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We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's $\mu$ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the so-called Milnor-Johnson correspondence: in particular, an analogue of the Birman-Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.
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arxiv:math/0402035
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We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant $V_2$, are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is applied to the theory of claspers for string links.
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arxiv:math/0402036
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In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope to be a space tiling zonotope, i.e. a zonotope which admits a face-to-face tiling of space by translations. Implicitly, he related space tiling zonotopes to a special class of oriented matroids (regular matroids). We will extend his result to give a new proof of Voronoi's conjecture for zonotopes using oriented matroids. This enables us to distinguish between combinatorial and metrical properties and to apply the fact that oriented matroids considered here have an essentially unique realization. Originally, this is a theorem due to Brylawski and Lucas. By using oriented matroid duality we interpret a part of McMullen's arguments as an elegant geometric proof of this theorem in the special case of real numbers.
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arxiv:math/0402053
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We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern-avoidance. Applying these results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.
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arxiv:math/0402063
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For an arbitrary finite Coxeter group W we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B we obtain, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky's construction of the normal fan as a "cluster fan." Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.
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arxiv:math/0402086
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We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.
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arxiv:math/0402100
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We investigate the relation between the backward uniqueness and the regularity of the coefficients for a parabolic operator. A necessary and sufficient condition for uniqueness is given in terms of the modulus of continuity of the coefficients.
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arxiv:math/0402138
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In 2002, D. Hrencecin and L.H. Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.
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arxiv:math/0402162
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In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his "reduction-prolongation" procedure. After Cartan's work the following questions remained open: first the geometric reason for existence of Cartan's tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan's tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev . In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n greater than 4. For n=5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In our next paper we show that in the case n=5 our fundamental form coincides with Cartan's tensor.
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arxiv:math/0402171
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We describe typical degenerations of quadratic differentials thus describing ``generic cusps'' of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from ``generic'' degenerations is often negligible in problems involving information on compactification of the moduli space. However, even for a typical degeneration one may have several short loops on the Riemann surface which shrink simultaneously. We explain this phenomenon, describe all rigid configurations of short loops, present a detailed description of analogs of desingularized stable curves arising here, and show how one can reconstruct a Riemann surface endowed with a quadratic differential which is close to a ``cusp'' by the corresponding point at the principal boundary.
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arxiv:math/0402197
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We study Lispchitz solutions of partial differential relations $\nabla u\in K$, where $u$ is a vector-valued function in an open subset of $R^n$. In some cases the set of solutions turns out to be surprisingly large. The general theory is then used to construct counter-examples to regularity of solutions of Euler-Lagrange systems satisfying classical ellipticity conditions.
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arxiv:math/0402287
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A general class, introduced in [Ekeland et al. 2003], of continuous time bond markets driven by a standard cylindrical Brownian motion $\wienerq{}{}$ in $\ell^{2},$ is considered. We prove that there always exist non-hedgeable random variables in the space $\derprod{}{0}=\cap_{p \geq 1}L^{p}$ and that $\derprod{}{0}$ has a dense subset of attainable elements, if the volatility operator is non-degenerated a.e. Such results were proved in [Bj\"ork et al. 1997] in the case of a bond market driven by finite dimensional B.m. and marked point processes. We define certain smaller spaces $\derprod{}{s},$ $s>0$ of European contingent claims, by requiring that the integrand in the martingale representation, with respect to $\wienerq{}{}$, takes values in weighted $\ell^{2}$ spaces $\ell^{s,2},$ with a power weight of degree $s.$ For all $s > 0,$ the space $\derprod{}{s}$ is dense in $\derprod{}{0}$ and is independent of the particular bond price and volatility operator processes. A simple condition in terms of $\ell^{s,2}$ norms is given on the volatility operator processes, which implies if satisfied, that every element in $\derprod{}{s}$ is attainable. In this context a related problem of optimal portfolios of zero coupon bonds is solved for general utility functions and volatility operator processes, provided that the $\ell^{2}$-valued market price of risk process has certain Malliavin differentiability properties.
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arxiv:math/0402364
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Let I be the defining ideal of a smooth irreducible complete intersection space curve C with defining equations of degrees a and b. We use the partial elimination ideals introduced by Mark Green to show that the lexicographic generic initial ideal of I has Castelnuovo-Mumford regularity 1+ab(a-1)(b-1)/2 with the exception of the case a=b=2, where the regularity is 4. Note that ab(a-1)(b-1)/2 is exactly the number of singular points of a general projection of C to the plane. Additionally, we show that for any term ordering tau, the generic initial ideal of a generic set of points in P^r is a tau-segment ideal.
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arxiv:math/0402418
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By a recent result of Livingston, it is known that if a knot has a prime power branched cyclic cover that is not a homology sphere, then there is an infinite family of non-concordant knots having the same Seifert form as the knot. In this paper, we extend this result to the full extent. We show that if the knot has nontrivial Alexander polynomial, then there exists an infinite family of non-concordant knots having the same Seifert form as the knot. As a corollary, no nontrivial Alexander polynomial determines a unique knot concordance class. We use Cochran-Orr-Teichner's recent result on the knot concordance group and Cheeger-Gromov's von Neumann rho invariants with their universal bound for a 3-manifold.
