id
string | text
string | len_category
string | source
string |
|---|---|---|---|
math/0301240
|
The purpose of this note is to suggest an analogue for genus 2 curves of part
of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2
curves with CM by a quartic CM field K, it appears that primes dividing the
denominators of the discriminants of the Igusa class polynomials all have the
property 1) that they are bounded by d, the absolute value of the discriminant
of K, and 2) that they divide d-x^2, for some integer x whose square is less
than d. A slightly stronger condition is given in Section 3. Such primes are
primes of bad reduction for the genus 2 curve and primes of supersingular
reduction for the Jacobian of the genus 2 curve.
|
lt256
|
arxiv_abstracts
|
math/0301241
|
We define a class of multivariate Laurent polynomials closely related to
Chebyshev polynomials, and prove the simple but somewhat surprising (in view of
the fact that the signs of the coefficients of the Chebyshev polynomials
themselves alternate) result that their coefficients are non-negative. We
further show that a Central Limit Theorem holds for our polynomials.
|
lt256
|
arxiv_abstracts
|
math/0301242
|
The purpose of this note is to show how some results from the theory of
partial differential equations apply to the study of pseudo-spectra of
non-self-adjoint operators, which is a topic of current interest in applied
mathematics.
|
lt256
|
arxiv_abstracts
|
math/0301243
|
We consider dynamical systems generated by partially hyperbolic surface
endomorphisms of class C^r with one-dimensional strongly unstable subbundle. As
the main result, we prove that such a dynamical system generically admits
finitely many ergodic physical measures whose union of basins of attraction has
total Lebesgue measure, provided that r>= 19.
|
lt256
|
arxiv_abstracts
|
math/0301244
|
The notion of locally finite part of the dual coalgebra of certain quantized
coordinate rings is introduced. In the case of irreducible flag manifolds this
locally finite part is shown to coincide with a natural quotient coalgebra V of
U_q(g). On the way the coradical filtration of V is determined. A graded
version of the duality between V and the quantized coordinate ring is
established. This leads to a natural construction of several examples of
quantized vector spaces.
As an application covariant first order differential calculi on quantized
irreducible flag manifolds are classified.
Keywords: quantum groups, quantized flag manifolds
|
lt256
|
arxiv_abstracts
|
math/0301245
|
The number of topologically different plane real algebraic curves of a given
degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available
upper bound for the constant $C$. This bound follows from Arnold inequalities
on the number of empty ovals. To evaluate its rate we show its equivalence with
the rate of growth of the number of trees half of whose vertices are leaves and
evaluate the latter rate.
|
lt256
|
arxiv_abstracts
|
math/0301246
|
It is not completely unreasonable to expect that a computable function
bounding the number of Pachner moves needed to change any triangulation of a
given 3-manifold into any other triangulation of the same 3-manifold exists. In
this paper we describe a procedure yielding an explicit formula for such a
function if the 3-manifold in question is a Seifert fibred space.
|
lt256
|
arxiv_abstracts
|
math/0301247
|
In this paper we describe the structure of a group of conjugating
automorphisms $C_n$ of free group and prove that this structure is similar to
the structure of a braid group $B_n$ with $n>1$ strings. We find the linear
representation of group $C_n$. Also we prove that the braid group $B_n(S^2)$ of
2--sphere, mapping class group M(0,n) of the $n$--punctured 2--sphere and the
braid group $B_3(P^2)$ of the projective plane are linear. Using result of J.
Dyer, E. Formanek, E. Grossman and the faithful linear representation of
Lawrence--Krammer of $B_4$ we construct faithful linear representation of the
automorphism group $Aut(F_2)$.
|
lt256
|
arxiv_abstracts
|
math/0301248
|
In this paper we consider dimonoids, which are sets equipped with two
associative binary operations. Dimonoids in the sense of J.-L. Loday are
xamples of duplexes. The set of all permutations, gives an example of a duplex
which is not a dimonoid. We construct a free duplex generated by a given set
via planar trees and then we prove that the set of all permutations form a free
duplex on an explicitly described set of generators. We also consider duplexes
coming from planar binary trees and vertices of the cubes. We prove that these
duplexes are free with one generator in appropriate variety of duplexes.
|
lt256
|
arxiv_abstracts
|
math/0301249
|
This paper proves a result on the existence of finite flat scheme covers of
Deligne-Mumford stacks. This result is used to prove that a large class of
smooth Deligne-Mumford stacks with affine moduli space are quotient stacks, and
in the case of quasi-projective moduli space, to reduce the question to one
concerning Brauer groups of schemes.
|
lt256
|
arxiv_abstracts
|
math/0301250
|
We consider vector fields on knot/link complements in $S^3$ which are
transverse to the fibres of a fibration of the complement over a circle. We
prove that a large class of fibred knots/links, including all non-torus fibred
2-bridge knots, has the following property: any vector field transverse to the
fibres of the fibration of the complement must possess periodic orbits
representing all possible knot and link types.
|
lt256
|
arxiv_abstracts
|
math/0301251
|
We examine how the Seshadri constant of an ample line bundle at a very
general point of an algebraic surface can carry important global geometric
information about the surface. In particular, we obtain a numerical criterion
for when a surface admits a dominant map to an algebraic curve.
|
lt256
|
arxiv_abstracts
|
math/0301252
|
In this paper we will study the p-divisibility of partial sums of multiple
zeta value series. In particular we provide some generalizations of the
classical Wolstenholme's Theorem.
|
lt256
|
arxiv_abstracts
|
math/0301253
|
Every monoidal functor G: C --> M has a canonical factorization through the
category of bimodules over some monoid R in M such that the factor U: C -->_R
M_R is strongly unital. Using this result and the characterization of the
forgetful functors M_A -->_R M_R of bialgebroids A over R given by Schauenburg
together with their bimonad description given by the author recently here we
characterize the "long" forgetful functors M_A -->_R M_R --> M of both
bialgebroids and weak bialgebras.
