Matematisk fysik

The latest volume in The New York Times bestselling physics series explains Einstein's masterpiece: the general theory of relativityHe taught us classical mechanics, quantum mechanics and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein's general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler).An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe's real structure.

Using everyday computational power to tackle complex—and seemingly unsolvable—math problemsConfronted by a math problem of seemingly impenetrable difficulty, what can you do? In The Delicate Art of Brute Force, Paul Nahin shows how even if you can’t solve such a problem, you can still get an answer. The computational power of your own computer—the desktop in your home office, the laptop on your coffee table—can be deployed to overwhelm a problem’s complexity through a massive number-crunching assault. Nahin presents a series of apparently intractable math problems and shows the thought process that allows computational solution. Most of the problems are not abstract constructions but originate in the real world—one chapter is titled “How Wi-Fi Coverage and Anti-Submarine Warfare Are the Same.” For each one, Nahin compares computational and analytical approaches, demonstrating how well the computer solutions agree with theory. As a bonus, sometimes the computational solution is better, giving more insight or greater flexibility.Along the way, Nahin traces the evolution of high-speed electronic computation, explaining that it upended the way analysts approach complex math problems. After the postwar debut of ENIAC (Electronic Numerical Integrator and Computer) and the ever more powerful electronic computers that came afterward, problems that were simply too difficult or messy for analytical treatment could be defeated by the application of pure brute force—the ability of computers to perform an enormous number of simple operations in just minutes. Today, we can harness that computational power from our couches. Accessible to anyone who’s mastered high school calculus, this engaging book gives both mathematicians and nonmathematicians plenty to think about.

What’s the point of calculating definite integrals since you can’t possibly do them all?What makes doing the specific integrals in this book of value aren’t the specific answers we’ll obtain, but rather the methods we’ll use in obtaining those answers; methods you can use for evaluating the integrals you will encounter in the future.This book, now in its second edition, is written in a light-hearted manner for students who have completed the first year of college or high school AP calculus and have just a bit of exposure to the concept of a differential equation. Every result is fully derived. If you are fascinated by definite integrals, then this is a book for you. New material in the second edition includes 25 new challenge problems and solutions, 25 new worked examples, simplified derivations, and additional historical discussion.

The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as earthquakes, global warming, demographic patterns, financial crises, and the failure of materials. In this book, Didier Sornette boldly applies his varied experience in these areas to propose a simple, powerful, and general theory of how, why, and when stock markets crash. Most attempts to explain market failures seek to pinpoint triggering mechanisms that occur hours, days, or weeks before the collapse. Sornette proposes a radically different view: the underlying cause can be sought months and even years before the abrupt, catastrophic event in the build-up of cooperative speculation, which often translates into an accelerating rise of the market price, otherwise known as a "bubble." Anchoring his sophisticated, step-by-step analysis in leading-edge physical and statistical modeling techniques, he unearths remarkable insights and some predictions--among them, that the "end of the growth era" will occur around 2050.Sornette probes major historical precedents, from the decades-long "tulip mania" in the Netherlands that wilted suddenly in 1637 to the South Sea Bubble that ended with the first huge market crash in England in 1720, to the Great Crash of October 1929 and Black Monday in 1987, to cite just a few. He concludes that most explanations other than cooperative self-organization fail to account for the subtle bubbles by which the markets lay the groundwork for catastrophe. Any investor or investment professional who seeks a genuine understanding of looming financial disasters should read this book. Physicists, geologists, biologists, economists, and others will welcome Why Stock Markets Crash as a highly original "scientific tale," as Sornette aptly puts it, of the exciting and sometimes fearsome--but no longer quite so unfathomable--world of stock markets.

Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving Mathematical Problems includes numerous exercises and model solutions throughout. Assuming only a basic level of mathematics, the text is ideal for students of 14 years and above in pure mathematics.

The instant New York Times bestsellerQuanta and Fields, the second book of Sean Carroll’s already internationally acclaimed series The Biggest Ideas in the Universe, is an adventure into the bare stuff of reality.   Sean Carroll is creating a profoundly new approach to sharing physics with a broad audience, one that goes beyond analogies to show how physicists really think. He cuts to the bare mathematical essence of our most profound theories, explaining every step in a uniquely accessible way.   Quantum field theory is how modern physics describes nature at its most profound level. Starting with the basics of quantum mechanics itself, Sean Carroll explains measurement and entanglement before explaining how the world is really made of fields. You will finally understand why matter is solid, why there is antimatter, where the sizes of atoms come from, and why the predictions of quantum field theory are so spectacularly successful. Fundamental ideas like spin, symmetry, Feynman diagrams, and the Higgs mechanism are explained for real, not just through amusing stories. Beyond Newton, beyond Einstein, and all the intuitive notions that have guided homo sapiens for millennia, this book is a journey to a once unimaginable truth about what our universe is.