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arxiv:math/0402425
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We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero.
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arxiv:math/0402430
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We establish a calculus for branched spines of 3-manifolds by means of branched Matveev-Piergallini moves and branched bubble-moves. We briefly indicate some of its possible applications in the study and definition of State-Sum Quantum Invariants.
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arxiv:math/0403014
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It is known that many constructions arising in the classical Gaussian infinite dimensional analysis can be extended to the case of more general measures. One such extension can be obtained through biorthogonal systems of Appell polynomials and generalized functions. In this paper, we consider linear continuous operators from a nuclear Frechet space of test functions to itself in this more general setting. We construct an isometric integral transform (biorthogonal CS-transform) of those operators into the space of germs of holomorphic functions on a locally convex infinite dimensional nuclear space. Using such transform, we provide characterization theorems and give biorthogonal chaos expansion for operators.
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arxiv:math/0403025
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The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang-Mills theory are given.
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arxiv:math/0403040
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Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with $\pi_1(M) = Z^4times Z/3$, so that the index invariant in the KO-theory of the reduced $C^*$-algebra of $\pi_1(M)$ is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifolds cannot carry a metric of positive scalar curvature. The existence of such a metric is predicted by the (unstable) Gromov-Lawson-Rosenberg conjecture.
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arxiv:math/0403063
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We construct new special Lagrangian submanifolds in complex Euclidean space using a pair of minimal Legendrian submanifolds in odd-dimensional spheres and certain Lagrangian surface belonging to a family that can be considered as a generalization of the special Lagrangian surfaces in complex Euclidean plane. Our examples include those invariant under the standard action of SO(p+1)xSO(q+1) on C^n = C^(p+1) x C^(q+1), n=p+q+2.
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arxiv:math/0403108
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According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve G, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of G. It is proven that, for a generic X, the Abel-Jacobi map of the family of elliptic sextics on X factors through the moduli space of rank-2 vector bundles with c_1=-K_X and deg c_2=6 and that the latter is birational to the singular locus of the theta divisor of J(X).
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arxiv:math/0403122
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Suppose $H$ is a hyperbolic subgroup of a hyperbolic group $G$. Assume there exists $n > 0$ such that the intersection of $n$ essentially distinct conjugates of $H$ is always finite. Further assume $G$ splits over $H$ with hyperbolic vertex and edge groups and the two inclusions of $H$ are quasi-isometric embeddings. Then $H$ is quasiconvex in $G$. This answers a question of Swarup and provides a partial converse to the main theorem of \cite{GMRS}.
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arxiv:math/0403125
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We show that if the Atiyah Jones conjecture holds for a surface $X,$ then it also holds for the blow-up of $X$ at a point. Since the conjecture is known to hold for ${\mathbb P}^2$ and for ruled surfaces, it follows that the conjecture is true for all rational surfaces.
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arxiv:math/0403138
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I describe the monodromy of smooth hypersurfaces $X$ of high degree in a fixed smooth variety $Y$ containing a fixed subvariety $W$ of $Y$. The cohomology of $X$ in middle degree spanned by the pull-back of the cohomology of $Y$ and by the classes of the irreducible components of $W$ is monodromy invariant. I show that the monodromy representation on the orthogonal of those classes is irreducible. The proof is essentially topological. Difficulties arise from the fact that $W$ may have arbitrary singularities.
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arxiv:math/0403151
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We find a Hermite-type basis for which the eigenvalue problem associated to the operator $H_{A,B}:=B(-\partial_x^2)+Ax^2$ acting on $L^2({\bf R};{\bf C}^2)$ becomes a three-terms recurrence. Here $A$ and $B$ are two constant positive definite matrices with no other restriction. Our main result provides an explicit characterization of the eigenvectors of $H_{A,B}$ that lie in the span of the first four elements of this basis when $AB\not= BA$.
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arxiv:math/0403187
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The group of affine transformations with rational coefficients, $aff(Q)$, acts naturally on the real line, but also on the $p$-adic fields. The aim of this note is to show that all these actions are necessary and sufficient to represent bounded $\mu$-harmonic functions for a probability measure $\mu$ on $aff(Q)$ that is supported by a finitely generated sub-group, that is to describe the Poisson boundary.
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arxiv:math/0403197
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This paper introduces a new approach to the study of certain aspects of Galois module theory by combining ideas arising from the study of the Galois structure of torsors of finite group schemes with techniques coming from relative algebraic $K$-theory.
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arxiv:math/0403200
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A family $\BA_\a$ of differential operators depending on a real parameter $\a$ is considered. The problem can be formulated in the language of perturbation theory of quadratic forms. The perturbation is only relatively bounded but not relatively compact with respect to the unperturbed form. The spectral properties of the operator $\BA_\a$ strongly depend on $\a$. In particular, for $\a<\sqrt2$ the spectrum of $\BA_\a$ below 1/2 is finite, while for $\a>\sqrt2$ the operator has no eigenvalues at all. We study the asymptotic behaviour of the number of eigenvalues as $\a\nearrow\sqrt2$. We reduce this problem to the one on the spectral asymptotics for a certain Jacobi matrix.
