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|---|---|---|---|
math/0301340
|
Thirty-three new definitions are presented, derived from neutrosophic set,
neutrosophic probability, neutrosophic statistics, and neutrosophic logic. Each
one is independent, short, with references and cross references like in a
dictionary style.
|
lt256
|
arxiv_abstracts
|
math/0301341
|
Let g be a scattering metric on a compact manifold X with boundary, i.e., a
smooth metric giving the interior of X the structure of a complete Riemannian
manifold with asymptotically conic ends. An example is any compactly supported
perturbation of the standard metric on Euclidean space. Consider the operator
$H = \half \Delta + V$, where $\Delta$ is the positive Laplacian with respect
to g and V is a smooth real-valued function on X vanishing to second order at
the boundary. Assuming that g is non-trapping, we construct a global parametrix
for the kernel of the Schroedinger propagator $U(t) = e^{-itH}$ and use this to
show that the kernel of U(t) is, up to an explicit quadratic oscillatory
factor, a class of `Legendre distributions' on $X \times X^{\circ} \times
\halfline$ previously considered by Hassell-Vasy. When the metric is trapping,
then the parametrix construction goes through microlocally in the non-trapping
part of the phase space.
We apply this result to obtain a microlocal characterization of the
singularities of $U(t) f$, for any tempered distribution $f$ and any fixed $t
\neq 0$, in terms of the oscillation of f near the boundary of X. If the metric
is non-trapping, then we obtain a complete characterization; more generally we
need to assume that f is microsupported in the nontrapping part of the phase
space. This generalizes results of Craig-Kappeler-Strauss and Wunsch.
|
256
|
arxiv_abstracts
|
math/0301342
|
In this article we describe three constructions of complex variations of
Hodge structure, proving the existence of interesting opposite filtrations that
generalize a construction of Deligne. We also analyze the relation between
deformations of Frobenius modules and certain maximally degenerate variations
of Hodge structures. Finally, under a certain generation hypothesis, we show
how to construct a Frobenius manifold starting from a deformation of a
Frobenius module.
|
lt256
|
arxiv_abstracts
|
math/0301343
|
Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If
$|F|^\delta < |A| < |F|^{1-\delta}$ for some $\delta > 0$, then we prove the
estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps =
\eps(\delta) > 0$. This is a finite field analogue of a result of Erdos and
Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem
in finite fields, and obtain a new estimate for the Erdos distance problem in
finite fields, as well as the three-dimensional Kakeya problem in finite
fields.
|
lt256
|
arxiv_abstracts
|
math/0301344
|
We study the equation E_fc of flat connections in a fiber bundle and discover
a specific geometric structure on it, which we call a flat representation. We
generalize this notion to arbitrary PDE and prove that flat representations of
an equation E are in 1-1 correspondence with morphisms f: E\to E_fc, where E
and E_fc are treated as submanifolds of infinite jet spaces. We show that flat
representations include several known types of zero-curvature formulations of
PDE. In particular, the Lax pairs of the self-dual Yang-Mills equations and
their reductions are of this type. With each flat representation we associate a
complex C_f of vector-valued differential forms such that its first cohomology
describes infinitesimal deformations of the flat structure, which are
responsible, in particular, for parameters in Backlund transformations. In
addition, each higher infinitesimal symmetry S of E defines a 1-cocycle c_S of
C_f. Symmetries with exact c_S form a subalgebra reflecting some geometric
properties of E and f. We show that the complex corresponding to E_fc itself is
0-acyclic and 1-acyclic (independently of the bundle topology), which means
that higher symmetries of E_fc are exhausted by generalized gauge ones, and
compute the bracket on 0-cochains induced by commutation of symmetries.
|
256
|
arxiv_abstracts
|
math/0301345
|
To any bimodule which is finitely generated and projective on one side one
can associate a coring, known as a comatrix coring. A new description of
comatrix corings in terms of data reminiscent of a Morita context is given. It
is also studied how properties of bimodules are reflected in the associated
comatrix corings. In particular it is shown that separable bimodules give rise
to coseparable comatrix corings, while Frobenius bimodules induce Frobenius
comatrix corings.
|
lt256
|
arxiv_abstracts
|
math/0301346
|
In this paper we give necessary and sufficient conditions for discreteness of
a group generated by a hyperbolic element and an elliptic one of odd order.
This completes the classification of discrete groups with non-$\pi$-loxodromic
generators in the class of two-generator groups with real parameters. The
criterion is given also as a list of all parameters that correspond to discrete
groups. An interesting corollary of the result is that the group of the minimal
known volume hyperbolic orbifold has real parameters.
|
lt256
|
arxiv_abstracts
|
math/0301347
|
We investigate how to compare Hochschild cohomology of algebras related by a
Morita context. Interpreting a Morita context as a ring with distinguished
idempotent, the key ingredient for such a comparison is shown to be the grade
of the Morita defect, the quotient of the ring modulo the ideal generated by
the idempotent. Along the way, we show that the grade of the stable
endomorphism ring as a module over the endomorphism ring controls vanishing of
higher groups of selfextensions, and explain the relation to various forms of
the Generalized Nakayama Conjecture for Noetherian algebras. As applications of
our approach we explore to what extent Hochschild cohomology of an invariant
ring coincides with the invariants of the Hochschild cohomology.
|
lt256
|
arxiv_abstracts
|
math/0301348
|
We first show that co-amenability does not pass to subgroups, answering a
question asked by Eymard in 1972. We then address co-amenability for von
Neumann algebras, describing notably how it relates to the former.
|
lt256
|
arxiv_abstracts
|
math/0301349
|
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian
manifold with piecewise smooth boundary and suppose that the billiard
associated to the geodesic flow reflecting on the boundary acording to the laws
of geometric optics is ergodic. We prove that the boundary value of the
eigenfunctions of the Laplace operator with reasonable boundary conditions are
asymptotically equidistributed in the boundary, extending previous results by
G\'erard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained
under the additional assumption of the convexity of $M$.
|
lt256
|
arxiv_abstracts
|
math/0301350
|
We present a conformal deformation involving a fully nonlinear equation in
dimension 4, starting with positive scalar curvature. Assuming a certain
conformal invariant is positive, one may deform from positive scalar curvature
to a stronger condition involving the Ricci tensor. We also give a new
conformally invariant condition for positivity of the Paneitz operator, which
allows us to give many new examples of manifolds admitting metrics with
constant $Q$-curvature.
|
lt256
|
arxiv_abstracts
|
math/0301351
|
The paper considers (a) Representations of measure preserving transformations
(``rotations'') on Wiener space, and (b) The stochastic calculus of variations
induced by parameterized rotations $\{T_\theta w, 0 \le \theta \le \eps\}$:
``Directional derivatives'' $(dF(T_\theta w)/d \theta)_{\theta=0}$, ``vector
fields'' or ``tangent processes'' $(dT_\theta w /d\theta)_{\theta=0}$ and flows
of rotations.
|
lt256
|
arxiv_abstracts
|
math/0301352
|
We give some necessary conditions and sufficient conditions for the
compactness of the embedding of Sobolev spaces $W^{1,p}(\Omega,w) \to
L^p(\Omega,w),$ where $w$ is some weight on a domain $\Omega \subset \Real^n$.
