Preface
Contents
Quasilinear Operator Equation at Resonance
1 Introduction
2 Preliminaries
3 Existence Theorem
4 Application
References
A Control Problem for a System of ODE with Nonseparated Multipoint and Integral Conditions
1 Introduction
2 An Analysis of the Problem under Investigation and Obtaining Basic Formulas
3 Numerical Scheme of Solution to the Problem
4 Analysis of the Results of the Computational Experiments
5 Conclusion
References
On Generalized Convexity and Superquadracity
1 Introduction
2 Superquadracity and Generalized ψ-Uniformly Convexity
3 Reversed and Refined Minkowski Inequality
References
Well-Posedness of Nonsmooth Lurie Dynamical Systems Involving Maximal Monotone Operators
1 Introduction
2 Lurie Systems with Maximal Monotone Operators
3 Background from Convex and Set-Valued Analysis
4 Well-Posedness of Nonsmooth Lurie Systems
References
Numerical Method for Calculation of Unsteady Fluid Flow Regimes in Hydraulic Networks of Complex Structure
1 Introduction
2 Statement of the Problem
3 Numerical Method of the Solution to the Problem
4 The Results of Numerical Experiments
5 Conclusion
References
Numerical Solution to Inverse Problems of Recovering Special-Type Source of a Parabolic Equation
1 Introduction
2 Study of the Inverse Problem of Determination a Source Depending on a Space Variable
3 Study of the Inverse Problem of Determination a Source Depending on a Time Variable
4 Results of Numerical Experiments
5 Conclusions
References
Using an Integrating Factor to Transform a Second Order BVP to a Fixed Point Problem
1 Introduction
2 Introducing a Term to Create a New Operator
3 Application of the New Fixed Point Theorem
References
Volterra Relatively Compact Perturbations of the Laplace Operator
1 Introduction
2 Compact Operators not in the Schatten Classes
3 Volterra Relatively Compact Perturbations of the Laplace Operator
References
Computational Aspects of the General Rodrigues Problem
1 Introduction
2 A Brief Review on Matrix Functions
3 The General Rodrigues Problem
4 Using the Hermite Interpolation Polynomial
4.1 The Complexity of the Rodrigues Problem
4.2 The Solution of the Rodrigues Problem When the Eigenvalues have Double Multiplicity
4.3 Example for n=4 and a0(f)(X)
References
Approximation by Max-Product Operators of Kantorovich Type
1 Introduction
2 Approximation Properties of LKn(M) Deduced from Those of Ln(M)
3 Max-Product Sampling Kantorovich Operators Based on Fejér Kernel
4 Max-Product Kantorovich Operators Based on (ϕ, ψ)-Kernels
5 Max-Product Sampling Kantorovich Operators Based on Generalized Kernels
6 Future Researches
References
Variational Inequalities and General Equilibrium Models
1 Introduction
2 General Equilibrium Economic Models
3 Different Variational Inequality Problems
4 Exchange Economy Model
4.1 Characterization by Means of a Variational Problem
4.2 Existence of Equilibria
4.3 Remark
5 Model with Nominal Assets
5.1 Variational Inequality Approach
5.2 Existence of Equilibrium
6 Model with Numeraire Assets
7 Model with Restricted Participation
References
The Strong Convergence of Douglas-Rachford Methods for the Split Feasibility Problem
1 Introduction
2 Preliminaries
3 Halpern-Type Algorithm
4 Haugazeau-Type Algorithm
5 Example Results
References
Some Triple Integral Inequalities for Functions Defined on Three-Dimensional Bodies Via Gauss-Ostrogradsky Identity
