Dataset Viewer (First 5GB)
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math/0212159 | I give a construction of compact group action on a finite dimensional space
Y, whose orbit space is infinite dimensional.
| lt256 | arxiv_abstracts |
math/0212160 | In this article, we give a theorem of reduction of the structure group of a
principal bundle P with regular structure group G. Then, when G is in the
classes of Lie groups defined by T.Robart [13], we define the closed holonomy
group of a connection as the minimal closed Lie subgroup of G for which the
previous theorem of reduction can be applied. We also prove an infinite
dimensional version of the Ambrose-Singer theorem: the Lie algebra of the
holonomy group is spanned by the curvature elements.
| lt256 | arxiv_abstracts |
math/0212161 | Let R be a standard graded ring over a commutative Noetherian ring with unity
and I a graded ideal of R. Let M be a finitely generated graded R-module. We
prove that there exist integers e and \rho_M(I) such that for all large n,
reg(I^nM)= \rho_M(I)n+e.
| lt256 | arxiv_abstracts |
math/0212162 | In this paper, we give an affirmative answer to a conjecture raised by Polini
and Ulrich.
| lt256 | arxiv_abstracts |
math/0212163 | Bivariate generating functions for various subsets of the class of
permutations containing no descending sequence of length three or more are
determined. The notion of absolute indecomposability of a permutation is
introduced, and used in enumerating permutations which have a block structure
avoiding 321 and whose blocks also have such structure (recursively).
Generalizations of these results are discussed.
| lt256 | arxiv_abstracts |
math/0212164 | We prove $L^p$ bounds in the range $1<p<\infty$ for a maximal dyadic sum
operator on $\rn$. This maximal operator provides a discrete multidimensional
model of Carleson's operator. Its boundedness is obtained by a simple twist of
the proof of Carleson's theorem given by Lacey and Thiele, adapted in higher
dimensions by Pramanik and Terwilleger. In dimension one, the $\lp$ boundedness
of this maximal dyadic sum implies in particular an alternative proof of Hunt's
extension of Carleson's theorem on almost everywhere convergence of Fourier
integrals.
| lt256 | arxiv_abstracts |
math/0212165 | We give new general formulas for the asymptotics of the number of spanning
trees of a large graph. A special case answers a question of McKay (1983) for
regular graphs. The general answer involves a quantity for infinite graphs that
we call "tree entropy", which we show is a logarithm of a normalized
determinant of the graph Laplacian for infinite graphs. Tree entropy is also
expressed using random walks. We relate tree entropy to the metric entropy of
the uniform spanning forest process on quasi-transitive amenable graphs,
extending a result of Burton and Pemantle (1993).
| lt256 | arxiv_abstracts |
math/0212166 | We consider a normalized basis in a Banach space with the following property:
any normalized block sequence of the basis has a subsequence equivalent to the
basis. We show that under uniformity or other natural assumptions, a basis with
this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. We
also address an analogous problem for spreading models.
| lt256 | arxiv_abstracts |
math/0212167 | For the hypoelliptic differential operators $P={\partial^2_ x}+(x^k\partial_
y -x^l{\partial_t})^2$ introduced by T. Hoshiro, generalizing a class of M.
Christ, in the cases of $k$ and $l$ left open in the analysis, the operators
$P$ also fail to be {\em{analytic}} hypoelliptic (except for $(k,l)=(0,1)$), in
accordance with Treves' conjecture. The proof is constructive, suitable for
generalization, and relies on evaluating a family of eigenvalues of a
non-self-adjoint operator.
| lt256 | arxiv_abstracts |
math/0212168 | We classify unital associative conformal algebras of linear growth and
provide new examples of such.
| lt256 | arxiv_abstracts |
math/0212169 | Let F denote a homogeneous degree 4 polynomial in 3 variables, and let s be
an integer between 1 and 5. We would like to know if F can be written as a sum
of fourth powers of s linear forms (or a degeneration). We determine necessary
and sufficient conditions for this to be possible. These conditions are
expressed as the vanishing of certain concomitants of F for the natural action
of SL_3.
| lt256 | arxiv_abstracts |
math/0212170 | We establish the central limit theorem for the number of groups at the
equilibrium of a coagulation-fragmentation process given by a parameter
function with polynomial rate of growth. The result obtained is compared with
the one for random combinatorial structures obeying the logarithmic condition.
| lt256 | arxiv_abstracts |
math/0212171 | We consider a nonlinear semi-classical Schrodinger equation for which it is
known that quadratic oscillations lead to focusing at one point, described by a
nonlinear scattering operator. If the initial data is an energy bounded
sequence, we prove that the nonlinear term has an effect at leading order only
if the initial data have quadratic oscillations; the proof relies on a
linearizability condition (which can be expressed in terms of Wigner measures).
When the initial data is a sum of such quadratic oscillations, we prove that
the associate solution is the superposition of the nonlinear evolution of each
of them, up to a small remainder term. In an appendix, we transpose those
results to the case of the nonlinear Schrodinger equation with harmonic
potential.
| lt256 | arxiv_abstracts |
math/0212172 | In this preprint the notion of deformation quantization of endomorphism
bundles over symplectic manifolds is defined and developed, including index
theory.
| lt256 | arxiv_abstracts |
math/0212173 | We characterize the class of ideals of a polynomial ring such that the
hilbert series of their graded local cohomology modules is maximal.
| lt256 | arxiv_abstracts |
math/0212174 | Given a Brownian motion $B_t$ and a general target law $\mu$ (not necessarily
centered or even integrable) we show how to construct an embedding of $\mu$ in
$B$. This embedding is an extension of an embedding due to Perkins, and is
optimal in the sense that it simultaneously minimises the distribution of the
maximum and maximises the distribution of the minimum among all embeddings of
$\mu$. The embedding is then applied to regular diffusions, and used to
characterise the target laws for which a $H^p$-embedding may be found.
| lt256 | arxiv_abstracts |
math/0212175 | We give a simple interpretation of the adapted complex structure of
Lempert-Szoke and Guillemin-Stenzel: it is given by a polar decomposition of
the complexified manifold. We then give a twistorial construction of an
SO(3)-invariant hypercomplex structure on a neighbourhood of $X$ in $TTX$,
where $X$ is a real-analytic manifold equipped with a linear connection. We
show that the Nahm equations arise naturally in this context: for a connection
with zero curvature and arbitrary torsion, the real sections of the twistor
space can be obtained by solving Nahm's equations in the Lie algebra of certain
vector fields. Finally, we show that, if we start with a metric connection,
then our construction yields an SO(3)-invariant hyperk\"ahler metric.
| lt256 | arxiv_abstracts |
math/0212176 | We examine the moduli of framed holomorphic bundles over the blowup of a
complex surface, by studying a filtration induced by the behavior of the
bundles on a neighborhood of the exceptional divisor.
