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

I. Measurements and Numbers.- 1. Mathematics and Reality.- 2. Measurements and Natural Numbers.- 3. Measurements and Rational Numbers.- 4. Real Numbers: Infinite Exactness of Measurements.- 5. On the Boundary of the Real Continuum.- 6. Finite Exactness and m-adic Numbers.- 7. Rings of m-adic Numbers.- 8. Ultrametric Spaces.- 9. Ultrametric Social Space.- 10. Non-Real Models of Space.- II. Fundamentals.- 1. Einstein-Podolsky-Rosen Paradox.- 2. Foundations of Quantum Mechanics.- 3. Foundations of Probability Theory.- 4. Statistical Interpretation of Quantum mechanics.- 5. Quantum Probabilities; Two Slit Experiment.- 6. Bell’s Inequality and the Death of Reality.- 7. Individual Realists Interpretation and Hidden Variables.- 8. Orthodox Copenhagen Interpretation.- 9. Einstein-Podolsky-Rosen Paradox and Interpretations of Quantum Mechanics.- III. Non-Archimedean Analysis.- 1. Exponential Function.- 2. Normed and Locally Convex Spaces.- 3. Locally Constant Functions.- 4. Kaplansky’s Theorem.- 5. Differentiate Functions.- 6. Analytic Functions.- 7. Complex Non-Archimedean Numbers.- 8. Mahler Basis.- 9. Measures on the Ring of p-adic Integers.- 10. Volkenborn Integral (Uniform Distribution).- 11. The Monna-Springer Integration Theory.- IV. The Ultrametric Hilbert Space Description of Quantum Measurements with a Finite Exactness.- 1. Critique of Interpretations of Quantum Mechanics.- 2. Preparation Procedures and State Spaces.- 3. Ultrametric (m-adic) Hilbert Space.- 4. m-adic (Ultrametric) Axiomatic of Quantum Measurements.- 5. Heisenberg Uncertainty and Inexactness Relations.- 6. Energy Representation for the Harmonic Oscillator.- 7. Einstein-Podolsky-Rosen Paradox and Infinite Exactness of Measurements.- 8. Fuzzy Reality.- 9. Quantum-Classical Heisenberg InexactnessRelation for the Harmonic Oscillator and Free Particle.- V. Non-Kolmogorov Probability Theory.- 1. Frequency Probability Theory.- 2. Measure and Probability.- 3. Densities.- 4. Integration Technique.- 5. Non-Kolmogorov Axiomatics.- 6. Products of Probabilities.- 7. Proportional and Classical Definitions of Probability.- 8. p-adic Asymptotic of Bernoulli Probabilities.- 9. More Complicated p-adic Asymptotics.- 10. p-adic Bernoulli Theorem.- 11. Non-symmetrical Bernoulli Distributions.- 12. The Central Limit Theorem.- VI. Non-Kolmogorov Probability and Quantum Physics.- 1. Dirac, Feynman, Wigner and Negative Probabilities.- 2. p-adic Stochastic Point of View of Bell’s Inequality.- 3. An Example of p-adic Negative Probability Behaviour.- 4. p-adic Stochastic Hidden Variable Model with Violations of Bell’s Inequality.- 5. Quadri Variate Joint Probability Distribution.- 6. Non-Kolmogorov Statistical Theory.- 7. Physical Interpretation of Negative Probabilities in Prugovecki’s Empirical Theory of Measurement.- 8. Experiments to Find p-adic Stochastics in the Two Slit Experiment.- VII. Position and Momentum Representations.- 1. Groups of Unitary Isometric Operators in a p-adic Hilbert Space.- 2. p-adic Valued Gaussian Integration and Spaces of Square Integrable Functions.- 3. A Representation of the Translation Group.- 4. Gaussian Representations for the Position and Momentum Operators.- 5. Unitary Isometric One Parameter Groups Corresponding to the Position and Momentum Operators.- 6. Operator Calculus.- 7. Exactness of a Measurement of Positions and Momenta.- 8. Spectrum of p-adic Position Operator.- 9. L2-space with respect to p-adic Lebesgue distributions.- 10. Fourier Transform of L2-maps and Momentum Representation.- 11. Schrödinger Equation.- VIII. p-adicDynamical Systems with Applications to Biology and Social Sciences.- 1. Roots of Unity.- 2. Dynamical Systems in Non-Archimedean Fields.- 3. Dynamical Systems in the Field of Complex p-adic Numbers.- 4. Dynamical Systems in the Fields of p-adic Numbers.- 5. Computer Calculations for Fuzzy Cycles.- 6. The Human Subconscious as a p-adic Dynamical System.- 7. Ultrametric on the Genealogical Tree.- 8. Social Dynamics.- 9. Human History as a p-adic Dynamical System.- 10. God as p-adic Dynamical System.- 11. Struggle of Civilizations.- 12. Economical and Social Effectiveness.- Open Problems.- 1. Newton’s Method (Hensel Lemma).- 2. Non-Real Reality.- 5. Quantum Mechanics of Vladimirov and Volovich.