Operator Theory and Differential Equations

Preface
Contents
Extinction in a Finite Time for Parabolic Equations of Fast Diffusion Type on Manifolds
1 Introduction
2 Proof of the Weighted Sobolev Inequality
3 Extinction in Finite Time: Proof of Theorem 1.3
References
Inverse Problems in the Multidimensional Hyperbolic Equation with Rapidly Oscillating Absolute Term
1 Introduction
2 Principal Symbols
3 The Auxiliary Results
3.1 The Results of V.A. Il'in [27]
3.2 The Problem 1
3.3 The Lemma of Krasnosel'skii et al. [28, Sec. 22.1]
4 The Main Results
4.1 The Problem 2
4.2 The Inverse Problem 3
5 Proofs of the Main Results
References
On the Brezis–Lieb Lemma and Its Extensions
1 Introduction
2 Brezis–Lieb Spaces
3 Operator Version of the Brezis–Lieb Lemma in Convergent Vector Spaces
References
On the Commutant of the Generalized Backward Shift Operator in Weighted Spaces of Entire Functions
1 Introduction
2 Preliminary Information
3 Auxiliary Results
4 Main Results
5 The Generalized Duhamel Product
References
Global Boundedness of Solutions of Continuous Social Stratification Model
1 Introduction
2 The Change of Variables
3 Spatially Homogeneous Solutions
4 Sufficient Conditions for Global Boundedness
5 Social Strain Localization in the Case of No External Influence
6 Concluding Remarks
References
Stochastic Test of a Minimal Surface
1 Introduction
2 Some Definitions from the Theory of Random Processes
3 Main Result
3.1 Auxiliary Statements
3.2 Proof of Main Result
References
On Isomorphism of Some Functional Spaces under Action of Two-dimensional Singular Integral Operators
1 Introduction and Formulation of Results
2 Auxiliary Statements
3 Proof of Theorems 1–3
3.1 Operator S1
3.2 Operator S2
References
On One Class of Solutions of the Darboux System
1 Linear System (0.2)
2 Reduction of the Darboux System
References
Boolean Valued Analysis: Background and Results
1 Introduction
2 Kantorovich's Heuristic Principle
3 Boolean Valued Reals
4 Boolean Valued Representation of Structures
5 Boolean Valued Representation of Operators
6 Problems and Solutions
References
Approximation Properties of Vallée Poussin Means for Special Series of Ultraspherical Jacobi Polynomials
1 Main Result
2 Proof of the Main Result
3 Auxiliary Information
4 Proof of Theorem 2.1
4.1 The Case 0 ≤x ≤1 - c2(m+n)2
4.1.1 Estimation of J-1/2x-m+1m+n1-x(x)
4.1.2 Estimation of Jx+m+1m+n1-x1(x)
4.1.3 Estimation of Jx-1m+n1-xx+1m+n1-x(x)
4.1.4 Estimation of Jx-m+1m+n1-xx-1m+n1-x(x) and Jx+1m+n1-xx+m+1m+n1-x(x)
4.2 The Case 1 - c2(m+n)2 ≤x ≤1
References
Laplace's Integrals and Stability of the Open Flows of Inviscid Incompressible Fluid
1 Introduction
2 Formulation of the Initial-Boundary Value Problems
3 The Liapunov Functionals
4 Elementary and Instructive Example
5 Lagrangian Coordinates: Non-Separated Flows
6 Harmonic Flows and Operator-Valued Laplace's Integral
7 Flow Through a Gap Between Two Cylinders
8 Flow Through a Gap Between Two Spheres
References
Spectral Properties of Killing Vector Fields of Constant Length and Bounded Killing Vector Fields
1 Introduction
2 KVFCL on General Riemannian Manifolds
3 KVFCL on Geodesic Orbit Spaces
4 Algebraic Properties of Bounded Killing Vector Fields
References
The Radon-Nikodým Theorem for Disjointness Preserving Orthogonally Additive Operators
1 Introduction and Preliminaries
2 Result
References
On the Continuous Linear Right Inverse for a Convolution Operator
1 Introduction
2 Preliminaries
3 Functional Criterion
4 Necessary Conditions
5 Sufficient Conditions
6 The Main Result
References
Long Wavelength Asymptotics of Self-Oscillations of Viscous Incompressible Fluid
1 Introduction
2 Recurrence Formulas for Asymptotics of w
2.1 First Terms of the Asymptotic Expansion of w
2.2 kth Term of w
3 Recurrence Formulas for Asymptotics of v
3.1 First Terms of the Asymptotic Expansion of v
3.2 kth Term of v
4 Conclusions
References
The Convergence of the Fourier–Jacobi Series in Weighted Variable Exponent Lebesgue Spaces
1 Introduction
2 The Hilbert Transform in Lp(x)(R)
3 Some Information About the Jacobi Polynomials
4 Main Result
5 The Proof of Lemma 4.2
References
Hyperbolic B-Potentials: Properties and Inversion
1 Introduction
2 Basic Definitions
3 Definition of the Hyperbolic B-Potentials
4 Method of Approximative Inverse Operators and General Poisson Kernel
5 Representation of the Kernel gα,δ
6 Inversion of the Hyperbolic B-Potentials
References
Some Properties of Sobolev Orthogonal Polynomials Associated with Chebyshev Polynomials of the Second Kind
1 Introduction
2 Some Properties of Jacobi and Chebyshev Polynomials
3 The Chebyshev–Sobolev Polynomials of the Second Kind
4 Recurrence Relations for Polynomials U1,n(x) and U2,n(x)
5 Asymptotic Properties of the Polynomials Ur,n(x)
References
Structure of Essential Spectrum and Discrete Spectrum of the Energy Operator of Five-Electron Systems in the Hubbard Model—Doublet States
1 Introduction
2 First Doublet State
3 Second Doublet State
References
On Coefficient Inverse Problems of Heat Conduction for Functionally Graded Materials
1 Introduction
2 Statement of the Problem
3 The Solution of the Direct Problem of Heat Conduction
4 The Solution of the Inverse Heat Conduction Problem Based on an Iterative Approach
5 The Solution of the Inverse Problem by the Algebraization Method
6 Results of Computational Experiments
7 Conclusions
References
A Study of the Waves Processes in Inhomogeneous Cylindrical Waveguides
1 Introduction
2 Problem for Inhomogeneous Waveguide
3 Problem for Inhomogeneous Waveguide with Annular Delamination
4 Inverse Problem
5 Conclusion
References
Solution of a Class of First-Order Quasilinear Partial Differential Equations
1 Introduction
2 Cauchy Problem
3 Analogy Between Solution and Divided Difference
4 Lagrangian Variables
5 Mass Transport by an Electric Field
References