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math/0212260
We consider a class of measures called autophage which was introduced and studied by Szekely for measures on the real line. We show that the autophage measures on finite-dimensional vector spaces over real or p-adic field are infinitely divisible without idempotent factors and are absolutely continuous with bounded continuous density. We also show that certain semistable measures on such vector spaces are absolutely continuous.
lt256
arxiv_abstracts
math/0212261
In arXiv math.MG/0207296 we introduced a product construction for locally compact, complete, geodesic hyperbolic metric spaces. In the present paper we define the hyperbolic product for general Gromov-hyperbolic spaces. In the case of roughly geodesic spaces we also analyse the boundary at infinity.
lt256
arxiv_abstracts
math/0212262
The aim of this paper is the determination of the largest $n$-dimensional polytope with $n+3$ vertices of unit diameter. This is a special case of a more general problem proposed by Graham.
lt256
arxiv_abstracts
math/0212263
We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrodinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of degree at most two. We describe the separate roles of the nonlinearity and of the potential, with tools which seem to be specific to the class of potentials that we consider. We also discuss the case of more general subquadratic potentials.
lt256
arxiv_abstracts
math/0212264
A list of known quantum spheres of dimension one, two and three is presented.
lt256
arxiv_abstracts
math/0212265
In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the following extension of Voiculescu's random matrix result: Let X_1^(n),...,X_r^(n) be a system of r stochastically independent n by n Gaussian self-adjoint random matrices as in Voiculescu's random matrix paper [V4], and let (x_1,...,x_r) be a semi-circular system in a C*-probability space. Then for every polynomial p in r noncommuting variables lim_{n->oo}||p(X_1^(n),...,X_r^(n))|| = ||p(x_1,...,x_r)||, for almost all omega in the underlying probability space. We use the result to show that the Ext-invariant for the reduced C*-algebra of the free group on 2 generators is not a group but only a semi-group. This problem has been open since Anderson in 1978 found the first example of a C*-algebra A for which Ext(A) is not a group.
256
arxiv_abstracts
math/0212266
This is an introduction to gerbes for topologists, with emphasis on non-abelian cohomology.
lt256
arxiv_abstracts
math/0212267
Define $I_n^k(\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_i$, for some $i \geq 2$, and define $I_n^k(\emptyset;\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which contain the pattern $\alpha \in S_i$, for some $i \geq 2$, exactly once. Let $i_n^k(\alpha)$ be the number of elements in $I_n^k(\alpha)$ and let $i_n^k(\emptyset;\alpha)$ be the number of elements in $I_n^k(\emptyset;\alpha)$. We investigate $I_n^k(\alpha)$ and $I_n^k(\emptyset;\alpha)$ for all $\alpha \in S_3$. In particular, we show that $i_n^k(132)=i_n^k(213)=i_n^k(321)$, $i_n^k(231)=i_n^k(312)$, $i_n^k(\emptyset;132) =i_n^k(\emptyset;213)$, and $i_n^k(\emptyset;231)=i_n^k(\emptyset;312)$ for all $0 \leq k \leq n$.
lt256
arxiv_abstracts
math/0212268
There exists a smooth foliation with 3 singular points on the two-dimensional torus such that any lifting of a leaf of this foliation on the universal covering of the torus is a dense subset of the covering.
lt256
arxiv_abstracts
math/0212269
In this note, it is proved that the noise (in the sense of Tsirelson) generated by a Brownian sticky flow (as defined in math.PR/0211387) is black.
lt256
arxiv_abstracts
math/0212270
We organize the nilpotent orbits in the exceptional complex Lie algebras into series using the triality model and show that within each series the dimension of the orbit is a linear function of the natural parameter a=1,2,4,8, respectively for f_4,e_6,e_7,e_8. We also obtain explicit representatives in a uniform manner. We observe similar regularities for the centralizers of nilpotent elements in a series and graded components in the associated grading of the ambient Lie algebra. More strikingly, for a greater than one, the degrees of the unipotent characters of the corresponding Chevalley groups, associated to these series through the Springer correspondance are given by polynomials which have uniform expressions in terms of a.
lt256
arxiv_abstracts
math/0212271
The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology (Ellis Horwood, 1988). This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these results, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader. It is expected that this material will be part of a new edition of the book.
256
arxiv_abstracts
math/0212272
We study Harish-Chandra representations of Yangian for gl(2). We prove an analogue of Kostant theorem showing that resterited Yangians for gl(2) are free modules over certain maximal commutative subalgebras. We also study the categories of generic Harish-Chandra modules, describe their simple modules and indecomposable modules in tame blocks.
lt256
arxiv_abstracts
math/0212273
The main motivation for this work was to find an explicit formula for a "Szego-regularized" determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold. The idea of the Szego-regularization was suggested by V. Guillemin and K. Okikiolu. They have computed the second term in a Szego type expansion on a Zoll manifold of an arbitrary dimension. In the present work we compute the third asymptotic term in any dimension. In the case of dimension 2, our formula gives the above mentioned expression for the Szego-redularized determinant of a zeroth order PsDO. The proof uses a new combinatorial identity, which generalizes a formula due to G.A. Hunt and F.J. Dyson. This identity is related to the distribution of the maximum of a random walk with i.i.d. steps on the real line. The full version of this paper is also available, math.FA/0212275.
lt256
arxiv_abstracts
math/0212274
We outline the main features of the definitions and applications of crossed complexes and cubical $\omega$-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain some non commutative results and compute some homotopy types.
