xmath60 , where @xmath61 is the polar coordinate momentum component as measured by an observer at infinity ( i.e. , in the blf ; see [ sec : model ] ) . [ note , the conserved energy @xmath62 and azimuthal angular momentum @xmath63 of the target nonequatorially confined test particle orbits are given by analytically derived expressions presented in williams ( 2004 , 2002 , 1995 ) . ] the accretion disk model , used , is discussed in detail in williams ( 2003 ; see figure 1 and table 1 of that reference ) . in general , the target electron orbits are assumed to be populated during instability phases , more or less , in both the thin disk and thin disk / ion corona ( or adaf ) . in [ sec : asymmetry ] in the discussion of a possible scenario for `` jet reversal , '' a brief description is included on populating the target electron orbits from the inner region of a thin disk ( novikov & thorne 1973 ) . * pcs photons that can populate the photon orbit ( as in item 1.@xmath64 ) have gravitationally blueshifted energies specifically in the range @xmath65 mev , where @xmath66 is the energy at the photon orbit due to prior pcs ( williams 2002 ) . the lower limits of @xmath66 are due to pcs by equatorially [ @xmath53 mev , @xmath67 and nonequatorially [ @xmath68 mev , @xmath69 confined electron targets , respectively : note , as in item 2.@xmath18 , these nonequatorially confined target electrons are assumed to come from an adaf ( @xmath70 kev ) . * ppp ( @xmath0 ) can convert infalling soft x - ray photons to relativistic @xmath9 pairs , escaping with energies in the range @xmath71 mev . note , the `` stability '' of a turning point being perpetual ( i.e. , bound ) at the photon orbit decreases with increasing energy of the incoming incident photons undergoing pcs by equatorially confined target electrons . these calculations show that the most stable orbits ( or turning points ) appear to be the ones in which the infalling incident photons and the orbiting target electrons are self - consistent , i.e. , of the same accretion disk phase ( e.g. , thin disk or adaf ) . note , whenever the thin disk is present , the processes described above in items 2.@xmath18@xmath522.@xmath72 will occur in addition to those described in items 1.@xmath18@xmath521.@xmath72 . \3 . with instabilities [ implying the thin disk / ion corona model or adaf plus eilek s particles ( eilek 1980 ; eilek & kafatos 1983 ) to populate the target particle orbits , of electrons and photons , particularly the large @xmath28-value orbits ; see williams ( 1995 ) ] : * pcs can convert infalling photons [email protected] mev to escaping energies in the range @xmath45 6@xmath5214 mev this is in addition to the energy distribution of escaping pcs photons given in item 2.@xmath18 . eilek s particles contribute to the ion corona , nonequatorially confined @xmath9 pairs with energies peak around @xmath73 mev . at the peak , such electrons with inward trajectories ( @xmath74 ) would have to satisfy conditions to have a turning point at @xmath75 ( williams 2004 ) , requiring , say for the scattering radius @xmath12 , @xmath76 and/or @xmath77 ; these electrons do not appear to be important in the pcs process . observations suggest , however , that pcs by eilek s nonequatorially confined @xmath9 pairs with @xmath78 mev , yielding escaping energies in the range given above , might be important , requiring , for turning points to exist at @xmath12 , @xmath79 and/or @xmath80 , respectively ( compare fig . 1@xmath18 ; see also williams 2003 ) . [ note , eilek s electrons @xmath81 mev may be an important source of synchrotron emission into the ir , for a magnetic field strength @xmath82 g ( see [ sec : asymmetry ] ) . ] * ppp ( @xmath0 ) can convert infalling soft x - rays to relativistic @xmath9 pairs , escaping with energies ranging from @xmath45 300 mev to as high @xmath83 gev [ for a proton maxwell - boltzman distribution ( see item iii above ) ] , with input photon energy @xmath84 mev from @xmath85 decays ( eilek & kafatos 1983 ) . that is , the input ( target ) photons are gravitationally blueshifted ( by factor @xmath86 ) at the photons orbit to energies @xmath87 , and are assumed to have a turning point at ( or near ) this scattering radius , with @xmath88 , respectively , where the subscript @xmath89 represents the orbiting target photon . these ppp ( @xmath0 ) processes occur in additions to those given in item 2.@xmath72 . the exact range of the ppp electrons will depend on which of the inwardly directed photons , after being gravitationally blueshifted , satisfy conditions to have a turning point at or near the photon orbit [ see williams ( 2002 ) for details ] . for completeness , if protons of eilek s particles have a power - law distribution ( mahadevan et al . 1997 ) as mentioned in item iii , the maximum energy attained , using the penrose - williams model , for the escaping ppp ( @xmath0 ) electrons , is @xmath3 gev , for input photon energy @xmath50 gev , after being gravitationally blueshifted to @xmath90 gev , with @xmath91 . note that , there will be a slight time delay between pcs and ppp ( @xmath0 ) in items 1.@xmath18@xmath521.@xmath72 and 2.@xmath18@xmath522.@xmath72 that might be consistent with the time offset ( @xmath92 min ) between x - ray and ir flares observed in microquasar grs 1915 + 105 , indicating that these flares are produced by the same event : the x - ray flares occur with the apparent disappearance of the inner x - ray emitting region of the accretion disk ; and the subsequent ir flares are proposed to be due to synchrotron emitting ejecta of relativistic plasma into the polar direction ( eikenberry et al . 1999a ; eikenberry et al . 1999b ) . before discussing the luminosity spectra produced by these penrose processes , we first discuss the `` characteristic voids , '' existing , in general , in observed spectra of all agns , more or less ( compare fig . 1@xmath18 ) , and how these penrose processes suggest an explanation for them . these observed voids appear to be caused by the `` transitional energy regime '' between thin disk ( @xmath93 kev ) and ion corona ( or adaf ) ( @xmath94 kev ) states ( see table 1 of williams 2003 ) : therefore , we expect the penrose process to be void of participating particles with energy in the range @xmath95 kev , i.e. , if we assume such particles are sufficiently short - lived , save the infalling disk electrons with @xmath96 kev that can satisfy conditions to populate the equatorially confined target electron orbits ( @xmath23 ) at radii @xmath97 ( see [ sec : asymmetry ] and paragraph below ) , where @xmath98 indicates a general particle energy , predicted theoretically by a particular phase of the accretion disk . the pcs photon energies @xmath99 produced by incident and target particles in the transitional energy regime , and the subsequent gravitationally blueshifted energy @xmath66 at the photon orbit , of incoming pcs photons satisfying conditions to have a turning point there ( williams 2002 ) , expected to undergo ppp ( @xmath0 ) , are found to give characteristic voids in the following regimes : * for thin disk / ion corona ( @xmath70 kev ) : @xmath100 mev and @xmath101 mev , where the upper limit originates from nonequatorially confined target electrons , consistent with the electron temperature in the ion corona ( see williams 2003 ) . compare with item 2.@xmath64 above . * for thin disk / ion corona ( @xmath102 kev ) : @xmath103 mev and @xmath104 mev . * for thin disk / ion corona ( @xmath105 kev ) : @xmath106 mev and @xmath107 mev . note , the pcs processes considered above are those occurring at or near @xmath12 : since the highest energy will be extracted from this scattering radius , and it appears that the orbits at this radius will be the first to be populated , as the disk temperature increases ( [ sec : asymmetry ] ) , i.e. , because of the larger energy blueshift factor acquired , and the smaller @xmath27 needed , relative to these parameters at @xmath13 . the disk electron energies @xmath108 kev but @xmath109 , being gravitationally blueshifted by a factor @xmath110 ( see [ sec : model ] ) , satisfying appropriate turning point conditions ( williams 2004 ) , with @xmath111 , are assumed to populate the nonequatorially confined target particle orbits for pcs ( see also [ sec : asymmetry ] ) , where @xmath112 mev is the rest mass energy of an electron . a relativistic four - momentum treatment of disk particle processes in thin disk / ion corona accretion , inside the ergosphere , appears to be needed to theoretically validate this plausible assumption : at present , however , we do not have such a model ; therefore , we must rely on what observations convey to us . moreover , pcs by equatorially confined electron targets ( @xmath23 ) , assuming to originate from `` mild '' instabilities in the thin disk ( that would cause the electron energy to increase to @xmath113 kev , however , while still predominantly in the thin disk phase ) and radially infalling photons confined along the equatorial plane , originating from the ion corona ( @xmath114 kev ) , are not included in the above consideration of the characteristic voids . the reason for the exclusion is that observations suggest such pcs may not be important ( compare fig . 1@xmath18 , curve between points 6 and 7 ) , which could mean that these target orbits are depopulated while the disk is in the thin disk or transient phase , and therefore not available for pcs in the ion corona ( adaf ) phase . further , for the subsequent gravitationally blueshift of such inward directed pcs photons , with @xmath115 mev , @xmath116 mev , @xmath117 mev , corresponding to potential turning point energies : @xmath118 mev , @xmath119 mev , @xmath120 mev , at the photon orbit , for the ion coronas in items ( @xmath18)@xmath52(@xmath72 ) , respectively , we find that most of these potential turning points , it seems , are `` highly '' uncertain ( based on whether @xmath121 for @xmath122 ) . nevertheless , as the energy of the trapped target electron is increased ( @xmath24 ) , consistent with the general ion corona electron temperature , the uncertainty of the turning point orbit , being true , decreases ; then @xmath123 and @xmath124 are as given above in items ( @xmath18)@xmath52(@xmath72 ) . finally , in the above characteristic voids , pcs involving thin disks with energies less than @xmath125 kev , and pcs involving eilek s nonequatorially confined @xmath9 pairs , possibly occurring in the ion corona or adaf ( particularly @xmath126 mev , the range , based on observations , that appears to be the regime satisfying turning point conditions ) , are not included . inclusion of these would slightly affect the voids , yet the distinctive characteristics would remain . we will return to this discussion of the characteristic voids later in this section . the luminosity spectrum due to penrose processes for the specific case of quasar 3c 273 is plotted in figure 1@xmath18 , along with the observed spectrum for comparison ( heavy solid curves superimposed with squares or dots ) . the outgoing ( escaping ) luminosity spectrum produced by the penrose scattered particles is given by ( williams 2003 ) @xmath127 where @xmath128 is the cosmological distance of the black hole source ; @xmath129 is the flux of escaping photons ; @xmath130 and @xmath131 are the emittance of incoming and captured photons , respectively ; the @xmath132 values define the total fraction of the particles that undergoes scattering [ @xmath133 for pcs and @xmath134 for ppp ( @xmath0 ) ] . the values of @xmath135 are the fitting factors , which can make the penrose calculated luminosities agree with observations for the specific case of 3c 273 , to account for in general our letting every particle scatter in the model calculations , since in a realistic situation every particle will not scatter . in short , the @xmath132 values , defined as somewhat free parameters , are probabilities , which are @xmath136 , but @xmath137 ; they are dependent on the cross sections for pcs and ppp ( @xmath138 , @xmath139 ) , the fraction of the luminosity from the disk intersecting the scattering radii ( @xmath140 , @xmath141 ) , and the expansion rate of the jet ( @xmath142 ) . note , from equation ( [ eq : lum ] ) we obtain the model calculated continuum emission given by the top curves on figure 1@xmath18 ( labeled with numbers for specific cases of target and incident particles ; see below ) if we allow @xmath143 , and set the remaining @xmath132 s equal 1 , where we are assuming that the polar angle subtending the bandwidth , @xmath144 , straddling the equatorial plane , impinged by the luminosity , is @xmath145 at @xmath12 and @xmath146 at @xmath147 ) . see williams ( 2003 ) for further details and complete definitions of the @xmath132 values . the spectrum resulting from the ppp ( @xmath0 ) is produced by letting the escaping pairs undergo `` secondary penrose compton scattering '' ( spcs ) with low energy ( @xmath148 mev ) radially infalling equatorial accretion disk photons ( @xmath149 ) . tables 1 and 2 give model parameters corresponding to some of the numbers on figure 1@xmath18 [ see williams ( 2003 ) for other numbers ] . on these tables the parameters are defined as follows : @xmath14 is the scattering radius ; @xmath150 is the target electron energy for pcs ; @xmath151 is the energy value where most of the ppp ( @xmath0 ) electrons , used as targets for the spcs , are created ; @xmath152 is the initial infalling incident photon frequency ; @xmath153 and @xmath154 correspond to the points ( solid squares or dots superimposed on the small - dotted or dashed curve , respectively ) which give the continuum luminosity resulting from several distributions of pcs or spcs events ( each distribution has 2000 scattering events ) ; @xmath155 is the observed luminosity at @xmath153 ( the average frequency of the interval @xmath156 where most of pcs or spcs photons are emitted per 2000 scattering events ) . each distribution of 2000 infalling photons have monochromatic energies normalized to the power - law distribution for 3c 273 based on observations . the @xmath132 values given in the brackets are values used to fit the general model spectra to agree with specific observations . overall , to produce the calculated penrose luminosity spectra of figure 1@xmath18 , 74,000 infalling photon scattering events are used . thus , as one can see from figure 1@xmath18 , the penrose - williams mechanism can generate the necessary luminosity observed , and the three model calculated regions of emission [ due to pcs by equatorially confined targets ( curve passing through nos . @xmath157 ) , by nonequatorially confined targets that cross the equatorial plane ( curve passing up from 6 through nos . @xmath158 ) , and ppp ( @xmath0 ) ( curve passing through nos . @xmath159 ) ] are consistent with the three major regions of emission in all quasars and agns . moreover , taking into consideration the characteristic voids , discussed earlier , proposed to be produced by the different phases of the accretion disk , the lack of participating particles for pcs would cause a void between points 5 and 7 ( @xmath160 mev ) on figure 1@xmath18 , suggesting a transitional energy regime between a thin disk ( @xmath161 kev ) and ion corona ( @xmath162 kev ) . comparing the observed spectrum of 3c 273 to the model calculated spectrum , it appears that this quasar has a similar accretion disk structure . this suggests that a second void should occur between @xmath163 mev , as it does , agreeing strikingly well with observations between the energies of points @xmath164 and @xmath165 . further , comparing figures 8@xmath18 and 8@xmath64 of williams ( 2003 ) , where figure 8@xmath64 is the same as figure 1@xmath18 of this present manuscript , figure 8@xmath18 ( quasar 3c 379 ) does not appear to have an appreciable inner region thin disk to effectively populate and depopulate equatorially confined target electron orbits for pcs , which would give energies up to @xmath166 kev , like that of figure 8@xmath64 . the lack of populating the equatorially confined target orbits suggests that @xmath24 for the disk electrons , implying the presence of an ion corona or adaf , and leads to speculation that perhaps the equatorially confined ( @xmath23 ) particles were `` lost '' in a prior thermal , yet cooler , instability phase . this interpretation for the accretion is consistent with observations of 3c 279 , displayed in figure 8@xmath18 : 3c 279 appears to have an effective radiating ion corona up to @xmath167 kev , similar to the case of item ( @xmath64 ) , in identifying the voids , but it lacks appreciable evidence of an inner region thin disk , at least at this time of observation i.e . , since 3c 279 is classified as an optically violently variable ( ovv ) quasar ( see williams 2003 ) . such adaf phase , presumed for 3c 279 , could produce particles to populate the nonequatorially confined target electron orbits for pcs and subsequent ppp ( @xmath0 ; see items 2.@xmath18@xmath522.@xmath72 above ) and could satisfy the conditions to produce eilek s high energy particles ( see items 3.@xmath18 and 3.@xmath64 above ; compare also table 1 of williams 2003 ) , giving rise to a model calculated spectrum consistent with the observed spectrum of 3c 279 ( compare fig . 8@xmath18 ; see williams 2003 for further details ; see also [ sec:3c273 ] ) . the observed spectra of microquasars ( or galactic black holes ) , in general , appear not to have pcs emission by the nonequatorially confined target electrons , neither the highest energy @xmath8-ray emission due to ppp ( @xmath0 ) , suggesting that these sources may not have an ion corona ( or adaf ) , which would be need to populate the orbits to generate such emission , at least in the highest energy regime ( compare figure 1@xmath18 ) . general calculated spectra resulting from a self - consistent thin disk penrose process model for stellar mass black holes ( @xmath168 ) appear like a scaled - down figure 1@xmath18 ( with photon luminosity @xmath169 for total energy range @xmath170 kev@xmath528 mev ) , without any appreciable curve labeled between points 6@xmath5213 ( williams & hjellming 2002 ) ; see [ sec : cyg ] . this is consistent with observations of galactic black holes ( liang 1998 ) . the gravitomagnetic ( gm ) force field is the gravitational analog of a magnetic field . it is the additional gravitational force that a rotating mass produces on a test particle . the gm force is produced by the gradient of @xmath171 , where @xmath172 is the frame dragging velocity ( bardeen et al . 1972 ) and @xmath173 is the gm potential ( thorne , price , & macdonald 1986 ) . analysis of the equations governing the trajectories of the penrose process particles shows that the gm force , which acts proportional to the momentum of a particle , alters the incoming and outgoing momentum parameters of the incident and scattered particles , resulting in asymmetrical polar distributions , and thus , appearing to break the reflection symmetry of the kerr metric , above and below the equatorial plane ( williams 2004 , 2002 , 2003 , 1999 ) . effects of the gm force acting on the ppp ( @xmath0 ) process can be discerned from comparing figures 1@xmath128 and 1@xmath174 . when half of the 2000 target photons are allowed to have initial polar coordinate momentum @xmath175 and the other half @xmath176 , of equal absolute values , with increasing @xmath177 , the @xmath9 `` jet ( @xmath178 ) to counter - jet ( @xmath179 ) '' ratio @xmath180 achieves a maximum @xmath181 , favoring @xmath182 ( williams 2002 ) , as seen in figure 1@xmath174 ( compare fig . 1@xmath128 ) . the corresponding polar angles of escape for cases of figures 1@xmath128 and 1@xmath174 are displayed in figures 2@xmath18 and 2@xmath64 , respectively . polar coordinate momentum distributions , @xmath183 , for escaping pcs photons are displayed in figure 3 , where the primes indicate final conditions . the corresponding polar angles of escape for the cases of figure 3 are given in figure 4 . notice the effects of the gm force field causing the ( photon jet to counter - jet ) ratio @xmath184 to vary from nearly symmetric to asymmetric for the different cases shown . of these cases the largest ratio achieved is @xmath185 ( figs . 3@xmath72 and 4@xmath72 ) . the direct cause of the asymmetry in the polar direction appears to be due to the severe inertial frame dragging in the ergosphere in which the gm field lines are spacetime dragged in the direction that the black hole is rotating [ see williams ( 2002 ) for details ; see also williams 2004 ] . the resulting gm force acting on the particles produces the asymmetry . in most cases , the distribution favors the @xmath186 direction ( see figs . 3 and 5@xmath128 ) ; however , at particularly low energies , the asymmetry appears to reverse . for example , in the case of ppp ( @xmath0 ) at the low initial energies @xmath187 kev and @xmath188 mev for the infalling and orbiting photons , respectively , producing escaping @xmath9 pairs with energies peak around @xmath189 mev , @xmath190 per 2000 events ( fig . 5@xmath18 ) , and after undergoing spcs ( williams 2003 ) per 2000 infalling disk photons ( @xmath191 kev ) , the asymmetry in the final photon polar distribution , for the spcs , is reverse , with the inverse of the number of particles scattered in the positive polar direction to that in the negative direction @xmath192^{-1}=402/165\simeq 2.44 $ ] , favoring the @xmath179 direction ( fig . 5@xmath64 ) . this would make the @xmath179 jet appear more energetic and , thus , brighter , since the ppp @xmath9 polar jets , in this case , are nearly symmetrical , as can be seen in fig . such behavior is consistent with hjellming and rupen s ( 1995 ) observations of gro j1655 - 40 . these authors concluded that the jets themselves must be intrinsically asymmetric , and the sense of the asymmetry must change from event to event . moreover , they found that the jets lie almost in the plane of the sky , so relativistic beaming can not explain the observed brightness ratios . [ note , the potential for `` jet reversal '' due to the gm force field can be seen in eq . ( 47 ) of williams ( 2002 ) and eq . ( 8) of williams ( 2004 ) : occurring for particle distributions with relatively large @xmath193 and/or relatively small @xmath194 ( corresponding to small @xmath195 ) . ] also , the jet space velocity lorentz factor found by these authors ( @xmath196^{-1/2}\approx 2.5{\rightarrow}e\sim 1.3 $ ] mev ) is consistent with the target electron energy , of the spcs , we have found here , displaying the jet reversal ( compare @xmath197 stated above and figs . 5@xmath18 and 5@xmath64 ) , where we are assuming that the bulk velocity of a `` blob '' is @xmath198 `` average '' space velocity of the individual ppp electrons per bulk distribution , i.e. , assuming @xmath199 , valid at least in the case of the small scale , fast varying galactic black holes . the model calculated space velocities of the ppp @xmath9 jet particles for initial energies consistent with observed microquasars ( with @xmath200 ) are in the range @xmath201 , for @xmath202 mev , implying @xmath203 , respectively ; compare fig . thus , the consistency of apparent jet reversal , of these penrose processes , with observations , gives more compelling evidence that it is probably the penrose - williams mechanism at work , close to the event horizon , within @xmath13 , extracting rotational gravitational energy momentum : in the form of a particle flux , as opposed to the so - called bz - type models , proposed to extract energy and momentum : in the form of electromagnetic poynting flux and alfvn waves , respectively ( with the major problem still existing of converting to the necessary particle flux to fuel the observed jets ) . note , a specific possible scenario for the jet reversal in the case of a @xmath205 microquasar , similar to that of gro j1655 - 40 ( hjellming & rupen 1995 ) , for a classical thin relativistic accretion disk ( novikov & thorne 1973 ) , is the following : as secular density and thermal instabilities begin occurring in the inner region of a time dependent accretion disk commonly referred to as the `` lightman instabilities '' ( lightman 1974a , 1974b ; williams 1995 , 2003 ) , @xmath206 increases to a `` reasonable '' maximum @xmath207 kev , being consistent with observations . the infalling disk particle electrons with energies @xmath208 , and satisfying conditions for a turning point to exist at specific radii ( williams 1995 ) between @xmath209 , respectively , will be gravitational blueshifted according to the blueshift factor : @xmath210 , respectively ( recall discussion in [ sec : model ] , first paragraph ) , populating the equatorially confined ( @xmath211 ) target electron orbits with @xmath212 mev , respectively , for pcs ( williams 1995 ) . this appears to be the catalyst to `` turn on '' the self - consistent penrose - williams mechanism . [ note , the above reasonable maximum energy means before the critical surface density @xmath213 is reached ( lightman 1974a , 1974b ) , which causes the ion coronal / torus two - temperature phase to set in , or before the inner `` hot '' region ( novikov & thorne 1973 ) extends to @xmath214 ( eardley & lightman 1975 ) , for @xmath215 , @xmath216 , and @xmath217 , where @xmath4 is the viscosity parameter , @xmath218 the kompaneets parameter , and @xmath219 the sub - eddington accretion rate ; see williams ( 2003 ) . ] the subsequent escaping pcs x - ray emission becomes more and more asymmetric , favoring the @xmath186 direction , as the infalling initial photon energy is increased , say due to disk instabilities ( compare figs . 3@xmath18 and 3@xmath64 ) . as pcs of infalling disk photons ( @xmath191 kev ) depopulates the equatorially confined target electron orbits , some of the photons with @xmath220 , @xmath221 , and @xmath222 , after being gravitationally blueshifted by factor @xmath86 , satisfy conditions to have a turning point at the photon orbit ( williams 1995 , 2002 ) , populating , and thus supplying target photons for ppp ( @xmath0 ) in the range of @xmath223 , respectively , for the range of @xmath99 above , where @xmath224 . for @xmath225 mev , @xmath226 , as given by the analytical derived expressions of the conserved energy @xmath40 and angular momentum @xmath36 of nonequatorially confined particle trajectories ( see williams 1995 , 2002 , 2004 ) , the subsequent ppp ( @xmath0 ) with infalling disk photons @xmath227 ( assuming negligible electrons are left in the equatorially confined orbits between @xmath13 , @xmath12 ) will produce slightly asymmetrical jets ( favoring the @xmath186 direction ; compare fig . 5@xmath18 ) . the total energetics due to pcs and ppp ( @xmath0 ) at this phase will favor @xmath178 , therefore , producing a `` brighter '' jet in this polar direction . however , when some of these ppp electrons subsequently interact with infalling disk photons through spcs , the final emitted escaping photon jets undergo apparent reversal ( favoring @xmath179 ; compare figs . 5@xmath64 and 5@xmath72 ) : thus , the total energetics will now favor the @xmath228 direction . compare figures 3@xmath18 , 3@xmath64 , and 5@xmath18@xmath525@xmath72 , considering the observed time delays between outbursts ( hjellming & rupen 1995 ; eikenberry et al . 1999a ) and those expected between the different penrose processes : pcs , ppp ( @xmath0 ) , spcs ; and the synchrotron emission ( @xmath229 ghz ) by the escaping ppp electrons ( of fig . 5@xmath18 ) : due to , perhaps , their expected intrinsic magnetic field ( or an external accretion disk magnetic field ) , according to @xmath230 ( burbidge , jones , & odell 1974 ) , for @xmath82 g ( williams 2003 ; this assumed value , although consistent with observations in many instances , needs further investigation ) . in addition , some of the ppp electrons ( of fig . 5@xmath18 ) will be created with @xmath231 mev , @xmath232 , @xmath233 , and @xmath234 ( recall that @xmath235 ) , satisfying the condition to have a turning point at the iso - energy orbit @xmath236 ( circular orbit of equal energy at constant radius @xmath237 ; see williams 1995 , 2004 ) , with @xmath238 and @xmath239 at radii @xmath240 ( the last bound orbit for a material particle , deep within the ergosphere ) , before escaping to infinity along vortical orbits ( [ sec : collimation ] ) , satisfying ( williams 1995 ) @xmath241 or @xmath242 , implying no turning point in @xmath243 , i.e. , @xmath244 , yet @xmath245 . note , this satisfying of the condition to have a turning point at @xmath240 , before escaping to infinity along vortical trajectories , is also true for the supermassive kbh ( williams 2004 ) . observations of gro j1655 - 40 ( hjellming & rupen 1995 ) suggest that after the jet outbursts : due to lightman instabilities , inner region disk depletion , penrose processes , and plunging orbit ( bardeen et al . 1972 ) population - depopulation processes , the disk settles back down to its low , `` initial '' state , to prepare once again to repeat the total _ disk instability - penrose emission cycle _ , as described above , indefinitely ( i.e. , as long as there exists available matter to accrete ) . moreover , the disk instabilities are expected to change the accretion rate , thereby causing the penrose processes to vary . so , in conclusion of this section , it appears that once the initial requirement has been met : of populating the equatorially confined target electron bound , unstable orbits , inside the ergosphere , between @xmath75 [ at @xmath246 for maximum pcs energy extraction ( williams 1995 ) ] , the kbh operates as a self - consistent system , emitting @xmath9 and photon jets , relying only on the accretion disk to supply the incident infalling photons , and to populate the initial equatorially confined electron target orbits [ i.e. , due to disk instabilities ( kafatos & leiter 1979)]indicating the beginning of the `` cycle . '' and within this cycle for particularly low particle initial energies , the gm field can cause the jet brightness asymmetry to reverse . [ note , see williams ( 2002 ) for a complete description of the relations between the gm field and the space momenta displayed in the figures shown here . ] in addition , in the case of quasars - type agns ( williams 1995 , 2003 ) , it appears that an adaf is needed to populate the relatively high energy nonequatorially confined target electron orbits for pcs ; and to populate the highest energy photons at the photon orbit for ppp ( @xmath0 ) , yielding maximum escaping energies @xmath247 gev ( as discussed in [ sec : energy - luminosity ] ) . it is found that the penrose scattered particles escape along vortical trajectories collimated about the polar axis ( williams 1995 , 2000 , 2003 , 2004 ) . these distributions are fluxes of coil - like trajectories of relativistic jet - type particles , escaping out from the equatorial plane at the scattering radius @xmath248 , concentric the polar axis . the highest energy particles have the largest @xmath249 values ( compare fig . 1@xmath72 ; compare also figs . 3@xmath64 and 4@xmath64 of williams 1995 ) . note , @xmath250 is negative ( inward toward the polar axis ) for many of the pcs photons ( williams 2002 ) , and positive for all of the @xmath9 pairs ( compare fig . 1@xmath64 ) . the helical angle of escape @xmath251 , of particle type @xmath252 , relative to the equatorial plane , for the highest energy scattered particles ranges from @xmath253 to @xmath2 for pcs ( compare fig . 4 ) and @xmath254 to @xmath255 for the @xmath9 pairs ( compare fig . 2 ) ; compare also figs . 6 , 7 , and 9 of williams ( 2002 ) . the above characteristics of the escaping particles , along with their @xmath256 values ( compare figs . 1@xmath128 , 1@xmath174 , 3 , and 5 ) , imply strong collimation about the polar axis , giving rise to relativistic jets with particle velocities up to @xmath6 ( compare fig . 1@xmath204 ) . note , such vortical trajectories and collimation are consistent with the findings of de felice et al . ( de felice & curir 1992 , de felice & carlotto 1997 , de felice & zanotti 2000 ) , from spacetime geometrical studies of general particle geodesics in a kerr metric . moreover , the gm force field , discussed in the last section , responsible for the inertial frame dragging and the asymmetrical jets , also serves to boost the jets into opposite polar directions ( williams 2002 ) . in addition to statements made in [ sec : energy - luminosity ] concerning the model calculated spectra of 3c 273 and 3c 279 , below i summarize some of the important features resulting from application of the penrose - williams mechanism to observations of both 3c 273 and 3c 279 ( see fig . 8@xmath18 of williams 2003 ) . the observed spectra of both these sources can be explained by these penrose processes and the assumed accretion model : specified in [ sec : model ] [ see williams ( 1995 , 2003 ) for further details ] . as we can see from figure 1@xmath18 , there is a striking similarity between the energy range of the observed spectrum of 3c 273 and the model spectra produced by these penrose processes . upon comparing the spectra of radio - loud quasars 3c 273 and 3c 279 , based on these penrose processes , we find the following ( williams 2003 ) : the shape of the observed spectrum of 3c 273 looks like the `` enhanced '' ( i.e. , the highest observed energetic state ) spectrum of 3c 279 , except for the higher luminosities in 3c 279 and the radio tail in 3c 273 . the higher luminosity and the apparent lack of a radio tail in 3c 279 is probably , largely , due to the radiation of 3c 279 being beamed more in the direction of the observer than the radiation of 3c 273 . therefore , the spectrum of 3c 279 has been doppler blueshifted to an _ observed _ higher energy interval ; and the _ apparent _ luminosity has been increased . this is consistent with radio observations which detect more superluminal motion ( or relativistic beaming near the line of sight of the observer ) in 3c 279 than in 3c 273 ( porcas 1987 ) . on the other hand , it seems that 3c 273 has a `` hotter '' inner accretion disk and is in a predominantly bimodal quasi - stable state : appearing to be in the most effective or `` extreme '' thin disk / ion corona state as opposed to 3c 279 : which appears to oscillates in a highly variable fashion between the thin disk and ion corona phases for this reason 3c 279 is classified as an ovv quasar . the hotter state of the accretion disk ( ion corona ) , which is heated by a runaway thermal instability ( shapiro , lightman , & eardley 1976 ) , would result in enhanced penrose processes [ pcs and ppp ( @xmath0 ) ] , and enhanced synchrotron radiation due to the presence of more relativistic electrons , particularly if eilek s ( eilek 1980 ; eilek & kafatos 1983 ) particle reactions ( @xmath257 ) occur , hence contributing to the prominent observed radio tail of 3c 273 . this ion corona / adaf state , existing in conjunction with the thin disk , appears to be the case always in the continuum emission of 3c 273 and sometimes in the emission spectrum of 3c 279 , with 3c 279 not quite achieving the full `` hot '' ion corona / adaf status of 3c 273 ( williams 2003 ) , neither achieving the full `` cool '' @xmath258 kev thin disk phase , where more penrose processes would occur to liberate trapped energy . thus in summary , the differences in the spectra of 3c 279 and 3c 273 are probably due to ( 1 ) the more beaming effect in 3c 279 , and ( 2 ) the predominantly extreme hot , cool phases of 3c 273 . now , again , based on the `` characteristic voids '' discussed in [ sec : energy - luminosity ] , 3c 279 appears to be similar to case ( b ) and 3c 273 to that of case ( c ) . compare models @xmath259 and @xmath260 on table 1 of williams ( 2003 ) : models @xmath259 are similar to 3c 273 , and model @xmath260 is similar to 3c 279 . recent observations of the bright seyfert 1 galaxy mcg6 - 30 - 15 [ particularly of the broad fe k@xmath4 emission line at @xmath5 kev , believed to be originating from the inner accretion disk plasma ( wilms et al . 2001 ) ] , and other such type agns , are consistent with these model calculations . a qualitative model calculated scenario to explain the observed spectral observations of mcg6 - 30 - 15 , by these penrose processes , is as follows . if we assume that the plunging orbits of the target electron , inside the ergosphere , have been populated by accretion disk instabilities ( as described in [ sec : asymmetry ] ) , self - consistent computer simulations of these penrose processes consistent with mcg6 - 30 - 15 have model parameters for radial infalling photons ( @xmath261 kev ) from a thin disk ( novikov & thorne 1973 ) , that either undergo pcs by equatorially confined orbiting target electrons ( @xmath53 mev ) at @xmath12 , or ppp ( @xmath262 ) at @xmath147 . the energies ( due to frame dragging ) attained by the @xmath263 up to @xmath264 escaping particles , returning to the disk to be reprocessed and/or escaping to infinity , are the following : for pcs photons , @xmath265 kev for equatorially confined orbiting target electrons , with relative incoming and outgoing photon luminosities @xmath266 , respectively , where @xmath267 . and for the relativistic ppp electrons ( with @xmath268 and @xmath269 mev ) , @xmath270 mev [ consistent with synchrotron radiation into the radio regime for @xmath271 g , and inverse compton scattering ( spcs of disk photons ) into the x - ray / soft @xmath8-ray regime with relative incoming and outgoing photon luminosities @xmath272 , for @xmath273 , at @xmath274 mev , respectively ] , suggesting relatively weak , less powerful and less prominent radio jets , i.e. , a radio quiet agn , like a seyfert galaxy ( compare figs . 1@xmath18 , 5@xmath18 and 5@xmath64 for similarities and dissimilarities ) . note , for self - consistency , @xmath177 is assumed based on prior pcs photons with @xmath220 that satisfy conditions for the existence of a turning point at the photon orbit ( williams 2002 ) . note also that , at these low energies for @xmath275 and @xmath177 , the spcs polar jets appear to `` flip , '' undergoing brightness jet reversal ( as discussed in [ sec : asymmetry ] ) , differing by a factor @xmath276 , in particle numbers , favoring @xmath277 ( compare fig . 3 and figs . 5@xmath278 ) , whereas the initial ppp target electron polar jets , differ by a factor @xmath50 favoring @xmath279 ( compare fig . 5@xmath18 ) . the pcs photon distribution in the range such as @xmath99 above , emitted from @xmath280 , with the highest energy photons concentrated in the equatorial plane ( compare fig . 4 ) , is expected to be consistent with the observed extremely steep emissivity profile @xmath281 of wilms et al . ( 2001 ) , indicating that most of the fe k@xmath4 line emission originates from the inner region of a relativistic accretion disk . specific details of the emissivity @xmath282 of these penrose processes , particularly of the pcs , will be presented in a future paper by the author . recent radio observations of active galaxy m87 ( junor et al . 1999 ) suggest that electromagnetic collimation becomes important at radii @xmath283 , wherein the initial `` open angle '' of the jet @xmath284 ( at radii @xmath285 ) is made smaller to @xmath286 by the electromagnetic field , where @xmath287 ( @xmath288 , the radius of the ergosphere at the equator ) . this is consistent with the penrose mechanism providing ( in addition to the relativistic particles ) the initial collimation at radii ( @xmath285 ) , i.e. , closer to the black hole . since m87 is a giant elliptical galaxy , this could mean that its geometric configuration is possibly helping to maintain the initial collimation by the black hole ( williams 2003 ) : which begins at @xmath289 , and must extend out to at least @xmath290i.e . , until , it appears , electromagnetic collimation takes over . however , before one can say for certain of the electromagnetic processes occurring , a time dependent mhd evolution of the penrose escaping particle plasma must be performed ( presently under investigation by the author ) . it should not be ruled out that the intrinsic collimation due to the black hole , of the escaping relativistic plasma : and any associated `` dynamo '' generated magnetic field , may be sufficient to maintain collimation . further , concerning m87 , its observed spectrum in general can be explain by the penrose mechanism presented in this paper . some observational properties of m87 are the following ( eilek 1997 ) : @xmath291 ; striking comparisons of radio ( very large array ) and optical ( hubble space telescope ) images of the jet ; optical and possibly x - ray emission believed to be of synchrotron origin ; and more recently , the mid - ir observations ( perlman et al . 2001 ) showing that the nuclear ir emission is entirely consistent with synchrotron radiation , and there is no evidence for thermal emission from a dusty nuclear torus . based on these properties the following scenario can be devised according to the penrose - williams mechanism . the jet is no doubt beamed , since observed superluminal motions give apparent velocities up to @xmath292 , implying line - of - sight angle @xmath293 , bulk lorentz factor @xmath294 , jet doppler factor @xmath295 , and jet brightness boost @xmath296 , respectively ( biretta , sparks , & macchetto 1999 ; see also williams 2003 ) . m87 is probably an evolve blazar - type agn ( ovv quasar and bl lac object ) . its luminosity spectrum ( although less powerful , less energetic ) most likely resembles that of 3c 279 ( [ sec:3c273 ] ; see also williams 2003 ) . the most noticeable change in the spectrum from times past is probably the lack of high energy @xmath8-rays : due to the lack of the availability of infalling low energy ( soft x - ray ) disk photons , or the lack of high energy ppp ( @xmath0 ) electrons , to undergo effective spcs , that would result in escaping trajectories for the scattered @xmath8-rays . since the jet of m87 is still seen prominently in the radio / optical / x - ray , an optically thin hot ion torus , pcs , ppp ( @xmath0 ) , and subsequently synchrotron radiation of the ppp electrons ( particularly into the optical : implying @xmath297 mev for @xmath298 g , respectively ) , are consistent with the observations . the parenthetical statement above suggests that the magnetic field producing the synchrotron radiation may be that of the escaping penrose plasma rather than that of the popular proposed large scale dipolar - like field of the accretion disk ( since large - scale , large - strength dipolar accretion disk fields are in practice difficult to create ) ; this however requires an investigation . moreover , besides coming from the inner region of a relativistic thin disk ( novikov & thorne 1973 ) , there are two possibilities for producing the observed soft x - ray emission , within the confinements of the penrose - williams mechanism : ( 1 ) a synchrotron origin requires electron energies @xmath113 gev ( @xmath299 ) for @xmath50 kev emission at @xmath271 g , and could very well be produced by the ppp ( @xmath0 ) , at least for ultrarelativistic @xmath9 pairs up to @xmath3 gev ( williams 2003 ) for @xmath300 as low as @xmath83 g ; and ( 2 ) the jet is beamed , and self - compton scattering of lower energy radio and ir synchrotron photons by the escaping , intrinsically polar collimated ppp electrons is occurring : the observed energies of the inverse / self - compton scattered photons are blueshifted due to doppler boosting into the optical and x - ray regimes , respectively , according to @xmath301 ( dermer , schlickeiser , & mastichiadis 1992 ) for @xmath302 , @xmath303 . now , both items above could equally occur , more or less ; however , since superluminal motion appears to be important in m87 , item ( 2 ) is most likely the dominant , somewhat ruling out the other item . if this dominance is true , then the energies of the jet electrons need only be as high as @xmath304 mev for @xmath271 g. this is consistent with the penrose processes described here , in the presence of a thin disk / ion corona accretion without the need of eilek s @xmath85 decays to populate the photon orbit ( see [ sec : energy - luminosity ] , items @xmath305 ) . note , such ion coronas or tori are poor radiators , and expected to be of relatively low density , with @xmath306 kev ; this may account for the lack of evidence for an inner `` dusty '' torus emitting thermal radiation in the mid - ir observations by perlman et al . however , it still seems unlikely that such a low electron energy and particle density ion torus ( or adaf ) can be jet fuel for the bz - type models near the event horizon , inside the ergosphere , as required by such models ( blandford & begelman 1999 ) . [ see williams ( 2003 ) for a complete description of the accretion disk model consistent with these penrose processes and observations . ] nevertheless , such bz - type models ( e.g. , punsly 1991 ; koide et al . 2000 ) might be important at @xmath307 , as suggested by observations ( junor et al . 1999 ) , particularly if the penrose - williams particles are used as fuel . the penrose - williams model presented here applies to all mass size kbhs , with the stellar mass black hole appearing as a scaled - down supermassive hole . when the parameters are expressed in gravitational units ( @xmath308 ) , the penrose process emission energy - momentum spectra ( @xmath309 vs. @xmath40 ; @xmath38 vs. @xmath40 ; @xmath310 vs. @xmath40 ) over the range of masses are approximately identical . the luminosity spectra of these penrose processes for the different masses , in general , span over a range @xmath311 ( compare figs . 1@xmath18 and 6 ) . in general , the differences of the penrose process output luminosities between supermassive kbhs and `` micro - massive '' kbhs are determined by the bolometric luminosity of the incoming photons ( eilek 1980 ; williams 2003 ) , directly dependent on the accretion rate , which is governed by the surrounding accretion disk environment . for example , the observations of the classical stellar / galactic black - hole candidate cygnus x-1 ( liang 1998 ) can be explained by these penrose processes : processes consistent with cyg x-1 have model parameters for radial infalling photons ( @xmath191 kev ) from a thin disk ( novikov & thorne 1973 ) , that either undergo pcs by equatorially confined orbiting target electrons ( @xmath53 mev ) at @xmath12 or ppp ( @xmath262 ) at @xmath147 . the energies ( due to frame dragging ) attained by the @xmath312 up to @xmath313 escaping particles , returning to the disk to be reprocessed and/or escaping to infinity , are the following : for the pcs photons , @xmath314 kev , with relative incoming and outgoing photon luminosities @xmath315 , respectively , where @xmath316 . and for the relativistic ppp electrons ( with @xmath317 and @xmath318 mev ) , @xmath319 mev [ consistent with synchrotron radiation into the radio regime for @xmath271 g , and inverse compton scattering ( spcs of disk photons ) into the hard x - rays / soft @xmath8-ray regime with relative incoming and outgoing photon luminosities @xmath320 , for @xmath321 , between @xmath322 kev@xmath323 mev ] ; compare figure 6 ; see williams & hjellming ( 2002 ) for further details . note , for self - consistency , @xmath177 is assumed based on prior pcs photons with @xmath220 that satisfy conditions for the existence of a turning point at the photon orbit ( williams 2002 ) . note also that , as in the cases of gro j1655 - 40 ( [ sec : asymmetry ] ) and mcg6 - 30 - 15 ( [ sec : mcg ] ) , at these low energies for @xmath275 and @xmath177 , the spcs polar jets undergo slight so - called jet reversal ( as discussed in [ sec : asymmetry ] ) , differing by a factor @xmath324 favoring @xmath277 , whereas the initial ppp target electron polar jets , differ by a factor @xmath325 favoring @xmath279 ( compare figs . 3@xmath18 , 3@xmath64 , and 5 ) . in the above model for cyg x-1 , the ppp electron energy @xmath197 can increase to @xmath326 mev , as the infalling thin disk photon energy for pcs by equatorially confined target electrons is increased to @xmath327 kev ( williams & hjellming 2002 ) , say due to disk instabilities ( compare [ sec : asymmetry ] ) . this appears to be the case for cyg x-1 when in its `` high '' state ( mcconnell et al . 1989 ) , and to explain the persistent power - law @xmath8-ray tail up to @xmath328 mev ( mcconnell et al . 1994 ) . now , concerning @xmath329 quasi - periodic oscillations ( qpos ) observed in galactic black holes ( strohmayer 2001 ; remillard et al . 2002 ; abramowicz et al . 2002 ) , such qpos can be predicted from the penrose scattering processes described here . the `` qpos '' of , say , a given local distribution of neighboring target electrons , responsible for pcs into the x - ray / soft @xmath8-ray regime , emitting from geodesic orbits at radii between @xmath13 and @xmath12 , can be obtained from ( bardeen et al . 1972 ) @xmath330 where @xmath331 is the coordinate angular velocity of a circular orbit ; @xmath332 is the inverse of the blueshift factor ( see [ sec : model ] ) , commonly referred to as the `` redshift '' factor ; @xmath333 is the orbital velocity in the azimuthal direction of the target particles relative to the lnrf , i.e. , as measured by a general observer at rest relative to this frame ( bardeen et al . 1972 ; see also williams 1995 ) ; @xmath334 is the radius of the circumference about the axis of symmetry ( thorne et al . so , with the frame dragging _ angular _ velocity given by @xmath335 , we find ( from eq . [ [ eq : qpo ] ] ) the predicted range to be between : @xmath336 hz , @xmath337 hz at @xmath13 , @xmath12 , respectively , corresponding to periods @xmath50 ms , as measured by an observer at infinity . note , the counterpart qpos for a supermassive ( @xmath338 ) kbh are @xmath339 hz ; this relatively low frequency is probably the reason these counterpart qpos have yet to be detected in sources harboring such massive kbhs ( see miller et al . 2002 and references therein ) . these calculations suggest that @xmath49 khz qpos may also be due to the inertial frame dragging of the nonequatorially confined target particles orbital `` ring '' at scattering radius @xmath14 ( williams 1995 ) , particular of the nodes ( points at which the orbit , in going between negative and positive latitudes , intersects the equatorial plane)which happens to be where the most effective penrose scattering processes would occur , as resulting emitting regions of neighboring particles sweep across the line of sight of the observer . in this case , the observed oscillation frequencies , given by @xmath172 ( bardeen et al . 1972 ) , might be slightly smaller ( between @xmath340 hz , @xmath341 hz at @xmath13 , @xmath12 , respectively ) and appear twice as fast as those given above or in pairs . see williams ( 2002 ) for a discussion of the nonequatorially confined spherical - like orbits , first proposed by wilkins ( 1972 ) . the above findings are consistent with the qpos proposed to originate from orbits within the radius of the marginal stable orbit @xmath13 ( zhang , shrohmayer , & swank 1997 ) , and the suggestion that the energy distribution of the energetic electrons must be oscillating at the qpo frequency ( morgan , remillard , & greiner 1997 ) . note , in the above qualitative , yet self - consistent models , for the radio quiet seyfert galaxy ( @xmath342 ; [ sec : mcg ] ) and the galactic black hole cyg x-1 ( @xmath343 ; present section ) , for the initial conditions used based on properties of the accretion disk , the main differences in the emitted spectra are the number of penrose produced @xmath9 pairs escaping , and the range of @xmath197 : for the seyfert galaxy @xmath197 is in the narrow range @xmath344 mev , and for the galactic black hole , @xmath345 mev . in both cases , most ( if not all ) of the ppp electrons have turning points in the nonequatorially confined ( spherical - like ) electron orbits at @xmath246 , as discussed in [ sec : asymmetry ] , indicating that these electrons escape along vortical trajectories collimated about the polar axis , without interacting appreciably with the inner edge of the bound stable accretion disk ( located at @xmath346 ; williams 2004 ) . from the penrose - williams model presented here , to extract energy momentum from a rotating black hole , we can conclude the following : pcs is an effective way to boost soft x - rays to hard x - rays and @xmath8-rays up to @xmath347 mev . ppp ( @xmath0 ) is an effective way to produce relativistic @xmath9 pairs up to @xmath348 gev : this is the probable mechanism producing the fluxes of relativistic pairs emerging from cores of agns ; and when relativistic beaming is included , apparent energies @xmath45 tev can be achieved ( williams 2003 ) . these penrose processes can operate for any size rotating black hole , from quasars to microquasars ( i.e. , galactic black holes ) . overall , the main features of quasars : ( a ) high energy particles ( x - rays , @xmath9 pairs , @xmath8-rays ) coming from the central source ; ( b ) large luminosities ; ( c ) collimated jets ; ( d ) one - sided ( or uneven ) polar jets which under certain conditions the asymmetry brightness appears to `` flip , '' can all be explained by these penrose processes . moreover , it is shown here that the geodesic treatment of individual particle processes close to the event horizon , as governed by the black hole , is sufficient to described the motion of the particles . this is consistent with mhd that the behavior of such individual particles on geometry ( or gravity)-induced trajectories is also that of the bulk of fluid elements in the guiding center approximation ( de felice & zanotti 2000 ) . in light of this , with some ease , mhd should be incorporated into these calculations , particularly to describe the flow of the penrose escaping particles away from the black hole , to perhaps further collimate and accelerate these jet particles out to the observed distances . importantly , we can conclude that , the difference between quasars , radio quiet and radio loud galaxies , and microquasars , appears to be the presence or the lack of a two - temperature adaf : with or without nuclear reactions ( @xmath349 ) in the inner region of the accretion disk ( see eilek 1980 ; eilek & kafatos 1983 ) . quasars appear to have thin disk / ion corona ( adaf ) with nuclear reactions . in the case of the radio quiet and radio loud galaxies the adaf may no longer be `` nuclear reactive , '' however just hot , and in some cases the disk may have evolved back to its cool thin disk phase , including the associated thermal - cycle lightman instabilities ( lightman 1974a , 1974b ) . the microquasars , on the other hand , appear in general not to satisfy conditions for the existence of an adaf , which is determined by the accretion rate ( williams & hjellming 2002 ) , but do appear to satisfy conditions to have a soft x - ray inner region and an apparent thermal - cycle instability , with disk temperature up to @xmath102 kev . finally , what makes the penrose mechanism described here so admirable is that it allows one to relate the macroscopic conditions , i.e. , of the global gravitational field of the kbh , to the microscopic world of particle physics . this description , which is progressively being proven by observations , to be the correct description , allows us to see directly how energy is extracted from a black hole . the physics used in this penrose analysis is that of special and general relativity . from this analysis and its consistency with observations , we arrive at the following conclusion : close to the event horizon , gravity and particle - particle interactions , in the ergosphere , of highly curved spacetime ( where the effect of the external accretion disk magnetic field is apparently negligible ) , are sufficient to described energy - momentum extraction from a rotating black hole . i first thank god for his thoughts and for making this research possible . next , i thank dr . fernando de felice and dr . henry kandrup for their helpful comments and discussions . also , i thank dr . roger penrose for his continual encouragement . i am grateful to the late dr . robert ( bob ) hjellming for his helpful discussions and cherished collaboration . part of this work was done at the aspen center for physics . this work was supported in part by a grant from nsf at nrao and an aas small research grant . associated problems with popular mhd models are described below : \1 . in order to explain observations of the seyfert 1 galaxy mcg6 - 30 - 15 , that copious photons are been extracted from the black hole from radii less than the marginal stable orbit @xmath13 ( @xmath350 , in gravitational units with @xmath235 , where @xmath351 is the mass of the black hole ) , it has been claimed that the force lines of the disk magnetic field @xmath352 couple with matter deep within the `` plunging '' region @xmath353 , thereby extracting rotational energy in the form of electromagnetic energy ( wilms et al . 2001 ; krolik 2000 ) . however , the first detailed numerical relativistic time - dependent mhd calculations in a kerr metric ( koide et al . 2000 ; meier & koide 2000 ; meier , koide , & uchida 2001 ) show that in order for magnetic field lines to extend inward to the numerical limited radius @xmath354being frozen to the plasma , of keplerian velocity , the disk material must be initially counter rotating : opposite the direction that the black hole is rotating . this appears inconsistent with the observations of zhang , cui , & chen ( 1997 ) and in general the physics occurring inside the ergosphere in which inertial frames are dragged in the direction that the black hole is rotating . even though we know that particles can have retrograde orbits inside the ergosphere , relative to an observer at infinity , it is highly improbable that the whole disk of matter will be counter rotating , at least in the general sense . further , it appears that the net rotational energy being `` extracted '' in the numerical simulation of these authors ( koide et al . 2000 ) in the form of electromagnetic energy over and above the gravitational binding energy released due to the hydrodynamic energy transported into the black hole is merely the rotational energy from the nonphysical initial condition that the accretion disk plasma is counter rotating as it falls into the ergosphere . ) ! not only is this an untrue statement , but it is a violation of the laws of physics . the frame dragging circular velocity inside the ergosphere as measured by an observer at infinity is @xmath355 ( see bardeen et al . 1972 ; misner et al . 1973 ; thorne et al . 1986 ; williams 1995 ; see also [ sec : cyg ] ) . ] on the other hand , for a co - rotating disk these authors found that the inward limiting radius is even larger ( @xmath356 ) , attributed to a centrifugal barrier ( koide et al . although this time - dependent mhd model is an excellent representation of subrelativistic ( @xmath357 ) jet formation in a kbh magnetosphere , the inconsistencies of this mhd model , as matter nears the event horizon ( @xmath358 ) , is probably an indication of the limitation , of such fluid dynamical models , in describing energy extraction from a rotating black hole : this being based on the guiding center approximation , wherein the single - particle approach is essential close to the black hole ( de felice & carlotto 1997 ; de felice & zanotti 2000 ) , i.e. , the behavior of individual particles is also that of the bulk of fluid elements . this means that gravitational - particle interactions , such as the penrose processes describe here ( in this paper ) , are required . note , in these penrose processes , which occur close to the event horizon , elementary electromagnetic and atomic forces dominate on the microscopic scale , while gravity is dominant on the macroscopic scale thus , as it should be in the strong gravitational potential well of the kbh ; but far away from @xmath359 electromagnetism appears to dominate macroscopically ( junor , biretta , & livio 1999 ) . moreover , stability of the co - rotating disk , falling inward to the limiting radius @xmath356 , at the keplerian velocity , when magnetic field lines are coupled to the infalling plasma , with the jet formation similar to that of the schwarzchild black hole case ( koide , shibata , & kudoh 1999 ) , suggests that the large scale magnetic field plays a dominant role at large distances from @xmath359 , irrespective of whether or not the black hole is rotating . in addition , these numerical inward limiting radii , at least in the case of the counter - rotating disk ( @xmath346 ) , may also be a display of the horizon being a `` vacuum infinity '' ( punsly & coroniti 1989 ; punsly 1991 ; williams 2003 ) : to the associated magnetic field charge neutral disk particle plasma , in accordance with the `` no - hair '' theorem ( carter 1973 ; misner , thorne , & wheeler 1973 ; williams 1995 ) , suggesting that the interaction of the disk magnetic field with particles in bound , trapped orbits at radii @xmath353 is negligible compared to the penrose gravitational - particle interactions described here . therefore , it appears that electromagnetic energy can not be effectively extracted from the so - called plunging region : where gravitational - particle interactions will clearly dominate if the magnetic flux of an axisymmetric @xmath360 , as it does in general upon nearing the vacuum infinity horizon [ punsly & coroniti 1989 ; punsly 1991 ; williams 2003 ; see also bik ( 2000 ) and bik & ledvinka ( 2000 ) for a detailed general relativistic calculation showing this ] . \2 . to convert the electromagnetic energy to particle energy at the event horizon , and to duplicate the observed luminosities from a poynting flux , it requires a large - scale magnetic field strength @xmath361 g ( wilms et al . 2001 ; blandford & znajek 1977 ) . in order to create @xmath9 pairs along the field lines , as in the case of pulsars , a field strength of at least @xmath362 g is needed ( sturrock 1971 ; sturrock , petrosian , & turk 1975 ) . ( the mechanism , however , for the generation of the pairs in an electromagnetic field to date is a subject of debate . ) the first of the large strengths required above appears to be achieved for supermassive kbhs , at present i.e . , with speculated assumptions . but for galactic black holes ( microquasars ) with masses @xmath363 , @xmath364 g seems highly impossible to generate from , in most cases , a binary system accretion disk plasma flow . an effective model for agns must also operate for microquasars as well . moreover , according to electrodynamics , in general , to lift the particles `` frozen '' to the magnetic field lines , from a disk , accelerating them to relativistic speeds , there has to be an electric field component @xmath365 ( lovelace 1976 ) . however , there exist problems in generating sufficient @xmath366 parallel to the polar direction ( @xmath367 axis ) ; none of the polar mhd models of this particular type adequately gets rid of this problem . magnetic reconnection may be a solution to some degree . \3 . to get around problems in items 1 and 2 ( specifically , the large strength field required and the vacuum infinity horizon ) it is assumed that a `` hot '' ion corona or torus - like accretion can provide the necessary jet particles : ( @xmath18 ) for the magnetosphere to act on , accelerating and collimating through centrifugal driving winds ( see below ) ; and ( @xmath64 ) to provide the hot ram pressure , to `` ram '' the magnetic field lines inward to the event horizon . however , now there appears to be a problem concerning how to liberate particles from trapped orbits inside the ergosphere ( particularly in the plunging regime ) onto escaping orbits . particles in the plunging regime , as defined by bardeen , press , and teukolsky ( 1972 ) , i.e. , massless and material particles ( with @xmath368 ) originating from infinity , can only escape , by being injected onto escaping orbits by some physically process near the black hole such as the penrose scattering processes described here since nothing can come out of the hole ( bardeen et al . therefore , the bz - type models are faced with yet another problem , as the magnetic field is assumed to get closer to the kbh : where general relativistic effects must be considered , i.e. , how do we get the necessary escaping particles in numbers out of the ergospheric region ( @xmath353 ) into the jets by such models ? moreover , with observations showing m87 not having the expected large `` dusty '' thermal ir - emitting torus ( perlman et al . 2001 ) that could have possibly served as particle jet `` fuel '' for a bz - type model , the penrose mechanism to extract energy momentum , as described by williams ( 1995 ) , the so - called penrose - williams mechanism , appears to be the only possible , plausible way to power this agn , and thus , generate its jets ( [ sec : m87 ] ) . so , in summary , in addition to the problems above associated with the bz - type mhd models , there still exists the historical problem : how does one convert from electromagnetic energy to the particle energies observed in the jets , emanating from the region where energy is observed to be extracted , i.e. , inside the ergosphere close to the event horizon ? none of the existing bz - type mhd models thus far adequately solves this `` age - old '' problem . \4 . in the centrifugal driven winds ( blandford & payne 1982 ) mentioned above , the following is assumed : if the disk magnetic field lines subtends an angle of more than @xmath369 to the rotation axis , the gas will be flung away from the disk into collimated jets with speeds a few times the escape velocity at the magnetic footprint on the disk . now , this may be true at @xmath370 , but near the event horizon @xmath359 , the escape conditions ( see williams 1995 ) must be adequately applied . recently , a general relativistic mhd treatment of evolving tori that includes in some degree features of the blandford & payne ( 1982 ) type - models , which allow for centrifugal driven winds to power the jets ( hirose et al . 2004 ; de villiers , hawley , & krolik 2004 ; de villiers , hawley , & krolik 2003 ) , found no such appreciable relativistic winds emerging from the horizon , nor the so - called plunging region , nor the ergospheric accretion disk that could be tied directly to rotational energy extraction from the black hole , although it was found that the lorentz force inside the ergosphere increased due to inertial frame dragging . the plunging region for @xmath371 lacked adequate resolution , suggesting perhaps the need for a general relativistic particle geodesic treatment , according to the guiding center approximation ( discussed in item 1 of this appendix ; see also williams 2003 ) . these mhd calculations ( hirose et al . 2004 ; de villiers et al . 2004 ; de villiers et al . 2003 ) did , however , confirm the existence of the predicted funnel region ( rees et al . 1982 ; see williams 2003 ) , and are consistent with the evolved magnetic field configuration found by bik ( 2000 ) and bik & ledvinka ( 2000 ) , i.e. , that radial lines are expelled from the surrounding equatorial region ( @xmath372 ) , but at the poles @xmath373 . this clearly shows that the classical bz - type models ( blandford & znajek 1997 ) , where magnetic field lines are proposed to anchor to the event horizon , thereby extracting rotational energy , could not be an important source , because @xmath372 in the region of importance for extracting rotational energy . also these mhd calculations seem to confirm the importance of magnetic fields on a large scale , in gravitational accretion processes , i.e. , in aiding mass outflows to large distances in the jets of black holes as well as those of protostars ( see williams 2004 ) . \5 . finally , to clear up any confusion , the authors of the historical paper ( wilms et al . 2001 ) loosely called the bz - type models the penrose effect the very name for years that had distinguished williams ( 1991 , 1995 , 1999 , 2001 ) internationally known successful four - dimensional penrose model ( see also piran & shaham 1977 ; leiter & kafatos 1978 ; kafatos & leiter 1979 ; kafatos 1980 ; wagh & dadhich 1989 ) from the bz - type models . strangely , these authors did not reference williams investigation . nevertheless , to set the record straight , the penrose mechanism [ as summarized here and described in detail in williams ( 1995 ) ] , which involves gravitational extraction of energy from a spinning black hole , based on that visualized by penrose ( 1969 ) , and that of the so - called bz mechanism , which involves electromagnetic extraction of energy ( blandford & znajek 1977 ) , are two very different models . so different that the statement made by the authors in wilms et al . ( 2001 ) , `` for parameters relevant to our discussion , the extra energy source is provided by the spin via the penrose effect occurring within the radius of marginal stability ( but outside of the stretched horizon ) , '' indeed requires a proper reference , since williams ( 1991 , 1995 ) model is popularly known as the only existing completely worked out model of the penrose mechanism : occurring within the radius of marginal stability @xmath13 . whatever the case may be , the recent observations of mcg6 - 30 - 15 ( wilms et al . 2001 ) and m87 ( perlman et al . 2001 ) introduce compelling evidence suggesting that perhaps it is the effects of williams black hole source model that is being observed ( as described in this paper ) , and hardly those of the bz - type models . the evidence presented here strongly suggests that observed black hole sources have a central energy generation similar the mechanism described in this present paper . so , to avoid any further confusion , it seems appropriate to refer to williams model as the penrose - 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in this paper , i present results from a theoretical and numerical ( monte carlo ) _
n - particle _ fully relativistic four - dimensional analysis of penrose scattering processes ( compton and @xmath0 ) in the ergosphere of a supermassive kerr ( rotating ) black hole .
these general relativistic model calculations surprisingly reveal that the observed high energies and luminosities of quasars and other active galactic nuclei , the collimated jets about the polar axis , and the asymmetrical jets ( which can be enhanced by relativistic doppler beaming effects ) _ all _ are inherent properties of rotating black holes . from this analysis
, it is shown that the penrose scattered escaping relativistic particles exhibit tightly wound coil - like cone distributions ( highly collimated vortical jet distributions ) about the polar axis , with helical polar angles of escape varying from @xmath1 to @xmath2 for the highest energy particles .
it is also shown that the gravitomagnetic ( gm ) field , which causes the dragging of inertial frames , exerts a force acting on the momentum vectors of the incident and scattered particles , causing the particle emission to be asymmetrical above and below the equatorial plane , thus appearing to break the equatorial reflection symmetry of the kerr metric .
when the accretion disk is assumed to be a two - temperature bistable thin disk / ion corona ( or torus , defining an advection - dominated accretion flow ) , energies as high as @xmath3 gev can be attained by these penrose processes alone ; and when relativistic beaming is included , energies in the tev range can be achieved , agreeing with observations of some bl lac objects . when this model is applied specifically to quasars 3c 279 and 3c 273 , and the seyfert 1 galaxy mcg6 - 30 - 15 , their observed high energy luminosity spectra in general can be explained .
this energy - momentum extraction model can be applied to any size black hole , irrespective of the mass , and therefore applies to microquasars as well . when applied to the classical galactic black hole source cygnus x-1 , the results are consistent with observations .
the consistency of these penrose model calculations with observations suggests that the external magnetic field of the accretion disk plays a negligible role in the extraction of energy momentum from a rotating black hole , inside the ergosphere , close to the event horizon where gravitational forces , and thus the dynamics of the black hole , appear to be dominant , as would be expected .
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the embedding of supersymmetric gauge theories in a string framework using systems of d - branes has been very fruitful and inspiring for many developments . for example , the famous ads / cft correspondence @xcite is rooted in the realization of the @xmath7 super yang - mills ( sym ) theory by means of d3-branes in flat space and in the profile of the supergravity bulk fields they induce in space - time . in less supersymmetric and/or in non - conformal cases ( like the @xmath8 gauge theories in four dimensions we will be interested in ) the corresponding gravitational profile depends on some transverse directions representing the energy scale thus accounting for the running of the gauge theory . this fact was explicitly checked long ago @xcite -@xcite at the perturbative level in @xmath8 sym theories realized by fractional d3 branes of type iib at non - isolated singularities , like for instance the @xmath9 orbifold . by studying the emission of closed string fields from such branes , the corresponding `` perturbative '' supergravity solutions were constructed and it was found that a scalar field from the twisted sector , which we will call @xmath10 , varies logarithmically in the internal complex direction @xmath11 transverse to the orbifold , matching precisely the perturbative logarithmic running of the gauge coupling with the energy scale . however , such perturbative solutions suffer from singularities at small values of @xmath11 , _ i.e. _ in the ir region of the gauge theory , and have to be modified by non - perturbative corrections . it is well - known that in @xmath8 gauge theories there is a whole series of non - perturbative contributions to the low - energy effective action that are due to instantons . in the last two decades tremendous advances have been made in the study of instanton effects within field theory ( for reviews , see for instance @xcite ) , and more recently also within string theory by means of d - instantons , _ i.e. _ d - branes with dirichlet boundary conditions in all directions @xcite-@xcite . in the seminal papers @xcite the exact solutions for the low - energy effective @xmath8 theories in the coulomb branch , including all instanton corrections , were found using symmetry and duality arguments . in particular it was shown that the effective sym dynamics in the limit of low energy and momenta can be exactly encoded in the so - called seiberg - witten ( sw ) curve which describes the geometry of the moduli space of the sym vacua . later these results were rederived from a microscopic point of view with the help of localization techniques @xcite that permit an explicit evaluation of the integrals over the multi - instanton moduli space . these techniques fit naturally in the string / d - brane context and indeed have been exploited for interesting generalizations of the sw results in many different directions . it is then natural to ask how the infinite tower of instanton effects is encoded in the dual holographic description of the gauge theory in terms of gravity . to answer this question one possibility is to exploit symmetry and duality arguments and determine the background geometry that incorporates the exact sw solution , like in the m - theory constructions based on configurations of d4 and ns5 branes @xcite . another possibility is to compute directly the multi - instanton corrections to the profiles of the gravitational bulk fields . this is what we will discuss in this contribution , which heavily relies on the content of @xcite and especially of @xcite . in particular we will briefly review how to derive the exact supergravity profile of the twisted field @xmath10 emitted by a system of fractional d3-branes at a @xmath12-orbifold singularity supporting a @xmath8 quiver gauge theory with unitary groups and bi - fundamental matter , and show how to obtain from it the exact running of the gauge coupling constant , including the non - perturbative contributions , in perfect agreement with the sw solution . we study the prototypical case of @xmath13 sym theories that are realized with fractional d3-branes at the non - isolated orbifold singularity @xmath14 . in this orbifold there are two types of fractional d3-branes , which we call types 0 and 1 , corresponding to the two different irreducible representations of @xmath12 . the most general brane configuration therefore consists of @xmath15 branes of type 0 and @xmath16 branes of type 1 , and corresponds to an @xmath17 quiver theory in four dimensions with gauge group u(@xmath15)@xmath18u(@xmath16 ) and with a matter content given by one hypermultiplet in the bi - fundamental representation @xmath19 and one hypermultiplet in the @xmath20 representation . the corresponding quiver diagram is represented in fig . [ fig : quiver ] . the branes of type 0 are represented by the blue circle while the branes of type 1 are represented by the red circle . the oriented lines connecting the two types of branes represent the hypermultiplets in the bifundamental representations.,title="fig : " ] # 1#2#3#4#5 @font ( 4706,1844)(1203,-3953 ) ( 1441,-3121)(0,0)[lb ] ( 5176,-3121)(0,0)[lb ] ignoring the gauge degrees of freedom on the @xmath16 branes , one obtains an @xmath17 u(@xmath15 ) sym theory with @xmath21 fundamental flavors and u(@xmath16 ) as global symmetry group . furthermore , we will decouple the u(1 ) factors and concentrate on the su(@xmath15)@xmath18 su(@xmath16 ) part of the symmetry group . in this contribution we focus on the case @xmath22 , representing an @xmath8 su(2 ) sym theory with @xmath3 flavors , but our results and methods apply to the general case as well @xcite . the su(2 ) @xmath3 sym theory has a vanishing @xmath23-function but , when the flavors are massive , the gauge coupling gets renormalized at 1-loop by terms proportional to the mass parameters of the hypermultiplets . this situation corresponds to placing the fractional d3-branes at different positions in the transverse plane , _ i.e. _ to giving non - vanishing vacuum expectation values to the adjoint scalars @xmath24 and @xmath25 of the vector multiplets on the two types of branes according to @xmath26 note that this brane configuration implies that the masses of the four flavors are given by @xmath27 in this case one finds that the perturbative part of gauge coupling constant @xmath28 is given by @xmath29 where @xmath30 is the bare coupling . besides these perturbative terms , there are also non - perturbative corrections due to instantons which can be explicitly computed using localization techniques ( see for instance @xcite for details ) . the first two instanton contributions turn out to be given by @xmath31 where @xmath32 is the instanton counting parameter . the complete effective coupling is therefore the sum of ( [ taupert ] ) and ( [ tauinst ] ) . for our future considerations it is convenient to rewrite it in terms of the gauge invariant quantity @xmath33 which parametrizes the moduli space of the effective theory at the quantum level . using the multi - instanton calculus and localization techniques , one can show that @xmath34 is related to the classical vacuum expectation value @xmath35 in the following way @xcite @xmath36 inverting this relation and substituting it into ( [ taupert ] ) and ( [ tauinst ] ) , after some simple algebra we find @xmath37 for simplicity , and also for later convenience , we have introduced a notation that explicitly exhibits only the dependence of @xmath38 on the gauge invariant parameter @xmath34 . when the flavors are massless , the effective coupling , which we denote by @xmath39 , is related to @xmath40 as follows @xmath41 it is interesting to observe that the inverse relation can be expressed in terms of modular functions . indeed , inverting ( [ tau0 m ] ) we obtain @xmath42 where the @xmath43 s are the jacobi @xmath43-functions in terms of the dedekind @xmath44-functions . this expression is amenable of interesting generalizations for superconformal field theories with gauge groups su(@xmath45 ) with @xmath46 @xcite . we also observe that our coupling is related by a t - duality transformation , @xmath47 , to the one usually considered in the literature @xcite for which the relation ( [ q0tau0 ] ) takes the form @xmath48 . ] . notice that even in this simple case , @xmath30 and @xmath39 are different and represent two different choices of effective couplings for the massless theory ( see also @xcite ) . let us now consider the same modular function appearing in ( [ q0tau0 ] ) , but instead of the massless coupling @xmath39 let us use as argument the massive coupling @xmath49 : @xmath50 taking the logarithm of @xmath51 , we then get @xmath52 this expression has a very nice interpretation . indeed , let us consider the sw curve for the su(2 ) @xmath3 sym which , when the flavor masses are as in ( [ masses ] ) , can be written as @xcite @xmath53 where @xmath54 with @xmath55 the curve ( [ curve ] ) describes a torus whose complex structure parameter is @xmath56 where @xmath35 and @xmath57 are the periods of the sw 1-form differential @xmath58 @xmath59 computed around a basis of dual cycles @xmath60 normalized in such a way that @xmath61 . the sw differential @xmath58 can be written as @xmath62 with @xmath63 using this information , one can compute @xmath38 and check that , when it is expanded in powers of @xmath40 , it coincides precisely with the effective gauge coupling ( [ taumv ] ) . other interesting quantities that characterize the curve ( [ curve ] ) are the anharmonic ratios of the four roots of the equation @xmath64 . it is quite easy to see that these roots are @xmath65 and that a corresponding anharmonic ratio is @xmath66 it is easy to see that the expansion of @xmath67 in powers of @xmath40 perfectly matches the expression found in ( [ tmv ] ) and thus we are led to the identification @xmath68 this is not surprising since the relation between the anharmonic ratio @xmath69 and the complex structure parameter @xmath38 of a curve like ( [ curve ] ) is precisely @xmath70 , namely the same relation between @xmath71 and @xmath72 implied by ( [ qtheta ] ) and ( [ tmv ] ) . we observe observe that the right hand side of ( [ ratio1 ] ) can be nicely written in terms of the polynomials @xmath73 and @xmath74 of the sw curve describing the theory at the so - called `` enhanon vacuum '' @xcite . this is the specific point of the quantum moduli space corresponding to @xmath75 which describes a classical extended brane configuration resembling that of the enhanon ring @xcite . in the enhanon vacuum we therefore have @xmath76 , and the polynomials @xmath73 and @xmath74 become @xmath77 then , from ( [ ratio1 ] ) and ( [ tlogz ] ) it is easy to realize that @xmath78 using the information encoded in the sw curve it is also possible to compute the exact quantum correlators @xmath79 forming the chiral ring elements of the gauge theory . these correlators are in fact given by the integral @xmath80 or alternatively , they can be obtained by expanding the generating functional @xcite @xmath81 integrating ( [ genphij ] ) with respect to @xmath11 , it is easy to find @xmath82 where the integration constant has been fixed in order to match the @xmath83 terms in the expansion for large @xmath11 in both sides . with some further straightforward algebra , we can rewrite the right hand side of ( [ trphi ] ) in the following form @xmath84 this expression will be essential in the next section to write the exact ( _ i.e. _ all order in the instanton expansion ) gravitational profile of the twisted scalar field @xmath10 emitted by the system of fractional d3 branes , and to relate it with the dual gauge theory coupling . the fractional d3-branes in the @xmath12 orbifold are gravitational sources for a non - trivial metric and a 4-form r - r potential from the untwisted sectors , and for two scalars , @xmath85 and @xmath86 , from the twisted ns - ns and r - r sectors respectively ( see for instance @xcite ) . while the emitted untwisted fields can propagate in all six directions transverse to the d3-branes , the twisted scalars only propagate in the complex plane transverse to the d3-brane world - volume which is not affected by the orbifold projection and which we parametrize with a complex coordinate @xmath87 . a system of fractional d3-branes distributed on this plane therefore generate a non - trivial dependence of the fields @xmath85 and @xmath86 on @xmath87 . the twisted scalars are conveniently combined in a complex field @xmath88 where here @xmath38 stands for the axio - dilaton of the type iib string theory . for simplicity we assume that the axion is trivial and that there are no branes other than the fractional d3 branes so that the dilaton does not run . thus , in this case we simply have @xmath89 where @xmath90 is the string coupling constant . the field @xmath10 is actually part of a chiral bulk superfield @xmath91 whose structure is schematically given by @xmath92 with dots denoting the supersymmetric descendants of @xmath10 and @xmath93 being the complex conjugate of @xmath10 . the profile of the twisted scalar @xmath10 emitted by a system of fractional d3-branes can be derived by solving the classical field equations that follow from the bulk action containing the kinetic terms and the source action describing the emission from the fractional d3-branes . at the perturbative level this profile was obtained long ago in ref.s @xcite -@xcite and for a system of @xmath15 branes of type 0 and @xmath16 branes of type 1 located at the origin is @xmath94 where @xmath95 and @xmath96 is an arbitrary length scale . it is convenient to introduce the quantities @xmath97 with mass dimension 1 , and rewrite the solution ( [ proft ] ) as follows @xmath98 note that in the conformal cases ( @xmath99 ) , we simply have @xmath100 let us now consider a more general configuration in which the d3-branes are not all at the origin . this amounts to giving the adjoint scalars non - vanishing vacuum expectation values as in ( [ am ] ) ( from now on we focus again only on the case @xmath22 ) . then , one can show that the @xmath10 profile corresponding to such a configuration is @xmath101 it is not difficult to realize that this @xmath10 field satisfies the following differential equation @xmath102 with @xmath103 where in the second step we introduced the momentum operator conjugate to @xmath11 , that is @xmath104 . the current @xmath105 has a nice interpretation in terms of disk diagrams describing the couplings among the closed string twisted fields and the massless open string excitations of the fractional d3-branes . indeed , by considering the interactions of the ns - ns scalar @xmath85 ( whose vertex operator we denote by @xmath106 ) with the scalar @xmath24 ( whose vertex we denote by @xmath107 ) , we find @xmath108 this result follows by computing the correlation functions of the vertex operators using standard cft techniques as discussed for example in @xcite and by frozing the scalars to their vacuum expectation values . a completely similar calculation can be performed with the scalar @xmath25 of the type 1 branes leading to @xmath109 where the extra sign comes from the fact that branes of type 1 have opposite @xmath85-charge with respect of those of type 0 . adding an analogous term describing the interactions of the r - r twisted scalar @xmath86 , we can write the total contribution to the effective action as @xmath110 supersymmetry requires that this interaction must be accompanied by other structures ( that could also be computed from string diagrams with extra fermionic insertions ) in such a way that the effective action follows from a holomorphic prepotential . as discussed in @xcite-@xcite such a prepotential is obtained simply by promoting the bulk and boundary scalars to the corresponding chiral superfields . denoting by @xmath111 the fluctuation part of @xmath91 , one finds in particular the following term @xmath112 where the dots represent interactions of higher orders in @xmath111 . the effective action follows upon integrating the prepotential over @xmath113 ; when all four @xmath43 s are taken from @xmath111 and the superfields @xmath114 and @xmath115 are frozen to their vacuum expectation values , we recover precisely the interaction ( [ intert ] ) . the classical current ( [ tcl0 ] ) therefore is associated to a source term for @xmath10 and is related to the prepotential ( [ f ] ) in the following way @xmath116 let us now investigate how the classical profile ( [ proftam ] ) changes when non - perturbative effects due to gauge instantons are taken into account . in our brane set - up , instantons are introduced by adding fractional d(1)-branes . since we neglect the dynamics on the branes of type 1 , we only consider the effects produced by adding @xmath117 d - instantons of type 0 . the physical excitations of the open strings with at least one end - point on the d(1)-branes account for the instanton moduli which we collectively denote as @xmath118 . they consist of the neutral sector , corresponding to d(1)/d(1 ) open strings that do not transform under the gauge group , and of the charged and flavored sectors arising respectively from the d(1)/d3@xmath119 and d(1)/d3@xmath120 open strings . the complete list of instanton moduli and their transformation properties can be found in @xcite . here we just recall that among the neutral moduli we have the bosonic and fermionic goldstone modes of the supertranslations of the d3-brane world - volume which are broken by the d - instantons and which are identified with the superspace coordinates @xmath121 and @xmath43 , and a complex scalar @xmath122 transforming in the adjoint representation of the instanton symmetry group u(@xmath117 ) , whose eigenvalues describe the position of the d - instantons in the un - orbifolded directions transverse to the fractional d3-branes . in order to find the non - perturbative @xmath10 profile we first compute the instanton induced prepotential @xmath123 from which the non - perturbative source current @xmath124 can be derived following a procedure similar to the one outlined for the classical current @xmath105 . the non - perturbative prepotential is defined as @xmath125 where the integral is performed over the centered moduli @xmath126 , which include all moduli except the superspace coordinates @xmath121 and @xmath43 . here @xmath127 is the instanton action , describing the interactions of the instanton moduli with the boundary and bulk superfields . as explained in @xcite , such an action is @xmath128 where @xmath129 is the part accounting for the interactions of the moduli among themselves and with the fields in the vector multiplet , and @xmath130 means trace over the u(@xmath117 ) indices . inserting ( [ sinst1 ] ) in ( [ fnp ] ) , to linear order in @xmath111 we find @xmath131 the integration over the moduli space can be explicitly performed using localization techniques and nekrasov s approach to the multi - instanton calculus @xcite . this amounts to first define the deformed instanton partition function @xmath132 where @xmath133 and @xmath134 are deformation parameters which in our string set - up can be introduced by putting the brane system in a graviphoton background @xcite , and then to compute the prepotential according to @xmath135 the integral appearing in ( [ fnp2 ] ) is related to the instanton part of the chiral ring elements @xmath79 of the gauge theory on the d3-branes , which can be computed as @xmath136 notice that the integrals in ( [ zinst ] ) and ( [ trphi0 ] ) are over all moduli including @xmath121 and @xmath43 , and that in the limit @xmath137 the factor @xmath138 in ( [ trphi0 ] ) compensates for the volume @xmath139 of the regularized four dimensional superspace . plugging ( [ trphi0 ] ) into ( [ fnp2 ] ) one gets @xmath140 which is nothing but the instanton completion of ( [ f ] ) . adding the classical and the instanton contributions we obtain the full source current for @xmath10 : @xmath141 where @xmath142 . the field equation satisfied by @xmath10 is therefore @xmath143\,\frac{\partial^{\ell}}{\partial z^{\ell}}\,\delta^2(z ) \label{fecom}\ ] ] which is solved by @xmath144 this explicit solution shows that all non - trivial information about the @xmath10 profile is contained in the ring of chiral correlators of the gauge theory defined on the d3-branes . this chiral ring accounts therefore for the full tower of d - instanton corrections to the gravity solution . the chiral correlators @xmath79 can be computed from ( [ trphi0 ] ) using nekrasov s approach to the multi - instanton calculus . equivalently ( and more efficiently ) , as we have explained in the previous section , they can be obtained from the sw curve describing the sym theory . in fact , inserting ( [ trphi1 ] ) in ( [ tauz ] ) and taking into account the explicit definition of @xmath74 given in ( [ pq ] ) , we can obtain the exact expression for the twisted scalar field emitted by the brane system , namely @xmath145 our result generalizes the one derived in @xcite for the pure su(@xmath45 ) sym theories using supergravity and m - theory considerations , and is also perfectly consistent with the findings of @xcite where the su(@xmath146 sym theory is realized in type iia using d4 branes stretched between two ns branes . we can therefore say that the methods we have developed provide a microscopic derivation of the supergravity profile for @xmath10 in which a direct relation with the chiral ring elements of the gauge theory on the source branes is clearly established and the non - perturbative effects are explicitly explained in terms of fractional d - instantons . we have considered a fractional d3-brane system in a @xmath0 orbifold supporting an @xmath147 sym theory with su@xmath2 gauge group and @xmath3 flavors . we have considered the scalar field @xmath10 from the twisted closed string sector emitted by such a configuration , which , at the tree level , plays the rle of the gauge coupling on the d3-branes . as it is well known , the fractional d3-branes act as sources for @xmath10 , so that @xmath10 has a logarithmic profile in the complex direction @xmath11 transverse to the orbifold ; this profile matches the perturbative running of the gauge coupling if the transverse space is identified with the coulomb branch of the gauge theory . we have taken into account the non - perturbative effects corresponding to the inclusion of ( fractional ) d - instantons and explicitly shown how they modify the source for @xmath10 and hence its profile . the moduli space integrals that determine the non - perturbative source terms are related to the ones appearing in the computation of the chiral ring operators of the gauge theory . through this relation , we can then express the profile of the twisted scalar as the quantum expectation value of its perturbative expression ( see ( [ tauz ] ) ) . this , in turn , can be written in terms of the sw curve that describes the effective dynamics of the gauge theory on the coulomb moduli space ( see ( [ tsugra ] ) ) . at the non - perturbative level , the gauge / gravity relation is deeply modified with respect to its perturbative standing . the twisted scalar @xmath10 can no longer be simply identified with the effective gauge coupling . however , if we consider the situation in which the source d3-branes sit at the `` enhanon '' vacuum , @xmath75 , the scalar @xmath148 is still directly , albeit non - trivially , related to the effective coupling @xmath49 when @xmath149 is identified with the quantum coulomb space variable @xmath34 . indeed , in this case @xmath148 is given by ( [ texact ] ) ; this expression corresponds , according to ( [ tlogz ] ) , to the logarithm of the anharmonic ratio @xmath69 which parametrizes the sw torus . the anharmonic ratio is related to the complex structure @xmath38 , namely to the effective gauge coupling , through the modular function appearing in ( [ qtheta ] ) . in these proceedings we focused on the conformal su(2 ) case , but in @xcite we showed that a similar pattern occurs for higher rank conformal gauge theories , and also , after decoupling some flavors , for asymptotically free cases : the twisted scalar emitted by the branes at the enhanon vacuum is related in the gauge / gravity correspondence to the low energy effective couplings via non - 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recently we provided a microscopic derivation of the exact supergravity profile for the twisted scalar field emitted by systems of fractional d3-branes at a @xmath0 orbifold singularity . in this contribution
we focus on a set - up supporting an @xmath1 sym theory with su@xmath2 gauge group and @xmath3 .
we take into account the tower of d - instanton corrections to the source terms for the twisted scalar and find that its profile can be expressed in terms of the chiral ring elements of the gauge theory .
we show how the twisted scalar , which at the perturbative level represents the gravity counterpart of the gauge coupling , at the non - perturbative level is related to the effective gauge coupling in an interestingly modified way .
* non - perturbative aspects of gauge / gravity duality * 0.5 cm marco bill@xmath4 , marialuisa frau@xmath4 , luca giacone@xmath5 and alberto lerda@xmath6 0.4 cm _ @xmath4 universit di torino , dipartimento di fisica and _ _ i.n.f.n . - sezione di torino _ _ via p.giuria 1 , i-10125 torino , italy _ 0.3 cm _ @xmath6 universit del piemonte orientale , _ _ dipartimento di scienze e innovazione tecnologica and _ _ i.n.f.n .
- gruppo collegato di alessandria - sezione di torino _ _ viale t. michel 11 , i-15121 alessandria , italy
. _ 0.3 cm _ e mail : billo , frau , giacone , [email protected]_ 0.5 cm
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You are an expert at summarizing long articles. Proceed to summarize the following text:
it is widely believed that accretion disks around kerr black holes exist in many astrophysical environments , ranging from active galactic nuclei to some stellar binary systems @xcite . people usually assume that the inner boundary of a thin keplerian disk around a kerr black hole is located at the marginally stable circular orbit , inside which the centrifugal force is unable to stably balance the gravity of the central black hole @xcite . in the disk region , particles presumably move on nearly circular orbits with a very small inward radial velocity superposed on the circular motion , the gravity of the central black hole is approximately balanced by the centrifugal force . as disk particles reach the marginally stable circular orbit , the gravity of the central black hole becomes to dominate over the centrifugal force and the particles begin to nearly free - fall inwardly . the motion of fluid particles in the plunging region quickly becomes supersonic then the particles loose causal contact with the disk , as a result the torque at the inner boundary of the disk is approximately zero ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . this is usually called the `` no - torque inner boundary condition '' of thin accretion disks . some recent studies on the magnetohydrodynamics ( mhd ) of accretion disks have challenged the `` no - torque inner boundary condition '' . magnetic fields have been demonstrated to be the most favorable agent for the viscous torque in an accretion disk transporting angular momentum outward ( * ? ? ? * and references therein ) . by considering the evolution of magnetic fields in the plunging region , krolik @xcite pointed out that magnetic fields can become dynamically important in the plunging region even though they are not so on the marginally stable circular orbit , and argued that the plunging material might exert a torque to the disk at the marginally stable circular orbit . with a simplified model , gammie @xcite solved maxwell s equations in the plunging region and estimated the torque on the marginally stable circular orbit . he demonstrated that the torque can be quite large and thus the radiation efficiency of the disk can be significantly larger than that for a standard accretion disk where the torque at the inner boundary is zero . furthermore , agol and krolik @xcite have investigated how a non - zero torque at the inner boundary affects the radiation efficiency of a disk . numerical simulations of mhd disks @xcite have greatly improved our understanding of disk accretion processes . these simulations show that the magneto - rotational instability effectively operates inside the disk and leads to accretion , though the accretion picture is much more complicated than that assumed in the standard theory of accretion disks . generally , the disk accretion is non - axisymmetric and strongly time - dependent . it is also found that , as disk material gets into the plunging region , the magnetic stress at the marginally stable circular orbit does not vanish but smoothly extends into the plunging region @xcite , though the effect is significantly reduced as the thickness of the disk goes down @xcite . furthermore , the specific angular momentum of particles in the plunging region does not remain constant , which implies that the magnetic field may be dynamically important in the plunging region @xcite . all these results are fascinating and encouraging . unfortunately , due to the limitation in space resolution and time integration , stationary and geometrically thin accretion disks are not accessible to the current 2-d and 3-d simulations . so it remains unclear how much insights we can get for stationary and geometrically thin accretion disks from these simulations @xcite . instead of small - scale and tangled magnetic fields in an accretion disk transporting angular momentum within the disk , a large - scale and ordered magnetic field connecting a black hole to its disk may also exist and play important roles in transportation of angular momentum and energy between the black hole and the disk @xcite . recent _ xmm - newton _ observations of some seyfert galaxies and galactic black hole candidates provide possible evidences for such a magnetic connection between a black hole and its disk @xcite . all these promote the importance of studying the evolution and the dynamical effects of magnetic fields around a kerr black hole . in this paper , we use a simple model to study the evolution of magnetic fields in the plunging region around a kerr black hole . we assume that around the black hole the spacetime metric is given by the kerr metric ; in the plunging region , which starts at the marginally stable circular orbit and ends at the horizon of the black hole , a stationary and axisymmetric plasma fluid flows inward along timelike geodesics in a small neighborhood of the equatorial plane . the plasma is perfectly conducting and a weak magnetic field is frozen to the plasma . the magnetic field and the velocity field have two components : radial and azimuthal . we will solve the two - dimensional maxwell s equations where the magnetic field depends on two variables : time and radius , and investigate the evolution of the magnetic field . this model is similar to that studied by gammie @xcite , but here we include the time variable . furthermore , we ignore the back - reaction of the magnetic field on the motion of the plasma fluid to make the model self - consistent , since if the dynamical effects of the magnetic field are important the strong electromagnetic force will make the fluid expand in the vertical direction . the ignorance of the back - reaction of the magnetic field will allow us to to analytically study the evolution of the magnetic field , but it will prevent us from quantitatively studying the dynamical effects of the magnetic field . however , we believe that the essential features of the evolution of the magnetic field in the plunging region is not sensitive to the details of the dynamical effects . to check the self - consistency of the model , we will estimate the dynamical effects of the magnetic field by considering the back - reaction of the magnetic field on the fluid motion . we will also discuss the self - consistent solutions to the coupled maxwell and dynamical equations , and look for implications to the `` no - torque inner boundary condition '' . the paper is organized as follows : in sec . [ sec2 ] we write down the maxwell s equations for an ideal mhd plasma . in sec . [ sec3 ] we write down the general forms of the magnetic field and the velocity field of the plasma fluid around a kerr black hole . in sec . [ sec4 ] we solve maxwell s equations with the approximations outlined above . in sec . [ sec5 ] , with the solutions obtained in sec . [ sec4 ] , we study the evolution of the magnetic field in the plunging region . in sec . [ sec6 ] , we check if magnetic fields can become dynamically important in the plunging region . in sec . [ sec7 ] we draw our conclusions . in the appendix we solve maxwell s equations in the disk region , since we want our solutions in the plunging region to be continuously joined to the solutions in the disk region . throughout the paper we use the geometrized units @xmath0 and the boyer - lindquist coordinates @xmath1 @xcite , except in the appendix where cylindrical coordinates are used . in a curved spacetime , maxwell s equations are @xmath2 } = 0\ ; , \hspace{0.5 cm } \label{max2}\end{aligned}\ ] ] where @xmath3 is the electromagnetic field tensor , @xmath4 is the current density 4-vector of electric charge . in eqs . ( [ max1 ] ) and ( [ max2 ] ) , @xmath5 denotes the covariant derivative operator that is compatible with metric : @xmath6 ; the square brackets `` [ ] '' denote antisymmetrization of a tensor . for an ideal mhd plasma fluid whose electric resistivity is zero , the electric field in the comoving frame is zero , i.e. @xmath7 where @xmath8 is the 4-velocity of the fluid . in other words , the magnetic field is frozen to the plasma fluid . then , the electromagnetic field tensor @xmath9 can be written as @xmath10 where @xmath11 is the magnetic field measured by an observer comoving with the fluid ( i.e. , having a 4-velocity @xmath8 ) , and @xmath12 is the totally antisymmetric tensor of the volume element that is associated with the metric @xmath13 . by definition , the magnetic field @xmath11 satisfies @xmath14 the corresponding electromagnetic stress - energy tensor is @xmath15 where @xmath16 . in the case of mhd , the electric current density @xmath4 is unknown , but it is only defined by eq . ( [ max1 ] ) . the fundamental variables are the electric field @xmath17 and the magnetic field @xmath11 . maxwell s equations are then reduced to eq . ( [ max2 ] ) . since in the comoving frame of an ideal mhd flow @xmath18 and @xmath11 is related to the electromagnetic tensor @xmath9 by eq . ( [ fab ] ) , the dual of eq . ( [ max2 ] ) gives @xcite @xmath19 } ) = 0 \;. \label{max3}\end{aligned}\ ] ] for an ideal mhd fluid eqs . ( [ max2 ] ) and ( [ max3 ] ) are equivalent . ( [ max3 ] ) can be expanded as @xmath20 the contraction of eq . ( [ max3a ] ) with @xmath21 leads to @xmath22 where @xmath23 is the acceleration of the fluid . in deriving eq . ( [ div ] ) we have used eq . ( [ bua ] ) and the identity @xmath24 . substituting eq . ( [ div ] ) into eq . ( [ max3a ] ) , we get @xmath25 the tensor @xmath26 is decomposed as @xcite @xmath27 where @xmath28 is the space - projection tensor , @xmath29 is the expansion , @xmath30 is the shear tensor , and @xmath31}$ ] is the vorticity tensor of the fluid . here the braces `` ( ) '' denote symmetrization of a tensor . it is easy to check that @xmath32 substituting eq . ( [ dab ] ) into eq . ( [ max3b ] ) , we obtain @xmath33 which shows that the evolution of the magnetic field is governed by the expansion , the shear , the vorticity , and the acceleration of the fluid . the contraction of eq . ( [ max3c ] ) with @xmath34 gives @xmath35 where we have used @xmath36 and eq . ( [ bua ] ) . we note that with the space - projection tensor @xmath37 , eq . ( [ div ] ) can be written as @xmath38 which says that the spatial divergence of the magnetic field is zero . in terms of differential forms , the tensor @xmath9 is a closed 2-form , then the maxwell equation ( [ max2 ] ) can be written as @xmath39 @xcite . using stokes theorem and eq . ( [ ezero ] ) , it can be shown that for a perfect conducting fluid the magnetic flux threading any 2-dimensional spatial surface @xmath40 is unchanged as the surface moves with the fluid @xmath41 this is the mathematical formulation for the statement that magnetic field lines are frozen to a perfectly conducting fluid . now let us assume that the background spacetime is the outside of a kerr black hole of mass @xmath42 and angular momentum @xmath43 , where @xmath44 . in boyer - lindquist coordinates , the kerr metric is @xcite @xmath45 where @xmath46 a kerr black hole usually has two event horizons : an inner event horizon and an outer event horizon , whose radii are given by the two roots of @xmath47 . , the kerr black hole becomes a schwarzschild black hole , then the inner event horizon disappears . when @xmath48 ( the case of an extreme kerr black hole ) , the inner event horizon coincides with outer event horizon . ] what is relevant in this paper is the outer event horizon , so whenever we talk about the `` event horizon '' we always mean the outer event horizon , whose radius is @xmath49 let us define an orthonormal tetrad attached to an observer comoving with the frame dragging of the kerr black hole , @xmath50 , by @xmath51\ ; , \hspace{4.3 cm } \nonumber \\ e_1^a \equiv \left(\frac{\delta}{\sigma}\right)^{1/2 } \left(\frac{\partial}{\partial r}\right)^a\ ; , \hspace{0.6 cm } e_2^a \equiv \frac{1}{\sigma^{1/2 } } \left(\frac{\partial}{\partial \theta}\right)^a\ ; , \hspace{0.6 cm } e_3^a \equiv \left(\frac{\sigma}{a}\right)^{1/2 } \frac{1 } { \sin\theta } \left(\frac{\partial}{\partial \phi}\right)^a\ ; , \label{ont2}\end{aligned}\ ] ] where @xmath52 which are respectively the lapse function and the frame dragging angular velocity . as @xmath53 , we have @xmath54 and @xmath55 , where @xmath56 is the angular velocity of the event horizon . then , the 4-velocity of the fluid , @xmath8 , can be decomposed as @xmath57 where @xmath58 are the components of the 3-velocity of the fluid relative to the observer comoving with the frame dragging , and @xmath59 is the lorentz factor . inserting eq . ( [ ont2 ] ) into eq . ( [ ua1 ] ) , we have @xmath60\ ; , \label{ua2}\end{aligned}\ ] ] where @xmath61 is the angular velocity of the fluid . the specific angular momentum of a fluid particle is @xmath62 obviously , @xmath63 when @xmath64 . thus , an observer comoving with the frame dragging of a kerr black hole has zero angular momentum @xcite . the specific energy of a fluid particle is @xmath65 in eq . ( [ sen ] ) , the term @xmath66 represents the coupling between the orbital angular momentum of the particle and the frame dragging of the kerr black hole . when the particle moves on a geodesic , @xmath67 and @xmath68 defined above are conserved @xcite . the magnetic field @xmath11 , which satisfies eq . ( [ bua ] ) , can be decomposed as @xmath69 from which we have @xmath70 inserting eq . ( [ ont2 ] ) into eq . ( [ ba1 ] ) , we have @xmath71 + b_{\hat{r } } \left(\frac{\delta}{\sigma } \right)^{1/2 } \left(\frac{\partial}{\partial r}\right)^a \nonumber\\ & & + \frac{b_{\hat{\theta}}}{\sigma^{1/2 } } \left(\frac { \partial}{\partial \theta}\right)^a + \frac{b_{\hat{\phi } } } { \sin\theta } \left(\frac{\sigma}{a}\right)^{1/2 } \left(\frac{\partial}{\partial \phi}\right)^a\ ; ; \label{ba2}\end{aligned}\ ] ] and , correspondingly @xmath72 the maxwell equations that we want to solve are given by eq . ( [ max3 ] ) . in terms of coordinate components , ( [ max3 ] ) takes a very simple form @xmath73 = 0 \ ; , \label{maxeq}\end{aligned}\ ] ] where @xmath74 , @xmath75 is the determinant of the metric tensor @xmath76 , @xmath77 and @xmath78 are respectively @xmath79 and @xmath80 because of the constraint eq . ( [ bua ] ) , among the four equations of ( [ maxeq ] ) only three are independent . since the background spacetime is stationary and axisymmetric , we can look for stationary and axisymmetric solutions with @xmath81 . however , since we are interested in the time evolution of magnetic fields , we will keep the @xmath82 terms on magnetic fields but adopt that @xmath83 . to simplify the problem , we further assume that in a small neighborhood of the equatorial plane ( i.e. , @xmath84 ) , @xmath85 ( i.e. , @xmath86 ) . this assumption , which has also been used by gammie @xcite , ensures that @xmath87 on the equatorial plane . we emphasize that , when @xmath88 , this assumption holds only if the fluid moves geodesically , which requires that the magnetic fields are weak and their dynamical effects are negligible . otherwise the electromagnetic force will make @xmath89 and @xmath90 non - zero except exactly on the equatorial plane . thus , hereafter we assume that fluid particles move on timelike geodesics in the plunging region . this assumption will be justified latter . now , let us focus on solutions on the equatorial plane ( @xmath91 ) . considering the fact that for the kerr metric @xmath92 on the equatorial plane , eq . ( [ maxeq ] ) is reduced to @xmath93 + \frac{\partial}{\partial r } \left[r^2 ( u^r b^\beta - u^\beta b^r)\right ] = 0 \;. \label{eq1}\end{aligned}\ ] ] for @xmath94 , eq . ( [ eq1 ] ) gives @xmath95 = 0 \ ; , \hspace{1 cm } \frac{\partial}{\partial r}\left[r^2 ( u^r b^t - u^t b^r)\right ] = 0 \ ; , \label{eq2}\end{aligned}\ ] ] respectively . the solution of eq . ( [ eq2 ] ) is @xmath96 where @xmath97 is a constant . with eq . ( [ flux ] ) , it can be checked that @xmath98 is the magnetic flux threading the 2-dimensional surface defined by @xmath99 , @xmath100 , @xmath101 and @xmath102 . for @xmath103 , eq . ( [ eq1 ] ) is automatically satisfied since @xmath104 in a small neighborhood of the equatorial plane everywhere and all the time . for @xmath105 , eq . ( [ eq1 ] ) gives @xmath106 + \frac{\partial}{\partial r } \left[r^2 ( u^r b^\phi - u^\phi b^r)\right ] = 0 \;. \label{eq3}\end{aligned}\ ] ] from the constraint eq . ( [ bua ] ) , we have @xmath107 substituting eq . ( [ btr ] ) for @xmath108 into eq . ( [ cons ] ) , we obtain @xmath109 \;. \label{bfr}\end{aligned}\ ] ] now , substitute eq . ( [ btr ] ) for @xmath108 and eq . ( [ bfr ] ) for @xmath110 into eq . ( [ eq3 ] ) , we obtain a first order partial differential equation @xmath111 where @xmath112 in deriving eq . ( [ eq4 ] ) we have used @xmath113 . ( [ eq4 ] ) simply says that @xmath114 is conserved along the worldline of a fluid particle : @xmath115 . let us define @xmath116 which is the coordinate time interval spent by a fluid particle to move from @xmath117 to @xmath118 . then eq . ( [ eq4 ] ) can be written as @xmath119 the solution of eq . ( [ eq5 ] ) is simply @xmath120 i.e. @xmath114 is a function of @xmath121 . ( [ psol ] ) gives a `` retarded '' solution to eq . ( [ eq5 ] ) : at any time @xmath122 the solution at radius @xmath118 is given by the solution at an earlier time @xmath123 at the radius @xmath124 . thus , a variation in the magnetic fields at any @xmath118 will propagate with the fluid motion ( fig . [ fig1 ] ) . the solution is unique if a suitable boundary or initial condition is imposed . for example , if a boundary condition is given on @xmath125 : @xmath126 , then the solution is @xmath127 . in order for the solution to exist for a region specified by @xmath128 in the @xmath129-space , the boundary function @xmath130 must be defined on the whole @xmath131-axis : @xmath132 . similarly , if a boundary condition is given on @xmath133 ( i.e. , @xmath134 ) : @xmath135 , then the solution is @xmath136 . in this case , in order for the solution to exist for a region specified by @xmath137 in the @xmath129-space , the boundary function @xmath138 must be defined on the whole @xmath122-axis : @xmath139 . we can also specify the boundary condition in another way @xmath140 then , the solution of eq . ( [ eq5 ] ) is @xmath141 i.e. , the value of @xmath114 in region @xmath142 ( region i ) is determined by the value of @xmath114 on the boundary @xmath143 ; the value of @xmath114 in region @xmath144 ( region ii ) is determined by the value of @xmath114 on the boundary @xmath145 ( see fig . [ fig2 ] ) . in order for the solutions to be smoothly matched on the diagonal line @xmath146 separating region i and region ii , @xmath130 and @xmath138 must be smoothly matched at @xmath147 : @xmath148 given the solution of @xmath114 in eq . ( [ psol ] ) , we can solve @xmath149 from eq . ( [ psi ] ) , then solve @xmath110 from eq . ( [ bfr ] ) . the results are @xmath150 \;. \label{fsol2}\end{aligned}\ ] ] using eq . ( [ btr ] ) , we obtain @xmath151 \;. \label{fsol2a}\end{aligned}\ ] ] note that @xmath108 , @xmath149 , and @xmath110 satisfy the constraint eq . ( [ cons ] ) , so among the three components only two are independent . from eqs . ( [ ualp ] ) , ( [ balp ] ) , ( [ fsol1 ] ) , ( [ fsol2 ] ) , and the fact that @xmath152 and @xmath153 , we can solve for @xmath154 and @xmath155 @xmath156 where eqs . ( [ sang ] ) and ( [ sen ] ) have been used to simplify the expression for @xmath155 . since we focus on the solutions on the equatorial plane , here and hereafter we set @xmath157 , and use @xmath158 , @xmath159 , and @xmath160 to refer their values at @xmath161 . note , in the solutions in eqs . ( [ fsol1 ] ) and ( [ fsol2 ] ) [ or , equivalently , eqs . ( [ br ] ) and ( [ bf ] ) ] all the dependence on time @xmath122 is contained in the function @xmath114 . since we assume that the fluid particles move on geodesics , the specific angular momentum @xmath67 and the specific energy @xmath68 are constants . if @xmath114 is also a constant , the combination @xmath162 is a constant , which we denote as @xmath163 . then , eqs . ( [ br ] ) and ( [ bf ] ) become @xmath164 which are stationary solutions of maxwell s equations . if the black hole is a schwarzschild black hole ( i.e. , the specific angular momentum @xmath165 ) , the stationary solutions are reduced to @xmath166 where @xmath167 , @xmath168 , and @xmath169 . the non - stationary solutions can be obtained by replacing @xmath163 with @xmath162 . we are interested in the evolution of magnetic fields in the plunging region in the equatorial plane between @xmath170 , the marginally stable circular orbit , and @xmath171 , the event horizon of the black hole . for direct circular orbits ( i.e. , corotating with @xmath172 ) around a kerr black hole , the radius of the marginally stable orbit is given by @xcite @xmath173^{1/2}\right\ } \ ; , \label{rms}\end{aligned}\ ] ] where @xmath174 ( @xmath175 ) if @xmath176 ( @xmath177 ) , and @xmath178 \;,\\ z_2 & \equiv & \left(z_1 ^ 2 + 3 \frac{a^2}{m^2}\right)^{1/2 } \;.\end{aligned}\ ] ] the angular velocity of a particle geodesically moving on the marginally stable circular orbit is @xmath179^{-1 } \;. \label{wr}\end{aligned}\ ] ] from eqs . ( [ angv ] ) , ( [ rms]-[wr ] ) , we can calculate the circular velocity on the marginally stable circular orbit by @xmath180 the corresponding lorentz factor by @xmath181 , specific angular momentum @xmath182 by eq . ( [ sang ] ) , and specific energy @xmath183 by eq . ( [ sen ] ) ( setting @xmath161 and @xmath170 ) . here and hereafter the subscript `` ms '' represents the values at @xmath184 . now let us choose the boundary radius @xmath185 , the conserved specific angular momentum and specific energy to be @xmath186 where @xmath187 . i.e. , we keep the specific energy the same as that on the marginally stable circular orbit , but decrease the specific angular momentum by a little amount . then , we can calculate the lorentz factor at @xmath170 , corresponding to the specific energy and the specific energy specified by eq . ( [ le0 ] ) , by @xmath188 and the corresponding boundary values of @xmath189 and @xmath190 at @xmath170 by @xmath191 from eqs . ( [ sang ] ) and ( [ sen ] ) , we can calculate @xmath192 , @xmath193 , @xmath189 , and @xmath190 at any @xmath118 by @xmath194^{1/2 } \;,\end{aligned}\ ] ] where @xmath68 and @xmath67 are given by eq . ( [ le0 ] ) . the parameter @xmath131 defined by eq . ( [ tau ] ) can then be calculated with @xmath195 with the above formulae at hands , we can calculate @xmath154 , @xmath155 , and @xmath196 at any @xmath118 by eqs . ( [ br ] ) and ( [ bf ] ) , giving the constant @xmath97 and the function @xmath197 . to determine @xmath97 and @xmath114 , we need to specify the boundary conditions for the magnetic field . we will consider stationary solutions and non - stationary solutions separately . it is straightforward to specify the boundary conditions for stationary solutions ( i.e. solutions with @xmath198 ) . to determine the solutions , we need only specify the values of @xmath154 and @xmath155 at @xmath170 : @xmath199 and @xmath200 . then , by eqs . ( [ br1 ] ) and ( [ bf1 ] ) , we have @xmath201 where @xmath202 , @xmath203 is the angular velocity corresponding to @xmath204 @xmath205 with the @xmath163 and @xmath114 determined above , we can calculate @xmath154 , @xmath155 , and @xmath196 at any @xmath118 by eqs . ( [ br1 ] ) and ( [ bf1 ] ) . since @xmath154 and @xmath155 linearly depend on @xmath199 and @xmath200 , it is sufficient to study the effects of @xmath199 and @xmath200 separately . the results for any linear combination of @xmath199 and @xmath200 are simply linear superpositions of the results for @xmath199 and @xmath200 separately . in fig . [ fig3 ] we show the evolution results of @xmath155 with the boundary condition @xmath206 and @xmath207 at @xmath170 , for different spinning status of the black hole and different values of @xmath208 that specify the kinetic boundary conditions of the fluid . all quantities are scaled to the mass of the black hole so we do not need to specify the value of @xmath42 . since @xmath154 does not depend on @xmath200 , @xmath154 is always zero . since @xmath209 and @xmath210 , @xmath155 evolves according to @xmath211 . though in the plunging region @xmath118 decreases , @xmath212 grows faster except at the neighbor of @xmath213 . so the evolution of @xmath155 is dominated by the variation of @xmath214 . thus , as fluid particles get into the plunging region , @xmath155 decreases quickly as clearly shown in fig . this radial expansion effect is not sensitive to the spin of the black hole , but very sensitive to the value of @xmath208 ( or , effectively , the initial value of @xmath214 ) . as @xmath208 decreases ( i.e. , @xmath215 decreases ) , the fluid expands more as it gets into the plunging region , so the value of @xmath155 decreases more . in fig . [ fig4 ] we show the evolution results of @xmath154 ( dashed lines ) and @xmath155 ( solid lines ) with the boundary condition @xmath216 and @xmath217 at @xmath170 . though at @xmath170 we have @xmath217 , in the plunging region @xmath155 becomes nonzero since @xmath155 depends on both @xmath199 and @xmath200 [ eqs . ( [ bf1 ] ) , ( [ c1 ] ) and ( [ c3 ] ) ] . this is the manifestation that the shear motion of the fluid in the plunging region generates @xmath155 from @xmath154 . the radial component of the magnetic field , @xmath154 , increases gradually as the fluid enters the plunging region , according to @xmath218 , and blows up on the black hole horizon where @xmath47 . the shear motion of the fluid does not amplify @xmath154 , which echos with the fact that @xmath154 is always zero if it is zero at @xmath170 ( fig . [ fig3 ] ) . since @xmath154 does not depend on @xmath214 , there is only one dashed line in each panel of fig . [ fig4 ] . in comparison , the toroidal component , @xmath155 , increases more quickly in the transition region , since the shear motion of the fluid magnifies @xmath155 . this is more prominent for small values of @xmath208 ( i.e. small @xmath219 ) , since @xmath220 [ eq . ( [ bf1 ] ) ] and @xmath190 is close to @xmath221 as the particles just leave the marginally stable circular orbit . for small values of @xmath208 , @xmath155 increases sharply as the fluid just gets into the plunging region , then decreases a little bit due to the radial expansion of the fluid . unlike @xmath154 , @xmath155 is always finite on the black hole horizon . we see that , @xmath154 and @xmath155 evolves in very different ways . from fig . [ fig4 ] we also see that the evolution of the magnetic field in the plunging region depends on the spin of the black hole , though not very sensitively . interestingly , the toroidal component and the poloidal component depend on the spin of the black hole in an opposite way . as the dimensionless spin parameter @xmath222 increases from zero to positive values , the shear amplification effect on the toroidal component of the magnetic field increases ( except for the case of @xmath223 for which the amplification effect is not prominent ) , while the amplification of the radial component caused by the compression in the azimuthal direction ( i.e. , the decrease in radius @xmath118 ) decreases . but , if @xmath222 decreases from zero to negative values , the shear amplification effect on the toroidal component of the magnetic field decreases , while the compression amplification of the radial component increases . the opposite dependence for positive and negative spins is probably due to the different coordinate distances from the marginally stable circular orbit to the black hole horizon for black holes of positive and negative spins . in fig . [ fig5 ] we show the evolution of @xmath224 [ defined by eq . ( [ bsq ] ) ] with the same boundary conditions in fig . as the fluid just enters the plunging region ( near the right ends of curves ) , @xmath224 sharply increases due to the small values of @xmath225 there , which is the manifestation of amplification effect caused by the shear rotation of the fluid . after that , i.e. , after the fluid obtains a large radial velocity ( the dashed lines in the figure ) , the amplification effect is reduced but the expansion effect becomes prominent [ see eq . ( [ evol ] ) ] . on the horizon of the black hole @xmath224 is always finite , so the boundary conditions on the horizon is satisfied ( * ? ? ? * and references therein ) . to specify the boundary conditions for non - stationary solutions is a little bit complicated . as discussed earlier , to determine the solutions in a region with @xmath226 and @xmath227 ( i.e. @xmath228 ) in the @xmath129-space , we need to specify the boundary conditions on the axes ( @xmath143 ) and ( @xmath229 ) , i.e. specify @xmath130 and @xmath138 [ see eq . ( [ bond ] ) ] . as an example , let us assume that @xmath230 , and @xmath217 on the axis ( @xmath143 ) . i.e. , the solution on the boundary @xmath133 is stationary for @xmath231 , and the @xmath232-component of the magnetic field in the plunging region is zero at @xmath125 . equivalently , we specify the boundary conditions as follows : @xmath233 then , if we apply eqs . ( [ br ] ) and ( [ bf ] ) to the boundary at @xmath170 and @xmath231 , we can solve for @xmath97 and @xmath234 . the results are @xmath235 \ ; , \label{c0s } \\ \psi_2 & = & - \gamma_0 b_{\hat{r}0 } a_{\rm ms}^{1/2 } ( \omega_0 - \omega_{\rm ms } v_{\hat{r}0}^2 ) \;. \label{psi2}\end{aligned}\ ] ] applying eq . ( [ bf ] ) to the boundary at @xmath125 and @xmath236 , and substituting eq . ( [ c0s ] ) for @xmath97 , we can solve for @xmath237 @xmath238 \ ; , \hspace{1 cm } \psi_1(r ) = \frac{(\omega - \omega v_{\hat{r}}^2 ) \gamma e c_0 } { ( \omega - \omega v_{\hat{r}}^2 ) \gamma l + \chi } \ ; , \label{psi1}\end{aligned}\ ] ] where @xmath239 is given by the inverse of @xmath240 . the values of @xmath154 on @xmath125 and @xmath241 is then determined by eq . ( [ br ] ) . with the @xmath237 and @xmath234 determined above , we can obtain @xmath114 by eq . ( [ phsol ] ) . this , together with the @xmath97 given by eq . ( [ c0s ] ) , allows us to calculate @xmath154 and @xmath155 with eqs . ( [ br ] ) and ( [ bf ] ) for any radius in the plunging region at any time @xmath128 . it is easy to check that @xmath242 , so the solutions are continuously matched on the line @xmath243 . since @xmath244 the solutions are smoothly matched on the line @xmath243 only in the limit @xmath245 since then we have @xmath246 at @xmath247 . if @xmath221 is not zero but small ( @xmath248 ) , the solutions are approximately smoothly matched on the line @xmath243 if @xmath249 is not large at @xmath170 . in figs [ fig6 ] [ fig8 ] we show the results for the non - stationary evolution of magnetic fields . each figure corresponds to a different spinning state of the black hole . in fig . [ fig6 ] @xmath250 , in fig . [ fig7 ] @xmath251 , while in fig . [ fig8 ] @xmath252 . the boundary conditions for the magnetic field are given by eqs . ( [ bond1 ] ) and ( [ bond2 ] ) . the kinetic boundary conditions of the fluid are given by eq . ( [ le0 ] ) , here we assume @xmath253 . in each figure , each panel corresponds to the state of the magnetic field at a particular moment . especially , the first ( left and up ) panel shows the initial ( @xmath125 ) state of the magnetic field [ i.e. eqs . ( [ bond1 ] ) and ( [ bond2 ] ) ] . in each panel , the thick dashed line shows @xmath154 , the thick solid line shows @xmath155 , and the thin lines show the corresponding stationary solutions . each curve starts from the marginally stable circular orbit ( right end ) and ends on the horizon of the black hole ( left end ) . from these figures we see that , though initially the magnetic field is deviated from the stationary state , it evolves to and finally saturates at the stationary state as time goes on . since we hold the magnetic field on the marginally stable circular orbit in a stationary state for @xmath231 [ eq . ( [ bond1 ] ) ] and the fluid moves from the marginally stable circular orbit toward the horizon of the black hole , the stationary state propagates from the marginally stable circular orbit toward the horizon of the black hole ( i.e. , from right to left in the figures ) . this is clear seen in figs . [ fig6 ] [ fig8 ] : the magnetic field at a radius closer to the right end of the curve gets into the stationary state earlier . in each panel , with a black dot we show the position of a particle that is at the marginally stable radius at @xmath125 , which clearly shows the propagation of the state of the magnetic field with the motion of the fluid . this suggests that the stationary state of the magnetic field in the plunging region is uniquely determined by the boundary conditions at the marginally stable circular orbit . the figures also show the dependence of the evolution of the magnetic field on the spin of the black hole . for a black hole with a negative @xmath254 a longer coordinate time is needed for most of the fluid in the plunging region to settle into the stationary state . this is apparently caused by the fact that a black hole with a negative @xmath254 has a larger coordinate radial distance from the marginally stable circular orbit to the horizon . in previous sections we have studied the evolution of magnetic fields in the plunging region and seen that magnetic fields can be amplified by the convergence ( the radial magnetic field ) and shear motion ( the toroidal magnetic field ) of the fluid . thus , we can ask if magnetic fields can become dynamically important in the plunging region assuming that their dynamical effects are negligible at the marginally stable circular orbit . we try to answer this question in this section . assuming that in the plunging region the gas pressure of the fluid is negligible , then the total stress - energy tensor of the fluid is @xmath255 where @xmath256 is the mass - energy density of the fluid matter measured by an observer comoving with the fluid , @xmath257 is the stress - energy tensor of the electromagnetic field as given by eq . ( [ tab ] ) . then , the equation of motion of the fluid is given by @xmath258 when maxwell s equations are satisfied we have @xmath259 @xcite . then , by eq . ( [ ezero ] ) , we have @xmath260 for a magnetic field frozen to the fluid . therefore , the contraction of @xmath21 with eq . ( [ dtab ] ) leads to the equation of continuity ) also holds if the motion of the fluid is locally adiabatic @xcite . ] @xmath261 where @xmath262 is the flux density vector of mass . the equation of continuity does not explicitly contain magnetic field variables . since @xmath263 and @xmath264 are killing vectors , eq . ( [ dtab ] ) leads to the conservation of angular momentum and energy @xmath265 where @xmath266 is the flux density vector of angular momentum , @xmath267 is the flux density vector of energy . here we look for stationary and axisymmetric solutions in the plunging region . then , with the assumptions adopted in this paper , i.e. @xmath268 , @xmath85 and @xmath269 , in the equatorial plane the equation of continuity is reduced to @xmath270 similarly , the equations of angular momentum and energy conservation are reduced to @xmath271\right\ } = 0 \ ; , \label{lcon1}\end{aligned}\ ] ] and @xmath272\right\ } = 0 \;. \label{econ1}\end{aligned}\ ] ] the solution of eq . ( [ cont2 ] ) is @xmath273 where @xmath274 is a constant to be determined by the boundary conditions , which measures the mass flux across radius @xmath118 . [ fig9 ] shows the variation of the mass density with radius in the plunging region , assuming that fluid particles move on geodesics . the kinetic boundary conditions are given by eq . ( [ le0 ] ) . in each panel ( corresponding to a different spinning state of the black hole ) , we show the ratio of @xmath275 corresponding to four different values of @xmath208 : @xmath276 , @xmath277 , @xmath278 and @xmath279 , where @xmath280 . we see that , the evolution of the mass density of the fluid sensitively depends on the value of @xmath208 ( or effectively , the value of @xmath281 ) . while for not very small @xmath208 ( e.g. , @xmath223 ) the variation of @xmath256 is not dramatic , for very small @xmath208 ( e.g. , @xmath282 ) the variation of @xmath256 is dramatic : as the fluid gets into the plunging region the mass density drops sharply . this is caused by the sharp increase in the ratio @xmath283 in the plunging region for small @xmath208 . from eqs . ( [ lcon1 ] ) and ( [ econ1 ] ) we see that the dynamical role played by the magnetic field is characterized by the following three dimensionless parameters @xmath284 the parameter @xmath285 measures the transfer of angular momentum and energy from the fluid particles to the magnetic field ; @xmath286 and @xmath287 measure the transportation of angular momentum and energy from one part to another by the magnetic tension in the rest frame of the fluid . to justify the assumption that the effects of the magnetic field on the dynamics of the fluid particles are negligible , we must require that @xmath288 in eq . ( [ eta ] ) , @xmath256 is given by eq . ( [ rhm ] ) , @xmath224 is calculated by eq . ( [ bsq ] ) ( setting @xmath289 ) , @xmath290 and @xmath291 are calculated by @xmath292 near the marginally stable circular orbit , we have @xmath293 , @xmath294 , and @xmath295 . therefore , from the definitions of @xmath286 and @xmath287 , we have @xmath296 at @xmath297 . from fig . [ fig9 ] we see that for small @xmath208 the mass density @xmath256 drops quickly as the fluid enters the plunging region . however , the evolution of @xmath224 in the plunging region sensitively depends on the orientation of the magnetic field on the marginally stable circular orbit . from fig . [ fig5 ] , the value of @xmath224 corresponding to an initially radial magnetic field increases as the fluid enters the plunging region , because of the convergence of the radial magnetic field and the shear amplification of the toroidal magnetic field . if the initial magnetic field is purely toroidal , on the other hand , from fig . [ fig3 ] we see that the magnetic field keeps purely toroidal in the plunging region and the dilution effect arising from the drop in the mass density makes @xmath155 ( and thus @xmath224 ) decrease in the plunging region . therefore , in the plunging region the variation of @xmath285 , @xmath286 , and @xmath287 sensitively depends on the boundary condition of the magnetic field on the marginally stable circular orbit . we can imagine that , if on the marginally stable circular orbit the magnetic field is purely radial , then in the plunging region @xmath224 increases so the ratio @xmath285 also increases as @xmath256 decreases . in such a case , @xmath285 may become close to or even greater than @xmath298 in the plunging region even if it is @xmath299 on the marginally stable circular orbit , then the dynamical effects of the magnetic field become important in the plunging region . on the other hand , if on the marginally stable circular orbit the magnetic field is purely toroidal , then in the plunging region @xmath224 decreases which causes @xmath285 to decrease also if @xmath224 decreases faster than @xmath256 . therefore , to correctly estimate the dynamical effects of magnetic fields in the plunging region , we must choose a sensible boundary condition for the magnetic field on the marginally stable circular orbit . the magnetic field on the marginally stable circular orbit is determined by the mhd processes in the disk region . in the disk region ( @xmath300 ) , particles move on nearly circular orbits with @xmath301 ) of turns around the central black hole . as a result , the magnetic field lines frozen to the fluid are wound up around the center an infinite number of times . therefore , in a stationary state , we expect that the magnetic field in the disk region is predominantly toroidal . in the appendix we solve maxwell s equations in the disk region and show that in the stationary state the magnetic field in the disk is likely to be parallel to the velocity field : @xmath302 . thus , in the stationary state , on the marginally stable circular orbit the magnetic field does not take any orientation but the one that satisfies the following condition @xmath303 on the marginally stable circular orbit , eq . ( [ pc ] ) implies that @xmath304 since @xmath305 . from eqs . ( [ fsol1 ] ) and ( [ fsol2 ] ) , we have @xmath306 then , eq . ( [ pc ] ) is satisfied if and only if @xmath307 on the marginally stable circular orbit . in the stationary state @xmath114 is a constant , so we have @xmath308 throughout the plunging region . thus , in the stationary state , eq . ( [ pc ] ) holds throughout the plunging region . ) [ or , equivalently , eq . ( [ pc ] ) ] is satisfied . however , we point out that this does not always mean that the corresponding electric field measured by an observer comoving with the frame dragging is zero , unless the black hole is a schwarzschild black hole . one can check that , for the solutions in eqs ( [ fsol1 ] ) and ( [ fsol2 ] ) , the corresponding electric field measured by an observer comoving with the frame dragging [ i.e. , with a 4-velocity @xmath309 given in eq . ( [ ont2 ] ) ] is @xmath310 . so , when @xmath307 , we have @xmath311 , which is zero only if @xmath312 ( then the frame - dragging frequency @xmath313 ) . ] inserting eqs . ( [ ualp ] ) and ( [ balp ] ) into eq . ( [ pc ] ) , we obtain @xmath314 we have calculated @xmath285 , @xmath286 , and @xmath287 in the plunging region , assuming that they all are @xmath299 at @xmath170 and the boundary condition ( [ psi0 ] ) is satisfied . at @xmath170 the toroidal magnetic field is assumed to be @xmath315 in units of @xmath316 , the corresponding radial magnetic field @xmath154 is then given by eq . ( [ pc1 ] ) . the parameter @xmath208 is taken to be @xmath276 , @xmath277 , @xmath278 and @xmath279 , alternatively . the results are shown in figs . [ fig10 ] and [ fig11 ] . from these figures we see that , in the plunging region the evolution of @xmath285 , @xmath286 and @xmath287 sensitively depends on the value of @xmath208 . for very small @xmath208 , @xmath285 , @xmath286 and @xmath287 quickly decrease as the fluid gets into the plunging region . this is caused by the fact that a very small @xmath208 ( i.e. , a very small @xmath317 ) corresponds to a very small @xmath318 according to the boundary condition ( [ pc ] ) , while the magnetic field in the plunging region is predominantly determined by @xmath318 instead of @xmath319 ( see figs . [ fig3 ] and [ fig4 ] ) . for a moderate @xmath208 ( e.g. , @xmath223 ) , @xmath285 , @xmath286 and @xmath287 may increase in the plunging region . however , even in this case , the conditions in eq . ( [ nc1 ] ) are always satisfied throughout the plunging region . if we insert the solutions of maxwell s equations that we obtained in sec . [ sec4 ] into eqs . ( [ lcon1 ] ) and ( [ econ1 ] ) , then apply eq . ( [ rhm ] ) , we obtain @xmath320 where @xmath321 when @xmath307 [ i.e. , eq . ( [ pc ] ) is satisfied in the plunging region ] , we have @xmath322 . then eq . ( [ de1 ] ) implies that @xmath68 keeps constant in the plunging region . however , since @xmath323 , by eq . ( [ dl1 ] ) @xmath67 varies in the plunging region . therefore , for the solutions satisfying the boundary condition ( [ pc ] ) [ or , equivalently , eq . ( [ psi0 ] ) ] , the specific energy of particles is conserved but the specific angular momentum changes . setting @xmath324 , @xmath325 , and @xmath307 , eqs . ( [ dl1 ] ) and ( [ de1 ] ) can be integrated to obtain @xmath326 where @xmath327 where @xmath328 and @xmath329 are constants measuring the angular momentum flux and the energy flux across radius @xmath118 , respectively . using @xmath330 , we can solve eqs . ( [ fl ] ) and ( [ fe ] ) for @xmath331 @xmath332 then , from @xmath333 and @xmath334 , we obtain the equation for @xmath335 @xmath336 ^ 2 \;. \label{wind}\end{aligned}\ ] ] though the factor @xmath337 appears in the denominators on the right - hand side of eq . ( [ wind ] ) , @xmath338 is not a singularity of the equation , since the factor @xmath339 disappears from the denominators if we expand the second term on the right - hand side of eq . ( [ wind ] ) , then combine with the first term . however , the factor @xmath340 $ ] represents a singularity of eq . ( [ wind ] ) at @xmath341 the differentiation of eq . ( [ wind ] ) with respect to @xmath335 gives rise to another singularity of the equation , which is at @xmath342 this singularity appears as one differentiates eq . ( [ wind ] ) with respect to @xmath118 to obtain a differential equation for @xmath335 . if we define the relativistic alfvn velocity by @xmath343 then we have @xmath344 and @xmath345 where @xmath346 . therefore , the singularities given by eqs . ( [ sing1 ] ) and ( [ sing2 ] ) are critical points related to the alfvn speed : the alfvn point [ eq . ( [ sing1 ] ) ] and the fast critical point [ eq . ( [ sing2 ] ) ] @xcite . since we have assumed that the gas pressure is zero , the slow critical point is at @xmath347 which is not relevant to us here . the alfvn point is not an x - type singularity and it does not impose any additional conditions on the solution for @xmath335 except setting the integral constant @xcite . therefore , what is really relevant here is the fast critical point given by eq . ( [ sing2 ] ) . for weak fields , we expect that the fast critical point is located at a radius close to the marginally stable radius , where the accretion flow transits from subsonic motion in the disk region to supersonic motion in the plunging region @xcite . any physical solutions must smoothly pass the critical points , which sets strict constraints on the integral constants . it is far beyond the scope of the current paper to fully explore the properties of critical points in detail . however , as a first order approximation , we can assume that @xmath348 , where @xmath349 is the radius at the fast critical point . then , from eq . ( [ sing2 ] ) we have the radial velocity of the fluid at @xmath184 @xmath350^{1/3 } \;. \label{ur0}\end{aligned}\ ] ] if we set @xmath170 in eq . ( [ wind ] ) , then substitute eq . ( [ ur0 ] ) into eq . ( [ wind ] ) , we can solve for the constant @xmath351 , as a function of @xmath352 and @xmath353 . therefore , for solutions that smoothly pass the critical points , among the three integral constants @xmath352 , @xmath353 , and @xmath351 , only two are independent . without explicitly solving the radial flow equation , i.e. eq . ( [ wind ] ) , we can also obtain some interesting results on the horizon of the black hole . since @xmath354 as @xmath53 , from eq . ( [ wind ] ) we have @xmath355 where @xmath356 . then , from eq . ( [ uf ] ) , we can obtain the specific angular momentum for particles at @xmath357 @xmath358 eq . ( [ fe ] ) says that , with the boundary condition given by eq . ( [ psi0 ] ) , the specific energy of fluid particles does not change in the plunging region . however , the specific angular momentum of fluid particles does change . the specific angular momentum for particles on the marginally stable circular orbit , @xmath359 , can be calculated from eq . ( [ uf ] ) by setting @xmath184 . then , by comparing @xmath360 to @xmath359 , we can see how much has changed in the specific angular momentum as particles move from the marginally stable circular orbit to the black hole horizon . we have calculated the ratio @xmath361 and presented the results in fig . [ fig12 ] . to make the solutions smoothly joined to those in the disk region , in our calculations we have adopted that the specific energy is equal to that of a particle moving on the marginally stable circular orbit . then , we have @xmath362 . from eqs . ( [ car ] ) and ( [ ur0 ] ) , on the marginally stable circular orbit there is a one - to - one correspondence between @xmath363 and @xmath364 therefore , we use @xmath285 at @xmath365 to specify the boundary value of the magnetic field , then @xmath352 is determined ( up to a sign which is not relevant here ) . then , with the above approach , @xmath351 is determined as a function of @xmath285 and @xmath353 . we have chosen @xmath285 to be @xmath276 , @xmath277 , and @xmath278 , alternatively . from fig . [ fig12 ] we see that , though @xmath67 is not constant in the plunging region , its variation is extremely small , if on the marginally stable circular orbit the dynamical effects of the magnetic field on the motion of the fluid particles is negligible ( then @xmath366 at @xmath170 ) . as @xmath222 increases from @xmath367 to @xmath298 , the variation in the specific angular momentum decreases , caused by the fact that the coordinate distance between the marginally stable circular orbit and the event horizon decreases with increasing @xmath222 . for a black hole with @xmath368 , we have @xmath369 if @xmath370 on @xmath170 . for the extreme case of @xmath371 , we have @xmath372 , i.e. the specific angular momentum does not change at all . for the extreme case of @xmath373 , which has the largest coordinate distance from the marginally stable circular orbit to the event horizon , the variation in @xmath67 is largest . however , even in this case , the variation in the specific angular momentum is also not big : @xmath374 if @xmath375 on @xmath170 . for the same models we have also calculated the ratios @xmath285 and @xmath376 at @xmath171 , and presented the results in fig . [ fig13 ] . because of eq . ( [ eta1 ] ) , the value of @xmath285 is given by the left - hand side of eq . ( [ car ] ) . the ratio @xmath376 can be calculated by @xmath377 from fig . [ fig13 ] we see that both @xmath378 ( solid curves ) and @xmath376 ( dashed curves ) are small at @xmath171 . for @xmath379 , both are @xmath380 if @xmath370 at @xmath170 . the ratios go down as @xmath222 increases . even for the extreme case of @xmath381 , the ratios are also not big at @xmath171 if @xmath370 at @xmath184 : @xmath382 and @xmath383 . eq . ( [ urh ] ) implies that @xmath384 is always finite . then , @xmath385 at @xmath386 since @xmath387 . then , eq . ( [ crr ] ) implies that @xmath388 also at @xmath171 since its right - hand side is finite . however , from eq . ( [ crur ] ) we have @xmath389 since @xmath390 and @xmath391 . since @xmath384 is finite , eqs . ( [ car ] ) and ( [ eta1 ] ) imply that @xmath285 is also finite at @xmath171 . then , since @xmath392 at @xmath171 , we must have @xmath393 eqs . ( [ crurh ] ) and ( [ crurh2 ] ) imply that fluid particles always supersonically fall into the black hole . the numerical results in fig . [ fig13 ] confirm this conclusion . from figs . [ fig10 ] [ fig13 ] , we can demonstrate that when the boundary condition ( [ pc ] ) [ or , equivalently , ( [ psi0 ] ) ] is satisfied , the dynamical effects of magnetic fields in the plunging region are unimportant if they are so on the marginally stable circular orbit . with a simple model we have studied the evolution of magnetic fields in the plunging region around a kerr black hole . the model contains the following assumptions : ( 1 ) the background spacetime is described by the kerr metric ; ( 2 ) the plasma is perfectly conducting so that the magnetic field is frozen to the plasma fluid ; ( 3 ) the kinematic approximation @xcite applies , i.e. the dynamical effects of the magnetic field on the fluid motion are negligible so that the plasma fluid flows along timelike geodesics toward the central black hole ; ( 4 ) in a small neighborhood of the equatorial plane ( i.e. , @xmath84 ) the magnetic field and the velocity field have only radial and azimuthal components . the assumption ( 4 ) implies that @xmath85 and @xmath394 on the equatorial plane . with above assumptions , we have exactly solved maxwell s equations for axisymmetric solutions ( i.e. @xmath83 ) . the solutions are given by eqs . ( [ fsol1 ] ) and ( [ fsol2 ] ) [ or , equivalently , eqs . ( [ br ] ) and ( [ bf ] ) ] , where @xmath97 is a constant measuring the magnetic flux through a circle in the equatorial plane , @xmath114 is a function determining the time - evolution of the magnetic field . both @xmath97 and @xmath114 are determined by the boundary conditions . the dependence of the solutions on the coordinate time @xmath122 is given by the function @xmath395 , where @xmath396 is defined by eq . ( [ tau ] ) . the function @xmath114 is uniquely determined by the initial and boundary conditions of the magnetic field . the general form of @xmath114 determines the retarded nature of the solutions : at any time the state of the magnetic field at a radius in the plunging region is determined by the state of the magnetic field at a larger radius and an earlier time ( figs . [ fig1 ] and [ fig2 ] ) . this suggests that the stationary state of the magnetic field in the plunging region is uniquely determined by the boundary conditions at the marginally stable circular orbit . examples for the evolution of magnetic fields in the plunging region are shown in figs . [ fig3 ] [ fig8 ] , for both the stationary case ( @xmath198 , figs . [ fig3 ] [ fig5 ] ) and the non - stationary case ( @xmath397 , figs . [ fig6 ] [ fig8 ] ) . from these figures we see that , the boundary value of the radial component of the magnetic field at the marginally stable circular orbit is more important in determining the strength of the magnetic field in the plunging region than the toroidal component . the initially toroidal component is attenuated in the plunging region by the radial expansion of the fluid ( fig . [ fig3 ] ) . the initially radial component is amplified by the azimuthal and vertical compression ( the convergence of the fluid ) , and a toroidal component is generated from the radial component by the shear motion of the fluid though the toroidal component is initially zero at the marginally stable circular orbit ( fig . [ fig4 ] ) . this leads to the amplification of the magnetic field in the plunging region ( fig . [ fig5 ] ) . the evolution of the magnetic field depends on the spinning state of the black hole , but is more sensitive to the initial value of the radial velocity on the marginally stable circular orbit [ or , equivalently the parameter @xmath208 defined by eq . ( [ le0 ] ) ] . the time - evolution of magnetic fields shown in figs . [ fig6 ] [ fig8 ] confirms our claim that the stationary state of the magnetic field in the plunging region is uniquely determined by the boundary conditions on the marginally stable circular orbit . if in the plunging region a magnetic field is initially deviated from the stationary solutions , it will evolve to and finally get saturated at the state given by the stationary solutions in a dynamical time scale determined by the free - fall motion of fluid particles in the plunging region . if the magnetic field on the marginally stable circular orbit is in a stationary state , the magnetic field in the plunging region will automatically settle into a stationary state . thus , the evolution of magnetic fields in the plunging region is very different from that in the disk region , in the latter case the state of the magnetic field is determined by the local mhd processes . the difference is caused by the following fact : in the disk region the fluid has a very small radial velocity so that local mhd processes have shorter time scales than the radial motion ; in the plunging region the fluid has a large radial velocity so that local mhd processes usually have longer time scales than the radial motion . in deriving the solutions we have assumed that the dynamical effects of the magnetic field in the plunging region are unimportant [ the kinematic approximation adopted in assumption ( 3 ) ] . to justify this assumption , we have studied the dynamical effects of magnetic fields in the stationary state in two ways . first , we estimate the dynamical effects of magnetic fields by considering the back - reaction : substituting the solutions of maxwell s equations we obtained , where we assumed that fluid particles move on geodesics , into the dynamical equations to check if the motion of the fluid is significantly affected by magnetic fields . in this way , we estimate the dynamical effects of magnetic fields on the motion of the fluid by calculating the parameters @xmath285 , @xmath286 and @xmath287 defined by eq . ( [ eta ] ) . the mass density of the fluid , which appears in the denominators of @xmath285 , @xmath286 and @xmath287 , drops quickly in the plunging region if @xmath305 on the marginally stable circular orbit ( fig . [ fig9 ] ) . however , the evolution of the magnetic field , which appears in the numerators of @xmath285 , @xmath286 and @xmath287 , sensitively depends on the orientation of the magnetic field on the marginally stable circular orbit , the latter is essentially determined by the mhd processes in the disk region . if we require that the solutions in the plunging region are smoothly joined to the solutions in the thin keplerian disk region , then it is reasonable to assume that on the marginally stable circular orbit , as well as in the disk region , the magnetic field is parallel to the velocity field : @xmath398 ( see the appendix ) . this implies that for the solutions in the plunging region we should have @xmath307 [ eq . ( [ psi0 ] ) ] , i.e. the orientation of the magnetic field follows the orientation of the velocity field of the fluid [ eq . ( [ pc ] ) ] . with such a boundary condition , we have calculated @xmath285 , @xmath286 and @xmath287 , assuming that particles move on geodesics . we see that , if they are @xmath299 on the marginally stable circular orbit , then they remain so in the plunging region ( figs . [ fig10 ] and [ fig11 ] ) . indeed , @xmath285 , @xmath286 and @xmath287 decrease in the plunging region for sufficiently small @xmath225 on the marginally stable circular orbit . second , we self - consistently solve the coupled maxwell and dynamical equations on the horizon of the black hole , check how much has changed in the specific angular momentum of fluid particles since they leave the marginally stable circular orbit . the specific energy of fluid particles is not changed by a magnetic field that satisfies @xmath307 [ see eq . ( [ fe ] ) ] . however , the specific angular momentum does change [ see eq . ( [ fl ] ) ] . we have calculated the specific angular momentum of fluid particles as they reach the horizon of the black hole , assuming that the fluid particles pass the fast critical point near the marginally stable circular orbit . the deviation from the specific angular momentum as the particles just leave the marginally stable circular orbit is shown in fig . [ fig12 ] . we see that , though the specific angular momentum of fluid particles is changed by the magnetic field , the effects are always small assuming that on the marginally stable circular orbit the dynamical effects of the magnetic field are not important . for example , for @xmath399 we have @xmath369 if @xmath400 on the marginally stable circular orbit . we have also calculated the ratios @xmath401 and @xmath402 on the horizon ( fig . [ fig13 ] ) , where @xmath403 is the radial component of the alfvn velocity . these two ratios are relevant to the critical points in the flow [ see eqs . ( [ crur ] ) , ( [ car ] ) , and ( [ eta1 ] ) ] and measure at what level the magnetic field affects the motion of the fluid . [ fig13 ] shows that their values on the horizon are quite small . both approaches ( back - reaction and self - consistent solutions ) confirm that the dynamical effects of the magnetic field are unimportant in the plunging region if they are so on the marginally stable circular orbit . our results differ from that of gammie @xcite , in which he claimed that in the plunging region the dynamical effects of magnetic fields can be important . the difference is caused by the fact that gammie used a boundary condition that is different from ours for the magnetic field . gammie assumed that @xmath404 , where @xmath405 is the angular velocity of the disk at the marginally stable circular orbit , while we assume that @xmath307 . gammie s boundary condition implies that @xmath406 at @xmath170 ( as clearly stated in his paper ) , which makes @xmath170 a singular point where @xmath407 ( to keep the mass flux @xmath408 nonzero ) . hence , gammie s solutions are not well - behaved at the marginally stable circular orbit . our boundary condition allows the solutions in the plunging region to be smoothly joined to the solutions in the disk region ( see the appendix ) , since in our solutions all physical quantities are finite at the marginally stable orbit . certainly , to precisely take care of the transition from the disk region to the plunging region , gas pressure and non - electromagnetic stress must be properly taken into account near the inner edge of the disk . as mentioned in the introduction , recently the `` no - torque inner boundary condition '' for thin accretion disks has been challenged by some authors ( including gammie ) based on their studies on the evolution of magnetic fields in the plunging region @xcite . the results in this paper suggest that in the plunging region the dynamical effects of the magnetic field are not important , if the solutions in the plunging region are smoothly joined to the the solutions in the thin keplerian disk region . thus , the argument against the `` no - torque inner boundary condition '' is not founded . finally , we note that in the paper we have neglected dissipative processes like magnetic reconnection and ohmic dissipation , and the evaporation effect arising from the magnetic buoyancy and mhd instabilities . these processes operate in disks to limit the amplification of magnetic fields ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . certainly it is conceivable that they can also operate in the plunging region to reduce the amplification effect of magnetic fields . if these processes are important , the results of this paper tend to overestimate the amplification of magnetic fields in the plunging region . then , the dynamical effects of magnetic fields should be weaker than that we have estimated without considering dissipative processes , which will strengthen our conclusion that the dynamical effects of magnetic fields are unimportant in the plunging region . in the disk region ( @xmath409 ) , viscosity plays an important role in the dynamics of disk particles . the viscosity transports angular momentum outward and dissipates energy , which leads to disk accretion . so , in the disk region , particles move on non - geodesic worldlines . in the vertical direction , the gravity of the black hole is balanced by the gradient of the total pressure (= gas pressure + radiation pressure + magnetic field pressure ) in the disk . therefore , in a small neighborhood of the disk central plane , disk particles are more likely moving on planes parallel to the equatorial plane , rather than moving radially as in the plunging region . to describe such a motion , cylindrical coordinates @xmath410 are more suitable , where @xmath122 and @xmath411 have the same meaning as those in the boyer - lindquist coordinates , to the first order @xmath118 also has the same meaning as that in the boyer - lindquist coordinates , but @xmath412 where @xmath413 . in the cylindrical coordinates , in a small neighborhood of the disk central plane the kerr metric can be written as @xcite @xmath414 where @xmath415 and @xmath160 are defined by eq . ( [ del ] ) with @xmath416 . the maxwell equations that we need to solve are again given by eq . ( [ maxeq ] ) , but now we have @xmath417 and @xmath418 . as in the case for the plunging region , we assume that the motion of fluid particles is stationary but the evolution of the magnetic field can depend on time , i.e. , we let @xmath419 but keep @xmath420 in the maxwell equations . furthermore , we assume that in a small neighborhood of the disk central plane @xmath421 so that @xmath422 on the equatorial plane . finally , we adopt @xmath423 because of the axisymmetry of the system . then , on the equatorial plane eq . ( [ maxeq ] ) is reduced to @xmath424 + \frac{\partial}{\partial r } \left[r ( u^r b^\beta - u^\beta b^r)\right ] = 0 \;. \label{eq1a}\end{aligned}\ ] ] eq . ( [ fsol1d ] ) can be written as @xmath428 assume that as @xmath429 the disk is keplerian so that @xmath430 and @xmath431 , and @xmath432 keeps finite , then we must have @xmath433 eq . ( [ limp ] ) has an important implication for stationary solutions . for stationary solutions we have @xmath434 , then by eq . ( [ limp ] ) we must have @xmath435 . so , for stationary solutions we have the retarded nature of the solution given by eq . ( [ psol ] ) . a time variation in the magnetic field at radius @xmath124 will lead to the same variation in the magnetic field at radius @xmath439 at a later time , assuming the fluid moves toward smaller radii ( i.e. toward the central black hole ) . the time delay is given by the time needed by a fluid particle moving from @xmath124 to @xmath440.,width=453 ] a sketch of the solution given by eq . ( [ phsol ] ) , corresponding to the boundary conditions given by eq . ( [ bond ] ) . the parameter @xmath131 is defined by eq . ( [ tau ] ) , which is the coordinate time needed by a particle moving from radius @xmath124 to radius @xmath118 . so , @xmath133 corresponds to @xmath117 , @xmath441 corresponds to the black hole horizon @xmath171 . the coordinate time is represented by @xmath122 . in this diagram , the worldlines of fluid particles are represented by straight lines @xmath442 . if the boundary conditions are imposed on the axes as shown in the diagram [ eq . ( [ bond ] ) ] , the solution in the region between the @xmath131-axis and the dashed line @xmath443 ( i.e. , region i ) is determined by the boundary condition on the @xmath131-axis , the solution in the region between the @xmath122-axis and the dashed line ( i.e. , region ii ) is determined by the boundary condition on the @xmath122-axis.,width=453 ] stationary evolution of an initially toroidal magnetic field in the plunging region around a kerr black hole . each panel corresponds to a different spinning state of the black hole , as indicated by the dimensionless parameter @xmath222 . each curve starts from the marginally stable circular orbit ( @xmath170 , right end ) , ends at the horizon of the black hole ( @xmath171 , left end ) . at @xmath170 , the magnetic field is purely toroidal and has a value @xmath200 , the fluid particles have specific energy identical to that of the marginally stable circular orbit , and specific angular momentum smaller than that of the marginally stable circular orbit by a tiny fraction @xmath208 [ eq . ( [ le0 ] ) ] . in each panel , each curve corresponds to different values of @xmath208 : @xmath276 , @xmath277 , @xmath278 and @xmath279 ( downward ) . the radial component of the magnetic field is always zero . , width=566 ] similar to fig . [ fig3 ] but the magnetic field is initially radial , i.e. @xmath216 and @xmath444 at @xmath170 . solid lines represent the toroidal component of the magnetic field , dashed lines represent the radial component . though initially the toroidal component of the magnetic field is zero , the shear motion of the fluid in the plunging region generates toroidal component from the radial component . the evolution of the radial component does not depend on the initial kinetic state of the fluid , so there is only one dashed line in each panel . the four solid lines representing the toroidal component of the magnetic field correspond to different values of @xmath208 : @xmath276 , @xmath277 , @xmath278 and @xmath279 ( upward).,width=566 ] stationary evolution of @xmath445 ( solid curves ) . the boundary conditions are the same as that in fig . the thin dashed line in each panel shows the absolute value of the radial velocity ( i.e. @xmath225 ) of the fluid , corresponding to @xmath282 . if we choose a different value of @xmath208 , the curve for the radial velocity will change slightly : the right end will change according to @xmath446 , the left end always approaches @xmath298 ( i.e. the speed of light ) . ( for brevity , @xmath447 $ ] and @xmath225 use the same scale as labeled on the left side of the box in each panel.),width=566 ] time evolution of magnetic fields around a schwarzschild black hole ( i.e. @xmath165 ) . each panel corresponds to a particular moment . thick solid lines show the toroidal component of the magnetic field ( i.e. , @xmath155 ) , thick dashed lines show the radial component ( i.e. , @xmath154 ) . the thin ( upper ) lines show the corresponding stationary state solutions . the fluid particles have specific energy identical to that of the marginally stable circular orbit , and specific angular momentum smaller than that of the marginally stable circular orbit by a factor of @xmath253 . the first ( left and up ) panel shows the initial and boundary conditions of the magnetic field : on @xmath170 ( right end ) , @xmath217 , @xmath216 for all time @xmath128 ; in the plunging region ( @xmath241 ) , @xmath217 at @xmath448 [ i.e. , eqs . ( [ bond1 ] ) and ( [ bond2 ] ) ; the corresponding @xmath154 at @xmath125 is automatically determined by the solutions ] . the figures show that , as time goes on , the magnetic field quickly approaches and saturates at the state given by the stationary solutions . the black dot in each panel shows the radial position of a fluid particle at each moment : initially the particle is at @xmath170.,width=566 ] stationary evolution of the mass density in the plunging region around a kerr black hole . each panel corresponds to a different spinning state of the black hole , as indicated by the values of @xmath222 . the right end of each curve corresponds to the marginally stable circular orbit ( @xmath449 ) . the left end of each curve corresponds to the horizon of the black hole ( @xmath171 ) . the kinetic boundary conditions are given by eq . ( [ le0 ] ) . in each panel the four curves correspond respectively to @xmath223 , @xmath277 , @xmath278 and @xmath279 ( downward ) . the mass density is in units of @xmath450 the mass density at the marginally stable circular orbit.,width=566 ] stationary evolution of the ratio @xmath451 in the plunging region . each panel corresponds to a different spinning state of the black hole , as indicated by the values of @xmath222 . the kinetic boundary conditions are given by eq . ( [ le0 ] ) . the boundary conditions for the magnetic field are : on @xmath452 we have @xmath315 in units of @xmath453 . the corresponding @xmath154 is calculated with eq . ( [ pc1 ] ) so that the magnetic field is always parallel to the velocity field . in each panel the four curves correspond respectively to @xmath223 , @xmath277 , @xmath278 and @xmath279 ( downward ) . each curve starts from the marginal stable circular orbit ( right end ) and ends at the horizon of the black hole ( left end).,width=566 ] the change in the specific angular momentum of fluid particles as they approach the event horizon of the black hole [ eq . ( [ dell ] ) ] , as a function of the spin of the black hole . it is assumed that the fast critical point is at @xmath456 . the magnetic field is parallel to the velocity field of the fluid , then the specific energy of fluid particles keeps constant in the plunging region , which we have assumed to be equal to the specific energy of a particle moving on the marginally stable circular orbit . the boundary value of the magnetic field at @xmath184 is specified by the ratio @xmath457 . the three curves corresponds to three different values of @xmath285 at @xmath170 : @xmath276 , @xmath277 , and @xmath278 ( downward ) . , width=566 ]
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the evolution of magnetic fields frozen to a perfectly conducting plasma fluid around a kerr black hole is investigated .
we focus on the plunging region between the black hole horizon and the marginally stable circular orbit in the equatorial plane , where the centrifugal force is unable to stably balance the gravitational force . adopting the kinematic approximation where the dynamical effects of magnetic fields on the fluid motion are ignored ,
we exactly solve maxwell s equations with the assumptions that the geodesic motion of the fluid is stationary and axisymmetric , the magnetic field has only radial and azimuthal components and depends only on time and radial coordinates .
we show that the stationary state of the magnetic field in the plunging region is uniquely determined by the boundary conditions at the marginally stable circular orbit .
if the magnetic field at the marginally stable circular orbit is in a stationary state , the magnetic field in the plunging region will quickly settle into a stationary state if it is not so initially , in a time determined by the dynamical time scale in the plunging region .
the radial component of the magnetic field at the marginally stable circular orbit is more important than the toroidal component in determining the structure and evolution of the magnetic field in the plunging region . even if at the marginally stable circular orbit the toroidal component is zero , in the plunging region a toroidal component
is quickly generated from the radial component by the shear motion of the fluid .
finally , we discuss the dynamical effects of magnetic fields on the motion of the fluid in the plunging region .
we show that the dynamical effects of magnetic fields are unimportant in the plunging region if they are negligible on the marginally stable circular orbit .
this supports the `` no - torque inner boundary condition '' of thin disks , contrary to the claim in the recent literature .
