Part of Cambridge Studies in Advanced Mathematics
Not yet published - available from May 2024
FORMAT: Hardback ISBN: 9781009344678
The finite generation theorem is a major achievement of modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic zero is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar-Cascini-Hacon-McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend and break method, vanishing theorems, positivity theorems and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.
Explain the basics of minimal model theory for higher dimensional algebraic varieties with boundaries
The first book to explain the proof of the finite generation theorem of the canonical ring
Includes self-contained explanations of terminology and theorems, assuming only the basics in algebraic geometry
Preface
1. Introduction
2. Algebraic varieties with boundaries
3. The minimal model program
4. The finite generation theorem
Bibliography
Index.
Part of Cambridge Studies in Advanced Mathematics
Not yet published - available from May 2024
FORMAT: Hardback ISBN: 9781009123181
Research in recent years has highlighted the deep connections between the algebraic, geometric, and analytic structures of a discrete group. New methods and ideas have resulted in an exciting field, with many opportunities for new researchers. This book is an introduction to the area from a modern vantage point. It incorporates the main basics, such as Kesten's amenability criterion, Coulhon and Saloff-Coste inequality, random walk entropy and bounded harmonic functions, the Choquet?Deny Theorem, the Milnor?Wolf Theorem, and a complete proof of Gromov's Theorem on polynomial growth groups. The book is especially appropriate for young researchers, and those new to the field, accessible even to graduate students. An abundance of examples, exercises, and solutions encourage self-reflection and the internalization of the concepts introduced. The author also points to open problems and possibilities for further research.
Suitable for beginning graduate students, requiring only an undergraduate mathematics background
Combines the fields of geometric group theory and probability, allowing researchers in either area to enter into the joint research of random walks on groups
Features more than 450 exercises with solutions included
Contains open problems to provide jumping-off points for future research
Part I. Tools and Theory:
1. Background
2. Martingales
3. Markov chains
4. Networks and discrete analysis
Part II. Results and Applications:
5. Growth, dimension, and heat kernel
6. Bounded harmonic functions
7. Choquet?Deny groups
8. The Milnor?Wolf theorem
9. Gromov's theorem
Appendices: A. Hilbert space background
B. Entropy
C. Coupling and total variation
References
Index.
Part of Encyclopedia of Mathematics and its Applications
Not yet published - available from June 2024
FORMAT: HardbackI SBN: 9781009123198
Higher special functions emerge from boundary eigenvalue problems of Fuchsian differential equations with more than three singularities. This detailed reference provides solutions for singular boundary eigenvalue problems of linear ordinary differential equations of second order, exploring previously unknown methods for finding higher special functions. Starting from the fact that it is the singularities of a differential equation that determine the local, as well as the global, behaviour of its solutions, the author develops methods that are both new and efficient and lead to functional relationships that were previously unknown. All the developments discussed are placed within their historical context, allowing the reader to trace the roots of the theory back through the work of many generations of great mathematicians. Particular attention is given to the work of George Cecil Jaffe, who laid the foundation with the calculation of the quantum mechanical energy levels of the ionised hydrogen molecule ion.
Gives an overview of historical developments and places mathematical discoveries in a historical context
Presents methods to significantly expand the scope of application of the special functions
Demonstrates how to reproduce already known results under a new and far more general aspect
1. Introduction
2. Singularities in action
3. Fuchsian differential equations: the cornerstones
4. Central two-point connection problems and higher special functions
5. Applications and examples
6. Afterword
A. Standard central two-point connection problem
B. Curriculum vitae of George Cecil Jaffe
References
Index.
Copyright 2024
Hardback
ISBN 9781032623856
Paperback
ISBN 9781032611846
224 Pages
May 8, 2024 by Chapman & Hall
The goal of this unique text is to provide an “experience” that would facilitate a better transition for mathematics majors to the advanced proof-based courses required for their major.
If you “love mathematics, but I hate proofs” this book is for you. Example-based courses such as introductory Calculus transition somewhat abruptly, and without a warning label, to proof-based courses, and may leave students with the unpleasant feeling that a subject they loved has turned into material they find hard to understand.
The book exposes students and readers to the fundamental nature and principles of constructing mathematical proofs and in the context of main courses required for the major, e.g., probability, linear algebra, real analysis, and abstract algebra.
Four short chapters, each chapter focusing on a particular course, provide a short but rigorous introduction. Students then get a preview of the discipline, its focus, language, mathematical objects of interests, and common methods of proof presented in those courses. Because which ideas apply to which future courses may not be obvious in many transition courses, this structure addresses this need.
The book may also be used as a review tool at the end of course and for readers who want to learn the language and scope of the broad disciplines of linear algebra, abstract algebra, real analysis, and probability, before transitioning to these courses.
