Andrei Khrennikov's Books

Quantum Social Science

By: Emmanuel Haven,Andrei Khrennikov

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Written by world experts in the foundations of quantum mechanics and its applications to social science, this book shows how elementary quantum mechanical principles can be applied to decision-making paradoxes in psychology and used in modelling information in finance and economics. The book starts with a thorough overview of some of the salient differences between classical, statistical and quantum mechanics. It presents arguments on why quantum mechanics can be applied outside of physics and defines quantum social science. The issue of the existence of quantum probabilistic effects in psychology, economics and finance is addressed and basic questions and answers are provided. Aimed at researchers in economics and psychology, as well as physics, basic mathematical preliminaries and elementary concepts from quantum mechanics are defined in a self-contained way.

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305

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Cambridge University Press

Published Date

ISBN 10

1139851497

ISBN 13

9781139851497

Open Quantum Systems in Biology, Cognitive and Social Sciences

By: Andrei Y. Khrennikov

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Open Quantum Systems in Biology, Cognitive and Social Sciences

By: Andrei Y. Khrennikov

This book mathematically analyzes the basic problems of biology, decision making and psychology within the framework of the theory of open quantum systems. In recent years there has been an explosion of interest in applications of quantum theory in fields beyond physics. The main areas include psychology, decision-making, economics, finance, social science as well as genetics and molecular biology. The corresponding models are referred to as quantum-like; they don’t concern any genuine physical processes in the human brain. Quantum-like models reflect the special features of information processing in biological, cognitive, and social systems which match well with the quantum formalism. This formalism gives rise to the quantum probability model (QP) which differs essentially from Kolmogorov's classical probability model. QP also serves as the basis for quantum information theory. Recently QP has been widely applied to the resolution of the basic paradoxes of decision making theory and to modeling experimental data stemming from cognition, psychology, economics, and finance thereby shedding light on probability fallacies and irrational behavior. In this book, the theory of quantum instruments and the quantum master equation are applied to the modeling of biological and cognitive processes, in particular, to the stability of complex biological and social systems interacting with their environment. An essential part of the book is devoted to the theory of the social laser and the Fröhlich condensate.

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371

Publisher

Springer Nature

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ISBN 10

3031290240

ISBN 13

9783031290244

p-Adic Valued Distributions in Mathematical Physics

By: Andrei Y. Khrennikov

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p-Adic Valued Distributions in Mathematical Physics

By: Andrei Y. Khrennikov

Numbers ... , natural, rational, real, complex, p-adic .... What do you know about p-adic numbers? Probably, you have never used any p-adic (nonrational) number before now. I was in the same situation few years ago. p-adic numbers were considered as an exotic part of pure mathematics without any application. I have also used only real and complex numbers in my investigations in functional analysis and its applications to the quantum field theory and I was sure that these number fields can be a basis of every physical model generated by nature. But recently new models of the quantum physics were proposed on the basis of p-adic numbers field Qp. What are p-adic numbers, p-adic analysis, p-adic physics, p-adic probability? p-adic numbers were introduced by K. Hensel (1904) in connection with problems of the pure theory of numbers. The construction of Qp is very similar to the construction of (p is a fixed prime number, p = 2,3,5, ... ,127, ... ). Both these number fields are completions of the field of rational numbers Q. But another valuation 1 . Ip is introduced on Q instead of the usual real valuation 1 . I· We get an infinite sequence of non isomorphic completions of Q : Q2, Q3, ... , Q127, ... , IR = Qoo· These fields are the only possibilities to com plete Q according to the famous theorem of Ostrowsky.

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271

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Springer Science & Business Media

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ISBN 10

9401583560

ISBN 13

9789401583565

Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena

By: Andrei Y. Khrennikov

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Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena

By: Andrei Y. Khrennikov

In this book we develop various mathematical models of information dynamics, I -dynamics (including the process of thinking), based on methods of classical and quantum physics. The main aim of our investigations is to describe mathematically the phenomenon of consciousness. We would like to realize a kind of Newton-Descartes program (corrected by the lessons of statistical and quantum mechanics) for information processes. Starting from the ideas of Newton and Descartes, in physics there was developed an adequate description of the dynamics of material systems. We would like to develop an analogous mathematical formalism for information and, in particular, mental processes. At the beginning of the 21st century it is clear that it would be impossible to create a deterministic model for general information processes. A deterministic model has to be completed by a corresponding statistical model of information flows and, in particular, flows of minds. It might be that such an information statistical model should have a quantum-like structure.