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a main lesson learnt from experimental data collected in ( d)au+au and pb+pb collisions at rhic and the lhc respectively is that bulk particle production in ion - ion collisions is very different from a simple superposition of nucleon - nucleon collisions . such is evident in terms of the measured charged particle multiplicities , which exhibit a strong deviation from the scaling with the number of nucleon - nucleon collisions : @xmath7 , and also from the non - trivial transverse momentum dependence of nuclear modification factors measured in d+au collisions . these observations lead to the conclusion that strong coherence effects among the constituent nucleons , or the relevant degrees of freedom at the sub - nucleon level , must be present during the collisions process . indeed , and regardless of the question whether the cgc is the most suited framework for their description there is broad consensus that coherence effects are essential for the interpretation of present data on heavy ion collisions . in fact most if not all of the different phenomenological approaches for the description of particle production both in the soft or hard sector. on physical grounds , coherence phenomena are related to the presence of high gluon densities in the wave function of the colliding nuclei at small values of bjorken-@xmath0 proven in a hadronic collision can be obtained using @xmath8 kinematics @xmath9 , with @xmath10 , @xmath11 and @xmath12 being the collision energy and the transverse momentum and rapidity of the produced hadron respectively . ] while a detailed discussion of the different prescriptions found in the literature to account for coherence effects is beyond the scope of this brief review , one can identify in different models coherence effects at the level of the wave function and also at the level of primary particle production , sketched in fig . 1 ( left ) . to the first category correspond the nuclear shadowing ( in a partonic language ) or the percolation and string fusion ( in non - perturbative approaches ) . in both cases , when different constituents , whichever the degrees of freedom chosen are , overlap in phase space according to some geometric criterium , recombination of such constituents happen , thus reducing the total number of scattering centers gluons entering the collision process . similar phase - space arguments motivate the implementation of energy - dependent cut - offs to regulate independent particle production from different sources , normally a working hypothesis in most monte carlo event generators for heavy ion collisions . in fig 1 ( right ) we show the collision energy dependence of the transverse momentum cutoff that separates _ hard _ from _ soft _ particle production in several event generators for p+p ( pythia ) and a+a ( hydjety and hijing ) collisions . its strong rise with increasing energy signals the increasing importance of collective effects in particle production processes . also frequent in the literature are the resummation of multiple scatterings , either in a coherent higher twist formalism or in a incoherent _ glauber like _ formalism . finally , the modification of parton distribution functions to allow for the presence of a normally energy dependent intrinsic transverse momentum constitute other common practice in phenomenological works . all these ingredients are akin , at least at a conceptual level , to those dynamically built in the cgc , although they are formulated in very different ways . thus the debate is now which theoretical framework is most suited for their description : whether the cgc ( at its present degree of accuracy ! ) or alternative approaches , typically rooted in the standard collinear factorization framework . . , title="fig:",scaledwidth=47.0% ] . , title="fig:",scaledwidth=51.0% ] briefly , it can be said that the cgc is now entering the next - to - leading ( nlo ) order era . actually , the kernel of the bk and jimwlk evolution equations are now known to nlo accuracy @xcite or , also , to running coupling accuracy through the resummation of a partial subset of nlo diagrams @xcite . in a similar fashion , the following calculations concerning particle production processes have recently become available at nlo accuracy : photon impact factors in deep inelastic scattering @xcite , full nlo @xcite and _ inelastic _ @xcite contributions to hybrid calculation of single inclusive particle production in dilute - dense scattering , running coupling corrections to the @xmath13-factorization formula @xcite and a proof of factorization of multiparticle production processes at nlo @xcite . furthermore , our knowledge of exclusive particle production and multi - particle correlations ( di - hadrons , hadron - photon etc ) has been advanced significantly through a series of recent works that establish the precise relation between the @xmath14-point functions of the nuclear wave function and the observables of interest @xcite . thus , and despite the intrinsic technical difficulty of higher order calculations in the cgc one should recall that they are performed by expanding on a strong background color field progress on the theoretical side has been steadily delivered over the last years . notwithstanding the progress brought by these works , the cgc framework is still far from the degree of sophistication and accuracy that characterizes standard pqcd methods based on the collinear factorization formalism and dglap evolution equations . the reasons for this are manifold : first , the new theory tools discussed above have not been yet fully implemented in a systematic way in phenomenological applications . current descriptions of data combine nlo with lo theory ingredients in a somehow uncontrolled theoretical scheme . for instance , the arguably most ambitious effort so far to determine the universal properties of nuclear and proton wave functions at small-@xmath0 in a systematic way is provided by the global aamqs fits to e+p data@xcite and their extension to the nuclear case through monte carlo methods @xcite . the aamqs fits are rely on the use of the running coupling bk equation to describe the small-@xmath0 evolution of the 2-point function , but then use the lo dipole formalism to calculate the @xmath15-proton cross section . related issues obscure the successful phenomenological descriptions of data in single @xcite and double @xcite inclusive production in p+p and d+au collisions or the widespread use of @xmath13-factorization in models for total multiplicities , known not to be valid in the case of _ dense - dense _ scattering . most of the nlo tools being available , the main obstacle to fix such insufficiencies of present phenomenological works is the difficulty of their technical implementation . nlo and exclusive particle production calculations typically require the knowledge of the @xmath0-dependence of @xmath14-point functions beyond the 2-point function . hence , the use of the ( running coupling ) bk - equation for the 2-point function , easy to solve numerically , is no longer sufficient and solutions of the full b - jimwlk hierarchy of coupled equations more demanding numerically are needed . a promising analytic method to solve b - jimwlk equations through a gaussian approximation that allows to obtain arbitrary @xmath14-point functions in terms of the 2-point one has been proposed recently @xcite , and its practical implementation would substantially reduce the uncertainties progress related to calculations on di - hadron correlations . other factor that blurs the predictive power of the cgc framework is the paucity of experimental data on high - energy ( equivalently , small-@xmath0 ) nuclear reactions . such information is needed to constrain the non - perturbative parameters of the theory , like the initial conditions for the evolution at some initial scale ( @xmath16 tipically taken to be @xmath17 in practical applications ) or the impact parameter dependence of the nuclear unintegrated nuclear distributions . lacking such information , some degree of modeling is unavoidable . mean field approaches for the description of the nuclear geometry have been recently superseded by monte carlo methods as the mc - kln or mc - rcbk ones @xcite , where the position of nucleons in the transverse plane are treated as a random variable , thus allowing to account for geometry fluctuations in the collision process . keeping in mind the present limitations of cgc phenomenological works discussed in this section , let me now briefly review current cgc predictions for different observables at the lhc in the next sections . the cgc offers a very economical description of the integrated hadron multiplicities produced in heavy ion collisions . based solely in dimensional analysis , the number of particles produced per unit of transverse area rises proportional to the saturation scale ( in symmetric collisions ) @xmath18 such approximate expression can be realized either via solutions of the classical equation of motion in the presence of sources ( projectile and target ) or less rigorously for a+a collisions through the use of @xmath13-factorization . it accounts well for two of the most remarkable features observed in rhic d+au and au+au data as well as in lhc pb+pb data : _ i ) _ approximate factorization of the collision energy and collision centrality of total multiplicities and _ ii ) _ power - law dependence on integrated multiplicities on collision energy . in fig 2 we show the predictions for the rapidity dependence of hadron multiplicities from different cgc models rooted in @xmath13-factorization , where particle production is given by the convolution of the unintegrated gluon distributions of projectile and target . these models differ mainly in their input for the nuclear unintegrated gluon distributions and in the treatment of the geometry dependence either monte carlo methods or mean field approaches but also in the implementation details like the ir regularization or the rapidity to pseudo - rapidity jacobian or the modeling of the large-@xmath0 component of the corresponding wave functions ( hence the large deviations among them at more forward / backward rapidities ) . they have been tested against d+au rhic data and then extrapolated to lhc energies without further adjustments . overall they predict a charged hadron multiplicity @xmath19 an observable that has centered much of the discussion on the relevance of cgc physics in heavy ion collisions is the nuclear modification factor measured in d+au collisions at rhic . at mid - rapidity rhic data show a moderate enhancement of single inclusive particle production tipically atributed to semi - classical multiple scatterings , whereas they are continuously depleted at more forward rapidities . a good quantitative description of this forward suppression is possible within the cgc @xcite , where it relates to the non - linear evolution of the nuclear ugd . however , rhic forward data falls very close to the kinematic limit , where large-@xmath0 effects such as energy loss neglected in the in the cgc approach may also be relevant . indeed , there are alternative explanations of data where energy loss is the main dynamical ingredient ( see e.g @xcite ) . moreover , the cgc description of the most forward data on neutral pions performed by the star collaboration requires the use of @xmath20-factors smaller than unity , hinting at the relevance of large-@xmath0 effects . although more exclusive observables like di - hadron or hadron - photon correlations are expected to better discriminate between different approaches , a first test for models of particle production in hic shall come from data on inclusive multiplicities and single particle distributions as they are much easier to obtain experimentally . generic arguments based on kinematics suggest that one should expect a similar suppression of the nuclear modification factors in lhc p+pb collisions at mid - rapidity as the one observed in forward rhic d+au data , since in both cases the nuclear wave function is probed at similar values of bjorken-@xmath0 . while this argument can be a somewhat misleading ( the proximity of rhic forward data to the kinematic limit limits strongly the transverse momentum of the gluons probed in the nucler ugd ) , it is approximately realized in the different cgc - predictions . fig 3 show the predictions for @xmath21 at rapidity 0 ( left ) and @xmath22 ( right ) based from the rcbk mc @xcite and ip - sat @xcite models . they predict a moderate suppression ( similar to the one observed at rhic @xmath23 at moderate to small transverse momentum @xmath24 gev , and a smooth approach to unity at larger transverse momentum . the error bands in the mc - rcbk calculation originate from the use of different initial conditions to solve the running coupling bk evolution and also from the variation of factorization scales and fragmentation functions in the @xmath13-factorization formula they rely upon . also , it should be note that the mc - rcbk predictions presented here correspond to the _ quenched _ approximation for the nuclear geometry . the details of this calculation will be presented in @xcite . remarkably , the cgc predictions for mid - rapidity overlap strongly with those obtained within the collinear factorization framework using the eps09 parametrization for the npdf s ( results from @xcite ) . one then concludes that only this observable does not suffice to discriminate among different approaches to particle production in high energy heavy ion collisions . the differences between cgc and collinear factorization formalisms start becoming apparent as one moves to more forward rapidities . cgc models predict a faster onset of the suppression at higher values of transverse momentum with respect to he eps09 results . the weak rapidity dependence of eps09 results can be traced back to the flatness of the gluon modification factor at small-@xmath0 in this parametrization . thus , a rapidity scan of the nuclear modification factors would offer a much larger discriminating power . cgc predictions at intermediate values of rapidity suffer of a systematic uncertainty : in the cgc there are two distinct but related approaches to hadron production in high energy asymmetric collisions . particle production processes in the central rapidity region probe the wave functions of both projectile and target at small values of @xmath0 . here , one may employ the @xmath25-factorization formalism where both the projectile and target are characterized in terms of their rcbk evolved unintegrated gluon distributions ( ugds ) . however , at more forward rapidities , the proton is probed at larger values of @xmath0 while the target nucleus is shifted deeper into the small-@xmath0 regime . here , @xmath13-factorization fails to grasp the dominant contribution to the scattering process . rather , the _ hybrid _ formalism proposed in ref . @xcite . in the hybrid formalism the large-@xmath0 degrees of freedom of the proton are described in terms of usual parton distribution functions ( pdfs ) of collinear factorization which satisfy the momentum sum rule exactly and which exhibit a scale dependence given by the dglap evolution equations . on the other hand , the small-@xmath0 glue of the nucleus is still described in terms of its ugd . the corresponding limits of applicability of each formalism equivalently the precise value of @xmath0 at which one should switch from one to the other have only been estimated on an empirical basis , and in practice it is taken to be @xmath26 . the leading order hybrid formalism yield a stronger suppression that the @xmath13-factorization one ( see fig 2 right ) . recently the hybrid formalism has been improved through the calculation of inelastic contributions and full nlo corrections that may become important at high transverse momentum @xcite . the inelastic terms were recently implemented in phenomenological work @xcite . there it was observed that the effect of the inelastic corrections is to slightly increase the value of the nuclear modification factors with respect to the lo result , thus bringing them closer to the results obtained within the @xmath13-factorization framework . a full phenomenological implementation of the hybrid and @xmath13-factorization at nlo is necessary to better asses this systematic uncertainty . at a strictly qualitative level , two generic features of cgc predictions can be highlighted : * : dissapearence of the cronin peak in @xmath21 at central rapidities . * : stronger suppression at forward rapidities : @xmath27 for @xmath28 . these two features of cgc predictions originate from generic properties of the non - linear small-@xmath0 evolution , regardless of the degree of accuracy of the evolution kernel , nlo or lo etc . therefore , the persistence of the cronin peak in @xmath21 at the lhc would be very difficult to accommodate in the cgc framework . similarly the continuos depletion of nuclear modification factors reflects the relative enhancement of non - linear correction to the evolution of nuclear wave function with respect to that of a proton . correspond to results obtained within the @xmath13-factorization or hybrid formalism respectively . also shown in red the predictions corresponding to the eps09 npdf set within collinear factorization @xcite ( courtesy of p. quiroga - arias ) . , title="fig:",scaledwidth=49.0% ] correspond to results obtained within the @xmath13-factorization or hybrid formalism respectively . also shown in red the predictions corresponding to the eps09 npdf set within collinear factorization @xcite ( courtesy of p. quiroga - arias ) . , title="fig:",scaledwidth=49.0% ] the study of forward di - hadron correlations in d+au azimuthal correlations at rhic has provided the most solid indication for the relevance of cgc effects in data so far . the disappearance of the away side peak also dubbed _ monojet _ production in more central collisions ( and its persistence in more peripheral collisions and p+p collisions ) can be related to the interplay between the transverse momenta of the produced hadrons and the one acquired during the interaction with the nucleus . in the cgc approach the interaction with the nucleus is realized in a fully coherent way , and the momentum broadening is parametrically controlled by the @xmath0-dependent saturation scale of the nucleus . the latter , in turn , is described by means of the rcbk equation . a first semi - quantitative description of the data was provided in @xcite . this work neglected some leading in @xmath1 terms which calculation demands knowledge of higher @xmath14-point functions and also the contribution of the gluon channel to the production process . these caveats were partially fixed in a later work @xcite , where the small momentum imbalance approximation was used . moreover , the role of multiparton interactions that may enhance the contribution of the uncorrelated component of the double inclusive cross section ( specially in the forward region ) @xcite has not been fully explored in present cgc calculations . nevertheless , the description of data is good in these two works is good , see fig 4 . recently , another description of data based in a higher - twist calculation that also includes nuclear shadowing and cold nuclear matter energy loss has become available @xcite . a full cgc analyses of data on di - hadron correlations including the missing ingredients in previous cgc analyses is now underway @xcite . it should serve as the reference to generate precise quantitative predictions for p+pb collisons at the lhc . so far , qualitative expectations indicate that analogous suppression of azimuthal correlations should be observed in at the lhc . generically the strength of the decorrelation is expected to be stronger with : _ i ) _ increasing rapidity of the produced pair ; _ ii ) _ increasing collision centrality and _ iii ) _ decreasing transverse momentum of the trigger and associated particle . it has been recently proposed that hadron - photon @xcite and hadron - dileptons @xcite correlations may exhibit similar azimuthal structure . these observables offer the advantage that they can be computed in terms of only the rcbk - evolved 2-point function . although their experimental determination may be more complicated , their measurement at the lhc would provide additional constraints to determine the underlying dynamics of multiparticle production in high - density qcd scattering . in summary , while it is fair to say that a large number of observables in different systems from e+p to a+a collisions that probe the small-@xmath0 component of the wave function of the projectile or target find their natural interpretation in terms of high gluonic densities and also a good quantitative description in terms of cgc - based calculations , no conclusive claim for the observation saturation physics can be performed yet . important steps have been taken over the last years in promoting the cgc framework to a predictive and quantitative phenomenological tool . such has been possible through the systematic implementation of global fit and monte carlo methods and , more importantly , through an intense theoretical work in the determination of higher order corrections to the formalism , including running coupling corrections to non - linear evolution equations and also to particle production processes . nevertheless , this program is far for complete and there is still a large margin for improvement in the cgc phenomenological works . the p+pb data will provide precious information to sharpen the cgc quantitative tool . first of all it will provide empiric information needed to constrain the non - perturbative parameters of the theory . next it will allow to test the generic cgc predictions and also whether the present degree of accuracy of the cgc effective theory is sufficient to quantitatively describe data . i would like to thank the organizers for their invitation to this very interesting conference . my research is supported by a fellowship from the thorie lhc france initiative funded by the in2p3 . n. armesto , ( ed . ) , et al . , heavy ion collisions at the lhc - last call for predictions , j. phys . g35 ( 2008 ) 054001 . http://arxiv.org/abs/0711.0974 [ ] , http://dx.doi.org/10.1088/0954-3899/35/5/054001 [ ] . g. a. chirilli , b .- w . xiao , f. yuan , one - loop factorization for inclusive hadron production in @xmath31 collisions in the saturation formalism , phys.rev.lett . 108 ( 2012 ) 122301 . 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marian , e. petreska , two - gluon correlations and initial conditions for small - x evolution , phys.rev . d84 ( 2011 ) 014018 . http://arxiv.org/abs/1105.4155 [ ] , http://dx.doi.org/10.1103/physrevd.84.014018 [ ] . j. l. albacete , n. armesto , j. g. milhano , c. a. salgado , non - linear qcd meets data : a global analysis of lepton- proton scattering with running coupling bk evolution , phys . d80 ( 2009 ) 034031 . http://arxiv.org/abs/0902.1112 [ ] , http://dx.doi.org/10.1103/physrevd.80.034031 [ ] . j. l. albacete , n. armesto , j. g. milhano , p. quiroga arias , c. a. salgado , aamqs : a non - linear qcd analysis of new hera data at small - x including heavy quarks , eur.phys.j . c71 ( 2011 ) 1705 , * temporary entry * . http://arxiv.org/abs/1012.4408 [ ] , http://dx.doi.org/10.1140/epjc/s10052-011-1705-3 [ ] . j. l. albacete , c. marquet , single inclusive hadron production at rhic and the lhc from the color glass condensate , phys . b687 ( 2010 ) 174179 . http://arxiv.org/abs/1001.1378 [ ] , http://dx.doi.org/10.1016/j.physletb.2010.02.073 [ ] . j. l. albacete , c. marquet , azimuthal correlations of forward di - 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i present a brief review of present cgc phenomenological applications and of the physics prospects for the forthcoming proton - lead run at the lhc color glass condensate , high - density qcd , lhc besides its crucial role as a reference experiment to calibrate initial state effects in heavy ion collisions , the forthcoming p+pb run at the lhc will provide access to kinematic regions never explored so far in nuclear collisions and thus carries great potential for discovery of new qcd phenomena on its own . in particular , the huge leap forward in collision energy with respect to previous high energy electron - nucleus or proton - nucleus experiments will probe the nuclear wave function at values of bjorken-@xmath0 smaller than ever before .
it is theoretically well established that at small enough values of bjorken-@xmath0 qcd enters a novel regime governed by large gluon densities and non - linear coherence phenomena .
the color glass condensate ( cgc ) effective theory provides a consistent framework to study qcd scattering at small-@xmath0 or high collision energies ( for a review see e.g. @xcite ) .
it is based on three main physical ingredients : first , high gluon densities correspond to strong classical fields , which permit ab - initio first principles calculation of `` wave functions '' at small @xmath0 through classical techniques .
next , quantum corrections are incorporated via non - linear renormalization group equations such as the b - jimwlk hierarchy or , in the large-@xmath1 limit , the bk equation @xcite that describe the evolution of the hadron wave function towards small @xmath0 .
the non - linear , density - dependent terms in the cgc evolution equations are ultimately related to unitarity of the theory and , in the appropriate frame and gauge , can be interpreted as due to gluon recombination processes that tame or _
saturate _ the growth of gluon densities for modes with transverse momenta below a dynamically generated scale known as the saturation scale , @xmath2 .
finally , the presence of strong color fields @xmath3 leads to breakdown of standard perturbative techniques to describe particle production processes based on a series expansion in powers of the strong coupling @xmath4 .
terms of order @xmath5 need to be resummed to all orders .
the cgc provides the tools to perform such resummation although the precise prescription for the resummation may vary from process to process or colliding system while the cgc has been successfully applied to the description of different observables in different collision systems ( from e+p to aa ) , the p+pb run at the lhc will provide an excellent and probably in the near future unique possibility to disentangle the presently inconclusive situation on the role of cgc effects and also to distinguish among different approaches to describe high energy scattering in nuclear reactions . on the one hand , the lhc shall bring us closer to the limit of asymptotically high energy in which the cgc formalism is developed , thus reducing theoretical uncertainties on its applicability .
equivalently , the value of the saturation scale is expected to be a factor @xmath6 times larger than at rhic , so saturation effects should be visible in a larger range of transverse momenta , deeper into the perturbative domain . on the other hand , the much extended reach in the lhc will allow measurements far from the kinematic limit up to very forward rapidities , thus minimizing the role of large-@xmath0 effects which obscured the interpretation of forward rhic data .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the problem of biological aging has attracted much attention in recent years . based on the data of human demography and experiments of other living organisms , many important phenomena of longevity have been found @xcite . for instance , the gompertz law was observed for intermediate ages , that is , the mortality function increases exponentially with age , while at old ages the mortality was found to decelerate or level off , and even decline for some organisms like flies , worms , and yeast @xcite . to reproduce and explain these phenomena , various models of senescence have been proposed , with genetic or nongenetic mechanisms @xcite . among them , the one widely used by physicists is the penna model @xcite , where one computer word is used to represent the inherited genome of one individual and each bit of the word corresponds to one age of the individual lifetime . a bit set to one represents a deleterious mutation and the suffering from an inherited disease from this age on , and the individual will die if the accumulation of these set bits exceeds a threshold . although the penna model has been well applied to many problems related to biological aging @xcite , there exists an important flaw in this model as pointed out by pletcher and neuhauser very recently @xcite . that is , the model predicts that for a genetically identical population all individuals have their genetic death at the same age , but this is inconsistent with the experimental results @xcite which also exhibited the exponential gompertz law and the deceleration of the old age mortality for the genetically homogeneous case . thus , a more complicated model has been proposed @xcite . in this paper we develop a simpler stochastic model bridging the gap between the standard ( deterministic ) penna model and the pletcher - neuhauser approach . the simulations and analytic results of this model are shown to agree with some features of the biological aging , e.g. , the exponential increase of the mortality function and the deceleration at advanced ages , and the flaw of the penna model mentioned above can be avoided . as in the standard penna model , here the genome of each individual is characterized by a string ( computer word ) of 32 bits , and each bit is expressed as a particular age in the life of the individual . a bit @xmath0 is set to @xmath1 if it represents a deleterious mutation , and from this age @xmath0 on this bit will continuously affect the survival probability of the individual . that is , at age @xmath2 ( @xmath3 ) the death probability contributed by the mutated bit @xmath0 is @xmath4 , with the corresponding survival probability @xmath5 . otherwise , this bit is set to zero and has no effect on death . thus our assumptions are very different from an earlier `` fermi '' function in another stochastic penna model @xcite . the individual s survival probability @xmath6 up to age @xmath2 is the product of the contributions from all the bits before @xmath2 : @xmath7 where @xmath8 or @xmath9 ( @xmath10 ) represents the @xmath0th bit . with the form of @xmath4 , one can obtain the mortality function by simulation or analytical work . in this work we simply assume that @xmath11 with the constant @xmath12 and the limit @xmath13 , which means that the contribution of death probability from bit @xmath0 ( if set to @xmath1 ) is assumed to increase linearly with the age . the other forms of @xmath4 , such as the exponential and the square root forms , have been tried , and we have also simulated the other probabilistic penna model with fermi function @xcite . although some phenomena for the genetically heterogeneous steady - state population can be reproduced , they can not give a good result for the genetically homogeneous populations . the alive individual will generate @xmath14 offsprings from the minimum reproduction age @xmath15 to the maximum one @xmath16 , and the genome of each offspring is the same as the parent one , except for @xmath17 mutations randomly occurring at birth . at each time step @xmath18 , a verhulst factor @xmath19 denoting the survival probability of the individual due to the space and food restrictions is introduced , where @xmath20 is the current population size and @xmath21 is the carrying capacity of the environment , usually set to @xmath22 . in the next section [ sec3 ] the simulations based on these rules are presented , while for genetically identical individuals , which have the same genotype randomly sampled from the simulated steady - state population , the analytic results can be derived , as shown in section [ sec4 ] . moreover , in this paper the mortality function @xmath23 at age @xmath2 is defined as @xmath24 where @xmath25 denotes the number of alive individuals with age @xmath2 , and @xmath26 is the survival rate . to eliminate the effect of the verhulst factor , the normalized mortality function is preferred @xcite , i.e. , @xmath27 . \label{mortfunc}\ ] ] in our simulations , initially the population size @xmath28 is @xmath29 and all bits of all the strings are set to zero , i.e. , free of mutations . one time step @xmath18 corresponds to one aging interval of the individuals , or reading one bit of all strings . the reproduction range is set from @xmath30 to @xmath31 with the birth rate @xmath32 , and the results are similar if using the maximum value of @xmath33 . @xmath34 mutation for each offspring genome is introduced at birth , and here only the bad mutations are taken into account , that is , the bit randomly selected for mutation is always set to @xmath1 . ( the good mutations have also been considered , e.g. , 10% good mutations and 90% bad ones , and similar results are found . ) [ fig - pop - mut ] shows the evolution of the whole population size @xmath20 until @xmath35 . similar to the standard penna model , the steady - state population is obtained at late timesteps , and as a result of evolution and selection , the frequency of deleterious bits ( set as @xmath1 ) for the individual of the steady - state population is low at early ages ( especially before the reproduction age ) and very high at old ones . this behavior of the frequency ( or the bad mutation rate ) is shown in the inset of fig . [ fig - pop - mut ] . the mortality function is calculated using eq . ( [ mortfunc ] ) and averaged over the steady - state population from timesteps 5000 to 10000 , as shown in fig . [ fig - mortality ] . the result is consistent with the experimental and empirical observations @xcite , that is , at intermediate ages the mortality function increases exponentially , exhibiting the gompertz law , and deceleration occurs for old ages . for comparison , the mortality simulated by the standard ( deterministic ) penna model is also shown in fig . [ fig - mortality ] , with the threshold of the accumulated bad mutations @xmath36 and the other parameters unchanged . the exponential gompertz law can also be obtained for the standard penna model @xcite , however , no deceleration is observed except for suitable modifications summarized in @xcite ; see also @xcite . to study the genetically homogeneous population , one can randomly sample an individual ( genotype ) from the simulated steady - state population , and then `` clone '' it to create the whole genetically identical population . according to the form of these bit - strings , the mortality function can be derived and calculated analytically . as in some experiments of fruit flies @xcite , reproduction is prevented during the aging of genetically homogeneous individuals . thus , for this population of single genotype , we have @xmath37 where @xmath25 , @xmath38 , is the number of individuals with age @xmath2 in the population , and the function @xmath39 is defined by eq . ( [ liveprob ] ) . then the survival rate is easily obtained : @xmath40 for the mortality function , the normalized formula ( [ mortfunc ] ) is used to be consistent with the simulations in sec . [ sec3 ] , and then we have @xmath41 . \label{hommort}\ ] ] different genotypes have been selected randomly from the stable population of sec . [ sec3 ] , and the corresponding mortality function of each type is calculated using eq . ( [ hommort ] ) . some examples are shown in fig . [ fig - clonemort ] for linear - log plots , where part of them obey the exponential gompertz law at the intermediate ages , similar to that of the above simulation ( sec . [ sec3 ] ) and experiments @xcite . moreover , all of these curves exhibit the deceleration for old ages . moreover , the analytic calculation is also available if the reproduction is allowed as in other experiments of genetically identical population , but for the case of no mutation . the details are shown in the appendix , and the mortality function derived is the same as eq . ( [ hommort ] ) . in this paper a stochastic genetic model of aging is developed based on the bit - string asexual penna model , and the results of the exponentially increasing mortality at intermediate ages and its deceleration at old ages are obtained for both the genetically heterogeneous steady - state population and the homogeneous individuals . however , the decrease of mortality for the oldest ages , observed in some experiments @xcite , can not be described by the mechanism of this model . although the properties for intermediate and old ages have been well simulated in this model , the behavior at early ages can not be well reproduced , which is also an artifact of the penna - type genetic models . from fig . [ fig - clonemort ] for genetically identical populations , it can be found that some populations have unrealistic zero mortality at some early ages . thus , the effects for the early ages studied in the experiments , such as the investigations of genetic variation for ln - mortality contributed by steady - state population or by new mutations @xcite , can not be produced in this model . more efforts should be made to avoid this difficulty , e.g. , by considering different kinds of genes before and after the reproduction age @xcite . we thank scott d. pletcher and naeem jan for very helpful discussions and comments . this work was supported by sfb 341 . here an example of the analytic solution for this stochastic model is presented , for the case where the reproduction is allowed in the aging process of genetically identical population , but no mutation occurs when generating the genomes of offsprings . thus , the individuals keep homogeneous , characterized by the same bit - string @xmath42 with @xmath43 the length of genome ( @xmath44 in above studies ) . when the system evolves to the steady state , the population size at timestep @xmath18 of this state @xmath45 as well as the verhulst factor @xmath46 can be considered as constant . thus , the numbers of individuals with ages from @xmath1 to @xmath43 at this step @xmath18 are @xmath47 where @xmath39 is the living probability of individual at age @xmath2 , as defined in eq . ( [ liveprob ] ) , and the individuals of age zero ( newly born ) are generated by the ones with reproducible age ( from age @xmath15 to @xmath16 ) , that is , @xmath48 \nonumber\\ & = & bv[n_{r_{\rm min}-1}(t-1)g(1,2, ... ,r_{\rm min } ) + n_{r_{\rm min}}(t-1)g(1,2, ... ,r_{\rm min}+1 ) \nonumber\\ & + & \cdots+n_{r_{\rm max}-1}(t-1)g(1,2, ... ,r_{\rm max } ) ] \label{a3}\end{aligned}\ ] ] with the birth rate @xmath14 . consequently , the number of individuals with certain age @xmath2 ( @xmath49 ) can be expressed as @xmath50 therefore , if @xmath51 is unchanged for the steady state , all the @xmath52 , @xmath53 , will also keep unchanged , i.e. , independent of timestep @xmath18 , and then the survival rate @xmath54 can be obtained from eq . ( [ a2 ] ) , that is , @xmath55 and @xmath56 the verhulst factor can be eliminated when calculating the normalized rate : @xmath57 and from the definition of eq . ( [ mortfunc ] ) one can obtain the normalized mortality function , which is the same as eq . ( [ hommort ] ) . the constant property of the population size @xmath20 and the number @xmath51 for age @xmath9 , as well as the above analytic result of the mortality function have been confirmed by the simulation . moreover , the steady state condition can be derived from eqs . ( [ a3 ] ) and ( [ a4 ] ) , which is @xmath58=1 , \label{a5}\end{aligned}\ ] ] depending on the parameters @xmath14 , @xmath15 , and @xmath16 . j. w. vaupel , j. r. carey , k. christensen , t. e. johnson , a. i. yashin , n. v. holm , i. a. iachine , v. kannisto , a. a. khazaeli , p. liedo , v. d. longo , y. zeng , k. g. manton , and j. w. curtsinger , _ science _ * 280 * , 855 ( 1998 ) .
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a stochastic genetic model for biological aging is introduced bridging the gap between the bit - string penna model and the pletcher - neuhauser approach .
the phenomenon of exponentially increasing mortality function at intermediate ages and its deceleration at advanced ages is reproduced for both the evolutionary steady - state population and the genetically homogeneous individuals .
: biological aging ; mortality ; penna model .
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a @xmath0-dimensional random walk that proceeds via a sequence of unit - length steps , each in an independent and uniformly random direction , is sometimes called a _ pearson rayleigh _ random walk ( prrw ) , after the exchange in the letters pages of _ nature _ between karl pearson and lord rayleigh in 1905 @xcite . pearson was interested in two dimensions and questions of migration of species ( such as mosquitoes ) @xcite , although carazza has speculated that pearson was a golfer @xcite ; rayleigh had earlier considered the acoustic ` random walks ' in phase space produced by combinations of sound waves of the same amplitude and random phases . the prrw can be represented via partial sums of sequences of i.i.d . random vectors that are uniformly distributed on the unit sphere @xmath4 in @xmath5 . clearly the increments have mean zero , i.e. , the prrw has _ zero drift_. the prrw has received some renewed interest recently as a model for microbe locomotion @xcite . chapter 2 of @xcite gives a general discussion of these walks , which have been well - understood for many years . in particular , it is well known that the prrw is recurrent for @xmath6 and transient if @xmath2 . suppose that we replace the spherically symmetric increments of the prrw by increments that instead have some _ elliptical _ structure , while retaining the zero drift . for example , one could take the increments to be uniformly distributed on the surface of an ellipsoid of fixed shape and orientation , as represented by the picture on the right of figure [ fig1 ] . more generally , one should view the ellipses in figure [ fig1 ] as representing the _ covariance _ structure of the increments of the walk ( we will give a concrete example later ; the uniform distribution on the ellipse is actually not the most convenient for calculations ) . ( -2.5,-2.5 ) rectangle ( 2.5,2.5 ) ; ( 0,-2 ) edge ( 0,2 ) ( -2,0 ) edge ( 2,0 ) ; /in -1/-1 , -1/1 , 1/-1 , 1/1 , 0/0 ( , ) ellipse ( 10pt and 10pt ) ; ( -2.5,-2.5 ) rectangle ( 2.5,2.5 ) ; ( 0,-2 ) edge ( 0,2 ) ( -2,0 ) edge ( 2,0 ) ; /in -1/-1 , -1/1 , 1/-1 , 1/1 , 0/0 ( , ) ellipse ( 18pt and 6pt ) ; a little thought shows that the walk represented by the picture on the right of figure [ fig1 ] is essentially no different to the prrw : an affine transformation of @xmath5 will map the walk back to a walk whose increments have the same covariance structure as the prrw . to obtain genuinely different behaviour , it is necessary to abandon spatial homogeneity . in this paper we consider a family of spatially _ non - homogeneous _ random walks with zero drift . these include generalizations of the prrw in which the increments are not i.i.d . but have a distribution supported on an ellipsoid of fixed size and shape but whose orientation depends upon the current position of the walk . figure [ fig2 ] gives representations of two important types of example , in which the ellipsoid is aligned so that its principal axes are parallel or perpendicular to the vector of the current position of the walk , which sits at the centre of the ellipse . ( -2.5,-2.5 ) rectangle ( 2.5,2.5 ) ; ( 0,-2 ) edge ( 0,2 ) ( -2,0 ) edge ( 2,0 ) ; in 22.5 , 67.5 , ... , 360 ( 0:1.5 ) ellipse ( 18pt and 8pt ) ; ( -2.5,-2.5 ) rectangle ( 2.5,2.5 ) ; ( 0,-2 ) edge ( 0,2 ) ( -2,0 ) edge ( 2,0 ) ; in 0 , 45 , ... , 359 ( 0:1.5 ) ellipse ( 6pt and 14pt ) ; the random walks represented by figure [ fig2 ] are no longer sums of i.i.d . variables . these modified walks can behave very differently to the prrw . for instance , one of the two - dimensional random walks represented in figure [ fig2 ] is _ transient _ while the other ( as in the classical case ) is recurrent . the reader who has not seen this kind of example before may take a moment to identify which is which . it is this _ anomalous recurrence behaviour _ that is the main subject of the present paper . in the next section , we give a formal description of our model and state our main results . we end this introduction with a brief comment on motivation . in biology , the prrw is more natural than a lattice - based walk for modelling the motion of microscopic organisms , such as certain bacteria , on a surface . experiment suggests that the locomotion of several kinds of cells consists of roughly straight line segments linked by discrete changes in direction : see , e.g. , @xcite . the generalization to elliptically - distributed increments studied here represents movement on a surface on which either radial or transverse motion is inhibited . in chemistry and physics , the trajectory of a finite - step prrw ( also called a ` random chain ' ) is an idealized model of the growth of weakly interacting polymer molecules : see , e.g. , 2.6 of @xcite . the modification to ellipsoid - supported jumps represents polymer growth in a biased medium . we work in @xmath5 , @xmath7 . our main interest is in @xmath3 , as we shall explain shortly . write @xmath8 for the standard orthonormal basis vectors in @xmath5 . write @xmath9 for the origin in @xmath5 , and let @xmath10 denote the euclidean norm and @xmath11 the euclidean inner product on @xmath5 . write @xmath12 for the unit sphere in @xmath5 . for @xmath13 , set @xmath14 ; also set @xmath15 , for convenience . for definiteness , vectors @xmath16 are viewed as column vectors throughout . we now define @xmath17 , a discrete - time , time - homogeneous markov process on a ( non - empty , unbounded ) subset @xmath18 of @xmath5 . formally , @xmath19 is a measurable space , @xmath18 is a borel subset of @xmath5 , and @xmath20 is the @xmath21-algebra of all @xmath22 for @xmath23 a borel set in @xmath5 . suppose @xmath24 is some fixed ( i.e. , non - random ) point in @xmath18 . write @xmath25 for the increments of @xmath26 . by assumption , given @xmath27 , the law of @xmath28 depends only on @xmath29 ( and not on @xmath30 ) ; so often we ease notation by taking @xmath31 and writing just @xmath32 for @xmath33 . we also use the shorthand @xmath34 = { { \mathbb p } } [ \ , \cdot \ , \ ! \mid x_0 = { { \mathbf{x}}}]$ ] for probabilities when the walk is started from @xmath35 ; similarly we use @xmath36 for the corresponding expectations . we make the following moments assumption : : : there exists @xmath37 such that @xmath38 < \infty$ ] . the assumption ensures that @xmath32 has a well - defined mean vector @xmath39 $ ] , and we suppose that the random walk has _ zero drift _ : : : suppose that @xmath40 for all @xmath35 . the assumption also ensures that @xmath32 has a well - defined covariance matrix , which we denote by @xmath41 , $ ] where @xmath32 is viewed as a column vector . to rule out pathological cases , we assume that @xmath32 is _ uniformly non - degenerate _ , in the following sense . : : there exists @xmath42 such that @xmath43 \geq v$ ] for all @xmath35 . note that assumption is weaker than _ uniform ellipticity _ , which in this context usually means , for some @xmath44 , @xmath45 \geq { \varepsilon}$ ] for all @xmath46 and all @xmath47 . our main interest is in a recurrence classification . first , we state the following basic ` non - confinement ' result . [ lem : lim_sup_infty ] suppose that @xmath26 satisfies assumptions , and . then @xmath48 we give the proof of proposition [ lem : lim_sup_infty ] in section [ sec : non - confinement ] ; we actually prove more , namely that the hypotheses of proposition [ lem : lim_sup_infty ] ensure that a version of kolmogorov s ` other ' inequality holds . the fact ensures that questions of the escape of trajectories to infinity are non - trivial . indeed , we will give conditions under which one or other of the following two behaviours ( which are not _ a priori _ exhaustive ) occurs : * @xmath49 , a.s . , in which case we say that @xmath26 is _ transient _ ; * @xmath50 , a.s . , for some constant @xmath51 , when we say @xmath26 is _ recurrent_. if @xmath26 is an irreducible time - homogeneous markov chain on a locally finite state - space , these definitions reduce to the usual notions of transience and recurrence ; in general state - spaces , our approach allows us to avoid unnecessary technicalities concerning irreducibility . in dimension @xmath52 , it is a consequence of the classical chung fuchs theorem ( see @xcite or chapter 9 of @xcite ) that a spatially _ homogeneous _ random walk with zero drift is necessarily recurrent . however , this is _ not _ true for a spatially non - homogeneous random walk : as observed by rogozin and foss @xcite , a counterexample is provided by a version of the ` oscillating random walk ' of kemperman @xcite in which the increment law is one of two distributions ( with mean zero but infinite second moment ) depending on the walk s present sign . our conditions exclude these heavy - tailed phenomena , so that in @xmath52 recurrence is assured in our setting . [ t : zero_drift_implies_recurrence ] suppose that @xmath52 . suppose that @xmath26 satisfies assumptions , , and . then @xmath26 is recurrent . theorem [ t : zero_drift_implies_recurrence ] is essentially contained in a result of lamperti ( * ? ? ? * theorem 3.2 ) ; we give a self - contained proof below . theorem [ t : zero_drift_implies_recurrence ] shows that in @xmath52 , under mild conditions , the classical chung fuchs recurrence classification for homogeneous zero - drift random walks extends to zero - drift non - homogeneous random walks . the purpose of the present paper is to demonstrate a natural family of examples in dimension @xmath3 where this extension fails , and hence exhibit the following . there exist spatially non - homogeneous random walks whose increments are non - degenerate , have uniformly bounded second moments , and have zero mean , which are -3 mm * transient in @xmath53 ; * recurrent in @xmath2 . although certainly appreciated by experts , this fact is perhaps not as widely known as it might be . zeitouni ( pp . 9192 of @xcite ) describes an example of a transient zero - drift random walk on @xmath54 , and states that the idea `` goes back to krylov ( in the context of diffusions ) '' . peres , popov and sousi @xcite investigate the minimal number of different increment distributions required for anomalous recurrence behaviour . we now introduce our family of non - homogeneous random walks . write @xmath55 for the matrix ( operator ) norm given by @xmath56 . the following assumption on the asymptotic stability of the covariance structure of the process along rays is central . : : suppose that there exists a positive - definite matrix function @xmath57 with domain @xmath4 such that , as @xmath58 , @xmath59 a little informally , says that @xmath60 as @xmath61 ; in what follows , we will often make similar statements , formal versions of which may be cast as in . note that and together imply that @xmath62 ; next we impose a key assumption on the form of @xmath57 that is considerably stronger . to describe this , it is convenient to introduce the notation @xmath63 that defines , for each @xmath46 , an inner product on @xmath5 via @xmath64 : : suppose that there exist constants @xmath65 and @xmath66 with @xmath67 such that , for all @xmath46 , @xmath68 informally , @xmath66 quantifies the total variance of the increments , while @xmath65 quantifies the variance in the radial direction ; necessarily @xmath69 . the assumption that @xmath70 excludes some degenerate cases . as we will see , one possible way to satisfy condition is to suppose that the eigenvectors of @xmath71 are all parallel or perpendicular to the vector @xmath72 , and that the corresponding eigenvalues are all constant as @xmath72 varies ; the level sets of the corresponding quadratic forms @xmath73 for @xmath46 are then ellipsoids like those depicted in figure [ fig2 ] . our main result is the following , which shows that both transience and recurrence are possible for _ any _ @xmath3 , depending on parameter choices ; as seen in theorem [ t : zero_drift_implies_recurrence ] , this possibility of anomalous recurrence behaviour is a genuinely multidimensional phenomenon under our regularity conditions . [ thm : recurrence ] suppose that @xmath26 satisfies , with constants @xmath74 as defined in . the following recurrence classification is valid . * if @xmath75 , then @xmath26 is transient . * if @xmath76 , then @xmath26 is recurrent . * if @xmath77 and holds with @xmath78 for some @xmath79 , then @xmath26 is recurrent . moreover , we show that in any of the above cases , @xmath26 is _ null _ in the following sense . [ thm : null ] suppose that @xmath26 satisfies , with constants @xmath74 as defined in . then , in any of the cases ( i)(iii ) in theorem [ thm : recurrence ] , for any bounded @xmath80 , @xmath81 [ rem : uequalsv ] theorems [ thm : recurrence ] and [ thm : null ] both remain valid if we permit @xmath82 in ; indeed , the condition @xmath83 is not used in the proof of theorem [ thm : recurrence ] given below , so this case is recurrent , by theorem [ thm : recurrence](ii ) . the condition @xmath83 is used at one point to simplify the proof of theorem [ thm : null ] given below , but a small modification of the argument also works in the case @xmath84 . the remainder of the paper is organised as follows . in section [ sec : ell - rws ] we describe a specific family of examples called _ elliptic random walk models _ that satisfy assumptions and exhibit both transient and recurrent behaviour dependent on the parameters of the model . we also present some simulated data that depicts the random walks in both cases . in section [ sec : non - confinement ] we prove a @xmath0-dimensional martingale version of kolmogorov s other inequality and use that to prove the non - confinement result ( proposition [ lem : lim_sup_infty ] ) . in section [ sec : rw ] we prove the recurrence classification ( theorem [ thm : recurrence ] ) , and in section [ sec : nullity ] we prove theorem [ thm : null ] . in the appendix we prove recurrence in the one - dimensional case ( theorem [ t : zero_drift_implies_recurrence ] ) . finally , we remark that in work in progress we investigate diffusive scaling limits for random walks of the type described in the present paper ; the diffusions that appear as scaling limits possess certain pathologies from the point of view of diffusion theory that make them interesting in their own right . let @xmath3 . we describe a specific model on @xmath85 where the jump distribution at @xmath16 is supported on an ellipsoid having one distinguished axis aligned with the vector @xmath47 . the model is specified by two constants @xmath86 . construct @xmath32 as follows . given @xmath87 , take @xmath88 uniform on @xmath4 and set @xmath89 for @xmath90 an orthogonal matrix representing a transformation of @xmath5 mapping @xmath91 to @xmath92 , and @xmath93 . see figure [ fig : delta ] . ( -2,-2.2 ) rectangle ( 10,2 ) ; ( -0.5,0 ) circle ( 1 ) ; ( -1.5,0 ) arc ( 180:360:1 and 0.4 ) ; ( 0.5,0 ) arc ( 0:180:1 and 0.4 ) ; at ( -0.3,0.65 ) @xmath88 ; ( 4,0 ) ellipse ( 1.2 and 2 ) ; ( 2.8,0 ) arc ( 180:360:1.2 and 0.6 ) ; ( 5.2,0 ) arc ( 0:180:1.2 and 0.6 ) ; at ( 4.2,1.2 ) @xmath94 ; ( 9,0 ) ellipse ( 1.2 and 1.8 ) ; ( 7.8,0 ) arc ( 180:360:1.2 and 0.75 ) ; ( 10.2,0 ) arc ( 0:180:1.2 and 0.75 ) ; ( -0.5,0 ) edge ( 0.1,0.6 ) ( 4,0 ) edge ( 4.72,1.2 ) [ rotate around=-30:(9,0 ) ] ( 9,0 ) edge ( 9.72,1.08 ) ; at ( 10,0.85 ) @xmath32 ; at ( 9.2,-0.3 ) @xmath47 ; at ( 7.2,-2.5 ) @xmath9 ; ( 9,0 ) edge ( 9,-3 ) ; ( 9,-3 ) circle ( 1pt ) ; ( 1,0 ) ( 2.4,0 ) ; ( 5.7,0 ) ( 7.1,0 ) ; ( recall that @xmath95 , so for @xmath96 we can take @xmath97 and @xmath98 . ) thus @xmath32 is a random point on an ellipsoid that has one distinguished semi - axis , of length @xmath99 , aligned in the @xmath92 direction , and all other semi - axes of length @xmath100 . note that the law of @xmath32 is well defined owing to the spherical symmetry of the uniform distribution on @xmath4 and the fact that only one axis of the ellipsoid is distinguished ( for this reason it is enough to take any @xmath90 satisfying @xmath101 in order to define @xmath32 ; see also remark [ rem : hx ] below ) . note also that @xmath32 is not chosen to be uniformly distributed on the surface of the ellipsoid ; this does not affect the range of asymptotic behaviour exhibited by the family of walks as @xmath102 and @xmath103 vary , but it does simplify the calculation of @xmath104 . indeed , we have @xmath105 = \operatorname{\mathbb{e } } [ q_{\hat{{{\mathbf{x } } } } } d { { \bm{\zeta}}}{{\bm{\zeta}}}^{{\scalebox{0.6}{$\top$}}}d q_{\hat{{{\mathbf{x}}}}}^{{\scalebox{0.6}{$\top$ } } } ] = q_{\hat{{{\mathbf{x } } } } } d \operatorname{\mathbb{e } } [ { { \bm{\zeta}}}{{\bm{\zeta}}}^{{\scalebox{0.6}{$\top$ } } } ] d q_{\hat{{{\mathbf{x}}}}}^{{\scalebox{0.6}{$\top$}}}= \frac{1}{d } q_{\hat{{{\mathbf{x } } } } } d^2 q_{\hat{{{\mathbf{x}}}}}^{{\scalebox{0.6}{$\top$}}},\ ] ] by linearity of expectation , and using the fact that @xmath106 = \frac{1}{d}i$ ] for @xmath88 uniformly distributed on @xmath4 . also , a calculation similar to the above confirms that @xmath40 for all @xmath16 , since @xmath107 = { { \mathbf{0}}}$ ] . since @xmath108 is bounded above by @xmath109 , assumption holds . clearly and hold , with @xmath110 for @xmath46 . it is also a simple matter to check that and hold : the matrix @xmath71 represented in coordinates for the orthonormal basis @xmath111 is diagonal with entries @xmath112 . indeed , @xmath113 q_{{\mathbf{u}}}^{{\scalebox{0.6}{$\top$}}}\\ & = a^2 { { \mathbf{u}}}{{\mathbf{u}}}^{{\scalebox{0.6}{$\top$}}}+ b^2 ( i - { { \mathbf{u}}}{{\mathbf{u}}}^{{\scalebox{0.6}{$\top$ } } } ) , \end{split}\ ] ] and therefore @xmath114 for all @xmath46 , and @xmath115 for all @xmath16 . [ rem : hx ] the seeming ambiguity in the definition of @xmath32 due to the choice of @xmath116 can be resolved by noting that @xmath32 can be rewritten as @xmath117 where @xmath118 is also uniform on @xmath4 ( this follows from the spherical symmetry of the uniform distribution on @xmath4 ) . moreover , the symmetric matrix @xmath119 is determined explicitly in terms of @xmath92 : @xmath120 consequently , we could choose to specify @xmath32 explicitly as @xmath121 with @xmath122 taken to be uniform on @xmath4 . as before , we find that @xmath123 = h_{\hat{{\mathbf{x } } } } \operatorname{\mathbb{e } } [ \tilde{{\bm{\zeta } } } ] = { { \mathbf{0}}}$ ] and @xmath124 = h_{\hat{{\mathbf{x } } } } \operatorname{\mathbb{e } } [ \tilde{{\bm{\zeta}}}\tilde{{\bm{\zeta}}}^{{\scalebox{0.6}{$\top$}}}]h_{\hat{{\mathbf{x } } } } = \textstyle\frac{1}{d}h_{\hat{{\mathbf{x}}}}^2 = a^2\hat{{\mathbf{x}}}\hat{{\mathbf{x}}}^{{\scalebox{0.6}{$\top$}}}+ b^2 ( i - \hat{{\mathbf{x}}}\hat{{\mathbf{x}}}^{{\scalebox{0.6}{$\top$}}}).\ ] ] recall that we assume our random walk to be time - homogeneous , so that equation in fact determines the distribution of @xmath28 for all @xmath125 . formally , we define @xmath126 a sequence of independent random variables uniformly distributed on @xmath4 , and for each @xmath125 we define @xmath28 conditional on @xmath127 via @xmath128 we call @xmath129 defined in this way an _ elliptic random walk_. as a corollary to theorems [ thm : recurrence ] and [ thm : null ] , we get the following recurrence classification for the elliptic random walk model . for this model the @xmath130 in is identically zero so we get a complete classification that includes the boundary case . [ cor : ellipsoid - lamperti ] let @xmath3 and @xmath131 . let @xmath26 be an elliptic random walk on @xmath5 . then @xmath26 is transient if @xmath132 and null - recurrent if @xmath133 . in two dimensions we can explicitly describe the random walk as follows . for @xmath134 , @xmath135 with @xmath136 in cartesian components , set @xmath137 . fix @xmath138 . let @xmath139 denote the ellipse with centre @xmath47 and principal axes aligned in the @xmath47 , @xmath140 directions , with lengths @xmath141 , @xmath142 respectively , given in parametrized form by @xmath143 \right\},\ ] ] and for @xmath144 set @xmath145 \right\}.\ ] ] the parameter @xmath146 in the parametrization ( [ param ] ) should be interpreted with caution : it is _ not _ , in general , the central angle of the parametrized point on the ellipse . given @xmath147 , @xmath148 is taken to be distributed on @xmath149 , ` uniformly ' with respect to the parametrization ( [ param ] ) . precisely , let @xmath150 be a sequence of independent random variables uniformly distributed on @xmath151 $ ] . then , on @xmath152 , @xmath153 while , on @xmath154 , @xmath155 figure [ fig : sim ] shows two sample paths of a simulation of the elliptic random walk in @xmath156 in the two cases of recurrence and transience . in each picture the walk starts at the origin at the centre of the picture ; time is represented by the variation in colour ( from red to yellow , or from dark to light if viewed in grey - scale ) . for the recurrent case @xmath157 ( _ left _ ) and the transient case @xmath158 ( _ right_).,title="fig : " ] for the recurrent case @xmath157 ( _ left _ ) and the transient case @xmath158 ( _ right_).,title="fig : " ] [ rem : ellipse ] a. the process @xmath26 reduces to the classical prrw when @xmath159 : in that case it is spatially homogeneous , i.e. , the distribution of the increment @xmath160 does not depend on @xmath29 . for @xmath161 the random walk is not spatially homogeneous , and the jump distribution depends upon the projection onto the unit sphere of the walk s current position . b. as mentioned earlier , we choose to take increments as defined at , rather than increments that are uniform on the ellipse with respect to one - dimensional lebesgue measure on @xmath139 , purely for computational reasons . in fact , in two dimensions , since the lebesgue measure on @xmath139 coincides with the measure induced by taking @xmath146 uniformly distributed on @xmath151 $ ] when @xmath159 , and the case @xmath162 is critically recurrent , the qualitative behaviour will be the same in either case : the walk will be transient for @xmath163 and recurrent for @xmath164 . for higher dimensions , taking increments that are uniform with respect to the lebesgue measure on @xmath165 will still specify a family of models that exhibit a phase transition , from transience ( for @xmath166 small ) to recurrence ( for @xmath166 large ) but the exact shape of the ellipsoid in the critical case ( i.e. , the smallest ratio @xmath166 for which the walk is recurrent ) may be different . c. it follows from that @xmath167 in particular , for this family of models @xmath168 is itself a markov process , since the distribution of @xmath169 depends only on @xmath170 and not @xmath29 ; however , in the general setting of section [ sec : rws ] , this need not be the case . + one - dimensional processes with evolutions reminiscent to that given by have been studied previously by kingman @xcite and bingham @xcite . those processes can be viewed , respectively , as the distance from its start point of a random walk in euclidean space , and the geodesic distance from its start point of a random walk on the surface of a sphere , but in both cases the increments of the random walk have the property that the distribution of the jump vector is a product of the independent marginal distributions of the length and direction of the jump vector . in contrast , for the elliptic random walk the laws of @xmath171 and @xmath172 are _ not _ independent ( except when @xmath162 ) . d. [ rem : stas ] the theory equally applies to the case where the ellipsoid specifying the jump distribution is oriented with some fixed angle @xmath173 with respect to the radial direction . if we define @xmath174 , where @xmath175 is an orthogonal matrix that maps @xmath176 to @xmath177 , then we find that transience of @xmath26 is equivalent to @xmath178 note that for @xmath53 , @xmath179 and therefore @xmath32 are well defined , but this is not so for higher dimensions . nevertheless , for _ any _ collection of matrices @xmath180 satisfying @xmath181 for all @xmath46 we get the same recurrence classification . this is because the distribution of @xmath182 given @xmath29 is determined through the angle @xmath183 via @xmath184 and therefore assumption holds with @xmath185 and @xmath186 . in this section we prove that the assumptions , , and imply that @xmath187 , a.s . we first present a general result for martingales on @xmath5 satisfying a `` uniform dispersion '' condition ; the result can be viewed as a @xmath0-dimensional martingale version of _ kolmogorov s other inequality _ ( see e.g. @xcite ) . [ lem : d - dim - koi ] let @xmath188 . suppose that @xmath189 is an @xmath5-valued process adapted to a filtration @xmath190 , with @xmath191 = 1 $ ] . suppose that there exist @xmath192 such that for all @xmath193 , a.s . , @xmath194 & \leq b ; \label{koi : moments}\\ \operatorname{\mathbb{e } } [ \| y_{n+1 } - y_n \|^2 \mid { { \mathcal g}}_n ] & \geq v ; \label{koi : unif - ellip } \\ \operatorname{\mathbb{e } } [ y_{n+1 } - y_n \mid { { \mathcal g}}_n ] & = { { \mathbf{0}}}. \label{koi : zero - drift } \end{aligned}\ ] ] then there exists @xmath195 , depending only on @xmath23 , @xmath196 , and @xmath197 , such that for all @xmath198 and all @xmath199 , @xmath200 \geq 1 - \frac{d(1+x)^2}{n } , { \ \text{a.s.}}\ ] ] let @xmath201 and set @xmath202 ; throughout the paper we adopt the usual convention @xmath203 . in analogy with previous notation , write @xmath204 for the jump distribution , and let @xmath205 where @xmath206 is a constant to be specified later . note that @xmath207 is @xmath208-measurable . now , on @xmath209 , @xmath210 and @xmath211 = { } & \operatorname{\mathbb{e } } [ \delta_n \mid { { \mathcal g}}_n ] \\ & { } + \operatorname{\mathbb{e } } [ \widehat{\delta}_n\left((a-1)(1+x ) - \|\delta_n\|\right){{\mathbf 1}{\{\|y_{n+1}\| > a(1+x)\ } } } \mid { { \mathcal g}}_n ] . \end{aligned}\ ] ] but @xmath212 implies that @xmath213 , and by , @xmath214 = { { \mathbf{0}}}$ ] . hence , on @xmath215 , @xmath216 \bigr\| & \leq \operatorname{\mathbb{e } } [ \|\delta_n\| { { \mathbf 1}{\ { \|\delta_n\| > ( a-1)(1+x)\ } } } \mid { { \mathcal g}}_n]\\ & \leq ( a-1)^{-1}(1+x)^{-1 } \operatorname{\mathbb{e } } [ \|\delta_n\|^2 \mid { { \mathcal g}}_n ] \\ & \leq b'(a-1)^{-1}(1+x)^{-1 } , { \ \text{a.s.}},\end{aligned}\ ] ] where , by and lyapunov s inequality , @xmath217 depends only on @xmath23 and @xmath196 . hence we can choose @xmath218 for some @xmath219 large enough so that @xmath220 \bigr\| \leq ( v/8)(1+x)^{-1 } , \text { on } \ { \| y_n \| \leq x \ } .\ ] ] [ eq : w - square ] also , on @xmath215 , by a similar argument , @xmath221 \bigr\| & = \operatorname{\mathbb{e } } [ \|\delta_n\|^2 \mid { { \mathcal g}}_n ] \nonumber\\ & \quad + \operatorname{\mathbb{e } } [ \left ( ( a-1)^2(1+x)^2 - \|\delta_n\|^2 \right){{\mathbf 1}{\ { \|y_{n+1}\| > a(1+x ) \ } } } \mid { { \mathcal g}}_n ] \nonumber\\ & \geq \operatorname{\mathbb{e } } [ \|\delta_n\|^2 \mid { { \mathcal g}}_n ] - \operatorname{\mathbb{e } } [ \|\delta_n\|^2 { { \mathbf 1}{\ { \| \delta_{n}\| > ( a-1)(1+x ) \ } } } \mid { { \mathcal g}}_n ] \nonumber\\ & \geq v - ( a-1)^{2-p}(1+x)^{2-p } \operatorname{\mathbb{e } } [ \|\delta_n\|^p \mid { { \mathcal g}}_n ] \nonumber\\ & \geq v/2,\end{aligned}\ ] ] for all @xmath222 and @xmath223 for sufficiently large @xmath224 , using and . now , set @xmath225 . then , on @xmath226 , by and , @xmath227 & = \operatorname{\mathbb{e } } [ \|w_{n+1}\|^2 - \|w_n\|^2 \mid { { \mathcal g}}_n ] \\ & = \operatorname{\mathbb{e } } [ \|w_{n+1 } - w_n\|^2 \mid { { \mathcal g}}_n ] + 2 \bigl{\langle}w_n , \operatorname{\mathbb{e } } [ w_{n+1 } - w_n \mid { { \mathcal g}}_n ] \bigr{\rangle}\\ & \geq \frac{v}{2 } - \frac { 2 \|w_n\| v } { 8(1+x ) } \geq \frac{v}{2 } - \frac { v x } { 4(1+x ) } \geq \frac{v}{4}.\end{aligned}\ ] ] hence @xmath228 is a @xmath208-adapted submartingale , where @xmath229 by construction , @xmath230 , so @xmath231 \leq \operatorname{\mathbb{e } } [ z_n \mid { { \mathcal g}}_0 ] - \sum_{k=0}^{n-1 } \operatorname{\mathbb{e } } [ v_k \mid { { \mathcal g}}_0 ] \leq a^2(1+x)^2 - \sum_{k=0}^{n-1 } \frac{v}{4 } { { \mathbb p } } [ n < \tau \mid { { \mathcal g}}_0 ] , \ ] ] which implies @xmath232 \leq a^2(1+x)^2 $ ] . in other words , @xmath233 \leq \frac{4a^2(1+x)^2}{v n } , { \ \text{a.s.}}\qedhere\ ] ] now we can give the proof of proposition [ lem : lim_sup_infty ] . it is enough to show that for all @xmath234 the event @xmath235 occurs infinitely often . for a given @xmath236 , we will apply lemma [ lem : d - dim - koi ] to @xmath237 with @xmath238 ; that result is applicable , since , and imply , and , respectively . thus lemma [ lem : d - dim - koi ] shows that , for some finite @xmath239 , @xmath240 } \geq \frac{1}{2 } , { \ \text{a.s.}},\ ] ] for all @xmath241 . for @xmath242 , define the event @xmath243 and filtration @xmath244 . then @xmath245 , and , by , @xmath246 } \geq \frac{1}{2}$ ] , a.s . , for all @xmath247 . an application of lvy s extension of the borel cantelli lemma ( see , e.g. , ( * ? ? ? * cor . 7.20 ) ) shows that @xmath248 occurs infinitely often , a.s . for each @xmath247 such that @xmath248 occurs , either * @xmath249 , or * @xmath250 and @xmath251 for some @xmath252 . since one of these cases must occur for infinitely many @xmath247 , we have that @xmath253 occurs infinitely often , as required . in this section we study the random walk @xmath29 and give the proof of the recurrence classification , theorem [ thm : recurrence ] . the method of proof is based on applying classical results of lamperti @xcite to the @xmath254-valued radial process given by @xmath255 . the method rests on an analysis of the increments @xmath256 given @xmath257 ; in general , @xmath258 is not itself a markov process . the following notation will be useful . given @xmath135 and @xmath259 , write @xmath260 so that @xmath261 is the component of @xmath262 in the @xmath177 direction , and @xmath263 is a vector perpendicular to @xmath177 . first we state a general result on the increments of @xmath258 for a markov process @xmath29 on @xmath18 . recall that we write @xmath264 , and let @xmath265 be the radial component of @xmath32 at @xmath87 in accordance with the notation described above ; no confusion should arise with our notation @xmath28 defined previously . we make an important comment on notation . when we write @xmath266 , and similar expressions , these are understood to be uniform in @xmath47 . that is , if @xmath267 and @xmath268 , we write @xmath269 to mean that there exist @xmath270 and @xmath271 such that @xmath272 [ lemma1 ] suppose that @xmath26 is a discrete - time , time - homogeneous markov process on @xmath273 satisfying for some @xmath37 . then , for @xmath255 , we have @xmath274 < \infty,\ ] ] and the radial increment moment functions satisfy @xmath275 = \operatorname{\mathbb{e}}_{{\mathbf{x } } } [ \delta_{{\mathbf{x } } } ] + \frac { \operatorname{\mathbb{e}}_{{\mathbf{x } } } [ \| \delta \|^2 - \delta^2_{{{\mathbf{x } } } } ] } { 2 \| { { \mathbf{x}}}\| } + o ( \| { { \mathbf{x}}}\|^{-1-\delta } ) , \\ \label{mu2 } \mu_2({{\mathbf{x } } } ) & : = \operatorname{\mathbb{e } } [ ( r_{n+1}- r_n)^2 \mid x_n = { { \mathbf{x } } } ] = \operatorname{\mathbb{e}}_{{\mathbf{x } } } [ \delta_{{\mathbf{x}}}^2 ] + o ( \|{{\mathbf{x}}}\|^{-\delta } ) , \end{aligned}\ ] ] as @xmath61 , for some @xmath276 . by time - homogeneity , it suffices to consider the case @xmath31 . by the triangle inequality , @xmath277 , so that follows from . we prove and by approximating @xmath278 \end{split}\ ] ] for large @xmath47 . let @xmath279 for some @xmath280 to be determined later . on the event @xmath281 we approximate using taylor s formula for @xmath282 , and on the event @xmath283 we bound using . indeed , for all @xmath284 , taylor s theorem with lagrange remainder shows that @xmath285 for some @xmath286 $ ] , so on the event @xmath281 , @xmath287 where the error terms follow from the fact that @xmath288 for @xmath289 . on the other hand , @xmath290 by the triangle inequality and the fact that @xmath291 on @xmath283 . since @xmath292 we can combine and to give @xmath293 \biggr|\\ & = \biggl| \| { { \mathbf{x}}}+ \delta \| - \|{{\mathbf{x}}}\| - \left[\delta_{{\mathbf{x}}}+ \left ( \frac{\| \delta \|^2 - \delta_{{\mathbf{x}}}^2 } { 2\| { { \mathbf{x}}}\| } \right ) \left ( 1 + o(\| { { \mathbf{x}}}\|^{\beta-1 } ) \right ) \right ] \biggr| { { \mathbf 1}{(a_{{\mathbf{x}}}^{{{\mathrm{c } } } } ) } } \\ & \leq \| \delta \|^p \| { { \mathbf{x}}}\|^{\beta(1-p ) } + \left| \delta_{{\mathbf{x}}}+ \left ( \frac{\| \delta \|^2 - \delta_{{\mathbf{x}}}^2 } { 2\| { { \mathbf{x}}}\| } \right ) \left ( 1 + o(\| { { \mathbf{x}}}\|^{\beta-1 } ) \right ) \right| { { \mathbf 1 } { ( a_{{\mathbf{x}}}^{{{\mathrm{c } } } } ) } } \\ & \leq 2 \| \delta \|^p \| { { \mathbf{x}}}\|^{\beta(1-p ) } + \frac{\| \delta \|^p}{2\| { { \mathbf{x}}}\| } \left ( 1 + o(\| { { \mathbf{x}}}\|^{\beta-1 } ) \right ) \|{{\mathbf{x}}}\|^{\beta(2-p ) } .\end{aligned}\ ] ] therefore , taking expectations and using , we obtain @xmath294 + \frac { \operatorname{\mathbb{e}}_{{\mathbf{x}}}[\| \delta \|^2 - \delta_{{\mathbf{x}}}^2 ] } { 2\| { { \mathbf{x}}}\| } + o ( \| { { \mathbf{x}}}\|^{\beta-2 } ) + o ( \| { { \mathbf{x}}}\|^{\beta(1-p ) } ) + o ( \| { { \mathbf{x}}}\|^{\beta(2-p ) - 1 } ) .\ ] ] taking @xmath295 makes all the error terms of size @xmath296 for some @xmath297 , namely for @xmath298 . for the second moment , we use the identity @xmath299 so that @xmath300 + \operatorname{\mathbb{e}}_{{\mathbf{x}}}[\|\delta\|^2 ] - 2\|{{\mathbf{x}}}\|\mu_1({{\mathbf{x } } } ) = \operatorname{\mathbb{e}}_{{\mathbf{x}}}[\delta_{{\mathbf{x}}}^2 ] + o(\|{{\mathbf{x}}}\|^{-\delta}),\ ] ] as required . with the additional assumptions , , and , we can use lemma [ lemma1 ] to prove the following result . [ lemma2 ] suppose that @xmath26 is a discrete - time , time - homogeneous markov process on @xmath273 satisfying , , , and . then , with @xmath301 defined at , , and @xmath130 defined at , there exists @xmath302 such that , as @xmath303 , @xmath304 by definition of @xmath130 at we have @xmath305 as @xmath303 . then implies that @xmath306 & = \operatorname{tr}{(m({{\mathbf{x}}}))}\\ & = \operatorname{tr}{(\sigma^2(\hat{{\mathbf{x } } } ) ) } + o({\varepsilon}(\|{{\mathbf{x}}}\|))\\ & = v + o({\varepsilon}(\|{{\mathbf{x}}}\| ) ) , \end{split}\ ] ] and @xmath307 & = { \langle}\hat{{\mathbf{x } } } , m({{\mathbf{x } } } ) \cdot \hat{{\mathbf{x}}}{\rangle}\\ & = { \langle}\hat{{\mathbf{x } } } , \sigma^2(\hat{{\mathbf{x } } } ) \cdot \hat{{\mathbf{x}}}{\rangle}+ o({\varepsilon}(\|{{\mathbf{x}}}\|))\\ & = u + o({\varepsilon}(\|{{\mathbf{x}}}\| ) ) , \end{split}\ ] ] and implies that @xmath308 = \operatorname{\mathbb{e}}_{{{\mathbf{x } } } } [ { \langle}\delta , \hat{{\mathbf{x}}}{\rangle } ] = { \langle}\mu({{\mathbf{x } } } ) , \hat{{\mathbf{x}}}{\rangle}= 0 $ ] . using these expressions in lemma [ lemma1 ] yields . now we can complete the proof of theorem [ thm : recurrence ] . we apply lamperti s @xcite recurrence classification to @xmath309 , the radial process . proposition [ lem : lim_sup_infty ] shows that @xmath310 , and lemma [ lemma1 ] tells us that is satisfied . because the error terms in are uniform in @xmath47 , lemma [ lemma2 ] shows that for all @xmath311 there exists @xmath312 such that @xmath313\ ] ] for all @xmath35 with @xmath314 . therefore , it follows from theorem 3.2 of @xcite that @xmath26 is transient if @xmath315 and recurrent if @xmath316 . for the boundary case , when @xmath317 , if @xmath78 then @xmath318 for @xmath319 , which implies that @xmath26 is recurrent , again by theorem 3.2 of @xcite . in this section we give the proof of theorem [ thm : null ] . in the transient case , this is straightforward . [ lem : null - trans ] in case ( i ) of theorem [ thm : recurrence ] , for any bounded @xmath80 , as @xmath320 , the null property holds . it is sufficient to prove in the case where @xmath321 . in case ( i ) , @xmath26 is transient , meaning that @xmath322 a.s . , so that @xmath323 , a.s . , for any @xmath271 . hence the cesro limit in is also @xmath324 , a.s . , and the @xmath325 convergence follows from the bounded convergence theorem . it remains to consider cases ( ii ) and ( iii ) , when @xmath26 is recurrent . thus there exists @xmath51 such that @xmath326 , a.s . let @xmath327 . it suffices to take @xmath328 , @xmath329 , so @xmath330 infinitely often . we make the following claim , whose proof is deferred until the end of this section , which says that if the walk has not yet entered a ball of radius @xmath331 ( for any @xmath332 big enough ) , the time until it reaches the ball of radius @xmath333 has tail bounded below as displayed . [ lem : null - estimate ] in cases ( ii ) and ( iii ) of theorem [ thm : recurrence ] , there exists a finite @xmath334 such that for any @xmath335 and @xmath336 there exists a finite positive @xmath337 such that @xmath338 \geq c m^{-1/2 } , \text { on } \ { n < \tau_{r } \ } , \ ] ] for all sufficiently large @xmath339 . assuming this result , we can complete the proof of theorem [ thm : null ] . in case ( i ) , the result is contained in lemma [ lem : null - trans ] . so consider cases ( ii ) and ( iii ) . fix @xmath333 and @xmath331 with @xmath340 , with @xmath341 as in lemma [ lem : null - estimate ] . note that @xmath342 , a.s . set @xmath343 and then define recursively , for @xmath344 , the stopping times @xmath345 with the convention that @xmath203 . since @xmath329 and @xmath346 ( by proposition [ lem : lim_sup_infty ] ) , for all @xmath344 we have @xmath347 and @xmath348 , a.s . , and @xmath349 in particular , @xmath350 , a.s . we now write @xmath351 . we use lemma [ lem : d - dim - koi ] to show that the process must exit from @xmath352 rapidly enough . indeed , if @xmath353 is any finite stopping time , set @xmath354 and @xmath355 . then the assumptions , and show that the hypotheses of lemma [ lem : d - dim - koi ] are satisfied , since , for example , @xmath356 = \operatorname{\mathbb{e } } [ \| x_{\kappa+n+1 } -x_{\kappa+n } \|^p \mid { { \mathcal f}}_{\kappa+n } ] = \operatorname{\mathbb{e}}_{x_{\kappa + n } } [ \| \delta \|^p ] , \ ] ] by the strong markov property for @xmath26 at the finite stopping time @xmath357 . in particular , another application of lemma [ lem : d - dim - koi ] , similarly to , shows that we may choose @xmath358 sufficiently large so that @xmath359 \geq \frac{1}{2 } , { \ \text{a.s.}},\ ] ] an event whose occurrence ensures that if @xmath360 , then @xmath26 exits @xmath352 before time @xmath361 . fix @xmath362 . then , an application of at stopping time @xmath363 shows that @xmath364 \leq { { \mathbb p}}\bigl [ \max_{0 \leq \ell \leq n(r ) } \| x_{{\gamma_k } + \ell } - x_{\gamma_k } \| < 2 r { \ ; \bigl| \;}{{\mathcal f}}_{\gamma_k } \bigr ] \leq \frac{1}{2 } , { \ \text{a.s.}}\ ] ] similarly , @xmath365 & = \operatorname{\mathbb{e}}\bigl [ { { \mathbf 1}{\ { \eta_k - \gamma_k > n(r ) \ } } } \operatorname{\mathbb{e } } [ { { \mathbf 1}{\ { \eta_k - \gamma_k > 2 n(r ) \ } } } \mid { { \mathcal f}}_{\gamma_k + n(r ) } ] { \ ; \bigl| \;}{{\mathcal f}}_{\gamma_k } \bigr ] \\ & \leq \frac{1}{2 } { { \mathbb p } } [ \eta_k - \gamma_k > n ( r ) \mid { { \mathcal f}}_{\gamma_k } ] \leq \frac{1}{4 } , \end{aligned}\ ] ] this time applying at stopping time @xmath366 as well . iterating this argument , it follows that @xmath367 \leq 2^{-m}$ ] , a.s . , for all @xmath368 . from here , it is straightforward to deduce that , for some constant @xmath312 , for any @xmath362 , @xmath369 \leq c , { \ \text{a.s.}}\ ] ] on the other hand , the tail estimate implies that @xmath370 \geq c m^{-1/2 } , { \ \text{a.s.}},\ ] ] for @xmath371 and all sufficiently large @xmath339 . for any @xmath372 , set @xmath373 , so that @xmath374 for @xmath375 . note @xmath376 and @xmath377 , a.s . then we claim @xmath378 this is easiest to see by considering two separate cases . first , if @xmath379 , @xmath380 which implies , since the set of @xmath247 less than @xmath30 for which @xmath381 is contained in the set @xmath382 . on the other hand , if @xmath383 , using the elementary inequality @xmath384 for non - negative @xmath385 with @xmath386 , we have @xmath387 which again gives . to estimate the growth rates of the numerator and denominator of the right - hand side of , we apply some results from @xcite . first , writing @xmath388 and @xmath389 , by we can apply theorem 2.4 of @xcite to the @xmath390-adapted process @xmath391 to obtain that for any @xmath44 , a.s . , for all but finitely many @xmath339 , @xmath392 on the other hand , writing @xmath393 and @xmath394 , by we can apply theorem 2.6 of @xcite to the @xmath390-adapted process @xmath391 to obtain that for any @xmath44 , for all @xmath339 sufficiently large , @xmath395 now gives the almost - sure version of the result . the @xmath325 version follows from the bounded convergence theorem . it remains to complete the proof of lemma [ lem : null - estimate ] . a more general , two - sided version of the inequality in lemma [ lem : null - estimate ] is proved in ( * ? * theorem 2.4 ) but under slightly different assumptions . because of this , we can not apply that result directly ; nevertheless , the proof techniques naturally transfer to our setting . in doing so , the arguments become simpler to apply , so we reproduce them here . by the markov property for @xmath26 it is enough to prove the statement for @xmath31 , namely that there exists finite @xmath396 such that for any @xmath397 and @xmath336 there exists a finite positive constant @xmath337 such that , if @xmath398 then @xmath399 \geq c m^{-1/2},\ ] ] for sufficiently large @xmath339 . we outline the two intuitive steps in the proof . first we show that the probability that @xmath400 exceeds some large @xmath236 is bounded below by a constant times @xmath401 . second , we show that if the latter event does occur , with probability at least @xmath402 it takes the process time at least a constant times @xmath403 to reach @xmath404 . combining these two estimates will show that with probability of order @xmath401 the walk takes time of order @xmath403 to reach @xmath404 , which gives the desired tail bound . roughly speaking , the first estimate ( reaching distance @xmath236 ) is provided by the optional stopping theorem and the fact that @xmath405 is a submartingale ( cf . * theorem 2.3 ) ) , and the second ( taking quadratic time to return ) is provided by a maximal inequality applied to an appropriate quadratic displacement functional ( cf . * lemma 4.11 ) ) . a technicality required for the first estimate is that to apply optional stopping , we need uniform integrability ; so we actually work with a truncated version of @xmath406 . we now give the details . recall that @xmath407 and let @xmath408 . lemmas [ lemma1 ] and [ lemma2 ] , with the fact that @xmath409 by , imply that @xmath410 \geq \frac{2{\varepsilon}}{r_k } + o(r_k^{-1 } ) \geq \frac{{\varepsilon}}{r_k},\ ] ] for all @xmath411 , for sufficiently large @xmath396 and some positive constant @xmath412 . now , suppose that @xmath333 and @xmath331 satisfy @xmath413 and fix @xmath236 with @xmath414 . set @xmath415 and @xmath416 . since @xmath417 is a martingale , we have that @xmath418 is a submartingale , as is the stopped process @xmath419 . in order to achieve uniform integrability , we consider the truncated process @xmath420 and show that this is a submartingale . for @xmath421 , we have @xmath422 so @xmath423 = 0 $ ] . for @xmath424 , @xmath425 and the last term can be bounded in absolute value : @xmath426 for @xmath427 as appearing in , since on @xmath428 we have @xmath429 and therefore @xmath430 implies that @xmath431 . applying from lemma [ lemma1 ] we obtain @xmath432 \leq bx^{1-p},\ ] ] for some @xmath433 not depending on @xmath236 . combining this with and again the fact that @xmath434 on @xmath435 , we have that @xmath436 \geq \frac{{\varepsilon}}{r_k } - bx^{1-p } \geq \frac{{\varepsilon}}{x } - bx^{1-p } \geq 0,\ ] ] for sufficiently large @xmath236 . hence , for sufficiently large @xmath236 , @xmath437 is a uniformly integrable submartingale and therefore , given @xmath438 , by optional stopping , @xmath439 & = \operatorname{\mathbb{e } } [ y_{\sigma_x}^x { { \mathbf 1}{\{\sigma_x < \tau_r\ } } } \mid x_0 ] + \operatorname{\mathbb{e } } [ y_{\tau_r}^x{{\mathbf 1}{\{\tau_r < \sigma_x\ } } } \mid x_0 ] \\ & \leq 2x { { \mathbb p } } [ \sigma_x < \tau_r \mid x_0 ] + r. \end{split}\ ] ] in other words , given @xmath438 , @xmath440 \geq \frac{r - r}{2x},\ ] ] for all sufficiently large @xmath236 . now , consider @xmath441 , adapted to @xmath442 . we have @xmath443 using the fact that @xmath418 is a submartingale together with the strong markov property for @xmath26 at the stopping time @xmath444 yields @xmath445 \geq 0 { \ \text{a.s.}}$ ] , and lemmas [ lemma1 ] and [ lemma2 ] again with the strong markov property imply that @xmath446 \leq c { \ \text{a.s . } } , $ ] for some constant @xmath447 ; hence @xmath448 \leq c { \ \text{a.s.}}$ ] , for some constant @xmath449 . then a maximal inequality ( * ? ? ? * lemma 3.1 ) similar to doob s submartingale inequality implies that , on @xmath450 , @xmath451 \leq \frac{cn}{y } , \text { for any } y > 0.\ ] ] in particular , we may choose @xmath44 small enough so that @xmath452 \leq \frac{1}{2 } , \text { on } \{\sigma_x < \infty\}.\ ] ] combining the inequalities and , we find that given @xmath438 , @xmath453\\ & = \operatorname{\mathbb{e}}\big [ { { \mathbf 1}{\{\sigma_x < \tau_r\ } } } { { \mathbb p}}\big [ \max_{0 \leq k \leq { \varepsilon}x^2 } |r_{\sigma_x+k } - r_{\sigma_x}| < x/2 { \ ; \bigl| \;}{{\mathcal f}}_{\sigma_x } \big ] { \ ; \bigl| \;}x_0 \big]\\ & \geq \frac{1}{2 } { { \mathbb p}}\big [ \max_{0 \leq k \leq \tau_r } r_k > x { \ ; \bigl| \;}x_0 \big ] \geq \frac{r - r}{4x } , \end{split}\ ] ] for sufficiently large @xmath236 , where the equality here uses the fact that @xmath454 . if both of the events @xmath455 and @xmath456 occur , then the process @xmath417 leaves the ball @xmath457 before time @xmath458 and takes more than @xmath459 steps to return to the ball @xmath460 , and therefore @xmath461 . setting @xmath462 and @xmath463 yields the claimed inequality . it is only in the proof of lemma [ lem : null - estimate ] that we use the condition @xmath83 from . in the case @xmath84 , inequality holds only for ( any ) @xmath464 , and not @xmath44 ; thus to obtain a submartingale one should look at @xmath465 for @xmath466 . the modified argument yields a weaker version of , with @xmath467 replaced by @xmath468 for any @xmath302 , but , as stated in remark [ rem : uequalsv ] , this is still comfortably enough to give theorem [ thm : null ] ( any exponent greater than @xmath469 in the tail bound will do ) . we omit these additional technical details , as the case @xmath84 is outside our main interest . we use a lyapunov function method with function @xmath470 . [ l : zero_drift_implies_recurrence ] suppose that @xmath26 is a discrete - time , time - homogeneous markov process on @xmath471 . suppose that for some @xmath427 and @xmath472 , @xmath473 & < \infty ; \\ \inf_{x \in { { \mathbb x } } } \operatorname{\mathbb{e } } [ ( x_{n+1 } - x_n ) ^2 \mid x_n = x ] & \geq v .\end{aligned}\ ] ] suppose also that for some bounded set @xmath474 , @xmath475 = 0 , \text { for all } x \in { { \mathbb x}}\setminus a .\ ] ] then there exists a bounded set @xmath476 for which @xmath477 \leq 0 , \text { for all } x \in { { \mathbb x}}\setminus a ' .\ ] ] write @xmath478 and @xmath479 . we compute @xmath480 & = \operatorname{\mathbb{e}}_x [ ( f(x + \delta ) - f ( x ) ) { { \mathbf 1 } { ( e_x ) } } ] \\ & { } ~~ { } + \operatorname{\mathbb{e}}_x [ ( f(x + \delta ) - f ( x ) ) { { \mathbf 1 } { ( e_x^{{\mathrm{c } } } ) } } ] .\end{aligned}\ ] ] on @xmath481 we have that @xmath236 and @xmath482 have the same sign , so @xmath483 \\ & = \operatorname{\mathbb{e}}_x \left [ \log \left ( \frac { 1 + | x + \delta |}{1+|x| } \right ) { { \mathbf 1 } { ( e_x ) } } \right ] \\ & = \operatorname{\mathbb{e}}_x \left [ \log \left ( 1 + \frac { \delta \operatorname{sgn}(x)}{1+|x| } \right ) { { \mathbf 1 } { ( e_x ) } } \right ] \\ & \leq \left ( \frac { \operatorname{sgn}(x)}{1+|x| } \right ) \operatorname{\mathbb{e}}_x [ \delta { { \mathbf 1 } { ( e_x ) } } ] - \frac{1}{6 } ( 1+|x| ) ^{-2 } \operatorname{\mathbb{e}}_x [ \delta^2 { { \mathbf 1 } { ( e_x ) } } ] , \end{aligned}\ ] ] using the inequality @xmath484 for all @xmath485 . here , since @xmath486 = 0 $ ] for @xmath487 , @xmath488 | \leq \operatorname{\mathbb{e}}_x [ | \delta | { { \mathbf 1 } { ( e^{{\mathrm{c}}}_x ) } } ] \leq \operatorname{\mathbb{e}}_x [ | \delta |^p |x|^{1-p } ] = o(|x|^{-1 } ) .\ ] ] similarly , @xmath489 \geq v - \operatorname{\mathbb{e}}_x [ \delta^2 { { \mathbf 1 } { ( e^{{\mathrm{c}}}_x ) } } ] \geq v- o(1 ) .\ ] ] ( note that here , and in what follows , our notation follows the convention as described by ; consequently , in one dimension the error terms are understood to be uniform as either @xmath490 , or @xmath491 . ) finally we estimate the term @xmath492 \right| \leq \operatorname{\mathbb{e}}_x \left [ \left ( \log ( 1 + | \delta | ) + \log ( 1 + 2 |\delta | ) \right ) { { \mathbf 1 } { ( e_x^{{\mathrm{c } } } ) } } \right ] .\ ] ] here , @xmath493 for all @xmath236 with @xmath494 greater than some @xmath495 sufficiently large , using the fact that @xmath496 is eventually decreasing . it follows that @xmath492 \right| \leq 2|x|^{-p } \log ( 1 + 2 |x| ) \operatorname{\mathbb{e}}_x[|\delta|^p ] = o(|x|^{-2}).\ ] ] combining these calculations we obtain @xmath480 & \leq \left ( \frac{\operatorname{sgn}(x)}{1 + |x| } \right ) o ( |x |^{-1 } ) - \frac{1}{6 } ( 1 + |x| ) ^{-2 } ( v - o(1 ) ) + o ( |x|^{-2 } ) \\ & \leq - \frac{v}{6 } ( 1+|x| ) ^{-2 } + o ( |x|^{-2 } ) , \end{aligned}\ ] ] which is negative for all @xmath236 with @xmath494 sufficiently large . under assumptions , and , the hypotheses of lemma [ l : zero_drift_implies_recurrence ] are satisfied , so that for some @xmath497 , @xmath477 \leq 0 , \text { for all $ x \in { { \mathbb x}}$ with $ |x| \geq x_0$},\ ] ] where @xmath498 . we note that assumption implies that @xmath499 < \infty$ ] for all @xmath30 , and therefore @xmath500 < \infty$ ] for all @xmath30 . let @xmath501 and set @xmath502 . let @xmath503 . then @xmath504 is a non - negative supermartingale , and hence there exists a random variable @xmath505 with @xmath506 , a.s . in particular , this means that @xmath507 setting @xmath508 , which satisfies @xmath509 , a.s . , since @xmath510 as @xmath511 , it follows that @xmath512 on @xmath513 . however , under assumptions , and , proposition [ lem : lim_sup_infty ] implies that @xmath514 , a.s . , so to avoid contradiction , we must have @xmath515 , a.s . in other words , @xmath516 = 1,\ ] ] and since @xmath517 was arbitrary , it follows that @xmath518 = 1,\ ] ] which gives the result . part of this work was supported by the engineering and physical sciences research council [ grant number ep / j021784/1 ] . an antecedent of this work , concerning only the elliptic random walk in two dimensions , was written down in 20089 by mm and aw , who benefited from stimulating discussions with iain macphee ( 7/11/195713/1/2012 ) . the present authors also thank stas volkov for a comment that inspired remark [ rem : ellipse ] . o. hryniv , i.m . macphee , m.v . menshikov , and a.r . wade , non - homogeneous random walks with non - integrable increments and heavy - tailed random walks on strips . _ electr . j. probab . _ * 17 * ( 2012 ) article 59 , 28pp .
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famously , a @xmath0-dimensional , spatially homogeneous random walk whose increments are non - degenerate , have finite second moments , and have zero mean is recurrent if @xmath1 but transient if @xmath2 . once spatial homogeneity is relaxed , this is no longer true .
we study a family of zero - drift spatially non - homogeneous random walks ( markov processes ) whose increment covariance matrix is asymptotically constant along rays from the origin , and which , in any ambient dimension @xmath3 , can be adjusted so that the walk is either transient or recurrent .
natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk s current position ; these _ elliptic random walks _ generalize the classical homogeneous pearson rayleigh walk ( the spherical case ) .
our proof of the recurrence classification is based on fundamental work of lamperti . _
key words : _ non - homogeneous random walk ; elliptic random walk ; zero drift ; recurrence ; transience .
_ ams subject classification : _
60j05 ( primary ) 60j10 , 60g42 , 60g50 ( secondary )
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although pulsar - like stars have many different manifestations , they are populated mainly by rotation - powered radio pulsars . a lot of information about pulsar radiative process is inferred from the integrated and individual pulses , the sub - pulses , and even the micro - structures of radio pulses . among the magnetospheric emission models , the user - friendly nature of ruderman & sutherland ( 1975 ; hereafter rs75 ) model is a virtue not shared by others @xcite . in rs75 and its modified versions ( e.g. , * ? ? ? * ) , a vacuum gap exists above polar cap of a pulsar , in which charged particles ( electrons and positrons ) are accelerated because of @xmath1 . these accelerated charged particles , moving along the curved magnetic field lines , radiate curvature or inverse - compton - scattering - induced high energy photons which are converted to @xmath2 while propagating in strong magnetic field . a follow - up breakdown of the vacuum gap produces secondary electron - positron pairs plasma that radiate coherent radio emission . these models with gap - sparking provide a good framework to analyze observational phenomena , especially the drifting @xcite and bi - drifting @xcite sub - pulses . however , the rs75-like vacuum gap models work only in strict conditions : strong magnetic field and low temperature on surface of pulsars ( e.g. , * ? ? ? * ; * ? ? ? the necessary binding energy of positive ions ( e.g. , @xmath0fe ) for rs75 model to work should be higher than @xmath3 kev , while calculations showed that the cohesive energy of @xmath0fe at the neutron star surface is @xmath4 kev @xcite . this binding energy problem could be solved within a partially screened inner gap model @xcite for normal neutron stars . alternatively , it is noted that the binding energy could be sufficiently high if pulsars are bare strange quark stars @xcite although strange stars were previously supposed to exist with crusts @xcite . certainly , it is very meaningful in the elementary strong interaction between quarks and the phases of cold quark matter that the binding energy problem could be solved by bare quark stars as pulsars @xcite . though the ideas of solving the binding energy problem in bss model were presented and discussed in some literatures , there is no comprehensive study with quantitative calculations up to now . in this paper , we are going to investigate the bss model in quantitative details and show the physical picture of binding of particles on bss s surface . our research results are that multi - accelerators could occur above the polar cap for ( and only for ) the curvature - radiation - induced ( cr - induced ) sparking normal pulsars ( nps ) , but for other cases , such as resonant inverse - compton - scattering - induced ( ics - induced ) sparking nps and both cr - induced and ics - induced millisecond pulsars ( msps ) , particles on surface of bsss are bound strongly enough to form vacuum gap and rs75-like models work well if pulsars are bsss . on a bss s surface , there are positively ( @xmath5-quarks ) and negatively ( @xmath6- and @xmath7-quarks and electrons ) charged particles . quarks are confined by strong color interaction , whose binding energy could be considered as infinity when compared with the electromagnetic interaction , while electrons are bound by electromagnetic interaction . therefore , in this paper we focus on the binding of electrons . let s discuss briefly the binding of electrons in the bss model at first . on one hand , assuming the electric potential at the top of rs75 vacuum gap is the same as that of the interstellar medium , one could then have a potential barrier for electrons by integrating the gap electric field from top to bottom in the vacuum gap . this potential barrier could then prevent electrons streaming into magnetosphere . on the other hand , electrons above the stellar surface of bss are described in the thomas - fermi model , in which the total energy of eletrons on fermi surface would be a constant , @xmath8 . in previous work ( e.g. alcock et al . 1986 ) , this constant is chosen to be zero , @xmath9 , because they did nt consider the effect of spinning bss with strong magnetic fields . due to the unipolar generator effect , potential drop between different magnetic field lines is set up from pole to equatorial plane . this potential drop could result in different @xmath8 , at different polar angle , @xmath10 , and the total energy of electrons would then be obtained by choosing certain zero potential magnetic field line ( i.e. , at @xmath11 or @xmath12 in fig . [ antipulsar ] ) . finally , comparing the total energy of electrons with the height of the potential barrier in vacuum gap , we can see whether eletrons can stream into magnetosphere freely or not . the distribution of electrons in bsss is described in the thomas - fermi model @xcite . in this model , equilibrium of electrons in an external electric field assures that the total energy of each electron on fermi surface is a constant , @xmath8 . for the case of extremely relativistic degenerate electron gas , it gives @xcite @xmath13 where @xmath14 is the total energy , @xmath15 is the fermi energy , @xmath16 is the electrostatic potential energy of electrons and @xmath17 is a constant , describing the potential energy of electrons in the thomas - fermi model at infinity . on the other hand , the potential distribution of electrons on the star s surface due to the electric field induced by the rotating , uniformly magnetized star , for the sake of simplicity , could be assumed and estimated as ( xu et al . 2006 , eq . 2 there ) @xmath18 where @xmath19 , and @xmath20 is the radius of a pulsar , @xmath21 is the pulsar period , @xmath10 is the polar angle and @xmath22 is another constant . in view of the distribution of electron above the surface of bss extends only thousands of femtometers , the macroscopic potential drop between different magnetic field lines could be thought to be at infinity in the thomas - fermi model . and the potential energy related to eq . [ ugp ] , @xmath23 , could be regarded as the constant , @xmath17 , in eq . [ fd ] . by choosing the certain zero potential magnetic field line , we could obtain the total energy of electrons , namely @xmath23 . two scenarios could be possible here . the first scenario is that we choose the critical field lines whose feet are at the same electric potential as the interstellar medium @xcite as the zero potential . we may also suggest a second choice that the zero potential should be at those magnetic field lines which separate annular and core regions determined by @xmath24@xmath25 , where @xmath26 and @xmath25 , are the stellar surface areas of annular region and core region , respectively . the second scenario is based on the idea that if particles with opposite charge stream into the magnetosphere with @xmath27 in both regions , areas of this two regions should approximately be equal in order to keep the star not charging . the feet of the critical field lines and the magnetic field lines determined by @xmath24@xmath25 are designated as c and b , respectively ( fig . [ antipulsar ] ) . for the above two scenarios , the total energy , @xmath28 , of electrons on the fermi surface are given by @xmath29 and @xmath30 respectively , where @xmath12 and @xmath11 are polar angles of c and b ( see fig . [ antipulsar ] ) . equations [ eea ] and [ eeb ] imply that the total energy of electrons is higher at the poles and decreases toward the equator for an antipular ( @xmath31 ) , which means that electrons in different regions above a polar cap may behave differently . in the following , we will consider the potential barrier of electrons in vacuum gap . unlike rs75 , we do calculations in situation of an antipulsar whose magnetic axis is parallel to its spin axis . a schematic representation for antipulsar is shown in fig . [ antipulsar ] . assuming the electric potential at the top of rs75 vacuum gap is the same as that of the interstellar medium , we could get a potential barrier for electrons by integrating the gap electric field from top to bottom in the vacuum gap . this potential barrier , in one - dimensional approximation , is ( rs75 ) @xmath32 where @xmath33 is the height of vacuum gap , @xmath34 is the space coordinate measuring height above the quark surface . this potential barrier may prevent electrons injecting into pulsar s magnetosphere . the height of this potential barrier mainly depends on the height of vacuum gap which is determined by cascade mechanics of sparking , i.e. , the cr - induced cascade sparking and the ics - induced cascade sparking . in cr - induced cascade sparking model , the gap height is ( rs75 ) @xmath35 and in ics - induced cascade sparking model , it is @xcite @xmath36 in previous work of gil et al . ( 2006 ) , the heights of the vacuum gap of both cr - induced and ics - induced sparking mechanism ( gil et al . 2006 , eqs . 21 and 22 there ) are different from what we used in this work . in the psg model , there was a partial flow of iron ions from the positively charged polar cap which coexist with the production of outflowing electron - positron plasmas . such a charge - depleted acceleration region is also highly sensitive to both the critical ion temperature and the actual surface temperature of the polar cap @xcite . differently , in our model , there is no flow of positively charged particles , namely quarks and also it is insensitive to the actual surface temperature . this means that there is no partial screened effect above polar cap of bare strange quark stars , namely the pure vacuum gap exists on polar cap of bare strange quark stars . that s the reason why we use eqs . [ hcr ] and [ hics ] in our calculation . whether this choice of height of vacuum gap could result in different driftrate of subpulses or not is a complicated problem . we will discuss this problem very briefly in 3 . the potential barrier of electrons in the gap for cr - induced cascade sparking model of typical normal pulsars ( nps ) is plotted in fig . [ pb ] , in which the total energy of electrons at the stellar surface , namely @xmath37 , is illustrated at different polar angles . the situation of cr - induced cascade sparking of typical millisecond pulsars ( msps ) is similar to that of nps but with greater height of potential barrier . , in vacuum gap of typical nps ( @xmath38 , @xmath39 g ) . the potential energy of electrons at stellar surface , namely @xmath40 , is illustrated with fixed polar angles , for example , with @xmath41 and @xmath42 , where @xmath43 is the polar angle of the feet of the last open field lines ( fig . [ antipulsar ] ) . ] comparing the potential barrier with total energy of electrons , we will explain behavior of electrons above polar cap . namely , only electrons with energy greater than the potential barrier can escape into pulsar s magnetosphere . it is known that energy of electrons is a function of polar angle ( eqs . [ eea ] and [ eeb ] ) . as a result , there may be a critical polar angle , @xmath44 , at which the energy of electrons equals the height of this potential barrier . comparison between the total energy of electrons and the height of potential barrier on stellar surface for typical nps of cr - induced sparking is shown in fig . [ pec ] ( @xmath44 does not exist for both ics - induced sparking of nps and msps , see table . [ pa ] ) . the results are as follows : free flow status stays in the region of [ 0 , @xmath44 ] and vacuum gap in [ @xmath44 , @xmath43 ] for antipulsars , where @xmath43 is polar angle of the feet of the last open field lines ( fig . [ antipulsar ] ) . we give the results of @xmath44 in table . [ pa ] for both pulsar and antipulsar , and find that for the special case of cr - induced sparking nps , free flow and vacuum gap could coexist above polar cap which differs from the previous scenario . the general case is that only vacuum gap exists . and @xmath47 , respectively . the solid horizontal line is the height of the potential barrier of electrons , namely @xmath48 . ] @lllllllll@ & & & & & & + + & @xmath43 ( rad ) & @xmath11 ( @xmath43 ) & @xmath12 ( @xmath43 ) & cr & ics & cr & ics & + & & & & 0.49 & ... @xmath49 & 0.58 & ... & @xmath31 + nps & 0.0145 & 0.69 & 0.76 & 0.84 & 2.76@xmath50 & 0.90 & 2.83@xmath50 & @xmath51 + & & & & ... & ... & ... & ... & @xmath52 + msps & 0.145 & 0.69 & 0.76 & 1.49@xmath50 & ... & 1.52@xmath50 & ... & @xmath53 + + @xmath49@xmath44 does not exit , which means that the whole polar cap region is vacuum gap . + @xmath50@xmath44 @xmath54 @xmath43 , which means that the whole polar cap region is vacuum gap . it follows from the previous argument that electrons inside bsss usually can not stream into magnetospheres . does any other process which may affect the existence of vacuum gap above polar cap ? in vacuum gap , except pulling electrons from the interior of bsss , two other processes which may also prevent vacuum gap from being formed are required to be investigated . one is the thermionic emission of electrons and another is the diffusion of electrons from the outer edge to the inner region of polar cap . for the first one , if the current density due to thermionic emission of electrons is much smaller than that of goldreich - julian charge density , the vacuum gap could be maintained as well . this current density is determined by the richard - dushman equation @xcite @xmath55 where @xmath56 is the mass of electron , @xmath57 is the boltzmann constant , @xmath58 is the temperature and @xmath59 is the work function of electrons . in the vacuum gap of bsss , the work function of thermionic electrons is the order of the difference between the height of the potential barrier and the total energy of electron at the surface of bsss . the order of the difference is about @xmath60 mev . at the same time , the surface temperature of polar caps of bsss is order of @xmath60 k. thus , the thermionic emission current density is @xmath61 , which means that the thermionic emission of electrons can not affect the existence of the vacuum gap . the second process is the diffusion of electrons whose distribution above bsss surface is @xcite @xmath62 eq . [ en ] implies that the number density of electrons ( so does the kinetic energy density , @xmath63 ) decreases rapidly with increasing of the distance from quark matter surface at which @xmath63 @xmath64 @xmath65 , where @xmath66 is the magnetic field energy density . as a result , there is a balance surface where the kinetic energy density equals the magnetic energy density . below this balance surface , electrons can cross magnetic field lines freely and above the balance surface , this motion is prevented . the physical picture of the diffusion of electrons is illustrated in fig . [ df ] . making use of @xmath67 , where @xmath68 ( @xmath69 is the fermi energy of degenerate electrons ) and @xmath70 , we can obtain the height of the balance surface . for nps , it is @xmath71@xmath72 and for msps , it is @xmath71@xmath73 , where @xmath74 . keep in mind that there is a directed outward surface electric field above the quark matter surface . this surface electric field is much stronger than the gap electric field but decreases rapidly with also increasing of the distance . which means that this surface electric field becomes smaller than the gap electric field above some certain distance , @xmath75 . for nps , it is @xmath75@xmath76 , and for msps , it is @xmath75@xmath77@xmath78 ( see fig . [ df ] ) . both have @xmath71@xmath79 @xmath75 for nps and msps . the diffusion of electrons beneath @xmath75 is still confined by the surface electric field meaning that only the diffusion of electrons above the surface with height of @xmath75 needs to be considered . the diffusion coefficient , @xmath80 , is given by @xcite @xmath81 where @xmath82 ( @xmath83 is the larmor radius ) is the cyclotron radius of relativistic electrons , and @xmath84 is the mean free flight time of electrons . the gradient of electrons along with the diffusion direction is approximately @xmath85 then , the diffusion rate is @xmath86 where @xmath87 . for both the nps and msps with different @xmath88 , we give the results of the diffusion rate @xmath89 and @xmath90 in table [ drv ] in which the flow with the goldreich - julian flux is @xmath91 @xmath92@xmath93c@xmath94 @xmath77@xmath95@xmath96@xmath97@xmath98 @xmath99 @xmath100 . we can know that both have @xmath89 @xmath79 @xmath90 for nps and msps from table [ drv ] . this means that the diffusion of electrons is also negligible which guarantees the existence of vacuum gap . @ccccc@ & & + + @xmath88 ( mev ) & @xmath89 ( @xmath101 ) & @xmath90 ( @xmath102 ) & @xmath89 ( @xmath103 ) & @xmath90 ( @xmath104 ) + 1 & @xmath1054.75 & & @xmath1057.52 & + 10&@xmath1054.91 & @xmath105@xmath106 & @xmath1057.52 & @xmath105@xmath107 + 20&@xmath1054.93 & & @xmath1057.53 & + in rs75 model , the binding energy problem is one of the most serious problems in the normal neutron star model of pulsars . arons and scharlemann ( 1979 ) developed an alternative model , the space - charge limited flow ( sclf ) model , in which the particles , both iron ions and electrons can be pulled out freely , and form a steady flow @xcite . in this sclf model , the drifting sub - pulse phenomenon which has been commonly observed in pulsars can rarely be reproduced . the prerequisite for understanding this phenomenon could be the existence of a vacuum gap . in a very special case , through our calculations , we find that there is a new physical scenario for cr - induced sparking of normal pulsars ( nps ) that free flow and vacuum gap may coexist above the polar cap . but in other cases , such as ics - induced sparking of nps and millisecond pulsars ( msps ) , only vacuum gap exists . in general , if a pulsar is not highly negatively charged @xcite , vacuum gap survives at polar cap as well . one limitation is that our calculation is based on one - dimensional approximation and it might fail in some cases of msps . as far as we find , it is very difficult to deal with the high - dimensional cases . the one - dimensional approximation provides a good understanding of the geometry of polar cap of bsss . in conclusion , the binding energy problem could be solved completely in the bss model of pulsar as long as bsss are neutral ( or not highly negative charged ) , and the structure of polar cap of bsss are very different with respect to that of nss . detailed information about the geometry of bss s polar cap is given in table [ ag ] . @llllll@ & & & + + & cr & ics & cr & ics & + & sclf & vg & vg & vg & @xmath31 + nps & vg & vg@xmath108 & sclf & vg@xmath108 & @xmath51 + & vg & vg & vg & vg & @xmath52 + msps & vg@xmath108 & vg & vg@xmath108 & vg & @xmath51 + + @xmath109@xmath44 represents @xmath110 while choosing @xmath111 and @xmath112 while choosing @xmath113 . + @xmath108for such cases , @xmath44 @xmath54 @xmath43 , which represents the structure of the whole polar cap region . a more interesting region from pole to equator may locate between that polar angle where the total energy of electron equals the potential barrier and the polar angle of the foot of zero potential magnetic field line ( i.e. , [ @xmath114 or [ @xmath115 , see fig . [ pec ] ) for cr - induced sparking nps . after the birth of a np , a vacuum gap exists at this region . when sparking starts , the potential in vacuum gap drops rapidly due to screen by electron - positron pairs and may become lower than that at the surface , namely @xmath116 . as a result , the sparking converts vacuum gap to free flow at this region until the sparking ends , i.e. , at [ @xmath117 or [ @xmath115 , vacuum gap and free flow work alternately . this argument may have profound implications for us to distinguish neutron stars and quark stars by pulsar s magnetospheric activities ( e.g. , the diversity pulse profiles ) . another issue to be discussed is about the drifting rate of subpulses when we use the height of pure vacuum gap in this work . the natural explanation of the drifting subpulse phenomena in vacuum gap is due to @xmath118 . unfortunately , these theoretical calculations gave higher drifting rate with respect to observations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? since it has been observed @xcite , the drifting subpulse phenomenon remains unclear which has been widely regarded as one of the most critical and potentially insightful aspects of pulsar emission @xcite . the psg mechanism ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) could be a way to understand lower drifting rates observed , but some complexities still exist which make the underlying physics of drifting subpulses keep complicated and far from knowing clearly . ( 1 ) in principle the drifting velocity of subpulses is the ratio of the drifting distance to the duration , while the expected velocity predicted by @xmath118 is only for electrons in separated emission units , namely the plasma filaments . these two velocities would not be the same if the plasma filaments may stop after sparking . when sparking starts , the electric field in the vacuum gap vanishes due to screen by plasmas ; while sparking ends , the electric field appears again . thus , the calculated drifting velocity with @xmath118 could be higher than that of observations . ( 2 ) the so - called aliasing effect : as one observes subpulses only once every rotation period , we can hardly determine their actual speed . the main obstacles in the aliasing problem are the under sampling of subpulse motion and our inability to distinguish between subpulses especially when the differences between subpulses formed by various subbeams are smaller than the fluctuations in subpulses from one single subbeam @xcite . anyway , detailed studies are very necessary in the future works . we assume that the potential energy related to eq . [ ugp ] , @xmath23 , to be the constant , @xmath17 , in eq . this assumption could be reasonable . for an uniformly magnetized , rotating conductor sphere , the unipolar generator will induce an electric field which is a function of polar angle , as described in eq . [ ugp ] . in the case of @xmath119 ( fig . [ antipulsar ] ) , the potential energy of electron is highest at the polar region which means that those electrons there could be easier to escape . alternatively , this conclusion could be quantitatively understood as following : because of lorentz force inside a star , more electrons locate at the polar region so that the fermi energy of electron is higher there and easier to escape into magnetosphere . we thank dr . kejia lee and other members in the pulsar group of peking university for their helpful and enlightened discussions . we also thank prof . janusz gil for his helpful comments and suggestions . junwei yu is grateful to dr . caiyan li for her helpful assistance . the work is supported by nsfc ( 10973002,10935001 ) , the national basic research program of china ( grant 2009cb824800 ) and the john templeton foundation . melikidze , g. & gil , j. 2009 , the eighth pacific rim conference on stellar astrophysics ed b soonthornthum , s. komonjinda , cheng , k. s. & leung k. c. ( san francisco : astronomical society of the pacific ) 73 , 131
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in ruderman & sutherland ( rs75 ) model , the normal neutron stars as pulsars bear a severe problem , namely the binding energy problem that both ions ( e.g. , @xmath0fe ) and electrons on normal neutron star surface can be pulled out freely by the unipolar generator induced electric field so that sparking on polar cap can hardly occur . this problem could be solved within the partially screened gap ( psg ) model in the regime of neutron stars . however , in this paper we extensively study this problem in a bare strange quark star ( bss ) model .
we find that the huge potential barrier built by the electric field in the vacuum gap above polar cap could usually prevent electrons from streaming into the magnetosphere unless the electric potential of a pulsar is sufficiently lower than that at infinite interstellar medium .
other processes , such as the diffusion and thermionic emission of electrons have also been included here .
our conclusions are as follows : both positive and negative particles on a bss s surface would be bound strongly enough to form a vacuum gap above its polar cap as long as the bss is not charged ( or not highly negative charged ) , and multi - accelerators could occur in a bss s magnetosphere .
our results would be helpful to distinguish normal neutron stars and bare quark stars through pulsar s magnetospheric activities .
