Tuesday, June 23rd, 2009

Time	Event
6:41a	Complex Monge-Ampere equations on Hermitian manifolds. (arXiv:0906.3548v1 [math.DG]) Complex Monge-Ampere equations on Hermitian manifolds. (arXiv:0906.3548v1 [math.DG]) Authors: Bo Guan, Qun Li We study complex Monge-Ampere equations on Hermitian manifolds, extending classical existence results of Yau and Aubin in the Kahler case, and those of Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As an application we generalize existing results on the Donaldson conjecture on geodesics in the space of Kahler metrics to the Hermitian setting. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Upper Semicontinuity of Random Attractors for Non-compact Random Dynamical Systems. (arXiv:0906.3536 Upper Semicontinuity of Random Attractors for Non-compact Random Dynamical Systems. (arXiv:0906.3536v1 [math.AP]) Authors: Bixiang Wang The upper semicontinuity of random attractors for non-compact random dynamical systems is proved when the union of all perturbed random attractors is precompact with probability one. This result is applied to the stochastic Reaction-Diffusion with white noise defined on the entire space R^n. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Freiman's theorem for solvable groups. (arXiv:0906.3535v1 [math.CO]) Freiman's theorem for solvable groups. (arXiv:0906.3535v1 [math.CO]) Authors: Terence Tao Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa to arbitrary abelian groups, where the controlling object is now a coset progression. We extend these results further to solvable groups of bounded derived length, in which the coset progressions are replaced by the more complicated notion of a ``coset nilprogression''. As one consequence of this result, any subset of such a solvable group of small doubling is is controlled by a set whose iterated products grow polynomially, and which are contained inside a virtually nilpotent group. As another application we establish a strengthening of the Milnor-Wolf theorem that all solvable groups of polynomial growth are virtually nilpotent, in which only one large ball needs to be of polynomial size. This result complements recent work of Breulliard-Green, Fisher-Katz-Peng, and Sanders. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	On the Distributions of Pseudoprimes, Carmichael Numbers, and Strong Pseudoprimes. (arXiv:0906.3533v On the Distributions of Pseudoprimes, Carmichael Numbers, and Strong Pseudoprimes. (arXiv:0906.3533v2 [math.NT]) Authors: Aran Nayebi Building upon the work of Carl Pomerance and others, the central purpose of this discourse is to discuss the distribution of base-2 pseudoprimes, as well as improve upon Pomerance's conjecture regarding the Carmichael number counting function. All conjectured formulas apply to any base $b \ge 2$ for $x \ge x_0(b)$. A table of base-2 pseudoprime, 2-strong pseudoprime, and Carmichael number counts up to $10^{15}$ is included in the Appendix. We also discuss strong pseudoprimes and probabilistic primality testing. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Decompositions into subgraphs of small diameter. (arXiv:0906.3530v1 [math.CO]) Decompositions into subgraphs of small diameter. (arXiv:0906.3530v1 [math.CO]) Authors: Jacob Fox, Benny Sudakov We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E) on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that |E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma, Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a constant depending only \epsilon. This shows that every dense graph can be partitioned into a small number of ``small worlds'' provided that few edges can be ignored. Improving on their result, we determine P(n,\epsilon,d) within an absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded for \epsilon < 1/4, P(n,\epsilon,3) = \Theta(1/\epsilon^2) for \epsilon > n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, R\"odl, Ruci\'nski, and Szemer\'edi. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Multivariate Gaussians, Semidefinite Matrix Completion, and Convex Algebraic Geometry. (arXiv:0906.3 Multivariate Gaussians, Semidefinite Matrix Completion, and Convex Algebraic Geometry. (arXiv:0906.3529v1 [math.ST]) Authors: Bernd Sturmfels, Caroline Uhler We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a spectrahedron, and to the problem of characterizing the image of the positive definite cone under an arbitrary linear projection. These problems at the interface of statistics and optimization are here examined from the perspective of convex algebraic geometry. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Constructive Field Theory in Zero Dimension. (arXiv:0906.3524v1 [math-ph]) Constructive Field Theory in Zero Dimension. (arXiv:0906.3524v1 [math-ph]) Authors: V. Rivasseau In this pedagogical note we propose to wander through five different methods to compute the number of connected graphs of the zero-dimensional $\phi^4$ field theory,in increasing order of sophistication. The note does not contain any new result but may be helpful to summarize the heart of constructive resummations, namely a replica trick and a forest formula. