id
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string | len_category
string | source
string |
|---|---|---|---|
math/0302353
|
We study a semilinear PDE generalizing the Fujita equation whose evolution
operator is the sum of a fractional power of the Laplacian and a convex
non-linearity. Using the Feynman-Kac representation we prove criteria for
asymptotic extinction versus finite time blow up of positive solutions based on
comparison with global solutions. For a critical power non-linearity we obtain
a two-parameter family of radially symmetric stationary solutions.
By extending the method of moving planes to fractional powers of the
Laplacian we prove that all positive steady states of the corresponding
equation in a finite ball are radially symmetric.
|
lt256
|
arxiv_abstracts
|
math/0302354
|
The purpose of this paper is to present a weighted kneading theory for
unidimensional maps with holes. We consider extensions of the kneading theory
of Milnor and Thurston to expanding discontinuous maps with holes and introduce
weights in the formal power series. This method allows us to derive techniques
to compute explicitly the topological entropy, the Hausdorff dimension and the
escape rate.
|
lt256
|
arxiv_abstracts
|
math/0302355
|
This is the third in a series of five papers math.DG/0211294,
math.DG/0211295, math.DG/0302356, math.DG/0303272 studying compact special
Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with
singularities x_1,...,x_n locally modelled on special Lagrangian cones
C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to
begin with the final paper math.DG/0303272 which surveys the series, gives
examples, and applies the results to prove some conjectures.
The first two papers math.DG/0211294, math.DG/0211295 studied the regularity
of X near its singular points, and the moduli space of deformations of X. In
this paper and the fourth math.DG/0302356 we construct desingularizations of X,
realizing X as a limit of a family of compact, nonsingular SL m-folds \tilde
N^t in M for small t>0. Suppose L_1,...,L_n are Asymptotically Conical SL
m-folds in C^m, with L_i asymptotic to the cone C_i at infinity. We shrink L_i
by a small t>0, and glue tL_i into X at x_i for i=1,...,n to get a 1-parameter
family of compact, nonsingular Lagrangian m-folds N^t for small t>0.
Then we show using analysis that when t is sufficiently small we can deform
N^t to a compact, nonsingular SL m-fold \tilde N^t via a small Hamiltonian
deformation. This \tilde N^t depends smoothly on t, and as t --> 0 it converges
to the singular SL m-fold X, in the sense of currents.
This paper studies the simpler cases, where by topological conditions on X
and L_i we avoid various obstructions to existence of \tilde N^t. The sequel
math.DG/0302356 will consider more complex cases when these obstructions are
nontrivial, and also desingularization in families of almost Calabi-Yau
m-folds.
|
256
|
arxiv_abstracts
|
math/0302356
|
This is the fourth in a series of five papers math.DG/0211294,
math.DG/0211295, math.DG/0302355, math.DG/0303272 studying compact special
Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with
singularities x_1,...,x_n locally modelled on special Lagrangian cones
C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to
begin with the final paper math.DG/0303272 which surveys the series, gives
examples, and applies the results to prove some conjectures.
The first paper math.DG/0211294 studied the regularity of X near its singular
points, and the second math.DG/0211295 the moduli space of deformations of X.
The third paper math.DG/0302355 and this one construct desingularizations of X,
realizing X as a limit of a family of compact, nonsingular SL m-folds \tilde
N^t in M for small t>0.
Let L_1,...,L_n be Asymptotically Conical SL m-folds in C^m, with L_i
asymptotic to C_i at infinity. We shrink L_i by t>0, and glue tL_i into X at
x_i for i=1,...,n to get a 1-parameter family of compact, nonsingular
Lagrangian m-folds N^t for small t>0. Then we show using analysis that for
small t we can deform N^t to a compact, nonsingular SL m-fold \tilde N^t via a
small Hamiltonian deformation. As t --> 0 this \tilde N^t converges to X, in
the sense of currents.
The third paper math.DG/0302355 studied simpler cases, where by topological
conditions on X and L_i we avoid obstructions to existence of \tilde N^t. This
paper considers more complex cases when these obstructions are nontrivial, and
also desingularization in smooth families of almost Calabi-Yau m-folds M^s for
s in F, rather than a single almost Calabi-Yau m-fold M.
|
256
|
arxiv_abstracts
|
math/0302357
|
We investigate the asymptotic behavior of the polynomials p, q, r of degrees
n in type I Hermite-Pade approximation to the exponential function, defined by
p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are
characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We
use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to
obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz),
and r(3nz) in every domain in the complex plane. An important role is played by
a three-sheeted Riemann surface and certain measures and functions derived from
it. Our work complements recent results of Herbert Stahl.
|
lt256
|
arxiv_abstracts
|
math/0302358
|
In this paper we provide examples of hypercomplex manifolds which do not
carry HKT structure. We also prove that the existence of HKT structure is not
stable under small deformations. Similarly we provide examples of compact
complex manifolds with vanishing first Chern class which do not admit a
Hermitian structure with restricted holonomy of its Bismut connection in SU(n).
Again we prove that such property is not stable under small deformations.
|
lt256
|
arxiv_abstracts
|
math/0302359
|
In this paper we introduce a model which provides a new approach to the
phenomenon of stochastic resonance. It is based on the study of the properties
of the stationary distribution of the underlying stochastic process. We derive
the formula for the spectral power amplification coefficient, study its
asymptotic properties and dependence on parameters.
|
lt256
|
arxiv_abstracts
|
math/0302360
|
This is a correction to the afore-mentioned paper in Duke Math. J. vol. 75
(1994), 99-119 by S. Keel, K. Matsuki, and J. McKernan. We completely rewrite
Chapter 6 according to the original manuscript of the second author, in order
to fix some crucial mistakes pointed out by Dr. Qihong Xie.
|
lt256
|
arxiv_abstracts
|
math/0303001
|
In his previous papers (J. reine angew. Math. 544 (2002), 91--110;
math.AG/0103203) the author introduced a certain explicit construction of
superelliptic jacobians, whose endomorphism ring is the ring of integers in the
$p$th cyclotomic field. (Here $p$ is an odd prime.)
In the present paper we discuss when these jacobians are mutually
non-isogenous. (The case of hyperelliptic jacobians was treated in author's
e-print math.NT/0301173 .)
|
lt256
|
arxiv_abstracts
|
math/0303002
|
We study in this paper some local invariants attached via multiplier ideals
to an effective divisor or ideal sheaf on a smooth complex variety. First
considered (at least implicitly) by Libgober and by Loeser and Vaquie, these
jumping coefficients consist of an increasing sequence of rational numbers
beginning with the log canonical threshold of the divisor or ideal in question.