lt256
arxiv_abstracts
math/0212275
This is a detailed version of the paper math.FA/0212273. The main motivation for this work was to find an explicit formula for a "Szego-regularized" determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold. The idea of the Szego-regularization was suggested by V. Guillemin and K. Okikiolu. They have computed the second term in a Szego type expansion on a Zoll manifold of an arbitrary dimension. In the present work we compute the third asymptotic term in any dimension. In the case of dimension 2, our formula gives the above mentioned expression for the Szego-redularized determinant of a zeroth order PsDO. The proof uses a new combinatorial identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This identity is related to the distribution of the maximum of a random walk with i.i.d. steps on the real line. The proof of this combinatorial identity together with historical remarks and a discussion of probabilistic and algebraic connections has been published separately.
lt256
arxiv_abstracts
math/0212276
Let X be a noetherian scheme defined over an algebraically closed field of positive characteristic p, and G be a finite group, of order divisible by p, acting on X. We introduce a refinement of the equivariant K-theory of X to take into account the information related to modular representation theory. As an application, in the 1-dimensional case, we generalize a modular Riemann-Roch theorem given by S.Nakajima, extending the link between Galois modules and wild ramification.
lt256
arxiv_abstracts
math/0212277
We describe a method of adding tails to C*-correspondences which generalizes the process used in the study of graph C*-algebras. We show how this technique can be used to extend results for augmented Cuntz-Pimsner algebras to C*-algebras associated to general C*-correspondences, and as an application we prove a gauge-invariant uniqueness theorem for these algebras. We also define a notion of relative graph C*-algebras and show that properties of these C*-algebras can provide insight and motivation for results about relative Cuntz-Pimsner algebras.
lt256
arxiv_abstracts
math/0212278
For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins that every algebraic curvature tensor has a structure which is very similar to that of the above Osserman curvature tensors. We verify our results by means of the Littlewood-Richardson rule and plethysms. For certain symbolic calculations we used the Mathematica packages MathTensor, Ricci and PERMS.
lt256
arxiv_abstracts
math/0212279
We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G is an irreducible Weyl group in GL(h) and V=h+ h^* then no smooth symplectic resolution of V/G exists unless G is of types A,B, or C.
lt256
arxiv_abstracts
math/0212280
We survey some results concerning finite group actions on products of spheres.
lt256
arxiv_abstracts
math/0212281
We consider the integral of fractional Brownian motion (IFBM) and its functionals $\xi_T$ on the intervals $(0,T)$ and $(-T,T)$ of the following types: the maximum $M_T$, the position of the maximum, the occupation time above zero etc. We show how the asymptotics of $P(\xi_T<1)=p_T, T\to \infty$, is related to the Hausdorff dimension of Lagrangian regular points for the inviscid Burgers equation with FBM initial velocity. We produce computational evidence in favor of a power asymptotics for $p_T$. The data do not reject the hypothesis that the exponent $\theta$ of the power law is related to the similarity parameter $H$ of fractional Brownian motion as follows: $\theta =-(1-H)$ for the interval $(-T,T)$ and $\theta =-H(1-H)$ for $(0,T)$. The point 0 is special in that IFBM and its derivative both vanish there.
lt256
arxiv_abstracts
math/0212282
We use $p$-adic families of automorphic forms for an unitary group in three variables, containing some non-tempered forms constructed by Rogawski, to prove some cases of the Bloch-Kato conjectures.
lt256
arxiv_abstracts
math/0212283
In this paper, we study the following problem $$ \{{ll} \Delta_{H^n} u-u+u^p=0 & in H^n u>0& in H^n u(x)\to 0 &\rho(x)\to\infty}. $$ where $1<p < \frac{Q+2}{Q-2}$, Q is the homogeneous dimension of Heisenberg group $H^n$. Our main result is that this problem has at least one positive solution.
lt256
arxiv_abstracts
math/0212284
We consider the class of groups called identity excluding which has the property that any non-trivial irreducible unitary representation restricted to a dense subgroup does not weakly contain the trivial representation. For adapted and strictly aperiodic probability measures on these groups, it is known that the averages of unitary representations converge strongly. We show that motion group of a totally disconnected nilpotent group and certain class of p-adic algebraic groups which includes gruops whose solvable radical is type R are identity excluding. We also prove the convergence of averages of unitary representations for split solvable algebraic groups, which are not necessarily identity excluding.
lt256
arxiv_abstracts
math/0212285
A Markov operator $P$ on a $\sigma$-finite measure space $(X, \Sigma, m)$ with invariant measure $m$ is said to have Krengel-Lin decomposition if $L^2 (X) = E_0 \oplus L^2 (X,\Sigma_d)$ where $E_0 = \{f \in L^2 (X) \mid ||P^n (f) || \ra 0 \}$ and $\Sigma_d$ is the deterministic $\sigma $-field of $P$. We consider convolution operators and we show that a measure $\lam$ on a hypergroup has Krengel-Lin decomposition if and only if the sequence $(\check \lam ^n *\lam ^n)$ converges to an idempotent or $\lam$ is scattered. We verify this condition for probabilities on Tortrat groups, on commutative hypergroups and on central hypergroups. We give a counter-example to show that the decomposition is not true for measures on discrete hypergroups which is in contrast to the discrete groups case.
lt256
arxiv_abstracts
math/0212286
The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper, we establish for O(p,2) an adjointness result between Borcherds' singular theta lift and the Kudla-Millson theta lift. We extend this result to arbitrary signature by introducing a new Borcherds lift for O(p,q). On the geometric side, this lift can be interpreted as a differential character in the sense of Cheeger and Simons.
lt256
arxiv_abstracts
math/0212287
In this paper an autonomous analytical system of ordinary differential equations is considered. For an asymptotically stable steady state x0 of the system a gradual approximation of the domain of attraction DA is presented in the case when the matrix of the linearized system in x0 is diagonalizable. This technique is based on the gradual extension of the "embryo" of an analytic function of several complex variables. The analytic function is the transformed of a Lyapunov function whose natural domain of analyticity is the DA and which satisfies a linear non-homogeneous partial differential equation. The equation permits to establish an "embryo" of the transformed function and a first approximation of DA. The "embryo" is used for the determination of a new "embryo" and a new part of the DA. In this way, computing new "embryos" and new domains, the DA is gradually approximated. Numerical examples are given for polynomial systems. For systems considered recently in the literature the results are compared with those obtained with other methods.
