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
Front matter
Download pp. i?iv
Contents
Download pp. v?vi
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
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
The European Congress of Mathematics, held every four years, is a well-established major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000), Stockholm (2004), Amsterdam (2008), Krakow (2012), and Berlin (2016), the Eighth European Congress of Mathematics (8ECM) took place in Portoro?, Slovenia, June 20?26, 2021, with about 1700 participants from all over the world, mostly online due to Covid pandemic.
Ten plenary and thirty invited lectures along with the special Abel and Hirzebruch lectures formed the core of the program. As in all the previous EMS congresses, ten outstanding young mathematicians received the EMS prizes in recognition of their research achievements. In addition, two more prizes were awarded: The Felix Klein Prize for a remarkable solution of an industrial problem and the Otto Neugebauer Prize for a highly original and influential piece of work in the history of mathematics. The program was complemented by five public lectures, several exhibitions, and 62 minisymposia with about 1000 contributions, spread over all areas of mathematics. A number of panel discussions and meetings were organized, covering a variety of issues ranging from the future of mathematical publishing and the role of the ERC to public awareness of mathematics.
These proceedings provide a permanent record of current mathematics of highest quality by presenting extended versions of seven plenary, six prize, and fourteen invited lectures as well as eleven lectures from minisymposia keynote speakers, all of which were delivered during the congress.
To what extent are the subjects of our thoughts and talk real? This is the question of realism. In this book, Justin Clarke-Doane explores arguments for and against moral realism and mathematical realism, how they interact, and what they can tell us about areas of philosophical interest more generally. He argues that, contrary to widespread belief, our mathematical beliefs have no better claim to being self-evident or provable than our moral beliefs. Nor do our mathematical beliefs have better claim to being empirically justified than our moral beliefs. It is also incorrect that reflection on the "genealogy" of our moral beliefs establishes a lack of parity between the cases. In general, if one is a moral antirealist on the basis of epistemological considerations, then one ought to be a mathematical antirealist as well. And, yet, Clarke-Doane shows that moral realism and mathematical realism do not stand or fall together ? and for a surprising reason. Moral questions, insofar as they are practical, are objective in a sense that mathematical questions are not. Moreover, the sense in which they are objective can be explained only by assuming practical anti-realism. One upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension. Another is that the objective questions in the neighborhood of questions of logic, modality, grounding, and nature are practical questions too. Practical philosophy should, therefore, take center stage.
Introduction
1:Realism, Ontology, and Objectivity
2:Self-Evidence, Proof, and Disagreement
3:Observation and Indispensability
4:Genealogical Debunking Arguments
5:Explaining our Reliability
6:Realism, Objectivity, and Evaluation
Conclusion