[ firstpage ] magnetospheric activities pulsars : general
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in recent years , cosmology has converged on a `` concordance '' model that continues to meet with remarkable success wherever new observational tests are introduced . the direct data span a large dynamic range of structural scales , including power spectrum probes over about a factor of @xmath0 in linear comoving scale . the microwave background spectrum and the products of cosmological nucleosynthesis confirm basic model assumptions and microphysics back to the first fraction of a second of the big bang . an important part of the standard scenario is inflation . the generic inflationary predictions of near global flatness and nearly scale - invariant , gaussian fluctuations are well confirmed by the data . we thus seem to have some direct information , on very large comoving scales , dating from some very early time , prior to the terascale ( or tev energies ) , when the hubble length was less than about a millimeter and the hubble time less than a few picoseconds . at the same time , advances in string theory have opened up a wide range of new possibilities for the initial conditions and the behavior of the universe at early times . seven extra dimensions seem mandatory for consistency ; they must be compactified , or gravity must be localized by some other means ; depending on the order of events , these possibilities may require extensions to the usual framework of inflation based on field theory in 3 + 1 dimensions . initial conditions , and even apparently fundamental parameters such as the shapes of effective potentials in the field theory lagrangian , may be strongly shaped by anthropic selection from a string - theory `` landscape '' , changing the character of arguments about what is natural for inflationary scenarios . branes of various dimensionalities introduce new behavior not present in a system of fields in 3d space , including new possibilities for inflation itself , for the generation of fluctuations , and for new kinds of macroscopic defects such as cosmic superstrings . such a large range of new possibilities can be introduced because the observational tests of concordance cosmology can not directly probe a very large swathe of early cosmic history . viewed in terms of logarithmic scales in energy space or cosmic scale factor , the unexplored region of our past light cone greatly exceeds the explored region . for example , information from particle content only survives from the decoupling of the relevant species ; for most dark matter candidates , this is well below a gev in temperature . the baryon content is a presumed relic of the pre - tev period , which sets the only firm lower limit on the end of inflation . and while it is likely true that the largest - scale perturbations date from very early , many ( possibly as many as about 27 ) orders of magnitude of expansion before the end of inflation , the direct data on these cover only a relatively brief period and a few orders of magnitude of comoving scale . we have almost no direct data on what happened during about 30 orders of magnitude of cosmic expansion ; for almost all of this period , we do not even have solid theoretical arguments to believe that we understand the dimensionality of space . advances in detection of high frequency gravitational waves are starting to provide a way to explore observationally this very large range of scales , epochs and phenomena not previously accessible . a new generation of ground - based detectors operating in the sub - kilohertz band@xcite , such as ligo , geo , and virgo , are now collecting data at their design sensitivity , and will be significantly upgraded in the years ahead . the space interferometer@xcite lisa , operating in the millihertz band , will have its core technology proven in a test flight in 2009 and will itself fly in about ten years . pulsar surveys are providing larger samples of clean pulsars for timing - based detection at frequencies of inverse years to decades@xcite . together , these techniques are maturing to the point where we will have cosmologically sensitive information over a range of frequencies from kilohertz to nanohertz . since gravitational waves penetrate all of cosmic history at sub - planckian densities , and even propagate some distance into extra dimensions , these techniques will access directly many phenomena that are currently hidden from view . this brief survey represents an introduction and update ; more detail on many of the ideas presented here appear in earlier reviews and surveys@xcite . also , since the emphasis is on classically - generated waves that might be directly detected in the near future ( and not on cosmic background techniques ) , inflationary quantum perturbations are little discussed , even though they dominate calculations in the cosmological literature ; direct detection of these waves might occur in the more distant future@xcite . an informative plot of all cosmic history is a `` redshifted hubble frequency diagram '' , shown in figure 1 . the inverse apparent horizon size at a scale factor @xmath1 , or hubble frequency @xmath2 , is redshifted today to a lower frequency , @xmath3 . we plot this observed frequency as a function of @xmath1 ( in units where the current scale factor @xmath4 ) . several simple scalings are : @xmath5 or slow roll inflation corresponds to @xmath6 ; matter - dominated evolution gives @xmath7 ; radiation - dominated evolution gives @xmath8 . a standard inflationary history with inflation at the gut scale is shown as the uppermost solid line . inflation ending at much lower energy is shown as other solid lines . the upper dashed lines show a reference universe that stays permanently radiation - dominated.@xcite for reference , note that a frequency of about 0.1 millihertz today corresponds to the terascale a temperature of about 1tev , and an apparent horizon size of about 1 mm : @xmath9 [ fig : figure1 ] the cross - hatched regions show areas about which we currently have at least some fairly direct data . at the lower right , we have the universe since nucleosynthesis ; at the lower left , standard gut scale inflation mapping onto the cmb anisotropies . the diagram has the virtue of expanding much intermediate history over a wide range of scales . the figure can be regarded as applying to extra dimensions as well , very early when these may be in play , as long as they can be described by a single scale factor . horizontal lines in this plot show constant observed frequency ; the universe lines cross these at observed hubble frequencies in a given model ; above this corresponds to frequencies within the horizon ( that is , less than @xmath10 ) at a given @xmath1 . lines are shown corresponding to frequency bands for ground - based interferometers , space - based interferometers , and pulsar timing techniques . the shaded regions correspond to areas where directly observable cosmic gravitational wave backgrounds might be generated . those to the right of the peak correspond to classical processes ; those to the left correspond to quantum processes . all the current probes revealing the character of relic structure via cmb , large scale structure , and quasar absorption extend only four or five orders of magnitude up from the bottom of the graph . constraints on cosmic history from the cmb spectrum and nucleosynthesis extend farther , back to weak decoupling . the formation of wimp - type dark matter and neutrino decoupling happen at about the same time , and axion dark matter condenses at the qcd epoch . from earlier times , microscopic information tends to be thermalized , apart from a few conserved quantities ; indeed , perhaps even baryon number does not survive from far above the terascale . events in the top two - thirds of the figure , including most of cosmic expansion history ( in log units ) , are currently invisible . that allows for a very wide range of possible new physics . gravitational waves have the potential to open up much of this unexplored region to direct observation . they will at least constrain models in a meaningful way , and may reveal new sources of gravitational waves . there is a possibility of obtaining detailed information about cosmic activity on mesoscopic scales , and about what physics was up to during those `` invisible ages '' . [ fig : figure2 ] = 3.5 in a schematic spectrum of the gravitational wave background is shown in figure 2 . an optimistic estimate of conventional inflationary waves is shown , saturating the current upper limit on tensor / scalar ratio , little damping , and a relativistic equation of state . the current limit on the tensor / scalar ratio limits these to about @xmath11 of the cmb energy density , but in many inflation models it is much less@xcite ; roughly speaking , the metric perturbation on the horizon is of the order of the inflationary expansion rate in planck units . it is not that surprising that the backgrounds are weak , since the production process corresponds to about a single graviton quantum per hubble 4-volume during inflation ; the occupation numbers are still at best of the order of unity when they arrive at our detectors . ( however , note that there are less conventional versions of inflation , particularly pre - big - bang scenarios , with quite different and sometimes quite intense predicted spectra at high frequencies , again illustrating how much there is to learn from the gravitational wave backgrounds ; see for example@xcite ) . the new classical sources discussed here are potentially much stronger in certain ranges of frequency ; the natural scale of energy density for these is the cmb density , times inefficiency factors which are smaller than unity by a factor depending on powers of the scale relative to the hubble length . note that lisa can detect backgrounds down to about six orders of magnitude less energy than the relativistic plasma , so it can detect sources with net gravitational radiation efficiency as small as about @xmath12 . more explanation of various parts of this figure can be found in reviews and summaries cited above ; a more detailed discussion for lisa in particular , exploiting the use of sagnac calibration for extracting broad band backgrounds , is in@xcite . during an early first order phase transition , the universe `` boils'' it nucleates bubble of new phase , and the growth of bubbles converts internal energy ( the latent heat of transition ) to relativistic flows on the nucleation scale . for a strong transition the separation between nucleated bubbles is macroscopically large , leading to flows that are coherent on scales about one to two orders of magnitude smaller than the horizon size , a dimensionless ratio which determines the characteristic peak frequency and radiation efficiency . the flows generate modestly relativistic bulk turbulent velocities and accelerate fluid mass , leading to generation of gravitational waves . the kinetic energy dissipates to smaller scales in a turbulent cascade , creating a power law of higher frequency radiation . gravitational wave spectra were estimated for qcd@xcite and electroweak@xcite phase transitions during the 1980 s . these papers estimated basic parameters such the characteristic frequency and intensity of the backgrounds , which depend mainly on the critical temperature and the degree of supercooling ( determined by the latent heat of transition and the nucleation process , e.g. @xcite ) . further work during the 1990 s down to the present @xcite made more detailed inroads into accurate models of the spectra , although these are still not realistic or definitive , as one might expect given the difficulties in understanding turbulent processes . figure 3 shows an enlarged view of the lisa band , with a typical peak frequency and amplitude shown . note the importance of understanding the astrophysical foregrounds in this frequency range with similar confusion - limited spectra , especially galactic and extragalactic dwarf binaries@xcite ; this will likely be the key limiting factor of the sensitivity to high redshift transitions @xcite . progress has continued on theoretical understanding of the physics of the transition itself.@xcite recent interest is especially prompted by the approach of real data on the terascale soon to come from cern s large hadron collider . lisa s band is well tuned to phase transitions at lhc scales since its peak sensitivity corresponds to about a tenth of the horizon scale at 1 tev . it has been proposed that the elecroweak transition is responsible for the departure from equilibrium that brought about baryogenesis ; these scenarios could be directly confirmed by a lisa detection . string theory requires many extra dimensions . the sizes of the dimensions , their shapes , and how they are stabilized are unknown . all we really know is that for string theory to be right , the extra dimensions must be small , or their radii of curvature must be small , so that they do not appear today in direct tests of the gravitational inverse - square law at submillimeter scale@xcite . ( the scales probed by standard model fields are of course much smaller than this , but they might be confined to a 3-brane living in a larger dimensional space . ) since it turns out the hubble length at the terascale is also about a millimeter , the current threshold of ignorance happens to be about the same in the laboratory gravity , particle / field , and cosmological realms . thus we find lab experiments , accelerator physics , and lisa cosmology converging on the same new regime in very different ways . it is even possible that new properties of gravity on this scale are related to cosmic dark energy ( whose energy density is about @xmath13 ) . the formation of extra stable dimensions introduces sources of free internal energy that might be released coherently on a macroscopic scale.@xcite moreover it introduces another kind of mechanism for generating gravitational waves : motion and curvature of our standard model brane in the extra dimensions . at the time of the localization of the graviton by curving extra dimensions , the position of our brane might be a random variable above the horizon / curvature scale , and in general , spatially inhomogeneous modes would be excited by the kibble mechanism . ( note that most braneworld scenarios concentrate instead on generation of quantum noise in an already - stabilized brane system ; see e.g. @xcite ) alternatively is also possible that the new scales are all much smaller than the hubble length when the stabilization occurs . in this case , the behavior of the extra dimensions can be described by a scalar order parameter as a function of 3 + 1d position , and the effect in 3 + 1d spacetime is similar to the phase transitions just discussed . there is no mandatory reason to assign an extremely high redshift to inflation . reheating ends as late as the terascale in many scenarios ( particularly those braneworlds where the planck scale is not far above the terascale ) , and could have ended as late as the qcd epoch or even electroweak decoupling without directly affecting current cosmological observables . if we are fortunate , it might have ended with enough classical gravitational noise to be accessible to direct gravitational wave detection . there is also no fundamental reason to assume a very quiet reheating . inflation itself is an extraordinarily coherent behavior of a scalar field ; reheating is a process that eventually converts its internal potential energy into a thermal mix of relativistic particles . in many scenarios ( especially `` hybrid '' ones ) , the conversion begins with macroscopically coherent but inhomogeneous motions that eventually cascade to microscopic scales . quantum coherent processes such as `` preheating '' transform into coherent classical motions which , like the phase transitions discussed above , generate backgrounds of the order of @xmath14 of the thermal plasma density@xcite . ( as with those transitions , the frequency of the background only matches the gravitational wave detectors if the final activity of this type occurs well below the gut scale ) . a closely related effect is the formation of nonlinear horizon - scale domains in axion- or higgs- like fields of very low mass ; in this case the field s internal energy is converted not into radiation , but into miniclusters of cosmic dark matter@xcite . the spectra from the above three sources resemble each other : a broad peak around a characteristic frequency , a power law tail at high frequency from the cascade , and a steep rolloff at low frequency from causality ( the redshifted hubble frequency ) , since low frequencies only come from small velocity flows which are inefficient radiators of gravitational waves . the above examples show that lisa is positioned to detect direct evidence of first - order phase transitions , or indeed any significant sharing of internal energy in sub - horizon - scale bulk motions near and somewhat above the terascale . cosmic strings have been studied for many years as a possible new form of mass - energy with new and distinctive astrophysical effects@xcite they were originally conceived in field theory as defects caused by broken u(1 ) symmetries in yang - mills theories . recently they have re - emerged as possible quasi - stable structures of fundamental string theory , sometimes called cosmic superstrings @xcite that tend to arise naturally as u(1 ) symmetries are broken in models of brane inflation @xcite . a tangled net of superstrings forms by a process of kibble quenching after cosmological inflation . the primordial net of long strings continually intersects with itself , spawning isolated , oscillating loops that ultimately radiate almost all of their energy into gravitational waves@xcite . although the fundamental physics differs widely for different types of strings , their quantitative gravitational effects are mainly governed by one fundamental parameter , the dimensionless mass per length or tension @xmath15 ( in planck units with @xmath16 ) . current limits on gravitational wave backgrounds ( from pulsar timing ) already suggest that if superstrings exist , they must be so light that they have no observable astrophysical effects other than their gravitational radiation . figure 4 shows predicted stochastic background spectra@xcite from strings for various values of @xmath15 . the current pulsar limits corresponding to @xmath17 already significantly constrain brane cosmology , and lisa will probe beyond this limit by several orders of magnitude , to @xmath18 . the spectrum from superstrings is clearly distinguishably different from that of phase transitions or any other predicted source : it is nearly flat ( in @xmath19 units ) over many decades at high frequencies , including the range where lisa is likely to observe it . there is a possibility , if the strings are not too much lighter than current limits , that occasional distinctive bursts might be seen from nearby loops that happen to beam gravitational waves in our direction from cusp catastrophes in the loop trajectory@xcite . these rare events , if they are intense enough to stand out above the background , are recognizable in principle from their universal waveform , which derives just from the geometry of the cusps . approaching time @xmath20 from the moment of the catastrophe , in units given by the fundamental mode of the loop , the metric strain amplitude due to radiation beamed from a cusp varies like @xmath21 , and is beamed within an angle @xmath22 . 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forthcoming advances in direct gravitational wave detection from kilohertz to nanohertz frequencies have unique capabilities to detect signatures from or set meaningful constraints on a wide range of new cosmological phenomena and new fundamental physics .
a brief survey is presented of the post - inflationary gravitational radiation backgrounds predicted in cosmologies that include intense new classical sources such as first - order phase transitions , late - ending inflation , and dynamically active mesoscopic extra dimensions .
lisa will provide the most sensitive direct probes of such phenomena near tev energies or terascale .
lisa will also deeply probe the broadband background , and possibly bursts , from loops of cosmic superstrings predicted to form in current models of brane inflation .
address= astronomy and physics departments , university of washington , seattle , washington 98195 - 1580
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You are an expert at summarizing long articles. Proceed to summarize the following text:
from the history of fundamental science everybody knows that cosmic rays physics as a part of astrophysics and particle physics especially based on accelerator studies have many common roots . in particular , many discoveries early in particle physics have been done in the study of cosmic rays . it is enough to remind that the researches in cosmic rays resulted in the discovery of such elementary particles as the positron @xmath0 in 1932 , the muon second charged lepton @xmath1 in 1937 , the charged and neutral pions @xmath2 , @xmath3 , the strange particles kaons @xmath4 , @xmath5 , @xmath6 and @xmath7-hyperon in 1947 , the antiproton , @xmath8 and @xmath9 in 1952 - 1955 . in the very beginning of the second half of xx century a period of divergence between cosmic rays physics and particle physics , both in methodology and in the places of interest , has been started . particle physicists have taken the path of building the big accelerators and large detectors . the experiments at the serpukhov accelerator , the isr and the s@xmath10s at cern , the tevatron collider at fnal allowed to learn the hadron interaction properties at high energies . the accelerator experiments together with theoretical efforts resulted in the construction of the `` standard model '' with clear understanding and power predictions at least in the electro - weak sector and with a number of new open questions . it is believed that new collider experiments such as the lhc project at cern and others might help to find the answers to the open questions in particle physics . at the same time it is quite clear that the new measurements at the accelerator experiments would be of great importance for cosmic rays physics . this is because high energy cosmic rays are usually measured indirectly by investigating the air showers they produce in the atmosphere of the earth . a correct interpretation of the air shower measurements with a necessity requires an improved understanding of the hadron interaction properties , explored by the accelerator experiments . there is a clear necessity to measure at accelerators the global characteristics of the high energy hadron interactions with a high accuracy , in order to accurately interpret the existing and newly data on the measurements of the highest energy air showers . certainly , this request , addressed to the physicists working on the hadron accelerators , is clearly and strongly motivated . the progress in cosmic rays physics during `` accelerator era '' has been less substantial compared to particle physics . probably the main reason of that is owing to above mentioned divergence between two branches in fundamental science . however , the recent results in cosmic ray studies and new astrophysical observations open a new page in particle physics . one of the most interesting results is the detection of cosmic ray particles with energies exceeded @xmath11ev . @xcite is the first article where the detection of a cosmic ray with energy @xmath12 ev has been published ( the volcano ranch experiment ) . at present time the total number of detected air showers with energy higher than @xmath12 ev is about 20 @xcite . the existence of cosmic ray particles with energies above @xmath13 ev has been confirmed by all experiments , regardless of the experimental technique used . why this result is so interesting for particle physics ? it is well known that soon after the discovery of the cosmic microwave background radiation ( cmbr ) by penzias and wilson @xcite almost simultaneously greisen in the usa @xcite and zatsepin&kuzmin @xcite in the ussr predicted that above @xmath12 ev the cosmic ray spectrum will steepen abruptly ( gzk effect ) . the cause of that catastrophic cutoff is the intense isotropic cmbr which is really a 2.7k thermal blackbody radiation produced at a very early stage of the universe evolution and confirmed by measurements of roll and wilkinson @xcite . cmbr photons with a 2.7k thermal spectrum fill the whole universe with a number density of @xmath14 400 @xmath15 . the physical mechanism of the gzk cutoff is quite clear , it is based on the interactions of ultra - high energy cosmic ray ( uhecr ) particles with cmbr photons . protons , photons , electrons , neutrinos , nuclei etc might be as such uhecr particles . mainly the process that cause the energy loss of uhecr particle , say proton , is the photo - production of pions on cmbr photons : @xmath16 . the threshold center of mass energy for photoproduction of one pion is @xmath17 1073 mev . in the cosmic rest frame ( crf , defined as the frame in which the cmbr is represented as isotropic photon gas ) one can estimate the proton threshold energy for pion photoproduction @xmath18 where @xmath19 is the proton energy in the cosmic rest frame . taking the average energy @xmath20 ev , the proton threshold energy is @xmath21ev . on the other hand , in the rest frame of a cosmic ray proton ( prf projectile rest frame ) a substantial fraction of the cmbr photons will look as @xmath22rays with energy above the threshold energy for pion photoproduction @xmath23 where @xmath24 is the cmbr photon energy in the projectile rest frame . the photoproduction cross section as a function of projectile photons for stationary protons is very well measured and studied at accelerator experiments @xcite . there is a detail information that is shown in fig . 1 . at low energies the cross section exhibits a pronounced resonance associated with the @xmath25 decaying into @xmath26 mode ; here the cross section exceeds 500 @xmath27b at the peak . the complicated range beyond the @xmath25resonance is essentially dominated by the higher mass resonances associated with multiple pion production @xmath28 . the whole resonance range is followed by the long tail with approximately constant cross section about 100 @xmath27b with a slow increase up to 1 tev . the photo - pion production cross section for neutrons is to a good approximation identical . as seen from fig . 1 at half - width of the @xmath25 resonance peak the total cross section @xmath29 300 @xmath27b = 3@xmath30 . taking into account the number density of the cmbr photons @xmath31 400 @xmath15 , for the mean free path @xmath32 in the cmbr photon gas one obtains @xmath33 the next important parameter is the proton inelasticity @xmath34 defining the fraction of energy that a proton loses in one collision . at threshold in each collision protons lose about 18% of their energy , and this energy loss fraction increases with increase of energy . in that way the energy loss length @xmath35 is estimated about a few tens mpc for protons with energy higher than @xmath13 ev . apart from photo - pion production , the process of pair production @xmath36 should be considered as well . while this process has a smaller threshold by a factor @xmath37 , it has a smaller cross section . thus , the estimated energy loss length @xmath38 is about @xmath39 mpc . nevertheless , this process might be important at sub gzk energies . the detailed analysis of the energy loss length of protons in interactions with the cmbr photons is presented in fig . 2 . for neutrons with @xmath40 ev , the dominant loss process is @xmath41decay @xmath42 . the neutron decay rate @xmath43 , with the laboratory lifetime @xmath44 sec , gives the neutron energy loss length @xmath45 obviously , uhecrs nuclei are expose to the same energy loss processes as uhecrs protons . so that , the respective threshold for the photo - pion production reaction , in particular , @xmath46 , is given by change of @xmath47 with the mass of the nucleus in eq . [ gzkproton ] @xmath48 from eq . [ gzknucleus ] it follows that the gzk cutoff energy for nuclei is shifted to larger values for heavier nuclei . however , it turns out that the dominant energy loss process for nuclei is photodisintegration , typically @xmath49 , happening due to giant resonances at about the same primary energy . in fact , here we have another excellent example of intersection between nuclear&particle physics and high - energy cosmic rays physics . recent detailed studies reveal that photodisintegration for nuclei leads to energy loss length of @xmath1410 mpc at energy @xmath50 ev , which is comparable with the energy loss length for nucleons . well , to resume the gzk effect physically means that isotropic cmbr photon gas makes the universe opaque to uhecrs particles whose energy is greater than @xmath13 ev . in terms of the energy loss length the gzk cutoff looks like a suppression of the uhecrs flux due to restriction of the propagation distance to a few tens of mpc . in that sense a notion of the gzk sphere arises : simply it is a sphere with the radius @xmath51 mpc within which a source has to locate to be the origin of the uhecrs particles with energy @xmath52 ev . as mentioned above the cosmic ray particles with energies exceeded @xmath13 ev have been detected . the data on the uhecrs spectra measured by fly s eye , agasa , hires i , and hires ii collaborations are collected and shown in fig . 3 . extracted from ref . @xcite ( see references therein ) . as seen from fig . 3 the combined uhecr spectrum does not exhibit the gzk cutoff at all , many events with @xmath53 ev have been observed . the strongest evidence for trans - gzk events comes from the agasa observations . the agasa group reported the detection of up to 17 events with energy @xmath54 ev and claimed that gzk cutoff effect is not observed . now a non - observation of the gzk effect is known as the gzk puzzle . of course , the gzk puzzle results in uncommonly profound consequences , raising questions to the nature of the primary uhecrs particles and their sources as well as the physical mechanisms responsible for endowing cosmic ray particles with such enormous energies or even to the particle physics in itself . that s why the gzk puzzle and all around of that are the targets of wide discussions in the literature at present time . many ideas and different models as solutions have been suggested @xcite , however , the true solution of the gzk puzzle is unknown so far . looking at figure 3 , one can see that the uhecr spectrum exhibits a dip structure at the energy about @xmath13 ev , i.e. the spectrum really has a minimum just at the gzk cutoff energy . thus the gzk puzzle ( irrespective of the source and accelerating mechanism for cosmic rays particles ) is transformed into the questions : what is the origin of this minimum , and how could one explain an appearance of the minimum in the uhecr spectrum in the framework of fundamental dynamics in particle physics . here we suggest a conjecture that the minimum in the uhecr spectrum might be related with nontrivial structure of the inelastic defect of total cross sections in scattering from nuclei . this point will be discussed in a more detail below . in the middle of xx century experimental and theoretical studies of high - energy particle interaction with deuterons have shown that even in the range of asymptotically high energies the total cross section in scattering from deuteron can not be treated as a simple sum of the proton and neutron total cross sections . glauber was the first to explain why a simple addition of the elementary free nucleon cross sections might be failed . using the methods of diffraction theory , the quasiclassical picture for scattering from composite systems and eikonal approximation for high - energy scattering amplitudes , he found fifty years ago @xcite that the deuteron total cross section can be expressed by the formula @xmath55 where @xmath56 here @xmath57 are the total cross sections in scattering from deuteron , proton and neutron , @xmath58 is the average value for the inverse square of the distance between the nucleons inside a deuteron , @xmath59 is the glauber shadow correction describing the effect of eclipsing or the screening effect in the recent terminology . the glauber shadow correction has quite a clear physical interpretation . this correction originates from elastic rescattering of an incident particle on the nucleons in a deuteron and corresponds to the configuration when the relative position of the nucleons in a deuteron is such that one casts its shadow " on the other . it is a genuine analog of the effect known for astronomers ; the decrease in luminosity of binary star systems during eclipses @xcite . soon after it was understood that in the range of high energies the shadow effects may arise due to inelastic interactions of an incident particle with the nucleons of a deuteron . therefore , an inelastic shadow correction had to be added to the glauber one . a simple formula for the total ( elastic plus inelastic ) shadow correction had been derived by gribov @xcite in the assumption of pomeron dominance in the dynamics of elastic and inelastic interactions @xmath60 where @xmath61 is the deuteron ( charge ) formfactor , @xmath62 is the sum of cross sections of all processes which take place in interaction of incident hadron with the nucleon at fixed transfer momentum @xmath63 . however , it was observed later on that the calculations performed by the gribov formula did not meet the experimental data : the calculated values of the shadow correction overestimated the experimental measurements . in that case the idea , that the pomeron dominance is not justified at the accelerator energies , becomes clear . it was believed sometime that the account of the triple - reggeon diagrams for six - point amplitude in addition to the triple - pomeron ones would allow to obtain a good agreement with the experiment . but careful analysis has shown that discrepancy between theory and experiment could not be eliminated by taking into account the triple - reggeon diagrams : in fact , it is needed to modify the dynamics of the six - point amplitude with more complicated diagrams than the triple regge ones @xcite . this indicates that up to now there is no a clear understanding , in the framework of regge phenomenology , the shadow corrections in elastic scattering from deuteron . the main difficulty , which the regge phenomenology faced with , was the problem to describe the cross section of the single diffraction dissociation processes . the latest experimental measurement of @xmath64 single diffraction dissociation at c.m.s . energies @xmath65 and @xmath66 , carried out by the cdf group at the fermilab tevatron collider @xcite , has shown that the most popular model of supercritical pomeron does not describe the existing experimental data . recent experimental results from hera @xcite lead us to the same conclusion . the soft pomeron phenomenology as currently developed can not incorporate the hera data on structure function @xmath67 at small @xmath68 and total @xmath69 cross section from @xmath67 measurements as a function of @xmath70 for different @xmath71 . such situation might be qualified as a `` super - crisis '' for the supercritical pomeron model . figure 1 extracted from paper @xcite demonstrates the `` super - crisis '' ( see details in ref . @xcite ) . meanwhile it s quite clear that the theoretical understanding of the shielding effects in scattering from any composite system is of fundamental importance , because the structure of shadow corrections is deeply related to the structure of the composite system itself . at the same time the structure of the shadow corrections displays new aspects for the fundamental dynamics . in the second half of 1970th we have concerned in the study of dynamics in three particles scattering in some details ( see recent review article @xcite and references therein ) . the bethe - salpeter - type equations reduced to one time have been used as an implement in our study of a dynamics for the three - body systems . it turned out that the three - body dynamics , under a consistent consideration of three - body problem in the framework of local quantum field theory , with a necessity contained new fundamental forces which the three - body forces are . the three - body forces in relativistic quantum theory appear as an inherent connected part of total three particle interaction which can not be represented by the sum of pair interactions . an existence of the three - body forces might be established even in the perturbation theory expansions . single - time formalism in quantum field theory used allows one to give a constructive definition of the three - body forces beyond the perturbation theory . on this way it was established that the fundamental three - body forces are related with specific inelastic interactions in two - body subsystems of the three - body system , and they govern the dynamics of special inelastic processes known as one - particle inclusive reactions . at the rather common assumptions we managed to calculate the contribution of the three - body forces to the deuteron total cross section and to derive the new , extremely simple and refined formula for defect of total cross section in scattering from deuteron with clear and transparent physical interpretation . the obtained structure of the shadow corrections to the deuteron total cross section has revealed new fundamental scaling laws @xcite in interaction of composite nuclear systems . here i would like to concern this point in a more detail . in our approach the defect of the deuteron total cross section is represented by the sum of two items @xmath72 where @xmath73 is elastic defect , and @xmath74 is inelastic one . for the elastic and inelastic defects one obtains @xmath75 where @xmath76 as seen from eq . [ partdef ] the elastic defect is proportional to the total elastic cross section , but the inelastic defect is proportional to the total single diffraction dissociation cross section in scattering from nucleon . the proportionality factors @xmath77 and @xmath78 are called the elastic and inelastic structure functions of a deuteron correspondingly . here @xmath79 is the deuteron radius defined by the deuteron formfactor , and scale variables @xmath80 and @xmath81 are defined through the slope of forward diffraction cone in elastic scattering @xmath82 and in single diffraction dissociation @xmath83 which are simply related to the effective radii of two - body @xmath84 and three - body @xmath85 forces . of course , it is supposed , that both at elastic and at inelastic interactions with nucleons of a deuteron , proton and neutron are dynamically indistinguishable , i.e. appropriate dynamic characteristics for a proton and neutron are identical @xmath86 , @xmath87 etc . such assumption is quite justified at enough high energies . structure functions have clear and quite a transparent physical meaning . the function @xmath88 is some kind of `` counter '' , which measures out a portion of events related with elastic rescattering of incident hadron on nucleons of a deuteron among of all the events during the interaction with a deuteron as whole , and this function attached to the total probability of elastic interaction of an incident particle with a separate nucleon in a deuteron . this function depends on a variable , which is effective radius of elastic interaction with a nucleon measured with the help of `` scale rule '' with a scale defined by the radius of a deuteron . at each value of this variable ( at a given value of energy ) the number of the function @xmath88 determines a weight , which the total cross section of elastic interaction with a nucleon at given energy enters the defect of the deuteron total cross section with . the same physical interpretation with obvious changes in the terms is transferred on the inelastic structure function . the function @xmath89 also represents some kind of `` instrument '' , but another , which count out a relative portion of other events among of possible interactions with a deuteron as a whole related with processes of inelastic interaction with nucleons inside a deuteron of inclusive type in the region of diffraction dissociation . the inelastic structure function depends on another scale variable , which is effective radius of inelastic interaction with a nucleon measured with the help of `` scale rule '' with the same scale defined by the radius of a deuteron . the number of the function @xmath89 at given value of energy determines a weight , which the total single diffraction dissociation cross section on a nucleon at the same energy enters the defect of the deuteron total cross section with . formulas ( [ defect ] ) and ( [ partdef ] ) may serve as toolkit for experimental study of structure functions @xmath88 and @xmath89 by measurement of the defect for total cross section in scattering from deuteron with usage of the experimental information about elastic cross section and total cross section of single diffraction dissociation on a nucleon . for these purposes , however , it would be extremely important to have a reliable substantiation of these formulas . it is remarkable that such theoretical substantiation can be really obtained . the formalism , which we have used , allowed us to carry out analytical calculations completely , if for these purposes to take advantage of the parameterizations , trustworthy established on experiment , for differential elastic cross sections and for one - particle inclusive cross sections in the range of diffraction dissociation @xmath90 , \qquad \frac{2s}{\pi}\frac{d\sigma_n^{sd}}{dtdm_x^2 } = a(s , m_x^2)\exp[b(s , m_x^2)t].\ ] ] in this way we managed to get the extremely simple formulas for structure functions @xmath77 and @xmath78 , which look like @xmath91 it should be especially emphasized once more an important element in our approach which consists that the inelastic defect in the deuteron total cross section appears as manifestation of the fundamental three - body forces , and at the same the three - body forces determine the dynamics of one - particle inclusive reactions . formula relating the three - body forces amplitude with the one - particle inclusive cross section has been derived as well ; see details in refs . @xcite and references therein . outcome of functions evaluations ( [ a ] ) and analysis of these functions , however , are worthy of separate discussion . at first , being returned to the formula ( [ defect ] ) , we shall remark that the glauber formula is followed if in this formula to neglect inelastic defect and for the elastic structure function to take approximation @xmath92 justified at @xmath93 , and to take into account that @xmath94 . secondly , it is necessary to pay attention that the structure functions @xmath77 and @xmath78 have quite different behavior : @xmath95 is the monotonic ( increasing ) function when argument vary on a semi - infinite interval @xmath96 , and the range of its values is limited to an interval @xmath97 , while the function @xmath78 at first increases , reaches a maximum at @xmath98 , and then decreases , disappearing at infinity , thus the range of its values is an interval @xmath99 . certainly , that such distinction in behavior of structure functions @xmath100 and @xmath101 results in far - reaching physical corollaries . for example , at superhigh energies corresponding @xmath102 , we discover the effect of weakening the inelastic screening i.e. the inelastic defect disappears ( taking into account that @xmath103 ) , the elastic defect tends to doubled value of the nucleon total elastic cross section , and the deuteron total cross section comes nearer to doubled value of the nucleon total absorption cross section . therefore , at superhigh energies the @xmath104dependence of the total cross sections in scattering from nuclei should be recovered with that only by odds that the fundamental value , standing at @xmath104 , is not the nucleon total cross section but the nucleon total absorption one . this means that the total absorption ( inelastic ) cross section manifests itself as a fundamental dynamical quantity for the constituents in a composite system at superhigh energies . of course , without any doubt , matching of the obtained theoretical outcomes with available experimental data on total cross sections in scattering of protons and antiprotons from deuterons represented for us the special interest . in figures 5 and 6 the preliminary results of such matching are shown . the curves in these figures correspond to the total cross sections in scattering of protons and antiprotons from deuterons calculated by the formulas ( [ sigmad ] , [ defect ] , [ partdef ] , [ a ] ) . there the global descriptions of @xmath105 and @xmath106 total cross sections ( see figs . 7,8 ) as well as of total single diffraction dissociation cross section in view of the latest experimental data obtained by cdf collaboration at fnal @xcite , made by us earlier @xcite , have been used . besides in the given occasion it should be necessary still to emphasize , that the matching with experimental data on total cross sections in scattering of protons and antiprotons from deuterons was carried out , as it were , in two stages . at the first stage the theoretical calculations were compared to experimental data on the antiproton - deuteron total cross section on the supposition , that @xmath107 is the single free parameter , which value should be determined from the fit to experimental data . as a result of a statistical analysis the following value for the @xmath108 was obtained : @xmath109 . here pertinently to pay attention to the following circumstance . the last experimental measurements of the deuteron matter radius testify @xmath110 @xcite whence follows that @xmath111 . the obtained value for @xmath112 satisfies equation @xmath113 . for entirety , the outcomes of theoretical calculations are represented in fig . 7 up to energies of tevatron at fnal . at the second stage the experimental data on the proton - deuteron total cross section were compared to theoretical calculations , in which the value of @xmath108 was fixed on the numerical value , which was obtained at the first stage from the analysis of the data on @xmath114 total cross section . in other words , the curve in fig . 8 corresponds to theoretical calculations made with the help of the formulas ( [ sigmad ] , [ defect ] , [ partdef ] , [ a ] ) , in which there was no free parameter . in this figure the outcomes of theoretical calculations are also represented up to energies of tevatron at fnal . as is seen , the figures 7 and 8 testify to the excellent agreement between the theory and experiment . in addition figures 9 and 10 demonstrate our global description of the proton - proton total cross section from the most low energies up to energies reachable in cosmic rays . in figure 11 the outcomes of the theoretical calculations of elastic and inelastic defects of total cross section in scattering of ( anti)protons from deuterons in energy range @xmath115 gev have been depicted . it follows from these calculations that the value of elastic defect makes about 10% from the value of nucleon nucleon total cross section , and the value of inelastic defect makes about 10% from the value of elastic defect i.e. approximately 1% from the value of nucleon nucleon total cross section . figuratively expressing , it would be possible to tell that if the elastic defect represents a fine structure of total cross section in scattering from deuteron then the inelastic defect should be referred to a hyperfine one . in our approach the inelastic defect is related to manifestation of fundamental three - body forces , therefore in this sense the three - body forces play a role of `` fine tuning '' in the dynamics of the relativistic three - particle system . it is necessary to render homage to the physicists - experimenters creating setups with the accuracy of measurements permitting to discriminate inelastic defects in total cross sections of particles scattering at high energies . in this connection the further experimental precise measurements of the hadron deuteron total cross sections at high energies seem to be extremely important as it was already mentioned above the maximum value of inelastic defect is achieved at @xmath116 ( @xmath117 ) . or else , the value of energy corresponding to maximum value of the inelastic defect is defined from the equation @xmath118 . the calculations made in view of our analysis of existing experimental data give @xmath119 . reevaluating c.m . energy @xmath120 to the lab . system one obtains @xmath121 . it is obvious , that such values of energies are not accessible on current and design accelerators . however it would be extremely interesting to look for manifestations of the given effect in phenomena related with extremely high energy cosmic rays . the theoretically calculated inelastic defect in the region of a maximum is shown in figure 12 . as it follows from eq . ( [ sigmad ] ) a maximum of the inelastic defect corresponds to a minimum of the total cross section . we have plotted in figure 13 the ( anti)proton deuteron total cross section scaled by the factor @xmath122 in the region of a maximum of the inelastic defect . the scale factor is selected _ ad arbitrium _ for a goal of illustration only to discern a minimum in the total cross section . let s remark , however , that the value @xmath123 has clear physical meaning , it separates two ranges on energy : the range of energies @xmath124 , at which effective radius of three - body forces does not exceed size of a deuteron or more exactly @xmath125 , and the range of energies @xmath126 , at which effective radius of three - body forces becomes more than size of a deuteron @xmath127 . the existence of boundary @xmath128 , since which there is a suppression of inelastic defect , seems to be the extremely important characteristic of fundamental dynamics . here we would like to make a conjecture that the observed structures in the cosmic rays spectra , in particular a minimum in the uhecr spectrum , might be related with the existence of such boundary . from the glauber formula ( [ deltasigmag ] ) it follows , that with decrease of inter - nucleon distance in a deuteron the value of elastic defect grows . but the configurations with small inter - nucleon distances in a deuteron are most favorable for a manifestation of purely three - particle interaction . when effective interaction radius of an incident hadron with a nucleon becomes comparable with inter - nucleon distance , the pattern of elastic rescattering on nucleons of a deuteron ceases to be adequate to complete pattern of interaction with a deuteron . in this case it is also necessary to take into account purely three - particle forces . it is obvious , that in a deuteron the configurations are dynamically probable , when the nucleons are close from each other , but the glauber theory does not allow to take into account such configurations . account of such configurations demands a more detailed study of the dynamics of processes of scattering from a deuteron . the technique of the dynamic equations in a quantum field theory , which we have used , just allows to carry out such detailed investigations . once again it should be emphasized , that the important role in our researches was assigned to conceptual notion of fundamental three - body forces which with necessity arise by consistent consideration of the dynamics of three particles system within the framework of a relativistic quantum theory . the relation of fundamental three - body forces with dynamics of one - particle inclusive reactions represents the important outcome obtained , as it were , by the way . this outcome especially is important , that can form the basis both for elaboration of methods of analytical calculations and for a different sort of phenomenological analysis . the comparison of the theory with experimental data on ( anti)proton deuteron total cross sections made shows , that for the description of particles scattering from a deuteron at high energies it is enough to take into account only nucleon degrees of freedom in a deuteron . the weakly bounded two - nucleon system the deuteron looks so , that the clusterization of quarks in nucleons is not broken even then , when the nucleons approach closely to each other . nucleons , being close from each other in a deuteron , do not lose of the individuality and consequently there is no necessity to introduce the six - quark configurations depersonalized in a deuteron . the structure derived for the defect of total cross section in scattering from a deuteron corresponds to such pattern . we managed to show , that the general formalism of quantum field theory admits a possibility of representation of dynamics of a particle scattering from composite system through the fundamental dynamics of a particle scattering from isolated constituents and structure of the composite system itself . though the dynamics of a particle scattering from two - particle bound system a deuteron was considered in details , the general formalism used admits a natural generalization and extension to more complex multiparticle compound nuclear systems . certainly , the complexity of consideration , at that , substantially increases . really , to consider the problem of scattering from nucleus consisting of @xmath104 nucleons we have to solve many - body problem for ( @xmath129)particle system . however , instead of solving this very complicated problem one could use a powerful reduction method . for this goal let s consider a nucleus consisting of @xmath104 nucleons as a bound system of one nucleon and nucleus consisting of ( @xmath130 ) nucleons . in that case the problem of scattering from a nucleus with @xmath104 nucleons is reduced to the problem of scattering from a two - body bound system which has been previously solved . of course such supposition is not unique and should be considered as a some sort of simplification . in the other way one could suppose that a nucleus consisting of @xmath104 nucleons may be represented as a two - body bound system of a nucleus with @xmath131 nucleons and other nucleus with @xmath132 nucleons so that @xmath133 . by this way the problem of scattering from a nucleus with @xmath104 nucleons is also reduced to the problem of scattering from a `` deuteron '' previously solved . anyway continuing a reduction in both cases we will come at a final stage to the expression for total cross section in scattering from a nucleus in terms of fundamental dynamics in scattering from a nucleon and the structure of a nucleus . formula ( [ sigmad ] ) for total cross section in a case of scattering from any nucleus with @xmath104 nucleons can be rewritten in the form @xmath134 where @xmath135 is the total cross sections in scattering from nucleon . the defect @xmath136 , in general , also contains two parts as in eq . ( [ defect ] ) @xmath137 where @xmath138 is elastic defect , and @xmath139 is inelastic one . from general point of view , as presented above , for the elastic and inelastic defects one can write @xmath140 where @xmath141 the functions @xmath142 and @xmath143 are called the elastic and inelastic structure functions of a nucleus . here we have included the combinatorial factors in the definition of the structure functions . @xmath144 is the nucleus radius defined by the nucleus formfactor . the scaled variables @xmath145 and @xmath146 are defined as @xmath80 and @xmath81 above but with another scale factor @xmath144 which is the radius of a nucleus as it should be . it is obviously that formulas ( [ defectnucl ] ) and ( [ partdefnucl ] ) may serve as a base to experimentally study the structure functions @xmath142 and @xmath143 by measuring the defect of total cross section in scattering from nuclei with using the experimental information about elastic cross section and total cross section of single diffraction dissociation in scattering from a nucleon . to calculate the structure functions @xmath142 and @xmath143 is a task of any theory or theoretical model . as mentioned above the calculation of the structure functions @xmath142 and @xmath143 in quantum field theory is a very complicated problem however , it would reasonably to use our experience acquired in solution of this problem for deuteron case . really , it seems to a good approximation , one could use the following expressions for the structure functions @xmath142 and @xmath143 of any nucleus @xmath147 here we have applied the identity @xmath148 and supposed that any many - fold rescattering in a nucleus feels one and the same structure function like for two - fold rescattering . it is clear that this supposition is a strong enough simplification , however , it might be precise one at ultra - high energies . at any rate , it would be very desired to test such simple pattern in ultra - high energy cosmic rays . for the effective radius of three - nucleon forces obtained in our previous investigations one can write the following analytical expression @xmath149\ , { \rm gev}^{-2},\quad ( { s_0})^{1/2}=20.74 \,{\rm gev}.\ ] ] then the equation @xmath150 , defining a value of energy at a maximum of the inelastic defect , has an obvious solution @xmath151^{1/2}.\ ] ] further , if we put as @xmath152 , then eq . ( [ smax_a ] ) can be rewritten in the form @xmath153^{1/2}\,{\rm gev},\ ] ] where the value for the deuteron radius @xmath154 mentioned above has been used . from eq . ( [ smax_an ] ) one obtains @xmath155 here , it is interesting to note that the gzk cutoff value @xmath156 gev appears for c.m . energy corresponding to maximum of the inelastic defect in a case of scattering from helium nucleus . the @xmath104-dependence in position of a maximum of the inelastic defect in scattering from nuclei may have a direct link to the question on chemical composition of the cosmic rays . an information about chemical composition of the cosmic rays can be elicited from detailed study of air showers development . it is quite clear , for instance , that air showers produced by heavy nuclei start the development in the earth atmosphere earlier compared to protons as primaries . the cosmic rays composition is often investigated by fitting the energy dependence of the depth into the atmosphere of maximum @xmath157 of the uhecrs - generated air showers . in fact , @xmath157 is the atmospheric depth at which the number of particles in a shower reaches its maximum . this quantity strongly depends on the primary energy and composition , that s why @xmath157 if often considered as the most useful observable of the air showers . at ground array detectors @xmath157 is mainly provided by measuring the muon content or more exactly the ratio of electrons to muons in air shower . in other case , optical fluorescent detectors allow to directly observe air shower development . just to say qualitatively , it should be mentioned that for a given primary energy a heavier nucleus creates air shower with a higher muon content and @xmath158 is higher up in the atmosphere compared to those for a proton - generated air shower . the higher muon content of air shower produced by heavy nucleus can be understood by the fact that it is relatively easier for charge pions to decay to muons before interacting with the medium when the shower develops higher up in the less dense atmosphere . besides , a less energetic pions generated from heavy nucleus have a higher decay probability , therefore the muon fraction is higher in air showers produced by heavy nuclei as well by this reason . this is clearly demonstrated in fig . 14 where the results of the kascade air shower experiment have been fitted , using the qgsjet model generator , with a composition dominated by helium nuclei and smaller contributions of proton , @xmath159o and @xmath160fe . as can be seen from figure 15 , the iron fraction gradually decreases when changing the energy from @xmath161 to @xmath13 ev , but the fraction of lighter nuclei increases in the same interval of energies i.e. there is trend from heavy toward lighter composition in the measurements of @xmath162 . it seems , the recent studies indicate that at the highest energies @xmath163 ev there is a significant fraction of nuclei with charge greater than unity , and less than 50% of the primary cosmic rays can be photons . in another words , the existing experimental data suggest that uhecrs are predominantly protons or light nuclei as for cosmic rays at much lower energies . however , due to poor statistics and large fluctuations from shower to shower the definite conclusions on the composition of the uhecrs have to await data from next generation experiments . here , we would like to emphasize that the simulation of shower development depends on the event generator used containing some model of hadronic interaction which results in further complication of data interpretation . the major uncertainties in air shower simulation stem from the hadronic interaction models which are usually represented by empirical parameterizations , and therefore almost all hadronic models are purely phenomenological . this is because one can not calculate soft hadronic interaction cross sections or hadronic multiparticle production within qcd from first principles . the second major source of uncertainty is the large extrapolation ( over 6 orders of magnitude in energy ) from accelerator experimental data to the uhecrs ones . in this place the reliable model and the precise accelerator data on fundamental , for example hadron - proton , total cross sections as well as on total cross sections in scattering from nuclei are needed to constrain uncertainties in the interpretation of cosmic rays data to accurately determine the energy spectrum and the composition of the uhecrs . additionally , theoretical understanding and description of the diffractive dissociation processes are of special importance for nuclei interactions and consequently for air shower development too . the most popular model which has been used to simulate interactions of nucleons and nuclei is based on regge phenomenology with the super - critical pomeron exchange . however , as mentioned above , this model faced with a serious difficulties in description of single diffractive dissociation in @xmath10 collisions . moreover , the super - critical pomeron model breaks the fundamental principles of relativistic quantum theory such as unitarity , and this fact is often overlooked . but only this pathology of the super - critical pomeron model is enough to reject the model from consideration . recent accurate and complete analysis of experimental data on hadron total cross sections rejects this model from statistical point of view . another , and sometime neglected , source of uncertainties is uncertainty provided by the measurements of @xmath10 total cross sections performed at tevatron . really , the cdf collaboration @xcite obtained @xmath164 mb which is considerably greater than those reported by e710 ( @xmath165 mb ) @xcite and e811 ( @xmath166 mb ) @xcite . such difference in the measurements permits of a wide range of different extrapolations . of course , the arising uncertainty directly transfers to predictions for air showers . namely , the main source of uncertainty of air shower predictions comes from differences in modelling hadronic interactions which can not be eliminated by existing accelerator data . that is why an accurate measurement of the @xmath105 total cross section at lhc is of great importance since it would allow to discriminate the different extrapolations and to make a selection among currently used models . the study of @xmath167 ( proton - nucleus ) collisions for the light nuclei at lhc is also of greatest interest , of course , not only for fundamental particle physics but for air shower physics as well . at the same time it s quite clear , that quantum field theory provides a sound theoretical basis with a definite guidelines how the fundamental interactions evolve with energy . in this respect the discussed here global description of the hadron total cross sections performed in the framework of general structures of local quantum field theory keeps a preferable place . thus an incorporation of the global pattern of hadronic interactions in generally used generators of events would be extremely desired . the next widely discussed subject is the question of origin of cosmic rays . although cosmic ray particles were discovered almost one hundred years ago since the first announcement of their observation in 1912 @xcite , the problem of origin of cosmic rays especially of uhecrs particles has no solution so far . the total cosmic rays spectrum is shown in fig . 16 . the commonly accepted point of view is that at energies below 1 gev the cosmic rays spectrum is dominated by particles coming from the sun because the intensities at such energies are correlated with the solar activity . at higher energies between 1 gev and up to the knee region ( see fig . 16 ) there are several arguments including energetics that an origin of the cosmic rays is outside the solar system but confines yet to the galaxy . at still higher energies between the knee and the ankle , and finally , beyond @xmath168 ev the situation becomes unclear , although the uhecrs are generally expected to have an extragalactic origin due to apparent isotropy , and the ankle is sometimes interpreted as a cross over from galactic to extragalactic component . at any rate , it is generally believed that the bulk of the cosmic rays observed at the earth is of extra - solar origin . here we would like to suggest quite another new idea that the bulk of the cosmic rays observed at the earth is of solely solar origin . in particular , the uhecrs particles coming to the atmosphere of the earth might be produced by reaction @xmath169 where we have used the notations : @xmath170 for galactic or extra - galactic uhecrs particle , @xmath171 for the sun , @xmath172 for uhecrs particle coming to the atmosphere of the earth from the sun . due to reaction ( [ uhecrprod ] ) almost all energy of @xmath170 particle is transferred to @xmath172 particle . the idea is based on the fact that the cross section of reaction ( [ uhecrprod ] ) is in 4 orders greater than the cross section of direct interaction of @xmath170 particle with the earth . of course , we did not concern of what is the source of the uhecrs in the universe , one only claims that uhecrs particles coming to the atmosphere of the earth are produced on the sun . in this respect we would like to remind an old idea suggested in @xcite to consider the virgo cluster as a source of the uhecrs . according to this idea the uhecrs particles , generated in m87 galaxy in virgo cluster , diffuse from the center of virgo in a postulated extragalactic field with the energy dependent diffusion coefficient , and they are focusing to the sun by galactic magnetic field . it is remarkable that there is no gzk cutoff , and there is no large anisotropy in the model . recently a new revival of this very interesting idea has been proposed . introducing a simple galactic wind in analogy to solar wind it has been shown @xcite that back - tracing the orbits of the highest energy cosmic rays events suggests that they may all come from the virgo cluster , probably from the active radio galaxy m87 . figure 17 shows the directions of the cosmic rays events at that point when they leave the halo of our galaxy in polar projection . in fig . 17 the direction to the active galaxy m87 ( virgo a ) , which is the dominant radio galaxy in the virgo cluster , is pointed out as well for reference . the two highest energy events are shown in fig . 17 twice : in assuming ( i ) that they are protons , and ( ii ) that they are helium nuclei ( filled black symbols ) . the shaded band in fig . 17 corresponds to the supergalactic plane . a remarkable observation made in the model calculations is that the directions of all tracks point north @xcite . with exception of two events with highest energy , all other 11 events can be traced to within less @xmath173 from virgo a. considering the uncertainty of the actual magnetic field distribution , it was found that all events are consistent with arising originally from virgo a. besides , if the two highest energy events are really helium nuclei , then all 13 events point within 20 degrees of virgo a. of course , it is very interesting that the simple model for a galactic wind rather similar to the solar wind may allow particle orbits at @xmath13 ev to be bent sufficiently to allow `` trans - gzk '' particles to arrive to the sun from virgo from different directions in agreement with the apparent isotropy in arrival directions . if the model assumptions might be confirmed then all powerful radio - galaxies might be considered as sources of the uhecrs . in that case the fantastic idea arises to use the powerful radio - galaxies as gigantic accelerators to set up particle interaction experiments in the sky @xcite . the sun may be used as a target , the atmosphere of the earth as a calorimeter to detect the highest energy events . we did not intend in this article to present a full understanding the cosmic rays observations . really we have only concerned very special but at the same time quite intriguing observation related to absence ( gzk puzzle ) of the predicted catastrophic cutoff ( gzk effect ) of the uhecrs spectrum at energy value about @xmath13 ev . without any doubt , an observation of a significant flux of uhecrs particles with energy above the expected gzk cutoff value is of great interest , and many attempts have been undertaken to explain the existence of such particles . as mentioned above an explanation of these particles requires the existence of extremely powerful sources within so called gzk sphere with the radius about a few tens mpc . there is evidence that such powerful radio - galaxies may really exist although this fact should be clearly confirmed by future experiments with the higher statistics . it should be fair to note that there is a controversial point of view which means that there are no particles with energy above the gzk cutoff value , but the present results of agasa and others , where such particles have been observed , are artefact of a combination of incorrect energy calibration , larger than predicted fluctuations in shower development , non gaussian tail in measurements etc . we did not touch this point of view at all . but here it should be pointed out the recent article of the hires collaboration @xcite where the hires measurement of the flux of ultrahigh energy cosmic rays with fluorescence technique shows a suppression at an energy of @xmath174ev , exactly the expected cutoff energy . the statistical significance of the break in the spectrum identified with the gzk cutoff is @xmath175 . the measured energy of the cutoff is @xmath176ev , where the first uncertainty is statistical and the second is systematic . at the same time teshima ( for agasa collaboration ) @xcite presented at the international conference on high energy physics ichep2006 ( moscow , russia , july 26august 2 , 2006 ) a new ( preliminary ) agasa reanalysis with recent corsika m.c . in which the number of events above @xmath13ev was reduced from 11 to 5@xmath146 , and the flux difference between agasa and hires became less significant . nevertheless the main conclusion in summary of the talk given by teshima has been unchanged : `` super gzk particles exist '' . this means that the gzk puzzle exists as well . in fact , the new ( preliminary ) agasa data extend beyond the gzk cutoff energy with no apparent suppression and probably without dip and bump . however , it should be noted that the combined agasa&hires data presented in fig . 18 ( fig . 3 from ref . @xcite ) clearly show the dip structure at the gzk cutoff energy as mentioned in the introduction . we also did not concern any exotic models to explain the existence of `` trans - gzk '' particles . among them there are the so called `` z - burst '' scenario , top - down models , and even super - exotic explanation due to violation of lorenz invariance ; see e.g. excellent review article @xcite and references therein . our conjecture is the attempt to find the solution of the gzk puzzle in the fundamental dynamics of scattering from nuclei . certainly , it needs to do much work else to convert the conjecture into strong statement . at the same time it is expected that next generation experiments in astrophysics , especially in cosmic rays physics , will be able to yield significant new information about fundamental processes in particle physics . here the measurement of the uhecrs spectrum beyond the gzk cutoff is of great importance because such measurements being a grand element of cosmic rays physics has a very deep relation with particle physics . it seems the upcoming researches of particle interactions ( in the sky and on the earth ) promise to represent the near future as an exciting epoch in science when the three branches of science cosmology , astrophysics and particle physics are very probably to be combined into unique common fundamental science such as cosmoastropartphysology or casp - 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59.pdf ; arxiv : hep - ph/0012349 ( 2000 ) ; in proceedings of xvth workshop on high energy physics and quantum field theory , tver , russia , september 7 - 13 , 2000 , eds . m. dubinin , v. savrin , institute of nuclear physics , moscow state university , russia , 2001 , pp . 241 - 257 . f. schmidt - kaler et al . , phys . rev . lett . * 70 * , 2261 ( 1993 ) . arkhipov , _ proton - proton total cross sections from the window of cosmic ray experiments _ , preprint ihep 2001 - 23 , protvino , 2001 ; e - print hep - ph/0108118 ; in proceedings of ixth blois workshop on elastic and diffractive scattering , pruhonice near prague , czech republic , june 9 - 15 , 2001 , eds . v. kundrat , p. zavada , institute of physics , prague , czech republic , 2002 , pp . 293 - 304 . arkhipov , _ on a manifestation of dibaryon resonances in the structure of proton - proton total cross section at low energies _ , preprint ihep 2001 - 44 , protvino , 2001 ; e - print hep - ph/0110399 ; in proceedings of the ninth international conference on hadron spectroscopy , protvino , russia , 25 august-1 september , 2001 , eds . d. amelin et al . , melville , new york , 2002 , aip conference proceedings , vol . 771 - 776 . f. abe et al . , ( cdf collab . ) phys . rev . d*50 * , 5550 ( 1994 ) . amos et al . , ( e710 collab . ) phys . b243 , 158 ( 1990 ) . c. avila et al . , ( e811 collab . ) b445 , 419 ( 1999 ) . hess , phys . z. * 13 * , 1804 ( 1912 ) . cronin , t.k . gaisser , s.p . swordy , sci . * 276 * , 44 ( 1997 ) . j. wdowczyk , a.w . wolfendale , nature * 281 * , 356 ( 1979 ) ; + m. giller , j. wdowczyk and a.w . wolfendale , j. phys . * 6 * , 1561 ( 1980 ) . eun - joo ahn et al . , _ the origin of the highest energy cosmic rays : do all roads lead back to virgo ? _ , arxiv : astro - ph/9911123 ; + nucl . b(proc . suppl . ) * 87 * , 417 ( 2000 ) . abbasi et al . , ( hires collab . ) _ observation of the gzk cutoff by the hires experiment _ , arxiv : astro - ph/0703099 ( 2007 ) ; see also g. thomson talk presented at the international conference on high energy physics ichep2006 , moscow , russia , july 26august 2 , 2006 , available at + http://ichep06.jinr.ru/reports/50_1s4_16p50_thomson.ppt . m. teshima , _ ultra high cosmic rays observed by agasa _ talk presented at the international conference on high energy physics ichep2006 , moscow , russia , july 26august 2 , 2006 , available at + http://ichep06.jinr.ru/reports/49_1s4_16p30_teshima.ppt . p. bhattacharjee , g. sigl , phys . * 327 * , 109 ( 2000 ) ; arxiv : astro - ph/9811011 . vs @xmath177 compared with the predictions of the renormalized pomeron flux model of goulianos @xcite ( solid line ) and of the model gostman , levin and maor @xcite ( dashed line , labelled glm).,title="fig:"][fig4 ] ( 288,188 ) ( 15,10 ) with the cosmic - ray data points from akeno observatory and fly s eye collaboration . solid line corresponds to our theory predictions @xcite.,title="fig:"][fig9 ] ( 144,0)@xmath178 ( -5,95 )
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the conjecture that the gzk puzzle might be related with nontrivial structure of the inelastic defect of total cross sections in scattering from nuclei has been suggested . *
the gzk puzzle and fundamental dynamics * + a. a. arkhipov + _ state research center
institute for high energy physics " + 142281 protvino , moscow region , russia _ +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the avogadro constant , @xmath0 , is determined by dividing the molar volume of a silicon single - crystal highly enriched with @xmath1si by its unit cell volume , obtained from the measured value of the lattice parameter @xcite . the lattice parameter is measured by means of combined x - ray and optical interferometry . recently , the effect of surface stress was brought to our attention as a possible error source in our measurement of the lattice parameter @xcite . indeed , this issue was already taken into account but never definitely investigated @xcite . an intrinsic surface - stress exists in solid , in a similar way as it occurs in liquids , which was first modelled by gibbs @xcite and shuttleworth @xcite , whose works were seminal in surface thermodynamics . though it is widely reported in review papers and textbooks on surface science @xcite , their theory is still a matter of debate @xcite . a continuous mechanics theory of solid surfaces has been developed by gurtin and murdoch @xcite , which is summarized in @xcite . when the surface relaxes , it compresses or pulls the underlying crystal ; since the resulting lattice strain depends on the crystal size and geometry , the lattice parameter of an x - ray interferometer is different from that of the si spheres used to determine the molar volume . after summarizing the operation of a combined x - ray and optical interferometer , this paper investigates the effect of surface stress on the interferometer operation . an elastic membrane model has been used to provide a surface load in a finite element analysis of the lattice strain . particular emphasis was given to the operation of the @xmath1si interferometer used to determine @xmath0 . eventually , a measurement of the lattice parameter with a variable - thickness interferometer is proposed in order to evidence the effect of surface stress on the underlying crystal . a review of the lattice parameter measurement can be found in @xcite . as shown in fig . [ xint ] , an x - ray interferometer consists of three si crystal slabs so cut that the \{220 } planes are orthogonal to the crystal surfaces . x - rays from a 17 kev mo k@xmath2 source are split by the first crystal and recombined , via two transmission crystals , by the third , called the analyser . when the analyser is moved in a direction orthogonal to the \{220 } planes , a periodic intensity - variation of the transmitted and diffracted x - rays is observed , the period being the diffracting - plane spacing , about 192 pm . the analyser embeds front and rear mirrors , so that its displacement and rotations can be measured by an optical interferometer . the measurement equation is @xmath3 , where @xmath4 pm is the spacing of the \{220 } planes and @xmath5 is the number of x - ray fringes observed in a displacement of @xmath6 optical fringes having period @xmath7 nm . to ensure the interferometer calibration , the laser source operates in single - mode and its frequency is stabilized against that of a transition of the @xmath8i@xmath9 molecule . to eliminate the adverse influence of the refractive index of air and to achieve millikelvin temperature uniformity and stability , the experiment is carried out in a thermovacuum chamber . continuous developments led the measurement sensitivity and accuracy to approach @xmath10 ; the most recent results are given in @xcite . similarly to liquid surfaces , solid surfaces exhibit a stress also if the bulk material is stress - free . this is the macroscopic consequence of atomic affects : roughly speaking , the atoms on the surface lose half of the bonds they would have in the bulk ; therefore , crystal surfaces are disturbances in the bonding of atoms . surface stress was first generalized to solids by gibbs and shuttleworth @xcite as the amount of reversible work per unit area needed to stretch a pre - existing surface . since a solid surface may change not only by exchange of matter with the bulk , but also by elastic deformation , the concept of surface stress must be derived from the molecular dynamics of the surface @xcite . continuum mechanics provides the framework to define the surface properties and the relationships among them . the elasticity theory of solid surfaces to which we refer was developed by gurtin and murdoch @xcite . a summary of the linearised theory which is valid when both the surface and bulk strains are small is given in @xcite . the key concept is that , for a surface attached to the underlying solid , a stress - free reference does not exist ; since the bonds between the surface atoms and between the bulk atoms are different , the surface of a stress - free crystal has residual strain , @xmath11 , and stress , @xmath12 . if the surface tends to shrink ( expand ) with respect to the bulk , the surface stress is positive ( negative ) and it is said to be tensile ( compressive ) . since a fully relaxed surface has no normal stress , surface strain and stress are second rank tensors , which are characterized by two principal orthogonal - components . if the surface has more than a 2-fold symmetry , the surface stress and strain tensors are isotropic . in this case , @xmath13 and the free surface - energy per unit area is @xcite @xmath14[free ] where @xmath15 are the strain component in the plane tangent to the surface , @xmath16 is the surface energy for a stress - free bulk , @xmath17 and @xmath18 are elastic constants , and the summation over the repeated indices @xmath19 is implied . the definition of surface stress is @xmath20 ; hence , by taking the derivatives , @xmath21 where @xmath22 is the intrinsic surface stress . @rrrll @xmath23/(n / m ) & @xmath24/(n / m ) & @xmath25/(n / m ) & year & reference + @xmath26 & @xmath27 & @xmath28 & 1988 & @xcite + @xmath29 & @xmath30 & @xmath31 & 1989 & @xcite + @xmath32 & @xmath33 & @xmath34 & 1990 & @xcite + @xmath35 & @xmath36 & @xmath37 & 1992 & @xcite + @xmath38 & @xmath39 & @xmath40 & 1993 & @xcite + @xmath29 & @xmath41 & @xmath42 & 1994 & @xcite + @xmath43 & @xmath44 & @xmath45 & 2005 & @xcite + @xmath46 & @xmath47 & @xmath48 & 2005 & @xcite + @xmath49 & @xmath50 & @xmath51 & 2008 & @xcite + the determination of the surface stress in solids is a challenge ; the measurement of the lattice parameter in small particles has been the main method @xcite . given the experimental difficulties and since the results are affected by the sample preparation and adsorbed impurities , the values obtained from _ ab - initio _ and molecular dynamics calculations seem more reliable . the results for the symmetric @xmath52 reconstruction of the si(001 ) surface are summarized in table [ s - stress ] ; the reconstructed surface is formed by domains of anisotropic stress , but , for symmetry reasons , the mean stress is isotropic . the scatter of data suggests a reassuring small surface stress . because of the anisotropic nature of the crystal , surface stress depends on the surface orientation . a measurement of the polar dependence of the relative surface - stress at high temperature ( 1373 k ) is given in @xcite , which allows the surface stress of any surface to be determined from that of the ( 001 ) reference surface . in the case of our x - ray interferometer , the @xmath53 surfaces are those of main interest ; because of their 2-fold symmetry , two values are necessary to represent the surface stress of the two perpendicular @xmath54 $ ] and @xmath55 $ ] components . as reported in @xcite , at 1373 k , the sought principal components are @xmath56 and @xmath57 . indicative @xmath25 values are given in table [ s - stress ] . the crystal s surfaces are normally covered by a few nanometre thick oxide - layer , which can further stress the surface . the stress changes of the si(001 ) @xmath52 surface during plasma oxidation were measured by using a thin silicon cantilever ; the bending caused by the oxidation of one face was optically measured and related to the surfaces stress difference by elasticity theory @xcite . initially , the oxidation induces a compressive stress , followed by a tensile stress ; eventually , the stress variation turns to compressive for oxide thickness greater than about 1.5 nm . the initial stress changes were explained in terms of the electron diffusion into the sample and the shrinkage caused by the oxygen - bridged dimer structure . the final compressive variation , @xmath58 n / m , was explained by the oxidation of si atoms at deep sites ; since the oxygen expands the si si bond , oxidation is expected to cause a compressive stress . the surface relaxes the intrinsic stress by compressing or pulling the underlying crystal . for instance , the ( isotropic ) lattice strain of a sphere of radius @xmath59 is @xmath60 where @xmath61 is the bulk modulus and the young - laplace approximation holds when the elastic constants can be neglected when compared to the @xmath62 product . by choosing @xmath63 n / m , we can use this formula where @xmath64 mm and @xmath65 gpa to estimate the lattice strain of the spheres used to determine @xmath0 . the result is @xmath66 nm / m , which is irrelevant for the @xmath0 determination . however , if @xmath67 mm which is a value equal to the thickness of the crystal used to measure the lattice parameter we obtain @xmath68 nm / m , which is significant with respect to the uncertainty of the lattice - parameter measurement . figure [ analyser ] ( left ) shows the analyser crystal of the @xmath1si x - ray interferometer used for the @xmath0 measurement and its crystallographic orientation . the end mirrors are parallel to the diffracting planes ; to avoid stress propagation , they are separated by vertical cuts . the lamella thickness , 1.2 mm , was chosen by a trade off between the increased lattice strain ( due to the surface stress ) of a thin lamella and the increased x - ray absorption of a thick lamella . to estimate the lattice parameter change caused by the surface stress , we used a commercial finite element analysis software @xcite . as shown in fig . [ analyser ] , the ( 220 ) planes are orthogonal to the lamella faces which are parallel to @xmath53 planes . owing the large uncertainty of the residual stress values , though the @xmath53 surfaces are anisotropic , the analyser surfaces were modelled as stretched by isotropic membranes attached to the underlying crystal lattice . since we are only concerned with infinitesimal strains , though the surface stress @xmath69 depends on the strain according to ( [ stress ] ) , we used an isotropic @xmath63 n / m value . this surface stress value allows the analysis results to be easily scaled if a different value has to be considered and , hopefully , it is larger than the actual stress value . hence , according the young - laplace equation , the surface applies the inward pressure @xmath70 , where @xmath71 is the mean curvature . as schematically shown in fig . [ analyser ] ( right ) , when considering flat surfaces , the surface stress was modelled by shear forces per unit length applied orthogonally to the surface edges . si interferometer . m : end mirrors , a : analyser lamella , b : base . the @xmath72 mm@xmath73 reliefs that support the crystal are coloured in black . the lattice parameter was measured along the indicated line , at @xmath74 mm . right : surface - stress modelling by shear forces.,title="fig:",width=245 ] si interferometer . m : end mirrors , a : analyser lamella , b : base . the @xmath72 mm@xmath73 reliefs that support the crystal are coloured in black . the lattice parameter was measured along the indicated line , at @xmath74 mm . right : surface - stress modelling by shear forces.,title="fig:",width=245 ] as strains are infinitesimal , we uses a linear model ; the mesh , of about @xmath75 tetrahedral anisotropic elements , was the result of successive refinements . the not null elements of the stiffness matrix are given table [ s : matrix ] in both the [ 100 ] , [ 010 ] , [ 001 ] and [ 110 ] , [ @xmath7610 ] , [ 001 ] crystal axes . as regards the model parameters , the si density and gravitational acceleration are 2330 kg / m@xmath73 and 9.81 m / s@xmath77 , respectively . @ccccccccc + & & + & & + + & @xmath78 & & @xmath79 & & @xmath80 + & 35.2 & & 165.7 & & 50.9 + the analyser rests on a silicon plate ; it stands on three @xmath72 mm@xmath73 reliefs carved out of the crystal base by chemical etching . in order to avoid stresses , it is held in position by a thin layer of high - viscosity silicone oil . since the oil applies only viscous forces , the contact with the support has been simulated by setting to zero the displacement of the contact nodes . we calculated the strain @xmath81 , where @xmath82 is the horizontal component of the displacement vector . both the effects of self - weight and surface strain have been considered ; the supports were so chosen as to minimize bending or sagging @xcite . the three dimensional map of the strain distribution in the analyser and a magnified image of the strained diffracting planes are shown in fig . [ strain:1 ] . the @xmath83 strain is trivially related to the lattice parameter by @xmath84 . hence , the relative variations of the @xmath85 values along measurement lines at different heights are identical to the @xmath83 values calculated along the same lines ; they are shown in fig . [ strain:2 ] ( left ) . the figure indicates that , in the measurement horizontal strip a couple of millimetre wide at a @xmath74 mm , the lattice is uniformly strained by about @xmath86 nm / m , as estimated by the previous `` spherical - cow '' model . in addition to the horizontal strain , there is a vertical strain gradient of about @xmath87 nm m@xmath88 cm@xmath88 , the bottom part being less strained , which could be the subject matter of future experimental verifications . component of the diffracting - plane strain for the optimal location of the supports , where blue is @xmath90 nm / m and red is @xmath91 nm / m . right : qualitative magnified image of the diffracting planes . a realignment of the analyser has been simulated by subtracting the average displacement and tilt . the displacement magnification factors is shown by the 1 nm bar.,title="fig:",width=245 ] component of the diffracting - plane strain for the optimal location of the supports , where blue is @xmath90 nm / m and red is @xmath91 nm / m . right : qualitative magnified image of the diffracting planes . a realignment of the analyser has been simulated by subtracting the average displacement and tilt . the displacement magnification factors is shown by the 1 nm bar.,title="fig:",width=245 ] variations in @xmath92 plane , at different distances from the analyser bottom - surface . right : relative @xmath85 variations along the @xmath74 mm measurement baseline ( figure [ analyser ] ) . different support - to - mirror distances have been considered , as indicated in the graph.,title="fig:",width=245 ] variations in @xmath92 plane , at different distances from the analyser bottom - surface . right : relative @xmath85 variations along the @xmath74 mm measurement baseline ( figure [ analyser ] ) . different support - to - mirror distances have been considered , as indicated in the graph.,title="fig:",width=245 ] figure [ strain:2 ] ( right ) updates the corresponding figure 6 of @xcite , which was used to assess the sole effect of self weigh in the lattice parameter measurement . it shows that , if the analyser surfaces are loaded by a tensile stress of 1 n / m , the @xmath85 value is about @xmath93 smaller than its value in an unstrained crystal . the figure also shows that the lattice parameter profile given in @xcite can not deliver clues about the surface contribution to the measured @xmath85 value because the only effect of the surface stress is to translate all the curves downwards by @xmath93 . since the uncertainty associated to the measured @xmath85 value is @xmath94 , the lattice strain induced by a 1 n / m stress impairs the measurement accuracy . the results indicate also that we can cope effectively the surface stress by a numerical determination of its effect . however , owing the large uncertainty of the sign and value of the surface stress , we do nt yet propose to reconsider the corrections and error - budget contributions given in @xcite . further investigations are under way to assess and to quantify the effect of surface stress , if any . in the next section we describe an analyser design to work a lattice parameter measurement out so that there is a measurable effect of the surface stress . variations are calculated along the indicated lines , at @xmath95 mm.,width=245 ] a way to evidence if the surface stress strains the analyser crystal up to an extent which is significant to the lattice parameter measurement is to compare the results of measurements carried out in crystals having a large thickness difference . since the induced @xmath85 changes are small , to avoid they are masked by lattice imperfections , measurements must be carried out in the same crystal . consequently , as shown in fig . [ new - xint ] , we designed an x - ray interferometer having a variable analyser thickness . the minimum 0.4 mm thickness is set by machining capabilities ; the maximum 1.5 mm thickness is set by x - ray absorption . after the optimization of the finite element analysis and the location of the supports so as to minimize the self - weight bending , we obtained the strain distribution shown in fig . [ new - strain:1 ] ( left ) ; a magnified image of the strained \{220 } planes is shown in fig . [ new - strain:1 ] ( right ) . the expected @xmath85 profile along the horizontal lines at @xmath95 mm height is shown in fig . [ new - strain:2 ] ( left ) . as shown in fig . [ new - strain:2 ] ( right ) , the surface stress and self - weight have opposite effects . the surface stress will be estimated from the @xmath85 gap between the thick and thin analyser parts . if the surface is loaded by a tensile stress of 1 n / m , @xmath85 decreases by about @xmath96 from the central ( thick ) to the outer ( thin ) parts . this variation is large enough to be detected . with a high crystal perfection , the present detection limit of @xmath97 corresponds to a minimum detectable stress of 0.2 n / m . to ensure that self - weight bending does not change in a significant way the @xmath85 gap between the thick and thin analyser parts , the height of the analyser base was increased to 19.8 mm . figure [ rejection ] shows that , when the analyser supports are randomly displaced ( in @xmath98 plane ) from their optimal positions up to @xmath99 cm , the @xmath85 gap between the thick and thin analyser - parts changes less than what can be experimentally detected . component of the diffracting - plane strain for the variable - thickness analyser , where blue is @xmath90 nm / m and red is @xmath91 nm / m . right : qualitative magnified image of the diffracting planes . a realignment of the analyser has been simulated by subtracting the average displacement and tilt . the displacement magnification factors is shown by the 1 nm bar.,title="fig:",width=245 ] component of the diffracting - plane strain for the variable - thickness analyser , where blue is @xmath90 nm / m and red is @xmath91 nm / m . right : qualitative magnified image of the diffracting planes . a realignment of the analyser has been simulated by subtracting the average displacement and tilt . the displacement magnification factors is shown by the 1 nm bar.,title="fig:",width=245 ] variations in @xmath92 plane , along the @xmath95 mm measurement lines ( figure [ new - xint ] ) . the supports are optimally located so as to minimize the self - weight bending . left : both the self - weight and surface stress are considered . right : only the self - weight is considered.,title="fig:",width=245 ] variations in @xmath92 plane , along the @xmath95 mm measurement lines ( figure [ new - xint ] ) . the supports are optimally located so as to minimize the self - weight bending . left : both the self - weight and surface stress are considered . right : only the self - weight is considered.,title="fig:",width=245 ] variations in @xmath92 plane , along the @xmath95 mm measurement lines , when the analyser supports are displaced ( in @xmath98 plane ) from their optimal positions up to @xmath99 cm.,width=245 ] the influence of the intrinsic surface - stress on the operation of the x - ray interferometer utilized to measure the lattice parameter of the @xmath1si crystal used to determine the avogadro constant has been investigated by finite element analysis . the crystal surfaces , modelled as membranes having a tensile stress of 1 n / m , relax by compressing the underlying crystal . though the induced strain is greater than the uncertainty associated to the measured value of the lattice parameter , since the stress magnitude is highly uncertain and the chosen 1 n / m value could be pessimistic , we do not propose a correction of the published value . to verify if the surface stress contributes or does not contribute to the lattice parameter measurement , a variable - thickness design of the x - ray interferometer has been investigated to induce detectable strain variations in the same interferometer crystal . the realization of such an interferometer is under way in collaboration at the physikalisch technische bundesanstalt in collaboration with the leibniz - institut fr oberflchenmodifizierung . we thank petr kren of the czech metrology institute for having brought to our attention this criticality of the lattice parameter measurement . this work was jointly funded by the european metrology research programme ( emrp ) participating countries within the european association of national metrology institutes ( euramet ) and the european union . 99 andreas b _ et al . _ 2011 determination of the avogadro constant by counting the atoms in a @xmath1si crystal _ phys . rev . lett . _ * 106 * 030801 andreas b _ et al . _ 2011 counting the atoms in a @xmath1si crystal for a new kilogram definition _ metrologia _ * 48 * s1 - 13 massa e , mana g , kuetgens u and ferroglio l 2011 measurement of the 220 lattice - plane spacing of a 28si x - ray interferometer _ metrologia _ * 48 * s37 - 43 kren p private communication ferroglio l , mana g , palmisano c and zosi g 2008 influence of surface stress in the determination of the ( 220 ) lattice spacing of silicon _ metrologia _ * 45 * 110 - 8 gibbs j w 1906 _ the scientific papers of j. willard gibbs , vol . 1 _ ( london : longmans - green ) p 55 shuttleworth r 1950 _ phys . soc . a _ * 63 * 444 - 57 cammarata r c 1994 surface and interface stress effects in thin films _ prog . surf . sci . _ * 46 * 1 - 38 desjonqures m c and spanjaard d 1996 _ concept in surface physics _ ( springer , berlin ) makkonen l 2012 misinterpretation of the shuttleworth equation _ scripta materialia _ * 66 * 627 - 9 gurtin m e and murdoch i 1975 a continuum theory of elastic material surfaces _ arch . * 57 * 291 - 323 gurtin m e and murdoch i 1978 surface stress in solids _ int . j. solids struct . _ * 14 * 431 - 40 wolfer w g 2011 elastic properties of surfaces on nanoparticles _ acta materialia _ * 59 * 7736 - 43 becker p and mana g 1994 the lattice parameter of silicon : a survey _ metrologia _ * 31 * 203 - 9 miyamoto 1994 comparative study of si(100 ) surface stresses with dimerized group - iv adatoms _ phys . b _ * 49 * 1947 - 51 hara s , izumi s , kumagai t and sakai s 2005 surface energy , stress and structure of well - relaxed amorphous silicon : a combination approach of ab initio and classical molecular dynamics _ surface science _ * 585 * 17 - 24 alerhand o l , vanderbilt d , meade r d and joannopoulos j d 1988 spontaneous formation of stress domains on crystal surfaces _ phys . lett . _ * 61 * 1973 - 6 payne m c , roberts n , needs r j , needels m and joannopoulos j d 1989 total energy and stress of metal and semiconductor surfaces _ surf . sci . _ * 211/212 * 1 - 20 meade r d and vanderbilt d 1990 _ proc . on the physics of semiconductors _ singapore : world scientific p 123 poon t w , yip s , ho p s and abraham f f 1992 ledge interactions and stress relaxations on si(001 ) stepped surfaces _ phys . b _ * 45 * 3521 - 31 garca a and northrup j e 1993 stress relief from alternately buckled dimers in si(100 ) _ phys . b _ * 48 * 17350 - 3 delph t j 2008 near - surface stresses in silicon(001 ) _ surf . sci . _ * 602 * 259 - 67 mtois j j , sal a and mueller p 2005 measuring the surface stress polar dependence _ nature materials _ * 4 * 238 - 42 itakura a n , narushima t , kitajima m , teraishi k , yamada a and miyamoto a 2000 surface stress in thin oxide layer made by plasma oxidation with applying positive bias _ applied surface science _ * 159 - 160 * 62 - 6 comsol multiphysics@xmath100 4.3a ferroglio l , mana g , and massa e 2011 the self - weight deformation of an x - ray interferometer _ metrologia _ * 48 * s50 - 4
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a stress exists in solids surfaces , similarly to liquids , also if the underlying bulk material is stress - free .
this paper investigates the surface stress effect on the measured value of the si lattice parameter used to determine the avogadro constant by counting si atoms .
an elastic - film model has been used to provide a surface load in a finite element analysis of the lattice strain of the x - ray interferometer crystal used to measure the lattice parameter .
eventually , an experiment is proposed to work a lattice parameter measurement out so that there is a visible effect of the surface stress .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the lightest nuclei can be accurately modeled as systems of nucleons interacting via effective two- and three - body forces motivated e.g. by meson exchange . this picture is expected to break down at a higher energy scale , where the physics is more efficiently described in terms of subnuclear degrees of freedom . few - body methods have been an essential tool for determining model hamiltonians that describe low - energy nuclear physics . they also have the potential to be a useful framework for testing the limitations of viewing the nucleus as a few nucleon system . the latter requires extending few - body models and calculations to higher energies . in order to successfully do this , a number of challenges need to be addressed . these include replacing the nonrelativistic theory with a relativistic one , overcoming limitations imposed by interactions fit to elastic nucleon - nucleon ( nn ) scattering data , including new degrees of freedom that appear above the pion production threshold , as well as numerical problems related to the proliferation of partial waves characteristic for scattering calculations at higher energies . thus the intermediate energy regime is a new territory for few - body calculations which waits to be explored . in this paper we address two of the challenges . we demonstrate that it is now possible to perform converged three - body scattering calculations at energies up to 2 gev laboratory kinetic energy . key elements are a consistent implementation of a poincar symmetric quantum theory @xcite , and the use of direct integration methods that avoid the partial wave decomposition , successfully applied below the pion - production threshold @xcite . in a series of publications @xcite the technique of solving the nonrelativistic momentum - space faddeev equation without partial waves has been mastered , for bound as well as scattering states . the relativistic faddeev equation , based on a poincar invariant mass operator , has been formulated in detail in @xcite , showing that it has both kinematical and dynamical differences with respect to the corresponding nonrelativistic equation . the formulation of the theory is given in a representation of poincar invariant quantum mechanics where the interactions are invariant with respect to kinematic translations and rotations @xcite . the model hilbert space is a three - nucleon hilbert space ( thus not allowing for absorptive processes ) . the method used to introduce the nn interactions in the unitary representation of the poincar group allows input of high - precision nn interactions @xcite in a way that reproduces the measured two - body observables . however in this study we use a simpler , spin - independent interaction consisting of a superposition of an attractive and a repulsive yukawa interaction that supports a bound state with the deuteron binding energy . this is mathematically equivalent to three - boson scattering . poincar invariance and @xmath0-matrix cluster properties dictate how the two - body interactions must be embedded in the three - body dynamical generators . scattering observables are calculated using the faddeev equation formulated with the mass casimir operator ( rest hamiltonian ) constructed from these generators . we want to point out that the relativistic faddeev equation with realistic spin - dependent interactions has been solved below the pion - production threshold in a partial wave basis @xcite . in order to estimate the size of relativistic effects the interactions employed in the nonrelativistic and relativistic calculations presented here are chosen to be two - body phase shift equivalent . this is achieved in this article by adding the interaction to the square of the mass operator @xcite . thus differences in relativistic and nonrelativistic calculations first appear in the three - body calculations . those differences are in the choice of kinematic variables ( jacobi momenta are constructed using lorentz boosts rather than galilean boosts ) and in the embedding of the two - body interactions in the three - body problem , which is a consequence of the non - linear relation between the two and three - body mass operators . these differences modify the permutation operators and the off - shell properties of the kernel of the faddeev equation . this article is organized as follows . in section ii the formulation of the poincar invariant faddeev equation is given and numerical aspects for computing the faddeev kernel are discussed . in sections iii and iv we present calculations for elastic and breakup processes in the intermediate energy regime from 0.2 to 1.5 gev . our focus here is the investigation of the convergence of the multiple scattering series as a function of projectile kinetic energy . we compare our calculations to selected breakup observables and investigate a simple approximation of the embedding of the two - body interaction into the three - body problem . a detailed formulation of poincar invariant three - body scattering has been given in @xcite , where the driving term in the relativistic faddeev equation ( first order in the two - body transition operator ) has been used to evaluate cross sections for elastic as well as break - up scattering . this is now being complemented by fully solving the relativistic faddeev equation based on the numerical techniques previously used to solve the non - relativistic faddeev equation @xcite . for the convenience of the reader essential equations are repeated , but for the detailed derivation of the expressions we refer to ref . @xcite . the symmetrized transition operators @xmath1 for elastic scattering and @xmath2 for breakup reactions can be expressed in terms of the solution @xmath3 of the symmetrized faddeev equations @xmath4 where @xmath5 is the invariant mass operator for three non - interacting particles and the permutation operator @xmath6 is given by @xmath7 . the operator @xmath3 is the solution to the symmetrized faddeev equation @xmath8 where the operator @xmath9 is the two - body transition operator embedded in the three - particle hilbert space and defined as the solution to @xmath10 here @xmath11 is the two - body interaction embedded in the three - body hilbert space and @xmath12 is the invariant mass operator for two interacting particles and a spectator . for calculating transition matrix elements , explicit basis states need to be introduced . the momenta of the three particles can be labeled either by single - particle momenta @xmath13 , @xmath14 , and @xmath15 , or the total momentum @xmath16 and the relativistic poincar - jacobi momenta @xcite @xmath17 and @xmath18 . the explicit relations between the three - body poincar jacobi momenta and the single particle momenta are @xmath19 then the poincar - jacobi momenta @xmath17 and @xmath18 are given as @xmath20 where @xmath21 . in addition , the transformation from the single particle momenta @xmath22 to the poincar - jacobi momenta has a jacobian given by @xmath23 where for @xmath24 the jacobian becomes @xmath25 in the above expression we chose , without loss of generality , particle 1 as the spectator . the poincar - jacobi momenta are relevant for the calculation of the permutation operator @xmath6 in eqs . ( [ eq:2.1 ] ) and ( [ eq:2.2 ] ) . the matrix elements of the permutation operator are then explicitly calculated as @xmath26 , \label{eq:2.7}\end{aligned}\ ] ] where the function @xmath27 contains the product of two jacobians and reads @xmath28{(e(\mathbf{q } ) + e(\mathbf{q } + \mathbf{q}'))^2 - \mathbf{q}'^2 } \;\ ; \sqrt[4]{(e(\mathbf{q } ' ) + e(\mathbf{q } + \mathbf{q}'))^2 - \mathbf{q}^2 } } { \sqrt{e(\mathbf{q})e(\mathbf{q } ' ) } } . \label{eq:2.8}\end{aligned}\ ] ] with @xmath29 . the function @xmath30 is calculated as @xmath31 these permutation operators , which change the order of coupling , are essentially racah coefficients for the poincar group . in the nonrelativistic case the functions @xmath27 and @xmath32 both reduce to the constant 1 and have the relatively compact form of the matrix elements of @xmath6 given in e.g. @xcite . in matrix form the faddeev equation , eq . ( [ eq:2.2 ] ) , reads @xmath33 where we have factored out a delta function in the total momentum and set @xmath34 . inserting complete sets of states and explicitly evaluating the permutation operator leads to @xmath35 the quantities @xmath36 and @xmath37 are determined by the laboratory kinetic energy @xmath38 of the incident nucleon , @xmath39 the nucleon rest mass is given by @xmath40 , the rest mass of the deuteron is @xmath41 , where @xmath42 is the deuteron binding energy . the poincar jacobi momentum between projectile and target , @xmath43 , is related to @xmath38 by @xmath44 the invariant parametric energy @xmath45 which enters the two - body t - matrix is given by @xmath46 . since we consider bosons , we introduce the symmetrized two - body transition matrix @xmath47 @xmath48 this two - body t - matrix has a simple pole at @xmath49 . thus , for the practical calculation we need to take this pole explicitly into consideration by defining @xmath50 and solving eq . ( [ eq:2.11 ] ) for @xmath51 @xmath52 for the explicit calculation we introduce the independent variables @xcite @xmath53 so that @xmath54 , is a function of 5 variables . in the variables of eq . ( [ eq:2.17 ] ) and defining @xmath55 , the final expression for eq . ( [ eq:2.11 ] ) reads @xmath56 while the deuteron pole can be numerically taken care of with a single subtraction in the @xmath57-integration , the free three - nucleon propagator in the 2nd term under the integral of eq . ( [ eq:2.19 ] ) contains singularities depending on @xmath57 as well as @xmath58 leading to a singular region in the @xmath59 plane . in order to simplify the calculation , we carry out the integration of the kernel in a frame in which the @xmath60-axis is along the direction of @xmath17 . in this frame @xmath61 and @xmath62 is the azimuthal angle of @xmath63 . with these definitions one has @xmath64 where @xmath65 and @xmath66 are the azimuthal angles of @xmath18 and @xmath37 in the frame described above . since there is a freedom in choosing the @xmath67-axis , we may place @xmath37 in the @xmath68-plane , this gives @xmath69 . with this choice @xmath65 is evaluated as @xmath70 the remaining variables in eq . ( [ eq:2.19 ] ) are explicitly evaluated as @xmath71 and @xmath72 and @xmath73 with @xmath74 for the integration of the 3n propagator , each singularity in the @xmath58-integration ( for fixed @xmath57 ) is explicitly taken into account by a subtraction . however , this leads to logarithmic singularities in @xmath57 at the boundaries @xmath75 . these we integrate in the semi - analytic fashion introduced in ref . @xcite by using cubic hermite splines . while using cubic hermite splines is advantageous in dealing with the logarithmic singularities , this method is not as effective as gauss - legendre quadrature when integrating over large , non - singular regions . thus , in order to make the most efficient use of both methods , we divide the interval of the @xmath57-integration into several integration regions , and use gauss - legendre quadrature in the non - singular integrals , while keeping the cubic hermite splines in the small regions around the singularities . with this procedure we are able to successfully integrate over the faddeev kernel with sufficient accuracy . for the final solution of eq . ( [ eq:2.19 ] ) the kernel is successively applied and the resulting terms are summed up as pad sums . at the higher energies we will also carry out the neumann sum . our explicit calculations are based on a simple interaction of malfliet - tjon type consisting of a superposition of an attractive and repulsive yukawa interaction that supports a bound state with the deuteron binding energy . the parameters of this nonrelativistic interaction are given in ref . @xcite . in order to obtain a relativistic interaction which is phase shift equivalent with the nonrelativistic one , we employ a scheme in which @xmath76 multiplied with the interaction is added to the square of the non - interacting two - body mass operator . this procedure was introduced by coester , pieper and serduke @xcite and used here in the form given in @xcite . it guarantees that differences in the relativistic and nonrelativistic calculations first appear in the three - body calculations . before entering a detailed study on relativistic effects , we want to present further details on the numerical quality of our solution of the relativistic faddeev equation . one internal consistency check of the solution is provided by the optical theorem , which states that the total cross section , being the sum of the total elastic cross section , @xmath77 , and the total breakup cross section , @xmath78 , must be equal to the imaginary part of the transition operator for elastic scattering @xmath79 in forward direction . in the center - of - momentum ( c.m . ) frame this relation reads @xmath80 listed in table [ table-1 ] are our fully relativistic calculations of the total cross sections for elastic scattering and breakup reaction for projectile energies from 0.1 to 2.0 gev , together with the total cross sections . the total cross sections are calculated as sum of the elastic and breakup cross sections , @xmath81 , and via the optical theorem , @xmath82 , from the imaginary part of the operator @xmath79 in forward direction , @xmath83 . a comparison of those two numbers for the total cross section shows , that our calculations fulfill the optical theorem to about 1% or better up to 1 gev . this error increases to about 3% at 2 gev . here we did not push the calculations any further , since our model potential is too simple to take it to much higher energies anyway . for the sake of showing the numerical quality of our calculations , we included 2 gev in table [ table-1 ] , but will not show any further observables at this energy . the transition amplitude of eq . ( [ eq:2.19 ] ) is a function of 5 variables , and is the solution of an integral equation in three dimensions . thus , in the calculation the dependence of the result on the various choices of grids has to be considered . as far as the momentum grids are concerned , the accuracy of the calculation is most sensitive to the @xmath84-grid , as already found in ref . @xcite . in fig . [ fig1 ] we show the dependence of the relative error @xmath85 in the optical theorem as function of the size of the @xmath84-grid , @xmath86 , for a calculation at 1 gev projectile laboratory kinetic energy . the slope of @xmath87 shows that indeed the accuracy of the calculation is strongly influenced by the size of this grid . for our calculation , @xmath86 = 50 is sufficient at 1 gev . next , we consider the sensitivity of the calculation to the size of different angle grids . in table [ table-2 ] we give the cross sections for elastic scattering and breakup reactions together with the total cross section @xmath82 extracted from the optical theorem when varying the size of the different angle grids . we can see , that the results are most sensitive with respect to the grids in @xmath88 and @xmath58 . it is common wisdom in calculations using an angular momentum basis that as the energy of the projectile increases , the number of partial waves needed to obtain a converged result increases rather quickly . in our 3d calculations , all partial waves are included . the increase in energy manifests itself in a two - body t - matrix acquiring a more pronounced peak structure in the forward and backward directions with respect to the angle between the two momentum vectors @xcite . this peak structure at @xmath89 must be adequately covered in calculations at higher energies to ensure converged results . in fig . [ fig2 ] we show the relative error @xmath90 in the optical theorem as function of the size of the @xmath88-grid for three different projectile laboratory kinetic energies . the necessity of increasing the @xmath88-grid with increasing projectile energy is clearly seen . whereas for 0.2 gev @xmath91 = 24 is clearly sufficient , at 0.5 gev one needs already at least 28 points , whereas at 1 gev a minimum of 36 points is required . this conclusion is also reached in our table [ table-3 ] , where we show the relativistic differential cross section at selected angles while varying the @xmath88-grid . it should be noted that the angle @xmath88 is related to the angular momentum of the relative motion between the spectator and the interacting pair . the angle @xmath92 , which is related to the angular momentum of the interacting pair is not nearly as sensitive as @xmath88 . in table [ table-2 ] we vary the @xmath92 grid from 20 to 24 points , and see hardly any difference . it is illustrative to contrast the computational algorithm for direct integration with the experience gained when using a partial wave basis in the 3n system . our experience tells us that at @xmath38 = 200 mev the total angular momentum of the 2n subsystem @xmath93 needs to be @xmath94 = 5 to reach convergence . furthermore , the maximum total angular momentum @xmath95 of the 3n system required to reach convergence is @xmath96 = 25/2 . let us assume that @xmath97 , where @xmath98 is the maximal angular momentum of the projectile nucleon with respect to the target pair , @xmath99 is the spin of the projectile and @xmath100 is the relative orbital angular momentum between the projectile and target pair . this leads to @xmath101 = 15/2 for a 3n scattering calculation at @xmath38 = 200 mev . disregarding the spin degree of freedom for the three nucleons , leading to the three - boson model under consideration here , we find that @xmath96 = 12 with @xmath102 = 5 and @xmath103 = 7 are necessary for a converged calculation at @xmath38 = 200 mev . in the three - boson case @xmath104 , the orbital momentum of the interacting pair , and @xmath100 take the role of @xmath93 and @xmath105 . in order to estimate the corresponding maximal number of angular momenta needed for @xmath38 = 1 gev , one needs the effective deuteron radius @xmath106 , which leads to @xmath103 = 7 at @xmath38 = 200 mev . nonrelativistically the 3n c.m . energy is given as @xmath107 , leading to @xmath108 400 mev / c at @xmath38 = 200 mev and @xmath108 900 mev / c at @xmath38 = 1 gev . if we roughly set @xmath109 , then we find at @xmath38 = 200 mev a value @xmath110 3.5 fm , which appears reasonable . applying the same value at @xmath38 = 1 gev then leads to @xmath103 = 15 . using our experience in calculating the nn system in the gev regime @xcite , where one needs for converged nn observables at least @xmath94 = 14 at 1 gev , we estimate that a converged partial wave 3n calculation of the three - boson system would need @xmath111 = 14 + 15 = 29 . let us now regard the two cases , ( a ) @xmath38 = 200 mev , @xmath102 = 5 , @xmath103 = 7 , @xmath96 = 12 , and ( b ) @xmath38 = 1 gev , @xmath102 = 14 , @xmath103 = 15 , @xmath96 = 29 in a partial wave decomposition . to illustrate the tremendous number of partial waves needed in ( b ) compared to the feasible case ( a ) it is sufficient to consider simple algebra for different values of @xmath95 . take for example @xmath95 = 5 . then simple counting yields 30 different @xmath112 combinations in case ( a ) and 125 in case ( b ) . for @xmath95 = 10 this number increases in case ( b ) to 160 . moreover , since the number of total @xmath95 s at 1 gev is more than twice the number of @xmath95 s at 200 mev , it appears quite unreasonable to enforce a partial wave decomposition at energies far above 200 mev in the 3-boson ( nucleon ) system . in addition , it would also be numerically very demanding to evaluate the various ingredients in the faddeev equation reliably for the very high angular momenta . another advantage of using direct integration of vector variables in the faddeev equation is the simplicity of the permuation operators . in the following we present our results for elastic and breakup scattering in the energy regime from about 200 to 1500 mev laboratory projectile kinetic energy . we start with a comparison of our model calculation to calculations based on a realistic nn force at lower energies in order to show that even though our model is very simple , we see similar features in the cross sections . then we study relativistic effects at higher energies . there are several questions we want to address . first , we want to identify scattering observables that are sensitive to the difference between the relativistic and non - relativistic formulations of the three - body problems and to study the size of those relativistic effects as function of increasing energy . this can at present only be done with our model interaction . second , we want to study the convergence properties of the faddeev multiple scattering series as function of the projectile kinetic energy . here the question of interest is , if , once the energy is high enough , it is sufficient to only consider the first few terms in the multiple scattering series . in addition , we also want to study some approximations to our relativistic scheme . the laboratory kinetic energy of 200 mev is a perfect energy to study if the features of the 3n system we find based on our model interaction are also present in calculations based on a realistic model of the nn interaction , which describes the nn observables with high accuracy . the so - called high - precision interactions are fitted up to about 350 mev , but strictly speaking only valid below the pion - production threshold . we also know @xcite that relativistic effects are already visible at 200 mev . we choose the cd - bonn interaction @xcite for this comparison . in fig . [ fig3 ] we show the @xmath113 total cross section extracted from the said database @xcite together with the total cross section obtained from the mt - iii interaction assuming bosonic symmetry . the parameters of the mt - iii interaction @xcite are adjusted such that a two - body bound state at @xmath114 = 2.23 mev is supported . [ fig3 ] shows that the experimental @xmath113 total cross section falls slightly below the two - body cross section predicted by our model at energies smaller than @xmath115 300 mev , is about equal between 300 and 400 mev , and then reaches a constant value from about 600 mev on , while our model prediction continues to decrease . the slight rise of the experimental value around 600 mev is a manifestation of the influence of the @xmath116(1232 ) resonance in the nn system . the cd - bonn interaction is fitted to nn observables to about 350 mev laboratory projectile energy and thus coincides with the said result up to that energy . in fig . [ fig4 ] we show a comparison of elastic and breakup cross sections at 200 mev projectile laboratory energy for the 3d calculations based on our mt - iii model interaction and calculations based on a partial wave decomposition employing the cd - bonn potential . the top row displays the differential cross section for elastic scattering . we see that in both cases the difference between the fully relativistic calculation and the nonrelativistic one is overall quite small , and mostly visible at the backward angles , an observation already made in @xcite . the differential cross section in the forward direction is much larger for our model interaction , which is consistent with the larger two - body total cross section . in addition , there are more diffraction minima in the bosonic case than in the fermionic case , however , the minimum at around 130@xmath117 is present in both calculations . in the middle row we display the cross section for inclusive breakup scattering as function of the laboratory kinetic energy of the ejected particle at fixed laboratory angle 18@xmath117 . both cross sections are qualitatively similar , the fully converged faddeev calculation gives a lower cross section than the first order calculation , indicating the importance of rescattering contributions at this low energy . the difference between the relativistic and nonrelativistic calculations is quite small in both cases . in the calculation based on the cd - bonn interaction the fsi peak is more pronounced due to the virtual bound state in the @xmath118s@xmath119 state . the latter is absent in the mt - iii model . the bottom row shows the five - fold differential cross section as function of the arc - length @xmath0 for a configuration in which the laboratory angles @xmath120 are given in the scattering plane ( @xmath121 ) . the position of the peaks is identical for both calculations , which is a manifestation that peak structures are given by the kinematics of the problem . in both cases the relativistic calculation gives a significantly larger cross section for the central peak at @xmath122 mev compared to the nonrelativistic result , an increase by a factor of @xmath1231.5 for the full partial wave calculation and by a factor @xmath123 2 for the full 3d calculation . this increase is already present in both first order calculations . for the mt - iii model this trend is the same for all other peaks , whereas for the cd - bonn model the nonrelativistic calculations give a slightly larger cross section compared to the relativistic one in the peaks at small and large values of arc - length s. summarizing , the comparison of cross section obtained from our model interaction mt - iii with those given by a realistic nn interaction like cd - bonn at 200 mev indicates that , despite our model being quite simple , the qualitative features of especially the breakup cross sections are very similar . the differences between the fully relativistic calculations and their nonrelativistic counterparts are still quite small at this low energy for elastic scattering and inclusive breakup . for the exclusive breakup , however , even at this energy complete configurations with large changes of the nonrelativistic cross section due to relativity can be found . this sensitivity of the complete breakup to relativistic effects has already been observed in @xcite . starting from our model interaction we now consider three - body scattering in the energy regime up to 1.5 gev . the total cross section for elastic scattering is related to the symmetrized transition operator @xmath79 of eq . ( [ eq:2.1 ] ) via @xmath124 in fig . [ fig5 ] we display the total cross section for elastic scattering as a function of the projectile kinetic energy up to 1.5 gev obtained from our fully converged relativistic faddeev calculation as well as the one obtained from the first order term . it is obvious that , especially for the energies below 300 mev , the contribution of rescattering terms is huge . since the logarithmic scale from the top panel is unsuited to extract detailed information about the size of relativistic effects , we show in the lower two panels the relative difference of the relativistic calculations with respect to their nonrelativistic counterparts . the bottom panel displays the relative difference between the relativistic first order term and its nonrelativistic counterpart as dotted line . essentially the first order calculation does not show any effect . this is theoretically consistent when having in mind that in first order ( @xmath125 ) only the two - body t - matrix enters into the cross section . the relativistic two - body t - matrix is constructed to be phase - shift equivalent to the non - relativistic one via the cps method @xcite . thus , seeing no difference between the fully relativistic and the corresponding nonrelativistic calculations indicates that relativistic effects are taken into account consistently at the two - body level . doing the same comparison with fully converged faddeev calculations ( solid line in the middle panel ) indicates that relativistic effects in the three - body problem increase the elastic scattering total cross section with increasing energy . at our highest energy , 1.5 gev , this increase is about 8.3% . often only effects due to relativistic kinematics are taken into account . here we have the opportunity to study consequences of such a simple approximation . for the calculations labeled @xmath126 we only consider the lorentz transformations from laboratory to center - of - momentum ( c.m . ) frame and the relativistic phase space factor of eq . ( [ eq:3.1 ] ) , whereas the matrix elements of the operator @xmath79 are calculated from the solution of the nonrelativistic faddeev equation . the relative difference between this calculation and a completely nonrelativistic calculation is indicated by the short - dashed line in the bottom panel of fig . [ fig5 ] , where only the first order term is considered . the triple - dotted curve in the middle panel is the same comparison , but now between fully converged faddeev calculations . for both , full and first order calculation , the effect is huge . to understand better which piece of the kinematics included is responsible for this large enhancement of the cross section , we also plot in fig . [ fig5 ] calculations ( labeled nr@xmath127 ) , which only contain the lorentz transformation between the laboratory and c.m . frame , but carry the nonrelativistic phase space factor in eq . ( [ eq:3.1 ] ) . the dash - dotted line in the lower panel show the 1st order calculation and the dotted line in the middle panel the full faddeev calculation . using a lorentz transformation in the change of frames has the effect that the two - body t - matrix is calculated at a slightly different c.m . momentum @xmath43 , and thus there is a small effect , about a 5% underestimation of the total cross section . the huge effect is entirely due to the relativistic phase space factor , and relativistic dynamics then has an equally large effect of the opposite sign . this interplay of increasing effects due to the relativistic phase - space factor and decreasing effects due to relativistic dynamics has been already observed in the partial - wave based faddeev calculations with realistic interactions @xcite . it led for elastic scattering cross section at 250 mev to relativistic effects which are relatively small and restricted to backward angles . recent measurements of the neutron - deuteron ( @xmath128 ) differential cross section at 248 mev @xcite indicate that for discrepancies of theoretical predictions in this observable , short range components of a three - nucleon force are equally important . the problem with approximating relativistic effects only through kinematics and phase space factors can be easily understood in the 2 + 1 body problem , where the phase equivalence is achieved by choosing the invariant mass as a function of the non - relativistic two - body hamiltonian , @xmath129 . the eigenvalues equation for the scattering problems @xmath130 are equivalent , but the replacement of @xmath131 by @xmath132 must be compensated by replacing the interaction @xmath133 by @xmath134 . including only kinematic relativistic effects is equivalent to making the replacement @xmath135 without making the compensating replacement @xmath136 . we also study a more sophisticated approximation to the relativistic dynamics . in ref . @xcite we described in detail how we obtain the transition amplitude of the 2n subsystem , @xmath137 , embedded in the three - particle hilbert space which enters the faddeev equation , eq . ( [ eq:2.10 ] ) . the fully off - shell amplitude is the solution of a first resolvent type equation @xcite given by @xmath138 here @xmath139 is taken to be right half - shell with @xmath140 . note that in this equation the unknown matrix element is to the _ left _ of the kernel . it was suggested in ref . @xcite that a reasonable approximation to this embedded 2n transition amplitude might by the born term of the above integral equation , which is @xmath141 in this approximation , the fully off - shell 2n transition amplitude is replaced by a half - shell amplitude . the effect of this approximation is not large in elastic scattering , as shown in fig . [ fig6 ] , where we plot the differential cross section in forward direction for the fully relativistic calculations and the ones containing the approximation of eq . ( [ eq:3.3 ] ) to the boost ( curves labeled h ) . consistently and independent of projectile energy , approximating the embedded two - body t - matrix by the half - shell t - matrix leads to an underprediction of the differential cross section in forward direction . though not plotted , this also leads to a smaller total cross section for elastic scattering . finally , we want to investigate the convergence of the multiple scattering series as a function of the projectile laboratory kinetic energy . one might expect that with increasing energy only a few terms in the multiple scattering series are sufficient for a converged result . our converged relativistic faddeev calculations now allow a detailed study . this is of particular interest , since recently relativistic calculations in the energy regime around 1 gev have been published @xcite , which are carried out in a multiple scattering expansion of the faddeev equations up to 2nd order , and which use the off - shell continuation of the experimental nn amplitudes as two - body input . first we want to consider the convergence of the faddeev multiple scattering series in the total cross sections for elastic scattering as well as breakup reactions as a function of projectile kinetic energy . in the bottom row of fig . [ fig7 ] the different orders ( successively summed up as neumann sum to the order indicated in the legend ) are shown as functions of the projectile laboratory energy . we see a distinct difference in the behavior of the elastic total cross section in comparison with the breakup total cross section . while the elastic total cross section converges very rapidly , the total breakup cross section does not . the left upper panel of fig . [ fig7 ] shows the elastic total cross section as a function of the order in the multiple scattering series ( the orders are successively summed up the one indicated on the x - axis ) . even at 200 mev there is very little change due to contributions from the 2nd or higher order rescattering terms . for the higher energies , the 1st order term already captures the essential physics this is very different for the total breakup cross section , where for 200 mev projectile energy the full solution of the faddeev equation is clearly necessary . for energies of 1 gev and higher , at least one rescattering contribution ( 2nd order in the multiple scattering series ) is necessary to come close to the full solution . since the total cross section for elastic scattering might be insensitive to higher orders in the faddeev multiple scattering series , we plot in fig . [ fig8 ] the differential cross section at forward and backward angles as a function of the order in the multiple scattering series for the same laboratory projectile energies . here we see that at the lowest energy , 0.2 gev , the convergence is not as fast as the total cross section suggests . in fact , at least 5 orders are necessary , which is consistent with the experience from nonrelativistic calculations at low energies @xcite . for energies of 1 gev and higher , the forward direction is converged at the 3rd order in @xmath142 , whereas the backward angle is not as sensitive ( it should be pointed out that the cross section in backward direction is about five orders of magnitude smaller than the one in forward direction ) . it seems accidental that the multiple scattering series converges faster at 0.5 gev compared to 1 gev . however , a similar finding was presented in ref . @xcite , where it was observed that polarization observables for elastic proton - deuteron ( @xmath143 ) scattering at 395 mev were described better than those at 1.2 gev , when calculating the faddeev multiple scattering series up to the 2nd order . the calculation of breakup cross sections requires knowledge of the matrix element @xmath144 in eq . ( [ eq:2.1 ] ) . for details of the derivation we refer to ref . @xcite and only give the final expressions here . the five - fold differential cross section for exclusive breakup is given in the laboratory frame as @xcite @xmath145 here @xmath146 is the total energy of the system and @xmath16 its total momentum . the subscripts @xmath147 and @xmath148 indicate the two outgoing particles . in inclusive breakup only one of the particles is detected , and thus one of the angles in eq . ( [ eq:3.4 ] ) is integrated out . this leads to the inclusive breakup cross section in the laboratory frame @xcite @xmath149 where @xmath150 is determined by the condition @xmath151 . in refs . @xcite we already pointed out and demonstrated that relativistic kinematics is essential to obtain the correct position of e.g. the peak for quasi - free scattering ( qfs ) , especially at higher energies . the difference between a nonrelativistic calculation of the breakup cross section and a relativistic one is quite large at higher energies . however one may argue that this difference is artificially large , since it is natural to use relativistic kinematics at higher energies . therefore , here we will _ not _ compare to entirely nonrelativistic calculations , but rather calculations where the three - body transition amplitude has been obtained from the solution of a nonrelativistic faddeev equation , but the transformations between the laboratory frame and the c.m . frame are lorentz transformations . this is equivalent to comparing the relativistic and non - relativistic calculations in the center - of - momentum frame . in addition we use the relativistic phase space factor for the cross sections . in fig . [ fig9 ] we show the inclusive breakup cross section as a function of the laboratory kinetic energy of the ejected particle at fixed angle @xmath152 for different projectile kinetic energies calculated from the full solution of the relativistic faddeev equations together with ` nonrelativistic ' calculations using the above defined relativistic kinematics . there is still a shift of the position of the qfs peak towards lower ejectile energies , which increases with increasing projectile energy . there is also a very visible effect of the relativistic phase space factor used together with the nonrelativistic three - body transition amplitude . at 1000 mev the size of the qfs peak is a factor of two larger compared to exact relativistic calculation . for the lower energies the 1st order calculation yields a significantly higher qfs peak compared to the full calculation , whereas for the higher energies , the peak height is almost the same for the 1st order and the full calculation . next we investigate in detail the convergence of the faddeev multiple scattering series in the region of the qfs peak as a function of the projectile energy . in fig . [ fig10 ] we display calculations at selected energies from 200 to 1000 mev . the solid line represents the solution of the relativistic faddeev equation , whereas the other curves show the neumann sum of the multiple scattering series containing the sum up to the order in the two - body t - matrix as indicated in the legend . for the lowest energy , 200 mev , it is obvious that the multiple scattering series does not converge rapidly . this changes considerably as the projectile kinetic energy grows . though the variation of the different orders is not as large anymore at 500 mev , the multiple scattering series must still be summed up to 4th order in the qfs peak to coincide with the full result , whereas at 800 mev already the 2nd order is almost identical with the full result , and even a 1st order calculation can be considered quite good . this trend continues as the energy grows . of course , 1st order calculations are never able to capture the fsi peak at the maximum energy of the ejectile , nor do they describe the high energy shoulder of the qfs peak . however , our study indicates that for energies in the gev regime it is very likely sufficient to consider only one rescattering term when studying inclusive breakup reactions in the vicinity of the qfs peak . finally , we also want to study the approximation suggested in eq . ( [ eq:3.3 ] ) , namely replacing the off - shell two - body transition amplitude embedded in the three - body hilbert space by the half - shell one . the calculations based on the approximation of eq . ( [ eq:3.3 ] ) and labeled ( h ) are plotted in fig . [ fig11 ] together with the exact solution . considering only the 1st order calculation we observe a similar trend as in the differential cross section for elastic scattering , the approximation slightly underpredicts the exact result , independent of the energy under consideration . however , when this approximate two - body transition amplitude is iterated to all orders in the faddeev equation , the deviations from the exact calculations become larger . at 800 and 1000 mev the iteration of the exact amplitude increases the cross section in the qfs peak , whereas it decreases for the approximation with respect to the 1st order term . at 200 and 500 mev the approximation does not only give a smaller cross section in the qfs peak but also fails to develop an fsi peak towards the maximum allowed ejectile energies . from this we conclude that eq . ( [ eq:3.3 ] ) does not provide a good approximation for inclusive breakup cross sections . our calculations indicate that at energies 1 gev or higher , it is important to carry out the poincar invariant aspects of the calculation exactly . they also indicate that it is sufficient to consider only one rescattering term to capture most features of the cross section . although these conclusions are based on the use of a simple model two - body interaction , we conjecture that calculations based on realistic interactions will have similar characteristics . for our study of exclusive breakup scattering in the intermediate energy regime we choose two different experimental situations where there are data available . first we consider the @xmath153h(p,2p)n reaction at 508 mev , where the two outgoing protons are measured for a given angle pair @xmath154 in the scattering plane @xcite . since the convergence of the multiple scattering series is already discussed in @xcite , we only want to investigate the effect of the approximations previously given in this reaction . in fig . [ fig12 ] selected angle configurations are shown . the left column of the figure shows the first order calculations and the right column the full solution of the faddeev equation . the exact 1st order calculation is given by the dotted line in the left column and the exact full solution by the solid line in the right column . the angle combination @xmath155 is a qfs configuration . first , we see that in a qfs configuration , the 1st order calculation is already almost identical to the full faddeev calculation @xcite , whereas this is not the case for the other configurations shown . if only relativistic kinematics is considered , namely the lorentz transformations between laboratory and c.m . frame together with the relativistic phase space factor , and a nonrelativistic three - body transition amplitude is employed , we obtain the double - dotted curve for the 1st order calculations and the dashed line for the full solution of the faddeev calculation . again , the qfs configuration is quite insensitive to this approximation . however , the deviation from the exact calculation is quite visible in the other two configurations shown . finally , we also consider the approximation suggested by eq . ( [ eq:3.3 ] ) , which is indicated by the dash - dotted line , labeled ` h ' in the left column ( 1st order calculation ) and the dotted line in the right column ( full solution of the faddeev equation ) . here we see that even in the qfs configuration there are already deviations of this approximation for the high energy shoulder . the approximation underpredicts the full solution . this tendency becomes stronger for the other two configurations . the interesting property of this approximation is that while it appears to be a reasonable approximation to the faddeev kernel , the errors in the approximation increase when the equation is iterated . thus we conclude that this approximation , though simplifying the calculation of the two - body t - matrix embedded in the three - body hilbert space , does not seem to capture essential structures of the two - body t - matrix . the failure of this approximation , which approximates the off shell two body transition operator in the faddeev equation with the half - shell transition operator , suggests that some care is necessary in modeling the off - shell behavior of the transition operators in more phenomenological schemes . for breakup reaction at a slightly higher energy we consider the @xmath118h(d,2p)n reaction at 2 gev deuteron kinetic energy @xcite . here the two outgoing protons are measured . energetically , this reaction would correspond to @xmath143 scattering at roughly 1 gev and thus is within the range of the calculations presented here . in fig . [ fig13 ] we show the five - fold differential cross section as a function of the angle of the second detected proton for four different momenta of the first detected proton . the full relativistic faddeev calculation is represented by the solid line . in order to investigate the convergence of the multiple scattering series we show the 1st order calculation as a dotted line , then successively add one ( 2nd order ) and two ( 3rd order ) rescattering terms to the leading order . in this reaction , the first two rescattering terms are about the same size , but have opposite sign , so that the 3rd order calculations are very close to the 1st order one . we also observe that the 3rd order calculation is already so close to the full faddeev calculation that the neumann series can be considered converged with three terms . in this work we demonstrated the feasibility of applying poincar invariant quantum mechanics to model three - nucleon reactions at energies up to 2 gev . this is an important first step for studying dynamical models of strongly interacting particles in the energy range where sub - nuclear degrees of freedom are thought to be relevant . at these energies the poincar invariance of the theory is an essential symmetry . at lower energies non - relativistic quantum mechanical models are powerful tools for understanding the dynamics of strongly interacting nucleons . at higher energies the physics is more complicated , but one can expect that it is still dominated by a manageable number of degrees of freedom . poincar invariant quantum mechanics is the only alternative to quantum field theory where it is possible to realize the essential requirements of poincar invariance , spectral condition , and cluster properties @xcite . it has the advantage that the faddeev equation provides a mathematically well - defined method for exactly solving the strong interaction dynamics . the faddeev equation in this framework is more complicated than the corresponding non - relativistic equation , due to the non - linear relation between the mass and energy in relativistic theories , but these difficulties can be overcome @xcite . an important advance that allows these calculations to be extended to energies in the gev range is the use of numerical methods based on direct integrations , rather than partial wave expansions @xcite . these have been successfully applied to the non - relativistic three - nucleon problem . this paper demonstrates that they can also be successfully applied to the relativistic problem , even with its additional complications . the model presented here involves three nucleons interacting with a spin - independent malfliet - tjon @xcite type of interaction . it differs from more realistic interactions @xcite in that it is spin independent and it does not give a high - precision fit to the two - body scattering data . in addition , the model is for fixed numbers of particles , not allowing pion production , which is an open channel at these energies . while these limitations must be addressed in realistic applications , the three - body faddeev calculations presented in this paper provide a powerful framework for both testing approximations and for examining the sensitivity of scattering observables to relativistic effects . in order to investigate relativistic effects , we treat the interaction as if it was determined by fitting the cross section obtained by solving the non - relativistic lippmann - schwinger equation to scattering data . when this is done with a realistic interaction the experimental differential cross section is properly transformed from the lab frame to the center of momentum frame before the fit is done . the result of this process is that the computed differential cross section agrees with the fully - relativistic experimental differential cross section in the center of momentum frame as a function of the relative momentum . thus , even though the two - body scattering observables are computed with a non - relativistic equation , there is nothing non - relativistic about the result . at the two - body level the corresponding relativistic lippmann - schwinger equation must be designed to give the same scattering observables . this can be achieved by expressing the relativistic mass operator as a simple function of the non - relativistic center of momentum hamiltonian @xcite . the important consequence of this is that it does not make sense to relate the relativistic and non - relativistic two - body models using @xmath156 expansions ; the prediction of the relativistic and non - relativistic two - body models are identical . real differences in the dynamics appear when the two - body dynamical operators are used to formulate the three - body dynamics . how this must be done in the two and three - body cases is dictated , up to three - body interactions , by cluster properties . the faddeev equation for the relativistic and non - relativistic system have identical operators forms . the permutation operators , two - body transition operators and free resolvents that are input to the faddeev equation have different forms in the relativistic and non - relativistic equations . these differences are responsible for differences in the relativistic and non - relativistic three - body calculations . the calculations presented in this paper have a number of consequences . the most important result is a demonstration that direct integration methods can be successfully applied to extend the energy range for converged solutions to faddeev equations to intermediate energies . our estimates of the number of partial waves needed for calculations at different energies suggest that it is not currently practical to extend existing partial wave calculations beyond a few hundred mev , while in this paper we have demonstrated convergence of the direct integration methods for laboratory energies up to 2 gev . while our model interaction is not realistic , when we compared the results of our calculations to relativistic calculations at 200 mev that have been performed with realistic interactions @xcite in a partial wave basis , we found that the qualitative features of the realistic model are reproduced in our simple model , suggesting that some of the conclusions derived from our model should be applicable to models with realistic interactions . having a model where it is possible to perform numerically exact solutions of scattering observables in the intermediate energy range provides us with a tool to test approximations that have been used in other calculations as well as to look for observables that are sensitive to the differences between the relativistic and non - relativistic models . one common approximation that we tested is the replacement of non - relativistic kinematic factors by the corresponding relativistic kinematic factors in a non - relativistic model . our tests clearly illustrated a big effect , but most of it is canceled by the associated dynamical corrections . this suggests that including only kinematic corrections can actually provide large relativistic effects . such an approach should never be used in the absence of a complete theory where relativistic effects can be rigorously estimated . a second important set of approximations are multiple scattering approximations . these are expected to improve at higher energies , but it is important to understand in the context of models based on realistic interactions how high these energies have to be for convergence . our conclusion is that the convergence of the multiple scattering series is non - uniform . even at 200 mev our calculations show that the first - order term reproduces the total elastic cross section ; for the total breakup cross section at least one more iteration is needed up to about 600 mev . both of these observations turn out to be misleading when one investigates the differential cross sections . while the total elastic cross section is reproduced at 200 mev by the first order term , the correct angular distribution requires at least five orders in the multiple scattering series . even at 1 gev the first order approximation is not accurate enough at forward angles . for inclusive breakup reactions our computations show that the first order calculation does not give the right size of the quasifree peak even at 1 gev , however for 800 mev and above the second order term is a good approximation . for exclusive breakup the convergence of the multiple scattering series even at 1 gev energy depends on a specific configuration . another type of approximation that is employed is to use on - shell transition operators with a phenomenological representation of the off - shell dependence . in our formulation of the three - body problem that off shell behavior needs to be computed by solving a singular integral equation . it was suggested in @xcite that simply replacing the off - shell two - body @xmath157 by its on - shell value might be a good approximation . this was based on the observation that the difference between on and off shell faddeev kernel is small . our calculations show that while this does not lead to a large effect in the elastic cross section , the off - shell effects lead to non - trivial modifications when one considers the breakup cross sections . this shows that such an approximation should not be used and also suggests that phenomenological parameterizations of the off - shell behavior of the two - body amplitudes need to be carefully tested , especially for breakup reactions . while a number of calculations have shown small relativistic effects for the three - body binding energy , non - trivial effects have already been observed in scattering observables at 200 mev @xcite . our model confirms these previously observed effects and indicates that they continue into the intermediate energy region . our calculations exhibit a number of sensitivities to relativistic effects in the breakup observables both the shape and size of the quasielastic peak differ from the non - relativistic quantities . this paper demonstrates both the need for a relativistic description of few - nucleon dynamics in the intermediate energy range and shows that the problem is amenable to a numerically exact solution , using direct integration , for laboratory energies up to 2 gev . in the future relativistic few - body calculations will be important tools for testing the validity of approximations , such as the eikonal approximation . obviously extensions to include spin - dependent interactions , meson channels , and interactions that are fit to higher energy data will be needed for realistic applications . the success of the calculations in this paper provide a strong motivation for continuing this program . this work was performed in part under the auspices of the u. s. department of energy , office of nuclear physics , under contract no . de - fg02 - 93er40756 with ohio university , contract no . de - fg02 - 86er40286 with the university of iowa , and contract no . de - ac02 - 06ch11357 with argonne national laboratory . it was also partially supported by the helmholtz association through funds provided to the virtual institute `` spin and strong qcd '' ( vh - vi-231 ) . we thank the ohio supercomputer center ( osc ) for the use of their facilities under grant phs206 . part of the numerical calculations were performed on the ibm regatta p690 + of the nic in jlich , germany . h. liu , ch . elster , and w. glckle , phys.rev . c*72 * , 054003 ( 2005 ) . h. liu , ch . elster , and w. glckle , few - body systems 33 , 241 ( 2003 ) . elster , w. schadow , a. nogga , and w. glckle , few - body systems * 27 * , 83 ( 1999 ) . t. lin , ch . elster , w. n. polyzou and w. glckle , phys . rev . c*76 * , 014010 ( 2007 ) . wiringa , v.g.j . stoks , r. schiavilla , phys . c*51 * , 38 ( 1995 ) . r. machleidt , phys . c*63 * , 024001 ( 2001 ) . stoks , r.a.m . klomp , c.p.f . terheggen , and j.j.de swart , phys . c*49 * , 2950 ( 1994 ) . . the total elastic and break - up cross sections together with the total cross section extracted via the optical theorem calculated from a malfliet - tjon type potential as a function of the projectile laboratory kinetic energy . [ cols="^,^,^",options="header " , ] in the optical theorem as a function of the grid points in the angle grid @xmath88 , when this grid is increased successively by 4 gauss - legendre points . the different curves correspond to the three different laboratory projectile energies in gev , indicated in the legend . [ fig2],width=302 ] = 200 mev . the left column shows results obtained from the malfliet - tjon - iii potential assuming boson symmetry and no partial wave decomposition , the right column shows the corresponding realistic calculations obtained with the cd - bonn @xcite potential where partial wave decomposition is applied . the top row displays the differential cross section for elastic scattering , the middle row shows the breakup cross section for inclusive scattering for the laboratory angle @xmath158 of the outgoing particle . the bottom row shows the five - fold differential for exclusive breakup reaction as a function of the arc - length s. the laboratory angles of the outgoing particles are @xmath159 , and @xmath160 . the fully relativistic converged faddeev calculations are given by the solid lines ( r ) , the corresponding nonrelativistic calculations by the long - dashed lines ( nr ) . in addition the relativistic ( dash - dot ) and nonrelativistic ( dotted ) first order calculations are shown . [ fig4],width=566 ] ) are given as dotted line for a full faddeev calculation and as short - dashed line for the first order term . calculations which only take into account the lorentz transformations between the laboratory and c.m . frame ( labeled nr@xmath127 ) are shown as dotted line for the full faddeev calculation and as dash - dotted line for the 1st order one . the middle panel shows the relative difference between the fully relativistic faddeev calculation and the nonrelativistic one ( solid line ) together with the difference to the nonrelativistic calculation if only relativistic kinematics is considered . the bottom panel shows the corresponding relative differences when only the first order term is taken into account . [ fig5],width=340 ] for selected laboratory kinetic energies . the converged solution of the relativistic faddeev equation is given as solid line . the dotted line shows the converged solution of the relativistic faddeev equation in which the fully off - shell 2n t - matrix is replaced by the half - shell t - matrix . the corresponding first order calculation are given by the short - dashed line and the dash - dotted line . for details see text . [ fig6],width=566 ] of the emitted particle at an emission angle @xmath152 . the incident laboratory kinetic energy for each cross section is indicated in each panel . the solid lines ( r full ) represent the converged relativistic faddeev calculation and the dotted line the corresponding first order calculations ( r 1st ) . the lines labeled r@xmath161 correspond to calculations in which only relativistic kinematics is taken into account . [ fig9],width=566 ] of the emitted particle at an emission angle @xmath152 . the incident laboratory kinetic energy for each cross section is indicated in each panel . the solid lines ( r ) represent the converged relativistic faddeev calculation . the triple - dotted line shows the 1st order calculation , for the short dashed line the 2nd order contribution is added to the previous , for the dash - dotted line the 3rd order is added , and for the dotted line the 4th order . [ fig10],width=566 ] of the emitted particle at an emission angle @xmath152 . the incident laboratory kinetic energy for each cross section is indicated in each panel . the solid lines ( r ) represent the converged relativistic faddeev calculation . the dotted line ( h ) displays the calculation in which the fully off - shell two - body t - matrix is replaced by the half - shell one . the calculations labeled 1st stand for the corresponding 1st order calculations . [ fig11],width=566 ] h(p,2p)n reaction at 508 mev laboratory projectile energy for different proton angle pairs @xmath162-@xmath163 with respect to the beam axis as a function of the laboratory kinetic energy of the first detected proton . the left column represents 1st order calculation , whereas the right column gives full solution of the faddeev equation . the curves labeled r ( solid for the full faddeev calculation and dotted for the 1st order one ) represent the full relativistic calculations , whereas for the curves labeled r@xmath161 ( dashed in the right column and double - dotted in the left ) only relativistic kinematics is taken into account ( see text ) , and for the curves labeled h ( dotted in the right column and dash - dotted in the left ) the fully off - shell t - matrix is replaced by the half - shell one . the data are taken from ref . @xcite . [ fig12],width=566 ] h(d,2p)n at 2 gev deuteron energy as a function of the angle @xmath163 of the second of the outgoing protons for a fixed first proton momentum indicated in the figure . the solid line represents the solution of the full relativistic faddeev equation . the dotted line gives the result of the first order calculation . for the dashed line the 2nd order term is added and for the dash - dotted line the 2nd and 3rd order terms are added . the data are taken from ref . [ fig13],width=566 ]
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the relativistic faddeev equation for three - nucleon scattering is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition .
the equation is solved through pad summation , and the numerical feasibility and stability of the solution is demonstrated .
relativistic invariance is achieved by constructing a dynamical unitary representation of the poincar group on the three - nucleon hilbert space .
based on a malfliet - tjon type interaction , observables for elastic and break - up scattering are calculated for projectile energies in the intermediate energy range up to 2 gev , and compared to their nonrelativistic counterparts .
the convergence of the multiple scattering series is investigated as a function of the projectile energy in different scattering observables and configurations .
approximations to the two - body interaction embedded in the three - particle space are compared to the exact treatment .