|
lt256
|
arxiv_abstracts
|
math/0301254
|
We present some further results on Liouville type theorems for some
conformally invariant fully nonlinear equations.
|
lt256
|
arxiv_abstracts
|
math/0301255
|
We study the algebraic varieties defined by the conditional independence
statements of Bayesian Networks. A complete algebraic classification is given
for Bayesian Networks on at most five random variables. Hidden variables are
related to the geometry of higher secant varieties.
|
lt256
|
arxiv_abstracts
|
math/0301256
|
One of the actual problems in the field of numerical optimisation, as is well
known, is the problem of the search for the global extremum of a multivariate
function [1-9,13,14,17-21]. Various versions of the random search methods
[6,8,9] are considered to be the most reliable to solve the problem of global
optimisation. In this work we present the little-known methods of Halton and
LP-search, which has been proved as one of the best practical solutions of the
global optimisation problem.
|
lt256
|
arxiv_abstracts
|
math/0301257
|
A purely combinatorial construction of the quantum cohomology ring of the
flag manifold $G/B$ is presented. We show that the ring we construct is
commutative, associative and satisfies the usual grading condition. By using
results of two of our previous papers, we obtain a presentation of this ring in
terms of generators and relations, as well as formulas for quantum Giambelli
polynomials. We show that these polynomials satisfy a certain orthogonality
property, which - for G=SL_n(C) - was proved previously by Fomin, Gelfand and
Postnikov.
|
lt256
|
arxiv_abstracts
|
math/0301258
|
The Poincare Problem can be reduced to a problem on fibered surfaces,
concretely, to bound the genus of the fibration by means of numerical
information of the canonical sheaf of the associated foliation. In this paper
we: 1. explain how this reduction can be carried out, 2. apply unipotent
reduction to obtain some possible answers to the problem, 3. obtain an answer
under some hypotheses on the eigenvalues of the singularities of the foliation
|
lt256
|
arxiv_abstracts
|
math/0301259
|
We introduce the notion of finite right (respectively left) numerical index
on a bimodule $X$ over C*-algebras A and B with a bi-Hilbertian structure. This
notion is based on a Pimsner-Popa type inequality. The right (respectively
left) index element of X can be constructed in the centre of the enveloping von
Neumann algebra of A (respectively B). X is called of finite right index if the
right index element lies in the multiplier algebra of A. In this case we can
perform the Jones basic construction. Furthermore the C*--algebra of bimodule
mappings with a right adjoint is a continuous field of finite dimensional
C*-algebras over the spectrum of Z(M(A)), whose fiber dimensions are bounded
above by the index. We show that if A is unital, the right index element
belongs to A if and only if X is finitely generated as a right module.
We show that bi-Hilbertian, finite (right and left) index C*-bimodules are
precisely those objects of the tensor 2-C*-category of right Hilbertian
C*-bimodules with a conjugate object, in the sense of Longo and Roberts, in the
same category.
|
256
|
arxiv_abstracts
|
math/0301260
|
We prove global existence and scattering for the defocusing, cubic nonlinear
Schr\"odinger equation in $H^s(\rr^3)$ for $s > {4/5}$. The main new estimate
in the argument is a Morawetz-type inequality for the solution $\phi$. This
estimate bounds $\|\phi(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the
well-known Morawetz-type estimate of Lin-Strauss controls
$\int_0^{\infty}\int_{\rr^3}\frac{(\phi(x,t))^4}{|x|} dx dt
|
lt256
|
arxiv_abstracts
|
math/0301261
|
We use Green's canonical syzygy conjecture for generic curves to prove that
the Green-Lazarsfeld gonality conjecture holds for generic curves of genus g,
and gonality d, if $g/3<d<[g/2]+2$.
|
lt256
|
arxiv_abstracts
|
math/0301262
|
Over a noetherian local ring certain minimal finite free resolutions possess
a property which we call stiffness. This calls to mind the Buchsbaum-Eisenbud
criterion for exactness. Yet we only prove stiffness over equicharacteristic
rings. However, Hochster's Canonical Element Conjecture is shown to be true for
every ring with a fixed prime residual characteristic, precisely when every
resolution over each Gorenstein ring of this type is stiff.
|
lt256
|
arxiv_abstracts
|
math/0301263
|
The infinite dimensional Clifford Algebra has a maze of irreducible unitary
representations. Here we determine their type -real, complex or quaternionic.
Some, related to the Fermi-Fock representations, have no real or quetrnionic
structures. But there are many on L(2) of the circle that do and which seem to
have analytic meaning.
|
lt256
|
arxiv_abstracts
|
math/0301264
|
We present a table containing the maximal number of rational points on a
genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks
on Frobenius non-classical quartics over finite fields are given.
|
lt256
|
arxiv_abstracts
|
math/0301265
|
We study the problem of existence of surfaces in ${\bf R}^3$ parametrized on
the sphere ${\mathbb S}^2$ with prescribed mean curvature $H$ in the
perturbative case, i.e. for $H=H_0+\epsilon H_1$, where $H_0$ is a nonzero
constant, $H_1$ is a $C^2$ function and $\epsilon$ is a small perturbation
parameter.
|
lt256
|
arxiv_abstracts
|
math/0301266
|
We determine the maximal gap between the optimal values of an integer program
and its linear programming relaxation, where the matrix and cost function are
fixed but the right hand side is unspecified. Our formula involves irreducible
decomposition of monomial ideals. The gap can be computed in polynomial time
when the dimension is fixed.
|
lt256
|
arxiv_abstracts
|
math/0301267
|
The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic
group if and only if the group is not virtually free.
|
lt256
|
arxiv_abstracts
|
math/0301268
|
We consider the problem of designing a set of computational agents so that as
they all pursue their self-interests a global function G of the collective
system is optimized. Three factors govern the quality of such design. The first
relates to conventional exploration-exploitation search algorithms for finding
the maxima of such a global function, e.g., simulated annealing. Game-theoretic
algorithms instead are related to the second of those factors, and the third is
related to techniques from the field of machine learning. Here we demonstrate
how to exploit all three factors by modifying the search algorithm's
exploration stage so that rather than by random sampling, each coordinate of
the underlying search space is controlled by an associated
machine-learning-based ``player'' engaged in a non-cooperative game.
