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Consider a random graph, having a pre-specified degree distribution F but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation the paper determines R_0 and tau_0, the basic reproduction number and the asymptotic final size in case of a major outbreak. Further, the paper looks at some different local vaccination strategies where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies the paper determines R_v: the reproduction number, and tau_v: the asymptotic final proportion infected in case of a major outbreak, after vaccinating a fraction v.
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arxiv:math/0702021
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We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is quasi-isomorphic to the Deligne sheaf.
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arxiv:math/0702087
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We give a mathematically rigorous analysis which confirms the surprising results in a recent paper of Benilov, O'Brien and Sazonov about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics.
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arxiv:math/0702122
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The author will prove that Drinfel'd's pentagon equation implies his two hexagon equations in the Lie algebra, pro-unipotent, pro-$l$ and pro-nilpotent contexts.
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arxiv:math/0702128
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We introduce certain polynomials, so-called H.Weyl and H.Minkowski polynomials, which have a geometric origin. The location of roots of these polynomials is studied.
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arxiv:math/0702139
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In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the unit tangent bundle of the surface. From this we also deduce a geometric description of the Hitchin component the symplectic group PSp(4,R).
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arxiv:math/0702184
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We present a new viewpoint (namely, reproducing kernels) and new proofs for several recent results of J. Geronimo and H. Woerdeman on orthogonal polynomials on the two dimenional torus (and related subjects). In addition, we show how their results give a new proof of Ando's inequality via an equivalent version proven by Cole and Wermer. A simple necessary and sufficient condition for two variable polynomial stability is also given.
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arxiv:math/0702203
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A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph $R$, the universal Urysohn metric space $\Ur$, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of $\Ur$ (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that $\Ur$ admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.
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arxiv:math/0702207
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This article deals with the so called GVF (Gradient Vector Flow) introduced by C. Xu, J.L. Prince . We give existence and uniqueness results for the front propagation flow for boundary extraction that was initiated by Paragios, Mellina-Gottardo et Ralmesh . The model combines the geodesic active contour flow and the GVF to determine the geometric flow. The motion equation is considered within a level set formulation to result an Hamilton-Jacobi equation.
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arxiv:math/0702255
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The complex affine quadric $Q^{m}=\{z\in {\Bbb C}^{m+1}\mid z_{1}^{2}+...+z_{m+1}^{2}=1\}$ deforms by retraction onto $S^{m}$; this allows us to identify $[Q^{k},Q^{n}]$ and $[S^{k},S^{n}]=\pi_{k}(S^{n})$. Thus one will say that an element of $\pi_{k}(S^{n})$ is complex representable if there exists a complex polynomial map from $Q^{k}$ to $Q^{n}$ corresponding to this class. In this Note we show that $\pi_{n+1}(S^{n})$ and $\pi_{n+2}(S^{n})$ are complex representable.
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arxiv:math/0702324
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Let H be the n-dimensional hyperbolic space of constant sectional curvature -1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space OG_n of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of OG_n and H. Moreover, we show that OG_3 is K\"{a}hler and find an orthogonal almost complex structure on OG_7.
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arxiv:math/0702365
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We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are generated by elements of the symmetric group $S_n$. We separately investigate the case $S_4$. In this case we solve the corresponding KZ-equation in the explicit form.
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arxiv:math/0702404
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We shown the rationality of the Taylor coefficients of the inverse of the Schwarz triangle functions for a triangle group about any vertex of the fundamental domain.
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arxiv:math/0702422
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Formulated problems concern the following topics: (1) Birationally nonequivalent linear actions; (2) Cayley degrees of simple algebraic groups; (3) Singularities of two-dimensional quotients.
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arxiv:math/0702533
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For split smooth Del Pezzo surfaces, we analyse the structure of the effective cone and prove a recursive formula for the value of alpha, appearing in the leading constant as predicted by Peyre of Manin's conjecture on the number of rational points of bounded height on the surface. Furthermore, we calculate alpha for all singular Del Pezzo surfaces of degree at least 3.
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arxiv:math/0702549
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We give counterexamples to the Kawamata-Viehweg vanishing theorem on ruled surfaces in positive characteristic, and prove that if there is a counterexample to the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface f:X-->C, then either C is a Tango curve or all of sections of f are ample.
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arxiv:math/0702554
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We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.
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arxiv:math/0702612
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Let $\mathfrak{g}$ be a solvable Lie algebra and $Q$ an $ad \mathfrak{g}$-stable prime ideal of the symmetric algebra $S(\mathfrak{g})$ of $\mathfrak{g}$. If $E$ is the set of non zero elements of $S(\mathfrak{g})/Q$ which are eigenvectors for the adjoint action of $\mathfrak{g}$ in $S(\mathfrak{g})/Q$, the localised algebra $(S(\mathfrak{g})/Q)_{E}$ has a natural structure of Poisson algebra. We study this algebra here.
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arxiv:math/0702615
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We show that the probability mass function of the riff-shuffle distribution, also known as the minimum negative binomial distribution, is unimodal, but in general not log-concave.
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arxiv:math/0702639
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The universal covering of the group PSU(1,1) acts naturally on H^m(\delta), the space of holomorphic differentials of order m on the Poincare disk. The purpose of this paper is to survey, as broadly as I am able, the basic sources and examples of invariant measures for this action.
