Contents -- Foreword -- Chapter 1. Elementary Aspects of Potential Theory in Mathematical Physics -- 1.1. Introduction -- 1.2. The Laplace Differential Operator and the Poisson-Dirichlet Potential Problem -- 1.3. The Dirichlet Problem in Connected Planar Regions: A Conformal Transformation Method for Green Functions in String Theory -- 1.4. Hilbert Spaces Methods in the Poisson Problem -- 1.5. The Abstract Formulation of the Poisson Problem -- 1.6. Potential Theory for the Wave Equation in R3 - Kirchho. Potentials (Spherical Means) -- 1.7. The Dirichlet Problem for the Diffusion Equation - Seminar Exercises -- 1.8. The Potential Theory in Distributional Spaces - The Gelfand-ChilovMethod -- References -- Appendix A. Light Deflection on de-Sitter Space -- A.1.The Light Deflection -- A.2.The Trajectory Motion Equations -- A.3. On the Topology of the Euclidean Space-Time -- Chapter 2. Scattering Theory in Non-Relativistic One-Body Short-Range Quantum Mechanics: Möller Wave Operators and Asymptotic Completeness -- 2.1. The Wave Operators in One-Body Quantum Mechanics -- 2.2. Asymptotic Properties of States in the Continuous Spectra of the Enss Hamiltonian -- 2.3. The Enss Proof of the Non-Relativistic One-Body QuantumMechanical Scattering -- References -- Appendix A -- Appendix B -- Appendix C -- Chapter 3. On the Hilbert Space Integration Method for the Wave Equation and Some Applications to Wave Physics -- 3.1. Introduction -- 3.2. The Abstract Spectral Method - The Nondissipative Case -- 3.3. The Abstract Spectral Method - The Dissipative Case -- 3.4. The Wave Equation "Path-Integral" Propagator -- 3.5. On The Existence of Wave-Scattering Operators -- 3.6. Exponential Stability in Two-Dimensional Magneto-Elasticity: A Proof on a Dissipative Medium -- 3.7. An Abstract Semilinear Klein Gordon Wave Equation - Existence and Uniqueness -- References.

Appendix A. Exponential Stability in Two-Dimensional Magneto-Elastic: Another Proof -- Appendix B. Probability Theory in Terms of Functional Integrals and theMinlos Theorem -- Chapter 4. Nonlinear Di.usion and Wave-Damped Propagation: Weak Solutions and Statistical Turbulence Behavior. -- 4.1. Introduction -- 4.2. The Theorem for Parabolic Nonlinear Diffusion -- 4.3. The Hyperbolic Nonlinear Damping -- 4.4. A Path-Integral Solution for the Parabolic Nonlinear Diffusion -- 4.5. Random Anomalous Diffuusion, A Semigroup Approach -- References -- Appendix A -- Appendix B -- Appendix C -- Appendix D. Probability Theory in Terms of Functional Integrals and the Minlos Theorem - An Overview -- Chapter 5. Domains of Bosonic Functional Integrals and Some Applications to the Mathematical Physics of Path-Integrals and String Theory -- 5.1. Introduction -- 5.2. The Euclidean Schwinger Generating Functional as a Functional Fourier Transform -- 5.3. The Support of Functional Measures - The Minlos Theorem -- 5.4. Some Rigorous Quantum Field Path-Integral in the Analytical Regularization Scheme -- 5.5. Remarks on the Theory of Integration of Functionals on Distributional Spaces and Hilbert-Banach Spaces -- References -- Appendix A -- Appendix B. On the Support Evaluations of Gaussian Measures -- Appendix C. Some Calculations of the Q.C.D. Fermion Functional Determinant in Two-Dimensions and (Q.E.D.)2 Solubility -- Appendix D. Functional Determinants Evaluations on the Seeley Approach -- Chapter 6. Basic Integral Representations in Mathematical Analysis of Euclidean Functional Integrals. -- 6.1. On the Riesz-Markov Theorem -- 6.2. The L. Schwartz Representation Theorem on C∞() (Distribution Theory) -- 6.3. Equivalence of Gaussian Measures in Hilbert Spaces and Functional Jacobians -- 6.4. On the Weak Poisson Problem in Infinite Dimension.

