This text is about spreading of information and influence in complex networks. Although previously considered similar and modeled in parallel approaches, there is now experimental evidence that epidemic and social spreading work in subtly different ways. While previously explored through modeling, there is currently an explosion of work on revealing the mechanisms underlying complex contagion based on big data and data-driven approaches. This volume consists of four parts. Part 1 is an Introduction, providing an accessible summary of the state of the art. Part 2 provides an overview of the central theoretical developments in the field. Part 3 describes the empirical work on observing spreading processes in real-world networks. Finally, Part 4 goes into detail with recent and exciting new developments: dedicated studies designed to measure specific aspects of the spreading processes, often using randomized control trials to isolate the network effect from confounders, such as homophily. Each contribution is authored by leading experts in the field. This volume, though based on technical selections of the most important results on complex spreading, remains quite accessible to the newly interested. The main benefit to the reader is that the topics are carefully structured to take the novice to the level of expert on the topic of social spreading processes. This book will be of great importance to a wide field: from researchers in physics, computer science, and sociology to professionals in public policy and public health.

The spin degree-of-freedom is o?ering a wide range of intriguing oppor- nities both in fundamental as well as in applied solid-state physics. When combined with the rich and fertile physics of low-dimensional semicondu- ingstructuresandwiththepossibilitytochange,forexample,carrierdensity, electric ?elds or coupling to other quantum systems in a controlled way, an extremely exciting and interesting research ?eld is opened. Most comm- cial electronic devices are based on spin-independent charge transport. In the last two decades, however, scientists have been focusing on the ambitious objective of exploiting the spin degree-of-freedom of the electron to achieve novel functionalities. Ferromagnetic semiconductors, spin transistors, sing- spin manipulations or spin-torque MRAMs (magnetoresistive random access memories) are some of the hot topics. The importance of spin phenomena for new applications was recognized by the Royal Swedish Academy of S- ences by awarding the 2007 Nobel Prize in Physics jointly to Albert Fert and Peter Grun .. berg "for the discovery of giant magnetoresistance". This - fect originates from spin-dependent scattering phenomena in a two-terminal ferromagnetic-paramagnetic-ferromagnetic junction leading to a new type of magnetic memory. The Hall e?ect and its applications remain fertile - search areas. The spin Hall e?ect, in analogy with the conventional Hall e?ect, occurs in paramagnetic systems as a result of spin-orbit interaction.

The second edition of this textbook presents the basic mathematical knowledge and skills that are needed for courses on modern theoretical physics, such as those on quantum mechanics, classical and quantum field theory, and related areas. The authors stress that learning mathematical physics is not a passive process and include numerous detailed proofs, examples, and over 200 exercises, as well as hints linking mathematical concepts and results to the relevant physical concepts and theories. All of the material from the first edition has been updated, and five new chapters have been added on such topics as distributions, Hilbert space operators, and variational methods. The text is divided into three parts: - Part I: A brief introduction to (Schwartz) distribution theory. Elements from the theories of ultra distributions and (Fourier) hyperfunctions are given in addition to some deeper results for Schwartz distributions, thus providing a rather comprehensive introduction to the theory of generalized functions. Basic properties and methods for distributions are developed with applications to constant coefficient ODEs and PDEs. The relation between distributions and holomorphic functions is considered, as well as basic properties of Sobolev spaces. - Part II: Fundamental facts about Hilbert spaces. The basic theory of linear (bounded and unbounded) operators in Hilbert spaces and special classes of linear operators - compact, Hilbert-Schmidt, trace class, and Schroedinger operators, as needed in quantum physics and quantum information theory - are explored. This section also contains a detailed spectral analysis of all major classes of linear operators, including completeness of generalized eigenfunctions, as well as of (completely) positive mappings, in particular quantum operations. - Part III: Direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators. The authors conclude with a discussion of the Hohenberg-Kohn variational principle. The appendices contain proofs of more general and deeper results, including completions, basic facts about metrizable Hausdorff locally convex topological vector spaces, Baire's fundamental results and their main consequences, and bilinear functionals. Mathematical Methods in Physics is aimed at a broad community of graduate students in mathematics, mathematical physics, quantum information theory, physics and engineering, as well as researchers in these disciplines. Expanded content and relevant updates will make this new edition a valuable resource for those working in these disciplines.

