Mathematical Analysis Differential Equhb: Mathematical Analysis, Differential Equations and Applications

"This comprehensive volume presents essential mathematical results devoted to topics of mathematical analysis, differential equations and their various applications. It focuses on differential operators, Wardowski maps, low-oscillation functions, Galois and Pataki connections, Hardy-type inequalities, to name just a few. Effort has been made for this unique title to have an interdisciplinary flavor and features several applications such as in tomography, elastic scattering, fluid mechanics, etc. This work could serve as a useful reference text to benefit professionals, academics and graduate students working in theoretical computer science, computer mathematics, and general applied mathematics"-- Provided by publisher Contents Preface About the Editors 1. Galois and Pataki Connections on Power Sets 1. Introduction 2. A Few Basic Facts on Relations 3. Some Important Relational Properties 4. Lower and Upper Bounds in Gosets 5. Some Further Important Basic Tools in Gosets 6. A Few Basic Facts on Increasing Functions 7. The Induced Order and Interior Relations 8. Extensive, Involutive and Idempotent Functions 9. Involution, Projection and Closure Operations 10. Galois-Type Connections between Gosets 11. Some Basic Properties of Increasingly Semiregular Functions 12. Some Basic Properties of Increasingly Seminormal Functions 13. Some Very Particular Properties of Increasingly Regular and Normal Functions 14. Characterizations of Increasingly Seminormal Functions 15. Some Further Characterizations of Increasingly Normal Functions 16. Characterizations of Semiregular Functions 17. Relational Characterizations of Increasingly Seminormal Functions 18. Relational Characterizations of Increasingly Semiregular Functions 19. Increasingly Seminormal Functions of Power Sets to Gosets 20. Characterizations of Increasing GF-Seminormalities 21. Some Further Characterizations of Increasing GF-Seminormalities 22. Increasingly Semiregular Functions of Power Sets to Gosets 23. Characterizations of ΦF-Semiregularities 24. Increasing Semiregularity and Seminormality Properties of Quasi-Increasing Functions 25. A Few Basic Fats on Union-Preserving Functions 26. Increasing Normality and Regularity of Union-Preserving Functions References 2. A Functional Equation Related to Inner Product Spaces in Šerstnev Probabilistic Normed Spaces 1. Introduction 2. Triangular Norms and Probabilistic Normed Spaces 3. Generalized Metrics and Fixed Point 4. Stability of Functional Equation (2): An Even Case 5. Stability of Functional Equation (2): An Odd Case 6. Stability of Functional Equation (2): A Mixed Case References 3. Hyers–Ulam–Rassias Stability of Set-Valued Functional Equations: A Fixed Point Approach 1. Introduction and Preliminaries 2. Main Results References 4. Two Heuristic Methods for Solving Generalized Nash Equilibrium Problems Using a Novel Penalty Function 1. Introduction 2. Mathematical Formulation of Generalized Nash Equilibrium Problems 3. Shadow Point Penalty Function 4. Evolutionary-Inspired Algorithm 4.1. Implementation on known examples 5. Stochastic Gradient Descent Algorithm 5.1. Implementation on known examples: SGD 6. Limitations and Future Work 6.1. EIA 6.2. SGD 7. Conclusion Acknowledgments References 5. The Finite Element Method with Applications to Fluid Mechanics 1. Basic Principles of Finite Elements 1.1. Introduction 1.2. Finite element theory 1.2.1. Basic concepts and definitions 1.2.2. Error estimates 1.2.3. Piecewise polynomial spaces 1.2.4. An application to the Duffing equation 2. The Stokes and Navier–Stokes Problems 2.1. The Stokes problem 2.1.1. A priori error estimates 2.1.2. The backward-facing step problem 2.2. The Navier–Stokes problem 2.2.1. An application to the Poiseuille flow 3. Discontinuous and Adaptive FEM 3.1. The discontinuous Galerkin FEM 3.1.1. Application to the Poisson and Stokes equations 3.2. Adaptive mesh refinement FEM Conclusions Acknowledgment References 6. Comparison between Two Descent SQP-ADMs for Structured Variational Inequalities 1. Introduction 2. SQP Alternating Direction Method 3. Basic Results 4. Convergence of the Proposed Method 5. Comparison of Two Methods 6. Preliminary Computational Results 7. Conclusions References 7. Solvability for a Class of Unilateral Contact Problems with Friction and Damage 1. Introduction 2. The Mathematical Formulation 3. Statement of the Result 4. Proof of Theorem 1 References 8. Vector Inequalities for Analytic