id
string | text
string | len_category
string | source
string |
|---|---|---|---|
math/0302253
|
This paper has been withdrawn by the authors because M. Aschenbrenner pointed
out that the proof of theorem 3.1 was incorrect.
|
lt256
|
arxiv_abstracts
|
math/0302254
|
We consider the dual billiard map with respect to a smooth strictly convex
closed hypersurface in linear 2m-dimensional symplectic space and prove that it
has at least 2m distinct 3-periodic orbits.
|
lt256
|
arxiv_abstracts
|
math/0302255
|
We obtain upper bounds on the heat content and on the torsional rigidity of a
complete Riemannian manifold M, assuming a generalized Hardy inequality for the
Dirichlet Laplacian on M.
|
lt256
|
arxiv_abstracts
|
math/0302256
|
We consider noncommutative line bundles associated with the Hopf fibrations
of SUq(2) over all Podles spheres and with a locally trivial Hopf fibration of
S^3_{pq}. These bundles are given as finitely generated projective modules
associated via 1-dimensional representations of U(1) with Galois-type
extensions encoding the principal fibrations of SUq(2) and S^3_{pq}. We show
that the Chern numbers of these modules coincide with the winding numbers of
representations defining them.
|
lt256
|
arxiv_abstracts
|
math/0302257
|
The act of a person juggling can be viewed as a Markov process if we assume
that the juggler throws to random heights. I make this association for the
simplest reasonable model of random juggling and compute the steady state
probabilities in terms of the Stirling numbers of the second kind. I also
explore several alternate models of juggling.
|
lt256
|
arxiv_abstracts
|
math/0302258
|
We construct anisotropic conductivities with the same Dirichlet-to-Neumann
map as a homogeneous isotropic conductivity. These conductivities are singular
close to a surface inside the body.
|
lt256
|
arxiv_abstracts
|
math/0302259
|
An error analysis for some Newton-Cotes quadrature formulae is presented.
Peano-like error bounds are obtained. They are generally, but not always,
better than the usual Peano bounds.
|
lt256
|
arxiv_abstracts
|
math/0302260
|
Detailed illustration of the method for calculating the Chow group of a
rational surface over a local field [math.AG/0302157 (th.~4)], applied to a
certain del Pezzo surface of degree~4. Involves the construction of a regular
integral model and the determination of the specialisation map.
|
lt256
|
arxiv_abstracts
|
math/0302261
|
We generalize the well-known mean value inequality of subharmonic functions
for a slightly more general function class. We also apply this generalized mean
value inequality to weighted boundary behavior and nonintegrability questions
of subharmonic and superharmonic functions.
|
lt256
|
arxiv_abstracts
|
math/0302262
|
We study the behaviour of the Stark conjecture for an abelian extension K/k
of totally real number fields as K varies in a cyclotomic Z_p-tower. We
consider possible strengthenings of the natural norm-coherence in the tower of
putative solution of the complex conjecture and,especially, the consequences
for the analogous p-adic conjecture. More precisely, under two principal
assumptions - (i) that these solutions are given by exterior powers of
norm-coherent sequences of global (or p-semilocal) units and (ii) that p splits
in k - we show how the values of p-adic twisted zeta-functions for K/k are
determined at any integer by a group-ring-valued regulator formed from the
appropriate Coates-Wiles homomorphisms.
|
lt256
|
arxiv_abstracts
|
math/0302263
|
A skew brane is an immersed codimension 2 submanifold in affine space, free
from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew
brane cannot lie on a quadratic hypersurface. We also prove that there are no
skew loops on embedded ruled developable discs in 3-space. The paper extends
recent work by M. Ghomi and B. Solomon.
|
lt256
|
arxiv_abstracts
|
math/0302264
|
We study in optimal control the important relation between invariance of the
problem under a family of transformations, and the existence of preserved
quantities along the Pontryagin extremals. Several extensions of Noether
theorem are provided, in the direction which enlarges the scope of its
application. We formulate a more general version of Noether's theorem for
optimal control problems, which incorporates the possibility to consider a
family of transformations depending on several parameters and, what is more
important, to deal with quasi-invariant and not necessarily invariant optimal
control problems. We trust that this latter extension provides new
possibilities and we illustrate it with several examples, not covered by the
previous known optimal control versions of Noether's theorem.
|
lt256
|
arxiv_abstracts
|
math/0302265
|
Let M be a compact manifold with a Hamiltonian T action and moment map Phi.
The restriction map in equivariant cohomology from M to a level set Phi^{-1}(p)
is a surjection, and we denote the kernel by I_p. When T has isolated fixed
points, we show that I_p distinguishes the chambers of the moment polytope for
M. In particular, counting the number of distinct ideals I_p as p varies over
different chambers is equivalent to counting the number of chambers.
|
lt256
|
arxiv_abstracts
|
math/0302266
|
Let $K$ be a field finitely generated over ${\Q}$, and $A$ an Abelian variety
defined over $K$. Then by the Mordell-Weil Theorem, the set of rational points
$A(K)$ is a finitely-generated Abelian group. In this paper, assuming Tate's
Conjecture on algebraic cycles, we prove a limit formula for the Mordell-Weil
rank of an arbitrary family of Abelian varieties $A$ over a number field $k$;
this is the Abelian fibration analogue of the Nagao formula for elliptic
surfaces $E$, originally conjectured by Nagao, and proven by Rosen and
Silverman to be equivalent to Tate's Conjecture for $E$. We also give a short
exact sequence relating the Picard Varieties of the family $A$, the parameter
space, and the generic fiber, and use this to obtain an isomorphism (modulo
torsion) relating the Neron-Severi group of $A$ to the Mordell-Weil group of
$A$.
|
lt256
|
arxiv_abstracts
|
math/0302267
|
We define the category of mixed Tate motives over the ring of S-integers of a
number field. We define the motivic fundamental group (made unipotent) of a
unirational variety over a number field. We apply this to the study of the
motivic fundamental group of the projective line punctured at zero, infinity
and all N-th roots of unity.
|
lt256
|
arxiv_abstracts
|
math/0302268
|
Poisson manifolds may be regarded as the infinitesimal form of symplectic
groupoids. Twisted Poisson manifolds considered by Severa and Weinstein
[math.SG/0107133] are a natural generalization of the former which also arises
in string theory. In this note it is proved that twisted Poisson manifolds are
in bijection with a (possibly singular) twisted version of symplectic
groupoids.
