Download or read book Selberg Zeta and Theta Functions written by Ulrich Bunke and published by De Gruyter Akademie Forschung. This book was released on 1995 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors give a self contained exposition of the theory of Selberg zeta and theta functions for bundles on compact locally symmetric spaces of rank 1. The connection between these functions and the spectrum of certain elliptic differential operators is provided by a version of the Selberg trace formula. The theta function is a regularized trace of the wave group. Originally defined geometrically, the Selberg zeta function has a representation in terms of regularized determinants. This leads to a complete description of its singularities. These results are employed in order to establish a functional equation and further properties of the Ruelle zeta function. A couple of explicit examples is worked out. Additional chapters are devoted to the theta function of Riemannian surfaces with cusps and to alternative descriptions of the singularities of the Selberg zeta function in terms of Lie algebra and group cohomology.
Download or read book Selberg Zeta Functions and Transfer Operators written by Markus Szymon Fraczek and published by Springer. This book was released on 2017-05-11 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a method for evaluating Selberg zeta functions via transfer operators for the full modular group and its congruence subgroups with characters. Studying zeros of Selberg zeta functions for character deformations allows us to access the discrete spectra and resonances of hyperbolic Laplacians under both singular and non-singular perturbations. Areas in which the theory has not yet been sufficiently developed, such as the spectral theory of transfer operators or the singular perturbation theory of hyperbolic Laplacians, will profit from the numerical experiments discussed in this book. Detailed descriptions of numerical approaches to the spectra and eigenfunctions of transfer operators and to computations of Selberg zeta functions will be of value to researchers active in analysis, while those researchers focusing more on numerical aspects will benefit from discussions of the analytic theory, in particular those concerning the transfer operator method and the spectral theory of hyperbolic spaces.
Download or read book Cohomological Theory of Dynamical Zeta Functions written by Andreas Juhl and published by Birkhäuser. This book was released on 2012-12-06 with total page 712 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dynamical zeta functions are associated to dynamical systems with a countable set of periodic orbits. The dynamical zeta functions of the geodesic flow of lo cally symmetric spaces of rank one are known also as the generalized Selberg zeta functions. The present book is concerned with these zeta functions from a cohomological point of view. Originally, the Selberg zeta function appeared in the spectral theory of automorphic forms and were suggested by an analogy between Weil's explicit formula for the Riemann zeta function and Selberg's trace formula ([261]). The purpose of the cohomological theory is to understand the analytical properties of the zeta functions on the basis of suitable analogs of the Lefschetz fixed point formula in which periodic orbits of the geodesic flow take the place of fixed points. This approach is parallel to Weil's idea to analyze the zeta functions of pro jective algebraic varieties over finite fields on the basis of suitable versions of the Lefschetz fixed point formula. The Lefschetz formula formalism shows that the divisors of the rational Hassc-Wcil zeta functions are determined by the spectra of Frobenius operators on l-adic cohomology.
Download or read book Spectral Theory of Infinite Area Hyperbolic Surfaces written by David Borthwick and published by Birkhäuser. This book was released on 2016-07-12 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added. Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution. The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields. Review of the first edition: "The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)
Download or read book Zeta Functions over Zeros of Zeta Functions written by André Voros and published by Springer Science & Business Media. This book was released on 2009-11-21 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this text, the famous zeros of the Riemann zeta function and its generalizations (L-functions, Dedekind and Selberg zeta functions)are analyzed through several zeta functions built over those zeros.
Download or read book Theta Functions Bowdoin 1987 Part 2 written by Leon Ehrenpreis and published by American Mathematical Soc.. This book was released on 1989 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Theta Functions Bowdoin 1987 written by Leon Ehrenpreis and published by American Mathematical Soc.. This book was released on 1989 with total page 730 pages. Available in PDF, EPUB and Kindle. Book excerpt: During his long and productive career, Salomon Bochner worked in a variety of different areas of mathematics. This four part set brings together his collected papers, illustrating the range and depth of his mathematical interests. The books are available either individually or as a set.