They encode interesting geometric and algebraic information, and we show that
they arise naturally in several different contexts. Given a polynomial f having
only isolated singularities, results of Varchenko, Loeser and Vaquie imply that
if \xi is a jumping number of f = 0 lying in the interval (0, 1], then -\xi is
a root of the Bernstein-Sato polynomial of f. We adapt an argument of Kollar to
show prove that this holds also when the singular locus of f has positive
dimension. In a more algebraic direction, we show that the number of such
jumping coefficients bounds the uniform Artin-Rees number of the principal
ideal (f) in the sense of Huneke: in the case of isolated singularities, this
in turn leads to bounds involving the Milnor and Tyurina numbers of f . Along
the way, we establish a general result relating multiplier to Jacobian ideals.
We also explore the extension of these ideas to the setting of graded families
of ideals. The paper contains many concrete examples.
|
256
|
arxiv_abstracts
|
math/0303003
|
Let M be a closed, connected manifold, and LM its loop space. In this paper
we describe closed string topology operations in h_*(LM), where h_* is a
generalized homology theory that supports an orientation of M. We will show
that these operations give h_*(LM) the structure of a unital, commutative
Frobenius algebra without a counit. Equivalently they describe a positive
boundary, two dimensional topological quantum field theory associated to
h_*(LM). This implies that there are operations corresponding to any surface
with p incoming and q outgoing boundary components, so long as q >0. The
absence of a counit follows from the nonexistence of an operation associated to
the disk, D^2, viewed as a cobordism from the circle to the empty set. We will
study homological obstructions to constructing such an operation, and show that
in order for such an operation to exist, one must take h_*(LM) to be an
appropriate homological pro-object associated to the loop space. Motivated by
this, we introduce a prospectrum associated to LM when M has an almost complex
structure. Given such a manifold its loop space has a canonical polarization of
its tangent bundle, which is the fundamental feature needed to define this
prospectrum. We refer to this as the "polarized Atiyah - dual" of LM . An
appropriate homology theory applied to this prospectrum would be a candidate
for a theory that supports string topology operations associated to any
surface, including closed surfaces.
|
256
|
arxiv_abstracts
|
math/0303004
|
By means of a penalization argument due to del Pino and Felmer, we prove the
existence of multi-spike solutions for a class of quasilinear elliptic
equations under natural growth conditions. Compared with the semilinear case
some difficulties arise, mainly concerning the properties of the limit
equation. The study of concentration of the solutions requires a somewhat
involved analysis in which a Pucci-Serrin type identity plays an important
role.
|
lt256
|
arxiv_abstracts
|
math/0303005
|
Every lattice is isomorphic to a lattice whose elements are sets of sets, and
whose operations are intersection and an operation extending the union of two
sets of sets A and B by the set of all sets in which the intersection of an
element of A and of an element of B is included. This representation spells out
precisely Birkhoff's and Frinks's representation of arbitrary lattices, which
is related to Stone's set-theoretic representation of distributive lattices.
|
lt256
|
arxiv_abstracts
|
math/0303006
|
By means of a penalization scheme due to del Pino and Felmer, we prove the
existence of single-peaked solutions for a class of singularly perturbed
quasilinear elliptic equations associated with functionals which lack of
smoothness. We don't require neither uniqueness assumptions on the limiting
autonomous equation nor monotonicity conditions on the nonlinearity. Compared
with the semilinear case some difficulties arise and the study of concentration
of the solutions needs a somewhat involved analysis in which the Pucci-Serrin
variational identity plays an important role.
|
lt256
|
arxiv_abstracts
|
math/0303007
|
This paper studies the behavior under iteration of the maps
T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and
bx if x<0. The orbits under iteration correspond to solutions of the nonlinear
difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This
family of maps has the parameter space (a,b)\in R^2. These maps are
area-preserving homeomorphisms of R^s that map rays from the origin into rays
from the origin. This paper shows the existence of special parameter values
where T_{ab} has every nonzero orbit an invariant circle with irrational
rotation number, and these invariant circles are piecewise unions of arcs of
conic sections. Numerical experiments suggest the possible existence of many
other parameter values having invariant circles.
|
lt256
|
arxiv_abstracts
|
math/0303008
|
In this paper we prove existence and multiplicity results of unbounded
critical points for a general class of weakly lower semicontinuous functionals.
We will apply a suitable nonsmooth critical point theory.
|
lt256
|
arxiv_abstracts
|
math/0303009
|
One generalizes the intuitionistic fuzzy logic (IFL) and other logics to
neutrosophic logic (NL). The distinctions between IFL and NL {and the
corresponding intuitionistic fuzzy set (IFS) and neutrosophic set (NS)
respectively} are presented.
|
lt256
|
arxiv_abstracts
|
math/0303010
|
By means of a perturbation method recently introduced by Bolle, we discuss
the existence of infinitely many solutions for a class of perturbed symmetric
higher order Schrodinger equations with non-homogeneous boundary data on
unbounded domains.
|
lt256
|
arxiv_abstracts
|
math/0303011
|
The prenex fragments of first-order infinite-valued Goedel logics are
classified. It is shown that the prenex Goedel logics characterized by finite
and by uncountable subsets of [0, 1] are axiomatizable, and that the prenex
fragments of all countably infinite Goedel logics are not axiomatizable.
|
lt256
|
arxiv_abstracts
|
math/0303012
|
We classify all knot diagrams of genus two and three, and give applications
to positive, alternating and homogeneous knots, including a classification of
achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a
proof of the 3- and 4-move conjectures, and the calculation of the maximal
hyperbolic volume for weak genus two knots. We also study the values of the
link polynomials at roots of unity, extending denseness results of Jones. Using
these values, examples of knots with unsharp Morton (weak genus) inequality are
found. Several results are generalized to arbitrary weak genus.
|
lt256
|
arxiv_abstracts
|
math/0303013
|
We prove a theorem which implies a quantum (multiplicative) analogue of the
Horn conjecture, and also of the saturation conjecture. We obtain
transversality statements for quantum schubert calculus in any characteristic
and also determine the smallest power of q in an arbitrary (small quantum)
product of Schubert varieties in a Grassmannian.