Experiments demonstrate that this modification improves SA by up to an order of
magnitude for bin-packing and for a model of an economic process run over an
underlying network. These experiments also reveal novel small worlds phenomena.
|
256
|
arxiv_abstracts
|
math/0301269
|
We extend the method of minimal vectors to arbitrary Banach spaces. It is
proved, by a variant of the method, that certain quasinilpotent operators on
arbitrary Banach spaces have hyperinvariant subspaces.
|
lt256
|
arxiv_abstracts
|
math/0301270
|
We study the free path length and the geometric free path length in the model
of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a
complete and rigorous proof for the existence of their distributions in the
small-scatterer limit and explicitly compute them.
As a corollary one gets a complete proof for the existence of the constant
term $c=2-3\ln 2+\frac{27\zeta(3)}{2\pi^2}$ in the asymptotic formula
$h(T)=-2\ln \eps+c+o(1)$ of the KS entropy of the billiard map in this model,
as conjectured by P. Dahlqvist.
|
lt256
|
arxiv_abstracts
|
math/0301271
|
In this paper we define a notion of gerbed tower, and use this notion to give
a geometric representation of cohomological classes.
|
lt256
|
arxiv_abstracts
|
math/0301272
|
Consider the Fulton-MacPherson configuration space of $n$ points on $\P^1$,
which is isomorphic to a certain moduli space of stable maps to $\P^1$. We
compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this
space. This yields a geometric interpretation of known asymptotic formulas for
the number of integral points of bounded height on compactifications of $\SL_2$
in the space of binary forms of degree $n\ge 3$.
|
lt256
|
arxiv_abstracts
|
math/0301273
|
We introduce the notion of N=1 supergeometric vertex operator superalgebra
motivated by the worldsheet geometry underlying genus-zero, two-dimensional,
holomorphic N=1 superconformal field theory. We then show, assuming the
convergence of certain projective factors, that the category of such objects is
isomorphic to the category of N=1 Neveu-Schwarz vertex operator superalgebras.
|
lt256
|
arxiv_abstracts
|
math/0301274
|
We show how to determine the $k$-th bit of Chaitin's algorithmically random
real number $\Omega$ by solving $k$ instances of the halting problem. From this
we then reduce the problem of determining the $k$-th bit of $\Omega$ to
determining whether a certain Diophantine equation with two parameters, $k$ and
$N$, has solutions for an odd or an even number of values of $N$. We also
demonstrate two further examples of $\Omega$ in number theory: an exponential
Diophantine equation with a parameter $k$ which has an odd number of solutions
iff the $k$-th bit of $\Omega$ is 1, and a polynomial of positive integer
variables and a parameter $k$ that takes on an odd number of positive values
iff the $k$-th bit of $\Omega$ is 1.
|
lt256
|
arxiv_abstracts
|
math/0301275
|
We show that for each positive integer $k$ there is a $k\times k$ matrix $B$
with $\pm 1$ entries such that putting $E$ to be the span of the rows of the
$k\times 2k$ matrix $[\sqrt{k}I_k,B]$, then $E,E^{\bot}$ is a Kashin splitting:
The $L_1^{2k}$ and the $L_2^{2k}$ are universally equivalent on both $E$ and
$E^{\bot}$. Moreover, the probability that a random $\pm 1$ matrix satisfies
the above is exponentially close to 1.
|
lt256
|
arxiv_abstracts
|
math/0301276
|
We obtain a discrete time analog of E. Noether's theorem in Optimal Control,
asserting that integrals of motion associated to the discrete time Pontryagin
Maximum Principle can be computed from the quasi-invariance properties of the
discrete time Lagrangian and discrete time control system. As corollaries,
results for first-order and higher-order discrete problems of the calculus of
variations are obtained.
|
lt256
|
arxiv_abstracts
|
math/0301277
|
The relationship between the Ohno relation and multiple polylogarithms are
discussed. Using this relationship, the algebraic reduction of the Ohno
relation is given.
|
lt256
|
arxiv_abstracts
|
math/0301278
|
We introduce a bond portfolio management theory based on foundations similar
to those of stock portfolio management. A general continuous-time zero-coupon
market is considered. The problem of optimal portfolios of zero-coupon bonds is
solved for general utility functions, under a condition of no-arbitrage in the
zero-coupon market. A mutual fund theorem is proved, in the case of
deterministic volatilities. Explicit expressions are given for the optimal
solutions for several utility functions.
|
lt256
|
arxiv_abstracts
|
math/0301279
|
We develop a new approach to the pulling back fixed point theorem of W.
Browder and use it in order to prove various generalizations of this result.
|
lt256
|
arxiv_abstracts
|
math/0301280
|
Let U^+ be the plus part of the quantized enveloping algebra of a simple Lie
algebra and let B^* be the dual canonical basis of U^+. Let b,b' be in B* and
suppose that one of the two elements is a q-commuting product of quantum flag
minors. We show that b and b' are multiplicative if and only if they q-commute.
|
lt256
|
arxiv_abstracts
|
math/0301281
|
In this article we study the tangent cones at first time singularity of a
Lagrangian mean curvature flow. If the initial compact submanifold is
Lagrangian and almost calibrated by Re\Omega in a Calabi-Yau n-fold (M,\Omega),
and T>0 is the first blow-up time of the mean curvature flow, then the tangent
cone of the mean curvature flow at a singular point (X,T) is a stationary
Lagrangian integer multiplicity current in R\sup 2n with volume density greater
than one at X. When n=2, the tangent cone consists of a finite union of more
than one 2-planes in R\sup 4 which are complex in a complex structure on R\sup
4.
|
lt256
|
arxiv_abstracts
|
math/0301282
|
Hexagonal circle patterns with constant intersection angles mimicking
holomorphic maps z^c and log(z) are studied. It is shown that the corresponding
circle patterns are immersed and described by special separatrix solutions of
discrete Painleve and Riccati equations. The general solution of the Riccati
equation is expressed in terms of the hypergeometric function. Global
properties of these solutions, as well as of the discrete z^c and log(z), are
established.
|
lt256
|
arxiv_abstracts
|
math/0301283
|
We deduce the Schaper formula for Hecke-algebras at root of unity from the
Jantzen conjecture in the LLT-setup. This explains an observation due to R.