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Jorgensen's Inequalities and Collars in n-dimensional Quaternionic Hyperbolic Space. (arXiv:0906.356 Jorgensen's Inequalities and Collars in n-dimensional Quaternionic Hyperbolic Space. (arXiv:0906.3562v1 [math.GT]) Authors: Wensheng Cao, Guijian Zhang In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Cao [4]. As applications, we use the quaternionic version of Jorgensen's inequalities to construct embedded collars about short, simple, closed geodesics in quaternionic hyperbolic manifolds. We show that these canonical collars are disjoint from each other. Our results give some improvement over earlier results of Markham and Parker and answer an open question posed in [15]. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations. (arXiv:0906.3565v Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations. (arXiv:0906.3565v1 [math-ph]) Authors: Lee-Peng Teo In this article, we classify the solutions of the dispersionless Toda hierarchy into degenerate and non-degenerate cases. We show that every non-degenerate solution is determined by a function $\mathcal{H}(z_1,z_2)$ of two variables. We interpret these non-degenerate solutions as defining evolutions on the space $\mathfrak{D}$ of pairs of conformal mappings $(g,f)$, where $g$ is a univalent function on the exterior of the unit disc, $f$ is a univalent function on the unit disc, normalized such that $g(\infty)=\infty$, $f(0)=0$ and $f'(0)g'(\infty)=1$. For each solution, we show how to define the natural time variables $t_n, n\in\Z$, as complex coordinates on the space $\mathfrak{D}$. We also find explicit formulas for the tau function of the dispersionless Toda hierarchy in terms of $\mathcal{H}(z_1, z_2)$. Imposing some conditions on the function $\mathcal{H}(z_1, z_2)$, we show that the dispersionless Toda flows can be naturally restricted to the subspace $\Sigma$ of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers the result of Zabrodin. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Sofic equivalence relations. (arXiv:0906.3619v1 [math.FA]) Sofic equivalence relations. (arXiv:0906.3619v1 [math.FA]) Authors: Gábor Elek, Gábor Lippner We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of L\"uck, Sauer and Wegner hold for treeable equivalence relations. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	A Variance Reduction Method for Parametrized Stochastic Differential Equations using the Reduced Bas A Variance Reduction Method for Parametrized Stochastic Differential Equations using the Reduced Basis Paradigm. (arXiv:0906.3600v1 [math.NA]) Authors: Sebastien Boyaval, Tony Lelievre In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized It\^o stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Approximate groups, I: the torsion-free nilpotent case. (arXiv:0906.3598v1 [math.CO]) Approximate groups, I: the torsion-free nilpotent case. (arXiv:0906.3598v1 [math.CO]) Authors: Emmanuel Breuillard, Ben Green We describe the structure of ``K-approximate subgroups'' of torsion-free nilpotent groups, paying particular attention to Lie groups. Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which independently address related issues. We comment briefly on some of the connections between these papers. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Non-relativistic conformal symmetries in fluid mechanics. (arXiv:0906.3594v1 [physics.flu-dyn]) Non-relativistic conformal symmetries in fluid mechanics. (arXiv:0906.3594v1 [physics.flu-dyn]) Authors: P.A. Horváthy, P.-M. Zhang The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schr\"odinger group, which also involves, in addition, Schr\"odinger expansions. The recently discussed Conformal Galilei group is not a symmetry. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Self-Triggered Control: trading actuation for computation. (arXiv:0906.3588v1 [math.OC]) Self-Triggered Control: trading actuation for computation. (arXiv:0906.3588v1 [math.OC]) Authors: Manuel Mazo Jr., Adolfo Anta, Paulo Tabuada Event-triggered and self-triggered control have recently been proposed as alternatives to the well established periodic implementation of control loops on digital platforms. In a self-triggered implementation, the control task is responsible for computing the new actuator values as well as the next instant of time at which the state should be sampled, the control law recomputed, and the actuator values updated. In between these time instants, the system under control requires no attention and can operate in open loop. Self-triggered implementations can be seen as double edged swords: on the one hand, by using the state of the plant to determine when the control law needs to be recomputed, desired levels of control performance are enforced while drastically reducing the resources (processor time, communication bandwidth, etc) used for control; on the other hand, by operating the plant in open loop for extended periods of time, robustness of self-triggered implementations becomes an honest concern. In this paper we propose a self-triggered implementation for linear systems achieving exponential input-to-state stability, hence addressing the robustness concerns. Moreover, the proposed implementation computes the largest possible times during which the plant can operate in open loop while meeting desired performance levels and subject to the computational capabilities of the digital platform. