Reflections on Quanta, Symmetries, and Supersymmetries

Author: V.S. Varadarajan

Publisher: Springer Science & Business Media

ISBN: 1441906673

Category: Mathematics

Page: 236

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This is a collection of essays based on lectures that author has given on various occasions on foundation of quantum theory, symmetries and representation theory, and the quantum theory of the superworld created by physicists. The lectures are linked by a unifying theme: how the quantum world and superworld appear under the lens of symmetry and supersymmetry. In the world of ultra-small times and distances such as the Planck length and Planck time, physicists believe no measurements are possible and so the structure of spacetime itself is an unknown that has to be first understood. There have been suggestions (Volovich hypothesis) that world geometry at such energy regimes is non-archimedian and some of the lectures explore the consequences of such a hypothesis. Ultimately, symmetries and supersymmetries are described by the representation of groups and supergroups. The author's interest in representation is a lifelong one and evolved slowly, and owes a great deal to conversations and discussions he had with George Mackey and Harish-Chandra. The book concludes with a retrospective look at these conversations.

Ultrametric Pseudodifferential Equations and Applications

Author: Andrei Yu. Khrennikov,Sergei V. Kozyrev,W. A. Zúñiga-Galindo

Publisher: Cambridge University Press

ISBN: 1107188822

Category: Mathematics

Page: 258

View: 9326

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Provides a novel interdisciplinary perspective on the state of the art of ultrametric pseudodifferential equations and their applications.

A Course in Complex Analysis and Riemann Surfaces

Author: Wilhelm Schlag

Publisher: American Mathematical Society

ISBN: 0821898477

Category: Mathematics

Page: 384

View: 2066

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Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag's treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces. The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the half-plane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the Riemann-Roch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level. This text is intended as a detailed, yet fast-paced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study.

Supersymmetry and String Theory

Beyond the Standard Model

Author: Michael Dine

Publisher: Cambridge University Press

ISBN: 1107048389

Category: Science

Page: 608

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This fully updated second edition provides a thorough overview of string theory and supersymmetry and includes the groundbreaking Higgs discovery.

The Mathematical Theory of Time-Harmonic Maxwell's Equations

Expansion-, Integral-, and Variational Methods

Author: Andreas Kirsch,Frank Hettlich

Publisher: Springer

ISBN: 3319110861

Category: Mathematics

Page: 337

View: 3833

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This book gives a concise introduction to the basic techniques needed for the theoretical analysis of the Maxwell Equations, and filters in an elegant way the essential parts, e.g., concerning the various function spaces needed to rigorously investigate the boundary integral equations and variational equations. The book arose from lectures taught by the authors over many years and can be helpful in designing graduate courses for mathematically orientated students on electromagnetic wave propagation problems. The students should have some knowledge on vector analysis (curves, surfaces, divergence theorem) and functional analysis (normed spaces, Hilbert spaces, linear and bounded operators, dual space). Written in an accessible manner, topics are first approached with simpler scale Helmholtz Equations before turning to Maxwell Equations. There are examples and exercises throughout the book. It will be useful for graduate students and researchers in applied mathematics and engineers working in the theoretical approach to electromagnetic wave propagation.

Reflection Groups and Invariant Theory

Author: Richard Kane

Publisher: Springer Science & Business Media

ISBN: 1475735421

Category: Mathematics

Page: 379

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Reflection groups and invariant theory is a branch of mathematics that lies at the intersection between geometry and algebra. The book contains a deep and elegant theory, evolved from various graduate courses given by the author over the past 10 years.

Patterns of Change

Linguistic Innovations in the Development of Classical Mathematics

Author: Ladislav Kvasz

Publisher: Springer Science & Business Media

ISBN: 3764388404

Category: Mathematics

Page: 262

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Kvasz’s book is a contribution to the history and philosophy of mat- matics, or, as one might say, the historical approach to the philosophy of mathematics. This approach is for mathematics what the history and philosophy of science is for science. Yet the historical approach to the philosophy of science appeared much earlier than the historical approach to the philosophy of mathematics. The ?rst signi?cant work in the history and philosophy of science is perhaps William Whewell’s Philosophy of the Inductive Sciences, founded upon their History. This was originally published in 1840, a second, enlarged edition appeared in 1847, and the third edition appeared as three separate works p- lished between 1858 and 1860. Ernst Mach’s The Science of Mech- ics: A Critical and Historical Account of Its Development is certainly a work of history and philosophy of science. It ?rst appeared in 1883, and had six further editions in Mach’s lifetime (1888, 1897, 1901, 1904, 1908, and 1912). Duhem’s Aim and Structure of Physical Theory appeared in 1906 and had a second enlarged edition in 1914. So we can say that history and philosophy of science was a well-established ?eld th th by the end of the 19 and the beginning of the 20 century. By contrast the ?rst signi?cant work in the history and philosophy of mathematics is Lakatos’s Proofs and Refutations, which was p- lished as a series of papers in the years 1963 and 1964.