256
arxiv_abstracts
math/0212288
We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded.
lt256
arxiv_abstracts
math/0212289
In the setting of $\R^d$ with an $n-$dimensional measure $\mu,$ we give several characterizations of Lipschitz spaces in terms of mean oscillations involving $\mu.$ We also show that Lipschitz spaces are preserved by those Calderon-Zygmund operators $T$ associated to the measure $\mu$ for which T(1) is the Lipschitz class $0.$
lt256
arxiv_abstracts
math/0212290
Soit F un corps local non archimedien de caracteristique residuelle p. On designe par R un corps algebriquement clos de caracteristique p et par Q une cloture algebrique du corps des nombres p-adiques. On classifie les modules simples de dimension finie de la R-algebre de Hecke-Iwahori du groupe lineaire Gl_3(F). Ils sont obtenus par reduction modulo p des modules simples de la Q-algebre de Hecke-Iwahori qui possedent une structure entiere.
lt256
arxiv_abstracts
math/0212291
The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell's equations written for differential forms over a 3-manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel time metric as well as the scalar wave impedance on the manifold. This invariant result leads also to a complete characterization of the non-uniqueness of the corresponding inverse problem in bounded domains of R^3. AMS-classifications: 35R30, 35L20, 58J45
lt256
arxiv_abstracts
math/0212292
Hilbert space representations of the cross product *-algebra of the Hopf *-algebra U_q(su_2) and its module *-algebras O(S^2_{qc}) of Podles' spheres are studied. Two classes of representations are described by explicit formulas for the actions of the generators.
lt256
arxiv_abstracts
math/0212293
For $\a,\b>0$ and for a locally integrable function (or, more generally, a distribution) $\f$ on $(0,\be)$, we study integral ooperators ${\frak G}^{\a,\b}_\f$ on $L^2(\R_+)$ defined by $\big({\frak G}^{\a,\b}_\f f\big)(x)=\int_{\R_+}\f\big(x^\a+y^\b\big)f(y)dy$. We describe the bounded and compact operators ${\frak G}^{\a,\b}_\f$ and operators ${\frak G}^{\a,\b}_\f$ of Schatten--von Neumann class $\bS_p$. We also study continuity properties of the averaging projection $\Q_{\a,\b}$ onto the operators of the form ${\frak G}^{\a,\b}_\f$. In particular, we show that if $\a\le\b$ and $\b>1$, then ${\frak G}^{\a,\b}_\f$ is bounded on $\bS_p$ if and only if $2\b(\b+1)^{-1}<p<2\b(\b-1)^{-1}$.
lt256
arxiv_abstracts
math/0212294
We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.
lt256
arxiv_abstracts
math/0212295
We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward'' dualities are also established: a Lambda duality over the Novikov Ring and a Topological Vector Space duality over the reals.
lt256
arxiv_abstracts
math/0212296
Stochastic processes on manifolds over non-Archimedean fields and with transition measures having values in the field $\bf C$ of complex numbers are defined and investigated. The analogs of Markov, Poisson and Wiener processes are studied. For Poisson processes the non-Archimedean analog of the L\`evy theorem is proved. Stochastic antiderivational equations as well as pseudodifferential equations on manifolds are investigated.
lt256
arxiv_abstracts
math/0212297
We utilize the obstruction theory of Galewski-Matumoto-Stern to derive equivalent formulations of the Triangulation Conjecture. For example, every closed topological manifold M^n with n > 4 can be simplicially triangulated if and only if the two distinct combinatorial triangulations of RP^5 are simplicially concordant.
lt256
arxiv_abstracts
math/0212298
Let O be a three-dimensional Nil-orbifold, with branching locus a knot Sigma transverse to the Seifert fibration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in (pi-epsilon, pi). We also study the space of Dehn filling parameters of O-Sigma. Surprisingly it is not diffeomorphic to the deformation space constructed from the variety of representations of O-Sigma. As a corollary of this, we find examples of spherical cone manifolds with singular set a knot that are not locally rigid. Those examples have large cone angles.
lt256
arxiv_abstracts
math/0212299
We consider the set of the power non-negative polynomials of several variables and its subset that consists of polynomials which can be represented as a sum of squares. It is shown in the classic work by D.Hilbert that it is a proper subset. Both sets are convex. In our paper we have made an attempt to work out a general approach to the investigation of the extremal elements of these convex sets. We also consider the class of non-negative rational functions. The article is based on the following methods: 1.We investigate non-negative trigonometrical polynomials and then with the help of the Calderon transformation we proceed to the power polynomials. 2.The way of constructing support hyperplanes to the convex sets is given in the paper.
lt256
arxiv_abstracts
math/0212300
We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size $L^2$, inverse temperature $\beta>\betac$ and overall magnetization conditioned to take the value $\mstar L^2-2\mstar v_L$, where $\betac^{-1}$ is the critical temperature, $\mstar=\mstar(\beta)$ is the spontaneous magnetization and $v_L$ is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when $v_L^{3/2} L^{-2}$ tends to a definite limit. Specifically, we identify a dimensionless parameter $\Delta$, proportional to this limit, a non-trivial critical value $\Deltac$ and a function $\lambda_\Delta$ such that the following holds: For $\Delta<\Deltac$, there are no droplets beyond $\log L$ scale, while for $\Delta>\Deltac$, there is a single, Wulff-shaped droplet containing a fraction $\lambda_\Delta\ge\lamc=2/3$ of the magnetization deficit and there are no other droplets beyond the scale of $\log L$. Moreover, $\lambda_\Delta$ and $\Delta$ are related via a universal equation that apparently is independent of the details of the system.