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arxiv:math/0403226
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A criterion is given which assures that two p-divisible groups X and Y over an algebraically closed field of characteristic p are isomorphic when their p-kernels X[p] and Y[p] are isomorphic.
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arxiv:math/0403256
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The second H. Weyl curvature invariant of a Riemannian manifold, denoted $h_4$, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of $h_4$ is that it is nonnegative for Einstein manifolds, hence it provides a geometric obstruction to the existence of Einstein metrics in dimensions $\geq 4$, independently from the sign of the Einstein constant. This motivates our study of the positivity of this invariant. Here in this paper, we prove many constructions of metrics with positive second H. Weyl curvature invariant, generalizing similar well known results for the scalar curvature.
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arxiv:math/0403286
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We consider an integral series f(X,t) which depends on the choice of a set X of labelled planar rooted trees. We prove that its inverse for composition is of the form f(Z,t) for another set Z of trees, deduced from X. The proof is self-contained, though inspired by the Koszul duality theory of quadratic operads.
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arxiv:math/0403316
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We introduce "noninvertible" generalization of statistics - semistatistics replacing condition when double exchanging gives identity to "regularity" condition. Then in categorical language we correspondingly generalize braidings and the quantum Yang-Baxter equation. We define the doubly regular R-matrix and introduce obstructed regular bialgebras.
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arxiv:math/0403343
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We show that for every $0 < \epsilon \leq 1$ and integer $k\geq 1$, there exists an integer $n = n(\epsilon,k)$ so that for all primes $p$, and integers $0 \leq a \leq p-1$, there exist integers $1 \leq x_1 < ... < x_n \leq p^\epsilon$ such that $a \equiv x_1^{-1} + ... + x_n^{-1} \pmod{p}$. This extends a result of I. Shparlinski.
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arxiv:math/0403360
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Let $G$ be the linear algebraic group $SL_3$ over a field $k$ of characteristic two. Let $A$ be a finitely generated commutative $k$-algebra on which $G$ acts rationally by $k$-algebra automorphisms. We show that the full cohomology ring $H^*(G,A)$ is finitely generated. This extends the finite generation property of the ring of invariants $A^G$. We discuss where the problem stands for other geometrically reductive group schemes.
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arxiv:math/0403361
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Let k be a field of characteristic 0, R = k[x_1, ..., x_d] be a polynomial ring, and m its maximal homogeneous ideal. Let I be a homogeneous ideal in R. In this paper we investigate asymptotic behaviour of the quotient between the length of local cohomology group H^0_m(R/I^n) and n^d. We show that this quantity always has a limit as n goes to infinity. We also give an example for which the limit is irrational; in particular, this proves that the length of H^0_m(R/I^n) is not asymptotically a polynomial in n.
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arxiv:math/0403370
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Let M be a compact, connected, C^2-smooth and globally minimal hypersurface M in P_2(C) which divides the projective space into two connected parts U^{+} and U^{-}. We prove that there exists a side, U^- or U^+, such that every continuous CR function on M extends holomorphically to this side. Our proof of this theorem is a simplification of a result originally due to F. Sarkis.
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arxiv:math/0403384
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We consider the inverse problem of the recovery of the gauge field in R^2 modulo gauge transformations from the non-abelian Radon transform.A global uniqueness theorem is proven for the case when the gauge field has a compact support.
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arxiv:math/0403447
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The functional relation of the Hurwitz zeta function is proved by using the connection problem of the confluent hypergeometric equation.
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arxiv:math/0403457
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For a knot K in $S^3$ and a regular representation $\rho$ of its group $G_K$ into SU(2) we construct a non abelian Reidemeister torsion on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion provides a volume form on the SU(2)-representation space of $G_K$. In another way, we construct according to Casson--or more precisely taking into account Lin's and Heusener's further works--a volume form on the SU(2)-representation space of $G_K$. Next, we compare these two apparently different points of view--the first by means of the Reidemeister torsion and the second defined ``a la Casson"--and finally prove that they define the same topological knot invariant.
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arxiv:math/0403470
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We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these determine the tree and the $d$ length of all edges. Here we consider distortions $\d$ of $d$ such that for all leaves $u$ and $v$ it holds that $|d(u,v) - \d(u,v)| < f/2$ if either $d(u,v) < M$ or $\d(u,v) < M$, where $d$ satisfies $f \leq d(e) \leq g$ for all edges $e$. Given such distortions we show how to reconstruct in polynomial time a forest $T_1,...,T_{\alpha}$ such that the true tree $T$ may be obtained from that forest by adding $\alpha-1$ edges and $\alpha-1 \leq 2^{-\Omega(M/g)} n$. Metric distortions arise naturally in phylogeny, where $d(u,v)$ is defined by the log-det of a covariance matrix associated with $u$ and $v$. of a covariance matrix associated with $u$ and $v$. When $u$ and $v$ are ``far'', the entries of the covariance matrix are small and therefore $\d(u,v)$, which is defined by log-det of an associated empirical-correlation matrix may be a bad estimate of $d(u,v)$ even if the correlation matrix is ``close'' to the covariance matrix. Our metric results are used in order to show how to reconstruct phylogenetic forests with small number of trees from sequences of length logarithmic in the size of the tree. Our method also yields an independent proof that phylogenetic trees can be reconstructed in polynomial time from sequences of polynomial length under the standard assumptions in phylogeny. Both the metric result and its applications to phylogeny are almost tight.