The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations.

Covering a wide range of material, this book has three appendices which provide background information required for some of the chapters. The authors organize fundamental results in a unified way and refine existing proofs. For instance in chapter three, a generalization of Jantzen filtration is reformulated in an algebraic manner, and geometric interpretation is provided. Statements, widely believed to be true, are collated, and results which are known but not verified are proven, such as the corrected structure theorem of Fock modules in chapter eight.

This book will be of interest to a wide range of mathematicians and physicists from the level of graduate students to researchers.

This book provides an elementary introduction to the ideas and methods of topology by the detailed study of certain topics. There are elegant but rigorous proofs of many of the basic theorems, and special attention is given to the results needed in the theory of functions.

Density functional theory (DFT) is by now a well-established method for tackling the quantum mechanics of many-body systems. Originally applied to compute properties of atoms and simple molecules, DFT has quickly become a work horse for more complex applications in the chemical and materials sciences. The present set of lectures, spanning the whole range from basic principles to relativistic and time-dependent extensions of the theory, is the ideal introduction for graduate students or nonspecialist researchers wishing to familiarize themselves with both the basic and most advanced techniques in this field.

This work provides the current theory and observations behind the cosmological phenomenon of dark energy. The approach is comprehensive with rigorous mathematical theory and relevant astronomical observations discussed in context. The book treats the background and history starting with the new-found importance of Einstein's cosmological constant (proposed long ago) in dark energy formulation, as well as the frontiers of dark energy. The authors do not presuppose advanced knowledge of astronomy, and basic mathematical concepts used in modern cosmology are presented in a simple, but rigorous way. All this makes the book useful for both astronomers and physicists, and also for university students of physical sciences.

"Pseudochaotic Kicked Oscillators: Renormalization, Symbolic Dynamics, and Transport" presents recent developments in pseudochaos, which is concerned with complex branching behaviors of dynamical systems at the interface between orderly and chaotic motion. Pseudochaos is characterized by the trapping of orbits in the vicinity of self-similar hierarchies of islands of stability, producing phase-space displacements which increase asymptotically as a power of time. This monograph is a thorough, self-contained investigation of a simple one-dimensional model (a kicked harmonic oscillator) which exhibits pseudochaos in its purest form. It is intended for graduate students and researchers in physics and applied mathematics, as well as specialists in nonlinear dynamics.
Dr. John H. Lowenstein is a Professor Emeritus in the Department of Physics at New York University, USA."

Begegnungen am CERN Menschen, die die Geheimnisse des Universums entschlusseln/Michael Krause stellt sie uns vor Eine "wissenschaftliche Sensation", ein "historischer Meilenstein", historisch so bedeutend wie die Mondlandung: Als Wissenschaftler des CERN im Juli 2012 die Existenz eines "Gottesteilchens", das allen anderen Teilchen Masse verleiht, mit einer Wahrscheinlichkeit von mehr als 99,999 Prozent bestatigten, waren die Reaktionen uberwaltigend. Schliesslich arbeiten die Forscher hier an nichts Geringerem als an der Entschlusselung der Geheimnisse des Universums. Die Suche nach dem Grund des Daseins und dem Ursprung aller Materie ist auch das Thema dieses Buches. Traume, Visionen, Forschungen: die Menschen stehen im Mittelpunkt Das "Gottesteilchen", die Erforschung der "Dunklen Materie" und der "Dunklen Energie" und der starkste je gebaute Teilchenbeschleuniger der Welt all das sind die spannenden Bestandteile des Buches. In den Mittelpunkt seiner Arbeit stellt Krause jedoch die Menschen am CERN, ihren Hintergrund, ihre Geschichte, ihre Arbeit, ihre Forschungen, Traume und Visionen sie sind das Hauptthema des Buches. Krause verleiht den wissenschaftlichen Forschungen am CERN somit ein personliches Gesicht und schreibt das Portrait einer ganzen Generation von Wissenschaftlern, die in den vergangenen Jahrzehnten an der sogenannten "Neuen Physik" gearbeitet haben.