|
lt256
|
arxiv_abstracts
|
math/0303014
|
Let $GL_M$ be general linear Lie group over the complex field. The
irreducible rational representations of the group $GL_M$ are labeled by pairs
of partitions $\mu$ and $\tilde\mu$ such that the total number of non-zero
parts of $\mu$ and $\tilde{\mu}$ does not exceed $M$. Let $U$ be the
representation of $GL_M$ corresponding to such a pair. Regard the direct
product $GL_N\times GL_M$ as a subgroup of $GL_{N+M}$. Let $V$ be the
irreducible rational representation of the group $GL_{N+M}$ corresponding to a
pair of partitions $\lambda$ and $\tilde{\lambda}$. Consider the vector space
$W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $GL_N$. Let $n$
be sum of parts of $\lambda$ less the sum of parts of $\mu$. Let $\tilde{n}$ be
sum of parts of $\tilde{\lambda}$ less the sum of parts of $\tilde{\mu}$. For
any choice of two standard Young tableaux of skew shapes $\lambda/\mu$ and
$\tilde{\lambda}/\tilde{\mu}$ respectively, we realize $W$ as a subspace in the
tensor product of $n$ copies of the defining $N$-dimensional representation of
$GL_N$, and of $\tilde{n}$ copies of the contragredient representation. This
subspace is determined as the image of a certain linear operator $F$ in the
tensor product, given by explicit multiplicative formula. When M=0 and $W=V$ is
an irreducible representation of $GL_N$, we recover the classical realization
of $V$ as a subspace in the space of all traceless tensors. Then the operator
$F$ can be regarded as the rational analogue of the Young symmetrizer,
corresponding to the chosen standard tableau of shape $\lambda$. Even in the
special case M=0, our formula for the operator $F$ is new. Our results are
applications of representation theory of the Yangian of the Lie algebra $gl_N$.
|
256
|
arxiv_abstracts
|
math/0303015
|
The mod 2 cohomology algebra of the holomorph of any finite cyclic group
whose order is a power of 2 is determined.
|
lt256
|
arxiv_abstracts
|
math/0303016
|
A Lie-Rinehart algebra consists of a commutative algebra and a Lie algebra
with additional structure which generalizes the mutual structure of interaction
between the algebra of functions and the Lie algebra of smooth vector fields on
a smooth manifold. Lie-Rinehart algebras provide the correct categorical
language to solve the problem whether Kaehler quantization commutes with
reduction which, in turn, may be seen as a descent problem.
|
lt256
|
arxiv_abstracts
|
math/0303017
|
In an earlier paper, we used the absolute grading on Heegaard Floer homology
to give restrictions on knots in $S^3$ which admit lens space surgeries. The
aim of the present article is to exhibit stronger restrictions on such knots,
arising from knot Floer homology. One consequence is that all the non-zero
coefficients of the Alexander polynomial of such a knot are $\pm 1$. This
information in turn can be used to prove that certain lens spaces are not
obtained as integral surgeries on knots. In fact, combining our results with
constructions of Berge, we classify lens spaces $L(p,q)$ which arise as
integral surgeries on knots in $S^3$ with $|p|\leq 1500$. Other applications
include bounds on the four-ball genera of knots admitting lens space surgeries
(which are sharp for Berge's knots), and a constraint on three-manifolds
obtained as integer surgeries on alternating knots, which is closely to related
to a theorem of Delman and Roberts.
|
lt256
|
arxiv_abstracts
|
math/0303018
|
We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive
Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the recently added
condition (by P. B\'alint, N. Chernov, D. Sz\'asz, and I. P. T\'oth, in order
to save this fundamental result) on the algebraic character of the smooth
boundary components of the configuration space is unnecessary. Having saved the
theorem in its original form by using additional ideas in the spirit of the
initial proof, the result becomes stronger and it applies to a larger family of
models.
|
lt256
|
arxiv_abstracts
|
math/0303019
|
Let R be an integral domain, h non-zero in R such that R/hR is a field, and
HA the category of torsionless (or flat) Hopf algebras over R. We call any H in
HA "quantized function algebra" (=QFA), resp. "quantized (restricted) universal
enveloping algebra" (=QrUEA), at h if H/hH is the function algebra of a
connected Poisson group, resp. the (restricted, if R/hR has positive
characteristic) universal enveloping algebra of a (restricted) Lie bialgebra.
We establish an "inner" Galois' correspondence on HA, via the definition of
two endofunctors, ()^\vee and ()', of HA such that:
(a) the image of ()^\vee, resp. of ()', is the full subcategory of all
QrUEAs, resp. all QFAs, at h; (b) if R/hR has zero characteristic, the
restriction of ()^\vee to QFAs and of ()' to QrUEAs yield equivalences inverse
to each other; (c) if R/hR has zero characteristic, starting from a QFA over a
Poisson group, resp. from a QrUEA over a (restricted) Lie bialgebra, the
functor ()^\vee, resp. ()', gives a QrUEA, resp. a QFA, over the dual Lie
bialgebra, resp. the dual Poisson group.
In particular, (a) yields a recipe to produce quantum groups of both types
(QFAs or QrUEAs), (b) gives a characterization of them within HA, and (c) gives
a "global" version of the "quantum duality principle" after Drinfeld. We then
apply our result to Hopf algebras defined over a field k and extended to the
polynomial ring k[h]: this yields quantum groups, hence "classical" geometrical
symmetries of Poisson type (via specialization) associated to the "generalized
symmetry" encoded by the original Hopf algebra over k. Both the main result and
the above mentioned application are illustrated via several examples of many
different kinds, which are studied in full detail.
|
256
|
arxiv_abstracts
|
math/0303020
|
A class of representations of a Lie superalgebra (over a commutative
superring) in its symmetric algebra is studied. As an application we get a
direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt
theorem, extending this theorem to a class of nilpotent Lie superalgebras.
Other applications are presented. Our results are new already for Lie algebras.
|
lt256
|
arxiv_abstracts
|
math/0303021
|
The survey is devoted to associative $\Z_{\ge0}$-graded algebras presented by
n generators and n(n-1)/2 quadratic relations and satisfying the so-called
Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such
algebras depending on two continuous parameters (namely, on an elliptic curve
and a point on this curve) which are flat deformations of the polynomial ring
in n variables. Diverse properties of these algebras are described, together
with their relations to integrable systems, deformation quantization, moduli
spaces and other directions of modern investigations.
|
lt256
|
arxiv_abstracts
|
math/0303022
|
This paper develops the reduction theory of implicit Hamiltonian systems
admitting a symmetry group at a singular value of the momentum map. The results
naturally extend those known for (explicit) Hamiltonian systems described by
Poisson brackets.
|
lt256
|
arxiv_abstracts
|
math/0303023
|
We consider non-selfadjoint perturbations of a self-adjoint
$h$-pseudodifferential operator 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 satisfies $h^{\delta_0} <\epsilon \le
\epsilon_0$ for some $\delta_0\in ]0,1/2[$ and a sufficiently small $\epsilon
_0>0$. We get a complete asymptotic description of all eigenvalues in certain
rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$. In particular we are able
to treat the case when $\epsilon >0$ is small but independent of $h$.