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arxiv:math/0403508
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In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat ``hat''. In our main theorem, we show that every C^infty-smooth CR diffeomorphism h: M to M' between two globally minimal real analytic hypersurfaces in C^n (n >1) is real analytic at every point of M if M' is holomorphically nondegenerate. More generally, we establish that the reflection function R_h' associated to such a C^infty-smooth CR diffeomorphism between two globally minimal hypersurfaces in C^n (n > 1) always extends holomorphically to a neighborhood of the graph of \bar{h} in M\times \bar{M}', without any nondegeneracy condition on M'. This gives a new version of the Schwarz symmetry principle in several complex variables. Finally, as an appendix, we show that every C^infty-smooth CR mapping h: M to M' between two real analytic hypersurfaces containing no complex curves is real analytic at every point of M, without any rank condition on h. This answers a conjecture, explicitely stated for C^infty-smooth maps at p.~328 of the survey article: S.M. Baouendi, P. Ebenfelt and L.-P. Rothschild, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 309--336. Importantly, notice that a much stronger result, valid for continuous maps, has been obtained by K. Diederich and S. Pinchuk in: Michigan Math. J. {\bf 51} (2003), no. 1, 111--140; no.~3, 667--668.
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arxiv:math/0403539
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Let $G$ be a group and assume that $(A_p)_{p\in G}$ is a family of algebras with identity. We have a {\it Hopf $G$-coalgebra} (in the sense of Turaev) if, for each pair $p,q\in G$, there is given a unital homomorphism $\co_{p,q}:A_{pq}\to A_p \ot A_q$ satisfying certain properties. Consider now the direct sum $A$ of these algebras. It is an algebra, without identity, except when $G$ is a finite group, but the product is non-degenerate. The maps $\co_{p,q}$ can be used to define a coproduct $\co$ on $A$ and the conditions imposed on these maps give that $(A,\co)$ is a multiplier Hopf algebra. It is $G$-cograded as explained in this paper. We study these so-called {\it group-cograded multiplier Hopf algebras}. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras.
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arxiv:math/0404026
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We propose a construction of some canonical bases for quantum loop algebras of Kac-Moody algebras. We consider a smooth projective curve X, a group of automorphism G of X such that X/G=P^1, and we consider some Quot schemes of G-equivariant coherent sheaves on X. We view these spaces as loop analogues of spaces of representations of quivers and following Lusztig, we consider a convolution algebra of (semisimple, equivariant) perverse sheaves on the collection of these Quot schemes. We relate this algebra to a quantum loop algebra of some Kac-Moody algebra. In particular, when X=P^1 and G is a subgroup of SL(2,C), we obtain a canonical basis of a positive part of a quantum affine algebra in the Drinfeld presentation, and relate it (in some examples) to the bases constructed by Lusztig and Kashiwara. When X is an elliptic curve, we will obtain in this way a canonical basis of quantum toroidal algebras of type D_4, E_6, E_7 and E_8.
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arxiv:math/0404032
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The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra structure. Such a description is rather imprecise: these dimensions may coincide for cohomology spaces of distinct algebras. We explicitely describe the Lie superalgebras on the space of cohomology of the maximal nilpotent subalgebra of any simple finite dimensional Lie algebra. We briefly review related results by Grozman, Penkov and Serganova, Poletaeva, and Tolpygo. We cite a powerful Premet's theorem complementary to the Borel-Weil-Bott theorem. We mention relations with the Nijenhuis bracket and nonholonomic systems.
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arxiv:math/0404139
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We establish the eventual periodicity of the spectrum of any monadic second-order formula where: (i) all relation symbols, except equality, are unary, and (ii) there is only one function symbol and that symbol is unary.
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arxiv:math/0404150
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As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is rather that of Lie group bundle of graded 2-step nilpotent Lie groups $GM$. We then construct the tangent groupoid of $(M,H)$ as a differentiable groupoid $\cG_{H} M$ encoding the smooth deformation of $M\times M$ to $GM$. In this construction a crucial use is made of a refined notion of privileged coordinates and of a tangent approximation result for Heisenberg diffeomorphisms.
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arxiv:math/0404174
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This is an informal discussion on one of the basic problems in the theory of empirical processes, addressed in our preprint "Combinatorics of random processes and sections of convex bodies", which is available at ArXiV and from our web pages.