|
lt256
|
arxiv_abstracts
|
math/0302269
|
We define an affine Jacquet functor and use it to describe the structure of
induced affine Harish-Chandra modules at noncritical levels, extending the
theorem of Kac and Kazhdan [KK] on the structure of Verma modules in the
Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined
with a vanishing result for certain extension groups to construct a block
decomposition of the categories of affine Harish-Chandra modules of Lian and
Zuckerman [LZ]. The latter provides an extension of the works of Rocha-Caridi,
Wallach [RW] and Deodhar, Gabber, Kac [DGK] on block decompositions of BGG
categories for Kac-Moody algebras. We also prove a compatibility relation
between the affine Jacquet functor and the Kazhdan-Lusztig tensor product. A
modification of this is used to prove that the affine Harish-Chandra category
is stable under fusion tensoring with the Kazhdan-Lusztig category (a case of
our finiteness result [Y]) and will be further applied in studying translation
functors for Kac-Moody algebras, based on the fusion tensor product.
|
256
|
arxiv_abstracts
|
math/0302270
|
Using a simple classical method we derive bilateral series identities from
terminating ones. In particular, we show how to deduce Ramanujan's 1-psi-1
summation from the q-Pfaff-Saalschuetz summation. Further, we apply the same
method to our previous q-Abel-Rothe summation to obtain, for the first time,
Abel-Rothe type generalizations of Jacobi's triple product identity. We also
give some results for multiple series.
|
lt256
|
arxiv_abstracts
|
math/0302271
|
A random walk on Z^d is excited if the first time it visits a vertex there is
a bias in one direction, but on subsequent visits to that vertex the walker
picks a neighbor uniformly at random. We show that excited random walk on Z^d,
is transient iff d>1.
|
lt256
|
arxiv_abstracts
|
math/0302272
|
We show that if a knot or link has n thin levels when put in thin position
then its exterior contains a collection of n disjoint, non-parallel, planar,
meridional, essential surfaces. A corollary is that there are at least n/3
tetrahedra in any triangulation of the complement of such a knot.
|
lt256
|
arxiv_abstracts
|
math/0302273
|
Let A be a separable unital nuclear purely infinite simple C*-algebra
satisfying the Universal Coefficient Theorem, and such that the K_0-class of
the identity is zero. We prove that every automorphism of order two of the
K-theory of A is implemented by an automorphism of A of order two. As a
consequence, we prove that every countable Z/2Z-graded module over the
representation ring of Z/2Z is isomorphic to the equivariant K-theory for some
action of Z/2Z on a separable unital nuclear purely infinite simple C*-algebra.
Along the way, we prove that every not necessarily finitely generated module
over the group ring of Z/2Z which is free as an abelian group has a direct sum
decomposition with only three kinds of summands, namely the group ring itself
and Z on which the nontrivial element of Z/2Z acts either trivially or by
multiplication by -1.
|
lt256
|
arxiv_abstracts
|
math/0302274
|
In this note, necessary and sufficient conditions are obtained for unilateral
weighted shifts to be near subnormal . As an application of the main results,
many answers to the Hilbert space problem 160 are presented at the end of the
paper.
|
lt256
|
arxiv_abstracts
|
math/0302275
|
The complete characterizations of the spectra and their various parts of
hyponormal unilateral and bilateral weighted shifts are presented respectively
in this paper. The results obtained here generalize the corresponding work of
the references.
|
lt256
|
arxiv_abstracts
|
math/0302276
|
Let F be a subfield of a commutative field extending R. Let phi_n:F^n \times
F^n ->F, phi_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say
that f:R^n->F^n preserves distance d>=0 if for each x,y \in R^n |x-y|=d implies
phi_n(f(x),f(y))=d^2. Let A_n(F) denote the set of all positive numbers d such
that any map f:R^n->F^n that preserves unit distance preserves also distance d.
Let D_n(F) denote the set of all positive numbers d with the property: if x,y
\in R^n and |x-y|=d then there exists a finite set S(x,y) with {x,y} \subseteq
S(x,y) \subseteq R^n such that any map f:S(x,y)->F^n that preserves unit
distance preserves also the distance between x and y. Obviously, {1} \subseteq
D_n(F) \subseteq A_n(F). We prove: A_n(C) \subseteq {d>0: d^2 \in Q} \subseteq
D_2(F).
Let K be a subfield of a commutative field Gamma extending C. Let psi_2:
Gamma^2 \times Gamma^2->Gamma,
psi_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:C^2->K^2
preserves unit distance if for each X,Y \in C^2 psi_2(X,Y)=1 implies
psi_2(f(X),f(Y))=1. We prove: if X,Y \in C^2, psi_2(X,Y) \in Q and X \neq Y,
then there exists a finite set S(X,Y) with {X,Y} \subseteq S(X,Y) \subseteq C^2
such that any map f:S(X,Y)->K^2 that preserves unit distance satisfies
psi_2(X,Y)=psi_2(f(X),f(Y)) and f(X) \neq f(Y).
|
256
|
arxiv_abstracts
|
math/0302277
|
For many classically chaotic systems it is believed that the quantum wave
functions become uniformly distributed, that is the matrix elements of smooth
observables tend to the phase space average of the observable. In this paper we
study the fluctuations of the matrix elements for the desymmetrized quantum cat
map. We present a conjecture for the distribution of the normalized matrix
elements, namely that their distribution is that of a certain weighted sum of
traces of independent matrices in SU(2). This is in contrast to generic chaotic
systems where the distribution is expected to be Gaussian. We compute the
second and fourth moment of the normalized matrix elements and obtain agreement
with our conjecture.
|
lt256
|
arxiv_abstracts
|
math/0302278
|
We show the existence of a unital subalgebra of the symmetric group algebra
linearly spanned by sums of permutations with a common peak set, which we call
the peak algebra. We show that this algebra is the image of the descent algebra
of type B under the map to the descent algebra of type A which forgets the
signs, and also the image of the descent algebra of type D. The peak algebra
contains a two sided ideal which is defined in terms of interior peaks. This
object was introduced in previous work by Nyman; we find that it is the image
of certain ideals of the descent algebras of types B and D introduced in
previous work of N. Bergeron et al. We derive an exact sequence involving the
peak ideal and the peak algebras of degrees $n$ and $n-2$. We obtain this and
many other properties of the peak algebra and its peak ideal by first
establishing analogous results for signed permutations and then forgetting the
signs. In particular, we construct two new commutative semisimple subalgebras
of the descent algebra by grouping permutations according to their number of
peaks or interior peaks. We discuss the Hopf algebraic structures that exist on
the direct sums of these spaces over $n\geq 0$ and explain the connection with
previous work of Stembridge; we also obtain new properties of his
descents-to-peaks map and construct a type B analog.