|
lt256
|
arxiv_abstracts
|
math/0301353
|
Rings form a bicategory [Rings], with classes of bimodules as horizontal
arrows, and bimodule maps as vertical arrows. The notion of Morita equivalence
for rings can be translated in terms of bicategories in the following way. Two
rings are Morita equivalent if and only if they are isomorphic objects in the
bicategory. We repeat this construction for von Neumann algebras. Von Neumann
algebras form a bicategory [W*], with classes of correspondences as horizontal
arrows, and intertwiners as vertical arrows. Two von Neumann algebras are
Morita equivalent if and only if they are isomorphic objects in the bicategory
[W*].
|
lt256
|
arxiv_abstracts
|
math/0301354
|
We study cubical sets without degeneracies, which we call square sets. These
sets arise naturally in a number of settings and they have a beautiful
intrinsic geometry; in particular a square set C has an infinite family of
associated square sets J^i(C), i=1,2,..., which we call James complexes. There
are mock bundle projections p_i:|J^i(C)|-->|C| (which we call James bundles)
defining classes in unstable cohomotopy which generalise the classical
James--Hopf invariants of Omega(S^2). The algebra of these classes mimics the
algebra of the cohomotopy of Omega(S^2) and the reduction to cohomology defines
a sequence of natural characteristic classes for a square set. An associated
map to BO leads to a generalised cohomology theory with geometric
interpretation similar to that for Mahowald orientation [M Mahowald, Ring
Spectra which are Thom complexes, Duke Math. J. 46 (1979) 549--559] and [B
Sanderson, The geometry of Mahowald orientations, SLN 763 (1978) 152--174].
|
lt256
|
arxiv_abstracts
|
math/0301355
|
Given n polynomials in n variables of respective degrees d_1,...,d_n, and a
set of monomials of cardinality d_1...d_n, we give an explicit
subresultant-based polynomial expression in the coefficients of the input
polynomials whose non-vanishing is a necessary and sufficient condition for
this set of monomials to be a basis of the ring of polynomials in n variables
modulo the ideal generated by the system of polynomials. This approach allows
us to clarify the algorithms for the Bezout construction of the resultant.
|
lt256
|
arxiv_abstracts
|
math/0301356
|
This is the third of three papers about the Compression Theorem: if M^m is
embedded in Q^q X R with a normal vector field and if q-m > 0, then the given
vector field can be straightened (ie, made parallel to the given R direction)
by an isotopy of M and normal field in Q X R. The theorem can be deduced from
Gromov's theorem on directed embeddings [M Gromov, Partial differential
relations, Springer--Verlag (1986) 2.4.5 C'] and the first two parts
(math.GT/9712235 and math.GT/0003026) gave proofs. Here we are concerned with
applications. We give short new (and constructive) proofs for immersion theory
and for the loops--suspension theorem of James et al and a new approach to
classifying embeddings of manifolds in codimension one or more, which leads to
theoretical solutions. We also consider the general problem of controlling the
singularities of a smooth projection up to C^0--small isotopy and give a
theoretical solution in the codimension > 0 case.
|
lt256
|
arxiv_abstracts
|
math/0301357
|
We first show that a Laplace isospectral family of Riemannian orbifolds,
satisfying a lower Ricci curvature bound, contains orbifolds with points of
only finitely many isotropy types. If we restrict our attention to orbifolds
with only isolated singularities, and assume a lower sectional curvature bound,
then the number of singular points in an orbifold in such an isospectral family
is universally bounded above. These proofs employ spectral theory methods of
Brooks, Perry and Petersen, as well as comparison geometry techniques developed
by Grove and Petersen.
|
lt256
|
arxiv_abstracts
|
math/0301358
|
This paper deals with the Nash problem, which claims that there are as many
families of arcs on a singular germ of surface $U$ as there are essential
components of the exceptional divisor in the desingularisation of this
singularity. Let $\mathcal{H}=\bigcup \bar{N_\alpha}$ be a particular
decomposition of the set of arcs on $U$, described later on. We give two new
conditions to insure that $\bar{N_\alpha}\not \subset \bar{N_\beta}$, $\alpha
\not = \beta$; more precisely,for the first one, we claim that if there exists
$f \in {\mathcal{O}}_{U}$ such that $ord_{E_\alpha}(f)<ord_{E_\beta}(f)$, where
$E_\alpha, E_\beta$ are exceptional divisors of the desingularisation, then
$\bar{N_\alpha}\not \subset \bar{N_\beta}$. The second condition, used when the
singularity is rational and of surface, is the following:let $(S,s)$ et
$(S',s')$ be two rational surface singularities so that there exist a dominant
and birational morphism $\pi$ from $(S,s)$ to $(S',s')$;then,let $E_\alpha,
E_\beta$ be two essential components of the exceptional divisors in the minimal
desingularisation of $(S,s)$, such that their image by $\pi$, $E'_\alpha$ and
$E'_\beta$, are exceptional divisor for $(S',s')$; if
$\bar{N'_\alpha}(S',s')\not \subset \bar{N'_\beta}(S',s')$ then
$\bar{N_\alpha}(S,s)\not \subset \bar{N_\beta}(S,s)$. These two conditions are
simple, but it allows us to prove quite directly the conjecture for the
rational minimal surface singularities, using the decomposition of minimal
suface singularities into cyclic quotient singularities of type $A_n$. A proof
of the conjecture for these singularities has already been given by Ana
Reguera.
|
256
|
arxiv_abstracts
|
math/0301359
|
We prove the Green conjecture for generic curves of odd genus. That is we
prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The
curves we consider are smooth curves $X$ on a K3 surface whose Picard group has
rank 2. This completes our previous work, where the Green conjecture for
generic curves of genus $g$ with fixed gonality $d$ was proved in the range
$d\geq g/3$, with the possible exception of the generic curves of odd genus.
|
lt256
|
arxiv_abstracts
|
math/0301360
|
We study the dynamics of $N$ point vortices on a rotating sphere. The
Hamiltonian system becomes infinite dimensional due to the non-uniform
background vorticity coming from the Coriolis force. We prove that a relative
equilibrium formed of latitudinal rings of identical vortices for the
non-rotating sphere persists to be a relative equilibrium when the sphere
rotates. We study the nonlinear stability of a polygon of planar point vortices
on a rotating plane in order to approximate the corresponding relative
equilibrium on the rotating sphere when the ring is close to the pole. We then
perform the same study for geostrophic vortices. To end, we compare our results
to the observations on the southern hemisphere atmospheric circulation.
|
lt256
|
arxiv_abstracts
|
math/0301361
|
Some forms of qKdV type equations are indicated which arise from Virasoro
considerations.
|
lt256
|
arxiv_abstracts
|
math/0301362
|
In this paper we study algebraic supergroups and their coadjoint orbits as
affine algebraic supervarieties. We find an algebraic deformation quantization
of them that can be related to the fuzzy spaces of non commutative geometry.
|
lt256
|
arxiv_abstracts
|
math/0301363
|
The jackknife variance estimator and the the infinitesimal jackknife variance
estimator are shown to be asymptotically equivalent if the functional of
interest is a smooth function of the mean or a trimmed L-statistic with Hoelder
continuous weight function.
|
lt256
|
arxiv_abstracts
|
math/0301364
|
We extend the problem of finding Hamiltonian-invariant volume forms on a
Poisson manifold to the problem of construction of Hamiltonian-invariant
generalized functions. For this we introduce the notion of generalized center
of a Poisson algebra, which is the space of generalized Casimir functions. We
study as the case when the set of test-objects for generalized functions is the
space of compactly supported smooth functions, so the case when the
test-objects are n-forms, where n is the dimension of the Poisson manifold. We
describe the relations of this problem with the homological properties of the
Poisson structure, with Bott connection for the corresponding symplectic
foliation and the modular class.
|
lt256
|
arxiv_abstracts
|
math/0301365
|
We consider partitions of a set with $r$ elements ordered by refinement. We
consider the simplicial complex $\bar{K}(r)$ formed by chains of partitions
which starts at the smallest element and ends at the largest element of the
partition poset. A classical theorem asserts that $\bar{K}(r)$ is equivalent to
a wedge of $r-1$-dimensional spheres. In addition, the poset of partitions is
equipped with a natural action of the symmetric group in $r$ letters.
