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A SHARP INVERSE LITTLEWOODOFFORD THEOREM
, 2009
"... Let ηi, i = 1,...,n be iid Bernoulli random variables. Given a multiset v of n numbers v1,..., vn, the concentration probability P1(v) of v is defined as P1(v): = sup x P(v1η1 +...vnηn = x). A classical result of LittlewoodOfford and Erdős from the 1940s asserts that if the vi are nonzero, then t ..."
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Cited by 12 (3 self)
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Let ηi, i = 1,...,n be iid Bernoulli random variables. Given a multiset v of n numbers v1,..., vn, the concentration probability P1(v) of v is defined as P1(v): = sup x P(v1η1 +...vnηn = x). A classical result of LittlewoodOfford and Erdős from the 1940s asserts that if the vi are non
BILINEAR AND QUADRATIC VARIANTS ON THE LITTLEWOODOFFORD PROBLEM
, 2009
"... If f(x1,..., xn) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical LittlewoodOfford problem: Given nonzero constants ..."
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Cited by 11 (0 self)
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If f(x1,..., xn) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical LittlewoodOfford problem: Given nonzero
From the LittlewoodOfford problem to the Circular Law: Universality of the spectral distribution of random matrices
 BULL. AMER. MATH. SOC
, 2009
"... The famous circular law asserts that if Mn is an n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix 1 √ Mn converges both in probability and n almost surely to the uniform distribution on the unit disk {z ∈ C: z  ≤1 ..."
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Cited by 53 (7 self)
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in the establishment of the circular law at this level of generality, in particular recent advances in understanding the LittlewoodOfford problem and its inverse.
The littlewoodofford problem and invertibility of random matrices
 Adv. Math
"... Abstract. We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n −1/2, which is optimal for Gaussian matrices. Moreover, we give a opti ..."
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Cited by 104 (18 self)
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optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the LittlewoodOfford problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum � k akXk lies near some number v. For arbitrary coefficients ak
LittlewoodOfford inequalities for random variables
 11] Nathan Linial, Avner Magen, and Michael
, 1994
"... The concentration of a realvalued random variable X is c(X) sup P(t < X < + 1). Given bounds on the concentrations of n independent random variables, how large can the concentration of their sum be? The main aim of this paper is to give a best possible upper bound for the concentration of the ..."
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Cited by 4 (0 self)
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of the sum of n independent random variables, each of concentration at most 1/k, where k is an integer. Other bounds on the concentration are also discussed, as well as the case of vectorvalued random variables. Key words. LittlewoodOfford problem, concentration, normed spaces AMS subject classifications.
Inverse Littlewood–Offord theorems and the condition number of random discrete matrices
, 2009
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An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
, 2008
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Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition
, 1998
"... Abstract. This paper is a survey of the inverse scattering problem for timeharmonic acoustic and electromagnetic waves at fixed frequency. We begin by a discussion of “weak scattering ” and Newtontype methods for solving the inverse scattering problem for acoustic waves, including a brief discussi ..."
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Cited by 1072 (45 self)
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discussion of Tikhonov’s method for the numerical solution of illposed problems. We then proceed to prove a uniqueness theorem for the inverse obstacle problems for acoustic waves and the linear sampling method for reconstructing the shape of a scattering obstacle from far field data. Included in our
Results 1  10
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