Format: Hardback, height x width: 235x155 mm, Approx. 640 p.,
Series: Springer Monographs in Mathematics
Pub. Date: 14-Mar-2024
Brooks' Theorem (1941) is one of the most famous and fundamental theorems in graph theory it is mentioned/treated in all general monographs on graph theory. It has sparked research in several directions. This book presents a comprehensive overview of this development and see it in context. It describes results, both early and recent, and explains relations: the various proofs, the many extensions and similar results for other graph parameters. It serves as a valuable reference to a wealth of information, now scattered in journals, proceedings and dissertations. The reader gets easy access to this wealth of information in comprehensive form, including best known proofs of the results described. Each chapter ends in a note section with historical remarks, comments and further results. The book is also suitable for graduate courses in graph theory and includes exercises. The book is intended for readers wanting to dig deeper into graph coloring theory than what is possible in the existing book literature. There is a comprehensive list of references to original sources.
1 Degree Bounds for the Chromatic Number.- 2 Degeneracy and Colorings.-
3 Colorings and Orientations of Graphs.- 4 Properties of Critical Graphs.-
5 Critical Graphs with few Edges.- 6 Bounding by and .- 7 Coloring of
Hypergraphs.- 8 Homomorphisms and Colorings.- 9 Coloring Graphs on Surface.-
Appendix A: Brooks Fundamental Paper.- Appendix B: Tuttes Lecture from
1992.- Appendix C: Basic Graph Theory Concepts.
Format: Hardback, 531 pages, height x width: 235x155 mm, 10 Illustrations, color; 10 Illustrations, black and white; X, 531 p. 20 illus., 10 illus. in color
Pub. Date: 07-Mar-2024
ISBN-13: 9783031500619
This textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups. Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the DrinfeldLusztig double of a group algebra, appearing for the first time in a textbook at this level. Based on over twenty years of teaching various aspects of algebra, mainly at the Ecole Normale Superieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level.
1 Prerequisites and Preliminaries.- Part I Rings and Modules.- 2 Rings,
Polynomials, Divisibility.- 3 Polynomial Rings in Several Indeterminates.- 4
More on Modules.- 5 On Representations of Finite Groups.- Part II Integral
Domains, Polynomials, Fields.- 6 Prime and Maximal Ideals, Integral Domains.-
7 Fields, Division Rings.- Part III Finitely Generated Modules.- 8
Integrality, Noetherianity.- 9 Finitely Generated Projective Modules.- 10
Finitely Generated Modules Over Dedekind Domains.- 11 Complement on Dedekind
Domains.- Part IV Characteristic Zero Linear Representations of Finite
Groups.- 12 Monoidal Categories: An Introduction.- 13 Characteristic 0
Representations.- 14 Playing With the Base Field.- 15 Induction and
Restriction: Some Applications to Finite Groups.- 16 Brauers Theorem and
Some Applications.- 17 Graded Representations and Characters.- 18 The
DrinfeldLusztig Double of a Group Algebra.
Format: Hardback, 524 pages, height x width: 235x155 mm, 66 Illustrations, black and white; XIV, 524 p. 66 illus.,
Series: University Texts in the Mathematical Sciences
Pub. Date: 15-Mar-2024
ISBN-13: 9789819986798
Designed for senior undergraduate and graduate courses in mathematics and engineering, this self-contained textbook discusses key topics in linear algebra with real-life applications. Split into two partstheory in part I and solved problems in part IIthe book makes both theoretical and applied linear algebra easily accessible. Topics such as sets and functions, vector spaces, linear transformations, eigenvalues and eigenvectors, normed spaces, and inner product spaces are discussed in part I; while in part II, over 500 meticulously solved problems show how to use linear algebra in real-life situations. A must-have book for linear algebra courses; it also serves as valuable supplementary material.
1.Preliminaries.-
2. Vector Spaces.-
3. Linear Transformations.-
4. Eigenvalues and Eigenvectors.-
5. Normed Spaces and Inner Product Spaces.-
6. Bounded Linear Maps.-
7. Solved Problems: Preliminaries.-
8. Solved Problems: Vector Spaces.-
9.Solved Problems: Linear Transformations.-
10.Solved Problems: Eigenvalues and Eigenvectors.-
11. Solved Problems: Normed Spaces and Inner Product Spaces.-
12.Solved Problems: Bounded Linear Maps.
