Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction
Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

Author: Alberto Parmeggiani

Publisher: Springer Science & Business Media

Published: 2010-04-22

Total Pages: 260

ISBN-13: 3642119212

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This volume describes the spectral theory of the Weyl quantization of systems of polynomials in phase-space variables, modelled after the harmonic oscillator. The main technique used is pseudodifferential calculus, including global and semiclassical variants. The main results concern the meromorphic continuation of the spectral zeta function associated with the spectrum, and the localization (and the multiplicity) of the eigenvalues of such systems, described in terms of “classical” invariants (such as the periods of the periodic trajectories of the bicharacteristic flow associated with the eiganvalues of the symbol). The book utilizes techniques that are very powerful and flexible and presents an approach that could also be used for a variety of other problems. It also features expositions on different results throughout the literature.

International Symposium on Mathematics, Quantum Theory, and Cryptography
International Symposium on Mathematics, Quantum Theory, and Cryptography

Author: Tsuyoshi Takagi

Publisher: Springer Nature

Published: 2020-10-22

Total Pages: 275

ISBN-13: 981155191X

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This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan. The international symposium MQC addresses the mathematics and quantum theory underlying secure modeling of the post quantum cryptography including e.g. mathematical study of the light-matter interaction models as well as quantum computing. The security of the most widely used RSA cryptosystem is based on the difficulty of factoring large integers. However, in 1994 Shor proposed a quantum polynomial time algorithm for factoring integers, and the RSA cryptosystem is no longer secure in the quantum computing model. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. This book is suitable for postgraduate students in mathematics and computer science, as well as for experts in industry working on post-quantum cryptography.

Noncommutative Geometry and Physics 3
Noncommutative Geometry and Physics 3

Author: Giuseppe Dito

Publisher: World Scientific

Published: 2013

Total Pages: 537

ISBN-13: 981442501X

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Noncommutative differential geometry has many actual and potential applications to several domains in physics ranging from solid state to quantization of gravity. The strategy is to formulate usual differential geometry in a somewhat unusual manner, using in particular operator algebras and related concepts, so as to be able to plug in noncommutativity in a natural way. Algebraic tools such as K-theory and cyclic cohomology and homology play an important role in this field.

Casimir Force, Casimir Operators and the Riemann Hypothesis
Casimir Force, Casimir Operators and the Riemann Hypothesis

Author: Gerrit Dijk

Publisher: Walter de Gruyter

Published: 2010-10-19

Total Pages: 295

ISBN-13: 3110226138

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This volume contains the proceedings of the conference "Casimir Force, Casimir Operators and the Riemann Hypothesis – Mathematics for Innovation in Industry and Science" held in November 2009 in Fukuoka (Japan). The motive for the conference was the celebration of the 100th birthday of Casimir and the 150th birthday of the Riemann hypothesis. The conference focused on the following topics: Casimir operators in harmonic analysis and representation theory Number theory, in particular zeta functions and cryptography Casimir force in physics and its relation with nano-science Mathematical biology Importance of mathematics for innovation in industry The latter topic was inspired both by the call for innovation in industry worldwide and by the fact that Casimir, who was the director of Philips research for a long time in his career, had an outspoken opinion on the importance of fundamental science for industry. These proceedings are of interest both to research mathematicians and to those interested in the role science, and in particular mathematics, can play in innovation in industry.

Mathematical Modelling for Next-Generation Cryptography
Mathematical Modelling for Next-Generation Cryptography

Author: Tsuyoshi Takagi

Publisher: Springer

Published: 2017-07-25

Total Pages: 363

ISBN-13: 9811050651

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This book presents the mathematical background underlying security modeling in the context of next-generation cryptography. By introducing new mathematical results in order to strengthen information security, while simultaneously presenting fresh insights and developing the respective areas of mathematics, it is the first-ever book to focus on areas that have not yet been fully exploited for cryptographic applications such as representation theory and mathematical physics, among others. Recent advances in cryptanalysis, brought about in particular by quantum computation and physical attacks on cryptographic devices, such as side-channel analysis or power analysis, have revealed the growing security risks for state-of-the-art cryptographic schemes. To address these risks, high-performance, next-generation cryptosystems must be studied, which requires the further development of the mathematical background of modern cryptography. More specifically, in order to avoid the security risks posed by adversaries with advanced attack capabilities, cryptosystems must be upgraded, which in turn relies on a wide range of mathematical theories. This book is suitable for use in an advanced graduate course in mathematical cryptography, while also offering a valuable reference guide for experts.

Topics in Algebraic and Topological K-Theory
Topics in Algebraic and Topological K-Theory

Author: Paul Frank Baum

Publisher: Springer

Published: 2010-10-28

Total Pages: 322

ISBN-13: 3642157084

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This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.

Blow-up Theories for Semilinear Parabolic Equations
Blow-up Theories for Semilinear Parabolic Equations

Author: Bei Hu

Publisher: Springer

Published: 2011-03-17

Total Pages: 137

ISBN-13: 364218460X

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There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while seeking to avoid massive technical computations. To reach this goal, we use the simplest equation to illustrate the methods; these methods very often apply to more general equations.

Lévy Matters I
Lévy Matters I

Author: Thomas Duquesne

Publisher: Springer Science & Business Media

Published: 2010-09-05

Total Pages: 216

ISBN-13: 3642140068

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Focusing on the breadth of the topic, this volume explores Lévy processes and applications, and presents the state-of-the-art in this evolving area of study. These expository articles help to disseminate important theoretical and applied research to those studying the field.

Arithmetic Geometry
Arithmetic Geometry

Author: Jean-Louis Colliot-Thélène

Publisher: Springer

Published: 2010-10-27

Total Pages: 251

ISBN-13: 3642159451

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Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties through arbitrary rings, in particular through non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory. A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry. This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties. The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thelene, Peter Swinnerton Dyer and Paul Vojta.