256
arxiv_abstracts
math/0212301
The calculation of volumes of polyhedra in the three-dimensional Euclidean, spherical and hyperbolic spaces is very old and difficult problem. In particular, an elementary formula for volume of non-euclidean simplex is still unknown. One of the simplest polyhedra is the Lambert cube Q(\alpha,\beta,\gamma). By definition, Q(\alpha,\beta,\gamma) is a combinatorial cube, with dihedral angles \alpha,\beta and \gamma assigned to the three mutually non-coplanar edges and right angles to the remaining. The hyperbolic volume of Lambert cube was found by Ruth Kellerhals (1989) in terms of the Lobachevsky function \Lambda(x). In the present paper the spherical volume of Q(\alpha,\beta,\gamma) is defined in the terms of the function \delta(\alpha,\theta) which can be considered as a spherical analog of the Lobachevsky function \Delta(\alpha,\theta)=\Lambda(\alpha + \theta) - \Lambda(\alpha - \theta)
lt256
arxiv_abstracts
math/0212302
The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get semi-local variants of the inequality holding in any dimension that involve domains with non-vanishing Gauss-Kronecker curvature. The paper also contains inequalities of isoperimetric type involving the total curvature, as well as a "reverse" isoperimetric inequality for spaces with constant curvature.
lt256
arxiv_abstracts
math/0212303
Let G be a reductive connected p-adic group. With help of the Fourier inversion formula used in [Une formule de Plancherel pour l'algebre de Hecke d'un groupe reductif p-adique - V. Heiermann, Comm. Math. Helv. 76, 388-415, 2001] we give a spectral decomposition on G. In particular we deduce from it essentially that a cuspidal representation of a Levi subgroup M is in the cuspidal support of a square integrable representation of G, if and only if it is a pole of Harish-Chandra's \mu-function of order equal to the parabolic rank of M. This result has been conjectured by A. Silberger in 1978. In more explicit terms, we show that this condition is necessary and that its sufficiency is equivalent to a combinatorical property of Harish-Chandra's \mu-function which appears to be a consequence of a result of E. Opdam. We get also identities between some linear combinations of matrix coeffieicients. These identities contain informations on the formel degree of square integrable representations and on their position in the induced representation.
256
arxiv_abstracts
math/0212304
Let f be a newform of weight at least 2 and squarefree level with Fourier coefficients in a number field K. We give explicit bounds, depending on congruences of f with other newforms, on the set of primes lambda of K for which the deformation problem associated to the mod lambda Galois representation of f is obstructed. We include some explicit examples.
lt256
arxiv_abstracts
math/0212305
We clarify the exposition of Phases 2 and 3a in "The Floyd-Warshall Algorithm, the AP and the TSP". We also improve and simplify theorem 3.6 . In line with clarifying the exposition, we change the matrices in examples 3.4 and 3.5 of "The Floyd-Warshall Algorithm, the AP and the TSP II".
lt256
arxiv_abstracts
math/0212306
We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative two-torus with real multiplication. These results are based on the techniques of math.QA/0211262 allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.
lt256
arxiv_abstracts
math/0212307
This paper studies state quantization schemes for feedback stabilization of control systems with limited information. The focus is on designing the least destabilizing quantizer subject to a given information constraint. We explore several ways of measuring the destabilizing effect of a quantizer on the closed-loop system, including (but not limited to) the worst-case quantization error. In each case, we show how quantizer design can be naturally reduced to a version of the so-called multicenter problem from locational optimization. Algorithms for solving such problems are discussed. In particular, an iterative solver is developed for a novel weighted multicenter problem which most accurately represents the least destabilizing quantizer design.
lt256
arxiv_abstracts
math/0212308
The connection between Riemann surfaces with boundaries and the theory of vertex operator algebras is discussed in the framework of conformal field theories defined by Kontsevich and Segal and in the framework of their generalizations in open string theory and boundary conformal field theory. We present some results, problems, conjectures, their conceptual implications and meanings in a program to construct these theories from representations of vertex operator algebras.
lt256
arxiv_abstracts
math/0212309
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein's Theorem. Our extension relates volumes of polytopes with the number of connected components of the complex zero set of a polynomial system, and allows any number of polynomials and/or variables. (2) A near optimal complexity bound for computing mixed area -- a quantity intimately related to counting complex roots in the plane.
lt256
arxiv_abstracts
math/0212310
We present a general framework for TQFT and related constructions using the language of monoidal categories. We construct a topological category C and an algebraic category D, both monoidal, and a TQFT functor is then defined as a certain type of monoidal functor from C to D. In contrast with the cobordism approach, this formulation of TQFT is closer in spirit to the classical functors of algebraic topology, like homology. The fundamental operation of gluing is incorporated at the level of the morphisms in the topological category through the notion of a gluing morphism, which we define. It allows not only the gluing together of two separate objects, but also the self-gluing of a single object to be treated in the same fashion. As an example of our framework we describe TQFT's for oriented 2D-manifolds, and classify a family of them in terms of a pair of tensors satisfying some relations.