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a _ @xmath0-graph _ is a graph which at each vertex has a bijection from the outgoing half - edges to the vertices of a cycle graph . half - edges which are mapped to adjacent vertices are ( formally ) _ adjacent_. half - edges are said to be _ opposite _ if they are mapped to vertices of maximal distance in the cycle graph . by an embedding of a @xmath0-graph @xmath2 into a surface @xmath3 we always mean an embedding of @xmath2 into @xmath3 such that the formal relation of being adjacent coincides with the relation of being adjacent induced by the embedding . an _ angle _ in a @xmath0-graph is a pair of adjacent half - edges at a vertex . a _ checkerboard embedding _ of a @xmath0-graph @xmath2 into @xmath3 is an embedding such that the cells of @xmath4 admit a 2-coloring under which any cells with a common edge have different colors . checkerboard embeddings are exactly those embeddings whose first @xmath1-homology class is zero . in @xcite the second named author ( v.o.m . ) gave a solution to the question of whether four - valent framed graphs are planar . in @xcite , he addressed the question of determining the genus of surfaces into which four - valent framed graphs can be embedded , in particular considering the special case of surfaces into which four - valent framed graphs may be checkerboard - embedded . in @xcite , the first named author ( t.f . ) introduced @xmath0-graphs as a generalization of four - valent framed graphs , and gave a planarity condition for @xmath0-graphs with each vertex of degree 4 or 6 . in @xcite , the authors characterized the genera of orientable surfaces into which @xmath0-graphs with each vertex of degree 4 or 6 may be checkerboard - embedded , generalizing some of the results of @xcite . in this paper , we continue the project of generalizing results about embeddability properties of framed four - valent graphs to @xmath0-graphs with each vertex of degree 4 or 6 , now considering checkerboard - embeddability into nonorientable surfaces . in theorem [ main ] we show that this is equivalent to a problem on matrices . our methods are close to those used in our previous paper @xcite , which were themselves based closely on those used by the second named author ( v.o.m . ) in @xcite for framed four - valent graphs . the goal of this paper is to provide a method for determining whether a given @xmath0-graph @xmath2 has a checkerboard embedding into a nonorientable surface of genus @xmath5 . we show that this is equivalent to a problem on matrices . to accomplish this , we fix a cycle @xmath6 in @xmath2 satisfying certain properties , called a rotating - splitting cycle . then we define a correspondence between checkerboard embeddings of @xmath2 and `` permissible separations '' of a chord diagram @xmath7 , where the result of a permissible separation is a pair of chord diagrams . we then show that the number of white ( black ) cells in the embedding is equal to the number of circles resulting from surgery of the first ( second ) of these chord diagrams . the circuit - nullity theorem allows us to calculate the number of circles resulting from surgery of each chord diagram in terms of the rank of their intersection matrices over @xmath1 . from this we have the total number of cells in the embedding , from which the genus @xmath5 of the surface can be easily calculated . the authors of this paper would like to thank victor anatolievich vassiliev and sergei vladimirovich chmutov for valuable discussions . a _ @xmath0-atom _ is a closed 2-surface @xmath3 into which a connected graph @xmath2 ( the _ skeleton _ of the @xmath0-atom ) is embedded in such a way that it divides @xmath3 into black and white cells so that cells sharing an edge have different colors . this embedding induces a @xmath0-structure on the skeleton . the @xmath0-structure at each vertex determines a set of @xmath8 angles among which we say that two angles are adjacent if they share a half - edge . two adjacent angles never have the same color . thus the angles around a vertex can be partitioned into two sets @xmath9 and @xmath10 in such a way that for any @xmath0-atom corresponding to the @xmath0-graph @xmath2 , either all angles in @xmath9 are black and all angles in @xmath10 are white , or all angles in @xmath9 are white and all angles in @xmath10 are black . thus given a connected @xmath0-graph @xmath2 , the @xmath0-atoms corresponding to @xmath2 are uniquely determined by a choice of one of the two possible colorings at each vertex . thus the main problem can be reformulated as follows : given a @xmath0-graph @xmath2 in which all vertices have degree 4 or 6 , choose a coloring for the angles around each vertex such that the genus of the resulting atom is minimal . if such a graph has @xmath11 vertices , there are @xmath12 corresponding @xmath0-atoms . an _ euler circuit _ @xmath6 of a @xmath0-graph @xmath2 is a surjective mapping @xmath13 which is one - to - one except at the vertices , and such that every vertex of degree @xmath8 has @xmath14 preimages . given an euler circuit @xmath6 of a @xmath0-graph @xmath2 , a 4-vertex @xmath15 is _ rotating _ with respect to @xmath6 if for every @xmath16 , @xmath17 and @xmath18 are on adjacent half - edges around @xmath19 . given an euler circuit @xmath6 of a @xmath0-graph @xmath2 , a 6-vertex @xmath15 is _ rotating _ with respect to @xmath6 if for every @xmath16 , @xmath17 and @xmath18 are on adjacent half - edges around @xmath19 . given an euler circuit @xmath6 of a @xmath0-graph @xmath2 , a 6-vertex @xmath15 is _ splitting _ with respect to @xmath6 if for some @xmath16 , @xmath17 and @xmath18 are on opposite half - edges around @xmath19 , and for the other two points @xmath20 , @xmath21 and @xmath22 are on adjacent half - edges , and @xmath23 and @xmath24 are on adjacent half - edges . a _ rotating - splitting circuit _ is a circuit with respect to which every vertex is rotating or splitting . a rotating - splitting circuit induces an orientation on the half - edges around a rotating 6-vertex . if the order of the edges containing @xmath17 and @xmath18 agrees with this orientation , then the angle is said be _ untwisted _ ; otherwise it is _ twisted_. see figures [ r0 t ] , [ r1 t ] , [ r2 t ] , [ r3 t ] . [ rscycle - exists ] if @xmath2 is a connected @xmath0-graph in which all vertices have degree 4 or 6 , then @xmath2 admits a rotating - splitting circuit . assign to each vertex any rotating or splitting structure . this gives a partition of the edges of @xmath2 into edgewise disjoint cycles . if there is only one such cycle , we are done . if there is more than one cycle , since the graph is connected , there must be a vertex @xmath19 shared by different cycles . if @xmath19 has degree 4 , it must be rotating , and we can join the two cycles meeting at @xmath19 by assigning to @xmath19 the other possible rotating structure . if @xmath19 has degree 6 , we consider the cycles given by starting at @xmath19 , exiting through one of its incident edges , and following the rotating - splitting structure until we come back to @xmath19 . there are three such cycles , up to a change in orientation . if each of these cycles contains a pair of adjacent edges at @xmath19 , we can assign to @xmath19 the rotating structure shown on the right side of figure [ r0t - intro ] , so that the three cycles are joined together , and @xmath19 becomes a rotating vertex with no twisted angles . note that before making the change , @xmath19 may have some structure other than that shown on the left side of figure @xmath25 ; the left side of the figure and the others referenced in this proof are merely examples . if exactly two of the three cycles contain a pair of adjacent edges around @xmath19 , then the third must contain a pair of opposite edges , and we can assign to @xmath19 the splitting structure shown on the right side of figure [ s0t - intro ] to join the cycles together . if exactly one of the three cycles contains a pair of adjacent edges around @xmath19 , then the other two must contain pairs of edges which are neither opposite nor adjacent . in this case we can join the cycles by assigning the rotating structure shown on the right side of figure [ r1t - intro ] , so that @xmath19 becomes a rotating vertex with one twisted angle . if none of the cycles contains a pair of adjacent edges , then we have two possibilities : each of the cycles contains a pair of opposite edges , or one of the cycles contains a pair of opposite edges and the other two contain a pair of edges which are neither opposite nor adjacent . if each of the cycles contains a pair of opposite edges , we can assign to @xmath19 the rotating structure shown on the right side of figure [ r3t - intro ] to join the cycles . if one of the cycles contains a pair of opposite edges and the other contains a pair of edges which are neither opposite nor adjacent , we can assign to @xmath19 the rotating structure shown on the right side of figure [ r2t - intro ] to join the cycles . . [ r0t - intro ] . [ s0t - intro ] . [ r1t - intro ] . [ r3t - intro ] . [ r2t - intro ] a _ chord diagram _ is a cubic graph @xmath26 with a distinguished hamiltonian cycle ; i.e. an embedding @xmath27 which covers all the vertices of @xmath26 . a _ signed chord diagram _ is a chord diagram in which each edge not in the distinguished cycle is assigned a positive or negative sign . a @xmath0-chord diagram is a graph @xmath26 with a distinguished simple cycle ( i.e. an embedding @xmath28 ) , such that every vertex in @xmath26 has degree 3 or 4 and for every edge @xmath29 in @xmath26 , one of the following holds : 1 . @xmath29 is in the distinguished cycle . 2 . both of the vertices on @xmath29 are in the distinguished cycle , and both have degree 3 one of the vertices on @xmath29 is in the distinguished cycle , the other is not , and both have degree 3 . both of the vertices on @xmath29 are in the distinguished cycle , one has degree 3 , and the other has degree 4 . a _ signed @xmath0-chord diagram _ is a @xmath0-chord diagram in which each edge not in the distinguished cycle is assigned a positive or negative sign . an _ arc _ of a @xmath0-chord diagram @xmath2 is an edge in the distinguished cycle of @xmath2 . a _ chord _ of a @xmath0-chord diagram @xmath2 is an edge not in the distinguished cycle of @xmath2 , connecting two vertices of degree 3 which are in the cycle . a _ triad _ of a @xmath0-chord diagram @xmath2 is a vertex @xmath19 not in the distinguished cycle of @xmath2 , together with the three edges incident to @xmath19 . the vertex @xmath19 is called a _ triad point_. a _ double chord _ of a @xmath0-chord diagram @xmath2 is a pair of edges not in the distinguished cycle of @xmath2 , which are incident to a shared vertex @xmath19 . the vertex @xmath19 is called the _ principal vertex _ of the double chord . given a @xmath0-graph @xmath2 with all vertices of degree 4 or 6 , and given a rotating - splitting circuit @xmath6 of @xmath2 , we define a @xmath0-chord diagram @xmath30 as follows : for each 4-vertex @xmath19 in @xmath2 , the two points in @xmath31 which are mapped to @xmath19 by @xmath6 are connected by a chord , whose sign is positive if and only if the two half - edges through which @xmath6 enters @xmath19 are not adjacent . for any rotating 6-vertex @xmath19 in @xmath2 , the three points in @xmath31 which are mapped to @xmath19 by @xmath6 are connected by a triad , and the edge connecting a vertex @xmath16 to the triad point has positive sign if and only the angle into which @xmath6 maps a neighborhood of @xmath32 is not twisted . for any splitting 6-vertex @xmath19 in @xmath2 , the three points in @xmath31 which are mapped to @xmath6 are connected by a double chord , whose principal vertex is @xmath16 such that @xmath18 and @xmath17 are in opposite half - edges around @xmath19 , where the sign of the edge connecting @xmath32 another vertex @xmath33 is positive if and only if @xmath21 is adjacent to @xmath18 [ figure ] . an _ expansion _ of as signed @xmath0-chord diagram @xmath30 a signed chord diagram @xmath7 such that 1 . for every chord in @xmath34 containing vertices @xmath32 and @xmath35 , there is a chord of the same sign in @xmath36 of the same sign connecting vertices @xmath32 and @xmath35 . 2 . for every triad @xmath29 in @xmath37 containing at least one edge with positive sign , for some labeling @xmath38 of the vertices of @xmath29 such that the edge connecting @xmath32 to the triad point is positive , @xmath36 contains a chord connecting @xmath39 to @xmath35 and a chord connecting @xmath40 to @xmath41 , with @xmath42 chosen in such a way that the chords are not linked . furthermore , the chords connecting @xmath32 to @xmath35 and @xmath32 to @xmath41 in @xmath36 have the same signs as the edges connecting the triad point of @xmath29 to @xmath35 and @xmath41 in @xmath37 , respectively . 3 . for every triad @xmath29 in @xmath37 in which all edges have negative sign , for some labeling @xmath38 of the vertices of @xmath29 , @xmath36 contains a chord connecting @xmath39 to @xmath35 and a chord connecting @xmath40 to @xmath41 , with @xmath42 chosen in such a way that the chords are linked . furthermore , the chords connecting @xmath32 to @xmath35 and @xmath32 to @xmath41 in @xmath36 have signs opposite to the edges connecting the triad point of @xmath29 to @xmath35 and @xmath41 in @xmath37 , respectively . 4 . for every double chord @xmath29 in @xmath37 with principal vertex @xmath32 , for some labeling @xmath43 of the nonprincipal vertices of @xmath29 , there is a chord in @xmath36 connecting @xmath44 to @xmath35 and a chord in @xmath36 connecting @xmath45 to @xmath41 . these chords are not linked . a _ permissible separation _ of a signed chord diagram @xmath7 arising as an expansion of a signed @xmath0-chord diagram @xmath30 is a pair of signed chord diagrams @xmath46 and @xmath47 such that 1 . every chord in @xmath7 is in exactly one of @xmath46 and @xmath47 . 2 . two chords in @xmath7 which come from the same triad in @xmath30 are both in @xmath46 , or both in @xmath47 . 3 . of any two chords in @xmath7 which come from the same double chord in @xmath30 , one chord is in @xmath46 and the other is in @xmath47 . suppose @xmath2 is checkerboard - embedded in a closed surface @xmath3 . then the rotating - splitting circuit @xmath6 gives a mapping from @xmath31 to @xmath3 which is one - to - one except at the preimages of vertices @xmath2 . this mapping can be smoothed to give an embedding of @xmath31 into @xmath3 , as in figure [ fig : smoothed_rscycle ] . observe that the circle @xmath48 divides the surface into a black part and a white part . we can draw the chords of @xmath7 as small edges lying in neighborhoods of vertices of @xmath2 , see figure [ fig : small_chord ] . 0.3 0.3 0.3 [ fig : small_chord ] the coloring of @xmath3 divides the chords of @xmath7 into two families : those lying in the white part and those lying in the black part . observe that the two chords in the neighborhood of a rotating vertex are in the same part , and the two chords in the neighborhood of a splitting vertex are in different parts . thus we have a permissible separation of @xmath7 . vice versa , given a @xmath0-graph @xmath2 with all vertices of degree 4 or 6 and which satisfies the source - sink condition , a rotating - splitting circuit @xmath6 of @xmath2 , and a permissible separation of @xmath7 , we can recover the coloring of the angles around each vertex of @xmath2 , and thus we can recover the surface @xmath3 . thus given a @xmath0-graph @xmath2 with all vertices of degree 4 or 6 and which satisfies the source - sink condition , and an expansion @xmath7 of its @xmath0-chord diagram , we have a one - to - one correspondence between atoms of @xmath2 and permissible separations of @xmath7 . note that the two chords to be drawn in the neighborhood of any rotating 6-vertex @xmath19 do not cross in @xmath3 , as shown in figures [ r0t - lifecycle ] , [ r1t - lifecycle ] , [ r2t - lifecycle ] , and [ r3t - lifecycle ] . thus we have an embedding of @xmath7 into @xmath3 . furthermore , since the embedding of @xmath2 divides @xmath3 into 2-cells , the embedding of @xmath7 does as well . given a chord diagram @xmath26 , _ surgery _ of @xmath26 is the following process : for each chord @xmath29 connecting points @xmath49 , delete a neighborhood of @xmath29 and connect the obtained endpoints @xmath50 to @xmath51 and @xmath52 to @xmath53 if @xmath29 is positive , and @xmath50 to @xmath53 and @xmath52 to @xmath51 if @xmath29 is negative . this produces a family of circles ; these are called the _ result of surgery _ of @xmath26 . to form the _ intersection matrix _ of a signed chord diagram @xmath26 with @xmath11 chords , first enumerate the chords @xmath54 . then the _ intersection matrix _ @xmath55 is an @xmath56 matrix over @xmath1 , such that @xmath57 if and only if the chord @xmath58 is negative , and @xmath59 for @xmath60 if and only if the chords @xmath58 and @xmath61 are linked . the number of components in the manifold obtained from a signed chord diagram @xmath26 by surgery of the circle is one plus the corank of @xmath55 . [ sep - to - genus ] given a @xmath0-graph @xmath2 in which all vertices have degree 4 or 6 and a rotating - splitting circuit @xmath6 of @xmath2 , and a checkerboard embedding of @xmath2 into a nonorientable surface @xmath3 , the nonorientable genus of @xmath3 is given by @xmath62 where @xmath46 and @xmath47 are the results of the permissible separation of @xmath7 induced by the embedding . consider the embedding @xmath63 described above . the number of 2-cells on the white side of the embedding is the number of circles resulting in surgery of @xmath46 . likewise , the number of 2-cells on the black side of the embedding is the number of circles resulting in surgery of @xmath47 . applying the circuit nullity theorem , the total number of 2-cells is @xmath64 . introducing the notation @xmath65 to represent the number of chords in @xmath7 , the number of arcs in @xmath7 is @xmath66 , so its total number of edges is @xmath67 . the number of vertices in @xmath7 is @xmath66 . thus the euler characteristic of @xmath3 is @xmath68 @xmath69 so the nonorientable genus of @xmath3 is @xmath70 . thus @xmath2 has admits an atom of genus @xmath5 if and only if some permissible separation of @xmath30 results in @xmath71 . this can be reduced to a problem on matrices , as follows . a _ permissible partition _ of the indices of @xmath72 is a partition of the indices of @xmath72 ( which are just the chords of @xmath7 ) into two parts , in such a way that chords arising from the same triad in @xmath30 are in the same part , and chords arising from the same double chord in @xmath30 are in different parts . clearly , @xmath46 and @xmath47 are a permissible separation of @xmath7 if and only if there exists a permissible partition @xmath73 of the indices of @xmath72 such that @xmath74 is the intersection matrix of @xmath46 and @xmath75 is the intersection matrix of @xmath47 . [ main ] for a @xmath0-graph @xmath2 which does not satisfy the source - target condition and which has rotating - splitting circuit @xmath6 , @xmath2 has a checkerboard embedding into a nonorientable surface of genus @xmath5 if and only if there is a permissible partition of the indices of @xmath76 into parts @xmath77 and @xmath78 such that @xmath79 . let @xmath7 be any expansion of @xmath30 . by lemma [ sep - to - genus ] , @xmath2 has a checkerboard embedding into a surface of genus @xmath5 if and only there is a permissible separation @xmath80 of @xmath7 such that @xmath81 . such a permissible separation exists if and only if there is a permissible partition of @xmath76 into parts @xmath77 and @xmath78 such that @xmath79 . thus , the problem of finding the minimal nonorientable genus into which a @xmath0-chord diagram with each vertex of degree 4 or 6 may be checkerboard - embedded , is equivalent to the problem of finding a permissible partition of the indices of a matrix @xmath82 into parts @xmath77 and @xmath78 which minimizes @xmath83 . a @xmath0-graph @xmath2 with rotating - splitting circuit @xmath6 is embeddable into the projective plane if and only if there exists a permissible separation @xmath80 of @xmath36 such that @xmath85 and @xmath86 . in other words , @xmath2 is @xmath84-embeddable if and only if there exists a permissible separation of @xmath7 into two chord diagrams , one of which consists of a family of pairwise - linked negative chords and a family of positive chords which are not linked to each other or to the negative chords , and the other of which consists of a family of pairwise - unlinked negative chords . we can test this condition by the following algorithm , which takes time quadratic in the number of chords of @xmath36 : first assign all negative chords to the same chord diagram . then for each assigned chord , assign all positive chords linked to it to the other chord diagram . if an assigned chord originates from a triad , assign the other chord coming from this triad to the same chord diagram , and if the assigned chord originates from a double chord , assign the other chord coming from this double chord to the other chord diagram . then , for each of the newly assigned chords , assign any unassigned linked chords or chords coming from the same triad or double chord , using the same rules described above . repeat this process until for every assigned chord , the linked chords and any chord coming from the same triad or double chord have been assigned . if not all chords have been assigned , take any unassigned chord and arbitrarily assign it to @xmath46 or @xmath47 , and repeat until all chords have been assigned . finally , check whether this is a permissible separation , and whether @xmath87 . @xmath2 is @xmath84-embeddable if and only if both of these conditions are true . a @xmath0-graph @xmath2 with rotating - splitting circuit @xmath6 is embeddable into the klein bottle if and only if there exists a permissible separation @xmath80 of @xmath36 such that @xmath88 . there are two possible cases in which this can occur : @xmath89 or @xmath90 and @xmath86 . we will first consider the case where @xmath89 . in this case , we have a permissible separation of @xmath7 , each of which consists of a family of pairwise - linked negative chords and a family of positive chords which are not linked to each other or to the negative chords . this condition also admits a quadratic - time test , as follows : assign one of the chords arbitrarily to @xmath46 or @xmath47 . if the chord is positive , assign all chords linked to it to the other chord diagram ; if it is negative , assign all positive linked chords and all negative unlinked chords to the other diagram . regardless of sign , if an assigned chord originates from a triad , assign the other chord coming from this triad to the same chord diagram , and if the assigned chord originates from a double chord , assign the other chord coming from this double chord to the other chord diagram . repeat this process until for every assigned chord , the linked chords and any chord coming from the same triad or double chord have been assigned . if not all chords have been assigned , take any unassigned chord and arbitrarily assign it to @xmath46 or @xmath47 , and repeat until all chords have been assigned . finally , check whether this is a permissible separation , and whether @xmath89 . these conditions are met if and only if there is an embedding of @xmath2 into the klein bottle so that a smoothing of @xmath6 divides the klein bottle into two mbius bands . if this test fails , there is still the possibility that @xmath2 has an embedding into the klein bottle where the smoothing of @xmath6 bounds a disc . to cover this possibility , we choose a negative chord @xmath41 of @xmath36 and perform surgery at that chord , in the manner shown in [ klein - surgery ] . since this reverses the orientation of part of the designated cycle , we should also change the sign of all chords which cross @xmath41 , producing a new chord diagram @xmath91 . then for any surface @xmath3 and any embedding of @xmath92 which respects the signs of the chords , there is a corresponding embedding @xmath93 , still respecting the signs of the chords . furthermore , if the distinguished cycle in the embedding @xmath92 into the klein bottle bounds a disc , then the distinguished cycle in the embedding @xmath93 bounds a mbius band . thus @xmath2 has an embedding into the klein bottle where the smoothing of @xmath6 bounds a disc if and only @xmath94 has an embedding into the klein bottle where the distinguished cycle bounds a mbius band . this condition can be checked using the algorithm in the previous paragraph .
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this paper considers @xmath0-graphs in which all vertices have degree 4 or 6 , and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded . in a previous paper @xcite by the authors , the problem of calculating whether a given @xmath0-graph in which all vertices have degree 4 or 6 admits a @xmath1-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices . here
we extend those results to nonorientable surfaces .
the embeddability condition that we obtain yields quadratic - time algorithms to determine whether a @xmath0-graph with all vertices of degree 4 or 6 admits a @xmath1-homologically trivial embedding into the projective plane or into the klein bottle .
* keywords : * graph , @xmath0-graph , surface , embedding , genus * ams subject classification : * primary 05c10 ; secondary 57c15 , 57c27
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in this article we consider markov processes that are stopped when reaching the boundary of a given set @xmath1 . these processes appear in a wide range of applications , such as population genetics @xcite , finance @xcite , neuroscience @xcite , physics @xcite and engineering @xcite . the vast majority of the papers in the literature deal with fully observed stopped processes and assume the parameters of the model are known . in this paper we address problems when this is not the case . in particular , bayesian inference for the model parameters is considered , when the stopped process is observed indirectly via data . we will propose a generic simulation method that can cope with many types of partial observations . to the best of our knowledge , there is no previous work in this direction . an exception is @xcite , where maximum likelihood inference for the model parameters is investigated for the fully observed case . in the fully observed case , stopped processes have been studied predominantly in the area of rare event simulation . in order to estimate the probability of rare events related to stopped processes , one needs to efficiently sample realisations of a process that starts in a set @xmath0 and terminates in the given rare target set @xmath1 before returning to @xmath0 or getting trapped in some absorbing set . this is usually achieved using importance sampling ( is ) or multi - level splitting ; see@xcite and the references in those articles for an overview . recently , sequential monte carlo ( smc ) methods based on both these techniques have been used in @xcite . in @xcite the authors also prove under mild conditions that smc can achieve same performance as popular competing methods based on traditional splitting . sequential monte carlo methods can be described as a collection of techniques used to approximate a sequence of distributions whose densities are known point - wise up to a normalizing constant and are of increasing dimension . smc methods combine importance sampling and resampling to approximate distributions . the idea is to introduce a sequence of proposal densities and to sequentially simulate a collection of @xmath4 samples , termed particles , in parallel from these proposals . the success of smc lies in incorporating a resampling operation to control the variance of the importance weights , whose value would otherwise increase exponentially as the target sequence progresses e.g. @xcite . applying smc in the context of fully observed stopped processes requires using resampling while taking into account how close a sample is to the target set . that is , it is possible that particles close to @xmath1 are likely to have very small weights , whereas particles closer to the starting set @xmath0 can have very high weights . as a result , the diversity of particles approximating longer paths before reaching @xmath1 would be depleted by successive resampling steps . in population genetics , for the coalescent model @xcite , this has been noted as early as in the discussion of @xcite by the authors of @xcite . later , in @xcite the authors used ideas from splitting and proposed to perform the resampling step only when each sample of the process reaches intermediate level sets , which define a sequence of nested sets from @xmath0 to @xmath1 . the same idea appeared in parallel in ( * ? ? ? * section 12.2 ) , where it was formally interpreted as an interacting particle approximation of appropriate multi - level feynman - kac formulae . naturally , the choice of the intermediate level sets is critical to the performance of such a scheme . that is , the levels should be set in a `` direction '' towards the set @xmath1 and so that each level can be reached from the previous one with some reasonable probability @xcite . this is usually achieved heuristically using trial simulation runs . also more systematic techniques exist : for cases where large deviations can be applied in @xcite the authors use optimal control and in @xcite the level sets are computed adaptively on the fly using the simulated paths of the process . the contribution of this paper is to address the issue of inferring the parameters of the law of the stopped markov process , when the process itself is a latent process and is only partially observed via some given dataset . in the context of bayesian inference one often needs to sample from the posterior density of the model parameters , which can be very complex . employing standard markov chain monte carlo ( mcmc ) methods is not feasible , given the difficulty one faces to sample trajectories of the stopped process . in addition , using smc for sequential parameter inference has been notoriously difficult ; see @xcite . in particular , due to the successive resampling steps the simulated past of the path of each particle will be very similar to each other . this has been a long standing bottleneck when static parameters @xmath5 are estimated online using smc methods by augmenting them with the latent state . these issues have motivated the recently introduced particle markov chain monte carlo ( pmcmc ) @xcite . essentially , the method constructs a markov chain on an extended state - space in the spirit of @xcite , such that one may apply smc updates for a latent process , i.e. use smc approximations within mcmc . in the context of parameter inference for stopped process this brings up the possibility of using the multi - level smc methodology as a proposal in mcmc . this idea to the best of our knowledge has not appeared previously in the literature . the main contributions made in this article are as follows : * when the sequence of level sets is fixed _ a priori _ , the validity of using multi - level smc within pmcmc is verified . * to enhance performance we propose a flexible scheme where the level sets are adapted to the current parameter sample . the method is shown to produce unbiased samples from the target posterior density . in addition , we show both theoretically and via numerical examples how the mixing of the pmcmc algorithm is improved when this adaptive strategy is adopted . this article is structured as follows : in section [ sec : problem - formulation ] we formulate the problem and present the coalescent as a motivating example . in section [ sec : stopping_time ] we present multi - level smc for stopped processes . in section [ sec : pmcmc ] we detail a pmcmc algorithm which uses multi - level smc approximations within mcmc . in addition , specific adaptive strategies for the levels are proposed , which are motivated by some theoretical results that link the convergence rate of the pmcmc algorithm to the properties of multi - level smc approximations . in section [ sec : numerical ] some numerical experiments for the the coalescent are given . the paper is concluded in section [ sec : summary ] . the proofs of our theoretical results can be found in the appendix . the following notations will be used . a measurable space is written as @xmath6 , with the class of probability measures on @xmath7 written @xmath8 . for @xmath9 , @xmath10 the borel sets are @xmath11 . for a probability measure @xmath12 we will denote the density with respect to an appropriate @xmath13-finite measure @xmath14 as @xmath15 . the total variation distance between two probability measures @xmath16 is written as @xmath17 . for a vector @xmath18 , the compact notation @xmath19 is used ; if @xmath20 @xmath19 is a null vector . for a vector @xmath21 , @xmath22 is the @xmath23norm . the convention @xmath24 is adopted . also , @xmath25 is denoted as @xmath26 and @xmath27 is the indicator of a set @xmath1 . let @xmath7 be a countable ( possibly infinite dimensional ) state - space , then @xmath28 denotes the class of stochastic matrices which possess a stationary distribution . in addition , we will denote as @xmath29 the @xmath30-dimensional vector whose @xmath31 element is @xmath32 and is @xmath33 everywhere else . finally , for the discrete collection of integers we will use the notation @xmath34 . let @xmath5 be a parameter vector on @xmath35 , @xmath36 with an associated prior @xmath37 . the stopped process @xmath38 is a @xmath39valued discrete - time markov process defined on a probability space @xmath40 , where @xmath41 is a probability measure defined for every @xmath42 such that for every @xmath43 , @xmath44 is @xmath45measurable . for simplicity will we will assume throughout the paper that the markov process is homogeneous . the state of the process @xmath38 begins its evolution in a non empty set @xmath0 obeying an initial distribution @xmath46 and a markov transition kernel @xmath47 . the process is killed once it reaches a non - empty target set @xmath48 such that @xmath49 . the associated stopping time is defined as @xmath50 where it is assumed that @xmath51 and @xmath52 , where @xmath53 is a collection of positive integer values related to possible stopping times . in this paper we assume that we have no direct access to the state of the process . instead the evolution of the state of the process generates a random observations vector , which we will denote as @xmath54 . the realisation of this observations vector is denoted as @xmath55 and assume that it takes value in some non empty set @xmath56 . we will also assume that there is no restriction on @xmath1 depending on the observed data @xmath55 , but to simplify exposition this will be omitted from the notation . in the context of bayesian inference we are interested in the posterior distribution : @xmath57 where @xmath58 is the stopping time , @xmath59 is the prior distribution and @xmath60 is the un - normalised complete - data likelihood with the normalising constant of this quantity being : @xmath61 the subscript on @xmath5 will be used throughout to explicitly denote the conditional dependance on the parameter @xmath5 . given the specific structure of the stopped processes one may write @xmath60 as @xmath62 where @xmath63 is the likelihood of the data given the trajectory of the process . throughout , it will be assumed that for any @xmath42 , @xmath64 , @xmath60 admits a density @xmath65 with respect to a @xmath66finite measure @xmath67 on @xmath68 and the posterior and prior distributions @xmath69 admit densities @xmath70 respectively both defined with respect to appropriate @xmath66finite dominating measures . note that ( [ eq : target ] ) is expressed as an inference problem for @xmath71 and not only @xmath5 . the overall motivation originates from being able to design a mcmc that can sample from @xmath72 , which requires one to write the target ( or an unbiased estimate of it ) up - to a normalizing constant @xcite . still , our primary interest lies in bayesian inference for the parameter and this can be recovered by the marginals of @xmath72 with respect to @xmath5 . as it will become clear in section [ sec : pmcmc ] the numerical overhead when augmenting the argument of the posterior is necessary and we believe that the marginals with respect to @xmath73 might also be useful by - products . the framework presented so far is rather abstract , so we introduce the coalescent model as a motivating example . in figure [ fig : coalgraph ] we present a particular realisation of the coalescent for two genetic types @xmath74 . the process starts at epoch @xmath75 when the most recent common ancestor ( mrca ) splits into two versions of itself . in this example @xmath1 is chosen to be the mcra and the process continues to evolve by split and mutation moves . at the stopping point ( here @xmath76 ) we observe some data @xmath55 , which corresponds to the number of genes for each genetic type . ( 1,1 ) ( .5,.75)(0,1)0.25 ( 0.25,.75)(1,0).5 ( 0.75,.75)(0,-1)0.75 ( 0.25,.75)(0,-1)0.25 ( 0.125,.5)(1,0)0.25 ( 0.125,.5)(0,-1)0.5 ( 0.375,.5)(0,-1)0.5 ( -0.1,0.0)(0.1,0)12(1,0)0.02 ( -0.1,0.2)(0.1,0)12(1,0)0.02 ( -0.1,0.5)(0.1,0)12(1,0)0.02 ( -0.1,0.5)(0.1,0)12(1,0)0.02 ( -0.1,0.6)(0.1,0)12(1,0)0.02 ( -0.1,0.75)(0.1,0)12(1,0)0.02 ( 0.56,.90)@xmath1 ( 0.76,0.62)@xmath77 ( -0.05,0.21)@xmath77 ( 0.1,-0.05)@xmath78 ( 0.35,-0.05)@xmath1 ( 0.73,-0.05)@xmath78 ( 1.1,0.0)@xmath79 stop @xmath80 ( 1.1,0.2)@xmath81 mutation ( 1.1,0.5)@xmath82 split ( 1.1,0.6)@xmath83 mutation ( 1.1,0.75)@xmath75 split in general we will assume there are @xmath30 different genetic types . the latent state of the process @xmath84 is composed of the number of genes of each type @xmath85 at epoch @xmath86 of the process and let also @xmath87 . the process begins by default when the first split occurs , so the markov chain @xmath88 is initialised by the density @xmath89 and is propagated using the following transition density : @xmath90 where @xmath91 is defined in section [ sub : notations ] . here the first transition type corresponds to individuals changing type and is called mutation , e.g. @xmath77 at @xmath92 in figure [ fig : coalgraph ] . the second transition is called a split event , e.g. @xmath93 in the example of figure [ fig : coalgraph ] . to avoid any confusion we stress that in figure [ fig : coalgraph ] we present a particular realisation of the process that is composed by a sequence of alternate split and mutations , but this is not the only possible sequence . for example , the bottom of the tree could have be obtained with @xmath78 being the possible mcra and a sequence of two consecutive splits and a mutation . the process is stopped at epoch @xmath94 when the number of individuals in the population reaches @xmath95 . so for the state space we define : @xmath96 and for the initial and terminal sets we have : @xmath97 the data is generated by setting @xmath98 , which corresponds to the counts of genes that have been observed . in the example of figure [ fig : coalgraph ] this corresponds to @xmath99 . hence for the complete likelihood we have : @xmath100\label{eq : gamma_y}\ ] ] as expected , the density is only non - zero if at time @xmath94 @xmath101 matches the data @xmath55 exactly . our objective is to infer the genetic parameters @xmath102 , where @xmath103 and @xmath104 and hence the parameter space can be written as @xmath105 . to facilitate monte carlo inference , one can reverse the time parameter and simulate backward from the data . this is now detailed in the context of importance sampling following the approach in @xcite . to sample realisations of the process for a given @xmath42 , importance sampling is adopted but with time reversed . first we introduce a time reversed markov kernel @xmath106 with density @xmath107 . this is used as an importance sampling proposal where sampling is performed backwards in time and the weighting forward in time . we initialise using the data and simulate the coalescent tree backward in time until two individuals remain of the same type . this procedure ensures that the data is hit when the tree is considered forward in time . the process defined backward in time can be interpreted as a stopped markov process with the definitions of the initial and terminal sets appropriately modified . for convenience we will consider the reverse event sequence of the previous section , i.e we posed the problem backwards in time with the reverse index being @xmath108 . the proposal density for the full path starting from the bottom of the tree and stopping at its root can be written as @xmath109 with reference to we have @xmath110 then the marginal likelihood can be obtained @xmath111 in @xcite the authors derive an optimal proposal @xmath112 with respect to the variance of the marginal likelihood estimator . for the sake of brevity we omit any further details . in the current setup where there is only mutation and coalescences , the stopped - process can be integrated out @xcite , but this is not typically possible in more complex scenarios . a more complicated problem , including migration , is presented in section [ sec : migration ] . finally , we remark that the relevance of the marginal likelihood above will become clear later in section [ sec : pmcmc ] as a crucial element in numerical algorithms for inferring @xmath5 . in this section we shall briefly introduce generic smc without extensive details . we refer the reader for a more detailed description to @xcite . to ease exposition , when presenting generic smc , we shall drop the dependence upon parameter @xmath5 . smc algorithms are designed to simulate from a sequence of probability distributions @xmath113 defined on state space of increasing dimension , namely @xmath114 . each distribution in the sequence is assumed to possess densities with respect to a common dominating measure : @xmath115 with each un - normalised density being @xmath116 and the normalizing constant being @xmath117 . we will assume throughout the article that there are natural choices for @xmath118 and that we can evaluate each @xmath119 point - wise . in addition , we do not require knowledge of @xmath120 smc algorithms approximate @xmath121 recursively by propagating a collection of properly weighted samples , called particles , using a combination of importance sampling and resampling steps . for the importance sampling part of the algorithm , at each step @xmath122 of the algorithm , we will use general proposal kernels @xmath123 with densities @xmath124 , which possess normalizing constants that do not depend on the simulated paths . a typical smc algorithm is given in algorithm [ alg : generic - smc - algorithm ] and we obtain the following smc approximations for @xmath125 @xmath126 and for the normalizing constant @xmath117 : @xmath127 initialisation , @xmath128 : for @xmath129 1 . sample @xmath130 . 2 . compute weights @xmath131 for @xmath132 , for @xmath129 , 1 . resampling : sample index @xmath133 , where @xmath134 . 2 . sample @xmath135 and set @xmath136 . 3 . compute weights @xmath137 in this paper we will use @xmath138 to be the multinomial distribution . then the resampled index of the ancestor of particle @xmath85 at time @xmath122 , namely @xmath139 , is also a random variable with value chosen with probability @xmath140 . for each time @xmath122 , we will denote the complete collection of ancestors obtained from the resampling step as @xmath141 and the randomly simulated values of the state obtained during sampling ( step 2 for @xmath142 ) as @xmath143 . we will also denote @xmath144 the concatenated vector of all these variables obtained during the simulations from time @xmath145 . note @xmath146 the is a vector containing all @xmath147 simulated states and should not be confused with the particle sample of the path @xmath148 . furthermore , the joint density of all the sampled particles and the resampled indices is @xmath149 the complete ancestral genealogy at each time can always traced back by defining an ancestry sequence @xmath150 for every @xmath151 and @xmath142 . in particular , we set the elements of @xmath150 using the backward recursion @xmath152 where @xmath153 . in this context one can view smc approximations as random probability measures induced by the imputed random genealogy @xmath154 and all the possible simulated state sequences that can be obtained using @xmath155 . this interpretation of smc approximations was introduced in @xcite and will be later used together with @xmath156 for establishing the complex extended target distribution of pmcmc . for different classes of problems one can find a variety of enhanced smc algorithms ; see e.g. @xcite . in the context of stopped processes , a multi - level smc implementation was proposed in @xcite and the approach was illustrated for the coalescent model of section [ sec : coal_model ] . we consider a modified approach along the lines of section 12.2 of @xcite which seems better suited for general stopped processes and can provably yield estimators of much lower variance relative to vanilla smc . introduce an arbitrary sequence of @xmath157nested sets @xmath158 with the corresponding stopping times denoted as @xmath159 note that the markov property of @xmath160 implies @xmath161 . the implementation of multi - level smc differs from the generic algorithm of section [ sec : smc_algo ] in that between successive resampling steps one proceeds by propagating in parallel trajectories of @xmath162 until the set @xmath163 is reached for each @xmath164 . for a given @xmath164 the path @xmath162 is `` frozen '' once @xmath165 , until the remaining particles reach @xmath163 and then a resampling step is performed . more formally denote for @xmath128 @xmath166 where @xmath167 is a realisation for the stopping time @xmath168 and similarly for @xmath169 we have @xmath170 multi - level smc is a smc algorithm which ultimately targets a sequence of distributions @xmath171 each defined on a space @xmath172 where @xmath173 , @xmath174 and @xmath175 are finite collections of positive integer values related to the stopping times @xmath176 respectively . in the spirit of generic smc define intermediate target densities @xmath177 w.r.t to an appropriate @xmath13-finite dominating measure @xmath178 . we will assume there exists a natural sequence of densities @xmath179 obeying the restriction @xmath180 so that the last target density @xmath181 coincides with @xmath182 in ( [ eq : unnormalised_likeli ] ) . note that we define a sequence of @xmath183 target densities , but this time the dimension of @xmath119 compared to @xmath184 grows with a random increment of @xmath185 . in addition , @xmath181 should clearly depend on the value of @xmath5 , but this suppressed in the notation . @xmath186the following proposition is a direct consequence of the markov property : [ prop : markov_level]assume @xmath51 . then the stochastic sequence defined @xmath187 forms a markov chain taking values in @xmath188 defined . in addition , for any bounded measurable function @xmath189 , then @xmath190 . the proof can be found in ( * ? ? ? * proposition 12.2.2 , page 438 ) , ( * ? ? ? * proposition 12.2.4 , page 444 ) and the second part is due to @xmath191 . we will present multi - level smc based as a particular implementation of the generic smc algorithm . firstly we replace @xmath192 with @xmath193 respectively . contrary to the presentation of algorithm [ alg : generic - smc - algorithm ] for multi - level smc we will use a homogeneous markov importance sampling kernel @xmath194 , where @xmath195 , @xmath196 by convention and @xmath112 is the corresponding density w.r.t . @xmath14 . to compute the importance sampling weights of step 3 for @xmath142 in algorithm [ alg : generic - smc - algorithm ] we use instead : @xmath197 and for step 2 at @xmath128 : @xmath198 to simplify notation from herein we write @xmath199 and given @xmath183 , for any @xmath169 we have @xmath200 where again we have suppressed the @xmath5-dependance of @xmath201 in the notation . we present the multi - level smc algorithm in algorithm [ alg : multilevel - smc - algorithm ] . note here we include a procedure whereby at each stage @xmath122 , particles that do not reach @xmath163 before time @xmath202 are rejected by assigning them a zero weight , whereas before it was hinted that resampling is performed when all particles reach @xmath163 . similar to , it is clear that the joint probability density of all the random variables used to implement a multi - level smc algorithm with multinomial resampling is given by : @xmath203 where @xmath204 is defined similarly to @xmath146 . finally , recall by construction @xmath205 so the approximation of the normalizing constant of @xmath60 for a fixed @xmath5 is @xmath206 initialisation , @xmath128 : for @xmath129 1 . for @xmath207 : 1 . sample @xmath208 . 2 . if @xmath209 set @xmath210 , @xmath211 and go to step 2 . compute weights @xmath212 for @xmath132 , for @xmath129 , 1 . resampling : sample index @xmath133 , where @xmath134 . 2 . for @xmath213 : 1 . sample @xmath208 . 2 . if @xmath214 set @xmath215 , @xmath216 and go to step 3 . 3 . set @xmath217 . 4 . compute weights @xmath218 we will begin by showing how the levels can be set for the coalescent example of section [ sec : coal_model ] . we will proceed in the spirit of section [ sec : like_comp ] and consider the backward process so that the `` time '' indexing is set to start from the bottom of the tree towards the root . we introduce a a collection of integers @xmath219 and define @xmath220 clearly we have @xmath221 , @xmath222 and @xmath223 . one can also write the sequence of target densities for the multi - level setting as : @xmath224 the major design problem that remains in general is that given _ any _ candidates for @xmath225 , how to set the spacing ( in some sense ) of the @xmath226 and how many levels are needed so that good smc algorithms can be constructed . that is , if the @xmath226 are far apart , then one can expect that weights will degenerate very quickly and if the @xmath226 are too close that the algorithm will resample too often and hence lead to poor estimates . for instance , in the context of the coalescent example of section [ sec : coal_model ] , if one uses the above construction for @xmath226 the importance weight at the @xmath122-th resampling time is @xmath227 now , in general for any @xmath228 and @xmath183 it is hard to know beforehand how much better ( or not ) the resulting multi - level algorithm will perform relative to a vanilla smc algorithm . whilst @xcite show empirically that in most cases one should expect a considerable improvement , there @xmath5 is considered to be fixed . in this case one could design the levels sensibly using offline heuristics or more advanced systematic methods using optimal control @xcite or adaptive simulation @xcite , e.g. by setting the next level using the median of a pre - specified rank of the particle sample . what we aim to establish in the next section is that when @xmath5 is varies as in the context of mcmc algorithms , one can both construct pmcmc algorithms based on multi - level smc and more importantly easily design for each @xmath5 different sequences for @xmath226 based on similar ideas . particle markov chain monte carlo ( pmcmc ) methods are mcmc algorithms , which use all the random variables generated by smc approximations as proposals . as in standard mcmc the idea is to run an ergodic markov chain to obtain samples from the distribution of interest . the difference lies that in order use the simulated variables from smc , one defines a complex invariant distribution for the mcmc on an extended state space . this extended target is such that a marginal of this invariant distribution is the one of interest . this section aims on providing insight to the following questions : 1 . is it valid in general to use multi - level smc within pmcmc ? 2 . given that it is , how can we use the levels to improve the mixing of pmcmc ? the answer to the first question seems rather obvious , so we will provide some standard but rather strong conditions for which multi - level pmcmc is valid . for the second question we will propose an extension to pmmh that adapts the level sets used to @xmath5 at every iteration of pmcmc . @xcite introduces three different and generic pmcmc algorithms : particle independent metropolis hastings algorithm ( pimh ) , particle marginal metropolis hastings ( pmmh ) and particle gibbs samplers . in the remainder of the paper we will only focus on the first two of these . 1 . sample @xmath229 from using the multi - level implementation of algorithm [ alg : generic - smc - algorithm ] detailed in section and compute @xmath230 . sample @xmath231 . 2 . set @xmath232 and @xmath233 3 . for @xmath234 : 1 . propose a new @xmath235 and @xmath236 as in step 1 and compute @xmath237 2 . accept this as the new state of the chain with probability @xmath238 if we accept , set @xmath239 and @xmath240 . otherwise reject , @xmath241 and @xmath242 . we will begin by presenting the simplest generic algorithm found in @xcite , namely the particle independent metropolis hastings algorithm ( pimh ) . in this case @xmath5 and @xmath183 are fixed and pimh is designed to sample from the pre - specified target distribution @xmath243 also considered in section [ sec : smc_algo ] . although pimh is not useful for parameter inference it is included for pedagogic purposes . one must bear in mind that pimh is the most basic of all pmcmc algorithms . as such it is easier to analyse but still can provide useful intuition that can be used later in the context of pmmh and varying @xmath5 . pimh is presented in algorithm [ alg : pimh ] . it can be shown , using similar arguments to @xcite , that the invariant density of the markov kernel above is exactly ( see the proof of proposition [ prop : stop_within_mcmc ] ) @xmath244 where @xmath245 is as in and as before we have @xmath246 and @xmath247 for every @xmath248 . note that @xmath249 admits the target density of interest , @xmath250 as the marginal , when @xmath251 and @xmath252 are integrated out . we commence by briefly investigating some convergence properties of pimh with multi - level smc . even though the scope of pimh is not parameter inference , one can use insight on what properties are desired by multi - level smc for pimh when designing other pmcmc algorithms used for parameter inference . we begin with posing the following mixing and regularity assumption : [ assump : a1 ] for every @xmath42 and @xmath253 there exist a @xmath254 such that for every @xmath255 : @xmath256 there exist a @xmath257 such that for @xmath222 and every @xmath258 : @xmath259 the stopping times are finite , that is for @xmath222 there exist a @xmath260 such that @xmath261 assumption ( a[assump : a1 ] ) is rather strong , but are often used in the analysis of these kind of algorithms @xcite because they simplify the proofs to a large extent . recall that @xmath42 and @xmath253 are fixed . we proceed by stating the following proposition : [ prop : conv_rate ] assume ( a[assump : a1 ] ) . then for @xmath262 algorithm [ alg : pimh ] generates a sequence @xmath263 that for any @xmath264 , @xmath265 , @xmath42 satisfies : @xmath266 the proof can be found in the appendix . the following remarks are generalised and do not always hold , but provide some intuition for the ideas that follow . the result shows intrinsically that as the supremum of the sum of the stopping times with respect to @xmath267 gets smaller , so does the convergence rate increase . this can be also linked to the variance of the estimator of @xmath230 , which is well known to increase linearly with @xmath183 ( * ? ? ? * theorem 12.2.2 , pages 451 - 453 ) . shorter stopping times will typically yield lower variance and hence better mcmc convergence properties . on the other hand often @xmath268 will be larger for longer @xmath183 and longer stopping times ( proposition [ prop : conv_rate ] is derived for a fixed @xmath183 ) . in addition , sampling a stopped process is easier using a higher number of levels . in summary , the tradeoff is that although it is more convenient to use more auxiliary variables for simulating the process , these will slow down the mixing of pmcmc . in practice one balances this by trying to use a moderate number of levels for which most the particles to reach @xmath1 . this tradeoff serves as a motivation for developing flexible schemes to vary @xmath269 with @xmath5 in the pmcmc algorithm presented later in section [ sec : adaptive_strat ] . in the remainder of this section we will focus on using a multi - level smc implementation within a pmmh algorithm . given the commentary in section [ sec : stopping_time_smc ] and our interest in drawing inference on @xmath42 , it seems that using multi - level smc within pmcmc should be highly beneficial . recall can be expressed in terms of densities as : @xmath270 and let the marginal density given by @xmath271 for the time being we will consider the case when @xmath183 is fixed . in the context of our stopped markov process , we propose a pmmh algorithm targeting @xmath272 in algorithm [ fig : stop_within_mcmc ] . 1 . sample @xmath273 . given @xmath274 sample @xmath275 using multi - level smc and compute @xmath276 . sample @xmath231 . 2 . set @xmath277 and @xmath278 . 3 . for @xmath234 : 1 . sample @xmath279 ; given @xmath280 propose a new @xmath281 and @xmath236 as in step 1 and compute @xmath282 . 2 . accept this as the new state of the chain with probability @xmath283 if we accept , set @xmath284 and @xmath285 . otherwise reject , @xmath241 and @xmath286 . we will establish the invariant density and convergence of this algorithm , under the following assumption : [ assump : a5 - 6 ] for any @xmath42 and @xmath253 we define the following sets for @xmath145 : @xmath287 and @xmath288 . for any @xmath42 we have that @xmath289 . in addition the ideal metropolis hastings targeting @xmath290 using proposal density @xmath291 is irreducible and aperiodic . this assumption contains assumptions 5 and 6 of @xcite modified to our problem with a simple change of notations . we proceed with the following result : [ prop : stop_within_mcmc ] assume ( a[assump : a5 - 6 ] ) ; then for any @xmath262 : 1 . the invariant density of the procedure described in algorithm [ fig : stop_within_mcmc ] , is on the space @xmath292 and has the representation @xmath293 where @xmath294 is as in and @xmath295 as in . in addition , admits @xmath290 as a marginal . algorithm [ fig : stop_within_mcmc ] generates a sequence @xmath296 such that @xmath297 where @xmath72 is as in . the proof of the result is in the appendix . the result is based on theorem 4 of @xcite . note that algorithm [ fig : stop_within_mcmc ] presented in a generic form of a `` vanilla '' pmmh algorithm , so it can be enhanced using various strategies . for example , it is possible to add block updating of the latent variables or backward simulation in the context of a particle gibbs version @xcite . in the next section , we propose a flexible scheme that allows to set a different number of levels after a new @xmath298 @xmath186is proposed . the remaining design issue for pmmh is how to tune multi - level smc by choosing @xmath183 and @xmath267 . whilst , for a fixed @xmath42 , one could solve the problem with preliminary runs , when @xmath5 varies this is not an option . in general the value of @xmath5 should dictate how small or large @xmath183 should be to facilitate an efficient smc algorithm . hence , to obtain a more accurate estimate of the marginal likelihood and thus an efficient mcmc algorithm , we need to consider adaptive strategies to propose randomly a different number of levels @xmath183 and levels sequence @xmath267 for each @xmath299 sampled at every pmmh iteration @xmath85 . to ease exposition we will assume that @xmath269 can be expressed as functions of an arbitrary auxiliary parameter @xmath300 . given @xmath299 is a random variable , the main questions we wish to address is how to perform such an adaptive strategy consistently . an important point , is the fact that since we are interested in parameter inference , it is required that the marginal of the pmmh invariant density is @xmath290 . this can be ensured ( see proposition [ prop : adap_stop ] ) by introducing at each pmmh iteration , the parameters that form the level sets @xmath301 as an auxiliary process , which given @xmath299 is conditionally independent of @xmath302 . this way we define an extended target for the mcmc algorithm , which includes @xmath183 and @xmath267 in the target variables . it should be noted that this scheme is explicitly different from proposition 1 of @xcite , where the mcmc transition kernel at iteration @xmath85 is dependent upon an auxiliary process . here one just augments the target space with more auxiliary variables . 1 . sample @xmath273 . given @xmath274 : sample @xmath303 from @xmath304 , then @xmath305 using multi - level smc and compute @xmath276 . sample @xmath231 . 2 . set @xmath306 and @xmath278 . 3 . for @xmath234 : 1 . sample @xmath279 ; sample @xmath307 from @xmath308 and @xmath309 @xmath236 as in step 1 and compute @xmath282 . 2 . accept this as the new state of the chain with probability @xmath283 if we accept , set @xmath310 and @xmath285 . otherwise reject , @xmath241 and @xmath286 . consider now that it is possible at every pmmh iteration @xmath85 to simulate the auxiliary process @xmath300 defined upon an abstract state - space @xmath311 . let this with associated random variable @xmath300 , be distributed according to @xmath312 , which is assumed to possess a density with respect to a. @xmath66finite measure @xmath313 written as @xmath314 . as hinted by the notation @xmath312 should depend on @xmath5 and @xmath300 is meant be used to determine the sequence of levels @xmath267 for each @xmath299 in pmmh . this auxiliary variable will induce for every @xmath42 : * a random number of level sets @xmath315 . * a sequence of level sets @xmath316 with @xmath317 . we will assume that for any @xmath42 , proposition [ prop : markov_level ] and will hold @xmath318almost everywhere , where this time @xmath183 should be replaced by @xmath319 . this implies that for every @xmath42 we have : @xmath320 where the expression holds @xmath318 almost everywhere . in algorithm [ fig : stop_within_mcmc1 ] we propose a pmmh algorithm , which at each step @xmath85 uses @xmath299 to adapt the levels @xmath321 . for algorithm [ fig : stop_within_mcmc1 ] we present the following proposition that verifies varying the level sets in this way is theoretically valid : [ prop : adap_stop ] assume ( a[assump : a5 - 6 ] ) and hold . then , for any @xmath262 : 1 . the invariant density of the procedure in algorithm [ fig : stop_within_mcmc1 ] is defined on the space @xmath322 and has the representation @xmath323 where @xmath294 is as in and @xmath295 is as in . in addition , admits @xmath290 as a marginal . the generated sequence @xmath324 satisfies @xmath325 where @xmath72 is as in . the proof is contained in the appendix . we are essentially using an auxiliary framework similar to @xcite . as in we included @xmath73 in the target posterior , when we were primarily interested in @xmath5 , this time we augment the target posterior with @xmath300 and the smc variables @xmath326 , which is a consequence of using pmcmc . the disadvantage is that as the space of the posterior increases it is expected that the mixing of the algorithm will be slower . this could be improved if we have opted @xmath73 and @xmath300 to be dependent on each other given @xmath5 , but this would need additional assumptions for the structure of @xmath60 . in addition , in many applications the parameters @xmath300 that determine @xmath267 appear naturally and @xmath300 often is low dimensional . also , in most applications it might seem easier to find intuition on how to construct and tune @xmath312 than computing the level sets directly from @xmath5 . for example , for the coalescent model of section [ sec : coal_model ] with the mutation matrix @xmath327 is fixed , one can envisage for a larger value of @xmath328 coalescent events are less likely and more level sets closer together are needed compared to smaller values of @xmath328 . we will illustrate the performance of pmmh using numerical examples on two models from population genetics . the first one deals with the coalescent model of section [ sec : coal_model ] when a low dimensional dataset is observed . this is meant as an academic / toy example suitable for comparing different pmmh implementations . the second example is a more realistic application and deals with a coalescent model that allows migration of individual genetic types from one sub - group to another @xcite . in both cases we will illustrate the performance of pmmh implemented with a simple intuitive strategy for adapting the level sets . we will use a known stochastic matrix @xmath327 with all entries equal to @xmath329 . in this example @xmath330 with and the dataset is @xmath331 . the parameter - space is set as @xmath332 $ ] and a uniform prior will be used . for @xmath106 we will use the optimal proposal distributions provided by @xcite . the pmmh proposal @xmath333 in algorithm [ fig : stop_within_mcmc1 ] is a log normal random walk , i.e. we use @xmath334 with @xmath335 . we will compare pmmh when implemented with a simple adaptive scheme for @xmath183 and when @xmath183 is fixed . in the latter case we set @xmath336 . when an adaptive strategy is employed we will sample each time @xmath183 directly using a multinomial distribution defined on @xmath337 with weights proportional to @xmath338 . in both cases given @xmath183 we place the levels almost equally spaced apart . the adaptive and normal versions were run with @xmath339 for @xmath340 iterations . in each case the algorithm took approximately @xmath341 , @xmath342 , @xmath343 hours to complete when implemented in matlab and run on a linux workstation using a intel core 2 quad q9550 cpu at 2.83 ghz . the results are shown in figure [ fig : plots_noadap ] and [ fig : plots_adap ] . we observed that when we varied the number of levels , this allowed the sampler to traverse through a bigger part of the state space compared to when a fixed number of levels is used . as a result the estimated pdf of the adaptive case manages to include a second mode that is not seen in the non adaptive case . in the fixed levels case we see a clear improvement with increasing @xmath344 , although the difference in the mixing between @xmath345 and @xmath346 is marginal . in the adaptive case the sampler performed well even with lower values of @xmath344 . level sets is used . left : estimated pdf of @xmath328 for @xmath339 . centre : the trace plot for @xmath345 . right : autocorrelation plots for @xmath339 . the average acceptance ratio was @xmath347 , @xmath348 and @xmath349 respectively.,title="fig:",scaledwidth=25.0% ] level sets is used . left : estimated pdf of @xmath328 for @xmath339 . centre : the trace plot for @xmath345 . right : autocorrelation plots for @xmath339 . the average acceptance ratio was @xmath347 , @xmath348 and @xmath349 respectively.,title="fig:",scaledwidth=25.0% ] level sets is used . left : estimated pdf of @xmath328 for @xmath339 . centre : the trace plot for @xmath345 . right : autocorrelation plots for @xmath339 . the average acceptance ratio was @xmath347 , @xmath348 and @xmath349 respectively.,title="fig:",scaledwidth=25.0% ] . far left : estimated pdf of @xmath328 for @xmath339 . central left : the trace plot for @xmath345 . central right : histogram of number of levels in the posterior for @xmath345 . far right : autocorrelation function plots for @xmath339 . the average acceptance ratio was @xmath349 , @xmath350 and @xmath351 respectively.,title="fig:",scaledwidth=25.0% ] . far left : estimated pdf of @xmath328 for @xmath339 . central left : the trace plot for @xmath345 . central right : histogram of number of levels in the posterior for @xmath345 . far right : autocorrelation function plots for @xmath339 . the average acceptance ratio was @xmath349 , @xmath350 and @xmath351 respectively.,title="fig:",scaledwidth=25.0% ] . far left : estimated pdf of @xmath328 for @xmath339 . central left : the trace plot for @xmath345 . central right : histogram of number of levels in the posterior for @xmath345 . far right : autocorrelation function plots for @xmath339 . the average acceptance ratio was @xmath349 , @xmath350 and @xmath351 respectively.,title="fig:",scaledwidth=25.0% ] . far left : estimated pdf of @xmath328 for @xmath339 . central left : the trace plot for @xmath345 . central right : histogram of number of levels in the posterior for @xmath345 . far right : autocorrelation function plots for @xmath339 . the average acceptance ratio was @xmath349 , @xmath350 and @xmath351 respectively.,title="fig:",scaledwidth=25.0% ] the model is similar to the one as described in section [ sec : coal_model ] . the major difference is that this time genetic types are of classified into sub - groups within which most activity happens . in addition , individuals are allowed to migrate from one group to another . we commence with a brief description of the model and refer the interested reader to @xcite for more details . as in section [ sec : coal_model ] we will consider the process forward in time . let @xmath352 be the number of groups and the state at time @xmath86 be composed as the concatenation of @xmath352 groups of different genetic types as : @xmath353 the process under - goes split , mutation and migration transitions as follows : @xmath354 where @xmath355 with @xmath356 and @xmath357 is a vector with a zero in every element except the @xmath358 -th one . similarly to the simpler model of section [ sec : coal_model ] the transition probabilities are parameterised by the mutation parameter @xmath328 , mutation matrix @xmath327 and a migration matrix @xmath359 . the latter is a symmetric matrix with zero values on the diagonal and positive values on the off - diagonals . finally the data is generated when at time @xmath94 the number of individuals in the population reaches @xmath95 , and @xmath360 . as for the model described in section [ sec : coal_model ] one can reverse time and employ an backward sampling forward weighting importance sampling method ; see @xcite for the particular implementation details . in our example we generated data with @xmath361 , @xmath362 and @xmath363 . this is quite a challenging set - up . as in the previous example we set the mutation matrix @xmath327 to be known and uniform and we will concentrate on inferring the @xmath364 . independent gamma priors with shape and scale parameters equal to @xmath32 were adopted for each of the parameters . we implemented pmmh using @xmath339 and a simple adaptive scheme for @xmath183 . we allow @xmath365 and use approximately equal spacing between the levels . we choose each @xmath183 with probability proportional to @xmath366 the proposals for the parameters were gaussian random - walks on the log - scale . the algorithm was implemented in c / c++ was run for @xmath340 iterations , which took approximately 3 , 6 and 12 hours to complete . whilst the run - time is quite long it can be improved by at least one order of magnitude if the smc is implemented on graphical processing units ( gpu ) as in @xcite . for the dataset plotted in figure [ fig : plots2 ] ( left ) the results are plotted in figures [ fig : plots2 ] ( right ) and [ fig : plots1 ] . the auto - correlation and trace plots indicate that the sampler mixes reasonably well for every @xmath344 . these results in this example are encouraging as to the best of our knowledge bayesian inference has not been attempted for this class of problems . we expect that practitioners with insight in the field of population genetics can come with more sophisticated mcmc proposals or adaptive schemes for the level sets , so that the methodology can be extended to realistic applications . in the resulting posterior for @xmath345.,title="fig:",scaledwidth=25.0% ] in the resulting posterior for @xmath345.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] + . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] + . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] . top row : estimated pdfs for @xmath328 , @xmath367 , @xmath368 , @xmath369 ( from left to right ) . middle : trace plots for @xmath345 . bottom : autocorrelation function plots . the acceptance rate was @xmath370 respectively.,title="fig:",scaledwidth=25.0% ] in this article we have presented a multi - level pmcmc algorithm which allows one to perform bayesian inference for the parameters of a latent stopped processes . in terms of methodology the main novelty of the approach is that uses auxiliary variables to adaptively compute the level sets with @xmath5 . the general structure of this auxiliary variable allows it to incorporate the use of independent smc runs with less particles to set the levels . in the numerical examples we demonstrated that the addition auxiliary variables slow down the convergence of pmcmc , but this seemed a reasonable compromise in terms of performance compared when fixed number of level sets were used . the proposed algorithm requires considerable amount of computation , but to the authors best knowledge for such problems there seems to be a lack of alternative approaches . also , recent developments gpu hardware can be adopted to speed up the computations even by orders of magnitude as in @xcite . there are several extensions to the work here , which may be considered . firstly , the scheme that is used to adapt the level sets relies mainly on intuition . we found simple adaptive implementations to work well in practice . in the rare events literature one may find more systematic techniques to design the level sets , based upon optimal control @xcite or simulation @xcite . although these methods are not examined here , they can be characterised using alternative auxiliary variables similar to the ones in proposition [ prop : adap_stop ] , so the auxiliary variable framework we use is quite generic . in addition , we emphasise that within a pmcmc framework one may also include multi - level splitting algorithms instead of smc , which might appeal practitioners familiar with multi - level splitting . secondly , one could seek to use these ideas within a smc sampler framework of @xcite as done in @xcite . as noted in the latter article , a sequential formulation can improve the sampling scheme , sometimes at a computational complexity that is the same as the original pmcmc algorithm . in addition , this article focuses on the pmmh algorithm , so clearly extensions using particle gibbs and block updates might prove valuable for many applications . finally , from a modelling perspective , it may be of interest to apply our methodology in the context of hidden markov models . in this context , one has @xmath371 with @xmath372 being the conditional likelihood of the observations . it would be important to understand , given a range of real applications , the feasibility of statistical inference , combined with the development of our methodology . an investigation of the effectiveness of such a scheme when applied to queuing networks is currently underway . we thank arnaud doucet and maria de iorio for many conversations on this work . the first author was supported by a ministry of education grant . the second author was kindly supported by the epsrc programme grant on control for energy and sustainability ep / g066477/1 . much of this work was completed when the first author was employed by the department of mathematics , imperial college london . [ proof of proposition [ prop : conv_rate ] ] the result is a straight forward application of theorem 6 of @xcite which adapted to our notation states : @xmath373\wedge\mathbb{e}_{\psi_{\theta}}\left[1\wedge\frac{\hat{z}_{p}(\xi)}{\hat{z}_{p}(\xi)}\bigg|\xi\right]\right)\bigg)^{i}\bigg],\ ] ] where the conditional expectation is the expectation w.r.t . the smc algorithm ( i.e. @xmath374 ) and the outer expectation is w.r.t . the pimh target ( i.e. @xmath375 ) . we also denote the estimate of the normalizing constant as @xmath376 with @xmath377 denoting which random variables generate the estimate . now , clearly via ( a[assump : a1 ] ) @xmath378^{\tau_{n}+\tau_{n-1}}\ ] ] with the convention that @xmath379 . thus , it follows that @xmath380^{\bar{\tau}_{n}+\bar{\tau}_{n-1}}\leq\bigg[\frac{1}{\rho\varphi}\bigg]^{2\sum_{n=1}^{p}\bar{\tau}_{n}}\ ] ] and we obtain : @xmath381 note that by assumption @xmath382 and thus we have @xmath383\bigg)^{i}\ ] ] given ( * ? ? ? * theorem 7.4.2 , equation ( 7.17 ) , page 239 ) and the fact that @xmath60 is defined to be strictly positive in ( a[assump : a1 ] ) we have that the smc approximation @xmath376 is an unbiased estimate of the normalizing constant @xmath384 @xmath385=z_{p},\label{eq : smc_unbiased}\ ] ] and we can easily conclude . [ proof of proposition [ prop : stop_within_mcmc ] ] the proof of parts 1 . and 2 . follows the line of arguments used in theorem 4 of @xcite , which we will adapt to our set - up . the main difference lies in the multi - level construction and second statement regarding the marginal of @xmath386 . for the validity of the multi - level set - up we will rely on proposition [ prop : markov_level ] . suppose we design a metropolis hastings kernel with invariant density @xmath386 and use a proposal @xmath387 . then @xmath388 where we denote the normalising constant of the posterior in as : @xmath389 therefore the metropolis - hastings procedure to sample from @xmath390 will be as in algorithm [ fig : stop_within_mcmc ] . alternatively using similar arguments one we may write @xmath391 summing over @xmath251 and using the unbiased property of the smc algorithm in equation it follows that @xmath392 admits @xmath393 as a marginal , so the proof of part 1 . is complete . [ proof of proposition [ prop : adap_stop ] ] the proof is the similar as that of proposition [ prop : stop_within_mcmc ] . for the proof of the first statement of part 1 . one repeats the same arguments as for proposition [ prop : stop_within_mcmc ] with difference being in the inclusion of @xmath394 for @xmath395 and @xmath396 . for the second statement , to get the marginal of @xmath386 , re - write the target as : @xmath397 let @xmath398 denote the marginal of @xmath392 obtained in proposition [ prop : stop_within_mcmc ] . using and the conditional independence of @xmath300 and @xmath326 , then for the marginal of @xmath399 w.r.t @xmath300 , @xmath326 , @xmath251 we have that @xmath400 where the summing over @xmath251 and integrating w.r.t . @xmath326 is as in proposition [ prop : stop_within_mcmc ] . for part 2 . note that the conditional density given @xmath251 and @xmath300 and @xmath5 of @xmath401 is @xmath402 hence the sequence @xmath403 satisfies the required property as direct consequence theorem 1 in @xcite and assumption ( a[assump : a5 - 6 ] ) . lee , a. , yau , c. , giles , m. , doucet , a. & holmes c.c . ( 2010 ) on the utility of graphics cards to perform massively parallel implementation of advanced monte carlo methods , _ j. comp . . statist . _ , * 19 * , 769789 .
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in this article we consider bayesian parameter inference associated to partially - observed stochastic processes that start from a set @xmath0 and are stopped or killed at the first hitting time of a known set @xmath1 .
such processes occur naturally within the context of a wide variety of applications .
the associated posterior distributions are highly complex and posterior parameter inference requires the use of advanced markov chain monte carlo ( mcmc ) techniques .
our approach uses a recently introduced simulation methodology , particle markov chain monte carlo ( pmcmc ) @xcite , where sequential monte carlo ( smc ) @xcite approximations are embedded within mcmc .
however , when the parameter of interest is fixed , standard smc algorithms are not always appropriate for many stopped processes . in @xcite the authors introduce smc approximations of multi - level feynman - kac formulae , which can lead to more efficient algorithms .
this is achieved by devising a sequence of nested sets from @xmath0 to @xmath1 and then perform the resampling step only when the samples of the process reach intermediate level sets in the sequence .
naturally , the choice of the intermediate level sets is critical to the performance of such a scheme . in this paper
, we demonstrate that multi - level smc algorithms can be used as a proposal in pmcmc .
in addition , we propose a flexible strategy that adapts the level sets for different parameter proposals .
our methodology is illustrated on the coalescent model with migration .
+ * key - words * : stopped processes , sequential monte carlo , markov chain monte carlo @xmath2department of statistics & applied probability , national university of singapore , singapore , 117546 , sg .
+ e-mail:`[email protected] ` + @xmath3department of electrical engineering , imperial college london , london , sw7 2az , uk .
+ e-mail:`[email protected] `
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the abundance of massive clusters of galaxies provides sensitive constraints on the cosmological parameters that govern structure growth in the universe . however , a prerequisite for this is reliable mass measurements for large samples of clusters with well - understood selection criteria . cluster mass measurements used have traditionally come from virial analysis of the velocity dispersion measurements of cluster galaxies ( e.g. , frenk et al . 1990 ; carlberg et al . 1997 ; borgani et al . 1999 ) , or x - ray temperature measurements of the hot intra - cluster gas under the assumption that the gas is in hydrostatic equilibrium ( for a review see * ? ? ? satellite observatories such as rosat , asca , xmm - newton and _ chandra _ have made increasingly accurate x - ray temperature measurements of clusters , and have produced well - defined cluster samples of sufficient size to accurately measure the x - ray temperature and luminosity functions ( e.g. * ? ? ? * ; * ? ? ? * ) . however , the relation between cluster mass and x - ray temperature and luminosity , respectively , must be determined to convert these into a reliable cluster mass function . x - ray luminosities are available for large samples of clusters , but the x - ray luminosity is highly sensitive to the complex physics of cluster cores , making it challenging to relate to cluster mass . measuring x - ray temperatures is observationally much more demanding , but the x - ray temperature is mainly determined by gravitational processes and is hence more directly related to cluster mass than x - ray luminosity . both from simulations and observations the intrinsic scatter in mass around the mass temperature relation is thus found to be much smaller ( @xmath7 , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) than the scatter in mass around the mass luminosity relation ( @xmath8 , * ? ? ? two different routes have been followed for determining the mass temperature relation . most studies have used a small sample ( up to about a dozen ) of supposedly well understood clusters for which the assumptions underlying the mass determination should be fulfilled to a high degree . the main concern for this approach is that the selected clusters may not be representative of the whole cluster population , and therefore the derived mass temperature relation may only apply to a subset of clusters . alternatively , the mass temperature relation may be determined from a large sample of more objectively selected clusters . this is more fruitful when comparing such a locally determined mass temperature relation to a sample of high - redshift clusters where the data quality does not allow a similar selection of the `` most suitable '' clusters . also , mass temperature relations derived from simulations are usually based on a large range of simulated clusters with no pre - selection . hence it is most appropriate to compare observationally obtained mass temperature relations determined from all available clusters to the relations from simulations . on the other hand , for some of the clusters in such a sample the hydrostatic assumption may be invalid , making x - ray based mass determinations unreliable for a subset of the clusters . a larger scatter ( which may not be symmetric ) around the mean mass temperature relation may be expected , when such clusters are included . there are still poorly understood systematic uncertainties associated with establishing the mass temperature relation . the normalization of the mass temperature relation based on cluster masses determined from x - ray data @xcite , tend to differ significantly between studies , and from the expectations based on numerical simulations ( e.g. , evrard et al . 1996 ; eke et al . 1998 ; pen 1998 ; borgani et al . 2004 ) . the determination of this normalization is currently the dominating source of discrepancies between the reported values for the power spectrum normalization on the scale of galaxy clusters , @xmath0 , derived from the observed cluster temperature function ( huterer & white 2002 ; seljak 2002 ; pierpaoli et al . 2003 ; henry 2004 ) . observations using x - ray based mass determinations have traditionally favored low normalizations ( and hence low values of @xmath0 ) , while simulations have favored somewhat higher normalizations . gravitational lensing provides an opportunity to measure cluster masses without invoking the assumption of hydrostatic equilibrium in the hot intra - cluster gas implicit in the x - ray based mass determinations . also , in this case the measurement of cluster mass is truly independent of the x - ray temperature measurement . hjorth , oukbir & van kampen ( 1998 ) used weak gravitational mass measurements for eight clusters drawn from the literature to find a relation between mass over cluster - centric radius , and temperature . they determined a normalization of this relation consistent with the value predicted by evrard et al . ( 1996 ) , but with a preference for somewhat higher cluster masses ( if the redshift scaling of equation [ eq : theorymt ] is assumed , see below ) . however , smith et al . ( 2005 ; hereafter s05 ) determined a mass temperature relation with a normalization significantly lower than indicated by the hjorth et al . ( 1998 ) study . s05 based their results on a sample of 10 clusters with weak lensing masses and temperatures determined from _ chandra _ data . here , we present a new weak gravitational lensing - based measurement of the normalization of the mass temperature relation . the main improvements with respect to the work of hjorth et al . ( 1998 ) and s05 is that we use a significantly larger cluster sample which represents a significant fraction of all the clusters in an even larger sample with well - defined objective selection criteria ( dahle et al . 2002 ; h. dahle 2006 , in preparation ) . an additional improvement over the work of hjorth et al . ( 1998 ) is that the weak lensing analysis has been performed in a consistent way for all clusters , using the same shear estimator and making the same assumptions about e.g. , the typical redshift of the lensed galaxy population and the degree of contamination by cluster galaxies . we note that the early data set of clusters with published weak lensing masses used by hjorth et al . ( 1998 ) is biased at some level towards systems that were observed because of `` extreme '' properties , such as being the hottest or most x - ray luminous system known at the time , or having a large number of strongly gravitationally lensed arcs . furthermore , we note that our gravitational lensing measurements are made at larger radii than probed by s05 , requiring smaller extrapolations to estimate the mass within e.g. , the virial radii of the clusters . the data set used for the analysis is described in [ sec : data ] , our results for the mass temperature relation and @xmath0 are presented in [ sec : results ] , and our results are compared to other work and the implications discussed in [ sec : discussion ] . except when specifically noted otherwise ( for easy comparison to previous results using different cosmologies ) , we assume a spatially flat cosmology with a cosmological constant ( @xmath5 , @xmath9 ) , and the hubble parameter is given by @xmath10 . our weak lensing data set is a sample of 30 clusters ( see table [ tab : dataset ] ) , of which 28 were included in the weak lensing cluster sample of dahle et al . data for two additional clusters come from a recent extension of this data set ( h. dahle 2006 , in preparation ) . the clusters targeted for these weak lensing studies were generally selected to lie above an x - ray luminosity limit @xmath11 ergs s@xmath12 ( this luminosity limit is for our chosen cosmology with @xmath13 ) and within a redshift range @xmath14 . the observed clusters were selected from the x - ray luminous cluster samples of briel & henry ( 1993 ) and ebeling et al . ( 1996;1998;2000 ) . the cluster samples of the first two of these papers are based on correlating an optically selected cluster sample ( abell 1958 ; abell , corwin , & olowin 1989 ) with x - ray sources from the rosat all - sky survey ( rass ; trmper et al . 1993 ) , while the two last papers contain x - ray flux limited cluster catalogs , also based on rass . of the total sample of 30 clusters , three ( , , and ) are drawn from the briel & henry ( 1993 ) sample and two ( and ) are drawn from the xbac sample of ebeling et al . ( 1996 ) . of the remaining 25 clusters , 22 are included in the x - ray brightest cluster sample ( bcs ) of ebeling et al . ( 1998 ) , while three ( , , and ) come from its low - flux extension ( ebcs ; ebeling et al . 2000 ) . of the bcs and ebcs clusters in our sample , 24 objects are included in a volume - limited sample of 35 clusters selected from the bcs and ebcs samples ( dahle 2006 ) . hence , while our sample is not strictly physically well - defined ( in the sense that the availability of an x - ray temperature measurement is one of the defining selection criteria ) , it still has significant overlap with a well - defined cluster sample . in a recent paper , stanek et al . ( 2006 ) discuss how a significant scatter around the mean mass - luminosity relation may cause a significant malmquist bias in x - ray flux - limited cluster samples , causing high - mass , low flux clusters to drop out at high redshifts . this would result in a bias in the mass - luminosity ( or mass - temperature ) relation derived based on such a sample . we note , however , that although the rass - based samples from which our cluster sample is drawn are flux - limited , the cluster sample discussed here quite closely approximates a volume - limited sample , and we therefore expect any such bias to be negligible . the observations were made with the @xmath15 uh8k mosaic ccd camera and the @xmath16 tek ccd camera at the 2.24 m university of hawaii telescope and with the @xmath16 alfosc ccd camera at the 2.56 m nordic optical telescope . all clusters were imaged in both the @xmath17- and @xmath18-band , with typical total exposure times of 3.5h in each passband for the uh8k data and @xmath19h for the data obtained with the more sensitive @xmath16 detectors . the seeing was in the range @xmath20 for all the imaging data used for the weak lensing analysis . the median seeing was 082 in the @xmath17-band and 09 in the @xmath18-band . this gave typically @xmath21 usable background galaxies per square arcminute , or a `` figure of merit '' value of @xmath22deg@xmath23 , as defined by kaiser ( 2000 ) . as noted below , the background galaxies were selected based on signal to noise ratio rather than magnitude , with limits corresponding to @xmath24 and @xmath25 for point sources . the observations and data reduction of the data set used for the weak lensing mass measurements are described in detail by dahle et al . ( 2002 ) . major efforts are being made to improve the methods for the estimation of weak gravitational lensing , particularly in connection with ongoing and future studies of `` cosmic shear '' based on wide - field optical surveys . the requirements for the precision of shear estimates in these surveys are substantially more stringent than for weak lensing observations of massive clusters , given the significantly weaker lensing effects measured in random fields . in this work , we have used the shear estimator of kaiser ( 2000 ) , which was `` blind - tested '' ( along with several other shear estimators ) by heymans et al . ( 2005 ) , using simulated lensing data . the shear estimator of kaiser ( 2000 ) is more mathematically rigorous than the currently most widely used shear estimator ( kaiser , squires , & broadhurst 1995 ) , but it displays a significant non - linear response to shear , unlike most other shear estimators . if we correct our shear values using a second order polynomial based on the test results of heymans et al . ( 2005 ) , we find that most cluster masses stay within @xmath26% of the mass calculated based on uncorrected shear values . furthermore , the change in average cluster mass is @xmath27% , i.e. , there is very little systematic shift in mass . in the end , we chose not to apply this correction , since it would , in a few cases , require extrapolations outside the range of shear values over which the shear estimator has been tested . for more details about the practical implementation of this shear estimator , see dahle et al . ( 2002 ) . to convert the measurements of weak gravitational shear into actual cluster masses , the distances to the background galaxies need to be known . the background galaxy redshifts were estimated from spectroscopic and photometric redshifts in the hubble deep field ( for details , see dahle et al . 2002 ) . for our data set and chosen cosmological model , the average value of the ratio between the lens - source and observer - source angular diameter distances , @xmath28/@xmath29 , is well approximated by the relation @xmath30 within the redshift range of our cluster sample . this then provides an effective critical surface density for lensing ( @xmath31 ; where @xmath32 is the angular diameter distance to the cluster ) , which is used for deriving cluster masses from the shear estimates . the quoted value of @xmath33 corresponds to the value at large cluster radii ; at smaller radii a correction term has to be employed to account for contamination by cluster galaxies , as discussed below and illustrated in figure [ fig : cluscont ] . the observable galaxy shape distortions caused by gravitational lensing provide a measurement of the reduced tangential shear , @xmath34 , where @xmath35 is the tangential component of the shear and @xmath36 is the convergence . we fit an nfw - type mass density profile , @xmath37 ( navarro , frenk , & white 1997 ) , to the observed reduced shear profile @xmath38 of each cluster . here , @xmath39 is the critical density of the universe at the redshift of the cluster , and @xmath40 we assumed a concentration parameter @xmath41 , corresponding to the median halo concentration predicted by bullock et al . ( 2001 ) for a @xmath42 cluster from simulations of dark matter halos in a @xmath4cdm universe . here , @xmath43 , and @xmath44 , where @xmath45 is defined as the radius within which the average mass density is 200 times the critical density @xmath46 , and @xmath47 is the virial radius of the cluster . the lensing properties of the nfw model have been calculated by bartelmann ( 1996 ) and wright & brainerd ( 2000 ) . from our fit , we calculated @xmath48 , the mass enclosed by the radius @xmath49 . the mass estimates are listed in table [ tab : dataset ] . the shear measurements used for the fit were made at clustercentric radii @xmath50 for the clusters that were observed with @xmath16 ccd cameras and @xmath51 for the clusters that were observed with the uh8k camera . by comparison , we find @xmath49 values typically in the range @xmath52 for the clusters we study here . in many cases , we need to extrapolate the nfw profile out to @xmath49 ( in table [ tab : dataset ] we list the ratio of the outermost radii of our shear measurements , @xmath53 , to @xmath49 , and note that @xmath54 for our chosen nfw model ) . in this extrapolation , we assume the median nfw concentration parameter given above . hence , any intrinsic scatter in @xmath55 will introduce an extra uncertainty in the cluster mass estimates . if we assume a random scatter around the mean value of @xmath55 at the level ( a @xmath56 @xmath57 ) predicted by bullock et al . ( 2001 ) , we find a corresponding scatter in the mass estimates of 20% for our data set . this additional scatter is not included in the uncertainties of the listed mass measurements in table [ tab : dataset ] , but is considered further in section 3.2 . the measured gravitational lensing signal is sensitive to the two - dimensional surface mass distribution , including mass associated with the cluster outside @xmath49 , and random structures seen in projection along the line of sight ( metzler , white & loken 2001 ; hoekstra 2001 ; clowe , de lucia & king 2004 ; de putter & white 2005 ) . this will introduce additional uncertainty ( and potentially a net bias ) to any lensing - based estimates of the cluster mass contained within a 3d volume . studies based on simulated clusters ( e.g. , clowe et al . 2004 ) indicate that the net bias is no more than a few percent when 3d cluster masses are estimated by fitting observations of @xmath38 to predictions from theoretical models of the mass distribution , such as the nfw model . however , the scatter in the mass estimates from projection effects amount to a weak lensing mass dispersion of @xmath58 for massive galaxy clusters , which should be added to the observational uncertainties of the lensing mass estimates . in this paper , we have assumed a lensing mass dispersion of 0.26 resulting from projection effects , corresponding to the value estimated by metzler , white & loken ( 2001 ) from their n - body simulations . although these authors considered a somewhat different mass estimator , more recent estimates indicate a similar mass dispersion for the nfw profile fitting method that we have used . this additional mass uncertainty has been added in quadrature to the uncertainties of @xmath48 values listed in table [ tab : dataset ] . the absence of reliable information about the individual redshifts of the faint galaxies used for the weak lensing measurement will inevitably result in some degree of confusion between lensed background galaxies and unlensed cluster galaxies . the magnitude of this effect will depend on the projected number density of cluster galaxies , and should thus have a strong dependency on cluster radius . hence , a radially dependent correction factor was applied to the shear measurements to correct for contamination from cluster galaxies in the faint galaxy catalogs that were used to measure the gravitational shear ( these catalogs included all galaxies in the cluster fields that were detected at a signal - to - noise ratio @xmath59 , with no additional selection based on e.g. , galaxy color ) . the magnitude of this correction was estimated from the radial dependence of the average faint galaxy density in two `` stacks '' of clusters observed with the uh8k camera , one at @xmath60 and the other at @xmath61 , assuming that the contamination is negligible at the edge of the uh8k fields , @xmath62 mpc from the cluster center . the estimated degree of contamination is shown in figure [ fig : cluscont ] . given the difference in cluster redshift , the similarity of the two curves in figure [ fig : cluscont ] may be somewhat surprising , as one would naively expect the more distant clusters to display a significantly lower surface density of cluster galaxies . however , there are several competing effects that affect the observed galaxy density at a fixed angular radius : firstly , if all cluster galaxies were detectable regardless of cluster redshift , the change in apparent image scale with redshift should increase the surface density by a factor given by the square of the ratio of the angular diameter distances . on the other hand , a fixed angular radius would correspond to a larger physical cluster - centric radius ( and hence lower galaxy density in physical units ) at the larger redshift , the difference depending on the slope of the radial galaxy density profile . at the radii probed in this study , both the radial surface mass density profile and the number density profile of bright cluster galaxies follow approximately the power law behavior of a singular isothermal sphere ( @xmath63 , with @xmath64 ) . hence , the physical number density ( in galaxies / mpc@xmath65 ) at a fixed angular radius should decrease as the inverse of the ratio of angular diameter distances . in addition , the faintest galaxies drop below the detection limit at higher redshift , the effect depending on the slope of the cluster luminosity function around @xmath66 . assuming a schechter ( 1976 ) luminosity function with a faint - end slope @xmath67 ( typical of rich clusters ) and @xmath68 , the luminosity function can be integrated down to the detection limit ( corresponding to @xmath69 and @xmath70 at @xmath71 and @xmath72 , respectively ) , to estimate the fraction of cluster galaxies that drop out at the higher redshift ( @xmath73% ) . finally , a redshift - dependence given by @xmath74 was assumed to account for galaxy evolution in the clusters . the combination of all these effects would predict a surface density of cluster galaxies which is 7% less at @xmath75 , compared to @xmath71 , for a fixed cluster richness . even this small difference would be erased by a slight decrease in the assumed values of the slopes @xmath76 and @xmath77 . a faint - end slope of the luminosity function of @xmath78 would be sufficient to remove the predicted difference in galaxy surface density at the two different redshifts . based on _ hubble space telescope _ ( hst ) wfpc2 imaging of the galaxy cluster ( which is similar to the clusters studied here in terms of optical richness , lensing mass and x - ray properties ) , pracy et al . ( 2004 ) find that the cluster core shows a relative depletion of dwarf galaxies , leading to a radial profile of faint galaxies which is significantly shallower than the sis prediction . for the `` intermediate '' dwarf population ( @xmath79 , similar to the range in absolute magnitude of cluster galaxies in our faint galaxy catalogs ) , these authors find a radial distribution with a slope @xmath80 . assuming a similar slope for the radial distribution of the faint cluster galaxies in our catalogs would also remove the predicted difference between the two curves in figure [ fig : cluscont ] . based on this figure , and the above discussion , we conclude that we are probably justified in ignoring the redshift - dependence in our cluster galaxy contamination correction . the level of cluster galaxy contamination for individual clusters will generally differ from the mean level calculated above , as there will be significant cluster - to - cluster variations in the abundances of cluster dwarf galaxies . based on the sample of clusters observed with the uh8k camera , the scatter in dwarf galaxy richness was estimated to be @xmath81 ( this estimate also includes variations in the field galaxy density caused by uncorrelated large - scale structures along the line of sight , and hence the true scatter in dwarf galaxy richness of the clusters is somewhat overestimated ) . by employing the mean contamination correction calculated above rather than an estimate appropriate for each cluster , we introduce an additional scatter of up to @xmath82 in our mass estimates . this additional scatter is not included in the uncertainties of the tabulated mass measurements in table [ tab : dataset ] , but is considered further in section 3.2 . eight of the clusters in our sample were also included in the combined strong and weak gravitational lensing study of s05 , based on observations of a sample of 10 x - ray luminous galaxy clusters at @xmath83 using hst wfpc2 . these authors estimated the projected cluster mass within a clustercentric radius of @xmath84 kpc , assuming an einstein - de sitter ( @xmath85 , @xmath86 ) cosmology . figure [ fig : s05comp ] shows a comparison of the mass values listed by s05 with our cluster mass estimates , using the best - fit nfw model to derive projected cluster masses , assuming the same cosmology as s05 . these authors assumed a spatially constant contamination of 20% cluster galaxies in their background galaxy catalogs at radii @xmath87 , while we find an average contamination of 30% for our data at these radii . hence , for the plot in figure [ fig : s05comp ] we have adjusted our radially dependent contamination correction such that the average contamination at small radii is consistent with that assumed by s05 . we find that our cluster mass estimates are generally consistent with those of s05 , although with a tendency for higher masses ( by about 30% ) . for clusters in the weak lensing data set , we compiled a list of corresponding x - ray temperatures from the literature . for many of these clusters , their global temperature , or even a temperature map , has been determined using data from _ chandra _ and/or xmm - newton . however , these temperatures constitute a rather heterogeneous sample , for which the systematics are not well established . consequently , x - ray temperatures were primarily drawn from the samples of @xcite , @xcite and @xcite , each providing a homogeneous measure of the global cluster temperature ( i.e. , temperature measured within a cluster - centric distance close to @xmath49 ) for a large fraction of the clusters in the weak lensing sample . all these authors derived temperatures based on analysis of asca spectra . for two clusters not in either of the samples mentioned above we extracted published temperatures from other sources . specifically , from the works of @xcite and @xcite we extracted temperatures estimated by these authors by fitting an isothermal plasma model with the galactic absorbing column density as a free parameter . the temperatures taken from @xcite were derived by fitting an isothermal plasma model with the nominal galactic absorbing column density fixed , but since only energies above 1 kev are used in the @xcite spectral fitting the fixed column density should not introduce systematic effects relative to the temperatures from the @xcite and the @xcite samples . for the remaining two clusters we extracted published temperatures obtained in a similar way ( see table [ tab : dataset ] ) . for the clusters in two or more samples , their derived temperatures agree within the uncertainties , and for any cluster the derived temperatures differ by less than 20% between samples . also , the mean of temperature differences between any two of the samples by @xcite , @xcite , and @xcite is less than 3% , indicating the low level of systematic temperature variance between different analyses . although the isothermal plasma model has proven too simplistic for nearby clusters , the global cluster temperature is straightforward to derive from observations as well as simulations , enabling a rather direct comparison between observations and theory . furthermore , for the majority of distant ( @xmath88 ) clusters only global isothermal temperatures can be obtained in the foreseeable future . hence , we refrain from going into the detailed spatial and spectral modeling of the intra - cluster gas . the effects of cluster dynamics , `` cooling cores '' , non - sphericity etc . generally affects the global temperatures only at the 10%-20% level ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? lcccccccc & 0.255 & @xmath89 & 0.30 & @xmath90 & & & + & 0.197 & @xmath91 & 0.50 & @xmath92 & & & + & 0.206 & @xmath93 & 1.62 & & & & @xmath94 + & 0.230 & @xmath95 & 1.32 & @xmath96 & & & + & 0.203 & @xmath97 & 1.20 & & @xmath98 & & + & 0.171 & @xmath99 & 0.21 & @xmath100 & @xmath101 & @xmath102 & + & 0.288 & @xmath103 & 0.59 & & & @xmath104 & + & 0.182 & @xmath105 & 0.36 & @xmath106 & @xmath107 & @xmath108 & + & 0.282 & @xmath109 & 0.39 & @xmath110 & & @xmath111 & + & 0.217 & @xmath112 & 0.30 & @xmath113 & @xmath114 & @xmath115 & + & 0.285 & @xmath116 & 1.45 & @xmath117 & & & + & 0.206 & @xmath118 & 1.51 & @xmath119 & @xmath120 & @xmath121 & + & 0.299 & @xmath122 & 1.58 & & & @xmath123 & + & 0.226 & @xmath124 & 0.55 & @xmath125 & & @xmath126 & + & 0.325 & @xmath127 & 2.58 & @xmath128 & & & + & 0.280 & @xmath129 & 0.33 & @xmath130 & & & + & 0.228 & @xmath131 & 0.45 & @xmath132 & & @xmath133 & + & 0.253 & @xmath134 & 0.39 & @xmath135 & @xmath136 & @xmath137 & + & 0.171 & @xmath138 & 0.44 & & & @xmath139 & + & 0.320 & @xmath140 & 1.23 & @xmath141 & & @xmath142 & + & 0.153 & @xmath143 & 0.23 & @xmath144 & & @xmath145 & + & 0.229 & @xmath146 & 0.47 & @xmath147 & & & + & 0.152 & @xmath148 & 0.27 & @xmath149 & @xmath150 & @xmath151 & + & 0.228 & @xmath152 & 0.45 & & @xmath153 & @xmath154 & + & 0.224 & @xmath155 & 0.32 & @xmath156 & @xmath157 & @xmath158 & + & 0.258 & @xmath159 & 0.58 & & @xmath160 & @xmath161 & + & 0.345 & @xmath162 & 0.44 & @xmath163 & & & + & 0.164 & @xmath164 & 0.39 & & & & @xmath165 + & 0.235 & @xmath166 & 0.38 & @xmath167 & & & + & 0.291 & @xmath168 & 0.47 & & @xmath169 & @xmath170 & + from the virial relation @xmath171 between cluster mass , @xmath172 , inside radius @xmath173 , galaxy velocity , @xmath174 , and gas temperature , @xmath175 , combined with the definition of mass within an over - density of 500 times the critical density @xmath176 where the term @xmath177 describes the evolution of the over - density for a given cosmology , the mass temperature relation is obtained as @xmath178 in this study , we have measured @xmath48 , relating to @xmath179 through @xmath180 for the average cluster redshift with our chosen nfw model . we take into account that the slope may deviate from the simple theoretical expectation @xmath181 and normalize the relation at 8 kev since our sample is dominated by massive clusters . hence , mass temperature relations were obtained by fitting the data in table [ tab : dataset ] using the bces(@xmath182 ) estimator of @xcite to the following parameterization of the mass temperature relation @xmath183 the redshift - dependent factor , @xmath184 , contained in eq.[eq : fittedmtrelation ] must be calculated individually for each cluster , as this would otherwise produce an artificial 15% variation in mass over the redshift range spanned by our cluster sample . in effect , the normalization of the relation refers to the present epoch ( @xmath185 ) . the fitting procedure of @xcite takes uncertainties in temperatures as well as in weak lensing masses into account , and makes no assumptions about the intrinsic scatter of both quantities . results from fitting sub - samples as well as the full sample are presented in table [ tab : tstar ] and figure [ fig : mtrel ] . for the full data set ( for those clusters with temperature from more than one sample the temperature was taken in prioritized order from @xcite , @xcite , and @xcite ) we find the following normalization of the mass temperature relation at 8 kev @xmath186 and a slope of @xmath187 . it is evident that the slope of the mass temperature relation is not well - determined since our data only span a modest range at the high mass / high temperature end of the cluster distribution . in fact , it is not obvious that there is a tight mass temperature relation at the high temperature end probed here . the concentration of clusters around @xmath188 kev enables a robust measurement of the normalization of the mass temperature relation at the high mass end . even though the slope of the mass temperature relation varies substantially between the three x - ray sub - samples @xcite the best fit normalizations agree within their statistical uncertainty . we note that for all fits , the four different regressions of @xcite all result in normalizations within 20% . the strongest constraints on the mass temperature relation normalization are obtained by taking advantage of previous studies of massive clusters @xcite , showing that the mass temperature relation slope is close to @xmath2 as expected from simple gravitational collapse models @xcite . hence , in order to express the normalization of the mass temperature relation in terms of the characteristic temperature @xmath189 @xcite we assume @xmath181 ( with a representative uncertainty of 10% , e.g. * ? ? ? as is custom for quoting @xmath190 values we adopt the redshift dependence factor @xmath191^{-3/2}$ ] @xcite where @xmath192 is the mean overdensity inside the virial radius in units of the critical density at the relevant redshift ( @xmath193 and @xmath184 differs by 7% at @xmath71 ) . we find @xmath194 for our full sample and results from calculations of @xmath190 based on various subsamples are listed in table [ tab : tstar2 ] . from the @xmath0 - @xmath190 relation plotted by @xcite in their figure 2 , we find @xmath195 , based on our full sample . we note that this relation is valid only for an intrinsic scatter in temperature of @xmath196 around the mean mass temperature relation . a larger intrinsic scatter will imply a lower value of @xmath0 . we provide our constraints on the intrinsic scatter below . the squared scatter in lensing mass , @xmath48 , around the best fit , @xmath197 , is the sum of the squared measurement error , the squared intrinsic scatter , and the squared systematic errors @xmath198 . the main systematic errors in the lensing mass ( see section 2.1 ) arise from extrapolating the assumed nfw mass profile out to @xmath49 ( due to cluster - to - cluster variations in the assumed concentration parameter @xmath199 ) and from the separation of cluster / background galaxies ( due to cluster - to - cluster richness variations ) . each of these introduces a scatter of 20% in the lensing mass , hence @xmath200 . for the full sample we find @xmath201 which is larger than expected from the mean lensing mass error , @xmath202 and the systematic errors , indicating either a sizable intrinsic scatter in mass or that the measurement / systematic errors are severely under - estimated . accounting for errors in both mass and temperature , we find an intrinsic scatter in @xmath190 of @xmath203 . there is a 70% probability that the scatter in temperature is larger than 10% , favoring somewhat lower values of @xmath0 than quoted above . however , most of the scatter is caused by the low mass clusters . we looked into whether relaxed clusters and non - relaxed clusters have the same normalization of the mass temperature relation . our `` relaxed '' cluster sample consists of a586 , a963 , a1835 , a1995 , a2204 , a2261 , rxj1720 , and rxj1532 . these are clusters with `` spherical '' optical and x - ray morphology , and no known cluster - scale dynamic disturbances . for the relaxed clusters we find a normalization of the mass temperature relation of @xmath204 while the normalization for the non - relaxed clusters is @xmath205 ( see figure [ fig : relax ] ) . the higher normalization of relaxed clusters is supported by the fact that the mean mass of relaxed clusters is a factor @xmath206 larger than the mean mass of `` non - relaxed '' clusters , although the relaxed and the non - relaxed clusters span roughly the same temperature range . the scatter in mass for the relaxed sample ( @xmath207 ) is similar to the scatter for the non - relaxed sample ( @xmath208 ) . the mean error for both samples is @xmath209 . either relaxed clusters spread as much around their mass temperature relation as clusters in general , or we have used a poor definition of `` relaxed '' clusters . however , the fact that the normalization of the mass temperature relation for relaxed clusters is higher than for non - relaxed clusters indicates that there is a physical difference between the two sub - samples . from the present study , it thus seems that relaxed clusters do not form a tighter mass temperature relation than clusters in general . for a given mass , non - relaxed clusters are found to be @xmath210% hotter than relaxed clusters . since we consider global , isothermal temperatures , the presence of `` cooling cores '' in relaxed clusters will result in a lower global temperature than the virial temperature . however , this effect is at the 10%-20% level ( e.g. , * ? ? ? * ; * ? ? ? * ) so this can not alone explain the temperature difference between relaxed and non - relaxed clusters . based on the mass temperature relation from 10 clusters ( 3 of which are considered relaxed ) , s05 also find that non - relaxed clusters are hotter than relaxed clusters . an objective classification of the degree of relaxation for a sizeable cluster sample is required for further quantifying the size of this effect . ccr @xmath211 & @xmath212 & ota & mitsuda ( 2004 ) + @xmath213 & @xmath214 & allen ( 2000 ) + @xmath215 & @xmath216 & white ( 2000 ) + @xmath217 & @xmath218 & all + @xmath219 & @xmath220 & `` relaxed '' + @xmath221 & @xmath222 & `` non - relaxed '' + ccr @xmath223 & @xmath224 & ota & mitsuda ( 2004 ) + @xmath225 & @xmath224 & allen ( 2000 ) + @xmath226 & @xmath224 & white ( 2000 ) + @xmath227 & @xmath224 & all + @xmath228 & @xmath224 & `` relaxed '' + @xmath229 & @xmath224 & `` non - relaxed '' + lcccr x - rays & 0.09 & @xmath230 & fitted & @xcite + & 0.09 & @xmath231 & fitted & @xcite + & & & + lensing & 0.23 & @xmath232 & fixed & s05 + & 0.23 & @xmath233 & fitted & this study + & & & + simulations & 0.04 & @xmath234 & fixed & @xcite + & 0.00 & @xmath235 & fitted & @xcite + based on the hitherto largest sample of x - ray luminous clusters with measured lensing masses , we derive a normalization of the mass temperature relation at the high mass end , @xmath236 . this value is higher than the lensing based mass temperature normalization of s05 , based on a smaller cluster sample , but is consistent with this within @xmath237 errors ; see table [ tab : mnorm ] . temperature relations with masses determined from x - ray data tend to have a lower normalization than lensing based relations , and they are only marginally consistent with our normalization . this is also the case for the two recent studies of @xcite and @xcite based on smaller samples of lower mass ( and hence cooler ) clusters . @xcite measured cluster masses inside @xmath49 from x - ray observations of a sample of 13 low redshift clusters with a median temperature of 5.0 kev while @xcite determined the normalization from x - ray derived masses of 10 nearby clusters with a mean temperature of 4.8 kev . the two studies agree on the same normalization , higher than previous x - ray mass based studies , but there still seems to be a @xmath238 discrepancy between x - ray and lensing derived mass temperature relations . we note , however , that the lensing based and x - ray based normalizations are made at different redshifts , and that this discrepancy would vanish if the redshift - dependence predicted by the self - similar collapse model in equation [ eq : theorymt ] were neglected . given the heterogeneous nature of these data sets , any claim of significant departures from self - similarity would be premature , but this clearly provides an interesting avenue for future research , involving even larger cluster samples spanning a wider interval in redshift . we confirm the result of smith et al . ( 2005 ) that non - relaxed clusters are on average significantly hotter than relaxed clusters . this is qualitatively consistent with n - body / hydrodynamical cluster simulations which show that major mergers can temporarily boost the x - ray luminosities and temperatures well above their equilibrium values ( e.g. * ? ? ? in contrast to several previous ( mainly x - ray mass based ) published mass temperature relations , the normalization derived in this study is in good agreement with the normalization derived from numerical simulations . however , the accuracy of the normalization is not good enough to discriminate between simulations including different physical processes . our results show that x - ray based measurements of the cluster abundances , after reducing the major systematic uncertainties associated with the mass temperature normalization , give an amplitude of mass fluctuations on cluster scales that is consistent with other methods . this lends additional support to the `` concordance model '' cosmology , and lends credence to the basic assumptions of gaussian density fluctuations . our determination of @xmath239 is higher than most @xmath0 determinations from cluster data ( for a compilation of these , see e.g. , * ? ? ? however , our finding is consistent with the value derived from weak gravitational lensing in the combined deep and wide cfht legacy survey ( @xmath240 ; semboloni et al . 2005 ) based on the halo model of density fluctuations ( smith et al . it is also consistent with the cmb+2dfgrs+ly@xmath241 forest result ( @xmath242 ) of spergel et al . ( 2003 ) , with the joint cmb + weak lensing analysis of contaldi , hoekstra , & lewis ( 2003 ) , which gave @xmath243 , and with cmb analyses @xcite yielding @xmath244 . however , the more recent 3-year wmap results ( spergel et al . 2006 ) give a significantly lower value of @xmath0 , and also a preference for a value of @xmath245 lower than 0.3 . also , results from the recent 100 square degree weak lensing survey ( benjamin et al . 2007 ) favor a lower value of @xmath246 for @xmath5 . we note that our quoted value of @xmath0 is based on the assumption that the intrinsic scatter about the mass temperature relation is @xmath196 , and that our @xmath0 estimate will be biased high if the true scatter significantly exceeds this value @xcite . the limiting factor of our measurement of the normalization of the mass temperature relation is the magnitude of the measurement errors ( dominating the systematic errors , estimated to be @xmath247 ) . in order for the mass temperature relation to be a competitive route for constraining cosmological parameters and to discriminate between simulations with different input physics , the normalization must be measured to better than @xmath248% accuracy . however , there are good prospects for improving on these results in the near future . firstly , the superior spectro - imaging capabilities of _ chandra _ and xmm - newton will allow the construction of large , homogeneous cluster temperature samples . a comparison to tailored simulations with realistic physics , analyzed in the same way as observations , will advance our understanding of systematics and the link between the mass temperature relation and structure formation ( c.b . hededal et al . 2007 , in preparation ) . secondly , more accurate weak lensing - based mass measurements of a larger sample of clusters are feasible as large mosaic ccd cameras that can probe intermediate - redshift clusters beyond their virial radii are now common , and the cluster sample could easily be doubled from a similar survey in the southern celestial hemisphere . finally , we note that a more direct measurement of @xmath0 from weak lensing by clusters is possible , provided that weak lensing mass estimates are available for a large , well - defined , volume - limited cluster sample . this makes it feasible to calculate the cluster mass function directly from the lensing masses , rather than indirectly via the x - ray temperature function ( dahle 2006 ) . since mass estimates based on baryonic tracers of the total cluster mass only enters indirectly as a selection criterion ( e.g. , clusters selected based on x - ray luminosity above a certain threshold ) , the method is less susceptible to systematic and random errors , as it does not require an accurate characterization of the scatter around the mean mass temperature relation . we thank per b. lilje and jens hjorth for valuable comments on a draft version of this paper . we also thank the anonymous referee for comments and suggestions that improved the presentation of our results . kp acknowledges support from the danish national research council , the carlsberg foundation , and instrument center for danish astrophysics . the dark cosmology centre is funded by the danish national research foundation . hd acknowledges support from the research council of norway and travel support from nordita .
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the main uncertainty in current determinations of the power spectrum normalization , @xmath0 , from abundances of x - ray luminous galaxy clusters arises from the calibration of the mass temperature relation .
we use our weak lensing mass determinations of 30 clusters from the hitherto largest sample of clusters with lensing masses , combined with x - ray temperature data from the literature , to calibrate the normalization of this relation at a temperature of 8 kev , @xmath1 .
this normalization is consistent with previous lensing - based results based on smaller cluster samples , and with some predictions from numerical simulations , but higher than most normalizations based on x - ray derived cluster masses .
assuming the theoretically expected slope @xmath2 of the mass temperature relation , we derive @xmath3 for a spatially - flat @xmath4cdm universe with @xmath5 .
the main systematic errors on the lensing masses result from extrapolating the cluster masses beyond the field - of - view used for the gravitational lensing measurements , and from the separation of cluster / background galaxies , contributing each with a scatter of 20% . taking this into account ,
there is still significant intrinsic scatter in the mass temperature relation indicating that this relation may not be very tight , at least at the high mass end .
furthermore , we find that dynamically relaxed clusters are @xmath6 hotter than non - relaxed clusters .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the main goal of this note is to provide a gentle introduction to the theory of feynman - jackson integrals , and to compute an example of such integrals as explicitly as possible . roughly speaking feynman - jackson integrals are the analogue in @xmath0-calculus of feynman integrals . the name of jackson is included since integration in @xmath0-calculus was introduced by f.h . jackson , see @xcite and @xcite . our computations are done in the one dimensional setting . extending our concepts to higher dimensions , in particular to infinite dimensions , is the main problem in this subject . in principle such extension is possible but making the notation easy to handle is hard , as the reader will learn by looking at the simple case of one dimensional integrals . + the other fundamental task in this subject matter is to find applications of feynman - jackson integrals in physics . although in this note we focus on the mathematical properties of such integrals , we believe that our formalism will find applications thanks to the following facts : * in this paper our emphasis is on feynman - jackson integrals which rely on the construction of a @xmath0-analogue of the gaussian measure . given the widespread range of applications in mathematics and physics alike of the gaussian measure , we expect its @xmath0-analogue to gradually find its natural set of applications . the subject of @xmath0-probability theory and @xmath0-random processes is still in a developing phase , but already a solid step forward has been taken by kupershmidt in @xcite . * @xmath0-calculus is adapted to work with arbitrary functions while the usual rules of calculus demand certain kind of regularity . the @xmath0-gaussian measure is likely to find applications in the context of highly non - regular phenomena . * jackson integrals are given by infinite sums . cutting off the number of terms appearing in such sums provides a natural regularization method for diverging jackson integrals . * classically , we think of the continuous as a limit of the discrete . indeed , in the limit @xmath1 we recover from feynman - jackson integrals the usual feynman integrals . thus our feynman - jackson integrals provide a new method for computing feynman integrals : first compute the corresponding feynman - jackson integral and then take the limit as @xmath0 goes to @xmath2 . * from a quantum perspective , it is the continuous that should be regarded as an approximation to the fundamental discrete quantum theory . in particular it has been argued , see @xcite , @xcite , that in the quantization of gravity discrete structures will emerge naturally . if that is so , then integration over discrete structures may become a fundamental issue . our formalism may shed some light as the form that such theory of discrete integration may take . we emphasize that the calculus itself in @xmath0-calculus is discrete while the variables involved remain continuous . let us recall some notions of @xmath0-calculus , see @xcite , @xcite,@xcite and @xcite for more information . fix a real number @xmath3 . let @xmath4 be a function and @xmath5 be a real number . the @xmath0-derivative of @xmath6 at @xmath7 is given by @xmath8 for example if @xmath9 then @xmath10_{q}x^{t-1}$ ] where @xmath11_{q}=\frac{q^{t}-1}{q-1}$ ] . + the @xmath0-integral , better known as the jackson integral , of @xmath6 between @xmath12 and @xmath13 of @xmath6 is given by @xmath14 we also define @xmath15 and @xmath16 notice that in the limit @xmath1 the @xmath0-derivative and the @xmath0-integral approach the usual derivative and the riemann integral , respectively . the @xmath0-analogues of the rules of derivation and integration by parts are @xmath17 @xmath18 the first goal of this note is to describe the @xmath0-analogue of the gaussian measure on @xmath19 . the moments of the gaussian measure are given by the integrals @xmath20{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}x^ndx.\ ] ] a remarkable property of the gaussian measure is that it provides a bridge between measure theory and combinatorics . indeed , the moments of the gaussian measure are @xmath21{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}x^{2n}dx= ( 2n-1)(2n-3) ... 7.5.3.1,\ ] ] and @xmath22{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}x^{2n+1}dx= 0.\ ] ] the number @xmath23 is often denoted by @xmath24 and is called the double factorial . the reader may consult @xcite for a natural generalization of such numbers . it can be shown that @xmath23 counts the number of pairings on the set @xmath25=\{1,2, ... ,2n\}.$ ] a pairing on @xmath25 $ ] is a partition of @xmath25 $ ] into @xmath26 blocks each of cardinality two . so for example we have that @xmath27{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}x^{4}dx= 3,\ ] ] since as shown in figure [ fig : pochha graphs ] there are @xmath28 pairings on a set with @xmath29 elements . thus we see that the gaussian measure has a clear cut combinatorial meaning . this simple fact explains the source of graphs in the computation of feynman integrals . in order to define the @xmath0-analogue of the gaussian measure we must find @xmath0-analogues for the objects appearing in the gaussian measure , namely @xmath30{2\pi},\infty , e^{-\frac{x^2}{2 } } , x^n$ ] and @xmath31 . the lebesgue measure @xmath31 agrees with riemann integration for good functions . thus it is only natural to replace @xmath31 by jackson integration @xmath32 while the factor @xmath33 remains unchanged , finding the @xmath0-analogue of @xmath34 is actually quite a subtle matter . first , we must find a @xmath0-analogue for the exponential function @xmath35 which is characterized by the properties @xmath36 and @xmath37 . so we look for a function @xmath38 such that @xmath39 and @xmath40 a solution to this couple of equations is @xmath41_{q}!},\ ] ] where @xmath42_{q}!=[n]_{q}[n-1]_{q}[n-2]_{q} ... [2]_q \mbox { \ \ and \ \ } [ n]_{q}= 1 + q + q^2 + q^3 + ... + q^{n-1}.\ ] ] the @xmath0-analogue of the identity @xmath43 is @xmath44 , where @xmath45_{q}!}.\ ] ] once we have obtained @xmath0-analogues for the exponential map and its inverse one might think that it is straightforward to generalized the term @xmath34 of the gaussian integrals . however , this is not the case and while our first impulse is to try @xmath46_q}}$ ] , the right answer @xcite is to replace @xmath34 by @xmath47_q}}= \sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n+1)}x^{2n}}{(1+q)^{n}[n]_{q^{2}}!}.\ ] ] now we consider the integration limits . it is amusing that whereas the classical gaussian measure is given by an improper integral , its @xmath0-analogue turns out to be a definite integral whose limits depend on @xmath0 and go to ( plus or minus ) infinity as @xmath0 approaches to @xmath2 . a similar situation occurs with the integral representations of the @xmath0-analogue of the gamma function , see @xcite and @xcite . indeed , without further motivation we shall take the boundary limits in the gaussian integrals to be @xmath48 and @xmath49 where @xmath50 to find the @xmath0-analogue @xmath51 of the @xmath52 appearing in gaussian integrals we must demand that @xmath51 be such that @xmath53_q}}d_qx = 1.\ ] ] thus @xmath51 is given by @xmath54_q}}d_qx = 2\int_{0}^{\nu } e_{q^{2}}^{\frac{-q^2x^2}{[2]_q}}d_qx = 2(1-q)\nu\sum_{n=0}^{\infty}q^n e_{q^{2}}^{\frac{-q^2(q^n\nu)^2}{[2]_q}},\ ] ] so @xmath55_{q^{2}}!}\ ] ] and interchanging the order of summation we get the identity @xmath56_{q^{2}}!}.\ ] ] since @xmath57 we obtain the amusing identity @xmath58_{q^{2}}!}.\ ] ] we now look for the @xmath0-analogue of identities @xmath59 and @xmath60 for the moments of gaussian integrals . a key result is that one can show the identities @xmath61_q}}x^{2n}d_{q}x= [ 2n-1]_{q}[2n-3]_{q} ... [7]_{q}[5]_{q}[3]_{q}[1]_q=[2n-1]_q!!,\ ] ] and @xmath62_q}}x^{2n+1}d_{q}x= 0.\ ] ] identity @xmath63 follows from the fact that @xmath64 is an odd function and @xmath65_q}}$ ] is an even function . identity @xmath66 is proved recursively . using formula @xmath67 , one shows that @xmath68_q}}x^{2n+2}d_{q}x= \frac{[2n+1]_{q}}{c(q)}\int_{-\nu}^{\nu}e_{q^{2}}^{\frac{-q^2x^2}{[2]_q}}x^{2n}d_{q}x.\ ] ] identities ( [ qg ] ) and ( [ qgc3 ] ) imply identity ( [ qgc1 ] ) . next we describe a combinatorial interpretation of the number @xmath69_q!!=[2n-1]_{q}[2n-3]_{q} ... [7]_{q}[5]_{q}[3]_{q}[1]_q.\ ] ] an ordered pairing @xmath70 on @xmath25=\{1,2, ... ,2n\}$ ] is a sequence @xmath71 ^ 2)^n$ ] such that * @xmath72 . * @xmath73 * @xmath74=\bigsqcup_{i=1}^{n } \{a_i , b_i\}.}$ ] we denote by @xmath75 $ ] the set of ordered pairings on @xmath25 $ ] . we are going to define a weight @xmath76 for each @xmath77 $ ] . let us introduce the following notation * @xmath78 : a_i < j < b_i\}$ ] for all @xmath79 . * @xmath80 . the weight of @xmath70 is defined by the rule @xmath81 figure [ fig:1 ] and figure [ fig:2 ] below show a couple of examples of pairings together with the corresponding weights .[fig:1],width=288 ] .[fig:2],width=288 ] using this language we can state the following result which is proved by induction @xmath82_q!!= \sum_{p \in p[2n]}w(p).\ ] ] notice that as @xmath83 we recover from @xmath84 the well - known identity @xmath85\}|.\ ] ] for example @xmath86_q = 1+q+q^{2}$ ] which agrees with the sum of the weights of the three pairings on @xmath87 $ ] shown in figure @xmath88 . _ q.$ ] [ fig : pochha graphs],width=528 ] in conclusion we have proved that @xmath89_q}}x^{2n}d_{q}x= \sum_{p \in p[2n]}w(p).\ ] ] we want to study the computation of feynman - jackson integrals of the form @xmath90_q}+g \frac{x^3}{[3]_q ! } } } d_qx.\ ] ] as usual in the theory of feynman integrals we regard @xmath91 as a formal power series in the formal variable @xmath92 , i.e. , @xmath93 $ ] . the first step in the computation of a feynman - jackson integrals is to reduce it to the computation of a countable number of gaussian - jackson integrals . this step is carry out with the help of following formula proved in @xcite @xmath94 where @xmath95_{q^{2}}![c - k]_{q^{2}}!}}$ ] . making the substitutions @xmath96_q}}$ ] and + @xmath97_q!}}$ ] we obtain @xmath98_q}+g\frac{x^{3}}{[3]_q!}}= e_{q^2}^{-\frac{q^{2}x^{2}}{[2]_q}}\sum_{c , d\geq 0 } \lambda_{c , d } \frac{(-1)^{c}q^{2c } x^{2c } x^{3d } } { [ 2]_q^{c}([3]_q!)^{d}}g^{d}=e_{q^2}^{-\frac{q^{2}x^{2}}{[2]_q}}\sum_{c , d\geq 0 } \lambda_{c , d } \frac{(-1)^{c}q^{2c } x^{2c+3d } } { [ 2]_q^{c}([3]_q!)^{d}}g^{d } } , \ ] ] so we get @xmath99_q}+g\frac{x^{3}}{[3]_q!}}=\sum_{c , d , k } \frac{(-1)^{2c - k } { d+k\choose k } q^{(d+k)(d+k-1)+2c}}{[2]_q^c([3]_q!)^{d}[d+k]_{q^{2}}![c - k]_{q^{2 } } ! } x^{2c+3d } g^d } .\ ] ] if we @xmath0-integrate both sides of equation ( [ 26 ] ) we get @xmath100_q}+g\frac{x^{3}}{[3]_q ! } } } d_qx&=&{\displaystyle\sum_{c , d , k } \frac{(-1)^{2c - k } { 2d+k\choose k } q^{(2d+k)(2d+k-1)+2c } [ 2c+6d-1]_q!!}{[2]_q^c([3]_q!)^{2d}[d+k]_{q^2 } ! [ c - k]_{q^2}!}g^{2d } } \\ \mbox { } & \mbox { } & \mbox { } \nonumber \\ & = & { \displaystyle \sum_{c , d , k } \frac{(-1)^{2c - k } { 2d+k\choose k } q^{(2d+k)(2d+k-1)+2c } [ 2c+6d-1]_q!!}{[2]_q^c([3]_q!)^{2d}[2d+k]_{q^2 } ! [ c - k]_{q^2}!}g^{2d}}\label{e2}\end{aligned}\ ] ] the second step in the computation of a feynman integral is to write such as integral as a sum of a countable number of contributions , where each summand is naturally associated to certain kind of graph , see @xcite . so we want to understand the right hand side of the equation ( [ e2 ] ) in terms of a summation of the weights of an appropriated set of isomorphism classes of graphs . consider the category @xmath101 whose objects are planar graphs @xmath102 such that 1 . @xmath103 , where @xmath104 and @xmath105 . 2 . @xmath106 . @xmath107 this data must satisfy the following axioms : 1 . @xmath108 and @xmath109 . 2 . @xmath110 and if @xmath111 then @xmath112 for @xmath113 . 3 . if @xmath114 , then @xmath115 or @xmath116 for some @xmath117 or@xmath118 . 4 . if @xmath119 then @xmath120 . 5 . @xmath121 . + to each graph @xmath122 as above we associate two polynomials in @xmath0 , @xmath123 and @xmath124 , given respectively , by 1 . @xmath125 . above @xmath70 is the natural pairing induced by @xmath126 on the flags of @xmath127 , i.e.,the set @xmath128 . @xmath129 is the weight of @xmath70 as given by formula ( [ peso ] ) . 2 . @xmath130_q^{|v_1|}([3]_q!)^{|v_2|}[|v_2|+|e_2|]_{q^2}![|v_1|-|e_2|]_{q^2}!$ ] . figure [ fig : wq ] and figure [ fig : wq1 ] show examples of graphs in @xmath101 together with the corresponding polynomials @xmath131 and @xmath132 . with @xmath133 and @xmath134_q^4[3]_q^3$].[fig : wq],width=192 ] with @xmath135 and @xmath134_q^2([3]_q!)^2[4]_q!$ ] . [ fig : wq1],width=192 ] now we are ready to state the main result of this paper , namely , we write the integral ( [ int24 ] ) as a sum over the weights of graphs as follows @xmath136_q}+g\frac{x^{3}}{[3]_q ! } } } d_qx=\sum_{(v , e , b)\in{\mathbf{graph}_q^3}/\sim } \frac{\omega_q(\gamma)}{a_q(\gamma)}g^{|v_2| } } \ ] ] where the sum runs over all isomorphisms classes of graphs in @xmath101 . identity ( [ ult ] ) follows directly from identity ( [ e2 ] ) and the definitions above . we thank bernardo uribe and the math department at uniandes where this work was finished . rafael daz and eddy pariguan , _ feynman - jackson integrals _ , journal of nonlinear mathematical physics 13 ( 2006 ) , no . 3 , 365376 . rafael daz and carolina teruel , _ q , k - generalized gamma and beta functions _ , journal of nonlinear mathematical physics 12 ( 2005 ) , no . 1 , 118134 . h.t . koelink and koornwinder , _ q - special functions , in deformation theory and quantum groups with applications to mathematical physics _ , edited by murray gerstenhaber and jim stasheff , amer . math soc 134 ( 1992 ) , 141142 . rovelli carlo and smolin lee , _ discreteness of area and volume in quantum gravity _ , nuclear phys , b442 , 3 , ( 1995 ) , 593 - 619 . smolin , lee , _ the physics of spin networks _ , the geometric universe , oxford univ . press , oxford , ( 1998 ) , 299 - 304 . a. de sole and v. kac , _ on integral representations of q - gamma and q - beta functions_,atti . lincei cl . fis . mat . lincei ( 9 ) . mat . 16 , 41 , ( 2005 ) , 11 - 29 .
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we review the construction of a @xmath0-analogue of the gaussian measure .
we apply that construction to obtain a @xmath0-analogue of feynman integrals and to compute explicitly an example of such integrals .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the analyses reported in this talk were performed using either a sample of @xmath9 @xmath7 events or a sample of @xmath10 @xmath8 events collected with the upgraded beijing spectrometer ( besii ) detector @xcite at the beijing electron - positron collider ( bepc ) . a new structure , denoted as @xmath0 and with mass @xmath11 gev/@xmath12 and width @xmath13 mev/@xmath12 , was observed by the babar experiment in the @xmath14 initial - state radiation process @xcite . this observation stimulated some theoretical speculation that this @xmath15 state may be an @xmath16-quark version of the @xmath17 since both of them are produced in @xmath18 annihilation and exhibit similar decay patterns @xcite . here we report the observation of the @xmath0 in the decays of @xmath19 , with @xmath20 , @xmath21 , @xmath22 . a four - constraint energy - momentum conservation kinematic fit is performed to the @xmath23 hypothesis for the selected four charged tracks and two photons . @xmath24 candidates are defined as @xmath25-pairs with @xmath26 gev/@xmath12 , a @xmath6 signal is defined as @xmath27 gev/@xmath12 , and in the @xmath28 invariant mass spectrum , candidate @xmath29 mesons are defined by @xmath30 gev/@xmath12 . the @xmath31 invariant mass spectrum for the selected events is shown in fig . [ draft - fit ] , where a clear enhancement is seen around 2.18 gev/@xmath12 . fit with a breit - wigner and a polynomial background yields @xmath32 signal events and the statistical significance is found to be @xmath33 for the signal . the mass of the structure is determined to be @xmath34 gev/@xmath12 , the width is @xmath35 gev/@xmath12 , and the product branching ratio is @xmath36 . the mass and width are consistent with babar s results . invariant mass distribution of the data ( points with error bars ) and the fit ( solid curve ) with a breit - wigner function and polynomial background ; the dashed curve indicates the background function.,scaledwidth=40.0% ] structures in the @xmath38 invariant - mass spectrum have been observed by several experiments both in the reaction @xmath39 @xcite and in radiative @xmath7 decays @xcite . the @xmath2 was first observed by the mark - iii collaboration in @xmath7 radiative decays @xmath40 . a fit to the @xmath38 invariant - mass spectrum gave a mass of 2.22 gev/@xmath12 and a width of 150 mev/@xmath12 @xcite . an angular analysis of the structure found it to be consistent with a @xmath41 assignment . it was subsequently observed by the dm2 collaboration , also in @xmath42 decays @xcite . we present results from a high statistics study of @xmath43 in the @xmath44 final state , with the @xmath45 missing and reconstructed with a one - constraint kinematic fit . after kinematic fit , we require both the @xmath46 and @xmath47 invariant masses lie within the @xmath6 mass region ( @xmath48 mev/@xmath12 and @xmath49 mev/@xmath12 ) . the @xmath38 invariant mass distribution is shown in fig . [ dalitz ] . there are a total of 508 events with a prominent structure around 2.24 gev/@xmath12 . invariant mass distribution for @xmath50 candidate events . the dashed histogram is the phase space invariant mass distribution , and the dotted curve indicates how the acceptance varies with the @xmath38 invariant mass.,scaledwidth=40.0% ] a partial wave analysis of the events with @xmath51 2.7 gev/@xmath12 was performed . the two - body decay amplitudes in the sequential decay process @xmath52 , @xmath53 and @xmath54 are constructed using the covariant helicity coupling amplitude method . the intermediate resonance @xmath55 is described with the normal breit - wigner propagator @xmath56 , where @xmath16 is the @xmath38 invariant mass - squared and @xmath57 and @xmath58 are the resonance s mass and width . when @xmath59 , @xmath60 is fitted with both the @xmath38 and @xmath61 systems in a @xmath62-wave , which corresponds to a pseudoscalar @xmath55 state , the fit gives @xmath63 events with mass @xmath64 gev/@xmath12 , width @xmath65 gev/@xmath12 , and a statistical significance larger than @xmath66 , and a product branching fraction of : @xmath67 . the presence of a signal around 2.24 gev/@xmath12 and its pseudoscalar character are confirmed , and the mass , width , and branching fraction are in good agreement with previous experiments . a pseudoscalar gluonium candidate , the so - called @xmath68 , was observed in @xmath69 annihilation in 1967 @xcite and in @xmath7 radiative decays in the 1980 s @xcite . the study of the decays @xmath70 \{@xmath5 , @xmath6}@xmath71 is a useful tool in the investigation of quark and possible gluonium content of the states around 1.44 gev/@xmath72 . here we investigate the possible structure in the @xmath71 final state in @xmath7 hadronic decays at around @xmath73 gev/@xmath72 . in this analysis , @xmath5 mesons are observed in the @xmath74 decay , @xmath6 mesons in the @xmath75 decay , and other mesons are detected in the decays : @xmath76 , @xmath77 . @xmath71 could be @xmath78 or @xmath79 . figures [ fig : w - x1440-recoiling ] and [ fig : x1440-phikksp ] show the @xmath80 and @xmath81 invariant mass spectra after @xmath5 selection ( @xmath82 gev / c@xmath83 ) or @xmath6 signal selection ( @xmath84 gev/@xmath72 ) . clear @xmath4 signal is observed recoiling against the @xmath5 , and there is no significant signal recoiling against a @xmath6 . the @xmath80 invariant mass distribution in @xmath85 ( fig . [ fig : w - x1440-recoiling](b ) ) is fitted with a bw function convoluted with a gaussian mass resolution function ( @xmath86 mev/@xmath72 ) to represent the @xmath4 signal and a third - order polynomial background function . the mass and width obtained from the fit are @xmath87 mev/@xmath72 and @xmath88 mev/@xmath72 , and the fit yields @xmath89 events . using the efficiency of @xmath90 determined from a uniform phase space mc simulation , we obtain the branching fraction to be @xmath91 , where the first error is statistical and the second one systematic . for @xmath92 mode , by fitting the @xmath81 mass spectrum in fig . [ fig : w - x1440-recoiling](c ) with same functions , we obtain the mass and width of @xmath93 mev/@xmath72 and @xmath94 mev/@xmath72 , and the number of events from the fit is @xmath95 . the efficiency is determined to be @xmath96 from a phase space mc simulation , and the branching fraction is @xmath97 , in good agreement with the isospin symmetry expectation from @xmath85 mode . the distribution of @xmath98 and @xmath99 invariant mass spectra recoiling against the @xmath6 signal are shown in fig . [ fig : x1440-phikksp ] , and there is no evidence for @xmath4 . the upper limits on the branching fractions at the @xmath100 c.l . are @xmath101 and @xmath102 . in conclusion , the mass and width of the @xmath4 are measured , which are in agreement with previous measurements ; the branching fractions we measured are also in agreement with the dm2 and mark - iii results . the significant signal in @xmath103 mode and the missing signal in @xmath104 mode may indicate the @xmath105 component in the @xmath4 is not significant . besides conventional meson and baryon states , qcd also predicts a rich spectrum of glueballs , hybrids , and multi - quark states in the 1.0 to 2.5 @xmath107 mass region . therefore , searches for the evidence of these exotic states play an important role in testing qcd . the radiative decays of @xmath106 to hadrons are expected to contribute about 1% to the total @xmath106 decay width @xcite . however , the measured channels only sum up to about 0.05% @xcite . we measured the decays of @xmath106 into @xmath108 , @xmath109 , @xmath110 , @xmath111 , @xmath112 , @xmath113 , @xmath114 , @xmath115 , @xmath116 , and @xmath117 , with the invariant mass of the hadrons ( @xmath118 ) less than 2.9 @xmath107 for each decay mode @xcite . the differential branching fractions are shown in fig . [ difbr ] . the branching fractions below @xmath119 @xmath120 are given in table [ tot - nev ] , which sum up to @xmath121 of the total @xmath106 decay width . we also analyzed @xmath122 and @xmath123 modes to study the resonances in @xmath124 and @xmath125 invariant mass spectrum . significant signals for @xmath126 and @xmath127 were observed , but the low statistics prevent us from drawing solid conclusion on the other resonances @xcite . differential branching fractions for @xmath128 , @xmath129 , @xmath111 , and @xmath110 . here @xmath118 is the invariant mass of the hadrons . for each point , the smaller longitudinal error is the statistical error , while the bigger one is the total error . , scaledwidth=48.0% ] .[tot - nev ] branching fractions for @xmath130 with @xmath119 @xmath107 , where the upper limits are determined at the 90% c.l . [ cols="<,<",options="header " , ] using the 58 m @xmath131 and 14 m @xmath106 events samples taken with the besii detector at the bepc storage ring , bes experiment provided many interesting results in charmonium decays , including the observation of the @xmath0 , @xmath2 , @xmath4 , and many @xmath106 radiative decays . these results shed light on the understanding of strong interaction sector of the standard model . a. etkin _ et al . _ , phys . b * 201 * , 568 ( 1988 ) . z. bai _ et al . _ [ mark - iii collaboration ] , phys . lett . * 65 * , 1309 ( 1990 ) . d. bisello _ et al . _ [ dm2 collaboration ] , phys . b * 179 * , 294 ( 1986 ) . d. bisello _ et al . _ [ dm2 collaboration ] , phys . b * 241 * , 617 ( 1990 ) . m. ablikim _ et al . _ [ bes collaboration ] , phys . d * 77 * , 032005 ( 2008 ) .
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we present recent results from the bes experiment on the observation of the @xmath0 in @xmath1 , study of @xmath2 in @xmath3 , and the production of @xmath4 recoiling against an @xmath5 or a @xmath6 in @xmath7 hadronic decays .
the observation of @xmath8 radiative decays is also presented .
