Advances in Deterministic and Stochastic Analysis

✍ Scribed by Nguyen Minh Chuong

• Publisher: World Scientific
• Year: 2007
• Tongue: English
• Leaves: 372
• Edition: Illustrated
• Category: Library

✦ Synopsis

Trent Duncan did a good job holding his family together after his dad died. Hed kept his little sister out of trouble and taught her about life. Its just too bad he couldnt do the same for himself. Now hes the man your momma always warned you about: charming, smooth talking--and jobless. Hes got a phony business card and a line for every situation--and every conquest. But the ultimate player is about to play himself right outta the game. Because a couple of Trents ex-girlfriends are about to make him wish hed listened to his momma. . . The only person Trent cant seem to get around anymore is his big brother, Wil. Wils got problems of his own. He thought he was happily married, until his wife, Diane, stopped being intimate with him. Shes got her reasons, but if she doesnt explain herself soon, she may lose her husband to his voluptuous--and lusty--new secretary. Meanwhile, little sister Melanie is all grown up and sure shes met her prince--literally. Prince may be a friend of Trents, but the two men are like night and day. Prince is the kind of man Melanie would like to have kids with. Trouble is, shes not alone. Pretty soon, these three very different siblings have something in common--theyre all in hot water. And they need to find a way to help themselves--and each other--before they get burned. . .