lt256
arxiv_abstracts
math/0212311
We analyze geometry of the second order differential operators, having in mind applications to Batalin--Vilkovisky formalism in quantum field theory. As we show, an exhaustive picture can be obtained by considering pencils of differential operators acting on densities of all weights simultaneously. The algebra of densities, which we introduce here, has a natural invariant scalar product. Using it, we prove that there is a one-to-one correspondence between second-order operators in this algebra and the corresponding brackets. A bracket on densities incorporates a bracket on functions, an ``upper connection'' in the bundle of volume forms, and a term similar to the ``Brans--Dicke field'' of the Kaluza--Klein formalism. These results are valid for even operators on a usual manifold as well as for odd operators on a supermanifold. For an odd operator $\Delta$ we show that conditions on the order of the operator $\Delta^2$ give an hierarchy of properties such as flatness of the upper connection and the Batalin--Vilkovisky master equation. In particular, we obtain a complete description of generating operators for an arbitrary odd Poisson bracket.
256
arxiv_abstracts
math/0212312
Marion Scheepers, in his studies of the combinatorics of open covers, introduced the property Split(U,V) asserting that a cover of type U can be split into two covers of type V. In the first part of this paper we give an almost complete classification of all properties of this form where U and V are significant families of covers which appear in the literature (namely, large covers, omega-covers, tau-covers, and gamma-covers), using combinatorial characterizations of these properties in terms related to ultrafilters on N. In the second part of the paper we consider the questions whether, given U and V, the property Split(U,V) is preserved under taking finite unions, arbitrary subsets, powers or products. Several interesting problems remain open.
lt256
arxiv_abstracts
math/0212313
We construct explicitly non-polynomial eigenfunctions of the difference operators by Macdonald in case $t=q^k$, $k\in{\mathbb Z}$. This leads to a new, more elementary proof of several Macdonald conjectures, first proved by Cherednik. We also establish the algebraic integrability of Macdonald operators at $t=q^k$ ($k\in {\mathbb Z}$), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including $BC_n$ case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of $A_n$ root system where the previously known methods do not work.
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arxiv_abstracts
math/0212314
We prove the Hodge-D-conjecture for general K3 and Abelian surfaces. Some consequences of this result, e.g., on the levels of higher Chow groups of products of elliptic curves, are discussed.
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arxiv_abstracts
math/0212315
Let Z be a general surface in P^3 of degree at least 5. Using a Lefschetz pencil argument, we give an elementary new proof of the vanishing of a regulator on K_1(Z).
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arxiv_abstracts
math/0212316
Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been used in the Mirror Principle [L-L-Y] that relates Gromov-Witten invariants and mirror symmetry computations. In this paper the A-twisted moduli stack for higher genus curves is defined and systematically studied. It is proved that such a moduli stack is an Artin stack. For genus 0, it has the A-twisted moduli space of [M-P] as the coarse moduli space. The detailed proof of the regularity of the collapsing morphism by Jun Li in [L-L-Y: I and II] can be viewed as a natural morphism from the moduli stack of genus 0 stable maps to the A-twisted moduli stack at genus 0. Due to the technical demand of stacks to physicists and the conceptual demand of supersymmetry to mathematicians, a brief introduction of each topic that is most relevant to the main contents of this paper is given in the beginning and the appendix respectively. Themes for further study are listed in the end.
256
arxiv_abstracts
math/0212317
This is a condensed write-up of a talk delivered at the Ramanujan International Symposium on Kac-Moody Lie algebras and Applications in Chennai in January 2002. The talk introduces special coideal subalgebras of quantum affine algebras which appear in physics when solitons are restricted to live on a half-line by an integrable boundary condition. We review how the quantum affine symmetry determines the soliton S-matrix in affine Toda field theory and then go on to use the unbroken coideal subalgebra on the half-line to determine the soliton reflection matrix. This gives a representation theoretic method for the solution of the reflection equation (boundary Yang-Baxter equation) by reducing it to a linear equation.
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arxiv_abstracts
math/0212318
In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov--Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U(gl(n)). As an application, we construct real forms of the quantum orbits and classify finite dimensional representations. We compute the non-commutative Connes index for basic homogenous vector bundles over the quantum orbits.
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arxiv_abstracts
math/0212319
We show that in the neighborhood of each ``finite type'' singular orbit of a real analytic integrable dynamical system (hamiltonian or not) there is a real analytic torus action which preserves the system and which is transitive on this orbit. We also show that the local automorphism group of the system near such an orbit is
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arxiv_abstracts
math/0212320
We briefly review and illustrate our procedure to 'decouple' by transformation of generators: either a Hopf algebra $H$ from a $H$-module algebra $A_1$ in their cross-product $A_1 >\triangleleft H$; or two (or more) $H$-module algebras $A_1,A_2$. These transformations are based on the existence of an algebra map $A_1 >\triangleleft H\to A_1$.
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arxiv_abstracts
math/0212321
The main result of this paper amounts to a complete evaluation of the integral cohomological structure of the stable mapping class group. In particular it verifies the conjecture of D.Mumford about the rational cohomology of the stable mapping class group.
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arxiv_abstracts
math/0212322
An isoperimetric upper bound on the resistance is given. As a corollary we resolve two problems, regarding mean commute time on finite graphs and resistance on percolation clusters. Further conjectures are presented.
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arxiv_abstracts
math/0212323
The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non--doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on euclidean space and their known extensions for doubling measures. We start by analyzing the images of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined in terms of the metric, play a basic role too. For a euclidean space equipped with one of these measures, we also consider the so-called `` regular\rq\rq $\bmo$ space introduced by X. Tolsa. We show that it contains the image of a Lebesgue space in the appropriate limit case and also that the image of the space `` regular\rq\rq $\bmo$ is contained in the adequate Lipschitz space.
