A continuous variant of the inverse Littlewood-Offord problem for quadratic forms

Inverse Littlewood-offord Problems and the Singularity of Random Symmetric Matrices

Let Mn denote a random symmetric n by n 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 [4], we show that Mn is nonsingular with probability 1 − O(n−C) for any positive constant C. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of inter...

Inverse Littlewood-offord Theorems and the Condition Number of Random Discrete Matrices

Consider a random sum η1v1 + . . . + ηnvn, where η1, . . . , ηn are i.i.d. random signs and v1, . . . , vn are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as P(η1v1+. . .+ηnvn = 0) subject to various hypotheses on the v1, . . . , vn. In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman’s inverse theory) in ...

Optimal Inverse Littlewood-offord Theorems

Let ηi, i = 1, . . . , n be iid Bernoulli random variables, taking values ±1 with probability 1 2 . Given a multiset V of n integers v1, . . . , vn, we define the concentration probability as ρ(V ) := sup x P(v1η1 + . . . vnηn = x). A classical result of Littlewood-Offord and Erdős from the 1940s asserts that, if the vi are non-zero, then ρ(V ) is O(n−1/2). Since then, many researchers have obt...

Erratum for “inverse Littlewood-offord Problem“

Haar Matrix Equations for Solving Time-Variant Linear-Quadratic Optimal Control Problems

‎In this paper‎, ‎Haar wavelets are performed for solving continuous time-variant linear-quadratic optimal control problems‎. ‎Firstly‎, ‎using necessary conditions for optimality‎, ‎the problem is changed into a two-boundary value problem (TBVP)‎. ‎Next‎, ‎Haar wavelets are applied for converting the TBVP‎, ‎as a system of differential equations‎, ‎in to a system of matrix algebraic equations‎...