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arxiv:math/0702672
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We study the relationship between the loop problem of a semigroup, and that of a Rees matrix construction (with or without zero) over the semigroup. This allows us to characterize exactly those completely zero-simple semigroups for which the loop problem is context-free. We also establish some results concerning loop problems for subsemigroups and Rees quotients.
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arxiv:math/0702691
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Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $\chi_orb$ with the usual free entropy $\chi$ is shown. Moreover, the dimension counterpart $\delta_{0,orb}$ of $\chi_orb$ is discussed, and we obtain the relation of $\delta_{0,orb}$ with the original free entropy dimension $\delta_0$ with applications to $\delta_0$ itself.
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arxiv:math/0702745
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We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.
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arxiv:math/0702749
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We obtain estimates in the corona theorem for the algebra of analytic functions in the unit disc whose nth derivative is bounded, and its subalgebras defined by the boundary continuity of the nth derivative. The corona theorem for such algebras is trivial and known, but corona theorem with estimates is new and much harder to prove. As an auxiliary result of independent interest we get the continuity of the best estimate in such algebras (including the classical case of the algebra $H^\infty$ of bounded analytic functions). We also show that for a fixed $n$ the best estimate is the same for all algebras we consider.
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arxiv:math/0702754
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In the present paper we study tensor C*-categories with non-simple unit realised as C*-dynamical systems (F,G,\beta) with a compact (non-Abelian) group G and fixed point algebra A := F^G. We consider C*-dynamical systems with minimal relative commutant of A in F, i.e. A' \cap F = Z, where Z is the center of A which we assume to be nontrivial. We give first several constructions of minimal C*-dynamical systems in terms of a single Cuntz-Pimsner algebra associated to a suitable Z-bimodule. These examples are labelled by the action of a discrete Abelian group (which we call the chain group) on Z and by the choice of a suitable class of finite dimensional representations of G. Second, we present a construction of a minimal C*-dynamical system with nontrivial Z that also encodes the representation category of G. In this case the C*-algebra F is generated by a family of Cuntz-Pimsner algebras, where the product of the elements in different algebras is twisted by the chain group action. We apply these constructions to the group G = SU(N).
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arxiv:math/0702775
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We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theorem, a result concerning the symmetries of knots is also obtained. This latter result can be interpreted as a characterisation of the trivial knot.
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arxiv:math/0702783
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In this paper we study the PBW filtration on irreducible integrable highest weight representations of affine Kac-Moody algebra $\gh$. The $n$-th space of this filtration is spanned with the vectors $x_1... x_s v$, where $x_i\in\gh$, $s\le n$ and $v$ is a highest weight vector. For the vacuum module we give a conjectural description of the corresponding adjoint graded space in terms of generators and relations. For $\g$ of the type $A_1$ we prove our conjecture and derive the fermionic formula for the graded character.
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arxiv:math/0702797
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We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible.
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arxiv:math/0702826
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Embedded principalization of ideals in smooth schemes, also known as Log-resolutions of ideals, play a central role in algebraic geometry. If two sheaves of ideals, say $I_1$ and $I_2$, over a smooth scheme $V$ have the same integral closure, it is well known that Log-resolution of one of them induces a Log-resolution of the other. On the other hand, in case $V$ is smooth over a field of characteristic zero, an algorithm of desingularization provides, for each sheaf of ideals, a unique Log-resolution. In this paper we show that algorithms of desingularization define the same Log-resolution for two ideals having the same integral closure. We prove this result here by using the form of induction introduced by W{\l}odarczyk. We extend the notion of Log-resolution of ideals over a smooth scheme $V$, to that of Rees algebras over $V$; and then we show that two Rees algebras with the same integral closure undergo the same constructive resolution. The key point is the interplay of integral closure with differential operators.
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arxiv:math/0702836
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Let $L$ be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces $L^{p}(R^{n};X)$ of $X$-valued functions on $R^n$. We characterize Kato's square root estimates $\|\sqrt{L}u\|_{p} \eqsim \|\nabla u\|_{p}$ and the $H^{\infty}$-functional calculus of $L$ in terms of R-boundedness properties of the resolvent of $L$, when $X$ is a Banach function lattice with the UMD property, or a noncommutative $L^{p}$ space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case $X=C$, we get a new approach to the $L^p$ theory of square roots of elliptic operators, as well as an $L^{p}$ version of Carleson's inequality.
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arxiv:math/0703012
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We compare convergence rates of Metropolis--Hastings chains to multi-modal target distributions when the proposal distributions can be of ``local'' and ``small world'' type. In particular, we show that by adding occasional long-range jumps to a given local proposal distribution, one can turn a chain that is ``slowly mixing'' (in the complexity of the problem) into a chain that is ``rapidly mixing.'' To do this, we obtain spectral gap estimates via a new state decomposition theorem and apply an isoperimetric inequality for log-concave probability measures. We discuss potential applicability of our result to Metropolis-coupled Markov chain Monte Carlo schemes.