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arxiv_abstracts
math/0212324
The "noncommutative geometry" of complex algebraic curves is studied. As first step, we clarify a morphism between elliptic curves, or complex tori, and C*-algebras T_t={u,v | vu=exp(2\pi it)uv}, or noncommutative tori. The main result says that under the morphism isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces.
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arxiv_abstracts
math/0212325
We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_\Phi associated to an associator \Phi using cohomological techniques. We then introduce a criterion ensuring that the ``left Hopf algebra'' of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This provides a construction of quantization functors, equivalent to the Etingof-Kazhdan construction.
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arxiv_abstracts
math/0212326
The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomology. Now let us consider a Hopf algebra A such that its Jacobson radical J is a nilpotent Hopf ideal and H:=A/J is a semisimple algebra. By using our homological results, we prove that the canonical projection of A on H has a section which is an H-colinear algebra map. Furthermore, if H is cosemisimple too, then we can choose this section to be an (H,H)-bicolinear algebra morphism. This fact allows us to describe A as a `generalized bosonization' of a certain algebra R in the category of Yetter-Drinfeld modules over H. As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. In this case, many results that we obtain hold true for a large enough class of H-module coalgebras, where H is a cosemisimple Hopf algebra.
256
arxiv_abstracts
math/0212327
It is shown that the cycle space of an arbitrary orbit of a non-Hermitian real form G in a flag manifold $Z=G^\mathbb C/Q$ of its complexification is naturally equivalent to a certain universal domain which depends only on G. This makes use of complex geometric methods which were recently developed for the purpose of handling the case of open orbits together with a better understanding of the connection to Schubert varieties and the related complex slices along lower-dimensional Gorbits.
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arxiv_abstracts
math/0212328
We present a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. This gives a simple combinatorial proof of recent results of Robertson, Saracino and Zeilberger, and the first author. We also show that our bijection preserves additional statistics, which extends the previous results.
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arxiv_abstracts
math/0212329
THEOREM. For every prime $p$ and each $n=2, 3, ... \infty$, there is an action of $G=\prod_{i=1}^{\infty}(Z/ pZ)$ on a two-dimensional compact metric space $X$ with $n$-dimensional orbit space. This theorem was proved in [DW: A.N. Dranishnikov and J.E. West, Compact group actions that raise dimension to infinity, Topology and its Applications 80 (1997), 101-114] with an error in one of the lemmas (Lemma 15). This paper presents a corrected version of Lemma 15 and it is identical with [DW] in the rest.
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arxiv_abstracts
math/0212330
In math.AG/0207028 we began the study of higher sheaf theory (i.e. stacks theory) on higher categories endowed with a suitable notion of topology: precisely, we defined the notions of S-site and of model site, and the associated categories of stacks on them. This led us to study a notion of \textit{model topos} (orginally due to C. Rezk), a model category version of the notion of Grothendieck topos. In this paper we treat the analogous theory starting from (1-)Segal categories in place of S-categories and model categories. We introduce notions of Segal topologies, Segal sites and stacks over them. We define an abstract notion of Segal topos and relate it with Segal categories of stacks over Segal sites. We compare the notions of Segal topoi and of model topoi, showing that the two theories are equivalent in some sense. However, the existence of a nice Segal category of morphisms between Segal categories allows us to improve the treatment of topoi in this context. In particular we construct the 2-Segal category of Segal topoi and geometric morphisms, and we provide a Giraud-like statement characterizing Segal topoi among Segal categories. As an example of applications, we show how to reconstruct a topological space up to homotopy from the Segal topos of locally constant stacks on it, thus extending the main theorem of Toen, "Vers une interpretation Galoisienne de la theorie de l'homotopie" (to appear in Cahiers de top. et geom. diff. cat.) to the case of un-based spaces. We also give some hints of how to define homotopy types of Segal sites: this approach gives a new point of view and some improvements on the \'etale homotopy theory of schemes, and more generally on the theory of homotopy types of Grothendieck sites as defined by Artin and Mazur.
256
arxiv_abstracts
math/0212331
$T$-semi-selfdecomposability and subclasses $L_m(b, Q)$ and $\tilde L_m(b, Q)$ of measures on complete separable metric vector spaces are introduced and basic properties are proved. In particular, we show that $\mu$ is $T$-semi-selfdecomposable if and only if $\mu = T(\mu) \nu$ where $\nu$ is infinitely divisible and $\mu$ is operator selfdecomposable if and only if $\mu \in L_0(b, Q)$ for all $0< b < 1$.
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arxiv_abstracts
math/0212332
In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime $p$ and any left 3-Engel element $x$ of finite $p$-power order in a group $G$, $x^p$ is in the Baer radical of $G$. Also it is proved that $<x,y>$ is nilpotent of class 4 for every two left 3-Engel elements in a group $G$.
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arxiv_abstracts
math/0212333
In this Note we study the groups $G$ satisfying condition $(\mathcal{N},n)$, that is, every subset of $G$ with $n+1$ elements contains a pair $\{x,y\}$ such that the subgroup $<x,y>$ is nilpotent.
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arxiv_abstracts
math/0212334
These notes form an extended version of a minicourse delivered in Universite de Montreal (June 2002) within the framework of a NATO workshop ``Normal Forms, Bifurcations and Finiteness Problems in Differential Equations''. The focus is on Poincare--Dulac theory of ``Fuchsian'' (logarithmic) singularities of integrable systems, with applications to problems on zeros of Abelian integrals in view.