Consequently, the associated homology modules are representations of the
symmetric groups. One observes that the $r-1$th homology modules of
$\bar{K}(r)$, where $r = 1,2,...$, are dual to the Lie representation of the
symmetric groups.
In this article, we would like to point out that this theorem occurs a
by-product of the theory of \emph{Koszul operads}. For that purpose, we improve
results of V. Ginzburg and M. Kapranov in several directions. More
particularly, we extend the Koszul duality of operads to operads defined over a
field of positive characteristic (or over a ring). In addition, we obtain more
conceptual proofs of theorems of V. Ginzburg and M. Kapranov.
|
256
|
arxiv_abstracts
|
math/0301366
|
Two algebroid branches are said to be equivalent if they have the same
multiplicity sequence. It is known that two algebroid branches $R$ and $T$ are
equivalent if and only if their Arf closures, $R'$ and $T'$ have the same value
semigroup, which is an Arf numerical semigroup and can be expressed in terms of
a finite set of information, a set of characters of the branch.
We extend the above equivalence to algebroid curves with $d>1$ branches. An
equivalence class is described, in this more general context, by an Arf
semigroup, that is not a numerical semigroup, but is a subsemigroup of $\mathbb
N^d$. We express this semigroup in terms of a finite set of information, a set
of characters of the curve, and apply this result to determine other curves
equivalent to a given one.
|
lt256
|
arxiv_abstracts
|
math/0301367
|
We study two questions posed by Johnson, Lindenstrauss, Preiss, and
Schechtman, concerning the structure of level sets of uniform and Lipschitz
quotient maps from $R^n\to R$. We show that if $f:R^n\to R$, $n\geq 2$, is a
uniform quotient map then for every $t\in R$, $f^{-1}(t)$ has a bounded number
of components, each component of $f^{-1}(t)$ separates $R^n$ and the upper
bound of the number of components depends only on $n$ and the moduli of
co-uniform and uniform continuity of $f$. Next we obtain a characterization of
the form of any closed, hereditarily locally connected, locally compact,
connected set with no end points and containing no simple closed curve, and we
apply it to describe the structure of level sets of co-Lipschitz uniformly
continuous mappings $f:R^2\to R$. We prove that all level sets of any
co-Lipschitz uniformly continuous map from $R^2$ to $R$ are locally connected,
and we show that for every pair of a constant $c>0$ and a function $\Omega$
with $\lim_{r\to 0}\Omega(r)=0$, there exists a natural number $M=M(c,\Omega)$,
so that for every co-Lipschitz uniformly continuous map $f:R^2\to R$ with a
co-Lipschitz constant $c$ and a modulus of uniform continuity $\Omega$, there
exists a natural number $n(f)\le M$ and a finite set $T_f\subset R$ with
$\card(T_f)\leq n(f)-1$ so that for all $t\in R\setminus T_f$, $f^{-1}(t)$ has
exactly $n(f)$ components, $R^2\setminus f^{-1}(t)$ has exactly $n(f)+1$
components and each component of $f^{-1}(t)$ is homeomorphic with the real line
and separates the plane into exactly 2 components. The number and form of
components of $f^{-1}(s)$ for $s\in T_f$ are also described - they have a
finite graph structure. We give an example of a uniform quotient map from
$R^2\to R$ which has non-locally connected level sets.
|
256
|
arxiv_abstracts
|
math/0301368
|
In this note we extend duality theorems for crossed products obtained by M.
Koppinen and C. Chen from the case of a base field or a Dedekind domain to the
case of an arbitrary noetherian commutative ground ring under fairly weak
conditions. In particular we extend an improved version of the celebrated
Blattner-Montgomery duality theorem to the case of arbitrary noetherian ground
rings.
|
lt256
|
arxiv_abstracts
|
math/0301369
|
We study $n$ dimensional Riemanniann manifolds with harmonic forms of
constant length and first Betti number equal to $n-1$ showing that they are
2-steps nilmanifolds with some special metrics. We also characterise, in terms
of properties on the product of harmonic forms, the left invariant metrics
among them. This allows us to clarify the case of equality in the stable
isosytolic inequalities in that setting. We also discuss other values of the
Betti number.
|
lt256
|
arxiv_abstracts
|
math/0301370
|
Soit l un entier et E_{c,l} la famille de Kubert des courbes elliptiques
definies sur Q munies d'un point rationnel A d'ordre l. On note F_{c,l} la
courbe elliptique quotient de E_{c,l} par le groupe engendre' par A, et f_l
l'isogenie de E_{c,l} sur F_{c,l}. Pour l = 3, 4, 5 et 6, nous construisons
explicitement, pour des parametrisations convenables de c, des elements
non-triviaux de F_{c,l}(Q) / (f_l(E_{c,l}(Q))), autrement dit, des points
explicites de F_{c,l}(Q) qui ne sont l'image par f_l d'aucun element de
E_{c,l}(Q). Ces points sont en general d'ordre infini. Nous donnons des
applications de cette methode a la construction d'extensions cycliques de Q de
degre' l, et retrouvons certains corps obtenus par Shanks et Gras. Dans un
article ulterieur, nous etudierons les proprietes arithmetiques de certaines
des extensions obtenues ici.
|
lt256
|
arxiv_abstracts
|
math/0301371
|
We prove that the homotopy class of a Morin mapping f: P^p --> Q^q with p-q
odd contains a cusp mapping. This affirmatively solves a strengthened version
of the Chess conjecture [DS Chess, A note on the classes [S_1^k(f)], Proc.
Symp. Pure Math., 40 (1983) 221-224] and [VI Arnol'd, VA Vasil'ev, VV Goryunov,
OV Lyashenko, Dynamical systems VI. Singularities, local and global theory,
Encyclopedia of Mathematical Sciences - Vol. 6 (Springer, Berlin, 1993)]. Also,
in view of the Saeki-Sakuma theorem [O Saeki, K Sakuma, Maps with only Morin
singularities and the Hopf invariant one problem, Math. Proc. Camb. Phil. Soc.