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arxiv:math/0404193
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We develop a theory of weak omega categories that will be accessible to anyone who is familiar with the language of categories and functors and who has encountered the definition of a strict 2-category. The most remarkable feature of this theory is its simplicity. We build upon an idea due to Jacques Penon by defining a weak omega category to be a span of omega magmas with certain properties. (An omega magma is a reflexive, globular set with a system of partially defined, binary composition operations which respects the globular structure.) Categories, bicategories, strict omega categories and Penon's weak omega categories are all instances of our weak omega categories. We offer a heuristic argument to justify the claim that Batanin's weak omega categories also fit into our framework. We show that the Baez-Dolan stabilization hypothesis is a direct consequence of our definition of weak omega categories. We define a natural notion of a pseudo-functor between weak omega categories and show that it includes the classical notion of a homomorphism between bicategories. In any weak omega category the operation of composition with a fixed 1-cell defines such a pseudo-functor. Finally, we define a notion of weak equivalence between weak omega categories which generalizes the standard definition of an equivalence between ordinary categories.
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arxiv:math/0404216
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We use the Blanchfield-Duval form to define complete invariants for the cobordism group C_{2q-1}(F_\mu) of (2q-1)-dimensional \mu-component boundary links (for q\geq2). The author solved the same problem in math.AT/0110249 via Seifert forms. Although Seifert forms are convenient in explicit computations, the Blanchfield-Duval form is more intrinsic and appears naturally in homology surgery theory. The free cover of the complement of a link is constructed by pasting together infinitely many copies of the complement of a \mu-component Seifert surface. We prove that the algebraic analogue of this construction, a functor denoted B, identifies the author's earlier invariants with those defined here. We show that B is equivalent to a universal localization of categories and describe the structure of the modules sent to zero. Taking coefficients in a semi-simple Artinian ring, we deduce that the Witt group of Seifert forms is isomorphic to the Witt group of Blanchfield-Duval forms.
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arxiv:math/0404229
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In this paper, we describe geometrical constructions to obtain triangulations of connected sums of closed orientable triangulated 3-manifolds. Using these constructions, we show that it takes time polynomial in the number of tetrahedra to check if a closed orientable 3-manifold, equipped with a minimal triangulation, is reducible or not. This result can easily be generalized to compact orientable 3-manifolds with non-empty boundary.
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arxiv:math/0404309
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Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum size of a binary code of length $n$ and minimum distance $d$. The well-known Gilbert-Varshamov bound asserts that $A_2(n,d) \geq 2^n/V(n,d-1)$, where $V(n,d) = \sum_{i=0}^{d} {n \choose i}$ is the volume of a Hamming sphere of radius $d$. We show that, in fact, there exists a positive constant $c$ such that $$ A_2(n,d) \geq c \frac{2^n}{V(n,d-1)} \log_2 V(n,d-1) $$ whenever $d/n \le 0.499$. The result follows by recasting the Gilbert- Varshamov bound into a graph-theoretic framework and using the fact that the corresponding graph is locally sparse. Generalizations and extensions of this result are briefly discussed.
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arxiv:math/0404325
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We use twisted Fourier-Mukai transforms to study the relation between an abelian fibration on a holomorphic symplectic manifold and its dual fibration. Our reasoning leads to an equivalence between the derived category of coherent sheaves on one space and the derived category of twisted sheaves on the other space.
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arxiv:math/0404365
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We provide necessary and sufficient conditions on the derived type of a vector field distribution $\Cal V$ in order that it be locally equivalent to a partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. This result fully generalises the classical Goursat normal form. Our proof is constructive: it is proven that if $\Cal V$ is locally equivalent to a partial prolongation of $\Cal C^{(1)}_q$ then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on the ambient manifold.
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arxiv:math/0404377
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Consider an operator equation (*) $B(u)+\ep u=0$ in a real Hilbert space, where $\ep>0$ is a small constant. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the equation $B(u)=0$. Existence of the unique solution is proved by the DSM for equation (*) with monotone hemicontinuous operators $B$ defined on all of$ If $\ep=0$ and equation (**) $B(u)=0$ is solvable, the DSM yields$ solution to (**).
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arxiv:math/0404437
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This paper proves that homology equivalences of cogenerating complexes induce homology equivalences of the cofree coalgebras in many interesting cases. We show that the underlying chain complex of any cofree coalgebra is naturally a direct summand of the underlying chain-complex of a cofree coalgebra over a free operad. This is combined with the previous result to prove the homology invariance of all cofree coalgebras.
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arxiv:math/0404470
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The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C^*-algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with infinite cyclic kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups.
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arxiv:math/0404486
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Let Omega be a domain in C^2. We prove the following theorem. If the envelope of holomorphy of Omega is schlicht over Omega, then the envelope is in fact schlicht. We provide examples showing that the theorem is not true in C^n, n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in C^2 whose envelopes are multiply sheeted.
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arxiv:math/0404497
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In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated K and L is nicely bounded. In this appendix, we achieve a polynomial bound on the diameter of that intersection (in the ratio of the dimensions of the sections).
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arxiv:math/0404502
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Ergodic properties of the signal-filtering pair are studied for continuous time finite Markov chains, observed in white noise. The obtained law of large numbers is applied to the stability problem of the nonlinear filter with respect to initial conditions. The Furstenberg-Khasminskii formula is derived for the top Lyapunov exponent of the Zakai equation and is used to estimate the stability index of the filter.