Much like our everyday moments of novelty, innovations often catalyse subsequent breakthroughs, creating a chain of progress that can be traced across history.The Science of the New explores the dynamics of novelty and innovation. It originates from the observation that these two interrelated phenomena that shape human experience, from the ordinary encounters we have in daily life to the groundbreaking advancements in science, technology, and society, are two faces of the same medal. In other words, novelties and innovation can be described by the same underlying processes and their emergence obeys the same general statistical laws.The construction of a solid mathematical framework to treat the phenomenologies related to the experience of the new represents a formidable challenge and a necessary one if we want to understand the world around us and be able to make predictions in an ever-changing environment and society. This book provides a minimal toolkit to face this problem. Organised into three distinct parts, it begins by outlining the fundamental theoretical tools necessary for analysing how novelties and innovations emerge. It then delves into classical and contemporary models that explain the processes behind innovation, illustrating the deep relationship between several ideas that took birth in different fields and influenced one another. The final section provides empirical case studies, applying the discussed frameworks to real-world systems and showing how mathematical and computational methods can help us understand innovation in various contexts.

Beginning graduate students in mathematical sciences and related areas in physical and computer sciences and engineering are expected to be familiar with a daunting breadth of mathematics, but few have such a background. This bestselling book helps students fill in the gaps in their knowledge. Thomas A. Garrity explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.

This book tells the story of the probability integral, the approaches to analyzing it throughout history, and the many areas of science where it arises. The so-called probability integral, the integral over the real line of a Gaussian function, occurs ubiquitously in mathematics, physics, engineering and probability theory. Stubbornly resistant to the undergraduate toolkit for handling integrals, calculating its value and investigating its properties occupied such mathematical luminaries as De Moivre, Laplace, Poisson, and Liouville. This book introduces the probability integral, puts it into a historical context, and describes the different approaches throughout history to evaluate and analyze it. The author also takes entertaining diversions into areas of math, science, and engineering where the probability integral arises: as well as being indispensable to probability theory and statistics, it also shows up naturally in thermodynamics and signal processing. Designed to be accessible to anyone at the undergraduate level and above, this book will appeal to anyone interested in integration techniques, as well as historians of math, science, and statistics.

The widespread availability of high-quality, high-frequency data has revolutionised the study of financial markets. By describing not only asset prices, but also market participants' actions and interactions, this wealth of information offers a new window into the inner workings of the financial ecosystem. In this original text, the authors discuss empirical facts of financial markets and introduce a wide range of models, from the micro-scale mechanics of individual order arrivals to the emergent, macro-scale issues of market stability. Throughout this journey, data is king. All discussions are firmly rooted in the empirical behaviour of real stocks, and all models are calibrated and evaluated using recent data from Nasdaq. By confronting theory with empirical facts, this book for practitioners, researchers and advanced students provides a fresh, new, and often surprising perspective on topics as diverse as optimal trading, price impact, the fragile nature of liquidity, and even the reasons why people trade at all.

This book – specifically developed as a novel textbook on elementary classical mechanics – shows how analytical and numerical methods can be seamlessly integrated to solve physics problems. This approach allows students to solve more advanced and applied problems at an earlier stage and equips them to deal with real-world examples well beyond the typical special cases treated in standard textbooks.Another advantage of this approach is that students are brought closer to the way physics is actually discovered and applied, as they are introduced right from the start to a more exploratory way of understanding phenomena and of developing their physical concepts.While not a requirement, it is advantageous for the reader to have some prior knowledge of scientific programming with a scripting-type language. This edition of the book uses Matlab, and a chapter devoted to the basics of scientific programming with Matlab is included. A parallel edition using Python instead of Matlab is also available.Last but not least, each chapter is accompanied by an extensive set of course-tested exercises and solutions.

