68,881 research outputs found
A continuous variant of the inverse Littlewood-Offord problem for quadratic forms
Motivated by the inverse Littlewood-Offord problem for linear forms, we study
the concentration of quadratic forms. We show that if this form concentrates on
a small ball with high probability, then the coefficients can be approximated
by a sum of additive and algebraic structures.Comment: 17 pages. This is the first part of http://arxiv.org/abs/1101.307
Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices
Let denote a random symmetric by matrix, whose upper diagonal
entries are iid Bernoulli random variables (which take value -1 and 1 with
probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show
that is non-singular with probability for any positive
constant . The proof uses an inverse Littlewood-Offord result for quadratic
forms, which is of interest of its own.Comment: Some minor corrections in Section 10 of v
Random doubly stochastic matrices: The circular law
Let be a matrix sampled uniformly from the set of doubly stochastic
matrices of size . We show that the empirical spectral distribution
of the normalized matrix converges almost surely
to the circular law. This confirms a conjecture of Chatterjee, Diaconis and
Sly.Comment: Published in at http://dx.doi.org/10.1214/13-AOP877 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An isogeometric analysis for elliptic homogenization problems
A novel and efficient approach which is based on the framework of
isogeometric analysis for elliptic homogenization problems is proposed. These
problems possess highly oscillating coefficients leading to extremely high
computational expenses while using traditional finite element methods. The
isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in
this paper is regarded as an alternative approach to the standard Finite
Element Heterogeneous Multiscale Method (FE-HMM) which is currently an
effective framework to solve these problems. The method utilizes non-uniform
rational B-splines (NURBS) in both macro and micro levels instead of standard
Lagrange basis. Beside the ability to describe exactly the geometry, it
tremendously facilitates high-order macroscopic/microscopic discretizations
thanks to the flexibility of refinement and degree elevation with an arbitrary
continuity level provided by NURBS basis functions. A priori error estimates of
the discretization error coming from macro and micro meshes and optimal micro
refinement strategies for macro/micro NURBS basis functions of arbitrary orders
are derived. Numerical results show the excellent performance of the proposed
method
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