This volume features selected papers from The Fifteenth International Conference on Order Analysis and Related Problems of Mathematical Modeling, which was held in Vladikavkaz, Russia, on 15 - 20th July 2019. Intended for mathematicians specializing in operator theory, functional spaces, differential equations or mathematical modeling, the book provides a state-of-the-art account of various fascinating areas of operator theory, ranging from various classes of operators (positive operators, convolution operators, backward shift operators, singular and fractional integral operators, partial differential operators) to important applications in differential equations, inverse problems, approximation theory, metric theory of surfaces, the Hubbard model, social stratification models, and viscid incompressible fluids. 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