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250

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Springer Science & Business Media

Published Date

ISBN 10

9401704791

ISBN 13

9789401704793

Ultrametric Pseudodifferential Equations and Applications

By: Andrei Yu. Khrennikov,Sergei V. Kozyrev,W. A. Zúñiga-Galindo

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Ultrametric Pseudodifferential Equations and Applications

By: Andrei Yu. Khrennikov,Sergei V. Kozyrev,W. A. Zúñiga-Galindo

Starting from physical motivations and leading to practical applications, this book provides an interdisciplinary perspective on the cutting edge of ultrametric pseudodifferential equations. It shows the ways in which these equations link different fields including mathematics, engineering, and geophysics. In particular, the authors provide a detailed explanation of the geophysical applications of p-adic diffusion equations, useful when modeling the flows of liquids through porous rock. p-adic wavelets theory and p-adic pseudodifferential equations are also presented, along with their connections to mathematical physics, representation theory, the physics of disordered systems, probability, number theory, and p-adic dynamical systems. Material that was previously spread across many articles in journals of many different fields is brought together here, including recent work on the van der Put series technique. This book provides an excellent snapshot of the fascinating field of ultrametric pseudodifferential equations, including their emerging applications and currently unsolved problems.

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255

Publisher

Cambridge University Press

Published Date

ISBN 10

1108100104

ISBN 13

9781108100106

P-adic Deterministic and Random Dynamics

By: Andrei Y. Khrennikov,Marcus Nilsson

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P-adic Deterministic and Random Dynamics

By: Andrei Y. Khrennikov,Marcus Nilsson

This book provides an overview of the theory of p-adic (and more general non-Archimedean) dynamical systems. The main part of the book is devoted to discrete dynamical systems. It presents a model of probabilistic thinking on p-adic mental space based on ultrametric diffusion. Coverage also details p-adic neural networks and their applications to cognitive sciences: learning algorithms, memory recalling.

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279

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Springer Science & Business Media

Published Date

ISBN 10

1402026609

ISBN 13

9781402026607

Superanalysis

By: Andrei Y. Khrennikov

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Superanalysis

By: Andrei Y. Khrennikov

defined as elements of Grassmann algebra (an algebra with anticom muting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used. Later, during the next twenty years, the algebraic apparatus de veloped by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G 1. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticom muting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanaly sis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" == maps of sets and not as elements of Grassmann algebras. In 1974, Salam and Strathdee proposed a very apt name for a set of super points. They called this set a superspace.

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359

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Springer Science & Business Media

Published Date

ISBN 10

9401146098

ISBN 13

9789401146098

Applied Algebraic Dynamics

By: Vladimir Anashin,Andrei Khrennikov

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Applied Algebraic Dynamics

By: Vladimir Anashin,Andrei Khrennikov

This monograph presents recent developments of the theory of algebraic dynamical systems and their applications to computer sciences, cryptography, cognitive sciences, psychology, image analysis, and numerical simulations. The most important mathematical results presented in this book are in the fields of ergodicity, p-adic numbers, and noncommutative groups. For students and researchers working on the theory of dynamical systems, algebra, number theory, measure theory, computer sciences, cryptography, and image analysis.

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558

Publisher

Walter de Gruyter

Published Date

ISBN 10

3110203014

ISBN 13

9783110203011

Quantum Adaptivity in Biology: From Genetics to Cognition

By: Masanari Asano,Andrei Khrennikov,Masanori Ohya,Yoshiharu Tanaka,Ichiro Yamato

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Quantum Adaptivity in Biology: From Genetics to Cognition

By: Masanari Asano,Andrei Khrennikov,Masanori Ohya,Yoshiharu Tanaka,Ichiro Yamato

This book examines information processing performed by bio-systems at all scales: from genomes, cells and proteins to cognitive and even social systems. It introduces a theoretical/conceptual principle based on quantum information and non-Kolmogorov probability theory to explain information processing phenomena in biology as a whole. The book begins with an introduction followed by two chapters devoted to fundamentals, one covering classical and quantum probability, which also contains a brief introduction to quantum formalism, and another on an information approach to molecular biology, genetics and epigenetics. It then goes on to examine adaptive dynamics, including applications to biology, and non-Kolmogorov probability theory. Next, the book discusses the possibility to apply the quantum formalism to model biological evolution, especially at the cellular level: genetic and epigenetic evolutions. It also presents a model of the epigenetic cellular evolution based on the mathematical formalism of open quantum systems. The last two chapters of the book explore foundational problems of quantum mechanics and demonstrate the power of usage of positive operator valued measures (POVMs) in biological science. This book will appeal to a diverse group of readers including experts in biology, cognitive science, decision making, sociology, psychology, and physics; mathematicians working on problems of quantum probability and information and researchers in quantum foundations.