| lt256 | arxiv_abstracts |
math/0212177 | We notice that for any positive integer $k$, the set of $(1,2)$-specialized
characters of level $k$ standard $A_{1}^{(1)}$-modules is the same as the set
of rescaled graded dimensions of the subspaces of level $2k+1$ standard
$A_{2}^{(2)}$-modules that are vacuum spaces for the action of the principal
Heisenberg subalgebra of $A_{2}^{(2)}$. We conjecture the existence of a
semisimple category induced by the "equal level" representations of some
algebraic structure which would naturally explain this duality-like property,
and we study potential such structures in the context of generalized vertex
operator algebras.
| lt256 | arxiv_abstracts |
math/0212178 | We prove that any pair of bivariate trinomials has at most 5 isolated roots
in the positive quadrant. The best previous upper bounds independent of the
polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate
roots) via a famous general result of Khovanski. Our bound is sharp, allows
real exponents, allows degeneracies, and extends to certain systems of
n-variate fewnomials, giving improvements over earlier bounds by a factor
exponential in the number of monomials. We also derive analogous sharpened
bounds on the number of connected components of the real zero set of a single
n-variate m-nomial.
| lt256 | arxiv_abstracts |
math/0212179 | Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of
supports. Our main result is an upper bound on the probability that the
condition number of f in a region U is larger than 1/epsilon. The bound depends
on an integral of a differential form on a toric manifold and admits a simple
explicit upper bound when the Newton polytopes (and underlying covariances) are
all identical.
We also consider polynomials with real coefficients and give bounds for the
expected number of real roots and (restricted) condition number. Using a Kahler
geometric framework throughout, we also express the expected number of roots of
f inside a region U as the integral over U of a certain {\bf mixed volume}
form, thus recovering the classical mixed volume when U = (C^*)^n.
| lt256 | arxiv_abstracts |
math/0212180 | We study the asymptotics of almost holomorphic sections $s \in H^0_J(M,
\omega)$ of an ample line bundle $L \to M$ over an almost complex symplectic
manifold in the sense of Boutet de Monvel-Guillemin. Such sections are defined
as the kernel of a complex which is analogous to the $\bar{\partial}$ complex
for a positive line bundle over a complex manifold. Our main result is the
scaling limit asymptotics of the Szego projectors $\Pi_N$ of powers $L^N$. The
Kodaira embedding theorem and Tian almost isometry theorem are almost immediate
consequences of the scaling limit. We also relate such almost holomorphic
sections to the asymptotically holomorphic sections in the sense of Donaldson
and Auroux.
| lt256 | arxiv_abstracts |
math/0212181 | We define a Gaussian measure on the space $H^0_J(M, L^N)$ of almost
holomorphic sections of powers of an ample line bundle $L$ over a symplectic
manifold $(M, \omega)$, and calculate the joint probability densities of
sections taking prescribed values and covariant derivatives at a finite number
of points. We prove that they have a universal scaling limit as $N \to \infty$.
This result completes our proof (with P. Bleher) that correlations between
zeros of sections in the almost-holomorphic setting have the same universal
scaling limit as in the complex case (see Universality and scaling of zeros on
symplectic manifolds, Random matrix models and their applications, 31--69,
Math. Sci. Res. Inst. Publ., 40)
| lt256 | arxiv_abstracts |
math/0212182 | We prove that an abelian category equipped with an ample sequence of objects
is equivalent to the quotient of the category of coherent modules over the
corresponding algebra by the subcategory of finite-dimensional modules. In the
Noetherian case a similar result was proved by Artin and Zhang.
| lt256 | arxiv_abstracts |
math/0212183 | In this paper we explicitly attach to a geometric classical r-matrix $r$ (not
necessarily unitary), a geometric (i.e., set-theoretical) quantum R-matrix $R$,
which is a quantization of $r$. To accomplish this, we use the language of
bijective cocycle 7-tuples, developed by A. Soloviev in the study of
set-theoretical quantum R-matrices. Namely, we define a classical version of
bijective cocycle 7-tuples, and show that there is a bijection between them and
geometric classical r-matrices. Then we show how any classical bijective
cocycle 7-tuple can be quantized, and finally use Soloviev's construction,
which turns a (quantum) bijective cocycle 7-tuple into a geometric quantum
R-matrix.
| lt256 | arxiv_abstracts |
math/0212184 | Suppose that f is a dominant morphism from a k-variety X to a k-variety Y,
where k is a field of characteristic 0 and v is a valuation of the function
field k(X). We allow v to be an arbitary valuation, so it may not be discrete.
We prove that there exist sequences of blowups of nonsingular subvarieties
from X' to X and from Y' to Y such that X', Y' are nonsingular and X' to Y' is
locally a monomial mapping near the center of v. This extends an earlier result
of ours (in Asterisque 260) which proves the above result with the restriction
that f is generically finite.
| lt256 | arxiv_abstracts |
math/0212185 | We give an analytic characterisation of the interpolating sequences for the
Nevanlinna and Smirnov classes. From this we deduce a necessary and a
sufficient geometric condition, both expressed in terms of a certain
non-tangential maximal function associated to the sequence. Some examples show
that the gap between the necessary and the sufficient condition cannot be
covered. We also discuss the relationship between our results and the previous
work of Naftalevic for the Nevanlinna class, and Yanagihara for the Smirnov
class. Finally, we observe that the arguments used in the previous proofs show
that interpolating sequences for ``big'' Hardy-Orlicz spaces are in general
different from those for the scale included in the classical Hardy spaces.
| lt256 | arxiv_abstracts |
math/0212186 | In this paper we extend the Balian--Low theorem, which is a version of the
uncertainty principle for Gabor (Weyl--Heisenberg) systems, to functions of
several variables. In particular, we first prove the Balian--Low theorem for
arbitrary quadratic forms. Then we generalize further and prove the Balian--Low
theorem for differential operators associated with a symplectic basis for the
symplectic form on ${\mathbb R}^{2d}$.
| lt256 | arxiv_abstracts |
math/0212187 | Novikov initiated the study of the algebraic properties of quadratic forms
over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom
transversality construction of a Seifert surface of a knot and the infinite
cyclic cover of the knot exterior. In this paper the analogy is applied to
explain the relationship between the Seifert forms over a ring with involution
and Blanchfield forms over the Laurent polynomial extension.
| lt256 | arxiv_abstracts |
math/0212188 | In this paper we study the asymptotic behaviour of the solutions of some
minimization problems for integral functionals with convex integrands, in
two-dimensional domains with cracks, under perturbations of the cracks in the
Hausdorff metric. In the first part of the paper, we examine conditions for the
stability of the minimum problem via duality arguments in convex optimization.