The essays in this book look at way in which the fundaments of physics might need to be changed in order to make progress towards a unified theory. They are based on the prize-winning essays submitted to the FQXi essay competition "Which of Our Basic Physical Assumptions Are Wrong?", which drew over 270 entries. As Nobel Laureate physicist Philip W. Anderson realized, the key to understanding nature's reality is not anything "magical", but the right attitude, "the focus on asking the right questions, the willingness to try (and to discard) unconventional answers, the sensitive ear for phoniness, self-deception, bombast, and conventional but unproven assumptions." The authors of the eighteen prize-winning essays have, where necessary, adapted their essays for the present volume so as to (a) incorporate the community feedback generated in the online discussion of the essays, (b) add new material that has come to light since their completion and (c) to ensure accessibility to a broad audience of readers with a basic grounding in physics. The Foundational Questions Institute, FQXi, catalyzes, supports, and disseminates research on questions at the foundations of physics and cosmology, particularly new frontiers and innovative ideas integral to a deep understanding of reality, but unlikely to be supported by conventional funding sources.

One of the mathematical challenges of modern physics lies in the development of new tools to efficiently describe different branches of physics within one mathematical framework. This text introduces precisely such a broad mathematical model, one that gives a clear geometric expression of the symmetry of physical laws and is entirely determined by that symmetry.The first three chapters discuss the occurrence of bounded symmetric domains (BSDs) or homogeneous balls and their algebraic structure in physics. The book further provides a discussion of how to obtain a triple algebraic structure associated to an arbitrary BSD; the relation between the geometry of the domain and the algebraic structure is explored as well. The last chapter contains a classification of BSDs revealing the connection between the classical and the exceptional domains.With its unifying approach to mathematics and physics, this work will be useful for researchers and graduate students interested in the many physical applications of bounded symmetric domains. It will also benefit a wider audience of mathematicians, physicists, and graduate students working in relativity, geometry, and Lie theory.

It is not uncommon to find engineers in test labs or design groups who have not had occasion to use the mathematical tools acquired in college. When suddenly faced with vibration issues they find themselves ill equipped to get a solid grasp of the vibration process. It is the intent of this technical reference to provide access to vibration theory, initially at a very elementary level, then progressing from basic analytical formulations toward the more mature mathematical representations associated with eigenvectors and the Fourier Transform. Mode shapes are introduced without any reference to the eigenvalue problem, but connected immediately to simple coordinate transformations in two and three dimensions. This allows a rather simple picture of operators, ultimately leading to a straight forward derivation of the Frequency Response Function (FRF) formula. It is hoped that many engineers will find their way back into a more analytical approach to vibration problems. providing fresh viewpoints from time to time, such as the development of modal force as a contravariant vector, providing a detailed view of the FRF as a superposition of modal FRFs.

Covering the years 2008-2012, this bookprofilesthe life and work of recent winners of the Abel Prize:
. John G. Thompson and Jacques Tits, 2008
. MikhailGromov, 2009
. John T. Tate Jr., 2010
. John W. Milnor, 2011
. Endre Szemeredi, 2012.
The profiles feature autobiographical information as well as a description of each mathematician's work. In addition, each profile contains a complete bibliography, a curriculum vitae, as well as photos old and new. As an added feature, interviews with the Laureates are presented on an accompanying web site (http: //extras.springer.com/).

The book also presents a history of the Abel Prize written by the historian Kim Helsvig, and includes a facsimile of aletter from Niels Henrik Abel, which is transcribed, translated into English, and placed into historical perspectiveby Christian Skau.

This book follows onThe Abel Prize: 2003-2007, The First Five Years(Springer, 2010), which profiles the work of the first Abel Prize winners.

"

This proceedings volume of the ISEA 2006 examines sports engineering, an interdisciplinary subject which encompasses and integrates not only sports science and engineering but also biomechanics, physiology and anatomy, and motion physics. This is the first title of its kind in the emerging field of sports technology.