1 Introduction
2 Some Preliminary Facts
3 Identities of Interest
4 Integral Inequalities
5 Applications for Three-Dimensional Balls
References
Optimal Emergency Evacuation with Uncertainty
1 Introduction
2 The Deterministic Model
3 Two-Stage Stochastic Model
4 Two-Stage Variational Inequality Formulation
5 Numerical Results
6 Conclusions
References
On Global Hyperbolicity of Spacetimes: Some Recent Advances and Open Problems
1 Introduction to the Terrain
2 Some Preliminaries
2.1 A Spell of Domain Theory and General Topology
2.2 Causal Relations in a Spacetime
2.3 Weighted Riemannian Manifolds and All That
3 Globally Hyperbolic Spacetimes
4 Different Candidates for a Topology: Which Is the Most Fruitful'' One?
5 The Klein–Gordon Equation in Globally Hyperbolic Manifolds
References
Spectrum Perturbations of Linear Operators in a Banach Space
1 Introduction
2 The Quasi-normed Ideal Γp
3 Particular Cases
3.1 Absolutely p-Summing Operators
3.2 Ideal Ep and Absolutely (p, 2)-Summing Operators
4 Additional Upper Bounds for Determinants
5 Hille–Tamarkin Integral OperatorsClose'' to Volterra Ones
6 Hille–Tamarkin Infinite Matrices ``Close'' to Triangular Ones
7 An Inequality Between Resolvents and Determinants for Nuclear Operators in a Banach Space
8 Perturbations of Nuclear Operators
9 Maximal Chains of Projections
10 Operators Having Continuous Maximal Chains
11 Norm Estimates for Resolvents
12 Perturbations of Triangularizable Operators
13 Powers of Volterra Operators in Lp
13.1 Hille–Tamarkin Volterra Operators
13.2 Volterra Operators in L1 and L∞
14 Triangularizable Operators in Lp
15 Integral Operators in Lp
15.1 The Case 1<p<∞
15.2 The Case p=1
16 Multiplicative Representations for Resolvents of Operators in a Banach Space
References
Perturbations of Operator Functions: A Survey
1 Introduction
2 Norm Estimates for Resolvents of Operators in a Hilbert Space
2.1 Properties of Singular Numbers
2.2 The Resolvent of a Hilbert–Schmidt Operator
2.3 The Resolvent of a Schatten–von Neumann Operator
2.4 The Resolvent of an Operator with a Hilbert–Schmidt Component
2.5 The Resolvent of an Operator with a Schatten–von Neumann Component
2.6 Resolvents of Finite-Dimensional and Nuclear Operators
Finite-Dimensional Operators
Nuclear Operators
3 Norm Estimates for Operator Functions Regular on the Convex Hull of Spectra
4 Functions Nonregular on the Convex Hull of the Spectrum
5 Representations of Commutators
6 Perturbations of Taylor Series with Operator Arguments
7 Entire Operator-Valued Functions
8 Perturbations of Operator Functions Regular on the Convex Hull of the Spectrum
8.1 The Finite-Dimensional Operators
8.2 Operators with Hilbert–Schmidt Components
9 Perturbations of Functions of Infinite Matrices
9.1 Statement of the Result
9.2 Proof of Theorem 14
10 Positivity Conditions for Operator Functions in a Hilbert Lattice
11 Proof of Theorem 15
12 Examples to Theorem 15
13 Perturbations of Operator Functions in a Hilbert Lattice
14 Perturbations of Operator Fractional Powers
15 Perturbations of the Operator Logarithm
15.1 Definition via Contour Integral
15.2 Definition via Improper Integrals
References
Representation Variety for the Rank One Affine Group
1 Introduction
2 General Background
2.1 Character Varieties
2.2 Representation Varieties of Orientable Surfaces
2.3 Mixed Hodge Structures
2.4 Grothendieck Ring of Algebraic Varieties
3 Geometric Method
3.1 Stratification Analysis and Computation of Virtual Classes
3.2 The Moduli Space of the Representations and the Character Variety
4 Arithmetic Method