|
256
|
arxiv_abstracts
|
math/0302279
|
Let T be the attractor of injective contractions f_1,...,f_m on R^2 that
satisfy the Open Set Condition. If T is connected, \partial T is arcwise
connected. In particular, the boundary of the Levy dragon is arcwise connected.
|
lt256
|
arxiv_abstracts
|
math/0302280
|
We consider various equivalence relations on the set of homotopy classes of
curves on a hyperbolic surface based on topological, algebraic, and geometric
structures. The purpose of this work is to determine the relationship between
these equivalences.
|
lt256
|
arxiv_abstracts
|
math/0302281
|
We study almost Kaehler manifolds whose curvature tensor satisfies the third
curvature condition of Gray. We show that the study of manifolds within this
class reduces to the study of a subclass having the property that the torsion
of the first canonical Hermitian connection has the simplest possible algebraic
form. This allows to understand the structure of the Kaehler nullity of an
almost Kaehler manifold with parallel torsion.
|
lt256
|
arxiv_abstracts
|
math/0302282
|
We show sensitive dependece on initial condition and dense periodic points
imply asymptotic sensitivity, a stronger form of sensitivity, where the
deviation happens not just once but infintely many times. As a consequence it
follows that all Devaney chaotic systems (e.g. logistic map) have this
property.
|
lt256
|
arxiv_abstracts
|
math/0302283
|
Globular CW-complexes and flows are both geometric models of concurrent
processes which allow to model in a precise way the notion of dihomotopy.
Dihomotopy is an equivalence relation which preserves computer-scientific
properties like the presence or not of deadlock. One constructs an embedding
from globular CW-complexes to flows and one proves that two globular
CW-complexes are dihomotopic if and only if the corresponding flows are
dihomotopic. This note is the first one presenting some of the results of
math.AT/0201252.
|
lt256
|
arxiv_abstracts
|
math/0302284
|
One proves that the category of globular CW-complexes up to dihomotopy is
equivalent to the category of flows up to weak dihomotopy. This theorem
generalizes the classical theorem which states that the category of
CW-complexes up to homotopy is equivalent to the category of topological spaces
up to weak homotopy. This note is the second one presenting some of the results
of math.AT/0201252.
|
lt256
|
arxiv_abstracts
|
math/0302285
|
Noncommutativity lays hidden in the proofs of classical dynamics. Modern
frameworks can be used to bring it to light: *-products, groupoids, q-deformed
calculus, etc.
|
lt256
|
arxiv_abstracts
|
math/0302286
|
Spectral boundary conditions for Laplace-type operators, of interest in
string and brane theory, are partly Dirichlet, partly Neumann-type conditions,
partitioned by a pseudodifferential projection. We give sufficient conditions
for existence of associated heat trace expansions with power and power-log
terms. The first log coefficient is a noncommutative residue, vanishing when
the smearing function is 1. For Dirac operators with general well-posed
spectral boundary conditions, it follows that the zeta function is regular at
0. In the selfadjoint case, the eta function has a simple pole at zero, and the
value of zeta as well as the residue of eta at zero are stable under
perturbations of the boundary projection of order at most minus the dimension.
|
lt256
|
arxiv_abstracts
|
math/0302287
|
We consider an integrable Hamiltonian system with n-degrees of freedom whose
first integrals are invariant under the symplectic action of a compact Lie
group G. We prove that the singular Lagrangian foliation associated to this
Hamiltonian system is symplectically equivalent, in a G-equivariant way, to the
linearized foliation in a neighborhood of a compact singular non-degenerate
orbit. We also show that the non-degeneracy condition is not equivalent to the
non-resonance condition for smooth systems.
|
lt256
|
arxiv_abstracts
|
math/0302288
|
We interpret magnetic billiards as Finsler ones and describe an analog of the
string construction for magnetic billiards. Finsler billiards for which the law
"angle of incidence equals angle of reflection" are described. We characterize
the Finsler metrics in the plane whose geodesics are circles of a fixed radius.
This is a magnetic analog of Hilbert's fourth problem asking to describe the
Finsler metrics whose geodesics are straight lines.
|
lt256
|
arxiv_abstracts
|
math/0302289
|
For a Dirac-type operator D with a spectral boundary condition, the
associated heat operator trace has an expansion in powers and log-powers of t.
Some of the log-coefficients vanish in the Atiyah-Patodi-Singer product case.
We here investigate the effect of perturbations of D, by use of a
pseudodifferential parameter-dependent calculus for boundary problems. It is
shown that the first k log-terms are stable under perturbations of D vanishing
to order k at the boundary (and the nonlocal power coefficients behind them are
only locally perturbed). For perturbations of D from the APS product case by
tangential operators commuting with the tangential part A, all the
log-coefficients vanish if the dimension is odd.
|
lt256
|
arxiv_abstracts
|
math/0302290
|
In this note we show that any real exact G-invariant (1,1)-form is the Ricci
form of a Kaehler metric on the complexification of an irreducible compact
symmetric space G/K.
|
lt256
|
arxiv_abstracts
|
math/0302291
|
We give upper and lower bounds for the order of the top Chern class of the
Hodge bundle on the moduli space of principally polarized abelian varieties.
|
lt256
|
arxiv_abstracts
|
math/0302292
|
Let (M,g,J) be a compact Hermitian manifold with a smooth boundary. Let
$\Delta_p$ and $D_p$ be the realizations of the real and complex Laplacians on
p forms with either Dirichlet or Neumann boundary conditions. We generalize
previous results in the closed setting to show that (M,g,J) is Kaehler if and
only if $Spec(\Delta_p)=Spec(2D_p)$ for p=0,1. We also give a characterization
of manifolds with constant sectional curvature or constant Ricci tensor (in the
real setting) and manifolds of constant holomorphic sectional curvature (in the
complex setting) in terms of spectral geometry.
|
lt256
|
arxiv_abstracts
|
math/0302293
|
We study deformations of the Plancherel measure of the symmetric group by
lifting them to the symmetric group and using combinatorics of card shuffling.
The existing methods for analyzing deformations of Plancherel measure are not
obviously applicable to the examples in this paper. The main idea of this paper
is to find and analyze a formula for the total variation distance between
iterations of riffle shuffles and iterations of "cut and then riffle shuffle".
