Singular Traces
Singular Traces

Author: Steven Lord

Publisher: Walter de Gruyter

Published: 2012-12-19

Total Pages: 468

ISBN-13: 311026255X

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This book is the first complete study and monograph dedicated to singular traces. The text mathematically formalises the study of traces in a self contained theory of functional analysis. Extensive notes will treat the historical development. The final section will contain the most complete and concise treatment known of the integration half of Connes' quantum calculus. Singular traces are traces on ideals of compact operators that vanish on the subideal of finite rank operators. Singular traces feature in A. Connes' interpretation of noncommutative residues. Particularly the Dixmier trace,which generalises the restricted Adler-Manin-Wodzicki residue of pseudo-differential operators and plays the role of the residue for a new catalogue of 'geometric' spaces, including Connes-Chamseddine standard models, Yang-Mills action for quantum differential forms, fractals, isospectral deformations, foliations and noncommutative index theory. The theory of singular traces has been studied after Connes' application to non-commutative geometry and physics by various authors. Recent work by Nigel Kalton and the authors has advanced the theory of singular traces.Singular traces can be equated to symmetric functionals of symmetric sequence or function spaces, residues of zeta functions and heat kernel asymptotics, and characterised by Lidksii and Fredholm formulas. The traces and formulas used in noncommutative geometry are now completely understood in this theory, with surprising new mathematical and physical consequences. For mathematical readers the text offers fundamental functional analysis results and, due to Nigel Kalton's contribution, a now complete theory of traces on compact operators. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to Dixmier traces and access to the deeper mathematical features of traces on ideals associated to the harmonic sequence. These features, not known and not discussed in general texts on noncommutative geometry, are undoubtably physical and probe to the fascinating heart of classical limits and quantization.

Theory
Theory

Author: Steven Lord

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2021-07-19

Total Pages: 416

ISBN-13: 3110378051

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This book is the second edition of the first complete study and monograph dedicated to singular traces. The text offers, due to the contributions of Albrecht Pietsch and Nigel Kalton, a complete theory of traces and their spectral properties on ideals of compact operators on a separable Hilbert space. The second edition has been updated on the fundamental approach provided by Albrecht Pietsch. For mathematical physicists and other users of Connes’ noncommutative geometry the text offers a complete reference to traces on weak trace class operators, including Dixmier traces and associated formulas involving residues of spectral zeta functions and asymptotics of partition functions.

The Memory Trace (PLE: Memory)
The Memory Trace (PLE: Memory)

Author: Erich Goldmeier

Publisher: Psychology Press

Published: 2014-05-09

Total Pages: 274

ISBN-13: 1317695402

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There was some agreement about what memory traces were not, but little about what actually did characterize the memory trace. Yet models and theories of memory at the time could not help making implicit and often unrecognized assumptions about the memory trace. Originally published in 1982, this title aimed to strengthen the meagre base on which memory theories rested at the time. It challenges old assumptions and introduces new concepts, foremost the notion of singularity, as they become necessary to understand traces adequately. Some research data of the past was found in need of reinterpretation. The result is a new theory of the memory trace.

Regularised Integrals, Sums and Traces
Regularised Integrals, Sums and Traces

Author: Sylvie Paycha

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 203

ISBN-13: 0821853678

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``Regularization techniques'' is the common name for a variety of methods used to make sense of divergent series, divergent integrals, or traces of linear operators in infinite-dimensional spaces. Such methods are often indispensable in problems of number theory, geometry, quantum field theory, and other areas of mathematics and theoretical physics. However arbitrary and noncanonical they might seem at first glance, regularized sums, integrals, and traces often contain canonical concepts, and the main purpose of this book is to illustrate and explain this. This book provides a unified and self-contained mathematical treatment of various regularization techniques. The author shows how to derive regularized sums, integrals, and traces from certain canonical building blocks of the original divergent object. In the process of putting together these ``building blocks'', one encounters many problems and ambiguities caused by various so-called anomalies, which are investigated and explained in detail. Nevertheless, it turns out that the corresponding canonical sums, integrals, sums, and traces are well behaved, thus making the regularization procedure possible and manageable. This new unified outlook on regularization techniques in various fields of mathematics and in quantum field theory can serve as an introduction for anyone from a beginning mathematician interested in the subject to an experienced physicist who wants to gain a unified outlook on techniques he/she uses on a daily basis.

