Part of Lecture Notes in Logic
Not yet published - available from June 2024
FORMAT: Hardback
ISBN: 9781009520713
Developing the theory up to the current state-of-the art, this book studies the minimal model of the Largest Suslin Axiom (LSA), which is one of the most important determinacy axioms and features prominently in Hugh Woodin's foundational framework known as the Ultimate L. The authors establish the consistency of LSA relative to large cardinals and develop methods for building models of LSA from other foundational frameworks such as Forcing Axioms. The book significantly advances the Core Model Induction method, which is the most successful method for building canonical inner models from various hypotheses. Also featured is a proof of the Mouse Set Conjecture in the minimal model of the LSA. It will be indispensable for graduate students as well as researchers in mathematics and philosophy of mathematics who are interested in set theory and in particular, in descriptive inner model theory.
Provides the first proof of the consistency of the Largest Suslin Axiom relative to large cardinals
Develops the core model induction, a universal technique for building models of determinacy axioms, up to the region of the Largest Suslin Axiom
Proves the Mouse Set Conjecture, one of the central conjectures of descriptive inner model theory
1. Introduction
2. Hybrid J-structures
3. Short tree strategy mice
4. A comparison theory of HOD mice
5. HOD mice revisited
6. The internal theory of LSA HOD mice
7. Analysis of HOD
8. Models of LSA as derived models
9. Condensing sets
10. Applications
11. A proof of square in LSA-small HOD mice
12. LSA from PFA
References
Index.
Part of Cambridge Tracts in Mathematics
Not yet published - available from November 2024
FORMAT: Hardback ISBN: 9781009562294
The theory of definable equivalence relations has been a vibrant area of research in descriptive set theory for the past three decades. It serves as a foundation of a theory of complexity of classification problems in mathematics and is further motivated by the study of group actions in a descriptive, topological, or measure-theoretic context. A key part of this theory is concerned with the structure of countable Borel equivalence relations. These are exactly the equivalence relations generated by Borel actions of countable discrete groups and this introduces important connections with group theory, dynamical systems, and operator algebras. This text surveys the state of the art in the theory of countable Borel equivalence relations and delineates its future directions and challenges. It gives beginning graduate students and researchers a bird's-eye view of the subject, with detailed references to the extensive literature provided for further study.
Surveys the state of the art in a currently very active area of research
Organizes in a systematic way material spread in numerous publications and clearly delineates the main future directions and challenges
Includes detailed references to the extensive literature for further study
1. Equivalence relations and reductions
2. Countable Borel equivalence relations
3. Essentially countable relations
4. Invariant and quasi-invariant measures
5. Smoothness, $\mathbf{E}_0$ and $\mathbf{E}_\infty$
6. Rigidity and incomparability
7. Hyperfiniteness
8. Amenability
9. Treeability
10. Freeness
11. Universality
12. The poset of bireducibility types
13. Structurability
14. Topological realizations
15. A universal space for actions and equivalence relations
16. Open problems
References
Index.
Copyright 2024
Paperback
Hardback
ISBN 9781032109497
346 Pages 79 B/W Illustrations
July 23, 2024
Bayesian Inference: Theory, Methods, Computations provides a comprehensive coverage of the fundamentals of Bayesian inference from all important perspectives, namely theory, methods and computations.
All theoretical results are presented as formal theorems, corollaries, lemmas etc., furnished with detailed proofs. The theoretical ideas are explained in simple and easily comprehensible forms, supplemented with several examples. A clear reasoning on the validity, usefulness, and pragmatic approach of the Bayesian methods is provided. A large number of examples and exercises, and solutions to all exercises, are provided to help students understand the concepts through ample practice.
The book is primarily aimed at first or second semester master students, where parts of the book can also be used at Ph.D. level or by research community at large. The emphasis is on exact cases. However, to gain further insight into the core concepts, an entire chapter is dedicated to computer intensive techniques. Selected chapters and sections of the book can be used for a one-semester course on Bayesian statistics.
Explains basic ideas of Bayesian statistical inference in an easily comprehensible form
Illustrates main ideas through sketches and plots
Contains large number of examples and exercises
Provides solutions to all exercises
Includes R codes
1. Introduction
2. Bayesian Modelling
3. Choice of Prior
4. Decision Theory
5. Asymptotic Theory
6. Normal Linear Models
7. Estimation
8. Testing and Model Comparison
9. Computational Techniques
10. Solutions
11. Appendix
Index
Copyright 2025
Hardback
ISBN 9781032816524
766 Pages 525 B/W Illustrations
August 14, 2024 by CRC Press
The original goal of writing this book was to introduce the reader to the tools of combinatorics from an applied point of view. This third edition of Applied Combinatorics was substantially rewritten. There are many new examples and exercises. References throughout the book to modern literature and real applications, a key feature of the book, have been updated and expanded. The exposition continues to be updated with each new edition, as the first edition was published 40 years ago.