4.1 Katz Theorem and E-Polynomials
4.2 Representation Variety for the Affine Group
4.3 Exhaustive Polynomial Count
5 Quantum Method
5.1 Definition of Topological Quantum Field Theories
5.2 Quantization of the Virtual Classes of Representation Varieties
5.3 Representation Varieties via the Quantum Method
5.4 Concluding Remarks
References
A Regularized Stochastic Subgradient Projection Method for an Optimal Control Problem in a Stochastic Partial DifferentialEquation
1 Introduction
2 Regularized Stochastic Subgradient Projection Method
3 Optimal Control for Stochastic PDEs
4 A Numerical Example
5 Concluding Remarks
References
A Survey on Interpolative and Hybrid Contractions
1 Introduction
2 Preliminaries
2.1 Simulation Functions
2.2 Comparison and c-Comparison Functions
2.3 Admissible Mappings
2.4 Branciari Distance Space
2.5 Partial Metric Spaces
2.6 b-Metric Spaces
3 Quasi-Metric Spaces
3.1 Significant Contractions in Metric Fixed Point Theory
4 Interpolative Contraction
4.1 Motivation
4.2 A Pioneering Notion: An Interpolative Kannan Type Contraction
4.3 An Interpolative Rus–Reich–Ćirić Type Contraction Int1
Consequences
Interpolative Rus–Reich–Ćirić Type Contractions on Branciari Distance Spaces
Interpolative Rus–Reich–Ćirić Type Contractions on Partial Metric Spaces
4.4 An Interpolative Hardy-Rogers Type Contraction
5 Hybrid Contractions
5.1 Hybrid Contractions in the Context of Quasi-metric Spaces
5.2 Jaggi Type Hybrid Contraction
6 Hybrid Contractions in b-Metric Spaces
7 Admissible Hybrid Z-Contractions in b-Metric Spaces
7.1 Hybrid Contractions in Branciari Distance Spaces
References
Identifying the Computational Problem in Applied Statistics
1 Introduction
2 Computational Difficulties: GLM
2.1 Iterative p2 Calculation
3 Difficulties: Adopting HF Calculation
3.1 Hypergeometric Functions (HF) in Statistics
3.2 Transformations Can Reduce Calculations
4 Discussion
Appendix 1: On Non-central Chi-Square
Appendix 2: Numerical Evaluation of Hypergeometric Function
References
Fractional Integral Operators in Linear Spaces
1 Introduction
2 Generalized Fractional Integral Operators and Fractional Area Balance Operators
3 Some Lemmas
4 Some Inequalities for Operator T25
5 Some Inequalities for Operator T28
References
Anisotropic Elasticity and Harmonic Functions in CartesianGeometry
1 Introduction
2 Basic Theory in Anisotropic Elasticity
3 Complete Isotropy in Elasticity and Fundamental Equation
4 Anisotropy in Elasticity and Fundamental Equation: The Cubic System
5 Anisotropic Harmonic Eigenfunctions
6 Conclusions and Discussion
References
Hyers–Ulam Stability of Symmetric Biderivations on BanachAlgebras
1 Introduction and Preliminaries
2 Hyers–Ulam Stability of Symmetric Biderivations and Skew-Symmetric Derivations on Banach Algebras: Direct Method
3 Hyers–Ulam Stability of Skew-Symmetric Biderivations on Banach -Algebras: Direct Method
4 Hyers–Ulam Stability of Symmetric Biderivations and Skew-Symmetric Derivations on Banach Algebras: Fixed Point Method
5 Hyers–Ulam Stability of Skew-Symmetric Biderivations on Banach -Algebras: Fixed Point Method
6 Conclusions
References
Some New Classes of Higher Order Strongly Generalized Preinvex Functions
1 Introduction
2 Preliminary Results
3 Main Results
4 Applications
5 Conclusion
References
Existence of Global Solutions and Stability Results for a Nonlinear Wave Problem in Unbounded Domains
1 Introduction: Preliminaries
2 Global Existence, Blow-Up Results, and Invariant Sets
3 Stability Results
References