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arxiv:math/0703021
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In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to 3CP^2#5(-CP^2) which has two essential Lagrangian tori with simply connected complement. These manifolds can be used to replace E(1) in many known theorems and constructions. Examples in this article include the smallest known minimal symplectic manifolds with abelian fundamental groups including symplectic manifolds with finite and infinite cyclic fundamental group and Euler characteristic 6.
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arxiv:math/0703065
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What are the prices of random variables? In this paper, we define the least-squares prices of coin-flipping games, which are proved to be minimal, positive linear, and arbitrage-free. These prices depend both on a set of games that are available for investing simultaneously and on a risk-free interest rate. In addition, we show a case where the mean-variance portfolio theory is inappropriate.
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arxiv:math/0703079
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On a complete noncompact K\"{a}hler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by $m^2$ if the Ricci curvature is bounded from below by $-2(m+1)$. Then we show that if this upper bound is achieved then the manifold has at most two ends. These results improve previous results on this subject proved by P. Li and J. Wang in \cite {L-W3} and \cite{L-W} under assumptions on the bisectional curvature.
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arxiv:math/0703098
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We present a general method of constructing an uncountable family of regular Borel measures on certain path spaces of Lipschitz functions having fixed Lipschitz constants. We use this method to give a definition of Lebesgue measure and uniform probability measre on these spaces.
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arxiv:math/0703117
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We extend Calabi ansatz over K\"ahler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat K\"ahler metric on K\"ahler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat K\"ahler metric on the toric K\"ahler cone manifold constructed from a toric diagram with a constant height.
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arxiv:math/0703138
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The fundamental combinatorial structure of a net in CP^2 is its associated set of mutually orthogonal latin squares. We define equivalence classes of sets of orthogonal Latin squares by label equivalences of the lines of the corresponding net in CP^2. Then we count these equivalence classes for small cases. Finally, we prove that the realization spaces of these classes in CP^2 are empty to show some non-existence results for 4-nets in CP^2.
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arxiv:math/0703142
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In this note, we reconcile two approaches that have been used to construct stringy multiplications. The pushing forward after pulling back that has been used to give a global stringy extension of the functors K_0,K^{top},A^*,H^* [CR, FG, AGV, JKK2], and the pulling back after having pushed forward, which we have previously used in our (re)-construction program for G-Frobenius algebras, notably in considerations of singularities with symmetries and for symmetric products. A similar approach was also used by [CH] in their considerations of the Chen-Ruan product in a deRham setting for Abelian orbifolds. We show that the pull-push formalism has a solution by the push-pull equations in two situations. The first is a deRham formalism with Thom push-forward maps and the second is the setting of cyclic twisted sectors, which was at the heart of the (re)-construction program. We go on to do formal calculations using fractional Euler classes which allows us to formally treat all the stringy multiplications mentioned above in the general setting. The upshot is the formal trivialization of the co-cycles of the reconstruction program using the presentation of the obstruction bundle of [JKK2].
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arxiv:math/0703209
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We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered hyperk\"ahler manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.
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arxiv:math/0703245
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We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated with quantum groups at roots of unity. We also show that in the case of p-groups, the corresponding pure braid group representations factor through a finite p-group, which answers a question asked of the first author by V. Drinfeld.
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arxiv:math/0703274
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Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove that for t=1 the transformation $\mathbb B_1$ coincides with the canonical bijection $\mathbb B : M \to M_{inf-div}$ discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M_{inf-div} stands for the set of probability distributions in M which are infinitely divisible with respect to free additive convolution. As a consequence, we have that $\mathbb B_t(\mu)$ is infinitely divisible with respect to free additive convolution for any for every $\mu$ in M and every t greater than or equal to one. On the other hand we put into evidence a relation between the transformations $\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations $\mathbb B_t$ as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to free multiplicative convolution, and always reaches infinite divisibility with respect to free additive convolution by the time t=1.
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arxiv:math/0703295
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In this paper we construct some invariants of spatial graphs by disk-summing the constituent knots and show the delta edge-homotopy invariance of them. As an application, we show that there exist infinitely many slice spatial embeddings of a planar graph up to delta edge-homotopy, and there exist infinitely many boundary spatial embeddings of a planar graph up to delta edge-homotopy.
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arxiv:math/0703319
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Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for rk_p(E/F), generalizing the results in [MR], which applied to dihedral extensions. If K is the (unique) quadratic extension of k in F, G = Gal(F/K), G^+ is the subgroup of elements of G commuting with a choice of involution of F over k, and rk_p(E/K) is odd, then we show that (under mild hypotheses) rk_p(E/F) \ge [G:G^+]$. As a very specific example of this, suppose A is an elliptic curve over Q with a rational torsion point of order p, and with no complex multiplication. If E is an elliptic curve over Q with good ordinary reduction at p, such that every prime where both E and A have bad reduction has odd order in F_p^\times, and such that the negative of the conductor of E is not a square mod p, then there is a positive constant B, depending on A but not on E or n, such that rk_p(E/Q(A[p^n])) \ge B p^{2n} for every n.
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arxiv:math/0703363
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The dissipated spaces form a class of compacta which contains both the scattered compacta and the compact LOTSes (linearly ordered topological spaces), and a number of theorems true for these latter two classes are true more generally for the dissipated spaces. For example, every regular Borel measure on a dissipated space is separable. A product of two compact LOTSes is usually not dissipated, but it may satisfy a weakening of that property. In fact, the degree of dissipation of a space can be used to distinguish topologically a product of n LOTSes from a product of m LOTSes.