Rouquier.
|
lt256
|
arxiv_abstracts
|
math/0301284
|
The deformation space of a simplicial G-tree T is the set of G-trees which
can be obtained from T by some collapse and expansion moves, or equivalently,
which have the same elliptic subgroups as T. We give a short proof of a
rigidity result by Forester which gives a sufficient condition for a
deformation space to contain an Aut(G)-invariant G-tree. This gives a
sufficient condition for a JSJ splitting to be invariant under automorphisms of
G. More precisely, the theorem claims that a deformation space contains at most
one strongly slide-free G-tree, where strongly slide-free means the following:
whenever two edges e_1, e_2 incident on a same vertex v are such that G_{e_1}
is a subset of G_{e_2}, then e_1 and e_2 are in the same orbit under G_v.
|
lt256
|
arxiv_abstracts
|
math/0301285
|
We deal with the representation theory of quantum groups and Hecke algebras
at roots of unity. We relate the philosophy of Andersen, Jantzen and Soergel on
graded translated functors to the Lascoux, Leclerc and Thibon-algorithm. This
goes via the Murphy standard basis theory and the idempotents coming from the
Murphy-Jucys operators. Our results lead to a guess on a tilting algorithm
outside the lowest $p^2$ alcove, which at least in the $SL_2$-case coincides
with Erdmann's results.
|
lt256
|
arxiv_abstracts
|
math/0301286
|
We consider a system of N phase oscillators having randomly distributed
natural frequencies and diagonalizable interactions among the oscillators. We
show that in the limit of N going to infinity, all solutions of such a system
are incoherent with probability one for any strength of coupling, which implies
that there is no sharp transition from incoherence to coherence as the coupling
strength is increased, in striking contrast to Kuramoto's (special) oscillator
system.
|
lt256
|
arxiv_abstracts
|
math/0301287
|
The paper contains the computation of the noncommutative A-ideal of the
figure-eight knot, a noncommutative generalization of the A-polynomial. We show
that if a knot has the same noncommutative A-ideal as the figure-eight knot,
then all colored Kauffman brackets are the same as those of the figure-eight
knot.
|
lt256
|
arxiv_abstracts
|
math/0301288
|
For a connected reductive group G and a finite-dimensional G-module V, we
study the invariant Hilbert scheme that parameterizes closed G-stable
subschemes of V affording a fixed, multiplicity-finite representation of G in
their coordinate ring. We construct an action on this invariant Hilbert scheme
of a maximal torus T of G, together with an open T-stable subscheme admitting a
good quotient. The fibers of the quotient map classify affine G-schemes having
a prescribed categorical quotient by a maximal unipotent subgroup of G. We show
that V contains only finitely many multiplicity-free G-subvarieties, up to the
action of the centralizer of G in GL(V). As a consequence, there are only
finitely many isomorphism classes of affine G-varieties affording a prescribed
multiplicity-free representation in their coordinate ring.
Final version, to appear in Journal of Algebraic Geometry
|
lt256
|
arxiv_abstracts
|
math/0301289
|
This note contains a correction to the paper, ``Local contribution to the
Lefschetz fixed point formula'', Inv. Math. 111 (1993), pp. 1-33.
|
lt256
|
arxiv_abstracts
|
math/0301290
|
We prove global existence for quasilinear wave equations outside of a wide
class of obstacles. The obstacles may contain trapped hyperbolic rays as long
as there is local exponential energy decay for the associated linear wave
equation. Thus, we can handle all non-trapping obstacles. We are also able to
handle non-diagonal systems satisfying the appropriate null condition.
|
lt256
|
arxiv_abstracts
|
math/0301291
|
We prove the existence of an automorphism-invariant coupling for the wired
and the free uniform spanning forests on Cayley graphs of finitely generated
residually amenable groups.
|
lt256
|
arxiv_abstracts
|
math/0301292
|
Three aspects of time series are uncertainty (dispersion at a given time
scale), scaling (time-scale dependence), and intermittency (inclination to
change dynamics). Simple measures of dispersion are the mean absolute deviation
and the standard deviation; scaling exponents describe how dispersions change
with the time scale. Intermittency has been defined as a difference between two
scaling exponents. After taking a moving average, these measures give
descriptive information, even for short heart rate records. For this data,
dispersion and intermittency perform better than scaling exponents.
|
lt256
|
arxiv_abstracts
|
math/0301293
|
Let $f\colon X\to Y$ be a perfect map between finite-dimensional metrizable
spaces and $p\geq 1$. It is shown that the space $C^*(X,\R^p)$ of all bounded
maps from $X$ into $\R^p$ with the source limitation topology contains a dense
$G_{\delta}$-subset consisting of $f$-regularly branched maps. Here, a map
$g\colon X\to\R^p$ is $f$-regularly branched if, for every $n\geq 1$, the
dimension of the set $\{z\in Y\times\R^p: |(f\times g)^{-1}(z)|\geq n\}$ is
$\leq n\cdot\big(\dim f+\dim Y\big)-(n-1)\cdot\big(p+\dim Y\big)$. This is a
parametric version of the Hurewicz theorem on regularly branched maps.
|
lt256
|
arxiv_abstracts
|
math/0301294
|
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
piecewise-linear maps has the parameter space (a,b)\in R^2. These maps are
area-preserving homeomorphisms of R^2 that map rays from the origin into rays
from the origin. The action on rays defines a map S_{ab} of the circle, which
has a well-defined rotation number. This paper characterizes the possible
behaviors of T_{ab} under iteration when the rotation number is rational. It
characterizes cases where the map T_{ab} is a periodic map.
|
lt256
|
arxiv_abstracts
|
math/0301295
|
A celebrated theorem of Harich-Chandra asserts that all invariant
eigendistributions on a semisimple Lie group are locally integrable functions.
