Sums, series, and products in Diophantine approximation

There is not much that can be said for all $x$ and for all $n$ about the sum $$\sum_{k=1}^n \frac{1}{|\sin k\pi x|}.$$ However, for this and similar sums, series, and products, we can establish results for almost all $x$ using the tools of continued fractions. We present in detail the appearance of these sums in the singular series for the circle method. One particular interest of the paper is the detailed proof of a striking result of Hardy and Littlewood, whose compact proof, which delicately uses analytic continuation, has not been written freshly anywhere since its original publication. This story includes various parts of late 19th century and early 20th century mathematics.Comment: 83 pages; thanks for correspondence with Christoph Aistleitne

The early historical roots of Lee-Yang theorem

A deep and detailed historiographical analysis of a particular case study concerning the so-called Lee-Yang theorem of theoretical statistical mechanics of phase transitions, has emphasized what real historical roots underlie such a case study. To be precise, it turned out that some well-determined aspects of entire function theory have been at the primeval origins of this important formal result of statistical physics.Comment: History of Physics case study. arXiv admin note: substantial text overlap with arXiv:1106.4348, arXiv:math/0601653, arXiv:0809.3087, arXiv:1311.0596 by other author

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Quantum Field Theory

This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction). From the reviews: "… Quantum field theory is one of the great intellectual edifices in the history of human thought. … This volume differs from other books on quantum field theory in its greater emphasis on the interaction of physics with mathematics. … an impressive work of scholarship." (William G. Faris, SIAM Review, Vol. 50 (2), 2008) "… it is a fun book for practicing quantum field theorists to browse, and it may be similarly enjoyed by mathematical colleagues. Its ultimate value may lie in encouraging students to enter this challenging interdisciplinary area of mathematics and physics. Summing Up: Recommended. Upper-division undergraduates through faculty." (M. C. Ogilvie, CHOICE, Vol. 44 (9), May, 2007

Comparison of random variables from a game-theoretic perspective

This work consists of four related parts, divided into eight chapters. A ¯rst part introduces the framework of cycle-transitivity, developed by De Baets et al. It is shown that this framework is ideally suited for describing and compar- ing forms of transitivity of probabilistic relations. Not only does it encompass most already known concepts of transitivity, it is also ideally suited to describe new types of transitivity that are encountered in this work (such as isostochas- tic transitivity and dice-transitivity). The author made many non-trivial and sometimes vital contributions to the development of this framework. A second part consists of the development and study of a new method to compare random variables. This method, which bears the name generalized dice model, was developed by De Meyer et al. and De Schuymer et al., and can be seen as a graded alternative to the well-known concept of ¯rst degree stochastic dominance. A third part involves the determination of the optimal strategies of three game variants that are closely related to the developed comparison scheme. The de¯nitions of these variants di®er from each other solely by the copula that is used to de¯ne the payo® matrix. It turns out however that the characterization of the optimal strategies, done by De Schuymer et al., is completely di®erent for each variant. A last part includes the study of some combinatorial problems that orig- inated from the investigation of the transitivity of probabilistic relations ob- tained by utilizing the developed method to compare random variables. The study, done by De Schuymer et al., includes the introduction of some new and interesting concepts in partition theory and combinatorics. A more thorough discussion, in which each section of this work is taken into account, can be found in the overview at the beginning of this manuscript. Although this work is oriented towards a mathematical audience, the intro- duced concepts are immediately applicable in practical situations. Firstly, the framework of cycle-transitivity provides an easy means to represent and compare obtained probabilistic relations. Secondly, the generalized dice model delivers a useful alternative to the concept of stochastic dominance for comparing random variables. Thirdly, the considered dice games can be viewed in an economical context in which competitors have the same resources and alternatives, and must choose how to distribute these resources over their alternatives. Finally, it must be noted that this work still leaves opportunities for future research. As immediate candidates we see, ¯rstly the investigation of the tran- sitivity of generalized dice models in which the random variables are pairwisely coupled by a di®erent copula. Secondly, the characterization of the transitivity of higher-dimensional dice models, starting with dimension 4. Thirdly, the study of the applicability of the introduced comparison schemes in areas such as mar- ket e±ciency, portfolio selection, risk estimation, capital budgeting, discounted cash °ow analysis, etc