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the acceleration of charged particles to high energies in the solar corona is related to flares , which reveal the dissipation of magnetically stored energy in complex magnetic field structures of the low corona , and to coronal mass ejections ( cmes ) , which are large - scale , complex magnetic - field - plasma structures ejected from the sun . cmes can drive bow shocks , and their perturbation of the coronal magnetic field can also give rise to magnetic reconnection , where energy can be released in a similar way as during flares . when several cmes are launched along the same path , a faster cme may overtake a slower preceding one , and the two cmes can merge into a single structure . for this phenomenon @xcite introduced the term _ cme cannibalism_. the cme - cme interaction was found associated with a characteristic low - frequency continuum radio emission . @xcite interpreted this type of activity as the radio signature of non - thermal electrons originating either during reconnection between the two cmes or as the shock of the second , faster cme travels through the body of the first ( see * ? ? ? * for a numerical study of two interacting coronal mass ejections ) . in this paper we use radio diagnostics to study electron acceleration during a complex solar event broadly consisting of two stages , each associated with a distinct episode of a flare and with a fast cme , which occurred in close temporal succession on 17 january 2005 . the cmes interacted at a few tens of r@xmath0 . both the flare / cme events and the cme interaction were accompanied by radio emission , which is used here to study electron acceleration scenarios . energetic electrons in the corona and interplanetary space are traced by their dm - to - km - wave radio emission , mostly excited at or near the electron plasma frequency . the emission provides a diagnostic of the type of the exciter and its path from the low corona ( cm - dm wavelengths ) to 1 au ( km wavelengths ) . radio emissions from exciters moving through the corona appear in dynamic spectra as structures exhibiting a drift in the time frequency domain . the drift rate depends on their speed and path , resulting in a variety of bursts . type iii bursts trace the path of supra thermal electrons guided by magnetic structures . they appear , on dynamic spectra , as short ( lasting from a fraction of a second at dm - waves to a few tens of minutes at km - waves ) structures with fast negative drift , ( @xmath1 ; see for example * ? ? ? this corresponds to anti - sunward propagation of the electrons through regions of decreasing ambient density at speeds @xmath2 ( e.g. , * ? ? ? sunward travelling beams produce reverse drift bursts ( rs bursts ) , and beams propagating in closed loops emit type u or j bursts comprising a succession of an initial drift towards lower frequencies and a more or less pronounced rs burst . type ii bursts are more slowly drifting bursts ( @xmath3 ; see , for example , table a.1 in * ? ? ? * ) excited by electrons accelerated at travelling shocks and emitting in their upstream region . finally broadband dm - m wave continuum emission that may last over several minutes or even hours ( type iv burst ) is ascribed to electrons confined in closed coronal magnetic structures . the reader is referred to the reviews in @xcite , @xcite , @xcite and @xcite for more detailed accounts of the radio emission by non thermal electrons in the corona . lllll * event * & * time * & * characteristics * & * remarks * + & * ut * & & + sxr start & 06:59 & & ar10720 ( n15@xmath4 w25@xmath4 ) + type iv & 08:40 & 3.0 - 630 mhz & ar10720 + cme@xmath5 & 09:00 & & lift - off + * sxr stage 1 * & 09:05 & & + first cm & 09:05 & & rstn 15400 mhz + burst start & & & + type iii@xmath5 & 09:07 - 09:28 & 0.2 - 630 mhz & ar10720 + type ii@xmath5 & 09:11 & 0.2 - 5 mhz & ar10720 + h@xmath6 start & 09:13 & 3b & kanz , ar10720 + cme@xmath5 & 09:30 & 2094 km sec@xmath7 & on c2 + hxr start & 09:35:36 & & rhessi number 5011710 + cme@xmath8 & 09:38 & & lift - off + * sxr stage 2 * & 09:42 & & end sxr stage 1 + second cm & 09:43 & & rstn 15400 mhz + burst start & & & + type iii@xmath8 & 09:43 - 09:59 & 0.2 - 630 & ar10720 + hxr peak & 09:49:42 & 7865 counts sec@xmath7 & + type ii@xmath8 & 09:48 & 0.2 - 8 mhz & ar10720 + sxr peak & 09:52 & x3.8 & end sxr stage 2 + cme@xmath8 & 09:54 & 2547 km sec@xmath7 & on c2 + first rise & 10:00 & 38 - 315 kev & ace / epam + electron flux & & & + sxr end & 10:07 & & ar720 + hxr end & 10:38:52 & 53152112 total counts & rhessi + second rise & 12:00 & 38 - 315 kev & ace / epam + electron flux & & & + type iii@xmath9 & 11:37 & 0.5 mhz & cme@xmath5 , cme@xmath8 merge at 37 r@xmath0 + & & & type ii@xmath8 overtakes type ii@xmath5 + h@xmath6 end & 11:57 & & kanz + type iv end & 15:24 & 3.0 - 630 mhz & ar10720 + line centre ( top left ) and in the wing , observed at kanzelhhe observatory ( courtesy m. temmer ) . solar north is at the top , west on the right . the two snapshots at the top show the active region before the flare under discussion , the two bottom images show two instants during the stages 1 and 2 , respectively . these stages were associated with the disappearance of the filaments labelled ` f1 ' and ` f2 ' . ] . bottom : two frames of the 09:54:05 halo cme with back - extrapolated lift off at 09:38:25 ut and plane - of - the - sky speed 2547 km sec@xmath7 . solar north is at the top , west on the right . ] the 17 january 2005 event consisted of a complex flare , two very fast coronal mass ejections ( cmes ) , and intense and complex soft x - ray ( sxr ) and radio emission . in all radiative signatures two successive stages can be distinguished . the cmes were launched successively from neighbouring regions of the corona and interacted in interplanetary space . the sequence of the observed energetic phenomena is summarized in table [ t ] and described , in detail , in the following subsections . figure [ fig_kanz ] displays snapshots in the h@xmath6 line obtained from the kanzelhhe solar observatory ( courtesy m. temmer ; see also @xcite , their figure 2 , for details on the evolution of the h@xmath6 flare ribbons ) . the only major active region on the disk is noaa 10720 in the north - western quadrant ( n15@xmath4 w25@xmath4 ) . it displayed nearly uninterrupted activity since the early hours of 17 january 2005 . the most conspicuous event was a 3b h@xmath6 flare reported by kanzelhhe 09:16 - 11:57 ut . this flare proceeded successively in two different parts of ar 10720 , as shown in the two snapshots of the bottom panel . the first part of the event , referred to as stage 1 " ( illustrative snapshot at 09:13 ut ) , is seen in the eastern part of the active region , close to the sunspots . it is associated with the temporary disappearance or eruption of the filament labelled ` f1 ' in the upper right panel . two major flare ribbons are distinguished in the snapshot at 09:13 ut : a narrow band essentially in the east - west direction and a broader north - southward oriented region . the significant offset of the two ribbons with respect to the neutral line shows the eruption of a strongly sheared magnetic field . after about 09:35 ut the brightest emission is seen in the western part of the active region ( stage 2 " ; see snapshot at 09:54 ut ) , together with the eruption of another filament ` f2 ' ( or of a different part of the filament whose northern section erupted before ) . the brightening consisted of two essentially parallel flare ribbons , which were connected by post flare loops in later snapshots ( not shown here ) . these two stages of the event were also found in the soft x - ray ( sxr ) and radio emissions , as will be discussed below . two cmes were observed in close succession . a sequence of difference images from the large angle and spectrometric coronagraph ( lasco ) aboard the soho spacecraft @xcite is displayed in figure [ cmes ] : the first cme ( henceforth cme@xmath5 ) is seen in the image at 09:30 ut in the north - western quadrant . while it travelled through the corona , the second , broader cme ( cme@xmath8 ) appeared underneath ( image at 09:54 ut ) . the most conspicuous features of both cmes are seen above the north - western limb , but both were labelled halo cmes in the lasco cme catalog @xcite . speeds of , respectively , 2094 and 2547 km s@xmath7 were derived from linear fits to the trajectories of their fronts published in the cme catalogue . formally the cme fronts described by the fits intersected near 12:32 ut at a heliocentric distance of about 38 r@xmath0 . the statistical error of the speeds of the cme fronts and their liftoff times , derived from the abovementioned linear least - squares fit to the measured heliocentric distances , leads to an uncertainty of @xmath103 h in the time of intersection . this uncertainty stems from the fact that the two height - time trajectories are nearly parallel . we will show in sect . [ cme2cme ] that cme interaction actually occurred well before the formal time of intersection . of course a single instant of interaction between two complex cmes is fictitious anyway . an overview of the complex radio event is given in figure [ f3 ] . there we present the dynamic flux density spectrum of the radio bursts in the 650 mhz-20 khz range ( heliocentric distance @xmath11 1.1 r@xmath0to 1 au ) using combined recordings of the _ appareil de routine pour le traitement et lenregistrement magntique de linformation spectrale _ ( artemis - iv ) solar radio - spectrograph @xcite and the _ wind_/waves experiment @xcite . several other time histories are superposed on the dynamic spectrum : * dashed lines display the approximate frequency - time trajectories of the two cme fronts , using the density model of @xcite , which describes well the coronal density behavior in the large range of distances from low corona to interplanetary space : @xmath12 + the linear fits to the height - time trajectories of the cme fronts in the lasco images were converted to frequency - time tracks of fundamental ( black line ) and harmonic ( red line ) plasma emission . * the solid blue curve displays the sxr time history ( 0.1 - 0.8 nm ) , using goes on line data ( http//www.sel.noaa.gov / ftpmenu / indices ) , describing thermal emission from the flare - heated plasma . * the red curve is the microwave time history at 15.4 ghz , produced by non thermal electrons ( energies @xmath13100 kev ) in magnetic fields of a few hundred g ; these were obtained from the san vito solar observatory of the radio solar telescope network ( rstn ) @xcite . the two stages of the flare identified in the h@xmath6 observations in figure [ fig_kanz ] correspond to two distinct events of energy release seen in the sxr and microwave time profiles ( figure [ f3 ] ) . the sxr time profile had an initial smooth increase between 06:59 ut and 09:05 ut . subsequently the sxr flux rose slightly faster until 09:45 ut ( stage 1 ) , and even faster ( stage 2 ) until the x3.8 peak at 09:52 ut . the gradual rise in stage 1 and the faster rise in stage 2 were each accompanied by strong microwave bursts . the second burst was also observed in hard x - rays by rhessi @xcite . the dominant features in the dynamic spectrum observed by _ wind_/waves at frequencies below 2 mhz are two groups of type iii bursts , labelled iii@xmath5 and iii@xmath8 . they occurred in association with the sxr and microwave emissions of stages 1 and 2 , respectively , and with the two different parts of the h@xmath6 flare . the two type iii groups occurred near the extrapolated liftoff times of the two cmes . radio images taken by the nanay radioheliograph ( nrh ; * ? ? ? * ) show that the sources are located in the north - western quadrant near the flaring active region . hence both flare episodes were efficient accelerators of electrons that escaped to the interplanetary space along open magnetic field lines rooted at or near the flare site . the second type iii group ( type iii@xmath8 ) was followed by a more slowly drifting narrow - band burst ( type ii , labelled ii@xmath8 ) produced by a coronal shock wave . upon closer inspection the spectrum suggests that similar drifting features can also be associated with the first flare episode , although the association is less evident . we label these bursts ii@xmath5 in figure [ f3 ] . since the two cmes are extremely fast , they are expected to drive shock waves in the corona . the observed type ii emission can be compared with the dashed curves in figure [ f3 ] , which track fundamental ( black ) and harmonic ( red ) emission expected from the trajectory of the cme front and the coronal density model . it is clear that this density model is only indicative , especially in the perturbed corona through which travels the second cme ( see discussion in subsection [ m ] ) . we therefore associate type ii@xmath5 and ii@xmath8 to the bow shocks of the two cmes , although other interpretations , like shocks on the flanks or shocks from a driver related to the flare , are not excluded . the dm - m wave emission consisted of a type iv continuum , the metre wave counterparts of the dekametre - hectometre ( dh ) type iii groups and of the type ii bursts . the type iv continuum started near 08:40 ut during the initial smooth increase of the sxr flux before stage 1 . it was first visible as a grey background in the dynamic spectrum , and became progressively more intense . it dominated the metre wave spectrum during and after type iii@xmath8 , and gradually penetrated to lower frequencies , down to 5 mhz . images in the eit 195 channel @xcite and in the 164 - 432 mhz range taken by the nrh indicate that the thermal ( soft x - rays ) and non thermal ( radio ) emissions all originated near noaa ar 10720 . in the time interval from the start of the type iv burst to the start of stage 1 a wealth of fine structures was recorded ( see * ? ? ? * ) . from the high - resolution observations in the 200 - 500 mhz range ( see figures [ fs1 ] , [ fs2 ] for example ) it appears that most bursts are broadband pulsations . other fine structures of type iv emission such as spikes , fiber bursts and zebra pattern appear occasionally ( see * ? ? ? * for a description of fine structure of type iv emission ) . during type iii@xmath5 the spectral character of the radio emission was clearly different at frequencies below and above the inferred frequency - time track of the cme@xmath5 ( see fig . [ f3 ] ) . on the low - frequency side of the track strong type iii bursts were prominent after about 09:22 ut . they were preceded by a less regular emission , which @xcite label complex type iii bursts " because of its varying flux density across the spectrum . the metre wave counterpart on the high - frequency side of the estimated cme track consisted of a succession of spectral fine structures on the time scale of seconds , with different spectral characteristics superposed on the type iv continuum , followed after 09:11 ut by the high - frequency extension of the dekametre - hectometre ( dh ) type iii group iii@xmath5 . a more detailed view of the difference spectrum is given in the top panel of figure [ f4 ] , while high resolution images of the fine structures are in figure [ fs1 ] . among these fine structures were broadband pulsations , bursts with ordinary and reverse drift , and fiber bursts due to whistlers travelling upwards in the corona ( see figure [ fs1 ] , _ e.g. _ , 09:16:20 - 09:16:45 ut ) . the variety of these bursts shows the acceleration and partial trapping of electron populations in the corona well behind the front of the cme . indeed , few of the well - identified bursts above 100 mhz seem to continue into the 30 - 70 mhz range . it was only near the end of type iii@xmath5 ( @xmath11 08:18 ut ) that metre wave type iii bursts appeared as systematic high - frequency extensions of the type iii bursts observed below 2 mhz . type iii@xmath8 started at 09:43 ut , together with the second microwave burst , near the back - extrapolated lift - off of cme@xmath8 ( 09:38 ut ) and the onset of stage 2 of the sxr burst . a close look at the dynamic spectrum ( bottom panel of figure [ f4 ] ) reveals negative overall drifts below 100 mhz , while burst groups with positive overall drift prevailed above 130 mhz . the high - resolution spectrogram in the 300 - 400 mhz range ( fig . [ fs2 ] ) shows a wealth of individual bursts with different drift rates and zebra pattern . these bursts show again , like in stage 1 of the event , that the high - frequency bursts are produced by an accelerator below the cme front , while low - frequency bursts show the start of the prominent dh type iii bursts . well after the decay of the sxr and microwave emission a third group of bursts ( iii@xmath9 in figure [ f3 ] ) is identified ( near 11:30 ut ) , with unusually low starting frequency ( 0.6 mhz ) , pointing to an acceleration of the emitting electrons at unusually great height . a more detailed view of the low - frequency radio spectrum of this burst group and the preceding groups iii@xmath5 and iii@xmath8 is given by the dynamic spectrum as observed by the _ thermal noise receiver _ ( tnr ) of _ wind_/waves in the left panel of figure [ fig_tnr_20050117 ] . the narrow - band short bursts near 32 khz are langmuir wave packets . together with the fainter continuous band on which they are superposed they indicate how the electron plasma frequency evolves at the _ wind _ spacecraft . at the time of iii@xmath9 it is about 35 khz . using a standard interplanetary density model , where the electron plasma frequency decreases as the inverse of the heliocentric distance @xmath14 , the starting frequency of iii@xmath9 implies @xmath15 r@xmath0 for fundamental plasma emission , and @xmath16 r@xmath0 for the harmonic . from the lasco observations and the uncertainties resulting from the straight - line fits to the cme front trajectories , the heliocentric distances of the cme fronts at 11:30 ut are , respectively , @xmath17 r@xmath0 and @xmath18 r@xmath0 . the burst group iii@xmath9 is hence consistent with harmonic emission from the vicinity of the cme fronts . this points to a close relationship of this episode of electron acceleration with the interaction of the two cmes . comparison of the three groups of type iii bursts in the tnr spectrum of figure [ fig_tnr_20050117 ] shows that type iii@xmath9 is much shorter than the previous type iii bursts . it has intrinsic structure that indicates a group of bursts . the low - frequency cutoff is near the plasma frequency at the spacecraft at that time . more details are seen in the selected time profiles in the right panel , plotted together with the peak times of the burst at each frequency in the 35 - 256 khz range ( open triangles ) . the time profiles show that the peak times are not distinguishable over a large part of the frequency spectrum with 1-min integrated data , but that the centre of gravity of the brightest feature shifts to later times at the lower frequencies . we determined the maximum of the burst at each of the tnr frequencies where it is well defined , using a parabolic interpolation between the observed maximum and its two neighbours . it is this interpolated time which is plotted by an open triangle . the peak time spectrum resembles a type iii burst especially at the lower frequencies . the peak time delay is merely 1 min between 250 and 50 khz , but becomes clear at frequencies below 50 khz . for comparison , the peak time delay between 50 and 250 khz is 42 min during the previous burst iii@xmath8 . the frequency drift rate is hence faster than 3 khz / s during iii@xmath9 , as compared to 0.08 khz / s during iii@xmath8 . because of the morphological similarity in the dynamic spectrum , and despite the different drift rates , we assume in the following that the type iii@xmath9 bursts are indeed produced by electron beams travelling in the anti - sunward direction from the acceleration region . since the emission extends rapidly to the plasma frequency at the spacecraft , we conclude that the electron beams do not travel within the cmes , but escape rapidly from the acceleration region in the vicinity of the cme fronts to 1 au . this means that they must travel along pre - existing open solar wind field lines . to the extent that drift rates reflect the speed of the exciter , the fast frequency drift of the type iii@xmath9 bursts implies that the exciter speed is higher than during the preceding groups iii@xmath5 and iii@xmath8 . the total and differential radio spectrum observed by _ wind_/waves are shown in figure [ iii3 ] . the spectrum shows a chain of narrow - band emissions with negative frequency drift , indicating the type ii bursts , followed by the high - frequency part of type iii@xmath9 between 11:28 and 11:40 ut . the spectrum in figure [ f3 ] leaves it open if this is the continuation of the first type ii burst ( ii@xmath5 ) , presumably associated with cme@xmath5 , or whether it contains contributions from both cmes . the starting frequency of the type iii bursts is similar to the type ii frequency when extrapolated to the time of the type iii bursts . this is consistent with the type iii electron beams radiating in the upstream region , like the shock - acclerated electrons emitting the type ii burst . one may go one step further and consider this coincidence as a hint that the electron beams are accelerated at the shock , as argued in cases where type iii bursts clearly emanate from type ii lanes ( see * ? ? ? * ; * ? ? ? we will come back to this problem in the discussion . the energetic particle data were obtained from the _ advanced composition explorer _ ( ace ) spacecraft . we use high - resolution intensities of magnetically deflected electrons ( de ) in the energy range 38 - 315 kev measured by the b detector of the ca60 telescope of the epam experiment ( _ electron , proton and alpha monitor _ ; * ? ? ? * ) on board ace , and measurements of the angular distributions in the energy range 45 - 312 kev detected by the sunward looking telescope lefs60 . in figure [ o1 ] ( top ) an overview of the 20-min averaged differential intensities of four channels is presented for the interval 15 - 20 january 2005 . ar 10720 produced numerous solar events prior to as well as on 17 january 2005 @xcite ; in response to this solar activity , a sequence of energetic electron intensity enhancements was observed . the electron intensities are observed to reach their maximum values during this period following the solar events on 17 january 2005 . figure [ o1 ] ( bottom ) shows 1-min averaged deflected electron intensities ( 38 - 315 kev ) for the time interval 04:00 - 20:00 ut on 17 january 2005 . the intensities measured during the time interval 04:00 - 08:00 ut for each electron channel have been averaged to obtain a pre - event background ( denoted by horizontal lines in figure [ o1 ] ) . we defined the onset time of the event at ace for all energy channels as the time when the intensities get @xmath19 above the background and continue to rise from then on . using this criterion , we found the first significant rise of the electron intensities to occur at 10:00 ut . no velocity dispersion was observed , probably because the high pre - event ambient intensities ( see top panel figure [ o1 ] ) mask the onset of the electron event ( see * ? ? ? * for a similar case ) . the spiky increase observed at about 10:40 ut is probably due to x - ray contamination . we found no evidence of a magnetic structure influencing the intensity profiles , which indicates the observed time intensity changes are not due to spatial structures crossing over the spacecraft , but are most likely dominated by temporal effects . twenty - minute averaged representative snapshots of pitch angle distributions ( pads ) are shown as inserts in the bottom panel of figure [ o1 ] . normalized differential electron intensity is plotted versus the cosine of the pitch angle . statistically significant pads are detected first at about 11:00 ut . the pad snapshot denoted as * a * in figure [ o1 ] indicates that immediately after the onset of the event unidirectional electron anisotropies are observed . based on the observations available we can not distinguish whether the electrons were directed sunward or antisunward , since the magnetic field ( not shown ) was directed dominantly transverse to the radial direction during this period ( see figure 7 in * ? ? ? * for a similar case ) . however , it is highly likely that the observed electrons are streaming away from the sun in response to the intense solar activity during this period . furthermore the type iii bursts clearly indicate that electrons stream away from the sun ( towards regions of lower density ) . we can not be certain that the electron population measured at ace / epam is the high - energy counterpart of the electron beams emitting the radio waves , yet the overall timing suggests this . in the work by @xcite a detailed analysis of the plasma and magnetic field measurements at 1 au by ace during the period 16 - 26 january 2005 was carried out . this includes the period under study in the present paper . a forward shock was detected at 07:12 ut on 17 january 2005 . we have denoted the arrival time of this shock by a vertical solid line in figure [ o1 ] ( bottom ) . the analysis has shown that after the passage of this shock an unusually extended region exhibiting sheath - like characteristics is observed for @xmath11 1.5 day with highly variable magnetic field magnitude and directions and typical to high proton temperatures ( see figure 3 of * ? ? ? * also ruth skoug , ace / swepam pi team , private communication ) . this region is probably related to two cmes ejected in close temporal sequence at the sun on 15 january ( see figure 2 of * ? ? ? subsequently , at @xmath11 23:00 ut on 18 january 2005 the arrival of an icme at earth is detected , ending at about 02:30 ut on 20 january . the energetic electrons observed at 1 au analyzed in this work were thus detected in the region with disturbed magnetic field characteristics following the shock on 17 january 2005 . for the purposes of this work , as an approximation , we calculated that the nominal parker spiral for the measured solar wind speed of 620 @xmath20 ( ace / swepam ) at the time of the rise of the electron intensity had a length of about 1.05 au and was rooted near w 37@xmath4 on the hypothetical solar wind source surface at 2.5 r@xmath0 . this longitude is not contradictory with an active region at w 25@xmath4 , because non - radial coronal field lines can easily establish a connection @xcite . supposing that the early rise of the intensities was produced by the faster electrons in an energy channel moving along the interplanetary field line with 0 pitch angle , we estimate a travel time of about 15 min , which indicates the electrons were released from about 09:45 ut at the sun . this corresponds to a photon arrival time at 09:53 ut . given that our estimate of the electron rise gives only an upper limit , we consider that this electron release is related to type iii@xmath5 and type iii@xmath8 ( table [ t ] ) , but can not give a more detailed identification . the electron intensities are subsequently observed to exhibit a significant and more abrupt rise at all energies . extrapolation of this second rise to the pre - event background intensities indicates a start at @xmath11 12:00 ut . the electron pads ( inset * b * in figure [ o1 ] ) indicate stronger unidirectional anisotropies are observed in association with this electron enhancement , which provides evidence for fresh injection of energetic electrons between the sun and the spacecraft . the outstanding radio emission near this time is the group of fast type iii bursts during the cme interaction , type iii@xmath9 . if the electrons are accelerated at a heliocentric distance of about 25 r@xmath0 , the path travelled to the spacecraft along the nominal parker spiral is 0.92 au for the solar wind speed measured at the time of type iii@xmath9 ( 800 km / s ) . the inferred upper limit of the solar release time is 11:46 ut for 100 kev electrons . photons released at that time at 25 r@xmath0(0.12 au ) will reach the earth about 7 min later . since the high background implies that our estimations of the electron rise times are upper limits , we consider that this timing is consistent with the type iii@xmath9 burst group near 11:37 ut ( table [ t ] ) . this process is hence accompanied by the acceleration of copious amounts of electrons that escape to the vicinity of the earth . on 17 january 2005 two flare / cme events occurred in close temporal succession in the same active region . both cmes had very high projected speed , above 2000 km s@xmath7 , but the second one was faster than the first and eventually overtook it . the cmes were associated with two successive filament eruptions and sxr enhancements in the same active region . since the filament eruptions occurred at neighbouring places in the parent active region , the cmes probably resulted from the eruption of neighbouring parts of the same overall magnetic configuration . the soft x - ray characteristics of the two successive events were different : a slow monotonic rise to moderate flux during the first event , and a more impulsive rise to the x3.8 level ( @xmath21 ) during the second . both events had a conspicuous microwave burst , but the first one was stronger than the second , contrary to the soft x - rays . the second burst was also seen in hard x - rays by rhessi , which was in the earth s shadow during the first burst . [ [ evidence - for - evolving - acceleration - regions - in - the - corona - during - the - flares ] ] evidence for evolving acceleration regions in the corona during the flares ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the decametre - to - hectometre wave spectra in the two stages looked similar , with bright groups of type iii bursts signalling the escape of electron beams at heliocentric distances beyond 2 @xmath22 . but radio emission from lower heights shows distinctive differences that point to an evolving acceleration region , _ i.e _ either an acceleration region which progresses through the corona or a number of acceleration sites activated in succession . shock acceleration was clearly at work during both stages of the flare , as shown by the type ii emission . the strong type iii bursts at the low - frequency side of the estimated spectral track of the cme front could also be ascribed to the acceleration at the type ii shock . the presence of type iii bursts with negative drift at higher frequencies and the fine structures of the type iv continuum show , however , that at the time when the shock traveled through the high corona , other acceleration regions were active at lower altitude , as is usually the case during complex type iii bursts at decametre and longer waves ( see * ? ? ? * and references therein ) . the type iii bursts ( iii@xmath5 ) might then not start near the cme front , but at higher frequencies , and be interrupted by interactions of the electron beams with the turbulence near the front of cme@xmath5 . this is a frequently quoted interpretation of complex features in type iii bursts , both at kilometric @xcite and decametric wavelengths ( e.g. * ? ? ? * ) . in the second type iii group ( iii@xmath8 ) the overall frequency drift of the low - resolution spectrum ( figure [ f4 ] ) was positive . the persistence of the metric type iv burst , which suggested acceleration in the lower corona rather than at the shock during the first stage , is again a likely indication of an accelerator that was distinct from the cme shock , and acted in addition to the shock , at lower altitude . this is consistent with an interpretation of the electron acceleration in terms of reconnection in the corona behind the cme @xcite . new evidence for this interpretation has recently been provided by @xcite using uv and white light coronographic diagnostics along with radio data . decametric - hectometric radio emission as a signature of cme interaction was discussed in some detail in two event studies @xcite . in both events the radio emission had a limited bandwidth , and was referred to as a continuum . in the present case this emission is likely a set of type iii bursts and was therefore labelled type iii@xmath9 . the starting frequency and the timing of these bursts are consistent with the idea that the electrons are accelerated while the faster following cme catches up with the slower preceding one . the association of cme interaction with particle acceleration has been ascribed by @xcite to acceleration by the shock of the second cme as it traverses the previous one . problems of this interpretation were discussed by @xcite . another important question is how the preceding cme could lead to a strengthening of the shock of the following one . this problem is still more evident in the 17 january 2005 event , because here the two cmes are already extremely fast and likely to drive strong shocks even in the ambient solar wind . their relative speed , however , is rather slow , so that efficient acceleration by the shock of the second cme is not expected in the first cme . another important feature is that the type iii@xmath9 bursts extend to the plasma frequency at the spacecraft . the electrons hence can not propagate in closed magnetic structures related to the cmes . the accelerator must release the electron beams onto open solar wind - type field lines . an alternative scenario to acceleration at the cme shock is again magnetic reconnection . one can surmise that these rapid cmes were preceded by sheath regions with strong magnetic fields of interplanetary origin , draped around the cme front . these regions are favourable for magnetic reconnection ( see the overview by * ? ? ? * and references therein ) . the high pressure in the sheath of the second cme will be further enhanced when its progression is slowed down by the previous cme . this makes the configuration favourable to magnetic reconnection involving open solar wind field lines and strong magnetic fields , and allows one to understand qualitatively why accelerated electrons escape immediately towards the outer heliosphere . in this scenario the type iii emission is expected to start close to the cme , in the upstream region . this can explain why the starting frequency is close to the frequency of the type ii burst , without implying that the electron beams were themselves accelerated at the shock . the flare / cme events under discussion were clearly related with enhanced fluxes of near - relativistic electrons at 1 au . the peak intensity measured by ace , of order @xmath23 in the 38 - 53 kev range , make the event comparable to the most intense ones of the sample studied by @xcite , as seen in their figure 3a . the cme speed is well above the speeds of the cmes identified in that sample @xcite . since the energetic electrons were observed at 1 au within the region exhibiting sheath - like characteristics following the shock on 17 january 2005 , it is difficult to estimate electron travel times and to relate the _ in situ _ measurements to solar processes . but the observations strongly suggest that successive intensity increases are first due to the coronal acceleration in the flare / cme event , and then to an episode during the interaction of the two cmes . the escape of these electrons to ace confirms the view discussed above that the electrons can not have been accelerated in the body of the first cme , even if a shock driven by the second one passed through it . neither can they originate from reconnection between closed magnetic field lines of the two cmes . the electrons must rather be accelerated in regions from where they have ready access to solar wind magnetic field lines . this is consistent with a common acceleration of the mildly relativistic electrons and the electron beams at lower energies that produce the type iii@xmath9 emission . this work was supported in part by the university of athens research center ( elke / ekpa ) . the authors appreciate discussions with and assistance of c. caroubalos , c. alissandrakis . and they would also like to thank an anonymous referee for many useful comments on the original manuscript h@xmath6 data were provided by the kanzelhhe observatory , university of graz , austria by m. temmer . the soho / lasco data used here are produced by a consortium of the naval research laboratory ( usa ) , max - planck - institut fuer aeronomie ( germany ) , laboratoire dastronomie ( france ) , and the university of birmingham ( uk ) . the soho / lasco cme catalog is generated and maintained at the cdaw data center by nasa and the catholic university of america in cooperation with the naval research laboratory . soho is a project of international cooperation between esa and nasa . klk acknowledges the kind hospitality of the solar radio astronomy group at the university of athens .
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on 17 january 2005 two fast coronal mass ejections were recorded in close succession during two distinct episodes of a 3b / x3.8 flare . both were accompanied by metre - to - kilometre type - iii groups tracing energetic electrons that escape into the interplanetary space and by decametre - to - hectometre type - ii bursts attributed to cme - driven shock waves .
a peculiar type - iii burst group was observed below 600 khz 1.5 hours after the second type iii group .
it occurred without any simultaneous activity at higher frequencies , around the time when the two cmes were expected to interact .
we associate this emission with the interaction of the cmes at heliocentric distances of about 25 r@xmath0 .
near - relativistic electrons observed by the epam experiment onboard ace near 1 au revealed successive particle releases that can be associated with the two flare / cme events and the low - frequency type - iii burst at the time of cme interaction .
we compare the pros and cons of shock acceleration and acceleration in the course of magnetic reconnection for the escaping electron beams revealed by the type iii bursts and for the electrons measured _ in situ_.
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phase transition phenomena in quantum field theories are typically of nonperturbative nature and thus naive perturbation theory based on an expansion in the coupling constant can not be employed . this is clearly the case of phase changes at high temperatures , where perturbation theory becomes unreliable because powers of the coupling constant become surmounted by powers of the temperature @xcite . problems with perturbation theory also happen in phenomena occurring close to critical points , because large fluctuations can emerge in the system due to infrared divergences , thus requiring nonperturbative methods as well in their studies . this is the case of studies involving second order phase transitions and also in weak first order phase transitions @xcite . typical examples where these problems can manifest are in studies of symmetry changing phenomena in a hot and dense medium , a subject of interest in quantum chromodynamics ( qcd ) in the context of heavy - ion collision experiments , and also in studies of the early universe . consequently , there is a great deal of interest in investigating thermal field theories describing matter under extreme conditions @xcite . familiar nonperturbative methods that have been used to study symmetry changing phenomena at finite temperatures are resummationlike techniques , such as the daisy and superdaisy schemes @xcite , composite operator methods @xcite , and field propagator dressing methods @xcite . other methods used include expansions in parameters not related to a coupling constant , like the @xmath1 expansion and the @xmath2-expansion @xcite . in addition , there are numerical methods , the most notably ones are those based on lattice monte carlo simulations @xcite . each method has its own advantages and disadvantages . for instance , in numerical methods there may be issues related to numerical precision , lattice spacing , and lattice sizes . in addition , there is the notorious problem of simulating fermions on the lattice at finite chemical potentials @xcite . in any nonperturbative method based on an expansion in some parameter one has to face the problem of higher order terms becoming increasingly cumbersome , so stalling further analysis . this is usually the case when carrying out calculations beyond leading order in the @xmath1 expansion . careless use of a nonperturbative method can also lead to problems like the lack of self - consistency or overcounting of effects . known examples of such problems are the earlier resummation works dealing with daisy and superdaisy schemes , that at some point were giving wrong results , e.g. predicting a first order transition @xcite for the @xmath3 theory , an unexpected result since the model belongs to the universality class of the ising model , which is second order . these methods also predicted a strong first order phase transition in the electroweak standard model , a result proved to be misleading @xcite . let us recall that the breakdown of perturbation theory at high temperatures and its poor convergence properties have been dealt with many different methods . examples are the use of self - consistent approximations @xcite , hard - thermal - loop ( htl ) resummation @xcite , perturbative expansions in the coupling constant with resummation implemented with the use of a variational mass parameter , also known as screened perturbation theory ( spt ) @xcite , and the use of two - particle irreducible ( 2pi ) effective actions @xcite . the 2pi method , in particular , leads to a much better convergence of thermodynamic quantities ( like the pressure ) as compared to some of the other methods @xcite . related to the 2pi method is the @xmath4-derivable technique , which has been used to study the thermodynamics of scalar and gauge theories @xcite . one difficulty with the 2pi effective actions is that the renormalization procedure is nontrivial @xcite . in addition , there seems that the @xmath4-derivable technique breaks down for a coupling beyond some value @xcite . in general , it is desirable that any analytical nonperturbative method obey two basic requirements . first , it should be self - consistent , and second , it should produce useful results already at lowest orders without the need for going to higher orders . that is , it should produce results that quickly converge at some order where calculations are still feasible analytically or semianalytically . though some of the cited methods may satisfy one , or to some extent both of these requirements , in the present paper we are particularly interested in the one known as the linear @xmath0 expansion ( lde ) @xcite , a nonperturbative method that has been used successfully in different contexts related to thermal field theories @xcite and in many other theories for a long , but far from complete list of references see refs . @xcite . in the lde , a linear interpolation on the original model lagrangian density is performed in terms of a fictitious expansion parameter @xmath0 , which is used only for bookkeeping purposes and set at the end equal to one . the standard application of the lde to a theory described by a lagrangian density @xmath5 starts with an interpolation defined by @xmath6\ ; , \label{opt}\end{aligned}\ ] ] where @xmath7 is the lagrangian density of a solvable theory , which is modified by the introduction of an arbitrary mass parameter ( or parameters ) @xmath8 . the lagrangian density @xmath9 interpolates between the solvable @xmath10 ( when @xmath11 ) and the original @xmath5 ( when @xmath12 ) . the procedure defined by eq . ( [ opt ] ) leads to modified feynman vertices , that become multiplied by @xmath0 , and modified propagators , that now depend on @xmath8 . all quantities evaluated at any finite order in the lde will then depend explicitly on @xmath8 , unless one could perform a calculation to all orders . up to this stage the results remain strictly perturbative and very similar to the ones obtained via an ordinary perturbative calculation . it is through the freedom in fixing @xmath8 that nonperturbative results can be generated in this method . since @xmath8 does not belong to the original theory , one may fix it requiring that a physical quantity @xmath13 , calculated perturbatively to order-@xmath14 , be evaluated at the value where it is less sensitive to this parameter . this criterion , known as the principle of minimal sensitivity ( pms ) , translates into the variational relation @xcite @xmath15 the optimum value @xmath16 which satisfies eq . ( [ pms ] ) is a function of the original parameters of the theory . in particular , @xmath16 is a nontrivial function of the couplings and because of this nonperturbative results are generated . another optimization procedure used is known as the fastest apparent convergence ( fac ) criterion @xcite . it requires from the @xmath17-th coefficient of the perturbative expansion @xmath18 that @xmath19\bigr|_{\delta=1 } = 0\ ; , \label{fac}\ ] ] which is just equivalent to taking the @xmath17-th coefficient ( at @xmath20 ) in eq . ( [ fac0 ] ) equal to zero . one should note that it is not at all guaranteed that the condition in eq . ( [ pms ] ) has a nontrivial solution . in cases where this may happen , the second criterion , eq . ( [ fac ] ) , may be more appropriate . one example where the condition given by eq . ( [ pms ] ) fails to produce a nontrivial solution was in the problem studied by the authors in ref . @xcite , who applied the lde to compute the effective potential in superspace . there , the authors found that while the pms condition was unable to give a nonperturbative solution to the effective potential , the fac criterion worked perfectly well . of course , in many situations both optimization criteria may work and in this case one may ask whether they lead to equivalent results . previous studies indicated that this is indeed so , but a full comparison of results obtained with both optimization criteria is still lacking . another issue associated with the lde is its convergence . rigorous lde convergence proofs have been obtained for the problem of the quantum anharmonic oscillator , at zero temperature , considered in ref . @xcite , while its partition function at finite temperatures was considered in @xcite . for quantum field theories , ref . @xcite has proved convergence for a particular perturbative series in an asymptotically - free , renormalizable model at zero temperature . for a critical @xmath3 @xmath21 theory in three dimensions the issue of convergence was studied in @xcite employing both pms and fac optimization criteria . finally , regarding the possible solutions that can emerge from the optimization criteria ( pms or fac ) , we must use a definite approach in selecting the optimum root @xmath16 from either eq . ( [ pms ] ) or ( [ fac ] ) . the problem of dealing with the many possible solutions for @xmath22 was treated in details in the first two papers cited in ref . @xcite , where the convergence of the lde was also studied in details . typically , the higher the order in @xmath0 , the more solutions can appear . as shown in those references , all solutions at each given order in @xmath0 can be classified into families . the optimum value for @xmath8 is chosen as follows : the trivial solutions for @xmath22 , e.g. @xmath23 and those that are not dependent on the coupling constant ( and thus can not lead to nonperturbative results ) are not considered . in addition to these , at first order there is only one nontrivial solution ( first family ) , consistent with all our approximations , ( like the high - temperature approximation , used later in our calculations ) . this family is then followed in the next orders and used in all our calculations . as proved in earlier references with the lde method , this is a consistent and unambiguous way for choosing the optimum value for @xmath8 . it is important to stress that in the method of the lde the selection and evaluation of feynman diagrams proceed in the same fashion as in ordinary perturbation theory , including the renormalization procedure @xcite . the results obtained are free from infrared divergences , even at the critical point and in its neighborhood , thus making it a particularly suitable method to study phase transition phenomena in quantum field theories . it is important to recall here that there are similarities between the lde and the spt methods . in particular , the implementation of the latter can be put in a form similar to the lde by means of a modified loop expansion @xcite , named optimized perturbation theory ( opt ) in this reference . but there are also some major differences between these methods . for instance , in the lde no assumption is made _ a priori _ for the parameter @xmath8 , while in the spt / opt it is assumed that such a mass term is already of some order in the coupling constant . the implication of this is that the order counting of loop expansion has to be readjusted accordingly . in the present paper we study the application of the lde to the @xmath24 theory . we will study the applicability of the pms and fac optimization criteria for the symmetric and broken phases of the theory , and compare results obtained with both methods . in addition , in the present work we choose to optimize the free energy , instead of the self - energy like in many other works employing the lde , particularly refs . there are several reasons for doing so @xcite , but an important one is that in some situations it might happen that the optimization of the self - energy does not lead to nontrivial solutions , while optimization of the free energy with pms or fac are seen to lead to nontrivial solutions already at first order in @xmath0 . the critical temperature @xmath25 , the pressure @xmath26 , and the background dependent free energy @xmath27 are obtained here in an explicit calculation up to order @xmath28 . calculations at this order require a calculation of vacuum terms up to three loops . since the thermodynamics of this model has been extensively studied before in the literature with a number of methods , our calculation here will be useful to benchmark the application of the lde and its two main optimization procedures against those previous applications . in addition , we compare our results with those obtained with standard perturbation theory . besides correctly reproducing the expected second order phase transition pattern for the model , our results at order @xmath28 are shown to be sufficient to obtain the thermodynamics of the model , in the sense that the results at @xmath29 are not much different from the ones at @xmath30 . the results point towards a quickly convergent lde , as already indicated in previous studies with different models under different conditions @xcite . this work is organized as follows . in the next section we introduce the interpolation procedure for the model . in sec . iii we compute the free energy in the symmetric and broken phases to @xmath29 . in sec . iv we present the results obtained from the optimization procedures . the pressure is evaluated and contrasted order by order with the one obtained within perturbation theory . the critical temperature , the temperature dependent vacuum expectation value of the scalar field and the free energy are determined to @xmath29 . our conclusions are presented in sec . the interpolation defined by eq . ( [ opt ] ) when applied for the standard @xmath31 model gives @xmath32 where @xmath33 and @xmath34 is the part of the lagrangian density carrying the renormalization terms needed to render the model finite . details about this renormalization procedure in the lde and the explicit form for @xmath34 are given e.g. in ref . @xcite for the case of background field dependent contributions ( broken symmetry phase ) , while the field independent contributions ( symmetric phase ) were given in ref . @xcite within the context of the spt , so we will not repeat those same renormalization details here . one should also note that the only new " terms introduced by the @xmath0-expansion interpolation are quadratic terms and so the renormalizability of the original theory is not changed . this means that the renormalization of the theory can be carried out in an analogous way as in ordinary perturbation theory @xcite . specifically , the interpolation procedure given by eq . ( [ interp ] ) introduces a new ( quadratic ) interaction term , with feynman rule @xmath35 . in addition to this modification , the original bare propagator , @xmath36 now becomes @xmath37 while the original quartic vertex is changed from @xmath38 to @xmath39 . in the next section we will show the results for the finite temperature free energy density using the interpolated model with the lde at @xmath40 . we will consider the cases of @xmath41 and @xmath42 in eq . ( [ l0 ] ) , corresponding to the symmetric and broken phases , respectively . we perform the standard derivation of the free energy @xcite up to @xmath40 . with the constant field introduced through the usual shift of the scalar field , @xmath43 , the lagrangian density is rewritten as @xmath44=\mathcal{l}_2[\phi(x),\varphi ] + \mathcal{l}_i[\phi(x),\varphi]\;,\end{aligned}\ ] ] where @xmath45 is the part of the lagrangian quadratic in the fields , @xmath46 = \frac{1}{2 } \left ( \partial _ { \mu } \phi\right ) ^{2}-\frac{1}{2}\omega ^{2}\phi^{2}\ ; , \label{l2}\end{aligned}\ ] ] while @xmath47 is @xmath48 = -\frac { \delta \lambda } { 6}\varphi \phi^{3}-\frac{\delta \lambda } { 4!}\phi^{4}\;,\end{aligned}\ ] ] where in eq . ( [ l2 ] ) @xmath49 is given by @xmath50 note that in all loop contributions the propagators will carry a mass term as given by eq . ( [ omega ] ) . these terms are then expanded in @xmath0 to the desired order , thus generating the insertions of @xmath51 that appear as a consequence of the quadratic vertex introduced in eq . ( [ interp ] ) . the free energy is @xmath52&=&f_0(\varphi ) + f_{\rm 1-loop } ( \varphi)\nonumber\\ & + & \frac{1}{{\cal v } } i \ln \bigg{\langle}\exp\left\ { i\int d^4x\ , \mathcal{l}_i[\phi(x),\varphi]\right\ } \bigg{\rangle}\ ; , \label{free energy}\end{aligned}\ ] ] where @xmath53 is the tree - level classical potential and @xmath54 is the one - loop contribution to the free energy ( @xmath55 is the space volume ) given by @xmath56 } \;.\end{aligned}\ ] ] higher loops are given by the last term in eq . ( [ free energy ] ) , with the average @xmath57 meaning @xmath58 } } { \int \mathcal{d}\phi \ ; e^ { i\int d^{4}x\,\mathcal{l}_{2}\left [ \phi\left ( x\right ) , \varphi \right ] } } \;.\ ] ] as said above , the scalar field propagators in the diagrams are obtained from @xmath59 $ ] , and the vertices are determined from @xmath60 $ ] , with both as given at the end of sec . our calculations are performed , as usual , in the imaginary - time formalism @xcite . thus , the scalar boson field has euclidean four - momentum @xmath61 , with @xmath62 , where @xmath63 are the discrete matsubara bosonic frequencies @xmath64 , with @xmath65 , and @xmath66 . loop diagrams involve sums over the matsubara frequencies and integrals over the space momentum @xmath67 . all space momentum integrals are performed in arbitrary dimension @xmath68 and renormalization is performed in the modified minimal subtraction scheme ( @xmath69 ) . the measure used in the sum - integrals is then defined as @xmath70 where @xmath71 is an arbitrary momentum scale in dimensional regularization . the factor @xmath72 is introduced so that , after minimal subtraction of the poles in @xmath2 due to ultraviolet divergences , @xmath71 coincides with the renormalization scale in the @xmath69 scheme . from eq . ( [ free energy ] ) , the free energy is expressed up to @xmath73 by expanding all appropriate terms in @xmath0 . considering the vacuum contributions to the free energy , this means that terms up to three - loops must be included . all bare ( unrenormalized ) contributions are shown in fig . [ loopdiagrams ] . , given by ( a ) vacuum diagrams and ( b ) background field ( external legs ) . the black dots indicate a @xmath74 insertion . ] the renormalization procedure for the symmetric phase was performed in detail in refs . the counterterms for the vacuum diagrams are given in ref . @xcite , while those for the field dependent diagrams are given in ref . we also note that the divergences in the broken phase can be removed by the same counterterms determined for the symmetric phase @xcite , so the renormalization for the broken phase does not require extra effort . the renormalization proceeds just as in standard perturbation theory and as shown in detail in ref . @xcite , only temperature independent counterterms are required and the temperature dependent divergent terms cancel out exactly . all diagrams of counterterms contributing to @xmath75 $ ] up to @xmath76 are shown in fig . [ ctdiagrams ] . : ( a ) vacuum contribution , ( b ) background field contribution . as in fig . 1 , the black dot indicates a @xmath74 insertion . the circle - cross denotes either insertion of a mass counterterm or of a vertex counterterm . ] the circle - cross in fig . [ ctdiagrams ] denotes either a mass counterterm vertex @xmath77 , or a vertex counterterm @xmath78 , given respectively by @xcite @xmath79 - \delta^2 \frac{\lambda^2}{(32 \pi^2)^2}\left(\frac{-2}{\epsilon^2}+ \frac{1}{\epsilon}\right)\left(m^2 + \eta^2\right)\ ; , \label{deltam } \\ & & \delta \lambda = -\delta^2 \frac{3\lambda^2}{32 \pi^2 \epsilon}\;. \label{deltalambda}\end{aligned}\ ] ] the final expression for the renormalized free energy @xmath75 $ ] , including all terms shown in figs . [ loopdiagrams ] and [ ctdiagrams ] becomes @xmath80 = f_{\rm{vacuum } } + f_{\varphi } \label{freeenergy}\end{aligned}\ ] ] where @xmath81 denotes the vacuum contributions , @xmath82 \mathcal{m}^{4 } -\frac{1 } { 2\left ( 4\pi ^{2}\right ) } j_{0}\left ( \beta \mathcal{m } \right ) t^{4 } \nonumber\\ & + & \delta \frac{\lambda } { 8\left ( 4\pi \right ) ^{4}}\left [ \left ( \ln \left ( \frac{\mu ^{2}}{\mathcal{m}^{2}}\right ) + 1\right ) \mathcal{m}^{2}-j_{1 } \left ( \beta \mathcal{m } \right ) t^{2}\right ] ^{2 } \nonumber \\ & + & \delta \frac{\eta ^{2}}{2\left ( 4\pi \right ) ^{2}}\left [ \left ( \ln \left ( \frac{\mu ^{2}}{\mathcal{m } ^{2}}\right ) + 1\right ) \mathcal{m } ^{2}-j_{1}\left ( \beta \mathcal{m } \right ) t^{2}\right ] \nonumber \\ & -&\delta ^{2}\frac{\eta ^{4}}{4\left ( 4\pi \right ) ^{2}}\left [ \ln \left ( \frac{\mu ^{2}}{\mathcal{m } ^{2}}\right ) + j_{2}\left ( \beta \mathcal{m } \right ) \right ] \nonumber \\ & -&\delta ^{2}\frac{\lambda } { 4\left ( 4\pi \right ) ^{4}}\eta ^{2 } \left [ \ln\left ( \frac{\mu ^{2}}{\mathcal{m } ^{2}}\right ) + j_{2 } \left ( \beta \mathcal{m } \right ) \right ] \left [ \left ( \ln \left ( \frac{\mu ^{2}}{\mathcal{m } ^{2}}\right ) + 1\right ) \mathcal{m}^{2}-j_{1}\left ( \beta \mathcal{m } \right ) t^{2}\right ] \nonumber \\ & -&\delta ^{2}\frac{\lambda ^{2}}{48\left ( 4\pi \right ) ^{6}}\left\ { \left [ 5\ln ^{3}\left ( \frac{\mathcal{m}^{2}}{\mu ^{2}}\right ) + 17\ln ^{2}\left ( \frac{\mathcal{m } ^{2}}{\mu ^{2}}\right ) + \frac{41}{2}\ln \left ( \frac{\mathcal{m } ^{2}}{\mu^{2}}\right ) -23-\frac{23}{12\pi ^{2}}\right . \right . \nonumber \\ & -&\left . \psi ^{\prime \prime } \left ( 1\right ) + c_{0 } + 3\left ( \ln \left ( \frac{\mathcal{m } ^{2}}{\mu ^{2}}\right ) + 1\right)^{2}j_{2}\left ( \beta\mathcal{m } \right ) \right ] \mathcal{m}^{4 } -\left [ 12\ln ^{2}\left ( \frac{\mathcal{m } ^{2}}{\mu ^{2}}\right ) + 28\ln \left ( \frac{\omega ^{2}}{\mu ^{2}}\right ) \right.\right . \nonumber \\ & -&\left . \left . 12-\pi ^{2}-4c_{1}+6\left ( \ln \left ( \frac{\mathcal{m}^{2}}{\mu^{2}}\right ) + 1\right ) j_{2 } \left ( \beta \mathcal{m } \right ) \right ] j_{1 } \left ( \beta \mathcal{m } \right ) \omega^{2}t^{2}\right . \nonumber \\ & + & \left . \left [ 3\left ( 3\ln \left ( \frac{\mathcal{m}^{2}}{\mu^{2}}\right ) + 4\right ) j_{1}^{2}\left ( \beta \mathcal{m } \right ) + 3j_{1}^{2}\left ( \beta \mathcal{m } \right ) j_{2}\left ( \beta \mathcal{m } \right ) + 6k_{2 } + 4k_{3}\right ] t^{4}\right\ } \ ; , \label{fphi0}\end{aligned}\ ] ] and @xmath83 denotes the background field dependent contributions , @xmath84 -\delta ^{2}\frac{\lambda \eta ^{2}}{32\pi ^{2}}\left [ \ln \left ( \frac { \mathcal{m}^{2}}{\mu ^{2}}\right ) -j_{2}\left ( \beta \mathcal{m}\right ) \right ] \right . \nonumber \\ & -&\delta ^{2}\frac{\lambda ^{2}\mathcal{m}^{2}}{2\left ( 32\pi ^{2}\right ) ^{2}}\left [ \left ( \ln \left ( \frac{\mathcal{m}^{2}}{\mu ^{2}}\right ) \right ) ^{2}+\frac{\pi ^{2}}{6}\right ] -\delta ^{2}\frac{3\lambda ^{2 } \mathcal{m}^{2}}{2\left ( 32\pi ^{2}\right ) ^{2}}\left [ \left ( \ln \left ( \frac{\mathcal{m}^{2}}{\mu ^{2}}\right ) -1\right ) ^{2}+1+\frac{\pi ^{2}}{6 } \right ] \nonumber \\ & + & \delta ^{2}\frac{\lambda ^{2}}{1024\pi ^{4}}\left [ \mathcal{m}^{2}\left ( 1+\frac{\pi ^{2}}{6}\right ) + 4\mathcal{m}^{2}\ln \left ( \frac{\mu } { \mathcal { m}}\right ) \left [ 1+j_{2}\left ( \beta \mathcal{m}\right ) \right ] + j_{2}\left ( \beta \mathcal{m}\right ) \mathcal{m}^{2}\right . \nonumber \\ & + & \left . 8\mathcal{m}^{2}\ln ^{2}\left ( \frac{\mu } { \mathcal{m}}\right ) -4\ln \left ( \frac{\mu } { \mathcal{m}}\right ) j_{1}\left ( \beta \mathcal{m } \right ) t^{2}-j_{2}\left ( \beta \mathcal{m}\right ) j_{1}\left ( \beta \mathcal{m}\right ) t^{2}\right ] \nonumber \\ & + & \delta ^{2}\frac{\lambda ^{2}t^{2}}{24\left ( 4\pi \right ) ^{2}}\left [ \ln \left ( \frac{\mathcal{m}^{2}}{t^{2}}\right ) + 5.3025\right ] + \delta ^{2}\frac { \lambda ^{2}\mathcal{m}^{2}}{256\pi ^{4}}\left [ \frac{\pi ^{2}}{24}-3\ln \left ( \frac{\mathcal{m}}{\mu } \right ) \right . \nonumber \\ & + & \left . \left . 2\ln ^{2}\left ( \frac{\mathcal{m}}{\mu } \right ) + 1.164032 \right ] \right\ } \frac{\varphi ^{2}}{2}-\delta ^{2}\frac{3\lambda ^{2 } } { 32\pi ^{2}}\left\ { \log \left ( \frac{\mathcal{m}^{2}}{\mu ^{2}}\right ) -j_{2}\left ( \beta \mathcal{m}\right ) \right\ } \frac{\varphi ^{4}}{4!}\;. \label{fphineq0}\end{aligned}\ ] ] with @xmath85 given by @xmath86 \varphi^2 + \delta \frac{\lambda}{24 } \varphi^4\;. \label{f0}\end{aligned}\ ] ] in eqs . ( [ fphi0 ] ) and ( [ fphineq0 ] ) , @xmath87 , and the constant terms appearing in eq . ( [ fphi0 ] ) are defined as follows : @xmath88 , where @xmath89 is the zeta function , @xmath90 and @xmath91 , while @xmath92 and @xmath93 are three - dimensional integrals that can be evaluated numerically @xcite . in the high - temperature limit , @xmath94 , they are given by @xcite @xmath95 - 372.65 \beta\mathcal{m}\left [ \ln \left(\beta\mathcal{m } \right)+ 1.4658\right]\;,\end{aligned}\ ] ] and @xmath96\;.\ ] ] in eqs . ( [ fphi0 ] ) and ( [ fphineq0 ] ) , we have also defined the temperature dependent integrals @xmath97 ( @xmath98 ) as follows , @xmath99 which can be expressed as a series expansion as follows @xcite @xmath100 @xmath101 + \frac { 4\pi ^{2}}{3 } \nonumber \\ & -&16\sum_{n=1}^\infty\left ( \frac{\left ( -1\right)^{n}\left ( 2n-1\right ) ! ! \zeta \left ( 2n+1\right ) a^{\left ( 2n+2\right ) } } { 4n!2^{n+1}\left ( n+1\right ) \left ( 2\pi \right)^{2n } } \right)\ ; , \label{j1}\end{aligned}\ ] ] and @xmath102 \;. \label{j2}\end{aligned}\ ] ] equations ( [ j0])-([j2 ] ) are all convergent in the high - temperature limit as can be easily checked by considering a few terms in the sums in these equations . we should note that when optimizing the free energy , since @xmath103 , @xmath104 and @xmath105 are dependent on @xmath8 , it is important to check the stability of the results when truncating the sums in eqs . ( [ j0])-([j2 ] ) . this is particularly critical for parameter values such that @xmath106 is not much smaller than 1 , a situation that requires a fairly large number of terms in the sums . in all results shown in the next section we have used enough terms in eqs . ( [ j0])-([j2 ] ) so to obtain stable results for all parameter and temperature values used . we now turn to the application of the optimization procedures in the lde and show the results obtained by implementing the pms , eq . ( [ pms ] ) , and fac , eq . ( [ fac ] ) . as we explained in the introduction , the optimization criteria are applied directly to the free energy . the results obtained with each optimization criterion are contrasted with each other and with those available from other methods . this will then allow us to gauge the performance of each optimization procedure regarding both reliability and convergence . we initially restrict our calculations to the symmetric phase ( with positive mass term in the classical potential ) and evaluate the pressure , @xmath107 . in fig . [ fig : p_pert ] we show our results for the pressure using the usual perturbation theory in @xmath108 up to @xmath109 and where we have restricted to the case of high temperatures ( @xmath110 ) . in this figure the behavior of the pressure is shown as a function of the renormalized coupling constant , @xmath111 , and @xmath112 is a reference energy scale chosen as @xmath113 , where @xmath114 is the renormalized mass . this is similar as done in ref . @xcite using the 2pi method . note that in ref . @xcite the authors define the quartic coupling differently from us . in their case , @xmath115 , and their results are plotted as a function of the renormalized coupling @xmath116 . we choose here the same scale as in ref . @xcite so to facilitate the comparison with their results for the pressure . it is clear in fig . [ fig : p_pert ] the typical alternating behavior of the perturbative calculation , which indicates its very poor convergence . up to @xmath109 . the parameters used are @xmath117 for the temperature and @xmath118 for the renormalization scale . ] next , we use the result for the free energy evaluated up to @xmath76 , given by eq . ( [ fphi0 ] ) . note that in the symmetric phase the pressure depends only on vacuum terms , since the free energy is evaluated at the vacuum expectation value for the field , @xmath119 . by optimizing the free energy using the pms criterion , eq . ( [ pms ] ) , we determine the root @xmath22 , which is then substituted back into the expression for the free energy , with the criterion used for choosing the optimum root as discussed below eq . ( [ fac ] ) . this naturally brings nonlinear @xmath108 contributions and generates nonperturbative results . the pressure obtained in this case is shown in fig . [ fig : p_pert+lde ] , where we show the results obtained up to orders @xmath0 and @xmath28 . in the same figure we also show the perturbative results of fig . [ fig : p_pert ] for comparison . it becomes evident here the convergence of the results with the lde - pms , with both @xmath30 and @xmath29 results not differing too much , in contrast to the perturbative ( in @xmath108 ) results . and @xmath28 using the pms optimization criterion - the perturbative results of fig . [ fig : p_pert ] are also shown . the parameters used are the same as those in fig . [ fig : p_pert ] . ] in fig . [ fig : p_lde ] we show again the results for the pressure , but now using the fac optimization criterion , eq . ( [ fac ] ) . we once again see the excellent convergence for the pressure when contrasted to the perturbative results . and @xmath28 using the fac optimization criterion . ] in fig . [ fig : p_fac+pms_d2 ] we plot side by side our results for the pressure at order @xmath28 using the pms and fac optimization criteria . it is seen as an excellent agreement between the two optimization criteria and it shows the equivalence of these two optimization procedures . using the pms and fac optimization criteria . ] a side by side comparison of the order @xmath28 result for the pressure ( from either the pms or fac ) with the 2pi two - loop result of ref . @xcite ( second panel of theirs fig . 1 ) shows an excellent agreement between the two results . since operationally the lde is much simpler to be implemented than the 2pi calculation and also when compared with other methods , like those based on the renormalization group and schwinger - dyson equations , this may be a great advantage of the lde . many previous applications of the lde to a large variety of problems ( cited previously ) also confirm the strength of the method . its strength comes basically from the fact that its implementation is similar to that of standard perturbation theory . the important and fundamental difference with standard perturbation theory resides in the optimization procedure that fixes an initial , _ a priori _ , arbitrary parameter , @xmath8 . it is then interesting to investigate what kind of role the optimum @xmath8 represents in the lde after optimization . this is partially clarified in the plot shown in fig . [ fig : eta2_lde_d1 ] , where we show the optimum @xmath8 as a function of the renormalized coupling constant . it shows that by increasing the order in @xmath0 , @xmath16 becomes closer and closer to the thermal mass @xmath120 , here computed at one - loop order for simplicity . in general , we can extrapolate this expectation and say that the expected optimum value for @xmath8 should be the ( quantum and ) thermal mass ( quadratic in the field ) corrections , as would be derived from a true gap equation . this is in fact confirmed by the many applications of the lde to the gross - neveu model @xcite , in which case exact results are known ( in the large-@xmath121 approximation ) and can then be readily compared with the results obtained from the lde method applied to that model . with respect to the renormalized coupling constant and evaluated at order @xmath0 and @xmath28 using the fac optimization criterion . the parameters used are the same as in the previous figures . ] let us now turn to the symmetry broken case ( with negative square mass term in the classical potential ) . for this case we found that only the fac optimization criterion leads to nontrivial solutions for @xmath8 . the fac criterion is applied to the free energy and the resulting nonlinear equation is solved simultaneously with the equation giving the minimum condition for the field ( thermal ) expectation value , @xmath122 , given by the condition @xmath123 } { d\ , \varphi } \right\vert _ { \varphi = \nu(t ) } = 0\;.\end{aligned}\ ] ] as it is well known , the phase transition in the pure scalar theory is second order @xcite , as required by universality reasons . our results for the free energy using the fac criterion indicate a second order phase transition . this is shown in fig . [ fig : veff ] , where the free energy for @xmath124 is plotted for different temperature values . the critical temperature obtained here is @xmath125 , consistent with the perturbative prediction @xcite and other nonperturbative calculations @xcite . another quantity that indicates that the transition is a continuous one is the temperature evolution of the minimum of free energy , @xmath122 . this is shown in fig . [ fig : nut ] for @xmath126 and @xmath127 . evaluated at order @xmath28 and using the fac optimization criterion , for three different temperatures : @xmath128 , @xmath129 and @xmath130 , where @xmath131 ( in units of the renormalization scale @xmath71 ) . here we have set @xmath132 . ] at order @xmath28 obtained with the fac optimization criterion . here @xmath133 is the tree - level minimum of the bare free energy . ] [ fig : nut ] finally , in fig . [ fig : mt2 ] we show the temperature dependence of the thermal mass , @xmath120 , as derived from the free energy , @xmath134}{d^2 \varphi}\bigr|_{\varphi=0}\;. \label{mt2}\ ] ] we once again can notice a continuous and smooth transition . we note that one can determine the critical temperature by looking at which value of @xmath135 @xmath136 changes sign and check whether this gives the same result for @xmath25 as obtained from @xmath137 ( as in fig . [ fig : nut ] ) . evaluated with the fac criterion for two values of the coupling constant @xmath108 . we use in this plot @xmath132 . ] one of the motivations for using the lde to study the thermodynamics of the scalar field theory at high temperatures , as done in this work , was its ease of implementation and renormalization , which is no different from those of the standard perturbation theory . one recalls that similar studies in the context of the 2pi and related methods typically face difficulties in the renormalization procedure , making their applicability a nontrivial task . in addition , the lde method , differently from other methods , like spt ( or opt - optimized perturbation theory ) , makes no assumption on the introduced mass parameter @xmath8 , thus we do not need to adjust the order of the loop expansion accordingly . by using the lde , we have studied the thermodynamics of the @xmath3 scalar field theory in the symmetric and broken phases . the lde is used with two popular optimization procedures , known as pms and fac . there are two major differences with the work we have done here and previous ones , like e.g. in refs . first , while in general the pms procedure is the favorite optimization criterion in the literature related to the lde and similar methods , we here have shown that the fac procedure leads to numerically indistinguishable results from the ones obtained with the pms . in addition , while there may be cases where the pms procedure leads to trivial results only , the same may not be the case for the fac ( here we have shown this to be the case in the broken phase ) . in this sense they can be used in a complementary way , when pms fails , one can try fac , or vice - versa . secondly , unlike in refs . @xcite , where the quantity optimized is the self - energy , here we choose to optimize the free energy . one advantage of this is that , while there is no solution for the lde at first order when optimizing the self - energy , we do find solutions when optimizing the free energy already at first order in @xmath0 . furthermore , as shown in ref . @xcite , the optimization of the free energy can be shown to immediately lead to the solution of the gap equation ( here verified numerically through the results for the optimum @xmath8 ) , while in optimizing the self - energy further constraints must be employed , as for example renormalization group equations . in the numerical studies performed in the present work , we have shown that the optimum @xmath8 carries both temperature and coupling constant contributions . thus , the lde with optimization of the free energy implements automatically a nonperturbative resummation of the thermal corrections , in conformity with analytical results produced when this method was used to study the gross - neveu model in ref . @xcite , from which exact solutions are available and a close comparison with the lde results is possible . by studying the behavior of the pressure and contrasting the results obtained with perturbation theory and the 2pi method , we have shown that the lde leads to convergent results already at lowest order in the lde expansion parameter @xmath0 , with the first and second order results changing only slightly and producing results consistent with the 2pi nonperturbative method . in addition , as already mentioned above , we have shown that both optimization procedures , fac and pms , lead to equivalent results for the pressure . another important result of our work is that the lde is shown to be adequate for studying phase transitions at high temperatures . in particular , when applied to the phase transition in the @xmath3 , the lde predicts the correct order of the phase transition , which is second order , in agreement with general results of statistical mechanics . besides this , since the lde method automatically produces an infrared cutoff , the results are shown to be valid and applicable _ below _ , _ above _ , and _ at _ the critical temperature @xmath25 , showing that the lde circumvents the usual problem seen in perturbation theory , namely , the appearance of infrared divergences close to critical points . we would like to thank u. reinosa for helpful discussions regarding their 2pi results and the renormalization issues in the method . we would like to thank f. gardim for discussions on related matters . this work was partially supported by cnpq , fapesp , and faperj ( brazilian agencies ) . 99 m. le bellac , _ thermal field theory _ ( cambridge university press , cambridge , 1996 ) . m. gleiser and r. o. ramos , phys . b300 * , 271 ( 1993 ) . j .- blaizot , e. iancu , and a. k. rebhan , aip conf . 739 , 63 ( 2004 ) . d. h. rischke , prog . phys.*52 * , 197 ( 2004 ) . u. kraemmer and a. k. rebhan , rept . prog . phys . * 67 * , 351 ( 2004 ) . j. o. andersen and m. strickland , ann . * 317 * , 281 ( 2005 ) . j. r. espinosa , m. quirs and f. zwirner , phys . b291 * , 115 ( 1992 ) . j. arafune , k. ogata and j. sato , prog . . phys . * 99 * , 119 ( 1998 ) . g. amelino - 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the thermodynamics of a scalar field with a quartic interaction is studied within the linear @xmath0 expansion ( lde ) method .
using the imaginary - time formalism the free energy is evaluated up to second order in the lde .
the method generates nonperturbative results that are then used to obtain thermodynamic quantities like the pressure .
the phase transition pattern of the model is fully studied , from the broken to the symmetry restored phase .
the results are compared with those obtained with other nonperturbative methods and also with ordinary perturbation theory .
the results coming from the two main optimization procedures used in conjunction with the lde method , the principle of minimal sensitivity ( pms ) and the fastest apparent convergence ( fac ) are also compared with each other and studied in which cases they are applicable or not .
the optimization procedures are applied directly to the free energy .
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m dwarfs comprise the bulk ( @xmath375% ) of the stellar population of our galaxy . moreover , their low masses and small radii , compared to sun - like stars , make planets of a given mass , size and orbital separation much easier to detect around them via both the transit and doppler methods . in addition , the habitable zone ( hz ) of such low luminosity red dwarfs lies considerably closer to the central star than in the case of solar - types ; this further enhances the transit and doppler signatures of hz planets , and also allows a larger number of planetary orbits to be observed in a given time , making the detection and characterisation of such planets even easier . consequently , the next generation of missions investigating exoplanets are aimed at later - type stars rather than the mainly solar - type ones targeted by _ kepler _ , and discussions of exoplanet habitability increasingly focus on m dwarf systems . there are many complications pertaining to the habitability of planets around m - dwarfs , even if they possess surface temperature and pressure conditions favourable to liquid water , such as tidal locking ( e.g. * ? ? ? * ; * ? ? ? * ) , run - away greenhouse effects ( e.g. * ? ? ? * ) and water loss @xcite . the first major uncertainty , however , is whether they can possess such conditions at all , given what we currently know about exoplanets ( e.g. * ? ? ? the last decade of exoplanet discoveries has dramatically altered our understanding of a `` standard '' low - mass planet . _ kepler _ has detected thousands of planetary candidates ( e.g. * ? ? ? * ) , the majority of which are small ( @xmath5 r@xmath6 : * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and close to their host stars . correcting for observational biases , one finds that most stars probably harbour at least one such `` kepler '' planet @xcite . masses have been now been obtained for a significant number ( albeit a small fraction ) of these exoplanets , by combining _ kepler _ transit data with either transit timing variations ( ttvs ; e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or radial velocity measurements ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? a very surprising result to emerge from this work is that a large fraction of low - mass , close - in kepler planets are enshrouded by voluminous h / he envelopes , which contain a non - negligible fraction of the planet s mass ( as demonstrated , for instance , by the extremely low densities of some : @xmath7 g @xmath8 ; e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) : very unlike the low - mass planets in our own solar system . in particular , statistical analyses of the properties of the low - mass exoplanet population reveal that the latter are inconsistent with a completely rocky composition @xcite . instead , comparisons of the observed exoplanets with measured masses and radii to structural models @xcite indicate that the dominant structure is a solid core overlaid with a @xmath31% ( by mass ) h / he envelope . since close - in planets are subject to intense irradiation by high energy stellar photons , which can drive an outflow by heating the upper layers of the atmosphere to close to the escape temperature ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the evaporation of such h / he envelopes becomes important for the planets evolution ( e.g. * ? ? ? * ; * ? ? ? forward modelling studies suggest that the majority of close - in exoplanets were born with h / he envelopes , but about half have subsequently lost these through evaporation @xcite . in summary , current exoplanet demographics lead us to infer that a dominant mode of low - mass planet formation produces a rocky earth - like core surrounded by a h / he envelope with a mass - fraction of @xmath31% @xcite . this observational result is supported by recent theoretical calculations within the framework of core accretion , which suggest that low - mass cores @xmath9m@xmath6 will acquire envelope mass - fractions of order a few percent @xcite . for example , the scalings of @xcite imply that a 1m@xmath6 core will accrete a h / he envelope mass - fraction of 0.1 - 1% during the gas disk s lifetime . we note that various studies of gas accretion onto earth - like cores find that the results are sensitive to a number of uncertain parameters , such as the opacity and disc properties ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and , especially , the assumed core accretion rate . thus , these studies suggest a range of initial h / he envelope masses for a given core mass . however , this theoretical prejudice should not blind us to the _ empirical _ fact that current exoplanet data indicate that envelope mass fractions @xmath10% are preferred . for a terrestrial - mass planet , orbiting either a solar - type star or an m dwarf and located within the classical hz ( defined here as the range of orbital separations where an earth - mass planet , with roughly earth - like composition and atmosphere , can harbour liquid surface water ; eg . , see * ? ? ? * ) , a h / he envelope with mass - fraction of order a percent would preclude habitability , by yielding very high surface temperatures and pressures incompatible with liquid water . however , habitable conditions may be achieved if evaporation can reduce the h / he envelope mass - fraction to @xmath11 ( e.g. * ? ? ? * ) ; strip away this primordial envelope entirely so that it is replaced by a tenuous secondary atmosphere , such as those of the solar system terrestrial planets ( e.g. * ? ? ? * ; * ? ? ? * ) ; or separate h and he in planets with a low initial h / he fraction to leave a habitable he atmosphere @xcite . in other words , evaporation may turn a large population of uninhabitable low - mass planets with voluminous h / he envelopes into habitable ones . such an effect is unlikely in the hz of sun - like stars , since their x - ray / uv fluxes over gyr timescales at @xmath31au are too low to remove a massive h / he envelope ( though several works have shown it is possible to remove a _ small _ h / he atmosphere ; e.g. , * ? ? * ; * ? ? ? * ; * ? ? ? m dwarfs , on the other hand , are far more active , and remain so for much longer , than solar - types , leading to much higher xuv fluxes within their hzs over gyr timescales ( e.g. * ? ? ? * ; * ? ? ? * ) . moreover , low - mass planets appear abundant around these red dwarfs , which are themselves the most common stars in the galaxy ; evaporation , if efficient enough , could thus lead to a plethora of habitable planets in m dwarf systems . in this work , therefore , we consider terrestrial - mass planets in the hz of m dwarfs , with an initial h / he envelope mass - fraction of @xmath31% , and investigate whether evaporation over a gyr can remove a sufficient portion of this atmosphere to render the planet habitable at the end . we account for both stellar evolution and the thermal evolution of the planet . in order to make advances in this important area , we also abjure various simplifying and , as it turns out , erroneous assumptions about the evaporative flow made in previous studies of this subject , as described below , in order to obtain a more rigorous and realistic estimate of the mass loss . given the _ kepler _ results above , and the interest in habitable planets , several recent studies have investigated the evaporation of voluminous h / he envelopes around terrestrial - mass solid cores . however , previous studies have typically made two key simplifying assumptions : _ ( i ) _ radiative cooling in the flow is either neglected , or accounted for by assuming a fixed energy efficiency for driving the flow - either in a global or local sense ; and _ ( ii ) _ evaporation is assumed to occur in the hydrodynamic limit at all times . both assumptions are likely to lead to an overestimation of the amount of h / he a planet can lose . first , radiative cooling becomes important when the timescale for a fluid element to advect its heat outwards ( the ` flow timescale ' ) becomes comparable to the timescale for radiative losses to cool that fluid element ( e.g. * ? ? ? this lowers the temperature in the flow , which in turn pushes the sonic point to greater heights ( lower densities ) and thus reduces the mass - loss rate . second , in order to be in the hydrodynamic limit , one requires that the gas remain collisional up to the sonic point ( e.g. , * ? ? ? if this is not satisfied , the flow will collapse and switch to jeans escape , which has a much reduced mass - loss rate ( as we demonstrate later , jeans escape can not remove a significant h / he atmosphere on gyr time - scales , and other non - thermal processes are also unlikely to play a significant role ) . finally , these two points are inter - related : as radiative cooling pushes the sonic point to higher heights and lower densities , it can also trigger the transition to jeans escape earlier than in calculations that do not include cooling . @xcite and @xcite studied the evolution of terrestrial mass cores in the hz of solar - type stars , using evaporation rates calculated in the `` locally energy - limited '' approach , where a fixed fraction of the absorbed photons energy is locally deposited into heat . this fraction is calibrated to detailed calculations by @xcite that explicitly solve the micro - physics of photon absorption ; as such , it is more accurate than the standard `` energy - limited '' formalism , wherein an ad hoc fraction of the total incoming radiative flux goes into heating the gas . however , these calculations still neglect radiative cooling , and as such will overestimate the mass - loss rate when radiative cooling is important . @xcite further argue that if the core accretion rate is very large ( mass doubling times @xmath12 years ) , resulting in a high - entropy bloated atmosphere , evaporation may be able to completely remove initially massive h / he envelopes from low - mass cores and leave behind a potentially habitable planet . however , such large core accretion rates require that the time of planet formation to be fine - tuned to occur just before disc dispersal , otherwise the core would end up with a mass @xmath13 m@xmath6 . with more reasonable accretion rates , they find cores are likely to retain a large fraction of their original envelope . @xcite , expanded on this work to demonstrate that earth - mass planets at 1au around solar - type stars can lose a massive 1% h / he envelope over a gyr only if the star is an unusually fast rotator ( i.e. , unusually active , with x - ray luminosity in the upper 90th percentile of the observed spread in x - ray luminosities in these stars ) . in the majority of cases , i.e. , for solar - type stars with more standard rotation rates and actitivity levels , they find that such planets can only lose h / he envelopes with initial mass - fractions @xmath140.1% . @xcite have recently investigated photoevaporation around m dwarfs . using the standard `` energy - limited '' formalism and further assuming a hydrodynamic flow at all times , they argue that evaporation can completely strip h / he envelopes with a mass - fraction @xmath1% from earth - mass cores in the hzs of m dwarfs . if true , this would imply a potentially vast number of worlds with habitable conditions around m dwarfs . however , the problems noted above with the adopted assumptions call this result into question . specifically , below an envelope mass - fraction of @xmath31% , the radius of the planet is an extremely strong function of the remaining envelope mass - fraction @xcite ; at the same time , the flow timescale increases strongly with decreasing planetary radius ( as the atmosphere falls deeper into the planet s gravity well ) . thus , one expects radiative cooling to increase significantly as the h / he mass - fraction descends below @xmath31% , resulting first in suppressed hydrodynamic mass - loss rates , and then a transition to jeans escape with greatly diminished loss rates . this process can quench mass loss from planets with a hefty h / he envelope still remaining ; indeed , the effect has already been identified at closer separations and higher planetary masses @xcite , and is the origin of the so - called `` evaporation valley '' @xcite . in this paper , therefore , we eschew the assumptions made in earlier studies . instead , we explicitly account for the effect described above , by : _ ( i ) _ smoothly transitioning from the regime where heating is balanced by outflow ( as in the locally energy - limited calculations of * ? ? ? * ) to the regime where heating is balanced by radiative cooling , using a parametrisation calibrated to detailed monte - carlo radiative transfer simulations ; and _ ( ii ) _ using our hydrodynamic models to determine when the transition to jeans escape occurs , triggering a strong suppression in mass - loss rates . in general , with our more appropriate treatment , we find that whether or not a planet can lose enough h / he to be considered habitable depends strongly on the core mass and orbital separation . in particular , the mass loss we derive for @xmath31 cores with initially @xmath31% he / he envelopes in the hz of m dwarfs is much lower than previous estimates , implying that such planets will not be habitable . stellar xuv - driven evaporation is particularly important around m dwarfs because these stars are extremely active : for instance , an average early- to mid - m dwarf remains at saturated levels of coronal activity , with @xmath15/ @xmath16 10@xmath1710@xmath18 , for a few 100myr , compared to only a few@xmath1910myr of saturated activity in an average solar - type star ( e.g. , * ? ? ? consequently , a planet with a given equilibrium temperature receives orders of magnitude more high - energy radiation , integrated over its lifetime , around an m dwarf than around a star like the sun . in this study , we are concerned with the evaporation of low - mass planets . @xcite demonstrated that at high x - ray fluxes , mass - loss from such planets is predominantly x - ray - driven , and euv heating can be neglected . we thus confine ourselves to x - ray - driven flows here . to study the latter , we need a fiducial x - ray spectrum ; we use that of ad leo , an m3.5 dwarf of mass @xmath30.4 , and one of the most active nearby stars . th very strong coronal activity on ad leo arises not because it is anomalous , but simply because it is relatively young compared to most nearby field stars . at a median age of @xmath3100myr ( estimated age range @xmath325300myr ; * ? ? ? * ) , it is still in its saturated phase of activity , with @xmath15/ = 10@xmath20 ( @xmath15 = 7@xmath1910@xmath21ergs@xmath22 and = 9@xmath1910@xmath23ergs@xmath22 ; * ? ? ? * ) . its high levels of activity , combined with its proximity ( @xmath24pc ) , have made ad leo a touchstone for understanding the coronae of m dwarfs . nevertheless , the intrinsic faintness of m dwarfs ( ad leo included : @xmath2510@xmath26 ) means that , even when @xmath15/ is very high , the star is still very faint in x - rays , and it is not yet possible to acquire a full high - quality x - ray spectrum . instead , one usually reconstructs the spectrum using the technique of emission measure distributions ( emds ) , based upon observations of bright emission lines . we use an emd - derived synthetic x - ray spectrum of ad leo supplied by j. sanz - forcada ( pvt.comm.2014 ) . the methodology for constructing it is described in detail by @xcite and @xcite ; this spectrum is also used by @xcite , who show that it agrees very well with current x - ray ( and uv ) data for ad leo . we plot the spectrum in fig.1 , and compare it to a scaled - solar spectrum ( representing a young solar - type star ) ; the comparison solar - like spectrum was calculated in @xcite using the method of @xcite , and was designed to be representative of a young , saturated solar - type star . it is immediately clear that the ad leo spectrum is considerably softer ; thus , since x - ray heating is mainly due to photo - electrons liberated by soft @xmath30.1 - 1kev photons , we expect significantly more heating using our realistic m dwarf spectrum than with a scaled - solar proxy for it ( e.g. , as done in * ? ? ? 0.11kev ) x - ray photons in ad leo relative to the young solar - type star . ] since ad leo is currently still in the saturated regime , we assume it has maintained the same @xmath15/ = 10@xmath20 since birth to the present ( the standard behaviour of saturated low - mass stars , before stellar rotation slows sufficiently for the activity to become unsaturated ) . however , its bolometric luminosity certainly _ has _ evolved with time , as the star has contracted towards the main sequence ; we assume that its follows the theoretical evolutionary track of @xcite for a 0.4 star , and scale @xmath15 accordingly in time . finally , for ease of computation , we assume that ad leo will remain saturated , i.e. , its @xmath15/ will stay unchanged , till 1gyr ( when we terminate our evaporation calculations ) . this is not physically strictly accurate , since a 0.4 star should start spinning down after a few 100myr ( e.g. , * ? ? ? * ) , thus entering the unsaturated regime and evincing decreasing @xmath15/ ; assuming saturation at these late times then formally implies that our mass - loss rates will be overestimated here . however , as we shall see , for the core - mass and h / he envelope mass - fraction regime we study in this paper , our calculations show that the planets are either completely stripped of their envelopes well before 100myr , or have entered the jeans escape regime , with tiny mass - loss rates , by this age . as such , our simplifying assumption of saturated activity beyond a few 100 myr has no discernible impact on our results . in order to derive evaporative mass - loss rates , we must specify the planetary structure . as our starting point , we assume planets with earth - like solid cores , composed of 2/3 rock + 1/3 iron , swathed in a h / he envelope with a mass - fraction ( defined as @xmath27/@xmath28 ) up to @xmath31% . as discussed above , this is consistent with both observations and theoretical calculations of the conditions at birth of `` kepler '' planets @xcite . current theories of planet formation , though , do not strongly constrain the initial entropy ( or equivalently , radius ) of such planets , with both `` hot start '' planets ( those with short initial cooling time scales ) and `` cold start '' ones ( with long initial cooling time scales ) remaining viable . we thus choose initial planetary radii ( entropies ) corresponding to cooling timescales in the range 10@xmath2910@xmath30 yr , to comfortably span the plausible range from `` hot '' to `` cold start '' scenarios ( further discussed in 5 ) . finally , we are interested in the potential habitability , under the effects of evaporation , of low - mass planets in the hz of m dwarfs . the `` classical '' hz is defined as the range of orbital separations around a star where an earth - mass planet with a co@xmath31h@xmath31o n@xmath31 atmosphere can sustain liquid water on the surface ; the inner edge of this zone is ( conservatively ) set by the moist greenhouse effect , and the outer edge ( again , conservatively ) by the maximum greenhouse effect @xcite . the position of the hz of course changes as the star evolves in temperature and luminosity ; what we are really interested in is the fate of planets located within the stable hz of an m dwarf on the main sequence ( ms ) . using the fitting equations supplied by @xcite for the hz boundaries , and the ms values of the stellar temperature and luminosity for a 0.4 dwarf from the @xcite tracks ( @xmath16 3500k , @xmath16 1.86@xmath1910@xmath26 ; the star arrives on the ms after @xmath3500myr ) , yields an inner edge of the classical hz for our m dwarf at 0.15au , and an outer edge at 0.28au . for ease of calculation , and discussions for m dwarfs with masses around 0.4 m@xmath32 , we choose here to parametrise the hz in terms of the blackbody temperature ( @xmath33 ) of the planet instead , which we define as the equilibrium temperature of a planet with zero albedo , orbiting a star of luminosity at a radial separation of @xmath34 : @xmath35 [ /@xmath36^{1/4}$ ] . we set the inner edge of the classical hz around our m dwarf to be at a ms blackbody temperature of @xmath37k , and the outer edge at @xmath38k . these imply radii of @xmath34 = 0.12au and 0.26au respectively on the ms : very close to the actual separations from the @xcite equations . since we are interested in framing the general theory of evaporation driven habitability here , the @xmath31020% difference has no appreciable effect on our conclusions . detailed calculations of the habitability of systems with specific parameters ( i.e. future detected exoplanet systems ) , can wait until a later date . we discuss our theory of x - ray driven evaporation , in both the hydrodynamic and ballistic limits , in 3 ; our numerical implementation of this in 4 ; and our treatment of thermal evolution in 5 . we present our results in 6 , and explore the implications for habitability around m dwarfs in 7 ; our main conclusions are summarised in 8 . our interest lies in low mass planets ; in these , atmospheric heating and evaporation are dominated by x - rays @xcite . it is well known that , in radiative equilibrium , heating due to high energy photons can be described using a relationship between temperature ( @xmath39 ) and ionization parameter ( @xmath40 , where @xmath41 is the received x - ray flux and @xmath42 the number density of particles ) @xcite . @xcite used this principle to calculate solutions to the evaporation problem . however , the precise form of the @xmath43 relation is sensitive to the input x - ray spectrum and the metallicity of the gas . the x - ray spectrum of m dwarfs is quite different from that of solar type stars , being in general much softer in the @xmath44 kev range important for photo - electric heating . we must therefore calculate a new @xmath43 relation , specific to our template m dwarf , to understand the behaviour of the x - ray irradiated planetary atmosphere . we use the radiative transfer code mocassin @xcite to solve for the @xmath43 relation appropriate to the ad leo x - ray spectrum discussed in 2.2 . we irradiate a plane parallel atmosphere , with densities spanning @xmath45 g @xmath8 , the expected range in the evaporative flow . for specificity , we scale the x - ray flux to that expected at a planet orbiting at @xmath37k ( @xmath30.12au ) ; however , since we are concerned with mapping out the @xmath43 profile , rather than with the x - ray flux itself , the exact flux chosen to perform the calculation does not matter very much . the resulting @xmath43 profile is shown in fig.[fig : t - xi ] , along with a functional fit which we adopt in our further calculations . the soft m dwarf x - ray spectrum has a higher fraction of @xmath30.1 - 1kev photons compared to solar - type spectra ( see fig.[fig : spec ] ) ; consequently , the m dwarf produces considerably higher gas temperatures at low ionization parameters ( @xmath46 ) . we first attempt to solve for the evaporative flow by inserting our @xmath43 relation into the semi - analytical methodology of @xcite , which assumes radiative equilibrium in the flow . this , however , leads to an inconsistency : over much of the parameter range of interest , the mechanical luminosity of the resulting evaporative wind becomes comparable to the radiative energy input rate , implying that radiative equilibrium is not a good approximation ; instead , energy losses due to @xmath47 work may dominate over radiative losses much of the time . in hindsight , this is not too surprising , given the higher heating efficiency of our m dwarf x - ray spectrum relative to that in the original @xcite calculations . as such , we must derive a full numerical solution to the radiation - hydrodynamic problem . nevertheless , the fact that , _ when _ radiative equilibrium holds , the gas temperature should equal that given by the @xmath43 relation enables us to simplify the numerics , fully span the pertinent parameter space and , crucially , smoothly transition to the situation where radiative losses _ do _ become important ( which , as discussed in 2.1 , ultimately determines when our hydrodynamic wind is quenched ) . our specific numerical technique for accomplishing this is detailed in 4.1 . to assess the importance of _ non_-hydrodynamic mass loss , we calculate the ballistic ( jeans escape ) mass - loss rate , given by @xmath48 where @xmath49 is the planetary radius , @xmath50 the density at the exobase and @xmath51 the distribution of particle velocities . we assume that the exobase is in local thermodynamic equilibrium with the x - ray irradiation , so that its temperature is specified by our @xmath43 relation ; @xmath51 is then the maxwell - boltzmann distribution at this temperature . the exobase density is @xmath52 where @xmath53 is the scale height of the atmosphere ; @xmath54 the mean molecular weight of the gas particles , set to 1.35 for an atomic solar - abundance h / he mixture ; and @xmath55 the collision cross - section , for which we adopt @xmath56 @xmath57 @xcite . equation ( [ eqn : mdot_jeans ] ) can be solved numerically for the mass - loss rate as a function of core mass . since we expect the jeans escape rates to be low , we assume the extent of the h / he atmosphere is small , and approximate the planetary radius @xmath49 by that of the solid core . the core mass - radius relation is from @xcite , for a core comprising 2/3 rock and 1/3 iron by mass . , @xmath58 , @xmath59 , @xmath60 in 1 gyr . ] the resulting @xmath61 are plotted in fig [ fig : jeans ] as a function of core mass and are compatible with those found previously for highly irradiated terrestrial mass planets ( e.g. * ? ? ? * ; * ? ? ? these mass - loss rates are indeed very low ; in particular , they are significantly below the rates one might expect from hydrodynamic evaporation ( as we will see ) . we find that jeans escape can only remove a h / he envelope mass - fraction ( @xmath62 ) of order 10@xmath63 over a gyr timescale : completely negligible compared to the @xmath31% initial h / he mass - fractions we are concerned with here . thus , in order to strip any significant h / he envelope from solid terrestrial - mass cores in the hz of an m dwarf , efficient hydrodynamic evaporation appears essential . as discussed in 3.1 , we must resort to full numerical hydrodynamic calculations to obtain the evaporation rates for the m dwarf case . furthermore , to avoid the problems faced by @xcite in finding steady - state solutions to the wind problem , we explicitly integrate the time - dependent hydrodynamic equations until a steady - state is reached after several sound - crossing times ( e.g * ? ? ? * ) : in all our calculations , we run the simulations for 30 crossing times to ensure that steady - state is achieved . we employ the zeus hydrodynamical code @xcite , which has been successfully tested and used for planetary evaporation problems in the past ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? our modified zeus algorithm includes heating by x - rays and their radiative transfer ( c.f . * ? ? ? * ; * ? ? ? as noted in 3.1 , our evaporative flow may be in a state far from radiative equilibrium . specifically , we expect the gas to behave as follows . in steady - state , since the flow is expanding and we do not anticipate any heating by shocks , the gas will always be cooler than the temperature at radiative equilibrium ( @xmath64 , given by the @xmath43 relation ) . when the gas is considerably cooler than @xmath64 - due to a large amount of @xmath47 work - the photo - heating rate will greatly exceed the radiative cooling rate , and we may ignore radiative cooling . as the gas temperature approaches @xmath64 , however , radiative cooling will become comparable to the photo - heating rate , and the temperature will asymptote to @xmath64 . with this qualitative picture in mind , we build a simplified thermal model for x - ray heating / radiative cooling , using a `` newtonian - heating '' approach . we begin by defining a _ local _ heating time as @xmath65 , where @xmath66 is the gas internal energy per unit mass at @xmath67 , and @xmath68 is the local heating rate per unit mass due to absorption of high energy photons , given by @xmath69\frac{\sigma_\nu}{\mu}\ , { \rm d}\nu\ ] ] where @xmath70 is the optical depth ( from above ) at a distance @xmath71 from the centre of the planet : @xmath72 here , @xmath73 represents the fraction of the photon energy that is converted into heating rather than used up in ionization . we caution that this is _ not to be confused _ with the efficiency parameter @xmath74 commonly deployed in `` energy - limited '' estimates of the mass - loss rate ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which ( arbitrarily ) sets how much of the photon energy goes into @xmath47 work instead of being radiatively lost . in our analysis , @xmath73 merely accounts for the energy diverted to ionization processes ; radiative energy losses , on the other hand , are explicitly accounted for by incorporating our derived @xmath43 relationship into the calculations ( via @xmath66 in the energy equation , as described further below ) , and _ not _ via an ad hoc efficiency parameter @xmath74 . integrating a frequency - dependent radiation - hydrodynamics problem is beyond the scope of this work . instead , we use the fact that the heating in our case is dominated by soft x - ray photons , mainly at energies of @xmath31kev @xcite . we can thus approximate our frequency - dependent integral in equation(3 ) by a monochromatic calculation at 1kev , with an absorption cross - section of @xmath75 @xmath57 . heating by @xmath31kev photons is primarily due to secondary photo - electrons ; @xmath76 then depends on the ionization state of the gas , with a value of @xmath30.1 in neutral gas and @xmath30.9 in highly ionized plasma @xcite . while we do not attempt to solve for the full ionization structure of the gas , our mocassin results indicate the gas is partially ionized ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , so we pick an intermediate value of @xmath77 consistent with the ionization fraction . our calculations should not be exceptionally sensitive to our choice , as the thermostatting of the flow towards the @xmath43 relationship will weaken the sensitivity of the flow to the choice of @xmath78 . these considerations enable us to include radiative heating and cooling explicitly in our calculations , by writing the internal energy equation as @xmath79 where @xmath80 is the specific internal energy of a fluid element and @xmath81 is the ratio of specific heats , set to 5/3 for atomic gas . the first term on the r.h.s . of equation(5 ) is the sink in internal energy due to @xmath47 cooling , while the second is the sum of the source and sink terms due to radiative heating and cooling respectively . when @xmath82 , this term is dominated by heating , with negligible cooling by radiation ; when @xmath83 , radiative heating and cooling rates begin to balance each other ( they must be equal when @xmath67 ) ; and finally , when @xmath84 , this term dominates over @xmath47 cooling and thermostats the gas temperature to @xmath67 . equation ( [ eqn : energy_evolve ] ) allows us to satisfy energy conservation , so that we do not overestimate the fraction of incoming x - ray flux that is converted to mechanical luminosity , while also accounting for the burgeoning amount of energy lost as radiation as the gas temperature draws closer to the radiative equilibrium value . this approach is much more satisfactory than the typical assumption of a constant `` heating efficiency '' everywhere ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , since it allows regions to locally account for radiative losses and thus varying the net heating efficiency in both a local and global manner . hence , in addition to a monochromatic ray - tracing calculation for 1kev photons , we solve equation ( [ eqn : energy_evolve ] ) along with the time - dependent equations of hydrodynamics . we note that our model explicitly assumes that @xmath85 , i.e. , that we approach thermal equilibrium from below . this is satisfied here since the flow is expanding ( and hence cooling via @xmath47 work ) everywhere . if this were not true e.g. if there was heating due to compression or shocks then equation ( [ eqn : energy_evolve ] ) would need to be generalised to include a cooling timescale , in addition to the heating timescale @xmath86 . we solve our evaporation problem in 1d along a streamline connecting the star and the planet , within a spherical co - ordinate system ( we thus implicitly assume that the velocity divergence is that of spherically symmetric outflow ) . we account for stellar gravity by using an effective gravitational potential of the form @xmath87 where @xmath88 and @xmath89 are the planetary and stellar masses respectively , @xmath71 is the radial distance from the centre of the planet , and @xmath34 the orbital semi - major axis . we do not however include a contribution from the coriolis force , since it is negligible in setting the mass - loss rates @xcite . while our code solves the hydrodynamic problem over the entire input parameter space , the applicability of the equations of hydrodynamics must be verified _ a posteriori _ , once the steady - state flow solution is obtained . whether a flow is hydrodynamic or ballistic depends on whether the gas particles are collisional or not on a length - scale shorter than the flow - scale ( e.g. * ? ? ? this property is characterised by the knudsen number : @xmath90 where @xmath91 is the mean free path of the gas particles . the relevant flow length - scale for the problem is the pressure scale - height , given by @xmath92 clearly , @xmath93 increases with distance from the planet ( since the particle number density @xmath42 decreases as an exponential function of the pressure scale - height ; note that the exobase is the radius at which @xmath94 ) . at the same time , our hydrodynamic flow solutions are valid only if the gas remains in the hydrodynamic limit ( @xmath95 ) all the way to the sonic surface ; otherwise , the flow loses causal contact with the planet ( and thus the ability to be influenced by upstream conditions ) before the sonic point is reached ( rendering the concept of a sonic point meaningless ) . putting these two facts together , we follow @xcite in defining the evaporation to occur in the hydrodynamic regime if @xmath95 at the sonic point . using the framework described above , we construct a grid of evaporative mass - loss rates as a function of planetary mass , planetary radius and stellar x - ray luminosity , for a specified orbital distance . in order to comfortably cover the desired parameter space , we span planet masses ranging from 0.5 to 20 ; radii ranging from a value corresponding to a density of 10g@xmath8 ( for a given planet mass ) to half a hill radius at the given orbital separation ; and x - ray luminosities ( evenly spaced logarithmically ) ranging from 10@xmath96 to 10@xmath97ergs@xmath22 . this grid is then coupled to the planetary thermal evolution code , to finally yield the detailed evolution of a given planet and its atmosphere . before proceeding to a discussion of thermal evolution ( 5 ) and the final results for individual planets ( 6 ) , however , it is instructive to examine a slice through our mass - loss grid for a fiducial orbit and stellar x - ray luminosity . fig.[fig : mdot_rates ] shows mass - loss rates as a function of planetary mass and radius , for an x - ray flux of 1.23@xmath98 erg s@xmath22 : the irradiation expected from ad leo at @xmath3100myr ( roughly its current age ) at an orbital separation of @xmath99au ( corresponding to @xmath100k , i.e. , the inner edge of the hz at 1gyr ) . over - plotted is the mass - radius relationship for a solid core comprising 2/3 rock and 1/3 iron ( _ dashed magenta line _ ) . a planet with a h / he envelope mass - fraction @xmath141% , for which the radius is dominated by the solid core , will lie only slightly to the right of this line ( @xmath3 factor 2 , @xcite ) , while a planet with a larger envelope mass - fraction , with radius dominated by the envelope , will lie much further to the right . erg s@xmath22 cm@xmath26 . the thick lines show contours of constant knudsen number at the sonic point ( kn@xmath102 ) of 10 ( _ red dot - dashed _ ) , 1.0 ( _ black solid _ ) & 0.1 ( _ blue dotted _ ) . the _ purple thin dashed line _ shows the mass - radius relation for a solid core comprising 2/3 rock and 1/3 iron , taken from @xcite . the _ white thin solid line _ shows a line of constant binding energy to thermal energy in the bolometrically heated atmosphere ; planets to the right of this line have underlying envelopes too loosely bound to be considered tethered to the core ( see text ) . ] also over - plotted are lines of constant knudsen number at the sonic point ( which we will denote by kn@xmath102 ) : kn@xmath102 = 10 ( _ red dot - dash line _ ) , 1 ( _ solid black line _ ) and 0.1 ( _ blue dotted line _ ) . for a fixed planetary mass , we see that kn@xmath102 decreases with increasing radius . thus ( recalling that hydrodynamic escape requires kn@xmath102 @xmath103 1 ) , it is easier to hydrodynamically evaporate a more distended atmosphere ; an intuitive result , since more puffed - up atmospheres have a lower gravitational binding energy for a given planetary mass . indeed , planets with radii to the right of the thin white line in the plot have atmospheres that are so loosely bound that they are unlikely to remain tethered to the planet for more than a myr : our hydrodynamic calculations indicate that the evaporative flows on these planets are driven entirely by stellar bolometric heating rather than by x - ray heating , leading to a rapid `` boil - off '' phase during which most of the atmosphere is lost shortly after disc dispersal ( see * ? ? ? * for a discussion of this process ) , see we are considering planets with relatively low envelope mass fractions this effect is only prominent in one case . finally , we also see that the lines of constant kn@xmath102 are shallower than the mass - radius relationship for our solid core , with the kn@xmath102=1 line intersecting the latter locus at @xmath104 . since hydrodynamic flows can only occur to the right of the kn@xmath102=1 line ( because they require kn@xmath102 @xmath103 1 ) , only planets less massive than @xmath32 can be plausibly stripped entirely of their h / he envelopes via such flows . higher mass planets can still undergo hydrodynamic evaporation as long as their envelopes remain very extended , but such flows will stall once the planetary radius shrinks sufficiently to hit the kn@xmath102=1 limit , which occurs while these planets still retain h / he mass - fractions @xmath1051% . subsequent mass loss only happens ballistically , which can hardly put a dent in such atmospheres over gyr timescales ( as demonstrated in 3.2 ) ; thus , planets more massive than @xmath32 will retain substantial h / he envelopes over their lifetimes . in short , it is easier to completely strip a lower mass planet of its h / he envelope than a higher mass one : not surprising , since escape velocity increases with planetary mass . the threshold mass of @xmath32 identified here , separating the envelope retention / stripping outcomes , is of course quantitatively valid only for the specific x - ray flux adopted in this snapshot plot , and only in the absence of thermal evolution ( which is not accounted for in this plot ) . the true threshold will depend on the integrated history of the x - ray irradiation ( since a higher x - ray flux , expected at earlier times , will push the sonic point deeper into the atmosphere , raising the threshold mass ) , as well as on the thermal evolution of the planet ( since the atmosphere , contracting as it cools , sinks ever deeper into the planet s gravity well , making hydrodynamic escape harder and decreasing the threshold mass ) . by coupling our full mass - loss grid ( of which fig.[fig : mdot_rates ] is only an instantaneous slice at one x - ray flux ) to our thermal evolution code , we will account for both of these effects in 5 and 6 below . nevertheless , the physical effects summarised here will remain qualitatively valid , and only particularly low - mass cores can be stripped entirely by evaporation . in order to fully model the evolution in evaporative mass loss , we must account for how the h / he envelope cools and contracts . we solve the coupled thermal evolution and evaporation problem for a planet with a solid core and h / he envelope using the mesa stellar evolution code adapted to planets @xcite . the method is described at length in @xcite ; here we only summarise the salient inputs to the code . as discussed in 2.2 , the stellar bolometric luminosity ( @xmath106 ) is assumed to follow the lyon evolutionary track @xcite for a 0.4m@xmath107 star ( appropriate for ad leo ) , while the planet s orbital distance is fixed at either @xmath108au or 0.26au , so that the blackbody temperature at the planet s location at 1gyr is @xmath37k or 200k respectively ( corresponding to the inner and outer edges of the classical hz around an m dwarf ; see 2.2 ) . for reference , the evolution in blackbody temperature at 0.12au is shown in fig.[fig : teq_time ] . ) for a planet at @xmath109 au around an m - star . ] the radius of the solid core is assumed to be constant throughout the planet s evolution , and is specified , for a given core mass and our adopted core composition of 2/3 rock + 1/3 iron , by the mass - radius relationship from @xcite . we do account for the thermal content of the core , due to both radioactive decay and thermal heat capacity , by adopting an earth - like value ( see * ? ? ? the effects of both external bolometric irradiation by the star and internal release of thermal energy from the core are accounted for in mesa , using the @xmath110 relationship formulated by @xcite for a gray atmospheric model with both internal and external heat fluxes , where one uses an incoming stream with a frequency set to the peak of the black - body spectrum for the star and an outgoing ir stream in local thermodynamic equilibrium with the upper atmosphere . current planet formation models are unable to stringently constrain the initial thermal properties of newborn planets . consequently , we consider planets with a large range of initial radii ( or , more accurately , initial entropies ) . the latter are parametrised in terms of an initial cooling ( i.e. , kelvin - helmholz ) timescale @xmath111 , defined as the ratio of a planet s initial internal energy to its initial luminosity . naively , one expects this kelvin - helmholz timescale to be of order the `` age '' of the planet at birth , i.e. , the time it takes to form , and thus comparable to the mean lifetime of primordial disks , @xmath310@xmath112 yr ( e.g. * ? ? ? a ten - fold uncertainty in this estimate is not infeasible , due to variations in disc lifetimes ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or post - formation processes ( e.g. * ? ? ? * ; * ? ? ? ; as such , we use starting models with @xmath111 in the range 10@xmath2910@xmath30 yr , to encompass a plausible maximal difference between `` hot start '' planets ( with short initial cooling times ) and `` cold start '' ones ( with long initial cooling times ) ( c.f . * ; * ? ? ? finally , evaporation is included in our mesa calculations by coupling the code to our grid of newly calculated mass - loss rates as a function of planetary mass , radius and stellar x - ray flux ( 4.3 ) . we present here our results for mass loss ( accounting for both evaporation and thermal evolution ) in the hz of our fiducial m dwarf , for three core masses @xmath113 = 0.8 , 0.9 and 1 at two different orbital separations : @xmath108au ( roughly the inner edge of the hz ) and 0.26au ( roughly the outer edge ) . we investigate initial h / he mass - fractions ( denoted henceforth by @xmath114 ) of @xmath31% and @xmath30.04% ( i.e. , @xmath115 and 4@xmath1910@xmath18 ) . we stop our calculations either after 1gyr of evolution , or once the envelope mass - fraction falls to @xmath59 , the amount that can be efficiently removed by jeans escape on less than gyr timescales ( see [ sec : jeans ] ) . we start by considering the evolution of the atmospheric temperature , density and pressure at the planetary surface ( @xmath116 , @xmath117 and @xmath118 respectively ) , over gyr timescales , in the _ absence _ of evaporation ( i.e. , with evaporation artificially turned off ) . these results constitute a baseline from which to judge whether evaporation will aid habitability or not . we first carry out this analysis for @xmath119 = 300k ( i.e. , at @xmath120au ) . figs.6 , 7 and 8 show our results for @xmath113 = 0.8 , 0.9 and 1.0 respectively , for a h / he envelope with @xmath121 . with evaporation turned off , of course , this mass fraction remains constant over time . in each case , we present results for both a relatively long initial cooling timescale ( @xmath111 ) for the planetary atmosphere , corresponding to a `` cold start '' planet formation scenario , and a relatively short @xmath111 , corresponding to a `` hot start '' ( see 5 ) . we see that , for all three core masses , the final surface temperature after a gyr is @xmath122k , and the final pressure @xmath118 is nearly 10@xmath123 bar ( with variations in @xmath111 making negligible difference to these results ) . these are well above the critical point of water ( @xmath124k , @xmath125 bar ) ; as such , any surface water would exist as a supercritical fluid , not as a liquid ( see phase diagram of water , fig.20 ; constructed from phase transition equations provided by martin chaplin ( pvt.comm.2015 ) ) . hence , as stated earlier , removal of a large fraction of the h / he envelope via hydrodynamic escape , which will reduce @xmath116 and @xmath118 , is _ required _ for habitability when @xmath114 is of order a percent . the degree to which this can happen is the focus of our analysis in 6.1 and 6.2 below , since , as discussed , such relatively high initial mass fractions are commensurate with current data . next , in figs.9 , 10 and 11 , we plot the evolution of surface conditions for a much lower initial atmospheric mass : @xmath114 @xmath16 4@xmath126 . again , in the absence of evaporation , this fraction remains constant . now the surface conditions after a gyr are much more benign , with @xmath127k and @xmath128bar . as the phase diagram in fig.20 shows , surface water should exist as a liquid under these conditions . however , realistic levels of evaporation will in fact remove most or all of this tenuous atmosphere : as we will show below , strong hydrodynamic escape very quickly reduces the atmospheric mass - fraction to @xmath310@xmath63 ; even jeans escape can then remove the remainder on gyr timescales . any potential long - term habitability will then depend on the quantity , properties and evolution of ( outgassed ) secondary atmospheres , similar to the case on earth and mars ( e.g. * ? ? ? as such , for these low initial atmospheric mass fractions , we will plot the evolution of the mass fraction down to 10@xmath63 , to make the point that the primordial h / he atmosphere is likely to be entirely lost , but will abstain from any further discussion of the final surface conditions , which will depend on secondary atmospheres whose modeling is beyond the scope of this paper ( although several authors have begun to investigate this stage theoretically ; e.g. , * ? ? ? * ; * ? ? ? * ) . nonetheless , since secondary atmospheres are at least _ present _ on all the terrestrial planets and massive moons in the solar system , we will consider planets whose h / he envelopes have been completely stripped to still be `` potentially habitable '' . the results for @xmath38k ( @xmath129au ) without evaporation ( not plotted ) are similar : while the surface temperatures after a gyr are now somewhat lower , @xmath116 and @xmath118 are still far too high for liquid surface water when @xmath121 ; for @xmath130 , the surface conditions are again conducive to liquid water , but with the same caveats as above . we now turn the evaporation on , and once again examine the results for core masses of 0.8 , 0.9 and 1.0 at @xmath37k . * 0.8 : * fig.12 shows the evolution of the atmospheric mass - fraction and planetary radius for @xmath28 = 0.8 , for @xmath132 ( top panels ) and 4@xmath126 ( bottom panels ) . in both cases , evaporation strips the atmosphere down to a mass fraction of 10@xmath63 extremely rapidly , in @xmath1410myr . this steep decline is due to runaway evaporation , wherein the cooling timescale exceeds the evaporation timescale : the atmosphere can not contract quickly enough to avoid evaporation by falling deeper into the planet s gravitational potential well ( c.f . * ? ? ? note that the runaway occurs much earlier than the median age of 10@xmath133myr estimated for ad leo ( and for other early to mid - m field dwarfs evincing similar saturated x - ray activity ; see 2.2 ) ; consequently , this evaporation result is unaffected by our simplistic assumption that activity remains saturated beyond a few 100 myr up to a gyr ( instead of declining at later times ) . for a 0.8 core , therefore , we expect any primordial h / he atmosphere with initial mass fraction @xmath141% to be completely lost : eroded down to 10@xmath63 in @xmath1410myr by runaway hydrodynamic evaporation , as shown , and the minuscule remainder efficiently peeled off by continuing hydrodynamic and/or jeans escape over a less than gyr timescale . * 0.9 : * fig.13 shows the evolution in atmospheric mass fraction and planetary radius for @xmath28 = 0.9 . for @xmath132 , we see that the `` hot '' and `` cold '' start cases behave very differently . the atmosphere in the former is initially relatively bloated , resulting in a large fraction of it being blown off almost instantaneously when evaporation is switched on ( a phenomenon termed `` boil - off '' by owen & wu 2015 ; see 4.3 ) . for reasons of clarity , this initial boil - off , which occurs over the first few time - steps of our calculations ( in @xmath141.5 myr ) , is masked out in fig.13 and subsequent figures . essentially one can not build stable models with envelope mass fractions and cooling times as short as desired they are not globally thermodynamically stable in the presence of mass - loss . specifically , in the pressure confining environment of the parent disc the planet can acquire a @xmath10% envelope , but once this pressure confinement is removed such a massive envelope can not be gravitationally bound to the planet ( see owen & wu , 2015 ) . therefore , the bolometrically driven mass - loss allows the planet to readjust to a stable state that is in quasi thermodynamic equilibrium . the plots trace the evolution of the planet after boil - off ends . the remainder then undergoes runaway hydrodynamic escape , reducing the mass fraction to 10@xmath63 in @xmath310myr . for a `` cold start '' , on the other hand , the initial atmosphere is much more compact , and the pace of hydrodynamic evaporation is thus far more leisurely ; by @xmath330myr , evaporation is too slight to make any significant difference , and the atmospheric mass fraction settles down at @xmath310@xmath17 . jeans escape , which only becomes efficient at mass fractions @xmath1410@xmath63 , has no discernible effect on the surviving atmosphere . note as well that the @xmath330myr it takes for the mass fraction to reach approximate steady - state is at the lower limit of the 10@xmath133myr age estimate for ad leo ; our result is thus minimally affected by our naive assumption that activity continues to be saturated beyond a few 100 myr instead of declining at such late times . finally , cores with @xmath13410@xmath18 are evaporated down to 10@xmath63 in only a few myr , regardless of initial cooling timescale , just as in the 0.8 case . the evolution of surface conditions for the @xmath132 case is shown in fig.14 . for a `` cold start '' 0.9 core , we see that @xmath116 and @xmath118 after a gyr are @xmath3 530k and 630bar respectively ( note that the slight continuing evolution of these quantities beyond @xmath3100myr is predominantly due to ongoing thermal evolution of the planet , and not any appreciable evaporative mass loss ) . this [ @xmath116 , @xmath118 ] combination is conducive to liquid surface water ( fig.20 ) , and within the range found in deep - ocean hydrothermals vents on earth . fig.14 also shows the `` hot start '' core with the same @xmath132 for comparison . the [ @xmath116,@xmath118 ] @xmath3 [ 350k , 10bar ] achieved at the end of our calculations is of no particular significance , since we expect the tiny remaining atmospheric mass fraction of 10@xmath63 to be completely removed subsequently by even jeans escape ; we only note that the surface temperature is beginning to flatten out since it can not fall below the equilibrium temperature ( with albedo assumed to be zero ) of 300k at this orbital radius , while the pressure continues to plummet , which will eventually lead to any liquid surface water boiling into vapour . of more physical interest is the fact that @xmath116 in the `` hot start '' case is _ lower _ than in the `` cold start '' one , once the rapid initial boil - off stage ( which is not plotted in these figures ) has ended . this is because hydrodynamic escape requires mass continuity at the sonic point ; when a substantial fraction of the atmosphere is blown off from the top , therefore , the underlying layers swiftly expand to take its place , and the upward advection of heat in this process causes the surface to cool ( see discussion in * ? ? ? * ) . in summary , at @xmath37k , we expect liquid water to survive on gyr timescales on the surface of a `` cold start '' 0.9 core with @xmath132 ; the partial evaporation of the h / he atmosphere engenders habitable conditions in this case . a `` hot start '' 0.9 core with the same @xmath114 , or a core with much smaller @xmath114 and either a `` hot '' or a `` cold '' start , however , is unlikely to retain any of its primordial h / he atmosphere ; habitability conditions in this case will depend on any secondary atmosphere that arises . therefore , a 0.9 m@xmath6 core appears to be at the _ transition _ between a planet becoming habitable due to evaporation , and remaining uninhabitable due to retention of a significant h / he envelope . * 1.0 : * finally , fig.15 shows the the evolution in atmospheric mass fraction and planetary radius for @xmath28 = 1.0 . there is relatively little evaporation when @xmath132 , with both `` hot '' and `` cold start '' cores equilibrating at a mass fraction of @xmath32@xmath1910@xmath17 ( there is some initial boil - off in the `` hot start '' case , but hardly enough to initiate runaway evaporation ; the planet s gravity is too strong ) . when @xmath13410@xmath18 , on the other hand , runaway evaporation reduces the fraction to 10@xmath63 in just a few myr , as in the 0.8 and 0.9 cases . in fig.16 we plot the evolution of surface conditions for @xmath132 . both `` hot '' and `` cold '' start cores end up with very similar surface temperatures and pressures after a gyr , with [ @xmath116 , @xmath118 ] @xmath3 [ 800k , 2@xmath1910@xmath136bar ] . fig.20 demonstrates that these conditions are too extreme for liquid water ; any surface water here can only exist as a supercritical fluid . we now investigate the effects of hydrodynamic evaporation at @xmath138k . the single case of a 0.8 core will serve to illustrate the main results here . * 0.8 : * we plot the evolution in atmospheric mass fraction and planetary radius for this core mass in fig.17 . there is hardly any evaporation for either @xmath132 or 4@xmath126 : in both cases , the final mass fraction after a gyr is very similar to the initial value . figs.18 and 19 show the evolution in surface conditions for these two initial mass fractions . for @xmath114 of 10@xmath26 , the final [ @xmath116 , @xmath118 ] after a gyr is @xmath3 [ 800k , 4@xmath1910@xmath136bar ] ; as the phase diagram in fig.20 indicates , surface water will exist as a supercritical fluid instead of a liquid under these conditions . for @xmath114 of 4@xmath1910@xmath139 , on the other hand , the equilibrium [ @xmath116 , @xmath118 ] after a gyr is @xmath3 [ 400k , 280bar ] , which _ is _ conducive to liquid water . we can easily extrapolate from these results , and the preceding ones , to deduce the implications for 0.9 and 1.0 cores at this radial separation . first , surface temperature and pressure always increase with core mass , for a given @xmath114 and orbital radius . thus , since a 0.8 core with an initial h / he atmospheric mass fraction of a percent can not harbour liquid water at @xmath38k , higher mass cores with the same initial mass fraction can not either . second , in the _ absence _ of evaporation , @xmath116 and @xmath118 are very similar for 0.81.0 cores at any fixed @xmath114 and orbital radius ( e.g. , see the no - evaporation cases plotted in fig.20 for @xmath37k : the @xmath118 for 0.81.0 cores are nearly identical , and their @xmath116 vary by @xmath140100k , for both @xmath114 of 10@xmath26 and 4@xmath1910@xmath18 ) . we found above that there is hardly any evaporation in the 0.8 case even with @xmath13410@xmath18 at this orbital radius ; therefore , since the evaporative rate decreases with increasing core mass ( as the gravitational potential well deepens ) , we can invoke the no - evaporation results here for the entire range 0.81.0 . this in turn implies that 0.9 and 1.0 cores with @xmath13410@xmath18 will have [ @xmath116 , @xmath118 ] very close to that of the 0.8 core with the same initial mass fraction , and can thus also harbour liquid surface water on gyr timescales . these inferences are indeed what our more detailed calculations show . = 300k ( inner edge of hz ) , with a core mass @xmath113 = 0.8m@xmath141 and an initial h / he envelope mass - fraction of @xmath142 @xmath16 10@xmath26 ( the latter remains constant with time in these ` no - evaporation ' calculations ) . the _ dashed vertical line _ shows the median age of ad leo ( @xmath3100myr ) . the _ red curve _ is for a `` hot - start '' model , and the _ blue curve _ for a `` cold - start '' model ; @xmath111 for each is noted in the left panel . ] , but for @xmath113 = 0.9m@xmath141 . ] , but for @xmath113 = 1.0m@xmath141 . ] , for the same core mass @xmath113 = 0.8m@xmath141 , but now for an initial h / he envelope mass - fraction of @xmath142 @xmath16 4@xmath1910@xmath18 . ] , but for @xmath113 = 0.9m@xmath141 . ] , but for @xmath113 = 1.0m@xmath141 . ] = 300k ( inner edge of hz ) , with a core mass @xmath113 = 0.8m@xmath6 . the top row is for planets with an initial h / he envelope mass - fraction of @xmath142 @xmath16 10@xmath26 , while the bottom row is for @xmath142 @xmath16 4@xmath1910@xmath18 . the evolution of the `` hot - start '' and `` cold - start '' models is essentially indistiguishable in the bottom row . ] , but for @xmath113 = 0.9m@xmath6 . note that the `` hot - start '' model ( _ red curve _ ) , for @xmath142 @xmath16 10@xmath26 ( top panels ) , undergoes `` boil - off '' @xcite at very early times , resulting in an almost instantaneous decrease in the initial envelope mass and planetary radius compared to the `` cold - start '' model ( the initial boil - off phase is not plotted for clarity ) - see text . ] ( i.e. , planets at @xmath119 = 300k , with @xmath113 = 0.9m@xmath6 and @xmath142 @xmath16 10@xmath26 ) . ] , but for @xmath113 = 1.0m@xmath6 . ] ( i.e. , planets at @xmath119 = 300k , with @xmath113 = 1.0m@xmath6 and @xmath142 @xmath16 10@xmath26 ) . ] .[fig:1.0e_300_0.01 ] , with the same core mass @xmath113 = 0.8m@xmath141 , but now for a radial location corresponding to @xmath119 = 200k ( i.e. , outer edge of hz ) . ] ( i.e. , planets at @xmath119 = 200k , with @xmath113 = 0.8m@xmath6 and @xmath142 @xmath16 10@xmath26 ) . ] ( i.e. , planets at @xmath119 = 200k , with @xmath113 = 0.8m@xmath6 and @xmath142 @xmath16 4@xmath1910@xmath18 ) . ] the central result of the preceding analysis is that solid cores made of 2/3 rock + 1/3 iron , with mass @xmath41 , can not lose enough of their envelopes to become habitable in the classical hz of m dwarfs , if they are born with significant h / he envelopes . the fundamental reason is that hydrodynamic escape is quenched in such planets while a sizeable portion of the h / he envelope still lingers , and subsequent jeans escape which can at best extract a fractional mass of @xmath310@xmath63 over a gyr is too feeble to remove this remainder . only cores @xmath1401 ( with the precise limiting mass depending on where in the hz one is ) can undergo hydrodynamic escape all the way down to their surface in spite of having initial h / he mass - fractions of @xmath31% , and may thus be habitable ( either due to a tenuous remnant h / he envelope , or a similarly wispy secondary atmosphere ) in the classical hz of m dwarfs . to summarise : _ kepler _ data imply that small ( terrestrial - size ) planets are ubiquitous at small orbital separations , with nearly every star ( statistically speaking ) hosting one such `` kepler '' planet ; around m dwarfs , a sizeable fraction of these close - in planets reside within the classical hz ( see also further below ) . concurrently , both observations and theory indicate that such planets are usually born with h / he envelopes with a mass - fraction @xmath41% , and our calculations imply that planets with cores comprising 2/3 rock and 1/3 iron , with core masses @xmath41 and initial h / he envelope mass - fractions @xmath41% , can not lose enough of their envelopes to be habitable within the classical hz of m dwarfs . with these considerations , we propose three possible classes of solid - core habitable planets around m dwarfs : _ ( 1 ) sub - earth mass rock / iron - core planets within the classical hz _ : we found above that sub - earth mass rock / iron cores with mass @xmath140.9 near the inner edge of the classical hz of m dwarfs , and @xmath1400.8 near the outer edge can be stripped of @xmath31% initial h / he envelopes within a gyr . such @xmath14venus - mass planets may therefore be habitable within the classical hz of m dwarfs , if they acquire secondary atmospheres like the solar system terrestrial planets ( the hz results of @xcite can then be applied to such planets ) . _ ( 2 ) sub - earth to super - earth mass ice - core planets within the classical hz _ : our discussion so far has focussed on rock / iron cores . naively , the results from _ ( 1 ) _ above should apply to ice cores as well ( but see below ) , except for a higher threshold mass ( the limiting planet mass below which @xmath31% initial h / he envelopes will be stripped ) . this is easily seen by examining fig.4 , where the threshold mass is defined as the intersection of the kn@xmath102=1 line and the locus of a core with fixed density . for a given core mass , an ice core , with a density @xmath310 times smaller than a rock / iron one , will have the latter locus shifted to the right ( i.e. , to a larger radius ) by a factor of 10@xmath1432 , resulting in a threshold mass larger by a factor of @xmath32 as well from fig.4 . thus , to zeroeth order , one expects ice - core planets , with a somewhat ( factor of @xmath32 ) larger limiting mass than rock / iron core ones , to also be habitable within the classical hz of m dwarfs . however , a correct analysis of evaporation here must also account for the opacity of steam ( liberated from the core s surface ) in addition to h / he , which may strongly affect the outcome ( since oxygen in this case within h@xmath31o molecules is a strong x - ray absorber ) . this will be the subject of future work . note also that icy cores can only form beyond the ice - line , situated at @xmath31.5au for a @xmath30.4 m dwarf at an age of order a myrau ; with a stellar bolometric luminosity of @xmath144@xmath145 for a 0.4 star at an age of 1.5myr , from the evolutionary tracks of @xcite ] . thus ice - cores must migrate inwards for the above discussion to be pertinent . _ ( 3 ) planets formed after gas - disk dispersal _ : finally , our entire analysis in this paper is for planets that form while a significant amount of gas still remains in the surrounding primordial disk , allowing them to accrete relatively massive h / he envelopes ; _ kepler _ data , as discussed , imply that such planets are ubiquitous . however , the terrestrial planets in our own solar system evince signatures of having coalesced after the gas disk dispersed ( as the radiometricly determined age of the earth @xcite is estimated to be @xmath146 myr younger than that of the solar - system , e.g. @xcite , well after the gas disc would have dispersed ) . if solid - core planets around m dwarfs can form in a similar fashion , then they might well be habitable in the classical hz of m dwarfs ( and slightly interior / exterior to it depending on planetary mass ) ; the work by @xcite , examining solely secondary atmospheres , is more directly applicable in that case than this paper . we note that there is , prima facie , another interesting possibility . while we find that rock / iron cores @xmath41 with @xmath31% initial h / he envelopes retain too large a fraction of these envelopes to be habitable within the classical hz of m dwarfs , they _ will _ be stripped of such atmospheres if their orbits are much smaller . simultaneously , @xcite show that the inner boundary of the hz ( set by either the moist greenhouse or runaway greehouse effect ) for such _ super_-earth mass planets will be closer to the star than for earth - mass ones ( where the latter are used , by definition , to calculate the classical hz limits ) . therefore , one might expect an overlap in space , interior to the classical hz , between where a super - earth of given mass can be stripped of a @xmath31% h / he envelope , and where it will be habitable with a secondary atmosphere before greenhouse effects become overwhelming ; super - earths would then be habitable within this overlap region . unfortunately , such an overlap is implausible . for solar - type stars , @xcite show that the inner boundary of the hz moves inwards from 0.99au for a 1 planet to 0.94au for a 10 one ; i.e. , only a very small shift of 0.05au for a factor of 10 increase in planetary mass . the effect should be similar around m dwarfs . this tiny decrease in orbital separation will have negligible impact on the incident x - ray flux and thus on the mass - loss rates ; as such , we do not expect any hz for super - earths interior to the classical hz around m dwarfs . conversely , for sub - earth mass planets , the inner boundary of the hz moves slightly _ outwards _ , as @xcite show ; thus , we do not expect any hz for these planets interior to the classical hz either . there are two main ingredients that determine the actual frequency of habitable planets around m dwarfs . the first , as we have shown , is a rigorous treatment of evaporation , using hydrodynamic models that account for radiative cooling as well as the transition to jeans escape , and including the thermal evolution of the planet . a simplistic `` energy - limited '' formalism , that moreover assumes that the escape is always hydrodynamic ( e.g. , @xcite for m dwarfs ) , leads to a gross overestimation of evaporative mass - loss rates , and thereby an overly optimistic appraisal of the fraction of planets that can be rendered habitable by stripping their primordial h / he envelopes . our results are in direct conflict with those of @xcite . the latter authors calculations suggest that in the majority of cases , m dwarfs can completely strip envelopes with mass fraction @xmath1410% ( and even larger ones in some cases ) at the inner edge of the hz , and mass fractions @xmath141% at the outer edge of the hz , from 12 m@xmath6 cores . in other words , they conclude that evaporation can generate a plethora of potentially habitable earth - mass planets around m - dwarfs . our results indicate that this is not the case : it is the simplifications made by @xcite thet lead them to conclude otherwise . the main reason is their choice of a constant efficiency energy - limited mass - loss prescription . as we have discussed throughout this work , at late times when the xuv flux declines and the envelope mass fraction drops below @xmath31% ( i.e. , when the planet s radius shrinks rapidly with decreasing envelope mass ) , the efficiency of hydrodynamic mass - loss decreases significantly ( c.f . * ) ; furthermore , the flow transitions to non - hydrodynamic ( jeans escape ) at late times . neglecting these effects , @xcite find that their planets can undergo runaway mass - loss at ages @xmath147 gyr ; including these effects , however , we find that a @xmath31m@xmath6 planet can only ever undergo runaway mass - loss at early times ( @xmath148 gyr ) , when the flux is high enough to permit hydrodynamic evaporation . in fact , if we adopt an energy - limited mass - loss rate with no hydrodynamic cut - off at low fluxes , we indeed recover the runaway mass - loss at late times found by @xcite . this is demonstrated in fig . [ fig : el_compare ] , where we compare the results of our calculations ( solid line ) to those for the energy - limited case with efficiencies ( @xmath74 ) of 5% ( dot - dashed ) , 10% ( dashed ) & 25% ( dotted ) ; note that @xcite s default choice is @xmath74 = 30% . the plot shows that , while our planet loses negligible mass at late times due to the hydrodynamic cut - off , the energy - limited cases can undergo unphysical runaway mass - loss at these ages . we note that while an efficiency of @xmath149 produces a good match to the final envelope mass - fraction at 10 gyr in this _ particular _ example , this is _ not _ a generally applicable result : indeed , the plot shows that the 10% case ( dashed line ) is also starting to enter a run - away phase at the end of the calculation . in general , there is no single efficiency value that can mimic the trends in evaporation in our results , especially the hydrodynamic cut - off , which turns out to be extremely important for evaporation and the question of potential habitability . using an energy - limited formalism to determine evaporation rates when the flow is close to transitioning from hydrodynamic to non - hydrodynamic can lead to rather poor predictions . finally , we note that our prediction that planets @xmath41 will retain most of their initial voluminous h / he envelopes , in the classical hz of m dwarfs , should be directly testable with tess , which will target a large number of m dwarfs . planet at the inner edge of the hz . the _ solid _ curve shows the evolution with the mass - loss rates calculated in this work , while the _ dot - dashed _ , _ dashed _ and _ dotted _ curves show mass - loss rates calculated in the `` energy - limited '' framework with efficiencies of @xmath74 = 5 , 10 and 25% respectively . note that we have assumed that the x - ray flux remains saturated over the entire evolution : even in this extreme case , the flow still becomes non - hydrodynamic ( ballistic ) at late times in our model , preventing runanway mass loss ; this is not true of the energy - limited cases . ] the second ingredient that controls the actual prevalence of habitable planets around m dwarfs is the frequency of their planets as a function of planetary mass , at the small end of the planetary spectrum . a recent careful analysis by @xcite indicates that the occurrence of m dwarf planets continues to increase to planetary radii below 1 , without any strong evidence for a downturn in the planet radius function that was suggested by previous studies . if this is true , then it bodes well for the frequency of potentially habitable planets orbiting m dwarfs , since the smallest ( lowest - mass ) planets will be most easily stripped of their initial h / he envelopes . indeed , applying their improved calculations to previous work by @xcite and @xcite , @xcite already show that the frequency of earth - size planets in the hz of m dwarfs calculated by @xcite to be @xmath30.15 planets per star increases to @xmath30.250.8 , a remarkable fraction . if this frequency continues to increase towards smaller planets , as @xcite suggest , then evaporation can indeed create a bonanza of potentially habitable planets around these cool red dwarfs . however , the latter results are based on a quite limited sample ; they need to be verified and sharpened with more data from the k2 mission , as well as upcoming ones such as tess . our key results are as follow : \1 ) jeans escape can only remove a very small h / he mass - fraction from low - mass exoplanets in the hz of m dwarfs ( @xmath310@xmath63 over a gyr ) : hydrodynamic evaporation is essential for removing significant amounts of h / he . \2 ) previously derived mass - loss rates , based on the `` energy - limited '' ( or `` locally energy - limited '' ) formalism , and assuming moreover that escape is always in the hydrodynamic limit , are significant overestimations . our improved model , accounting for both radiative losses and the transition from hydrodynamic to jeans escape , and including the thermal evolution of the planet , yields much lower mass loss rates . \3 ) in particular , for cores made of 2/3 rock + 1/3 iron , we find that only sub - earth mass cores @xmath140.9 near the inner edge of the classical hz of m dwarfs , and @xmath1400.8 near the outer edge can be stripped of @xmath31% initial h / he envelopes . rock / iron cores with mass @xmath41 , and born with h / he envelope mass - fractions @xmath41% , can not lose sufficient envelope mass to become habitable within the classical hz of m dwarfs . our prediction that @xmath41 cores in the hz of m dwarfs should still be shrouded by their voluminous natal h / he envelopes over gyr timescales will be directly testable with tess . \4 ) we propose three classes of potentially habitable planets in the hz of m dwarfs : _ ( i ) _ planets with sub - earth mass rock / iron cores : since they can be stripped of 1% h / he envelopes within a gyr , they may be habitable if they can acquire suitable secondary atmospheres , like solar system terrestrial planets ; _ ( ii ) _ planets with ice - cores , with an upper limit to the core mass somewhat higher than the sub - earth mass for rock / iron cores , may also be able to lose their primordial h / he envelopes and thus become habitable ; however , steam opacity must be included in the evaporation calculations to verify this possibility ; and _ ( iii ) _ planets ( possibly like the terrestrial ones in the solar system ) that form after gas disk dispersal , and thus have only tenuous secondary atmospheres , with very little primordial h / he . we thank the referee for an instructive report . we are grateful to barbara ercolano , ren heller and renyu hu form comments on the manuscript . we are indebted to jorge sanz - forcada for supplying the synthetic xuv spectrum of ad leo , which enabled a more realistic analysis of evaporation around m dwarfs than possible with the scaled - solar spectra used in previous work , and for comments on the manuscript . we are also very grateful to martin chaplin for providing the equations for the phase diagram of water . jeo acknowledges support provided by nasa through hubble fellowship grant hst - hf2 - 51346.001-a awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , for nasa , under contract nas 5 - 26555 . sm acknowledges the support of stfc - uk consolidated grant st / k001051/1 .