124 (1998) 501-511] on the Hopf invariant one problem and Morin mappings, this
implies that a manifold P^p with odd Euler characteristic does not admit Morin
mappings into R^{2k+1} for p > 2k not equal to 1,3 or 7.
|
lt256
|
arxiv_abstracts
|
math/0301372
|
Each labeled rooted tree is associated with a hyperplane arrangement, which
is free with exponents given by the depths of the vertices of this tree. The
intersection lattices of these arrangements are described through posets of
forests. These posets are used to define coalgebras, whose dual algebras are
shown to have a simple presentation by generators and relations.
|
lt256
|
arxiv_abstracts
|
math/0301373
|
Let $M$ be a closed symplectic manifold and suppose $M\to P\to B$ is a
Hamiltonian fibration. Lalonde and McDuff raised the question whether one
always has $H^*(P;\mathbb Q)=H^*(M;\mathbb Q)\otimes H^*(B;\mathbb Q)$ as
vector spaces. This is known as the c--splitting conjecture. They showed, that
this indeed holds whenever the base is a sphere. Using their theorem we will
prove the c--splitting conjecture for arbitrary base $B$ and fibers $M$ which
satisfy a weakening of the Hard Lefschetz condition.
|
lt256
|
arxiv_abstracts
|
math/0301374
|
Let $G$ be a connected semi-simple group defined over and algebraically
closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set
of simple reflections associated to the simple positive roots corresponding to
$(T,B)$, and let ${\cal B}\cong G/B$ denote the Borel variety. For any $s_i\in
S$, $1\leq i\leq n$, let $\bar{O}(s_1,..., s_n)= \{(B_0,..., B_{n})\in {\cal
B}^{n+1} | (B_{i-1},B_{i})\in \bar{O(s_i)}, 1\leq i\leq n\}$, where $O(s)$
denotes the subvariety of pairs of Borels in ${\cal B}^2$ in relative position
$s$. We show that such varieties are smooth and indicate why this result is, in
one sense, best possible. Our main results assume that $k$ has characteristic
0.
|
lt256
|
arxiv_abstracts
|
math/0301375
|
We associate a cohomological invariant to each outer action of a group on a
factor, and classify them by the invariant in the case that the group is a
countable discrete amenable group and the factor is appoximately finite
dimensional. The invariant defined for the group Out(M)=Aut(M)/Int(M) is called
the intrinsic modular obstruction. The invariant for an outer action alpha is
given as the pull back of the intrinsic modular obstruction, which is called
the modular obstruction of alpha and denoted by Ob_m(alpha). This is the first
part of the theory and presents general theory. In the case that the factor is
not of type III_0, the invariant is substantially simplified. These cases and
examples will be discussed in forthcoming paper.
|
lt256
|
arxiv_abstracts
|
math/0301376
|
We generalize Sunada's method to produce new examples of closed, locally
non-isometric manifolds which are isospectral. In particular, we produce pairs
of isospectral, simply-connected, locally non-isometric normal homogeneous
spaces. These pairs also allow us to see that in general group actions with
discrete spectra are not determined up to measurable conjugacy by their
spectra. In particular, we show this for lattice actions.
|
lt256
|
arxiv_abstracts
|
math/0301377
|
Let $(1) Rh=f$, $0\leq x\leq L$, $Rh=\int^L_0 R(x,y)h(y) dy$, where the
kernel $R(x,y)$ satisfies the equation $QR=P\delta(x-y)$. Here $Q$ and $P$ are
formal differential operators of order $n$ and $m<n$, respectively, $n$ and $m$
are nonnegative even integers, $n>0$, $m\geq 0$, $Qu:=q_n(x)u^{(n)} +
\sum^{n-1}_{j=0} q_j(x) u^{(j)}$, $Ph:=h^{(m)} +\sum^{m-1}_{j=0} p_j(x)
h^{(j)}$, $q_n(x)\geq c>0$, the coefficients $q_j(x)$ and $p_j(x)$ are smooth
functions defined on $\R$, $\delta(x)$ is the delta-function, $f\in
H^\alpha(0,L)$, given. Here $\dot H^{-\alpha}(0,L)$ is the dual space to
$H^\alpha(0,L)$ with respect to the inner product of $L^2(0,L)$.
Under suitable assumptions it is proved that $R:\dot H^{-\alpha}(0,L) \to
H^\alpha(0,L)$ is an isomorphism.
Equation (1) is the basic equation of random processes estimation theory.
Some of the results are generalized to the case of multidimensional equation
(1), in which case this is the basic equation of random fields estimation
theory.
$\alpha:=\frac{n-m}{2}$, $H^\alpha$ is the Sobolev space.
An algorithm for finding analytically the unique solution $h\in\dot
H^{-\alpha} (0,L)$ to (1) of minimal order of singularity is
|
256
|
arxiv_abstracts
|
math/0301378
|
Consider an operator equation $F(u)=0$ in a real Hilbert space.
The problem of solving this equation is ill-posed if the operator $F'(u)$ is
not boundedly invertible, and well-posed otherwise.
A general method, dynamical systems method (DSM) for solving linear and
nonlinear ill-posed problems in a Hilbert space is presented.
This method consists of the construction of a nonlinear dynamical system,
that is, a Cauchy problem, which has the following properties:
1) it has a global solution,
2) this solution tends to a limit as time tends to infinity,
3) the limit solves the original linear or non-linear problem. New
convergence and discretization theorems are obtained. Examples of the
applications of this approach are given. The method works for a wide range of
well-posed problems as well.
|
lt256
|
arxiv_abstracts
|
math/0301379
|
A new understanding of the notion of regularizer is proposed. It is argued
that this new notion is more realistic than the old one and better fits the
practical computational needs. An example of the regularizer in the new sense
is given. A method for constructing regularizers in the new sense is proposed
and justified.
|
lt256
|
arxiv_abstracts
|
math/0301380
|
Several questions of approximation theory are discussed: 1) can one
approximate stably in $L^\infty$ norm $f^\prime$ given approximation $f_\delta,
\parallel f_\delta - f \parallel_{L^\infty} < \delta$, of an unknown smooth
function $f(x)$, such that $\parallel f^\prime (x) \parallel_{L^\infty} \leq
m_1$?
2) can one approximate an arbitrary $f \in L^2(D), D \subset \R^n, n \geq 3$,
is a bounded domain, by linear combinations of the products $u_1 u_2$, where
$u_m \in N(L_m), m=1,2,$ $L_m$ is a formal linear partial differential operator
and $N(L_m)$ is the null-space of $L_m$ in $D$, $3) can one approximate an
arbitrary $L^2(D)$ function by an entire function of exponential type whose
Fourier transform has support in an arbitrary small open set? Is there an
analytic formula for such an approximation? N(L_m) := \{w: L_m w=0 \hbox{in\}
D\}$?
|
lt256
|
arxiv_abstracts
|
math/0301381
|
Let $F$ be a nonlinear Frechet differentiable map in a real Hilbert space.