✦ Table of Contents

CONTENTS
Preface
Chapter I Integral and Pseudodifferential Operators
$1. Pseudodifferential Operators of Second Order with Sign-Changed Characteristic Form Nguyen Minh Tri
References
$2. A Semilinear Nonclassical Pseudodifferential Boundary Value Problem in Sobolev Spaces 1 < p < 00 Nguyen Minh Chuong and Dang Anh Tuan
1. Introduction
2. Sobolev Spaces
3. Pseudodifferential Operators in R”
4. Pseudodifferential Operator in Ry
5. Pseudodifferential Operator on a Compact Domain C2
Ellipticity Condition
6. A Linear Non-classical Pseudodifferential Boundary Value Problem
7. A Semilinear Non Classical Pseudodifferential Boundary Value Problem
References
$3. Singular Integral Operators in Functional Spaces of Morrey Type Lubomira Softova
1. Definitions and Preliminary Results
2. Singular Integral Estimates
References
$4. Classification of Integral Transforms Vu Kim Tuan
1. Introduction
2. Convolution Transforms
2.1. Fourier convolution transforms
2.2. Laplace convolution transforms
2.3. Mellin convolution transforms
2.3.1. Mellin convolution transforms of Watson type
2.3.2. Mellin convolution transforms of Riemann-Liouville type
2.3.3. Mellin convolution transforms of Laplace type
2.4. Fourier cosine and sine convolution transforms
3. Nonconvolution Transforms
3.1. W i m p nonconvolution G-transform.
3.2. Transforms of Kontorovich-Lebedev type
3.3. The Buchholt nonconvolution G-transform
3.4. The cherry nonconvolution G-transform
3.5. Other nonconvolution G-transforms
References
Chapter II Partial Differential Equations
$5. Unified Minimax Methods Martin Schechter
1. Introduction
2. Definitions and Theorems
3. Linking Subsets
4. A Method Using Homeomorphisms
5. A Method Using Metric Spaces
6. A Method Using Homotopy Stable Families
7. Some Applications
References
$6. Some Remarks on Single Conservation Laws Mikio Tsuji and Peter Wagner
1. Introduction
2. Existence Domains of Solutions for Burgers Equation
3. A Single Conservation Law without Convexity Condition
4. Behavior of the Shock S1
5 . Behavior of the Shock Sz
References
$7. Iterative Method for Solving a Mixed Boundary Value Problem for Biharmonic Type Equation Dung Quang A and Le Tung Son
1. Introduction
2. Reduction of the Problem to Boundary - Domain Operator Equation
3. Construction of Approximate Solution of the Original Problem Via a Perturbed Problem
4. Iterative Method for Solving the Perturbed Problem
5 . Numerical Experiments
References
$8. Numerical Solution to a Non-Linear Parabolic Boundary Control Problem Dinh Nho Hao, Nguyen Trung Thanh and H. Sahli
1. Introduction
2. The Control Problem and Its dc Representation
3. DCA and Continuation Techniques
3.1. DCA
3.1.1.
3.1.2.
3.1.3.
3.1.4. Let g be a conwex function.
3.1.5. Let g E I'o(X) and lower semi-continuous (lsc).
3.1.6. Consider the DC program
3.1.7. DC algorithm:
3.2. The Continuation Technique
4. Numerical Results
4.1. Numerical Examples
References
$9. A Class of Solutions to Maxwell’s Equations in Matter and Associated Special Functions Peter Massopust
1. Introduction and Preliminaries
1.1. Basic Magnetostatics
1.2. Coordinate Systems
1.2.1. Cartesian Coordinates
1.2.2. Circular Cylindrical Coordinates
1.2.3. Spherical Coordinates
1.2.4. Parabolic Coordinates
1.2.5. Elliptic Cylindrical Coordinates
2. Cartesian V
3. Cylindrical V
4. Spherical V
4.1. Semi-spherical V
5. Parabolic V
5.1. Circular Parabolic V
5.2. Elliptic Parabolic V
References
$10. On the Cauchy Problem for a Quasilinear Weakly Hyperbolic System in Two Variables and Applications to that for Weakly Hyperbolic Classical Monge- Ampkre Equations Ha Tien Ngoan and Nguyen Thi Nga
1. Introduction
2. Hyperbolicity
3. Reduced System
4. Diagonalization
5. Application to the Classical Weakly Hyperbolic Monge- AmpGre Equation
6. Examples
References
$11. Some Singular Perturbation Problems Related to the Navier- Stokes Equations Makram Hamouda and Roger Temam
1. Introduction
2. Correctors of Order 0 and 1
3. The Corrector of Order N ( N 2 2)
3.1. Preliminary results
4. Convergence Result
5. The Nonlinear Case
5.1. Corrector of order zero
5.2. Corrector of order one
5.2.1. Construction of the corrector 8'9'
5.2.2. Convergeme result: th,e main theorem
Appendix
Acknowledgements
References
Chapter III Geometric Analysis
$12. Monotone Invariants and Embeddings of Statistical Manifolds Le Hong Van
1. Introduction
2. Statistical Models and Statistical Manifolds
2.1. Example of a Weak Fisher Metric
2.2. The Fisher Metric on the Space (CapN)+ of all positive probability distributions on ilN (see also Refs. 1,496)
2.3. Divergence Potential (see 1 , l d )
2.4. Chentsov-Amari connections
2.5. Statistical Submanif olds
2.6. Statistical Models and Statistical Manifolds
3. Embeddings of Linear Statistical Spaces
3.1. Race Type of a Symmetric 3-tensor
3.2. Commasses as Monotone Invariants
4. Monotone Invariants and Obstructions to Embeddings of Statistical Manifolds
4.1. Examples
4.2. Diameters of Statistical Manifolds
5. Existence of Isostatistical Embeddings into CupN
Acknowledgement
References
$13. Graded Cech Cohomology in Noncommutative Geometry Do Ngoc Diep
1. Introduction
2. Preparation: Differential Systems and the Cyclic Theory
2.1. Differential Systems, Following Kashiwara
2.2. Fields of C*-algebras and Sheafification
3. Z2-graded c e c h Cohomology
3.1. Grothendieclc Topos
3.2. The Standard Cosimplicial Complex of a Continuous Functor
3.3. The Periodic Cyclic Bicomplex
4. Homotopy Invariance and Morita Invariance
5. Comparison with the Classical Cech Cohomology Theory
Acknowledgments
References
$14. Sobolev Spaces with Weight on Riemannian Manifolds Nguyen Manh Chuong and Le DUC Thinh
1. Introduction
2. Sobolev Spaces with Weighted Norm on Riemannian Manifold
References
Chapter IV Stochastic and Infinite-Dimentional Analysis
$15. Stochastic Population Control and RSDE with Jumps Situ Rong
1. A Deterministic Population Dynamical System
2. RSDE for Stochastic Population Dynamical System
3. Existence of Solutions to RSDE with Jumps
4. Stochastic Population Solutions and Their Properties
5. The Optimal Stochastic Population Control
References
$16. Noncausal Stochastic Calculus Revisited - Around the So-called Ogawa Integral Shigeyoshi Ogawa
1. Introduction
2. Noncausal Problems in Stochastic Analysis
3. Review of the Noncausal Stochastic Calculus
3.1. Causal functions and the B-diflerentiability
3.2. Noncausal stochastic integral
3.3. Equivalent expressions and variants
3.4. Condition for the integrability - in the framework of the Homogeneous Chaos theory
3.5. Relation between symmetric and noncausal integrals
4. Applications to the Noncausal SDEs
4.1. Noncausal Cauchy problem
4.2. Discussions for the more general cases
5. Question of Uniqueness - Noncausal It8 Formula
References
$17. Infinite-Dimensional Stochastic Analysis and Foundations of Quantum Mechanics Andrei Khrennikov
1. Introduction
2. Infinite-dimensional analysis
3 . Dequantization
3.1. Classical and quantum statistical models
3.2. Asymptotic Gaussian analysis
4. Gaussian underground for pure states
5. Pure states as one-dimensional projections of spatial white-noise
5.1. The finite-dimensional case
5.2. Prequantum white noise field
6. Unbounded operators
References
$18. Noncommutative Trigonometry and Quantum Mechanics Karl Gustafson
1. Introduction, Background, and Summary
2. Essentials of the Noncommutative Trigonometry
3. Quick Summary of Applications to Date
4. The Bell Inequalities of Quantum Theory
5. The Zeno Problem of Quantum Theory
Acknowledgements
References