Congestion Control and Optimal Maintenance of Communication Networks with Stochastic Cost Functions: A VariationalFormulation
1 Introduction
2 The Congestion Control Model and Its Stochastic Variational Inequality Formulation
3 The Optimal Network Improvement Model
4 Numerical Experiments
Appendix
References
Nonlocal Problems for Hyperbolic Equations
1 Introduction
2 Nonlocal Problems for Hyperbolic Equations with Integral Conditions
2.1 A Problem with Second-Kind Integral Conditions
2.2 A Problem with First-Kind Integral Conditions
3 A Problem with Dynamical Nonlocal Conditions
References
On the Solution of Boundary Value Problems for Loaded Ordinary Differential Equations
1 Introduction
2 Formulation of the Problem
3 Main Results
4 Applications
4.1 Three-Point BVP with Nonhomogeneous Boundary Conditions
4.2 Two-Point BVP with Point and Integral Boundary Conditions
4.3 Four-Point BVP with Integral Boundary Conditions
5 Conclusions
References
Set-Theoretic Properties of Generalized Topologically Open Sets in Relator Spaces
1 Introduction
2 A Few Basic Facts on Relations
3 A Few Basic Facts on Relators
4 Structures Derived from Relators
5 Further Structures Derived from Relators
6 Reflexive, Non-partial, and Non-degenerated Relators
7 Topological and Quasi-Topological Relators
8 A Few Basic Facts on Filtered Relators
9 A Few Basic Facts on Quasi-Filtered Relators
10 Some Further Theorems on Topologically Filtered Relators
11 Some More Particular Theorems on Topologically Filtered Relators
12 Some Generalized Topologically Open Sets
13 The Duals of the Families TRκ with ==q, ps, s, and p
14 The Duals of the Families TRκ with ==γ, δ, α, β, a, and b
15 Topologically Regular Open Sets
16 Some Further Theorems on the Family TRr
17 Characterizations of the Families TRκ with ==s, p, α, β, and b
18 Intrinsic Characterizations of the Families TR and TRκ with ==q, ps, s, p
19 Intrinsic Characterizations of the Families TRκ with ==γ, δ, α, β, a, b
20 Intrinsic Characterizations of the Family TRr
21 Conditions in Order That Could Be in TRκ
22 Conditions in Order That a Singleton Could Be in TRκ
23 Some Further Conditions in Order That a Singleton Could Be in TRκ
24 Conditions in Order That X Could Be in TRκ
25 Union Properties of the Families TRκ
26 An Illustrating Example to Theorem 124
27 Intersection Properties of the Families TRq and TRs
28 Intersection Properties of the Families TRps and TRp
29 Intersection Properties of the Family TRr
30 Intersection Properties of the Families TRα and with TRβ
31 Intersection Properties of the Families TRκ with ==γ, δ, a, and b
32 A Further Intersection Property of the Families TRs and TRα
33 Minimality Properties of the Families TRκ
34 Maximality Property of the Families TRκ
References
On Degenerate Boundary Conditions and Finiteness of the Spectrum of Boundary Value Problems
1 Introduction
2 On Degenerate Boundary Conditions in the Sturm–Liouville Problem
3 Degenerate Boundary Conditions for the Diffusion Operator
4 The Degenerate Boundary Conditions for Boundary Value Problems with an Odd-Order Differential Equation
5 The Degenerate Boundary Conditions for Boundary Value Problems with a Third-Order Differential Equation
6 On Degenerate Boundary Conditions for Operator D4
7 Degenerate Boundary Conditions for the Sturm–Liouville Problem on a Geometric Graph
8 Finiteness of the Spectrum of Boundary Value Problems
References
Deceptive Systems of Differential Equations
1 Introduction
2 Some Lemmas
3 Proofs
References
Some Certain Classes of Combinatorial Numbers and Polynomials Attached to Dirichlet Characters: Their Construction by p-Adic Integration and Applications to Probability Distribution Functions