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arxiv:math/0703429
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A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W. A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions of a Coxeter group having special factors and special amalgamated subgroup are easily recognized from the presentation of the Coxeter group. If a Coxeter group is a subgroup of the fundamental group of a given graph of groups, then the Coxeter group is also the fundamental group of a graph of special subgroups, where each vertex and edge group is a subgroup of a conjugate of a vertex or edge group of the given graph of groups. A vertex group of an arbitrary graph of groups decomposition of a Coxeter group is shown to split into parts conjugate to special groups and parts that are subgroups of edge groups of the given decomposition. Several applications of the main theorem are produced, including the classification of maximal FA-subgroups of a finitely generated Coxeter group as all conjugates of certain special subgroups.
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arxiv:math/0703439
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The probleme des menages (married couples problem) introduced by E.Lucas in 1891 is a classical problem that asks the number of ways to arrange n married couples around a circular table, so that husbands and wives are in alternate places but no couple is seated together. In this paper we present a new version of the Menage Problem that carries the constraints consistent with the Muslim culture.
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arxiv:math/0703444
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Let $T$ be a torus acting on $\CC^n$ in such a way that, for all $1\leq k\leq n$, the induced action on the grassmannian $G(k,n)$ has only isolated fixed points. This paper proposes a natural, elementary, explicit description of the corresponding $T$-equivariant Schubert calculus. In a suitable natural basis of the $T$-equivariant cohomology, seen as a module over the $T$-equivariant cohomology of a point, it is formally the same as the ordinary cohomology of a grassmann bundle. The main result, useful for computational purposes, is that the $T$-equivariant cohomology of $G(k,n)$ can be realized as the quotient of a ring generated by derivations on the exterior algebra of a free module of rank $n$ over the $T$-equivariant cohomology of a point.
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arxiv:math/0703445
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The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index r extends the well-known polar duality for reflexive polytopes (case r=1). We consider the Borisov duality between two nef-partitions as a duality between two Gorenstein polytopes P and P^* of index r together with selected special (r-1)-dimensional simplices S in P and S' in P^*. Different choices of these simplices suggest an interesting relation to Homological Mirror Symmetry.
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arxiv:math/0703456
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The radial probability measures on $R^p$ are in a one-to-one correspondence with probability measures on $[0,\infty[$ by taking images of measures w.r.t. the Euclidean norm mapping. For fixed $\nu\in M^1([0,\infty[)$ and each dimension p, we consider i.i.d. $R^p$-valued random variables $X_1^p,X_2^p,...$ with radial laws corresponding to $\nu$ as above. We derive weak and strong laws of large numbers as well as a large deviation principle for the Euclidean length processes $S_k^p:=\|X_1^p+...+X_k^p\|$ as k,p\to\infty in suitable ways. In fact, we derive these results in a higher rank setting, where $R^p$ is replaced by the space of $p\times q$ matrices and $[0,\infty[$ by the cone $\Pi_q$ of positive semidefinite matrices. Proofs are based on the fact that the $(S_k^p)_{k\ge 0}$ form Markov chains on the cone whose transition probabilities are given in terms Bessel functions $J_\mu$ of matrix argument with an index $\mu$ depending on p. The limit theorems follow from new asymptotic results for the $J_\mu$ as $\mu\to \infty$. Similar results are also proven for certain Dunkl-type Bessel functions.
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arxiv:math/0703520
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A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.
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arxiv:math/0703537
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We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.
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arxiv:math/0703575
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We study the stable norm on the first homology of a closed, non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater than two.
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arxiv:math/0703667
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We establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. These expectations turn out to be the generating series of certain paths in the Cayley graph of the symmetric group. We then compute the asymptotic distribution of a random unitary matrix under the heat kernel measure on the unitary group U(N) as N tends to infinity, and prove a result of asymptotic freeness result for independent large unitary matrices, thus recovering results obtained previously by Xu and Biane. We give an interpretation of our main expansion in terms of random ramified coverings of a disk. Our approach is based on the Schur-Weyl duality and we extend some of our results to the orthogonal and symplectic cases.
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arxiv:math/0703690
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This article contains two main theoretical results on neural spike train models. The first assumes that the spike train is modeled as a counting or point process on the real line where the conditional intensity function is a product of a free firing rate function s, which depends only on the stimulus, and a recovery function r, which depends only on the time since the last spike. If s and r belong to a q-smooth class of functions, it is proved that sieve maximum likelihood estimators for s and r achieve essentially the optimal convergence rate (except for a logarithmic factor) under L_1 loss. The second part of this article considers template matching of multiple spike trains. P-values for the occurrences of a given template or pattern in a set of spike trains are computed using a general scoring system. By identifying the pattern with an experimental stimulus, multiple spike trains can be deciphered to provide useful information.
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arxiv:math/0703829
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Using the identification of sections of the Galois group of the ground field into the arithmetic fundamental group with neutral fiber functors of the category of finite connections, we define the "packets" in Grothendieck's section conjecture and show their properties predicted by him.