The emphasis on applications from computer science, genetics, experimental design, chemistry, scheduling, voting, and other topics remains a central feature of the book. Unique to the literature is that entire sections focus on applications such as switching functions, the use of enzymes to uncover unknown RNA chains, searching and sorting problems of information retrieval, construction of error-correcting codes, counting of chemical compounds, calculation of power in voting situations, and uses of Fibonacci numbers. There are entire sections on applications of recurrences involving convolutions, applications of eulerian chains, and applications of generating functions.
The book continues to be based on the authorsf philosophy that the best way to learn mathematics is through problem solving. Combinatorics can be a wonderful mechanism for introducing students to proofs. However, the book is not designed for an introduction to proofs course. The authors treat proofs as rather informal, and many of the harder proofs in the book are optional.
Applied Combinatorics, Third Edition is divided into four parts. The first part introduces the basic tools of combinatorics and their applications. The remaining three parts are organized around the three basic problems of combinatorics: the counting problem, the existence problem, and the optimization problem.
Most of the book is written for a first course on the topic at the undergraduate level. On the other hand, at a fast pace, there is more than enough material for a challenging graduate course. This book first appeared when courses on combinatorics were rare. We are pleased to think that, through its use, the book has helped to establish a key course in many colleges and universities throughout the world. We hope that this new edition will remain a valuable tool for instructors and students alike.
Chapter 1: What Is Combinatorics? THE BASIC TOOLS OF COMBINATORICS
Chapter 2: Basic Counting Rules;
Chapter 3: Introduction to Graph Theory;
Chapter 4 Relations; THE COUNTING PROBLEM
Chapter 5: Generating Functions and Their Applications;
Chapter 6: Recurrence Relations;
Chapter 7: The Principle of Inclusion and Exclusion;
Chapter 8: The Polya Theory of Counting; THE EXISTENCE PROBLEM
Chapter 9: Combinatorial Designs;
Chapter 10: Coding Theory;
Chapter 11: Existence Problems in Graph Theory; COMBINATORIAL OPTIMIZATION
Chapter 12: Matching and Covering;
Chapter 13: Optimization Problems for Graphs and Networks; Appendix: Answers
to Selected Exercises; Author Index; Subject Index; References appear at
the end of each chapte
Format: Paperback / softback, 700 pages, height x width: 240x170 mm, weight: 1118 g,
101 Illustrations, color; 43 Tables, black and white; 134 Illustrations, black and white
Series: De Gruyter Textbook
Pub. Date: 20-May-2024
ISBN-13: 9783111331799
This text offers a unique balance of theory and a variety of standard and new applications along with solved technology-aided problems.
The book includes the fundamental mathematical theory, as well as a wide range of applications, numerical methods, projects, and technology-assisted problems and solutions in Maple, Mathematica, and MATLAB. Some of the applications are new, some are unique, and some are discussed in an essay. There is a variety of exercises which include True/False questions, questions that require proofs, and questions that require computations.
The goal is to provide the student with is a solid foundation of the mathematical theory and an appreciation of some of the important real-life applications. Emphasis is given on geometry, matrix transformations, orthogonality, and least-squares.
Designed for maximum flexibility, it is written for a one-semester/two semester course at the sophomore or junior level for students of mathematics or science.
Format: Hardback, 231 pages, height x width: 240x168 mm, 82 Illustrations, color;
41 Illustrations, black and white; X, 230 p., 1 Hardback
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 10-Sep-2024
ISBN-13: 9783031599743
This book provides a concise but thorough introduction to partial differential equations which model phenomena that vary in both space and time. The author begins with a full explanation of the fundamental linear partial differential equations of physics. The text continues with methods to understand and solve these equations leading ultimately to the solutions of Maxwellfs equations. The author then addresses nonlinearity and provides examples of separation of variables, linearizing change of variables, inverse scattering transform, and numerical methods for select nonlinear equations. Next, the book presents rich sources of advanced techniques and strategies for the study of nonlinear partial differential equations. This second edition includes updates, additional examples, and a new chapter on reaction?diffusion equations. Ultimately, this book is an essential resource for readers in applied mathematics, physics, chemistry, biology, and engineering who are interested in learning about the myriad techniques that have been developed to model and solve linear and nonlinear partial differential equations.
Introduction.- The Equations of Maxwell.- Laplace's Equation.- Fourier
Series, Bessel Functions, and Mathematical Physics .- The Fourier Transform,
Heat Conduction, and the Wave Equation.- The ThreeDimensional Wave
Equation.- An Introduction to Nonlinear Partial Differential Equations.-
Raman Scattering and Numerical Methods.- ReactionDiffusion Equations.- The
HartmanGrobman Theorem.