Functions of Operators in Hilbert Spaces and Applications for Numerical Radius and p-Schatten Norm 1. Introduction 2. Vector Inequalities 3. Norm and Numerical Radius Inequalities 4. Inequalities for Trace of Operators 5. Applications for Complex Perspectives 6. Two Examples References 9. General Equivariant Minimax Principle and Fountain Theorem in the Presence of Nonadmissible Representations 1. Introduction 2. Isometric Representation on a Banach Space 3. An Equivariant Deformation Theorem 4. An Equivariant Minimax Theorem 5. A Nonsmooth Version of Fountain Theorem 6. Admissible Representations References 10. On a Prey–Mesopredator–Predator System 1. Introduction 2. Preliminaries 2.1. The model of Malthus 2.2. Verhulst model with a constant rate predation 2.3. Verhulst model with variable rate predation 3. Three-Compartment Model References 11. Differential Operator Associated with the (q, k)-Symbol Raina’s Function 1. Introduction 2. Procedures 2.1. Geometric techniques 2.2. Jackson calculus 2.3. Raina’s function 2.4. Arguments 3. Findings 4. Conclusion References 12. Analysis and Solvability of Complex K-Symbol Liu Fractional Dynamical Systems 1. Introduction 2. K-Symbol Methods 2.1. K-symbol Mittag–Leffler function 3. Results 3.1. Stabilizing the origin 3.2. Computational outcome 3.3. Synchronizing at the origin 3.4. Solvability 3.5. Upper solution 4. Conclusion References 13. Elastic Scattering by an Inhomogeneous Medium with Unknown Buried Obstacles 1. Introduction 2. The Direct Scattering Problem 3. The Inverse Scattering Problem and the Factorization Method References 14. Some New Bounds of Gauss–Jacobi and Hermite–Hadamard-Type Integral Inequalities 1. Introduction 2. Some New Bounds of the Quadrature Formula of Gauss–Jacobi Type 3. Some New Bounds of Hermite–Hadamard Type via k-Fractional Integral Inequalities 4. Applications to Special Means Acknowledgements References 15. Payoff-Independent Action Update for Continuous Action Social Dilemmas: A Preliminary Investigation 1. Introduction 2. Preliminaries 2.1. Differential equations 2.2. Bimatrix games 3. Social Dilemmas 3.1. Two players, two actions 3.2. Two players, continuous actions 4. The Basic Action Update Model 4.1. The update equations 4.2. Equilibria 4.3. Attraction basins 4.3.1. Initial conditions (x1(0), x2(0)) ∈ (1/2, 1) × (1/2, 1) 4.3.2. Initial conditions (x1(0), x2(0)) ∈ (1/2, 1) × (0, 1/2) 4.3.3. Initial conditions (x1(0), x2(0)) ∈ (0, 1/2) × (1/2, 1) 4.3.4. Initial conditions (x1(0), x2(0)) ∈ (0, 1/2) × (0, 1/2) 4.3.5. Some initial conditions belonging to the set {0, 1/2, 1} 4.3.6. Summary and discussion 5. Extensions 5.1. Applicability to general continuous action social dilemmas 5.2. Symmetric update equations with unitary coefficients 5.3. Asymmetric update equations with unitary coefficients 5.4. General update equations 5.5. More than two players 6. Achieving a Prescribed NE 7. Conclusion References 16. Applying Logarithm Sobolev Inequalities to Probability and Statistics 1. Introduction 2. Sobolev Inequalities 3. Adopting LSI to Statistics 4. Statistical Extensions 5. Discussion A.1. Proof of Proposition 1 A.2. The “Normal Equations” for Nγ(μ, σ2I) A.3. Sketch of the Proof References 17. Some Certain Families of Catalan-Type Numbers and Polynomials: Analysis of Their Generating Function with Their Functional Equations 1. Preliminaries 1.1. Catalan numbers and their generating functions 1.2. Some notations and formulas for the beta and gamma functions and factorial polynomials 1.3. Some certain families of special numbers and polynomials 2. Some Formulas for the Catalan Numbers 2.1. Some formulas for the Catalan numbers derived from the Euler’s Gamma function and Beta function 2.2. Riemann integral representations for the Catalan numbers 2.3. p-adic integral representations for the Catalan numbers 2.4. A formula covering the Catalan numbers, the Stirling numbers of the second kind and the Bernoulli numbers of negative order 2.5. Inequalities and asymptotic expansions including the Catalan numbers 2.6. Approximation for the Catalan numbers 3. Some Catalan-Type Numbers 4. Further Classes of Catalan-Type Numbers and Polynomials 4.1. Further properties of the Catalan-type numbers Vn(ω) 4.2. Asymptotic behaviour of the Catalan-type numbers Vn(ω) 4.3. Derivative formulas for the Catalan-type polynomials Vn(z; ω) 4.4. Integral formulas for the Catalan-type numbers Vn(ω) and Catalan-type polynomials Vn(z; ω) 4.4.1. Formulas for Riemann integral of the polynomials Vn(z; ω) 4.4.2. Formulas for p-adic integrals of the