Maximum Entropy, Information Without Probability and Complex Fractals

1. Front Matter

   Pages I-XIX

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2. Introduction

   • Guy Jumarie

   Pages 1-8

3. Summary of Information Theory

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   Pages 9-28

4. Path Entropies of Non-Random Functions

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   Pages 29-59

5. Path Entropies of Random Functions and of Non-Random Distributed Functions

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   Pages 60-82

6. Quantum Entropies of Non-Probabilistic Square Matrices

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   Pages 83-128

7. Complex-Valued Fractional Brownian Motion of Order n. Part I

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   Pages 129-155

8. Complex-Valued Fractional Brownian Motion of Order n. Part II

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   Pages 156-181

9. Information Thermodynamics and Complex-Valued Fractional Brownian Motion of Order n

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   Pages 182-204

10. Fractals, Path Entropy, and Fractional Fokker-Planck Equation

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    Pages 205-229

11. Outline of Applications

    • Guy Jumarie

    Pages 230-265

12. Back Matter

    Pages 267-270

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Every thought is a throw of dice. Stephane Mallarme This book is the last one of a trilogy which reports a part of our research work over nearly thirty years (we discard our non-conventional results in automatic control theory and applications on the one hand, and fuzzy sets on the other), and its main key words are Information Theory, Entropy, Maximum Entropy Principle, Linguistics, Thermodynamics, Quantum Mechanics, Fractals, Fractional Brownian Motion, Stochastic Differential Equations of Order n, Stochastic Optimal Control, Computer Vision. Our obsession has been always the same: Shannon's information theory should play a basic role in the foundations of sciences, but subject to the condition that it be suitably generalized to allow us to deal with problems which are not necessarily related to communication engineering. With this objective in mind, two questions are of utmost importance: (i) How can we introduce meaning or significance of information in Shannon's information theory? (ii) How can we define and/or measure the amount of information involved in a form or a pattern without using a probabilistic scheme? It is obligatory to find suitable answers to these problems if we want to apply Shannon's theory to science with some chance of success. For instance, its use in biology has been very disappointing, for the very reason that the meaning of information is there of basic importance, and is not involved in this approach.

• Brownian motion
• communication
• entropy
• fractional Brownian motion
• information
• information theory
• random function
• random walk

• Department of Mathematics, University of Quebec at Montreal, Montreal, Canada

  Guy Jumarie

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