6.5. The Path-Integral Triviality Argument -- 6.6. The Loop Space Argument for the Thirring Model Triviality -- References -- Appendix A. Path-Integral Solution for a Two-Dimensional Model with Axial-Vector-Current-Pseudoscalar Derivative Interaction -- Appendix B. Path Integral Bosonization for an Abelian Nonrenormalizable Axial Four-Dimensional FermionModel -- B.1. Introduction -- B.2. The Model -- B.3. Complement -- Chapter 7. Nonlinear Diffusion in RD and Hilbert Spaces: A Path-Integral Study -- 7.1. Introduction -- 7.2. The Nonlinear Diffusion -- 7.3. The Linear Diffusion in the Space L2(.) -- References -- Appendix A. The Aubin-Lion Theorem -- Appendix B. The Linear Diffusion Equation -- Chapter 8. On the Ergodic Theorem -- 8.1. Introduction -- 8.2. On the Detailed Mathematical Proof of the RAGE Theorem -- 8.3. On the Boltzmann Ergodic Theorem in Classical Mechanics as a Result of the RAGE Theorem -- 8.4. On the Invariant Ergodic Functional Measure for Some NonlinearWave Equations -- 8.5. An Ergodic Theorem in Banach Spaces and Applications to Stochastic-Langevin Dynamical Systems -- 8.6. The Existence and Uniqueness Results for Some Nonlinear Wave Motions in 2D -- References -- Appendix A. On Sequences of Random Measureson Functional Spaces -- Appendix B. On the Existence of Periodic Orbits in a Class of Mechanical Hamiltonian Systems -An Elementary Mathematical Analysis -- B.1. Elementary May be Deep - T. Kato -- Chapter 9. Some Comments on Sampling of Ergodic Process: An Ergodic Theorem and Turbulent Pressure Fluctuations. -- 9.1. Introduction -- 9.2. A Rigorous Mathematical Proof of the Ergodic Theorem for Wide-Sense Stationary Stochastic Process -- 9.3. A Sampling Theorem for Ergodic Process -- 9.4. A Model for the Turbulent Pressure Fluctuations (Random Vibrations Transmission) -- References.

Appendix A. Chapters 1 and 9 - On the Uniform Convergence of Orthogonal Series - Some Comments -- A.1. Fourier Series -- A.2. Regular Sturm-Liouville Problem -- Appendix B. On the Müntz-Szasz Theorem on Commutative Banach Algebras -- B.1. Introduction -- Appendix C. Feynman Path-Integral Representations for the Classical Harmonic Oscillator with Stochastic Frequency -- C.1. Introduction -- C.2. The Green Function for External Forcing -- C.3. The Homogeneous Problem -- Appendix D. An Elementary Comment on the Zeros of the Zeta Function (on the Riemann's Conjecture) -- D.1. Introduction - "Elementary May be Deep" -- D.2.On the Equivalent Conjecture D.1 -- Chapter 10. Some Studies on Functional Integrals Representations for Fluid Motion with Random Conditions -- 10.1. Introduction -- 10.2. The Functional Integral for Initial Fluid Velocity RandomConditions -- 10.3. An Exactly Soluble Path-Integral Model for Stochastic Beltrami Fluxes and its String Properties -- 10.4. A Complex Trajectory Path-Integral Representation for the Burger-Beltrami Fluid Flux -- References -- Appendix A. A Perturbative Solution of the Burgers Equation Through the Banach Fixed Point Theorem -- Appendix B. Some Comments on the Support of Functional Measures in Hilbert Space -- Chapter 11. The Atiyah-Singer Index Theorem: A Heat Kernel (PDE's) Proof -- References -- Appendix A. Normalized Ricci Fluxes in Closed Riemann Surfaces and the Dirac Operator in the Presence of an Abelian Gauge Connection -- Index.