Lecture notes in applied differential equations of mathematical physics /

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Bibliographic Details
Author / Creator:	Botelho, Luiz C. L.
Imprint:	Singapore ; Hackensack, NJ : World Scientific Pub., c2008.
Description:	xiv, 324 p. ; 24 cm.
Language:	English
Subject:	Differential equations. Functional analysis. Mathematical physics. Differential equations. Functional analysis. Mathematical physics.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/7478639

Hidden Bibliographic Details
Varying Form of Title:	Applied differential equations of mathematical physics
ISBN:	9789812814579 (hbk.) 9812814574 (hbk.)
Notes:	Includes index.
Summary:	"Functional analysis is a well-established powerful method in mathematical physics, especially those mathematical methods used in modern non-perturbative quantum field theory and statistical turbulence. This book presents a unique, modern treatment of solutions to fractional random differential equations in mathematical physics. It follows an analytic approach in applied functional analysis for functional integration in quantum physics and stochastic Langevin-turbulent partial differential equations."--BOOK JACKET.

Table of 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 R[superscript 3] - Kirchhoff Potentials (Spherical Means)
1.7. The Dirichlet Problem for the Diffusion Equation - Seminar Exercises
1.8. The Potential Theory in Distributional Spaces - The Gelfand-Chilov Method
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: Moller 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 Quantum Mechanical 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 the Minlos Theorem
Chapter 4. Nonlinear Diffusion 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 Diffusion, 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.)[subscript 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[superscript infinity] ([Omega]) (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 Fermion Model
Chapter 7. Nonlinear Diffusion in R[superscript D] and Hilbert Spaces: A Path-Integral Study
7.1. Introduction
7.2. The Nonlinear Diffusion
7.3. The Linear Diffusion in the Space L[superscript 2]([Omega])
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 Nonlinear Wave 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 Muntz-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 Random Conditions
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

Lecture notes in applied differential equations of mathematical physics /

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