Condition sufficient for existence of a solution to the equation $F(u)=0$ is
given, and a method (dynamical systems method, DSM) to calculate the solution
as the limit of the solution to a Cauchy problem is justified under suitable
assumptions.
|
lt256
|
arxiv_abstracts
|
math/0301382
|
Several methods for solving efficiently the one-dimensional deconvolution
problem are proposed. The problem is to solve the Volterra equation ${\mathbf
k} u:=\int_0^t k(t-s)u(s)ds=g(t),\quad 0\leq t\leq T$. The data, $g(t)$, are
noisy. Of special practical interest is the case when the data are noisy and
known at a discrete set of times. A general approach to the deconvolution
problem is proposed: represent ${\mathbf k}=A(I+S)$, where a method for a
stable inversion of $A$ is known, $S$ is a compact operator, and $I+S$ is
injective. This method is illustrated by examples: smooth kernels $k(t)$, and
weakly singular kernels, corresponding to Abel-type of integral equations, are
considered. A recursive estimation scheme for solving deconvolution problem
with noisy discrete data is justified mathematically, its convergence is
proved, and error estimates are obtained for the proposed deconvolution method.
|
lt256
|
arxiv_abstracts
|
math/0301383
|
Inequalities for the transformation operator kernel $A(x,y)$ in terms of
$F$-function are given, and vice versa. These inequalities are applied to
inverse scattering on half-line. Characterization of the scattering data
corresponding to the usual scattering class $L_{1,1}$ of the potentials, to the
class of compactly supported potentials, and to the class of square integrable
potentials is given. Invertibility of each of the steps in the inversion
procedure is proved.
|
lt256
|
arxiv_abstracts
|
math/0301384
|
The Newton-Sabatier method for solving inverse scattering problem with
fixed-energy phase shifts for a sperically symmetric potential is discussed. It
is shown that this method is fundamentally wrong: in general it cannot be
carried through, the basic ansatz of R.Newton is wrong: the transformation
kernel does not have the form postulated in this ansatz, in general, the method
is inconsistent, and some of the physical conclusions, e.g., existence of the
transparent potentials, are not proved. A mathematically justified method for
solving the three-dimensional inverse scattering problem with fixed-energy data
is described. This method is developed by A.G.Ramm for exact data and for noisy
discrete data, and error estimates for this method are obtained. Difficulties
of the numerical implementation of the inversion method based on the
Dirichlet-to-Neumann map are pointed out and compared with the difficulty of
the implementation of the Ramm's inversion method.
|
lt256
|
arxiv_abstracts
|
math/0301385
|
This paper is a continuation of the paper (A.G.Ramm, Amer. Math. Monthly,
108, N 9, (2001), 855-860), where bounded Fredholm operators are studied. The
theory of bounded linear Fredholm-type operators is presented in many texts.
This paper is written for a broad audience and the author tries to give simple
and short arguments.
|
lt256
|
arxiv_abstracts
|
math/0301386
|
Cubature formulas, asymptotically optimal with respect to accuracy, are
derived for calculating multidimensional weakly singular integrals. They are
used for developing a universal code for calculating capacitances of conductors
of arbitrary shapes.
|
lt256
|
arxiv_abstracts
|
math/0301387
|
Let L/F be a dihedral extension of degree 2p, where p is an odd prime. Let
K/F and k/F be subextensions of L/F with degrees p and 2, respectively. Then we
will study relations between the p-ranks of the class groups Cl(K) and Cl(k).
|
lt256
|
arxiv_abstracts
|
math/0302001
|
Assume that $$ Au=f,\quad (1) $$ is a solvable linear equation in a Hilbert
space, $||A||<\infty$, and $R(A)$ is not closed, so problem (1) is ill-posed.
Here $R(A)$ is the range of the linear operator $A$. A DSM (dynamical systems
method) for solving (1), consists of solving the following Cauchy problem: $$
\dot u= -u +(B+\ep(t))^{-1}A^*f, \quad u(0)=u_0, \quad (2) $$ where $B:=A^*A$,
$\dot u:=\frac {du}{dt}$, $u_0$ is arbitrary, and $\ep(t)>0$ is a continuously
differentiable function, monotonically decaying to zero as $t\to \infty$.
A.G.Ramm has proved that, for any $u_0$, problem (2) has a unique solution for
all $t>0$, there exists $y:=w(\infty):=\lim_{t\to \infty}u(t)$, $Ay=f$, and $y$
is the unique minimal-norm solution to (1). If $f_\d$ is given, such that
$||f-f_\d||\leq \d$, then $u_\d(t)$ is defined as the solution to (2) with $f$
replaced by $f_\d$. The stopping time is defined as a number $t_\d$such that
$\lim_{\d \to 0}||u_\d (t_\d)-y||=0$, and $\lim_{\d \to 0}t_\d=\infty$. A
discrepancy principle is proposed and proved in this paper. This principle
yields $t_\d$ as the unique solution to the equation: $$
||A(B+\ep(t))^{-1}A^*f_\d -f_\d||=\d, \quad (3) $$ where it is assumed that
$||f_\d||>\d$ and $f_\d\perp N(A^*)$. For nonlinear monotone $A$ a discrepancy
principle is formulated and justified.
|
256
|
arxiv_abstracts
|
math/0302002
|
A nonlinear operator equation $F(x)=0$, $F:H\to H,$ in a Hilbert space is
considered. Continuous Newton's-type procedures based on a construction of a
dynamical system with the trajectory starting at some initial point $x_0$ and
becoming asymptotically close to a solution of $F(x)=0$ as $t\to +\infty$ are
discussed. Well-posed and ill-posed problems are investigated.
|
lt256
|
arxiv_abstracts
|
math/0302003
|
We give a definition of a noncommutative torsor by a subset of the axioms
previously given by Grunspan. We show that noncommutative torsors are an
equivalent description of Hopf-Galois objects (without specifying the Hopf
algebra). In particular, this shows that the endomorphism of a torsor featuring
in Grunspan's definition is redundant.
|
lt256
|
arxiv_abstracts
|
math/0302004
|
We obtain Gaussian upper and lower bounds on the transition density q_t(x,y)
of the continuous time simple random walk on a supercritical percolation
cluster C_{\infty} in the Euclidean lattice. The bounds, analogous to Aronsen's
bounds for uniformly elliptic divergence form diffusions, hold with constants
c_i depending only on p (the percolation probability) and d. The irregular
nature of the medium means that the bound for q_t(x,\cdot) holds only for t\ge
S_x(\omega), where the constant S_x(\omega) depends on the percolation
configuration \omega.
|
lt256
|
arxiv_abstracts
|
math/0302005
|
We show that for every morphism f between nonsingular hypersurfaces of
dimension at least 3 and of general type in projective space, there is an
everywhere defined endomorphism F of projective space that restricts to f.
As a corollary, we see that if X,Y are nonsingular hypersurfaces of general
type of dimension at least 3 such that there is a nonconstant morphism f from X
to Y, then degY divides degX with quotient q, and moreover the endomorphism F
of projective space is given by polynomials of degree q.
|
lt256
|
arxiv_abstracts
|
math/0302006
|
We show that every morphism from a degree 5 hypersurface in 4-dimensional
projective space to a nonsingular degree 3 hypersurface in 4-dimensional
projective space is necessarily constant. In the process, we also classify
morphisms from the projective plane to nonsingular cubic threefolds given by
degree 3 polynomials.
|
lt256
|
arxiv_abstracts
|
math/0302007
|
In this paper we construct abelian extensions of the group of diffeomorphisms
of a torus. We consider the jacobian map, which is a crossed homomorphism from
the group of diffeomorphisms into a toroidal gauge group. A pull-back under
this map of a central 2-cocycle on a gauge group turns out to be an abelian
cocycle on the group of diffeomorphisms. We show that in the case of a circle,
the Virasoro-Bott cocycle is a pull-back of the Heisenberg cocycle. We also
give an abelian generalization of the Virasoro-Bott cocycle to the case of a
manifold with a volume form.
|
lt256
|
arxiv_abstracts
|
math/0302008
|
In this note we study Morita contexts and Galois extensions for corings. For
a coring $\QTR{cal}{C}$ over a (not necessarily commutative) ground ring $A$ we
give equivalent conditions for $\QTR{cal}{M}^{\QTR{cal}{C}}$ to satisfy the
weak. resp. the strong structure theorem. We also characterize the so called
\QTR{em}{cleft}$C$\QTR{em}{-Galois extensions} over commutative rings. Our
approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft)
$H$-Galois extensions.
|
lt256
|
arxiv_abstracts
|
math/0302009
|
The Hanna Neumann conjecture states that if F is a free group, then for all
nontrivial finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <=
[rank(H)-1] [rank(K)-1]. Where most papers to date have considered a direct
graph theoretic interpretation of the conjecture, here we consider the use of
monomorphisms. We illustrate the effectiveness of this approach with two
results. First, we show that for any finitely generated groups H,K <= F either
the pair H,K or the pair H^{-}, K satisfy the Hanna Neumann conjecture--Here
{-} denotes the automorphism which sends each generator of F to its inverse.