Noncommutative Geometry and the Standard Model of Elementary Particle Physics
Noncommutative Geometry and the Standard Model of Elementary Particle Physics

Author: Florian Scheck

Publisher: Springer

Published: 2008-01-11

Total Pages: 352

ISBN-13: 3540460829

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The outcome of a close collaboration between mathematicians and mathematical physicists, these lecture notes present the foundations of A. Connes noncommutative geometry as well as its applications in particular to the field of theoretical particle physics. The coherent and systematic approach makes this book useful for experienced researchers and postgraduate students alike.

Advances in Analysis, Probability and Mathematical Physics
Advances in Analysis, Probability and Mathematical Physics

Author: Sergio Albeverio

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 255

ISBN-13: 9401584516

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In 1961 Robinson introduced an entirely new version of the theory of infinitesimals, which he called `Nonstandard analysis'. `Nonstandard' here refers to the nature of new fields of numbers as defined by nonstandard models of the first-order theory of the reals. This system of numbers was closely related to the ring of Schmieden and Laugwitz, developed independently a few years earlier. During the last thirty years the use of nonstandard models in mathematics has taken its rightful place among the various methods employed by mathematicians. The contributions in this volume have been selected to present a panoramic view of the various directions in which nonstandard analysis is advancing, thus serving as a source of inspiration for future research. Papers have been grouped in sections dealing with analysis, topology and topological groups; probability theory; and mathematical physics. This volume can be used as a complementary text to courses in nonstandard analysis, and will be of interest to graduate students and researchers in both pure and applied mathematics and physics.

Advances in Noncommutative Geometry
Advances in Noncommutative Geometry

Author: Ali Chamseddine

Publisher: Springer Nature

Published: 2020-01-13

Total Pages: 753

ISBN-13: 3030295974

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This authoritative volume in honor of Alain Connes, the foremost architect of Noncommutative Geometry, presents the state-of-the art in the subject. The book features an amalgam of invited survey and research papers that will no doubt be accessed, read, and referred to, for several decades to come. The pertinence and potency of new concepts and methods are concretely illustrated in each contribution. Much of the content is a direct outgrowth of the Noncommutative Geometry conference, held March 23–April 7, 2017, in Shanghai, China. The conference covered the latest research and future areas of potential exploration surrounding topology and physics, number theory, as well as index theory and its ramifications in geometry.

Trace Formulas
Trace Formulas

Author: Steven Lord

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2023-04-03

Total Pages: 4197

ISBN-13: 3110700247

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This volume introduces noncommutative integration theory on semifinite von Neumann algebras and the theory of singular traces for symmetric operator spaces. Deeper aspects of the association between measurability, poles and residues of spectral zeta functions, and asymptotics of heat traces are studied. Applications in Connes’ noncommutative geometry that are detailed include integration of quantum differentials, measures on fractals, and Connes’ character formula concerning the Hochschild class of the Chern character.

Quantum and Stochastic Mathematical Physics
Quantum and Stochastic Mathematical Physics

Author: Astrid Hilbert

Publisher: Springer Nature

Published: 2023-04-02

Total Pages: 390

ISBN-13: 3031140311

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Sergio Albeverio gave important contributions to many fields ranging from Physics to Mathematics, while creating new research areas from their interplay. Some of them are presented in this Volume that grew out of the Random Transformations and Invariance in Stochastic Dynamics Workshop held in Verona in 2019. To understand the theory of thermo- and fluid-dynamics, statistical mechanics, quantum mechanics and quantum field theory, Albeverio and his collaborators developed stochastic theories having strong interplays with operator theory and functional analysis. His contribution to the theory of (non Gaussian)-SPDEs, the related theory of (pseudo-)differential operators, and ergodic theory had several impacts to solve problems related, among other topics, to thermo- and fluid dynamics. His scientific works in the theory of interacting particles and its extension to configuration spaces lead, e.g., to the solution of open problems in statistical mechanics and quantum field theory. Together with Raphael Hoegh Krohn he introduced the theory of infinite dimensional Dirichlet forms, which nowadays is used in many different contexts, and new methods in the theory of Feynman path integration. He did not fear to further develop different methods in Mathematics, like, e.g., the theory of non-standard analysis and p-adic numbers.

The Origin and Development of the Bengali Language
The Origin and Development of the Bengali Language

Author: Sunita Kumar Chatterji

Publisher: Taylor & Francis

Published: 2024-05-01

Total Pages: 749

ISBN-13: 104003022X

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First published in 1970, The Origin and Development of the Bengali Language (Vol. 1) is the first systematic and detailed history of a Modern Indo-Aryan Language written by an Indian, and incidentally, as it is comparative in its treatment, taking into consideration facts in other Indo-Aryan speeches, it is an invaluable contribution to the scientific study of the Modern Indo-Aryan languages as a whole. This book will be of interest to students of language, linguistics and South Asian studies.