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This book develops a new physical/mathematical model for the functioning of the human brain, based, not on the modern Newton-Einstein view of physical reality, but on 'information reality'. The work is devoted to the physical-mathematical modeling of (conscious) cognitive phenomena. The most important distinguishing feature of the theory presented here is a new model of mental space, the so-called p-adic hierarchic tree space, and the development of mental analogs of classical and quantum mechanics. Mental processes and more general information processes are handled as a kind of new physical processes. In particular, the procedure of information quantization and an information analog of Bohmian mechanics are developed. Here, mind is a singularity in the mental pilot wave. Applications to neurophysiology, localization of mental function and brain ablations, and psychology (in particular, Freud's psychoanalysis) are considered.
List of Figures. Introduction.
1: Processing Information on p-Adic Trees. 1.1. Ultrametric Spaces. 1.2. m-adic Geometry. 1.3. Geometry of Information Spaces. 1.4. Dynamical Processing of Information. 1.5. Role of Hierarchical Structure. 1.6. Role of Chance in Processing of Cognitive Information. 1.7. Information Reductionism.
2: Hierarchy of Information. 2.1. Hierarchical Coding of Information. .2.2. Flows of Associations and Ideas. 2.3. How Can the Brain Play Dice? 2.4. Constraints on Information Spaces.
3: p-Adic Dynamical Systems. 3.1. p-Adic Numbers. 3.2. Roots of Unity. 3.3. Dynamical Systems in Non-Archimedian Fields. 3.4. Dynamical Systems in the Field of Complex p-adic Numbers. 3.5. Dynamical Systems in the Fields of p-adic Numbers. 3.6. p-adic Ergodicity. 3.7. Newton's Method (Hensel's Lemma). 3.8. Computer Calculations for Fuzzy Cycles.
4: Random Processing of Information. 4.1. Random Dynamical Systems. 4.2. Long-term Behaviour, Dynamics on the Attractor, Examples. 4.3. Consequences for Cognitive Sciences.
5: Information Quantum Mechanics. 5.1. Quantum-Like Formalism for a One-Layer Brain. 5.2. Motivation Observable. 5.3. Neuron Activation Observable. 5.4. Complex Cognitive Systems: Evolution.
6: Bohmian Mechanics on Information Spaces. 6.1. Newton Laws for Information Processes. 6.2. Bohmian Mechanics for Hierarchical Information. 6.3. Interactions between Information Systems. 6.4. Hamiltonian Equations and Active Information. 6.5. Information Mass. 6.6. Wave Functions Taking Values in p-adic Fields. 6.7. Information Waves on p-adic Trees. 6.8. p-adic Bohmian Mechanics and Waves of Brain Activation. 6.9. Conservation Laws. 6.10. Mechanics of a System of Information Transformers, Constraints on Information Spaces. 6.11. Social and Anomalous Phenomena.
7: Abstract Ultrametric Information Spaces. 7.1. Abstract Ultrametric Spaces. 7.2. Hierarchy of Associations. 7.3. Topology and Materialism. 7.4. Existence of Universal Mental Space. 7.5. Towers of Associations. 7.6. Infinite Information Towers.
8: Pathway Representation of Cognitive Information. 8.1. Model: Thinking on a Cognitive Tree. 8.2. Dynamics in the Information Space. 8.3. Diffusion Model for Dynamics of a Mental State. 8.4. Information Phase Space. 8.5. Mental State as the Distribution of a p-adic Random Walk. 8.6. Discussion of the Neural Pathways Thinking Model.
9: Contextual Approach to Quantum Theory. 9.1. The Växjö Interpretation of Quantum Mechanics. 9.2. Contextual Viewpoint of Quantum Shastics. 9.3. Law of Statistical Balance in Nature. 9.4. Experiments on Quantum-Like Behaviour of the Mind. 9.5. Experimental Confirmation.
10: Frequency Analysis of Foundations of Quantum Mechanics. 10.1. Classification of Transforms of Probability. 10.2. Classical, Quantum and Non-Classical-Quantum Physics. 10.3. Hyperbolic Probabilistic Behaviour. 10.4. Linear Space Representation of the Trigonometric Rule. 10.5. Linear Space Representation of the Hyperbolic Rule. 10.6. Conclusions.
11: Bohmian Mechanics for Financial Progresses. 11.1. Price Phase Space. 11.2. Hamiltonian Price Dynamics and the Sk Market. 11.3. Financial Pilot Waves. 11.4. The Dynamics of Prices Guided by the Financial Pilot Wave.