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arxiv_abstracts
math/0212335
A MAD (maximal almost disjoint) family is an infinite subset A of the infinite subsets of {0,1,2,..} such that any two elements of A intersect in a finite set and every infinite subset of {0.1.2...} meets some element of $\aa$ in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G-delta set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology in inherits a subset of the Power set of {0,1,2,..}, ie the Cantor set.
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arxiv_abstracts
math/0212336
A subset X of the Cantor space, 2^\omega, is a lambda-prime-set iff for every countable subset Y of the Cantor space Y is relatively G-delta in X union Y. In this paper we prove two forcing results about lambda-prime-sets. First we show that it is consistent that every lambda-prime-set is a gamma-set. Secondly we show that is independent whether or not every dagger-lambda-set is a lambda-prime-set.
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arxiv_abstracts
math/0212337
In this paper we provide a criterion for the quasi-autonomous Hamiltonian path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds $(M,\omega)$ to be length minimizing in its homotopy class in terms of the spectral invariants $\rho(G;1)$ that the author has recently constructed (math.SG/0206092). As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in {\it its homotopy class} with fixed ends, when it has no contractible periodic orbits {\it of period one}, has a maximum and a minimum point which are generically under-twisted and all of its critical points are nondegenerate in the Floer theoretic sense. This is a sequel to the papers math.SG/0104243 and math.SG/0206092.
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arxiv_abstracts
math/0212338
We give an alternative proof of the existence of the scaling limit of loop erased random walk which does not use Lowner's differential equation.
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arxiv_abstracts
math/0212339
Let $(A,\frak m)$ be an excellent normal local ring with algebraically closed residue class field. Given integrally closed $\frak m$-primary ideals $I\supset J$, we show that there is a composition series between $I$ and $J$, by integrally closed ideals only. Also we show that any given integrally closed $\fm$-primary ideal $I$, the family of integrally closed ideals $J\subset I, l_A(I/J)=1$ forms an algebraic variety with dimension $\dim A -1$.
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arxiv_abstracts
math/0212340
Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several counter-examples. We consider the subadditivity theorem for monomial ideals on toric rings, and construct a counter-example on a three-dimensional toric ring.
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arxiv_abstracts
math/0212341
A basic point about hyperbolic groups is that they have "spaces at infinity" which are spaces of homogeneous type in the sense of Coifman and Weiss, and with a lot of self-similarity coming from the group. This short survey deals with some of the notions involved.
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arxiv_abstracts
math/0212342
We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in R^n such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in R^n can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.
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arxiv_abstracts
math/0212343
In this article we generalize packing density problems from permutations to patterns with repeated letters and generalized patterns. We are able to find the packing density for some classes of patterns and several other short patterns.
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arxiv_abstracts
math/0212344
In this paper we study the average $\NL_{2\alpha}$-norm over $T$-polynomials, where $\alpha$ is a positive integer. More precisely, we present an explicit formula for the average $\NL_{2\alpha}$-norm over all the polynomials of degree exactly $n$ with coefficients in $T$, where $T$ is a finite set of complex numbers, $\alpha$ is a positive integer, and $n\geq0$. In particular, we give a complete answer for the cases of Littlewood polynomials and polynomials of a given height. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.
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arxiv_abstracts
math/0212345
We prove that some ergodic linear automorphisms of $\T^N$ are stably ergodic, i.e. any small perturbation remains ergodic. The class of linear automorphisms we deal with includes all non-Anosov ergodic automorphisms when N=4 and so, as a corollary, we get that every ergodic linear automorphism of $\T^N$ is stably ergodic when $N\leq 5$.
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arxiv_abstracts
math/0212346
A class of high-order lowpass filters, the discrete singular convolution (DSC) filters, is utilized to facilitate the Fourier pseudospectral method for the solution of hyperbolic conservation law systems. The DSC filters are implemented directly in the Fourier domain (i.e., windowed Fourier pseudospectral method), while a physical domain algorithm is also given to enable the treatment of some special boundary conditions. By adjusting the effective wavenumber region of the DSC filter, the Gibbs oscillations can be removed effectively while the high resolution feature of the spectral method can be retained. The utility and effectiveness of the present approach is validated by extensive numerical experiments.
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arxiv_abstracts
math/0212347
We obtain the fermionic formulas for the characters of (k, r)-admissible configurations in the case of r=2 and r=3. This combinatorial object appears as a label of a basis of certain subspace $W(\Lambda)$ of level-$k$ integrable highest weight module of $\hat{sl}_{r}$. The dual space of $W(\Lambda)$ is embedded into the space of symmetric polynomials. We introduce a filtration on this space and determine the components of the associated graded space explicitly by using vertex operators. This implies a fermionic formula for the character of $W(\Lambda)$.
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arxiv_abstracts
math/0212348
For all $k$, we construct a bijection between the set of sequences of non-negative integers ${\bf a}=(a_i)_{i\in{\bf Z}_{\geq0}}$ satisfying $a_i+a_{i+1}+a_{i+2}\leq k$ and the set of rigged partitions $(\lambda,\rho)$. Here $\lambda=(\lambda_1,...,\lambda_n)$ is a partition satisfying $k\geq\lambda_1\geq...\geq\lambda_n\geq1$ and $\rho=(\rho_1,...,\rho_n)\in{\bf Z}_{\geq0}^n$ is such that $\rho_j\geq\rho_{j+1}$ if $\lambda_j=\lambda_{j+1}$. One can think of $\lambda$ as the particle content of the configuration ${\bf a}$ and $\rho_j$ as the energy level of the $j$-th particle, which has the weight $\lambda_j$. The total energy $\sum_iia_i$ is written as the sum of the two-body interaction term $\sum_{j<j'}A_{\lambda_j,\lambda_{j'}}$ and the free part $\sum_j\rho_j$. The bijection implies a fermionic formula for the one-dimensional configuration sums $\sum_{\bf a}q^{\sum_iia_i}$. We also derive the polynomial identities which describe the configuration sums corresponding to the configurations with prescribed values for $a_0$ and $a_1$, and such that $a_i=0$ for all $i>N$.
