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

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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This work can be recommended as an extensive course on p-adic mathematics, treating subjects such as a p-adic theory of probability and stochastic processes; spectral theory of operators in non-Archimedean Hilbert spaces; dynamic systems; p-adic fractal dimension, infinite-dimensional analysis and Feynman integration based on the Albeverio-Hoegh-Kr?hn approach; both linear and nonlinear differential and pseudo-differential equations; complexity of random sequences and a p-adic description of chaos.
Also, the present volume explores the unique concept of using fields of p-adic numbers and their corresponding non-Archimedean analysis, a p-adic solution of paradoxes in the foundations of quantum mechanics, and especially the famous Einstein-Podolsky-Rosen paradox to create an epistemological framework for scientific use.
Audience: This book will be valuable to postgraduate students and researchers with an interest in such diverse disciplines as mathematics, physics, biology and philosophy.

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This work can be recommended as an extensive course on p-adic mathematics, treating subjects such as a p-adic theory of probability and stochastic processes; spectral theory of operators in non-Archimedean Hilbert spaces; dynamic systems; p-adic fractal dimension, infinite-dimensional analysis and Feynman integration based on the Albeverio-Hoegh-Kr?hn approach; both linear and nonlinear differential and pseudo-differential equations; complexity of random sequences and a p-adic description of chaos.
Also, the present volume explores the unique concept of using fields of p-adic numbers and their corresponding non-Archimedean analysis, a p-adic solution of paradoxes in the foundations of quantum mechanics, and especially the famous Einstein-Podolsky-Rosen paradox to create an epistemological framework for scientific use.
Audience: This book will be valuable to postgraduate students and researchers with an interest in such diverse disciplines as mathematics, physics, biology and philosophy.
Content:
Front Matter....Pages i-xvii
Measurements and Numbers....Pages 1-41
Fundamentals....Pages 43-100
Non-Archimedean Analysis....Pages 101-129
The Ultrametric Hilbert Space Description of Quantum Measurements with Finite Precision....Pages 131-167
Non-Kolmogorov Probability Theory....Pages 169-219
Non-Kolmogorov Probability and Quantum Physics....Pages 221-247
Position and Momentum Representations....Pages 249-282
p-adic Dynamical Systems with Applications to Biology and Social Sciences....Pages 283-328
Back Matter....Pages 329-374

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This work can be recommended as an extensive course on p-adic mathematics, treating subjects such as a p-adic theory of probability and stochastic processes; spectral theory of operators in non-Archimedean Hilbert spaces; dynamic systems; p-adic fractal dimension, infinite-dimensional analysis and Feynman integration based on the Albeverio-Hoegh-Kr?hn approach; both linear and nonlinear differential and pseudo-differential equations; complexity of random sequences and a p-adic description of chaos.
Also, the present volume explores the unique concept of using fields of p-adic numbers and their corresponding non-Archimedean analysis, a p-adic solution of paradoxes in the foundations of quantum mechanics, and especially the famous Einstein-Podolsky-Rosen paradox to create an epistemological framework for scientific use.
Audience: This book will be valuable to postgraduate students and researchers with an interest in such diverse disciplines as mathematics, physics, biology and philosophy.
Content:
Front Matter....Pages i-xvii
Measurements and Numbers....Pages 1-41
Fundamentals....Pages 43-100
Non-Archimedean Analysis....Pages 101-129
The Ultrametric Hilbert Space Description of Quantum Measurements with Finite Precision....Pages 131-167
Non-Kolmogorov Probability Theory....Pages 169-219
Non-Kolmogorov Probability and Quantum Physics....Pages 221-247
Position and Momentum Representations....Pages 249-282
p-adic Dynamical Systems with Applications to Biology and Social Sciences....Pages 283-328
Back Matter....Pages 329-374
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