In the second part, we study the limit problem in some special cases when there
is no stability, using the tool of $\Gamma$-convergence.
| lt256 | arxiv_abstracts |
math/0212189 | The density conjecture of Bers, Sullivan and Thurston predicts that each
complete hyperbolic 3-manifold M with finitely generated fundamental group is
an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove
that the conjecture obtains for each complete hyperbolic 3-manifold with no
cusps and incompressible ends.
| lt256 | arxiv_abstracts |
math/0212190 | Theories of classification distinguish classes with some good structure
theorem from those for which none is possible. Some classes (dense linear
orders, for instance) are non-classifiable in general, but are classifiable
when we consider only countable members. This paper explores such a notion for
classes of computable structures by working out several examples. One
motivation is to see whether some classes whose set of countable models is very
complex become classifiable when we consider only computable members.
We follow recent work by Goncharov and Knight in using the degree of the
isomorphism problem for a class to distinguish classifiable classes from non-
classifiable. For some classes (undirected graphs, fields of fixed
characteristic, and real closed fields) we show that the isomorphism problem is
\Sigma^1_1 complete (the maximum possible), and for others it is of relatively
low complexity. For instance, for algebraically closed fields, archimedean real
closed fields, and vector spaces, we show that the isomorphism problem is
\Pi^0_3 complete.
| 256 | arxiv_abstracts |
math/0212191 | We explore the structure of the p-adic automorphism group Gamma of the
infinite rooted regular tree. We determine the asymptotic order of a typical
element, answering an old question of Turan.
We initiate the study of a general dimension theory of groups acting on
rooted trees. We describe the relationship between dimension and other
properties of groups such as solvability, existence of dense free subgroups and
the normal subgroup structure. We show that subgroups of Gamma generated by
three random elements are full-dimensional and that there exist finitely
generated subgroups of arbitrary dimension. Specifically, our results solve an
open problem of Shalev and answer a question of Sidki.
| lt256 | arxiv_abstracts |
math/0212192 | We provide an analog of the Drinfeld quantum double construction in the
context of crossed Hopf group coalgebras introduced by Turaev. We prove that,
provided the base group is finite, the double of a semisimple crossed Hopf
group coalgebra is both modular and unimodular.
| lt256 | arxiv_abstracts |
math/0212193 | A compact Lie group G and a faithful complex representation V determine a
Sato-Tate measure, defined as the direct image of Haar measure on G with
respect to the character of V. We give a necessary and sufficient condition for
a Sato-Tate measure to be an isolated point in the set of such measures,
regarded as a subset of the space of distributions. In particular we prove that
the Sato-Tate measure of a connected and semisimple group with respect to an
irreducible representation is an isolated point.
| lt256 | arxiv_abstracts |
math/0212194 | We investigate the continuity properties of the solution operator to the wave
map system from the flat Minkowski space to a general nonflat target of
arbitrary dimension, and we prove by an explicit class of counterexamples that
this map is not uniformly continuous in the critical norms on any neighbourhood
of zero.
| lt256 | arxiv_abstracts |
math/0212195 | Coxeter decompositions of hyperbolic simplices where studied in
math.MG/0212010 and math.MG/0210067. In this paper we use the methods of these
works to classify Coxeter decompositions of bounded convex pyramids and
triangular prisms in the hyperbolic space H^3.
| lt256 | arxiv_abstracts |
math/0212196 | Fiber cones of 0-dimensional ideals with almost minimal multiplicity in
Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of
ideals with respect to another ideal is introduced. This is used to find a
bound on the reduction number with respect to an ideal. Rossi's bound on
reduction number in terms of Hilbert coefficients is obtained as a consequence.
Sufficient conditions are provided for the fiber cone of 0-dimensional ideals
to have almost maximal depth. Hilbert series of such fiber cones are also
computed.
| lt256 | arxiv_abstracts |
math/0212197 | We give a proof avoiding spectral sequences of Deligne's decomposition
theorem for objects in a triangulated category admitting a Lefschetz
homomorphism.
| lt256 | arxiv_abstracts |
math/0212198 | Functions, uniformly bounded in $BV$ norm in some bounded open set $U$ in
$R^n$, are compact in $L_1(U)$. This result is known when $U$ has Lipschitz
boundary [EG Th. 4 p. 176], [G 1.19 Th. p. 17], [Z 5.34 Cor. p. 227]; the proof
for general $U$ here, after identifying the operator theoretic definition of
bounded $BV$ norm with that of the Tonelli variation, appeals to the standard
compactness criterion in $L_1$ [DS 21 TH. p. 301] [Y, p. 275] (For
completeness, these two auxiliary results are also presented).
| lt256 | arxiv_abstracts |
math/0212199 | Motivated by the study of the local extrema of sin(x)/x we define the
\emph{Amplitude Modulation} transform of functions defined on (subsets of) the
real line. We discuss certain properties of this transform and invert it in
some easy cases.
| lt256 | arxiv_abstracts |
math/0212200 | We prove that if f is a self-map of an algebraic variety over a field K, then
under certain conditions on X, f and K the set of possible periods of K-valued
periodic points of f is finite.
| lt256 | arxiv_abstracts |
math/0212201 | This paper is devoted to a detailed study of a p-spins interaction model with
external field, including some sharp bounds on the speed of self averaging of
the overlap as well as a central limit theorem for its fluctuations, the
thermodynamical limit for the free energy and the definition of an
Almeida-Thouless type line. Those results show that the external field
dominates the tendency to disorder induced by the increasing level of
interaction between spins, and our system will share many of its features with
the SK model, which is certainly not the case when the external magnetic field
vanishes.
| lt256 | arxiv_abstracts |
math/0212202 | We illustrate the principle: rational generating series occuring in
arithmetic geometry are motivic in nature.
| lt256 | arxiv_abstracts |
math/0212203 | This paper deals with valuations of fields of formal meromorphic functions
and their residue fields. We explicitly describe the residue fields of the
monomial valuations. We also classify all the discrete rank one valuations of
fields of power series in two and three variables, according to their residue
fields. We prove that all our cases are possible and give explicit
constructions.
| lt256 | arxiv_abstracts |
math/0212204 | We derive an explicit formula for the Jacobi field that is acting in an
extended Fock space and corresponds to an ($\R$-valued) L\'evy process on a
Riemannian manifold. The support of the measure of jumps in the
L\'evy--Khintchine representation for the L\'evy process is supposed to have an
infinite number of points. We characterize the gamma, Pascal, and Meixner
processes as the only L\'evy processes whose Jacobi field leaves the set of
finite continuous elements of the extended Fock space invariant.
| lt256 | arxiv_abstracts |
math/0212205 | We prove existence and regularity of entire solutions to Monge-Ampere
equations invariant under an irreducible action of a compact Lie group.
| lt256 | arxiv_abstracts |
math/0212206 | We introduce the notion of a braid group parametrized by a ring, which is
defined by generators and relations and based on the geometric idea of painted
braids. We show that the parametrized braid group is isomorphic to the
semi-direct product of the Steinberg group (of the ring) with the classical
braid group. The technical heart of the proof is the Pure Braid Lemma, which
asserts that certain elements of the parametrized braid group commute with the
pure braid group.