256
arxiv_abstracts
math/0212349
Let $X$ be the exterior of connected sum of knots and $X_i$ the exteriors of the individual knots. In \cite{morimoto1} Morimoto conjectured (originally for $n=2$) that $g(X) < \sigma_{i=1}^n g(X_i)$ if and only if there exists a so-called \em primitive meridian \em in the exterior of the connected sum of a proper subset of the knots. For m-small knots we prove this conjecture and bound the possible degeneration of the Heegaard genus (this bound was previously achieved by Morimoto under a weak assumption \cite{morimoto2}): $$\sigma_{i=1}^n g(X_i) - (n-1) \leq g(X) \leq \sigma_{i=1}^n g(X_i).$$
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arxiv_abstracts
math/0212350
In this paper we will discuss a procedure to improve the usual estimator of a linear functional of the unknown regression function in inverse nonparametric regression models. In Klaassen, Lee, and Ruymgaart (2001) it has been proved that this traditional estimator is not asymptotically efficient (in the sense of the H\'{a}jek - Le Cam convolution theorem) except, possibly, when the error distribution is normal. Since this estimator, however, is still root-n consistent a procedure in Bickel, Klaassen, Ritov, and Wellner (1993) applies to construct a modification which is asymptotically efficient. A self-contained proof of the asymptotic efficiency is included.
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arxiv_abstracts
math/0212351
Call {\em i-hedrite} any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and $p_2+p_3=i$. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its {\em central circuits}, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any i-hedrite is a projection of an alternating link, whose components correspond to its central circuits. Call an i-hedrite {\em irreducible}, if it has no {\em rail-road}, i.e. a circuit of 4-gonal faces, in which every 4-gon is adjacent to two of its neighbors on opposite edges. We present the list of all i-hedrites with at most 15 vertices. Examples of other results: (i) All i-hedrites, which are not 3-connected, are identified. (ii) Any irreducible i-hedrite has at most i-2 central circuits. (iii) All i-hedrites without self-intersecting central circuits are listed. (iv) All symmetry group of i-hedrites are listed.
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arxiv_abstracts
math/0212352
A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3-valent plane graph and, especially, of simple two-faced polyhedra, i.e., 3-valent 3-polytopes with only $a$-gonal and $b$-gonal faces, where $3 \le a < b \le 6$; the main cases are $(a,b)=(3,6)$, $(4,6)$ and $(5,6)$ (the fullerenes). We completely describe the zigzag structure for the case $(a,b)$=$(3,6)$. For the case $(a,b)$=$(4,6)$ we describe symmetry groups, classify all tight graphs with simple zigzags and give the upper bound 9 for the number of zigzags in general tight graphs. For the remaining case $(a,b)$=$(5,6)$ we give a construction realizing a prescribed zigzag structure.
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arxiv_abstracts
math/0212353
Given a lattice $L$, a full dimensional polytope $P$ is called a {\em Delaunay polytope} if the set of its vertices is $S\cap L$ with $S$ being an {\em empty sphere} of the lattice. Extending our previous work \cite{DD-hyp} on the {\em hypermetric cone} $HYP_7$, we classify the six-dimensional Delaunay polytopes according to their {\em combinatorial type}. The list of 6241 combinatorial types is obtained by a study of the set of faces of the polyhedral cone $HYP_7$.
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arxiv_abstracts
math/0212354
We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and differential forms on a purely even Lagrangian submanifold. We establish a criterion of ``normality'' of a volume form on an odd symplectic supermanifold in terms of the canonical odd Laplacian acting on semidensities.
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arxiv_abstracts
math/0212355
Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon on hyperideal polyhedra, and those of Rousset on fuchsian hyperideal polyhedra. Our results can also be stated in terms of circle configurations on $\dr M$, they provide an extension of the Koebe theorem on circle packings. The proof uses some elementary properties of the hyperbolic volume, in particular the Schl\"afli formula and the fact that the volume of (truncated) hyperideal simplices is a concave function of the dihedral angles.
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arxiv_abstracts
math/0212356
For any pair of integers $n\geq 1$ and $q\geq 2$, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class $q[F]$ of an elliptic surface E(n), where $[F]$ is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedded into a more general class of symplectic 4-manifolds.
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arxiv_abstracts
math/0212357
We extend the setup in our previous paper to deal with the case in which more than one steady state may exist in feedback configurations. This provides a foundation for the analysis of multi-stability and hysteresis behaviour in high dimensional feedback systems.
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arxiv_abstracts
math/0212358
This paper explains the conjectured algebraic duality between genus zero Gromov-Witten theory and genus zero "Closed String topology". This duality in another perspective is discussed on page 87 of the book "Frobenius manifold, quantum cohomology, and moduli spaces" (by Yuri Manin). This paper also discusses Fulton MacPherson strata.
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arxiv_abstracts
math/0212359
We analyze the structure of co-invariant subspaces for representations of the Cuntz algebras O_N for N = 2,3,..., N < infinity, with special attention to the representations which are associated to orthonormal and tight-frame wavelets in L^2(R) corresponding to scale number N.
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arxiv_abstracts