Format: Paperback / softback, 307 pages, height x width: 235x155 mm, V, 307 p.,
Series: Universitext
Pub. Date: 16-Mar-2024
ISBN-13: 9789819992546
This book provides a foundation for arithmetic topology, a new branch of mathematics that investigates the analogies between the topology of knots, 3-manifolds, and the arithmetic of number fields. Arithmetic topology is now becoming a powerful guiding principle and driving force to obtain parallel results and new insights between 3-dimensional geometry and number theory. After an informative introduction to Gauss' work, in which arithmetic topology originated, the text reviews a background from both topology and number theory. The analogy between knots in 3-manifolds and primes in number rings, the founding principle of the subject, is based on the etale topological interpretation of primes and number rings. On the basis of this principle, the text explores systematically intimate analogies and parallel results of various concepts and theories between 3-dimensional topology and number theory. The presentation of these analogies begins at an elementary level, gradually building to advanced theories in later chapters. Many results presented here are new and original. References are clearly provided if necessary, and many examples and illustrations are included. Some useful problems are also given for future research. All these components make the book useful for graduate students and researchers in number theory, low dimensional topology, and geometry. This second edition is a corrected and enlarged version of the original one. Misprints and mistakes in the first edition are corrected, references are updated, and some expositions are improved. Because of the remarkable developments in arithmetic topology after the publication of the first edition, the present edition includes two new chapters. One is concerned with idelic class field theory for 3-manifolds and number fields. The other deals with topological and arithmetic DijkgraafWitten theory, which supports a new bridge between arithmetic topology and mathematical physics.
Chapter 1. Introduction.
Chapter 2. Preliminaries - Fundamental Groups and Galois Groups.-
Chapter 3. Knots and Primes,
3-Manifolds and Number Rings.
Chapter 4. Linking Numbers and Legendre Symbols.
Chapter 5. Decompositions of Knots and Primes.
Chapter 6. Homology Groups and Ideal Class Groups I Genus Theory.
Chapter 7. Idelic Class Field Theory for 3-Manifolds and Number Fields.
Chapter 8. Link Groups and Galois Groups with Restricted Ramification.
Chapter 9. Milnor Invariants and Multiple Power Residue Symbols.
Chapter 10. Alexander Modules and Iwasawa Modules.
Chapter 11. Homology Groups and Ideal Class Groups II Higher Order Genus Theory.-
Chapter 12. Homology Groups and Ideal Class Groups III Asymptotic Formulas.
Chapter 13. Torsions and the Iwasawa Main Conjecture.
Chapter 14. Moduli Spaces of Representations of Knot and Prime Groups.
Chapter 15. Deformations of Hyperbolic Structures and of p-Adic Ordinary Modular Forms.
Chapter 16. DijkgraafWitten Theory for 3-Manifolds and Number Rings.
Format: Hardback, 158 pages, height x width: 235x155 mm, 1 Illustrations, black and white; XI, 158 p. 1 illus.,
Series: Infosys Science Foundation Series
Pub. Date: 05-Mar-2024
ISBN-13: 9789819992577
This book provides an up-to-date introduction to the theory of manifolds, submanifolds, semi-Riemannian geometry and warped product geometry, and their applications in geometry and physics. It then explores the properties of conformal vector fields and conformal transformations, including their fixed points, essentiality and the Lichnerowicz conjecture. Later chapters focus on the study of conformal vector fields on special Riemannian and Lorentzian manifolds, with a special emphasis on general relativistic spacetimes and the evolution of conformal vector fields in terms of initial data. The book also delves into the realm of Ricci flow and Ricci solitons, starting with motivations and basic results and moving on to more advanced topics within the framework of Riemannian geometry. The main emphasis of the book is on the interplay between conformal vector fields and Ricci solitons, and their applications in contact geometry. The book highlights the fact that Nil-solitons and Sol-solitons naturally arise in the study of Ricci solitons in contact geometry. Finally, the book gives a comprehensive overview of generalized quasi-Einstein structures and Yamabe solitons and their roles in contact geometry. It would serve as a valuable resource for graduate students and researchers in mathematics and physics as well as those interested in the intersection of geometry and physics
Preface
1. Manifolds and Submanifolds Reviewed
2. Lie Group and Lie Derivative
3. Conformal Transformations
4. Conformal Vector Fields
5. Integral Formulas and Conformal Vector Fields
6. Conformal Vector Fields on Lorentzian Manifolds
7. Ricci Solitons
8. Conformal Vector Fields and Ricci Solitons in Complex and Contact Geometries
9. Quasi-Einstein Manifolds and Conformal Vector Fields
Format: Paperback / softback, 215 pages, height x width: 240x168 mm, 43 Tables, color; XIV, 215 p.
Series: Oberwolfach Seminars 53
Pub. Date: 01-Mar-2024
ISBN-13: 9783031514616
Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.
After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.
This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry.
Preface.- Historical Snapshot.- Critical Equations.- Computations.-
Polar Degrees.- Wasserstein Distance.- Curvature.- Reach and Offset.- Voronoi
Cells.- Condition Numbers.- Machine Learning.- Maximum Likelihood.- Tensors.-
Computer Vision.- Volumes of Semialgebraic Sets.- Sampling.- References.