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Additive Number Theory : Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson / edited by David Chudnovsky, Gregory Chudnovsky
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: New York, NY : Springer Science+Business Media, LLC, 2010Edition: 1Description: Online-Ressource (XI, 361p. 23 illus., 4 illus. in color, digital)ISBN:- 9780387683614
- 1282982540
- 9781282982543
- 512.7
- QA241-247.5
- QA241
Contents:
Summary: Addictive Number Theory -- Sum-Product Theorems and Applications -- Can You Hear the Shape of a Beatty Sequence? -- Variance of Signals and Their Finite Fourier Transforms -- Sparse Sets in Time and Frequency Related to Diophantine Problems and Integrable Systems -- Addition Theorems in Acyclic Semigroups -- Small Sumsets in Free Products of $$\mathbb{Z}/2\mathbb{Z}$$ -- A Combinatorial Approach to Sums of Two Squares and Related Problems -- A Note on Elkin’s Improvement of Behrend’s Construction -- Distinct Matroid Base Weights and Additive Theory -- The Postage Stamp Problem and Essential Subsets in Integer Bases -- A Universal Stein-Tomas Restriction Estimate for Measures in Three Dimensions -- On the Exact Order of Asymptotic Bases and Bases for Finite Cyclic Groups -- The Erd?s–Turán Problem in Infinite Groups -- A Tiling Problem and the Frobenius Number -- Sumsets and the Convex Hull -- Explicit Constructions of Infinite Families of MSTD Sets -- An Inverse Problem in Number Theory and Geometric Group Theory -- Cassels Bases -- Asymptotics of Weighted Lattice Point Counts Inside Dilating Polygons -- Support Bases of Solutions of a Functional Equation Arising From Multiplication of Quantum Integers and the Twin Primes Conjecture -- Exponential Sums and Distinct Points on Arcs -- New Vacca-Type Rational Series for Euler’s Constant ? and Its “Alternating” Analog $$\ln \frac{4}{\pi }$$ -- Mixed Sums of Primes and Other Terms -- Classes of Permutation Polynomials Based on Cyclotomy and an Additive Analogue.Summary: This impressive volume is dedicated to Mel Nathanson, a leading authoritative expert for several decades in the area of combinatorial and additive number theory. Nathanson's numerous results have been widely published in top notch journals and in a number of excellent graduate textbooks (GTM Springer) and reference works. For several decades, Mel Nathanson's seminal ideas and results in combinatorial and additive number theory have influenced graduate students and researchers alike. The invited survey articles in this volume reflect the work of distinguished mathematicians in number theory, and represent a wide range of important topics in current research.PPN: PPN: 1650244711Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Additive Number Theory; Preface; Contents; Addictive Number Theory; A True Story; Remarks on Some of My Articles; References; Sum-Product Theorems and Applications; Introduction; 0 Sum-Product Theorem in Fp; 1 Preliminaries from Additive Combinatorics; 2 Some Tools from Graph Theory: The Balog--Szemerédi--Gowers Theorem; 3 Exponential Sum Estimate; 4 Additive Relations in Multiplicative Groups; 5 Multilinear Exponential Sums; 6 Extensions to 'Almost Groups'; 7 Sum-Product Theorem and Gauss Sums in Arbitrary Finite Fields; 8 The Case of General Polynomial (mod p); 9 The Sum-Product in Zq =Z/qZ
10 Exponential Sums in Finite Commutative Rings11 Euclidean Algorithm in Algebraic Number Fields; 12 Application to QUE; References; Can You Hear the Shape of a Beatty Sequence?; 1 Introduction; 2 Proofs; 2.1 Proof of Theorem 1; 2.2 Proof of Theorem 2; 2.3 Proof of Theorem 3; 2.4 Rasmussen's Approach to Conjecture 1; 2.5 Proof of Theorem 4; 3 Open Questions Concerning Generalized Polynomials; References; Variance of Signals and Their Finite Fourier Transforms; 1 Eigenvalue and Eigenvectors of the Finite Fourier Matrix; 1.1 McClellan Basis; 1.2 Carlitz/Morton Basis
1.3 Dickinson--Steiglitz or Hofstatder Basis2 Discrete Analogs; 3 Theta Function Expressions for the Fourier Eigenvectors; 4 Variational Principles for the Determination of Eigenfunctions of the Discrete Fourier Transform; 5 Discrete Uncertainty Principle; 6 Explicit expressions for the matrix Mw2 in the DiscreteVersion of the Uncertainty Principle; 7 Theta Function Bounds for Minimal Eigenvaluesin the Discrete Uncertainty Principle; 8 Numerical Evaluation of the Minimal Eigenvalues in the Discrete Uncertainty Principle
9 Extensions of the Heisenberg-Weyl Inequality in the Continuous and Discrete Cases9.1 The Dickenson-Steiglitz Basis as Derived from Variational Principles; References; Sparse Sets in Time and Frequency Related to Diophantine Problems and Integrable Systems; The Hilbert Matrix and Related Operators; General Prolate Functions; General Prolate Functions and Commuting Differential Operators; Szego Problem and Concentrated Polynomials; Hilbert Matrix and a Commuting Differential Operator; Szego Problem and Arbitrary Unions of Intervals; Garnier Isomonodromy Deformation Equations
Explicit Expressions for the Non-Linear Darboux TransformDarboux Transformation for m = 3 Case; Generalized Prolate Functions and Another Isomonodromy Problem; Generalized Prolate Matrices; Eigenvalue Problems for Hankel Matrices and Fourth Order Differential Equations; Variational Principles and q: Difference Equations; References; Addition Theorems in Acyclic Semigroups; 1 Introduction; 2 Cayley Graphs on Semigroups; 3 Vosper's Theorem; References; Small Sumsets in Free Products of z/2z; 1 Introduction; 2 The Function kG(r,s); 3 Free products of groups; 4 Proof of m G(r,s) k G(r,s)
5 Optimality
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