More generally, we define, for any crystallographic root system, a braid
group and a parametrized braid group with parameters in a commutative ring. The
parametrized braid group is expected to be isomorphic to the semi-direct
product of the corresponding Steinberg group with the braid group. The first
part of the paper (described above) treats the case of the root system $A_n$;
in the second part, we handle the root system {$D_n$}. Other cases will be
treated in the sequel.
| lt256 | arxiv_abstracts |
math/0212207 | The purpose of this note is to give explicit criteria to determine whether a
real generalized Cartan matrix is of finite type, affine type or of hyperbolic
type by considering the principal minors and the inverse of the matrix. In
particular, it will be shown that a real generalized Cartan matrix is of finite
type if and only if it is invertible and the inverse is a positive matrix. A
real generalized Cartan matrix is of hyperbolic type if and only if it is
invertible and the inverse is non-positive.
| lt256 | arxiv_abstracts |
math/0212208 | In this note we discuss some arithmetic and geometric questions concerning
self maps of projective algebraic varieties.
| lt256 | arxiv_abstracts |
math/0212209 | We establish a criterion for a complex number to be algebraic over Q of
degree at most two. It requires that, for any sufficiently large real number X,
there exists a non-zero polynomial with integral coefficients, of degree at
most two and height at most X, whose absolute value at that complex number is
at most (1/4)X^{-(3+sqrt{5})/2}. We show that the exponent (3+sqrt{5})/2 in
this condition is optimal, and deduce from this criterion a result of
simultaneous approximation of a real number by conjugate algebraic numbers.
| lt256 | arxiv_abstracts |
math/0212210 | We construct three compatible quadratic Poisson structures such that generic
linear combination of them is associated with Elliptic Sklyanin algebra in n
generators. Symplectic leaves of this elliptic Poisson structure is studied.
Explicit formulas for Casimir elements are obtained.
| lt256 | arxiv_abstracts |
math/0212211 | We use intersection theory, degeneration techniques and jet schemes to study
log canonical thresholds. Our first result gives a lower bound for the log
canonical threshold of a pair in terms of the log canonical threshold of the
image by a suitable smooth morphism. This in turn is based on an inequality
relating the log canonical threshold and the Samuel multiplicity, generalizing
our previous result from math.AG/0205171. We then give a lower bound for the
log canonical threshold of an affine scheme defined by homogeneous equations of
the same degree in terms of the dimension of the non log terminal locus (this
part supersedes math.AG/0105113). As an application of our results, we prove
the birational superrigidity of every smooth hypersurface of degree N in P^N,
if 4\leq N\leq 12.
| lt256 | arxiv_abstracts |
math/0212212 | This paper presents control and coordination algorithms for groups of
vehicles. The focus is on autonomous vehicle networks performing distributed
sensing tasks where each vehicle plays the role of a mobile tunable sensor. The
paper proposes gradient descent algorithms for a class of utility functions
which encode optimal coverage and sensing policies. The resulting closed-loop
behavior is adaptive, distributed, asynchronous, and verifiably correct.
| lt256 | arxiv_abstracts |
math/0212213 | We describe a variety of symplectic surgeries (not a priori compatible with
Kahler structures) which are obtained by combining local Kahler degenerations
and resolutions of singularities. The effect of the surgeries is to replace
configurations of Lagrangian spheres with symplectic submanifolds. We discuss
several examples in detail, relating them to existence questions for symplectic
manifolds with $c_1>0, c_1=0, c_1<0$ in four and six dimensions.
| lt256 | arxiv_abstracts |
math/0212214 | We find stability conditions ([Do], [Br]) on some derived categories of
differential graded modules over a graded algebra studied in [RZ], [KS]. This
category arises in both derived Fukaya categories and derived categories of
coherent sheaves. This gives the first examples of stability conditions on the
A-model side of mirror symmetry, where the triangulated category is not
naturally the derived category of an abelian category. The existence of
stability conditions, however, gives many such abelian categories, as predicted
by mirror symmetry.
In our examples in 2 dimensions we completely describe a connected component
of the space of stability conditions as the universal cover of the
configuration space of $(k+1)$ distinct points with centre of mass zero in
$\C$, with deck transformations the braid group action of [KS], [ST]. This
gives a geometric origin for these braid group actions and their faithfulness,
and axiomatises the proposal for stability of Lagrangians in [Th] and the
example proved by mean curvature flow in [TY].
| 256 | arxiv_abstracts |
math/0212215 | The main result in this paper is a one term Szego type asymptotic formula
with a sharp remainder estimate for a class of integral operators of the
pseudodifferential type with symbols which are allowed to be non-smooth or
discontinuous in both position and momentum. The simplest example of such
symbol is the product of the characteristic functions of two compact sets, one
in real space and the other in momentum space. The results of this paper are
used in a study of the violation of the area entropy law for free fermions in
[18]. This work also provides evidence towards a conjecture due to Harold
Widom.
| lt256 | arxiv_abstracts |
math/0212216 | The Hodge conjecture is shown to be equivalent to a question about the
homology of very ample divisors with ordinary double point singularities. The
infinitesimal version of the result is also discussed.
| lt256 | arxiv_abstracts |
math/0212217 | A locally free resolution of a subscheme is by definition an exact sequence
consisting of locally free sheaves (except the ideal sheaf) which has
uniqueness properties like a free resolution. The purpose of this paper is to
characterize certain locally Cohen-Macaulay subschemes by means of locally free
resolutions. First we achieve this for arithmetically Buchsbaum subschemes.