|
lt256
|
arxiv_abstracts
|
math/0303024
|
We define a smooth functional calculus for a non-commuting tuple of
(unbounded) operators $A_j$ on a Banach space with real spectra and resolvents
with temperate growth, by means of an iterated Cauchy formula. The construction
is also extended to tuples of more general operators allowing smooth functional
calculii. We also discuss the relation to the case with commuting operators.
|
lt256
|
arxiv_abstracts
|
math/0303025
|
We prove a recent conjecture of Lassalle about positivity and integrality of
coefficients in some polynomial expansions. We also give a combinatorial
interpretation of those numbers. Finally, we show that this question is closely
related to the fundamental problem of calculating the linearization
coefficients for binomial coefficients.
|
lt256
|
arxiv_abstracts
|
math/0303026
|
In this article, we prove under some hypothesis of non ramification, a
conjecture of Kottwitz and Rapoport giving the existence of crystals with
additional structures.
|
lt256
|
arxiv_abstracts
|
math/0303027
|
We prove that the bar construction of an $E_\infty$ algebra forms an
$E_\infty$ algebra. To be more precise, we provide the bar construction of an
algebra over the surjection operad with the structure of a Hopf algebra over
the Barratt-Eccles operad. (The surjection operad and the Barratt-Eccles operad
are classical $E_\infty$ operads.)
|
lt256
|
arxiv_abstracts
|
math/0303028
|
We study the number of solutions of the general semigroup equation in one
variable, $X^\al=X^\be$, as well as of the system of equations $X^2=X, Y^2=Y,
XY=YX$ in $H\wr T_n$, the wreath product of an arbitrary finite group $H$ with
the full transformation semigroup $T_n$ on $n$ letters. For these solution
numbers, we provide explicit exact formulae, as well as asymptotic estimates.
Our results concerning the first mentioned problem generalize earlier results
by Harris and Schoenfeld (J. Combin. Theory Ser. A 3 (1967), 122-135) on the
number of idempotents in $T_n$, and a partial result of Dress and the second
author (Adv. in Math. 129 (1997), 188-221). Among the asymptotic tools employed
are Hayman's method for the estimation of coefficients of analytic functions
and the Poisson summation formula.
|
lt256
|
arxiv_abstracts
|
math/0303029
|
The classical HKR-theorem gives an isomorphism of the n-th Hochschild
cohomology of a smooth algebra and the n-th exterior power of its module of
K\"ahler differentials. Here we generalize it for simplicial, graded and
anticommutative objects in ``good pairs of categories''. We apply this
generalization to complex spaces and noetherian schemes and deduce two
decomposition theorems for their (relative) Hochschild cohomology (special
cases of those were recently shown by Buchweitz-Flenner and Yekutieli). The
first one shows that Hochschild cohomology contains tangent cohomology:
$\HH^n(X/Y,\sM)=\coprod_{i-j=n}\Ext^i(\dach^j\LL(X/Y),\sM)$. The left side is
the n-th Hochschild cohomology of $X$ over $Y$ with values in $\sM$. The right
hand-side contains the $n$-th relative tangent cohomology
$\Ext^n(\LL(X/Y),\sM)$ as direct factor. The second consequence is a
decomposition theorem for Hochschild cohomology of complex analytic manifolds
and smooth schemes in characteristic zero:
$\HH^n(X)=\coprod_{i-j=n}H^i(X,\dach^j\sT_X).$ On the right hand-side we have
the sheaf cohomology of the exterior powers of the tangent complex.
|
256
|
arxiv_abstracts
|
math/0303030
|
This is a very basic introduction to some notions related to logic and
complexity.
|
lt256
|
arxiv_abstracts
|
math/0303031
|
A global real analytic regularity theorem for a quasilinear sum of squares of
vector fields of Hormander rank 2 is given. A related local result for a
special case was proved recently by the second author and L. Zanghirati in a
paper titled "Local Real Analyticity of Solutions for sums of squares of
non-linear vector fields".
|
lt256
|
arxiv_abstracts
|
math/0303032
|
We show that all smooth solutions of model non-linear sums of squares of
vector fields are locally real analytic. A global result for more general
operators is presented in a paper by Makhlouf Derridj and the first author
under the title "Global Analytic Hypoellipticity for a Class of Quasilinear
Sums of Squares of Vector Fields".
|
lt256
|
arxiv_abstracts
|
math/0303033
|
Let M be a manifold, and G a Lie group which satisfies the unique extension
property. An (M,G) manifold N is a manifold endowed with an atlas (U_i,f_i)
where f_i is a diffeomorphism between U_i and an open set of M such that the
coordinates change defined by this atlas are restriction of elements of G. We
define the notion of geometric structures for toposes, and apply it to fields
theory. We also interpret the Beyli theorem in this setting.
|
lt256
|
arxiv_abstracts
|
math/0303034
|
We initiate the study of classical knots through the homotopy class of the
n-th evaluation map of the knot, which is the induced map on the compactified
n-point configuration space. Sending a knot to its n-th evaluation map realizes
the space of knots as a subspace of what we call the n-th mapping space model
for knots. We compute the homotopy types of the first three mapping space
models, showing that the third model gives rise to an integer-valued invariant.
We realize this invariant in two ways, in terms of collinearities of three or
four points on the knot, and give some explicit computations. We show this
invariant coincides with the second coefficient of the Conway polynomial, thus
giving a new geometric definition of the simplest finite-type invariant.
Finally, using this geometric definition, we give some new applications of this
invariant relating to quadrisecants in the knot and to complexity of polygonal
and polynomial realizations of a knot.
|
lt256
|
arxiv_abstracts
|
math/0303035
|
In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended)
Rees algebras in terms of some invariants of the base ring. Also, we give an
explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over
Veronese subrings.
|
lt256
|
arxiv_abstracts
|
math/0303036
|
We describe an efficient algorithm to write any element of the alternating
group A_n as a product of two n-cycles (in particular, we show that any element
of A_n can be so written -- a result of E. A. Bertram). An easy corollary is
that every element of A_n is a commutator in the symmetric group S_n.
|
lt256
|
arxiv_abstracts
|
math/0303037
|
We show that the projectivization of the exceptional rank 2 vector bundle on
an arbitrary smooth V14 Fano threefold after a certain natural flop turns into
the projectivization of an instanton vector bundle on a smooth cubic threefold.
And vice versa, starting from a smooth cubic threefold with an instanton vector
bundle of charge 2 on it we reconstruct V14 threefold.