Next, using particular monomorphisms from F to F_2, we obtain that if the Hanna
Neumann conjecture is false then there is a counterexample H,K < F_2 having the
additional property that all the branch vertices in the foldings of H and K are
of degree 3, and all degree 3 vertices have the same local structure or
``type''.
|
lt256
|
arxiv_abstracts
|
math/0302010
|
We obtain new omega results for the error terms in two classical lattice
point problems. These results are likely to be the best possible.
|
lt256
|
arxiv_abstracts
|
math/0302011
|
Functions of several quaternion variables are investigated and integral
representation theorems for them are proved. With the help of them solutions of
the $\tilde \partial $-equations are studied. Moreover, quaternion Stein
manifolds are defined and investigated.
|
lt256
|
arxiv_abstracts
|
math/0302012
|
We consider a two-dimensional convection model augmented with the rotational
Coriolis forcing, $U_t + U\cdot\nabla_x U = 2k U^\perp$, with a fixed $2k$
being the inverse Rossby number. We ask whether the action of dispersive
rotational forcing alone, $U^\perp$, prevents the generic finite time breakdown
of the free nonlinear convection. The answer provided in this work is a
conditional yes. Namely, we show that the rotating Euler equations admit global
smooth solutions for a subset of generic initial configurations. With other
configurations, however, finite time breakdown of solutions may and actually
does occur. Thus, global regularity depends on whether the initial
configuration crosses an intrinsic, ${\mathcal O}(1)$ critical threshold, which
is quantified in terms of the initial vorticity, $\omega_0=\nabla \times U_0$,
and the initial spectral gap associated with the $2\times 2$ initial velocity
gradient, $\eta_0:=\lambda_2(0)-\lambda_1(0), \lambda_j(0)= \lambda_j(\nabla
U_0)$. Specifically, global regularity of the rotational Euler equation is
ensured if and only if $4k \omega_0(\alpha) +\eta^2_0(\alpha) <4k^2, \forall
\alpha \in \R^2$ . We also prove that the velocity field remains smooth if and
only if it is periodic. We observe yet another remarkable periodic behavior
exhibited by the {\em gradient} of the velocity field. The spectral dynamics of
the Eulerian formulation reveals that the vorticity and the eigenvalues (and
hence the divergence) of the flow evolve with their own path-dependent period.
We conclude with a kinetic formulation of the rotating Euler equation.
|
256
|
arxiv_abstracts
|
math/0302013
|
We consider a two-dimensional weakly dissipative dynamical system with
time-periodic drift and diffusion coefficients. The average of the drift is
governed by a degenerate Hamiltonian whose set of critical points has an
interior. The dynamics of the system is studied in the presence of three time
scales. Using the martingale problem approach and separating the time scales,
we average the system to show convergence to a Markov process on a stratified
space. The averaging combines the deterministic time averaging of periodic
coefficients, and the stochastic averaging of the resulting system. The
corresponding strata of the reduced space are a two-sphere, a point and a line
segment. Special attention is given to the description of the domain of the
limiting generator, including the analysis of the gluing conditions at the
point where the strata meet. These gluing conditions, resulting from the
effects of the hierarchy of time scales, are similar to the conditions on the
domain of skew Brownian motion and are related to the description of spider
martingales.
|
256
|
arxiv_abstracts
|
math/0302014
|
We study generating functions for the number of even (odd) permutations on n
letters avoiding 132 and an arbitrary permutation $\tau$ on k letters, or
containing $\tau$ exactly once. In several interesting cases the generating
function depends only on k and is expressed via Chebyshev polynomials of the
second kind.
|
lt256
|
arxiv_abstracts
|
math/0302015
|
Let $w_{n+2}=pw_{n+1}+qw_{n}$ for $n\geq0$ with $w_0=a$ and $w_1=b$. In this
paper we find an explicit expression, in terms of determinants, for
$\sum_{n\geq0} w_n^kx^n$ for any $k\geq1$. As a consequence, we derive all the
previously known results for this kind of problems, as well as many new
results.
|
lt256
|
arxiv_abstracts
|
math/0302016
|
In this paper we consider a class of nonparametric estimators of a
distribution function F, with compact support, based on the theory of IFSs. The
estimator of F is tought as the fixed point of a contractive operator T defined
in terms of a vector of parameters p and a family of affine maps W which can be
both depend of the sample (X_1, X_2, ...., X_n). Given W, the problem consists
in finding a vector p such that the fixed point of T is ``sufficiently near''
to F. It turns out that this is a quadratic constrained optimization problem
that we propose to solve by penalization techniques. If F has a density f, we
can also provide an estimator of f based on Fourier techniques. IFS estimators
for F are asymptotically equivalent to the empirical distribution function
(e.d.f.) estimator. We will study relative efficiency of the IFS estimators
with respect to the e.d.f. for small samples via Monte Carlo approach.
For well behaved distribution functions F and for a particular family of
so-called wavelet maps the IFS estimators can be dramatically better than the
e.d.f. (or the kernel estimator for density estimation) in presence of missing
data, i.e. when it is only possibile to observe data on subsets of the whole
support of F.
This research has also produced a free package for the R statistical
environment which is ready to be used in applications.
|
256
|
arxiv_abstracts
|
math/0302017
|
We prove that if $R$ is a commutative Noetherian local pro-$p$domain of
characteristic 0 then every finitely generated$R$-standard group is $R$-linear.
|
lt256
|
arxiv_abstracts
|
math/0302018
|
We prove the rationality of some zeta functions associated tocharacters of
pro-p groups of finite rank.
|
lt256
|
arxiv_abstracts
|
math/0302019
|
We determine the isomorphism class of the Brauer groups of certain
nonrational genus zero extensions of number fields. In particular, for all
genus zero extensions E of the rational numbers Q that are split by
Q(\sqrt{2}), Br(E) is isomorphic to Br(Q(t)).
|
lt256
|
arxiv_abstracts
|
math/0302020
|
Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert
space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant
subspace of $\mathbf{A}$. Assuming that $\sup\spec(A_0)\leq \inf\spec(A_1)$,
where $A_0$ and $A_1$ are restrictions of $\mathbf{A}$ onto the subspaces
$\mathfrak{H}_0$ and $\mathfrak{H}_1=\mathfrak{H}_0^\perp$, respectively, we
study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded
self-adjoint perturbations $\mathbf{V}$ that are off-diagonal with respect to
the decomposition $\mathfrak{H} = \mathfrak{H}_0\oplus\mathfrak{H}_1$. We
obtain sharp two-sided estimates on the norm of the difference of the
orthogonal projections onto invariant subspaces of the operators $\mathbf{A}$
and $\mathbf{B}=\mathbf{A}+\mathbf{V}$. These results extend the celebrated
Davis-Kahan $\tan 2\Theta$ Theorem. On this basis we also prove new existence
and uniqueness theorems for contractive solutions to the operator Riccati
equation, thus, extending recent results of Adamyan, Langer, and Tretter.