This leads to the notion of an $\Omega$-resolution and extends a result of
Chang. Second we characterize quasi-Buchsbaum subschemes by means of weak
$\Omega$-resolutions. Finally, we describe the weak $\Omega$-resolutions which
belong to arithmetically Buchsbaum surfaces of codimension two. Various
applications of our results are given.
| lt256 | arxiv_abstracts |
math/0212218 | In previous works we analysed conditions for linearization of hermitian
kernels. The conditions on the kernel turned out to be of a type considered
previously by L. Schwartz in the related matter of characterizing the real
space generated by positive definite kernels. The aim of this note is to find
more concrete expressions of the Schwartz type conditions: in the Hamburger
moment problem for Hankel type kernels on the free semigroup, in dilation
theory (Stinespring type dilations and Haagerup decomposability), as well as in
multi-variable holomorphy. Among other things, we prove that any hermitian
holomorphic kernel has a holomorphic linearization, and hence that holomorphic
kernels automatically satisfy L. Schwartz's boundedness condition.
| lt256 | arxiv_abstracts |
math/0212219 | A 2-group is a `categorified' version of a group, in which the underlying set
G has been replaced by a category and the multiplication map m: G x G -> G has
been replaced by a functor. A number of precise definitions of this notion have
already been explored, but a full treatment of their relationships is difficult
to extract from the literature. Here we describe the relation between two of
the most important versions of this notion, which we call `weak' and `coherent'
2-groups. A weak 2-group is a weak monoidal category in which every morphism
has an inverse and every object x has a `weak inverse': an object y such that x
tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak
2-group in which every object x is equipped with a specified weak inverse x*
and isomorphisms i_x: 1 -> x tensor x*, e_x: x* tensor x -> 1 forming an
adjunction. We define 2-categories of weak and coherent 2-groups and construct
an `improvement' 2-functor which turns weak 2-groups into coherent ones; using
this one can show that these 2-categories are biequivalent. We also internalize
the concept of a coherent 2-group. This gives a way of defining topological
2-groups, Lie 2-groups, and the like.
| 256 | arxiv_abstracts |
math/0212220 | For any prime p we consider the density of elements in the multiplicative
group of the finite field F_p having order, respectively index, congruent to
a(mod d). We compute these densities on average, where the average is taken
over all finite fields of prime order. Some connections between the two
densities are established. It is also shown how to compute these densities with
high numerical accuracy.
| lt256 | arxiv_abstracts |
math/0212221 | In this paper we prove that among the permutations of length n with i fixed
points and j excedances, the number of 321-avoiding ones equals the number of
132-avoiding ones, for all given i,j<=n. We use a new technique involving
diagonals of non-rational generating functions. This theorem generalizes a
recent result of Robertson, Saracino and Zeilberger, for which we also give
another, more direct proof.
| lt256 | arxiv_abstracts |
math/0212222 | This paper is an introduction to the theory of multivector functions of a
real variable. The notions of limit, continuity and derivative for these
objects are given. The theory of multivector functions of a real variable, even
being similar to the usual theory of vector functions of a real variable, has
some subtle issues which make its presentation worhtwhile.We refer in
particular to the derivative rules involving exterior and Clifford products,
and also to the rule for derivation of a composition of an ordinary scalar
function with a multivector function of a real variable.
| lt256 | arxiv_abstracts |
math/0212223 | In this paper we develop with considerable details a theory of multivector
functions of a $p$-vector variable. The concepts of limit, continuity and
differentiability are rigorously studied. Several important types of
derivatives for these multivector functions are introduced, as e.g., the $A$%
-directional derivative (where $A$ is a $p$-vector) and the generalized
concepts of curl, divergence and gradient. The derivation rules for different
types of products of multivector functions and for compositon of multivector
functions are proved.
| lt256 | arxiv_abstracts |
math/0212224 | In this paper we introduce the concept of \emph{multivector functionals.} We
study some possible kinds of derivative operators that can act in interesting
ways on these objects such as, e.g., the $A$-directional derivative and the
generalized concepts of curl, divergence and gradient. The derivation rules are
rigorously proved. Since the subject of this paper has not been developed in
previous literature, we work out in details several examples of derivation of
multivector functionals.
| lt256 | arxiv_abstracts |
math/0212225 | We introduce perfect resolving algebras and study their fundamental
properties. These algebras are basic for our theory of differential graded
schemes, as they give rise to affine differential graded schemes. We also
introduce etale morphisms. The purpose for studying these, is that they will be
used to glue differential graded schemes from affine ones with respect to an
etale topology.
| lt256 | arxiv_abstracts |
math/0212226 | We construct a 2-category of differential graded schemes. The local affine
models in this theory are differential graded algebras, which are graded
commutative with unit over a field of characteristic zero, are concentrated in
non-positive degrees and have perfect cotangent complex. Quasi-isomorphic
differential graded algebras give rise to 2-isomorphic differential graded
schemes and a differential graded algebra can be recovered up to
quasi-isomorphism from the differential graded scheme it defines. Differential
graded schemes can be glued with respect to an etale topology and fibered
products of differential graded schemes correspond on the algebra level to
derived tensor products.
| lt256 | arxiv_abstracts |
math/0212227 | For every finitely generated recursively presented group G we construct a
finitely presented group H containing G such that G is (Frattini) embedded into
H and the group H has solvable conjugacy problem if and only if G has solvable
conjugacy problem. Moreover G and H have the same r.e. Turing degrees of the
conjugacy problem. This solves a problem by D. Collins.
| lt256 | arxiv_abstracts |
math/0212228 | We study noncommutative generalizations of such notions of the classical
symplectic geometry as degenerate Poisson structure, Poisson submanifold and
quotient manifold, symplectic foliation and symplectic leaf for associative
Poisson algebras. We consider these structures for the case of the endomorphism
algebra of a vector bundle, and give the full description of the family of
Poisson structures for this algebra.
| lt256 | arxiv_abstracts |
math/0212229 | We consider linear operators $T$ mapping a couple of weighted $L_p$ spaces
$\{L_{p_0}(U_0), L_{p_1}(U_1)\}$ into $\{L_{q_0}(V_0), L_{q_1}(V_1)\}$ for any
$1\le p_0$, $ p_1$, $q_0$, $q_1\le\infty$, and describe the interpolation orbit
of any $a\in L_{p_0}(U_0)+ L_{p_1}(U_1)$ that is we describe a space of all
$\{Ta\}$, where $T$ runs over all linear bounded mappings from $\{L_{p_0}(U_0),
L_{p_1}(U_1)\}$ into $\{L_{q_0}(V_0),L_{q_1}(V_1)\}$. We present in this paper
the proofs of results which were announced in V.I.Ovchinnikov, C. R. Acad. Sci.
Paris Ser. I 334 (2002) 881--884.
