Spectral Analysis in Geometry and Number Theory

Spectral Analysis in Geometry and Number Theory
Author: Motoko Kotani
Publisher: American Mathematical Soc.
Total Pages: 363
Release: 2009
Genre: Mathematics
ISBN: 0821842692

This volume is an outgrowth of an international conference in honor of Toshikazu Sunada on the occasion of his sixtieth birthday. The conference took place at Nagoya University, Japan, in 2007. Sunada's research covers a wide spectrum of spectral analysis, including interactions among geometry, number theory, dynamical systems, probability theory and mathematical physics. Readers will find papers on trace formulae, isospectral problems, zeta functions, quantum ergodicity, random waves, discrete geometric analysis, value distribution, and semiclassical analysis. This volume also contains an article that presents an overview of Sunada's work in mathematics up to the age of sixty.

Schrödinger Operators, Spectral Analysis and Number Theory

Schrödinger Operators, Spectral Analysis and Number Theory
Author: Sergio Albeverio
Publisher: Springer Nature
Total Pages: 316
Release: 2021-06-03
Genre: Mathematics
ISBN: 3030684903

This book gives its readers a unique opportunity to get acquainted with new aspects of the fruitful interactions between Analysis, Geometry, Quantum Mechanics and Number Theory. The present book contains a number of contributions by specialists in these areas as an homage to the memory of the mathematician Erik Balslev and, at the same time, advancing a fascinating interdisciplinary area still full of potential. Erik Balslev has made original and important contributions to several areas of Mathematics and its applications. He belongs to the founders of complex scaling, one of the most important methods in the mathematical and physical study of eigenvalues and resonances of Schrödinger operators, which has been very essential in advancing the solution of fundamental problems in Quantum Mechanics and related areas. He was also a pioneer in making available and developing spectral methods in the study of important problems in Analytic Number Theory.

Discrete Harmonic Analysis

Discrete Harmonic Analysis
Author: Tullio Ceccherini-Silberstein
Publisher: Cambridge University Press
Total Pages: 589
Release: 2018-06-21
Genre: Mathematics
ISBN: 1107182336

A self-contained introduction to discrete harmonic analysis with an emphasis on the Discrete and Fast Fourier Transforms.

Spectral Analysis on Graph-like Spaces

Spectral Analysis on Graph-like Spaces
Author: Olaf Post
Publisher: Springer Science & Business Media
Total Pages: 444
Release: 2012-01-06
Genre: Mathematics
ISBN: 3642238394

Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.

Number Theory, Fourier Analysis and Geometric Discrepancy

Number Theory, Fourier Analysis and Geometric Discrepancy
Author: Giancarlo Travaglini
Publisher: Cambridge University Press
Total Pages: 251
Release: 2014-06-12
Genre: Mathematics
ISBN: 1139992821

The study of geometric discrepancy, which provides a framework for quantifying the quality of a distribution of a finite set of points, has experienced significant growth in recent decades. This book provides a self-contained course in number theory, Fourier analysis and geometric discrepancy theory, and the relations between them, at the advanced undergraduate or beginning graduate level. It starts as a traditional course in elementary number theory, and introduces the reader to subsequent material on uniform distribution of infinite sequences, and discrepancy of finite sequences. Both modern and classical aspects of the theory are discussed, such as Weyl's criterion, Benford's law, the Koksma–Hlawka inequality, lattice point problems, and irregularities of distribution for convex bodies. Fourier analysis also features prominently, for which the theory is developed in parallel, including topics such as convergence of Fourier series, one-sided trigonometric approximation, the Poisson summation formula, exponential sums, decay of Fourier transforms, and Bessel functions.

Spectral Theory and Analytic Geometry over Non-Archimedean Fields

Spectral Theory and Analytic Geometry over Non-Archimedean Fields
Author: Vladimir G. Berkovich
Publisher: American Mathematical Soc.
Total Pages: 181
Release: 2012-08-02
Genre: Mathematics
ISBN: 0821890204

The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and -adic analysis.

Fractal Geometry, Complex Dimensions and Zeta Functions

Fractal Geometry, Complex Dimensions and Zeta Functions
Author: Michel L. Lapidus
Publisher: Springer Science & Business Media
Total Pages: 583
Release: 2012-09-20
Genre: Mathematics
ISBN: 1461421764

Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.

Numerical Analysis of Spectral Methods

Numerical Analysis of Spectral Methods
Author: David Gottlieb
Publisher: SIAM
Total Pages: 167
Release: 1977-01-01
Genre: Technology & Engineering
ISBN: 0898710235

A unified discussion of the formulation and analysis of special methods of mixed initial boundary-value problems. The focus is on the development of a new mathematical theory that explains why and how well spectral methods work. Included are interesting extensions of the classical numerical analysis.

Spectral Action in Noncommutative Geometry

Spectral Action in Noncommutative Geometry
Author: Michał Eckstein
Publisher: Springer
Total Pages: 165
Release: 2018-12-18
Genre: Science
ISBN: 3319947885

What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry à la Connes, deliberately unveiling the answers to these questions. After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given. Chapter 2 is designed to serve as a toolkit for computations. The third chapter offers an in-depth view into the subtle links between the asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the spectral action under fluctuations by gauge potentials. A subjective list of open problems in the field is spelled out in the fifth Chapter. The book concludes with an appendix including some auxiliary tools from geometry and analysis, along with examples of spectral geometries. The book serves both as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.

A View from the Top

A View from the Top
Author: Alex Iosevich
Publisher: American Mathematical Soc.
Total Pages: 154
Release: 2007
Genre: Mathematics
ISBN: 0821843974

Based on a capstone course that the author taught to upper division undergraduate students with the goal to explain and visualize the connections between different areas of mathematics and the way different subject matters flow from one another, this book is suitable for those with a basic knowledge of high school mathematics.