Topics in Model Theory

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【簡介】 This book has two chapters. The first is a modern or contemporary account of stability theory. A focus is on the local (formula-by-formula) theory, treated a little differently from in the author's book Geometric Stability Theory. There is also a survey of general and geometric stability theory, as well as applications to combinatorics (stable regularity lemma) using pseudofinite methods. The second is an introduction to "continuous logic" or "continuous model theory," drawing on the main texts and papers, but with an independent point of view. This chapter includes some historical background, including some other formalisms for continuous logic and a discussion of hyperimaginaries in classical first order logic. These chapters are based around notes, written by students, from a couple of advanced graduate courses in the University of Notre Dame, in Autumn 2018, and Spring 2021. 【目錄】 Sample Chapter(s) Preface Chapter 1: Stability Theory Contents: Preface Stability Theory: Introduction Preliminaries Stability Continuous Logic: Introduction Background "Official" Continuous Logic Stability in Continuous Logic Index Readership: Graduate students and researchers in mathematics and related subjects interested in model theory and its applications.

Advanced Topics in Quantum Mechanics (1版)

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Quantum mechanics is one of the most successful theories in science, and is relevant to nearly all modern topics of scientific research. This textbook moves beyond the introductory and intermediate principles of quantum mechanics frequently covered in undergraduate and graduate courses, presenting in-depth coverage of many more exciting and advanced topics. The author provides a clearly structured text for advanced students, graduates and researchers looking to deepen their knowledge of theoretical quantum mechanics. The book opens with a brief introduction covering key concepts and mathematical tools, followed by a detailed description of the Wentzel–Kramers–Brillouin (WKB) method. Two alternative formulations of quantum mechanics are then presented: Wigner's phase space formulation and Feynman's path integral formulation. The text concludes with a chapter examining metastable states and resonances. Step-by-step derivations, worked examples and physical applications are included throughout. Covers many advanced mathematical techniques in quantum mechanics, each illustrated with detailed examples Presents two alternative formulations of quantum mechanics; the path integral and phase space formulations, which are useful in advanced applications Provides a pedagogical and thorough overview of the exact WKB method, which plays an increasingly important role in many areas of modern physics

TOPICS IN STATIS MECH (2版)

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Building on the material learned by students in their first few years of study, Topics in Statistical Mechanics (Second Edition) presents an advanced level course on statistical and thermal physics. It begins with a review of the formal structure of statistical mechanics and thermodynamics considered from a unified viewpoint. There is a brief revision of non-interacting systems, including quantum gases and a discussion of negative temperatures. Following this, emphasis is on interacting systems. First, weakly interacting systems are considered, where the interest is in seeing how small interactions cause small deviations from the non-interacting case. Second, systems are examined where interactions lead to drastic changes, namely phase transitions. A number of specific examples is given, and these are unified within the Landau theory of phase transitions. The final chapter of the book looks at non-equilibrium systems, in particular the way they evolve towards equilibrium. This is framed within the context of linear response theory. Here fluctuations play a vital role, as is formalised in the fluctuation–dissipation theorem. The second edition has been revised particularly to help students use this book for self-study. In addition, the section on non-ideal gases has been expanded, with a treatment of the hard-sphere gas, and an accessible discussion of interacting quantum gases. In many cases there are details of Mathematica calculations, including Mathematica Notebooks, and expression of some results in terms of Special Functions.