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the ubiquity of m dwarfs , combined with the relative ease of detecting terrestrial - mass planets around them , has made them prime targets for finding and characterising planets in the `` habitable zone '' ( hz ) .
however , _ kepler _ finds that terrestrial - mass exoplanets are often born with voluminous h / he envelopes , comprising mass - fractions ( @xmath0 ) @xmath1% . if these planets retain such envelopes over gyr timescales , they will not be `` habitable '' even within the hz . given the strong x - ray / uv fluxes of m dwarfs , we study whether sufficient envelope mass can be photoevaporated away for these planets to become habitable .
we improve upon previous work by using hydrodynamic models that account for radiative cooling as well as the transition from hydrodynamic to ballistic escape . adopting a template active m dwarf xuv spectrum , including stellar evolution , and considering both evaporation and thermal evolution ,
we show that : _ ( 1 ) _ the mass - loss is ( considerably ) lower than previous estimates that use an `` energy - limited '' formalism and ignore the transition to jeans escape ; _ ( 2 ) _ at the inner edge of the hz , planets with core mass @xmath2 can lose enough h / he to become habitable if their initial envelope mass - fraction is @xmath31% ; _ ( 3 ) _ at the outer edge of the hz , evaporation can not remove a @xmath31% h / he envelope even from cores down to 0.8 .
thus , if planets form with bulky h / he envelopes , only those with low - mass cores may eventually be habitable .
cores @xmath4 1 , with @xmath41% natal h / he envelopes , will _ not _ be habitable in the hz of m dwarfs .
ps .
p_surf 1t_bb , ms
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You are an expert at summarizing long articles. Proceed to summarize the following text:
according to the standard hierarchical structure formation theory based on the @xmath0cdm cosmology model , density perturbations grow into small haloes where the first stars and galaxies form ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? these astrophysical sources produce ultraviolet ( uv ) ionizing photons that escape into the intergalactic medium ( igm ) which is eventually reionized . the epoch when this process occurs is called the `` epoch of reionization '' ( eor ) . the detailed astrophysics of the eor is currently poorly understood because no observation has yet probed the early and middle stages of the eor although current observations have provided fruitful information on late stage of the eor . for example , the absorption spectra of high-@xmath1 quasars indicate that reionization was complete by @xmath2 ( e.g. * ? ? ? * ) and the number density of ly@xmath3 emitter galaxies at @xmath4 implies that the neutral hydrogen fraction increases at @xmath5 ( e.g. * ? ? ? * ) . on cosmological scales , reionization induces thomson scattering of cmb photons off free electrons . the optical depth of thomson scattering measured by planck is @xmath6 which corresponds , using an instantaneous reionization toy model , to a reionization at @xmath7 @xcite . to further improve our understanding of the eor , observations targeted on the cosmological redshifted 21 cm signal from the eor are on - going . the 21 cm signal emission is due to the hyperfine structure of neutral hydrogen atoms and is expected to be a powerful tool to probe the neutral igm , yielding both astrophysical and cosmological information such as matter density fluctuations and the ionization state and thermal history of the igm at high redshift @xcite . recently , some first - generation radio interferometers have been attempting to detect statistically the 21 cm signal from the eor , such as the murchison wide field array ( mwa ) @xcite , the low frequency array ( lofar ) @xcite and the precision array for probing the epoch of reionization ( paper ) @xcite . these observational efforts have resulted in upper limits of the 21 cm power spectrum and on the ionized state of the igm at @xmath8 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? furthermore , future instruments such as the square kilometre array ( ska ) @xcite and hydrogen epoch of reionization array ( hera ) @xcite are designed to detect the 21 cm signal power spectrum with higher signal to noise ratio and at higher redshift , during the cosmic dawn . the ska should also be able to image the signal in 3d , which requires high sensitivity on small enough scale ( that is sufficient collecting area in a large enough core ) . so we expect a wealth of 21 cm data in the near future . then we face the fundamental question : what we can learn from the data ? from theoretical and numerical works , we have already some insights on the process of reionization . for instance , considering galaxies in relatively massive host haloes results in larger and more uniform ionized bubbles and imprints a peak at larger scales in the 21 cm power spectrum ( e.g. * ? ? ? * ; * ? ? ? on the other hand , abundant and small minihaloes , which serve as absorption systems by self - shielding from ionizing photons , result in small and disjointed ionized regions ( e.g. * ? ? ? * ) . to extract information on the eor from the observed 21 cm signal , we need to be able compute the 21 cm signal from the basic physics of reionization ( e.g. * ? ? ? the most self - consistent method is to run numerical simulations . there are currently two approaches . the first is to implement the full radiation hydrodynamics ( rhd ) . this type of simulations is relevant if small scales , where feedback effects such as photo - heating play an important role , are resolved ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the second approach is to run large scale simulations where radiative transfer ( rt ) is computed in post - processing ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? this type of simulations are able to account for large scales fluctuations in the signal but they require sub - grid modelling to treat processes on galaxy scales ( feedback , ionizing photons escape fraction , etc . ) . although numerical study based on simulations is the most consistent method , it implies a large computational cost . this disadvantage is somewhat alleviated by using a semi - numerical approach instead of rhd or rt simulations . in most semi - numerical simulations , the production of ionizing photons is calculated based on the excursion set formalism , using analytically derived halo mass functions , and density fluctuations are obtained with linear perturbation theory @xcite . however , the growth of hii regions is simply evaluated by considering the balance between the production of ionizing photons and the number of neutral hydrogen atoms @xcite . recently , the effect of recombination is also taken into account ( e.g. * ? ? ? the results produced by the semi - numerical approach shows good agreement with those by rt simulations on large scale ( @xmath9 mpc ) @xcite . to maximize the scientific return of the upcoming observations it is important to establish systematic procedures to derive constraints on the eor modeling parameters from the observed 21-cm data recently , several works have studied how such constraints can be obtained by exploring the eor parameter space with semi - numerical simulations . for example , fisher analysis @xcite and bayesian parameter inference such as the markov chain monte carlo ( mcmc ) approach @xcite have been applied . in this work , we suggest a new approach for parameter reconstruction based on a machine learning method . machine learning is one of the hot topics in data science as a method to deal with big data . it is currently applied to many fields such as pattern recognition or search engine . the main purpose of machine learning is to find approximate functions that , given the input produce the desired outputs . this is achieved by learning " from training datasets with known inputs and outputs ( e.g. * ? ? ? * ; * ? ? ? recently , machine learning methods have been applied in the field of astronomy . for example , learning from huge galaxy image catalogs helps with morphological classification of galaxies @xcite . machine learning can also help with the selection and classification of transients @xcite . applying machine learning to a large sample of spectroscopic and photometric galaxies data can improve the accuracy of estimates of photometric redshifts . using simulated gravitational wave data with noise as learning sample , machine learning can help us search for gravitational wave signals from noisy real data @xcite . in the context of cosmology , machine learning is used to model galaxy formation @xcite or to make templates of nonlinear matter power spectrum @xcite . closer to our field of interest , there is a study applying a machine learning method to estimate the escape fraction of ionizing photons during the eor @xcite . in this study , they show how machine learning can estimate the lyman continuum escape fraction by using mock spectroscopic simulation data . in our work , using a simple astrophysical parameterization of the eor , we apply artificial neural network ( ann ) , which is one of the machine learning methods , to reconstruct the parameter values from the 21 cm power spectrum data . this paper is organised as follows . in section [ sec:21 cm ] , we introduce the cosmological 21 cm signal and the eor parameterization we focus on . in section [ sec : ann ] , we describe our artificial neural network and test the impact of its chosen architecture . in section [ sec : result ] we show our main results , and we give a summary and discussion in section [ sec : summary ] . throughout this paper , we employ the best fit cosmological parameters obtained by . the brightness temperature for the @xmath10 cm signal is given by ( e.g. * ? ? ? * ) : @xmath11 . \label{eq : brightness}\end{aligned}\ ] ] here , @xmath12 and @xmath13 represent the local spin temperature of the igm and the cmb temperature , respectively . @xmath14 is the local optical depth in the 21 cm rest frame frequency @xmath15 , @xmath16 is the local neutral fraction of the hydrogen gas , @xmath17 is the evolved matter overdensity , @xmath18 is the local gradient of the gas velocity along the line of sight and @xmath19 is the hubble parameter . all quantities are evaluated at redshift @xmath20 . as we can see , the 21 cm signal includes both astrophysical and cosmological information . thus , we can hope to use the 21 cm signal to disentangle and quantify them @xcite . let us now introduce the power spectrum of the 21 cm fluctuations . we define the 21 cm power spectrum as @xmath21 in our context , we use the _ dimensional _ 21 cm power spectrum , @xmath22.the 21 cm ps would describe the statistical properties of the 21 cm fluctuations perfectly if they were a gaussian random field . however , the 21 cm fluctuations are expected to deviate from a gaussian behaviour because of astrophysical effects such as ionization and x - ray heating . thus it is useful to compute higher order statistics and one - point statistics such as the bispectrum , the variance and the skewness to estimate non - gaussian features in the 21 cm fluctuations . in order to generate the 21 cm ps for a given set of astrophysical parameters , we use the publicly available code * 21cmfast * @xcite . this code is based on a semi - numerical model of cosmic reionization and thermal history of the igm . it quickly generates maps and ps of the brightness temperature , matter density , velocity , spin temperature and ionization fraction at designated redshifts . we performed simulations in a @xmath23 @xmath24 comoving box with @xmath25 grid cells for a wide range of eor parameters described in the next section . in our calculation , we use the 21 cm power spectrum in the range @xmath26 divided into 14 bins . it is common to characterize eor models with parameters and then examine the effect of changing the parameters on the 21 cm signal . we employ three key parameters which are often used . let us briefly define these three parameters : + 1 . @xmath27 , _ the ionizing _ _ efficiency _ : @xmath27 is the combination of several parameters related to ionizing photons escaping from high redshift galaxies and is defined as @xmath28 @xcite . here , @xmath29 is the fraction of ionizing photons escaping from galaxies into the igm and @xmath30 is the fraction of baryons locked into stars . these parameters are extremely uncertain at high redshift @xcite . @xmath31 is the number of ionizing photons produced per baryon in stars and @xmath32 is the mean recombination rate per baryon . in our calculation , we explore a range of @xmath33 . @xmath34 , _ the minimum _ _ virial _ _ temperature _ _ of _ _ haloes _ _ producing _ _ ionizing _ _ photons _ : @xmath34 parameterizes the minimum mass of haloes producing ionizing photons during the eor . typically , @xmath35 is chosen to be @xmath36 , corresponding to the temperature above which atomic cooling becomes effective . @xmath35 parameterizes the physics of star formation in high redshift galaxies . in haloes with virial temperature @xmath37 atomic cooling is sufficient to trigger gravothermal instability and thus star formation . however , star formation is quenched if agn or supernovae feedback is effective and the igm is heated up . this leads to effective minimum virial temperature larger than @xmath38[k ] . in haloes with viral temperature @xmath39 , hydrogen molecule cooling is necessary . however , if stars begin to form in a halo , radiative feedback such as the photodissociation of @xmath40 by lyman - werner photons may become effective and prevent the gas from cooling @xcite . conversely , positive feedback , such as the enhancement of @xmath40 molecules formation due to an increase in the free electrons density , tends to push the minimum virial temperature to lower value because cooling becomes more effective . thus @xmath35 parameterizes the uncertainty in the efficiency of radiative feedback . in our work , we explore @xmath35 ranging from @xmath41 to @xmath42 . @xmath43 , _ the mean free path of ionizing photons _ : the propagation of ionizing photons through the ionized igm strongly depends on the presence of absorption systems and the sizes of ionized regions are determined by the balance between sinks and sources of ionizing photons ( e.g. * ? ? ? this process is modelled by the maximum mean free path of ionizing photons , @xmath43@xcite . physically , the mean free path of ionizing photons corresponds to the typical distance traveled by photons within ionized regions before they are abosorbed and is determined by the number density and the optical depth of lyman - limit systems . in our calculation , we explore @xmath43 from 10 mpc to 60 mpc . in this section , we introduce artificial neural networks ( ann ) . anns are one of the machine learning methods and are a mathematical model inspired by the natural neuron network in our brain . the main purpose of anns is to construct approximate functions which associate input data with output data . in order to construct such a function , the ann has to learn from `` _ training data _ '' . the architecture of a simple class of ann consists of three layers : the input layer , the hidden layer and the output layer . each of them has a number of neurons as shown in fig.[fig : fig1 ] . in a more general case , we could choose the number of hidden layers and the number of neurons at each layer arbitrarily . in our study , we use 1 hidden layer . note that it is mathematically proven that neural networks with only 1 hidden layer can approximate any function with any accuracy if we use a large enough number of neurons @xcite . let us briefly describe the architecture of our ann . the input data @xmath44 is fed to the @xmath45-th neurons in the input layer . each neuron in the input layer is connected to the @xmath46-th neuron in the hidden layer and a weight @xmath47 is associated with the connection . the input to the @xmath46-th neuron in the hidden layer @xmath48 is a linear combination of all the input neurons with weight @xmath47 : @xmath49 here , @xmath50 is the number of input data . in the hidden layer , the @xmath46-th neuron is activated by an activation function @xmath51 such as its output is @xmath52 . generally , the activation function is a nonlinear function . we use the sigmoid function @xmath53 . the properties of the sigmoid function are such that it saturates and returns a constant output when the absolute value of the input is large and that it is a smooth and differentiable function . thanks to the nonlinear activation function , a trained ann can express any function . in the output layer , we compute linear combinations of the activated outputs of the neurons in the hidden layer with weights @xmath54 and obtain the output vector : @xmath55 here @xmath56 is the number of neurons in the hidden layer . note that we do not activate the output value . the aim of training the ann is to find a set of weights that ensures the output vectors produced by the ann for a set of input vectors is sufficiently close to the desired output vectors . once we adjust the weights to reach this goal on a training sample , we can make predictions for output vectors for arbitrary input vectors outside of the training sample ( for example , new observational data ) . a popular algorithm to compute the trained weights is the `` _ back propagation algorithm _ '' we will describe this algorithm briefly in the following section . in this section , we present the back propagation algorithm for the 1 hidden layer case . we also show the back propagation algorithm for multiple hidden layers case in appendix . in order to quantify how well the output obtained by the ann approximates the desired output for the training data set , we define the ( total ) cost function as : @xmath57 , \label{eq : cost}\ ] ] where @xmath58 is the number of training input vectors and @xmath59 is the number of neurons at the output layer . @xmath60 and @xmath61 are outputs of the ann and the ( desired ) training output data , respectively . our purpose is to find the weight set that minimises the cost function . in order to find this weights set , we need to compute the partial derivative of @xmath62 with respect to the individual weights @xmath63 and find the local minimum of @xmath62 using gradient descent . the weights are updated by gradient descent following the formula : @xmath64 here , @xmath65 is a learning coefficient which controls how fast the weights are updated . we used @xmath66 . we only need to calculate the derivative of the cost function for each training input vector and then sum over all input vectors as shown in eq.[eq : gradient ] . first , let us consider the derivative with respect to the weights between output layer and hidden layer . in this case ( @xmath67=2 ) , we can simply calculate the derivative of @xmath62 as @xmath68 next , we calculate the derivative of @xmath62 with respect to the weights between the hidden layer and the input layer . in this case ( @xmath67=1 ) , the derivative of @xmath62 is @xmath69 in the second line , we use the chain rule for derivative because @xmath62 depends on the activated neuron @xmath70 in the hidden layer only through the output neuron @xmath60 . here , @xmath71 denotes the derivative of the activation function with respect to @xmath72 . using eqs.([eq : gradient ] ) , ( [ eq : derivative1 ] ) and ( [ eq : derivative2 ] ) , we can iterate on the gradient descent until the outputs obtained by the ann converge to the desired outputs ( minimum of the cost function ) . the back - propagation algorithm can be summarised as follows : 1 . starting with random weights , compute the output of the ann using eq.([eq : hidden ] ) and eq.([eq : output ] ) for all input vectors in the training set ( _ forward propagation _ ) + 2 . compute the cost function + 3 . compute the derivative of the cost function with respect to the weights between output layer and hidden layer with eq.([eq : derivative1 ] ) and then the derivative with respect to the weights between input layer and hidden layer with eq.([eq : derivative2 ] ) ( _ back - propagation _ ) . update the weights with eq.([eq : gradient ] ) . go back to ( i ) and iterate until the cost function converges to a minimum . in our case , we prepared 70 training data sets . each set consist of the 21 cm ps @xmath73 obtained with 21cmfast as input data and the corresponding eor parameters used in the simulation , @xmath74 as output data . the architecture of the ann is the following : ( _ i _ ) 14 neurons in the input layer , ( _ ii _ ) 14 neurons in the hidden layer , ( _ iii _ ) 3 neurons in the output layer . as we mentioned in section [ sec:21cmps ] , we use 14 bins for the 21 cm ps , and we use 3 eor parameters . this is why the number of neurons in the input layer and in the output layers are 14 and 3 , respectively . note that the number of redshifts used to train will always match the number of redshifts being fit . to analyse in details how the ann performs we look at partial cost functions : we use the normalised root mean square error ( rmse ) defined by @xmath75 here , @xmath76 ( with @xmath77 one of @xmath78 or , @xmath79 . @xmath80 and @xmath81 are the eor parameters evaluated by the ann and from the training data , respectively . ( red ) , @xmath27 ( green ) and @xmath35 ( blue ) as functions of the number of iterations for the learning process , for a network with 14 neurons . ] ( solid line ) , @xmath27 ( dashed line ) and @xmath35 ( dot - dashed line ) as functions of the number of neurons for @xmath82 iterations in the learning process . ] first , we study how the rmse depends on the number of iteration for each eor parameter . the result is plotted in fig.[fig : fig2 ] . in this figure , we fix the number of neurons in the hidden layer at 14 and perform calculation with @xmath83 iterations . after initial fluctuations , the rmse decreases and converges for @xmath84 iterations . next , we show how the rmse depends on the number of neurons in the hidden layer after @xmath85 iterations . the rmse is only weakly dependent on the number of neurons in the explored range . in the following section , we perform @xmath82 iterations for the back - propagation algorithm with 14 neurons in the hidden layer unless specifically mentioned we apply the trained ann to 54 test datasets . as for the training datasets , these test datasets consist of the 21 cm ps and the corresponding eor parameters . we use the 21 cm ps data as input for the ann and obtain estimated values for the eor parameters . then , we compare the estimated values with the true values that were used to compute the ps in the 21cmfast simulation , and evaluate how the ann performs . first , we apply the ann to the 21 cm ps at a single redshift without including the effect thermal noise or sample variance . in figs.[fig : fig4 ] and [ fig : fig5 ] , we plot the eor parameters estimated with the ann from the 21 cm ps data against the true values . in fig.[fig : fig4 ] , we show the result at @xmath86 . as we can see , @xmath27 and @xmath35 estimated with the ann are in relatively good agreement with the true values . on the other hand , for @xmath43 , the correct values are not recovered . this is because the 21 cm ps is not sensitive at @xmath86 to @xmath43 . on the other hand , the 21 cm ps becomes sensitive to @xmath43 at lower redshift when reionization advances much more ( since @xmath43 physically expresses the maximum bubble size , the effect of @xmath43 on the 21 cm ps is remarkable mostly at lower redshift ) . thus the cost function is insensitive to changes in @xmath43 values ; the learning process of the ann is incomplete and systematic deviations remain . note that the reason why the estimated value of @xmath43 seems to be constant at @xmath87 is that @xmath43=30 occurred more often than other values in the training datasets . in fig.[fig : fig5 ] we show the same plots obtained from the ps at @xmath88 . the agreement between recovered and expected values extends over a larger range than at @xmath1=12 , in particular for @xmath43 . indeed the 21 cm ps at @xmath88 is sensitive to @xmath43 and thus the ann learning process works better . in the previous two cases , we considered the 21 cm ps without any source of noise . we will now consider both the contribution of thermal noise and sample variance . we model the thermal noise as a gaussian random field characterized by its power spectrum . in an annulus in fourier space with @xmath89 cells , the thermal noise power spectrum is ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) : @xmath90 where @xmath77 is the angle between * k * and line of sight , @xmath91 is the effective area of antenna , @xmath92 is the observed wavelength and @xmath93 are factors converting @xmath94 space units into comoving wavenumber units and are determined by cosmology . @xmath95 is expressed by @xmath96 where @xmath97 is the total system temperature , @xmath98 is the bandwidth and @xmath99 is the effective observing time . the spherically averaged noise can be obtained by summing over @xmath77 in a shell with radius @xmath56 shown in eq.[eq : noise_average ] . @xmath100^{2}\right\}^{-1/2}. \label{eq : noise_average}\ ] ] we recommend to read mcquinn et al . ( 2006 ) . in our case , we assume ska specification @xcite . we estimate the sample variance from 10 simulations using different realizations of the initial conditions . in the general case , we would include the noise at the level of the visibilities and compute the resulting noisy power spectrum . assuming that the noise is a random gaussian field , we simply generate a noisy power spectrum by using the following formula : @xmath101 where @xmath102 is the noisy power spectrum , @xmath103 is the power spectrum produced by the simulation for a given set of parameters generically labeled by the index j , and @xmath104 is _ a random draw _ from a gaussian probability distribution with variance equal to the thermal noise power spectrum or the sample variance . a different and independent draw is required for each value of @xmath56 . for the learning sample we include only the sample variance . for each parameter set ( 70 possible values for @xmath46 ) , we generate 50 realizations of the noise ( labeled by @xmath105 ) . this means that we use 3500 different ps for the learning sample . this is necessary as we are not aware of a standard technique for directly including an uncertainty in the inputs of an ann . for the test datasets , we add both thermal noise and sample variance to the 21 cm ps , using the same procedure . note that the result for the virial temperature is plotted in log scale . it is the same for the following figures . ] = 12 . in this case , we include both thermal noise and sample variance . ] in fig.[fig : fig6 ] , we show the eor parameters found by the ann as functions of the values used in the simulations , using the 21 cm signal ps at @xmath88 , including both thermal noise and sample variance . the difference between fig.[fig : fig5 ] and fig.[fig : fig6 ] is not obvious at a glance . in order to quantify the difference , we compute the _ mean chi - square value _ , @xmath106 : @xmath107 @xmath80 is the @xmath43 , @xmath35 , @xmath27 reconstructed by the ann and @xmath108 is the value of the corresponding parameter used in the simulation . in table.[table : chi ] , we compare the @xmath106 values for each of the parameters , with and without noise . as we can see , the @xmath106 values for the 21 cm ps with noises are worse than those without noises . noise alters the efficiency of the learning process for the ann . [ my - label ] [ cols="<,<,<,<,<,<",options="header " , ] here , we show the back propagation algorithm in the case of multiple hidden layers . by analogy with eq.([eq : derivative2 ] ) , we can express the derivative of @xmath109 with respect to the weight in the @xmath67-th layer , @xmath110 @xmath111 as @xmath112 where @xmath113 . since the changes in @xmath114 are transmitted to @xmath109 through each @xmath115 in the ( @xmath67 + 1)-th layer , the derivative of @xmath109 with respect to @xmath114 can be expressed as @xmath116 remember that @xmath115 can be expressed with the activation function @xmath51 as @xmath117 , @xmath118 here , we define @xmath119 , then we can re - write eq.([eq : derivativel2 ] ) as @xmath120 combining eq.([eq : delta ] ) with @xmath121 , the eq.([eq : derivativel ] ) can be re - written simply as @xmath122 this form tells us that we can easily obtain the derivative of the cost function with respect to @xmath110 , which connects the neuron @xmath45 in the ( @xmath67 - 1 ) th layer to the neuron @xmath46 in the @xmath67 th layer , as the product of @xmath123 and @xmath124 . as shown in eq.([eq : delta ] ) , we start to calculate @xmath123 from the output layer ( @xmath67=l ) to the input layer . this is why this algorithm is called `` _ back propagation _ '' . if we use eq.([eq : cost ] ) as the cost function , we easily calculate @xmath125 . this work is benefited from a grant from the french anr funded project orage ( anr-14- ce33 - 0016 ) . we thank to g.mellema , a. fialkov , s. majumdar , s.giri and k. hasegawa for their useful comments and thank to s. yoshiura for providing the thermal noise data . abel , t. , bryan , g. l. , & norman , m. l. 2002 , science , 295 , 93 agarwal , s. , abdalla , f. b. , feldman , h. a. , lahav , o. , & thomas , s. a. 2012 , , 424 , 1409 agarwal , s. , abdalla , f. b. , feldman , h. a. , lahav , o. , & thomas , s. a. 2014 , , 439 , 2102 beardsley , a. p. , hazelton , b. j. , sullivan , i. s. , et al . 2016 , , 833 , 102 bernardi , g. , zwart , j. t. l. , price , d. , et al . 2016 , , 461 , 2847 bloom , j. s. , & richards , j. w. 2012 , advances in machine learning and data mining for astronomy , 89 bromm , v. , coppi , p. s. , & larson , r. b. 2002 , , 564 , 23 bromm , v. 2013 , reports on progress in physics , 76 , 112901 furlanetto . p. s and briggs . f , phys . rept . * 433 * ( 2006 ) 181 [ astro - ph/0608032 ] . deboer , d. r. , parsons , a. r. , aguirre , j. e. , et al . 2016 , arxiv:1606.07473 deboer , d. r. , parsons , a. r. , aguirre , j. e. , et al . 2016 , arxiv:1606.07473 fialkov , a. , barkana , r. , visbal , e. , tseliakhovich , d. , & hirata , c. m. 2013 , , 432 , 2909 fialkov , a. , cohen , a. , barkana , r. , & silk , j. 2017 , , 464 , 3498 folkes , s. r. , lahav , o. , & maddox , s. j. 1996 , , 283 , 651 greig , b. , & mesinger , a. 2015 , , 449 , 4246 greig , b. , mesinger , a. , & koopmans , l. v. e. 2015 , arxiv:1509.03312 greig , b. , mesinger , a. , & pober , j. c. 2016 , , 455 , 4295 gnedin , n. y. , kravtsov , a. v. , & chen , h .- w . 2008 , , 672 , 765 - 775 hornik , k. , stinchcombe , m . , white , h. , _ neural networks _ vol.2 pp359 - 366 , 1989 jacobs , d. c. , pober , j. c. , parsons , a. r. , et al . 2015 , , 801 , 51 jensen , h. , zackrisson , e. , pelckmans , k. , et al . 2016 , , 827 , 5 kamdar , h. m. , turk , m. j. , & brunner , r. j. 2016 , , 455 , 642 kamdar , h. m. , turk , m. j. , & brunner , r. j. 2016 , , 457 , 1162 kim , k. , harry , i. w. , hodge , k. a. , et al . 2015 , classical and quantum gravity , 32 , 245002 mellema , g. , koopmans , l. v. e. , abdalla , f. a. , et al . 2013 , experimental astronomy , 36 , 235 mesinger . a and furlanetto . s , arxiv:0704.0946 [ astro - ph ] . . a , furlanetto . r,2011 , mnras , 411 , 955 mesinger , a. , ferrara , a. , & spiegel , d. s. 2013 , , 431 , 621 yoshida . n , omukai . k , hernquist . l , & t. abel 2006 , apj , 652 , 6 yoshiura.s , shimabukuro.h , takahashi.k , et al . 2015 , mon . not . 451 , 4785 zahn , o. , lidz , a. , mcquinn , m. , et al . 2007 , , 654 , 12 zahn , o. , mesinger , a. , mcquinn , m. , et al . 2011 , , 414 , 727
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the 21 cm signal from the epoch of reionization should be observed within the next decade .
while a simple statistical detection is expected with ska pathfinders , the ska will hopefully produce a full 3d mapping of the signal . to extract from the observed data constraints on the parameters describing the underlying astrophysical processes ,
inversion methods must be developed .
for example , the markov chain monte carlo method has been successfully applied . here
we test another possible inversion method : artificial neural networks ( ann ) .
we produce a training set which consists of 70 individual sample .
each sample is made of the 21 cm power spectrum at different redshifts produced with the 21cmfast code plus the value of three parameters used in the semi - numerical simulations that describe astrophysical processes . using this set we train the network to minimize the error between the parameter values it produces as an output and the true values .
we explore the impact of the architecture of the network on the quality of the training .
then we test the trained network on the new set of 54 test samples with different values of the parameters .
we find that the quality of the parameter reconstruction depends on the sensitivity of the power spectrum to the different parameters at a given redshift , that including thermal noise and sample variance decreases the quality of the reconstruction and that using the power spectrum at several redshifts as an input to the ann improves the quality of the reconstruction .
we conclude that anns are a viable inversion method whose main strength is that they require a sparse exploration of the parameter space and thus should be usable with full numerical simulations .
[ firstpage ] cosmology : theory intergalactic medium epoch of reionization 21 cm line
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merging the exquisite tunability of electronic nanostructures with ferromagnetic materials in nanospintronic devices bears great potential for applications and fundamental investigations . electronic devices using the electron spin in magnetic field sensing are very successful , for example in hard disks of computers . however , to use the electron spin directly , for example in a spin - transistor @xcite or as quantum bits @xcite , it is necessary to fabricate nanostructures with the required long coherence times and electrical tunability . carbon nanotubes ( cnts ) and graphene are in principle ideally suited for spintronic devices due to the large intrinsic coherence times , tunable electron density and large maximum current densities . early electrically tunable spin valves on carbon nanotubes @xcite , or nonlocal spin - accumulation experiments on graphene @xcite demonstrate the great potential of carbon based nanostructures . to obtain an electrically tunable spin signal , one strategy is to fabricate a nanostructure with a gate - tunable conductance , e.g. a quantum dot ( qd ) @xcite . however , this electrical tunability introduces additional complexity to the data analysis , since the signal now depends on the position , amplitude and broadening of a conductance feature , which all can vary with the magnetisations of the contacts , as will be discussed in the results sections of this paper . while in nonlocal measurements the spin signal can in principle be separated from the charge signal , this is difficult in two - terminal qd devices because both signals are detected at the same contacts . despite considerable efforts , spin injection and detection in qd spin valves are not yet reproducible enough for more complex experiments or applications , e.g. as detectors of electron spin entanglement @xcite . this lack of reproducibility can have several reasons . the most fundamental spin transport device is a spin valve with two ferromagnetic ( f ) contacts to a non - magnetic material in - between . in fig . 1a such a device is shown schematically with a carbon nanotube ( cnt ) between the f - contacts . ideally , the contacts are either magnetized parallel or anti - parallel to each other , adjustable by an external magnetic field . the normalized difference between the electrical conductance ( or resistance ) of these two configurations is called magnetoresistance ( mr ) . here we address two more technical problems . the first is limitations in the device design . compared to other carbon based nanoscale devices with normal metal @xcite or superconducting leads @xcite , the contact material has to be chosen from a very limited range of readily available and processable magnetic metals , which limits the optimization of the contacts . in addition , most ferromagnetic materials form oxides when exposed to air , which diminishes the electrical contact yield . to obtain low - ohmic contacts with non - magnetic materials , one often chooses large contact areas , which , however , is in conflict with using narrow contact geometries to control the shape anisotropy and thus the magnetic field at which the magnetisation is reversed ( switching field ) @xcite . even the thickness of the deposited material is limited to avoid the formation of vertical , more complex magnetic domains . no adhesion or contact layer can be used because the equilibrium spin polarization decays very rapidly in non - magnetic metals ( on the scale of the exchange interaction , typically @xmath0 nm ) . our choice of contact material is the well - studied ni@xmath1/fe@xmath2 alloy permalloy ( py ) , for which one can obtain single - domain contacts and control over the magnetic easy axis by the shape of the contacts @xcite . we demonstrate that the same electrical and magnetic characteristics for sub - micrometer scale py contacts can be obtained by sputter deposition and for thermal evaporation . this opens up the large field of magnetic multi - layer structures to be used in carbon based nano spintronic device fabrication , e.g. anti - ferromagnetic exchange - bias layers @xcite . the second technical problem we propose a solution to is resist residues and the unwanted formation of py nanoparticles near and on top of the device , which can strongly alter the device characteristics . in nanospintronics the interface area between the ferromagnetic contact and the non - magnetic structure , for example a cnt , is usually very small , which makes it very susceptible to resist residues . this not only compromises the spin and charge transport properties , but also the electrical stability of a device due to dielectric charge traps . here we report the fabrication of cnt spin valves by electron beam lithography using an essentially residue free ( on sio@xmath3 ) low - density polymer that allows the fabrication of optimal polymer mask cross sections without resorting to multiple resist layers . using this recipe we obtain electrical contacts with a significantly increased electrical stability and yield , for both , thermal and sputter deposition of py . we present measurements from two devices to discuss the need for an extended data analysis in nanospintronic devices . our cnts are grown by chemical vapor deposition at a temperature of @xmath4c using methane as source gas and fe / ru catalyst nano particles @xcite . the substrate is a heavily doped si wafer acting as a backgate and a @xmath5 nm thermal oxide top layer . as shown schematically in fig . 1a our approach to obtain reproducible magnetic domains and switching characteristics for the ferromagnetic contacts is to fabricate @xmath6 nm thick ferromagnetic permalloy ( py ) strips with a large aspect ratio ( @xmath7 ) @xcite . these strips are fabricated by electron beam lithography with an electron sensitive polymer resist , followed by metal deposition and a lift - off procedure . to deposit py we use two techniques : 1 ) thermal evaporation of py by an electron gun in a uhv chamber at a base pressure of @xmath8 mbar , sample cooling to @xmath9c and a deposition rate of @xmath10 / s . 2 ) dc sputter deposition using an ar plasma at the power of @xmath11w and an ar pressure of @xmath12 mbar in a uhv chamber with a base pressure of @xmath8 mbar . the fabrication of nanostructures by sputter deposition is often difficult because the sputtered material is scattered at gas particles in the chamber , which leads to a large angular spread that can fill the lithographically defined polymer trench and lead to lift - off problems . we obtain sufficient directionality for the sputter deposition of py by working with a relatively low ar pressure and no sample rotation . the sample resides directly above the py target at a distance of @xmath13 cm from the plasma at room temperature . we use a deposition rate of @xmath14 / s . we have systematically investigated the morphology and magnetic properties of py strips fabricated by electron beam lithography with different resist systems and beam acceleration voltages and identify two fundamental problems rather specific to nanospintronic devices : 1 ) py nanoparticles form at the side walls of polymer structures with insufficient under - cuts and are deposited nearby or on the py strips in the lift - off procedure , 2 ) resist residues lead to a significant decrease in the yield of obtaining low - ohmic electrical contacts to cnts ( with @xmath15m@xmath16 two - terminal resistance at room temperature ) . both problems are illustrated in scanning electron microscopy ( sem ) images in fig . subfigure ( i ) shows a tilted side view of a @xmath17 nm thick poly(methyl methacrylate ) ( pmma ) mask of a strip after lithography and thermal py deposition . the py strip forms at the bottom of the polymer trench . however , due to the large beam acceleration voltage of @xmath18kv used for this structure , the polymer trench is v - shaped with a thin metal film deposited also on the side walls , which often leads to a bad lift - off and large ferromagnetic residues . subfigure ( ii ) shows a top view of the resulting py strip . while the strip appears well defined , we reproducibly find a large number of py nanoparticles on top and around the strip , as indicated by the black arrow . such particles can be magnetic with very large characteristic fields , leading to seemingly non - symmetric low - field mr curves . the polymer profile can be improved significantly by either reducing the sem acceleration voltage , e.g. to @xmath19kv , or by using an additional more sensitive resist layer , which both lead to a larger under - cut . we have tested the copolymer system pmma / ma ( ma : methacrylic acid ) and pmma(950k)/pmma(50k ) with different pmma chain lengths . all three methods can be optimised to obtain better undercuts and a significantly reduced number of py particles on the surface . from this finding we conclude that the particles form at the side walls of v - shaped profiles , which is therefore an essentially geometric effect and independent of the polymer . the red ( bright ) arrow in subfigure ( ii ) of fig . 1b points out polymer residues , which we identify by the smaller sem contrast and the much shorter oxygen dry etching times required to remove them in test samples ( not shown ) . we reproducibly find polymer residues for trenches fabricated in the polymer systems pmma , pmma / ma @xcite and pmma(950k)/pmma(50k ) , with methyl isobutyl ketone ( mibk ) and ipa ( 1:3 ) as developer and lift - off in warm acetone . this is a well - known issue not specific to nanospintronic devices @xcite . in semiconductor device fabrication , or when contacting metallic parts of the device , the residues can be removed before the next metal deposition by standard cleaning procedures like oxygen plasma etching or ar sputtering . however , most of these procedures also remove or damage the cnt and graphene parts of a device . we note that also the post - deposition structuring of py films frequently used for the fabrication of nanometer scaled magnetic devices , e.g. ion milling or chemical etching , also remove carbon with a large rate . a close to ideal polymer mask cross section and negligible residues can be obtained using the copolymer resist zep 520a ( zep ) @xcite and @xmath20s development in n - amylacetate , stopped in a 9:1 solution of mibk and ipa , followed by rinsing in ipa . after the metalization , a good lift - off is achieved in a @xmath21min n - methyl-2-pyrrolidone ( nmp ) bath at @xmath22c , followed by @xmath18min in acetone at @xmath23c and rinsing in ipa . we use a @xmath24 nm thick zep layer , an electron acceleration voltage of @xmath25kv and a typical dose of @xmath26c/@xmath27 , for which we obtain undercuts with a narrow opening at the top of the polymer film , as demonstrated in fig . 1c ( i ) . this undercut can be tuned systematically by the dose and acceleration voltage . subfigure ( ii ) shows a resulting py strip obtained by thermal py evaporation . we find no metallic particles or metal flakes and could not detect any resist residues . we obtain similarly clean strips with a slightly increased surface roughness using the zep recipe and sputter deposition of py , as demonstrated in fig . 1d . we fabricate long ( @xmath28 m ) , thin ( @xmath6 nm ) py strips with a small width @xmath29 , which forces the magnetisation of the ferromagnetic contacts to lie along the strip axis . the direction can be inverted by an external magnetic field along the axis that switches the magnetisation to the opposite orientation at a characteristic switching field @xmath30 tunable by the width @xmath29 of the strip @xcite . to assess the magnetic properties and material quality of an individual py strip , it is contacted by pd contacts to measure the anisotropic magnetoresistance ( amr ) . an example curve is shown in the inset of fig . 2 , where the resistance @xmath31 of a @xmath32 nm wide strip of sputtered py is plotted as a function of the external magnetic field @xmath33 along the strip axis . sharp characteristic changes in the resistance at @xmath34mt indicate the reversal of the magnetisation @xcite . the smooth background variation we attribute to a small ( @xmath35 ) misalignment between the field and the strip axis , which mixes in the large continuous mr signals obtained when the field is applied perpendicular to the strip axis . figure 2 shows the switching fields @xmath36 as a function of @xmath29 for strips obtained by different fabrication techniques . we find that the sputtered and thermally evaporated contacts defined using zep exhibit the same dependence on @xmath29 as the pmma processed and thermally evaporated py contacts @xcite . the switching fields can be distinguished reliably for widths @xmath37 nm , for which @xmath36 increases strongly for smaller @xmath29 . while the amr curves of individual py strips are very reproducible , the resulting mr in a spin valve are more problematic , as will be discussed below . we note already here that amr experiments are sensitive to the bulk of the material , while in spin valve configurations the last few atomic layers are crucial . in this section we demonstrate the need of extended data acquisition and analysis for magnetoresistance devices with non - trivial conductance characteristics . the magnetoresistance ( mr ) of a spin valve device is defined in terms of its conductances @xmath38 and @xmath39 when the magnetisations in the two contact strips are either parallel ( p ) or anti - parallel ( ap ) . similar expressions are easily obtained using the device resistances . here we define @xmath40 which is more symmetric than the usual definitions and leads to smaller mr values of maximally @xmath41% . this definition is more adequate for our purpose because it provides an equal measure for positive and negative mr . in a qd spin valve , the conductance depends on the gate voltage , which tunes the qd level energies . the charging energy and level separation lead to characteristic coulomb blockade ( cb ) conductance maxima , with a strongly reduced conductance in between . in the color scale image in fig . 3 the qd conductance is plotted as a function of the backgate voltage @xmath42 and an increasing magnetic field @xmath33 ( up - sweep ) for a qd fabricated with standard pmma - based lithography . the base temperature in all experiments presented here is @xmath43mk . the qd conductance has a maximum at @xmath44v and decays rapidly away from this value . a cross section at constant magnetic field is plotted in white . when the magnetic field is increased from negative values beyond @xmath45 , a first sharp change ( @xmath46mt ) in the conductance pattern occurs at @xmath47mt , and another at @xmath48mt . these fields correspond well to the contact switching fields of the two py strips . at @xmath49 the amplitude of the cb resonance increases by a factor of @xmath50 and the peak position shifts by about @xmath51mv , which corresponds to an energy shift of @xmath52ev or to almost the resonance width . while the amplitude of the cb resonance increases by almost a factor of 2 at @xmath49 starting at the low field side , it does not change at @xmath53 and is reduced slightly only at higher fields . at @xmath53 the resonance position switches back roughly to the same gate position as for @xmath54 . in a standard magnetoresistance ( mr ) measurement the conductance is recorded as a function of @xmath33 alone , which corresponds to cross sections in fig . 3 at a fixed gate voltage . two examples for slightly off - resonance voltages are shown on top and below the main figure : at a more negative gate voltage ( green dashed line , 2 ) we find a decrease in conductance for the anti - parallel magnetisations , @xmath55 , which corresponds to an increased resistance and a positive mr of @xmath56% . off - resonance for a more positive gate voltage ( blue dashed line , 1 ) the mr at fixed voltage is negative , @xmath57% . these large values are almost exclusively due to the large shift of the resonance position . in the simplest model by jullire for tunneling mr @xcite one would expect @xmath58% when using @xmath59 for the tunneling polarizations in the two f - contacts @xcite . these values rather correspond to the amplitude modulation ( @xmath60% ) than to the mr observed in cross sections . we will discuss shifts of the conductance features in the mr in more detail in the next section and only point out that while the mr at @xmath49 and @xmath53 might be described by a simple spin valve model , the increase of @xmath61 for @xmath62 with respect to @xmath63 is more difficult to explain since it suggests a difference between the two parallel configurations , a phenomenon possibly related to the single switching behavior reported before @xcite . the electrical stability and reproducibility of the qd spin valve signals is considerably improved for devices fabricated using the zep recipe introduced above . we analyze in more detail the data shown in fig . 4 measured on a sample with sputtered py contacts . in fig . 4a the qd spin valve conductance @xmath61 is plotted for a large backgate voltage interval at a base temperature of @xmath64mk . the cb peaks occur in groups of four consistent with the spin and valley degeneracy of a cnt orbital . such a pattern suggests that the cnt segment forming the qd is relatively clean @xcite . from charge stability diagrams ( not shown ) we find the lever arm of the backgate to the qd @xmath65 , a charging energy of @xmath66mev and a level spacing of @xmath67mev . we estimate the source , drain and backgate capacitances as @xmath68af , @xmath69af and @xmath70af . from the cb maxima of @xmath71 , the average broadening of the peaks @xmath72mev and using the breit - wigner form for resonant tunneling at low temperatures ( @xmath73 ) , we find for the tunnel couplings of the qd to source and drain @xmath74mev and @xmath75mev , which gives a relatively small coupling asymmetry of @xmath76 ( we chose the larger value as @xmath77 ) . we now focus on the four cb peaks highlighted in fig . 4a by the red rectangle , which originate from the same four - fold degenerate qd orbital . in fig . 4b and 4c the qd conductance @xmath61 is plotted for this gate voltage interval and as a function of an external magnetic field @xmath33 along the py contact strips . in fig . 4b the field is increased from negative values , while in 4c it is decreased , starting from positive values . the magnetisations were saturated at @xmath78mt before the respective sweep . in the up sweep in fig . 4b , @xmath61 is larger for @xmath79mt , which usually is identified as the anti - parallel configuration of the contact magnetisations . in the down sweep the magnetisation switching occurs at negative fields and we find an increased conductance for @xmath80mt . the variation between the absolute values of the switching fields in the up and down sweeps are compatible with the variation observed in the corresponding amr curves . in fig . 4d shows the up and down sweeps at a fixed backgate voltage , indicated by the dashed lines in figs . 4b and 4c . we find a sharp switching of the conductance at the py strip switching fields , which corresponds to a mr of @xmath81% . the mr is negative for all gate voltages , which we now discuss in more detail . in devices with a variable conductance , e.g. as a function of the backgate voltage , the origin of the mr signal can lie in changes of the width , position and amplitude of the conductance feature . in fig . 5a we plot the cb oscillations indicated in fig . 4a as a function of @xmath42 for the different magnetisation configurations . the two parallel configurations lead to identical conductances , which demonstrates the reproducibility of both , the magnetic and electronic structures in the device . the anti - parallel configuration , however , deviates significantly from the parallel . the resulting mr vs @xmath42 curve is plotted in fig . 5b ( full red line ) . the mr is negative for almost all backgate voltages and shows a mr modulation of @xmath82% on an offset of about @xmath83% . the modulation is correlated with the gradient @xmath84 of @xmath61 , i.e. it is largest at the gate voltages where @xmath61 has the largest slopes , which suggests that the mr is caused mainly by a shift of the cb resonances . in the next step we fit the data with multiple lorenzians to extract the amplitude , width and position of the individual cb peaks ( no background is subtracted ) . the resulting parameters are plotted in figs . 5c - e for the third cb peak highlighted by an asterisk in fig . compared to the parallel magnetisation configurations , the anti - parallel shows an increase in amplitude and width by @xmath85% and @xmath86% , respectively , and a shift of @xmath87mv , which corresponds to @xmath88ev or @xmath89% of the peak width . we obtain similar values for the other cb peaks . all peaks are shifted by the same absolute value within experimental errors . the extracted parameters allow us to investigate the respective impact on the mr , for example by calculating the mr from the measured curve for the parallel magnetisations and a shifted curve in the anti - parallel case . the result is plotted in fig . 5b as blue dashed line ( cor . mr ) and has mr maxima at gate voltages where also @xmath61 has maxima , as expected if the shifts were corrected precisely enough . the mr variation on this curve is only @xmath90% with a slightly smaller negative offset than in the original data . we generally find a better electrical stability and larger contact yield for qds fabricated with the presented recipes . we note , however , that also with this method we obtain samples that show gate - dependent or temporal charge rearrangements , which we tentatively attribute to surface impurities on the substrate . nevertheless , these methods lead to devices with reproducible mr with a yield of @xmath91% , a clear improvement compared to previous methods that yielded useful devices only rarely ( @xmath92% ) . a periodic modulation of the mr with the cb oscillations was observed already earlier and modeled by spin dependent effective tunnel rates @xcite . in this simple model one can construct negative mr signals for a strongly asymmetric qd coupling to the contacts , which can lead to a negative offset for strongly overlapping resonances . the change of the effective tunnel couplings at the switching fields could in principle also result in a change of the resonance widths . however , this model requires strongly asymmetric tunnel couplings and predicts that the mr maxima occur near the conductance minima , both in contrast to our observations . this model does not produce shifts in the cb resonance energies , either . more elaborate models @xcite predict major contributions to the mr from shifts of the cb resonances in an effective magnetic field , caused either by spin dependent electron scattering at the qd - contact interfaces or by a spin - dependent renormalization of the qd energy levels . characteristic for both mechanisms is that the sign of the shifts depends on the spin state of the cb resonance . specifically , of the four states in a cnt orbital , two should be shifted in energy opposite to the other two . none of the models predict identical shifts for all four peaks , nor a negative offset of the mr . another mechanism that results in a constant shift in the anti - parallel configuration is the magneto - coulomb effect ( mce ) @xcite . the opposite zeemann shifts and the different density of states at the fermi energy of the majority and minority bands leads to a rearrangement of electrons between the bands , which is compensated by a change in the electrical potential . we estimate an mce shift of the qd resonances in an external magnetic field @xmath33 of @xmath93v / t . in the last step we used @xmath94 as an upper limit of the ( thermodynamic ) py polarization in both leads , the land @xmath95-factor in thick ( @xmath96 nm ) py films of @xmath97 @xcite and the bohr magneton @xmath98 . with the same parameters one obtains a total change in position of @xmath99v when sweeping the field beyond both switching fields . the negligible slope observed for the peak positions is consistent with the small value obtained in these estimates . however , the same parameters also predict a negligible change at the switching fields . we note that also the qualitative curve shape observed in the experiments does not follow the triangular characteristics at the switching fields of the mce @xcite . a quite natural explanation of our experimental findings is that one py strip does not couple directly to the qd , but rather to an anti - ferromagnetically coupled contact area . such effects could occur at the chemical bonds between the metal / cnt interface @xcite , or in oxidized layers of the magnetic material that are strongly coupled to the bulk . the latter coupling could depend on the thickness of the oxide and explain why both contacts are not coupled identically to the cnt , a phenomenon well known from non - magnetic metal contacts to cnts . this scenario explains the sign reversal of the mr gate modulation and offset , but not the peak shifts at the switching fields . a physically more intriguing scenario would be a spatially varying spin susceptibility ( rkky interaction ) in the cnt segments connecting the qd @xcite , which could lead to a rotation of the injected spins depending on the distance from the ferromagnetic contact - an effect that can persist over large length scales due to the low dimension of the cnt and the strong electron - electron interactions . we report a recipe for essentially residue free electron beam lithography , useful to improve the fabrication of carbon based nanospintronic devices . we obtain very reproducible magnetic and electrical contact properties for sputtered and thermally deposited permalloy when both processes are optimised individually . using these recipes , we obtain an improved yield of electrical contacts and more reproducible features in the mr of carbon nanotube quantum dot spin valves . since the mr in nanospintronic devices not only depends on the magnetic orientation of the contacts , but also on the electrostatic environment ( e.g. gates ) , it is necessary to expand the standard magnetic field sweeps to three - dimensional maps that also contain a variable gate voltage to track the origin of the observed mr . we demonstrate this idea with two devices and show that the major contribution to the mr can be due to a shift in the conductance features . from the discussion of several mechanisms specific to nanospintronic devices we tentatively conclude that interface properties might be crucial to explain the presented magnetoresistance characteristics . we gratefully acknowledge useful discussions with b. hickey and m. elkin . this work is financially supported by the swiss national science foundation ( snf ) , the swiss nccr qsit and nccr nano , the erc project quest and the fp7 project se2nd .
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we report an improved fabrication scheme for carbon based nanospintronic devices and demonstrate the necessity for a careful data analysis to investigate the fundamental physical mechanisms leading to magnetoresistance .
the processing with a low - density polymer and an optimised recipe allows us to improve the electrical , magnetic and structural quality of ferromagnetic permalloy contacts on lateral carbon nanotube ( cnt ) quantum dot spin valve devices , with comparable results for thermal and sputter deposition of the material .
we show that spintronic nanostructures require an extended data analysis , since the magnetisation can affect all characteristic parameters of the conductance features and lead to seemingly anomalous spin transport .
in addition , we report measurements on cnt quantum dot spin valves that seem not to be compatible with the orthodox theories for spin transport in such structures .
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among the extremely remarkable properties of the fractional quantum hall ( fqh ) effect @xcite a major role is played by the emergence of anyonic excitations carrying fractional charge and statistics @xcite . in particular , quasiparticle ( qp ) excitations for states belonging to the laughlin @xcite and jain sequence @xcite are predicted to have abelian exchange statistics . more intriguingly , some of the proposed models for the filling factor @xmath0 @xcite predict the emergence of excitations with charge @xmath1 , and multiples , with possible non - abelian properties @xcite . these predictions paved the way to possible applications of non - abelian anyons in fault - tolerant topological quantum computation ( see @xcite and references therein ) . unfortunately , as far as we know , a direct confirmation of fractional statistics is still lacking even if different proposal are reported in the literature @xcite . so far , evidence of the fractional statistics is indirect , essentially based on the evidence of the existence of fractional charges @xcite . a great experimental effort has been devoted in the last years to access fractional charges through shot noise measurements in quantum point contact ( qpc ) geometry starting with the seminal works of refs . @xcite . in this direction , composite states ( for example @xmath2 ) showed a quite universal phenomenology leading to a crossover between two different value of the effective charge as a function of temperature or bias @xcite . this behaviour has been explained in terms of competition of two charge carriers : the agglomerates and the single - qp @xcite . similar arguments hold as well for the state at @xmath0 @xcite . other interpretations based on edge state reconstruction @xcite , local filling factor effects @xcite or tunnelling amplitude non - linearities @xcite have also been proposed . unfortunately , in the discussed measurements , the contributions associated to the various excitations are typically mixed because these studies are conducted at very low frequencies ( almost dc ) . therefore , it is useful to find alternative methods to address the excitations separately . a possible way is to consider the noise at finite frequency ( f.f . ) indeed , this quantity presents resonant singular behaviour ( such as peaks or dips ) in correspondence of the josephson frequency @xmath3 associated to each charge carrier @xmath4 with @xmath5 the applied bias . this is an independent method to measure the charge of the fractional excitations in the system that has not yet be experimentally explored for fqh states so far . indeed , for sufficiently low temperatures @xmath6 , it allows to separate the different charge contributions realising a sort of qp spectroscopy . intriguingly enough , f.f . noise could efficiently address also other properties , like the scaling dimensions associated to each qp excitation using only a bias scan at fixed frequency as we will show . this combination of information has the potential to give further constraints on the edge state model @xcite , and finally to address the topological order of the bulk ground state @xcite . first theoretical steps in this direction were done on symmetrised noise @xcite , typical quantity considered at low frequencies . in such case the expected features associated to the state at @xmath0 @xcite as well as the possibility to access the contribution associated to the different qps predicted by theoretical models @xcite have been investigated . however , at high frequencies @xmath7 , quantum effects become relevant and the symmetrised f.f . noise is only one possible choice among different experimental quantities addressed by different protocols . indeed , in such regime , one has to identify the relevant quantity measured with the specific setup under investigation @xcite . hereafter we get inspiration by the proposal of lesovik and loosen @xcite where a model based on a resonant lc circuit is discussed in order to extract non - symmetrised current - current correlations . recent experiments carried out for a two dimensional electron gas qpc in absence of magnetic field fully agree with theoretical predictions @xcite . since the resonant circuit coupled to a qpc is the prototypical measurement scheme in fqh experiments @xcite it appears quite obvious to explore the same physics in this contest . we have recently investigated the same setup for an abelian fqh states @xcite , and here we consider it to the case @xmath0 analysing the signatures of the different non - abelian phases ( pfaffian and anti - pfaffian ) . the goal of the present paper is to analyse in details the expected f.f . measured noise for a realistic situation . the effects associated to the temperature of the fqh qpc system and the lc detector on the visibility are also carefully considered . the paper is divided as follows : in section 2 we discuss the non - abelian models for edge states pointing out similarities and differences in term of the most dominant fractionally charged excitations . in section 3 we discuss the definition of the noise properties for a qpc in the weak - backscattering regime and the definition of the noise power measured in the proposed setup . in section 4 we discuss the result for the measured noise power obtained for two considered non - abelian models and we also compare them with the well know symmetrised noise . finally we inspect the effect of changing the detector temperature @xmath8 . we start recalling the two more accredited models of composite edge states at filling factor @xmath0 @xcite : the pfaffian ( p ) @xcite and the disorder dominated anti - pfaffian ( ap ) @xcite . the associated lagrangian densities are given by the sum of charged ( @xmath9 ) and neutral contributions(@xmath10 ) , namely @xmath11 , with ( @xmath12 ) _ c=- _ x _ c(_t_c+v_c _ x _ c ) [ lagrangian_charged ] and _ n =- i(_t + v_n _ x)- _ x _ n ( _ t _ n + v_n _ x _ ) . [ lagrangian_neutral ] they both describe an hall fluid at filling factor @xmath13 with two additional filled landau levels , playing the role of the _ vacuum _ of the theory , according to the conventional decomposition @xmath14 . the charged bosonic field @xmath15 is related to the electron number density through @xmath16 , while @xmath17 is a bosonic neutral field and @xmath18 represents a neutral majorana fermion in the ising sector @xcite . the parameter @xmath19 denotes the direction of propagation of neutral modes with respect to the charged ones . in particular , for @xmath20 the modes are co - propagating , while for @xmath21 they are counter - propagating . the two models differ in the neutral sector @xmath10 @xcite , with @xmath22 and @xmath23 for p , @xmath24 and @xmath21 for ap . the propagation velocities of the charged and neutral modes are indicated with @xmath25 and @xmath26 respectively . due to the hidden symmetry of the neutral sector of ap model @xcite in the disorder dominated phase , one has the same velocity @xmath26 both for the bosonic @xmath27 and the fermionic neutral modes @xmath28 . moreover one may reasonably assume a larger charge velocity @xmath29 @xcite . the quantization of the above bosonic fields is given by the commutation relation = i_c / n sgn(x - y ) , with @xmath30 and @xmath31 , while the majorana fermion commutes with both . operators destroying an excitation along the edge can be written as @xcite @xmath32}\ , , \label{psi}\end{aligned}\ ] ] with @xmath33 integer numbers and where the operator in the ising sector @xmath34 can be the identity operator @xmath35 , the majorana fermion @xmath18 or the spin operator @xmath36 . they are associated to the fact that the excitation charge is even or odd multiple of the fundamental charge of the model @xmath37 . indeed all the excitations described by previous operators have charge @xmath38 and we call them @xmath39-agglomerates @xcite . the single - valuedness properties of the phase acquired by an @xmath39-agglomerate with respect to the operation of encircling an electron in the bulk , force @xmath39 and @xmath40 to be : even integers for @xmath41 or @xmath28 , and odd integers for @xmath42 @xcite . notice that the presence of @xmath43 in the operator leads to non - abelian statistical properties important for fault - tolerant quantum computation as determined by the fusion rules @xcite . the zero temperature time dependent green s functions associated to the operators of the ising sector and the charged and neutral bosonic fields are @xcite ( 0,t ) ( 0,0 ) & = ( 1+i _ t)^-_,&[gfising ] + _ s(0,t ) _ s(0,0 ) & = -|_s| & s= , [ gfbosonic ] with @xmath44 , @xmath45 and @xmath46 the conformal weights of the field in the ising sector @xcite and we introduced the energy bandwidths @xmath47 , with @xmath48 a finite length cut - off . in the following we will assume @xmath49 as the largest energy scale of the model . from the long - time behaviour of the imaginary time two - point green s function and in the scaling @xmath50 . from now on indeed we will consider for any @xmath39-agglomerate only the qp operators with the minimal scaling dimension compatible with the single - valuedness requirement . ] t_^(m)_l ( ) _ l^(m)^(0)||^-2^(m)_ll= we can extract the scaling dimensions @xcite of the @xmath39-agglomerates _ p^(m)=_+ m^2;_ap^(m)=_+ m^2+n^2 [ delta ] which depends on the model considered . therefore , for the single - qp with minimal charge @xmath1 ( @xmath51 , @xmath42 and only for ap @xmath52 ) one has respectively _ p^(1)=;_ap^(1)= , [ 1_8 ] while the @xmath53-agglomerate excitation with charge @xmath54 ( @xmath55 , @xmath56 , @xmath41 ) , with a scaling dimension driven by the charged mode contribution only with _ p^(2)=_ap^(2)=. [ deltaagg ] these values indicate the single - qp as the most dominant excitation at low energy in the p case , while in the ap case single - qp and 2-agglomerate have equal relevance with the same scaling dimensions @xcite . the latter situation is quite general and valid for all anti - read - rezayi states @xcite . all other excitations , with higher charges , have higher scaling dimensions and can be safely neglected in what follows . it is worth to mention that interactions with the external environment can lead to renormalizations of the scaling parameters with remarkable consequences on the transport properties ( see ref . @xcite for a better discussion ) . in the following , for sake of simplicity , we will focus on the unrenormalised case only despite the method may be generalised to the renormalised case . once characterised the excitations of the considered models for @xmath0 , we can investigate the associated f.f . backscattering noise in the qpc geometry shown in fig . a similar measurement scheme was proposed for the first time by lesovik and loosen in ref . @xcite . here , the qpc is subjected to a bias voltage v and coupled to a resonant lc circuit , playing the role of the detector ( with measurement frequency @xmath57 ) , via an impedance matching circuit ( see dashed box in fig . [ fig1 ] ) . with strong magnetic field the impedance matching in the system is a challenging technological problem @xcite therefore it is advantageous to suppose to work at fixed resonant frequency @xmath58 assuming a very high quality factor of the detector . is applied to the qpc and @xmath39-agglomerate excitations can tunnel between the edges . the two circuits are matched in impedance via a coupling circuit ( inside dashed line ) . here we assume they are kept at two different temperatures @xmath59 and @xmath8 as indicated in the picture . in the figure we explicitly show also with grey boxes a low - pass ( lp ) and high - pass ( hp ) filters which make possible to directly couple the dc bias to qpc and , at the same time , to deviate the high - frequency components toward the lc circuit . ] we focus on the simple two terminal geometry in the weak backscattering limit , where @xmath39-agglomerate tunnelling processes can be treated separately . in real systems a four terminal version of this setup is required , however this does nt change the main result obtained with this simplified version . the point - like tunnelling of a generic @xmath39-agglomerate between the right - propagating ( @xmath60 ) and the left - propagating ( @xmath61 ) edge can be described through the tunnelling hamiltonian ( @xmath62 ) [ tun ] h^(m)_t , l= t_m _ l , + ^(m)(0)^(m)_l , -^(0)+h.c . , where @xmath63 is the @xmath39-agglomerate tunnelling amplitude ( assumed energy independent ) . the finite bias @xmath5 between the two edges can be included in our formalism through the gauge transformation @xmath64 , where , @xmath65 is the josephson frequency associated to the fundamental charge @xmath66 @xcite . from the tunnelling hamiltonian in ( [ tun ] ) one can easily derive the backscattering current operator associated to the @xmath39-agglomerate @xcite i^(m)_b , l(t)=i me^ * ( t_m e^im_0t_l,+^(m)(0,t)^(m)_l,-^(0,t)-h.c . ) . the contribution to the averaged backscattering current of the @xmath39-agglomerate at lowest order in the tunnelling can be easily written in terms of the tunnelling rates @xmath67 @xcite i^(m)_b , l = m e^*(1- e^-m_0/t)^(m)_l ( _ 0 ) , [ current_bal ] with the average @xmath68 taken over the quantum statistical ensemble . the rates can be also evaluated analytically at low temperatures @xmath69 @xcite ^(m)_l ( _ 0 ) & = & t^_m+_l , -1e^ + & & b(_m+_l , -i ; _ m+_l , + i ) [ eq : gamma ] where @xmath70 is the euler beta function . here , @xmath71 and @xmath72 depend on the variables @xmath73 and @xmath40 which characterise the tunnelling excitation for the specific p or ap model considered ( see discussion around ( [ 1_8 ] ) and ( [ deltaagg ] ) ) . note that for @xmath74 asymptotic expansion shows that @xmath75 , with the expected power law dependences of the rates from the bias energy @xmath76 and the scaling dimension @xmath50 as usually happen in the luttinger liquid theory . for higher bias value @xmath77 the power - law does not depend anymore on the neutral components and one finds @xmath78 where the power - law scaling is determined only by the charge of the @xmath39-agglomerates . the low - energy analytical result presented corresponds to the standard golden rule rate for the tunnelling processes and one may eventually calculate it also with numerical methods following the prescription of ref . @xcite . the proper quantity to consider in order to investigate the current fluctuations of the qpc coupled to the resonant circuit is the non - symmetrised noise @xcite s^(m)_+()=^+_- dt e^i t i^(m)_b , l(0 ) i^(m)_b , l(t ) , where we have introduced the back - scattering current fluctuation @xmath79 . on the noise power since its definition is exactly the same for the two models . ] this quantity represents , for @xmath80 , the noise power emitted by the system into the detector . the corresponding absorptive part is given by s^(m)_-()=^+_- dt e^i t i^(m)_b , l(t ) i^(m)_b , l(0 ) = s_+^(m)(- ) [ s_minus ] . with these quantities it is easy to calculate the f.f . symmetrised noise @xcite usually considered in literature [ eq : ssym ] s^(m)_sym()&= & ^+_- d t e^i t \ { i^(m)_b , l(t ) , i^(m)_b , l(0)}= s^(m)_+()+s^(m)_- ( ) [ eq : s_sym ] having indicated with @xmath81 the anticommutator . at lowest order in the tunnelling amplitudes and using standard keldysh formalism also the non - symmetrised noise can be expressed in terms of the qpc tunnelling rates @xcite s^(m)_+(,_0)= , [ s_plus_rate ] with a peculiar combination of the frequency @xmath58 and the bias voltage @xmath76 in the arguments of the golden rule rates . the detector of fig . [ fig1 ] represents a concrete measurement scheme to access current fluctuations at high frequencies . in the following we will focus on the regime where the qpc temperature @xmath59 is lower than the frequency ( quantum limit ) and the bias ( shot noise limit ) , i.e. @xmath82 . this allow to investigate the fractional qp contributions via a sort of spectroscopy . the measurable quantity is the spectral power measured in the amplifier chain ( grey area in fig . [ fig1 ] ) , which is proportional to the variation of the energy stored in the lc before and after the switching on of the lc - qpc coupling . from now on we will indicate it as measured noise @xmath83 where with @xmath58 we indicate the frequency of the lc circuit and with @xmath76 the qpc bias . at lowest order in the coupling @xmath84 it can be expressed as @xcite s^(m)_meas(,_0)= k \{s^(m)_+(,_0)+n_b ( ) } , [ s_meas ] where the non - symmetrised noise qpc spectra for the @xmath39-agglomerate are @xmath85 of ( [ s_plus_rate ] ) . here , @xmath86 $ ] the bose distribution describing the equilibrium state of the lc detector and @xmath87 the detector inverse temperature . in general @xmath8 can be different from the system temperature @xmath59 since system and detector are _ weakly _ coupled . we wish finally recall that this quantity can be also investigated using a strategy similar to the definition of the excess noise which further simplify the impedance matching problem at the level of the lc - qpc coupling ( see ref . @xcite for details ) . to consider all contributions due to tunnelling of different @xmath39-agglomerate , in weak tunnelling , one can directly sum them ( @xmath88 ) s_i(_0)= _ m s^(m)_i(_0 ) , [ noise_sum ] where from now on we suppress the explicit dependence on the lc frequency @xmath58 since in the following discussion is always kept fixed . we conclude this part noting that these results suggest that f.f . noise is a spectroscopy tool for different tunnelling charges . ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model at @xmath0 as a function of the bias @xmath90 and at fixed frequency @xmath58 . the qpc temperature associated to each curve is indicated in the legend . ( bottom panels ) corresponding derivatives @xmath91 . the curve at the lowest temperature ( @xmath92 mk ) has been omitted for better visibility . other parameters expressed in temperature scale are : @xmath93 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model at @xmath0 as a function of the bias @xmath90 and at fixed frequency @xmath58 . the qpc temperature associated to each curve is indicated in the legend . ( bottom panels ) corresponding derivatives @xmath91 . the curve at the lowest temperature ( @xmath92 mk ) has been omitted for better visibility . other parameters expressed in temperature scale are : @xmath93 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] + ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model at @xmath0 as a function of the bias @xmath90 and at fixed frequency @xmath58 . the qpc temperature associated to each curve is indicated in the legend . ( bottom panels ) corresponding derivatives @xmath91 . the curve at the lowest temperature ( @xmath92 mk ) has been omitted for better visibility . other parameters expressed in temperature scale are : @xmath93 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model at @xmath0 as a function of the bias @xmath90 and at fixed frequency @xmath58 . the qpc temperature associated to each curve is indicated in the legend . ( bottom panels ) corresponding derivatives @xmath91 . the curve at the lowest temperature ( @xmath92 mk ) has been omitted for better visibility . other parameters expressed in temperature scale are : @xmath93 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] the results concerning the measured f.f . noise @xmath98 in ( [ s_meas ] ) and ( [ noise_sum ] ) at @xmath0 are shown in fig . [ fig2 ] ( upper panels ) as a function of the qpc bias @xmath76 for different qpc temperatures @xmath59 , keeping fixed the resonant circuit temperature @xmath8 . we discuss only the behaviour at positive bias @xmath99 since the noise is a symmetric function of the qpc bias @xmath76 . analogies and differences between the p ( upper left panel ) and ap ( upper right panel ) models become evident from a direct comparison . starting from the lowest temperature case ( @xmath92 mk , black curves ) we observe that both models show a flat behaviour at @xmath100 , which is a clear signature of the lack of contribution of ground states fluctuations in the considered measurement scheme @xcite . the little deviation from zero are associated to the mismatch between the systems and detector temperature which can be always cancelled when @xmath101 @xcite . steep jumps associated to the @xmath53-agglomerate contribution appear at @xmath102 showing an identical profile in both models , which reflects the same scaling dimension of the two model for that excitation ( see ( [ deltaagg ] ) ) . different is the spike associated with the single - qp occuring at @xmath103 . indeed , they are much more high and sharp in the p case with respect to the ap reflecting a lower scaling dimension of qp excitation for the p model . this feature could quite clearly distinguish between the two models . however the temperature should be kept quite low , since increasing it the differences are progressively less marked . eventually some signature survive only by considering the bias derivative of this quantity ( see bottom panels of fig . [ fig2 ] ) . the previous behaviours can be explained in a simple way in the quantum limit ( @xmath104 ) for the detector and the shot - noise limit ( @xmath105 ) for the system . in this case one has the contributions of single and double excitations for the two models ( @xmath62 ) @xcite [ shift ] s_meas(_0)_1 * * _ l^(1)(_0-)+_2 * * _ l^(2)(2_0- ) with @xmath106 and @xmath107 constant prefactors and the explicit expression of the rates are reported in ( [ eq : gamma ] ) for @xmath108 . this result confirms the same scaling for the @xmath53-agglomerate in the two models , but different behaviours for the single - qp contributions ( see ( [ 1_8 ] ) and ( [ deltaagg ] ) ) . as we observed after ( [ eq : gamma ] ) for high biases @xmath109 the scaling dimensions are determined only by the charged part that is the same for the two models . this is another reason why conventional scaling analysis , which is typically done in asymptotic regime , would fails in detecting the differences between the two models especially when the neutral mode bandwidth @xmath110 is quite small . ] . by increasing the system temperature @xmath59 , keeping fixed the one of the detector ( @xmath93 mk ) , the peaked structure become progressively smoothened due to a rounding of the singularities . in this regime , the differences in the measured f.f . noise become clear only by looking the derivative with respect to the qpc bias @xmath91 as shown in the bottom panels of fig . [ fig2 ] . by focusing on the blue curves , and keeping in mind the different scale in the ordinates between the two panels , one can observe again the similarity of peaks associated to the @xmath53-agglomerate ( @xmath102 ) . however concerning the single - qp ( @xmath103 ) , the difference in the scaling leads to a pronounced peak followed by a stronger dip in the p case with respect to the ap case . ( dashed lines ) and the symmetrised noise and @xmath111 ( solid lines ) for different values of the ratio @xmath112 ( indicated in the legend ) . ( left panel ) the f.f . noises @xmath113 with @xmath114 in units of @xmath89 . note that @xmath115 for the symmetrised noise . ( right panel ) the bias derivatives @xmath116 in units @xmath117 . other parameters are : @xmath118 mk , @xmath119 mk , @xmath95 mk and @xmath96 k. , title="fig : " ] ( dashed lines ) and the symmetrised noise and @xmath111 ( solid lines ) for different values of the ratio @xmath112 ( indicated in the legend ) . ( left panel ) the f.f . noises @xmath113 with @xmath114 in units of @xmath89 . note that @xmath115 for the symmetrised noise . ( right panel ) the bias derivatives @xmath116 in units @xmath117 . other parameters are : @xmath118 mk , @xmath119 mk , @xmath95 mk and @xmath96 k. , title="fig : " ] until now all the plots were done for fixed ratio @xmath120 . however this parameter is unknown and may change for any specific experimental realization . for this reason we present in fig . [ fig3 ] with dashed lines @xmath98 ( right panel ) and its bias derivative @xmath121 ( left panel ) as a function of bias with changing @xmath122 values . we concentrate mainly on the ap model but similar considerations can be repeated in the p case . as expected , increasing this parameter progressively enhances the @xmath53-agglomerate contribution with respect to the single - qp but still leave both visible at different bias values . this is particularly true looking the bias derivative . this result shows the convenience of proposed setup in order to address the presence of the two different charged excitations also when eventually one of the contribution is deeply suppressed in comparison of the other due to non - universal effects . ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model for the hall state at @xmath0 as a function of the voltage and at fixed frequency ( @xmath123 ) varying the detector temperature ( see legend ) . ( bottom panels ) corresponding derivatives @xmath91 . other parameters are ( in temperature units where necessary ) : @xmath118 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model for the hall state at @xmath0 as a function of the voltage and at fixed frequency ( @xmath123 ) varying the detector temperature ( see legend ) . ( bottom panels ) corresponding derivatives @xmath91 . other parameters are ( in temperature units where necessary ) : @xmath118 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] + ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model for the hall state at @xmath0 as a function of the voltage and at fixed frequency ( @xmath123 ) varying the detector temperature ( see legend ) . ( bottom panels ) corresponding derivatives @xmath91 . other parameters are ( in temperature units where necessary ) : @xmath118 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] ( in units of @xmath89 ) for the p ( left ) and the ap ( right ) model for the hall state at @xmath0 as a function of the voltage and at fixed frequency ( @xmath123 ) varying the detector temperature ( see legend ) . ( bottom panels ) corresponding derivatives @xmath91 . other parameters are ( in temperature units where necessary ) : @xmath118 mk , @xmath94 mk , @xmath95 mk , @xmath96 k and @xmath97.,title="fig : " ] in order to make this statement more quantitative , in fig . [ fig3 ] , we compare @xmath98 with the f.f . symmetrised noise @xmath124 ( solid lines ) in ( [ eq : ssym ] ) and their bias derivatives ( in right panel ) @xcite . here , again , we keep fixed the frequency @xmath58 , changing the bias @xmath76 that is , by far , the most convenient protocol at the investigated ghz range . we see that the @xmath124 , as a function of the bias , is unable to detect the two singularities associated to the two fractional charges even in the bias derivatives . in particular , changing the tunnelling amplitude ratio @xmath122 , the quantity seems only affected with a common multiplicative factor demonstrating that the signature of the two excitations is mainly mixed in that quantity . this supports the idea that in this bias dependent protocol @xmath124 is not useful especially in comparison to @xmath98 . this statement can be easily verified by looking at both the left and right panel of fig . [ fig3 ] . finally , in order to find a signature of the charged excitations without identifying their scaling dimension , we could vary the lc detector temperature @xmath8 in order to increase the sensibility for charge detection @xcite . in fig . [ fig4 ] we show that this approach works for both the two non - abelian edge models . increasing @xmath8 increases the height of the jump in @xmath98 ( top panels ) which correspond also to an increase of the height of the peak in the derivatives ( bottom ) . note that since the coupling with the detector is assumed weak ( no poisoning from the detector ) the width of the peaks is only slightly influenced by the detector temperature , preserving the resolving power of the discussed bias spectroscopy . then the crucial limiting factor to the bias spectroscopy is the qpc temperature @xmath59 . as shown in fig . [ fig4 ] the power to distinguish between the ap and p model is not essentially compromised by the detector temperature since all the feature characterising the model in terms of @xmath98 seems mutually amplified . increasing @xmath8 is an interesting resource in order to increase detection efficiency , in this perspective . we have investigated the behaviour of f.f . emitted power @xmath98 of a resonant lc circuit weakly coupled to a qpc built in a quantum hall bar at filling factor @xmath0 . we showed that the emitted power is represented in terms of the non - symmetrised noise components of the quantum hall qpc weighted by the bosonic distribution of the resonant lc circuit . we have inspected the different predictions of the pfaffian and anti - pfaffian non - abelian edge states models for this quantity . we showed that this setup can detect and discriminate between the dominant and sub - dominant fractionally charged excitations looking at the bias dependence at fixed ghz frequencies . we also discussed the advantage to use this measurement protocol in comparison to the f.f . symmetrised noise . finally we demonstrated how the sensibility of the proposed setup can be increased varying the lc detector temperature @xmath8 . we thank c. glattli , w. belzig and p. solinas for useful discussions . we acknowledge the support of the miur - 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we investigate the finite frequency noise of a quantum point contact at filling factor @xmath0 using a weakly coupled resonant lc circuit as a detector .
we show how one could spectroscopically address the fractional charged excitations inspecting separately their charge and scaling dimensions .
we thus compare the behaviour of the pfaffian and the anti - pfaffian non - abelian edge states models in order to give possible experimental signatures to identify the appropriate model for this fractional quantum hall states .
finally we investigate how the temperature of the lc resonant circuit can be used in order to enhance the sensibility of the measurement scheme .
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it is well established that interactions and/or galaxy collisions represent an important stage in the evolution of galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? numerous studies show clear indications of the importance of those external mechanisms for the enhancement of star - formation and its effects in the chemical enrichment of galaxies . in particular , in major merger of massive galaxies the preexisting gas metallicity can be substantially diluted by the inflow of metal poor gas from the outskirts to the nucleus ( e.g. * ? ? ? * ; * ? ? ? * ) . in the case of low - mass , low - metallicity ( 7.0 @xmath6 12 + log(o / h ) @xmath6 8.4 ; * ? * ) and star - forming dwarf galaxies the effects of tidal interactions and/or mergers ( e.g. * ? ? ? * ; * ? ? ? * ) also has a huge impact in their evolution . observational evidences suggest that hii / bcd galaxies arise from the interactions or accretion of extended hi cloud complexes ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , the triggering mechanisms of the current burst of star - formation in those objects is not yet clear since most are , apparently , isolated systems ( e.g. * ? ? ? * ; * ? ? ? thus , if not triggered by external agents star - formation is likely produced by internal processes ( e.g. gravitational cloud collapse , infall of gas in conjunction with small perturbations ) and/or minor mergers ( see * ? ? ? * and references therein ) . as described above , a considerable fraction of these galaxies has been associated with hi clouds @xcite or low - mass and undetected companions in the optical ( e.g. * ? ? ? * ) , which could rule out the idea of bcd galaxies as isolated systems @xcite . in fact , a significant fraction of bcds do show signs of extensions or tails in their outer envelopes , suggesting a tidal origin . many of these low - metallicity galaxies that show cometary " or elongated shapes show values of 12+log(o / h ) @xmath7 7.6 ( e.g. * ? ? ? * ) . within this subsample of bcds or extremely metal poor ( xmp ) bcd galaxies we found the least chemically evolved galaxies in the local universe @xcite . this particular morphology has been interpreted for high redshift galaxies in the hubble deep field as the result of weak tidal interactions @xcite , gravitational instabilities in gas - rich and turbulent galactic disks in formation at high redshift @xcite and stream - driven accretion of metal - poor gas from the cosmic web @xcite . @xcite argue that weak interactions between low - mass stellar or gaseous companions , or propagating shock waves , lead to a bar - like gas distribution triggering the star - formation that by propagation could subsequently produce a cometary morphology in xmp bcds . recently , @xcite interpret the metallicity variation in a sample of low metallicity galaxies with cometary morphology as a sign of external gas accretion / infall of metal poor gas . they argue that these results are consistent with the local tadpole " galaxies being disks in early stages of assembling , with their star - formation sustained by pristine gas infall . in any case , dwarf galaxies tend to show flat abundance ( o / h , n / o ) gradients ( e.g. * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , suggesting efficient dispersion and mixing of metals in the interstellar medium ( ism ) by expanding starburst - driven superbubbles ( e.g. * ? ? ? * ) , afterward the gas begins to cool down by radiation and gravity , and/or external gas infall ( e.g. * ? ? ? * ; * ? ? ? these mechanisms have been put forth as potential causes for the observed flat metal distributions in local dwarf galaxies . while in massive star - forming and/or interacting galaxies , bar - induced rotation or shear ( e.g. * ? ? ? * ) and merger - induced gas flows ( e.g. * ? ? ? * ) could produce the metal dispersal and mixing . as expressed above , local hii / bcd and xmp bcd galaxies are considered chemically homogeneous and only in a few isolated cases we observed indications of variation of 12+log(o / h ) over the ism ( e.g. sbs 0335 - 052e , haro 11 , hs 2236 + 1344 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in addition to the expected low metal content in young galaxies at high redshift , according to theoretical models these objects should produce strong ly@xmath0 ( 1216 @xmath8 ) emission as the result of their intense star - formation activity ( e.g. * ? ? ? * ; * ? ? ? however , the absence and/or diminished ly@xmath0 emission in these galaxies , which is significantly lower than the theoretical recombination ratio , indicate that the ly@xmath0 photons are likely redistributed by multiple scattering in the hi envelope , or are absorbed by dust . examples of the detection of ly@xmath0 halos produced by hi scattering envelopes can be seen in the literature ( e.g. * ? ? ? * ; * ? ? ? it has been suggested in the literature ( e.g. * ? ? ? * ) that there should be an increase in the ly@xmath0/h@xmath9 flux ratio as the metallicity of the galaxy decreases , since presumably low - metallicity objects contain less dust and hence suffer less ly@xmath0 photon destruction . ly@xmath0 can also be enhanced at low gas metallicity due to collisional excitation @xcite . as pointed by @xcite , the fact that there is no ly@xmath0 emission in the two most metal - deficient bcds known , i zw 18 @xcite and sbs 0335 - 052 @xcite , and also in tol 65 , argues against the existence of some correlation between the ly@xmath0 emission , metallicity and dust @xcite . therefore , an important issue in the understanding of ly@xmath0 emission in galaxies is the study of the spatial distribution of properties in the ism of those objects in order to see the different regulation mechanisms involved in the detectability of ly@xmath0 emission . although during the last years some progress has been made in this field , many questions remain about the metal content , the mechanism involved in the transport and mixing of metals and the star - formation activity in hii / bcd and xmp galaxies . the morphologically diverse nature of hii / bcd , and xmp bcds , galaxies allows us to consider the role played by galaxy interactions and the feedback between the star - formation and the ism in the observed metal distributions . our main objective in this paper is to carry out a spatial investigation of the warm gas properties in the cometary " and ly@xmath0 absorbing xmp hii / bcd galaxy tol 65 . these include , the spatial distribution of emission lines , equivalent width ew(h@xmath9 ) , extinction c(h@xmath9 ) , ionization ratios ( [ oiii]@xmath105007/h@xmath9 , [ sii]@xmath116717,6731/h@xmath0 and [ nii]@xmath106584/h@xmath0 ) , kinematics , the chemical pattern ( e.g. o / h , n / h and n / o , etc . ) and also the possible dependence between these properties . our aim is to search for the existence of metallicity inhomogeneities as expected if the ongoing star - formation activity is sustained by the infall or accretion of metal - poor gas . to this end , we use high resolution integral field unit ( ifu ) spectroscopy observations . the paper is organized as follows : sect . [ sect_tol65 ] describes the most important properties of our analyzed galaxy . [ sect_obser_reduc ] contains the technical details regarding observations and the data reduction . [ sect_results ] describes our results : the ionized gas structure as well as the physical and kinematic properties of the ionized gas . [ sect_discussion ] discusses the results . finally , sect . [ conclusions ] itemizes the conclusions . [ cols="<,^,^,^,^,^,^,>,<,>",options="header " , ] in this context , an estimate of the mass of emitting ionized hydrogen in the tail of the galaxy , is an important indicator of the amount of material present in this structure . this can be made using the h@xmath0 emission - line luminosity l(h@xmath0)=7.14@xmath210@xmath12 erg s@xmath13 and the expression for the mass of the ionized gas from @xcite ; @xmath14 where m@xmath15 is the mass of the proton and n@xmath16 is the electron density . we use the effective recombination coefficient @xmath17 for h@xmath0 and case b emission at t=2@xmath210@xmath18 k @xcite . thus , assuming an average electron density of @xmath510 @xmath19 , we find a total mass of ionized hydrogen in the tail of m(hii)@xmath51.70@xmath210@xmath3 m@xmath20 . this mass corresponds with @xmath524 per cent of the total ionized mass flux of the galaxy summing all the spaxels over the fov . thus , f(h@xmath0)/f(h@xmath9)=3.4899 with f(h@xmath9)=47.48@xmath210@xmath21 erg @xmath22 s@xmath13 . ] of the galaxy ( or @xmath53 per cent if we consider @xmath5100 @xmath19 ) with m(hii)@xmath57.00@xmath210@xmath3 m@xmath20 . although this mass estimate is approximate , it could clearly indicate that there is a large amount of hi in this extended structure . the star - formation rate ( sfr ) inferred using the @xcite formula after correction for a kroupa imf @xcite , is 0.152 m@xmath4yr@xmath13 for the main body of the galaxy and 0.090 m@xmath4yr@xmath13 and 0.016 m@xmath4yr@xmath13 for regions nos . 1 and 2 , respectively . this global sfrs are low in absolute terms . however , its starbursting nature appears when one computes the specific sfr , i.e. the sfr per unit of mass . somewhat equivalently , their sfr per unit of area ( @xmath23 ) . we can translate the integrated sfr of the main body into an integrated @xmath23 assuming an aperture of @xmath51.95 kpc@xmath24 that corresponds to the total area of this region . thus , we obtain that @xmath250.078 m@xmath4 yr@xmath13kpc@xmath26 . if we compare this result with the one obtained in our previously analyzed xmp bcd galaxy , using ifu spectroscopy , ( hs 2236 + 1344 , * ? ? ? * ) we found that in tol 65 the @xmath23 is 2 times lower , while the @xmath27 is comparable to the star - formation per unit of area of local tadpole galaxies @xcite . on the other hand , the @xmath23 in tol 65 , and also in hs 2236 + 1344 , is above the sfrs found in normal " or more metal rich hii / bcd galaxies . from fig . 15 in @xcite it is evident that the sfr per unit of area in our xmp galaxies studied so far are higher than the ones found in most of their sample of hii galaxies . this is explained by the high concentration of hi gas found in xmps ( e.g. * ? ? ? * ; * ? ? ? * ) . using the results obtained in sect . [ sect_abundances ] we show , in fig . [ figure_sfr_oh ] , the spatially resolved relation between the @xmath23 and the oxygen abundance 12+log(o / h ) in tol 65 . even this relation at spaxel scales is dubious , our findings show a marginal anticorrelation between these quantities in the sense that high spatial star - formation is found in regions of lower metallicities ( see the linear fit to this relation in fig [ figure_sfr_oh ] ) . the calculation of pearson s correlation for these data gives a value of -0.41 , that indicate a moderate monotonically decreasing relationship between these quantities . if we interpret this result as the consequence of an ongoing infall of metal - poor gas from the outskirts the scatter ( @xmath28=0.13 dex ) in fig . [ figure_sfr_oh ] clearly indicates that the metals in the ism are almost fully diluted . therefore , from an statistical point of view the ism of the galaxy can be considered chemically homogeneous . we will come back on this discussion later in sect . [ sect_disc_properties ] . and oxygen abundance 12+log(o / h ) at spaxel scales . the cyan line represents the linear fit to this relation . the 12+log(o / h)=7.56 obtained in the main body of the galaxy is represented by the continuous line , while the errors at 1@xmath29 level are in dotted lines.,width=321 ] it is however worth noting that the age ( @xmath53 - 5 myr ; see sect . [ sect_i_c_ew ] ) of the current burst of tol 65 agrees with the idea proposed by @xcite , in the sense that this galaxy is a young starburst embedded in a static hi cloud , which produces a damped ly@xmath0 absorption . @xcite found even with the poor quality of their spectrum , that the oi @xmath101302 absorption line in tol 65 is blue shifted by @xmath5200 km s@xmath13 with respect to the emission lines and the siii@xmath101304 absorption line . they argue that if this is true , the oi absorption would be produced in gas shells moving outward from the central star clusters @xcite . the spatial distribution of electron density n@xmath16(sii ) , in fig . [ figure_temp_den_galaxies ] , shows a clear area of constant values in region no . 1 with n@xmath16(sii)@xmath5200 @xmath19 surrounded by relatively higher density values @xmath30200 @xmath19 . in particular , the area in between the two regions shows spaxels which reach values of n@xmath16(sii)@xmath5500 @xmath19 and the [ sii]@xmath116717,6731/h@xmath0 ratio ( see fig . [ figure_ratios_galaxies ] ) is slightly enhanced in this inter - cluster region which would be consistent with an enhanced contribution to the ionization / excitation by shocks . therefore , the gas ejected by massive stars within the star clusters grows close to the centre creating a high density region at the same time that the gas expands with velocities close or higher than the supersonic values ( with @xmath3110 km s@xmath13 ) into the surrounding region , then compressing the gas into high - density condensations and creating a low density region between the centre and the expanding high density shells . we found high fwhm values in the same area of high electron density in between the ghiirs ( see fig . [ figure_velocity_field_tol65 ] ) indicating that the turbulent motions in the ionized gas could be produced by winds from the main star cluster complexes . likewise , the expanding gas would cause the expansion in the surrounding material ( hi ) , so moving it out of the galactic plane . this agrees with the arguments proposed by @xcite and the evolutionary models by @xcite in order to produce a damped ly@xmath0 absorption . in addition , if the wind material contains enough dust , this outflow of gas would give rise to blueshifted velocities ( compared to the clusters ) observed in between the ghiirs because redshifted emission would be preferentially extinguished by the dust . in fact , the lack of ly@xmath0 emission ( see fig . 2 in * ? ? ? * ) and the detection of relatively high c(h@xmath9 ) associated with the most intense star - forming region in this galaxy , region no . 1 , suggest that the ly@xmath0 photons have likely been destroyed by dust . this is in good agreement with previous findings by @xcite in the sense that dust extinction is an important ly@xmath0 escape regulator . a number of recent studies ( e.g. * ? ? ? * ; * ? ? ? * ) have suggested that local galaxies with cometary or tadpole morphology could have a variety of origins . @xcite suggest that most local tadpoles are bulge - free galaxy disks with lopsided star - formation , likely from environmental effects ( e.g , ram pressure , disk impacts or random collapse of local disk gas with an unstable jeans length ) . alternatively , propagating star - formation along the tail has been proposed by @xcite as one of the mechanisms that lead to the formation of the elongated underlying component or stellar tail of cometary hii / bcds . given that the starburst produces enough energy from stellar winds and supernovae ( sne ) , could the stellar feedback from the starburst have created the stellar tail in this galaxy ? to create a @xmath51.5 kpc long extension in @xmath55 myr ( age of the current burst ) would require an outflow velocity @xmath5293 km s@xmath13 , while the difference between the velocities found in the field is @xmath5 50 km s@xmath13 . therefore , the current star - formation activity appears to be too young to have pulled out the surrounding gas and stars , driven a galactic outflow , to form the tail . on the other hand , if the global starburst take longer @xmath510@xmath32 yr ( e.g. * ? ? ? * ) as compared to the current burst , certainly it may promote the formation of gas outflows . after that some part of the gas could decelerate and collapse to form the stellar tail . if so , the close similarity between @xmath33 and @xmath34 agrees with this scenario , given that the diffuse gas will acquire the kinematical information of the system which is transferred into the ism by the star - formation activity ( e.g. * ? ? ? @xcite found that the relatively red colors of the underlying stellar component , in tol 65 , suggest a stellar population with a mean age of @xmath351 gyr . this stellar age is compatible with this idea , but the intrinsic youth of xmps is questionable @xcite and the evidence of such strong winds and the presence of extended ( @xmath361 kpc ) super - shells in low luminosity dwarf galaxies is also sparse . finally , we did not observe evidences of propagating star - formation along the galaxy and the presence of star clusters in the tail . thus , there seems to be no correlation between the star - forming activity and the formation of the tail in this galaxy , but from our present data we can not rule out completely this possibility . in many respects , the tidal hypothesis is an attractive one for the origin of the tail in tol 65 , given that tidal interactions enhance lopsidedness in low mass galaxies @xcite . however , hii / bcds tend to populate low - density environments and are not associated with massive galaxies ( e.g. * ? ? ? * ; * ? ? ? * ) suggesting that tidal interactions with massive companions could not be the dominant starburst triggering mechanism . thus , the cometary or tadpole shapes among these may not generally be major mergers . we used the nasa / ipac extragalactic database ( ned ) to search for nearby objects with measured systemic velocities . figure [ figure_field_tol65 ] shows the field on the plane of the sky of the galaxy . we find two galaxies , with measured velocities , in the field of tol 65 . the first one j122531.50 - 360715.3 is an irregular ( dirr ) and likely dwarf galaxy with a difference between their respective systemic velocities of @xmath37v = -4 km s@xmath13 . eso digitized sky survey ( dss ) images show that this object is highly disturbed with an apparent size of @xmath530@xmath38 equivalent with @xmath56.2 kpc , which is two times the apparent size of tol 65 . the projected distance between these galaxies is r@xmath39100 kpc . the other object in the field is the spiral galaxy eso 380-g029 . for this galaxy we found a @xmath37v = 1249 km s@xmath13 , thus it is unlikely that this galaxy is producing the tidally - disturbed tail present in the galaxy , because companions with @xmath37v@xmath30 500 km s@xmath13 have not a significant dynamical influence @xcite . it is interesting to note that the tail of the galaxy is oriented , in the plane of the sky , to this spiral galaxy . possibly eso 380-g029 had an stronger influence over tol 65 in the past ( several gyr ago ) , thus the tail in the latter could be a remnant of that tidal interaction . however , there is no observational evidence showing that this galaxy had an interaction in the past with tol 65 to produce a tidal tail . @xcite using vlt images suggested the presence of a low - surface - brightness companion galaxy of tol 65 , but it was discarded by @xcite . they found this object is in fact a background galaxy . thus , if this extended structure has a tidal origin with a close companion(s ) it may be related with a faint and massive one that have not been identified by optical surveys . in any case , j122531.50 - 360715.3 remains as the only potential perturber . in this sense , the slight enhancement of the sfr in region no . 1 with respect to region no . 2 and the lopsidedness of those star - forming regions could be the consequence of this tidal interaction . although it is unlikely that another dwarf galaxy of low mass , as it seems , could have a tidal effect at this distance . from the present observations we can not say much about the impact of j122531.50 - 360715.3 over the physical properties of tol 65 but we speculate based on the aforementioned arguments that the co - evolution on these systems play an important role in the enhancement of star - formation ( e.g. * ? ? ? * ) and evolution in many of these galaxies @xcite . several studies have shown that a number of star - forming dwarf galaxies present filamentary hi and/or extended structures , which may indicate a recent cold gas infall / accretion ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and/or minor merger / interaction ( e.g. * ? ? ? * ; * ? ? ? * ) . in this sense , the extended structure of this galaxy could be a remnant of one of these processes . @xcite argue that if the stars are being formed as a consequence of the infall they will be formed along the path of the accretion . but , we did not observe the presence of star clusters in the tail of the galaxy . then , alternatively , the effect of these interactions has carried away in the recent past an important fraction of gas and stars to form the stellar tail . the fact that @xmath40 ( see sect . [ sect_velocity_field ] ) implies that the ionized gas still , on a global scale , retains the kinematic memory of its parental cloud and likely a common origin . while extended regions of diffuse gas are highly disturbed , probably due to unresolved expanding shells and the effects of massive star evolution . therefore , we interpreted the observed properties of the ism and the morphology of the galaxy as due to a late - stage minor merger and/or inflow of metal - poor gas in the recent past of the galaxy . if the current star - forming episode in xmps is the consequence of the infall of metal - poor gas , the less enriched gas dilutes the preexisting nuclear gas to produce a lower metallicity than would be obtained prior to the accretion . @xcite argue that such accretion flows could explain all the major xmp properties such as , i.e. , isolation , lack of interaction / merger signatures , metal - poor hi gas , metallicity inhomogeneities , etc . in this sense , @xcite argue that the variation of oxygen abundance along the major axis of two xmp star - forming dwarf galaxies can be interpreted as an early stage of assembling in disk galaxies with the star - formation sustained by external metal - poor gas accretion likely from the cosmic web @xcite . in @xcite we studied one of the objects analyzed by @xcite , the xmp galaxy hs 2236 + 1344 . in this study , we found indications of variation of o and n / o associated with the less luminous star - formation region of that galaxy . but given the uncertainties related to those measurements we considered the abundances across the galaxy as fairly uniform . since most of our hii / bcd galaxies @xcite , and xmp / bcds from the literature ( * ? ? ? * see references therein ) studied so far are chemically homogeneous it seems that these galaxies have mixed up the metals with the pre - existing ism before the current starburst appears . the newly synthesized metals from the current star - formation episode reside in a hot gas phase ( t @xmath41 10@xmath42 k ) and those will be dispersed probably in the whole galaxy by the expansion of starburst - driven outflows . the energy injected by stellar winds and sne may be able to eject part of the enriched gas into the intergalactic medium , but in most cases the expanding velocity ( e.g. * ? ? ? * and references therein ) are not sufficient to allow the gas to escape from the potential well of the galaxies . in addition , inflows or accretion of relatively low metallicity gas could possibly dilute more easily the abundance of the gas on large scales . assuming a sound speed of @xmath512 km s@xmath13 ( see sect . [ sect_velocity_field ] ) as the expected speed at which mixing occurs in tol 65 , we found that over 1.5 - 2.0 kpc it yields @xmath510@xmath32yr , which is similar to the timescale required for cooling and dispersal of metals produced by massive stars ( e.g. * ? ? ? * ) as observed in compact hii / bcd galaxies ( e.g. um 408 ; * ? ? ? * ) . consequently , the predominant mechanism for metal transport may not be resolved in star - forming dwarf galaxies . however , if we interpreted the observed properties of the ism as due to a late stage - merger and/or inflow of metal - poor gas , the accreted gas must be almost fully diluted as seen in fig . [ figure_sfr_oh ] . therefore , the final mixture of gas thus depends on the relative enrichment of the acquired and pre - enriched gas . @xcite showed that the neutral gas metallicity , in a sample of bcds , is equal or lower than the ones found in the hii region . this agree with the results obtained by @xcite , using the cosmic origin spectrograph ( cos ) onboard hst , in the sense that the hi gas abundances is factor @xmath52 lower than the warm gas abundance in i zw18 indicating that infall of , non fully pristine , metal - poor gas could be the responsible for its low metallicity . therefore , we can not exclude that the tail in this xmp galaxy could be made from a gas with much lower metallicity than that in the warm gas phase . however , we still lack an adequate explanation about the evolutionary status of tol 65 since no spatially resolved hi observations are available , but the infall / accretion of cold gas from the outskirts of the galaxy and/or minor merger / interaction with an small companion recently in the past of the galaxy could explain the main properties of the galaxy . as mentioned above , the tail and the _ cometary morphology _ could be the remnant of these interactions in which an appreciable fraction of gas has been carried away as the result of the tidal torques , then producing the mixing of metals through the ism at large scales . in this sense , the scatter in fig . [ figure_sfr_oh ] indicates that the metals in the ism were diluted in a relatively small timescale . if the past infall of metal poor gas do not explain the low metallicity of tol 65 , alternatively , it could be merely a consequence of the low star - formation efficiency due by the cosmic downsizing . in this work we have presented vimos - ifu spectroscopy of the xmp hii / bcd galaxy tol 65 . we studied the spatial distribution of properties ( i.e. , emission lines , abundances , kinematics , etc ) through the ism of the galaxy in an extended area of 13@xmath213@xmath38 encompassing the two ghiirs and most of the extended stellar tail . below , we summarize our results and conclusions : 1 . this galaxy shows a clear cometary shape with a bright main body and an extended and diffuse stellar tail . we found that the current star - formation activity in tol 65 started recently about 3 - 5 myr ago . our observations show the presence of an extended h@xmath0 emission in the tail of the galaxy . the mass of the ionized gas in the tail corresponds to @xmath524 per cent of the total mass of the ionized gas in the galaxy . + 2 . the integrated properties in the main body of the galaxy , it to say , the extinction coefficient c(h@xmath9)=0.29 , oxygen abundance 12+log(o / h)[email protected] , velocity dispersion @[email protected] kms@xmath13 obtained in this study agree , within the uncertainties , with the ones determined in the literature ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the velocity field v@xmath44 in tol 65 is highly disturbed and it shows no global rotation pattern . we found @xmath45 suggesting that the gas still retains the kinematic memory of its parental cloud and a common origin . while extended regions of diffuse gas are highly disturbed , probably due to unresolved expanding shells and the effects of massive star evolution . 4 . we did not observe a spatial variation of metal ( o , n , ne , ar and s ) in the ism of the galaxy . this agrees with previous observations in other hii galaxies in the literature ( * ? ? ? * and references therein ) . although the evidence is far from conclusive , we favour the idea that the most likely mechanisms to produce the flat abundance gradient and the cometary morphology in this xmp hii / bcd is the infall / accretion of cold gas from the outskirts of the galaxy or minor merger / interaction with an small companion recently in the past of the galaxy . this is in agreement with the scatter found between the gas metallicity ( 12+log(o / h ) ) and star - formation rate at spaxel scales , in fig . [ figure_sfr_oh ] , which clearly indicates that the metals in the ism are almost fully diluted . + our vimos - ifu observation of tol 65 provides a picture in which a late - stage minor merger and/or infall of gas explain most of the observed properties over the ism . therefore , we suggest that these global effects might be attributed as the main mechanism diminishing the preexisting metal of the galaxy ( e.g. * ? ? ? * ) , then keeping the oxygen abundance ( and other @xmath0 elements ) constant through the ism at large scales . clearly , hi observations of tol 65 with high spatial resolution and hst cos observations are needed in order to study the cool gas component and check whether the proposed mechanism are capable to produce the homogeneity of the ism , the difference between the neutral and warm gas metallicity and the morphology of this xmp galaxy . we would like to thank the referee , dr . daniel kunth , for his very valuable comments and suggestions . this work was supported by fundao para a cincia e a tecnologia ( fct ) through the research grant uid / fis/04434/2013 . is supported by a post - doctoral grant sfrh / bpd/72308/2010 , funded by fct ( portugal ) . r.d . gratefully acknowledges the support provided by the basal center for astrophysics and associated technologies ( cata ) , and by fondecyt grant n. 1130528 . , n.r . and j.m.g . acknowledge support by the fct under project fcomp-01 - 0124-feder-029170 ( reference fct ptdc / fis - ast/3214/2012 ) , funded by the feder program . is supported by fct through the investigador fct contract no . if/01220/2013 and poph / fse ( ec ) by feder funding through the program compete . is supported by a postdoctoral grant sfrh / bpd/66958/2009 , funded by fct ( portugal ) . we acknowledge support by the exchange programme ` study of emission - line galaxies with integral- field spectroscopy ' ( selgifs , fp7-people-2013-irses-612701 ) , funded by the eu through the irses scheme . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . filho , m. e. , winkel , b. , snchez almeida , j. , aguerri , j. a. , amorn , r. , ascasibar , y. , elmegreen , b. g. , elmegreen , d. m. , gomes , j. m. , humphrey , a. , lagos , p. , morales - luis , a. b. , muoz - tun , c. , papaderos , p. , vlchez , j. m. 2013 , a&a , 558 , 18
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in this study we present high - resolution visible multi - object spectrograph integral field unit spectroscopy ( vimos - ifu ) of the extremely metal - poor hii / blue compact dwarf ( bcd ) galaxy tol 65 .
the optical appearance of this galaxy shows clearly a cometary morphology with a bright main body and an extended and diffuse stellar tail .
we focus on the detection of metallicity gradients or inhomogeneities as expected if the ongoing star - formation activity is sustained by the infall / accretion of metal - poor gas .
no evidences of significant spatial variations of abundances were found within our uncertainties . however , our findings show a slight anticorrelation between gas metallicity and star - formation rate at spaxel scales , in the sense that high star - formation is found in regions of low - metallicity , but the scatter in this relation indicates that the metals are almost fully diluted .
our observations show the presence of extended h@xmath0 emission in the stellar tail of the galaxy .
we estimated that the mass of the ionized gas in the tail m(hii)@xmath11.7@xmath210@xmath3 m@xmath4 corresponds with @xmath524 per cent of the total mass of the ionized gas in the galaxy .
we found that the h@xmath0 velocity dispersion of the main body and the tail of the galaxy are comparable with the one found in the neutral gas by previous studies .
this suggests that the ionized gas still retains the kinematic memory of its parental cloud and likely a common origin . finally , we suggest that the infall / accretion of cold gas from the outskirts of the galaxy and/or minor merger / interaction may have produced the almost flat abundance gradient and the cometary morphology in tol 65 .
[ firstpage ] galaxies : dwarf galaxies : individual : tol 65 ( tol 1223 - 359 , eso380-g027 ) galaxies : ism galaxies : abundances .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
it has long been known that the wave function of an electron moving in a random potential becomes spatially localized . this effect was first predicted by anderson @xcite and is termed _ anderson localization_. in one and two dimensions _ all _ quantum states are localized in the presence of _ any _ amount of disorder while in three dimensions localization occurs only above some critical disorder . while this phenomenon is now fairly well understood in a single - particle picture , the inclusion of interactions in the disordered many - body system is a non - trivial problem . insight can be gained by studying the simpler case of just two interacting particles in a random potential . in this context , it has recently been claimed that the interaction can actually lead to a _ delocalization _ effect , in the sense that the spatial extent of the two - body wave function is larger than the single - particle localization length @xcite . this delocalization was found for both attractive and repulsive interactions , at least for some of the eigenstates in the continuous spectrum of energy eigenvalues . some authors objected to this finding @xcite and this point is still being debated . furthermore , the relevance of these results for the many - body problem has not yet been clarified . in this paper our main concern is the case of an _ attractive _ interaction . the concept of a two - particle bound state ( the cooper pair ) plays an important role in the theory of superconductivity . there is theoretical and experimental evidence for the existence of superconductor - insulator transitions , where localized states combine coherently into a superconducting condensate with a finite superfluid density . this finding motivates the question addressed in this paper : will the bound state of two attracting particles be extended over distances larger than the single - particle localization length ? we find that a result analogous to the anderson theorem for localized superconductors is valid for this problem in the limit of large single - particle localization lengths . by increasing the disorder we find a transition from a regime in which the interaction _ increases _ the localization length to a regime in which the interaction _ reduces _ the localization length . the hamiltonian for a disordered conventional ( @xmath0-wave ) superconductor is @xmath1 where @xmath2 represents the single - electron part including a spatially random external potential and the interaction parameter @xmath3 is taken as positive . the attractive interaction could be due to exchange of phonons or purely electronic mechanisms but its origin does not concern us here . depending on the disorder , the eigenstates @xmath4 of @xmath2 can be extended or localized ( with localization length @xmath5 at the fermi level ) . at t=0 the system described by ( [ bcs ] ) is a superconductor with spatially constant order parameter @xmath6 when the condition @xmath7 holds @xcite . @xmath8 is then given by the same expression as that for a clean superconductor . this is the anderson theorem . the important message of equation ( [ valanderson ] ) is that if the disorder is strong enough to localize the single - particle states , the superconducting order parameter @xmath8 and critical temperature @xmath9 will remain unaffected as long as the number of single - particle states in the energy range @xmath8 contained in the localization volume @xmath10 is still large . ( the superfluid density , on the other hand , is greatly reduced . ) if the amount of disorder is further increased so that ( [ valanderson ] ) is no longer valid the superconducting order parameter will fluctuate strongly in space and the critical temperature will be lowered . note that because @xmath11 is the wave function of the condensate , the anderson theorem tells us that the attractive interaction is delocalizing the cooper pairs . keep in mind , however , that in this derivation the cooper pairs are strongly interacting ( and actually overlapping ) with each other , forming a correlated liquid . in bcs theory the concept of cooper pair only has the formal significance that strong pair correlations exist between the particles in phase space . in the regime of validity of the anderson theorem @xmath8 is given by @xmath12 the denominator of this expression is the quasi - particle excitation energy . the binding energy of an isolated cooper pair is given by an expression of the same form as this one ( see eq.([cond ] ) below ) but with a different denominator . this is because equation ( [ delta ] ) takes into account the interactions between cooper pairs . experimentally , it has been found that superconductivity persists up to the anderson metal - insulator transition @xcite . these systems exhibit activated conductivity above @xmath9 . the coexistence of superconductivity and localization has also been observed in underdoped high-@xmath9 superconductors @xcite . in what follows the problem of two interacting electrons in a random potential will be addressed for the specific case of an attractive interaction . the aim is to see how the interplay of disorder and interaction affects the coherent propagation of the electrons in the ground state . the attractive interaction will be assumed to be short - ranged . we can describe this system by an anderson - hubbard hamiltonian @xmath13 with @xmath14 representing the site energies randomly distributed over a width @xmath15 , and @xmath16 . the single - particle eigenfunctions @xmath17 are assumed to be localized by the disorder with a localization length @xmath18 . we search for the two - particle ground state @xmath19 where @xmath20 and @xmath21 are the coordinates of the electrons . because the bound state is a spin singlet , @xmath22 is a symmetric function of its arguments and has the general form @xmath23 where @xmath17 are normalized single - particle eigenfunctions of ( [ ham ] ) with energy eigenvalues @xmath24 . in order to simulate the presence of a fermi sea , it is assumed that the two electrons can only be paired in states @xmath17 which lie above the fermi surface . according to the values of the parameters in ( [ ham ] ) we recognize several different regimes . if @xmath25 ( strong attraction ) then the electrons are tightly bound and move together like a heavy boson with hopping amplitude @xmath26 in an environment with disorder w. this boson would then become easily localized . in the remainder of the paper we concentrate on the regime in which the interaction is not strong ( @xmath27 ) . in this case @xmath28 is essentially the result of electron pairing in time - reversed single - particle eigenstates of ( [ ham ] ) for not too strong disorder . this can be seen as follows . the schrdinger equation for the cooper pair wave function is @xmath29 it admits the solution @xmath30 if the condition @xmath31 holds . ( here @xmath32 denotes the local density of states at the point @xmath33 ) . thus a result analogous to the anderson theorem is valid for the function @xmath34 . in such a case the solution to ( [ sch ] ) is @xmath35 and @xmath28 has no overlap with the state @xmath36 if @xmath37 . the condition ( [ cond ] ) can be satisfied if the integral of the local density of states @xmath38 over an energy interval of order @xmath39 does not depend on @xmath33 . this is possible even if the disorder is strong enough to localize the single - particle states as long as the condition @xmath40 holds . the binding energy @xmath39 will then be the same as that for a clean system . so we reach the conclusion that the attractive interaction can delocalize the pair or , at least , increase its localization length . the delocalization by interactions due to the attractive interaction can be understood within the block - scaling picture of localization introduced by thouless @xcite . in a recent paper @xcite , imry has used the block - scaling picture in order to argue that interactions ( attractive or repulsive ) should , in some cases , delocalize some of the eigenstates of the continuous spectrum of the pair of electrons . in what follows we extend the argument to the case of a cooper pair . suppose the system is divided into blocks of linear size @xmath18 ( measured in units of the lattice spacing ) so that the mean level spacing in a block is @xmath41 with @xmath42 equal to the band width . we can then solve the cooper problem , as above , for each block by pairing the two electrons in time - reversed states localized inside the block . if the fluctuation of the binding energy is smaller than the effective scattering amplitude of the pair between blocks then the pair will be extended over many blocks . we denote the cooper pair in the @xmath43-th block by @xmath44 . next , we estimate the scattering amplitude @xmath45 , _ due to the interaction _ , of the cooper pair between two neighbouring blocks as @xmath46 we now note that @xmath47 depends smoothly on @xmath48 and has no nodes because it is the wave function of a ground state , namely @xmath49 the normalization of @xmath44 implies @xmath50 and the number of terms in the sum in ( [ hopp ] ) is large , of the order of @xmath51 . all those terms interfere constructively yielding @xmath52 the fluctuation of the binding energy @xmath39 due to the randomness in @xmath24 is of the order of @xmath53 ( see the appendix ) . thus the condition for delocalization is obtained as @xmath54 the ratio @xmath55 is the bcs product @xmath56 . the following points should be noted : @xmath57 there are no phase correlations to consider in the sum ( [ hopp ] ) because all the terms are real and negative ; @xmath58 the large effective hopping amplitude @xmath45 resulted from @xmath47 being a smooth function of @xmath48 with no nodes . in other words , _ because we have been considering the ground state_. @xmath59 if the sign of the interaction @xmath60 is reversed , to make it repulsive , we do not expect any delocalization to occur in the ground state . the reason for this is that @xmath44 for each block ( using time - reversed pairing ) would now essentially involve only the single - particle eigenstate @xmath61 with the lowest energy : @xmath62 . then @xmath63 if @xmath64 . if we remove the constraint of time - reversed pairing then imry s argument would predict delocalization only if the pair has a certain excitation energy above the fermi level @xcite . so it would not be in the ground state . we have also performed numerical calculations of the cooper pair wave function for 1d systems of up to @xmath65 sites using the lanczos algorithm . in order to impose the constraint of a filled fermi sea the single - particle eigenstates for a given realization of the disorder potential are calculated and the eigenenergies of the states below the fermi energy are shifted by a large amount . thus only the one - particle states above the fermi surface are accessible for the cooper pair . before doing the lanczos diagonalization of the two - particle problem the hamiltonian is transformed back to real space representation where the number of nonzero matrix elements is much smaller than in the basis of the eigenstates of the non - interacting disordered system . in order to determine the spatial extent of the cooper pair wave function with respect to both the relative and the center - of - mass coordinates we have calculated two different quantities . the first one is the participation ratio @xmath66 where @xmath34 is normalized to unity . since only the diagonal part @xmath67 of the wave function is involved in the calculation the participation ratio can also be interpreted as the localization length of the cooper pair . the second quantity , which is related to the relative coordinate of the two electrons , is the average size @xmath68 of the cooper pair defined as @xmath69 our results are obtained for a system of 100 sites and each data point represents the average over 200 realizations of the disorder potential . since the fluctuations of the participation ratio @xmath70 are very large we found it more convenient to average @xmath71 instead of @xmath70 itself . 1 shows the behavior of @xmath71 as a function of @xmath72 for different values of the disorder strength @xmath73 . while in the case of strong disorder the interaction leads to a decrease of the participation ratio , for smaller values of @xmath73 an enhancement of @xmath71 is observed , at least for not too large @xmath72 . this is in qualitative agreement with our analytical arguments . one should however keep in mind that the calculations are done for a rather small system and that finite size effects can be of importance , especially in the weak disorder region where the localization length becomes large . in fig . 2 we see that the size of the cooper pair decreases rapidly with increasing interaction strength . the values for @xmath74 which should be of the order of the one - particle localization length are considerably reduced due to finite size effects . the interplay of disorder and interaction is a complex problem and therefore it is useful to start with the two - body problem in a disordered medium . the effect of a repulsive interaction seems to be very different from that of an attractive interaction since the latter can induce propagation of the bound state of two particles ( their ground state ) . the binding energy and spatial extent of the bound state is insensitive to disorder beyond the point where single - particle states become localized . this result for the cooper pair corresponds to the familiar anderson theorem for the many - body problem of dirty superconductors and is valid in a regime where the attraction and the disorder are not too strong so that electrons are paired in time - reversed single - particle eigenstates . these localized states can then be combined coherently into an extended pair wave function ( eq.([extended ] ) ) . in analogy with the case of dirty superconductors discussed in @xcite where the superfluid density is greatly reduced by the disorder , we also expect the energy of the wave function ( [ extended ] ) to be much less sensitive to the boundary conditions than that in a clean system . in other words , disorder has a stronger effect on the sensitivity to boundary conditions than on the localization length . it is possible to go away from this anderson regime in two ways . if the interaction is increased then both the binding energy and the effective mass of the pair increase , so the pair becomes more localized . on the other hand , if the interaction is kept not too strong but the disorder is increased , the binding energy and the localization length are reduced . we want to prove that @xmath75 . the binding energy e is obtained from the equation @xmath76 the density of states is @xmath77 where @xmath78 is the average density of states and @xmath79 the noise due to disorder . then @xmath80 where @xmath81 is the binding energy derived from @xmath78 . expanding the right - hand side of the above equation for small @xmath82 and taking into account the definition of @xmath81 we arrive at @xmath83 we make the following assumptions about the moments ( averages ) of @xmath79 : @xmath84 and @xmath85 . this latter condition is only intended to express the fact that the correlation persists over an energy of the order of the single - particle level spacing . taking the square of ( [ de ] ) and averaging we obtain r. a. rmer and m. schreiber , phys . lett * 78 * , 515 ( 1997 ) and phys . lett . * 78 * , 4890 ( 1997 ) ; k. frahm et al . lett . * 78 * , 4889 ( 1997 ) ; t. vojta , r. a. rmer and m. schreiber , cond - mat/9702241
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we discuss the effect of disorder on the coherent propagation of the bound state of two attracting particles .
it is shown that a result analogous to the anderson theorem for dirty superconductors is also valid for the cooper problem , namely , that the pair wave function is extended beyond the single - particle localization length if the latter is large .
a physical justification is given in terms of the thouless block - scaling picture of localization .
these arguments are supplemented by numerical simulations .
with increasing disorder we find a transition from a regime in which the interaction delocalizes the pair to a regime in which the interaction enhances localization .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
describing the properties of infinite nuclear matter has long been an important benchmark for realistic models of the nuclear force and the applied many - body methods . recent calculations @xcite have shown that the ( goldstone ) linked - diagram expansion ( up to at least second order ) can provide an adequate description of the zero - temperature equation of state when realistic two - nucleon and three - nucleon forces are employed . in the present work we study nuclear matter from the perspective of landau s fermi liquid theory @xcite , which is a framework for describing excitations of strongly - interacting normal fermi systems in terms of weakly - interacting quasiparticles . although the complete description of the interacting many - body ground state lies beyond the scope of this theory , various bulk equilibrium and transport properties are accessible through the quasiparticle interaction . the interaction between two quasiparticles can be obtained microscopically within many - body perturbation theory by functionally differentiating the total energy density twice with respect to the quasiparticle distribution function . most previous studies using realistic nuclear forces have computed only the leading - order contribution to the quasiparticle interaction exactly , while approximately summing certain classes of diagrams to all orders @xcite . in particular , the summation of particle - particle ladder diagrams in the brueckner @xmath3-matrix was used to tame the strong short - distance repulsion present in most realistic nuclear force models , and the inclusion of the induced interaction of babu and brown @xcite ( representing the exchange of virtual collective modes between quasiparticles ) was found to be essential for achieving the stability of nuclear matter against isoscalar density oscillations . to date , few works have studied systematically the order - by - order convergence of the quasiparticle interaction using realistic models of the nuclear force . in ref.@xcite the pion - exchange contribution to the quasiparticle interaction in nuclear matter was obtained at one - loop order , including also the effects of @xmath4-exchange with intermediate @xmath5-isobar states . in the present work we derive general expressions for the second - order quasiparticle interaction in terms of the partial wave matrix elements of the underlying realistic nucleon - nucleon ( nn ) potential . the numerical accuracy of the second - order calculation in this framework is tested with a scalar - isoscalar - exchange potential as well as a ( modified ) pion - exchange interaction , both of which allow for exact analytical solutions at second order . we then study the idaho n@xmath0lo chiral nn interaction @xcite and derive from this potential a set of low - momentum nucleon - nucleon interactions @xcite , which at a sufficiently coarse resolution scale ( @xmath6@xmath7 ) provide a model - independent two - nucleon interaction and which have better convergence properties when employed in many - body perturbation theory @xcite . we extract the four components of the isotropic ( @xmath1 ) quasiparticle interaction of which two are related to the nuclear matter incompressibility @xmath8 and symmetry energy @xmath9 . the @xmath2 fermi liquid parameters , associated with the angular dependence of the quasiparticle interaction , are used to obtain properties of the quasiparticles themselves , such as their effective mass @xmath10 and the anomalous orbital @xmath11-factor . our present treatment focuses on the role of two - nucleon interactions . it does not treat the contribution of the three - nucleon force to the quasiparticle interaction but sets a reliable framework for future calculations employing also the leading - order chiral three - nucleon interaction @xcite . in the present work , we therefore seek to identify deficiencies that remain when only two - nucleon forces are included in the calculation of the quasiparticle interaction . the paper is organized as follows . in section [ qpisec ] we describe the microscopic approach to landau s fermi liquid theory and relate the @xmath1 and @xmath2 landau parameters to various nuclear matter observables . we then describe in detail our complete calculation of the quasiparticle interaction to second order in perturbation theory . in section [ calres ] we first apply our scheme to analytically - solvable model interactions ( scalar - isoscalar boson exchange and modified pion exchange ) in order to assess the numerical accuracy . we then employ realistic low - momentum nucleon - nucleon interactions and make contact to experimental quantities through the landau parameters . the paper ends with a summary and outlook . the physics of ` normal ' fermi liquids at low temperatures is governed by the properties and interactions of quasiparticles , as emphasized by landau in the early 1960 s . since quasiparticles are well - defined only near the fermi surface ( @xmath12 ) where they are long - lived , landau s theory is valid only for low - energy excitations about the interacting ground state . the quantity of primary importance in the theory is the interaction energy between two quasiparticles , which can be obtained by functionally differentiating the ground - state energy density twice with respect to the quasiparticle densities : @xmath13 where @xmath14 and @xmath15 are spin and isospin quantum numbers . the general form of the central part of the quasiparticle interaction in nuclear matter excluding tensor components , etc . , is given by @xmath16 { \vec \sigma}_1 \cdot { \vec \sigma}_2 \ , , \label{ffunction}\ ] ] where @xmath17 and @xmath18 are respectively the spin and isospin operators of the two nucleons on the fermi sphere @xmath19 . for notational simplicity we have dropped the dependence on the quantum numbers @xmath20 and @xmath21 , which is introduced through the matrix elements of the operators : @xmath22 and @xmath23 . as it stands in eq . ( [ ffunction ] ) , the quasiparticle interaction is defined for any nuclear density @xmath24 , but the quantities of physical interest result at nuclear matter saturation density @xmath25@xmath26 ( corresponding to @xmath27@xmath7 ) . for two quasiparticles on the fermi surface @xmath19 , the remaining angular dependence of their interaction can be expanded in legendre polynomials of @xmath28 : @xmath29 where @xmath30 represents @xmath31 or @xmath32 , and the angle @xmath33 is related to the relative momentum @xmath34 through the relation @xmath35 it is conventional to factor out from the quasiparticle interaction the density of states per unit energy and volume at the fermi surface , @xmath36 , where @xmath10 is the nucleon effective mass ( see eq . ( [ effmasseq ] ) ) and @xmath37@xmath7 . this enables one to introduce an equivalent set of dimensionless fermi liquid parameters @xmath38 and @xmath39 through the relation @xmath40 p_l({\rm cos}\ , \theta ) . \label{ffunction2}\ ] ] provided the above series converges quickly in @xmath41 , the interaction between two quasiparticles on the fermi surface is governed by just a few constants which can be directly related to a number of observable quantities as we now discuss . the quasiparticle effective mass @xmath10 is related to the slope of the single - particle potential at the fermi surface and can be obtained from the landau parameter @xmath42 by invoking galilean invariance . the relation is found to be @xmath43 where @xmath44 mev is the free nucleon mass . the compression modulus @xmath8 of symmetric nuclear matter can be obtained from the isotropic ( @xmath1 ) spin- and isospin - independent component of the quasiparticle interaction @xmath45 the compression modulus of infinite nuclear matter can not be measured directly , but its value @xmath46 mev can be estimated from theoretical predictions of giant monopole resonance energies in heavy nuclei @xcite . the nuclear symmetry energy @xmath9 can be computed from the isotropic spin - independent part of the isovector interaction : @xmath47 global fits of nuclear masses with semi - empirical binding energy formulas provide an average value for the symmetry energy of @xmath48 mev over densities in the vicinity of saturated nuclear matter @xcite . the quasiparticle interaction provides also a link to the properties of single - particle and collective excitations . in particular , the orbital @xmath11-factor for valence nucleons ( i.e. , quasiparticles on the fermi surface ) is different by the amount @xmath49 from that of a free nucleon : @xcite : @xmath50 one possible mechanism for the anomalous orbital @xmath11-factor @xmath49 are meson exchange currents @xcite , which arise in the isospin - dependent components of the nucleon - nucleon interaction . according to eq . ( [ aog ] ) , the renormalized isoscalar and isovector orbital @xmath11-factors are @xmath51 are different . the former receives no correction , while the latter is sizably enhanced by the ( reduced ) effective mass @xmath10 as well as by the ( positive ) landau parameter @xmath52 . it receives a large contribution from one - pion exchange . nuclear matter allows for a rich variety of collective states , including density ( breathing mode ) , spin ( magnetic dipole mode ) , isospin ( giant dipole mode ) , and spin - isospin ( giant gamow - teller mode ) excitations . as previously discussed , the breathing mode is governed by the incompressibility @xmath8 of nuclear matter @xcite . the energy of the ( isovector ) giant dipole mode is correlated with the nuclear symmetry energy @xmath9 @xcite , while the dipole sum rule @xcite @xmath53 is connected to the anomalous orbital @xmath11-factor @xmath49 with @xmath54 . experimental results @xcite are consistent with a value of the anomalous orbital @xmath11-factor of @xmath55 . finally , the giant gamow - teller resonance has been widely studied due to its connection to the nuclear spin - isospin response function and for ruling out pion condensation in moderately - dense nuclear matter . an analysis of the experimental excitation energies and transition strengths @xcite leads to a value for the parameter @xmath56 which is used to model the spin - isospin interaction in nuclei as a zero - range contact interaction . as a convention it is related to the dimensionless landau parameter @xmath57 by @xmath58 where @xmath59 is the strong @xmath60 coupling constant . it is well - known that the giant gamow - teller resonances receive important contributions from the coupling to @xmath5-hole excitations @xcite . such dynamical effects due to non - nucleonic degrees of freedom are reflected in the leading - order , @xmath4-exchange three - nucleon interaction to be included in future work @xcite . expanding the energy density to second - order in the ( goldstone ) linked - diagram expansion and differentiating twice with respect to the nucleon distribution function , one obtains for the first two contributions to the quasiparticle interaction @xmath61 and @xmath62 in eqs . ( [ order1qp ] ) and ( [ order2qp ] ) the quantity @xmath63 denotes the antisymmetrized two - body potential ( with units of @xmath64 ) given by @xmath65 in the partial wave basis , and in eq . ( [ order2qp ] ) the summation is over intermediate - state momenta , spins and isospins . we specify our sign and normalization conventions through the perturbative relation between diagonal two - body matrix elements and phase shifts : @xmath66 . the first - order term of eq . ( [ order1qp ] ) is just the diagonal matrix element of the antisymmetrized two - body interaction , while the second - order term ( eq . ( [ order2qp ] ) ) has been separated into particle - particle , hole - hole , and particle - hole terms depicted diagrammatically in fig . [ pphhph ] . the distribution function @xmath67 is the usual step function for the nuclear matter ground state : @xmath68 in the following , we discuss the general evaluation of eqs . ( [ order1qp ] ) and ( [ order2qp ] ) for interactions given in the partial - wave basis . we first define the spin - averaged quasiparticle interaction @xmath69 , which is obtained from the full quasiparticle interaction by averaging over the spin - substates : @xmath70 where @xmath71 , and in @xmath72 the spins and isospins of the two quasiparticles are coupled to total spin @xmath73 and total isospin @xmath74 . we take an isospin - symmetric two - body potential and thus the quasiparticle interaction is independent of @xmath75 . the first - order contribution to the central part of the quasiparticle interaction is then obtained by summing over the allowed partial wave matrix elements : @xmath76 note that there is an additional factor of @xmath77 in eq . ( 41 ) in ref . @xcite and eq . ( 28 ) in ref . @xcite due to a different normalization convention . from eq . ( [ qpdec1 ] ) we can project out the individual components of the quasiparticle interaction using the appropriate linear combinations of @xmath78 with @xmath73 and @xmath74 : @xmath79 the leading - order expressions , eqs . ( [ qpdec1 ] ) and ( [ lincomb ] ) , give the full @xmath80-dependence ( i.e. , angular dependence ) of the quasiparticle interaction , and therefore one can project out the density - dependent landau parameters for arbitrary @xmath41 : @xmath81 for the second - order contributions to the quasiparticle interaction , the complete @xmath80-dependence is in general not easily obtained ( e.g. , for the particle - hole term ) . we instead compute the landau parameters for each @xmath41 separately , choosing the total momentum vector to be aligned with the @xmath82-axis . in the following , the two quasiparticle momenta are labeled @xmath83 and @xmath84 , while the intermediate - state momenta are labeled @xmath85 and @xmath86 . for the particle - particle contribution one finds @xmath87 where @xmath88 are associated legendre functions , @xmath89 , @xmath90 , @xmath91 and @xmath92 with @xmath93 . similarly , for the hole - hole diagram one obtains @xmath94 averaging over the spin substates and employing eq . ( [ lincomb ] ) with the substitution @xmath95 again yields the individual spin and isospin components of the quasiparticle interaction . the evaluation of the particle - hole diagram proceeds similarly ; however , in this case the coupling of the two quasiparticles to total spin ( and isospin ) requires an additional step . coupling to states with @xmath96 is achieved by taking the combinations ( neglecting isospin for simplicity ) @xmath97 \frac{n_3 ( 1-n_4)}{\epsilon_1 + \epsilon_3 - \epsilon_2 - \epsilon_4}. \label{phcomb}\end{aligned}\ ] ] we provide the expression for @xmath98 ( corresponding to @xmath1 ) in uncoupled quasiparticle spin and isospin states , appopriate for evaluating the first term in eq . ( [ phcomb ] ) , which can be easily generalized in order to obtain the second term . we find @xmath99 where now @xmath100 , @xmath101 , and the total momentum @xmath102 . the angle @xmath103 between @xmath104 and @xmath105 is fixed ( via @xmath106 together with @xmath107 ) by the relation @xmath108 , and analogously for the angle @xmath109 between @xmath110 and @xmath105 . the combination of spin clebsch - gordan coefficients that arises in the above expression is denoted by @xmath111 and likewise for the combination of isospin clebsch - gordan coefficients . in computing the particle - hole term for @xmath112 , we use @xmath113 and employ the addition theorem for spherical harmonics to write @xmath114 in terms of @xmath115 , @xmath116 and an azimuthal angle @xmath117 . the involved integral @xmath118 gives different selection rules for the @xmath119 values of the associated legendre functions in eq . ( [ ph2nd ] ) . in deriving eqs . ( [ pp2nd])([ph2nd ] ) , we have assumed that the intermediate - state energies in eq . ( [ order2qp ] ) are those of free particles : @xmath120 . later we will include the first - order correction to the dispersion relation arising from the in - medium self - energy , which leads to the substitution @xmath121 in the above equations . the numerical computation of the quasiparticle interaction at second order is obviously quite intricate , and truncations in the number of included partial waves and in the momentum - space integrations are necessary . in such a situation it is very helpful to have available analytical results for simple model interactions in order to test the accuracy of the numerical calculations . for that purpose we derived in this subsection analytical expressions for the quasiparticle interaction up to second order arising from ( i ) massive scalar - isoscalar boson exchange and ( ii ) pion exchange modified by squaring the static propagator . we omit all technical details of these calculations which can be found ( for @xmath1 ) in ref . @xcite for the case of tree - level and one - loop ( i.e. , second - order ) pion - exchange . in the present treatment the second - order quasiparticle interaction is organized differently than in eq . ( [ order2qp ] ) . the explicit decomposition of the in - medium nucleon propagator into a particle and hole propagator is replaced by the sum of a `` vacuum '' and `` medium insertion '' component : @xmath122 and the organization is now in the number of medium insertions rather than in terms of particle and hole intermediate states . the central parts of the quasiparticle interaction are constructed for any @xmath41 through an angle - averaging procedure @xmath123 in this equation @xmath124 represents the effective model interaction computed up to one - loop order ( second order ) . we first consider as a generic example the exchange of a scalar - isoscalar boson with mass @xmath125 and coupling constant @xmath126 ( to the nucleon ) . in momentum and coordinate space it gives rise to central potentials of the form @xmath127 for the first - order contributions to the @xmath128 landau parameters one finds @xmath129 \ , , \label{sciso1}\ ] ] @xmath130 \,,\ ] ] where @xmath131 and @xmath132 are short - hand notations for the spin - spin and isospin - isospin operators . the dimensionless variable @xmath133 denotes the ratio of the fermi momentum @xmath134 to the scalar boson mass @xmath125 . note that in this approach both direct and crossed diagrams can contribute . the crossed diagrams have to be multiplied by the negative product of the spin- and isospin - exchange operators @xmath135 . at second order there are five classes of diagrammatic contributions , shown in fig . [ qp2norbert ] , to the quasiparticle interaction . the direct terms from iterated ( second order ) boson exchange , see fig . [ qp2norbert](a ) , read @xmath136 @xmath137 \,,\ ] ] whereas the corresponding crossed terms ( b ) have the form @xmath138 @xmath139 \ , .\ ] ] the coupling of the exchanged boson to nucleon - hole states , fig . [ qp2norbert](c ) , gives rise to nonvanishing crossed terms which read @xmath140 \,,\ ] ] @xmath141 \ , .\ ] ] pauli blocking occurs in the planar- and crossed - box diagrams , fig . [ qp2norbert](d)(e ) , and for the sum of their direct terms one finds the forms @xmath142 \,,\ ] ] @xmath143 \ , . \end{aligned}\ ] ] on the other hand , the crossed terms of the planar - box diagram with pauli blocking , see fig . [ qp2norbert](f ) yield @xmath144 @xmath145 \bigg\}\ , , \end{aligned}\ ] ] with the auxiliary polynomial @xmath146 . finally , the density - dependent vertex correction to one - boson exchange , fig . [ qp2norbert](g ) , provides a nonzero contribution only in the crossed diagram . the corresponding expressions for the @xmath128 landau parameters read @xmath147 @xmath148 since at most double integrals over well - behaved functions are involved in the expressions in eqs . ( [ sciso1])([sc2 g ] ) , they can be evaluated easily to high numerical precision . after summing them together , they provide a crucial check for our calculation of the second - order quasiparticle interaction in the partial wave basis ( see section [ qp2n ] ) . we set the scalar boson mass @xmath149 mev and coupling constant @xmath150 and work with the partial wave matrix elements following from the central potential @xmath151 in eq . ( [ scisop ] ) . table [ scis ] shows the dimensionful fermi liquid parameters ( labeled ` exact ' ) as obtained from the above analytical formulas at nuclear matter saturation density ( @xmath152 @xmath7 ) . due to the simple spin and isospin dependence of the underlying interaction the constraint @xmath153 holds . for comparison we show also the first- and second - order results obtained with the general partial wave expansion . the second - order terms are further subdivided into particle - particle , hole - hole and particle - hole contributions . we find agreement between both methods to within 1% or better for all @xmath128 fermi liquid parameters . in order to achieve this accuracy , the expansions must be carried out through at least the lowest 15 partial waves . & @xmath154 [ @xmath64 ] & @xmath156 [ @xmath64 ] & @xmath157 [ @xmath64 ] & @xmath158 [ @xmath64 ] & @xmath42 [ @xmath64 ] & @xmath159 [ @xmath64 ] & @xmath160 [ @xmath64 ] & @xmath161 [ @xmath64 ] + 1st & @xmath162 & 0.164 & 0.164 & 0.164 & 0.060 & 0.060 & 0.060 & 0.060 + 2nd(pp ) & @xmath163 & 0.056 & 0.056 & 0.056 & 0.038 & @xmath164 & @xmath164 & @xmath164 + 2nd(hh ) & @xmath165 & 0.010 & 0.010 & 0.010 & 0.042 & @xmath166 & @xmath166 & @xmath166 + 2nd(ph ) & 0.198 & 0.061 & 0.061 & 0.061 & 0.100 & 0.085 & 0.085 & 0.085 + total & @xmath167 & 0.291 & 0.291 & 0.291 & 0.240 & 0.127 & 0.127 & 0.127 + exact & @xmath167 & 0.292 & 0.292 & 0.292 & 0.242 & 0.127 & 0.127 & 0.127 + a feature of all realistic nn interactions is the presence of a strong tensor force , which results in mixing matrix elements between spin - triplet states differing by two units of orbital angular momentum . at second order these mixing matrix elements generate substantial contributions to the @xmath128 fermi liquid parameters . in order to test the numerical accuracy of our partial wave expansion scheme for the additional complexity arising from tensor forces , we consider now the quasiparticle interaction in nuclear matter generated by ( modified ) `` pion '' exchange . to be specific we take a nucleon - nucleon potential in momentum space of the form @xmath168 where @xmath11 is a dimensionless coupling constant and @xmath169 a variable `` pion '' mass . the isovector spin - spin and tensor potentials in coordinate space following from @xmath170 read @xmath171 the basic motivation for squaring the propagator in eq . ( [ mopem ] ) is to tame the tensor potential at short distances , and thereby one avoids the linear divergence that would otherwise occur in iterated ( second - order ) one - pion - exchange . in the presence of non - convergent loop integrals , analytical and numerical treatments become difficult to match properly . let us now enumerate the contributions at first and second order to the @xmath128 landau parameters as they arise from modified `` pion '' exchange . the first - order contributions read @xmath172 \ , , \label{pm1}\ ] ] @xmath173 \,,\ ] ] with the abbreviation @xmath174 . for the second - order contributions we follow the labeling @xmath175 introduced previously for scalar - isoscalar boson exchange : @xmath176 \,,\ ] ] @xmath177 \,,\ ] ] @xmath178 \bigg\ } \,,\end{aligned}\ ] ] @xmath179 \nonumber \\ & & + ( \boldsymbol\sigma-3 ) \bigg[{1\over 1 + 2u^2}-{1\over u^2}+{2+u^2\over 2u^4}\ln{1 + 2u^2\over 1+u^2 } \nonumber \\ & & + \int_0^u\!\ ! dx\ , { u^2 - 2x^2 \over u^4(1 + 2x^2)^3}(1 + 4x^2 + 8x^4 ) ( \arctan 2x-\arctan x)\bigg ] \bigg\ } \ , . \end{aligned}\ ] ] @xmath180 \,,\ ] ] @xmath181 \ , .\ ] ] @xmath182 \,,\ ] ] @xmath183 \ , , \end{aligned}\ ] ] @xmath184 \,,\ ] ] @xmath185 \ , .\ ] ] we split the crossed terms from the planar - box diagram with pauli blocking , see fig . [ qp2norbert](f ) , into factorizable parts : @xmath186 ^ 2 \,,\ ] ] @xmath187\,,\ ] ] @xmath188 @xmath189 and non - factorizable parts : @xmath190 } \nonumber \\ & & + { 1\over r^{3/2}}(u^2-x^2-y^2 + 8x^2y^2)\ln{u \sqrt{r } + ( 1 - 4x y)(x - y ) \over u \sqrt{r } + ( 4x y-1)(x - y ) } \bigg\ } \ , , \end{aligned}\ ] ] @xmath191 ^ 2 + { 32x^3 \over ( 1 + 4x^2)^2 } \bigg [ u \ln{4(u - x)\over u+x}-x \ln{u^2-x^2 \over x^2 } \nonumber \\ & & + \int_0^{u - x } \!\!\!dy\,\bigg({4u(x - y)(1 + 4x y ) ( 1 + 2u^2 - 4x^2)\over r\,[4(x - y)^2+r ] } + \big(2(1 + 2u^2 - 4x^2 ) \nonumber \\ & & \times ( u^2-x^2-y^2 + 8x^2y^2 ) + r\big ) { 1\over r^{3/2}}\ln{u \sqrt{r } + ( 1 - 4x y)(x - y ) \over u \sqrt{r } + ( 4x y-1)(x - y ) } \bigg ) \bigg]\bigg\ } \ , , \end{aligned}\ ] ] with auxiliary polynomial @xmath146 . these two pieces , @xmath192 and @xmath193 , are distinguished by whether the remaining nucleon propagator can be cancelled or not by terms from the product of ( momentum - dependent ) @xmath60 interaction vertices in the numerator . finally , the density - dependent vertex corrections to modified `` pion '' exchange have nonzero crossed terms , which we split again into factorizable parts : @xmath194 \bigg[1-{1 + 2u^2 \over 4u^2}\ln(1 + 4u^2)\bigg ] \,,\ ] ] @xmath195\nonumber \\ & & \times \bigg [ 3u^2+{u^2\over 1 + 4u^2}-(1+u^2 ) \ln(1 + 4u^2 ) \bigg]\ , , \end{aligned}\ ] ] and non - factorizable parts : @xmath196 \nonumber \\ & & \times \bigg\ { { 2ux ( 1 + 4u^2)^{-1 } \over 1 + 4u^2 - 4x^2}+{u^2-x^2\over ( 1 + 4u^2 - 4x^2)^{3/2 } } \ln { ( u\sqrt{1 + 4u^2 - 4x^2 } + x ) ^2 \over(1 + 4u^2)(u^2-x^2 ) } \bigg\ } \ , , \end{aligned}\ ] ] @xmath197 \nonumber \\ & & \times \bigg\ { { 4ux ( 1 + 2u^2)\over ( 1 + 4u^2)(1 + 4u^2 - 4x^2 ) } -\ln{u+x\over u - x } \nonumber \\ & & + { 1+(u^2-x^2)(6 + 4u^2 ) \over ( 1 + 4u^2 - 4x^2)^{3/2 } } \ln { ( u\sqrt{1 + 4u^2 - 4x^2 } + x ) ^2 \over(1 + 4u^2 ) ( u^2-x^2 ) } \bigg\ } \ , . \label{pm2g}\end{aligned}\ ] ] together with the coupling constant @xmath198 we choose a large `` pion '' mass @xmath199mev in order to suppress partial wave matrix elements from the model interaction @xmath170 beyond @xmath200 in the numerical computations based on the partial wave expansion scheme . we show in table [ mpi ] the @xmath128 fermi liquid parameters ( at @xmath27 @xmath7 ) for the modified `` pion '' exchange interaction up to second order in perturbation theory . the summed results from the analytic formulas eqs . ( [ pm1])([pm2 g ] ) are labeled `` exact '' and compared to the results obtained by first evaluating the interaction in the partial wave basis and then using eqs . ( [ pp2nd])-([ph2nd ] ) . as in the case of scalar - isoscalar exchange , we find excellent agreement between the two ( equivalent ) methods . & @xmath154 [ @xmath64 ] & @xmath156 [ @xmath64 ] & @xmath157 [ @xmath64 ] & @xmath158 [ @xmath64 ] & @xmath42 [ @xmath64 ] & @xmath159 [ @xmath64 ] & @xmath160 [ @xmath64 ] & @xmath161 [ @xmath64 ] + 1st & 0.244 & @xmath2010.081 & @xmath2010.081 & 0.027 & @xmath2010.079 & 0.026 & 0.026 & @xmath2010.009 + 2nd(pp ) & @xmath2010.357 & @xmath2010.062 & 0.269 & 0.104 & 0.018 & @xmath2010.005 & 0.027 & 0.009 + 2nd(hh ) & @xmath2010.017 & @xmath2010.002 & 0.009 & 0.003 & 0.029 & 0.003 & @xmath2010.014 & @xmath2010.005 + 2nd(ph ) & 0.146 & @xmath2010.023 & 0.027 & 0.008 & 0.008 & 0.010 & 0.036 & @xmath2010.003 + total & 0.017 & @xmath2010.169 & 0.224 & 0.142 & @xmath2010.024 & 0.035 & 0.075 & @xmath2010.009 + exact & 0.017 & @xmath2010.169 & 0.224 & 0.142 & @xmath2010.023 & 0.035 & 0.074 & @xmath2010.009 + after having verified the numerical accuracy of our partial wave expansion scheme , we extend in this section the discussion to realistic nuclear two - body potentials . we start with the idaho n@xmath0lo chiral nn interaction @xcite and employ renormalization group methods @xcite to evolve this ( bare ) interaction down to a resolution scale ( @xmath6 @xmath7 ) at which the nn interaction becomes universal . the quasiparticle interaction in nuclear matter has been studied previously with such low - momentum nuclear interactions @xcite , but a complete second - order calculation has never been performed . given the observed better convergence properties of low - momentum interactions in nuclear many - body calculations , we wish to study here systematically the order - by - order convergence of the quasiparticle interaction derived from low - momentum nn potentials . a complete treatment of low - momentum nuclear forces requires the consistent evolution of two- and three - body forces together . we postpone the inclusion of contributions to the quasiparticle interaction from the ( chiral ) three - nucleon force to upcoming work @xcite . in table [ n3loc ] we compare the @xmath128 fermi liquid parameters obtained from the bare chiral n@xmath0lo potential to those of low - momentum interactions obtained by integrating out momenta above a resolution scale of @xmath202@xmath7 and @xmath203@xmath7 . the intermediate - state energies in the second - order diagrams are those of free nucleons @xmath120 , and we include partial waves up to @xmath200 which result in well - converged @xmath128 fermi liquid parameters . comparing the results at first - order , we find a large decrease in the isotropic spin- and isospin - independent landau parameter @xmath154 as the decimation scale decreases . this enhances the ( apparent ) instability of nuclear matter against isoscalar density oscillations . the effect results largely from integrating out some short - distance repulsion in the bare n@xmath0lo interaction . a repulsive contact interaction @xmath204 ( contributing with equal strength @xmath205 in singlet and triplet @xmath206-waves ) gives rise to a first - order quasiparticle interaction of the form @xmath207 and no contributions for @xmath208 . thus , integrating out the short - distance repulsion in the chiral n@xmath0lo potential yields a large decrease in @xmath154 and a ( three - times ) weaker increase in @xmath209 , and @xmath210 . the increase in @xmath211 gives rise to an increase in the nuclear symmetry energy at saturation density by approximately 20% for interactions evolved down to @xmath6 @xmath7 . overall , the scale dependence of the first - order @xmath2 landau parameters is weaker , and in particular the two isospin - independent ( @xmath42 and @xmath159 ) components of the @xmath2 quasiparticle interaction are almost scale independent . however , the parameter @xmath212 increases as the cutoff scale is lowered , which results according to eq . ( [ aog ] ) in an increase in the anomalous orbital @xmath11-factor by 1015% . & @xmath154 [ @xmath64 ] & @xmath156 [ @xmath64 ] & @xmath157 [ @xmath64 ] & @xmath158 [ @xmath64 ] & @xmath42 [ @xmath64 ] & @xmath159 [ @xmath64 ] & @xmath160 [ @xmath64 ] & @xmath161 [ @xmath64 ] + 1st & @xmath2011.274 & 0.298 & 0.200 & 0.955 & @xmath2011.018 & 0.529 & 0.230 & 0.090 + 2nd(pp ) & @xmath2011.461 & 0.023 & 0.686 & 0.255 & 0.041 & @xmath2010.059 & 0.334 & 0.254 + 2nd(hh ) & @xmath2010.271 & 0.018 & 0.120 & 0.041 & 0.276 & 0.041 & @xmath2010.144 & @xmath2010.009 + 2nd(ph ) & 1.642 & @xmath2010.057 & 0.429 & 0.162 & 0.889 & @xmath2010.143 & 0.130 & 0.142 + total & @xmath2011.364 & 0.281 & 1.436 & 1.413 & 0.188 & 0.367 & 0.550 & 0.477 + + & @xmath154 [ @xmath64 ] & @xmath156 [ @xmath64 ] & @xmath157 [ @xmath64 ] & @xmath158 [ @xmath64 ] & @xmath42 [ @xmath64 ] & @xmath159 [ @xmath64 ] & @xmath160 [ @xmath64 ] & @xmath161 [ @xmath64 ] + 1st & @xmath2011.793 & 0.357 & 0.394 & 1.069 & @xmath2010.996 & 0.493 & 0.357 & 0.152 + 2nd(pp ) & @xmath2010.974 & @xmath2010.098 & 0.594 & 0.185 & @xmath2010.129 & @xmath2010.003 & 0.252 & 0.193 + 2nd(hh ) & @xmath2010.358 & 0.030 & 0.169 & 0.075 & 0.338 & 0.028 & @xmath2010.180 & @xmath2010.042 + 2nd(ph ) & 2.102 & 0.095 & 0.588 & 0.254 & 1.512 & 0.003 & 0.204 & 0.329 + total & @xmath2011.023 & 0.385 & 1.744 & 1.583 & 0.725 & 0.521 & 0.634 & 0.632 + + & @xmath154 [ @xmath64 ] & @xmath156 [ @xmath64 ] & @xmath157 [ @xmath64 ] & @xmath158 [ @xmath64 ] & @xmath42 [ @xmath64 ] & @xmath159 [ @xmath64 ] & @xmath160 [ @xmath64 ] & @xmath161 [ @xmath64 ] + 1st & @xmath2011.919 & 0.327 & 0.497 & 1.099 & @xmath2011.034 & 0.475 & 0.409 & 0.178 + 2nd(pp ) & @xmath2010.864 & @xmath2010.079 & 0.507 & 0.164 & @xmath2010.130 & 0.011 & 0.236 & 0.174 + 2nd(hh ) & @xmath2010.386 & 0.022 & 0.195 & 0.085 & 0.355 & 0.034 & @xmath2010.195 & @xmath2010.049 + 2nd(ph ) & 2.033 & 0.164 & 0.493 & 0.292 & 1.620 & 0.098 & 0.234 & 0.412 + total & @xmath2011.135 & 0.434 & 1.692 & 1.640 & 0.812 & 0.617 & 0.684 & 0.715 + considering the three parts comprising the second - order quasiparticle interaction , we find large contributions from both the particle - particle ( @xmath213 ) and particle - hole ( @xmath214 ) diagrams . in particular , the @xmath214 term is quite large , which suggests the need for an exact treatment of this contribution which until now has been absent in the literature . as the decimation scale is lowered , the @xmath213 contribution is generally reduced while the hole - hole ( @xmath215 ) and @xmath214 contributions are both increased . in previous studies , the @xmath215 diagram has often been neglected since it was assumed to give a relatively small contribution to the quasiparticle interaction . however , we learn from our exact calculation that its effects are non - negligible for all of the spin - independent landau parameters . at second order , the contributions to @xmath154 are sizable and approximately cancel each other for the bare n@xmath0lo chiral nn interaction , but they become more strongly repulsive as the resolution scale @xmath216 is decreased . this reduces the large decrease at leading - order in @xmath154 effected through the renormalization group decimation , so that after including the second - order corrections , the spread in the values of @xmath154 for all three potentials ( bare n@xmath0lo and its decimations to @xmath202@xmath7 and @xmath217@xmath7 ) is much smaller than at first order . for each of the three different potentials , the second - order terms are strongly coherent in both the @xmath211 and @xmath210 channels . in the former case , this change alone would give rise to a dramatic increase the nuclear symmetry energy @xmath9 . this effect will be partly reduced through the increase in the quasiparticle effective mass @xmath10 , which we see is now close to unity for the bare n@xmath0lo chiral interaction but which is strongly scale - dependent and enhanced above the free mass @xmath218 as the decimation scale is lowered . the parameter @xmath210 , related to the energy of giant gamow - teller resonances , is increased by approximately 50% after inclusion of the second - order diagrams . in fig . [ dendepflp ] we plot the fermi liquid parameters of @xmath219 as a function of density @xmath220 from @xmath221 to @xmath222 . we see that all of the @xmath1 parameters , together with @xmath42 , are enhanced at lower densities . . here @xmath223@xmath26 is the nuclear matter saturation density.,height=453 ] we now discuss the leading - order ( hartree - fock ) contribution to the nucleon single - particle energy . the second - order contributions to the quasiparticle interaction get modified through the resulting change in the energy - momentum relation for intermediate - state nucleons . for a nucleon with momentum @xmath224 , the first - order ( in - medium ) self - energy correction reads @xmath225 where @xmath226 . in fig . [ effmassfig ] we plot the single - particle energy as a function of the momentum @xmath227 . in the left figure we show the results for all three nn interactions considered in the previous section at a fermi momentum of @xmath27 @xmath7 . in the right figure we consider only the low - momentum nn interaction with @xmath202 @xmath7 for three different densities . in all cases one can fit the dispersion relation with a parabolic form @xmath228 with @xmath10 the effective mass and @xmath5 the depth of the single - particle potential . from the figure one sees that this form holds well across the relevant range of momenta @xmath227 . in fig . [ spefig ] we show the extracted effective mass and potential depth for the three different interactions as a function of the density . the energy shift @xmath5 shows more sensitivity to the decimation scale @xmath216 than the effective mass @xmath10 . at saturation density @xmath229@xmath26 , the variation in @xmath5 is about 30% while the spread in @xmath230 is less than 10% . overall , the effective mass extracted from a global fit to the momentum dependence of the single - particle energy is in good agreement with the local effective mass at the fermi surface @xmath231 , encoded in the landau parameter @xmath42 . the largest difference occurs for the bare idaho n@xmath0lo potential owing to the larger momentum range over which eq . ( [ disp ] ) is fit to the spectrum . ) and fit ( lines ) with the form eq . ( [ disp ] ) characterized by an effective mass plus energy shift.,height=642 ] as a function of density for three different nn potentials . , height=529 ] we employ the quadratic parametrization of the single - particle energy in the second - order contributions to the quasiparticle interaction eqs . ( [ pp2nd])([ph2nd ] ) . this greatly simplifies the inclusion of the ( first - order ) in - medium nucleon self energy . the second - order quasiparticle interaction is effectively multiplied by the same factor @xmath230 , since the constant shift @xmath5 cancels in the energy differences . we then compute the dimensionless fermi liquid parameters by factoring out the density of states at the fermi surface @xmath232 . in table [ finaltable ] we show the results at @xmath27 @xmath7 for the idaho n@xmath0lo potential as well as @xmath233 for @xmath202@xmath7 and @xmath217@xmath7 . in addition , we have tabulated the theoretical values of the different nuclear observables that can be obtained from the fermi liquid parameters . the quasiparticle effective mass of the bare n@xmath0lo chiral nn interaction is @xmath234 , but this ratio increases beyond 1 for the low - momentum nn interactions . the inclusion of self - consistent single - particle energies in the second - order diagrams reduces the very large enhancement in the effective mass seen previously in table [ n3loc ] . compared to the other three @xmath1 landau parameters , the spin - spin interaction in nuclear matter is relatively small ( @xmath235 ) . despite the strong repulsion in @xmath236 that arises from the second - order @xmath214 diagram , we see that nuclear matter remains unstable against isoscalar density fluctuations ( @xmath237 ) , and this behavior is enhanced in evolving the potential to lower resolution scales . the nuclear symmetry energy @xmath9 is weakly scale dependent and we find that the predicted value is within the experimental errors @xmath238mev . partly due to the rather large effective mass @xmath230 at the fermi surface , the anomalous orbital @xmath11-factor comes out too small compared to the empirical value of @xmath239 . the @xmath1 spin - isospin landau parameter @xmath57 is quite large for the low - momentum nn interactions . using the conversion factor @xmath240 @xmath64 one gets the values @xmath241 , and 0.77 for the bare n@xmath0lo chiral nn interaction and evolved interactions @xmath242 and @xmath243 respectively . these numbers are in good agreement with values of @xmath244 obtained by fitting properties of giant gamow - teller resonances . the above results highlight the necessity for including three - nucleon force contributions to the quasiparticle interaction both for the bare and evolved potentials . in fact it has been shown that supplementing the ( low - momentum ) potentials considered in this work with the leading chiral three - nucleon force produces a realistic equation of state for cold nuclear matter @xcite . the large additional repulsion arising in the three - nucleon hartree - fock contribution to the energy per particle should remedy the largest deficiency observed in present calculation , namely the large negative value of the compression modulus @xmath8 . a detailed study of the effects of chiral three - forces ( or equivalently the density - dependent nn interactions derived therefrom @xcite ) on the fermi liquid parameters is presently underway . .sum of the first- and second - order contributions to the dimensionless fermi liquid parameters for the idaho n@xmath0lo potential and two low - momentum nn interactions @xmath233 at @xmath152 @xmath7 . hartree - fock self - energy insertions , as parameterized in eq . ( [ disp ] ) , are included in the second - order diagrams . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] we have performed a complete calculation up to second - order for the quasiparticle interaction in nuclear matter employing both the idaho n@xmath0lo chiral nn potential as well as evolved low - momentum nn interactions . the numerical accuracy of our results is on the order of 1% or better . this precision is tested using analytically - solvable ( at second order ) schematic nucleon - nucleon potentials emerging from scalar - isoscalar boson exchange and modified `` pion '' exchange . we have found that the first - order approximation to the full quasiparticle interaction exhibits a strong scale dependence in @xmath154 , @xmath211 , and @xmath212 , which decreases the nuclear matter incompressibility @xmath8 and increases the symmetry energy @xmath9 and anomalous orbital @xmath11-factor @xmath49 as the resolution scale @xmath216 is lowered . our second - order calculation reveals the importance of the hole - hole contribution in certain channels as well as the strong effects from the particle - hole contribution for the @xmath154 and @xmath42 landau parameters . the total second - order contribution has a dramatic effect on the quasiparticle effective mass @xmath10 , the nuclear matter incompressibility @xmath8 and symmetry energy @xmath9 , as well as the landau - migdal parameter @xmath210 that governs the nuclear spin - isospin response . in contrast , the components of the spin - spin quasiparticle interaction ( @xmath245 ) are dominated by the first - order contribution . we have included also the hartree - fock contribution to the nucleon single - particle energy , which reduces the second - order diagrams by about 30% ( as a result of the replacement @xmath246 ) . the final set of @xmath128 landau parameters representing the quasiparticle interaction in nuclear matter provides a reasonably good description of the nuclear symmetry energy @xmath9 and spin - isospin collective modes . our calculations demonstrate , however , that the second - order quasiparticle interaction , generated from realistic _ two_-body forces only , still leaves the nuclear many - body system instable with respect to scalar - isoscalar density fluctuations . neither the incompressibility of nuclear matter @xmath8 nor the anomalous orbital @xmath11-factor @xmath49 could be reproduced satisfactorily ( without the inclusion of three - nucleon forces ) . a detailed study of the expected improvements in the quasiparticle interaction resulting from the leading - order chiral three - nucleon force is the subject of an upcoming investigation @xcite . s. fritsch , n. kaiser and w. weise , nucl a750 * ( 2005 ) 259 . s. k. bogner , a. schwenk , r. j. furnstahl , and a. nogga , _ nucl . _ * a763 * ( 2005 ) 59 . siu , j. w. holt , t. t. s. kuo and g. e. brown , phys . c * 79 * ( 2009 ) 054004 . k. hebeler , s. k. bogner , r. j. furnstahl , a. nogga and a. schwenk , phys . c * 83 * ( 2011 ) 031301 . l. d. landau , sov . jetp , * 3 * ( 1957 ) 920 ; 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we employ landau s theory of normal fermi liquids to study the quasiparticle interaction in nuclear matter in the vicinity of saturation density .
realistic low - momentum nucleon - nucleon interactions evolved from the idaho n@xmath0lo chiral two - body potential are used as input potentials .
we derive for the first time exact results for the central part of the quasiparticle interaction computed to second order in perturbation theory , from which we extract the @xmath1 and @xmath2 landau parameters as well as some relevant bulk equilibrium properties of nuclear matter .
the accuracy of the intricate numerical calculations is tested with analytical results derived for scalar - isoscalar boson exchange and ( modified ) pion exchange at second order .
the explicit dependence of the fermi liquid parameters on the low - momentum cutoff scale is studied , which provides important insight into the scale variation of phase - shift equivalent _ two - body _ potentials .
this leads naturally to explore the role that three - nucleon forces must play in the effective interaction between two quasiparticles .