Similar results are given for affine shuffles, which allow us to determine
their convergence rate to randomness.
|
lt256
|
arxiv_abstracts
|
math/0302294
|
We describe an explicit geometric Littlewood-Richardson rule, interpreted as
deforming the intersection of two Schubert varieties so that they break into
Schubert varieties. There are no restrictions on the base field, and all
multiplicities arising are 1; this is important for applications. This rule
should be seen as a generalization of Pieri's rule to arbitrary Schubert
classes, by way of explicit homotopies. It has a straightforward bijection to
other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's
puzzles.
This gives the first geometric proof and interpretation of the
Littlewood-Richardson rule. It has a host of geometric consequences, described
in the companion paper "Schubert induction". The rule also has an
interpretation in K-theory, suggested by Buch, which gives an extension of
puzzles to K-theory. The rule suggests a natural approach to the open question
of finding a Littlewood-Richardson rule for the flag variety, leading to a
conjecture, shown to be true up to dimension 5. Finally, the rule suggests
approaches to similar open problems, such as Littlewood-Richardson rules for
the symplectic Grassmannian and two-flag varieties.
|
256
|
arxiv_abstracts
|
math/0302295
|
Let N_1, N_2, M be smooth manifolds with dim N_1 + dim N_2 +1 = dim M$ and
let phi_i, for i=1,2, be smooth mappings of N_i to M with Im phi_1 and Im phi_2
disjoint. The classical linking number lk(phi_1,phi_2) is defined only when
phi_1*[N_1] = phi_2*[N_2] = 0 in H_*(M).
The affine linking invariant alk is a generalization of lk to the case where
phi_1*[N_1] or phi_2*[N_2] are not zero-homologous. In arXiv:math.GT/0207219 we
constructed the first examples of affine linking invariants of
nonzero-homologous spheres in the spherical tangent bundle of a manifold, and
showed that alk is intimately related to the causality relation of wave fronts
on manifolds.
In this paper we develop the general theory. The invariant alk appears to be
a universal Vassiliev-Goussarov invariant of order < 2. In the case where
phi_1*[N_1] and phi_2*[N_2] are 0 in homology it is a splitting of the
classical linking number into a collection of independent invariants.
To construct alk we introduce a new pairing mu on the bordism groups of
spaces of mappings of N_1 and N_2 into M, not necessarily under the restriction
dim N_1 + dim N_2 +1 = dim M. For the zero-dimensional bordism groups, mu can
be related to the Hatcher-Quinn invariant. In the case N_1=N_2=S^1, it is
related to the Chas-Sullivan string homology super Lie bracket, and to the
Goldman Lie bracket of free loops on surfaces.
|
256
|
arxiv_abstracts
|
math/0302296
|
We describe a Schubert induction theorem, a tool for analyzing intersections
on a Grassmannian over an arbitrary base ring. The key ingredient in the proof
is the Geometric Littlewood-Richardson rule, described in a companion paper.
Schubert problems are among the most classical problems in enumerative
geometry of continuing interest. As an application of Schubert induction, we
address several long-standing natural questions related to Schubert problems,
including: the "reality" of solutions; effective numerical methods; solutions
over algebraically closed fields of positive characteristic; solutions over
finite fields; a generic smoothness (Kleiman-Bertini) theorem; and monodromy
groups of Schubert problems. These methods conjecturally extend to the flag
variety.
|
lt256
|
arxiv_abstracts
|
math/0302297
|
This is the first in a series of works devoted to small non-selfadjoint
perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2.
In the present work we treat the case when the classical flow of the
unperturbed part is periodic and the strength $\epsilon$ of the perturbation is
$\gg h$ (or sometimes only $\gg h^2$) and bounded from above by $h^{\delta}$
for some $\delta>0$. We get a complete asymptotic description of all
eigenvalues in certain rectangles $[-1/C, 1/C]+ i\epsilon [F_0-1/C,F_0+1/C]$.
|
lt256
|
arxiv_abstracts
|
math/0302298
|
We construct compact polyhedra with $m$-gonal faces whose links are
generalized 3-gons. It gives examples of cocompact hyperbolic bildings of type
$P(m,3)$. For $m=3$ we get compact spaces covered by Euclidean buildings of
type $A_2$.
|
lt256
|
arxiv_abstracts
|
math/0302299
|
The multi-symplectic form for Hamiltonian PDEs leads to a general framework
for geometric numerical schemes that preserve a discrete version of the
conservation of symplecticity. The cases for systems or PDEs with dissipation
terms has never been extended. In this paper, we suggest a new extension for
generalizing the multi-symplectic form for Hamiltonian systems to systems with
dissipation which never have remarkable energy and momentum conservation
properties. The central idea is that the PDEs is of a first-order type that has
a symplectic structure depended explicitly on time variable, and decomposed
into distinct components representing space and time directions. This suggest a
natural definition of multi-symplectic Birkhoff's equation as a
multi-symplectic structure from that a multi-symplectic dissipation law is
constructed. We show that this definition leads to deeper understanding
relationship between functional principle and PDEs. The concept of
multi-symplectic integrator is also discussed.
|
lt256
|
arxiv_abstracts
|
math/0302300
|
We study some aspects of the geometric representation theory of the Thompson
and Neretin groups, suggested by their analogies with the diffeomorphism groups
of the circle. We prove that the Burau representation of the Artin braid groups
extends to a mapping class group $A_T$ related to Thompson's group $T$ by a
short exact sequence $B_{\infty}\hookrightarrow A_T\to T$, where $B_{\infty}$
is the infinite braid group. This {\it non-commutative} extension abelianises
to a central extension $0\to \Z\to A_T/[B_{\infty},B_{\infty}]\to T\to 1$
detecting the {\it discrete} version $\bar{gv}$ of the
Bott-Virasoro-Godbillon-Vey class. A morphism from the above non-commutative
extension to a reduced Pressley-Segal extension is then constructed, and the
class $\bar{gv}$ is realised as a pull-back of the reduced Pressley-Segal
class. A similar program is carried out for an extension of the Neretin group
related to the {\it combinatorial} version of the Bott-Virasoro-Godbillon-Vey
class.
|
lt256
|
arxiv_abstracts
|
math/0302301
|
Let $A_n\subseteq S_n$ denote the alternating and the symmetric groups on
$1,...,n$. MacMahaon's theorem, about the equi-distribution of the length and
the major indices in $S_n$, has received far reaching refinements and
generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and
followers. Our main goal is to find analogous statistics and identities for the
alternating group $A_{n}$. A new statistic for $S_n$, {\it the delent number},
is introduced. This new statistic is involved with new $S_n$ equi-distribution
identities, refining some of the results of Foata-Schutzenberger and
Garsia-Gessel. By a certain covering map $f:A_{n+1}\to S_n$, such $S_n$
identities are `lifted' to $A_{n+1}$, yielding the corresponding $A_{n+1}$
equi-distribution identities.