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185

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Springer

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ISBN 10

9401798192

ISBN 13

9789401798198

Ubiquitous Quantum Structure

From Psychology to Finance

By: Andrei Y. Khrennikov

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Ubiquitous Quantum Structure

From Psychology to Finance

By: Andrei Y. Khrennikov

Quantum-like structure is present practically everywhere. Quantum-like (QL) models, i.e. models based on the mathematical formalism of quantum mechanics and its generalizations can be successfully applied to cognitive science, psychology, genetics, economics, finances, and game theory. This book is not about quantum mechanics as a physical theory. The short review of quantum postulates is therefore mainly of historical value: quantum mechanics is just the first example of the successful application of non-Kolmogorov probabilities, the first step towards a contextual probabilistic description of natural, biological, psychological, social, economical or financial phenomena. A general contextual probabilistic model (Växjö model) is presented. It can be used for describing probabilities in both quantum and classical (statistical) mechanics as well as in the above mentioned phenomena. This model can be represented in a quantum-like way, namely, in complex and more general Hilbert spaces. In this way quantum probability is totally demystified: Born's representation of quantum probabilities by complex probability amplitudes, wave functions, is simply a special representation of this type.

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226

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Springer Science & Business Media

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ISBN 10

3642051014

ISBN 13

9783642051012

Ultrametric Pseudodifferential Equations and Applications

By: Andrei Yu. Khrennikov,Sergei V. Kozyrev,W. A. Zúñiga-Galindo

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Ultrametric Pseudodifferential Equations and Applications

By: Andrei Yu. Khrennikov,Sergei V. Kozyrev,W. A. Zúñiga-Galindo

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256

Publisher

Cambridge University Press

Published Date

ISBN 10

1108102905

ISBN 13

9781108102902

Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models

By: Andrei Y. Khrennikov

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Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models

By: Andrei Y. Khrennikov

N atur non facit saltus? This book is devoted to the fundamental problem which arises contin uously in the process of the human investigation of reality: the role of a mathematical apparatus in a description of reality. We pay our main attention to the role of number systems which are used, or may be used, in this process. We shall show that the picture of reality based on the standard (since the works of Galileo and Newton) methods of real analysis is not the unique possible way of presenting reality in a human brain. There exist other pictures of reality where other num ber fields are used as basic elements of a mathematical description. In this book we try to build a p-adic picture of reality based on the fields of p-adic numbers Qp and corresponding analysis (a particular case of so called non-Archimedean analysis). However, this book must not be considered as only a book on p-adic analysis and its applications. We study a much more extended range of problems. Our philosophical and physical ideas can be realized in other mathematical frameworks which are not obliged to be based on p-adic analysis. We shall show that many problems of the description of reality with the aid of real numbers are induced by unlimited applications of the so called Archimedean axiom.

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386

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Springer Science & Business Media

Published Date

ISBN 10

9400914830

ISBN 13

9789400914834

Probability and Randomness

Quantum Versus Classical

By: Andre? I?U?r?evich Khrennikov

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Probability and Randomness

Quantum Versus Classical

By: Andre? I?U?r?evich Khrennikov

Creating a rigorous mathematical theory of randomness is far from being complete, even in the classical case. Interrelation of Classical and Quantum Randomness rectifies this and introduces mathematical formalisms of classical and quantum probability and randomness with brief discussion of their interrelation and interpretational and foundational issues. The book presents the essentials of classical approaches to randomness, enlightens their successes and problems, and then proceeds to essentials of quantum randomness. Its wide-ranging and comprehensive scope makes it suitable for researchers in mathematical physics, probability and statistics at any level"--

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299

Publisher

World Scientific

Published Date

ISBN 10

1783267976

ISBN 13

9781783267972

Probability and Randomness

Quantum Versus Classical

By: Andrei Khrennikov

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Probability and Randomness

Quantum Versus Classical

By: Andrei Khrennikov

Foundations of probability -- Randomness -- Supplementary notes on measure-theoretic and frequency -- Approaches -- Introduction to quantum formalism -- Quantum and contextual probability -- Interpretations of quantum mechanics and probability -- Randomness: quantum versus classical -- Probabilistic structure of Bell's argument -- Quantum probability outside of physics: from molecular biology to cognition

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282

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ISBN 10

1783267968

ISBN 13

9781783267965

Quantum Methods In Social Science: A First Course

By: Emmanuel Haven,Andrei Yu Khrennikov,Terry R Robinson

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Quantum Methods In Social Science: A First Course

By: Emmanuel Haven,Andrei Yu Khrennikov,Terry R Robinson

Shown here is how basic concepts of physics can be used to improve models in finance, economics, psychology and biology. Readers are introduced to how physical theory can inform non-physical phenomena in the social sciences, thereby improving decision making and modeling capabilities in research-based and professional settings.Consisting of three parts, the first part deals with the application of quantum operator methods to financial transactions and population dynamics. Part two develops physical concepts, working from classical Lagrangian and Hamiltonian mechanics and leading to an introduction of quantum information and its application to decision making. The final part treats classical and quantum probability theory in some detail and deals, at a more advanced level, with the impact of quantum probabilities on common knowledge and common beliefs between agents in systems.Quantum Methods in Social Science is a high level textbook for advanced undergraduate or graduate students of economics, finance and business, while also being of interest to those with a background in physics.

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