If there were no Coulomb interaction among electrons, it would be relatively straightforward to solve the many-electron Schrödinger equation. It is, however, precisely this interaction that is at the heart of numerous fascinating phenomena in condensed matter physics such as superconductivity, Kondo physics, magnetism, etc. Due to the large number of electrons in a material being of the order of Avogadro's number, it is at present — and perhaps in the foreseeable future — not feasible or even desirable to solve the Schrödinger equation to obtain the many-electron wavefunction. Fortunately, a large number of important physical properties can be calculated without explicit knowledge of the wavefunction.Two of the most important formalisms for dealing with the many-electron problem which avoid a direct use of the many-electron wavefunction are the Green function and the density functional theory. Within the Kohn-Sham scheme the latter is used to calculate ground-state properties whereas the former for excitation spectra. The book presents the fundamentals of these two theories in detail with essential many-body tools, such as the occupation number representation and Grassmann algebra developed from scratch. Prior knowledge of many-body theory is not a prerequisite so that it is readable for final-year undergraduates and graduate students in physics and chemistry as well as researchers in the field of electronic structure and many-body theory. The book includes in the last chapter an exposition of a density-functional path for determining the Green function, a new formalism recently proposed by the author. The book should be a valuable companion for those embarking in the field of many-electron physics.

Suitable for advanced undergraduate and graduate students, this new textbook contains an introduction to the mathematical concepts used in physics and engineering. The entire book is unique in that it draws upon applications from physics, rather than mathematical examples, to ensure students are fully equipped with the tools they need. This approach prepares the reader for advanced topics, such as quantum mechanics and general relativity, while offering examples, problems, and insights into classical physics. The book is also distinctive in the coverage it devotes to modelling, and to oft-neglected topics such as Green's functions.

The first edition of this book was originally published in 1985 under the ti­ tle "Probabilistic Properties of Deterministic Systems. " In the intervening years, interest in so-called "chaotic" systems has continued unabated but with a more thoughtful and sober eye toward applications, as befits a ma­ turing field. This interest in the serious usage of the concepts and techniques of nonlinear dynamics by applied scientists has probably been spurred more by the availability of inexpensive computers than by any other factor. Thus, computer experiments have been prominent, suggesting the wealth of phe­ nomena that may be resident in nonlinear systems. In particular, they allow one to observe the interdependence between the deterministic and probabilistic properties of these systems such as the existence of invariant measures and densities, statistical stability and periodicity, the influence of stochastic perturbations, the formation of attractors, and many others. The aim of the book, and especially of this second edition, is to present recent theoretical methods which allow one to study these effects. We have taken the opportunity in this second edition to not only correct the errors of the first edition, but also to add substantially new material in five sections and a new chapter.

A Guardian Best Science Book of the YearThe first biography - 'a stunning achievement' (Kai Bird, American Prometheus) - of the dazzling and painful life of Nobel Prize-winning physicist Roger PenroseWhen he was six years old, Roger Penrose discovered a sundial in a clearing near his house. Through that machine made of light, shadow, and time, Roger glimpsed a "world behind the world" of transcendently beautiful geometry. It spurred him on a journey to become one of the world's most influential mathematicians, philosophers, and physicists. Penrose would prove the limitations of general relativity, set a new agenda for theoretical physics, and astound colleagues and admirers with the elegance and beauty of his discoveries. However, as Patchen Barss documents in The Impossible Man, success came at a price: He was attuned to the secrets of the universe, but struggled to connect with loved ones, especially the women who care for or worked with him.Both erudite and poetic, The Impossible Man draws on years of research and interviews, as well as previously unopened archives to present a moving portrait of Penrose the Nobel Prize-winning scientist and Roger the human being. It reveals not just the extraordinary life of Roger Penrose, but asks who gets to be a genius, and who makes the sacrifices that allow one man to be one.

Using everyday computational power to tackle complexand seemingly unsolvablemath problemsConfronted by a math problem of seemingly impenetrable difficulty, what can you do? In The Delicate Art of Brute Force, Paul Nahin shows how even if you can't solve such a problem, you can still get an answer. The computational power of your own computerthe desktop in your home office, the laptop on your coffee tablecan be deployed to overwhelm a problem's complexity through a massive number-crunching assault. Nahin presents a series of apparently intractable math problems and shows the thought process that allows computational solution. Most of the problems are not abstract constructions but originate in the real worldone chapter is titled ';How Wi-Fi Coverage and Anti-Submarine Warfare Are the Same.' For each one, Nahin compares computational and analytical approaches, demonstrating how well the computer solutions agree with theory. As a bonus, sometimes the computational solution is better, giving more insight or greater flexibility.Along the way, Nahin traces the evolution of high-speed electronic computation, explaining that it upended the way analysts approach complex math problems. After the postwar debut of ENIAC (Electronic Numerical Integrator and Computer) and the ever more powerful electronic computers that came afterward, problems that were simply too difficult or messy for analytical treatment could be defeated by the application of pure brute forcethe ability of computers to perform an enormous number of simple operations in just minutes. Today, we can harness that computational power from our couches. Accessible to anyone who's mastered high school calculus, this engaging book gives both mathematicians and nonmathematicians plenty to think about.