|
lt256
|
arxiv_abstracts
|
math/0302302
|
We enumerate all ternary length-l square-free words, which are words avoiding
squares of words up to length l, for l<=24. We analyse the singular behaviour
of the corresponding generating functions. This leads to new upper entropy
bounds for ternary square-free words. We then consider ternary square-free
words with fixed letter densities, thereby proving exponential growth for
certain ensembles with various letter densities. We derive consequences for the
free energy and entropy of ternary square-free words.
|
lt256
|
arxiv_abstracts
|
math/0302303
|
Entringer, Jackson, and Schatz conjectured in 1974 that every infinite
cubefree binary word contains arbitrarily long squares. In this paper we show
this conjecture is false: there exist infinite cubefree binary words avoiding
all squares xx with |x| >= 4, and the number 4 is best possible. However, the
Entringer-Jackson-Schatz conjecture is true if "cubefree" is replaced with
"overlap-free".
|
lt256
|
arxiv_abstracts
|
math/0302304
|
In spite of physics terms in the title, this paper is purely mathematical.
Its purpose is to introduce triangulated categories related to singularities of
algebraic varieties and establish a connection of these categories with
D-branes in Landau-Ginzburg models.
|
lt256
|
arxiv_abstracts
|
math/0302305
|
We survey finite energy inverse results in N-body scattering, and we also
sketch the proof of the extension of our recent two-cluster to two-cluster
three-body result to the many-body case: this requires only minor
modifications. We also indicate how to extend a free-to-free inverse result in
three-body scattering to the many-body case.
|
lt256
|
arxiv_abstracts
|
math/0302306
|
P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space
of Riemann surfaces of genus s, of the least conformal length of a
nonseparating loop, is logarithmic in s. We present an application of
(polynomially) dense Euclidean packings, to estimates for an analogous
2-dimensional conformal systolic invariant of a 4-manifold X with indefinite
intersection form. The estimate turns out to be polynomial, rather than
logarithmic, in \chi(X), if the conjectured surjectivity of the period map is
correct. Such surjectivity is targeted by the current work in gauge theory. The
surjectivity allows one to insert suitable lattices with metric properties
prescribed in advance, into the second de Rham cohomology group of X, as its
integer lattice. The idea is to adapt the well-known Lorentzian construction of
the Leech lattice, by replacing the Leech lattice by the Conway-Thompson
unimodular lattices which define asymptotically dense packings. The final step
can be described, in terms of the successive minima \lambda_i, as deforming a
\lambda_2-bound into a \lambda_1-bound.
|
256
|
arxiv_abstracts
|
math/0302307
|
This work presents a hybrid approach to solve the maximum stable set problem,
using constraint and semidefinite programming. The approach consists of two
steps: subproblem generation and subproblem solution. First we rank the
variable domain values, based on the solution of a semidefinite relaxation.
Using this ranking, we generate the most promising subproblems first, by
exploring a search tree using a limited discrepancy strategy. Then the
subproblems are being solved using a constraint programming solver. To
strengthen the semidefinite relaxation, we propose to infer additional
constraints from the discrepancy structure. Computational results show that the
semidefinite relaxation is very informative, since solutions of good quality
are found in the first subproblems, or optimality is proven immediately.
|
lt256
|
arxiv_abstracts
|
math/0302308
|
These informal notes, initially prepared a few years ago, look at various
questions related to infinite processes in several parts of mathematics, with
emphasis on examples.
|
lt256
|
arxiv_abstracts
|
math/0302309
|
Let W be a finite Coxeter group. In this paper, we show that the properties
of the Solomon algebra homomorphism Phi (from the Solomom descent algebra to
the algebra of class functions) are strongly related to enumerative results:
certain joint statistics on W may be enumerated by the scalar products of
appropriate characters.
|
lt256
|
arxiv_abstracts
|
math/0302310
|
Let $\ell$ be a length function on a group G, and let $M_{\ell}$ denote the
operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following
Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It
defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state
space of $C_r^*(G)$. We show that if G is a hyperbolic group and if $\ell$ is a
word-length function on G, then the topology from this metric coincides with
the weak-* topology (our definition of a ``compact quantum metric space''). We
show that a convenient framework is that of filtered $C^*$-algebras which
satisfy a suitable `` Haagerup-type'' condition. We also use this framework to
prove an analogous fact for certain reduced free products of $C^*$-algebras.
|
lt256
|
arxiv_abstracts
|
math/0302311
|
We show that any set containing a positive proportion of the primes contains
a 3-term arithmetic progression. An important ingredient is a proof that the
primes enjoy the so-called Hardy-Littlewood majorant property. We derive this
by giving a new proof of a rather more general result of Bourgain which,
because of a close analogy with a classical argument of Tomas and Stein from
Euclidean harmonic analysis, might be called a restriction theorem for the
primes.
|
lt256
|
arxiv_abstracts
|
math/0302312
|
This paper is withdrown
|
lt256
|
arxiv_abstracts
|
math/0302313
|
Via the BGG-correspondence a simplicial complex D on [n] is transformed into
a complex of coherent sheaves L(D) on the projective space n-1-space. In
general we compute the support of each of its cohomology sheaves.
When the Alexander dual D* is Cohen-Macaulay there is only one such non-zero
cohomology sheaf. We investigate when this sheaf can be an a'th syzygy sheaf in
a locally free resolution and show that this corresponds exactly to the case of
D* being a+1-Cohen-Macaulay as defined by K.Baclawski.
By putting further conditions on the sheaves we get nice subclasses of a+1-
Cohen-Macaulay simplicial complexes whose f-vector depends only on a and the
invariants n,d, and c. When a=0 these are the bi-Cohen-Macaulay simplicial
complexes, when a=1 and d=2c cyclic polytopes are examples, and when a=c we get
Alexander duals of the Steiner systems S(c,d,n).
We also show that D* is Gorenstein* iff the associated coherent sheaf of D is
an ideal sheaf.
|
lt256
|
arxiv_abstracts
|
math/0302314
|
The W_3 algebra of central charge 6/5 is realized as a subalgebra of the
vertex operator algebra V_{\sqrt{2}A_2} associated with a lattice of type
\sqrt{2}A_2 by using both coset construction and orbifold theory. It is proved
that W_3 is rational. Its irreducible modules are classified and constructed
explicitly. The characters of those irreducible modules are also computed.
|
lt256
|
arxiv_abstracts
|
math/0302315
|
In this paper we give an algorithm for computing the mth base-b digit (m=1 is
the least significant digit) of an integer n (actually, it finds sharp
approximations to n/b^m mod 1), where n is defined as the last number in a
sequence of integers s1,s2,...,sL=n, where s1=0, s2=1, and each successive si
is either the sum, product, or difference of two previous sj's in the sequence.