Relying on the geometric properties of the above correspondence we prove that
the orthogonals to the exceptional pairs in the bounded derived categories of
coherent sheaves on a smooth V14 threefold and on the corresponding cubic
threefold are equivalent as triangulated categories.
|
lt256
|
arxiv_abstracts
|
math/0303038
|
Let $Z=X_1\times...\times X_n$ be a product of Drinfeld modular curves. We
characterize those algebraic subvarieties $X \subset Z$ containing a
Zariski-dense set of CM points, i.e. points corresponding to $n$-tuples of
Drinfeld modules with complex multiplication (and suitable level structure).
This is a characteristic $p$ analogue of a special case of the Andr\'e-Oort
conjecture. We follow closely the approach used by Bas Edixhoven in
characteristic zero, see math.NT/0302138. Note that in this paper we assume
that the characteristic $p$ is odd, and we only treat the case of Drinfeld
$F_q[T]$-modules.
|
lt256
|
arxiv_abstracts
|
math/0303039
|
In this paper, we consider the problem of prescribing the scalar curvature
under minimal boundary conditions on the standard four dimensional half sphere.
We provide an Euler-Hopf type criterion for a given function to be a scalar
curvature to a metric conformal to the standard one. Our proof involves the
study of critical points at infinity of the associated variational problem.
|
lt256
|
arxiv_abstracts
|
math/0303040
|
We define a notion of quasi-static evolution for the elliptic approximation
of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we
prove that this regular evolution converges to a quasi-static growth of brittle
fractures in linearly elastic bodies.
|
lt256
|
arxiv_abstracts
|
math/0303041
|
We discuss a special class of solutions to the minimal surface system. These
are vector-valued functions that "decrease area" and are natural generalization
of scalar functions. After defining area-decreasing maps, we show several
classical results for the minimal surface equation can be generalized. We also
conjecture the solvability of Dirichlet problems within the category of
area-decreasing maps.
|
lt256
|
arxiv_abstracts
|
math/0303042
|
A left orderable completely metrizable topological group is exhibited
containing Artin's braid group on infinitely many strands. The group is the
mapping class group (rel boundary) of the closed unit disk with a sequence of
interior punctures converging to the boundary. This resolves an issue suggested
by work of Dehornoy.
|
lt256
|
arxiv_abstracts
|
math/0303043
|
\medskip\noindent\textbf{R\'esum\'e.} Soit $l$ un entier et $\ors=(s_1,
\dots, s_l)$ une s\'equence d'entiers positifs. Dans ce document, nous
\'etudierons les propri\'et\'es arithm\'etique de sommes harmoniques multiples
$H(\ors; n)$, qui est le $n$-\`eme somme partielle de la valeur de la s\'erie
multiple zeta $\zeta(\ors)$. On conjecture que pour tout $\ors$ et de tous les
premiers $p$, il n'y a que de nombreux finitely $p$-partie int\'egrante sommes
$H(\ors,n)$. Ceci g\'en\'eralise une conjecture de Eswarathasan et Levine et
Boyd pour la s\'erie harmonique. Nous fournissons beaucoup d'\'el\'ements de
preuve pour cette conjecture g\'en\'erale ainsi que certaines heuristiques
argument soutenir. Ce document fait suite \`a \emph{Wolstenholme Type Theorem
for multiple harmonic sums}, Intl.\ J.\ of Number Theory \textbf{4}(1) (2008)
73-106.
|
lt256
|
arxiv_abstracts
|
math/0303044
|
We construct examples of elliptic fibrations of orbifold general type (in the
sense of Campana) which have no etale covers dominating a variety of general
type.
|
lt256
|
arxiv_abstracts
|
math/0303045
|
It is shown that if one keeps track of crossings, Feynman diagrams can be
used to compute $q$-Wick products and normal products in terms of each other.
|
lt256
|
arxiv_abstracts
|
math/0303046
|
A survey of the applications of the noncommutative Cohn localization of rings
to the topology of manifolds with infinite fundamental group, with particular
emphasis on the algebraic K- and L-theory of generalized free products.
|
lt256
|
arxiv_abstracts
|
math/0303047
|
This paper contains a long summary of the basic properties of higher FR
torsion. An attempt is made to simplify the constructions from my book Higher
Franz-Reidemeister Torsion (IP/AMS Studies in Advanced Math 31). Some new basic
theorems are also proved such as the Framing Principle in full generality. This
is used to compute the higher torsion for bundles with closed even dimensional
fibers. We construct a higher complex torsion for bundles with almost complex
fibers. This is shown to generalize the real even dimensional higher FR
torsion. We also show that the higher complex torsion is a multiple of
generalized Miller-Morita-Mumford classes.
|
lt256
|
arxiv_abstracts
|
math/0303048
|
We construct a compact nonpositively curved squared 2-complex whose universal
cover contains a flat plane that is not the limit of periodic flat planes.
|
lt256
|
arxiv_abstracts
|
math/0303049
|
We solve the problem of constructing all chiral genus-one correlation
functions from chiral genus-zero correlation functions associated to a vertex
operator algebra satisfying the following conditions: (i) the weight of any
nonzero homogeneous elements of V is nonnegative and the weight zero subspace
is one-dimensional, (ii) every N-gradable weak V-module is completely reducible
and (iii) V is C_2-cofinite. We establish the fundamental properties of these
functions, including suitably formulated commutativity, associativity and
modular invariance. The method we develop and use here is completely different
from the one previously used by Zhu and other people. In particular, we show
that the $q$-traces of products of certain geometrically-modified intertwining
operators satisfy modular invariant systems of differential equations which,
for any fixed modular parameter, reduce to doubly-periodic systems with only
regular singular points. Together with the results obtained by the author in
the genus-zero case, the results of the present paper solves essentially the
problem of constructing chiral genus-one weakly conformal field theories from
the representations of a vertex operator algebra satisfying the conditions
above.
|
256
|
arxiv_abstracts
|
math/0303050
|
In 1988, Brown and Ellis published [3] a generalised Hopf formula for the
higher homology of a group. Although substantially correct, their result lacks
one necessary condition. We give here a counterexample to the result without
that condition. The main aim of this paper is, however, to generalise this
corrected result to derive formulae of Hopf type for the n-fold Cech derived
functors of the lower central series functors Z_k. The paper ends with an
application to algebraic K-theory.
|
lt256
|
arxiv_abstracts
|
math/0303051
|
(On the fundamental group of rationnally connected varieties.) I show that
the fundamental group of a normal variety which is rationally chain connected
is finite. The proof holds in non-zero characteristic.