A Beautiful Question

Finding Nature's Deep Design

Author: Frank Wilczek

Publisher: Penguin

ISBN: 0143109367

Category: Science

Page: 448

View: 8067

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A respected physicist presents a survey of related discoveries, from Plato and Pythagoras up to the present, that explore how intertwined ideas about beauty and art are with scientific understandings of the cosmos.

Mathematical Analysis II

Author: V. A. Zorich

Publisher: Springer

ISBN: 3662489937

Category: Mathematics

Page: 720

View: 8115

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This second English edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems, and fresh applications to areas seldom touched on in textbooks on real analysis. The main difference between the second and first English editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. This second volume presents classical analysis in its current form as part of a unified mathematics. It shows how analysis interacts with other modern fields of mathematics such as algebra, differential geometry, differential equations, complex analysis, and functional analysis. This book provides a firm foundation for advanced work in any of these directions.

Modern Problems in Insurance Mathematics

Author: Dmitrii Silvestrov,Anders Martin-Löf

Publisher: Springer

ISBN: 3319066536

Category: Business & Economics

Page: 385

View: 5716

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This book is a compilation of 21 papers presented at the International Cramér Symposium on Insurance Mathematics (ICSIM) held at Stockholm University in June, 2013. The book comprises selected contributions from several large research communities in modern insurance mathematics and its applications. The main topics represented in the book are modern risk theory and its applications, stochastic modelling of insurance business, new mathematical problems in life and non-life insurance and related topics in applied and financial mathematics. The book is an original and useful source of inspiration and essential reference for a broad spectrum of theoretical and applied researchers, research students and experts from the insurance business. In this way, Modern Problems in Insurance Mathematics will contribute to the development of research and academy–industry co-operation in the area of insurance mathematics and its applications.

Not Even Wrong

The Failure of String Theory and the Search for Unity in Physical Law for Unity in Physical Law

Author: Peter Woit

Publisher: Basic Books

ISBN: 046500363X

Category: Science

Page: 320

View: 9182

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When does physics depart the realm of testable hypothesis and come to resemble theology? Peter Woit argues that string theory isn't just going in the wrong direction, it's not even science. Not Even Wrong shows that what many physicists call superstring “theory” is not a theory at all. It makes no predictions, not even wrong ones, and this very lack of falsifiability is what has allowed the subject to survive and flourish. Peter Woit explains why the mathematical conditions for progress in physics are entirely absent from superstring theory today, offering the other side of the story.

Regression, ANOVA, and the General Linear Model

A Statistics Primer

Author: Peter Vik

Publisher: SAGE Publications

ISBN: 1483316017

Category: Social Science

Page: 344

View: 3741

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Peter Vik's Regression, ANOVA, and the General Linear Model: A Statistics Primer demonstrates basic statistical concepts from two different perspectives, giving the reader a conceptual understanding of how to interpret statistics and their use. The two perspectives are (1) a traditional focus on the t-test, correlation, and ANOVA, and (2) a model-comparison approach using General Linear Models (GLM). This book juxtaposes the two approaches by presenting a traditional approach in one chapter, followed by the same analysis demonstrated using GLM. By so doing, students will acquire a theoretical and conceptual appreciation for data analysis as well as an applied practical understanding as to how these two approaches are alike.

Superstrings

A Theory of Everything?

Author: P. C. W. Davies,Julian Brown

Publisher: Cambridge University Press

ISBN: 9780521437752

Category: Science

Page: 234

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Discusses the background of the superstring theory and shares interviews with some of the physicists working on a unified theory of nature

The Fabric of the Cosmos

Space, Time, and the Texture of Reality

Author: Brian Greene

Publisher: Vintage

ISBN: 9780307428530

Category: Science

Page: 592

View: 1284

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From Brian Greene, one of the world’s leading physicists and author of the Pulitzer Prize finalist The Elegant Universe, comes a grand tour of the universe that makes us look at reality in a completely different way. Space and time form the very fabric of the cosmos. Yet they remain among the most mysterious of concepts. Is space an entity? Why does time have a direction? Could the universe exist without space and time? Can we travel to the past? Greene has set himself a daunting task: to explain non-intuitive, mathematical concepts like String Theory, the Heisenberg Uncertainty Principle, and Inflationary Cosmology with analogies drawn from common experience. From Newton’s unchanging realm in which space and time are absolute, to Einstein’s fluid conception of spacetime, to quantum mechanics’ entangled arena where vastly distant objects can instantaneously coordinate their behavior, Greene takes us all, regardless of our scientific backgrounds, on an irresistible and revelatory journey to the new layers of reality that modern physics has discovered lying just beneath the surface of our everyday world.