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	The quantum differential equation of the Hilbert scheme of points in the plane. (arXiv:0906.3587v1 [ The quantum differential equation of the Hilbert scheme of points in the plane. (arXiv:0906.3587v1 [math.AG]) Authors: Andrei Okounkov, Rahul Pandharipande We discuss here basic properties of the quantum differential equation of the Hilbert scheme of points in the plane. Our emphasis is on intertwining operators (which shift equivariant parameters) and their applications. In particular, we obtain an exact solution to the connection problem from the Donaldson-Thomas point q=0 to the Gromov-Witten point q=-1. The paper is an expanded version of what was once the last Section of arXiv:math/0411210. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Magma Proof of Strict Inequalities for Minimal Degrees of Finite Groups. (arXiv:0906.3574v1 [math.GR Magma Proof of Strict Inequalities for Minimal Degrees of Finite Groups. (arXiv:0906.3574v1 [math.GR]) Authors: Scott H. Murray, Neil Saunders The minimal faithful permutation degree of a finite group $G$, denote by $\mu(G)$ is the least non-negative integer $n$ such that $G$ embeds inside the symmetric group $\Sym(n)$. In this paper, we outline a Magma proof that 10 is the smallest degree for which there are groups $G$ and $H$ such that $\mu(G \times H) < \mu(G)+ \mu(H)$. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	A Topology for Galois Types in AECs. (arXiv:0906.3573v1 [math.LO]) A Topology for Galois Types in AECs. (arXiv:0906.3573v1 [math.LO]) Authors: Michael Lieberman We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	On monochromatic arm exponents for 2D critical percolation. (arXiv:0906.3570v2 [math.PR]) On monochromatic arm exponents for 2D critical percolation. (arXiv:0906.3570v2 [math.PR]) Authors: Vincent Beffara (UMPA-ENSL), Pierre Nolin (CIMS) We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well-understood) polychromatic exponents. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Critical homogenization of Levy process driven SDEs in random medium. (arXiv:0906.3569v1 [math.PR]) Critical homogenization of Levy process driven SDEs in random medium. (arXiv:0906.3569v1 [math.PR]) Authors: Rémi Rhodes (CEREMADE), Bamba A. Sow (LERSTAD) We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Cascades and e-invisibility. (arXiv:0906.3567v1 [math.DS]) Cascades and e-invisibility. (arXiv:0906.3567v1 [math.DS]) Authors: Yulij Ilyashenko, Denis Volk We consider statistical attractors of locally typical dynamical systems and their ``e-invisible'' subsets: parts of the attractors whose neighborhoods are visited by orbits with an average frequency of less than e << 1. For extraordinarily small values of e (say, smaller than 2^(-10^6)), an observer virtually never sees these parts when following a generic orbit. A trivial reason for e-invisibility in a generic dynamical system may be either a high Lipshitz constant (~1/e) of the mapping (i.e. it badly distorts the metric) or its proximity (~e) to the structurally unstable dynamical systems. However Ilyashenko and Negut [IN] provided a locally typical example of dynamical systems with an e-invisible set and a uniform moderate (<100) Lipshitz constant independent on e. These dynamical systems from [IN] are also |ln e|-distant from structurally unstable dynamical systems (in the class S of skew products). We further develop the example of [IN] to provide a better rate of invisibility while staying at the same distance away from the structurally unstable dynamical systems. We give an explicit example of C^1-balls in the space of ``step'' skew products over the Bernoulli shift such that for each dynamical system from this ball a large portion of the statistical attractor is invisible. Systems that are c/n-distant from structurally unstable ones (in the class S) have rate of invisibility e = 2^(-n^k) where 3k is the Hausdorff dimension of the phase space. read more at math updates on arXiv.org rss2lj (Comment on this)
6:41a	Perturbations of completely positive maps and strong NF algebras. (arXiv:0906.3510v1 [math.OA]) Perturbations of completely positive maps and strong NF algebras. (arXiv:0906.3510v1 [math.OA]) Authors: Caleb Eckhardt Let $\phi:M_n\to B(H)$ be an injective, completely positive contraction with $\V\phi^{-1}:\phi(M_n)\to M_n\V_{cb}\leq1+\delta(\epsilon).$ We show that if either (i) $\phi(M_n)$ is faithful modulo the compact operators or (ii) $\phi(M_n)$ approximately contains a rank 1 projection, then there is a complete order embedding $\psi:M_n\to B(H)$ with $\V\phi-\psi\V_{cb}<\epsilon.$ We also give examples showing that such a perturbation does not exist in general. As an application, we show that every $C^*$-algebra $A$ with $\mathcal{OL}_\infty(A)=1$ and a finite separating family of primitive ideals is a strong NF algebra, providing a partial answer to a question of Junge, Ozawa and Ruan. read more at math updates on arXiv.org rss2lj (Comment on this)

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