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arxiv:math/0703877
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Wiebe's criterion, which recognizes complete intersections of dimension zero among the class of noetherian local rings, is revisited and exploited in order to provide information on what we call C.I.0-ideals (those such that the corresponding quotient is a complete intersection of dimension zero) and also on chains of C.I.0-ideals. A correspondence is established between C.I.0-ideals and a certain kind of matrices which we call $x$-nice, and a chain of C.I.0-ideals corresponds to a factorization of some $x$-nice matrix. When the local ring $A$ itself is a complete intersection of dimension zero, a C.I.0-ideal is necessarily of the form $(0:bA)$ for some $b\in A$. Some criteria are provided to recognize whether an ideal $(0:bA)$ is C.I.0 or not. When $y$ is a minimal generator of the maximal ideal of $A$, it is also proved that the ideals $yA$ and $(0:yA)$ are C.I.0 simultaneously and that this is the case exactly when the ideal $(0:yA)$ is principal. These C.I.0-ideals of the form $(0:yA)$, $y$ being a minimal generator of the maximal ideal, are investigated. They are of interest because the smallest nonnull C.I.0-ideal in a strict chain of C.I.0-ideals of the maximal length is necessarily of that form, and their existence has some implications for a realization of the ring, i.e. for the way the ring can can be viewed as a quotient of a regular local ring.
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arxiv:math/0703880
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We compute the equivariant zeta function for bundles over infinite graphs and for infinite covers. In particular, we give a ``transfer formula'' for the zeta function of infinite graph covers. Also, when the infinite cover is given as a limit of finite covers, we give a formula for the limit of the zeta functions.
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arxiv:math/0703891
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We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers.
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arxiv:math/0703907
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It is shown that the weak $L^p$ spaces $\ell^{p,\infty}, L^{p,\infty}[0,1]$, and $L^{p,\infty}[0,\infty)$ are isomorphic as Banach spaces.
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arxiv:math/9201237
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A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Fr\"olicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of "traceless" vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed.
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arxiv:math/9201255
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Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-fractional and their nonlinearities tend to cancel leaving only a small total, then they can all be replaced with affine maps with the same domains and images and the resulting composition is a very good approximation of the original one. These technical tools are then applied to prove a theorem about critical circle maps.
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arxiv:math/9201274
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We establish that every formal critical portrait (as defined by Goldberg and Milnor), can be realized by a postcritically finite polynomial.
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arxiv:math/9201296
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We study how equivalent nonisomorphic models of unsuperstable theories can be. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues [HySh:474].
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arxiv:math/9202205
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Let $j:V_\lambda---> V_\lambda$ be an elementary embedding, with critical point $\kappa$, and let $f(n)$ be the number of critical points of embeddings in the algebra generated by $j$ which lie between $j^n(\kappa)$ and $j^{n+1}(\kappa)$. It is shown that $f(n)$ is finite for all $n$.
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arxiv:math/9204204
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In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of Riemann hypothesis will hold, (ii) that if in addition the function has a simple pole at the point s=1, then it must be a product of the Riemann zeta-function and another Dirichlet series with similar properties, and (iii) that a type of converse theorem holds, namely that all such Dirichlet series can be obtained by considering Mellin transforms of automorphic forms associated with arithmetic groups.
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arxiv:math/9204217
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We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold.
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arxiv:math/9207215
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Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<\l<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $\l$ and its existence is not a result of a trivial cardinal arithmetical reason, then $\l$ ``resembles'' $\aleph_1$ --- a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the non-existence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).
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arxiv:math/9209201
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This will is an expository description of quadratic rational maps. Sections 2 through 6 are concerned with the geometry and topology of such maps. Sections 7--10 survey of some topics from the dynamics of quadratic rational maps. There are few proofs. Section 9 attempts to explore and picture moduli space by means of complex one-dimensional slices. Section 10 describes the theory of real quadratic rational maps. For convenience in exposition, some technical details have been relegated to appendices: Appendix A outlines some classical algebra. Appendix B describes the topology of the space of rational maps of degree \[d\]. Appendix C outlines several convenient normal forms for quadratic rational maps, and computes relations between various invariants.\break Appendix D describes some geometry associated with the curves \[\Per_n(\mu)\subset\M\]. Appendix E describes totally disconnected Julia sets containing no critical points. Finally, Appendix F, written in collaboration with Tan Lei, describes an example of a connected quadratic Julia set for which no two components of the complement have a common boundary point.
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arxiv:math/9209221
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It is proved that the free $m$-generated Burnside groups $\Bbb{B}(m,n)$ of exponent $n$ are infinite provided that $m>1$, $n\ge2^{48}$.
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arxiv:math/9210221
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Bounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show that there are only two possibilities for the structure of the boundary of such a disk: either the boundary admits a nice decomposition onto a circle, or it is an indecomposable continuum.
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arxiv:math/9210225
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In this paper, we completely settle several of the open questions regarding the relationships between the three most fundamental forms of set convergence. In particular, it is shown that Wijsman and slice convergence coincide precisely when the weak star and norm topologies agree on the dual sphere. Consequently, a weakly compactly generated Banach space admits a dense set of norms for which Wijsman and slice convergence coincide if and only if it is an Asplund space. We also show that Wijsman convergence implies Mosco convergence precisely when the weak star and Mackey topologies coincide on the dual sphere. A corollary of these results is that given a fixed norm on an Asplund space, Wijsman and slice convergence coincide if and only if Wijsman convergence implies Mosco convergence.