We show that interpolation orbit is obtained by the Lions--Peetre method of
means with functional parameter as well as by the K-method with a weighted
Orlicz space as a parameter.
| lt256 | arxiv_abstracts |
math/0212230 | We study different ways of determining the mean distance $ < r_n >$ between a
reference point and its $n$-th neighbour among random points distributed with
uniform density in a $D$-dimensional Euclidean space. First we present a
heuristic method; though this method provides only a crude mathematical result,
it shows a simple way of estimating $ < r_n >$. Next we describe two
alternative means of deriving the exact expression of $<r_n>$: we review the
method using absolute probability and develop an alternative method using
conditional probability. Finally we obtain an approximation to $ < r_n >$ from
the mean volume between the reference point and its $n$-th neighbour and
compare it with the heuristic and exact results.
| lt256 | arxiv_abstracts |
math/0212231 | In this article, we consider a class of bi-stable reaction-diffusion
equations in two components on the real line. We assume that the system is
singularly perturbed, i.e. that the ratio of the diffusion coefficients is
(asymptotically) small. This class admits front solutions that are
asymptotically close to the (stable) front solution of the `trivial' scalar
bi-stable limit system $u_t = u_{xx} + u(1-u^2)$. However, in the system these
fronts can become unstable by varying parameters. This destabilization is
either caused by the essential spectrum associated to the linearized stability
problem, or by an eigenvalue that exists near the essential spectrum. We use
the Evans function to study the various bifurcation mechanisms and establish an
explicit connection between the character of the destabilization and the
possible appearance of saddle-node bifurcations of heteroclinic orbits in the
existence problem.
| lt256 | arxiv_abstracts |
math/0212232 | Let $E$ be a holomorphic vector bundle. Let $\theta$ be a Higgs field, that
is a holomorphic section of $End(E)\otimes\Omega^{1,0}_X$ satisfying
$\theta^2=0$. Let $h$ be a pluriharmonic metric of the Higgs bundle
$(E,\theta)$. The tuple $(E,\theta,h)$ is called a harmonic bundle.
Let $X$ be a complex manifold, and $D$ be a normal crossing divisor of $X$.
In this paper, we study the harmonic bundle $(E,\theta,h)$ over $X-D$. We
regard $D$ as the singularity of $(E,\theta,h)$, and we are particularly
interested in the asymptotic behaviour of the harmonic bundle around $D$. We
will see that it is similar to the asymptotic behaviour of complex variation of
polarized Hodge structures, when the harmonic bundle is tame and nilpotent with
the trivial parabolic structure. For example, we prove constantness of general
monodromy weight filtrations, compatibility of the filtrations, norm estimates,
and the purity theorem.
For that purpose, we will obtain a limiting mixed twistor structure from a
tame nilpotent harmonic bundle with trivial parabolic structure, on a punctured
disc. It is a partial solution of a conjecture of Simpson.
| 256 | arxiv_abstracts |
math/0212233 | The maximality of Abelian subgroups play a role in various parts of group
theory. For example, Mycielski has extended a classical result of Lie groups
and shown that a maximal Abelian subgroup of a compact connected group is
connected and, furthermore, all the maximal Abelian subgroups are conjugate.
For finite symmetric groups the question of the size of maximal Abelian
subgroups has been examined by Burns and Goldsmith in 1989 and Winkler in 1993.
We show that there is not much interest in generalizing this study to infinite
symmetric groups; the cardinality of any maximal Abelian subgroup of the
symmetric group of the integers is 2^{aleph_0}. Our purpose is also to examine
the size of maximal Abelian subgroups for a class of groups closely related to
the the symmetric group of the integers; these arise by taking an ideal on the
integers, considering the subgroup of all permutations which respect the ideal
and then taking the quotient by the normal subgroup of permutations which fix
all integers except a set in the ideal. We prove that the maximal size of
Abelian subgroups in such groups is sensitive to the nature of the ideal as
well as various set theoretic hypotheses.
| 256 | arxiv_abstracts |
math/0212234 | Let A and B be two first order structures of the same relational vocabulary
L. The Ehrenfeucht-Fraisse-game of length gamma of A and B denoted by
EFG_gamma(A,B) is defined as follows: There are two players called for all and
exists. First for all plays x_0 and then exists plays y_0. After this for all
plays x_1, and exists plays y_1, and so on. Eventually a sequence
<(x_beta,y_beta): beta<gamma> has been played. The rules of the game say that
both players have to play elements of A cup B. Moreover, if for all plays his
x_beta in A (B), then exists has to play his y_beta in B (A). Thus the sequence
< (x_beta,y_beta):beta<gamma > determines a relation pi subseteq AxB. Player
exists wins this round of the game if pi is a partial isomorphism. Otherwise
for all wins. The game EFG_gamma^delta (A,B) is defined similarly except that
the players play sequences of length<delta at a time.
Theorem 1: The following statements are equiconsistent relative to ZFC:
(A) There is a weakly compact cardinal.
(B) CH and EF_{omega_1}(A,B) is determined for all models A,B of cardinality
aleph_2 .
Theorem 2: Assume that 2^omega <2^{omega_3} and T is a countable complete
first order theory. Suppose that one of (i)-(iii) below holds. Then there are
A,B models T of power omega_3 such that for all cardinals 1<theta<=omega_3,
EF^theta_{omega_1}(A,B) is non-determined.
[(i)] T is unstable.
[(ii)] T is superstable with DOP or OTOP.
[(iii)] T is stable and unsuperstable and 2^omega <= omega_3.
| 256 | arxiv_abstracts |
math/0212235 | This paper deals with the problem of representing the matching independence
system in a graph as the intersection of finitely many matroids. After
characterizing the graphs for which the matching independence system is the
intersection of two matroids, we study the function mu(G), which is the minimum
number of matroids that need to be intersected in order to obtain the set of
matchings on a graph G, and examine the maximal value, mu(n), for graphs with n
vertices. We describe an integer programming formulation for deciding whether
mu(G)<= k. Using combinatorial arguments, we prove that mu(n)is in Omega(loglog
n). On the other hand, we establish that mu(n) is in O(log n / loglog n).
Finally, we prove that mu(n)=4 for n=5,...,12, and mu(n)=5 for n=13,14,15.
| lt256 | arxiv_abstracts |
math/0212236 | This article shows that under general conditions, p-adic orbital integrals of
definable functions are represented by virtual Chow motives. This gives an
explicit example of the philosophy of Denef and Loeser, which predicts that all
naturally occurring p-adic integrals are motivic.
| lt256 | arxiv_abstracts |
math/0212237 | This paper introduces the notion of a stability condition on a triangulated
category. The motivation comes from the study of Dirichlet branes in string
theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a
mathematical point of view, the most interesting feature of the definition is
that the set of stability conditions $\Stab(\T)$ on a fixed category $\T$ has a
natural topology, thus defining a new invariant of triangulated categories.
After setting up the necessary definitions I prove a deformation result which
shows that the space $\Stab(\T)$ with its natural topology is a manifold,
possibly infinite-dimensional.
| lt256 | arxiv_abstracts |
math/0212238 | It is an open question whether tight closure commutes with localization in
quotients of a polynomial ring in finitely many variables over a field. Katzman
showed that tight closure of ideals in these rings commutes with localization
at one element if for all ideals I and J in a polynomial ring there is a linear
upper bound in q on the degree in the least variable of reduced Groebner bases
in reverse lexicographic ordering of the ideals of the form J + I^{[q]}.
