Fundamental Mathematical Structures of Quantum Theory
Fundamental Mathematical Structures of Quantum Theory

Author: Valter Moretti

Publisher: Springer

Published: 2019-06-20

Total Pages: 345

ISBN-13: 3030183467

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This textbook presents in a concise and self-contained way the advanced fundamental mathematical structures in quantum theory. It is based on lectures prepared for a 6 months course for MSc students. The reader is introduced to the beautiful interconnection between logic, lattice theory, general probability theory, and general spectral theory including the basic theory of von Neumann algebras and of the algebraic formulation, naturally arising in the study of the mathematical machinery of quantum theories. Some general results concerning hidden-variable interpretations of QM such as Gleason's and the Kochen-Specker theorems and the related notions of realism and non-contextuality are carefully discussed. This is done also in relation with the famous Bell (BCHSH) inequality concerning local causality. Written in a didactic style, this book includes many examples and solved exercises. The work is organized as follows. Chapter 1 reviews some elementary facts and properties of quantum systems. Chapter 2 and 3 present the main results of spectral analysis in complex Hilbert spaces. Chapter 4 introduces the point of view of the orthomodular lattices' theory. Quantum theory form this perspective turns out to the probability measure theory on the non-Boolean lattice of elementary observables and Gleason's theorem characterizes all these measures. Chapter 5 deals with some philosophical and interpretative aspects of quantum theory like hidden-variable formulations of QM. The Kochen-Specker theorem and its implications are analyzed also in relation BCHSH inequality, entanglement, realism, locality, and non-contextuality. Chapter 6 focuses on the algebra of observables also in the presence of superselection rules introducing the notion of von Neumann algebra. Chapter 7 offers the idea of (groups of) quantum symmetry, in particular, illustrated in terms of Wigner and Kadison theorems. Chapter 8 deals with the elementary ideas and results of the so called algebraic formulation of quantum theories in terms of both *-algebras and C*-algebras. This book should appeal to a dual readership: on one hand mathematicians that wish to acquire the tools that unlock the physical aspects of quantum theories; on the other physicists eager to solidify their understanding of the mathematical scaffolding of quantum theories.

Discrete Mathematical Structures
Discrete Mathematical Structures

Author: D. S. Malik

Publisher: Course Technology Ptr

Published: 2004

Total Pages: 905

ISBN-13: 9780619212858

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Teaches students the mathematical foundations of computer science, including logic, Boolean algebra, basic graph theory, finite state machines, grammars and algorithms, and helps them understand mathematical reasoning for reading, comprehension and construction of mathematical arguments.

Elementary Overview Of Mathematical Structures, An: Algebra, Topology And Categories
Elementary Overview Of Mathematical Structures, An: Algebra, Topology And Categories

Author: Marco Grandis

Publisher: World Scientific

Published: 2020-08-12

Total Pages: 393

ISBN-13: 9811220336

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'The presentation is modeled on the discursive style of the Bourbaki collective, and the coverage of topics is rich and varied. Grandis has provided a large selection of exercises and has sprinkled orienting comments throughout. For an undergraduate library where strong students seek an overview of a significant portion of mathematics, this would be an excellent acquisition. Summing up: Recommended.'CHOICESince the last century, a large part of Mathematics is concerned with the study of mathematical structures, from groups to fields and vector spaces, from lattices to Boolean algebras, from metric spaces to topological spaces, from topological groups to Banach spaces.More recently, these structured sets and their transformations have been assembled in higher structures, called categories.We want to give a structural overview of these topics, where the basic facts of the different theories are unified through the 'universal properties' that they satisfy, and their particularities stand out, perhaps even more.This book can be used as a textbook for undergraduate studies and for self-study. It can provide students of Mathematics with a unified perspective of subjects which are often kept apart. It is also addressed to students and researchers of disciplines having strong interactions with Mathematics, like Physics and Chemistry, Statistics, Computer Science, Engineering.

Introduction · to Mathematical Structures and · Proofs
Introduction · to Mathematical Structures and · Proofs

Author: Larry Gerstein

Publisher: Springer Science & Business Media

Published: 2013-11-21

Total Pages: 355

ISBN-13: 1468467085

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This is a textbook for a one-term course whose goal is to ease the transition from lower-division calculus courses to upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology, combinatorics, and so on. Without such a "bridge" course, most upper division instructors feel the need to start their courses with the rudiments of logic, set theory, equivalence relations, and other basic mathematical raw materials before getting on with the subject at hand. Students who are new to higher mathematics are often startled to discover that mathematics is a subject of ideas, and not just formulaic rituals, and that they are now expected to understand and create mathematical proofs. Mastery of an assortment of technical tricks may have carried the students through calculus, but it is no longer a guarantee of academic success. Students need experience in working with abstract ideas at a nontrivial level if they are to achieve the sophisticated blend of knowledge, disci pline, and creativity that we call "mathematical maturity. " I don't believe that "theorem-proving" can be taught any more than "question-answering" can be taught. Nevertheless, I have found that it is possible to guide stu dents gently into the process of mathematical proof in such a way that they become comfortable with the experience and begin asking them selves questions that will lead them in the right direction.