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arxiv:math/0404515
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Mean square estimates for $Z_2(s) = \int_1^\infty |\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, and some related Mellin transforms of quantities connected with the fourth power moment of $|\zeta(1/2+ix)|^4$.
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arxiv:math/0404524
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We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional L\'{e}vy's modulus of continuity and many other results are its particular cases. Applications to approximation theory are discussed.
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arxiv:math/0405009
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In this article a complete set of invariants for ordinary quartic curves in characteristic 2 is computed.
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arxiv:math/0405052
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An analysis of several important aspects of competition or conflict in games, social choice and decision theory is presented. Inherent difficulties and complexities in cooperation are highlighted. These have over the years led to a certain marginalization of studies related to cooperation. The significant richness of cooperation possibilities and the considerable gains which my lie there hidden are indicated. Based on that, a reconsideration of cooperation is suggested, as a more evolved form of rational behaviour. As one of the motivations it is shown that the paradigmatic non-cooperative Nash equilibrium itself rests on a strong cooperation assumption in the case of $n \geq 3$ players.
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arxiv:math/0405065
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We obtain a Kaehler Einstein structure on the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained Kaehler Einstein structure cannot have constant holomorphic sectional curvature and is not locally symmetric.
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arxiv:math/0405123
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In this paper one constructs a function $\eta$ with the property that if $n$ is non-null then $\eta(n)$ is the smallest integer such that $\eta(n)!$ is divisible by $n$. In order to calculate it one considers, for each prime $p$, the associated function $\eta_{p}(n)$ in a power base.
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arxiv:math/0405143
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We prove that no finite time blow up can occur for nonlinear Schroedinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to use continuity arguments and to control the nonlinear effects.
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arxiv:math/0405197
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Let A be a unital simple direct limit of recursive subhomogeneous C*-algebras with no dimension growth. We give criteria which specify exactly when A has real rank zero, and exactly when A has the Property (SP): every nonzero hereditary subalgebra of A contains a nonzero projection. Specifically, A has real rank zero if and only if the natural map from K_0 (A) to the continuous affine functions on the tracial state space has dense range, A has the Property (SP) if and only if the range of this map contains strictly positive functions with arbitrarily small norm. By comparison with results for unital simple direct limit of homogeneous C*-algebras with no dimension growth, one might hope that weaker conditions might suffice. We give examples to show that several plausible weaker conditions do not suffice for the results above. If A has real rank zero and at most countably many extreme tracial states, we apply results of H. Lin to show that A has tracial rank zero and is classifiable.
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arxiv:math/0405265
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Algebraic basics on Temperley-Lieb algebras are proved in an elementary and straightforward way with the help of tensor categories behind them.
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arxiv:math/0405267
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Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains from the literature, analyzing the running time when possible. We present bounding chains for the transposition chain on permutations, the hard core gas model, proper colorings of a graph, the antiferromagnetic Potts model and sink free orientations of a graph.
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arxiv:math/0405284
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We study the rate of convergence of linear two-time-scale stochastic approximation methods. We consider two-time-scale linear iterations driven by i.i.d. noise, prove some results on their asymptotic covariance and establish asymptotic normality. The well-known result [Polyak, B. T. (1990). Automat. Remote Contr. 51 937-946; Ruppert, D. (1988). Technical Report 781, Cornell Univ.] on the optimality of Polyak-Ruppert averaging techniques specialized to linear stochastic approximation is established as a consequence of the general results in this paper.
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arxiv:math/0405287
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We define a poset of partitions associated to an operad. We prove that the operad is Koszul if and only if the poset is Cohen-Macaulay. In one hand, this characterisation allows us to compute the homology of the poset. This homology is given by the Koszul dual operad. On the other hand, we get new methods for proving that an operad is Koszul.
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arxiv:math/0405312
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We characterize the oriented Seifert-fibered three-manifolds which admit positive, transverse contact structures.
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arxiv:math/0405329
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This paper has been withdrawn by the author(s). The material contained in the paper will be published in a subtantially reorganized form, part of it is now included in math.QA/0510174
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arxiv:math/0405332
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This paper continues the work Glasner-Tsirelson-Weiss, ArXiv math.DS/0311450. For a Polish group G the notions of G-continuous functions and whirly actions are further exploited to show that: (i) A G-action is whirly iff it admits no nontrivial spatial (= pointwise) factors. (ii) Every action of a Polish Levy group is whirly. (iii) There exists a Polish monothetic group which is not Levy but admits a whirly action. (iv) In the Polish group Aut(X,\mu), for the generic automorphism T, the action of the Polish group \Lambda(T) = closure {T^n: n \in Z} \subset Aut(X,\mu) on the Lebesgue space (X,\mu) is whirly. (v) The Polish additive group underlying a separable Hilbert space admits both spatial and whirly faithful actions. (vi) When G is a non-archimedean Polish group then every G-action is spatial.
