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CLT for spectra of submatrices of Wigner random matrices II. Stochastic evolution
CLT submatrices Wigner random matrices II Stochastic evolution
2010/11/22
We show that the global fluctuations of spectra of GOE and GUE matrices and their principal submatrices executing Dyson's Brownian motion are Gaussian in the limit of large matrix dimensions. For nest...
Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds
Gromov Cauchy causal boundaries Riemannian
2010/11/11
Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary $\partial V$ for a standard conformally stationary spacetime V = ...
Introduction to the non-asymptotic analysis of random matrices
the non-asymptotic analysis of random matrices math
2010/11/19
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices wi...
The Harnack inequality and related properties for solutions to elliptic and parabolic equations with divergence-free lower-order coefficients
The Harnack inequality related properties elliptic
2010/11/15
We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, H...
Using Hook Schur Functions to Compute Matrix Cocharacters
Using Hook Schur Functions Compute Matrix Cocharacters
2010/11/12
We use hook Schur functions to compute trace cocharacters of matrices in hooks. As an example we compute the cocharacters of 3 x 3 matrices in the one by two hook
Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic dist...
We study the Schr\"odinger-Poisson system on the unit sphere $\SS^2$ of $\RR^3$, modeling the quantum transport of charged particles confined on a sphere by an external potential. Our first results c...
Finding approximately rank-one submatrices with the nuclear norm and l1 norm
rank-one submatrices the nuclear norm and l1 norm
2010/11/15
We propose a convex optimization formulation with the nuclear norm and $\ell_1$-norm to find a large approximately rank-one submatrix of a given nonnegative matrix. We develop optimality conditions f...
Short loop decompositions of surfaces and the geometry of Jacobians
Short loop decompositions the geometry of Jacobians
2010/11/19
Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directi...
Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra
coefficients weighted Ehrhart quasi-polynomials rational polyhedra
2010/11/12
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algori...
Five lectures on optimal transportation: Geometry, regularity and applications
optimal transportation Geometry regularity applications
2010/11/19
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Con...
Period Doubling Renormalization for Area-Preserving Maps and Mild Computer Assistance in Contraction Mapping Principle
Period Doubling Renormalization Area-Preserving Maps Mild Computer Assistance
2010/11/29
A universal period doubling cascade analogous to the famous Feigenbaum-Coullet-Tresser period doubling has been observed in area-preserving maps of R2. Existence of the “universal” map with orbits of ...
A refined Agler decomposition and geometric applications
refined Agler decomposition geometric applications
2010/12/13
We prove a refined Agler decomposition for bounded analytic functions on the bidisk and show how it can be used to reprove an interesting result of Guo et al. related to extending
holomorphic functio...
Drawing planar graphs of bounded degree with few slopes
planar graphs bounded degree few slopes
2010/12/1
settle a problem of Dujmović, Eppstein, Suderman, and Wood by showing that there exists
a function f with the property that every planar graph G with maximum degree d admits a drawing
with nonc...
雅虎研究员将圆周率精确到小数点后2千万亿位（图）
雅虎研究员 圆周率 小数点后2千万亿位
2010/9/21
据英国广播公司（BBC）2010年9月17日报道，数学家热衷于计算圆周率的小数位数，将此视为不错的消遣。不久前，日本一名男子刚刚创下将圆周率精确到小数点后5万亿位的新纪录。日前，雅虎公司的一名研究员采用“云计算”技术，将圆周率精确到小数点后2千万亿位。