|
256
|
arxiv_abstracts
|
math/0302021
|
In this paper we prove that for any commutative (but in general
non-associative) algebra $A$ with an invariant symmetric non-degenerate
bilinear form there is a graded vertex algebra $V = V_0 \oplus V_2 \oplus
V_3\oplus ...$, such that $\dim V_0 = 1$ and $V_2$ contains $A$. We can choose
$V$ so that if $A$ has a unit $e$, then $2e$ is the Virasoro element of $V$,
and if $G$ is a finite group of automorphisms of $A$, then $G$ acts on $V$ as
well. In addition, the algebra $V$ can be chosen with a non-degenerate
invariant bilinear form, in which case it is simple.
|
lt256
|
arxiv_abstracts
|
math/0302022
|
Suppose that two compact manifolds $X, X'$ are connected by a sequence of
Mukai flops. In this paper, we construct a ring isomorphism between cohomology
ring of $X$ and $X'$. Using the local mirror symmetry technique, we prove that
the quantum corrected products on $X, X'$ are the ordinary intersection
products. Furthermore, $X, X'$ have isomorphic Ruan cohomology. i.e. we proved
the cohomological minimal model conjecture proposed by Ruan.
|
lt256
|
arxiv_abstracts
|
math/0302023
|
We study invariants of Calabi-Yau varieties in positive characteristic,
especially the height of the Artin-Mazur formal group. We illustrate these
results by Calabi-Yau varieties of Fermat and Kummer type.
|
lt256
|
arxiv_abstracts
|
math/0302024
|
We study the geometric structure of Lorentzian spin manifolds, which admit
imaginary Killing spinors. The discussion is based on the cone construction and
a normal form classification of skew-adjoint operators in signature $(2,n-2)$.
Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces
and certain warped product structures. Exceptional cases with decomposable
holonomy of the cone are possible.
|
lt256
|
arxiv_abstracts
|
math/0302025
|
In this article, we study local models associated to certain Shimura
varieties. In particular, we present a resoultion of their singularities. As a
consequence, we are able to determine the alternating semisimple trace of the
geometric Frobenius on the sheaf of nearby cycles.
|
lt256
|
arxiv_abstracts
|
math/0302026
|
For every N > 0 there exists a group of deficiency less than -N that arises
as the fundamental group of a smooth homology 4-sphere and also as the
fundamental group of the complement of a compact contractible submanifold of
the 4-sphere. A group is the fundamental group of the complement of a
contractible submanifold of the n-sphere, n > 4, if and only if it is the
fundamental group of a homology n-sphere. There exist fundamental groups of
homology n-spheres, n > 4, that cannot arise as the fundamental group of the
complement of a contractible submanifold of the 4-sphere.
|
lt256
|
arxiv_abstracts
|
math/0302027
|
The Lie algebra gl(\lambda) with \lambda \in \mathbb C, introduced by
B.L.Feigin, can be embedded into the Lie algebra of differential operators on
the real line. We give an explicit formula of the embedding of gl(\lambda) into
the algebra D_{\lambda} of differential operators on the space of tensor
densities of degree \lambda on \mathbb R. Our main tool is the notion of
projectively equivariant symbol of a differential operator.
|
lt256
|
arxiv_abstracts
|
math/0302028
|
We prove nonlinear stability for finite amplitude perturbations of plane
Couette flow. A bound of the solution of the resolvent equation in the unstable
complex half-plane is used to estimate the solution of the full nonlinear
problem.The result is a lower bound, including Reynolds number dependence,of
the threshold amplitude below which all perturbations are stable. Our result is
an improvement of the corresponding bound derived in \cite{KLH:1}.
|
lt256
|
arxiv_abstracts
|
math/0302029
|
We determine the complete conjugate locus along all geodesics parallel or
perpendicular to the center (Theorem 2.3). When the center is 1-dimensional we
obtain formulas in all cases (Theorem 2.5), and when a certain operator is also
diagonalizable these formulas become completely explicit (Corollary 2.7). These
yield some new information about the smoothness of the pseudoriemannian
conjugate locus. We also obtain the multiplicities of all conjugate points.
|
lt256
|
arxiv_abstracts
|
math/0302030
|
We give a complete classification in canonical forms on finite-dimensional
vector spaces over the real numbers.
|
lt256
|
arxiv_abstracts
|
math/0302031
|
In this paper we have proposed a semi-heuristic optimization algorithm for
designing optimal plant layouts in process-focused manufacturing/service
facilities. Being a semi-heuristic search, our algorithm is likely to be more
efficient in terms of computer CPU engagement time as it tends to converge on
the global optimum faster than the traditional CRAFT algorithm - a pure
heuristic.
|
lt256
|
arxiv_abstracts
|
math/0302032
|
We study q-integral representations of the q-gamma and the q-beta functions.
This study leads to a very interesting q-constant. As an application of these
integral representations, we obtain a simple conceptual proof of a family of
identities for Jacobi triple product, including Jacobi's identity, and of
Ramanujan's formula for the bilateral hypergeometric series.
|
lt256
|
arxiv_abstracts
|
math/0302033
|
The Airy process is characterized by its finite-dimensional distribution
functions. We show that each finite-dimensional distribution function is
expressible in terms of a solution to a system of differential equations.
|
lt256
|
arxiv_abstracts
|
math/0302034
|
In the present text we discuss basic aspects of the Seiberg - Witten theory
mainly focusing the attantion on some geometrical details which could make the
introduction to the subject more illustrative. At the same time we list there
natural problems arise in this framework mostly interesting to the author. This
text could be regarded as additional remarks to any comlete course on the
Seiberg - Witten invariants.
|
lt256
|
arxiv_abstracts
|
math/0302035
|
A general method is developed for deriving Quantum First and Second
Fundamental Theorems of Coinvariant Theory from classical analogs in Invariant
Theory, in the case that the quantization parameter q is transcendental over a
base field. Several examples are given illustrating the utility of the method;
these recover earlier results of various researchers including Domokos,
Fioresi, Hacon, Rigal, Strickland, and the present authors.
|
lt256
|
arxiv_abstracts
|
math/0302036
|
We compute the Poisson cohomology of the one-parameter family of
SU(2)-covariant Poisson structures on the homogeneous space
S^{2}=CP^{1}=SU(2)/U(1), where SU(2) is endowed with its standard Poisson--Lie
group structure,thus extending the result of Ginzburg \cite{Gin1} on the
Bruhat--Poisson structure which is a member of this family. In particular, we
compute several invariants of these structures, such as the modular class and
the Liouville class. As a corollary of our computation, we deduce that these
structures are nontrivial deformations of each other in the direction of the
standard rotation-invariant symplectic structure on S^{2}; another corollary is
that these structures do not admit smooth rescaling.
|
lt256
|
arxiv_abstracts
|
math/0302037
|
Kazhdan and Lusztig have shown that the partition of the symmetric group of
degree $n$ into left cells is given by the Robinson-Schensted correspondence.