Mathematical Analysis I

Author: V. A. Zorich

Publisher: Springer

ISBN: 3662487926

Category: Mathematics

Page: 616

View: 7207

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This second edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems, and fresh applications to areas seldom touched on in textbooks on real analysis. The main difference between the second and first editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. The first volume constitutes a complete course in one-variable calculus along with the multivariable differential calculus elucidated in an up-to-date, clear manner, with a pleasant geometric and natural sciences flavor.

Financial Modeling, Actuarial Valuation and Solvency in Insurance

Author: Mario V. Wüthrich,Michael Merz

Publisher: Springer Science & Business Media

ISBN: 3642313922

Category: Mathematics

Page: 432

View: 1465

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Risk management for financial institutions is one of the key topics the financial industry has to deal with. The present volume is a mathematically rigorous text on solvency modeling. Currently, there are many new developments in this area in the financial and insurance industry (Basel III and Solvency II), but none of these developments provides a fully consistent and comprehensive framework for the analysis of solvency questions. Merz and Wüthrich combine ideas from financial mathematics (no-arbitrage theory, equivalent martingale measure), actuarial sciences (insurance claims modeling, cash flow valuation) and economic theory (risk aversion, probability distortion) to provide a fully consistent framework. Within this framework they then study solvency questions in incomplete markets, analyze hedging risks, and study asset-and-liability management questions, as well as issues like the limited liability options, dividend to shareholder questions, the role of re-insurance, etc. This work embeds the solvency discussion (and long-term liabilities) into a scientific framework and is intended for researchers as well as practitioners in the financial and actuarial industry, especially those in charge of internal risk management systems. Readers should have a good background in probability theory and statistics, and should be familiar with popular distributions, stochastic processes, martingales, etc.

Computational Linear and Commutative Algebra

Author: Martin Kreuzer,Lorenzo Robbiano

Publisher: Springer

ISBN: 3319436015

Category: Mathematics

Page: 321

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This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to present it in their lively and humorous style, interspersing core content with funny quotations and tongue-in-cheek explanations.

Warped Passages

Unraveling the Mysteries of the Universe's Hidden Dimensions

Author: Lisa Randall

Publisher: Harper Collins

ISBN: 0060531088

Category: Science

Page: 512

View: 4086

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The universe has its secrets. It may even hide extra dimensions, different from anything ever imagined. A whole raft of remarkable concepts now rides atop the scientific firmament, including parallel universes, warped geometry, and threedimensional sink-holes. We understand far more about the world than we did just a few short years ago -- and yet we are more uncertain about the true nature of the universe than ever before. Have we reached a point of scientific discovery so advanced that the laws of physics as we know them are simply not sufficient? Will we all soon have to accept explanations that previously remained in the realm of science fiction? Lisa Randall is herself making these extraordinary breakthroughs, pushing back the boundaries of science in her research to answer some of the most fundamental questions posed by Nature. For example, why is the gravitational field from the entire Earth so defenseless against the small tug of a tiny magnet? Searching for answers to such seemingly irresolvable questions has led physicists to postulate extra dimensions, the presence of which may lead to unimaginable gains in scientific understanding. Randall takes us into the incredible world of warped, hidden dimensions that underpin the universe we live in, describing how we might prove their existence, while examining the questions that they still leave unanswered. Warped Passages provides an exhilarating overview that tracks the arc of discovery from early twentieth-century physics to the razor's edge of today's particle physics and string theory, unweaving the current debates about relativity, quantum mechanics, and gravity. In a highly readable style sure to entertain and elucidate, Lisa Randall demystifies the science and beguilingly unravels the mysteries of the myriad worlds that may exist just beyond the one we are only now beginning to know.

The Complete Idiot's Guide to String Theory

Author: George Musser

Publisher: Penguin

ISBN: 1101029358

Category: Science

Page: 368

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Everything is connected... We''re living in the midst of a scientific revolution that''s captured the general public''s attention and imagination. The aim of this new revolution is to develop a "theory of everything"- -- a set of laws of physics that will explain all that can be explained, ranging from the tiniest subatomic particle to the universe as a whole. Here, readers will learn the ideas behind the theories, and their effects upon our world, our civilization, and ourselves.

Quantum Theory, Groups and Representations

An Introduction

Author: Peter Woit

Publisher: Springer

ISBN: 3319646125

Category: Science

Page: 668

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This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.