Je d\'emontre que le groupe fondamental d'une vari\'et\'e normale
rationnellement connexe par cha\^{\i}nes est fini. La d\'emonstration est
valable en caract\'eristique diff\'erente de z\'ero.
|
lt256
|
arxiv_abstracts
|
math/0303052
|
In this talk, I report on three theorems concerning algebraic varieties over
a field of characteristic $p>0$. a) over a finite field of cardinal $q$, two
proper smooth varieties which are geometrically birational have the same number
of rational points modulo $q$ (cf. Ekedahl, 1983). b) over a finite field of
cardinal $q$, a proper smooth variety which is rationally chain connected, or
Fano, or weakly unirational, has a number of rational points congruent to 1
modulo $q$ (Esnault, 2003). c) over an algebraic closed field of caracteristic
$p>0$, the fundamental group of a proper smooth variety which is rationally
chain connected, or Fano, or weakly unirational, is a finite group of order
prime to $p$ (cf. Ekedahl, 1983). The common feature of the proofs is a control
of the $p$-adic valuations of Frobenius and is best explained within the
framework of Berthelot's rigid cohomology. I also explain its relevant
properties.
|
lt256
|
arxiv_abstracts
|
math/0303053
|
An affine Cartan calculus is developed. The concepts of special affine
bundles and special affine duality are introduced. The canonical isomorphisms,
fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the
affine setting are proved.
|
lt256
|
arxiv_abstracts
|
math/0303054
|
Gel'fand triples of test and generalized functionals in Gaussian spaces are
constructed and characterized.
|
lt256
|
arxiv_abstracts
|
math/0303055
|
The aim of this paper is to prove an important generalization of the
construction of the Incidence Divisor given in [BMg]. Let Z be a complex
manifold and (X_{s})_{s\in S}an family of n-cycles (not necessarily compact) in
Z parametrized by reduced complex space S. Then, to any n+1- codimensional
cycle Y in Z wich satisfies the following condition : the analytic set (S\times
|Y|)\cap |X| in S\times Z is S-proper and generically finite on its image
|\Sigma_{Y}| wich is nowhere dense in S, is associated a Cartier Divisor
\Sigma_{Y} with support |\Sigma_{Y}|. Nice functorial properties of this
correspondance are proven and we deduce the intersection number of this divisor
with a curve in S.
|
lt256
|
arxiv_abstracts
|
math/0303056
|
The relation between differential geometry of surfaces and some Heisenberg
ferromagnet models is considered.
|
lt256
|
arxiv_abstracts
|
math/0303057
|
In this issue we announce a fascinating series of works on the comparison of
various types of convergence of sequences of functions. Some of these
properties are provably related to some of the properties which were introduced
in the earlier issues of the SPM Bulletin, and many problems remain open.
Section 2, written by Lev Bukovsk\'y, contains a brief survey of some of the
major open problems in this area.
This issue gives the first example of the importance of the transmission of
knowledge between the recipients of this bulletin: One of the announcements
implies a solution to one of the problems posed in an independent paper
announced here. looking forward to receive more announcements from other
recipients and readers of the bulletin.
|
lt256
|
arxiv_abstracts
|
math/0303058
|
We show that the non-trivially associated tensor category constructed from
left coset representatives of a subgroup of a finite group is a modular
category. Also we give a definition of the character of an object in a ribbon
category which is the category of representations of a braided Hopf algebra in
the category. The definition is shown to be adjoint invariant and
multiplicative. A detailed example is given. Finally we show an equivalence of
categories between the non-trivially associated double D and the category of
representations of the double of the group D(X).
|
lt256
|
arxiv_abstracts
|
math/0303059
|
Hasegawa and Petz introduced the notion of dual statistically monotone
metrics. They also gave a characterisation theorem showing that
Wigner-Yanase-Dyson metrics are the only members of the dual family. In this
paper we show that the characterisation theorem holds true under more general
hypotheses.
|
lt256
|
arxiv_abstracts
|
math/0303060
|
Jensen's trace inequality is established for every multivariable, convex
function and every trace or trace-like functional on a C*-algebra.
|
lt256
|
arxiv_abstracts
|
math/0303061
|
A hypergeometric identity equating a triple sum to a single sum, originally
found by Gelfand, Graev and Retakh [Russian Math. Surveys 47 (1992), 1-88] by
using systems of differential equations, is given hypergeometric proofs. As a
bonus, several $q$-analogues can be derived.
|
lt256
|
arxiv_abstracts
|
math/0303062
|
Our aim is to construct fibrewise localizations in model categories. For
pointed spaces, the general idea is to decompose the total space of a fibration
as a diagram over the category of simplices of the base and replace it by the
localized diagram. This of course is not possible in an arbitrary category. We
have thus to adapt another construction which heavily depends on Mather's cube
theorem. Working with model categories in which the cube theorem holds, we
characterize completely those who admit a fibrewise nullification. As an
application we get fibrewise plus-construction and fibrewise Postnikov sections
for algebras over an operad.
|
lt256
|
arxiv_abstracts
|
math/0303063
|
We study the asymptotic behavior of the simple random walk on oriented
version of $\mathbb{Z}^2$. The considered latticesare not directed on the
vertical axis but unidirectional on the horizontal one, with symmetric random
orientations which are positively correlated. We prove that the simple random
walk is transient and also prove a functionnal limit theorem in the space of
cadlag functions, with an unconventional normalization.
|
lt256
|
arxiv_abstracts
|
math/0303064
|
The paper is related to the following question of P.~L.~Ul'yanov: is it true
that for any $2\pi$-periodic continuous function $f$ there is a uniformly
convergent rearrangement of its trigonometric Fourier series? In particular, we
give an affirmative answer if the absolute values of Fourier coefficients of
$f$ decrease. Also, we study a problem how to choose $m$ terms of a
trigonometric polynomial of degree $n$ to make the uniform norm of their sum as
small as possible.
|
lt256
|
arxiv_abstracts
|
math/0303065
|
This paper is devoted to a detailed study of certain remarkable posets which
form a natural partition of all abelian ideals of a Borel subalgebra. Our main
result is a nice uniform formula for the dimension of maximal ideals in these
posets. We also obtain results on ad-nilpotent ideals which complete the
analysis started in \cite{CP2}, \cite{CP3}.
|
lt256
|
arxiv_abstracts
|
math/0303066
|
This survey text deals with irrationality, and linear independence over the
rationals, of values at positive odd integers of Riemann zeta function. The
first section gives all known proofs (and connections between them) of
Ap\'ery's Theorem (1978) : $\zeta(3)$ is irrational. The second section is
devoted to a variant of the proof, published by Rivoal and Ball-Rivoal, that
infinitely many $\zeta(2n+1)$ are irrational. The end of this text deals with
more quantitative statements.