1 Introduction and Preliminaries
1.1 Basic Properties of Dirichlet Characters
1.2 Some Basic Properties of p-Adic Integration Method
1.3 Other Needed Notations and Definitions
2 A Certain Class of Combinatorial Numbers Yn,χ(λ,q) and Polynomials Yn,χ(z;λ,q) Attached to Dirichlet Characters
3 Illustrations of the Generating Functions for the Numbers Yn,χ(λ) by Dirichlet Characters with Different Conductors d
4 Derivative and Integral Formulas for the Polynomials Yn, χ(z;λ,q)
5 Reduction to the Numbers Yn(λ) and the Polynomials Yn(x;λ)
6 Some Properties of the Numbers Yn(λ) and the Polynomials Yn(x;λ) with Their Generating Functions
7 Illustrations for the Numbers Yn(λ) and the Polynomials Yn(x;λ)
8 Some Identities Derived from Derivative Formulas of the Generating Functions for the Numbers Yn(λ) and the Polynomials Yn(x;λ)
9 Other Relations of the Numbers Yn(λ) and the Polynomials Yn(x;λ) with Some Special Numbers and Polynomials
10 Some Relations on Hypergeometric Functions Derived from the Integral of the Numbers Yn(λ) and the Polynomials Yn(x;λ)
11 Some Infinite Series Containing the Numbers Yn(λ)
12 Positive Higher-Order Extension of the Numbers Yn(λ) and the Polynomials Yn(x;λ) with Their Generating Functions
13 Negative Higher-Order Extension of the Numbers Yn(λ) and the Polynomials Yn(x;λ) with Their Generating Functions
13.1 Derivative Formulas and Recurrence Relations Derived from Partial Derivatives of the Functions G(t,k;λ) and G(t,x,k;λ)
14 Some Applications to the Probability Distribution Functions
15 Further Remarks and Observations
References
Pathwise Stability and Positivity of Semi-Discrete Approximations of the Solution of Nonlinear Stochastic Differential Equations
1 Introduction
2 Setting and Main Results
3 Proof of Convergence
3.1 Convergence of (ynΔ)
4 Proof of Stability
4.1 Stability of (ynΔ)
5 Numerical Illustration
6 Discussion and Future Work
References
Solution of Polynomial Equations
1 Introduction
2 General Description of the Solution of Our Problem
2.1 First Stage
2.2 Second Stage
2.3 Third Stage
3 Complete Description of Our Method
3.1 First Stage
3.2 Second Stage
3.3 Third Stage
4 Prerequisites
4.1 An Algorithm for the Multiplicity of a Root
Appendix
References
Meir–Keeler Sequential Contractions and Pata Fixed Point Results
1 Introduction
2 Dependent Choice Principles
3 Conv-Cauchy Structures
4 Meir–Keeler Relations
5 Statement of the Problem
6 Main Result
7 Pata Fixed Point Results
References
Existence and Stability of Equilibrium Points Under the Influence of Poynting–Robertson and Stokes Drags in the Restricted Three-Body Problem
1 Introduction
2 Equations of Motion with Dissipative Forces
3 Existence and Locations of the Equilibrium Points
4 Stability of the Non-collinear Equilibrium Points
5 Discussion and Conclusion
References
Nearest Neighbor Forecasting Using Sparse Data Representation
1 Introduction
2 A Modification of Classical Nearest Neighbors Forecasting
3 Sparse Data Representation
3.1 The Mathematics of SDR
3.2 Classification
3.3 Encoding
3.4 Transformations of SDR
3.5 Classification and Prediction Using SDR
4 Experimental Results
5 Conclusions
References
On Two Kinds of the Hardy-Type Integral Inequalities in the Whole Plane with the Equivalent Forms
1 Introduction
2 An Example and Two Lemmas
3 Main Results and Particular Cases
4 Conclusions
References
Product Formulae for Non-Autonomous Gibbs Semigroups
1 Introduction and Main Result
2 Preliminaries
3 Proof of Theorem 1.4
References