This textbook gradually introduces the reader to several topics related to black hole physics with a didactic approach. It starts with the most basic black hole solution, the Schwarzschild metric, and discusses the basic classical properties of black hole solutions as seen by different probes. Then it reviews various theorems about black hole properties as solutions to Einstein gravity coupled to matter fields, conserved charges associated with black holes, and laws of black hole thermodynamics. Next, it elucidates semiclassical and quantum aspects of black holes, which are relevant in ongoing and future research. The book is enriched with many exercises and solutions to assist in the learning.The textbook is designed for physics graduate students who want to start their research career in the field of black holes; postdocs who recently changed their research focus towards black holes and want to get up-to-date on recent and current research topics; advanced researchers intending to teach (or learn) basic and advanced aspects of black hole physics and the associated mathematical tools. Besides general relativity, the reader needs to be familiar with standard undergraduate physics, like thermodynamics, quantum mechanics, and statistical mechanics. Moreover, familiarity with basic quantum field theory in Minkowski space is assumed. The book covers the rest of the needed background material in the main text or the appendices.

Die Beobachtungsmöglichkeiten der modernen Astrophysik werden immer zahlreicher. Sie reichen von optischen Teleskopen über Weltraumteleskope im Infraroten bis zu Radioteleskopen und von Gravitationswellendetektoren bis hin zu Neutrinoobservatorien. Während die einen im Weltraum ein Augenmerk auf das Licht werfen, nehmen andere unter Wasser kleinste Teilchen unter die Lupe oder messen Störungen der Raumzeit, die kleiner sind als der Durchmesser eines Protons. Das Buch bietet einen spannenden umfassenden Überblick über unsere technischen Möglichkeiten, das Universum zu erforschen. Als Grundlage für eine Vorlesung in Astronomie bzw. Astrophysik ist es ebenso geeignet wie als Lektüre für naturwissenschaftlich Interessierte.

This book gives a comprehensive introduction to those parts of the theory of elliptic integrals and elliptic functions which provide illuminating examples in complex analysis, but which are not often covered in regular university courses. These examples form prototypes of major ideas in modern mathematics and were a driving force of the subject in the eighteenth and nineteenth centuries. In addition to giving an account of the main topics of the theory, the book also describes many applications, both in mathematics and in physics. For the reader’s convenience, all necessary preliminaries on basic notions such as Riemann surfaces are explained to a level sufficient to read the book.For each notion a clear motivation is given for its study, answering the question ‘Why do we consider such objects?’, and the theory is developed in a natural way that mirrors its historical development (e.g., ‘If there is such and such an object, then you would surely expect this one’). This feature sets this text apart from other books on the same theme, which are usually presented in a different order. Throughout, the concepts are augmented and clarified by numerous illustrations. Suitable for undergraduate and graduate students of mathematics, the book will also be of interest to researchers who are not familiar with elliptic functions and integrals, as well as math enthusiasts.

Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations. Topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory. "Thoroughly recommended." — Physics Bulletin. 1983 edition.

This compact guide presents the key features of general relativity, to support and supplement the presentation in mainstream, more comprehensive undergraduate textbooks, or as a re-cap of essentials for graduate students pursuing more advanced studies. It helps students plot a careful path to understanding the core ideas and basics of differential geometry, as applied to general relativity, without overwhelming them. While the guide doesn't shy away from necessary technicalities, it emphasises the essential simplicity of the main physical arguments. Presuming a familiarity with special relativity (with a brief account in an appendix), it describes how general covariance and the equivalence principle motivate Einstein's theory of gravitation. It then introduces differential geometry and the covariant derivative as the mathematical technology which allows us to understand Einstein's equations of general relativity. The book is supported by numerous worked exampled and problems, and important applications of general relativity are described in an appendix.