Katzman conjectured that this property would always be satisfied. In this paper
we prove several cases of Katzman's conjecture. We also provide an experimental
analysis (with proofs) of asymptotic properties of Groebner bases connected
with Katzman's conjectures.
| lt256 | arxiv_abstracts |
math/0212239 | We consider a compact manifold whose boundary is a locally trivial fiber
bundle and an associated pseudodifferential algebra that models fibered cusps
at infinity. Using trace-like functionals that generate the 0-dimensional
Hochschild cohomology groups, we express the index of a fully elliptic fibered
cusp operator as the sum of a local contribution from the interior and a term
that comes from the boundary. This answers the index problem formulated by
Mazzeo and Melrose. We give a more precise answer in the case where the base of
the boundary fiber bundle is the circle. In particular, for Dirac operators
associated to a "product fibered cusp metric", the index is given by the
integral of the Atiyah-Singer form in the interior minus the adiabatic limit of
the eta invariant of the restriction of the operator to the boundary.
| lt256 | arxiv_abstracts |
math/0212241 | This paper explores the effect of various graphical constructions upon the
associated graph $C^*$-algebras. The graphical constructions in question arise
naturally in the study of flow equivalence for topological Markov chains.
We prove that out-splittings give rise to isomorphic graph algebras, and
in-splittings give rise to strongly Morita equivalent $C^*$-algebras. We
generalise the notion of a delay as defined by Drinen to form in-delays and
out-delays. We prove that these constructions give rise to Morita equivalent
graph $C^*$-algebras. We provide examples which suggest that our results are
the most general possible in the setting of the $C^*$-algebras of arbitrary
directed graphs.
| lt256 | arxiv_abstracts |
math/0212242 | To a directed graph $E$ is associated a $C^*$-algebra $C^* (E)$ called a
graph $C^*$-algebra. There is a canonical action $\gamma$ of ${\bf T}$ on $C^*
(E)$, called the gauge action. In this paper we present necessary and
sufficient conditions for the fixed point algebra $C^* (E)^\gamma$ to be
simple. Our results also yield some structure theorems for simple graph
algebras.
| lt256 | arxiv_abstracts |
math/0212243 | A famous conjecture attributed to Kodaira asks whether any compact Kaehler
manifold can be approximated by projective manifolds. We confirm this
conjecture on projectivized direct sums of three line bundles on
three-dimensional complex tori which appears rather surprising in view of
expected dimensions of certain families of tori. We also discuss possible
counter examples.
| lt256 | arxiv_abstracts |
math/0212244 | Let $P$ be a simplex in $S^n$ and $G_P$ be a group generated by the
reflections with respect to the facets of $P$. We are interested in the case
when the group $G_P$ is discrete. In this case we say that $G$ generates the
discrete reflection group $G_P$. We develop the criteria for a simplex
generating a discrete reflection group. We also describe all indecomposable
spherical simplices generating discrete reflection groups.
| lt256 | arxiv_abstracts |
math/0212245 | Assuming the purity conjecture for the affine Springer fibers which has been
formulated by Goresky, Kottwitz and MacPherson, we prove a geometric analog of
the fundamental lemma for unitary groups. Our approach is similar to the one of
Goresky, Kottwitz and MacPherson. Our main new ingredient is the link between
affine Springer fibers and compactified Jacobians which is described in
math.AG/0204109.
| lt256 | arxiv_abstracts |
math/0212246 | Differentiable real function reproducing primes up to a given number and
having a differentiable inverse function is constructed. This inverse function
is compared with the Riemann-Von Mangoldt exact expression for the number of
primes not exceeding a given value. Software for computation of the direct and
inverse functions and their derivatives is developed. Examples of approximate
solution of Diophantine equations on the primes are given.
| lt256 | arxiv_abstracts |
math/0212247 | Starting from some considerations we make about the relations between certain
difference statistics and the classical permutation statistics we study
permutations whose inversion number and excedance difference coincide. It turns
out that these (so-called bi-increasing) permutations are just the 321-avoiding
ones. The paper investigates their excedance and descent structure. In
particular, we find some nice combinatorial interpretations for the
distribution coefficients of the number of excedances and descents,
respectively, and their difference analogues over the bi-increasing
permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This
yields a connection between restricted permutations, parallelogram polyominoes,
and lattice paths that reveals the relations between several well-known
bijections given for these objects (e.g. by Delest-Viennot,
Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an
application, we enumerate skew diagrams according to their rank and give a
simple combinatorial proof for a result concerning the symmetry of the joint
distribution of the number of excedances and inversions, respectively, over the
symmetric group.
| 256 | arxiv_abstracts |
math/0212248 | A new relation between a class of complex polynomials with a good behavior at
infinity studied by A. N\'emethi and A. Zaharia and the cohomology groups of
affine complex hyperplane arrangement complements with rank one local system
coefficients is introduced and explored.
This approach gives in particular new upper-bounds for the dimension of the
twisted cohomology groups of line arrangement complements in the complex affine
plane.
| lt256 | arxiv_abstracts |
math/0212249 | This is a slightly corrected version of an old work.
Under certain cardinal arithmetic assumptions, we prove that for every large
enough regular $\lambda$ cardinal, for many regular $\kappa < \lambda$, many
stationary subsets of $\lambda$ concentrating on cofinality $\kappa$ have super
BB. In particular, we have the super BB on $\{\delta < \lambda \colon
cf(\delta) = \kappa\}$. This is a strong negation of uniformization.
We have added some details. Works continuing it are [Sh:898] and [Sh:1028].
We thank Ari Brodski and Adi Jarden for their helpful comments.
In this paper we had earlier used the notion ``middle diamond" which is now
replaced by ``super BB'', that is, ``super black box'', in order to be
consistent with other papers (see [Sh:898]).
| lt256 | arxiv_abstracts |
math/0212250 | Our first motivation was the question: can a countable structure have an
automorphism group, which a free uncountable group? This is answered negatively
in [Sh:744]. Lecturing in a conference in Rutgers, February 2001, I was asked
whether I am really speaking on Polish groups. We can prove this using a more
restrictive condition on the set of equations. Parallel theorems, hold for semi
groups and for metric algebras, e.g. with non-isolated unit. Here we do the
general case. For instance we show that there is no Polish group which as a
group is free and uncountable.
| lt256 | arxiv_abstracts |
math/0212251 | This note proposes a method for pricing high-dimensional American options
based on modern methods of multidimensional interpolation. The method allows
using sparse grids and thus mitigates the curse of dimensionality. A framework
of the pricing algorithm and the corresponding interpolation methods are
discussed, and a theorem is demonstrated that suggests that the pricing method
is less vulnerable to the curse of dimensionality. The method is illustrated by
an application to rainbow options and compared to Least Squares Monte Carlo and
other benchmarks.