In many cases, the algorithm will find this mth digit using far less memory
than it takes to write down all the base-b digits of n, while the number of bit
operations will grow only slighly worse than linear in the number of digits.
One consequence of this result is that the mth base-10 digit of 2^t can be
found using O(t^{2/3} log^C t) bits of storage (for some C>0), and O(t log^C t)
bit operations.
The algorithm is also highly parallelizable, and an M-fold reduction in
running time can be achieved using M processors, although the memory required
will then grow by a factor of M.
|
lt256
|
arxiv_abstracts
|
math/0302316
|
For any finite group G we define the moduli space of pointed admissible
G-covers and the concept of a G-equivariant cohomological field theory
(G-CohFT), which, when G is the trivial group, reduce to the moduli space of
stable curves and a cohomological field theory (CohFT), respectively. We prove
that by taking the "quotient" by G, a G-CohFT reduces to a CohFT. We also prove
that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization
of a Frobenius algebra, and that the "quotient" by G agrees with the obvious
Frobenius algebra structure on the space of G-invariants, after rescaling the
metric.
We also introduce the moduli space of G-stable maps into a smooth, projective
variety X with G action. Gromov-Witten-like invariants of these spaces provide
the primary source of examples of G-CohFTs. Finally, we explain how these
constructions generalize (and unify) the Chen-Ruan orbifold Gromov-Witten
invariants of the global quotient [X/G] as well as the ring H*(X,G) of Fantechi
and Goettsche.
|
lt256
|
arxiv_abstracts
|
math/0302317
|
The theory of character sheaves on a reductive group is extended to a class
of varieties which includes the strata of the De Concini-Procesi completion of
an adjoint group.
|
lt256
|
arxiv_abstracts
|
math/0302318
|
We present existence results for certain singular 2-dimensional foliations on
4-manifolds. The singularities can be chosen to be simple, e.g. the same as
those that appear in Lefschetz pencils. There seems to be a wealth of such
creatures on most 4-manifolds. In certain cases, one can prescribe surfaces to
be transverse or be leaves of these foliations.
The purpose of this paper is to offer objects, hoping for a future theory to
be developed on them. For example, foliations that are taut might offer genus
bounds for embedded surfaces (Kronheimer's conjecture).
|
lt256
|
arxiv_abstracts
|
math/0302319
|
We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic
$\mathbf{Z}_p$-extension of the CM field, where $p$ is a prime of good,
ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to
even order, the two variable main conjecture of Rubin implies that the
Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension
is a torsion Iwasawa module. When the order of vanishing is odd, work of
Greenberg shows that it is not a torsion module. In this paper we show that in
the case of odd order of vanishing the dual of the Selmer group has rank
exactly one, and we prove a form of the Iwasawa main conjecture for the torsion
submodule.
|
lt256
|
arxiv_abstracts
|
math/0302320
|
Several new transformations for q-binomial coefficients are found, which have
the special feature that the kernel is a polynomial with nonnegative
coefficients. By studying the group-like properties of these positivity
preserving transformations, as well as their connection with the Bailey lemma,
many new summation and transformation formulas for basic hypergeometric series
are found. The new q-binomial transformations are also applied to obtain
multisum Rogers--Ramanujan identities, to find new representations for the
Rogers--Szego polynomials, and to make some progress on Bressoud's generalized
Borwein conjecture. For the original Borwein conjecture we formulate a
refinement based on a new triple sum representations of the Borwein
polynomials.
|
lt256
|
arxiv_abstracts
|
math/0302321
|
In 1985 Xiao Gang proved that the bicanonical system of a complex surface $S$
of general type with $p_2(S)>2$ is not composed of a pencil [Bull. Soc. Math.
France, 113 (1985), 23--51]. When in the end of the 80's it was finally proven
that $| 2K_S|$ is base point free, whenever $p_g\geq 1$, the part of this
theorem concerning surfaces with $p_g\geq 1$ became trivial.
In this note a new proof of this theorem for surfaces with $p_g=0$ is
presented.
|
lt256
|
arxiv_abstracts
|
math/0302322
|
In this paper we extend previous studies of selection principles for families
of open covers of sets of real numbers to also include families of countable
Borel covers.
The main results of the paper could be summarized as follows:
1. Some of the classes which were different for open covers are equal for
Borel covers -- Section 1;
2. Some Borel classes coincide with classes that have been studied under a
different guise by other authors -- Section 4.
|
lt256
|
arxiv_abstracts
|
math/0302323
|
We present a few general results on foliations of 4-manifolds by surfaces:
existence, tautness, relations to minimal genus of embedded surfaces; as well
as some open problems. We hope to stimulate interest in this area.
|
lt256
|
arxiv_abstracts
|
math/0302324
|
A detailed presentation of the results obtained during my Ph.D. research. The
main investigations concern explicit descriptions of classes of finite
dimensional pointed Hopf algebras and their quasi-isomorphism types.
|
lt256
|
arxiv_abstracts
|
math/0302325
|
Motivated by the study of depth 2 Frobenius extensions we introduce a new
notion of Hopf algebroid. It is a 2-sided bialgebroid with a bijective antipode
which connects the two, left and right handed, structures. While all the
interesting examples of the Hopf algebroid of J.H. Lu turn out to be Hopf
algebroids in the sense of this paper, there exist simple examples showing that
our definition is not a special case of Lu's. Our Hopf algebroids, however,
belong to the class of $\times_L$-Hopf algebras proposed by P. Schauenburg.
After discussing the axioms and some examples we study the theory of
non-degenerate integrals in order to obtain duals of Hopf algebroids.
|
lt256
|
arxiv_abstracts
|
math/0302326
|
We present a unified approach to improved $L^p$ Hardy inequalities in $\R^N$.
