Construction of Non-Linear Local Quantum Processes: I
About: This article is published in Annals of Mathematics.The article was published on 1970-11-01. It has received 123 citations till now. The article focuses on the topics: Quantum.
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05 Jun 2014TL;DR: This text gives a thorough overview of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry, and includes a "highlight application" such as Arrow's theorem from economics.
Abstract: Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.
867 citations
580 citations
TL;DR: In this paper, the problem of finding the sharp bound of G as an operator from L p(Rp) to L q(Rn) was addressed, and it was shown that the functions that saturate the bound are necessarily Gaussians.
Abstract: A Gaussian integral kernel G(x, y) on Rn x Rn is the exponential of a quadratic form in x and y; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound of G as an operator from L p(Rp) to L q(Rn) and to prove that the L P(Rn) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for 1 q in some special cases. Besides greatly extending previous results in this area, the proof technique is also essentially different from earlier ones. A corollary of these results is a fully multidimensional, multilinear generalization of Young’s inequality.
303 citations
01 Jan 1993
282 citations