This unique book gives a modern account of particle physics and gravity based on supersymmetry and supergravity, two of the most significant developments in theoretical physics since general relativity. The book begins with a brief overview of the history of unification and then goes into a detailed exposition of both fundamental and phenomenological topics. The topics in fundamental physics include Einstein gravity, Yang-Mills theory, anomalies, the standard model, supersymmetry and supergravity, and the construction of supergravity couplings with matter and gauge fields, as well as computational techniques for SO(10) couplings. The topics of phenomenological interest include implications of supergravity models at colliders, CP violation, and proton stability, as well as topics in cosmology such as inflation, leptogenesis, baryogenesis, and dark matter. The book is intended for graduate students and researchers seeking to master the techniques for building grand unified models.

A FINANCIAL TIMES AND TLS BOOK OF THE YEARAn exhilarating new biography of John von Neumann: the lost genius who invented our world'A sparkling book, with an intoxicating mix of pen-portraits and grand historical narrative. Above all it fizzes with a dizzying mix of deliciously vital ideas. . . A staggering achievement' Tim HarfordThe smartphones in our pockets and computers like brains. The vagaries of game theory and evolutionary biology. Self-replicating moon bases and nuclear weapons. All bear the fingerprints of one remarkable man: John von Neumann.Born in Budapest at the turn of the century, von Neumann is one of the most influential scientists to have ever lived. His colleagues believed he had the fastest brain on the planet - bar none. He was instrumental in the Manhattan Project and helped formulate the bedrock of Cold War geopolitics and modern economic theory. He created the first ever programmable digital computer. He prophesied the potential of nanotechnology and, from his deathbed, expounded on the limits of brains and computers - and how they might be overcome.Taking us on an astonishing journey, Ananyo Bhattacharya explores how a combination of genius and unique historical circumstance allowed a single man to sweep through so many different fields of science, sparking revolutions wherever he went.Insightful and illuminating, The Man from the Future is a thrilling intellectual biography of the visionary thinker who shaped our century.

Over the course of the past decade, decoherence has become a ubiquitous scienti?c term popular in all kinds of research, from fundamental theories of quantum physics to applications in nanoengineering. Decoherence has been hailed as the solution to long-standing foundational problems dating back to the beginnings of quantum mechanics. It has been cursed as the key obstacle to next-generation technologies, such as quantum computers (another se- ingly omnipresent ?eld of research). And while decoherence has been directly observed in various experiments, its scope and meaning have often been m- understood and misrepresented. Decoherence makes a fantastic subject of research, as it touches upon many di?erent facets of physics, from phi- sophically inclined questions of interpretation all the way to down-to-earth problems in experimental settings and engineering applications. This book will introduce the reader, in an accessible and self-contained manner, to these various fascinating aspects of decoherence. It will focus in particularontherelationofdecoherencetotheso-calledquantum-to-classical transition, i. e. , the question of how decoherence may explain the emergence of the classical appearance of the macroscopic world around us from the underlying quantum substrate. Thescopeofthisbookisrelativelybroadinordertofamiliarizethereader withthemanyfacetsofdecoherence,inboththetheoreticalandexperimental domains. Throughout the book, I have sought to maintain a healthy balance betweentheconceptualideasassociatedwiththedecoherenceprogramonthe one hand and the formal and mathematical details on the other hand. This book will establish a proper understanding of decoherence as a pure quantum phenomenon and will emphasize the importance of the correct interpretation of theconsequences and achievements of decoherence.

Itisaspecialpleasureformetowritethisforewordforaremarkablebookbya remarkableauthor.MarcoPettiniisadeepthinker,whohasspentmanyyears probing the foundations of Hamiltonian chaos and statistical mechanics, in particular phase transitions, from the point of view of geometry and topology. Itisinparticularthequalityofmindoftheauthorandhisdeepphysical,as well as mathematical insights which make this book so special and inspiring. It is a “must” for those who want to venture into a new approach to old problems or want to use new tools for new problems. Although topology has penetrated a number of ?elds of physics, a broad participationoftopologyintheclari?cationandprogressoffundamentalpr- lems in the above-mentioned ?elds has been lacking. The new perspectives topology gives to the above-mentioned problems are bound to help in their clari?cation and to spread to other ?elds of science. The sparsity of geometric thinking and of its use to solve fundamental problems, when compared with purely analytical methods in physics, could be relieved and made highly productive using the material discussed in this book. It is unavoidable that the physicist reader may have then to learn some new mathematics and be challenged to a new way of thinking, but with the author as a guide, he is assured of the best help in achieving this that is presently available.