We consider Hardy potentials that involve either the distance from a point, or
the distance from the boundary, or even the intermediate case where distance is
taken from a surface of codimension $1<k<N$. In our main result we add to the
right hand side of the classical Hardy inequality, a weighted $L^p$ norm with
optimal weight and best constant. We also prove non-homogeneous improved Hardy
inequalities, where the right hand side involves weighted L^q norms, q \neq p.
|
lt256
|
arxiv_abstracts
|
math/0302327
|
We consider a general class of sharp $L^p$ Hardy inequalities in $\R^N$
involving distance from a surface of general codimension $1\leq k\leq N$. We
show that we can succesively improve them by adding to the right hand side a
lower order term with optimal weight and best constant. This leads to an
infinite series improvement of $L^p$ Hardy inequalities.
|
lt256
|
arxiv_abstracts
|
math/0302328
|
This article is a continuation of work on construction and calculation
various of modifications of invariant based on the use Euclidean metric values
attributed to elements of manifold triangulation. We again address the well
investigated lens spaces as a standard tool for checking the nontriviality of
topological invariants.
|
lt256
|
arxiv_abstracts
|
math/0302329
|
In this paper, we answer a question posed by Kurt Johansson, to find a PDE
for the joint distribution of the Airy Process. The latter is a continuous
stationary process, describing the motion of the outermost particle of the
Dyson Brownian motion, when the number of particles get large, with space and
time appropriately rescaled. The question reduces to an asymptotic analysis on
the equation governing the joint probability of the eigenvalues of coupled
Gaussian Hermitian matrices. The differential equations lead to the asymptotic
behavior of the joint distribution and the correlation for the Airy process at
different times t_1 and t_2, when t_2-t_1 tends to infinity.
|
lt256
|
arxiv_abstracts
|
math/0302330
|
For a bounded convex domain \Omega in R^N we prove refined Hardy inequalities
that involve the Hardy potential corresponding to the distance to the boundary
of \Omega, the volume of $\Omega$, as well as a finite number of sharp
logarithmic corrections. We also discuss the best constant of these
inequalities.
|
lt256
|
arxiv_abstracts
|
math/0302331
|
We obtain Sobolev inequalities for the Schrodinger operator -\Delta-V, where
V has critical behaviour V(x)=((N-2)/2)^2|x|^{-2} near the origin. We apply
these inequalities to obtain pointwise estimates on the associated heat kernel,
improving upon earlier results.
|
lt256
|
arxiv_abstracts
|
math/0302332
|
There is an interpretation of open string field theory in algebraic topology.
An interpretation of closed string field theory can be deduced from this open
string theory to obtain as well the interpretation of open and closed string
field theory combined.
|
lt256
|
arxiv_abstracts
|
math/0302333
|
Two philosophical applications of the concept of program-size complexity are
discussed. First, we consider the light program-size complexity sheds on
whether mathematics is invented or discovered, i.e., is empirical or is a
priori. Second, we propose that the notion of algorithmic independence sheds
light on the question of being and how the world of our experience can be
partitioned into separate entities.
|
lt256
|
arxiv_abstracts
|
math/0302334
|
We obtain formulas for the first and second cohomology groups of a general
current Lie algebra with coefficients in the "current" module, and apply them
to compute structure functions for manifolds of loops with values in compact
Hermitian symmetric spaces.
|
lt256
|
arxiv_abstracts
|
math/0302335
|
We define united KK-theory for real C*-algebras A and B such that A is
separable and B is sigma-unital, extending united K-theory in the sense that
KK\crt(\R, B) = K\crt(B). United KK-theory contains real, complex, and
self-conjugate KK-theory; but unlike unaugmented real KK-theory, it admits a
universal coefficient theorem. For all separable A and B in which the
complexification of A is in the bootstrap category, KK\crt(A,B) can be written
as the middle term of a short exact sequence whose outer terms involve the
united K-theory of A and B. As a corollary, we prove that united K-theory
classifies KK-equivalence for real C*-algebras whose complexification is in the
bootstrap category.
|
lt256
|
arxiv_abstracts
|
math/0302336
|
Let G be the Sylow 2-subgroup of the unitary group $SU_3(4)$. We find two
essential classes in the mod-2 cohomology ring of G whose product is nonzero.
In fact, the product is the ``last survivor'' of Benson-Carlson duality. Recent
work of Pakianathan and Yalcin then implies a result about connected graphs
with an action of G. Also, there exist essential classes which cannot be
written as sums of transfers from proper subgroups.
This phenomenon was first observed on the computer. The argument given here
uses the elegant calculation by J. Clark, with minor corrections.
|
lt256
|
arxiv_abstracts
|
math/0302337
|
In this note we develop a coalgebraic approach to the study of solutions of
linear difference equations over modules and rings. Some known results about
linearly recursive sequences over base fields are generalized to linearly
(bi)recursive (bi)sequences of modules over arbitrary commutative ground rings.
|
lt256
|
arxiv_abstracts
|
math/0302338
|
In this paper a first order analytical system of difference equations is
considered. For an asymptotically stable fixed point x0 of the system a gradual
approximation of the domain of attraction DA is presented in the case when the
matrix of the linearized system in x0 is a contraction. This technique is based
on the gradual extension of the "embryo" of an analytic function of several
variables. The analytic function is a Lyapunov function whose natural domain of
analyticity is the DA and which satisfies an iterative functional equation. The
equation permits to establish an "embryo" of the Lyapunov function and a first
approximation of the DA. The "embryo" is used for the determination of a new
"embryo" and a new part of the DA. In this way, computing new "embryos" and new
domains, the DA is gradually approximated. Numerical examples are given for
polynomial systems.
|
lt256
|
arxiv_abstracts
|
math/0302339
|
We study the nonlinear Schrodinger equations with a linear potential. A
change of variables makes it possible to deduce results concerning finite time
blow up and scattering theory from the case with no potential.
|
lt256
|
arxiv_abstracts
|
math/0302340
|
We show that for a complete complex algebraic variety the pure component of
homology coincides with the image of intersection homology. Therefore pure
homology is topologically invariant. To obtain slightly more general results we
introduce "image homology" for noncomplete varieties.
|
lt256
|
arxiv_abstracts
|
math/0302341
|
We formulate and prove a free quantum analogue of the first fundamental
theorems of invariant theory. More precisely, the polynomial functions algebras
are replaced by free algebras, while the universal cosovereign Hopf algebras
play the role of the general linear group.
|
lt256
|
arxiv_abstracts
|
math/0302342
|
Spectral analysis of a certain doubly infinite Jacobi operator leads to
orthogonality relations for confluent hypergeometric functions, which are
called Laguerre functions. This doubly infinite Jacobi operator corresponds to
the action of a parabolic element of the Lie algebra $\mathfrak{su}(1,1)$. The
Clebsch-Gordan coefficients for the tensor product representation of a positive
and a negative discrete series representation of $\mathfrak{su}(1,1)$ are
determined for the parabolic bases. They turn out to be multiples of Jacobi
functions. From the interpretation of Laguerre polynomials and functions as
overlap coefficients, we obtain a product formula for the Laguerre polynomials,
given by a discontinuous integral over Laguerre functions, Jacobi functions and
continuous dual Hahn polynomials.