Download or read book Dynamical Spectral and Arithmetic Zeta Functions written by Michel Laurent Lapidus and published by American Mathematical Soc.. This book was released on 2001 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: The original zeta function was studied by Riemann as part of his investigation of the distribution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the development of $L$-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results. This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeting of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analogies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function. Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a comprehensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.
Download or read book Algebraic Groups written by Yuri Tschinkel and published by Universitätsverlag Göttingen. This book was released on 2007 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Selberg s Zeta L and Eisensteinseries written by U. Christian and published by Springer. This book was released on 2006-11-15 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Ergodic Theory Analysis and Efficient Simulation of Dynamical Systems written by Bernold Fiedler and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 816 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presenting very recent results in a major research area, this book is addressed to experts and non-experts in the mathematical community alike. The applied issues range from crystallization and dendrite growth to quantum chaos, conveying their significance far into the neighboring disciplines of science.
Download or read book Some Musings on Theta Eta and Zeta written by Floyd L. Williams and published by Springer Nature. This book was released on with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Zeta Functions of Graphs written by Audrey Terras and published by Cambridge University Press. This book was released on 2010-11-18 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.
Download or read book Path Integrals Hyperbolic Spaces and Selberg Trace Formulae written by Christian Grosche and published by World Scientific. This book was released on 2013 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this second edition, a comprehensive review is given for path integration in two- and three-dimensional (homogeneous) spaces of constant and non-constant curvature, including an enumeration of all the corresponding coordinate systems which allow separation of variables in the Hamiltonian and in the path integral. The corresponding path integral solutions are presented as a tabulation. Proposals concerning interbasis expansions for spheroidal coordinate systems are also given. In particular, the cases of non-constant curvature Darboux spaces are new in this edition. The volume also contains results on the numerical study of the properties of several integrable billiard systems in compact domains (i.e. rectangles, parallelepipeds, circles and spheres) in two- and three-dimensional flat and hyperbolic spaces. In particular, the discussions of integrable billiards in circles and spheres (flat and hyperbolic spaces) and in three dimensions are new in comparison to the first edition. In addition, an overview is presented on some recent achievements in the theory of the Selberg trace formula on Riemann surfaces, its super generalization, their use in mathematical physics and string theory, and some further results derived from the Selberg (super-) trace formula.
Download or read book Compactifications of Symmetric and Locally Symmetric Spaces written by Armand Borel and published by Springer Science & Business Media. This book was released on 2006-07-25 with total page 477 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces uniform constructions of most of the known compactifications of symmetric and locally symmetric spaces, with emphasis on their geometric and topological structures Relatively self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to analysis, number theory, algebraic geometry and algebraic topology
Download or read book Advances on Superelliptic Curves and Their Applications written by L. Beshaj and published by IOS Press. This book was released on 2015-07-16 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book had its origins in the NATO Advanced Study Institute (ASI) held in Ohrid, Macedonia, in 2014. The focus of this ASI was the arithmetic of superelliptic curves and their application in different scientific areas, including whether all the applications of hyperelliptic curves, such as cryptography, mathematical physics, quantum computation and diophantine geometry, can be carried over to the superelliptic curves. Additional papers have been added which provide some background for readers who were not at the conference, with the intention of making the book logically more complete and easier to read, but familiarity with the basic facts of algebraic geometry, commutative algebra and number theory are assumed. The book is divided into three sections. The first part deals with superelliptic curves with regard to complex numbers, the automorphisms group and the corresponding Hurwitz loci. The second part of the book focuses on the arithmetic of the subject, while the third addresses some of the applications of superelliptic curves.
Download or read book Explicit Formulas written by Jay Jorgenson and published by Springer. This book was released on 2006-11-14 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of explicit formulas for regularized products and series forms a natural continuation of the analytic theory developed in LNM 1564. These explicit formulas can be used to describe the quantitative behavior of various objects in analytic number theory and spectral theory. The present book deals with other applications arising from Gaussian test functions, leading to theta inversion formulas and corresponding new types of zeta functions which are Gaussian transforms of theta series rather than Mellin transforms, and satisfy additive functional equations. Their wide range of applications includes the spectral theory of a broad class of manifolds and also the theory of zeta functions in number theory and representation theory. Here the hyperbolic 3-manifolds are given as a significant example.
