Applied Analysis (3rd Ed)

✍ Scribed by Takashi Suzuki

• Publisher: World Scientific Pub Co Inc
• Year: 2022
• Tongue: English
• Leaves: 688
• Edition: 3
• Category: Library

✦ Synopsis

This book is to be a new edition of Applied Analysis. Several fundamental materials of applied and theoretical sciences are added, which are needed by the current society, as well as recent developments in pure and applied mathematics. New materials in the basic level are the mathematical modelling using ODEs in applied sciences, elements in Riemann geometry in accordance with tensor analysis used in continuum mechanics, combining engineering and modern mathematics, detailed description of optimization, and real analysis used in the recent study of PDEs. Those in the advance level are the integration of ODEs, inverse Strum Liouville problems, interface vanishing of the Maxwell system, method of gradient inequality, diffusion geometry, mathematical oncology. Several descriptions on the analysis of Smoluchowski-Poisson equation in two space dimension are corrected and extended, to ensure quantized blowup mechanism of this model, particularly, the residual vanishing both in blowup solution in finite time with possible collision of sub-collapses and blowup solutions in infinite time without it.

✦ Table of Contents

Contents
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Chapter 1 Mathematical Modeling
1.1 Mathematics for Modeling
1.1.1 Differentiation
1.1.2 Linear and Nonlinear Equations
1.1.3 Exponential Function of Matrices
1.1.4 Unique Solvability
1.2 Models in Chemistry
1.2.1 Self-Resolution
1.2.2 Mass Action
1.2.3 Michaelis Menten Reduction
1.2.4 Polymerization
1.2.5 Reaction Network
1.3 Systems of Differential Equations
1.3.1 Dynamical Systems
1.3.2 Gradient Systems
1.3.3 Hamilton Systems
1.4 Models in Ecology
1.4.1 Population
1.4.2 Prey Predator Systems
1.4.3 Competitive Systems
1.4.4 Hamilton Structure of Ecological Models
1.5 Models in Cell Biology
1.5.1 Infection
1.5.2 Invasion
1.5.3 Immunity
1.6 Models in Physiology
1.6.1 Heart Beating
1.6.2 Nerve Impulse Transmission
Chapter 2 Field Formation
2.1 Classical Mechanics
2.1.1 Outer Product
2.1.2 Particle Motion
2.2 Basic Notions of Vector Analysis
2.2.1 Gradient
2.2.2 Divergence
2.2.3 Rotation
2.2.4 Material Derivatives
2.3 Continuum Mechanics
2.3.1 Fluid Motion
2.3.2 Tensors
2.3.3 Solid Deformation
2.3.4 Viscous Fluids
Chapter 3 Objects and Coordinates
3.1 Curvature
3.1.1 Surfaces
3.1.2 Curves
3.1.3 Curves on Surfaces
3.2 Integral Formulae
3.2.1 Integrations
3.2.2 Integration on Surfaces
3.2.3 Circulations
3.3 Transformation of Coordinates
3.3.1 Orthogonal Coordinates
3.3.2 Curved Coordinates
3.3.3 Co-variant Derivatives
Chapter 4 Languages of Modern Geometry
4.1 Differential Forms
4.1.1 Differential Forms in R3
4.1.2 Differential Forms in Rn
4.1.3 Minkowski Spaces
4.2 Notions of Metric
4.2.1 Differential Forms on Surfaces
4.2.2 Tangent and Co-tangent Spaces
4.2.3 Ortho-normal Frames
4.2.4 Connections
4.2.5 Riemann Surfaces
4.3 Riemann Manifolds
4.3.1 Manifolds
4.3.2 Vector Fields on Manifolds
4.3.3 Riemann Metrics
Chapter 5 Optimizations
5.1 Extremals
5.1.1 Local Maxima and Minima
5.1.2 Gradient Method
5.1.3 Newton Method
5.1.4 Adjoint Gradient Method
5.1.5 Optimization with Constraints
5.2 Linear Programmings
5.2.1 Method of Simplices
5.2.2 Duality Theorem
5.2.3 Matrix Games
5.3 Convex Analysis
5.3.1 Convex Functions
5.3.2 Kuhn Tucker Duality
5.3.3 Duality Theorem Revisited
5.4 Notion of Graphs
5.4.1 Graphs
5.4.2 Connected Graphs
5.5 Statistical Inferences
5.5.1 Statistical Optimizations
5.5.2 Statistical Optimization with Constraints
5.5.3 Classifications of the Data
Chapter 6 Calculus of Variation