Preface
Chapter 1. Mathematical Logic and Methods of Proof
Chapter 2. Probability
Chapter 3. Real Analysis
Chapter 4. Linear Algebra
Chapter 5. Abstract Algebra
Appendix
Paperback
ISBN 9781032653761
260 Pages
June 25, 2024 by CRC Press
Request Inspection Copy
Classification and Examples of Differential Equations and their Applications is the sixth book within Ordinary Differential Equations with Applications to Trajectories and Vibrations, Six-volume Set. As a set, they are the fourth volume in the series Mathematics and Physics Applied to Science and Technology. This sixth book consists of one chapter (chapter 10 of the set). It contains 20 examples related to the preceding five books and chapters 1 to 9 of the set. It includes two recollections: the first with a classification of differential equations into 500 standards and the second with a list of 500 applications.
The ordinary differential equations are classified in 500 standards concerning methods of solution and related properties, including: (i) linear differential equations with constant or homogeneous coefficients and finite difference equations; (ii) linear and non-linear single differential equations and simultaneous systems; (iii) existence, unicity and other properties; (iv) derivation of general, particular, special, analytic, regular, irregular, and normal integrals; (v) linear differential equations with variable coefficients including known and new special functions.
The theory of differential equations is applied to the detailed solution of 500 physical and engineering problems including: (i) one- and multidimensional oscillators, with damping or amplification, with non-resonant or resonant forcing; (ii) single, non-linear, and parametric resonance; (iii) bifurcations and chaotic dynamical systems; (iv) longitudinal and transversal deformations and buckling of bars, beams, and plates; (v) trajectories of particles; (vi) oscillations and waves in non-uniform media, ducts, and wave guides.
Provides detailed solution of examples of differential equations of the types covered in tomes l-5 of the set (Ordinary Differential Equations with Applications to Trajectories and Vibrations, Six -volume Set)
Includes physical and engineering problems that extend those presented in the tomes 1-6 (Ordinary Differential Equations with Applications to Trajectories and Vibrations, Six-volume Set)
Includes a classification of ordinary differential equations and their properties into 500 standards that can serve as a look-up table of methods of solution
Covers a recollection of 500 physical and engineering problems and sub-cases that involve the solution of differential equations
Presents the problems used as examples including formulation, solution, and interpretation of results
10. Examples 10.1 to 10.20. Classification 10.1. Classification 10.2
Paperback
ISBN 9781032653358
428 Pages
June 25, 2024 by CRC Press
An Introduction to Integral Transforms is meant for students pursuing graduate and post graduate studies in Science and Engineering. It contains discussions on almost all transforms for normal users of the subject. The content of the book is explained from a rudimentary stand point to an advanced level for convenience of its readers. Pre‐requisite for understanding the subject matter of the book is some knowledge on the complex variable techniques.
1. Fourier Transform, 2. Finite Fourier Transform, 3. The Laplace Transform, 4. The Inverse Laplace Transform and Application, 5. Hilbert and Stieltjes Transforms, 6. Hankel Transforms, 7. Finite Hankel Transforms, 8. The Mellin Transform, 9. Finite Laplace Transforms, 10. Legendre Transforms, 11. The Kontorovich-Lebedev Transform, 12. The Mehler-Fock Transform, 13. Jacobi, Gegenbauer, Laguerre and Hermite Transforms, 14. The Z-Transform
Discusses fundamental concepts such as unigraphs, reconstruction, k-dimensional graphs, degree sequences, line graphs and line hypergraphs, helping the readers to understand them
and get insight into related fundamental problems.
Provides detailed descriptions of some decomposition methods in graph theory, presenting significant results in the field, many of which cannot be found in other books.
Contains a large number of illustrations enhancing the understanding of many important concepts of graph theory.
In general terms, a graph decomposition is a partition of a graph into parts satisfying some special conditions. Methods of Graph Decompositions discusses some state-of-the-art decomposition methods of graph theory, which are highly instrumental when dealing with a number of fundamental concepts such as unigraphs, isomorphism, reconstruction conjectures, k-dimensional graphs, degree sequences, line graphs and line hypergraphs.
The first part of the book explores the algebraic theory of graph decomposition, whose major idea is to define a binary operation that turns the set of graphs or objects derived from graphs into an algebraic semigroup. If an operation and a class of graphs are appropriately chosen, then, just as for integers, each graph has a unique factorization (or canonical decomposition) into a product of prime factors. The unique factorization property makes this type of decomposition especially efficient for problems associated with graph isomorphism, and several such examples are described in the book. Another topic is devoted to Krausz-type decompositions, that is, special coverings of graphs by cliques that are directly associated with representation of graphs as line graphs of hypergraphs. The book discusses various algorithmic and structural results associated with the existence, properties and applications of such decompositions.
In particular, it demonstrates how Krausz-type decompositions are directly related to topological dimension, information complexity and self-similarity of graphs, thus allowing to establish links between combinatorics, general topology, information theory and studies of complex systems. The above topics are united by the role played in their development by Professor Regina Tyshkevich, and the book is a tribute to her memory. The book will be ideal for researchers, engineers and specialists, who are interested in fundamental problems of graph theory and proof techniques to tackle them.
1.:Introduction
2.:Decomposition of Graphical Sequences and Unigraphs
3.:Matrogenic, Matroidal and Threshold Graphs
4.:Further Applications of Operator Decomposition
5.:Line Graphs and Hypergraphs
6.:Dimensionality of Graphs