|
lt256
|
arxiv_abstracts
|
math/0303067
|
The aim of this paper is to apply the work of Morris on Eisenstein series
over global function fields to the study of the asymptotic behavior of the
points of bounded height on a generalized flag variety defined as the quotient
of a semi-simple algebraic group by a reduced parabolic subgroup over such a
field. The formula obtained for the height zeta function has an interpretation
similar to the one known over a number field.
|
lt256
|
arxiv_abstracts
|
math/0303068
|
We consider Hopf crossed products of the the type $A#_\sigma \mathcal{H}$,
where $\mathcal{H}$ is a cocommutative Hopf algebra, $A$ is an
$\mathcal{H}$-module algebra and $\sigma$ is a "numerical" convolution
invertible 2-cocycle on $\mathcal{H}$. we give an spectral sequence that
converges to the cyclic homology of $A#_\sigma \mathcal{H}$ and identify the
$E^1$ and $E^2$ terms of the spectral sequence.
|
lt256
|
arxiv_abstracts
|
math/0303069
|
We review recent progress in the study of cyclic cohomology of Hopf algebras,
Hopf algebroids, and invariant cyclic homology starting with the pioneering
work of Connes-Moscovici.
|
lt256
|
arxiv_abstracts
|
math/0303070
|
In this paper, we study the local spectral properties for both unilateral and
bilateral weighted shift operators.
|
lt256
|
arxiv_abstracts
|
math/0303071
|
Bernoulli sieve is a recursive construction of a random composition (ordered
partition) of integer $n$. This composition can be induced by sampling from a
random discrete distribution which has frequencies equal to the sizes of
component intervals of a stick-breaking interval partition of $[0,1]$. We
exploit Markov property of the composition and its renewal representation to
derive asymptotics of the moments and to prove a central limit theorem for the
number of parts.
|
lt256
|
arxiv_abstracts
|
math/0303072
|
We propose a A.G.M. algorithm for the determination of the characteristic
polynomial of an ordinary non hyperelliptic curve of genus 3 over F_{2^N}.
|
lt256
|
arxiv_abstracts
|
math/0303073
|
In the present paper we study the Lie sphere geometry of Legendre surfaces by
the method of moving frame and we prove an existence theorem for real-analytic
Lie-minimal Legendre surfaces.
|
lt256
|
arxiv_abstracts
|
math/0303074
|
These are the notes for the lecture given by the author at the "Current
Events" Special Session of the AMS meeting in Baltimore on January 17, 2003.
Topics reviewed include the Langlands correspondence for GL(n) in the function
field case and its proof by V.Drinfeld and L.Lafforgue, the geometric Langlands
correspondence for GL(n) and its proof by D.Gaitsgory, K.Vilonen and the
author, and the work of A.Beilinson and V.Drinfled on the quantization of the
Hitchin system and the Langlands correspondence for an arbitrary semisimple
algebraic group.
|
lt256
|
arxiv_abstracts
|
math/0303075
|
We study the structure of abelian subgroups of Galois groups of function
fields of surfaces.
|
lt256
|
arxiv_abstracts
|
math/0303076
|
Stasheff showed that if a map between H-spaces is an H-map, then the
suspension of the map is extendable to a map between cprojective planes of the
H-spaces. Stahseff also proved the converse under the assumption that the
multiplication of the target space of the map is homotopy associative. We show
by giving an example that the assumption of homotopy associativity of the
multiplication of the target space is necessary to show the converse. We also
show an analogous fact for maps between higher homotopy associative H-spaces.
|
lt256
|
arxiv_abstracts
|
math/0303077
|
We study virtual isotopy sequences with classical initial and final diagrams,
asking when such a sequence can be changed into a classical isotopy sequence by
replacing virtual crossings with classical crossings. An example of a sequence
for which no such virtual crossing realization exists is given. A conjecture on
conditions for realizability of virtual isotopy sequences is proposed, and a
sufficient condition for realizability is found. The conjecture is reformulated
in terms of 2-knots and knots in thickened surfaces.
|
lt256
|
arxiv_abstracts
|
math/0303078
|
We show the existence of (non-Hermitian) strict quantization for every almost
Poisson manifold.
|
lt256
|
arxiv_abstracts
|
math/0303079
|
We deal with the ``nonrelativistic limit'', i.e. the limit c to infinity,
where c is the speed of light, of the nonlinear PDE system obtained by coupling
the Dirac equation for a 4-spinor to the Maxwell equations for the
self-consistent field created by the ``moving charge'' of the spinor. This
limit, sometimes also called ``Post-Newtonian'' limit, yields a
Schroedinger-Poisson system, where the spin and the magnetic field no longer
appear. However, our splitting of the 4-spinor into two 2-spinors preserves the
symmetry of "electrons'' and "positrons''; the latter obeying a Schroedinger
equation with ``negative mass'' in the limit.
We rigorously prove that in the nonrelativistic limit solutions of the
Dirac-Maxwell system converge in the energy space $C([0,T];H^{1})$ to solutions
of a Schroedinger-Poisson system, under appropriate (convergence) conditions on
the initial data.
We also prove that the time interval of existence of local solutions of
Dirac-Maxwell is bounded from below by log(c). In fact, for this result we only
require uniform $H^{1}$ bounds on the initial data, not convergence.
Our key technique is "null form estimates'', extending the work of Klainerman
and Machedon and our previous work on the nonrelativistic limit of the
Klein-Gordon-Maxwell system.
|
256
|
arxiv_abstracts
|
math/0303080
|
We consider the parabolic equation $$u_t-\Delta u=F(x,u),\quad
(t,x)\in\R_+\times\R^n\tag{P}$$ and the corresponding semiflow $\pi$ in the
phase space $H^1$. We give conditions on the nonlinearity $F(x,u)$, ensuring
that all bounded sets of $H^1$ are $\pi$-admissibile in the sense of
Rybakowski. If $F(x,u)$ is asymptotically linear, under appropriate
non-resonance conditions, we use Conley's index theory to prove the existence
of nontrivial equilibria of (P) and of heteroclinic trajectories joining some
of these equilibria. The results obtained in this paper extend earlier results
of Rybakowski concerning parabolic equations on {\it bounded} open subsets of
$\R^n$.
|
lt256
|
arxiv_abstracts
|
math/0303081
|
The twisted face-pairing construction of our earlier papers gives an
efficient way of generating, mechanically and with little effort, myriads of
relatively simple face-pairing descriptions of interesting closed 3-manifolds.
The corresponding description in terms of surgery, or Dehn-filling, reveals the
twist construction as a carefully organized surgery on a link.