We show that this result is a consequence of an algebraic property of a
holonomic D-module defined by Kashiwara and Hotta.
|
lt256
|
arxiv_abstracts
|
math/0301296
|
The invariant eigendistributions on a reductive Lie algebra are solutions of
a holonomic D-module which has been proved to be regular by Kashiwara-Hotta. We
solve here a conjecture of Sekiguchi saying that in the more general case of
symmetric pairs, the corresponding module are still regular.
|
lt256
|
arxiv_abstracts
|
math/0301297
|
We prove the following result, conjectured by Alan Weinstein: every smooth
proper Lie groupoid near a fixed point is locally linearizable, i.e. it is
locally isomorphic to the associated groupoid of a linear action of a compact
Lie group. In combination with a slice theorem of Weinstein, our result implies
the smooth linearizability of a proper Lie groupoid in the neighborhood of an
orbit under a mild condition.
|
lt256
|
arxiv_abstracts
|
math/0301298
|
We develop a symbol calculus for normal bimodule maps over a masa that is the
natural analogue of the Schur product theory. Using this calculus we are able
to easily give a complete description of the ranges of contractive normal
bimodule idempotents that avoids the theory of J*-algebras. We prove that if
$P$ is a normal bimodule idempotent and $\|P\| < 2/\sqrt{3}$ then $P$ is a
contraction. We finish with some attempts at extending the symbol calculus to
non-normal maps.
|
lt256
|
arxiv_abstracts
|
math/0301299
|
The convolution of indicators of two conjugacy classes on the symmetric group
S_q is usually a complicated linear combination of indicators of many conjugacy
classes. Similarly, a product of the moments of the Jucys--Murphy element
involves many conjugacy classes with complicated coefficients. In this article
we consider a combinatorial setup which allows us to manipulate such products
easily and we show that it very closely related to the combinatorial approach
to random matrices. Our formulas are exact (in a sense that they hold not only
asymptotically for large q). This result has many interesting applications, for
example it allows to find precise asymptotics of characters of large symmetric
groups and asymptotics of the Plancherel measure on Young diagrams.
|
lt256
|
arxiv_abstracts
|
math/0301300
|
In a flat 2-torus with a disk of diameter $r$ removed, let $\Phi_r(t)$ be the
distribution of free-path lengths (the probability that a segment of length
larger than $t$ with uniformly distributed origin and direction does not meet
the disk).
We prove that $\Phi_r(t/r)$ behaves like $\frac{2}{\pi^2 t}$ for each $t>2$
and in the limit as $r\to 0^+$, in some appropriate sense.
We then discuss the implications of this result in the context of kinetic
theory.
|
lt256
|
arxiv_abstracts
|
math/0301301
|
In a one-parameter study of a noninvertible family of maps of the plane
arising in the context of a numerical integration scheme, Lorenz studied a
sequence of transitions from an attracting fixed point to "computational
chaos." As part of the transition sequence, he proposed the following as a
possible scenario for the breakup of an invariant circle: the invariant circle
develops regions of increasingly sharper curvature until at a critical
parameter value it develops cusps; beyond this parameter value, the invariant
circle fails to persist, and the system exhibits chaotic behavior on an
invariant set with loops [Lorenz, 1989]. We investigate this problem in more
detail and show that the invariant circle is actually destroyed in a global
bifurcation before it has a chance to develop cusps. Instead, the global
unstable manifolds of saddle-type periodic points are the objects which develop
cusps and subsequently "loops" or "antennae." The one-parameter study is better
understood when embedded in the full two-parameter space and viewed in the
context of the two-parameter Arnold horn structure. Certain elements of the
interplay of noninvertibility with this structure, the associated invariant
circles, periodic points and global bifurcations are examined.
|
256
|
arxiv_abstracts
|
math/0301302
|
The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in
the simplicial category, are submonoids of monoids one finds in Temperley-Lieb
algebras, and as the monoids of Temperley-Lieb algebras are linked to
situations where an endofunctor is adjoint to itself, so the monoids of
simplicial endomorphisms are linked to arbitrary adjoint situations. This link
is established through diagrams of the kind found in Temperley-Lieb algebras.
Results about these matters, which were previously prefigured up to a point,
are here surveyed and reworked. A presentation of monoids of simplicial
endomorphisms by generators and relations has been given a long time ago. Here
a closely related presentation is given, with completeness proved in a new and
self-contained manner.
|
lt256
|
arxiv_abstracts
|
math/0301303
|
In a previous article, we proved tight lower bounds for the coefficients of
the generalized $h$-vector of a centrally symmetric rational polytope using
intersection cohomology of the associated projective toric variety. Here we
present a new proof based on the theory of combinatorial intersection
cohomology developed by Barthel, Brasselet, Fieseler and Kaup. This theory is
also valid for nonrational polytopes when there are no corresponding toric
varieties. So we can establish our bounds for centrally symmetric polytopes
even without requiring them to be rational.
|
lt256
|
arxiv_abstracts
|
math/0301304
|
Assuming the Hodge conjecture for abelian varieties of CM-type, one obtains a
good category of abelian motives over the algebraic closure of a finite field
and a reduction functor to it from the category of CM-motives. Consequentely,
one obtains a morphism of gerbes of fibre functors with certain properties. We
prove unconditionally that there exists a morphism of gerbes with these
properties, and we classify them.
