Graph Theory and Additive Combinatorics: Exploring Structure and Randomness (1版)

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【簡介】 Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, the structure of set addition, and the sum-product problem. Important combinatorial, graph theoretic, analytic, Fourier, algebraic, and geometric methods are highlighted. Students will appreciate the chapter summaries, many figures and exercises, and freely available lecture videos on MIT OpenCourseWare. Meant as an introduction for students and researchers studying combinatorics, theoretical computer science, analysis, probability, and number theory, the text assumes only basic familiarity with abstract algebra, analysis, and linear algebra. 【目錄】

STOCHAST PARTIAL DIFFERENT EQUATION ADDITIVE GAUSSIAN NOISE

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The stochastic partial differential equations (SPDEs) arise in many applications of the probability theory. This monograph will focus on two particular (and probably the most known) equations: the stochastic heat equation and the stochastic wave equation. The focus is on the relationship between the solutions to the SPDEs and the fractional Brownian motion (and related processes). An important point of the analysis is the study of the asymptotic behavior of the p-variations of the solutions to the heat or wave equations driven by space-time Gaussian noise or by a Gaussian noise with a non-trivial correlation in space. The book is addressed to public with a reasonable background in probability theory. The idea is to keep it self-contained and avoid using of complex techniques. We also chose to insist on the basic properties of the random noise and to detail the construction of the Wiener integration with respect to them. The intention is to present the proofs complete and detailed. Sample Chapter(s) Preface Chapter 1: Gaussian processes Contents: Gaussian Processes and Sheets: Gaussian Processes Fractional and Bifractional Brownian Motion Multiparameter Gaussian Processes Isonormal Processes and Wiener Integral Stochastic Heat and Wave Equations with Additive Gaussian Noise: The Stochastic Heat Equation with Space-Time White Noise The Stochastic Heat Equation with Correlated Noise in Space The Stochastic Wave Equation with Space-Time White Noise Power Variation and Statistical Inference for Solutions to SPDEs: Variations of the Solution to the Stochastic Heat Equation Parameter Estimation for the Stochastic Heat Equation via Power Variations Power Variations and Inference for Stochastic the Wave Equation Readership: Undergraduate, graduate students and researchers working in various areas of probability theory and mathematical statistics.