In this paper, we work out the relationship between the twisted face-pairing
description of closed 3-manifolds and the more common descriptions by surgery
and Heegaard diagrams. We show that all Heegaard diagrams have a natural
decomposition into subdiagrams called Heegaard cylinders, each of which has a
natural shape given by the ratio of two positive integers. We characterize the
Heegaard diagrams arising naturally from a twisted face-pairing description as
those whose Heegaard cylinders all have integral shape. This characterization
allows us to use the Kirby calculus and standard tools of Heegaard theory to
attack the problem of finding which closed, orientable 3-manifolds have a
twisted face-pairing description.
|
256
|
arxiv_abstracts
|
math/0303082
|
This paper extends to dimension 4 the results in the article "Second Order
Families of Special Lagrangian 3-folds" by Robert Bryant. We consider the
problem of classifying the special Lagrangian 4-folds in C^4 whose fundamental
cubic at each point has a nontrivial stabilizer in SO(4). Points on special
Lagrangian 4-folds where the SO(4)-stabilizer is nontrivial are the analogs of
the umbilical points in the classical theory of surfaces. In proving existence
for the families of special Lagrangian 4-folds, we used the method of exterior
differential systems in Cartan-Kahler theory. This method is guaranteed to tell
us whether there are any families of special Lagrangian submanifolds with a
certain symmetry, but does not give us an explicit description of the
submanifolds. To derive an explicit description, we looked at foliations by
submanifolds and at other geometric particularities. In this manner, we settled
many of the cases and described the families of special Lagrangian submanifolds
in an explicit way.
|
256
|
arxiv_abstracts
|
math/0303083
|
Based on the monoid classifier, we give an alternative axiomatization of
Freyd's paracategories, which can be interpreted in any bicategory of partial
maps. Assuming furthermore a free-monoid monad T in our ambient category, and
coequalisers satisfying some exactness conditions, we give an abstract envelope
construction, putting paramonoids (and paracategories) in the more general
context of partial algebras. We introduce for the latter the crucial notion of
saturation, which characterises those partial algebras which are isomorphic to
the ones obtained from their enveloping algebras. We also set up a
factorisation system for partial algebras, via epimorphisms and (monic) Kleene
morphisms and relate the latter to saturation.
|
lt256
|
arxiv_abstracts
|
math/0303084
|
The support of a matrix M is the (0,1)-matrix with ij-th entry equal to 1 if
the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose
adjacency matrix is the support of M is said to be the digraph of M. This paper
observes some structural properties of digraphs of unitary matrices.
|
lt256
|
arxiv_abstracts
|
math/0303085
|
Let $F \hookrightarrow X \to B$ be a fibre bundle with structure group $G$,
where $B$ is $(d{-}1)$-connected and of finite dimension, $d \geq 1$. We prove
that the strong L-S category of $X$ is less than or equal to $m + \frac{\dim
B}{d}$, if $F$ has a cone decomposition of length $m$ under a compatibility
condition with the action of $G$ on $F$. This gives a consistent prospect to
determine the L-S category of non-simply connected Lie groups. For example, we
obtain $\cat{PU(n)} \leq 3(n{-}1)$ for all $n \geq 1$, which might be best
possible, since we have $\cat{\mathrm{PU}(p^r)}=3(p^r{-}1)$ for any prime $p$
and $r \geq 1$. Similarly, we obtain the L-S category of $\mathrm{SO}(n)$ for
$n \leq 9$ and $\mathrm{PO}(8)$. We remark that all the above Lie groups
satisfy the Ganea conjecture on L-S category.
|
lt256
|
arxiv_abstracts
|
math/0303086
|
Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We
give a condition for $R$ to have a non-free module of G-dimension zero. We
shall also construct a family of non-isomorphic indecomposable modules of
G-dimension zero with parameters in an open subset of projective space. We
shall finally show that the subcategory consisting of modules of G-dimension
zero over $R$ is not necessarily a contravariantly finite subcategory in the
category of finitely generated $R$-modules.
|
lt256
|
arxiv_abstracts
|
math/0303087
|
We define geometric crystals and unipotent crystals for arbitrary Kac-Moody
groups and describe geometric and unipotent crystal structures on the Schubert
varieties.
|
lt256
|
arxiv_abstracts
|
math/0303088
|
Tropical R is the birational map that intertwines products of geometric
crystals and satisfies the Yang-Baxter equation. We show that the D^{(1)}_n
tropical R introduced by the authors and its reduction to A^{(2)}_{2n-1} and
C^{(1)}_n are equivalent to a system of bilinear difference equations of Hirota
type. Associated tropical vertex models admit solutions in terms of tau
functions of the BKP and DKP hierarchies.
|
lt256
|
arxiv_abstracts
|
math/0303089
|
It is shown that if there is a measurable cardinal above n Woodin cardinals
and M_{n+1}^# doesn't exist then K exists. K is not fully iterable, though, but
only iterable with respect to stacks of certain trees living between the Woodin
cardinals. However, it is still true that if M is an omega-closed iterate of V
then K^M is an iterate of K.
|
lt256
|
arxiv_abstracts
|
math/0303090
|
We show that the Voiculescu-Brown entropy of a noncommutative toral
automorphism arising from a matrix S in GL(d,Z) is at least half the value of
the topological entropy of the corresponding classical toral automorphism. We
also obtain some information concerning the positivity of local
Voiculescu-Brown entropy with respect to single unitaries. In particular we
show that if S has no roots of unity as eigenvalues then the local
Voiculescu-Brown entropy with respect to every product of canonical unitaries
is positive, and also that in the presence of completely positive CNT entropy
the unital version of local Voiculescu-Brown entropy with respect to every
non-scalar unitary is positive.
|
lt256
|
arxiv_abstracts
|
math/0303091
|
We continue the analysis, started by Abreu, McDuff and Anjos, of the topology
of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the
areas of the two spheres lies in the interval (1,2]. We express the group, up
to homotopy, as the amalgam of certain of its compact Lie subgroups. We use
this to compute the homotopy type of the classifying space of the group of
symplectomorphisms and the corresponding ring of characteristic classes for
symplectic fibrations.
|
lt256
|
arxiv_abstracts
|
math/0303092
|
We show that any closed biquotient with finite fundamental group admits
metrics of positive Ricci curvature. Also, let M be a closed manifold on which
a compact Lie group G acts with cohomogeneity one, and let L be a closed
subgroup of G which acts freely on M. We show that the quotient N := M/L
carries metrics of nonnegative Ricci and almost nonnegative sectional
curvature. Moreover, if N has finite fundamental group, then N admits also
metrics of positive Ricci curvature. Particular examples include infinite
families of simply connected manifolds with the rational cohomology rings and
integral homology of complex and quaternionic projective spaces.
|
lt256
|
arxiv_abstracts
|
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