|
lt256
|
arxiv_abstracts
|
math/0301305
|
We reduce the Abundance Conjecture in dimension 4 to the following numerical
statement: if the canonical divisor K is nef and has maximal nef dimension,
then K is big. From this point of view, we ``classify'' in dimension 2 nef
divisors which have maximal nef dimension, but which are not big.
|
lt256
|
arxiv_abstracts
|
math/0301306
|
In this paper, we associate canonically to every imaginary quadratic field
$K=\Bbb Q(\sqrt{-D})$ one or two isogenous classes of CM abelian varieties
over $K$, depending on whether $D$ is odd or even ($D \ne 4$). These abelian
varieties are characterized as of smallest dimension and smallest conductor,
and such that the abelian varieties themselves descend to $\Bbb Q$. When $D$ is
odd or divisible by 8, they are the `canonical' ones first studied by Gross and
Rohrlich. We prove that these abelian varieties have the striking property that
the vanishing order of their $L$-function at the center is dictated by the root
number of the associated Hecke character. We also prove that the smallest
dimension of a CM abelian variety over $K$ is exactly the ideal class number of
$K$ and classify when a CM abelian variety over $K$ has the smallest dimension.
|
lt256
|
arxiv_abstracts
|
math/0301307
|
We characterize the relationship between the singular values of a complex
Hermitian (resp., real symmetric, complex symmetric) matrix and the singular
values of its off-diagonal block. We also characterize the eigenvalues of an
Hermitian (or real symmetric) matrix C=A+B in terms of the combined list of
eigenvalues of A and B. The answers are given by Horn-type linear inequalities.
The proofs depend on a new inequality among Littlewood-Richardson coefficients.
|
lt256
|
arxiv_abstracts
|
math/0301308
|
With any integral lattice \Lambda in n-dimensional euclidean space we
associate an elementary abelian 2-group I(\lambda) whose elements represent
parts of the dual lattice that are similar to \Lambda. There are corresponding
involutions on modular forms for which the theta series of \Lambda is an
eigenform; previous work has focused on this connection. In the present paper
I(\Lambda) is considered as a quotient of some finite 2-subgroup of O_n(\R). We
establish upper bounds, depending only on n, for the order of I(\Lambda), and
we study the occurrence of similarities of specific types.
|
lt256
|
arxiv_abstracts
|
math/0301309
|
Any two compact, complete, one-dimensional geodesic spaces with identical
marked length spectrum have isometric $\pi_1$-hull.
The present version contains errors, notably in Lemmas 2.2 and 2.3 (path
cancellations can be more complicated), which then propagate through the paper.
The main result is correct as stated, and a proof can be found in the joint
paper with David Constantine, arXiv:1209.3709
|
lt256
|
arxiv_abstracts
|
math/0301310
|
We describe a numerical method to simulate an elastic shell immersed in a
viscous incompressible fluid. The method is developed as an extension of the
immersed boundary method using shell equations based on the Kirchhoff-Love and
the planar stress hypotheses. A detailed derivation of the shell equations used
in the numerical method is presented. This derivation as well as the numerical
method, use techniques of differential geometry in an essential way. Our main
motivation for the development of this method is its use in the construction of
a comprehensive three-dimensional computational model of the cochlea (the inner
ear). The central object of study within the cochlea is the ``basilar
membrane'', which is immersed in fluid and whose elastic properties rather
resemble those of a shell. We apply the method to a specific example, which is
a prototype of a piece of the basilar membrane and study the convergence of the
method in this case. Some typical features of cochlear mechanics are already
captured in this simple model. In particular, numerical experiments have shown
a traveling wave propagating from the base to the apex of the model shell in
response to external excitation in the fluid.
|
256
|
arxiv_abstracts
|
math/0301311
|
We describe a perfect group whose localization is not perfect.
|
lt256
|
arxiv_abstracts
|
math/0301312
|
We present an easy example of mutant links with different Khovanov homology.
The existence of such an example is important because it shows that Khovanov
homology cannot be defined with a skein rule similar to the skein relation for
the Jones polynomial.
|
lt256
|
arxiv_abstracts
|
math/0301313
|
The purpose of this paper is to point out a relation between the canonical
sheaf and the intersection complex of a singular algebraic variety. We focus on
the hypersurface case. Let $M$ be a complex manifold, $X\subset M$ a singular
hypersurface. We study residues of top-dimensional meromorphic forms with poles
along $X$. Applying resolution of singularities sometimes we are able to
construct residue classes either in $L^2$-cohomology of $X$ or in the
intersection cohomology. The conditions allowing to construct these classes
coincide. They can be formulated in terms of the weight filtration. Finally,
provided that these conditions hold, we construct in a canonical way a lift of
the residue class to cohomology of $X$.
|
lt256
|
arxiv_abstracts
|
math/0301314
|
We describe the weight filtration in the cohomology of toric varieties. We
present a role of the Frobenius automorphism in an elementary way. We prove
that equivariant intersection homology of an arbitrary toric variety is pure.
We obtain results concerning Koszul duality: nonequivariant intersection
cohomology is equal to the cohomology of the Koszul complex $IH_T^*(X)\otimes
H^*(T)$. We also describe the weight filtration in $IH^*(X)$.
|
lt256
|
arxiv_abstracts
|
math/0301315
|
We prove that the introduction of the class of geometrically atomic bundle
maps by Harvey and Lawson in their theory of singular connections is not
necessary because an arbitrary map satisfies the conditions of geometric
atomicity.
|
lt256
|
arxiv_abstracts
|
math/0301316
|
We construct a model of the Zero Point Field in terms of an infinite
collection of oscillators. This has relevance because of the recent
identification of Dark energy with such a Zero Point Field.
|
lt256
|
arxiv_abstracts
|
math/0301317
|
In this paper we will see deductive systems for classical propositional and
predicate logic in the calculus of structures. Like sequent systems, they have
a cut rule which is admissible. In addition, they enjoy a top-down symmetry and
some normal forms for derivations that are not available in the sequent
calculus. Identity axiom, cut, weakening and also contraction can be reduced to
atomic form. This leads to rules that are local: they do not require the
inspection of expressions of unbounded size.
