Part of Cambridge Mathematical Library
available from February 2025format: Paperbackisbn: 9781009602112
The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This book, reissued with a new Foreword, presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.
Very concrete approach; should appeal to physicists as well as mathematicians
Requires only a first acquaintance with complex function theory
Ideal introduction to the subject for graduate students and researchers in mathematics and physics
Foreword: Preface
1. First ideas: complex manifolds, Riemann surfaces, and projective curves
2. Elliptic integrals and functions
3. Theta functions
4. Modular groups and modular functions
5. Ikosaeder and the quintic
6. Imaginary quadratic number fields
7. Arithmetic of elliptic curves
References
Index.
In the series De Gruyter Textbook
This book introduces quadratic ideal numbers as objects of study with applications to binary quadratic forms and other topics. The text requires only minimal background in number theory, much of which is reviewed as needed. Computational methods are emphasized throughout, making this subject appropriate for individual study or research at the undergraduate level or above.
The definition of ideal numbers as objects in their own right distinguishes the book from others.
Emphasizes practical computations throughout.
Python code for calculations included at the end of each chapter.
Frontmatter
Publicly Available I
Preface
Publicly Available VII
Acknowledgments
Publicly Available IX
licly Available XI
1 Introduction to ideal numbers 1
2 Congruence classes of ideal numbers 21
3 Composition of ideal numbers 39
4 Classes of ideal numbers 60
5 Genera of ideal numbers 78
6 Ideal numbers of negative discriminant 96
7 Ideal numbers of positive discriminant 120
A Appendix: Review of number theory 149
B Appendix: Review of groups 172
C Appendix: Ideals of quadratic domains 185
D Appendix: Continued fractions 215
References 233
Index
Volume 64 in the series De Gruyter Studies in Mathematics
The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderon-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincare, and Robin boundary conditions in regular Semmes-Kenig-Toro domains.
The 1-st edition of the “Hodge-Laplacian”, De Gruyter Studies in Mathematics,
Volume 64, 2016, is a trailblazer of its kind, having been written at a time when new results in Geometric Measure Theory have just emerged, or were still being developed. In particular, this monograph is heavily reliant on the bibliographical items. The latter was at the time an unpublished manuscript which eventually developed into the five-volume series “Geometric Harmonic Analysis” published by Springer 2022-2023. The progress registered on this occasion greatly impacts the contents of the “Hodge-Laplacian” and warrants revisiting this monograph in order to significantly sharpen and expand on previous results. This also allows us to provide specific bibliographical references to external work invoked in the new edition.
Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals.
New developments are addressed in a largely self-contained manner
Offers also a panoramic view of the topics covered
Frontmatter
Publicly Available I
Preface to the first edition
Publicly Available V
Preface to the second edition
Publicly Available VII
Contents
Publicly Available IX
Acknowledgement
Publicly Available XIII
1 Introduction and statement of main results 1
2 Geometric concepts and tools 49
3 Harmonic layer potentials associated with the Hodge?de Rham formalism on UR domains 116
4 Harmonic layer potentials associated with the Levi?Civita connection on UR domains 149
5 Dirichlet and Neumann problems for the Hodge?Laplacian in infinitesimally flat AR domains 193
6 Fatou theorems and integral representations for the Hodge?Laplacian in infinitesimally flat AR domains 229
7 Solvability of boundary problems for the Hodge?Laplacian in the Hodge?de Rham formalism 277
8 Additional results and applications 321
9 Tools from differential geometry, harmonic analysis, geometric measure theory, functional analysis, partial differential equations and Clifford analysis 384
Bibliography 593
Author index 599
Subject index 601
Symbol index
In the series De Gruyter Textbook
The first two parts of this book focus on developing standard analysis concepts in the extended complex plane. We cover differentiation and integration of functions of one complex variable. Famous Cauchy formulas are established and applied in the frame of residue theory. Taylor series is used to investigate analytic functions, and they are connected to harmonic functions. Laurent series theory is developed.
The third part of the book finds applications of the earlier chapter in conformal mappings and the Laplace transform. Special functions solving ordinary differential equations are studied extensively, along with their asymptotic behavior. A highlight of the book is the elliptic function of Weierstrass and Jacobi. Finally, we present Laplace’s method, which is applied to find large arguments asymptotic of some special functions.
The book is filled with examples, exercises, and problems of varying degrees of difficulty. This makes it useful to all students in mathematics, physics, and related fields.
The book combines the classical treatment of complex analysis with a wide array of problems and exercises.
Special functions and elliptic functions are covered.
Large argument asymptotic for special functions are developed using Laplace’s method.