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arxiv:math/9302212
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It is an English translation of the paper originally published in Russian and Ukrainian in 1987. In the appendix of his book S.Banach introduced the following definition Let $X$ be a Banach space and $\Gamma$ be a subspace of the dual space $X^*$. The set of all limits of $w^{*}$-convergent sequences in $\Gamma $ is called the $w^*${\it -derived set} of $\Gamma $ and is denoted by $\Gamma _{(1)}$. For an ordinal $\alpha$ the $w^{*}$-{\it derived set of order} $\alpha $ is defined inductively by the equality: $$ \Gamma _{(\alpha )}=\bigcup _{\beta <\alpha }((\Gamma _{(\beta )})_{(1)}. $$
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arxiv:math/9303203
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The condition onto pair ($F,G$) of function Banach spaces under which there exists a integral operator $T:F\to G$ with analytic kernel such that the inverse mapping $T^{-1}:$im$T\to F$ does not belong to arbitrary a priori given Borel (or Baire) class is found.
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arxiv:math/9303205
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Using the standard square--function method (based on the Poisson semigroup), multiplier conditions of H\"ormander type are derived for Laguerre expansions in $L^p$--spaces with power weights in the $A_p$-range; this result can be interpreted as an ``upper end point'' multiplier criterion which is fairly good for $p$ near $1$ or near $\infty $. A weighted generalization of Kanjin's \cite{kan} transplantation theorem allows to obtain a ``lower end point'' multiplier criterion whence by interpolation nearly ``optimal'' multiplier criteria (in dependance of $p$, the order of the Laguerre polynomial, the weight).
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arxiv:math/9307203
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The techniques for polynomial interpolation and Gaussian quadrature are generalized to matrix-valued functions. It is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision.
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arxiv:math/9307215
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A category used by de Paiva to model linear logic also occurs in Vojtas's analysis of cardinal characteristics of the continuum. Its morphisms have been used in describing reductions between search problems in complexity theory. We describe this category and how it arises in these various contexts. We also show how these contexts suggest certain new multiplicative connectives for linear logic. Perhaps the most interesting of these is a sequential composition suggested by the set-theoretic application.
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arxiv:math/9309208
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In this paper the following result, which allows one to decouple U-Statistics in tail probability, is proved in full generality. Theorem 1. Let $X_i$ be a sequence of independent random variables taking values in a measure space $S$, and let $f_{i_1...i_k}$ be measurable functions from $S^k$ to a Banach space $B$. Let $(X_i^{(j)})$ be independent copies of $(X_i)$. The following inequality holds for all $t \ge 0$ and all $n\ge 2$, $$ P(||\sum_{1\le i_1 \ne ... \ne i_k \le n} f_{i_1 ... i_k}(X_{i_1},...,X_{i_k}) || \ge t) \qquad\qquad$$ $$ \qquad\qquad\le C_k P(C_k||\sum_{1\le i_1 \ne ... \ne i_k \le n} f_{i_1 ... i_k}(X_{i_1}^{(1)},...,X_{i_k}^{(k)}) || \ge t) .$$ Furthermore, the reverse inequality also holds in the case that the functions $\{f_{i_1... i_k}\}$ satisfy the symmetry condition $$ f_{i_1 ... i_k}(X_{i_1},...,X_{i_k}) = f_{i_{\pi(1)} ... i_{\pi(k)}}(X_{i_{\pi(1)}},...,X_{i_{\pi(k)}}) $$ for all permutations $\pi$ of $\{1,...,k\}$. Note that the expression $i_1 \ne ... \ne i_k$ means that $i_r \ne i_s$ for $r\ne s$. Also, $C_k$ is a constant that depends only on $k$.
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arxiv:math/9309211
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We give a model where there is a ccc Souslin forcing which does not satisfy the Knaster condition. Next, we present a model where there is a sigma-linked not sigma-centered Souslin forcing such that all its small subsets are sigma-centered but Martin Axiom fails for this order. Furthermore, we construct a totally nonhomogeneous Souslin forcing and we build a Souslin forcing which is proper but not ccc that does not contain a perfect set of mutually incompatible conditions. Finally we show that ccc Sigma^1_2-notions of forcing may not be indestructible ccc.
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arxiv:math/9310224
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According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional Teichm\"{u}ller space. This is a basic idea in Sullivan's approach to the Renormalization Conjecture. One of its principle ingredients is to supply ${\cal E}$ with the Teichm\"{u}ller metric. To have such a metric one has to know, first of all, that all maps of ${\cal E}$ are quasi-symmetrically conjugate. This was proved [Ji] and [JS] for some classes of non-renormalizable maps (when the critical point is not too recurrent). Here we consider a space of non-renormalizable unimodal maps with in a sense fastest possible recurrence of the critical point (called Fibonacci). Our goal is to supply this space with the Teichm\"{u}ller metric.
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arxiv:math/9311213
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We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence spaces with few surjective isometries. We also give a new example of a class of real sequence spaces with few surjective isometries.