|
lt256
|
arxiv_abstracts
|
math/0301318
|
We give a constructive proof that the Regge symmetry is a scissors congruence
in hyperbolic space. The main tool is Leibon's construction for computing the
volume of a general hyperbolic tetrahedron. The proof consists of identifying
the key elements in Leibon's construction and permuting them.
|
lt256
|
arxiv_abstracts
|
math/0301319
|
We discuss recent results on decay of correlations for non-uniformly
expanding maps. Throughout the discussion, we address the question of why
different dynamical systems have different rates of decay of correlations and
how this may reflect underlying geometrical characteristics of the system.
|
lt256
|
arxiv_abstracts
|
math/0301320
|
Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings
and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram
of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then
$1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are shape in the sense
that the upper bound of $b(D)$ is achieved by alternating knots and the lower
bound of $b(D)$ is achieved by torus knots. The second inequality becomes an
equality only when the knot is an alternating knot. We prove that the first
inequality becomes an equality only when the knot is a torus knot.
|
lt256
|
arxiv_abstracts
|
math/0301321
|
The human cochlea is a remarkable device, able to discern extremely small
amplitude sound pressure waves, and discriminate between very close
frequencies. Simulation of the cochlea is computationally challenging due to
its complex geometry, intricate construction and small physical size. We have
developed, and are continuing to refine, a detailed three-dimensional
computational model based on an accurate cochlear geometry obtained from
physical measurements. In the model, the immersed boundary method is used to
calculate the fluid-structure interactions produced in response to incoming
sound waves. The model includes a detailed and realistic description of the
various elastic structures present.
In this paper, we describe the computational model and its performance on the
latest generation of shared memory servers from Hewlett Packard. Using compiler
generated threads and OpenMP directives, we have achieved a high degree of
parallelism in the executable, which has made possible several large scale
numerical simulation experiments that study the interesting features of the
cochlear system. We show several results from these simulations, reproducing
some of the basic known characteristics of cochlear mechanics.
|
256
|
arxiv_abstracts
|
math/0301322
|
We introduce two classes of "egg type" domains, built on general bounded
symmetric domains, for which we compute the Bergmann kernel in explicit form.
We use the characterization of bounded symmetric domains through Jordan triple
systems. The egg type domains are defined using the generic norm. A
generalization of the Hua integral is computed; the result shows the existence
of a special polynomial with integer or half-integer coefficients, attached to
each irreducible bounded symmetric domain.
|
lt256
|
arxiv_abstracts
|
math/0301323
|
A differential algebra of finite type over a field k is a filtered algebra A,
such that the associated graded algebra is finite over its center, and the
center is a finitely generated k-algebra. The prototypical example is the
algebra of differential operators on a smooth affine variety, when char k = 0.
We study homological and geometric properties of differential algebras of
finite type. The main results concern the rigid dualizing complex over such an
algebra A: its existence, structure and variance properties. We also define and
study perverse A-modules, and show how they are related to the Auslander
property of the rigid dualizing complex of A.
|
lt256
|
arxiv_abstracts
|
math/0301324
|
The purpose of this paper is to exhibit a natural construction between
complex geometry and symplectic geometry following the idea of mirror symmetry.
Suppose we are given a family of pairs of 2-dimensional K\"ahler tori and
stable holomorphic vector bundles on them $(\hat M_{\ep}, E_{\ep})$, \ep \in
(0, 1]$, and each has structure of a Lagrangian torus fibration $\pi:\hat
M_{\ep} \to B$ whose fibers are of diameter $O(\ep)$, and let $A_{\ep}$ be a
family of hermitian Yang-Mills(HYM) connections on $E_{\ep}$. As $\ep$ goes to
zero, $A_{\ep}$ will, modulo possible bubbles, converge to a connection which
is flat on each fiber. Since each fiber is a torus, limit connection will
determine elements of the dual torus, which are points of the fiber of the
mirror $M_1$. These points gather to make up (special) Lagrangian variety.
|
lt256
|
arxiv_abstracts
|
math/0301325
|
In this paper we answer negatively a question posed by Casacuberta, Farjoun,
and Libman about the preservation of perfect groups under localization
functors. Indeed, we show that a certain $P$-localization of Berrick and
Casacuberta's universal acyclic group is not perfect. We also investigate under
which conditions perfectness is preserved: For instance, we show that if the
localization of a perfect group is finite then it is perfect.
|
lt256
|
arxiv_abstracts
|
math/0301326
|
We present an approach to solvable pseudo-Riemannian symmetric spaces based
on papers of M.Cahen, M.Parker and N.Wallach. Thereby we reproduce the
classification of solvable symmetric triples of Lorentzian signature $(1,n-1)$
and complete the case of signature $(2,n-2)$. Moreover we discuss the topology
of non-simply-connected symmetric spaces.
|
lt256
|
arxiv_abstracts
|
math/0301327
|
We give a detailed account of the classical Van Kampen method for computing
presentations of fundamental groups of complements of complex algebraic curves,
and of a variant of this method, working with arbitrary projections (even with
vertical asymptotes).
|
lt256
|
arxiv_abstracts
|
math/0301328
|
Quillen's plus construction is a topological construction that kills the
maximal perfect subgroup of the fundamental group of a space without changing
the integral homology of the space. In this paper we show that there is a
topological construction that, while leaving the integral homology of a space
unaltered, kills even the intersection of the transfinite lower central series
of its fundamental group. Moreover, we show that this is the maximal subgroup
that can be factored out of the fundamental group without changing the integral
homology of a space.