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the evolution of cooperation and altruism are fundamental scientific challenges highlighted by their role in the major transitions in life s history , when natural selection acted simultaneously on several competing levels @xcite . in this context , the relevance of basic concepts , including group selection and hamilton s rule remain controversial @xcite . here we address these problems by studying a framework for evolution in group structured populations that incorporates inter- and intra - group competition and migration . combining group - centric with gene - centric perspectives in a constructive group / kin selection approach , we build methodology that allows for the analysis of arbitrary non - linear fitness functions , resulting from complex multi - individual interactions across life cycles . we obtain the conditions for a rare social allele to invade the population . this is obtained in a mathematically rigorous way , by analyzing the stability of the equilibrium in which this allele is absent . this analysis is done for arbitrary strength of selection , but when selection is weak and groups are large the condition for invasion simplifies significantly into a form that is easy to apply and provides substantial intuition . in the case of linear fitness functions , the condition for invasion is identical to hamilton s rule , and it is natural to regard the more general non - linear cases as generalizations of that rule . our results also show that one of the most widely used approaches to analyzing kin selection models , @xcite , @xcite ( condition ( 6.7 ) ) , and @xcite ( box 6 ) , yields incorrect results in some biologically relevant situations . our results reveal biologically realistic conditions under which altruism can evolve when rare , but genetic relatedness in groups is modest . in this way we challenge a common understanding according to which inter - group selection favoring altruism could only override intra - group selection favoring selfishness under exceptional conditions , namely small group size and very low migration rates @xcite . we identify the emphasis on linear public goods games in the literature , including most of these papers , as having supported this belief . in contrast , we show that for iterated public goods games , in which altruists cooperate or not in each round based on previous outcomes @xcite , altruism can spread even when groups are large , selection is weak and migration rates are substantially larger than the inverse of group size . this result corrects @xcite , who predicted that large group size would not allow cooperation to spread when rare in this model . for species that live in groups , several vital group activities repeat themselves periodically and behavior changes as feedback is obtained from previous iterations . the iterated public goods game that we study is therefore often more realistic than a simple one shot public goods game . a proper analysis of this model fills therefore an important gap in the literature . to obtain our result we show that in the absence of selection , when groups are large , the fraction of group members that are close relatives of a randomly chosen individual has a non - gaussian distribution with a fat tail . as a consequence , even when altruistic alleles are rare in the population , they have a significant probability of concentrating in some groups , accruing substantial reproductive gains through multi - individual synergy . when members of a species live in groups , their reproductive success depends on the behavior of all group members . more efficient groups may grow faster and split , outcompeting the less efficient ones that die out . on the other hand , individuals may free ride on the cooperation of other members of their group , and in this way outcompete them . this picture is further complicated by migration among groups . the _ two - level fisher - wright framework with selection and migration _ ( 2lfw ) captures all these elements , in a simplified fashion . in 2lfw haploid individuals live in a large number @xmath0 of groups of size @xmath1 , and are of two genetically determined phenotypic types , a or n. generations do not overlap , reproduction is asexual and the type is inherited by the offspring . ( mutations will be considered briefly later . ) the relative fitness ( @xmath2 ) of a type a , and that of a type n , in a group that has @xmath3 types a , are , respectively , @xmath4 and @xmath5 , with the convention that @xmath6 , i.e. , @xmath7 . the quantities @xmath8 and @xmath9 represent life - cycle payoffs derived from behavior , physiology , etc . the parameter @xmath10 indicates the strength of selection . [ fig1 ] describes the creation of a new generation in the 2lfw through inter- and intra- group competition , followed by migration at rate @xmath11 . cases in which types a behave in some altruistic fashion are of particular interest @xcite . most of the literature concerns the very special case of a _ linear _ public goods game ( pg ) , defined by @xmath12 , @xmath13 , @xmath14 , in which each type a cooperates , at a cost @xmath15 to herself , providing a benefit @xmath16 shared by the other members of her group . the need to consider more complex intra - group interactions and non - linear payoff functions is , nevertheless , well known . non - linearities appear naturally whenever activities involve many group members simultaneously . they result from threshold phenomena , increasing returns to scale , saturation , etc . for instance , to hunt large prey may require a large minimum number of hunters , the likelihood of success may first increase rapidly with the number of hunters , but it may plateau when this number becomes very large . allowing for the analysis of such synergistic multi - individual interactions and activities is a central feature of our approach , distinguishing it from theoretical frameworks based on pairwise interactions , or single actors benefiting a group @xcite . . * ( center ) fw intragroup competition : * if a group descends from a group with @xmath3 types a , then it will have @xmath17 types a with probability @xmath18 , where the binomial probability @xmath19 is the probability of @xmath17 successes in @xmath1 independent trials , each with probability @xmath20 of success . * ( right ) migration : * once the new @xmath0 groups have been formed according to the two - level competition process , a random fraction @xmath11 of the individuals migrates . migrants are randomly shuffled . * note : * the assignment of relative fitness to the groups in the fashion done above is a necessary and sufficient condition @xcite for individuals in the parental generation to have each an expected number of offspring proportional to their personal relative fitness . , scaledwidth=50.0% ] the 2lfw framework can be seen as a generalization of the trait - group framework ( see sec . 2.3.2 of @xcite ) , which corresponds to the case @xmath21 . one can interpret @xmath22 as an assortment parameter . because migration is completely random in 2lfw , this assortment represents a worst case scenario , abstracting away additional assortment caused by kin recognition , greenbeard effects , selective acceptance of migrants , joint migration of individuals , etc . it is well known @xcite that even when @xmath21 non - linearities in fitness functions allow for coexistence of cooperators and defectors . but under the strong altruism condition @xmath23 ( meaning that each type a would be better off mutating into a type n ) , this is not the case @xcite . one of our goals is to determine the level of migration compatible with invasion by rare strong altruists . the 2lfw also generalizes the `` typical kin selection model '' of @xcite , where the payoffs were those of pg , and the analysis relied on this and on the assumption of weak selection . that paper was a response to @xcite , where group selection was argued to be an important mechanism for the evolution of cooperation , and a multilevel selection model based on moran s model was introduced . our analysis of 2lfw with non - linear fitness functions highlights the importance of combining group - centric with gene - centric perspectives , and shows that group selection can be an important force in evolution under realistic conditions . it also shows that mathematically rigorous analysis can be carried out even when selection is strong and fitness functions are non - linear . and it shows that one has to be very careful in applying mathematically non - rigorous methodology , as it can produce substantially incorrect results , even when selection is weak . non - linearities in life - cycle payoffs can result from activities repeating themselves during a lifetime , and behavior being contingent on previous outcomes . a basic example is the iterated public goods game ( ipg ) @xcite . in ipg a pg is repeated an average of @xmath24 times in a life - cycle . we will suppose that types n never cooperate , while types a cooperate in the first round and later cooperate only if at least a fraction @xmath25 of group members cooperated in the previous round . mathematically , this model generalizes the iterated prisoner dilemma and tit - for - tat , from the dyadic setting of @xcite and @xcite to the multi - individual setting . but while direct or indirect reciprocity requires the identification of individuals in the group , this is not the case here . the behavior of types a in the ipg can be triggered by individuals simply discontinuing cooperative behavior when previous cooperation produced negative feedback to them . in other words , allele a can predispose individuals to cooperate , but as they do it and obtain feedback from that behavior , they may continue it or discontinue it . the ipg is in this sense closely related to generalized reciprocity mechanisms @xcite with low cognitive requirements . negative feedback from cooperation should occur if the fraction of group members that cooperated was less than @xmath26 , but not if it was larger than that threshold , since in the former case the payoff to a cooperator is negative , while in the latter case it is positive . this gives a special role to the value @xmath27 . if @xmath28 , the behavior of types a is altruistic in the strong sense that each type a individual would increase its fitness if it behaved as a type n , everything else being equal , i.e. , @xmath29 ( see si appendix ( section 8) for a detailed discussion ) . moreover , types n always free ride and have greater fitness than types a in the same group , regardless of the values of @xmath25 and @xmath24 , i.e. , @xmath30 . [ fig2 ] displays a detailed analysis of some instances of the ipg , giving conditions for allele a to spread when rare . for many species that live and interact in groups for many years , several vital activities , including collective hunting and food sharing , can repeat themselves hundreds or thousands of times in a life - cycle , giving plausibility to the values of @xmath24 in fig . ( the assumption that individuals discontinue behavior after a single unsuccessful participation is a simplification . when this is not a realistic assumption , one can interpret the parameter @xmath24 as the ratio between the typical number of repetitions of the activity and the typical number of unsuccessful attempts before cooperation is discontinued by a type a. ) panel c , in which selection is weak and groups are large , shows two important contrasting results . when @xmath31 , and the ipg is identical to the pg , allele a can only invade under hamilton s condition @xmath32 . but as @xmath24 increases , the level of relatedness needed for invasion drops substantially , so that for modest values of @xmath33 , allele a can invade under @xmath34 significantly lower than @xmath35 , compatible with levels observed in several species , including humans @xcite ( table 8.3 ) , @xcite , @xcite ( tables 6.4 and 6.5 ) , @xcite , @xcite ( table 4.9 ) . the corresponding number of migrants per group per generation , @xmath36 , can be of the order of 10 . further examples showing the spread of altruism and cooperation under high levels of gene flow and low levels of relatedness are provided in figs . 5 , 6 , 7 , 8 , 20 and 21 in the si appendix . times in a life - cycle . in each round each individual can cooperate at a cost @xmath15 to herself , producing a benefit @xmath16 shared by the other members of the group . types n never cooperate , while types a cooperate in the first round and later cooperate only if at least a fraction @xmath25 of group members cooperated in the previous round . in all panels @xmath27 ( types a are strongly altruistic ) and curves correspond to @xmath37 ( black , this case is identical to pg ) , 10 ( blue ) , 100 ( magenta ) , 1000 ( green ) ( bottom to top in panel a , top to bottom in panels b and c ) . * panel a : * @xmath38 , @xmath39 , @xmath40 . curves give the critical migration rate @xmath41 below which types a proliferate when rare , and that solves @xmath42 , or equivalently @xmath43 in ( [ gen ] ) . ( the subscript ` s ' stands for ` survival ' . ) the dependence of @xmath41 on the strength of selection @xmath44 indicates the relevance of studying both weak and strong selection . short horizontal red lines indicate value of @xmath41 under weak selection , obtained from setting @xmath43 in ( [ ws ] ) ( note the excellent agreement ) . * panel b : * again , @xmath38 , @xmath39 , @xmath40 . curves give the critical relatedness @xmath45 above which types a proliferate . here @xmath46 is the relatedness obtained from neutral genetic markers . short horizontal red lines are again from @xmath47 in ( [ ws ] ) . * panel c : * limit of large @xmath1 under weak selection . critical values of relatedness @xmath48 , as function of @xmath26 . solid lines provide the solution to the equation @xmath49 derived from setting @xmath47 in ( [ wsln ] ) . dashed lines give its approximation ( [ rcblnt ] ) . red vertical line corresponds to @xmath50 , while horizontal red lines are at the same level of those from panel b. their intersections illustrate the fact that both the solid and dashed lines in panel c are good approximations to weak - selection values of critical relatedness , @xmath51 , when @xmath52 . , to analyze the 2lfw , denote by @xmath53 , @xmath54 the fraction of groups in generation @xmath55 that have exactly @xmath3 types a. denote by @xmath56 the frequency of types a in the population . the state of the population in generation @xmath55 is described by the vector @xmath57 , since @xmath58 . we will suppose that @xmath59 , so that , by the law of large numbers , @xmath60 evolves as a deterministic ( non - linear ) dynamical system in dimension @xmath1 . here we will study its linearization close to the fixed point @xmath61 , with no types a. this means that we are restricting ourselves to the case in which @xmath62 , and studying the conditions for allele a to invade the population when rare . with notation introduced in fig . [ fig1 ] , we have then @xmath63 , where @xmath64 , @xmath65 and @xmath66 , if @xmath67 , @xmath68 if @xmath69 . matrix @xmath70 represents the production of groups in the new generation , through the two - level competition . matrix @xmath71 represents the effect of types a migrating out of groups , and matrix @xmath16 represents the effect of these migrant types a joining groups that previous to migration had no types a. ( when @xmath62 , the migrant types a are a small fraction of the migrant population , and therefore each one is likely to settle in a different group that had no types a before migration . ) a standard application of the perron - frobenius theorem implies that when @xmath72 , we have , in good approximation , @xmath73 , where @xmath15 is a constant that depends on @xmath74 , @xmath75 is the leading eigenvalue of @xmath76 and @xmath77 is its corresponding left - eigenvector normalized as a probability vector . this means that , regardless of the initial distribution @xmath74 , with @xmath78 , demographics and natural selection drive @xmath60 towards multiples of @xmath77 , in what can be seen as self - organization of copies of a in the optimal stable way for them to spread . once this has happened , @xmath79 grows at rate @xmath80 . consequently , allele a will proliferate , when rare , if the _ viability condition _ @xmath81 holds , and it will vanish if @xmath82 . ( see fig . [ fig2 ] and si appendix ( sections 1 and 2 ) for applications , illustrations and further explanations . ) when @xmath81 , even if allele a is initially absent , a small rate of mutation will introduce it , allowing it to then invade the population . in the terminology of evolutionary game theory ( see , e.g. , @xcite chapter 7 ) , phenotype n is an evolutionary stable strategy ( ess ) when @xmath82 and n is not an ess when @xmath81 . the viability condition @xmath81 has a gene - centric ( kin - selection ) interpretation in terms of average ( neighbor modulated ) fitnesses . for this purpose , define @xmath83 . then it is well known that @xmath84 , where @xmath85 and @xmath86 are the average fitnesses of types a and n , and @xmath87 is the average fitness of all individuals . if we choose a random type a , it will have probability proportional to @xmath88 of being in a group with exactly @xmath3 types a ( bayesian sampling bias ) . therefore @xmath89 . when @xmath62 , if we choose a random individual , it is likely to be in a group with no types a. therefore , in good approximation , @xmath90 . since @xmath60 is driven towards multiples of @xmath77 , we obtain p = p , [ gen ] provided @xmath62 and @xmath72 ( the error term is of order @xmath91 + @xmath92 , with @xmath93 ) . the viability condition @xmath81 can also be stated as @xmath94 , in ( [ gen ] ) . it is important to observe that @xmath79 does not need to be monotone , and that @xmath94 is the proper condition for invasion only when , as in ( [ gen ] ) , one is considering the stationary regime , @xmath72 . if selection is weak , i.e. , @xmath95 , migration acts much faster than selection , providing a separation of time scales @xcite . this allows us to replace @xmath77 in ( [ gen ] ) with @xmath96 , obtained by assuming @xmath97 . algebraic simplifications ( presented in si appendix ( section 5 ) ) allow us then to rewrite the neighbor modulated fitness relation ( [ gen ] ) in the form p = p _ k=1^n _ k v^a_k [ ws ] ( the error term is of order @xmath98 ) , where @xmath99 and @xmath100 , with @xmath101 the markov transition matrix @xmath102 and its invariant distribution @xmath103 have natural interpretations in terms of identity by descent ( ibd ) under neutral genetic drift , as we explain next when we provide a second derivation of ( [ ws ] ) . two individuals are said to be ibd if following their lineages back in time , they coalesce before a migration event affects either one . the separation of time scales implies that when selection acts , the demographic distribution is well approximated by that obtained in equilibrium with @xmath97 . this means that in good approximation @xmath104 , where @xmath105 is the @xmath106 equilibrium probability that in the group of a randomly chosen focal type a there are exactly @xmath3 types a ( focal included ) . but because we are supposing that types a are rare , the only individuals that are type a in this group are those that are ibd to the focal , so that @xmath105 is also the probability that exactly @xmath3 individuals in this group are ibd to the focal . as in the derivation of ( [ gen ] ) , since types a are rare , we have @xmath107 and hence @xmath108 , which is ( [ ws ] ) . to compute @xmath103 , we will use the standard kronecker notation @xmath109 if @xmath110 and @xmath111 if @xmath112 . now , the probability @xmath113 that the focal is ibd to exactly @xmath114 other members of her group is @xmath115 if the focal is a migrant ( probability @xmath11 ) , while if she is not a migrant ( probability @xmath22 ) , then we have to consider how many individuals in her mother s group were ibd to her mother . if , counting her mother , that number was @xmath17 ( probability @xmath116 , assuming demographic equilibrium ) then the probability that the focal is ibd to exactly @xmath114 other members of her group is equal to the probability that of the @xmath117 other members of her group , exactly @xmath114 are non - migrants who chose for mother one of the @xmath17 candidates ( among @xmath1 possibilities ) that were ibd to the focal s mother ( probability @xmath118 ) . combining these pieces , we have @xmath119 this is exactly the same as the set of equations @xmath120 . the ibd distribution @xmath103 contains all the relevant information about genetic relatedness in the groups , including and exceeding that given by the average relatedness between group members , @xmath121 , obtained from lineages , regression coefficients , or wright s @xmath122 statistics . specifically , we can define @xmath123 as the probability that a second member chosen from the focal s group is ibd to the focal and then obtain ( from linearity of expected values ) that @xmath124 is a linear function of @xmath103 s first moment . when @xmath8 is a non - linear function of @xmath3 , more information contained in @xmath103 , including its higher moments , are needed to decide whether @xmath94 in ( [ ws ] ) . it is important to also stress that ( [ ws ] ) can be easily used for applications in which even the knowledge of all the moments of @xmath103 ( see @xcite ) would be cumbersome to apply , as for instance in the computation of the short horizontal red lines in fig . [ fig2 ] , panels a and b. the stationarity condition @xmath125 allows for a recursive computation of all the moments of @xmath103 ( see si appendix ( section 5 ) ) . these moments can then be used to show the powerful result that if @xmath1 is large and @xmath11 is small , then @xmath103 , when properly rescaled , is close to a beta distribution , with mean @xmath34 ( see si appendix ( section 6 ) ) . in this case , if in addition to the assumption of weak selection , also @xmath8 is well approximated by @xmath126 , for some piecewise continuous function @xmath127 , then ( [ ws ] ) takes the easy to apply form p = p ( - 1 ) _ 0 ^ 1 ( 1-x)^ - 2 v^a_x dx , [ wsln ] where @xmath128 . equations ( [ gen ] ) and ( [ wsln ] ) play complementary roles in the analysis of 2lfw . both provide the condition for invasion by allele a ; ( [ gen ] ) holds in full generality , while ( [ wsln ] ) requires special assumptions ( small @xmath44 , large @xmath1 ) , but is computationally much simpler and provides a great deal of intuition , as we discuss next . equation ( [ wsln ] ) should be contrasted with what @xcite predicted by supposing that the number of individuals in a group that are ibd to a focal individual would be well approximated by a binomial with @xmath117 attempts and probability @xmath123 of success . that would lead to a normal distribution , narrowly concentrated close to its mean @xmath123 , in place of the beta distribution above . our result reveals a strong dependency structure among lineages , producing the beta distribution , with a standard deviation comparable to its mean , and a tail that decays slowly compared to a gaussian distribution . as a consequence , fitness functions that are large only when the fraction of types a in a group is above a threshold value , as in the ipg , will allow for proliferation of types a under levels of relatedness substantially lower than that predicted under the assumption in @xcite . we will refer to the fact that the fraction of group members that are ibd to a focal individual has a non - vanishing standard deviation , even when selection is weak and groups are large , as _ persistence of variability_. this phenomenon poses a severe limitation to the applicability of covariance - regression methods in which regression of fitness on genotype is replaced with derivatives , as in @xcite , @xcite ( box 6 ) , @xcite ( condition ( 6.7 ) ) . both the assumptions in @xcite , or in @xcite applied to the ipg would have implied incorrectly that when selection is weak and groups are large , types a could only invade the population when rare if @xmath129 ( these computations are presented in the si appendix ( sections 8 and 9 ) ) . in a companion paper @xcite we show that methodologies in which one expresses the fitness of a focal individual in terms of partial derivatives with respect to the focal individual s phenotype and the phenotype of the individuals with whom the focal interacts , as in @xcite , require @xmath130 to be a linear function of @xmath131 . for the pg , ( [ ws ] ) and ( [ wsln ] ) clearly reduce to the well known @xmath132 . the same is also true for the more general ( [ gen ] ) , as was shown in @xcite , where in case of strong selection the relatedness @xmath123 depends on the payoff functions . in contrast , if we are under the conditions of ( [ wsln ] ) with @xmath133 , then p = p ( -c + br + b_2 r_2 + ... + b_l r_l ) , [ wspowers ] where @xmath134 $ ] is the @xmath135-th moment of the beta distribution . for the ipg , @xmath136 , if @xmath137 , and @xmath138 , if @xmath139 . the viability condition derived from ( [ wsln ] ) can be analyzed in detail , by simple , but long , computations , presented in the si appendix ( section 8) . in the case @xmath27 , the viability condition reads @xmath140 . when @xmath24 is large , this yields the following approximation for the critical relatedness @xmath34 : r = . [ rcblnt ] if also @xmath141 , then r = = . [ rcblnt+ ] the simplicity and transparency of ( [ rcblnt ] ) and ( [ rcblnt+ ] ) illustrate the power of ( [ wsln ] ) , and fig . [ fig2 ] shows how well they compare to the more general , but less transparent ( [ gen ] ) . note also how ( [ rcblnt ] ) and ( [ rcblnt+ ] ) provide a direct grasp on the effect of the number of repetitions in the game , and a nice comparison between the pg and the ipg . both fig . [ fig2 ] and ( [ rcblnt+ ] ) show that alleles that promote contingent cooperative behavior , which is discontinued when participation is low , can spread under levels of genetic relatedness ( @xmath142 ) more than 5 times smaller than @xmath26 . this mechanism should , therefore , be seriously investigated as a possible route for the proliferation of altruistic / cooperative behavior . 1 . natural selection in group structured populations is best analyzed by a combination of group - centric and gene - centric perspectives and methods . both shed light , carry intuition and provide computational power , in different ways . rigorous mathematical analysis of models is necessary , especially when fitness functions are non - linear , to assess the validity of non - rigorous approaches . contingent forms of group altruism that are discontinued when participation is low can proliferate under biologically realistic conditions . their role in the spread of altruism should be empirically investigated . 3 . natural selection can promote traits that ( in net terms over a full life - cycle ) are costly to the actors and beneficial to the other members of their group , under demographic conditions that are not stringent . this can happen in large groups and with realistically high levels of gene flow . excessive focus on one - shot linear public goods games in the literature has obscured this fact . the authors thank rob boyd for many hours of stimulating and informative conversations on the subjects in this paper . thanks are given to marek biskup for assistance in exploring the nature of the distribution @xmath103 . we are also grateful to clark barrett , maciek chudek , daniel fessler , kevin foster , willem frankenhuis , bailey house , laurent lehmann , glauco machado , sarah mathew , diogo meyer , cristina moya , peter nonacs , karthik panchanathan , susan perry , joan silk , jennifer smith and ming xue for nice conversations and feedback on various aspects of this project and related subjects . this project was partially supported by cnpq , under grant 480476/2009 - 8 . wenseleers t , gardner a , foster , kr ( 2010 ) in _ social behaviour : genes , ecology and evolution _ , eds . szekely t , moore aj , komdeur j ( cambridge university press , cambridge , uk , cambridge , uk ) , pp . 132158 . lessard s ( 2009 ) diffusion approximations for one - locus multi - allele kin selection , mutation and random drift in group - structured populations : a unifying approach to selection models in population genetics . , 659696 .
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the ways in which natural selection can allow the proliferation of cooperative behavior have long been seen as a central problem in evolutionary biology .
most of the literature has focused on interactions between pairs of individuals and on linear public goods games .
this emphasis led to the conclusion that even modest levels of migration would pose a serious problem to the spread of altruism in group structured populations . here
we challenge this conclusion , by analyzing evolution in a framework which allows for complex group interactions and random migration among groups .
we conclude that contingent forms of strong altruism can spread when rare under realistic group sizes and levels of migration .
our analysis combines group - centric and gene - centric perspectives , allows for arbitrary strength of selection , and leads to extensions of hamilton s rule for the spread of altruistic alleles , applicable under broad conditions .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
laser ranging to earth artificial satellites ( slr ) was initiated in 1964 after launch of the first geodetic / geodynamical satellite beacon - b . since that time satellite laser ranging ( slr ) technique have being widely used for geodynamical and geophysical researches . the primary fields of investigations used slr observations are earth rotation , maintenance of the terrestrial reference frame , tectonic motion , earth crust deformations , geopotential with its spatial and temporal variations , tides , movement of geocenter , support of satellite geophysical missions ( such as satellite altimetry ) , global time transfer , and others . detailed analysis of slr contribution to earth sciences can be found in ( tapley _ et al . _ 1993 ) . after the launching of the lageos satellite in may 1976 , slr became one of the main techniques for investigations of the earth rotation , and during over twenty years slr technique have being used for determination of erp . most of results contributed to iers are based on analysis of observations of lageos 1&2 satellites collected at the global tracking network of about 40 stations . slr provides high precision series of xp , yp , and lod . some analysis centers compute also ut that allow to densify universal time series in combination with vlbi data . importance of ilrs as one of the main method to study the earth led , naturally , to the establishment of the international laser ranging service ( ilrs ) in 1998 that coordinates now scientific activity slr ( and llr ) , chiefly in the framework of iag and iers projects . in this paper we will focus only on slr contribution to investigation of polar motion in accordance with topics of the conference . slr observations of lageos and erp derived from these observations are available from may 1976 . however , only in 198081 after significant improvement of range technique slr erp series achieved accuracy required for investigation of polar motion ( pm ) . figure [ fig : ac_contr ] summarizes information from iers annual reports for 19781997 concerning use of submitted slr series in iers yearly solutions . one can see that beginning from 1983 slr erp series is regularly used for iers combination . it s also seen that the center for space research , university of texas at austin ( csr ) provides most long - time spanned and stable pm series . = at present 5 analysis centers submit operational solutions with 2 - 15 days delay , and about 10 analysis centers yearly contribute final ( up to 23 years ) erp series . most long series available for analysis are listed in table [ tab : long_series ] . during long history of using slr technique to study pm drastic improvement in range precision was achieved . it s interesting to see how this technology development affected accuracy of pm series obtained from slr observations . figure [ fig : csriaa ] shows differences between slr series obtained at the csr and the institute of applied astronomy , st.petersburg ( iaa ) and iers combined solution eop(iers)c04 . one can clearly see in figure [ fig : csriaa ] that replacement of the first generation ranging equipment by the second generation one about 1980 and its further replacement by the third generation units leaded to significant improvement of accuracy of pm solution . however , in spite of improvement of ranging technique and implementation the fourth generation equipment , precision of pm series derived from slr observations remains the same during last 1012 years . we will try to discuss the problem below . in 1998 iers central bureau had derived final iers combination in two step : on the first stage combined vlbi , slr , and gps series was computed and then these was used for final iers combination . table [ tab : acc_c04 ] copied from the 1997 iers annual report shows that precision of slr and gps combinations is practically the same . unfortunately , this is not so for operational solutions . table [ tab : acc_usno ] compiled on basis of ` gpspol.asc ` files produced by usno along with bulletin a issues contains statistics of series used for iers rapid service . of course , errors in pm obtained there depend on weighting applied to combined series , but it already clear that accuracy of individual slr series is worse than vlbi and gps ones , and delay of slr contribution is much worse than gps one . nevertheless , one can see that at least two centers can produce operational solution with delay about 2 days . if all slr analysis centers would provide such a solution on regular basis , it will allow to have , in principle , combined slr solution with accuracy 0.10.15 mas and delay 1.52 days . during long time slr was the one of the main methods of determination of pm and densification of pm series based on vlbi observations . however , lately slr erp series are inferior to gps ones in quality ( accuracy , density , delay of operational solution ) of results . it would be very important to understand existing problems in slr observations and analysis procedures and discuss possible ways to solve them . the first problem with slr observations is that slr technique is an one - object one . this means that , unlike gps , slr station can observe only one object in time . hence , planning of observations and priority politics plays substantial role in acquisition of observations of satellites on which pm determination is based , especially keeping in mind that slr observations are rather expensive and number of ranging units are limited . at present four operational satellites seem most suitable for investigation of earth rotation ( as well as tectonic motion and long - term temporal variations in various geophysical parameters ) two lageos satellites and two etalon satellites . both lageos and etalon satellite was launched to long - time stable orbits and have a low area to mass ratio . their description is presented in figure [ fig : satellit ] . figure [ fig : observ ] shows number of observations of these satellites during last 12 years and ilrs priorities . one can see that number of observations of lageos satellites remains approximately the same during these years and number of observations of etalon satellite is too small to contribute seriously to analysis . = 0.75 let us see in more details distribution of slr observations in stations and time . figure [ fig : sta_map ] shows distribution of observations in stations for the period from sep 1983 till aug 1999 . table [ tab : obs_sta ] contains list of stations contributed more than 2% of total number of observations for whole period and during the last year ( in parenthesis ) . = 0.8 .distribution of observations in stations . [ cols="^,<,<,^,<,<",options="header " , ] [ tab : daily_int ] it should be mentioned also that producing daily solution ( independently which method i d used ) provide operational solution with steady delay about 2 days , which solves the second problem mentioned above . other serious problem is determination of ut from slr ( and other satellite observations ) . it is well known that ut1 can not be separated from longitude of node of satellite orbit during parameter solution . three methods are being used to solve this problem : * fixing longitude of node during parameter solution ( usually , during last iteration ) . * analysis of node longitude series , forecasting it and use predicted values for operational solution . * integrating lod series to obtain independent free - running ut series with its possible correction for high - frequency variations from comparison with vlbi series . evidently , only the latter method can provide ( in principle ) independent result . since that is not a subject of this paper , we will not stay on detailed analysis of this problem . however , it is worth to mention here that significant improvement of slr ut series is also impossible without increasing of number of satellites involved in determination of erp . at last , use the same terrestrial reference frame for all slr solutions seems evident to achieve uniform solutions for combination . use of itrf as terrestrial reference frame for by all analysis centers for their slr solutions provides more homogeneous series for slr combined solution . realization of this or alternative analysis strategy could provide more uniform , accurate and operative slr erp series . after that , combining of all submitted series to final ilrs slr product seems reasonable and useful for further use for iers and other purposes . we have not mention here such a serious problem as dependence of slr erp results on a priori values . this is worth to perform special investigation for each method used for computation of erp at various analysis centers . satellite laser ranging technique made and make a very valuable contribution to earth dynamics . in particular , very valuable contribution was made in investigation of pm . during many years slr was one of the main methods of determination of polar motion and main method of densification of erp series obtained with vlbi . at present 5 analysis centers submit operational solutions with 2 - 15 days delay , and about 10 analysis centers yearly contribute final ( up to 23 years ) erp series . however , due to principal peculiarity of this method ( relatively expensive experiment , limited number of units , lack of capability of multi - satellite ranging , etc . ) quality of slr erp data remains the same during the last decade in spite of ranging precision improved by a factor of a thousand from a few meters to a few millimeters since the first slr experiments ( and by factor of about 10 during last 10 - 12 years ) . this leads to decreasing of weight of slr solutions in the combined iers eop series . it is evident that only substantial increasing of number of observations and satellites involved in investigation of earth rotation and improvement of distribution of observations in stations and time can help in improvement of the slr erp accuracy . however , capacity of tracking network is practically exhausted . on the other hand , in spite of gps provides determination of erp with impressive accuracy and delay , slr results are very important for combined iers solution for improvement of systematic accuracy of the final iers product . analysis of precision of individual slr series shows that its combination can provide combined slr much more accurate and rapid series . to achieve highest accuracy of combined slr erp product is necessary to solve problems discussed in section [ sect : analysis ] . realization of this opportunities by ilrs would be very important for investigation of earth rotation because allow to save independent method of determination of pm and velocity of the earth rotation . author is very grateful to scientific organizing committee for invitation to the meeting and financial support of this trip .
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slr technique has being used for determination of erp during over twenty years .
most of results contributed to iers are based on analysis of observations of lageos 1&2 satellites collected at the global tracking network of about 40 stations .
now 5 analysis centers submit operative ( with 2 - 15 days delay ) solutions and about 10 analysis centers yearly contribute final ( up to 23 years ) erp series .
some statistics related to slr observations and analysis is presented and analyzed .
possible problems in slr observations and analysis and ways of its solution are discussed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a thermal counterflow in he ii is internal convection of two fluids , namely the normal fluid and the superfluid . when the counterflow velocity exceeds a critical value , a self - sustaining tangle of quantized vortices appears to form superfluid turbulence . in low aspect ratio channels , superfluid turbulence makes the mysterious transition . the increase in the counterflow velocity is observed to change the laminar state to the first turbulent state ti , and next to the second turbulent state tii @xcite . melotte and barenghi suggested that the transition from the ti to tii state is caused by the transition of the normal fluid from laminar to turbulent @xcite . the recent developments of the visualization technique have enabled us to confirm the scenario . guo @xmath0 have followed the motion of seeded metastable @xmath1 molecules by a laser - induced - fluoresence technique to observe that the normal fluid can be turbulent at relatively high velocities @xcite . to understand the mysterious transition of counterflow quantum turbulence , it is necessary to address the coupled dynamics of the two fluids ; the superfluid is described by the vortex filament model , the normal fluid by the navier - stokes equation , and they are coupled through the mutual friction @xcite . however , it is difficult to solve fully the coupled dynamics . as the first essential step , we address the ti state in a square channel with prescribing the velocity field of the normal fluid to a poiseuille profile . our simulation obtains a statically steady state of an inhomogeneous vortex tangle . baggaley @xmath0 @xcite studied numerically a thermal counterflow between two plates . they prescribed a poiseuille and turbulent profiles for the velocity field of the normal fluid . an inhomogeneous vortex tangle was obtained , where vortices concentrated near the solid boundaries . they suggested that their results supported the scenario proposed by melotte and barenghi . the better understanding of the ti and tii states would be obtained by studying the flow in a low aspect ratio channel where all boundaries are solid except for the flow direction . this is because the ti and tii states are actually observed in low aspect ratio channels and another turbulence tiii state is observed in high aspect ratio channels @xcite . in a vortex filament model @xcite a quantized vortex is represented by a filament passing through a fluid and has a definite vorticity . this approximation is very suitable in he ii , since the core size of a quantized vortex is much smaller than any other characteristic length scale . at zero temperature the vortex filament moves with the superfluid velocity @xmath2 where @xmath3 is the velocity field produced by vortex filaments , @xmath4 by solid boundaries , and @xmath5 is the applied superfluid velocity . the velocity field @xmath3 is given by the biot - savart law ; this work addresses the full biot - savart integral @xcite . since @xmath5 represents the laminar flow of the superfluid , @xmath5 requires irrotational condition , which is supposed to be uniform . the velocity field @xmath4 is obtained by a simple procedure ; it is just the field produced by an image vortex which is constructed by reflecting the filament into the surface and reversing its direction . taking into account the mutual friction , the velocity of a point @xmath6 on the filament is given by @xmath7,\ ] ] where @xmath8 and @xmath9 are the temperature - dependent coefficients , and the prime denotes derivatives of @xmath10 with respect to the coordinate @xmath11 along the filament . the velocity field of the normal fluid is prescribed to be a poiseuille profile . in a rectangular channel the poiseuille profile is given by @xmath12 \frac { \cos(m \pi y / 2 a ) } { m^3 } , \ ] ] where @xmath13 and @xmath14 are coordinates normal to the flow direction @xmath15 , and @xmath16 and @xmath17 are halves of the channel width along the @xmath13- and @xmath14- axes @xcite . in this study , all simulations are performed under the following conditions . we study thermal counterflow of he ii at temperatures _ k , 1.6 k and 1.9 k. the computing box is @xmath18 . periodic boundary conditions are used along the flow direction @xmath15 , while solid boundary conditions are applied to the channel walls at @xmath19 and @xmath20 . all simulations start with eight randomly oriented vortex rings of radius @xmath21 . the vortex line density ( vld ) is defined as @xmath22 , where the integral is performed along all vortices in the sample volume @xmath23 . the vortex tangle reaches the statistically steady state . figure 1 ( a ) shows the time evolution of vld . fluctuations are larger than those in a uniform counterflow @xcite , which is attributable to the mechanism characteristic of this system discussed in section 4.1 . . ( b ) vortex line density averaged over the statistically steady state as a function of @xmath24 . , title="fig:",scaledwidth=100.0% ] ( a ) . ( b ) vortex line density averaged over the statistically steady state as a function of @xmath24 . , title="fig:",scaledwidth=100.0% ] ( b ) the statistically steady state is known to exhibit the characteristic relation @xmath25 @xcite with the parameters @xmath26 and @xmath27 . we regard the counterflow velocity @xmath24 as the spatially averaged amplitude of @xmath28 . figure 1 ( b ) shows the vld temporally averaged over the statistically steady state , which almost satisfies the relation . table 1 shows the comparison of @xmath26 among the present work @xmath29 , the simulation @xmath30 under the periodic boundary condition @xcite and a typical experiment @xmath31 @xcite . the values of @xmath29 are lower than the values of @xmath30 obtained under the uniform counterflow . the difference of @xmath26 comes from the difference of the mechanism sustaining the vortex tangle . the origin of the discrepancy between @xmath29 and @xmath31 is not clear , but this may be attributable to neglecting the effect of the vortex tangle on the poiseuille flow of the normal fluid through mutual friction . . line density coefficients @xmath26 . numerical results @xmath29 by this work , @xmath30 by adachi _ @xcite and experimental results @xmath31 by childers and tough @xcite . [ cols="^,^,^,^ " , ] in order to estimate the inhomogeneity of the vortex tangle , we divide the computational box to @xmath32 sub - volumes and define the vld at a sub - volume as the local vld . figure 2 ( a ) shows the spatially dependence of @xmath33 , which is obtained by averaging the local vld spatially over the flow direction and temporally over the statistically steady state . we estimate the inhomogeneity of the vortex tangle by a spatial variance @xmath34 of @xmath35 . figure 2 ( b ) shows the characteristic dependence of @xmath34 on @xmath24 and @xmath36 . firstly , @xmath34 of 1.6k is the largest among three temperatures . secondly , @xmath34 at 1.3k and 1.6k increases with @xmath24 , while @xmath34 at 1.9 k decreases with @xmath24 . the dependence on @xmath36 is understood as discussed in section 4.2 , but the dependence on @xmath24 is not known . ) . one can see the concentration near the solid boundaries . ( b ) dependence of the spatially variance @xmath34 of @xmath35 on @xmath24 and @xmath36 . , title="fig:",scaledwidth=100.0% ] ( a ) ) . one can see the concentration near the solid boundaries . ( b ) dependence of the spatially variance @xmath34 of @xmath35 on @xmath24 and @xmath36 . , title="fig:",scaledwidth=100.0% ] ( b ) section 4.1 addresses the mechanism for sustaining the inhomogeneous vortex tangle . in section 4.2 we discuss how the vortex tangle becomes inhomogeneous depending on temperature . in section 4.3 we consider how the aspect ratio of the channel cross section affects the two fluids . as shown in fig . 1 ( a ) , the vld shows the non - linear oscillation with large amplitude in the statistical steady state , which is much different from the case of the uniform counterflow @xcite . the non - linear oscillation comes from the space - time oscillation of the vortex tangle through the mutual friction under the poiseuille flow . figure 3 shows the space - time pattern of the vortex tangle at 1.9 k. the period of the non - linear oscillation is about 0.6 s , consisting of four stages ( a)-(d ) . in fig . 3 ( a ) corresponding to the minimum of the vld , vortices are dilute , where vortices remain only near the solid walls . then the vortices near the walls invade to the central region in fig . 3 ( b ) . these vortices make lots of reconnections in the central region subject to the large counterflow in fig . hence the vld increases significantly to the maximum . eventually in fig . 3 ( d ) the poiseuille flow excludes the vortex tangle from the central region toward the solid walls . thus the vld around the central region decreases , and the vortices are absorbed by the solid boundaries . then the vortex tangle returns to the stage of fig . therefore the vortex tangle repeats the periodic motion , resulting in the non - linear oscillation of fig . the vortex tangle sustained by this mechanism is more dilute than the case of the uniform counterflow for the same @xmath36 and @xmath24 . 1.9 k , @xmath37 ) , corresponding to the results of fig . 1 ( a ) . , title="fig:",scaledwidth=100.0% ] ( a ) depletion ( 2.00 s ) 1.9 k , @xmath37 ) , corresponding to the results of fig . 1 ( a ) . , title="fig:",scaledwidth=100.0% ] ( b ) invasion ( 2.14 s ) 1.9 k , @xmath37 ) , corresponding to the results of fig . 1 ( a ) . , title="fig:",scaledwidth=100.0% ] ( c ) maximum ( 2.30 s ) 1.9 k , @xmath37 ) , corresponding to the results of fig . 1 ( a ) . , title="fig:",scaledwidth=100.0% ] ( d ) exclusion ( 2.42 s ) temperature dependence of @xmath34 in fig . 2 ( b ) is caused by the temperature dependent parameters , namely the density ratio @xmath38 of the two fluids and the mutual friction coefficients @xmath8 and @xmath9 . the increase in temperature causes two competitive effects as described in the following . one effect is that the increase of @xmath39 decreases @xmath40 . the thermal counterflow requires the conservation of mass , yielding @xmath41 where @xmath42 is spatially averaged @xmath43 over the cross section of the channel . the ratio @xmath44 increases with temperature . thus the counterflow velocity @xmath45 becomes more uniform at higher temperature , since @xmath5 is uniform and @xmath46 is non - uniform . figure 4 shows the temperature dependence of the counterflow velocity profile . the other effect is that the enhancement of the mutual friction increases @xmath47 . the mutual friction makes the vortex tangle inhomogeneous under the non - uniform @xmath48 . therefore the increase in temperature renders the profiles of @xmath48 uniform but enhances this action of the mutual friction . these two competitive effects maximize the inhomogeneity @xmath47 at some @xmath36 . ( a ) @xmath36=1.3 k ( @xmath39=0.052 ) ( b ) @xmath36=1.9 k ( @xmath39=0.738 ) the aspect ratio of the channel plays an important role in the density and the profile of the vortex tangle . the increase of the aspect ratio from unity reduces the counterflow velocity gradient along the long side of the cross section . thus the exclusion of the vortex tangle shown in fig . 3 ( d ) does not work so much along the long side of the cross section . hence the vld should increase compared to the vld in a low aspect ratio channel . it would be meaningful to consider how the aspect ratio affects the normal fluid , though it is prescribed in this formulation . the linear stability analysis of the navier - stokes equation shows that the critical reynolds number for turbulence transition of a viscous fluid increases significantly with decreasing the aspect ratio @xcite . according to this result , the normal fluid in counterflow could remain laminar in a low aspect ratio channel even if the superfluid becomes turbulent and the vortex tangle disturbs the normal fluid , which may correspond to the ti state . in order to understand the ti state , therefore , the studies in a low aspect ratio channel like this work will be indispensable . this work was supported by jsps kakenhi grant number 26400366 and mext kakenhi `` fluctuation & structure '' grant number 26103526 . 99 tough j t 1982 _ progress in low temperature physics _ , ed d f brewer ( amsterdam : north - holland ) , vol . 8 melotte d l and barenghi d f 1998 _ phys . lett . _ * 80 * 4181 guo w , cahn s b , nikkel j a , vinen w f , and mckinsey d n 2010 _ phys . lett . _ * 105 * 045301 kivotides d 2007 _ phys . _ b * 76 * 054503 baggaley a w and laizet s 2013 _ phys . fluids _ * 25 * 115101 schwarz k w 1985 _ phys . _ b * 31 * 5782 adachi h , fujiyama s and tsubota m 2010 _ phys . rev . _ b * 81 * 104511 1998 _ the handbook of fluid dynamics _ , ed r w johnson ( boca raton : crc press ) vinen w f 1957 _ proc . london _ ser . a * 242 * 493 childers r k and tough j t 1976 _ phys . _ b * 13 * 1040 tatsumi t and yoshimura t 1990 _ j. fluid mech . _ * 212 * 437
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we perform a numerical analysis of superfluid turbulence produced by thermal counterflow in he ii by using the vortex filament model .
counterflow in a low aspect ratio channel is known to show the transition from laminar flow to the two turbulent states ti and tii .
the present understanding is that the ti has the turbulent superfluid and the laminar normal fluid but both fluids are turbulent in the tii state .
this work studies the vortex tangle in the ti state .
solid boundary condition is applied to walls of a square channel , and the velocity field of the normal fluid is prescribed to be a laminar poiseuille profile .
an inhomogeneous vortex tangle , which concentrates near the solid boundaries , is obtained as the statistically steady state .
it is sustained by its characteristic space - time oscillation .
the inhomogeneity of the vortex tangle shows the characteristic dependence on temperature , which is caused by two competitive effects , namely the profile of the counterflow velocity and the mutual friction .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
particles moving in a viscous fluid induce a local flow field that affects other particles . these long - range , many - body interactions , mediated by the solvent are commonly called hydrodynamic interactions ( hi ) . the presence of hi is known to affect the dynamic properties of soft matter : they modify the values of diffusion coefficients in colloidal suspensions @xcite , affect the characteristics of the coil - stretch transition in polymers @xcite , change the kinetic pathways of phase separation in binary mixtures @xcite , alter the kinetics of macromolecule adsorption on surfaces @xcite or cause the polymer migration in microchannels @xcite . they are also important in the dynamics of biological soft matter , such as dna @xcite , proteins @xcite or lipid membranes @xcite . the proper account of hydrodynamic interactions is thus essential in simulation studies of soft matter in the flow . unfortunately , hi depend in a complicated nonlinear way on the instantaneous positions of all particles in the system . for a system of spheres , exact explicit expressions for the hydrodynamic interaction tensors exist in the form of the power series in interparticle distances , which may be incorporated into the simulation scheme @xcite . these are however relatively expensive numerically , thus various approximations are resorted to in order to make the computations more tractable . the simplest one is based on the oseen tensor , which assumes that the particles can be regarded as point force sources in the fluid . however , the diffusion matrix constructed in this way is not suitable for the brownian dynamics simulations , because it becomes non - positive definite when separations between the particles become small . this is not only unphysical ( since the positivity of diffusion is a consequence of second law of thermodynamics ) but also leads to numerical problems in the brownian dynamics simulations , where a square root of diffusion matrix is needed . another commonly used approximation is the rotne - prager - yamakawa tensor @xcite , which takes into account all the hi terms up to @xmath1 in the expansion in the inverse distance between the particles ( where @xmath2 is the particle radius ) . nevertheless , if the particles overlap , @xmath3 , the rpy tensor again looses its positive definiteness . to avoid this , a regularization for @xmath3 has been proposed by @xcite , which is not singular at @xmath4 and positive definite for all the particle configurations . the rotne - prager - yamakawa tensor with this regularization is by far the most popular method of accounting for hi in soft matter modelling @xcite . notably , rpy tensor is divergence - free , which considerably simplifies the application of brownian dynamics algorithm @xcite . however , when one goes beyond rpy approximation and includes many - body effects in hydrodynamic interactions , the divergence of mobility matrix becomes is non - zero and needs to be taken into account in brownian dynamics simulation schemes @xcite . the present paper takes a close look at the rotne - prager - yamakawa approximation and generalizes it in a number of ways . first , we re - derive the original rpy tensor using the direct integration of force densities over the sphere surfaces . when the spheres overlap then this method gives us automatically the regularization correction . in this way we derive the rpy regularizations not only for the translational degrees of freedom ( already obtained by @xcite ) but also for rotational degrees of freedom , as well as for the shear disturbance matrix @xmath5 - another hydrodynamic tensor , which gives the response of the particles to the external shear flow . the mobility evaluated using our technique may be applied for calculation of the diffusion tensor of complex molecules @xcite using bead models which include overlapping spheres . finally , we show how these results can be generalized for other boundary conditions and corresponding propagators . we consider a suspension of @xmath6 identical spherical particles of radius @xmath2 , in an incompressible fluid of viscosity @xmath7 at a low reynolds number . the particles are immersed in a linear shear flow @xmath8 where @xmath9 is the constant velocity gradient matrix , e.g. for a simple shear flow @xmath10,\qquad\dot{\gamma}=\mathrm{{const.}}\label{eq : shear_matrix}\ ] ] due to the linearity of the stokes equations , the forces and torques exerted by the fluid on the particles ( @xmath11 and @xmath12 ) depend linearly on translational and rotational velocities of the particles ( @xmath13 , @xmath14 ) . this relation defines the generalized friction matrix @xmath15 @xmath16 where @xmath17 ( with @xmath18 and @xmath19 ) are the cartesian tensors and the superscripts @xmath20 , @xmath21 and @xmath22 correspond to the translational , rotational and dipolar components , respectively . the tensor @xmath23 is the symmetric part of @xmath9 in ( [ eq : shear_matrix ] ) and @xmath24 is the vorticity of the incident flow . finally @xmath25 corresponds to the position of particle @xmath26 . the reciprocal relation giving velocities of particles moving under external forces / torques in external flow @xmath27 is determined by generalized mobility matrix @xmath28 written after @xcite @xmath29 + \left ( \begin{array}{c } \mathbf{c}^{t}_{i } \\ \mathbf{c}^{r}_{i } \end{array } \right ) : \mathbf{e}_{\infty } , \label{eq : generalmobilitymatrixseparatedc}\ ] ] where the shear disturbance tensor @xmath30 elements are defined as [ eq : ctensor ] @xmath31 in the case of single particle the mobility matrixes reduce to @xmath32 where the friction coefficients for a spherical particle are given by @xmath33 and @xmath34 . finding the mobility matrix ( or the associated diffusion matrix , @xmath35 ) is the problem of a fundamental importance in constructing the numerical algorithms for tracking the motion of the particles in viscous fluid . the two main numerical methods used for this purpose are the _ stokesian dynamics _ , which corresponds to the numerical integration of [ eq : generalmobilitymatrixseparatedc ] and the _ brownian dynamics _ , used whenever the brownian motion of the particles can not be neglected @xcite . in the latter , the random displacements of the particles , @xmath36 , need to be added on top of the deterministic displacements governed by eq . [ eq : generalmobilitymatrixseparatedc ] . the fluctuation - dissipation theorem implies that the covariance of @xmath37 is connected to the mobility matrix , e.g. for the translational displacements @xmath38 hence the calculation of @xmath36 requires finding a matrix @xmath39 such that @xmath40 . this is possible only when the mobility matrix is positively defined . any valid approximation scheme for the hydrodynamic interactions should then not only correctly reproduce the particle mobilities but also guarantee the positive definiteness of the mobility tensors . in principle the hydrodynamic interactions tensors can be calculated with arbitrary precision , following e.g. the multipole expansion or boundary integral method @xcite . in practice , however , the exact approach turns out to be too demanding computationally , so various approximation procedures have to be resorted to . the most commonly used is the rotne - prager - yamakwa approximation @xcite , based on the following idea : when a force ( or torque ) is applied to particle @xmath26 , that particle begins to move inducing the flow in the bulk of the fluid . the extent to which this additional flow affects translational and rotational velocities of another particle ( @xmath41 ) is then calculated using faxen s laws @xcite . in that way one neglects not only the multi - body effects ( involving three and more particles ) but also the higher order terms in two - particle interactions ( e.g. we do not consider the impact of the movement of particle @xmath41 back on particle @xmath26 ) . below , we follow this procedure to derive in a systematic way hydrodynamic tensors for both translational and rotational degrees of freedom . the stokes flow generated by a point force in the unbounded space is given by the oseen tensor @xcite @xmath43 since @xmath44 is a green function for stokes equations , one can use it to calculate the translational @xmath45 and rotational @xmath46 flows generated by the sphere situated at @xmath47 , to which we apply force @xmath48 and/or torque @xmath49 : @xmath50\cdot\bm{\mathcal{f}},\qquad \rho_j > a,\\ \,\\ \frac{1}{\zeta^{tt}}\bm{\mathcal{f}},\qquad \rho_j \leq a , \end{array}\right.\label{eq : v0t}\end{aligned}\ ] ] @xmath51 where @xmath52 is the distance from the sphere centre , @xmath53 denotes integration variable , @xmath54 is the unit normal vector to the sphere at point @xmath53 and @xmath55 denotes an integral over the surface of the sphere situated at @xmath56 . the curl of a tensor is defined in the following way @xmath57 where the greek letters denote the cartesian components . the faxen laws @xcite allow to express the velocity @xmath13 and angular velocity @xmath58 of a sphere @xmath26 immersed in an external flow @xmath59 placed in @xmath25 @xmath60 @xmath61 where the integration is performed over the sphere surface @xmath62 . thus substituting ( [ eq : v0t],[eq : v0r ] ) into equations ( [ eq : faxen_laws ] ) we obtain the contribution to velocity @xmath63 and angular velocity @xmath64 of a sphere @xmath26 due to the force / torque acting on a sphere @xmath41 @xmath65,\label{eq : u_final}\end{aligned}\ ] ] @xmath66.\label{eq : omega_final}\end{aligned}\ ] ] at this stage let us introduce tensors @xmath67 where @xmath68 . above tensors multiplied by force @xmath69 and torque @xmath70 have the interpretation of the force densities on the surface of the sphere due to the force and torque acting on the sphere . we can now write down the following general formulae for the mobility matrix @xmath71 where we use the bra - ket notation defined in the following way @xmath72^{t}\cdot\mathbf{t}_{0}\left(\mathbf{r}'-\mathbf{r}''\right)\cdot\mathbf{w}_{j}^{q}\left(\mathbf{r}''\right),\label{eq : ypq}\ ] ] with @xmath73 and @xmath74 - tensor transposition . the method of calculation of the integrals in ( [ eq : u_final])-([eq : omega_final ] ) is presented in the supplementary material . here , we simply quote the final results denoting @xmath75 . fortunately there is no need to integrate explicitly for non overlapping spheres . for the translational - translational mobility , we get : @xmath76,\quad r_{ij}>2a,\\ \\ \frac{1}{\zeta^{tt}}\left[\left(1-\frac{9r_{ij}}{32a}\right)\mathbf{1}+\frac{3r_{ij}}{32a}\hat{\mathbf{r}}_{ij}\hat{\mathbf{r}}_{ij}\right],\qquad r_{ij}\leq2a , \end{array}\right.\label{eq : mutt_final}\ ] ] which , in the limit of @xmath77 , yields the self mobility @xmath78 next , for the rotational degrees of freedom @xmath79,\qquad r_{ij}\leq2a , \end{array}\right.\label{eq : murr_final}\ ] ] with the self mobility given by @xmath80 finally , the translational - rotational mobility is described by the following tensor @xmath81^{t}=\left\ { \begin{array}{c } \frac{1}{2}\nabla\times\left(\mathbf{1}+\frac{a^{2}}{6}\nabla^{2}\right)\mathbf{t}_{\mathrm{0}}\left(\mathbf{r}_{ij}\right)=\frac{1}{2}\nabla\times\mathbf{t}_{0}\left(\mathbf{r}_{ij}\right)=\frac{1}{8\pi\eta r_{ij}^{2}}\boldsymbol{\epsilon}\cdot\hat{\mathbf{r}}_{ij},\qquad r_{ij}>2a,\\ \\ \frac{1}{16\pi\eta a^{2}}\left(\frac{r_{ij}}{a}-\frac{3}{8}\frac{r_{ij}^{2}}{a^{2}}\right)\boldsymbol{\epsilon}\cdot\hat{\mathbf{r}}_{ij},\qquad r_{ij}\leq2a , \end{array}\right.\label{eq : murt_final}\ ] ] with @xmath82 note that the formulae for the translational mobility matrix , both for @xmath83 and for @xmath84 were derived earlier by @xcite and @xcite and are known as rotne - prager - yamakawa mobility approximation . the expressions for the other components of the mobility matrix @xmath85 and @xmath86 are also known @xcite but only for @xmath83 . however , to our knowledge , the regularizing corrections for @xmath85 and @xmath86 for the overlapping particles ( @xmath84 ) have not been derived so far . importantly , as we will demonstrate in the section ( [ sec : pos_def ] ) , only with the use of these corrections the mobility matrix @xmath28 remains positive definite for all configurations of the particles . contrastingly , in the point - force ( stokeslet ) model which is sometimes used for modelling the dynamics of colloidal suspensions @xcite , the mobility matrix , defined as follows @xmath87 is not positive definite even for non overlapping spheres and does not possess the property ( [ eq : self_tt ] ) . the formula for the 3rd rank convection tensor @xmath30 can be obtained in the following way . @xcite provide a solution for the excess flow @xmath88 , produced by a freely moving sphere situated at @xmath56 in the ambient shear flow @xmath89 , which is a difference between total flow @xmath90 and ambient flow @xmath91\overleftarrow{\bm{\nabla}}\right\ } : \mathbf{e}_{\infty},\label{eq : kimkarrila_flow-1}\ ] ] where @xmath92_{\alpha\beta\gamma}=\partial_{\gamma}t_{\alpha\beta}(\mathbf{r})$ ] . the contribution to the surface force density due to the straining fluid motion is : @xmath93 , thus introducing tensor @xmath94 @xmath95 and using the green s formula we may express the excess flow over the shear flow @xmath89 in the following way @xmath96\overleftarrow{\bm{\nabla}}\right\ } : \mathbf{e}_{\infty},\quad \rho_j > a,\\ \\ -\mathbf{e}_{\infty}\cdot\bm{\rho}_j,\qquad \rho_j \leq a. \end{array}\right.\label{eq : vc}\ ] ] now , by the faxen laws ( [ eq : faxen_laws ] ) the contribution to velocity and angular velocity of another sphere ( say number @xmath26 ) immersed in such flow is @xmath97 @xmath98 for the case of @xmath83 , the form of @xmath99 , @xmath100 is expressed using ( [ eq : faxen_laws ] ) and ( [ eq : vc ] ) in terms of differential operators @xmath101\overleftarrow{\bm{\nabla}}\right\}:\mathbf{e}_{\infty},\label{eq : c_hasimoto_def-1}\ ] ] @xmath102\overleftarrow{\bm{\nabla}}\right\}:\mathbf{e}_{\infty},\label{eq : c_r_definition-1}\ ] ] where @xmath103 denotes derivation with respect to @xmath104 and @xmath105_{\alpha\beta\gamma}=\partial_{\gamma}t_{\alpha\beta}(\mathbf{r}_{ij})$ ] . this allows to write down the final results in the following form @xmath106 _ { \alpha\beta\gamma}=\left\ { \begin{array}{c } \frac{5}{6 } a \left[-\frac{16}{5}\frac{a^{4}}{r_{ij}^{4}}\hat{r}_{ij}^{\gamma}\delta_{\alpha\beta}+\left(-3\frac{a^{2}}{r_{ij}^{2}}+8\frac{a^{4}}{r_{ij}^{4}}\right)\hat{r}_{ij}^{\alpha}\hat{r}_{ij}^{\beta}\hat{r}_{ij}^{\gamma}\right],\qquad r_{ij}>2a,\\ \\ \frac{5}{6 } a \left[\left(-\frac{3}{5}\frac{r_{ij}}{a}+\frac{1}{4}\frac{r_{ij}^{2}}{a^{2}}\right)\hat{r}_{ij}^{\gamma}\delta_{\alpha\beta}-\frac{1}{16}\frac{r_{ij}^{2}}{a^{2}}\hat{r}_{ij}^{\alpha}\hat{r}_{ij}^{\beta}\hat{r}_{ij}^{\gamma}\right],\qquad r_{ij}\leq2a , \end{array}\right.\label{eq : ct}\ ] ] with the respective limit in the self case @xmath107 @xmath108 _ { \alpha\beta\gamma}=\left\ { \begin{array}{c } -\frac{5}{2}\left(\frac{a}{r_{ij}}\right)^{3}\epsilon_{\alpha\beta\zeta}\hat{r}_{ij}^{\zeta}\hat{r}_{ij}^{\gamma},\qquad r_{ij}>2a,\\ \\ -\frac{5}{2 } \left(\frac{3}{16}\frac{r_{ij}}{a}-\frac{1}{32}\frac{r_{ij}^{3}}{a^{3}}\right)\epsilon_{\alpha\beta\zeta}\hat{r}_{ij}^{\zeta}\hat{r}_{ij}^{\gamma},\qquad r_{ij}\leq2a , \end{array}\right.\label{eq : cr}\ ] ] and in the self case as limit @xmath109 the expressions for @xmath84 in ( [ eq : ct ] ) and ( [ eq : cr ] ) vanish for @xmath110 and match with the @xmath83 expressions at @xmath111 . note that ( [ eq : c_hasimoto_def-1 ] ) and ( [ eq : c_r_definition-1 ] ) do not determine @xmath112 and @xmath113 uniquely , since they define only the symmetric and traceless parts of mobility matrix . given this freedom , in ( [ eq : ct])-([eq : mu_rdself ] ) we take the matrices in the simplest algebraic form . this completes our derivation making all the terms in mobility equation ( [ eq : generalmobilitymatrixseparatedc ] ) directly computable under the rotne - prager - yamakawa approximation . it is now a straightforward task to demonstrate the positive definiteness of the mobility matrix given by ( [ eq : ypq ] ) . @xcite provide a simple proof of positive definiteness of a quadratic form such as in ( [ eq : ypq ] ) , which we will now summarize . consider the following quadratic form @xmath114 where @xmath115 is a complex valued function and the upper star is complex conjugation . we will show that from positive definiteness of @xmath116 follows that @xmath117 is positive definite . let @xmath118 where @xmath119 denotes an arbitrary vector . now we write @xmath120 which ends the proof . note that the above proof of positivity does not hold for the point - force model ( [ eq : point_force ] ) . in this case the off - diagonal ( @xmath121 ) terms of the mobility matrix can be cast in the form ( [ eq : ypq ] ) using @xmath122 . the diagonal terms , however , would then become infinite due to the singularity at @xmath123 . this problem is circumvented in the formulation ( [ eq : point_force ] ) by using single - particle mobilities @xmath124 for the diagonal terms . however , the resulting point - force mobility matrix is not positive definite for arbitrary configuration , thus can not be used in brownian dynamics simulations . in this section we consider a general case of particles interacting hydrodynamically e.g. in confined geometry , periodic boundary conditions or in the presence of interfaces . we assume that for a given geometry a positive - definite green s function , @xmath125 , can be derived . such solutions have indeed been constructed , e.g. for systems bounded by a cylinder and a sphere @xcite , for periodic system @xcite as well as for the system bounded by one @xcite and two walls @xcite . we define the rotne - prager - yamakawa approximation for the positive definite mobility matrix in analogous way to ( [ eq : ypq ] ) @xmath126^{\mathrm{t}}\cdot \mathbf{t(r',r''})\cdot \mathbf{w}_{j}^{q}(\mathbf{r '' } ) . \label{011}\ ] ] to clarify notation we introduce differential operators @xmath127 @xmath128 _ { \alpha \beta } = -\frac{1}{2}\epsilon _ { \alpha \beta \gamma } \frac{\partial } { \partial r_{\gamma } } , \quad \left [ \overleftarrow{\mathbf{d}^{r}}(\mathbf{r})\right ] _ { \alpha \beta } = \frac{1}{2}\epsilon _ { \alpha \beta \gamma } \overleftarrow{\frac{\partial } { \partial r_{\gamma } } } . \label{008}\ ] ] where arrow points to the direction of action of differentiation operator . we rewrite ( [ eq : v0 t ] ) and ( [ eq : v0r ] ) using these operators @xmath129 for the external flow @xmath130 which is regular ( has no sources within sphere @xmath26 ) , by the use of the definition of @xmath131 ( [ eq : wtr ] ) , the faxen laws may be written in analogy to ( [ eq : faxen_laws ] ) @xmath132^{t } \cdot \mathbf{v}_0 ( \mathbf{r } ' ) d \sigma ' = \overrightarrow{\mathbf{d}^{t}}(\mathbf{r}_{i})\cdot \mathbf{v}_0 ( \mathbf{r}_i ) , \quad \bm{\omega}_i = \int_{s_i } \left [ \mathbf{w}^{r}_i ( \mathbf{r } ' ) \right]^{t } \cdot \mathbf{v}_0 ( \mathbf{r } ' ) d \sigma ' = \overrightarrow{\mathbf{d}^{r}}(\mathbf{r}_{i})\cdot \mathbf{v}_0 ( \mathbf{r}_i ) . \label{eq : faxen_laws_general}\ ] ] we can now write down the rotne - prager - yamakawa mobilities for the unbounded space ( for oseen propagator @xmath133 for@xmath134 using the differential operators @xmath135 now we decompose the arbitrary propagator @xmath136 as follows @xmath137 + \mathbf{t}_{0}(\mathbf{r}'-\mathbf{r}'')=\mathbf{t}^{\prime } ( \mathbf{r}',\mathbf{r}'')+\mathbf{t}_{0}(\mathbf{r}'-\mathbf{r } '' ) . \label{012}\ ] ] the operator @xmath138 has no singularities at @xmath139 , thus see ( [ eq : faxen_laws_general ] ) , it has the property @xmath140 using ( [ 012 ] ) and ( [ 013 ] ) we can cast the mobility @xmath141 in the following form @xmath142 \nonumber \\ & = & \overrightarrow{\mathbf{d}^{p}}(\mathbf{r}_{i})\cdot \mathbf{t}(\mathbf{r}_{i},\mathbf{r}_{j})\cdot \overleftarrow{\mathbf{d}^{q}}(\mathbf{r}_{j})+\mathbf{y}^{pq}\mathbf{(r}_{ij } ) . \label{014}\end{aligned}\ ] ] the correction @xmath143 is non zero only for @xmath144 and is independent of the propagator @xmath145 . we write down explicitly the corrections for all components of the mobility matrix ( [ eq : mutt_final]),([eq : murr_final]),([eq : murt_final ] ) @xmath146 \right . \\ & & \left . -\frac{1}{8\pi\eta r_{ij}}\left[\left(1+\frac{2a^{2}}{3r_{ij}^{2}}\right)\mathbf{1}+\left(1-\frac{2a^{2}}{r_{ij}^{2}}\right)\hat{\mathbf{r}}_{ij}\hat{\mathbf{r}}_{ij}\right ] \right\ } , \end{array}\ ] ] @xmath147 \right . \\ & & \left . + \frac{1}{16\pi\eta r_{ij}^{3}}\left(\mathbf{1}-3\hat{\mathbf{r}}_{ij}\hat{\mathbf{r}}_{ij}\right ) \right\ } , \end{array}\ ] ] @xmath148 for the self case , @xmath149 the mobility @xmath150 is obtained from eq . ( [ 014 ] , upper line ) in the limit @xmath151 @xmath152 + \frac{1}{\mathbf{\varsigma } _ { 0}^{pq}}. \nonumber\end{aligned}\ ] ] to sum up , we have shown how to evaluate the rotne - prager - yamakawa approximation for an arbitrary propagator @xmath145 by applying to @xmath145 the differential operators in order to avoid the explicit and often infeasible surface integration . this allows one to construct the positive definite hydrodynamic tensors in systems with non - trivial geometry ( e.g. in the presence of a wall , in a channel or in periodic systems ) . for example , taking in ( [ 014 ] ) the green s function for a stokeslet in the presence of a wall @xcite leads ( for non - overlapping spheres ) to the rotne - prager - blake tensor derived before by @xcite , see also @xcite , and @xcite . however , these authors did not derive the regularizing correction for this tensor , which also prevented them from obtaining the self - term in a manner analogous to our eq . ( [ 015 ] ) . on a final note , let us stress that the regularizing correction ( [ eq : y_correction ] ) has the same simple analytical form in all cases , independently of the particular green s function @xmath145 . in this paper , we have re - visited the problem of constructing rotne - prager - yamakawa approximation for mobility and shear disturbance matrices . a systematic method was presented which allows one to derive the rpy approximation in a systematic way , for translational , rotational and dipolar components of the generalized mobility matrix , both for non - overlapping and overlapping particles . the regularization corrections for translational - rotational and rotational - rotational mobility tensors have not been previously derived . these regularizations are crucial in obtaining positive - definite hydrodynamic matrices , which is essential for the brownian dynamics simulations . the positive definiteness also allows for the evaluation of the diffusion tensor and mobility for the bead models ( including overlapping beads ) of complicated molecules . additionally , we have shown how our approach can be generalized to other boundary conditions and corresponding propagators . + ew and km acknowledge the support of the polish national science centre ( grant no 2012/05/b / st8/03010 ) . pjz acknowledges support of the foundation for polish science ( fnp ) through team/2010 - 6/2 project co - financed by the eu european regional development fund . ps acknowledges the support of the polish ministry of science and higher education ( grant no n n202 055440 ) . 35 natexlab#1#1 2012 fibrinogen conformations and charge in electrolyte solutions derived from dls and dynamic viscosity measurements . _ j. colloid interface sci . _ * 385 * , 244257 . 2013 on the importance of hydrodynamic interactions in lipid membrane formation . _ biophys . j. _ * 104 * , 96105 . 2005 hydrodynamic interactions of spherical particles in suspensions confined between two planar walls . _ j. fluid mech . _ * 541 * , 263292 . 1971 a note on the image system for a stokeslet in a no - slip boundary . _ camb . philos . soc . _ * 70 * , 303 . 1991 stokesian dynamics simulations of particle trajectories near a plane . fluids a _ * 3 * , 1853 . 1988 stokesian dynamics . fluid mech . _ * 20 * , 111157 . 1994 friction and mobility of many spheres in stokes flow . _ j. chem . phys . _ * 100 * , 37803790 . 2000 friction and mobility for colloidal spheres in stokes flow near a boundary : the multipole method and applications . _ j. chem . phys . _ * 112 * , 25482561 . 1996 _ an introduction to dynamics of colloids_. elsevier science . 1978 brownian dynamics with hydrodynamic interactions . _ j. chem . phys . _ * 69 * , 1352 . 1988 many - 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a position vector with respect to the centre of the sphere @xmath41 . to demonstrate the method it is enough to calculate explicitly one of the integrals appearing in ( 3.6)-(3.7 ) and ( 3.21)-(3.22 ) . we take the first one , @xmath156 the integration is performed over the surface of sphere @xmath26 , thus @xmath155 is expressed in spherical coordinates @xmath157 associated with sphere @xmath26 which leads to @xmath158+\frac{1}{2\zeta^{tt}}\mathbf{1}\left(1-\frac{r_{ij}}{2a}\right),\label{eq : mu_general_appendix}\ ] ] where @xmath159 is the meridional angle at which the two spheres intersect ( see figure 1 ) , defined by @xmath160 and the vector @xmath161 in the cartesian basis has the form @xmath162 the last term in ( [ eq : mu_general_appendix ] ) results from integration of the @xmath163 expression in ( 3.2 ) from @xmath164 to @xmath165 . all the azimuthal integrals in ( [ eq : mu_general_appendix ] ) are easily calculated to yield @xmath166\nonumber \\ & & + \frac{3}{16\zeta^{tt}}\hat{\mathbf{r}}_{ij}\hat{\mathbf{r}}_{ij}\int_{0}^{\theta_{0}}\mathrm{d}\theta\sin\theta\left(\frac{a^{3}}{r^{\prime3}}-\frac{a^{5}}{r^{\prime5}}\right)\left[2\left(\frac{r_{ij}}{a}+\cos\theta\right)^{2}-\sin^{2}\theta\right],\label{eq : app_mu_tt}\end{aligned}\ ] ] and for the choice of coordinate axes as in figure 1 we have @xmath167.\label{eq : rr}\ ] ] the calculation of @xmath168 is now straightforward . since @xmath169 @xmath170 @xmath171 @xmath172 @xmath173 @xmath174 the @xmath175 component of the mobility matrix takes the form @xmath176,\label{eq : app_mu_tt_final}\ ] ] as in ( 3.11 ) .
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rotne - prager - yamakawa approximation is one of the most commonly used methods of including hydrodynamic interactions in modelling of colloidal suspensions and polymer solutions .
the two main merits of this approximation is that it includes all long - range terms ( i.e. decaying as @xmath0 or slower in interparticle distances ) and that the diffusion matrix is positive definite , which is essential for brownian dynamics modelling . here , we extend the rotne - prager - yamakawa approach to include both translational and rotational degrees of freedom , and derive the regularizing corrections to account for overlapping particles .
additionally , we show how the rotne - prager - yamakawa approximation can be generalized for other geometries and boundary conditions .
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the question about how a piece of information ( a virus , a rumor , an opinion , etc . , ) is globally spread over a network , and which ingredients are necessary to achieve such a success , has motivated much research recently . the reason behind this interest is that identifying key aspects of spreading phenomena facilitates the prevention ( e.g. , minimizing the impact of a disease ) or the optimization ( e.g. the enhancement of viral marketing ) of diffusion processes that can reach system wide scales . in the context of political protest or social movements , information diffusion plays a key role to coordinate action and to keep adherents informed and motivated @xcite . understanding the dynamics of such diffusion is important to locate who has the capability to transform the emission of a single message into a global information cascade , affecting the whole system . these are the so - called `` privileged or influential spreaders '' . beyond purely sociological aspects , some valuable lessons might be extracted from the study of this problem . for instance , current viral marketing techniques ( which capitalizes on online social networks ) could be improved by encouraging customers to share product information with their acquaintances . since people tend to pay more attention to friends than to advertisers , targeting privileged spreaders at the right time may enhance the efficiency of a given campaign . the prominence ( importance , popularity , authority ) of a node has however many facets . from a static point of view , an authority may be characterized by the number of connections it holds , or the place it occupies in a network . this is the idea put forward in @xcite , where the authors seek the design of efficient algorithms to detect particular ( sub)graph structures : hierarchies and tree - like structures . turning to dynamics , a node may become popular because of the attention it receives in short intervals of time @xcite but that is a rather volatile way of being important , because it depends on activity patterns that change in the scale of hours or even minutes . a more lasting concept of influence comprises both a topological enduring ingredient and the dynamics it supports ; this is the case of centola s `` reinforcing signals '' @xcite or the @xmath0-core @xcite , which we follow here . in this paper , we approach the problem of influential spreaders taking into consideration data from the spanish `` 15 m movement '' @xcite . this pacific civil movement is an example of the social mobilizations from the `` arab spring '' to the `` occupy wall - street '' movement that have characterized 2011 . although whether osns have been fundamental instruments for the successful organization and evolution of political movements is not firmly established , it is increasingly evident @xcite that at least they have been nurtured mainly in osns ( facebook , twitter , etc . ) before reaching classic mass media . data from these grassroots movements but also from less conflictive phenomena in the web 2.0 provide a unique opportunity to observe system - wide information cascades . in particular , paying attention to the network structure allows for the characterization of which users have outstanding roles for the success of cascades of information . our results complement some previous findings regarding dynamical influence both at the theoretical @xcite and the empirical @xcite levels . besides , our analysis of activity cascades reveals distinctive traits in different phases of the protests , which provides important hints for future modeling efforts . the `` 15 m movement '' is a still ongoing civic initiative with no party or union affiliation that emerged as a reaction to perceived political alienation and to demand better channels for democratic representation . the first mass demonstration , held on sunday may 15 ( @xmath1 from now on ) , was conceived as a protest against the management of the economy in the aftermath of the financial crisis . after the demonstrations on day @xmath1 , hundreds of participants decided to continue the protests camping in the main squares of several cities ( puerta del sol in madrid , plaa de catalunya in barcelona ) until may 22 , the following sunday and the date for regional and local elections . from a dynamical point of view , the data used in this study are a set of messages ( tweets ) that were publicly exchanged through _ www . twitter.com_. the whole time - stamped data collected comprises a period of one month ( between april 25th , 2011 at 00:03:26 and may 26th , 2011 at 23:59:55 ) and it was archived by _ cierzo development ltd . _ , a start - up company . to filter out the whole sample and choose only those messages related to the protests , 70 keywords ( _ hashtags _ ) were selected , those which were systematically used by the adherents to the demonstrations and camps . the final sample consists of 535,192 tweets . on its turn , these tweets were generated by 85,851 unique users ( out of a total of 87,569 users of which 1,718 do not show outgoing activity , i.e. , they are only receivers ) . see @xcite for more details . twitter is most frequently used as a broadcasting platform . users subscribe to what other users say building a `` who - listens - to - whom '' network , i.e. , that made up of followers and followings in twitter . this means that any emitted message from a node will be immediately available to anyone following him , which is of utmost importance to understand the concept of activity cascade in the next sections . such relationships offer an almost - static view of the relationships between users , the `` follower network '' for short . to build it , data for all the involved users were scrapped directly from _ www.twitter.com_. the scrap was successful for the 87,569 identified users , for whom we also obtained their official list of followers restricted to those who had some participation in the protests . the resulting structure is a directed network , direction indicates who follows who in the online social platform . in practice , we take this underlying structure as completely static ( does not change through time ) because its time scale is much slower , i.e. , changes occur probably in the scale of weeks and months . in - degree @xmath2 expresses the amount of users a node is following ; whereas out - degree represents the amount of users who follow a node . this network exhibits a high level of reciprocity : a typical user holds many reciprocal relationships ( with other users who the node probably knows personally ) , plus a few unreciprocated nodes which typically point at hubs . . in yellow , the cumulative proportion of emitted messages as a function of time . note that the two lines evolve in almost the same way . according to this evolution , we have distinguished two sub - periods : one of them characterized as `` slow growth '' due to the low activity level and the other one tagged as `` explosive '' or `` bursty '' due to the intense information traffic within it . ] the main topological features of the follower network fit well in the concept of `` small - world '' @xcite , i.e. , low average shortest path length and high clustering coefficient . furthermore , both in- and out - degree distribute as a power - law , indicating that connectivity is extremely heterogenous . thus , the network supporting users interactions is scale - free with some rare nodes that act as hubs @xcite . an activity cascade or simply `` cascade '' , for short , starting at a _ seed _ , occurs whenever a piece of information or replies to it are ( more or less unchanged ) repeatedly forwarded towards other users . if one of those who `` hear '' the piece of information decides to reply to it , he becomes a _ spreader _ , otherwise he remains as a mere _ listener_. the cascade becomes global if the final number of affected users @xmath3 ( including the set of spreaders and listeners , plus the seed ) is comparable to the size of the whole system @xmath4 . intuitively , the success of an activity cascade greatly depends on whether spreaders have a large set of followers or not ( figure [ example ] ) ; remarkably , the seed is not necessarily very well connected . this fact highlights the entanglement between dynamics and the underlying ( static ) structure . note that the previous definition is too general to attain an _ operative _ notion of cascade . one possibility is to leave time aside , and consider only identical pieces of information traveling across the topology ( a _ retweet _ , in the twitter jargon ) . this may lead to inconsistencies , such as the fact that a node decides to forward a piece of information long after receiving it ( perhaps days or weeks ) . it is impossible to know whether his action is motivated by the original sender , or by some exogenous reason , i.e. , invisible to us . one may , alternatively , take into consideration time , thus considering that , regardless of the exact content of a message , two nodes belong to the same cascade as consecutive spreaders if they are connected ( the latter follows the former ) and they show activity within a certain ( short ) time interval , @xmath5 . the probability that exogenous factors are leading activation is in this way minimized . also , this concept of cascade is more inclusive , regarding dialogue - like messages ( which , we emphasize , are typically produced in short time spans ) . this scheme exploits the concept of spike train from neuroscience , i.e. , a series of discrete action potentials from a neuron taken as a time series . at a larger scale , two brain regions are identified as functionally related if their activation happens in the same time window . consequently , message chains are reconstructed assuming that activity is contagious if it takes place in short time windows . we apply the latter definition to explore the occurrence of information cascades in the data . in practice , we take a seed message posted by @xmath6 at time @xmath7 and mark all of @xmath6 s followers as listeners . we then check whether any of these listeners showed some activity at time @xmath8 . this is done recursively until no other follower shows activity , see figure [ example ] . in our scheme , a node can only belong to one cascade ; this constraint introduces a bias in the measurements , namely , two nodes sharing a follower may show activity at the same time , so their follower may be counted in one or another cascade ( with possible important consequences regarding average cascades size and penetration in time ) . to minimize this degeneration , we perform calculations for many possible cascade configurations , randomizing the way we process data . we distinguish information cascades ( or just cascades , for short ) from spreader - cascades . in information cascades we count any affected user ( listeners and spreaders ) , whereas in spreader - cascades only spreaders are taken into account . we measure cascades and spreader - cascades size distributions for three different scenarios : one in which the information intensity is low ( slow growth phase , from @xmath9 to @xmath10 ) , one in which activity is bursty ( explosive phase , @xmath11 to @xmath12 ) and one that considers all available data ( which spans a whole month , and includes the two previous scenarios plus the time in - between , @xmath9 to @xmath13 ) . figure [ growth ] illustrates these different periods . the green line represents the cumulative proportion of nodes in the network that had shown some activity , i.e. , had sent at least one message , measured by the hour . we tag the first 10 days of study as `` slow growth '' because , for that period , the amount of active people grew less than 5% of the total of users , indicating that recruitment for the protests was slow at that time . the opposite arguments apply in the case of the bursty or `` explosive '' phase : in only 8 days the amount of active users grew from less than 10% up to over an 80% . the same can be said about global activity ( in terms of the total number of emitted directed messages the activity network ) , which shows an almost exact growth pattern . besides , within the different time periods slow growth , explosive and total , different time windows have been set to assess the robustness of our results . our proposed scheme relies on the contagious effect of activity , thus large time windows , i.e. , @xmath14 hours , are not considered . the @xmath0-core decomposition of a network consists of identifying particular subsets of the network , called @xmath0-cores , each obtained by recursively removing all the vertices of degree less than @xmath0 , where @xmath15 indicates the total number of in- and out - going links of a node , until all vertices in the remaining graph have degree at least @xmath0 . in the end , each node is assigned a natural number ( its coreness ) , the higher the coreness the closer a node is to the nucleus or core of the network . the main advantage of this centrality measure is , in front of other quantities , its low computational cost that scales as @xmath16 , where @xmath4 is the number of vertices of the graph and @xmath17 is the number of links it contains @xcite . this decomposition has been successfully applied in the analysis of the internet and the autonomous systems structure @xcite . in the following section , we will use the @xmath0-core decomposition as a means to identify influence in social media . in particular , we discuss which , degree or coreness , is a better predictor of the extent of an information cascade . the upper panels ( @xmath18 ) of figure [ fig3 ] reflect that a cascade of a size @xmath19 can be reached at any activity level ( slow growth , explosive or both ) . as expected , these large cascades occur rarely as the power - law probability distributions evidence . this result is robust to different temporal windows up to 24h . in contrast , lower ( @xmath20 ) panels show significant differences between periods . specifically , the distribution of involved spreaders in the different scenarios changes radically from the `` slow growth '' phase ( figure [ fig3]d ) to the `` explosive '' period ( figure [ fig3]f ) ; the distribution that considers the whole period of study just reflects that the bursty period ( in which most of the activity takes place ) dominates the statistics . the importance of this difference is that one may conclude that , to attain similar results a proportionally much smaller amount of spreaders is needed in the slow growth period . going to the detail , however , it seems clear ( and coherent with the temporal evolution of the protests , fig . [ growth ] ) that although cascades in the slow period ( panel a ) affect as much as @xmath21 of the population , the system is in a different dynamical regime than in the explosive one : indeed , distributions suggest that there has been a shift from a subcritical to a supercritical phase . the previous conclusions raise further questions : is there a way to identify `` privileged spreaders '' ? are they placed randomly throughout the network s topology ? or do they occupy key spots in the structure ? and , will these influential users be more easily detected in a bursty period ( where large cascades occur more often ) ? in what context will influential spreaders single out ? to answer these questions , we capitalize on previous work suggesting that centrality ( measured as the @xmath0-core ) enhances the capacity of a node to be key in disease spreading processes @xcite . the authors in @xcite discussed whether the degree of a node ( its total number of neighbors , @xmath0 ) or its @xmath0-core ( a centrality measure ) can better predict the spreading capabilities of such node . note that the @xmath0-shell decomposition splits a network in a few levels ( over a hundred ) , while node degrees can range from one or two up to several thousands . we have explored the same idea , but in relation to activity cascades which are the object of interest here . the upper left panel of fig . [ coredegree ] shows the spreading capabilities as a function of classes of @xmath0-cores . specifically , we take the seed of each particular cascade and save its coreness and the final size of the cascade it triggers . having done so for each cascade , we can average the success of cascades for a given core number . remarkably , for every scenario under consideration ( slow , explosive , whole ) , a higher core number yields larger cascades . this result supports the ideas developed in @xcite , but it is at odds with those reported in @xcite , which shows that the @xmath0-core of a node is not relevant in rumor dynamics . exactly the same conclusion ( and even more pronounced ) can be drawn when considering degree ( lower left panel ) , which appears to be in contradiction with the mentioned previous evidence @xcite . at a first sight , our findings seem to point out that if privileged spreaders are to be found , one should simply identify the individuals who are highly connected . however , this procedure might not be the best choice . the right panels in figure [ coredegree ] show the @xmath0-core ( upper ) and degree ( lower ) distributions , indicating the number of nodes which are seeds at one time or another , classified in terms of their coreness or degree . unsurprisingly , many nodes belong to low cores and have low degrees . the interest of these histograms lies however in the tails of the distributions , where one can see that , while there are a few hundred nodes in the high cores ( and even over a thousand in the last core ) , highest degrees account only for a few dozen of nodes . in practice , this means that by looking at the degree of the nodes , we will be able to identify quite a few influential spreaders ( the ones that produce the largest cascades ) . however , the number of such influential individuals are far more than a few . as a matter of fact , high cascading capabilities are distributed over a wider range of cores , which in turn contain a significant number of nodes . focusing on fig . [ coredegree ] , note that triggering cascades affecting over @xmath22 of the network s population demands nodes with @xmath23 . checking the distribution of degrees ( right - hand side ) , it is easy to see that an insignificant amount of nodes display such degree range . in the same line , we may wonder what it takes to trigger cascades affecting over @xmath22 of the network s population , from the @xmath0-core point of view . in this case , nodes with @xmath0-core around 125 show such capability . a quick look at the core distribution yields that over 1500 nodes accomplish these conditions , i.e. , they belong to the 125th @xmath0-shell or higher . we may now distinguish between scenarios in figure [ coredegree ] . while any of the analyzed periods shows a growing tendency , i.e. , cascades are larger the larger is the considered descriptor , we highlight that it is in the slow growth period ( black circles ) where the tendency is more clear , i.e. , results are less noisy . between the other two periods , the explosive one ( red squares ) is distinctly the less robust , in the sense that cascade sizes oscillate very much across @xmath0-cores , and the final plot shows a smaller slope than the other two . this subtle fact is again of great importance : it means that during `` information storms '' a large cascade can be triggered from anywhere in the network ( and , conversely , small cascades may have begun in important nodes ) . the reason for this is that in periods where bursty activity dominates the system suffers `` information overflow '' , the amount of noise flattens the differences between nodes . for instance , in these periods a node from the periphery ( low coreness ) may balance his unprivileged situation by emitting messages very frequently . this behavior yields a situation in which , from a dynamical point of view , nodes become increasingly indistinguishable . the plot corresponding to the whole period analyzed ( green triangles ) lies consistently between the other two scenarios , but closer to the relaxed period . this is perfectly coherent , the study spans for 30 days and the explosive period represents only 25% of it , whereas the relaxed period stands for over 33% . furthermore , those days between @xmath10 and @xmath11 , and beyond @xmath12 , resemble the relaxed period as far as the flow of information is concerned . online social networks are called to play an ever increasing role in shaping many of our habits ( be them commercial or cultural ) as well as in our position in front of political , economical or social issues not only at a local , country - wide level , but also at the global scale . it is thus of utmost importance to uncover as many aspects as possible about topological and dynamical features of these networks . one particular aspect is whether or not one can identify , in a network of individuals with common interests , those that are influentials to the rest . our results show that the degree of the nodes seems to be the best topological descriptor to locate such influential individuals . however , there is an important caveat : the number of such privileged seeds is very low as there are quite a few of these highly connected subjects . on the contrary , by ranking the nodes according to their @xmath0-core index , which can be done at a low computational cost , one can safely locate the ( more abundant in number ) individuals that are likely to generate large ( near to ) system - wide cascades . the results here presented also lead to a surprising conclusion : periods characterized by explosive activity are not convenient for the spreading of information throughout the system using influential individuals as seeds . this is because in such periods , the high level of activity mainly coming from users which are badly located in the network introduces noise in the system . consequently , influential individuals lose their unique status as generators of system wide cascades and therefore their messages are diluted . on more general grounds , our analysis of real data remarks the importance of empirical results to validate theoretical contributions . in particular , fig . [ coredegree ] , together with the observations in @xcite , raises some doubts about rumor dynamics as a good proxy to real information diffusion . we hypothesize that such models approach information diffusion phenomena in a too simplistic way , thus failing to comprise relevant mechanisms such as complex activity patterns @xcite . finally , although the underlying topology may be regarded as constant , any modeling effort should also contemplate the time evolution of the dynamics . indeed , fig . [ fig3 ] suggests that the system is in a sub - critical phase when activity level is low , and critical or supercritical during the explosive period . this is related to the rate at which users are increasingly being recruited as active agents , i.e. the speed at which listeners become spreaders . this work has been partially supported by micinn through grants fis2008 - 01240 and fis2011 - 25167 , and by comunidad de aragn ( spain ) through a grant to the group fenol .
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social media have provided plentiful evidence of their capacity for information diffusion .
fads and rumors , but also social unrest and riots travel fast and affect large fractions of the population participating in online social networks ( osns ) .
this has spurred much research regarding the mechanisms that underlie social contagion , and also who ( if any ) can unleash system - wide information dissemination .
access to real data , both regarding topology the network of friendships and dynamics the actual way in which osns users interact , is crucial to decipher how the former facilitates the latter s success , understood as efficiency in information spreading . with the quantitative analysis that stems from complex network theory ,
we discuss who ( and why ) has privileged spreading capabilities when it comes to information diffusion .
this is done considering the evolution of an episode of political protest which took place in spain , spanning one month in 2011 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the interaction of ultraintense lasers with jets of molecular clusters or nanodroplets ( with typical size in the range @xmath0 nm and containing @xmath1 particles ) is a central research topic @xcite , with important applications , such as tabletop nuclear fusion for compact neutron sources @xcite , or the laboratory investigations of nucleosynthesis reactions , relevant to astrophysical scenarios @xcite . clustered media can be regarded as sparse distributions of tiny solid targets , a peculiar configuration that allows for both a deep penetration of the laser radiation and a strong laser - matter coupling with many individual , overdense targets , thus providing extremely efficient energy absorption @xcite . when hit by an ultraintense laser beam , the neutral atoms in a cluster are promptly ionized ( cf . ref . @xcite for a detailed analysis of the concurring ionization mechanisms in different laser / cluster configurations ) and a dense `` nanoplasma '' @xcite is formed . the free electrons then absorb energy from the laser pulse @xcite and start expanding , causing the formation of strong electric fields , which lead to efficient ion acceleration , as first predicted by dawson @xcite . when the energy transferred to the electrons is much smaller than the electrostatic energy stored in the ion core , charge separation is localized to regions much smaller than the cluster @xcite , which then remains quasi - neutral and undergoes a hydrodynamic - like expansion @xcite ; in opposite conditions ( e.g. with small deuterium clusters exposed to extremely intense laser radiation ) the electrons suddenly escape from the cluster and the remaining bare - ion distribution undergoes a pure coulomb explosion ( ce ) @xcite . in intermediate situations , the expansion dynamics is a mixture of the phenomenology of the two limits , with the expansion process being strongly dependent on the self - consistent dynamics of ions and trapped electrons @xcite . when increasing the laser energy , or when lowering the cluster size and density , the expansion conditions vary smoothly from quasi - neutral , hydrodynamic - like regimes to pure ce regimes , as confirmed by particle - in - cell ( pic ) simulations @xcite of the self - consistent laser - cluster interaction and by kinetic or fluid modeling of the expansion of finite - size , non - quasi - neutral plasma expansions @xcite . therefore , the expansion regime can be controlled by regulating the amount of energy transferred to the electrons @xcite , which can be obtained with appropriately shaped laser beams . an important example is the irradiation of homonuclear deuterium clusters with two sequential laser pulses having different intensities @xcite . in this way , one can taylor the expansion dynamics so as to induce overrunning between ions and the consequent formation of expanding shells containing multiple ion flows . within such structures ( here denoted as `` shock shells '' , following the terminology in @xcite ) , the relative velocities between deuterium ions from different flows can be sufficiently high for energetic collisions and _ dd _ fusion reactions to occurr @xcite . since these intracluster reactions occur early in the expansion , and before different exploding clusters overlap , they are expected to produce a time resolved burst of fusion neutrons before the bulk neutron signal due to intercluster reactions @xcite . in the case of heteronuclear clusters , e.g. deuterium - tritium clusters , ion species having different charge - to - mass ratios expand with different velocities making intracluster , interspecies reactions possible also with standard single - pulse irradiation @xcite . this paper provides an organic review of the work on controlled expansions of clusters and nanoplasmas published in @xcite , complemented with novel simulation results obtained with a recently developed molecular - dynamics technique @xcite . in the following , the transition to the ce regime is analyzed with a self - consistent kinetic model ( section [ sec : expansion ] ) , the concept of shock shell is briefly reviewed ( section [ sec : shocks ] ) , and the possibility of achieving intracluster nuclear reactions in homonuclear clusters @xcite by driving a slow expansion followed by a sudden ce @xcite is explored ( section [ sec : reactions ] ) . in order to analyze the influence of the electrons on the expansion dynamics , a kinetic model for the collisionless expansion of a spherically symmetric nanoplasma has been developed @xcite , based on the assumptions that the electrons are nonrelativistic and resorting to the large mass disparity between electrons and ions . the expansion process is divided in two stages : an initial charging transient with frozen ions , and the long term expansion of both ions and electrons . in the second stage , the expansion dynamics is described self - consistently by following the radial motion of the cold ions , while accounting for the three - dimensional dynamics of the hot electrons using a sequence of `` ergodic '' equilibrium configurations @xcite , represented by stationary solutions of the vlasov equation that depend on the total energy @xmath2 only . the electron density instantaneously in equilibrium with the electrostatic potential @xmath3 can be written as @xmath4 where @xmath5 indicates functional dependence on @xmath3 , @xmath6 is the energy distribution of the electrons , and @xmath7^{\frac{1}{2}}{\mathrm{d}r}^{\prime}$ ] is the probability density of finding an electron having total energy @xmath8 at the radial position @xmath9 . the self - consistent potential @xmath3 satisfies the nonlinear poisson equation @xmath10 $ ] . by resorting to the theory of adiabatic invariants for time varying hamiltonians @xcite , a closed set of equations is then obtained in the form @xmath11 where @xmath12 and @xmath13 are the ion mass and charge , respectively . equations and , determining the ion trajectory @xmath14 ( @xmath15 : initial radius ) and the evolution of the electron energy @xmath16 ( @xmath17 : initial energy ) , are coupled with the nonlinear poisson equation . in eq . , the relation @xmath18 has been used , where @xmath19 is the energy distribution of the electrons resulting from the initial charging transient , a fast process that can not be described by equations the self - consistent shape of @xmath19 is determined by a virtual charging transient , in which the ions stay immobile , while an external potential barrier , initially confining the electrons , is gradually moved from @xmath20 to infinity , with a series of small radial displacements ( the validity of this procedure has been confirmed by particle - in - cell simulations @xcite and ad - hoc solutions of the vp model @xcite ) . the initial , nonequilibrium electron distribution is assumed to be a maxwellian , with temperature @xmath21 , so that the whole expansion dynamics is fully determined by the single dimensionless parameter @xmath22 ( @xmath23 : maximum ion energy attainable from the ce of a uniform ion sphere with radius @xmath20 and total charge @xmath24 ; @xmath25 : initial electron debye length ) , which accounts for both the initial electron temperature and the cluster size and density . the positive charge buildup at the ion front , @xmath26 , and the mean kinetic energy of the trapped electrons , @xmath27 , as functions of @xmath28 , are displayed in fig . [ fig : qeq_teq ] . simple fits for @xmath26 and @xmath27 are found as @xmath29 and @xmath30 $ ] , where @xmath31 . for @xmath32 ( i.e. @xmath33 ) , the fits reduce to @xmath34 , thus recovering the theoretical results for planar expansions @xcite , and to @xmath35 . when the initial equilibrium is reached and the ions are allowed to expand , gaining kinetic energy , the electrons cool down and the charge buildup decreases ( until a ballistic regime is reached for both species @xcite ) . depending on the value of @xmath28 , such behavior strongly affects the ion dynamics and the resulting distribution of ion energy @xcite : for @xmath36 , the spectrum is monotonic as in a ce ; for @xmath37 , it exhibits a local maximum far from the cutoff energy , thus being qualitatively different from the ce case . in this respect , the transition value @xmath38 can be considered as the lower bound for the validity of the ce approximation . the cutoff energy is fit by @xmath39 , which , for @xmath40 , reduces to @xmath41 ; for @xmath37 , the location of the maximum is fit by the power - law @xmath42 . these scaling laws , valid for any combination of @xmath20 , @xmath43 , and @xmath21 , can be useful to interpret experimental data @xcite . in the absence of electrons , the explosion dynamics of a spherical distribution of cold ions is described by equation , which reduces to @xmath44 , where @xmath45 is the number of ions enclosed by a sphere of radius @xmath9 at time @xmath46 . if the initial profile of ion density is uniform and step - like ( i.e. @xmath47 for @xmath48 ) , the repulsive electric field grows linearly for @xmath49 , reaching its maximum at the outer boundary , which causes the outer ions to be always faster than the inner ones . thus , ions never overtake each other , @xmath45 is conserved along the ion trajectories ( so is the total energy of each ion ) , and the equation of motion can be integrated analytically @xcite , yielding , for ions initially at rest , the expansion velocity @xmath50^{1/2}r_0\omega{_{\mathrm{pi}}}$ ] , where @xmath51 is the initial ion plasma frequency . the radial trajectory is then given , in implicit form , by @xmath52^{1/2 } + \log[\xi{_{\mathrm{i}}}^{1/2}+(\xi{_{\mathrm{i}}}-1)^{1/2 } ] = ( 2/3)^{1/2}\omega{_{\mathrm{pi}}}t$ ] , where @xmath53 is the expansion factor . in this solution , @xmath54 is independent of @xmath15 , the @xmath55 phase - space profile is always a straight line with equation @xmath56^{1/2}\omega{_{\mathrm{pi}}}r$ ] , and the ion density maintains its step - like form , decreasing in time as @xmath57 . the asymptotic ion energy distribution is @xmath58 ( cf . [ fig : ionspectrum_t0 ] ) . if , more realistically , nonuniform density profiles are considered , the expansion features change qualitatively @xcite : when the initial density is a decreasing function of @xmath9 , the repulsive coulomb field reaches its maximum within the ion sphere @xcite , leading to ion overtaking and to the formation of multiple - flow regions ( shock shells ) with characteristic multi - branched phase - space profile as in fig . [ fig : dp ] ( since the ion trajectories are no longer independent of one another , the above analytical solution no longer holds @xcite ) . the physical interest of these shock shells resides in the appearance of multiple flows with large relative velocities within a single exploding cluster , which can lead to energetic ion - ion collisions and to intracluster reactions in homonuclear clusters @xcite . an effective strategy to produce large - scale shock shells in a controlled fashion is combining different expansion regimes so that a slow hydrodynamic - like expansion , providing a smoothly decreasing ion - density profile , is followed by an abrupt ce . this is achievable with a double - pump irradiation scheme as in fig . [ fig : dp ] , wherein a weak pulse ( intensity : @xmath59 ; duration : @xmath60 ) is followed by a strong pulse ( intensity : @xmath61 ; duration : @xmath62 ) , with suitable time delay @xmath63 : @xmath59 must be sufficiently high to ionize the atoms , creating a nanoplasma , but not so high as to expel a significant fraction of the electrons from the cluster , whereas @xmath64 must be high enough to drive a sudden ce . in these conditions , a pronounced shock shell is formed , whose features are determined by the key double - pump parameters , namely @xmath59 and @xmath63 , which strongly affects the density profile of the cluster when being hit by the second laser . the effectiveness of the technique has been demonstrated @xcite by resorting to two- and three - dimensional pic simulations performed using the osiris 2.0 framework @xcite , closely matching realistic physical scenarios . the ability of producing multiple flows with high relative velocities within single exploding clusters makes the phenomenon attractive as a possible way to induce intracluster fusion reactions in homonuclear deuterium clusters . as pointed out in @xcite , the rates for intracluster reactions can be significantly higher than those for intercluster reactions , because the typical densities within a shock shell ( @xmath65 @xmath66 ) are higher than those within the hot plasma filament resulting from the exploded clusters ( @xmath67 @xmath66 ) . however , it is crucial to consider also that the expanding shock - shell stays appreciably dense only for a very brief time ( @xmath68 fs ) , much shorter than the typical disassembly time of the plasma filament ( @xmath69 ps ) : as shown in @xcite , the rates of intracluster reactions exhibit a sharp , time - resolved peak right after the shock - shell formation . once the expansion dynamics is known , the number of intracluster fusion reactions is obtained by summing over all possible contributions from collisions between ions belonging to different velocity branches in a shock shell ( cf . fig . [ fig : dp ] ) . the number of reactions per unit time and volume , @xmath70 , is given by @xmath71 where @xmath72 is the cross section for _ dd _ fusion , while @xmath73 and @xmath74 indicate , respectively , the ion density and velocity on the @xmath75th branch . the number of intracluster fusion reactions , @xmath76 , is obtained by integrating @xmath70 over time and space , as @xmath77 where @xmath78 , @xmath79 , and @xmath80 are the shock - shell boundaries and formation time , respectively . in @xcite , the influence of the double - pump parameters on @xmath76 was analyzed using a simple 1d model wherein the laser field of the first pulse gradually strips the initially neutral cluster of a part of its electrons according to a cluster barrier suppression ionization ( cbsi ) model ( cf . ref . this provided preliminary estimates for the optimal combinations of delay and intensities that maximize @xmath76 , and suggested that , for very large clusters ( @xmath81 nm ) , the intracluster reaction yield can become comparable with the intercluster neutron yield , with @xmath82 of the fusion reactions arising from intracluster collisions @xcite . accurate calculations of the intracluster reaction yields achievable with the double - pump irradiation of homonuclear deuterium clusters are currently being performed resorting to three - dimensional molecular - dynamics simulations based on the recently developed scaled electron and ion dynamics ( seid ) technique @xcite , for different values of laser intensities , durations , and delays . preliminary results indicate that , with @xmath59 in the range @xmath83 w@xmath84 , @xmath60 in the range @xmath85 fs , @xmath86 w@xmath84 , @xmath87 fs , and @xmath63 in the range @xmath88 fs , the number of intracluster reaction per cluster can be as high as @xmath89 with a cluster size @xmath90 nm . these values are comparable with the estimate in @xcite ; furthermore , as predicted in @xcite , it is found that the optimal intensities of the first pulse are significantly lower , typically by an order of magnitude , than those estimated with the cbsi model . calculations based on the seid technique are being performed also in the case of heteronuclear deuterium - tritium clusters and nanodroplets irradiated by a single laser pulse , for either homogenous mixtures or layered targets composed by a deuterium core surrounded by a tritium shell . owing to the higher value of the deuterium - tritium fusion cross section for the energy range considered , higher number of intracluster reactions , on the order of @xmath91 , are obtained with respect to the pure - deuterium case . detailed results of these numerical simulations will be presented in a future publication @xcite . the ergodic model has been used to simulate the collisionless expansion of spherical nanoplasmas driven by energetic electrons over a wide range of values of plasma size , plasma density , and initial electron thermal energy , combined in the single dimensionless parameter @xmath28 : a qualitative change in the shape of the asymptotic energy spectrum of the ions when approaching the coulomb explosion regime has been identified and accurate fit laws for the relevant expansion features have been provided . the douple - pump irradiation technique to induce the formation of multiple ion flows during the expansion of large homonuclear deuterium clusters or nanodroplets has been described , and its applicability to produce intracluster _ dt _ fusion reactions has been proved resorting to three - dimensional numerical simulations performed with a recently developed scaled molecular dynamics technique . work partially supported by fct ( portugal ) through grants pdct / poci/66823/2006 , sfrh / bd/39523/2007 , and sfrh / bpd/34887/2007 , and by the european community - new and emerging science and technology activity under the fp6 `` structuring the european research area '' programme ( project euroleap , contract number 028514 ) . part of the simulations discussed here were performed using the ist cluster ( ist / lisbon ) . 99 ditmire t , tisch jwg , springate e , mason mb , hay n , smith ra , marangos j and hutchinson mhr 1997 _ nature _ * 386 * 54 ditmire t , zweiback jwg , yanovsky vp , cowan te , hays g and wharton kb 1999 _ nature _ * 398 * 489 . zweiback j , smith ra , cowan te , hays g , wharton kb , yanovsky vp and ditmire t 2000 _ phys . lett . _ * 84 * 2634 . zweiback j , cowan te , smith ra , hartley jh , howell r , steinke ca , hays g , wharton kb , crane jk and ditmire t 2000 _ phys . lett . _ * 85 * 3640 . madison kw , patel pk , price d , edens a , allen m , cowan te , zweiback j and ditmire t 2004 _ phys . plasmas _ * 11 * 270 . grillon g , balcou p , chambaret jp _ _ 2002 _ phys . lett . _ * 89 * 065005 . madison kw , patel pk , allen m , price d , fitzpatrick r and ditmire t 2004 _ phys . rev . a _ * 70 * 053201 . last i and jortner j 2006 _ phys . lett . _ * 97 * 173401 . heidenreich a , jortner j and last i 2006 _ proc . _ * 103 * 10589 . last i and jortner j 2008 _ phys . a _ * 77 * 033201 . ditmire t , smith ra , tisch jwg and hutchinson mhr 1997 _ phys . lett . _ * 78 * 3121 . last i and jortner j 2004 _ j. chem . phys . _ * 120 * 1336 . ditmire t , donnelly t , rubenchik am , falcone rw , and perry md 1996 _ phys . a _ * 53 * 3379 . mulser p , kanapathipillai m , and hoffmann dhh 2005 _ phys . lett . _ * 95 * 103401 . dawson jm 1964 _ phys . fluids _ * 7 * 981 . fukuda y , kishimoto y , masaki t and yamakawa k 2006 _ phys . a _ * 73 * 031201(r ) . mora p 2003 _ phys . lett . _ * 90 * 185002 . mora p 2005 _ phys . plasmas _ * 12 * 112102 . mora p 2005 _ phys . e _ * 72 * 056401 . murakami m , basko mm 2006 _ phys . plasmas _ * 13 * 012105 . ditmire t , springate e , tisch jwg , shao yl , mason mb , hay n , marangos jp and hutchinson mhr 1998 _ phys . a _ * 57 * 369 . milchberg hm , mcnaught sj and parra e 2001 _ phys . rev . e _ * 64 * 056402 . _ 1998 _ phys . lett . _ * 80 * 261 . zweiback j , cowan te , hartley jh , howell r , wharton kb , crane jk , yanovsky vp , hays g , smith ra , and ditmire t 2002 _ phys . plasmas _ * 9 * 3108 . liu cs and tripathi vk 2003 _ phys . plasmas _ * 10 * 4085 . kaplan ae , dubetsky by and shkolnikov pl 2003 _ phys . * 91 * 143401 . peano f , peinetti f , mulas r , coppa g and silva lo 2006 _ phys . _ * 96 * 1750021 . kishimoto y , masaki t and tajima t 2002 _ phys . plasmas _ * 9 * 589 . peano f 2005 _ laser - induced coulomb explosion of large deuterium clusters _ ( turin : politecnico di torino ) . peano f , coppa g , peinetti f , mulas r and silva lo 2007 _ phys . e _ * 75 * 066403 . peano f , martins jl , fonseca ra , silva lo , coppa g , peinetti f and mulas 2007 _ phys . plasmas _ * 14 * 056704 . peano f , fonseca ra , and silva lo 2005 _ phys . rev _ * 94 * 033401 . peano f , fonseca ra , martins jl and silva lo 2006 _ phys . rev . * 73 * 053202 . li h , liu j , wang ch , ni g , kim chj , li r and xu zh 2007 _ j. phys . b _ * 40 * 3941 . last i and jortner j 2007 _ phys . a _ * 75 * 042507 . ott e 1979 _ phys . lett . _ * 42 * 1628 . t. grismayer , p. mora , j. c. adam , and a. hron 2008 _ phys . e _ * 77 * 066407 . je , auer pl and allen je 1975 _ j. plasma phys . _ * 14 * 89 . manfredi g , mola s and feix mr 1993 _ phys . fluids _ b * 90 * 388 . sakabe s , shimizu s , hashida m _ _ 2004 _ phys . a _ * 69 * 023203 . hirokane m , shimizu s , hashida m , okada s , okihara s , sato f , iida t and sakabe s 2004 _ phys . a _ , * 69 * 063201 . sakabe s , shirai k , shimizu s , hashida m and masuno s 2006 _ phys . a _ * 74 * 043205 . kovalev vf , popov ki , bychenkov vy and rozmus w 2007 _ phys . plasmas _ * 14 * 053103 . parks pb , cowan te , stephens rb and campbell em 2001 _ phys . a _ * 63 * 063203 . li h , liu j , wang c , ni g , li r and xu z 2006 _ phys . a _ * 74 * 023201 . fonseca ra _ notes comp . sci . _ * 2331 * ( heilderberg : springer - 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the expansion of laser - irradiated clusters or nanodroplets depends strongly on the amount of energy delivered to the electrons and can be controlled by using appropriately shaped laser pulses . in this paper ,
a self - consistent kinetic model is used to analyze the transition from quasineutral , hydrodinamic - like expansion regimes to the coulomb explosion ( ce ) regime when increasing the ratio between the thermal energy of the electrons and the electrostatic energy stored in the cluster .
it is shown that a suitable double - pump irradiation scheme can produce hybrid expansion regimes , wherein a slow hydrodynamic expansion is followed by a fast ce , leading to ion overtaking and producing multiple ion flows expanding with different velocities .
this can be exploited to obtain intracluster fusion reactions in both homonuclear deuterium clusters and heteronuclear deuterium - tritium clusters , as also proved by three - dimensional molecular - dynamics simulations .
