Double roots of random littlewood polynomials

We consider random polynomials whose coefficients are independent and uniform on {-1, 1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^-2) when n+1 is not divisible by 4 and asymptotic to \(1/\sqrt 3 \) otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on {-1, 0, 1} and whose largest atom is strictly less than \(\frac{{8\sqrt 3 }}{{\pi {n^2}}}\). In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^-2) factor and we find the asymptotics of the latter probability.     

Encoding and Processing of Alphanumeric Information by Chemical Mixtures

A novel infochemical device that is based on ¹H NMR readout of chemical information is presented. This chemical encoding system utilizes two measurable parameters of homogeneous mixtures, chemical shift and peak integration, for three different applications: 1) a text‐encoding device that is based on spectral representation of a sequence of symbols, 2) encoding of 21‐digit binary numbers, each represented by an NMR spectrum, and their algebraic manipulations, such as addition and subtraction, and 3) encoding of 21‐digit decimal numbers. The first application enables molecular information storage and encryption. The relative concentration of each component, as measured by the relevant peak integration, can represent a symbol. The second application of this system, in addition to its obvious memory capability, enables mathematical operations. The NMR spectrum of a given mixture represents a 21‐digit binary number where each of the peaks encodes for a specific digit. In any of the input mixtures (numbers) each compound is either present or absent, representing either 1 or 0, respectively. We used the various binary numbers to carry out addition operations by combining two or more solutions (numbers). Subtraction operations were also preformed by digital processing of the information. The third application is the representation of decimal numbers. As before, each of the peaks encodes for a specific digit. In any of the input mixtures each compound is present in one of 10 different relative concentrations, representing the 10 digits of a decimal number.     

Tightness of the recentered maximum of the two‐dimensional discrete Gaussian free field

We consider the maximum of the discrete two‐dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at its mean, is tight, settling a longstanding conjecture. The proof combines a recent observation by Bolthausen, Deuschel, and Zeitouni with elements from Bramson's results on branching Brownian motion and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two‐dimensional torus, are also discussed. © 2011 Wiley Periodicals, Inc.     

Support convergence in the single ring theorem

We study the eigenvalues of non-normal square matrices of the form A _n ?=?U _n T _n V _n with U _n , V _n independent Haar distributed on the unitary group and T _n real diagonal. We show that when the empirical measure of the eigenvalues of T _n converges, and T _n satisfies some technical conditions, all these eigenvalues lie in a single ring.     

Onsager Machlup functionals for non trace class SPDE's

Summary An Onsager Machlup functional limit is derived for a class of SPDE's whose principal part is not trace class. Both nondegenerate and degenerate limits are obtained, and are illustrated by examples. The proof uses FKG type inequalities.The work of this author was partially supported by the Bernstein Fund for the promotion of research at the TechnionThe work of this author was partially supported by the Center for Intelligent Control Systems at MIT under US Army research office grant DAAL03-86-K0171