دانلود کتاب Advances in Number Theory and Applied Analysis

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Contents
Preface
About the Editors
Acknowledgments
Chapter 1. Open Problems in Number Theory
1. Introduction
2. Twin Prime Conjecture
3. The Goldbach Conjecture
4. The Riemann Hypothesis
5. The Mersenne Primes
6. The Fermat Numbers
7. Some Other Famous Open Problems
Acknowledgments
References
Chapter 2. Arithmetic and Congruence Properties of Partition Function
1. Introduction
2. Partitions
2.1. Unrestricted partitions
2.2. Restricted partitions
2.3. Perfect partitions
2.4. Plane and solid partitions
2.5. l-regular partitions
2.6. Overpartitions
3. Identities
3.1. Ramanujan’s identities
3.2. Rogers–Ramanujan identities
4. Generating Function
4.1. Generating function for unrestricted partitions
4.2. Generating function for overpartitions
4.3. Generalizations of p(n)
5. Congruences
5.1. Ramanujan’s Conjecture
5.2. The Parity of p(n)
5.3. Congruences for pj(n)
6. Generalization of the Congruences Modulo for All pj(n)
6.1. New infinite families of congruences modulo 11 for pj(n)
6.2. Ramanujan-type congruences modulo 8 for the overpartition function p0(n)
6.3. Some new identities for p(n) modulo 11, 17, 19 and 23
7. Conclusion
Acknowledgments
References
Chapter 3. Collatz Hypothesis and Kurepa’s Conjecture
1. Introduction
2. The Collatz Conjecture
3. Planck’s Radiation
4. Collatz Generalization
5. Kurepa’s Hypothesis
6. Diophantine Equation
7. Numerical Analysis for Kurepa’s Hypothesis
8. Conclusions
Acknowledgment
References
Chapter 4. On Zeta Functions and Allied Theta Functions
1. Introduction and Ramified RHB Correspondence
1.1. Periodic Dirichlet series
2. Genesis of the Lerch–Chowla–Selberg Formula
2.1. Chowla–Selberg integral formula
2.2. Genesis of the formula
3. Product Expansion of Theta-Allied Functions
Acknowledgments
References
Chapter 5. Sylvester Sums on the Frobenius Set in Arithmetic Progression with Initial Gaps
1. Introduction
2. Main Result
2.1. Examples
2.2. Special patterns
3. Bigger Gaps
3.1. General case
3.2. Special patterns
4. Comments
Acknowledgments
References
Chapter 6. Arithmetic Properties of Minimal Excludants of Partitions of Integers
1. Introduction
2. Congruences for σMex(n)
3. Congruence for σMoex(n)
4. Conclusion
Acknowledgments
References
Chapter 7. On Certain ℓ-regular Partitions: A Brief Survey
1. Introduction
2. ℓ-Regular Partitions
3. ℓ-Regular Overpartitions
Acknowledgments
References
Chapter 8. Some Applications of an Algorithm in Number Theory
1. Introduction and Preliminaries
2. An Application of the Algorithm M to the Computation of the Integers x and y Satisfying the Equation n = x2 + Ny2
3. Application of the Algorithm M to an Open Question
4. Conclusion and Future Works
Acknowledgments
References
Chapter 9. The Components Exponential Function in Scator Hypercomplex Space: Planetary Elliptical Motion and Three-Body Choreographies
1. Introduction
2. Scator Algebra Preamble
3. Scators in S1+2
3.1. Scator projections
4. Elliptical Trajectory-Curve-Motion
4.1. Eccentric anomaly
4.2. Trueanomaly
4.3. Dynamics
4.4. Hyperbolae
5. Three Bodies in the C13 Curve
5.1. Three bodies in the C13 curve
5.2. Curve projection in two-dimensional planes
6. Trifolium Rose
7. Trisectrix Limaçon
8. Lemniscatae
8.1. Lemniscata with constant additive director component
8.2. Lemniscatafrom ex, ey projection with φy = 3φx
9. Dynamics of the Gerono-Type Lemniscata
9.1. Angular momentum
10. Conclusions
Acknowledgments
References
Chapter 10. Caristi-Type Nonunique Fixed-Point Results and Fixed-Circle Problem on bυ(s)-Metric Spaces
1. Introduction and Preliminaries
2. Main Results
3. An Application to Activation Functions
4. Conclusion
Acknowledgments
References
Chapter 11. Extended Interpolative Hardy–Rogers–Geraghty–Wardowski Contractions and an Application
1. Introduction
2. Main Results
3. An Application to Integral Equations
4. Conclusions
Acknowledgments
References
Chapter 12. (η, ψ)-Rational F-Contractions and Weak-Wardowski Contractions in a Triple-Controlled Modular-Type Metric Space
1. Introduction
2. Main Results
3. Weak-Wardowski Contraction
4. Triple-Controlled Fuzzy ħ-Metric Spaces
5. Application
Acknowledgments
References
Chapter 13. Sufficient Conditions for Mittag-Leffler Function Associated with Conic Regions
1. Introduction
2. Preliminaries
3. Main Results
4. Conclusion
Acknowledgments
References
Chapter 14. Results on the Existence of Solutions for Some Controlled Optimization Problems
1. Introduction
2. Problem Description
3. Main Results
4. Conclusions
Acknowledgments
References
Chapter 15. The Theory of Nonabsolute Integrable Function Spaces over R∞ and Its Various Applications
1. Introduction
2. Preliminaries
3. Measurable Functions
4. R∞I and Its Integration Theory
5. Kuelbs–Steadman Spaces Kp(R∞I ) 1 ≤ p ≤ ∞
6. GD2 Gross Steadman–Hilbert Spaces
7. The Family Kp(R∞I)
8. The Space M[R∞I]
9. Kp(R∞I) and Fourier and Convolution Operators
Acknowledgments
References
Chapter 16. Subclass of Analytic Functions Involving Mittag-Leffler Type Borel Distribution Series
1. Introduction
1.1. Mittag-Leffler type Borel distribution
2. Coefficient Inequality
3. Distortion and Covering Theorem
4. Radii of Starlikeness, Convexity and Close-to-Convexity
5. Extreme Points
6. Hadamard Product
7. Closure Theorems
8. Concluding Remarks
Acknowledgments
References
Chapter 17. Hybrid Accelerated Conjugate Gradient Method for Solving Nonlinear System of Equations
1. Introduction
2. Hybrid Accelerated Conjugate Gradient Algorithm
3. Convergence Analysis
4. Numerical Results
5. Conclusion
Acknowledgments
References
Chapter 18. A Closed Form of Integral Transforms in terms of Lauricella Function F(n)A and Their Numerical Simulations
1. Introduction and Preliminaries
2. Main Results
3. Special Cases
3.1. Case of μ = −ν = 1/2
3.2. Case of μ = ν = 1/2
3.3. Case of s = ∓r and ξ = −r − 1/2
3.4. Case of s = ±(1/2 − r) and ξ = r
3.5. Case of s = ±(a+1/2 + b) and ξ = a/2
3.6. Case of s = 0, ξ = r
3.7. Case of s = 0 and ξ = 1/2
3.8. Case of s = a + 1/4 and ξ = −1/4
3.9. Case of s = a + 3/4 and ξ = 1/4
3.10. Case of s = r − 1/2 and ξ = r
3.11. Case of s = −1/4 and ξ = 1/4
4. Numerical Simulations
5. Conclusion
6. Appendix
Acknowledgments
References
Index