Frontmatter
Publicly Available I
Preface
Valery Serov and Markus Harju
Publicly Available V
Contents
Publicly Available VII
Part I
1 Complex numbers and their properties
1
2 Functions of complex variable
19
3 Analytic functions (differentiability)
34
4 Integration of functions of complex variable (curve integration)
50
5 Cauchy theorem and Cauchy integral formulae
57
Exercises
73
Part II
6 Fundamental theorem of integration 79
7 Harmonic functions and mean value formulae 83
8 Liouville’s theorem and the fundamental theorem of algebra 98
9 Representation of analytic functions via the power series 101
10 Laurent’s expansions 117
11 Residues and their calculus 135
12 The principle of the argument and Rouche’s theorem 144
13 Calculation of integrals by residue theory 149
14 Calculation of series by residue theory 170
15 Entire functions 175
Exercises 193
Part III
16 Conformal mappings 199
17 Laplace transform 220
18 Special functions 247
Exercises 340
Bibliography 349
Index
Volume 45 in the series De Gruyter Series in Nonlinear Analysis and Applications
The book contains eleven chapters introduced by an introductory description. Qualitative properties for the semilinear dissipative wave equations are discussed in Chapter 2 and Chapter 3 based on the solutions with compactly supported initial data. The purpose of Chapter 4 is to present results according to the well-possednes and behavior f solutions the nonlinear viscoelastic wave equations in weighted spaces. Elements of theory of Kirchhoff problem is introduced in Chapter 5. It is introduced same decay rate of second order evolution equations with density. Chapter 6 is devoted on the original method for Well posedness and general decay for wave equation with logarithmic nonlinearities. In Chapter 7, it is investigated the uniform stabilization of the Petrovsky-Wave nonlinear coupled system. The question of well-posedness and general energy decay of solutions for a system of three wave equations with a nonlinear strong dissipation are investigated in chapter 8 using the weighied. In sofar as Chapter 9 and chapter 10 are concerned with damped nonlinear wave problems in Fourier spaces. The last Chapter 11 analysis the existence/ nonexistence of solutions for structural damped wave equations with nonlinear memory terms in Rn.
The first book is devoted to the qualitative properties of problems in Rn.
Provided classes of physical systems governed by equations with different dissipative mechanisms.
Equipped with more details that do not exist in the literature.
Frontmatter
Publicly Available I
Preface
Khaled Zennir and Svetlin G. Georgiev
Publicly Available V
Contents
Publicly Available VII
1 Introduction 1
2 Semilinear dissipative wave equations in ?n 29
3 Viscoelastic wave equation in ?n 48
4 Nonlinear viscoelastic wave equations in weighted spaces 71
5 Wave equation of Kirchhoff type with density 103
6 Wave equation with logarithmic nonlinearities in Kirchhoff type 130
7 Petrowsky?Petrowsky system in ?n 147
8 System of three wave equations 161
9 Damped wave problems with memory term in Fourier spaces 207
10 Degenerate evolution equations in ?n 230
11 Structural damped wave equations with nonlinear memory terms 245
Bibliography 287
Index
Volume 77 in the series De Gruyter Expositions in Mathematics
Tropical Mathematics built on Idempotent Semi-Rings and Dioids permits an extension of the usual Linear methods to Non-Linear problems and provides powerful analyzing and computing in Theoretical Physics and Applied Mathematics. Until recently, solutions in mathematics and physics were organized around algebraic structures such as groups, rings, and fields. These techniques are not well-suited to modeling and solving non-linear problems.
This book covers how Idempotent Mathematics when applied appropriately can be a versatile and powerful way to transform non-linear problems into linear ones and can provide solutions to complicated theoretical physics problems.
Frontmatter
Publicly Available I
Preface
Publicly Available VII
Contents
Publicly Available XIII
Basic notations
Publicly Available XIX
Part I: Elements of tropical mathematics
Introduction 1
1 Elements of tropical algebras 5
2 Elements of tropical topology 20
3 Elements of tropical analysis 28
4 Elements of tropical functional analysis 47
Part II: Some applications of tropical bi-wavelets to signal processing
Introduction 81
5 Tropical bi-wavelets for Holder exponents and fractal dimensions calculations 83
6 Tropical bi-wavelets for multi-fractal analysis 93
7 Tropical bi-wavelets for image processing 105
Part III: Tropical intervals
Introduction 113
8 Tropical intervals 115
9 Tropical inclusion functions 135
10 Tropical probabilist set inversion 139
Part IV: Some applications of tropical mathematics in theoretical physics
Introduction 145
11 Numerical applications with tropical intervals 149
12 Tropical transforms for theoretical physics 150
13 Contributions of tropical mathematics to statistical physics 159
14 (min, +)-Path integral 163
15 Tropical complex variational calculus 175
Postface
Introduction 185
A Tropical intervals numerical implementation 187
B Tropical probabilist set inversion 191
C Tropical intervals functions optimization examples 197
D Tropical intervals linear algebra examples 203
Bibliography 209
Index