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arxiv:math/9403207
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The polynomial invariants $q_d$ for a large class of smooth 4-manifolds are shown to satisfy universal relations. The relations reflect the possible genera of embedded surfaces in the 4-manifold and lead to a structure theorem for the polynomials. As an application, one can read off a lower bound for the genera of embedded surfaces from the asymptotics of $q_d$ for large $d$. The relations are proved using moduli spaces of singular instantons.
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arxiv:math/9404232
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The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3 as a first restriction on the possible nerves of such packings. We show that the supremum k of the average kissing number for all packings satisfies 12.566 ~ 666/53 <= k < 8 + 4*sqrt(3) ~ 14.928 We obtain the upper bound by a resource exhaustion argument and the upper bound by a construction involving packings of spherical caps in S^3. Our result contradicts two naive conjectures about the average kissing number: That it is unbounded, or that it is supremized by an infinite packing of congruent spheres.
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arxiv:math/9405218
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{Let ${\Cal X}$ be a self-dual P-polynomial association scheme. Then there are at most 12 diagonal matrices $T$ such that $(PT)^3=I$. Moreover, all of the solutions for the classical infinite families of such schemes (including the Hamming scheme) are classified.
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arxiv:math/9406225
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We find the adjoint of the Askey-Wilson divided difference operator with respect to the inner procuct on L^2(-1,1,(1-x^2)^-1/2 dx) defined as a Cauchy principle value and show that the Askey-Wilson polynomials are solutions of a q-Sturm-Liouville problem. From these facts we deduce various properties of the polynomials in a simple and straightforward way. We also provide an operator theoretic description of the Askey-Wilson operator.
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arxiv:math/9408209
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Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.
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arxiv:math/9409226
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We address the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs), both directed and undirected. The problem turns out to be closely related to the problem of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. In particular, we find the largest Abelian graphs with 2 generators (dimensions) and a given diameter. (The results for 2 generators are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with 3 generators and a given diameter which we conjecture to be as large as possible; for the directed case, we obtain partial results, which lead to unusual lattice coverings of 3-space. We discuss the asymptotic behavior of the problem for large numbers of generators. The graphs obtained here are substantially better than traditional toroidal meshes, but, in the simpler undirected cases, retain certain desirable features such as good routing algorithms, easy constructibility, and the ability to host mesh-connected numerical algorithms without any increase in communication times.
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arxiv:math/9412223
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A class K of finite structures is said to have the extension property for automorphisms (EP) if for every A in K there exists an extension B in K such that every partial isomorphism on the structure A extends to an automorphism of B. Hrushovski proved EP for the class of all finite graphs. The main problem is to keep B finite. Hodges, Hodkinson, Lascar and Shelah showed in their paper that in certain cases the EP for the class K implies the Small Index Property (SIP) for the "generic" countable structure determined by K. E.g. Hrushovskis result yields the SIP for the random graph. In this preprint we prove the EP for the class of all finite K_n - free graphs (i.e. graphs with no complete subgraph of given size n), which implies SIP for the generic K_n - free graph. Also we prove EP for a certain family of classes of directed graphs, which implies SIP for the "Henson digraphs". This is a family of continuum many non isomorphic countable directed omega-categorical graphs. Finally we state EP for a more general family of classes, which covers all the cases mentioned above.
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arxiv:math/9502206
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We prove the following conjecture of Narayana: there are no dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime.
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arxiv:math/9502212
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We know extensions of first order logic by quantifiers of the kind "there are uncountable many ...", "most ..." with new axioms and appropriate semantics. Related are operations such as "set of x, such that ...", Hilbert's $\epsilon$-operator, Churche's $\lambda$-notation, minimization and similar ones, which also bind a variable within some expression, the meaning of which is however partly defined by a translation into the language of first order logic. In this paper a generalization is presented that comprises arbitrary variable-binding symbols as non-logical operations. The axiomatic extension is determined by new equality-axioms; models assign functionals to variable-binding symbols. The completeness of this system of the so called "Functional Logic of 1st Order" will be proved.
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arxiv:math/9503206
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We prove that, for any jointly stable random variables $X_1, \dots, X_k$ with zero mean, any $m<k,$ and any even continuous positive definite functions $f$ and $g$ on $\Bbb R^m$ and $\Bbb R^{k-m},$ the random variables $f(X_1,\dots,X_m)$ and $g(X_{m+1}, \dots,X_k)$ are non-negatively correlated. We also show another result that is related to an old question of whether $$P(\max_{1\le i\le k} |X_i|<t) \ge P(\max_{1\le i\le m} |X_i|<t) \ P(\max_{m+1\le i\le k} |X_i|<t)$$ where $X_1,\dots,X_k$ are jointly Gaussian random variables with zero mean, and $m<k.$ We show that the quantity in the left-hand side has a local minimum at the point where the random variables $X_i$ and $X_j$ are independent for any choice of $1\le i\le m$ and $m+1\le j\le k.$
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arxiv:math/9503212
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This work studies combinatorics and geometry of the Yoccoz puzzle for quadratic polynomials. It is proven that the moduli of the ``principal nest'' of annuli grow at linear rate. As a corollary we obtain complex a priori bounds and local connectivity of the Julia set for many infinitely renormalizable quadratics.