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let @xmath8 be the following singularly perturbed laguerre weight @xmath9 with @xmath10 the hankel determinant is defined as @xmath11=\det(\mu_{i+j})_{i , j=0}^{k-1},\ ] ] where @xmath12 is the @xmath13-th moment of @xmath8 , namely , @xmath14 note that when @xmath15 , the integral in the above formula is convergent so that the hankel determinant @xmath16=d_k[w(x;t)]$ ] in is well - defined . moreover , it is well - known that the hankel determinant can be expressed as @xmath17 = \prod_{j=0}^{k-1 } \gamma_{j , n}^{-2}(t);\]]see @xcite , where @xmath18 is the leading coefficient of the @xmath5-th order polynomial orthonormal with respect to the weight function in . or , let @xmath19 be the @xmath5-th order monic orthogonal polynomial , then @xmath18 appears in the following orthogonal relation @xmath20 for fixed @xmath3 . moreover , the monic orthogonal polynomials @xmath19 satisfy a three - term recurrence relation as follows : @xmath21with @xmath22 and @xmath23 , where the appearance of @xmath3 and @xmath4 in the coefficients indicates their dependence on @xmath3 and the parameter @xmath4 in the varying weight . in this paper , however , we will focus on the case when @xmath1 . since all the above integrals on @xmath24 become divergent for negative @xmath4 , we need to deform the integration path from the positive real axis to certain curves in the complex plane . consequently , the orthogonality will be converted to the _ non - hermitian orthogonality _ in the complex plane . more precisely , let us define the following new weight function on @xmath25 : @xmath26where @xmath27 is a complex constant , the curves @xmath28 , @xmath29 and @xmath30 ; see figure [ contour - ortho ] , @xmath31 being a positive constant . the potential is defined in the cut plane @xmath32 $ ] as @xmath33 the orthogonality relation now takes the form @xmath34 . , width=340 ] with the weight function @xmath35 given in , the corresponding hankel determinant @xmath36 $ ] in is well - defined . however , since @xmath35 is not positive on @xmath37 , the orthogonal polynomials @xmath38 in may not exist for some @xmath5 , and only makes sense if all polynomials @xmath39 for @xmath40 exist . it is worth mentioning that as part of our results , we will show that there exists a @xmath41 , such that @xmath42 exists for @xmath3 large enough and @xmath43 ; cf . section [ sec - rhp - ops ] . the recurrence relation still makes sense for such @xmath4 if all of @xmath44 , @xmath19 and @xmath45 exist . note that in the literature , the polynomials with non - hermitian orthogonality have been studied in several different contexts ; see for example @xcite , where the cubic and quartic potentials are considered . one of the main motivations of this paper comes from the wigner time - delay in the study of quantum mechanical scattering problem . to describe the electronic transport in mesoscopic ( coherent ) conductors , wigner @xcite introduced the so - called time - delay matrix @xmath46 ; see also eisenbud @xcite and smith @xcite . the eigenvalues @xmath47 of @xmath46 , called the proper delay times , are used to describe the time - dependence of a scattering process . the joint distribution of the inverse proper delay time @xmath48 was found , by brouwer et al.@xcite , to be @xmath49 then the probability density function of the average of the proper time delay , namely the wigner time - delay distribution , is defined as @xmath50 the moment generating function is the laplace transformation of the wigner time - delay distribution @xmath51 which is closely related to the hankel determinant as follows : @xmath52}{d_n[w(x;0)]}.\ ] ] recently , texier and majumdar @xcite studied the wigner time - delay distribution by using a coulomb gas method . they showed that @xmath53\}\qquad \textrm{for large } n,\ ] ] where @xmath54 is the unique minimizer for an energy problem with the external field @xmath55 in , and @xmath56 $ ] is the minimum energy . moreover , the density @xmath57 is computed explicitly in @xcite , namely , @xmath58,~ \mbox{with}~0<a < b,~c = t/\sqrt{ab}.\ ] ] here positive @xmath59 and @xmath60 are independent of @xmath61 and implicitly determined by @xmath4 as follows : @xmath62 one may notice that @xmath54 is a probability measure on @xmath63 $ ] as long as @xmath64 is non - negative . since @xmath64 is a continuous function of @xmath4 , we see that @xmath57 in is non - negative for @xmath65 , where @xmath66 is the critical value of @xmath4 corresponding to the case @xmath67 ; see theorem [ theorem : asymptotic of hankel ] . it is very interesting to observe that , for this @xmath41 , we have @xmath68 and @xmath69 where a phase transition emerges at the left endpoint @xmath70 . here the critical values @xmath71 , @xmath72 and @xmath73 are explicitly given in and . it is also interesting to look at our problem from another point of view . due to the term @xmath74 in the exponent of , we may consider the origin as an essential singular point of the weight function . in recent years , orthogonal polynomials whose weights possess essential singularities have been studied extensively . for example , chen and its @xcite consider orthogonal polynomials associated with the weight @xmath75 they show that , for fixed degree @xmath3 , the recurrence coefficient satisfies a particular painlev iii equation with respect to the parameter @xmath4 , and the hankel determinant of fixed size @xmath76 $ ] equals to the isomonodromy @xmath6-function of the painlev iii equation with parameters depending on @xmath3 . the matrix model and hankel determinants @xmath36 $ ] associated with the weight in were also encountered by osipov and kanzieper @xcite in bosonic replica field theories . later , the large @xmath3 asymptotics of the hankel determinants @xmath76 $ ] associated with the weight function in is studied by the current authors in @xcite and @xcite . for @xmath77 $ ] , the asymptotics of the hankel determinants are derived and expressed in terms of certain painlev iii transcendents . the asymptotics of the recurrence coefficients are also obtained therein . in the case of the gaussian weight perturbed by essential singularity @xmath78 the double scaling limit of the hankel determinants are also characterized in terms of painlev iii transcendents by brightmore et al.in @xcite . recently , atkin , claeys and mezzadri @xcite extend the results to the case of laguerre and gaussian weight perturbed by a pole of higher order at the origin , they obtain the double scaling asymptotics of the hankel determinants in terms of a hierarchy of higher order analogs to the painlev iii equation . the main objective of this paper is to study the hankel determinant @xmath16 $ ] with respect to the weight in the region @xmath1 . first , for fixed degree @xmath5 , we will show that the recurrence coefficient @xmath79 satisfies a painlev iii equation , and the hankel determinant @xmath16 $ ] equals to the isomonodromy @xmath6-function of the painlev iii equation . then , we will derive the double scaling limit of the hankel determinant @xmath36 $ ] , the recurrence coefficients and leading coefficients of the associated orthogonal polynomials . our results are described in terms of a certain tronque solution of the painlev i equation . to state our results , we need certain special solutions to the painlev i equation @xmath80 the reader is referred to ( * ? ? ? * ch.32 ) for properties of the painlev i equation , as well as the other painlev equations . in @xcite , kapaev formulates the following model riemann - hilbert ( rh , for short ) problem for @xmath81 , associated with the painlev i equation . this model rh problem will play a crucial role later in the construction of a local parametrix in the steepest descent analysis . * @xmath82 is analytic for @xmath83 , where @xmath84 are illustrated in figure [ contour - p1-model ] . + associated with the painlev i equation , width=283 ] * let @xmath85 denote the limiting values of @xmath82 as @xmath86 tends to the contour @xmath87 from the left and right sides , respectively . then , @xmath82 satisfies the following jump conditions @xmath88 \left ( \begin{array}{cc } 1 & 0 \\ i & 1 \\ \end{array } \right ) , & z \in \gamma_k , \ k=\pm 2 ; \\[.4 cm ] \left ( \begin{array}{cc } 0 & -i\\ -i&0 \\ \end{array } \right ) , & z\in \gamma^ * , \end{array}\right .\ ] ] where @xmath89 and @xmath90 , with @xmath27 being a complex constant . * as @xmath91 , @xmath82 satisfies the asymptotic condition @xmath92 for @xmath93 , where @xmath94 @xmath95 and @xmath96 are the pauli matrices @xmath97 it is known that , for each @xmath98 , @xmath99 is a solution of the painlev i equation . as a consequence , the above rh problem for @xmath82 has a solution if and only if @xmath100 is not a pole of @xmath101 . due to the meromorphic property of the painlev i transcendents , one also see that the solution of the above rh problem for @xmath82 is meromorphic in the parameter @xmath100 . moreover , it is shown in kapaev @xcite that @xmath101 is the so - called _ tronque _ solution of painlev i whose asymptotic behavior is given by @xmath102\left ( 1+o(z^{-\frac 38})\right ) \ ] ] as @xmath103 and @xmath104 $ ] . here @xmath105 is the _ tritronque _ solution satisfying @xmath106 \quad \textrm{as } z \to \infty , \ -\frac{\pi}{5}<\arg z < \frac{7\pi}{5},\ ] ] where the coefficients @xmath107 can be determined recursively ; see for example joshi and kitaev @xcite . the solution @xmath108 will appear in our main results below . we mention several known facts about the coefficients in in addition to . for example , the explicit formulas of @xmath109 and @xmath110 are given in @xcite as @xmath111 where @xmath112 is the hamiltonian of painlev i. first of all , when the degree @xmath5 is fixed , we show that the recurrence coefficient @xmath113 satisfies a particular painlev iii equation with certain initial conditions . moreover , we prove that the hankel determinant @xmath114 $ ] is related to the @xmath6-function of the painlev iii equation . similar results for the weight in been obtained by chen and its @xcite . [ theorem : hankel as tau function ] for fixed non - negative integer @xmath5 , let @xmath79 be the recurrence coefficient in , and @xmath115 then @xmath116 satisfies the following painlev iii equation @xmath117 with the initial conditions @xmath118 , @xmath119 . moreover , we have @xmath120=\mathrm{const } \cdot \tau(t)\ ; e^ { { n^2t}/{2}}\ ; t^ { { k(k+n)}/{2}},\ ] ] where @xmath121 is the jimbo - miwa - ueno isomonodromy @xmath6-function of the above painlev iii equation . next , we let @xmath7 and consider the double scaling limit when @xmath122 and @xmath123 simultaneously . we show that the asymptotics of the hankel determinant @xmath124 $ ] associated with the weight in can be expressed in terms of the tronque solution @xmath101 of painlev i equation given in . [ theorem : asymptotic of hankel ] let the constants @xmath71 , @xmath72 and @xmath73 be defined as @xmath125and @xmath126 for @xmath127 and @xmath128 in a way such that @xmath129 remains bounded . suppose @xmath130 is fixed and @xmath131 is not a pole of the tronque solution @xmath101 , then an asymptotic approximation of the logarithmic derivative of the hankel determinant @xmath132 $ ] associated with the weight function is given by @xmath133 .5 cm we would also derive the double scaling limit of the recurrence coefficients and the leading coefficients of the orthonormal polynomials . [ theorem : asymptotic of recurrence coff ] under the same conditions as in the previous theorem , the monic polynomial @xmath42 defined in exists for large enough @xmath3 and @xmath4 close to @xmath71 . moreover , we have the asymptotics of the recurrence coefficients @xmath134 @xmath135 and @xmath136 where @xmath137 is the hamiltonian of painlev i given in . it is well - known that the tronque solutions of painlev i are meromorphic functions and possess infinitely many poles in the complex plane . therefore , to make the results valid in the above theorems , we require the @xmath131 in is bounded away from the poles of @xmath101 . recently , through a more delicate _ triple scaling limit _ , bertola and tovbis @xcite successfully obtain the asymptotics near the poles of @xmath101 . similar results near the poles of @xmath101 might be derived by using their ideas in @xcite . however , we do not pursue that part . instead , we focus on the main task of the present paper to demonstrate that the painlev i asymptotics can also occur for the weight with negative @xmath4 . the rest of the paper is arranged as follows . in section [ finite determinants ] , we provide a rh problem for the orthogonal polynomials with respect to the weight . a transformed version of the solution is shown to fulfill a lax pair , which is closely related to the painlev iii equation . several differential identities are stated and justified . theorem [ theorem : hankel as tau function ] is also proved in this section . section [ sec - e - measure ] is devoted to the determination of equilibrium measures , involving a positive measure and a signed measure . in section [ sec - rh - analysis ] , we carry out a nonlinear steepest descent analysis of the rh problem for the orthogonal polynomials . particular attention will be paid to the construction of the local parametrix at the critical endpoint @xmath138 , where the painlev i transcendents are involved . then , the proofs of theorems [ theorem : asymptotic of hankel ] and [ theorem : asymptotic of recurrence coff ] are given in the last section , section [ sec - proofs ] . in this section , we state the rh problem for the perturbed laguerre orthogonal polynomials . then we show that after some elementary transformations , the rh problem is transformed into a rh problem for the painlev iii equation . as a consequence , we derive a painlev iii equation satisfied by the recurrence coefficient @xmath79 up to a translation , and establish a relation between the finite hankel determinant of the perturbed laguerre weight in with the @xmath6-function of this painlev iii equation . several differential identities for the hankel determinants and the recurrence coefficients of the perturbed laguerre orthogonal polynomials are also derived . the identities are important in the asymptotic analysis in later sections . although our calculations are similar to those in chen and its @xcite , we think it is convenient for the reader to have more details . we state the rh problem for the perturbed laguerre orthogonal polynomials as follows : @xmath139 is analytic in @xmath140 , @xmath141 ; see figure [ contour - ortho ] ; @xmath139 satisfies the jump condition @xmath142 where @xmath143 is the weight function piecewise - defined in ; the asymptotic behavior of @xmath139 at infinity is @xmath144 as @xmath145 , @xmath146 . using a by now standard argument , originally due to fokas , its , and kitaev @xcite , the solution of the above rh problem , if it exists , is uniquely given by @xmath147 -2\pi i \gamma_{k-1}^2 \;\pi_{k-1}(z ) & - \gamma_{k-1}^2\ ; \int_{\gamma } \frac { \pi_{k-1}(s ) w(s ; t ) } { s - z } ds \end{array } \right ) , \ ] ] where @xmath148 is the monic perturbed laguerre orthogonal polynomials defined in and @xmath149 is the leading coefficient for the orthonormal polynomial of degree @xmath5 . to show the existence of @xmath150 when @xmath3 is large enough , we will apply a series of invertible transformations to transform the original rh problem @xmath151 to a new rh problem for @xmath152 , which is solvable for sufficiently large @xmath3 , @xmath4 close to @xmath71 as in , and @xmath131 is not a pole of the tronque solution @xmath101 . tracing back the invertible transformations , we will see that the rh problem is solvable under the same conditions . indeed , it is also possible to prove the solvability for @xmath43 . however , since we are interested in the phase transition near @xmath71 , we do nt consider the case when @xmath153 in the subsequent analysis . thus , the perturbed laguerre orthogonal polynomials are well - defined for @xmath7 large enough . next , we derive some differential identities for the recurrence coefficients and the logarithmic derivative of the hankel determinant associated with the perturbed laguerre weight @xmath154 in . the results are expressed in terms of the entries of @xmath139 . [ lem : differential identity ] assume that @xmath155 . let @xmath113 and @xmath156 be the recurrence coefficients in , and @xmath16 $ ] be the hankel determinant in . define @xmath157 and @xmath158.\ ] ] then we have @xmath159 @xmath160\ ] ] and @xmath161 since @xmath155 , the orthogonal polynomials @xmath162 exist for all nonnegative @xmath5 and positive @xmath3 . first , we consider the recurrence coefficient @xmath113 . based on the three - term recurrence relation and the orthogonality condition , we get @xmath163 using the fact that @xmath164 and integrating by part once , the above formula gives us @xmath165 then follows from a partial fraction decomposition of @xmath166 , the orthogonality condition , and the explicit expression of @xmath139 in . next , we consider the hankel determinant . recall that the hankel determinant can be expressed in terms of the leading coefficients as @xmath167 = \prod_{j=0}^{k-1 } \gamma_{j , n}^{-2}(t);\]]see . taking logarithmic derivative of both sides of the above equation with respect to @xmath4 and using the integral representation of the leading coefficients in , we get @xmath168 differentiating the above formula again , we get from @xmath169 let @xmath170 be the coefficient of the @xmath171 term in @xmath38 , i.e. , @xmath172 comparing the @xmath173 powers in the recurrence relation , we obtain @xmath174 to derive @xmath175 , one can see from and that it is sufficient to obtain @xmath176 . this can be done by taking derivative of the following orthogonal formula with respect to the parameter @xmath4 @xmath177 more precisely , taking into account the orthogonal relation and the fact that @xmath178 , we have @xmath179 then , follows from a combination of , and , as well as the definition of @xmath139 in . finally , let us study @xmath156 . using the ideas leading to , we have @xmath180 where @xmath170 is introduced in . the first term on the right - hand side is @xmath181 ; cf . . an expression for the term on the extreme right can be obtained by deriving @xmath182 from , and using and . this completes the proof of our lemma . for later use , we need the differential identities of lemma [ lem : differential identity ] in the case when @xmath7 is large and @xmath183 . they can be obtained through an analytic continuation argument . indeed , @xmath139 determined by rh problem exists in this case , and is related to the @xmath184-function of the third painlev equation after an elementary transformation given in . thus @xmath139 is meromorphic with respect to @xmath4 in the cut plane @xmath185 . in particular , both @xmath151 and the hankel determinant are analytic in a domain containing the interval @xmath155 and a neighborhood of @xmath186 . note that the identities - hold for @xmath155 , then , by analytic continuation , they also hold for @xmath4 close to @xmath71 . we conclude that for @xmath7 large and @xmath183 , the identities - are also true . similar argument has previously been used in bleher and deao @xcite . introduce a purely imaginary parameter @xmath187 , and define @xmath188 where @xmath189 is the rescaled contour . then , @xmath190 solves the following rh problem with constant jumps : * @xmath191 is analytic for @xmath192 . as @xmath193 and @xmath37 only differ by a scale , one may refer to figure [ contour - ortho ] to see the properties of the contour @xmath193 . * @xmath191 satisfies the jump condition @xmath194 where @xmath195 ; cf . . * the asymptotic behavior of @xmath191 at infinity is @xmath196 where @xmath197 in the above formula , @xmath198 and @xmath199 are , respectively , the leading coefficient of the @xmath5-th orthonormal polynomial , and the coefficient of the @xmath171 term in the @xmath5-th monic orthogonal polynomial introduced in , with respect to the varying perturbed laguerre weight in and . * the asymptotic behavior of @xmath191 at @xmath200 is @xmath201 where @xmath202 with @xmath203 now , from the above rh problem , we derive the following lax pair for the function @xmath204 , which is exactly the same as the lax pair for painlev iii ; see @xcite . [ prop - phi - piii ] for the matrix function @xmath204 given in , we have @xmath205 where @xmath206 here , the coefficients in the above formula are given below @xmath207 @xmath208 and @xmath209 note that the jump matrices in are independent of @xmath210 and @xmath100 . this implies that both @xmath211 are analytic functions of @xmath210 with only possible isolated singularities at the origin and at infinity . using the asymptotic expansions in - , we find that @xmath212 , \qquad a_{-2}=\frac { is}{2}\phi(0)\sigma_3\phi(0)^{-1}\ ] ] and @xmath213 , \qquad b_{-1}=-\frac { a_{-2}}{s},\]]where @xmath214=xy - yx$ ] is the commutator . then direct computations give us the results . it is known in several circumstances that the hankel determinants admit an interpretation as the jimbo - miwa - ueno isomonodromic @xmath6-function for the rank 2 linear system of differential equations ; see @xcite for the hankel determinants associated with the exponential weight and @xcite for more general semi - classical weights . now we have established the relation between the perturbed laguerre orthogonal polynomials and the lax pair for the painlev iii equation . naturally , the associated hankel determinant is also expected to relate to the @xmath6-function of the painlev iii equation . thus we are in a position to prove our first result for fixed degree @xmath5 . _ proof of theorem [ theorem : hankel as tau function ] . _ according to proposition [ prop - phi - piii ] , @xmath204 satisfies the same lax pair as painlev iii . then , applying an argument in ( * ? ? ? * ( 5.3.4),(5.3.7 ) ) , we see that the function @xmath215 solves the painlev iii equation @xmath216 with the parameters @xmath217 and @xmath218 . by , we have @xmath219 substituting into gives us . next , we consider the hankel determinant @xmath16 $ ] . denote by @xmath220 and @xmath221 the series in the expansions and , namely , @xmath222 with @xmath223 qiven in and @xmath224 cf . and , where @xmath225 denotes the off - diagonal entries independent of @xmath210 . by the general theory of jimbo - miwa - ueno @xcite , the isomonodromy @xmath6-function for the lax pair in - is defined by the formula @xmath226 where @xmath227 see ( * ? ? ? * eq.(1.23 ) ) . substituting the definition of @xmath220 and @xmath221 into , we obtain @xmath228 now a combination of , , and gives @xmath229see for the definition of @xmath230 and @xmath231 . thus , we obtain from and that @xmath232 here use has been made of the relation @xmath233 . in view of the formula , and integrating both sides of , we arrive at the following relation between the hankel determinant @xmath16 $ ] and the @xmath6-function of the painlev iii equation : @xmath167=\mathrm{const } \cdot \tau(s ) e^ { { n^2t}/{2}}t^ { { k(k+n)}/{2}},\]]which is . this completes the proof of theorem [ theorem : hankel as tau function ] . the equilibrium measure with the external field @xmath55 in is given recently in texier and majumdar @xcite . to obtain a double scaling limit at the critical time , we need a modified equilibrium problem , which will involve a _ signed _ measure . this signed measure will be used to construct the important @xmath234-function and @xmath235-function in the riemann - hilbert analysis . the idea of considering a modified equilibrium problem has been successfully applied to study similar double scaling limits in several different problems , such as varying quartic potentials by claeys and kuijlaars @xcite and duits and kuijlaars @xcite , and a cubic potential by bleher and deao @xcite . in this section , we will go back to the weight , consider a regular equilibrium problem first , and see how the critical time occurs . then , to facilitate our future riemann - hilbert analysis near the critical time , we will consider a modified equilibrium problem by fixing the left endpoint . this will give us the signed measure we need . consider the extremal problem minimizing the energy with the external field @xmath55 in : @xmath236 according to the general potential theory @xcite , there exists a unique minimizer @xmath237 of @xmath238 among all borel probability measures @xmath239 on @xmath240 , such a probability measure is called the equilibrium measure . for the potential @xmath241 in , the equilibrium measure @xmath237 can be computed explicitly . the equilibrium measures for @xmath65 and @xmath128 have been computed explicitly in texier and majumdar @xcite . to make the present paper self - contained , we sketch the proof below , which differs from that in @xcite . inspired by @xcite , and in view of the measure for the positive-@xmath4 case , we assume that the support of @xmath242 has only one piece . also , for fixed @xmath4 , the behavior of the density @xmath243 is expected to demonstrate a weak singularity at the endpoints since the contour is deformed to keep away from the possible singularity at the origin . we derive the equilibrium measure by solving a scalar rh problem , based on the euler - lagrange equation . let @xmath243 be the density function of the equilibrium measure @xmath244 supported on an interval @xmath245 , such that @xmath246 . then for @xmath65 we have @xmath247 where @xmath248 and @xmath249 are determined by . moreover , when @xmath250 as in , we have @xmath251 where the critical endpoints @xmath72 and @xmath73 are given in . from , it is known that the equilibrium measure @xmath244 satisfies the euler - lagrange equation @xmath252 where @xmath253 is the lagrange multiplier . differentiating with respect to @xmath61 , we get @xmath254 where the integral is taken as the cauchy principle value . this is an integral equation for the density function @xmath243 , which can be solved explicitly . indeed , one can define @xmath255.\ ] ] then it follows from the plemelj formula that @xmath256 where the integral is the cauchy principle value . it is readily verified that @xmath257 satisfies the following scalar riemann - hilbert problem : * @xmath257 is analytic for @xmath258 $ ] , having at most weak singularities at @xmath259 ; * @xmath260 for @xmath261 ; * @xmath262 as @xmath103 . solving this rh problem yields @xmath263where @xmath264 , @xmath265 and @xmath266 $ ] . here attention should be paid to the fact that @xmath257 is analytic outside the interval @xmath267 $ ] , especially at @xmath268 . now expanding in powers of @xmath269 , the large-@xmath2 behavior of @xmath257 ensures that @xmath270which are indeed . furthermore , a combination of and yields . moreover , in the critical case when @xmath271 a straightforward computation gives us - . of course , the formulas for the density function and endpoints in and hold when @xmath272 . for any @xmath15 , the density function is supported on @xmath63 $ ] with @xmath273 and vanishes like square roots at both endpoints . as a consequence , one will obtain usual airy - type and sine - type asymptotic expansions for the orthogonal polynomials near the endpoints and inside the support , respectively . the critical case when @xmath186 is termed a _ freezing transition _ ; see @xcite . one can see that , when @xmath186 , the density function @xmath243 in vanishes like a @xmath274 root at @xmath72 . this suggests that the local behavior of the orthogonal polynomials near @xmath72 is described in terms of the painlev i transcendents ; see @xcite . to precisely construct a local parametrix near the endpoint @xmath72 by using the painlev i transcendents , a delicate study near @xmath72 is needed in our subsequent nonlinear steepest descent analysis for the rh problem . therefore , technically it is more convenient to have a measure whose left endpoint of the support is exactly located at @xmath72 . note that in the case when only positive measures are involved as in section [ sec - e - measure - regular ] , both endpoints @xmath59 and @xmath60 in vary when the value of parameter @xmath4 changes . so we need to minimize the same energy functional as there is no symmetry as in @xcite , the right endpoint @xmath60 may depend on @xmath4 . a similar treatment is also employed in @xcite . by a similar argument performed in section [ sec - e - measure - regular ] , we find the new minimizer explicitly . let @xmath275 be the signed density function of the minimizer of the minimizer of the energy functional in . then we have @xmath276,\ ] ] where @xmath277 and @xmath60 is determined by the equation @xmath278 .5 cm it is worth noting that for @xmath186 , the density of the modified equilibrium measure @xmath275 is reduced to @xmath279 in . moreover , near the critical time @xmath186 , we have @xmath280 & d_0=a_{cr}^2-\sqrt{\frac{a_{cr}}{b_{cr } } } \frac{2b_{cr } - a_{cr}}{2(b_{cr } - a_{cr})}(t - t_{cr})+o\left ( ( t - t_{cr})^2\right ) , \\[.2 cm ] & d_1=-2a_{cr}+\frac 12\sqrt{\frac{b_{cr}}{a_{cr}}}(b_{cr}-a_{cr})^{-1}(t - t_{cr})+o\left ( ( t - t_{cr})^2\right ) . \end{array}\ ] ] based on the signed measure obtained above , we define several auxiliary functions which will be used in our further analysis . the @xmath234-function is defined as @xmath281,\ ] ] where @xmath282 , and the equilibrium density function @xmath275 is given in . we also define the following @xmath235-functions @xmath283 where the branches are chosen such that @xmath284 , @xmath285 and @xmath286 . from the above definitions , it is immediately seen that @xmath287 satisfies the euler - lagrange equation @xmath288 and the variational inequality @xmath289where @xmath253 is the lagrange multiplier introduced in . moreover , the @xmath234-function and the @xmath235-function are related by @xmath290 note that @xmath291 and @xmath292 are close to each other when @xmath4 approaches @xmath71 . if we rewrite @xmath291 as @xmath293 then , in view of - , we have @xmath294 where @xmath295 is analytic in a neighborhood of @xmath138 and @xmath296 . we also need some local information of the functions @xmath292 and @xmath291 at critical points @xmath138 and @xmath268 . from their definitions in and , we have @xmath297 where @xmath298 , and @xmath299 where @xmath300 . from the above formula , one can see that @xmath301 if @xmath1 and @xmath2 approaches the origin such that @xmath302 ; see figures [ contour - phi - two ] and [ contour - phi - cr ] . in particular , we see that @xmath303 is exponentially small as @xmath145 , @xmath304 or @xmath305 ; cf . figure [ contour - ortho ] for the contours . in view of , one can see that the same holds for @xmath306 . it is worth mentioning that on @xmath307 and @xmath308 with @xmath309 , we have @xmath310 . . the left and right pictures correspond to cases when @xmath1 and @xmath155 , respectively.,width=585 ] . note that this figure is not the exact one for @xmath311 defined in . here we have rescaled the figure , especially near @xmath72 , for better illustration : because the exact value of @xmath72 is too small as compared with @xmath73 ; see . , width=415 ] in this section , we apply the nonlinear steepest descent method developed by deift and zhou et al.@xcite to the rh problem for @xmath151 . the idea is to obtain , via a series of invertible transformations @xmath312 , the rh problem for @xmath152 whose jump matrices are close to the identity ones . we make use of the @xmath234-function defined in to normalize the rh problem for @xmath151 in section [ sec - rhp - ops ] when @xmath7 . as @xmath314 for large @xmath315 , we introduce the first transformation @xmath316 as follows : @xmath317 where @xmath253 is the lagrange multiplier in . then , @xmath318 solves the following rh problem . ( t1 ) @xmath319 is analytic in @xmath320 ; see figure [ contour - ortho ] for @xmath25 ; ( t2 ) the jump condition is @xmath321 for @xmath322 , @xmath141 , where @xmath323 is defined in , and @xmath324 , @xmath325 , @xmath326 ; ( t3 ) the asymptotic behavior of @xmath319 at infinity is @xmath327 appealing to the properties of @xmath287 and @xmath328 in and , the jump matrices in can be expressed in terms of the function @xmath328 as follows : @xmath329 \left ( \begin{array}{cc } e^{2n\left ( \phi_t\right ) _ + ( z ) } & 1 \\ 0 & e^{2n\left ( \phi_t\right ) _ -(z ) } \\ \end{array } \right ) , & z\in(a_{cr},b ) . \end{array } \right .\ ] ] since @xmath331 are purely imaginary on @xmath332 , the jump matrix for @xmath319 on @xmath333 possesses highly oscillatory diagonal entries . to remove the oscillation , we open the lens near @xmath332 and introduce the second transformation : @xmath334 t(z ) \left ( \begin{array}{cc } 1 & 0 \\ -e^{2n\phi_t(z ) } & 1 \\ \end{array } \right ) , & \mbox{for $ z$ in the upper lens region;}\\[.4 cm ] t(z ) \left ( \begin{array}{cc } 1 & 0 \\ e^{2n\phi_t(z ) } & 1 \\ \end{array } \right ) , & \mbox{for $ z$ in the lower lens region . } \end{array}\right .\ ] ] . the shaded region is the region where @xmath335.,width=453 ] then @xmath336 solves the rh problem ( s1 ) @xmath336 is analytic in @xmath337 ; see figure [ contour - s ] for the contours ; ( s2 ) the jump conditions are @xmath338 \left ( \begin{array}{cc } 0 & 1 \\ -1 & 0 \\ \end{array } \right ) , & z\in(a_{cr},b ) , \\ [ .4 cm ] \left ( \begin{array}{cc } 1 & \alpha e^{-2n \phi_t(z ) } \\ 0 & 1 \\ \end{array } \right ) , & z\in \gamma_2,\\[.4 cm ] \left ( \begin{array}{cc } 1 & ( 1-\alpha)e^{-2n \phi_t(z ) } \\ 0 & 1 \\ \end{array } \right ) , & z\in \gamma_3,\\[.4 cm ] \left ( \begin{array}{cc } 1 & e^{-2n \phi_t(z ) } \\ 0 & 1 \\ \end{array}\right ) , & z\in ( b,+\infty ) . \end{array}\right.\ ] ] ( s3 ) the asymptotic behavior at infinity is @xmath339 to study the asymptotic behavior of @xmath336 for large @xmath3 , we may exam the signs of @xmath340 , to see if the jumps are of the form @xmath341 plus exponentially small terms . special attention should be paid in the present case since we are dealing with the signed measure . fortunately , when @xmath3 is large and @xmath342 is small enough , we can still determine the signs of @xmath340 near the endpoint @xmath72 . similar discussions can be found in @xcite where modified equilibrium problems are also addressed . [ prop - phi - sign ] let @xmath343 be a neighbourhood of @xmath72 . then , for any @xmath344 , there exists a @xmath345 such that for all @xmath346 with @xmath347 , we have @xmath348 on the upper and lower lips of the lens on the outside of @xmath343 , namely , @xmath349 . moreover , there exists a positive @xmath350 , such that @xmath351 on @xmath352 and on @xmath353 . in view of , we see that the factor @xmath354 in possesses a pair of zeros @xmath355 one can choose @xmath356 small enough , so that @xmath357 . similar to those conducted in ( * ? ? * prop.4.2 ) and @xcite , we can prove that the jumps on the portions of @xmath358 and @xmath359 , outside of @xmath343 and keeping a distance from the soft edge @xmath60 , are of the form @xmath341 plus an exponentially small term . next , we estimate @xmath340 on @xmath360 in a straightforward manner , using the explicit representations of the @xmath235-functions , and . in view of , we need only check the critical case for @xmath311 , and it is readily seen from that @xmath361 for @xmath2 on the semi - circle @xmath307 ; cf . figure [ contour - ortho ] , we take the integration path to be the arc from @xmath72 to @xmath2 , and adapt the parametrization @xmath362 , where @xmath363 and @xmath364 . as a result , we have @xmath365 where @xmath366 such that @xmath367 . what is more , we have @xmath368\subset [ 3\pi/4 , \pi]$ ] on @xmath307 , where @xmath369 so that @xmath370 . hence the argument of the integrand lies in the interval @xmath371 , so long as @xmath372 . therefore , from we see that @xmath373 , and @xmath374 is strictly monotonically increasing as @xmath304 goes away from @xmath72 . the same result holds for @xmath305 . it is worth mentioning that @xmath375 as @xmath145 , @xmath376 ; see the formula and the discussion that follows . we note that the estimates on the lens boundaries @xmath358 and @xmath359 can also be obtained from the representations , and . having had proposition [ prop - phi - sign ] , we see from that the jump matrix for @xmath377 is the identity matrix , plus an exponentially small term , for fixed @xmath2 bounded away from the interval @xmath332 . neglecting the exponentially small terms , we arrive at an approximating rh problem for @xmath378 as follows : ( n1 ) @xmath378 is analytic in @xmath379 $ ] ; ( n2 ) @xmath380 ( n3 ) @xmath381 the solution to the above rh problem is constructed explicitly as @xmath382 where @xmath383 and @xmath384 with @xmath385 and @xmath386 . the jump matrices of @xmath387 are not uniformly close to the identical matrix @xmath341 near the endpoints @xmath72 and @xmath60 , thus local parametrices have to be constructed in neighborhoods of these endpoints . the local parametrix at the right endpoint @xmath388 is the same as that of the laguerre polynomials at the soft edge . more precisely , the parametrix is to be constructed in @xmath389 , @xmath350 being a fixed positive number , such that ( a ) @xmath390 is analytic in @xmath391 ; see figure [ contour - s ] for the contour @xmath392 ; ( b ) in @xmath393 , @xmath390 satisfies the same jump conditions as @xmath336 does ; cf . ; ( c ) @xmath390 fulfils the following matching condition on @xmath394 : @xmath395 the parametrix can be constructed , out of the airy function and its derivative , as in ( * ? ? ? * ( 3.74 ) ) ; see also @xcite . now we focus on the construction of the parametrix at the endpoint @xmath72 . we seek a parametrix in @xmath396 , @xmath350 being a fixed positive number , such that the following rh problem is satisfied : ( a ) @xmath397 is analytic in @xmath398 ; cf . figure [ contour - s ] ; ( b ) in @xmath399 , @xmath397 satisfies the same jump conditions as @xmath336 does ; cf . ; ( c ) @xmath397 fulfils the following matching condition on @xmath400 : @xmath401 first we make a transformation to convert all the jumps of the rh problem for @xmath397 to constant jumps . let us define @xmath402 then it is readily verified that @xmath403 satisfies a rh problem as follows : ( a ) @xmath404 is analytic in @xmath398 ; ( b ) in @xmath405 , @xmath404 satisfies the jump conditions @xmath406 \left ( \begin{array}{cc } 1 & 1- \alpha \\ 0 & 1 \\ \end{array } \right ) , & z \in \gamma_3 , \\[.4 cm ] \left ( \begin{array}{cc } 0 & 1\\ -1&0 \\ \end{array } \right ) , & z\in ( a_{cr},a_{cr}+r ) , \\[.4 cm ] \left ( \begin{array}{cc } 1 & 0 \\ 1 & 1 \\ \end{array } \right ) , & z \in \gamma_4 \cup \gamma_5 . \end{array}\right .\ ] ] we are now in a position to construct a solution for the above rh problem by using the @xmath184-function associated with the painlev i equation , as introduced in section [ sec - model riemann - hilbert problem ] . we note that @xmath407 shares the same jumps with the @xmath184-function . then , we establish the following conformal mapping : @xmath408and @xmath350 being sufficiently small ; cf . . indeed , in view of , one can improve to obtain @xmath409 as @xmath410 . moreover , we have @xmath411 where the fractional power takes the principle branch . we also define @xmath412 where the square root takes the principal branch again . it follows from the definitions of @xmath413 and @xmath414 in and that @xmath415 is analytic in a neighborhood of @xmath138 and @xmath416 moreover , we have @xmath417 where @xmath418 is the function defined in . with all these preparations , the parametrix near the endpoint @xmath72 can be constructed explicitly as @xmath419 where @xmath420 is defined as @xmath421 it is ready to see that @xmath420 is analytic in @xmath405 and the function @xmath397 in indeed satisfies the matching condition . [ rmk - matching ] as we have discussed in section [ sec - model riemann - hilbert problem ] , the solution @xmath82 exists if and only if @xmath100 is not a pole of @xmath101 . then , to make @xmath397 in well - defined , we choose @xmath422 as @xmath122 , and require @xmath423 is not a pole of @xmath101 . moreover , from the large-@xmath86 behavior of @xmath82 in , we can verify the desired matching condition on @xmath400 in . with all the parametrices constructed , let us introduce the final transformation @xmath425 s(z ) ( p^{(a)})^{-1}(z ) , & z\in u(a , r)\backslash \sigma_{s } ; \\[.1 cm ] s(z ) ( p^{(b)})^{-1}(z ) , & z\in u(b , r)\backslash \sigma_{s } . \end{array}\right .\ ] ] then @xmath426 satisfies a rh problem as follows : ( r1 ) @xmath426 is analytic in @xmath427 ; see figure [ contour - r ] ; ( r2 ) @xmath426 satisfies the jump conditions @xmath428 where @xmath429 p^{(b)}(z)n^{-1}(z ) , & z\in\partial u(b , r),\\[.1 cm ] n(z)j_s(z)n^{-1}(z ) , ~ & \sigma_r\setminus \partial ( u(a_{cr},r)\cup u(b , r ) ) ; \end{array}\right .\ ] ] ( r3 ) @xmath426 satisfies the following behavior at infinity : @xmath430 , width=377 ] based on the matching conditions , and the properties of the function @xmath291 stated in proposition [ prop - phi - sign ] , we have the following estimates : @xmath431 i+o\left ( \frac 1n\right ) , & z\in\partial u(b , r),\\[.1 cm ] i+o(e^{-cn } ) , ~ & z\in \sigma_r\setminus ( \partial u(a , r)\cup \partial u(b , r ) ) , \end{array}\right .\ ] ] where @xmath432 is a positive constant , and the error term is uniform for @xmath2 on the corresponding contours . here we also require that @xmath433 ; see remark [ rmk - matching ] . then , applying the standard argument of integral operator and using the technique of deformation of contours , see for example @xcite , we can see that @xmath426 exits when @xmath3 is large enough and @xmath4 is close to @xmath71 . moreover , we have @xmath434 uniformly for @xmath2 in the whole complex plane . this completes the nonlinear steepest descent analysis . [ sec - proofs ] to prove our main results , theorems [ theorem : asymptotic of hankel ] and [ theorem : asymptotic of recurrence coff ] , we need more refined asymptotic approximations for @xmath426 than the one we get in . for this purpose , we derive an asymptotic approximation for @xmath426 . to simplify the statement of our result , we denote @xmath435 let @xmath436 and @xmath437 ; see . for @xmath127 , @xmath438 as in , and @xmath131 is not a pole of the tronque solution @xmath101 , we have @xmath439 where @xmath440 and @xmath441 the constants @xmath442 and @xmath443 in the above formulas are given as @xmath444and @xmath445 , cf . . from and , we derive an asymptotic expansion for the jump @xmath446 @xmath447 where @xmath448 for @xmath449 , @xmath450 is expanded in terms of @xmath451 for non - negative integers @xmath5 . thus , we have for large @xmath3 . to derive the explicit formulas of @xmath452 and @xmath453 , we combine , with . then one can see that @xmath452 and @xmath453 satisfy rh problems as follows : * @xmath452 and @xmath453 are analytic for @xmath454 ; * @xmath452 and @xmath453 satisfy the jump conditions @xmath455 and @xmath456 * as @xmath457 , both @xmath452 and @xmath453 are of order @xmath458 . it follows from the plemelj formula that @xmath452 and @xmath453 can be represented as the cauchy - type integrals on the circular contour @xmath400 of the right - hand terms in and , respectively . the above rh problems can then be solved by conducting residue calculations of the cauchy - type integrals . as a direct consequence we obtain and . now we are ready to derive the large-@xmath3 asymptotics for the recurrence coefficients and the hankel determinant , as stated in our main theorems , theorems [ theorem : asymptotic of hankel ] and [ theorem : asymptotic of recurrence coff ] . _ proof of theorems [ theorem : asymptotic of hankel ] and [ theorem : asymptotic of recurrence coff ] . _ tracing back the transformations @xmath459 in , and , we have @xmath460 for @xmath2 close to the origin . to apply the differential identities in lemma [ lem : differential identity ] , we need to extract the asymptotics of @xmath461 when @xmath7 , @xmath127 , @xmath438 as in , and @xmath131 is not a pole of the tronque solution @xmath101 . from the explicit formula of @xmath378 in and the approximation of @xmath426 in , we have @xmath462 where @xmath463 and the constants @xmath464 are defined in . now in lemma [ lem : differential identity ] implies that @xmath465 then the asymptotic formula for the hankel determinant follows immediately from the identities and . this completes the proof of theorem [ theorem : asymptotic of hankel ] . to derive the asymptotics of the recurrence coefficient @xmath466 and the leading coefficient @xmath467 , we use the relations @xmath468 where @xmath469 is the coefficient of the @xmath470 term in the asymptotic expansion of @xmath471 , that is , @xmath472 therefore , we also need to expand @xmath378 and @xmath426 as @xmath473 again , using the explicit formula of @xmath378 in and the asymptotic approximation of @xmath426 in , we have @xmath474 and @xmath475 where @xmath476 recalling the values of @xmath442 and @xmath443 in , and the identities @xmath477 and @xmath478 , we have from and that @xmath479this is . similarly , the asymptotics for the leading coefficient can be obtained . indeed , we have @xmath480 which gives us . finally , we derive the asymptotics for @xmath481 . note that , from , we have @xmath482 to find the asymptotic behavior of @xmath483 , we have from and that @xmath484 then , the asymptotic approximation for @xmath481 in follows from the above formula and . this completes the proof of theorem [ theorem : asymptotic of recurrence coff ] . the authors are grateful to the referee for the valuable comments and suggestions to improve the rigorousness of the paper . the work of shuai - xia xu was supported in part by the national natural science foundation of china under grant numbers 11201493 and 11571376 , guangdong natural science foundation under grant numbers s2012040007824 and 2014a030313176 , and the fundamental research funds for the central universities under grant number 13lgpy41 . dan dai was partially supported by grants from the research grants council of the hong kong special administrative region , china ( project no . cityu 11300115 , 11300814 ) . yu - qiu zhao was supported in part by the national natural science foundation of china under grant numbers 10871212 and 11571375 . m. bertola and a. tovbis , asymptotics of orthogonal polynomials with complex varying quartic weight : global structure , critical point behaviour and the first painlev equation , _ constr . _ , * 41*(2015 ) , 529 - 587 . p. deift , t. kriecherbauer , k.t .- r . mclaughlin , s. venakides and x. zhou , uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory , _ comm . pure appl . _ , * 52*(1999 ) , 1335 - 1425 . p. deift , t. kriecherbauer , k.t .- r . mclaughlin , s. venakides and x. zhou , strong asymptotics of orthogonal polynomials with respect to exponential weights , _ comm . pure appl . _ , * 52*(1999 ) , 1491 - 1552 . fokas , a.r . its , a.a . kapaev and v.yu . novokshenov , _ painlev transcendents : the riemann - hilbert approach _ , ams mathematical surveys and monographs , vol . 128 , amer . soc . , providence r.i . , 2006 . m. jimbo , t. miwa and k. ueno , monodromy preserving deformation of linear ordinary differential equations with rational coefficients . i. general theory and @xmath6-function , _ phys . d _ , * 2*(1981 ) , 306 - 352 .
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in this paper , we consider polynomials orthogonal with respect to a varying perturbed laguerre weight @xmath0 for @xmath1 and @xmath2 on certain contours in the complex plane . when the parameters @xmath3 , @xmath4 and the degree @xmath5 are fixed , the hankel determinant for the singular complex weight is shown to be the isomonodromy @xmath6-function of the painlev iii equation .
when the degree @xmath7 , @xmath3 is large and @xmath4 is close to a critical value , inspired by the study of the wigner time delay in quantum transport , we show that the double scaling asymptotic behaviors of the recurrence coefficients and the hankel determinant are described in terms of a boutroux tronque solution to the painlev i equation .
our approach is based on the deift - zhou nonlinear steepest descent method for riemann - hilbert problems .
2010 _ mathematics subject classification _ : primary 33e17 , 34m55 , 41a60 . _ keywords and phrases _ : asymptotics ; hankel determinants ; painlev i equation ; painlev iii equation ; riemann - hilbert approach .
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the application of cross - correlation techniques to measure velocity shifts has a long history ( simkin 1972 , 1974 ; lacy 1977 ; tonry & davis 1979 ) , and with the advent of massive digital spectroscopic surveys of galaxies and stars , the subject has renewed interest . the recently completed sloan digital sky survey ( sdss ) has collected spectra for more than 600,000 galaxies and 90,000 quasars ( adelman - mccarthy et al . 2007 , york et al . 2000 ) . the sdss has also obtained spectra for about 200,000 galactic stars , and it is now being extended at lower galactic latitudes by segue with at least as many spectra ( rockosi 2005 , yanny 2005 ) . another ongoing galactic survey , rave , is expected to collect high - resolution spectra for a million stars by 2011 ( steinmetz et al . 2006 ) , and the plans for the gaia satellite include measuring radial velocities for 10@xmath0 stars by 2020 ( katz et al . 2004 ) . extracting the maximum possible information from these spectroscopic surveys requires carefully designed strategies . cross - correlation has been the target of numerous developments in recent years ( see , e.g. , mazeh & zucker 1994 , statler 1995 , torres , latham & stefanik 2007 , zucker 2003 ) , but several practical aspects of its implementation would benefit from further research . these include the selection of templates ( e.g. , observed vs. synthetic libraries ) , how to combine measurements from multiple templates , the method to determine the maximum of the cross - correlation function , data filtering , and error determination . some of these issues are briefly addressed in this paper , but our focus is on how the requirement of coherence among all entries in a radial velocity data base can be used to improve the original measurements . a different but plausible approach has been recently proposed by zucker & mazeh ( 2006 ) . the doppler shifts of targets in a spectroscopic survey are determined one at a time . each object s projected velocity is measured independently , not counting a possible common set of cross - correlation templates . for a given template , from any pair of ( projected ) velocity measurements , we can derive a relative velocity between the two objects involved . however , that figure will likely be numerically different from the value inferred from the direct cross - correlation between their spectra , even if the two objects are of the same class . in this paper , we argue that it is possible to improve the original determinations by imposing consistency among all available measurements . our discussion is oriented to the case of a homogeneous sample : multiple observations of the same or similar objects . in the following section i introduce cross - correlation , with a brief discussion about error evaluation . section [ basic ] presents the notion of _ self - improvement _ and section [ general ] extends the method to the more realistic scenario in which the spectra in a given data set have varying signal - to - noise ratios . in [ sdss ] we explore an application of the proposed technique involving low - resolution spectra , concluding the paper with a brief discussion and reflections about future work . the most popular procedure for deriving relative velocities between a stellar spectrum and a template is the cross - correlation method ( tonry & davis 1979 ) . this technique makes use of all the available information in the two spectra , and has proven to be far superior than simply comparing the doppler shifts between the central wavelengths of lines when the signal - to - noise ratio is low . the cross - correlation of two arrays ( or spectra ) * t * and * s * is defined as a new array * c * @xmath1 if the spectrum * t * is identical to * s * , but shifted by an integer number of pixels @xmath2 , the maximum value in the array * c * will correspond to its element @xmath3 . cross - correlation can be similarly used to measure shifts that correspond to non - integer numbers . in this case , finding the location of the maximum value of the cross - correlation function can be performed with a vast choice of algorithms . the most straightforward procedure to estimate realistic uncertainties involves an accurate noise model and monte - carlo simulations , and that is the method we use in section [ sdss ] . we employ gaussians and low - order polynomials to model the peak of the cross - correlation function . for these simple models , implemented in a companion idl code , it is possible to derive analytical approximations that relate the uncertainty in the location of the maximum of the cross - correlation function to the covariance matrix [ u@xmath4 . digital cross - correlation , introduced in section [ xcorr ] , is commonly employed to derive doppler radial velocities between two spectra . the discussion in this section is , nonetheless , more general , and deals with the statistical improvement of a set of relative velocity measurements . if three spectra of the same object are available and we refer to the relative radial velocity between the first two as @xmath5 , an alternative estimate of @xmath5 can be obtained by combining the other relative velocity measurements , @xmath6 . assuming uniform uncertainties , the error - weighted average of the two values is @xmath7 . for a set of @xmath8 spectra , we can obtain an improved relative radial velocity determination between the pair @xmath9 by generalizing this expression @xmath10 it can be seen from eq . [ ci ] that the correlation of * t * and * s * is equal to the reverse of the correlation between * s * and * t*. thus , when the relative velocities between two spectra is derived from cross - correlation and the spectra have a common sampling , it will be satisfied that @xmath11 , but this will not be true in general . for example , if we are dealing with grating spectroscopy in air , changes in the refraction index with time may alter the wavelength scale and the spectral range covered by any particular pixel , requiring interpolation . if our choice is to interpolate the second spectrum ( * s * ) to the scale of the first ( * t * ) , this may introduce a difference between @xmath12 and @xmath13 due to different interpolation errors . we can accommodate the general case by writing @xmath14 note that this definition ensures that @xmath15 , and @xmath16 . if the quality of the spectra is uniform , and all measured radial velocities @xmath12 have independent uncertainties of the same size @xmath17 , the primed values would have an uncertainty @xmath18 . despite @xmath12 may be numerically different from @xmath13 , @xmath19 will be highly correlated with @xmath20 , and thus the uncertainty in the primed velocities will not be reduced that fast . in addition , all @xmath12 are also correlated with all @xmath21 , driving the improvement farther away from the ideal @xmath22 behavior . we can expect that after a sufficient number of spectra are included , either random errors will shrink below the systematic ones or all the available information will already be extracted , and no further improvement will be achieved . the case addressed in section [ basic ] corresponds to a set of spectra of the same quality . if the uncertainties in the measured relative radial velocities differ significantly among pairs of spectra , eq . [ vprime ] can be generalized by using a weighted average @xmath23 where @xmath24 and the uncertainty is @xmath25 in the common case in which @xmath26 , the counterpart of eq . [ symmetry ] for dealing with spectra of varying signal - to - noise ratios reduces to @xmath27 where @xmath28 in the next section we use simulated spectra for a case study : multiple observations of the same object or massive surveys involving large numbers of very similar objects at intermediate spectral resolution . the sdss spectrographs deliver a resolving power of @xmath29 , over the range 381910 nm . these two fiber - fed instruments are attached to a dedicated 2.5 m telescope at apache point observatory ( gunn et al . each spectrograph can obtain spectra for 640 targets simultaneously . as a result of a fixed exposure time in sdss spectroscopic observations , the flux in a stellar spectrum at a reference wavelength of 500 nm , @xmath30 , correlates well with the @xmath31 magnitude of the star and with the signal - to - noise ratio at 500 nm ( @xmath32 ) . on average , we find @xmath33 at @xmath34 mag . to build a realistic noise model , we used the fluxes and uncertainties for 10,000 spectra publicly released as part of dr2 ( abazajian et al . 2004 ) to derive , by least - squares fitting , a polynomial approximation . when @xmath30 is expressed in erg @xmath35 s@xmath36 @xmath36 , which are the units used in the sdss data base , we can write @xmath37 where @xmath38 . this relationship holds in the range @xmath39 . the uncertainties in the sdss fluxes for stars mostly relatively bright calibration sources are not dominated by photon noise , but by a _ floor _ noise level of 23% associated with a combination of effects , including imperfect flat - fielding and scattered light corrections . errors are highly variable with wavelength , but the noise at any given wavelength depends linearly on the signal . based on the same set of sdss spectra used for eq . [ snr500 ] , we determine the coefficients in the relation @xmath40 which we use here for our numerical experiments . for a given choice of @xmath32 , we interpolate the table of coefficients @xmath41 and @xmath42 derived from sdss data , and by inverting eq . [ snr500 ] we derive the flux at 500 nm . finally , we scale the spectrum fluxes and calculate the expected errors at all wavelengths using eq . gaussian noise is introduced for each pixel position , according to the appropriate error , simulating multiple observations of the same star to create an entire library of spectra . we employed a spectrum of hd 245 , a nearby g2 star@xmath43 ) , surface gravity ( @xmath44 ) , and kinematics , make this object a prototypical thick - disk turn - off star . ] , to produce spectra that resemble sdss observations with various signal - to - noise ratios . radial velocities are also artificially introduced . the spectrum of hd 245 used here has a resolving power of @xmath45 at 660 nm to 7700 at 480 nm . this variation is , however , irrelevant when smoothing the data to @xmath46 as we do in these experiments . ] and is included in the elodie.3 database ( moultaka et al . 2004 , prugniel & soubiran 2001 ) . as the rest of the library , this spectrum was obtained with the 1.9 m telescope and the elodie spectrograph at haute provence . the original fluxes are resampled to @xmath47 , and then smoothed to @xmath46 by gaussian convolution . the output fluxes are sampled with 12 pixels per resolution element . the doppler shift due to the actual radial velocity of hd 245 has already been corrected in the elodie library . new values for the radial velocity in the library of simulated sdss observations are drawn from a normal distribution with a @xmath48 km s@xmath36 , as to approximate the typical range found in f- and g - type stellar spectra included in the sdss ( mostly thick - disk and halo stars ) . the wavelength scale is then doppler shifted , changed to vacuum ( @xmath49 ) , and the spectrum resampled with a step of @xmath50 in @xmath51 ( approximately 2.17 pixels per resolution element ) . the elodie spectra only cover the range @xmath52 nm , and therefore a similar range is finally kept for the sdss - style files , which include 2287 pixels . we employed a set of 40 test spectra with @xmath53 , measuring the relative radial velocities for all possible pairs . [ xcf ] illustrates two sample spectra and their cross - correlation function . to avoid very large or small numbers , the input arrays are simply divided by their maximum values before cross - correlation . we used second and third order polynomials , as well as a gaussian to model the cross - correlation function and estimate the location of its maximum by least - squares fitting . the solid line in the lower panels of the figure are the best - fitting models . we experimented varying the number of pixels involved in the least - squares fittings ( @xmath54 ) . with the sampling used , the measured relative shifts in pixel space ( @xmath55 ) correspond to a velocity @xmath56 , where @xmath57 is the speed of light in vacuum ; one pixel corresponds to 69 km s@xmath36 . we compare the relative velocities between all pairs of spectra derived from the measurement of the location of the cross - correlation peaks with the _ known _ , randomly drawn , relative velocities . the average difference for the 1600 velocities ( 40 spectra ) @xmath58 and the rms scatter ( @xmath59 ) are used to quantify systematic and random errors , respectively . our experiments exhibit no systematic errors in the derived velocity when the number of points entering the fit @xmath54 was an odd number , i.e. , when we use the same number of data points on each side from the pixel closest to the peak of the cross - correlation function . modest offsets ( @xmath60 ) , however , are apparent when fitting polynomials to an even number of data points , despite we enforce the maximum to be bracketed by the two central data points . random errors increase sharply with the number of data points involved in the fittings for the polynomial models , but not for the gaussian model . the best results for the polynomials are obtained when the lowest possible orders are used . using less than 11 points for the gaussian did not produce reliable results , as there was not enough information to constrain all the parameters of the model , which includes a constant base line . the best performance @xmath61 km s@xmath36 was obtained using a second order polynomial and @xmath62 . using a gaussian model achieved a minimum @xmath63 km s@xmath36 , fairly independent of @xmath54 . the third order polynomial provided the poorest performance , @xmath64 km s@xmath36 at best . the cross - correlation can be computed in fourier space , taking advantage of the correlation theorem ( brigham 1974 ) . this fact is usually exploited to speed up the calculation dramatically , as fast fourier transforms can be calculated with a number of operations proportional to @xmath65 , compared to @xmath66 required by eq . note , however , that for medium - resolution surveys of galactic stars , the velocity offsets , limited by the galactic escape velocity , usually correspond to a limited number of pixels . therefore , it is only necessary to compute the values of * c * in the vicinity of the center of the array , rendering the timing for a direct calculation similar to one performed in transformed space ) took @xmath67 seconds in fourier space ( arrays padded to @xmath68 or @xmath69 ) , while in pixel space , with a lag range restricted to @xmath70 pixels ( @xmath71 km s@xmath36 ) , it took @xmath72 seconds . ] . spectra and @xmath33 . the solid line represents the original distribution , and the dashed line the result after applying self - improvement . the error distributions are symmetric because the array * v * is antisymmetric . , width=317 ] to test the potential of the proposed self - improvement technique we repeat the same exercise described in [ sdss1 ] , but using increasingly larger datasets including up to 320 spectra , and adopting three different values for the @xmath73 per pixel at 500 nm : 50 , 25 , and 12.5 . we calculated the cross - correlation between all pairs of spectra ( matrix * v * ) , and performed quadratic fittings to the 3 central data points , cubic polynomial fittings to the central 4 points , and gaussian fittings involving the 11 central points . we estimated the uncertainties in our measurements by calculating the rms scatter between the derived and the known relative velocities for all pairs . then we applied eq . [ symmetry ] to produce a second set of _ self - improved _ velocities . ( because the array of wavelengths , @xmath74 , is common to all spectra , the matrix * v * is antisymmetric and we can use eq . [ symmetry ] instead of eq . [ vprime ] . ) a first effect of the transformation from * v * to * v * , is that the systematic offsets described in [ sdss1 ] when using polynomial fittings with even values of @xmath54 disappear ( the same systematic error takes place for measuring @xmath12 and @xmath75 , canceling out when computing @xmath76 ) . more interesting are the effects on the width of the error distributions . fig . [ dist ] illustrates the case when a quadratic model is used for @xmath77 and @xmath33 . the solid line represents the original error distribution and the dashed line the resulting distribution after self - improvement . [ sigma ] shows the rms scatter as a function of the number of spectra for our three values of the @xmath73 ratio at 500 nm . the black lines lines show the original results , and the red lines those obtained after self - improvement . each panel shows three sets of lines : solid for the quadratic model , dotted for the cubic , and dashed for the gaussian . extreme outliers at @xmath78 km s@xmath36 , if any , were removed before computing the width of the error distribution ( @xmath59 ) . note the change in the vertical scale for the case with @xmath79 . for the experiments with @xmath80 , several runs were performed in order to improve the statistics , and the uncertainty ( standard error of the mean ) is indicated by the error bars . these results are based on the gaussian random - number generation routine included in idl 6.1 , but all the experiments were repeated with a second random number generator and the results were consistent . as described in [ sdss1 ] , the quadratic model performs better on the original velocity measurements for @xmath81 and @xmath82 . at the lowest considered @xmath73 value of 12.5 , however , the gaussian model delivers more accurate measurements . self - improvement reduces the errors in all cases . although a second order polynomial fitting works better than third order for the original measurements , the two models deliver a similar performance after self - improvement . interestingly , the impact of self improvement is smaller on the results from gaussian fittings than on those from polynomial fittings . as expected , the errors in the original measurements are nearly independent of the number of spectra in the test , but there is indication that at low signal - to - noise the errors after self - improvement for the polynomial models decrease as the sample increases in size , until they plateau for @xmath83 . from these experiments , we estimate that the best accuracy attainable with the original cross - correlation measurements are about 3 , 6 , and 15 km s@xmath36 at @xmath84 , 25 , and 12.5 , respectively . our results also indicate that by applying self - improvement to samples of a few hundred spectra , these figures could improve to roughly 2.5 , 4 , and 9 km s@xmath36 at @xmath84 , 25 , and 12.5 , respectively . we obtained an independent estimate of the precision achievable by simply measuring for 320 spectra the wavelength shift of the core of several strong lines ( h@xmath85 , h@xmath86 , h@xmath87 , and h@xmath88 ) relative to those measured in the solar spectrum ( see allende prieto et al . 2006 ) , concluding that radial velocities can be determined from line wavelength shifts with a @xmath89 uncertainty of 3.8 km s@xmath36 at @xmath90 , 7.2 km s@xmath36 at @xmath91 , and 15.9 km s@xmath36 at @xmath92 only 1020% worse than straight cross - correlation but these absolute measurements can not take advantage of the self - improvement technique . allende prieto et al . ( 2006 ) compared radial velocities determined from the wavelength shifts of strong lines for sdss dr3 spectra of g and f - type dwarfs with the sdss pipeline measurements based on cross - correlation . the derived @xmath89 scatter between the two methods was 12 km s@xmath36 or , assuming similar performances , a precision of 8.5 km s@xmath36 for a given method . the spectra employed in their analysis have a @xmath32 distribution approximately linear between @xmath93 and 65 , with @xmath94 and with mean and median values of 22 and 18 , respectively . their result is in line with the expectations based on our numerical tests that indicate a potential precision of 67 km s@xmath36 at @xmath95 . independent estimates by the sdss team are also consistent with these values ( rockosi 2006 ; see also www.sdss.org ) . after correcting for effects such as telescope flexure , the wavelength scale for stellar spectra in dr5 is accurate to better than 5 km s@xmath36 ( adelman - mccarthy et al . 2007 ) . this value , derived from the analysis of repeated observations for a set of standards and from bright stars in the old open cluster m67 , sets an upper limit to the accuracy of the radial velocities from sdss spectra , but random errors prevail for @xmath96 . provided no other source of systematic errors is present , our tests indicate that self - improvement could reduce substantially the typical error bars of radial velocities from low signal - to - noise sdss observations . this paper deals with the measurement of relative doppler shifts among a set of spectra of the same or similar objects . if random errors limit the accuracy of the measured relative velocity between any two spectra , there is potential for improvement by enforcing self - consistency among all possible pairs . this situation arises , for example , when a set of spectroscopic observations of the same object are available and we wish to co - add them to increase the signal - to - noise ratio . the spectra may be offset due to doppler velocity offsets or instrumental effects , the only difference being that in the former case the spectra should be sampled uniformly in velocity ( or @xmath97 ) space for cross - correlation , while in the latter a different axis may be more appropriate . another application emerges in the context of surveys that involve significant numbers of spectra of similar objects . radial velocities for individual objects can be derived using a small set of templates and later _ self - improved _ by determining the relative velocities among all the survey targets and requiring consistency among all measurements . the potential of this technique is illustrated by simulating spectra for a fictitious survey of g - type turn - off stars with the sdss instrumentation . our simulations show that applying self - improvement has a significant impact on the potential accuracy of the determined radial velocities . the tests performed dealt with relative velocities , but once the measurements are linked to an absolute scale by introducing a set of well - known radial velocity standards in the sample , the relative values directly translate into absolute measurements . the ongoing segue survey includes , in fact , large numbers of g - type stars , and therefore our results have practical implications for this project . the proposed scheme handles naturally the case when multiple templates are available . templates and targets are not treated differently . relative velocities are measured for each possible pair to build @xmath12 , and consistency is imposed to derive @xmath76 by using eqs . [ vprime ] or [ vprimegen ] . if , for example , the templates have been corrected for their own velocities and are the first 10 spectra in the sample , the velocity for the @xmath98th star ( @xmath99 ) can be readily obtained as the weighted average of the @xmath76 elements , where @xmath100 runs from 1 to 10 . the final velocities would take advantage of all the available spectra , not just the radial velocity templates , with differences in signal - to - noise among spectra already accounted for automatically . very recently , zucker & mazeh ( 2006 ) have proposed another approach with the same goals as the method discussed here . their procedure determines the relative velocities of a set of @xmath8 spectra by searching for the doppler shifts that maximize the value of the parameter @xmath101 , where @xmath102 is the maximum eigenvalue of the correlation matrix a two - dimensional array whose @xmath9 element is is the cross - correlation function between spectra @xmath100 and @xmath103 . zucker & mazeh s algorithm is quite different from the self - improvement method presented here . it involves finding the set of velocities that optimally aligns the sample spectra , whereas self - improvement consists on performing very simple algebraic operations on a set of radial velocities that have already been measured . self - improvement is obviously more simple to implement , but a detailed comparison between the performance of the two algorithms in practical situations would be very interesting . this paper also touches on the issue of error determination for relative radial velocities derived from cross - correlation , and convenient analytical expressions are implemented in an idl code available online . we have not addressed many other elements that can potentially impact the accuracy of doppler velocities from cross - correlation , such as systematic errors , filtering , sampling , or template selection . the vast number of spectra collected by current and planned spectroscopic surveys should stimulate further thought on these and other issues with the goal of improving radial velocity determinations . there is certainly an abundance of choices that need to be made wisely .
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the measurement of doppler velocity shifts in spectra is a ubiquitous theme in astronomy , usually handled by computing the cross - correlation of the signals , and finding the location of its maximum .
this paper addresses the problem of the determination of wavelength or velocity shifts among multiple spectra of the same , or very similar , objects .
we implement the classical cross - correlation method and experiment with several simple models to determine the location of the maximum of the cross - correlation function .
we propose a new technique , _ self - improvement _ , to refine the derived solutions by requiring that the relative velocity for any given pair of spectra is consistent with all others . by exploiting all available information , spectroscopic surveys involving large numbers of similar objects
may improve their precision significantly .
as an example , we simulate the analysis of a survey of g - type stars with the sdss instrumentation . applying _ self - improvement
_ refines relative radial velocities by more than 50% at low signal - to - noise ratio .
the concept is equally applicable to the problem of combining a series of spectroscopic observations of the same object , each with a different doppler velocity or instrument - related offset , into a single spectrum with an enhanced signal - to - noise ratio .
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the use of migdal eliashberg ( me ) theory for the study and analysis of electron - phonon systems is widespread . for conventional superconductors such as lead , where the electron - phonon coupling is higher than can be treated with bcs theory , me theory has been extremely successful for understanding the superconducting properties . recently , a lot of researchers have been interested in an apparent kink in the electronic dispersion of cuprate supercondutors as determined from angle resolved photo - emission spectroscopy ( arpes ) , and their analysis / interpretation typically uses the related engelsberg schrieffer ( es ) result ( i.e. just the lowest order fock diagram ) @xcite . the es result divides excitations into long - lived ( coherent ) low - energy excitations , and rapidly decaying high - energy excitations , with a kink at the phonon energy . es based analysis of arpes results suggests a large electron - phonon coupling in the cuprates , with estimates of the dimensionless coupling constant lying between @xmath0 and @xmath1 ( depending on doping ) and very large phonon frequencies of @xmath2 mev @xcite . there has also been a development of a maximum entropy technique for analysing cuprate superconductors which makes use of eliashberg theory @xcite . the analysis in ref . determines a dimensionless electron - phonon coupling of @xmath3 , but this is well above the @xmath4 value where one would normally expect perturbation theory to fail . the region of intermediate electron - phonon coupling is also relevant to a number of other materials . for example , electron paramagnetic resonance measurements on the manganites support a strong `` electron - phonon '' coupling leading to jahn teller polarons @xcite . in light of the current experimental situation , and the importance of the conclusions of refs . , a full re - analysis of the perturbation theory is of high significance . the neglect of vertex corrections suggested by migdal leads to a theory where an infinite set of feynman diagrams may be summed . according to an analysis carried out by migdal , this theory should be valid in the physical regime of electron - phonon problems , where the phonon energy is significantly smaller than the intersite hopping ( `` migdal s theorem '' ) @xcite specifically that the vertex corrections are small when @xmath5 ( @xmath6 is the phonon frequency and @xmath7 the fermi energy ) . on the basis of migdal s analysis , it is often believed that migdal eliashberg theory is applicable above @xmath4 in the adiabatic limit because of the apparently small size of the vertex corrections , even though in general perturbative approaches break down ( i.e. the functional form of the self - energy becomes incorrect ) when the coupling constant becomes large . in this article , we revisit migdal s analysis by examining the large-@xmath8 limit ( local approximation ) . the large-@xmath8 limit has been an effective workshop for determining the validity of approximate schemes . the quantum monte - carlo ( qmc ) solution in the large-@xmath8 limit @xcite has been compared with a number of different diagrammatic approaches @xcite demonstrating that self - consistent second - order perturbation theory worked better than migdal - eliashberg theory at intermediate phonon frequencies and strong coupling ( @xmath9 , @xmath10 ) . the qmc solution is difficult for very low phonon frequency and low temperature . qmc self - energies and green functions are generated along the matsubara axis . as such , it is not easy to make quantitative conclusions about the point of breakdown of me theory as small differences at individual matsubara frequencies may result in relatively large differences in the spectral functions . using an alternative approach , benedetti _ et al . _ @xcite found the formation of more than one extremum in the path - integral formulation at very low phonon frequencies and intermediate coupling , leading to a breakdown of me theory . work on the finite - dimensional holstein model also shows deficiencies in me theory . on the basis of a comparison with exact diagonalization results , alexandrov _ et al . _ @xcite have shown that me theory may break down at intermediate couplings even in the adiabatic limit . in this article , we investigate the holstein model of electron - phonon interactions @xcite , which employs a number of approximations . the phonon dispersion is flat corresponding to independently moving ions and phonon anharmonicity is neglected , resulting in a hamiltonian , @xmath11 where @xmath12 @xmath13 create ( annihilate ) electrons at site @xmath14 with spin @xmath15 , @xmath16 is the local ion displacement , @xmath17 the ion momentum , @xmath18 the ion mass , @xmath19 the electron hopping parameter and @xmath20 the chemical potential . a mean - field coulomb pseudopotential may also be included , but in the normal state , it is just absorbed into the chemical potential . in the following , @xmath21 and all energies ( including temperature ) are measured in units of @xmath22 . an expression for the effective interaction between electrons can be obtained by performing a linear transformation in the phonon variable to remove the electron - phonon term @xcite . the resulting interaction is retarded , with fourier components : @xmath23 where @xmath24 are the matsubara frequencies for bosons . taking the limit @xmath25 , @xmath26 , while keeping the ratio @xmath27 finite , leads to an attractive hubbard model @xcite with instantaneous attraction of magnitude @xmath28 , where @xmath29 is the half band - width ( note that @xmath30 is related to the bipolaron binding energy for the strong coupling two - electron problem ) . in the opposite limit ( @xmath31 , @xmath32 , keeping @xmath33 finite ) the phonon kinetic - energy vanishes , leaving only a static variable @xmath16 representing the phonon subsystem in the hamiltonian , @xmath34 . thus the problem looks like that of a single electron in a disordered potential . the large-@xmath8 limit of this model was exactly solved by millis _ _ @xcite and extended to deal with long range order by ciuchi _ et al _ @xcite . thus , the phonon frequency may be thought of as a parameter for tuning the level of electronic correlation @xcite . in this paper , we investigate the validity of the migdal eliashberg approach within the dynamical mean - field theory formalism . as we carry out the self - consistency on the real - axis self - energies , we can investigate the behaviour of me theory over a full range of temperatures and phonon frequencies . we begin by calculating self - energies , spectral functions , the quasi - particle weight and renormalized phonon frequency . to determine the validity of the theory , we calculate an expression for the lowest - order correction to the vertex function , and evaluate its magnitude . we find that the theory can also break down in the low frequency regime , contrary to the standard interpretation of migdal s analysis . finally , we calculate resistivity curves in the regime where the lowest - order corrections are small . in this article , we use the local approximation or dynamical mean - field theory ( dmft ) as a way of analysing the migdal - eliashberg theory . the self - energy of correlated - electron systems is momentum independent in limit of large dimensions @xcite , and approximately momentum independent in 3d . in large-@xmath8 , lattice models map onto anderson impurity models with a self - consistent hybridisation @xcite . the self - consistent dmft equations may be obtained by rewriting the action for the model to be considered in terms of an effective single - site action @xcite @xmath35 here @xmath36 plays the rle of the host green function in the equivalent impurity model . if one assumes that correlations carried via the bath between electrons entering and exiting a single site can be neglected ( true in the case of large coordination number ) , the degrees of freedom associated with all but one site can be integrated out . this leads to the following self - consistent equation @xcite , @xmath37},\ ] ] where @xmath38 is the site - local ( impurity ) green function , and is itself a functional of @xmath39 . @xmath40 are the fermionic matsubara frequencies . @xmath41 $ ] is the reciprocal function of the hilbert transform , defined as @xmath42=\xi$ ] . the hilbert transform of the non - interacting density of states , @xmath43 , is defined as @xmath44 . when electrons move on a tight - binding hypercubic lattice , the bare dos takes the form of a gaussian @xcite , @xmath45 and there is no simple expression for the reciprocal function . introducing the modified dyson equation , @xmath46 ( where @xmath47 is the electron self - energy ) , equation ( [ eqn : selfconsistent ] ) may be rewritten as , @xmath48=i\omega _ { n}+\mu -\sigma ( i\omega _ { n}),\ ] ] and inverted to give an expression for the green function in terms of the self - energy , @xmath49 this also allows the approximation to be interpreted as a course graining of the momentum space @xcite . to complete the scheme , a form for the electronic self - energy must be calculated in terms of the non - interacting weiss field . this is normally approximate . then a self - consistent procedure is followed : compute the green function from equation ( [ eqn : greensfn ] ) , the weiss field from equation ( [ eqn : weissfield ] ) and then re - calculate the self - energy until convergence is reached . the application of me theory within dmft corresponds to computing the self - energy and phonon propagator from the diagrammatic equations in figure [ fig : feynmandiag](a and b ) . the self - consistent solution of such a self - energy corresponds to the summation of all feynman diagrams which contain no vertex corrections . in the low ( non - zero ) phonon - frequency limit , migdal s analysis indicates a condition , @xmath50 , for the neglect of corrections to the vertex function @xcite . the solution of the dyson equation seen in figure [ fig : feynmandiag](a ) results in the full phonon propagator , @xmath51},\ ] ] which is then used for the calculation of the electron self - energy . the phonon polarisation bubble ( dimensionless self - energy ) @xmath52 is given from perturbation theory , @xmath53 and may be analytically continued by introducing the spectral representation , @xmath54 where @xmath55/\pi$ ] and performing the sum over matsubara frequencies to give , @xmath56=-2u\int _ { 0}^{\omega } dx\rho ( -x)\rho ( \omega -x),\ ] ] at absolute zero , and @xmath57=-u\sinh ( \frac{\omega } { 2t})\int _ { -\infty } ^{\infty } dx\ , \frac{\rho ( -x)}{\cosh ( \frac{x}{2t})}\frac{\rho ( \omega -x)}{\cosh ( \frac{\omega -x}{2t})},\ ] ] at finite @xmath58 . the full spectral function is used so that all diagrams with no vertex corrections are included , consistent with the proper interpretation of migdal s analysis . the lowest - order skeleton diagram shown in figure [ fig : feynmandiag](b ) , @xmath59 may be analytically continued in the same way as the polarisation bubble to give , @xmath60=u\int _ { 0}^{\omega } dx\rho ( x)\sigma ( \omega -x),\ ] ] with @xmath61/\pi$ ] . a more complicated expression applies at finite temperature , @xmath62 = u\cosh ( \frac{\omega } { 2t})\int _ { -\infty } ^{\infty } \frac{dx\ , \rho ( \omega -x)\sigma ( x)}{2\cosh ( \frac{\omega -x}{2t})\sinh ( \frac{x}{2t})}.\ ] ] at each stage , the kramers kronig relation is used to compute the real parts of the electron and phonon self - energies , for instance @xmath63=\mathrm{p}\int dx \mathrm{im}[\pi _ { 0}(x)]/\pi(\omega -x)$ ] , where p denotes the principal integral . after the self - consistent procedure has converged , physical properties are calculated , including the quasi - particle weight ( inverse effective mass ) , @xmath64 the effective phonon frequency @xmath65 from , @xmath66)=0,\ ] ] and the optical conductivity @xcite , @xmath67=\frac{\pi } { \omega } \int _ { -\infty } ^{\infty } d\varepsilon { \mathcal{d}}(\varepsilon ) \int _ { -\infty } ^{\infty } d\nu \ , \rho ( \varepsilon , \nu ) \rho ( \varepsilon , \nu + \omega ) [ f(\nu ) -f(\nu + \omega ) ] , \ ] ] where @xmath68 is the fermi - dirac distribution and @xmath69/\pi$ ] ( taking the limit , @xmath70 , the dc conductivity is recovered . ) although dmft is approximate , the formalism can be expected to work in the non - interacting limit , where course graining will give the exact non - interacting dos for the tight - binding model ( regardless of dimension ) . in the opposite limit ( @xmath71 ) , the neglect of loops through the host is justified , and the formalism should be exact . such propagation of correlation is also small when the coordination number is high ( e.g. fcc lattices ) . we have solved the dmft equations using the self - energies in equations ( [ eqn : selfenergyeqn ] ) and ( [ eqn : selfenergyt ] ) to find electron spectral functions . we show the evolution of these functions with coupling for @xmath72 in figure [ fig : spectralfunctions](a ) and @xmath73 in figure [ fig : spectralfunctions](b ) . the spectral functions show two features . there is a low energy peak with a width defined by the phonon frequency , and a high energy shoulder . this behaviour can be related to the two limits of the holstein model . at low energy scales ( @xmath74 ) , electrons interact via virtual processes and the behaviour is essentially hubbard - like ( correlated ) . at frequencies greater than @xmath75 , the spectral weight is reduced , phonons may be created , and static behaviour emerges ( in the static limit , phonons may always be created as no energy is required to deform the lattice ) . the central peak narrows with increased coupling , until a critical value is reached . here , the theory has a brinkman rice - like transition with a diverging effective mass @xcite . in figure [ fig : selfenergy ] , we plot the imaginary part of the self - energy . for low coupling strengths ( small @xmath30 and energy scales ( @xmath76 ) , @xmath77 $ ] is small because electrons can not create phonons , consistent with the englesberg schrieffer analysis ) . however , as the coupling increases , electrons can be scattered by bipolaron resonances and @xmath78 $ ] rises sharply on either side of the fermi - energy . the gap in the weight of the self - energy is frequently assumed in the analysis of arpes data , but as we can see here , when the electron - phonon coupling approaches the band width ( i.e. @xmath4 ) , the self - energy has a more complicated form , without the simple picture of coherent and incoherent quasi - particles . indeed , it has been demonstrated that in low dimensions , vertex corrections are required to achieve a sharp discrimination between coherent and incoherent dressed electrons @xcite . the onset of this `` transition '' can be studied by examining the inverse quasi - particle mass ( quasi - particle weight ) , shown in figure [ fig : qpweight ] for two values of phonon frequency . as the coupling increases , @xmath79 becomes smaller and eventually vanishes at a critical value of coupling ( the effective mass diverges ) . we also examine the phonon spectral function in figure [ fig : phonspec ] for @xmath80 and various couplings . as the coupling is increased , the phonon modes soften ( figure [ fig : phonfreq ] ) . the effective frequency does not tend to zero as quickly as the quasi - particle weight . we will revisit this point later in this article . it is clearly the case that me theory does not correctly describe the strong coupling limit . it is known from the exact solution of the static limit @xcite and from approximate `` iterated - perturbation theory '' @xcite and qmc @xcite solutions of the hubbard model , that sub - bands should form at strong coupling . as our calculations demonstrate , this sub - band formation is not properly reproduced within me theory and this makes it likely that effects of vertex corrections become significantly more important at strong coupling . since this is at variance with the traditional interpretation of migdal s analysis , we are motivated to re - examine the vertex function in the next section . the neglect of the lowest - order vertex correction shown in figure [ fig : feynmandiag](c ) and all higher - order corrections is central to migdal eliashberg theory . in this section , we compute the lowest - order correction and use this to define the region of validity for me theory . in this sense , we are revisiting migdal s analysis to understand why there is a contradiction between low frequency results computed with me theory and advanced numerical methods . the ratio of first to zeroth order vertices at finite temperature may be written as @xmath81 as temperature tends to zero , this sum becomes an integral which may easily be evaluated ( @xmath82 and @xmath83 are defined as in figure [ fig : feynmandiag ] ) , @xmath84 in order for vertex corrections to be unimportant , the ratio @xmath85 must be small . as an example , we show the lowest order correction to the vertex function for ( a ) @xmath86 and @xmath87 ( b ) @xmath88 and @xmath89 in figure [ fig : vertexcorrection ] . for the larger phonon frequencies , the ratio is clearly greater than 10% and the approximation is expected to be significantly changed by the inclusion of vertex corrections . for the weak coupling @xmath30 shown here , the corrections are much less pronounced for small phonon frequencies . in figure [ fig : diverge ] we show the magnitude of the central peak of the vertex correction ( always the largest part of the function ) for a range of couplings at @xmath90 . between @xmath91 and @xmath92 the vertex correction diverges , and me theory clearly breaks down . as it is necessary to pass through the divergence to reach @xmath93 , the theory is not applicable for strong couplings . figure [ fig : breakdown ] shows the lowest value of @xmath94 for given @xmath95 at which the ratio @xmath96 exceeds 0.1 ( the primed quantities are chosen so that @xmath97 represents an unprimed value @xmath98 , and a primed value of @xmath99 represents an unprimed value @xmath100 ) . below this value me theory can be expected to give accurate results , while the theory begins to break down above this line . for @xmath101 ( @xmath102 ) the standard migdal criterion works well and correctly predicts the region in which me theory is applicable . however for frequencies @xmath103 ( @xmath104 ) , the migdal criterion misses the divergence in the vertex correction associated with the divergence in the effective mass . this breakdown happens at @xmath105 or @xmath4 . the line of breakdown due to this vertex divergence levels off as the phonon frequency approaches zero and does not scale as @xmath106 ( migdal s criterion ) . these results imply that there is a breakdown in me theory in the adiabatic limit . we note that at high phonon frequencies ( @xmath107 ) , our line does not tend to zero . this is because the contribution @xmath108 does not tend to infinity . instead , the function gets wider as the phonon frequency increases ( see figure [ fig : vertexcorrection ] , panel b ) , and some combination of the magnitude and width of the vertex is probably a better measure of the importance of vertex corrections . at large @xmath6 , the breakdown is quite different to that at low phonon frequencies as the second order diagrams ( one of which contains a vertex correction ) develop the same functional form and magnitude . however the breakdown at @xmath109 ( @xmath110 ) is not too bad an estimate . at this value of @xmath30 , there is only a small mass renormalisation due to vertex corrections . our results are consistent with that of benedetti and zeyher @xcite , which was obtained with a different technique using a semi - circular density of states ( @xmath111 ) , and predicted breakdown for @xmath112 in the extreme adiabatic limit . we have included their result in the figure , choosing the bandwidth parameter by matching the bare dos at the fermi energy for the two cases ( the conversion factor is @xmath113 ) which is reasonable when considering low energy excitations at half - filling , and corresponds to a critical @xmath114 for the gaussian dos used here ( see the diamond in figure [ fig : breakdown ] ) . @xcite also predicted a breakdown in me theory at @xmath105 from their solution of the static limit of the infinite - dimensional holstein model , consistent with the breakdown that we have found . the solution of the static limit predicts the formation of sub - bands , which is something not reproduced in the me solution . the lack of subbands in me theory demonstrates that higher order diagrams are essential for the description of the sub - band formation , and therefore for the description of the strong coupling limit . we suggest that the breakdown of me theory should be understood in the following heuristic manner . as the coupling increases , the host spectral function ( @xmath115 ) narrows and spectral weight is shifted towards lower energy scales . there is then an effective bandwidth , @xmath116 , for quasi - particles close to the fermi surface . the condition for the applicability of me theory then becomes @xmath117 . as seen in figure [ fig : phonspec ] , there is also a reduction in the effective phonon frequency with increasing interaction strength . although the renormalisation of the phonon frequency helps to drive against the transition , we note that the band narrowing effect is much stronger , as can be seen by comparing figures [ fig : qpweight ] and [ fig : phonfreq ] . using the finite temperature form of the self - energy , it is possible to calculate resistivity curves . these are shown in figure [ fig : resistivityudep ] at various electron - phonon coupling and phonon frequency . at high temperatures , the tendency is to linear behavior , with a gradient depending on @xmath30 , but independent of @xmath118 . the effect of phonon frequency is most dramatically seen at @xmath119 , where a point of inflection is seen , before the curve tends to low temperature @xmath120 behavior , consistent with a weakly renormalized electron gas . we note that long range order has not been considered in this study , so no superconducting transition is seen . the negative curvature of the resistivity curve at intermediate temperatures is of interest , since that curvature has recently been interpreted as an onset of the mott limit , but may have a different interpretation as an intermediate coupling phenomenon @xcite . we have computed spectral functions , resistivity curves , self - energies , quasi - particle weight and effective phonon frequency for the holstein model at half - filling using migdal eliashberg theory , within the dmft framework , and revisited migdal s analysis of the strength of the vertex correction . by analysing the first order vertex correction , we have defined the region of applicability of me theory . we find that me theory breaks down at intermediate coupling in the adiabatic limit , showing that migdal eliashberg theory should only be trusted at weak coupling within this framework . the coupling at which breakdown occurs corresponds to a divergence in the effective mass , indicating that both long range order and vertex corrections should be included to correctly describe the strong coupling regime . the question remains : what was lacking from the analysis of migdal , which estimated the magnitude of the vertex corrections . this can be understood from the magnitude of the kinetic energy . in the calculations , the kinetic energy can be seen to be decreasing due to the localisation at the non - analyticities . thus in migdal s estimate , the renormalised kinetic energy , and not the bare kinetic energy should have featured . the renormalised phonon frequency should also have been used , but it tends to zero more slowly than the kinetic energy . at the non - analyticity , where the kinetic energy tends to zero , it can be seen that a vertex divergence is expected from the modified version of migdal s estimate . our results are timely because of a recent upsurge of interest in the role of electron - phonon coupling in the cuprate superconductors . there are many experimental results which have been analysed and interpreted using es / me theory , in spite of the large coupling constant that has been predicted from that analysis . many groups are using migdal s analysis to justify their claims , without a careful consideration of the self - consistent effect of coupling on the kinetic energy of the electrons ( polarons ) . exact numerical analysis shows that large electron - phonon couplings significantly reduce the kinetic energy of electrons @xcite and thus affect the internal consistency of migdal s analysis . therefore , the any very large @xmath121 determined from es style analysis of experimental results is not consistent with a theory that neglects vertex corrections ( i.e. a more detailed analysis with higher order effects included is necessary ) . without that analysis , the only conclusion that can be reached is that the coupling is large , but no reliable value for that coupling can be determined . the main conclusion of a breakdown is not expected to change in 1d , 2d or 3d . in three dimensional systems , the role of spatial fluctuations should only be significant very close to the bi - polaron instability . the role of spatial fluctuations on the spectral properties in 2d has been analysed at weak coupling , and is found to lead to quantitative differences in the coherent excitations @xcite . spatial fluctuations have a major role in the superconducting state , and it has also been shown that the eliashberg approach to superconductivity is inadequate for optical phonon mediated @xmath8-wave superconductivity @xcite , and incomplete for @xmath122-wave superconductivity @xcite . we therefore urge researchers to carefully analyse the internal consistency of their theories when dealing with strong coupling materials . the authors would like to thank f.gebhard and f.essler for useful discussions . jph thanks the epsrc for partial funding of this work . is the frequency of the emitted phonon and @xmath123 is the frequency of the incoming electron . neglect of this diagram is central to migdal s theorem . thick and thin lines represent the full and bare green functions respectively . wavy lines represent phonons and straight lines electrons.[fig : feynmandiag],width=491 ] ( panel a ) and @xmath124 ( panel b ) at @xmath125 . a bi - polaronic resonance forms at zero frequency . no upper and lower sub - bands are formed but spectral weight is shifted away from the fermi - energy . the central peak narrows with increasing @xmath30 , corresponding to a divergence in the effective mass . the general form of the curves is similar at all frequencies.[fig : spectralfunctions],title="fig:",height=377 ] ( panel a ) and @xmath124 ( panel b ) at @xmath125 . a bi - polaronic resonance forms at zero frequency . no upper and lower sub - bands are formed but spectral weight is shifted away from the fermi - energy . the central peak narrows with increasing @xmath30 , corresponding to a divergence in the effective mass . the general form of the curves is similar at all frequencies.[fig : spectralfunctions],title="fig:",height=377 ] ( panel a ) and @xmath124 ( panel b ) at @xmath125 . as the coupling , @xmath126 ( see discussion after eqn 2 ) increases , hubbard - like behaviour is seen at low energy scales , and fk - like behaviour at @xmath127 . note how the gap in the self - energy at weak coupling ( which is central to the engelsberg schrieffer approach ) fills up as coupling is increased . this is due to the increasing importance of higher order diagrams at stronger couplings . this shows that one should be careful about interpreting results from the cuprates ( @xmath128 ) using an es approach ( this type of analysis is typically used on arpes results ) , since the es form for the self - energy ( i.e. gapped at @xmath129 ) can only be seen for very small @xmath121 . [ fig : selfenergy],title="fig:",height=377 ] ( panel a ) and @xmath124 ( panel b ) at @xmath125 . as the coupling , @xmath126 ( see discussion after eqn 2 ) increases , hubbard - like behaviour is seen at low energy scales , and fk - like behaviour at @xmath127 . note how the gap in the self - energy at weak coupling ( which is central to the engelsberg schrieffer approach ) fills up as coupling is increased . this is due to the increasing importance of higher order diagrams at stronger couplings . this shows that one should be careful about interpreting results from the cuprates ( @xmath128 ) using an es approach ( this type of analysis is typically used on arpes results ) , since the es form for the self - energy ( i.e. gapped at @xmath129 ) can only be seen for very small @xmath121 . [ fig : selfenergy],title="fig:",height=377 ] , computed using me theory as a function of @xmath30 ( results for phonon frequencies @xmath130 and @xmath131 are shown ) . @xmath132 is strongly renormalised for intermediate @xmath30.[fig : qpweight],height=377 ] . the effective phonon frequency ( location of maximum ) and particle lifetime ( inverse width of peak ) are reduced with increasing coupling . when @xmath133 the curve is skewed in such a way that it is not lorenzian , and the excitation no longer has the properties of a single phonon.[fig : phonspec],height=377 ] ) = 0 $ ] . the renormalisation of the effective phonon frequency is not as strong as that of the electronic inverse mass ( quasi - particle weight @xmath132 ) shown in figure [ fig : qpweight ] , which goes to zero at the transition . the inset shows the quasiparticle lifetime given by @xmath134}\protect $ ] . when the value of @xmath121 tends to 1 , the phonons can no longer be treated as single particle excitations . [ fig : phonfreq],height=377 ] and @xmath87 ( b ) @xmath88 and @xmath89 . the central maximum in ( b ) shows a ratio of over @xmath135 and one should expect significant corrections to quantities computed within me theory . [ fig : vertexcorrection],width=529 ] and @xmath136 , chosen so that a value of 1 represents @xmath98 , etc . also shown is the result of benedetti and zeyher for very low phonon frequency ( filled diamond ) . above the line , the ratio of the first order vertex correction to the bare vertex , @xmath137 , exceeds 10% and me theory is no longer strictly valid . at low frequencies ( adiabatic limit ) , breakdown occurs at smaller coupling strength than expected as a result of the divergence found in the vertex corrections . this is in contrast to the interpretation of migdal s analysis which is often used when low frequency experimental data is analysed , and is the main result of this article . [ fig : breakdown],height=377 ]
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we investigate the applicability of migdal eliashberg ( me ) theory by revisiting migdal s analysis within the dynamical mean - field theory framework .
first , we compute spectral functions , the quasi - particle weight , the self energy , renormalised phonon frequency and resistivity curves of the half - filled holstein model .
we demonstrate how me theory has a phase - transition - like instability at intermediate coupling , and how the engelsberg schrieffer ( es ) picture is complicated by low - energy excitations from higher order diagrams ( demonstrating that es theory is a very weak coupling approach ) . through consideration of the lowest - order vertex correction ,
we analyse the applicability of me theory close to this transition .
we find a breakdown of the theory in the intermediate coupling adiabatic limit due to a divergence in the vertex function .
the region of applicability is mapped out , and it is found that me theory is only reliable in the weak coupling adiabatic limit , raising questions about the accuracy of recent analyses of cuprate superconductors which do not include vertex corrections .
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cross sections of high - energy nuclear reactions are expressed in terms of nuclear parton distribution functions ( npdfs ) , so that precise npdfs are essential for finding any new phenomena in the high - energy reactions . recently , this topic is becoming important in heavy - ion collisions for investigating properties of quark - hadron matters @xcite and also in neutrino reactions for investigating neutrino - oscillation physics @xcite . determination of precise npdfs is valuable for studying various phenomena in heavy - ion reactions such as color glass condensate @xcite , @xmath8 suppression @xcite , and parton - energy loss @xcite . the npdf studies should be also important for heavy - ion collisions at lhc ( large hadron collider ) @xcite . in neutrino oscillation experiments , most data are taken at small @xmath7 ( @xmath91 gev@xmath10 ) . we could approach such a kinematical region from the high - energy deep inelastic one by using quark - hadron duality @xcite . however , there are still unresolved issues in neutrino deep inelastic scattering . for example , an anomalous @xmath11 value was reported in the neutrino - iron scattering by the nutev collaboration @xcite . it could be related to a nuclear modification difference between the parton distributions @xmath12 and @xmath13 @xcite because the iron target is used in the nutev measurements . there is also an issue that nuclear corrections are different from the ones expected from electron and muon scattering experiments according to recent nutev data @xcite . in these high - energy nuclear reactions , nucleonic pdfs rather than the nuclear ones are often used in calculating cross sections by neglecting nuclear modifications although it is well known that nuclear corrections could be as large as 20% in medium - size nuclei @xcite . these nuclear modifications have been experimentally investigated mainly by the measurements of structure - function ratios @xmath0 and drell - yan cross - section ratios @xmath1 . physical mechanisms of the nuclear corrections are , for example , summarized in ref . @xcite . in the small-@xmath6 region , the npdfs become smaller than the corresponding nucleonic ones , which is called shadowing . there are depletions at medium @xmath6 , which is related to the nuclear binding mechanism and possibly to a nucleonic modification inside a nuclear medium @xcite . at large @xmath6 , the nucleon s fermi motion gives rise to positive corrections . because the pdfs are related to the nonperturbative aspect of quantum chromodynamics ( qcd ) , theoretical calculations have been done by lattice qcd or phenomenological models . however , such calculations are not accurate enough at this stage . one would like to have accurate npdfs , which are obtained in a model - independent way , for calculating precise nuclear cross sections . we should inevitably rely on experimental data for determining them . studies of nucleonic pdfs have a long history with abundant experimental data in a wide kinematical region @xcite . however , determination of npdfs is still at a premature stage with the following reasons . first , available experimental data are limited . the experiments of the hadron - electron ring accelerator ( hera ) provided data for structure functions at small @xmath6 in a wide range of @xmath7 ; however , such data do not exist for nuclei . because of final - state interactions , hadron - production data may not be suitable for the npdf determination , whereas they are used in the nucleonic analysis . second , the analysis technique is not established . parametrization studies for the npdfs started only recently . the npdfs are expressed in terms of a number of parameters which are then determined by a @xmath14 analysis of the nuclear data . however , it is not straightforward to find functional forms of mass - number ( @xmath15 ) and bjorken-@xmath6 dependencies in the npdfs . furthermore , higher - twist effects could be important in the small-@xmath7 region . a useful parametrization was investigated in ref . @xcite by analyzing @xmath16 structure functions and drell - yan data ; however , the distributions were obtained by simply assigning appropriate parameter values by hand in the versions of 1998 and 1999 . the first @xmath14 analysis was reported in ref . @xcite , and then uncertainties of the npdfs were obtained @xcite . all of these analyses are done in the leading order ( lo ) of the running coupling constant @xmath2 . a next - to - leading - order ( nlo ) analysis was recently reported @xcite . the lo @xmath14 analysis with the uncertainties was also investigated in the 2007 version of ref . there are related studies on the nuclear shadowing @xcite and a global analysis of structure functions @xcite . in this way , the parametrization studies have been developed recently for the npdfs , and they are not sufficient . here , we extend our studies in refs . @xcite by focusing on the following points : * nlo analysis with npdf uncertainties together with a lo one , * roles of nlo terms on the npdf determination by comparing lo and nlo results , * better determination of @xmath6 and @xmath15 dependence , * nuclear modifications in the deuteron by including @xmath17 data , * flavor asymmetric antiquark distributions . this article is organized as follows . in sec . [ analysis ] , our analysis method is described for determining the npdfs . analysis results are explained in sec . [ results ] . nuclear modifications in the deuteron are discussed in sec . [ deuteron ] . the results are summarized in sec . [ summary ] . the optimum npdfs are determined by analyzing experimental data of the @xmath16 structure functions and drell - yan cross sections for nuclear targets . details of our analysis method are described in refs . @xcite , so that only the outline is explained in the following . the parton distribution functions are expressed by two variables @xmath6 and @xmath7 . the variable @xmath7 is defined @xmath18 with the virtual photon momentum @xmath19 in the lepton scattering , and the scaling variable @xmath6 is given by @xmath20 with the nucleon mass @xmath21 and the energy transfer @xmath22 . the variables for the drell - yan process are momentum fractions , @xmath23 and @xmath24 for partons in the projectile and target , respectively , and @xmath7 defined by the dimuon mass as @xmath25 . in our analysis , the npdfs are expressed in terms of corresponding nucleonic pdfs multiplied by weight functions : @xmath26 the functions @xmath27 and @xmath28 indicate type-@xmath29 npdf and nucleonic pdf , respectively , and @xmath30 is a weight function which indicates a nuclear modification for the type-@xmath29 parton distribution . the function @xmath30 generally depends on not only @xmath6 and @xmath15 but also the atomic number @xmath31 . it should be noted that this expression sacrifices the large-@xmath6 ( @xmath32 ) nuclear distributions . finite distributions exist even at @xmath33 in nuclei , whereas the distributions should vanish in the nucleon . flavor symmetric antiquark distributions are assumed for @xmath34 , @xmath35 , and @xmath36 in the previous analyses @xcite . from the violation of the gottfried sum rule and drell - yan measurements , it is now well known that these antiquark distributions are different @xcite . it is more natural to investigate modifications from the flavor asymmetric antiquark distributions in the nucleon . in this work , flavor asymmetric antiquark distributions are used in nuclei , and the distribution type @xmath29 indicates @xmath12 , @xmath13 , @xmath34 , @xmath35 , @xmath36 , and @xmath37 : @xmath38 whereas the relation @xmath39 is assumed in refs . the number of flavor is three ( four ) at @xmath40 ( @xmath41 ) @xcite . the bottom and top quark distributions are neglected . the strange - quark distributions are assumed to be symmetric ( @xmath42 ) although there are recent studies on possible asymmetry @xmath43 @xcite . the charm - quark distributions are created by @xmath7 evolution effects @xcite . as for the nucleonic pdfs in the lo and nlo , the mrst ( martin , roberts , stirling , and thorne ) parametrization of 1998 is used @xcite in this analysis , where the charm - quark mass is @xmath44=1.35 gev and scale parameters are @xmath45=0.174 and 0.300 gev for the lo and nlo , respectively . in our previous analysis @xcite , the mrst-2001 version was employed . since the nlo gluon distribution is negative at @xmath46 and @xmath471 gev@xmath10 in the 2001 version , we use the 1998 parametrization in this work . negative gluon distributions in nuclei could affect our analysis inappropriately in the shadowing region . furthermore , other researchers may use our npdfs at small @xmath6 ( @xmath48 ) with @xmath491 gev@xmath10 for calculating cross sections , for example , in lhc experiments . we tested various pdfs of the nucleon , but overall results are not significantly changed . since we are interested in obtaining the distributions at @xmath501 gev@xmath10 in comparison with other distributions and also our previous distributions , we decided to use the mrst distributions . in the analyses of the cteq ( coordinated theoretical / experimental project on qcd phenomenology and tests of the standard model ) collaboration , the initial scale @xmath7=(1.3)@xmath10 gev@xmath10 is used . the nuclear modification is assumed to have the following functional form : @xmath51 where @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 , and @xmath57 are parameters . the parameter @xmath53 controls the shadowing part , @xmath54 , @xmath55 , and @xmath56 determine a minute functional form , and @xmath57 is related to the fermi - motion part at large @xmath6 . the parameter @xmath57 is fixed at @xmath57=0.1 because it can not be determined from a small number of data in the fermi - motion part . as it will be shown in the result section , the antiquark and gluon modifications can not be determined from the present data at @xmath58 . if a large @xmath57 , for example @xmath57=1 , is taken , the antiquark and gluon distributions could become unrealistically large at large @xmath6 . in order to avoid such an issue , @xmath59 is used . the overall @xmath15 dependence in eq . ( [ eqn : wi ] ) is taken @xmath60 simply by considering nuclear volume and surface contributions @xcite . there are three constraints for the parameters by the nuclear charge @xmath31 , baryon number @xmath15 , and momentum conservation @xcite : @xmath61 , \nonumber \\ a & = \int dx \ , \frac{a}{3 } \left [ u_v^a ( x , q_0 ^ 2 ) + d_v^a ( x , q_0 ^ 2 ) \right ] , \label{eqn:3conserv } \\ a & = \int dx \ , a \ , x \big [ u_v^a ( x , q_0 ^ 2 ) + d_v^a ( x , q_0 ^ 2 ) \nonumber \\ & + 2 \ , \big \ { \bar u^a(x , q_0 ^ 2 ) + \bar d^a(x , q_0 ^ 2 ) + \bar s^a(x , q_0 ^ 2 ) \big \ } + g^a ( x , q_0 ^ 2 ) \big ] . \nonumber\end{aligned}\ ] ] we selected three parameters , @xmath62 , @xmath63 , and @xmath64 , which are determined by these conditions . following improvements are made from the previous analysis @xcite . first , the parametrization of @xmath6 dependence is modified . the meaning of the parameters @xmath54 , @xmath55 , and @xmath56 is not obvious , so that it is difficult to limit the ranges of these parameters in the analysis . here , we take @xmath6 points ( @xmath65 , @xmath66 ) of extreme values for the function @xmath67 as the parameters instead of @xmath54 and @xmath55 . they are related with each other by @xmath68 then , the values of @xmath65 and @xmath66 become transparent at least for the valence - quark distributions . from the measurements of the ratios @xmath69 , where @xmath70 indicates the deuteron , we have a rough idea that the extreme values should be @xmath71 and @xmath72 . the drell - yan measurements indicate @xmath73 for the antiquark distributions . the value @xmath74 and the extreme values for the gluon distribution are not obvious . second , the @xmath15 dependence @xmath75 is too simple . from gross nuclear properties , the leading @xmath15 dependence could be described by @xmath76 . in order to describe more details , the parameters @xmath13 , @xmath77 , @xmath78 , and @xmath79 are taken to be @xmath15 dependent : @xmath80 because the extreme values @xmath65 and @xmath66 are almost independent of @xmath15 according to the @xmath16 data and also from the previous analyses @xcite , they are assumed to be independent of @xmath15 . in addition , the parameters @xmath81 and @xmath82 are fixed in the antiquark and gluon distributions as follows . because the gluon distributions can not be well determined from the present data , the parameter @xmath83 is taken @xmath84 as assumed in the previous analysis @xcite . it means that @xmath82 and @xmath85 are related by @xmath86 from eq . ( [ eqn : bici ] ) . there are still six parameters for the antiquark distributions and they should be too many in comparison with four parameters for the valence - quark distributions . we decided to fix the parameter @xmath81 , which is sensitive to the gluon shadowing ratio to the antiquark one because of the momentum conservation . we found that the gluon shadowing can not be well determined even in the nlo analysis ; therefore , the value of @xmath81 is taken so that the gluon shadowing is similar to the antiquark shadowing . after all , the following twelve parameters are used for expressing the nuclear modifications : @xmath87 these parameters are determined by the following global analysis . most of used experimental data are explained in ref . first , the data for @xmath69 are from european muon collaboration ( emc ) @xcite , the slac ( stanford linear accelerator center)-e49 , e87 , e139 , and e140 collaborations @xcite , the bologna - cern - dubna - munich - saclay ( bcdms ) collaboration @xcite , the new muon collaboration ( nmc ) @xcite , the fermilab ( fermi national accelerator laboratory)-e665 collaboration @xcite , and the hermes @xcite . second , the ratios @xmath88 ( @xmath89 ) are from the nmc @xcite . third , the drell - yan data are from the fermilab - e772 @xcite and e866/nusea @xcite collaborations . additional data to the hkn04 ( hirai , kumano , nagai in 2004 ) analysis are the ones for the deuteron - proton ratio @xmath5 . these deuteron data are added because precise nuclear modifications are needed for the deuteron which is , for example , used in heavy - ion experiments at the relativistic heavy ion collider ( rhic ) @xcite . the @xmath5 data are taken from the measurements by the emc @xcite , the bcdms @xcite , the fermilab - e665 @xcite , and the nmc @xcite . these data are used for extracting information on the flavor asymmetric antiquark distributions @xmath90 in the nucleon @xcite . therefore , the flavor asymmetric antiquark distributions in eq . ( [ eqn : wpart ] ) are essential for a successful fit and for extracting information on modifications in the deuteron . because the dglap ( dokshitzer - gribov - lipatov - altarelli - parisi ) evolution can be applied only in the perturbative qcd region , the data with small @xmath7 values can not be used in the analysis . however , the data in a relatively small-@xmath7 region ( @xmath7=1@xmath913 gev@xmath10 ) are valuable for determining antiquark distributions at small @xmath6 ( @xmath92 ) . as a compromise of these conflicting conditions , only the data with @xmath93 gev@xmath10 are used in the analysis . however , one should note that the data in the range , @xmath7=1@xmath913 gev@xmath10 , may contain significant contributions of higher - twist effects which are not considered in our leading - twist analysis . the parameters are determined by fitting experimental data for the ratios of the structure functions @xmath94 and drell - yan cross sections . the total @xmath14 @xmath95 is minimized to obtain the optimum parameters . the ratio @xmath96 indicates experimental data for @xmath0 and @xmath97 , and @xmath98 is a theoretical ratio calculated by the parametrized npdfs . the initial scale @xmath99 is taken @xmath99=1 gev@xmath10 , and the distributions in eq . ( [ eqn : wpart ] ) are evolved to experimental @xmath7 points to calculate the @xmath14 by the dglap evolution equations @xcite . all the calculations are done in the lo or nlo , and the modified minimal subtraction ( @xmath100 ) scheme is used in the nlo analysis . the structure function @xmath101 is expressed in terms of the npdfs and coefficient functions : @xmath102 \nonumber \\ & + c_g(x,\alpha_s ) \otimes g^a ( x , q^2 ) \bigg\ } , \label{eqn : f2}\end{aligned}\ ] ] where @xmath103 and @xmath104 are the coefficient functions @xcite , and @xmath105 is a quark charge . the symbol @xmath106 denotes the convolution integral : @xmath107 the proton - nucleus drell - yan cross section is given by the summation of @xmath108 annihilation and compton processes @xcite : @xmath109 they are expressed in terms of the pdfs and subprocess cross sections : @xmath110 \nonumber \\ & \ \times [ q_i ( y_1,q^2 ) \bar q_i^a ( y_2,q^2 ) + \bar q_i ( y_1,q^2 ) q_i^a ( y_2,q^2 ) ] , \label{eqn : dyqqbar}\end{aligned}\ ] ] @xmath111 \nonumber \\ & \ \ \ + \frac{d\hat\sigma_{qg}}{dq^2 dx_f } [ q_i ( y_1,q^2 ) + \bar q_i ( y_1,q^2 ) ] g^a ( y_2,q^2 ) \bigg ] . \label{eqn : dyqg}\end{aligned}\ ] ] the cross sections @xmath112 and @xmath113 indicate subprocess cross sections for @xmath108 annihilation processes in the lo and nlo , respectively . the @xmath114 indicates the cross section for @xmath115 and @xmath116 processes . the nlo expressions of these cross sections are , for example , found in ref . effects of possible parton - energy loss in the drell - yan process @xcite are neglected in this analysis . [ cols="^,^,^,^,^,^,^",options="header " , ] using these expressions for the structure functions @xmath94 and drell - yan cross sections , we calculate the theoretical ratios @xmath98 in eq . ( [ eqn : chi2 ] ) . the total @xmath14 is minimized by the cern program library minuit . from this analysis , an error matrix which is the inverse of a hessian matrix , is obtained . npdf uncertainties are estimated by using the hessian matrix as @xmath117 ^ 2=\delta \chi^2 \sum_{i , j } \left ( \frac{\partial f^a(x,\xi)}{\partial \xi_i } \right ) _ { \xi=\hat{\xi } } h_{ij}^{-1 } \left ( \frac{\partial f^a(x,\xi)}{\partial \xi_j } \right ) _ { \xi=\hat{\xi } } , \label{eq : dnpdf}\ ] ] where @xmath118 is the hessian matrix , @xmath119 is a parameter , and @xmath120 indicates the optimum parameter set . the @xmath121 value determines the confidence region , and it is calculated so that the confidence level @xmath122 becomes the one-@xmath123-error range ( @xmath124 ) for a given number of parameters ( @xmath125 ) by assuming the normal distribution in the multiparameter space . in the analysis with the twelve parameters , it is @xmath126 . this hessian method has been used for estimating polarized pdfs and fragmentation functions @xcite as well as nuclear pdfs in our previous version @xcite . the details of the uncertainty analysis are discussed in refs . @xcite as well as in the nucleonic pdf articles @xcite . determined parameters are listed in table [ table : parameters ] for both lo and nlo . three parameters are fixed by the constraints from baryon - number , charge , and momentum conservations in eq . ( [ eqn:3conserv ] ) , and we chose @xmath62 , @xmath63 , and @xmath64 for these parameters . the values of the obtained parameters and their errors are similar in the lo and nlo . however , the errors indicate that there are slight nlo improvements in comparison with the lo results . c@c@c @c@c nucleus & reference & # of data & @xmath14 & @xmath14 + & & & ( lo ) & ( nlo ) + d / p & @xcite & 290 & 375.5 & 322.5 + @xmath127he / d & @xcite & 35 & 60.9 & 51.8 + li / d & @xcite & 17 & 36.9 & 36.4 + be / d & @xcite & 17 & 39.3 & 53.0 + c / d & @xcite & 43 & 105.8 & 78.7 + n / d & @xcite & 162 & 136.3 & 121.7 + al / d & @xcite & 35 & 45.5 & 44.9 + ca / d & @xcite & 33 & 43.3 & 34.1 + fe / d & @xcite & 57 & 108.0 & 97.4 + cu / d & @xcite & 19 & 12.1 & 13.2 + kr / d & @xcite & 144 & 115.1 & 115.9 + ag / d & @xcite & 7 & 12.5 & 9.1 + sn / d & @xcite & 8 & 13.3 & 14.1 + xe / d & @xcite & 5 & 2.2 & 2.3 + au / d & @xcite & 19 & 55.6 & 32.3 + pb / d & @xcite & 5 & 5.7 & 4.5 + @xmath69 total & & 606 & 792.4 & 709.3 + be / c & @xcite & 15 & 12.6 & 11.9 + al / c & @xcite & 15 & 5.0 & 5.1 + ca / c & @xcite & 39 & 29.9 & 29.1 + fe / c & @xcite & 15 & 8.0 & 8.3 + sn / c & @xcite & 146 & 204.0 & 172.1 + pb / c & @xcite & 15 & 15.7 & 12.2 + c / li & @xcite & 24 & 67.4 & 64.9 + ca / li & @xcite & 24 & 69.0 & 65.3 + @xmath88 total & & 293 & 411.6 & 369.0 + c / d & @xcite & 9 & 9.3 & 8.1 + ca / d & @xcite & 9 & 5.8 & 13.8 + fe / d & @xcite & 9 & 12.6 & 17.9 + w / d & @xcite & 9 & 27.8 & 29.6 + fe / be & @xcite & 8 & 3.3 & 3.6 + w / be & @xcite & 8 & 14.9 & 12.1 + drell - yan total & & 52 & 73.8 & 85.1 + total & & 1241 & 1653.3 & 1485.9 + ( @xmath14/d.o.f . ) & & & ( 1.35 ) & ( 1.21 ) + each @xmath14 contribution is listed in table [ tab : chi2 ] . the values suggest that medium and large nuclei should be well explained by the current lo and nlo parametrizations . however , small nuclei are not so well reproduced . the lo fit ( @xmath14/d.o.f.=1.35 ) is better than the previous analysis with @xmath14/d.o.f.=1.58 @xcite , which is partly due to the introduction of new parameters for the @xmath15 dependence in eq . ( [ eqn : more - a ] ) . if the @xmath14 values of the lo analysis are compared with the ones in table iii of ref . @xcite , we find that much improvements are obtained for the nuclear ratios , @xmath128 , @xmath129 , @xmath130 , @xmath131 , @xmath132 , @xmath133 , @xmath134 , @xmath135 , @xmath136 , @xmath137 , and @xmath138 , whereas the fit becomes worse for @xmath139 and @xmath140 . the @xmath14/d.o.f . is further reduced in the current nlo analysis . according to table [ tab : chi2 ] , the nlo results with @xmath14/d.o.f.=1.21 reproduce the data better than the lo ones with @xmath14/d.o.f.=1.35 , especially in the following data sets : @xmath141 , @xmath142 , @xmath143 , @xmath144 , @xmath145 , @xmath146 , @xmath147 , @xmath133 , @xmath134 , and @xmath148 , whereas it becomes worse in @xmath149 , @xmath136 , and @xmath139 . the deuteron - proton data @xmath5 are added in this analysis to the data set of the previous version @xcite , and they should provide a valuable constraint on pdf modifications in the deuteron . because the @xmath5 data are sensitive to @xmath150 asymmetry @xcite , flavor asymmetric antiquark distributions should be used in our analysis . if the flavor symmetric distributions are used as initial ones , the fit produces a significantly larger @xmath14 . ( color online ) comparison with experimental ratios @xmath151 and @xmath5 . the rational differences between experimental and theoretical values [ @xmath152 are shown . the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data . theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=159 ] ( color online ) comparison with experimental ratios @xmath151 and @xmath5 . the rational differences between experimental and theoretical values [ @xmath152 are shown . the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data . theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=147 ] + ( color online ) comparison with experimental ratios @xmath151 and @xmath5 . the rational differences between experimental and theoretical values [ @xmath152 are shown . the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data . theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=159 ] ( color online ) comparison with experimental ratios @xmath151 and @xmath5 . the rational differences between experimental and theoretical values [ @xmath152 are shown . the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data . theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=147 ] ( color online ) comparison with experimental data of @xmath153 . the ratios @xmath154 are shown . the theoretical ratios and their uncertainties are calculated in the nlo . the notations are the same as fig . [ fig : rd].,title="fig:",width=159 ] ( color online ) comparison with experimental data of @xmath153 . the ratios @xmath154 are shown . the theoretical ratios and their uncertainties are calculated in the nlo . the notations are the same as fig . [ fig : rd].,title="fig:",width=147 ] + ( color online ) comparison with drell - yan data of @xmath155 . the ratios @xmath154 are shown . the theoretical ratios and their uncertainties are calculated in the nlo . the theoretical ratios are calculated at the @xmath7 points of the experimental data . the uncertainties are estimated at @xmath7=20 and 50 gev@xmath10 for the the @xmath156 type and @xmath157 one , respectively.,title="fig:",width=159 ] ( color online ) comparison with drell - yan data of @xmath155 . the ratios @xmath154 are shown . the theoretical ratios and their uncertainties are calculated in the nlo . the theoretical ratios are calculated at the @xmath7 points of the experimental data . the uncertainties are estimated at @xmath7=20 and 50 gev@xmath10 for the the @xmath156 type and @xmath157 one , respectively.,title="fig:",width=147 ] + the fit results of the nlo are compared with the used data in figs . [ fig : rd ] , [ fig : ra ] , and [ fig : dy ] for the ratios @xmath69 , @xmath88 , and @xmath158 , respectively . the rational differences between experimental and theoretical values @xmath154 , where @xmath159 is @xmath151 , @xmath88 , or @xmath158 , are shown . for the theoretical values , the nlo results are used and they are calculated at the experimental @xmath7 points . the uncertainty bands are also shown in the nlo , and they are calculated at @xmath7=10 gev@xmath10 for the structure function @xmath16 and at @xmath7=20 or 50 gev@xmath10 for the drell - yan processes . the scale @xmath7=10 gev@xmath10 is taken because the average of all the @xmath16 data is of the order of this value . the scale is @xmath7=50 gev@xmath10 for the drell - yan ratios of the @xmath157 type , and the lower scale 20 gev@xmath10 is taken for the ratio of the @xmath156 type because experimental @xmath7 values are smaller . these figures indicate that the overall fit is successful in explaining the used data . we notice that the @xmath14 values , 53.0 , 64.9 , and 29.6 in the nlo , are especially large for @xmath160 , @xmath161 , and @xmath140 in comparison with the numbers of their data , 17 , 24 , and 9 , according to table [ tab : chi2 ] . these large @xmath14 values come from deviations from accurate e139 , nmc , and e772 data ; however , such deviations are not very significant in figs . [ fig : rd ] , [ fig : ra ] , and [ fig : dy ] . there are general tendencies that medium- and large - size nuclei are well explained by our parametrization , whereas there are slight deviations for small nuclei . because any systematic deviations are not found from the experimental data , our analyses should be successful in determining the optimum nuclear pdfs . ratio @xmath162 and the drell - yan ratio @xmath136 . in the upper figures , the theoretical curves and uncertainties are calculated at @xmath7=10 gev@xmath10 for the @xmath16 ratio and at @xmath7=50 gev@xmath10 for the drell - yan ratio . the dashed and solid curves indicate lo and nlo results , and the lo and nlo uncertainties are shown by the dark- and light - shaded bands , respectively . the lower figures indicate the ratios @xmath154 where @xmath159 indicates @xmath144 or @xmath136 . here , the theoretical ratios are calculated at the experimental @xmath7 points . for comparison , the lo curves and their uncertainties are also shown by @xmath163 and @xmath164.,title="fig:",width=160 ] ratio @xmath162 and the drell - yan ratio @xmath136 . in the upper figures , the theoretical curves and uncertainties are calculated at @xmath7=10 gev@xmath10 for the @xmath16 ratio and at @xmath7=50 gev@xmath10 for the drell - yan ratio . the dashed and solid curves indicate lo and nlo results , and the lo and nlo uncertainties are shown by the dark- and light - shaded bands , respectively . the lower figures indicate the ratios @xmath154 where @xmath159 indicates @xmath144 or @xmath136 . here , the theoretical ratios are calculated at the experimental @xmath7 points . for comparison , the lo curves and their uncertainties are also shown by @xmath163 and @xmath164.,title="fig:",width=154 ] + next , actual data are compared with the lo and nlo theoretical ratios and their uncertainties for the calcium nucleus as an example in fig . [ fig : f2-dy ] . in the upper figures , the theoretical curves and the uncertainties are calculated at fixed @xmath7 points , @xmath7=10 gev@xmath10 and 50 gev@xmath10 for the @xmath16 and the drell - yan , respectively , whereas the experimental data are taken at various @xmath7 values . the rational differences @xmath154 are shown together with the difference between the lo and nlo curves , @xmath163 , in the lower figures . the comparison suggests that both lo and nlo parametrizations should be successful in explaining the @xmath6 dependence of the calcium data . it is noteworthy that the nlo error band of the @xmath16 ratio becomes slightly smaller in comparison with the lo one at small @xmath6 ; however , magnitudes of both uncertainties are similar in the region , @xmath165 . the nlo improvement is not clearly seen in the drell - yan ratio @xmath136 in the range of @xmath166 . there are discrepancies between the theoretical curves and the @xmath162 data at @xmath167 ; however , they are simply due to @xmath7 differences . if the theoretical ratios are calculated at the same experimental @xmath7 points , they agree as shown in the @xmath162 part of fig . [ fig : rd ] . these lo and nlo results indicate that the available data are taken in the limited @xmath6 range without small-@xmath6 data , and they are not much sensitive to nlo corrections . this fact leads to a difficulty in determining nuclear gluon distributions because the gluonic effects are typical nlo effects through the coefficient functions and in the @xmath7 evolution equations . one of possible methods for determining the gluon distribution in the nucleon is to investigate @xmath7 dependence of the structure function @xmath16 @xcite . because @xmath7 dependent data exist in the @xmath168 ratios , it may be possible to find nuclear modifications of the gluon distribution . -dependent data of @xmath169 by the hermes collaboration . the dashed and solid curves indicate lo and nlo results , and the nlo uncertainties are shown by the shaded bands.,scaledwidth=41.0% ] -dependent data of @xmath170 by the nmc . the notations are the same as fig . [ fig : krd - q2].,scaledwidth=41.0% ] the lo and nlo parametrization results are compared with @xmath7 dependent data for @xmath169 and @xmath170 measured by the hermes and nmc collaborations , respectively , in figs . [ fig : krd - q2 ] and [ fig : snc - q2 ] . the uncertainties are shown by the shaded bands in the nlo . because the lo and nlo uncertainties are similar except for the small-@xmath6 region , the lo ones are not shown in these figures . they are compared later in this subsection . the results indicate that overall @xmath7 dependencies are well explained by our parametrizations in both lo and nlo . the comparison suggests that the experimental data are not accurate enough to probe the details of the @xmath7 dependence . furthermore , @xmath7 dependencies of the hermes and nmc results are different . the hermes ratio @xmath171 tends to decrease with increasing @xmath7 at @xmath6=0.035 , 0.045 , and 0.055 , whereas the nmc ratio @xmath133 increases with @xmath7 at the same @xmath6 points , although the nuclear species are different . this kind of difference together with inaccurate @xmath7-dependent measurements makes it difficult to extract precise nuclear gluon distributions within the leading - twist dglap approach . it is reflected in large uncertainties in the gluon distributions as it becomes obvious in sec . [ npdfs ] . in our previous versions @xcite , the experimental shadowing in @xmath133 is underestimated at small @xmath6 ( @xmath172 ) partly because of an assumption on a simple @xmath15 dependence . as shown in fig . [ fig : snc - q2 ] , the shadowing is still slightly underestimated at @xmath173 ; however , the deviations are not as large as before . if the experimental errors and the npdf uncertainties are considered , our parametrization is consistent with the data . dependence of the ratio @xmath162 is compared in the lo and nlo at @xmath6=0.001 , 0.01 , 0.01 , and 0.7 . the dashed and solid curves indicate lo and nlo results , and lo and nlo uncertainties are shown by the dark- and light - shaded bands , respectively.,scaledwidth=40.0% ] the nlo uncertainties are compared with the lo ones in fig . [ fig : cad - q2 ] for the ratio @xmath162 . the lo and nlo ratios and their uncertainties are shown at @xmath6=0.001 , 0.01 , 0.1 , and 0.7 . the differences between both uncertainties are conspicuous at small @xmath6 ( = 0.001 and 0.01 ) ; however , they are similar at larger @xmath6 . the lo and nlo slopes are also different at small @xmath6 . these results indicate that the nlo effects become important at small @xmath6 ( @xmath174 ) , and the determination of the npdfs is improved especially in this small-@xmath6 region . because the nlo contributions are obvious only in the region , @xmath167 , it is very important to measure the @xmath7 dependence to pin down the nlo effects such as the gluon distributions . the possibilities are measurements at future electron facilities such as erhic @xcite and elic @xcite . nuclear modifications of the pdfs are shown for all the analyzed nuclei and @xmath175 at @xmath7=1 gev@xmath10 in fig . [ fig : npdfs - all ] . it should be noted that the modifications of @xmath12 are the same as the ones of @xmath13 in isoscalar nuclei , but they are different in other nuclei . the modifications increase as the nucleus becomes larger , and the dependence is controlled by the overall @xmath75 factor and the @xmath15 dependence in eq . ( [ eqn : more - a ] ) . the extreme values ( @xmath65 , @xmath66 ) are assumed to be independent of @xmath15 in our current analysis as explained in sec . [ paramet ] , so that they are the same in fig . [ fig : npdfs - all ] . although the oxygen data are not used in our global analysis , its pdfs are shown in the figure because they are useful for an application to neutrino oscillation experiments @xcite . our code is supplied at the web site in ref . @xcite for calculating the npdfs and their uncertainties at given @xmath6 and @xmath7 . ( @xmath176 , @xmath13 , @xmath177 , and @xmath37 ) are shown in the nlo for all the analyzed nuclei and @xmath175 at @xmath7=1 gev@xmath10 . as the mass number becomes larger in the order of d , @xmath127he , li , ... , and pb , the curves deviate from the line of unity ( @xmath178).,width=302 ] = 1 gev@xmath10 . the dashed and solid curves indicate lo and nlo results , and lo and nlo uncertainties are shown by the dark- and light - shaded bands , respectively.,title="fig:",width=151 ] = 1 gev@xmath10 . the dashed and solid curves indicate lo and nlo results , and lo and nlo uncertainties are shown by the dark- and light - shaded bands , respectively.,title="fig:",width=151 ] as examples of medium and large nuclei , we take the calcium and lead and show their distributions and uncertainties at @xmath7=1 gev@xmath10 in fig . [ fig : npdfs - ca - pb ] . because the deuteron is a special nucleus and it needs detailed explanations , its results are separately discussed in sec . [ deuteron ] . the figure indicates that valence - quark distributions are determined well in the wide range , @xmath179 because the uncertainties are small . it is also interesting to find that the lo and nlo uncertainties are almost the same . there are following reasons for these results . the valence - quark modifications at @xmath58 are determined by the accurate measurements of @xmath16 modifications . the antishadowing part in the region , @xmath180 , is also determined by the @xmath16 data because there is almost no nuclear modification in the antiquark distributions according to the drell - yan data . if the valence - quark distributions are obtained at @xmath181 , the small-@xmath6 behavior is automatically constrained by the baryon - number and charge conservations in eq . ( [ eqn:3conserv ] ) . because of these strong constraints , the lo and nlo results are not much different . in the near future , the jlab ( thomas jefferson national accelerator facility ) measurements will provide data which could constraint the nuclear valence - quark distribution especially at medium and large @xmath6 @xcite . furthermore , future neutrino measurements such as the miner@xmath22a experiment at fermilab @xcite and the one at a possible neutrino factory @xcite should provide important information at small @xmath6 ( @xmath182 ) . the antiquark distributions are also well determined except for the large-@xmath6 region , @xmath4 , because there is no accurate drell - yan data and the structure functions @xmath94 are dominated by the valence - quark distributions . the lo and nlo uncertainties are similar except for the large-@xmath6 region . because of gluon contributions in the nlo , the antiquark shadowing modifications are slightly different between the lo and nlo . possible large-@xmath6 drell - yan measurements such as j - parc ( japan proton accelerator research complex ) @xcite and fermilab - e906 @xcite should be valuable for the nuclear antiquark distributions in the whole-@xmath6 range . in the similar energy region , there is also the gsi - fair ( gesellschaft fr schwerionenforschung -facility for antiproton and ion research ) project @xcite . the gluon distributions contribute to the @xmath16 and drell - yan ratios as higher - order effects . therefore , they should be determined more accurately in the nlo analysis than the lo one . such tendencies are found in fig . [ fig : npdfs - ca - pb ] because the nlo uncertainties are smaller than the lo ones in both carbon and lead . however , these nlo improvements are not as clear as the cases of polarized pdf @xcite and fragmentation functions @xcite . it is because the @xmath7-dependent data are not accurate enough to probe such higher - order effects as discussed in sec . [ q2-dependence ] . in order to fix the gluon distributions , accurate measurements are needed for the scaling violation of @xmath0 @xcite . the gluon distributions should be also probed by production processes of such as charged hadrons @xcite , heavy flavor @xcite , @xmath183 @xcite , low - mass dilepton @xcite , and direct photon @xcite . there is a recent study on the gluon shadowing from the hera diffraction data @xcite . nuclear gluon distributions play an important role in discussing properties of quark - hadron matters in heavy - ion reactions , so that they need to be determined experimentally . = 100 gev@xmath10.,width=302 ] we also show the nuclear modifications at @xmath7=100 gev@xmath10 for the calcium in fig . [ fig : w - ca-100 ] . the nuclear modifications are not very different from those at @xmath7=1 gev@xmath10 for the valence - quark distributions . however , the shadowing corrections become smaller in the antiquark and gluon distributions in comparison with the ones at @xmath7=1 gev@xmath10 , and the modifications tend to increase at medium and large @xmath6 . the distribution functions themselves and their uncertainties are shown in figs . [ fig : npdf - ca ] and [ fig : npdf - pb ] for the calcium and lead , respectively . both lo and nlo distributions are shown . here , the uncertainties from the nucleonic pdfs are not included in the uncertainty bands . the calcium is an isoscalar nucleus , so that @xmath184 and @xmath185 are equal to @xmath186 and @xmath187 . however , they are different in the lead nucleus because of the neutron excess . in order to see nuclear modification effects , we show the distributions without the nuclear modifications . for example , the distribution @xmath188 is shown in the figure of @xmath184 by using the mrst distributions for @xmath12 and @xmath13 . although the uncertainties are large in the antiquark and gluon distributions at medium and large @xmath6 , they are not very conspicuous in figs . [ fig : npdf - ca ] and [ fig : npdf - pb ] because the distributions themselves are small . = 1 gev@xmath10 . no modification " indicates , for example , the distribution @xmath188 in the figure of @xmath184.,width=275 ] = 1 gev@xmath10 . no modification " indicates , for example , the distribution @xmath188 in the figure of @xmath184.,width=275 ] , is shown for the proton , lithium , aluminum , iron , and lead at @xmath7=1 gev@xmath10 . in the isoscalar nuclei , the distributions vanish ( @xmath189).,width=226 ] the flavor asymmetric antiquark distributions are assumed in this analysis as they are defined in eq . ( [ eqn : wpart ] ) . from this definition , it is obvious that @xmath185 and @xmath187 are equal in isoscalar nuclei such as carbon and calcium . in fig . [ fig : ub - db ] , the ratio @xmath190 is shown for the proton ( p ) , lithium ( li ) , aluminum ( al ) , iron ( fe ) , and lead ( pb ) at @xmath7=1 gev@xmath10 . because the nuclear corrections are assumed to be equal for the antiquark distributions at @xmath99 in eq . ( [ eqn : wpart ] ) , they are almost independent of nuclear species except for the isoscalar nuclei . it is interesting to investigate possible nuclear modifications on the distribution @xmath191 at the future facilities @xcite . there are noticeable differences between our nlo analysis results and the ones in ref . @xcite especially in the strange - quark and gluon modifications . these differences come from various sources . first , the analyzed experimental data sets are slightly different . second , the strange - quark distributions are created by the dglap evolution by assuming @xmath192 at the initial @xmath7 scale , and the charm distributions are neglected in ref . these differences lead to the discrepancies of the gluon modifications . the determined npdfs and their uncertainties can be calculated by using our code , which is supplied on our web site @xcite . by providing a kinematical condition for @xmath6 and @xmath7 and also a nuclear species , one can calculate the npdfs . it is explained in appendix [ library ] . if one needs analytical expressions of the npdfs at the initial scale @xmath99 , one may read instructions in appendix [ appen - a ] . nuclear densities are usually independent of the mass number , which indicates that the average nucleon separation is constant in nuclei . however , the deuteron is a special nucleus in the sense that its radius is about 4 fm , which is much larger than the average nucleon separation in ordinary nuclei ( @xmath193 fm ) . because it is a dilute system , nuclear modifications are often neglected . in fact , corrections to nucleonic structure functions and pdfs are small , namely within a few percentages according to theoretical estimates @xcite even if they are taken into account . c@c@c @c reference & @xmath7 ( gev@xmath10 ) & @xmath6 & modification ( % ) + @xcite & 4 & 0.015 & 1.1 + @xcite & 4 & 0.010 & 2.5 + @xcite & 4 & 0.010 & 2.0 + @xcite & 4 & 0.010 & [email protected] + @xcite & 4 & 0.010 & 1@xmath912 ( 3.5 ) + theoretical modifications in the deuteron depend much on models . the shadowing of ref . @xcite , which was used in the mrst analysis , is 1.1% at @xmath6=0.015 and @xmath7=4 gev@xmath10 ; however , other model calculations are different , [email protected]% at @xmath194 as shown in table [ tab : d - modification ] . furthermore , the modifications at medium @xmath6 could be as large as or larger than the small-@xmath6 shadowing ( for example , 5% at @xmath195 according to ref . @xcite ) . such deuteron corrections are becoming important recently although the magnitude itself may not be large . for example , precise nuclear modifications need to be taken into account for investigating quark - hadron matters in heavy - ion collisions such as deuteron - gold reactions in comparison with deuteron - deuteron ones at rhic @xcite . they are also valuable for discussing gottfried - sum - rule violation and flavor - asymmetric antiquark distributions because deuteron targets are used @xcite . in our previous versions on the npdfs @xcite , the data of @xmath5 are not included in the used data set . obtained nuclear modifications tend to be large in comparison with the theoretical model estimations . for example , the hkn04 analysis @xcite indicates about 6 and 8% corrections in antiquark and gluon distributions , respectively , at small @xmath6 ( @xmath196 ) with @xmath7=1 gev@xmath10 and about 5% in valence - quark ones at medium @xmath6 ( @xmath197 ) . they are possibly overestimations in the sense that typical theoretical models have corrections within the order of a few percentages . in order to obtain reasonable deuteron modifications from experimental data , we added @xmath5 measurements into the data set in our @xmath14 analysis . these deuteron data were already included in the analysis results in sec . [ results ] . in addition to the analysis of sec . [ results ] , two other analyses have been made by modifying eq . ( [ eqn : wi ] ) : @xmath198 an additional factor @xmath199 is introduced . the analysis results in sec . [ results ] correspond to the @xmath199=1 case . the other analyses have been made by taking @xmath199=0 and by taking it as a free parameter . we call them analyses 1 , 2 , and 3 , respectively : * analysis 1 : @xmath199=1 , * analysis 2 : @xmath199=0 , * analysis 3 : @xmath199=free parameter . the @xmath14 values of these analyses are listed in table [ tab : chi2-d ] . c@c@c @c@c nucleus & # of data & @xmath14 ( @xmath200 ) & @xmath14 ( @xmath201 ) & @xmath14 ( free ) + d / p & 290 & 322.5 & 282.6 & 284.9 + @xmath69 & 606 & 709.3 & 704.8 & 703.8 + @xmath88 & 293 & 369.0 & 381.6 & 375.3 + drell - yan & 52 & 85.1 & 84.5 & 85.4 + total & 1241 & 1485.9 & 1453.4 & 1449.4 + ( @xmath14/d.o.f . ) & & ( 1.21 ) & ( 1.18 ) & ( 1.18 ) + the deuteron modifications are terminated in the analysis 2 by taking @xmath201 . although such an assumption does not seem to make sense , we found a smaller @xmath14 value from the @xmath5 data than the one of the first analysis as shown in table [ tab : chi2-d ] . there are three major reasons for this result . first , the deuteron modifications obtained by the overall @xmath15 dependence in eq . ( [ eqn : wi ] ) are too large because the deuteron is a loosely bound system which is much different from other nuclei . smaller modifications are expected from the large nucleon separation . second , deuteron data are used for determining the pdfs in the nucleon " @xcite by considering nuclear shadowing modifications of ref . the modifications are calculated in a vector - meson - dominance mechanism and the shadowing in the deuteron is about 1% at @xmath202 according to this model . if this shadowing is not a realistic correction , the nucleonic pdfs of the mrst should partially contain nuclear effects at small @xmath6 . the medium- and large-@xmath6 regions are not corrected , so that some deuteron effects could be also included in the nucleonic pdfs in these regions . however , the corrections are not experimentally obvious in such @xmath6 regions as we find in the actual data of fig . [ fig : f2dp ] . in this way , the nucleonic pdfs could contain some deuteron modification effects . third , the nuclear effects could be absorbed into the @xmath150 asymmetry because it is determined partially by the @xmath5 data . these are the possible reasons why analysis 2 produces the smaller @xmath14 value . . the solid , dotted , and dashed curves indicate the nlo results of the analyses 1 ( @xmath200 ) , 2 ( @xmath201 ) , and 3 ( @xmath199=free ) , respectively . the shaded bands indicate uncertainties of the analysis-3 curves.,width=302 ] in the third analysis , the additional parameter @xmath199 is determined from the global analysis . as mentioned , the internucleon separation is exceptionally large in the deuteron . it leads to small nuclear corrections , which are much smaller than a smooth @xmath15-dependent functional form , as calculated in various models @xcite in table [ tab : d - modification ] . these models could be used for estimating an appropriate value for @xmath199 . however , we try to determine the nuclear pdfs without relying on specific theoretical models . the modification parameter @xmath199 is determined from the experimental data , and our analysis 3 indicates @xmath203 . if the diffuse deuteron system is considered , the 70% reduction may make sense . nonetheless , it should be noted that this factor may not reflect a realistic modification because it is likely that the deuteron effects are contained in the nucleonic pdfs . we show actual comparisons with the experimental data for @xmath5 by the nmc in fig . [ fig : f2dp ] . three global analysis results are shown , and the shaded areas indicate uncertainty bands in analysis 3 . the figure suggests that all the analyses are successful in explaining the data . however , as indicated in the @xmath14 reductions in the analyses 2 and 3 , it is clear that their curves are closer to the experimental data at small @xmath6 such as @xmath6=0.005 and 0.008 . there is a tendency that deviations from the data become larger as the deuteron modifications are increased . all the analysis results are more or less similar in the medium- and large-@xmath6 regions . all the curves of the analysis 1 ( @xmath200 ) and 2 ( @xmath201 ) are within the uncertainty bands of the analysis 3 ( @xmath199=free ) , although the analysis-1 curves are at the edges of the error bands at small @xmath6 . it means that all these analysis are consistent with each other and they explain the experimental data . = 1 gev@xmath10 . the solid and dashed curves are obtained by analyses 1 and 3 , respectively , and their uncertainties are shown by the shaded bands.,width=302 ] modifications of the pdfs are shown for the deuteron in fig . [ fig : npdfs - d ] at @xmath7=1 gev@xmath10 . the results of analyses 1 and 3 are shown with their uncertainty estimation . there is no deuteron modification in analysis 2 as obvious from the definition in eq . ( [ eqn : wi - d ] ) . the uncertainty bands of analysis 3 shrink at the points , where the nuclear modifications vanish ( @xmath204 ) , for example , in the figure of the valence - quark modification @xmath205 . this is caused by the error from the parameter @xmath199 . its contributions to the uncertainties are large and the derivative @xmath206 is proportional to @xmath207 , which vanishes at the same points as the function @xmath208 in eq . ( [ eqn : wi - d ] ) . it leads to the gourd - shaped uncertainty band in @xmath205 because other terms are small . such an error shape does not appear in analysis 1 because the error term of @xmath199 does not exist . here , the derivative @xmath209 is also a cubic polynomial and vanishes at three @xmath6 points . however , @xmath210 and @xmath211 are quadratic functions , which do not vanish at the same @xmath6 points , and their contributions to the uncertainties are of the same order of the @xmath212 and @xmath213 terms . this is the reason why such a gourd - shaped function does not appear in the uncertainties of the @xmath200 analysis . the antiquark shadowing is about 2% at @xmath214 and the valence - quark modification is about 1% in the analysis 1 according to fig . [ fig : npdfs - d ] . we should note that the antiquark shadowing is reduced about 30% at @xmath7=4 gev@xmath10 as shown in fig . [ fig : cad - q2 ] in comparing it with the theoretical values in table [ tab : d - modification ] . in analysis 3 , the antiquark shadowing becomes 0.5% , which could be slightly smaller than the theoretical ones . both corrections ( 2% and 0.5% ) are slightly different from the assumed correction ( 1% at @xmath6=0.01 and @xmath7=4 gev@xmath10 ) in the mrst fit . the uncertainty bands depend on the initial functional form or assignment of the parameters in the @xmath14 analysis . the uncertainties are generally larger in analysis 3 , and the line of @xmath204 is within the bands . it suggests that precise deuteron modifications can not be determined at this stage . the modifications in fig . [ fig : npdfs - d ] may not be seriously taken because the deuteron effects could be partially included in the nucleonic pdfs . it is difficult to judge what the realistic deuteron modifications are at this stage . actual modifications are possibly in between these analysis results . in order to obtain realistic modifications , the nucleonic pdfs should be determined by considering the deuteron modifications , for example , of our analysis results . then , using new nucleonic pdfs , we redetermine our nuclear pdfs including the ones in the deuteron . realistic deuteron modifications should be obtained by repeating this step . nuclear pdfs have been determined by the global analyses of experimental data for the ratios of the @xmath16 structure functions and drell - yan cross sections . the uncertainties of the determined npdfs are estimated by the hessian method . the first important point is that the uncertainties were obtained in both lo and nlo so that we can discuss the nlo improvement on the determination . we found slight nlo improvements for the antiquark and gluon distributions at small @xmath6 ( @xmath215 ) ; however , they are not significant at larger @xmath6 . accurate experimental measurements , especially on the @xmath7 dependence at small @xmath6 , should be useful for determining higher - order effects such as the nuclear gluon distributions . the valence - quark distributions are well determined . the antiquark distributions are also determined at @xmath3 ; however , they have large uncertainties at @xmath4 . the gluon modifications are not precisely determined in the whole @xmath6 region . future measurements are needed to determine accurate nuclear distributions . nuclear modifications were discussed for the deuteron in comparison with the experimental data of @xmath5 . however , it is difficult to find accurate modifications at this stage because deuteron effects could be partially contained in the nucleonic pdfs . an appropriate nucleonic pdf analysis is needed in addition to accurate measurements on the ratio @xmath5 . our npdfs and their uncertainties can be calculated by using the codes in ref . @xcite . m.h . and s.k . were supported by the grant - in - aid for scientific research from the japanese ministry of education , culture , sports , science , and technology . was supported by japan society for the promotion of science . the determined parameters are listed in table [ table : parameters ] . there are other parameters , @xmath62 , @xmath63 , and @xmath64 , which are automatically calculated from the tabulated values by the conservation conditions of nuclear charge , baryon number , and momentum in eq . ( [ eqn:3conserv ] ) . because they depend on nuclear species , namely on @xmath15 and @xmath31 , their calculation method is explained in the following . the flavor asymmetric antiquark distributions in eq . ( [ eqn : wpart ] ) are used in this analysis , whereas the flavor symmetric ones are used in the previous versions in refs . @xcite , so that relations are slightly different from the ones in appendix of ref . one can obtain values of these parameters for any nuclei by calculating the following integrals : @xmath216 , \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! & \nonumber \\ i_8 & = \int dx \frac{x}{(1-x)^{\beta_g } } g(x ) , & \end{aligned}\ ] ] where @xmath217 and @xmath218 is @xmath219 here , the parameters @xmath13 , @xmath77 , @xmath78 , and @xmath220 depend on @xmath15 according to eq . ( [ eqn : more - a ] ) . it is noteworthy that the integrals @xmath221 , @xmath222 , and @xmath223 depend on @xmath15 . using these integrals , one obtains parameter values by the conservations in eq . ( [ eqn:3conserv ] ) : @xmath224 @xmath225 @xmath226 . \label{eqn : ag}\end{aligned}\ ] ] from these values together with the parameters in table [ table : parameters ] and the nucleonic pdfs at @xmath7=1 gev@xmath10 of the mrst parametrization @xcite , one obtains the npdfs at @xmath7=1 gev@xmath10 for a given nucleus . if distributions are needed at different @xmath7 , one needs to evolve the npdfs by using own evolution code . if one does not have such an evolution code , one had better use the code in appendix b. numerical values of the npdfs can be obtained for given @xmath6 , @xmath7 , and @xmath15 . in particular , if one is interested in estimating the uncertainties of the npdfs , this practical code needs to be used . codes for calculating nuclear pdfs and their uncertainties can be obtained from the web site @xcite . a general guideline for usage is explained in appendix b of ref . @xcite , and the conditions are the same in the current version , hkn07 ( hirai , kumano , nagai in 2007 ) . therefore , the details should be found in ref . the npdfs can be calculated for nuclei at given @xmath6 and @xmath7 , for which the kinematical ranges should be in @xmath227 and 1 gev@xmath228 gev@xmath10 . input parameters for running the code are explained in the beginning of the file npdf07.f . a sample program , sample.f , is supplied as an example for calculating the nuclear pdfs and uncertainties . the uncertainties of the npdfs are estimated by using the hessian matrix and grid data for derivatives with respect to the parameters . the usage is explained in the sample program . j. adams _ et al . lett . * 97 * , 152302 ( 2006 ) ; 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nuclear parton distribution functions ( npdfs ) are determined by global analyses of experimental data on structure - function ratios @xmath0 and drell - yan cross - section ratios @xmath1 .
the analyses are done in the leading order ( lo ) and next - to - leading order ( nlo ) of running coupling constant @xmath2 .
uncertainties of the npdfs are estimated in both lo and nlo for finding possible nlo improvement .
valence - quark distributions are well determined , and antiquark distributions are also determined at @xmath3 . however , the antiquark distributions have large uncertainties at @xmath4 .
gluon modifications can not be fixed at this stage .
although the advantage of the nlo analysis , in comparison with the lo one , is generally the sensitivity to the gluon distributions , gluon uncertainties are almost the same in the lo and nlo .
it is because current scaling - violation data are not accurate enough to determine precise nuclear gluon distributions .
modifications of the pdfs in the deuteron are also discussed by including data on the proton - deuteron ratio @xmath5 in the analysis .
a code is provided for calculating the npdfs and their uncertainties at given @xmath6 and @xmath7 in the lo and nlo .
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