Part of Cambridge Studies in Advanced Mathematics
Not yet published - available from July 2023
FORMAT: Paperback ISBN: 9781009262484
Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of volume two covers the composition of generating functions, in particular the exponential formula and the Lagrange inversion formula, labelled and unlabelled trees, algebraic, D-finite, and noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course and focusing on combinatorics, especially the Robinson?Schensted?Knuth algorithm. An appendix by Sergey Fomin covers some deeper aspects of symmetric functions, including jeu de taquin and the Littlewood?Richardson rule. The exercises in the book play a vital role in developing the material, and this second edition features over 400 exercises, including 159 new exercises on symmetric functions, all with solutions or references to solutions.
Preface to Second Edition
Preface
5. Trees and the Composition of Generating Functions
6. Algebraic Generating Functions
7. Symmetric Functions
Appendices: References
Index.
Part of Lecture Notes in Logic
Not yet published - available from November 2023
FORMAT: HardbackISBN: 9781009437189
How should we treat the liar and kindred paradoxes? A Theory of Truth argues that we should diverge from classical logic, and presents a new formal theory of truth. The theory does not incorporate contradictions and is not substructural, but deviates from classical logic significantly, and endorses principles like 'No sentence is both true and false' and 'No sentence is neither true nor false'. The book starts with an introduction to the paradoxes, suitable for newcomers to the subject, before presenting its approach. Four versions of the theory are covered, extending the theory to a determinacy operator and to a full first-order language with quantifiers. Each includes all Tarskian biconditionals that can be formulated in its language. The author uses original methods to prove the consistency of each version and compares the theory to alternative non-classical theories, including Field's paracomplete approach, Ripley's nontransitive system and Zardini's contraction-free calculus.
Presents theories of truth using non-classical logic, from both formal and philosophical viewpoints
Contains a new and original framework for the theory of truth, as well as original ways of proving key theorems
Assumes no previous familiarity with the semantic paradoxes and begins with an introduction to the subject
1. Aspects of paradox
2. Against classical logic
3. Ambiguity and indexicality
4. A propositional theory of truth
5. Proving central theorem 1
6. Truth and determinacy
7. A first-order logic and theory of truth
8. Proving central theorem 4
9. Another first-order theory of truth
10. Truth in different non-classical logics
Afterword
References
Index.
Quaternions are non-commutative generalizations of the complex numbers, invented by William Rowan Hamilton in 1843. Their number-theoretical aspects were first investigated by Rudolf Lipschitz in the 1880s, and, in a streamlined form, by Adolf Hurwitz in 1896.
This book contains an English translation of his 1919 textbook on this topic as well as his famous 1-2-3-4 theorem on composition algebras. In addition, the reader can find commentaries that shed historical light on the development of this number theory of quaternions, for example, the classical preparatory works (of Fermat, Euler, Lagrange and Gauss to name but a few), the different notions of quaternion integers in the works of Lipschitz and Hurwitz, analogies to the theory of algebraic numbers, and the further development (including Dickson’s work in particular).
The authors have implemented parts of the book in stand-alone courses, and they further believe that the present book can also complement well a course on algebraic number theory (with respect to a non-commutative extension of the rational numbers).
This book provides a self-contained introduction to optimal transport, and it is intended as a starting point for any researcher who wants to enter into this beautiful subject.
The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto’s calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given.
Suitable for a course at the graduate level, the book also includes an appendix with a series of exercises along with their solutions. The present second edition contains a number of additions, such as a new section on the Brunn?Minkowski inequality, new exercises, and various corrections throughout the text.
1 Introduction
Download pp. 1?16
2 Optimal transport
pp. 17?58
3 Wasserstein distances and gradient flows
pp. 59?81
4 Differential viewpoint of optimal transport
pp. 83?101
5 Further reading
pp. 103?118
A Exercises on optimal transport (with solutions)
pp. 119?139
B Disintegrating the disintegration theorem
pp. 141?142
References
pp. 143?146
Overview / Contents
Bayesian Non-linear Statistical Inverse Problems cover
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Bayesian methods based on Gaussian process priors are frequently used in statistical inverse problems arising with partial differential equations (PDEs). They can be implemented by Markov chain Monte Carlo (MCMC) algorithms. The underlying statistical models are naturally high- or infinite-dimensional and the present book presents a rigorous mathematical analysis of the statistical performance, and algorithmic complexity, of such methods in a natural setting of non-linear random design regression.
Due to the non-linearity present in many of these inverse problems, natural least squares functionals are non-convex and the Bayesian paradigm presents an attractive alternative to optimisation-based approaches. This book develops a general theory of Bayesian inference for non-linear forward maps and rigorously considers two PDE model examples arising with Darcy’s problem and a Schrodinger equation. The focus is initially on statistical consistency of Gaussian process methods, and then moves on to study local fluctuations and approximations of posterior distributions by Gaussian or log-concave measures whose curvature is described by PDE mapping properties of underlying ‘information operators’. Applications to the algorithmic runtime of gradient-based MCMC methods are discussed as well as computation time lower bounds for worst case performance of some algorithms.
1 Non-linear statistical inverse problems
pp. 1?22
2 Global stability and posterior consistency
pp. 23?40
3 Information operators and curvature
pp. 41?66
4 Bernstein--von Mises theorems
pp. 67?90
5 Posteriors are probably log-concave
pp. 91?124
A Analytical background
pp. 125?138
B Further auxiliary results
pp. 139?150
References
pp. 151?158
Index
p. 159