polynomials Vn(z; ω) 4.4.3. Contour integral formulas for the numbers Vn(ω) 4.5. Infinite series representations for the Catalan-type numbers 4.6. Some inequalities for the Catalan-type numbers 4.7. Relations of the Catalan numbers with other special numbers References 18. On the Generalizations of the Cauchy–Schwarz–Bunyakovsky Inequality with Applications to Elasticity 1. Introduction 2. Cauchy–Bunyakovsky–Schwarz Inequality 2.1. Cauchy inequality 2.2. Cauchy–Bunyakovsky–Schwarz inequality 2.3. Geometrical interpretation of the CBS inequality 2.4. Triangular inequality 3. A Cauchy–Bunyakovsky–Schwarz-Type Inequality 4. Strengthened Cauchy–Bunyakovsky–Schwarz Inequality for Elasticity Appendix References 19. Lie Bracket Derivations in Banach Lie Algebras 1. Introduction and Preliminaries 2. Hyers–Ulam Stability of Lie Bracket Derivations in Complex Banach Lie Algebras: Fixed Point Method 3. Hyers–Ulam Stability of Lie Bracket Derivations in Banach Lie Algebras: Direct Method References 20. Aboodh Transform and Ulam Stability of Second-Order Linear Differential Equations 1. Introduction 2. Preliminaries 3. Ulam Stabilities for (1) 4. Ulam Stabilities for (2) 5. Examples and Remarks References 21. Some Classes of Extended General Variational Inequalities 1. Introduction 2. Preliminaries and Basic Concepts 3. Projection Methods 4. Wiener–Hopf Equations Technique 5. Dynamical System Technique 6. Auxiliary Principle Technique 7. General Equilibrium Problems 8. Generalizations and Extensions Acknowledgements References 22. Biconvex Functions and Bivariational Inequalities 1. Introduction 2. Preliminary Results 3. Properties of Biconvex Functions 4. Bivariational Inequalities 5. Conclusion Acknowledgements References 23. Some Properties of a Class of Network Games with Strategic Complements or Substitutes 1. Introduction 2. Basics on Network Games and Variational Inequalities 3. The Parametric Quadratic Model 4. Numerical Experiments 5. Conclusions and Future Research Perspectives Acknowledgements References 24. Chebyshev Polynomials of the First Kind and Applications in Tomography 1. Introduction 2. The Two-Dimensional Radon Transform and Its Inversion 3. Numerical Implementation via Chebyshev Polynomials 3.1. Chebyshev polynomials of the first kind 3.2. Inverse Radon transform via Chebyshev polynomials of the first kind 3.3. Numerical implementation of the inversion of the Radon transform via Chebyshev polynomials of the first kind: Analytic examples References 25. Bernstein-Type Polynomials Associated with Characteristic Function, Moment Generating Functions of Beta-Type Distribution and Their Approximation Applications 1. Introduction 2. Beta Distributions and Their Properties 3. Beta-Type Distributions and Their Properties 3.1. Series representations 4. Moment Generating Functions and Characteristic Function 4.1. Moment generating function for beta-type distributions 4.2. Formulas expected value of beta-type distributions 4.3. Formula for moment generating function for the distribution F(x; a, b; n, k) 5. Characteristic Function for Beta-Type Distributions 6. Approximation Properties of the Bernstein Polynomials References 26. On Removing Diverse Data for Training Machine Learning Models 1. Introduction 2. The Maximum Diversity Problem 3. Instance Selection by Maximum Diversity 4. Experimental Methodology 4.1. K-nearest neighbors 4.2. Baseline methods 4.3. The α parameter 5. Computational Experiments 5.1. Datasets 5.2. Evaluation 5.3. Cross-validation 6. Results and Discussion 6.1. Classification performance results 6.2. Wilcoxon tests 7. Concluding Remarks Acknowledgments References 27. Wardowski Maps Modeled by Low-Oscillation Functions 1. Introduction 2. Matkowski Associated Maps 3. Main Results 4. Particular Aspects 5. Wardowski–Dung Maps References 28. Contractive Maps on Relational MC-Quasimetric Spaces 1. Introduction 2. Dependent Choice Principles 3. Classes of Pseudometric Spaces 4. Statement of the Problem 5. Main Result 6. Ahmed–Fulga Approach 7. Nonlinear Aspects References 29. Hyers–Ulam–Rassias Stability of the Nonlinear Fractional Differential Equations with ρ-Fractional Derivative 1. Introduction 2. Preliminary 3. Main Results References 30. Out-of-Plane Equilibrium Points in the Restricted Three-Body Problem with Radiation Pressure, Poynting–Robertson Drag and Angular Velocity Variation 1. Introduction 2. Equations of Motion 3. Positions and Stability of the Out-of-Plane Equilibrium Points 4. Numerical Application 5. Discussion and Conclusion Acknowledgment References Index