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arxiv:math/0405352
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Using Talagrand's concentration inequality on the discrete cube {0,1}^m we show that given a real-valued function Z(x)on {0,1}^m that satisfies certain monotonicity conditions one can control the deviations of Z(x) above its median by a local Lipschitz norm of Z(x) at the point x. As one application, we give a simple proof of a nearly optimal deviation inequality for the number of k-cycles in a random graph.
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arxiv:math/0405355
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Recently Michel Talagrand gave a rigorous proof of the Parisi formula in the Sherrington-Kirkpatrick model. In this paper we build upon the methodology developed by Talagrand and extend his result to the class of SK type models in which the spins have arbitrary prior distribution on a bounded subset of the real line.
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arxiv:math/0405362
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Kuratowski's 14-set theorem says that in a topological space, 14 is the maximum possible number of distinct sets which can be generated from a fixed set by taking closures and complements. In this article we consider the analogous questions for any possible subcollection of the operations {closure, complement, interior, intersection, union}, and any number of initially given sets. We use the algebraic "topological calculus" to full advantage.
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arxiv:math/0405401
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In this paper we prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall's estimates between the Wiener sausage and the Brownian intersection local times.
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arxiv:math/0405413
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we study the exponential map for A_n = R^2^n, the Cayley_Dickson algebras for n bigher than 1,wich generalize the Complex exponential map to Quaternions,Octonions and so forth. As application,we show that the self-map of the unit sphere in A_n, S^(2^n)-1,given by taking k-powers has topological degree k for k an integer number,from this we derive a suitable "Fundamental theorem of algebra" for A_n for n bigher than 1.
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arxiv:math/0405424
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We investigate the modularity of three non-rigid Calabi-Yau threefolds with bad reduction at 11 which arise as fibre products of rational elliptic surfaces. For this purpose, we apply a method by Serre to compare two-dimensional 2-adic Galois representations with uneven trace. Hereby, we associate two of the threefolds (or, more precisely, a two-dimensional piece of their middle cohomology group) to newforms of weight 4 and level 22 and 55, respectively. This enables us to compute the zeta-functions of these varieties up to the Euler factors of the bad primes.
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arxiv:math/0405450
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J.P. Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the $\bar{\mu}$-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ${\Bbb Z}[t,t^{-1}]$. In addition, we give a relation between the Taylor expansion of a linking pairing around $t=1$ and derivation on links which is invented by T.D. Cochran. In fact, the coefficients of the powers of $t-1$ will be the linking numbers of certain derived links in $S^3$. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in $S^3$. This generalizes a result of J. Hoste.
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arxiv:math/0405481
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Some reverses for the generalised triangle inequality in complex inner product spaces that improve the classical Diaz-Metcalf results and applications are given.
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arxiv:math/0405497
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In this paper we first give a one-move version of Markov's braid theorem for knot isotopy in $S^3$ that sharpens the classical theorem. Then a relative version of Markov's theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markov's theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3--manifold.
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arxiv:math/0405498
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The distribution of the spectral numbers of an isolated hypersurface singularity is studied in terms of the Bernoulli moments. These are certain rational linear combinations of the higher moments of the spectral numbers. They are related to the generalized Bernoulli polynomials. We conjecture that their signs are alternating and prove this in many cases. One motivation for the Bernoulli moments comes from the comparison with compact complex manifolds.
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arxiv:math/0405501
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In this paper we study the semi-stable reduction of $p$ and $p^2$-cyclic covers of curves in equal characteristic $p>0$. The main tool we use is the classical Artin-Schreier-Witt theory for $p^n$-cyclic covers in characteristic $p$. Although the results of this paper concern only the cases of degree $p$ and $p^2$-cyclic cover we develop the techniques and the framework in which the general cyclic case can be studied.
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arxiv:math/0405529
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We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if D(L) is the order complex of a rank (r+1) geometric lattice L, then for all i \leq r/2 the h-vector of D(L) satisfies h(i-1) \leq h(i) and h(i) \leq h(r-i). We also obtain several inequalities for the flag h-vector of D(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling-Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of h(i-1) \leq h(i) when i \leq (2/7)(r + 5/2).
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arxiv:math/0405535
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We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps $C(X\times \T)$ onto $C(X\times \T)$.
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arxiv:math/0406007
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The article is devoted to the investigation of transformation groups of polynomials over Cayley-Dickson algebras and their manifolds of zeros. The problems about expressibility of zeros with the help of roots and decomposibility of polynomials as products of linear terms are studied.
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arxiv:math/0406048
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The numerical implementation of finite element discretization method for the stream function formulation of a linearized Navier-Stokes equations is considered. Algorithm 1 is applied using Argyris element. Three global orderings of nodes are selected and registered in order to conclude the best banded structure of matrix and a fluid flow calculation is considered to test a problem which has a known solution. Visualization of global node orderings, matrix sparsity patterns and stream function contours are displayed showing the main features of the flow.