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arxiv:math/9503215
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Let $F\in W^{1,n}_{\text{loc}}(\Omega; \Bbb R^n)$ be a mapping with nonnegative Jacobian $J_F(x)=\det DF(x)\ge 0$ for a.e. $x$ in a domain $\Omega\subset\Bbb R^n$. The {\it dilatation} of $F$ is defined (almost everywhere in $\Omega$) by the formula $$K(x)=\frac{|DF(x)|^n}{J_F(x)}\cdot$$ Iwaniec and \v Sver\' ak \ncite{IS} have conjectured that if $p\ge n-1$ and $K\in L^{p}_{\text{loc}}(\Omega)$ then $F$ must be continuous, discrete and open. Moreover, they have confirmed this conjecture in the two-dimensional case $n=2$. In this article, we verify it in the higher- dimensional case $n\ge 2$ whenever $p>n-1$.
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arxiv:math/9504225
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We study the determinacy of the game G_kappa (A) introduced in [FKSh:549] for uncountable regular kappa and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that G_kappa (A) is undetermined. For the class of linear orders, the existence of such A depends on the size of kappa^{< kappa}. In particular we obtain a characterization of kappa^{< kappa}= kappa in terms of determinacy of the game G_kappa (L) for linear orders L .
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arxiv:math/9505212
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We present four generalized small cancellation conditions for finite presentations and solve the word- and conjugacy problem in each case. Our conditions $W$ and $W^*$ contain the non-metric small cancellation cases C(6), C(4)T(4), C(3)T(6) (see [LS]) but are considerably more general. $W$ also contains as a special case the small cancellation condition $W(6)$ of Juhasz [J2]. If a finite presentation satisfies $W$ or $W^*$ then it has a quadratic isoperimetric inequality and therefore solvable word problem. For the class $W$ this was first observed by Gersten in [G7] which also contains an idea of the proof. Our main result here is the proof of the conjugacy problem for the classes $W$ and $W^*$ which uses the geometry of non-positively curved piecewise Euclidean complexes developed by Bridson in [Bri]. The conditions $V$ and $V^*$ generalize the small cancellation conditions C(7), C(5)T(4), C(4)T(5), C(3)T(7). If a finite presentation satisfies the condition $V$ or $V^*$, then it has a linear isoperimetric inequality and hence the group is hyperbolic.
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arxiv:math/9509206
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The renormalization of a quadratic-like map is studied. The three-dimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the {\sl a priori} complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan's sector theorem is discussed and is used to prove his result that the Feigenbaum quadratic polynomial has the {\sl a priori} complex bounds and is unbranched. A dense subset on the boundary of the Mandelbrot set is constructed so that for every point of the subset, the corresponding quadratic polynomial is unbranched and has the {\sl a priori} complex bounds.
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arxiv:math/9511208
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We study the regularity results of holomorphic correspondences. As an application, we combine it with certain recently developed methods to obtain the extension theorem for proper holomorphic mappings between domains with real analytic boundaries in the complex 2-space.
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arxiv:math/9512213
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Examples are given of degenerate elliptic operators on smooth, compact manifolds that are not globally regular in $C^\infty$. These operators degenerate only in a rather mild fashion. Certain weak regularity results are proved, and an interpretation of global irregularity in terms of the associated heat semigroup is given.
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arxiv:math/9512216
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In the original version of this paper, we assume a theory $T$ that the logic $\mathbb L_{\kappa, \aleph_{0}}$ is categorical in a cardinal $\lambda > \kappa$, and $\kappa$ is a measurable cardinal. There we prove that the class of model of $T$ of cardinality $<\lambda$ (but $\geq |T|+\kappa$) has the amalgamation property; this is a step toward understanding the character of such classes of models. In this revised version we replaced the class of models of $T$ by $\mathfrak k$, an AEC (abstract elementary class) which has LS-number ${<} \, \kappa,$ or at least which behave nicely for ultrapowers by $D$, a normal ultra-filter on $\kappa$. Presently sub-section \S1A deals with $T \subseteq \mathbb L_{\kappa^{+}, \aleph_{0}}$ (and so does a large part of the introduction and little in the rest of \S1), but otherwise, all is done in the context of AEC.
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arxiv:math/9602216
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This paper has been withdrawn by the authors, due a crucial error.
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arxiv:math/9604232
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We say that an even continuous function $H$ on the unit sphere $\Omega$ in $R^n$ admits the Blaschke-Levy representation with $q>0$ if there exists an even function $b\in L_1(\Omega)$ so that $H^q(x)=\int_\Omega |(x,\xi)|^q b(\xi)\ d\xi$ for every $x\in \Omega.$ This representation has numerous applications in convex geometry, probability and Banach space theory. In this paper, we present a simple formula (in terms of the derivatives of $H$) for calculating $b$ out of $H.$ We use this formula to give a sufficient condition for isometric embedding of a space into $L_p$ which contributes to the 1937 P.Levy's problem and to the study of zonoids. Another application gives a Fourier transform formula for the volume of $(n-1)$-dimensional central sections of star bodies in $R^n.$ We apply this formula to find the minimal and maximal volume of central sections of the unit balls of the spaces $\ell_p^n$ with $0<p<2.$
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arxiv:math/9605212
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