|
lt256
|
arxiv_abstracts
|
math/0302343
|
Through the study of some elliptic and parabolic fully nonlinear PDEs, we
establish conformal versions of quermassintegral inequality, the Sobolev
inequality and the Moser-Trudinger inequality for the geometric quantities
associated to the Schouten tensor on locally conformally flat manifolds.
|
lt256
|
arxiv_abstracts
|
math/0302344
|
We first prove that the set of domino tilings of a fixed finite figure is a
distributive lattice, even in the case when the figure has holes. We then give
a geometrical interpretation of the order given by this lattice, using (not
necessarily local) transformations called {\em flips}.
This study allows us to formulate an exhaustive generation algorithm and a
uniform random sampling algorithm.
We finally extend these results to other types of tilings (calisson tilings,
tilings with bicolored Wang tiles).
|
lt256
|
arxiv_abstracts
|
math/0302345
|
We consider strictly convex hypersurfaces which are evolving by the
non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary
condition. Solutions are shown to converge smoothly to hypersurfaces moving by
translation. In particular, for bounded domains we prove that convex functions
with prescribed normal derivative satisfy a uniform oscillation estimate.
|
lt256
|
arxiv_abstracts
|
math/0302346
|
A p-compact group, as defined by Dwyer and Wilkerson, is a purely
homotopically defined p-local analog of a compact Lie group. It has long been
the hope, and later the conjecture, that these objects should have a
classification similar to the classification of compact Lie groups.
In this paper we finish the proof of this conjecture, for p an odd prime,
proving that there is a one-to-one correspondence between connected p-compact
groups and finite reflection groups over the p-adic integers. We do this by
providing the last, and rather intricate, piece, namely that the exceptional
compact Lie groups are uniquely determined as p-compact groups by their Weyl
groups seen as finite reflection groups over the p-adic integers. Our approach
in fact gives a largely self-contained proof of the entire classification
theorem.
|
lt256
|
arxiv_abstracts
|
math/0302347
|
Motivated by work of Stembridge, we study rank functions for Viennot's heaps
of pieces. We produce a simple and sufficient criterion for a heap to be a
ranked poset and apply the results to the heaps arising from fully commutative
words in Coxeter groups.
|
lt256
|
arxiv_abstracts
|
math/0302348
|
We study the Boltzmann equation for a space-homogeneous gas of inelastic hard
spheres, with a diffusive term representing a random background forcing. Under
the assumption that the initial datum is a nonnegative $L^2$ function, with
bounded mass and kinetic energy (second moment), we prove the existence of a
solution to this model, which instantaneously becomes smooth and rapidly
decaying. Under a weak additional assumption of bounded third moment, the
solution is shown to be unique. We also establish the existence (but not
uniqueness) of a stationary solution. In addition we show that the
high-velocity tails of both the stationary and time-dependent particle
distribution functions are overpopulated with respect to the Maxwellian
distribution, as conjecturedby previous authors, and we prove pointwise lower
estimates for the solutions.
|
lt256
|
arxiv_abstracts
|
math/0302349
|
The theory of degenerate parabolic equations of the forms \[
u_t=(\Phi(u_x))_{x} \quad {\rm and} \quad v_{t}=(\Phi(v))_{xx} \] is used to
analyze the process of contour enhancement in image processing, based on the
evolution model of Sethian and Malladi. The problem is studied in the framework
of nonlinear diffusion equations. It turns out that the standard initial-value
problem solved in this theory does not fit the present application since it it
does not produce image concentration. Due to the degenerate character of the
diffusivity at high gradient values, a new free boundary problem with singular
boundary data can be introduced, and it can be solved by means of a non-trivial
problem transformation. The asymptotic convergence to a sharp contour is
established and rates calculated.
|
lt256
|
arxiv_abstracts
|
math/0302350
|
Let M be a Kaehler manifold with a free, holomorphic and Hamiltonian action
of the standard n-torus T. We give a simple, explicit and canonical formula for
the Kaehler potential on the Kaehler reduction of M. As a consequence we can
derive improvements of several classical results known for more general
Hamiltonian reductions. Among these are a forms-level proof of the
Duistermaat-Heckman theorem; an elementary proof of Atiyah's proof of the
convexity of the moment image of a complexified T-orbit; another formula due to
Biquard-Gauduchon for the Kaehler potential; and a formula in terms of moment
data for the Kaehler metric on a toric variety, due originally to the second
author.
|
lt256
|
arxiv_abstracts
|
math/0302351
|
Multiplier ideals, and the vanishing theorems they satisfy, have found many
applications in recent years. In the global setting they have been used to
study pluricanonical and other linear series on a projective variety. More
recently, they have led to the discovery of some surprising uniform results in
local algebra.
The present notes aim to provide a gentle introduction to the
algebraically-oriented local side of the theory. They follow closely a short
course on multiplier ideals given in September 2002 at the Introductory
Workshop of the program in commutative algebra at MSRI.
|
lt256
|
arxiv_abstracts
|
math/0302352
|
Let G_R be a Lie group acting on an oriented manifold M, and let $\omega$ be
an equivariantly closed form on M. If both G_R and M are compact, then the
integral $\int_M \omega$ is given by the fixed point integral localization
formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the
acting Lie group G_R is not compact: there simply may not be enough fixed
points present. A proposed remedy is to modify the action of G_R in such a way
that all fixed points are accounted for.
Let G_R be a real semisimple Lie group, possibly noncompact. One of the most
important examples of equivariantly closed forms is the symplectic volume form
$d\beta$ of a coadjoint orbit $\Omega$. Even if $\Omega$ is not compact, the
integral $\int_{\Omega} d\beta$ exists as a distribution on the Lie algebra
g_R. This distribution is called the Fourier transform of the coadjoint orbit.
In this article we will apply the localization results described in [L1] and
[L2] to get a geometric derivation of Harish-Chandra's formula (9) for the
Fourier transforms of regular semisimple coadjoint orbits. Then we will make an
explicit computation for the coadjoint orbits of elements of G_R* which are
dual to regular semisimple elements lying in a maximally split Cartan
subalgebra of g_R.
|
256
|
arxiv_abstracts
|
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