| lt256 | arxiv_abstracts |
math/0212252 | We provide an analog of the Joyal-Street center construction and of the
Kassel-Turaev categorical quantum double in the context of the crossed
categories introduced by Turaev. Then, we focus or attention to the case of
categories of representation. In particular, we introduce the notion of a
Yetter-Drinfeld module over a crossed group coalgebra H and we prove that both
the category of Yetter-Drinfeld modules over H and the center of the category
of representations of H as well as the category of representations of the
quantum double of H are isomorphic as braided crossed categories.
| lt256 | arxiv_abstracts |
math/0212253 | Let $\g$ be an affine Kac-Moody Lie algebra. Let $\U^+$ be the positive part
of the Drinfeld-Jimbo quantum enveloping algebra associated to $\g$. We
construct a basis of $\U^+$ which is related to the Kashiwara-Lusztig global
crystal basis (or canonical basis) by an upper triangular matrix (with respect
to an explicitly defined ordering) with 1's on the diagonal and with above
diagonal entries in $q_s^{-1} \Z[q_s^{-1}]$. Using this construction we study
the global crystal basis $\B(\Um)$ of the modified quantum enveloping algebra
defined by Lusztig. We obtain a Peter-Weyl like decomposition of the crystal
$\B(\Um)$ (Theorem 4.18), as well as an explicit description of two-sided cells
of $\B(\Um)$ and the limit algebra of $\Um$ at $q=0$ (Theorem 6.45).
| lt256 | arxiv_abstracts |
math/0212254 | We study inequalities between general integral moduli of continuity of a
function and the tail integral of its Fourier transform. We obtain, in
particular, a refinement of a result due to D. B. H. Cline [2] (Theorem 1.1
below). We note that our approach does not use a regularly varying comparison
function as in [2]. A corollary of Theorem 1.1 deals with the equivalence of
the two-sided estimates on the modulus of continuity on one hand, and on the
tail of the Fourier transform, on the other (Corollary 1.5). This corollary is
applied in the proof of the violation of the so-called entropic area law for a
critical system of free fermions in [4,5].
| lt256 | arxiv_abstracts |
math/0212255 | We demonstrate that the extended Kalman filter converges locally for a broad
class of nonlinear systems. If the initial estimation error of the filter is
not too large then the error goes to zero exponentially as time goes to
infinity. To demonstrate this, we require that the system be $C^2$ and
uniformly observable with bounded second partial derivatives.
| lt256 | arxiv_abstracts |
math/0212256 | This is meant to be a survey article for the Cubo Journal. We discuss the
existence and number of rational points over a finite field, the Hodge type
over the complex numbers, and the motivic conjectures which are controlling
those invariants. We present a geometric version of it.
| lt256 | arxiv_abstracts |
math/0212257 | Frenkel and Reshetikhin introduced q-characters to study finite dimensional
representations of quantum affine algebras. In the simply laced case Nakajima
defined deformations of q-characters called q,t-characters. The definition is
combinatorial but the proof of the existence uses the geometric theory of
quiver varieties which holds only in the simply laced case. In this article we
propose an algebraic general (non necessarily simply laced) new approach to
q,t-characters motivated by our deformed screening operators. The
t-deformations are naturally deduced from the structure of the quantum affine
algebra: the parameter t is analog to the central charge c. The q,t-characters
lead to the construction of a quantization of the Grothendieck ring and to
general analogues of Kazhdan-Lusztig polynomials in the same spirit as Nakajima
did for the simply laced case.
| lt256 | arxiv_abstracts |
math/0212258 | We classify trigonometric solutions to the associative Yang-Baxter equation
(AYBE) for A = Mat_n, the associative algebra of n-by-n matrices. The AYBE was
first presented in a 2000 article by Marcelo Aguiar and also independently by
Alexandre Polishchuk. Trigonometric AYBE solutions limit to solutions of the
classical Yang-Baxter equation. We find that such solutions of the AYBE are
equal to special solutions of the quantum Yang-Baxter equation (QYBE)
classified by Gerstenhaber, Giaquinto, and Schack (GGS), divided by a factor of
q - q^{-1}, where q is the deformation parameter q = exp(h). In other words,
when it exists, the associative lift of the classical r-matrix coincides with
the quantum lift up to a factor. We give explicit conditions under which the
associative lift exists, in terms of the combinatorial classification of
classical r-matrices through Belavin-Drinfeld triples. The results of this
paper illustrate nontrivial connections between the AYBE and both classical
(Lie) and quantum bialgebras.
| lt256 | arxiv_abstracts |
math/0212259 | We associate to a pair $(X,D)$, consisting of a smooth scheme with a divisor
$D\in \text{Div}(X)\otimes \mathbb{Q}$ whose support is a divisor with normal
crossings, a canonical Deligne--Mumford stack over $X$ on which $D$ becomes
integral. We then reinterpret the Kawamata--Viehweg vanishing theorem as
Kodaira vanishing for stacks.
| lt256 | arxiv_abstracts |
End of preview. Expand
in Data Studio
Common Pile v0.1 — Parquet Consolidated
Description
This dataset bundles all “raw” corpora from the Common Pile v0.1 Raw Data collection, converted to Apache Parquet and consolidated in a single repository.
Nothing has been filtered or modified; the only changes are:
- Format: original JSON → Parquet
- Layout: many repositories → one consolidated dataset
- Extra column: a
len_categorybucket for quick length-based filtering
Only the three original columns (id, text, source) are carried over; len_category is derived from text length.
Dataset Schema
| Column | Type | Notes |
|---|---|---|
id |
string | Original document ID |
text |
string | UTF-8 plain text |
source |
string | Name of the originating corpus (e.g. library_of_congress) |
len_category |
string | Bucketed document length (bytes) |
License Issues
Licensing follows the individual corpora.
While we aim to produce datasets with completely accurate licensing information, license laundering and inaccurate metadata can cause us to erroneously assign the incorrect license to some documents (for further discussion of this limitation, please see our paper). If you believe you have found an instance of incorrect licensing in this dataset, please start a discussion on this repository.
Other Versions
Citation
If you use this dataset, please cite:
@article{kandpal2025common,
title = {{The Common Pile v0.1: An 8TB Dataset of Public Domain and Openly Licensed Text}},
author = {Nikhil Kandpal and Brian Lester and Colin Raffel and Sebastian Majstorovic and
Stella Biderman and Baber Abbasi and Luca Soldaini and Enrico Shippole and
A. Feder Cooper and Aviya Skowron and Shayne Longpre and Lintang Sutawika and
Alon Albalak and Zhenlin Xu and Guilherme Penedo and Loubna Ben and Elie Bakouch and
John David and Honglu Fan and Dashiell Stander and Guangyu Song and Aaron Gokaslan and
John Kirchenbauer and Tom Goldstein and Brian R and Bhavya Kailkhura and Tyler Murray},
journal = {arXiv preprint},
year = {2025}
}
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