|
lt256
|
arxiv_abstracts
|
math/0301329
|
A K3-surface is a (smooth) surface which is simply connected and has trivial
canonical bundle. In these notes we investigate three particular pencils of
K3-surfaces with maximal Picard number. More precisely the general member in
each pencil has Picard number 19 and each pencil contains four surfaces with
Picard number 20. These surfaces are obtained as the minimal resolution of
quotients X/G, where $G\subset SO(4,\R)$ is some finite subgroup and $X\subset
\P_3(\C)$ denotes a G-invariant surface. The singularities of X/G come from fix
points of G on X or from singularities of X. In any case the singularities on
X/G are A-D-E surface singularities. The rational curves which resolve them and
some extra 2-divisible sets, resp. 3-divisible sets of rational curves generate
the Neron-Severi group of the minimal resolution.
|
lt256
|
arxiv_abstracts
|
math/0301330
|
Eigenfunctions of the Askey-Wilson second order $q$-difference operator for
$0<q<1$ and $|q|=1$ are constructed as formal matrix coefficients of the
principal series representation of the quantized universal enveloping algebra
$U_q(sl(2,\mathbb{C}))$. The eigenfunctions are in integral form and may be
viewed as analogues of Euler's integral representation for Gauss'
hypergeometric series. We show that for $0<q<1$ the resulting eigenfunction can
be rewritten as a very-well-poised ${}_8\phi_7$-series, and reduces for special
parameter values to a natural elliptic analogue of the cosine kernel.
|
lt256
|
arxiv_abstracts
|
math/0301331
|
The paper addresses questions of existence and regularity of solutions to
linear partial differential equations whose coefficients are generalized
functions or generalized constants in the sense of Colombeau. We introduce
various new notions of ellipticity and hypoellipticity, study their
interrelation, and give a number of new examples and counterexamples. Using the
concept of $\G^\infty$-regularity of generalized functions, we derive a general
global regularity result in the case of operators with constant generalized
coefficients, a more specialized result for second order operators, and a
microlocal regularity result for certain first order operators with variable
generalized coefficients. We also prove a global solvability result for
operators with constant generalized coefficients and compactly supported
Colombeau generalized functions as right hand sides.
|
lt256
|
arxiv_abstracts
|
math/0301332
|
Via a non degenerate symmetric bilinear form we identify the coadjoint
representation with a new representation and so we induce on the orbits a
simplectic form. By considering Hamiltonian systems on the orbits we study some
features of them and finally find commuting functions under the corresponding
Lie-Poisson bracket
|
lt256
|
arxiv_abstracts
|
math/0301333
|
Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its
vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed
into convex polytopes without adding any vertex. Then $P$ is infinitesimally
rigid. More generally, let $P$ be a polyhedron bounding a domain which is the
union of polytopes $C_1, ..., C_n$ with disjoint interiors, whose vertices are
the vertices of $P$. Suppose that there exists an ellipsoid which contains no
vertex of $P$ but intersects all the edges of the $C_i$. Then $P$ is
infinitesimally rigid. The proof is based on some geometric properties of
hyperideal hyperbolic polyhedra.
|
lt256
|
arxiv_abstracts
|
math/0301334
|
We prove that a suitably adjusted version of Peter Jones' formula for
interpolation by bounded holomorphic functions gives a sharp upper bound for
what is known as the constant of interpolation. We show how this leads to
precise and computable numerical bounds for this constant.
|
lt256
|
arxiv_abstracts
|
math/0301335
|
In previous papers we have introduced a sufficient condition for uniform
attractivity of the origin for a class of nonlinear time-varying systems which
is stated in terms of persistency of excitation (PE), a concept well known in
the adaptive control and systems identification literature. The novelty of our
condition, called uniform delta-PE, is that it is tailored for nonlinear
functions of time and state and it allows us to prove uniform asymptotic
stability. In this paper we present a new definition of u-delta-PE which is
conceptually similar to but technically different from its predecessors and
give several useful characterizations. We make connections between this
property and similar properties previously used in the literature. We also show
when this condition is necessary and sufficient for uniform (global) asymptotic
stability for a large class of nonlinear time-varying systems. Finally, we show
the utility of our main results on some control applications regarding
feedforward systems and systems with matching nonlinearities.
|
256
|
arxiv_abstracts
|
math/0301336
|
We study the semicrossed product of a finite dimensional C^*-algebra by two
types of commuting automorphisms, and identify them with matrix algebras of
analytic functions in two variables. We look at the connections with
semicrossed products by Z_+ actions.
|
lt256
|
arxiv_abstracts
|
math/0301337
|
We obtain a characterization in terms of dynamical systems of those
r-discrete groupoids for which the groupoid C*-algebra is approximately
finite-dimensional (AF). These ideas are then used to compute the K-theory for
AF algebras by utilizing the actions of these partial homeomorphisms, and these
K-theoretic calculations are applied to some specific examples of AF algebras.
Finally, we show that, for a certain class of dimension groups, a groupoid can
be obtained directly from the dimension group's structure whose associated
C*-algebra has K_0 group isomorphic to the original dimension group.
|
lt256
|
arxiv_abstracts
|
math/0301338
|
This paper gives a partial confirmation of a conjecture of P. Agarwal, S.
Har-Peled, M. Sharir, and K. Varadarajan that the total curvature of a shortest
path on the boundary of a convex polyhedron in the 3-dimensional Euclidean
space cannot be arbitrarily large. It is shown here that the conjecture holds
for a class of polytopes for which the ratio of the radii of the circumscribed
and inscribed ball is bounded. On the other hand, an example is constructed to
show that the total curvature of a shortest path on the boundary of a convex
polyhedron can exceed 2 \pi. Another example shows that the spiraling number of
a shortest path on the boundary of a convex polyhedron can be arbitrarily
large.
|
lt256
|
arxiv_abstracts
|
math/0301339
|
The classical Center-Focus problem posed by H. Poincare in 1880's asks about
the classification of planar polynomial vector fields such that all their
integral trajectories are closed curves whose interiors contain a fixed point
(which is called a center). In this paper we describe a new general approach to
the Center Problem.
|
lt256
|
arxiv_abstracts
|
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