The aim of this paper is to provide a similar description of the left cells in
type $B_n$ for a special class of choices of unequal parameters. This is based
on a generalization of the Robinson-Schensted correspondence for type $B_n$. We
also give an explicit description of the left cell representations and show
that they are irreducible and constructible.
|
lt256
|
arxiv_abstracts
|
math/0302038
|
Relying on recent advances in the theory of entropy solutions for nonlinear
(strongly) degenerate parabolic equations, we present a direct proof of an L^1
error estimate for viscous approximate solutions of the initial value problem
for \partial_t w+\mathrm{div} \bigl(V(x)f(w)\bigr)= \Delta A(w) where V=V(x) is
a vector field, f=f(u) is a scalar function, and A'(.) \geq 0. The viscous
approximate solutions are weak solutions of the initial value problem for the
uniformly parabolic equation \partial_t w^{\epsilon}+\mathrm{div} \bigl(V(x)
f(w^{\epsilon})\bigr) \Delta \bigl(A(w^{\epsilon})+\epsilon w^{\epsilon}\bigr),
\epsilon>0. The error estimate is of order \sqrt{\epsilon}.
|
lt256
|
arxiv_abstracts
|
math/0302039
|
Let X_1 and X_2 be mixing connected algebraic dynamical systems with the
Descending Chain Condition. We show that every equivariant continuous map X_1
to X_2 is affine (that is, X_2 is topologically rigid) if and only if the
system X_2 has finite topological entropy.
|
lt256
|
arxiv_abstracts
|
math/0302040
|
We discuss computational superstructures that, using repeated, appropriately
initialized short calls, enable temporal process simulators to perform
alternative tasks such as fixed point computation, stability analysis and
projective integration. We illustrate these concepts through the acceleration
of a gPROMS-based Rapid Pressure Swing Adsorption simulation, and discuss their
scope and possible extensions.
|
lt256
|
arxiv_abstracts
|
math/0302041
|
For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers,
denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that
within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic
sequence $\{x_{1},x_{2},...,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i
\leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as
well as upper and lower bounds on this function. In particular, we show that
this function exists for all $k$ if $S$ is an odd translate of the set of
primes and $r=2$.
|
lt256
|
arxiv_abstracts
|
math/0302042
|
In Shephard-Todd classification of finite (complex) reflection groups, the
group $G_{31}$ appears to be the unique one in rank 4 of order 46080. We
provide here an elementary construction starting from the Weyl group of type
$B_6$.
|
lt256
|
arxiv_abstracts
|
math/0302043
|
Visual cryptography schemes have been introduced in 1994 by Naor and Shamir.
Their idea was to encode a secret image into $n$ shadow images and to give
exactly one such shadow image to each member of a group $P$ of $n$ persons.
Whereas most work in recent years has been done concerning the problem of
qualified and forbidden subsets of $P$ or the question of contrast optimizing,
in this paper we study extended visual cryptography schemes, i.e. shared secret
systems where any subset of $P$ shares its own secret.
|
lt256
|
arxiv_abstracts
|
math/0302044
|
Pseudo-Riemannian manifolds of balanced signature which are both spacelike
and timelike Jordan Osserman nilpotent of order 2 and of order 3 have been
constructed previously. In this short note, we shall construct
pseudo-Riemannian manifolds of signature (2s,s) for any s (which is at least 2)
which are spacelike Jordan Osserman nilpotent of order 3 but which are not
timelike Jordan Osserman. Our example and techniques are quite different from
known previously both in that they are not in neutral signature and that the
manifolds constructed will be spacelike but not timelike Jordan Osserman.
|
lt256
|
arxiv_abstracts
|
math/0302045
|
In this article we classify quadruple Galois canonical covers of smooth
surfaces of minimal degree. The classification shows that they are either
non-simple cyclic covers or bi-double covers.
If they are bi-double then they are all fiber products of double covers. We
construct examples to show that all the possibilities in the classification do
exist. There are implications of this classification that include the existence
of families with unbounded geometric genus and families with unbounded
irregularity, in sharp contrast with the case of double and triple canonical
covers. Together with the results of Horikawa and Konno for double and triple
covers, a pattern emerges that motivates some general questions on the
existence of higher degree canonical covers, some of which are answered in this
article.
|
lt256
|
arxiv_abstracts
|
math/0302046
|
Shot-noise and fractional Poisson processes are instances of filtered Poisson
processes. We here prove Girsanov theorem for this kind of processes and give
an application to an estimate problem.
|
lt256
|
arxiv_abstracts
|
math/0302047
|
We construct the basis of a stochastic calculus for so-called Volterra
processes, i.e., processes which are defined as the stochastic integral of a
time-dependent kernel with respect to a standard Brownian motion. For these
processes which are natural generalization of fractional Brownian motion, we
construct a stochastic integral and show some of its main properties:
regularity with respect to time and kernel, transformation under an absolutely
continuous change of probability, possible approximation schemes and Ito
formula.
|
lt256
|
arxiv_abstracts
|
math/0302048
|
We characterize, in a purely algebraic manner, certain linear forms, called
stable, on a Lie algebra. As an application, we determine the index of a Borel
subalgebra of a semi-simple Lie algebra. Finally, we give an example of a
parabolic subalgebra of a semi-simple Lie algebra which does not admit any
stable linear form.
|
lt256
|
arxiv_abstracts
|
math/0302049
|
For supercritical multitype branching processes in continuous time, we
investigate the evolution of types along those lineages that survive up to some
time t. We establish almost-sure convergence theorems for both time and
population averages of ancestral types (conditioned on non-extinction), and
identify the mutation process describing the type evolution along typical
lineages. An important tool is a representation of the family tree in terms of
a suitable size-biased tree with trunk. As a by-product, this representation
allows a `conceptual proof' (in the sense of Kurtz, Lyons, Pemantle, Peres
1997) of the continuous-time version of the Kesten-Stigum theorem.
|
lt256
|
arxiv_abstracts
|
math/0302050
|
Let M be a riemannian manifold. The existence of a spin structure on M,
enables to study the topology of M. The obstruction to the existence of the
spin structure is given by the second Stiefel-Whitney class. This class is the
classifying cocycle of a gerbe. One may expect that the study of this gerbe may
have topological applications, for example, one may try to generalize the
spinors Lichnerowicz theorem in this setting. On this purpose, we must first
prove an Atiyah-Singer theorem for gerbes which is the main goal of this paper.
|
lt256
|
arxiv_abstracts
|
math/0302051
|
We prove the existence of at least two solutions for a fourth order equation,
which includes the vortex equations for the U(1) and CP(1) self-dual
Maxwell-Chern-Simons models as special cases. Our method is variational, and it
relies on an "asymptotic maximum principle" property for a special class of
supersolutions to this fourth order equation.
|
lt256
|
arxiv_abstracts
|
math/0302052
|
We say that a Hopf algebra H is semicocommutative if the right adjoint
coaction factorizes through the tensor product of H with the center of H. For
instance the commutative and the cocommutative Hopf algebras are
semicocommutative. The quasitriangular Hopf algebras generalize the
cocommutative Hopf algebras. In this paper we introduce and begin the study of
a similar generalization for the semicocommutative ones. These algebras, which
we call semiquasitriangular Hopf algebras have many of the basic properties of
the quasitriangular ones. In particular, they have associated braided
categories of representations in a natural way.
|
lt256
|
arxiv_abstracts
|
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