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arxiv:math/0406070
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For each k >= 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce m(k) = max[(k-1)!/ 2, 1] packings of h(k) equal disks inside a circle which we call "the curved hexagonal packings". The curved hexagonal packing of 7 disks (k = 1, m(1)=1) is well known and the one of 19 disks (k = 2, m(2)=1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the densest we obtained on a computer using a so-called "billiards" simulation algorithm. A curved hexagonal packing pattern is invariant under a 60 degree rotation. For k tending to infinity, the density (covering fraction) of curved hexagonal packings tends to pi*pi/12. The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we present new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ``tightness'' of the curved hexagonal pattern for k =< 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality.
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arxiv:math/0406098
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Bowen's formula relates the Hausdorff dimension of a conformal repeller to the zero of a `pressure' function. We present an elementary, self-contained proof which bypasses measure theory and the Thermodynamic Formalism to show that Bowen's formula holds for $C^1$ conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class. We show that Bowen's formula generalizes to such a repeller and that if the sequence is picked at random then the Hausdorff dimension of the repeller almost surely agrees with its upper and lower Box dimensions and is given by a natural generalization of Bowen's formula. For a random uniformly hyperbolic Julia set on the Riemann sphere we show that if the family of maps and the probability law depend real-analytically on parameters then so does its almost sure Hausdorff dimension.
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arxiv:math/0406114
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Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(\Lambda)$ be the number of congruence subgroups of $\Lambda$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$ The proof is based on the Bombieri-Vinogradov `Riemann hypothesis on the average' and on the solution of a new type of extremal problem in combinatorial number theory. Similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple Lie groups. If $G$ is such a Lie group and $\Gamma$ is an irreducible lattice of $G$ it turns out that the subgroup growth of $\Gamma$ is independent of the lattice and depends only on the Lie type of the direct factors of $G$. It can be calculated easily from the root system. The most general case of this result relies on the Generalized Riemann Hypothesis but many special cases are unconditional. The proofs use techniques from number theory, algebraic groups, finite group theory and combinatorics.
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arxiv:math/0406163
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After more than thirty years, the only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. It is also important to note that the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the (rational) Lie algebra of the Mal'cev completion of the corresponding lattice which is hyperbolic and unimodular. These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we classify Anosov Lie algebras of dimension less or equal than 8, which also classify nilmanifolds admitting an Anosov diffeomorphism in those dimensions. As a corollary we obtain that if an infranilmanifold of dimension n<9 admits an Anosov diffeomorphism f and it is not a torus or a compact flat manifold (i.e. covered by a torus), then n=6 or 8 and the signature of f necessarily equals {3,3} or {4,4}, respectively. We had to study the set of all rational forms up to isomorphism for many real Lie algebras, which is a subject on its own and it is treated in a section completely independent of the rest of the paper.
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arxiv:math/0406199
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Let $\Gamma$ denote the modular group $SL(2,\Bbb Z)$ and $C_n(\Gamma)$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$ We also present a very general conjecture giving an asymptotic estimate for $C_n(\Gamma)$ for general arithmetic groups. The lower bound of the conjecture is proved modulo the generalized Riemann hypothesis for Artin-Hecke L-functions, and in many cases is also proved unconditionally.
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arxiv:math/0406249
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Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrodinger equations when the noise converges to zero are presented. The noise is a complex additive gaussian noise. It is white in time and colored space wise. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.
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arxiv:math/0406362
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Let X be a metric space with metric d and T,S be two commutative measure-preserving maps of X. In this paper we obtain numerical results about multiple recurrence of almost every point of this dynamical system. On other words we study the question about convergence to zero of max{d(T^n x,x), d(S^n x,x)}.
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arxiv:math/0406413
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We propose a novel approach to modeling advertising dynamics for a firm operating over distributed market domain based on controlled partial differential equations of diffusion type. Using our model, we consider a general type of finite-horizon profit maximization problem in a monopoly setting. By reformulating this profit maximization problem as an optimal control problem in infinite dimensions, we derive sufficient conditions for the existence of its optimal solutions under general profit functions, as well as state and control constraints, and provide general characterization of the optimal solutions. Sharper, feedback-form, characterizations of the optimal solutions are obtained for two variants of the general problem.
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arxiv:math/0406435
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Relations between so-called harness processes and initial enlargements of the filtration of a Levy process with its positions at fixed times are investigated.
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arxiv:math/0406563
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Let r_m and r_M be the least and greatest finite boundary slopes of a hyperbolic knot K in S^3. We show that any cyclic surgery slopes of K must lie in the interval (r_m - 1/2, r_M + 1/2).
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arxiv:math/0407025
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This survey paper examines the work of J. von Neumann and M.H. Stone as it relates to the abstract theory of wavelets. In particular, we discuss the direct integral theory of von Neumann and how it can be applied to representations of certain discrete groups to study the existence of normalized tight frames in the setting of Gabor systems and wavelets, via the use of group representations and von Neumann algebras. Then the extension of Stone's theorem due to M. Naimark, W. Ambrose and R. Godement is reviewed, and its relationship to the multiresolution analyses of S. Mallat and Y. Meyer and the generalized multiresolution analyses of L. Baggett, H. Medina, and K. Merrill. Finally, the paper ends by discussing some recent work due to the author, Baggett, P. Jorgensen and Merrill, and its relationship to operator theory.
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arxiv:math/0407037
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