6.1 Isoperimetric Inequalities
6.1.1 Analytic Proof
6.1.2 Geometric Proof
6.2 Indirect Methods
6.2.1 Euler Equations
6.2.2 Minimal Surfaces
6.2.3 Analytic Mechanics
6.2.4 Quantum Mechanics
6.3 Direct Methods
6.3.1 Vibrating Strings
6.3.2 Minimizing Sequences
6.3.3 Sobolev Spaces
6.3.4 Lower Semi-Continuity
6.4 Numerical Schemes
6.4.1 Finite Difference Methods
6.4.2 Finite Element Methods
6.4.3 Error Analysis
Chapter 7 Infinite-Dimensional Analysis
7.1 Hilbert Spaces
7.1.1 Bounded Linear Operators
7.1.2 Representation Theorem of Riesz
7.1.3 Complete Orthonormal Systems
7.2 Fourier Analysis
7.2.1 Historical Notes
7.2.2 Completeness
7.2.3 Uniform Convergences
7.2.4 Pointwise Convergences
7.2.5 Fourier Transformations
7.3 Eigenvalue Problems
7.3.1 Vibrating Membranes
7.3.2 Gel’fand Triples
7.3.3 Self-Adjoint Operators
7.3.4 Symmetric Forms
7.3.5 Compact Operators
7.3.6 Eigenfunction Expansions
7.3.7 Minimax Principles
7.3.8 Hilbert Schmidt Operators
7.4 Distributions
7.4.1 Delta Functions
7.4.2 Locally Convex Spaces
7.4.3 Fréchet Spaces
7.4.4 Inductive Limits
7.4.5 Bounded Sets
7.4.6 Definitions and Examples
7.4.7 Fundamental Properties
7.4.8 Supports
7.4.9 Convergences
7.4.10 Fourier Transformations Revisited
Chapter 8 Scattering
8.1 Direct Theory
8.1.1 Jost Solution
8.1.2 S-Matrix
8.1.3 GLM Equation
8.2 Reverse Theory
8.2.1 GLM Equation Continued
8.2.2 Reconstruction
8.2.3 Consistency
Chapter 9 Random Motion of Particles
9.1 Mean Field Limits
9.1.1 Master Equation
9.1.2 Einstein’s Formula
9.1.3 Local Information Model
9.1.4 Smoluchowski Equation
9.1.5 Multiscale Models
9.2 Kinetic Models
9.2.1 Transport Equation
9.2.2 Boltzmann Equation
9.2.3 Semiconductor Device Equation
9.2.4 Drift Diffusion Model
Chapter 10 Linear PDE
10.1 Well-Posedness
10.1.1 Heat Equations
10.1.2 Uniqueness
10.1.3 Existence
10.2 Fundamental Solutions
10.2.1 Cauchy Problems
10.2.2 Gaussian Kernel
10.2.3 Semigroups
10.3 Laplace Equations
10.3.1 Harmonic Functions
10.3.2 Poisson Integrals
10.3.3 Perron Solutions
10.3.4 Boundary Regularities
10.3.5 Green’s Function
10.4 Potentials
10.4.1 Newton Potentials
10.4.2 Layer Potentials
10.4.3 Fredholm Theories
10.4.4 Poisson Equations
10.4.5 Schauder Estimates
Chapter 11 Real Analysis in PDE
11.1 Hölder Regularities
11.1.1 Dirichlet Principles
11.1.2 Moser’s Iteration Schemes
11.1.3 Local Minimum Principles
11.2 Hardy Spaces and BMO
11.2.1 John Nirenberg Inequality
11.2.2 Maximal Functions
11.2.3 Div Rot Lemma
11.2.4 Estimates of the Jacobian
11.2.5 A Harmonic Map
Chapter 12 Nonlinear PDE
12.1 Perturbations
12.1.1 Duhamel Principles
12.1.2 Semilinear Heat Equations
12.1.3 Global Existence
12.1.4 Blowup of the Solution
12.2 Energies
12.2.1 Lyapunov Functions
12.2.2 Global-in-Time Solutions
12.2.3 Unbounded Solutions
12.2.4 Stable and Unstable Sets
12.3 Rescaling
12.3.1 ODE Parts
12.3.2 Variational Structure
12.3.3 Scaling Invariance
12.3.4 Forward Self-Similar Transformations
12.3.5 Backward Self-Similar Transformations
12.4 Chemotaxis
12.4.1 Cellular Slime Molds
12.4.2 Symplified Systems
12.4.3 Free Energy
12.4.4 Smoluchowski Poisson Equation
12.4.5 Quantized Blowup Mechanism
12.4.6 Mass Quantization
12.4.7 Recursive Hierarchy
Appendix A Catalogue of Mathematical Analysis
A.1 Basic Analysis
A.2 Topological Spaces
A.3 Complex Analysis
A.4 Real Analysis
A.5 Abstract Analysis
Appendix B An Elliptic-Parabolic System
B.1 The System
B.2 Linearized System
B.3 The Mapping F
B.4 Unique Solvability
Appendix C Commentaries
C.1 Systems of ODEs
C.2 Sturm Liouville Problems
C.3 Elliptic Equations
C.4 Parabolic Equations
C.5 Diffusion Geometry
C.6 Self-Interacting Particles
C.7 Models in Theoretical Biology
C.8 Maxwell Systems
Bibliography
Index