Modern Algebra and the Rise of Mathematical Structures
Modern Algebra and the Rise of Mathematical Structures

Author: Leo Corry

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 463

ISBN-13: 3034879172

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This book describes two stages in the historical development of the notion of mathematical structures: first, it traces its rise in the context of algebra from the mid-1800s to 1930, and then considers attempts to formulate elaborate theories after 1930 aimed at elucidating, from a purely mathematical perspective, the precise meaning of this idea.

Mathematical Structures of Natural Intelligence
Mathematical Structures of Natural Intelligence

Author: Yair Neuman

Publisher: Springer

Published: 2017-12-01

Total Pages: 179

ISBN-13: 3319682466

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This book uncovers mathematical structures underlying natural intelligence and applies category theory as a modeling language for understanding human cognition, giving readers new insights into the nature of human thought. In this context, the book explores various topics and questions, such as the human representation of the number system, why our counting ability is different from that which is evident among non-human organisms, and why the idea of zero is so difficult to grasp. The book is organized into three parts: the first introduces the general reason for studying general structures underlying the human mind; the second part introduces category theory as a modeling language and use it for exposing the deep and fascinating structures underlying human cognition; and the third applies the general principles and ideas of the first two parts to reaching a better understanding of challenging aspects of the human mind such as our understanding of the number system, the metaphorical nature of our thinking and the logic of our unconscious dynamics.

An Introduction to Algebraic Structures
An Introduction to Algebraic Structures

Author: Joseph Landin

Publisher: Courier Corporation

Published: 2012-08-29

Total Pages: 275

ISBN-13: 0486150410

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This self-contained text covers sets and numbers, elements of set theory, real numbers, the theory of groups, group isomorphism and homomorphism, theory of rings, and polynomial rings. 1969 edition.

The Logical Structure of Mathematical Physics
The Logical Structure of Mathematical Physics

Author: Joseph D. Sneed

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 325

ISBN-13: 9401030669

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This book is about scientific theories of a particular kind - theories of mathematical physics. Examples of such theories are classical and relativis tic particle mechanics, classical electrodynamics, classical thermodynamics, statistical mechanics, hydrodynamics, and quantum mechanics. Roughly, these are theories in which a certain mathematical structure is employed to make statements about some fragment of the world. Most of the book is simply an elaboration of this rough characterization of theories of mathematical physics. It is argued that each theory of mathematical physics has associated with it a certain characteristic mathematical struc ture. This structure may be used in a variety of ways to make empirical claims about putative applications of the theory. Typically - though not necessarily - the way this structure is used in making such claims requires that certain elements in the structure play essentially different roles. Some playa "theoretical" role; others playa "non-theoretical" role. For example, in classical particle mechanics, mass and force playa theoretical role while position plays a non-theoretical role. Some attention is given to showing how this distinction can be drawn and describing precisely the way in which the theoretical and non-theoretical elements function in the claims of the theory. An attempt is made to say, rather precisely, what a theory of mathematical physics is and how you tell one such theory from anothe- what the identity conditions for these theories are.

Mathematical Structures of Epidemic Systems
Mathematical Structures of Epidemic Systems

Author: Vincenzo Capasso

Publisher: Springer Science & Business Media

Published: 2008-08-06

Total Pages: 291

ISBN-13: 3540565264

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The dynamics of infectious diseases represents one of the oldest and ri- est areas of mathematical biology. From the classical work of Hamer (1906) and Ross (1911) to the spate of more modern developments associated with Anderson and May, Dietz, Hethcote, Castillo-Chavez and others, the subject has grown dramatically both in volume and in importance. Given the pace of development, the subject has become more and more di?use, and the need to provide a framework for organizing the diversity of mathematical approaches has become clear. Enzo Capasso, who has been a major contributor to the mathematical theory, has done that in the present volume, providing a system for organizing and analyzing a wide range of models, depending on the str- ture of the interaction matrix. The ?rst class, the quasi-monotone or positive feedback systems, can be analyzed e?ectively through the use of comparison theorems, that is the theory of order-preserving dynamical systems; the s- ond, the skew-symmetrizable systems, rely on Lyapunov methods. Capasso develops the general mathematical theory, and considers a broad range of - amples that can be treated within one or the other framework. In so doing, he has provided the ?rst steps towards the uni?cation of the subject, and made an invaluable contribution to the Lecture Notes in Biomathematics. Simon A. Levin Princeton, January 1993 Author’s Preface to Second Printing In the Preface to the First Printing of this volume I wrote: \ . .

Discrete Mathematical Structures for Computer Science
Discrete Mathematical Structures for Computer Science

Author: Bernard Kolman

Publisher: Prentice Hall

Published: 1987

Total Pages: 488

ISBN-13:

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This text has been designed as a complete introduction to discrete mathematics, primarily for computer science majors in either a one or two semester course. The topics addressed are of genuine use in computer science, and are presented in a logically coherent fashion. The material has been organized and interrelated to minimize the mass of definitions and the abstraction of some of the theory. For example, relations and directed graphs are treated as two aspects of the same mathematical idea. Whenever possible each new idea uses previously encountered material, and then developed in such a way that it simplifies the more complex ideas that follow.