This book provides a broad introductory survey of this remarkable field, aiming to establish and clearly differentiate its physical principles, and also to provide a snapshot portrait of many of the most prominent current applications. Primary emphasis is placed on developing an understanding of the fundamental photonic origin behind the mechanism that operates in each type of effect. To this end, the first few chapters introduce and develop core theory, focusing on the physical significance and source of the most salient parameters, and revealing the detailed interplay between the key material and optical properties. Where appropriate, both classical and photonic (quantum mechanical) representations are discussed. The number of equations is purposely kept to a minimum, and only a broad background in optical physics is assumed. With copious examples and illustrations, each of the subsequent chapters then sets out to explain and exhibit the main features and uses of the various distinct types of mechanism that can be involved in optical nanomanipulation, including some of the very latest developments. To complete the scene, we also briefly discuss applications to larger, biological particles. Overall, this book aims to deliver to the non-specialist an amenable introduction to the technically more advanced literature on individual manipulation methods. Full references to the original research papers are given throughout, and an up-to-date bibliography is provided for each chapter, which directs the reader to other selected, more specialised sources.

This book provides a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems as important applications of the theory. The text is written to be used in the traditional way or in a more applied way. In addition to its use in a traditional one or two semester graduate course in mathematics, the book is organized to be used for interdisciplinary courses in applied mathematics, physics, and engineering.

Supernovae, their bearing on cosmology and their connection to gamma-ray bursts are now at the center of astrophysical research programs. This volume deals with astronomical observations of supernovae and their relation to nuclear and particle astrophysics. All known aspects of supernovae explosions are investigated in articles specifically written for researchers and advanced graduate students. It also includes recent numerical "experiments" related to the question of hydrodynamical instability in two and three dimensions and to problems concerning the complexity of radiation transport in the models. Other contributions discuss the possible energy sources needed to drive these powerful stellar explosions.

The fourth of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self-contained and accessible to the non-specialist, and topics covered include applications to mechanics, elasticity, plasticity, hydrodynamics, thermodynamics, statistical physics, and special and general relativity including cosmology. The book contains a detailed physical motivation of the relevant basic equations and a discussion of particular problems which have played a significant role in the development of physics and through which important mathematical and physical insight may be gained. It combines classical and modern ideas to build a bridge between the language and thoughts of physicists and mathematicians. Many exercises and a comprehensive bibliography complement the text.

The 1996 NATO Advanced Study Institute (ASI) followed the international tradi tion of the schools held in Cargese in 1976, 1979, 1983, 1987 and 1991. Impressive progress in quantum field theory had been made since the last school in 1991. Much of it is connected with the interplay of quantum theory and the structure of space time, including canonical gravity, black holes, string theory, application of noncommutative differential geometry, and quantum symmetries. In addition there had recently been important advances in quantum field theory which exploited the electromagnetic duality in certain supersymmetric gauge theories. The school reviewed these developments. Lectures were included to explain how the "monopole equations" of Seiberg and Witten can be exploited. They were presented by E. Rabinovici, and supplemented by an extra 2 hours of lectures by A. Bilal. Both the N = 1 and N = 2 supersymmetric Yang Mills theory and resulting equivalences between field theories with different gauge group were discussed in detail. There are several roads to quantum space time and a unification of quantum theory and gravity. There is increasing evidence that canonical gravity might be a consistent theory after all when treated in. a nonperturbative fashion. H. Nicolai presented a series of introductory lectures. He dealt in detail with an integrable model which is obtained by dimensional reduction in the presence of a symmetry."

This book begins by introducing magnetism and discusses magnetic properties of materials, magnetic moments of atoms and ions, and the elements important to magnetism. It covers magnetic susceptibilities and electromagnetic waves in anisotropic dispersive media among other topics. There are problems at the end of each chapter, many of which serve to expand or explain the material in the text. The bibliographies for each chapter give an entry to the research literature.

This book is devoted to one of the most interesting and rapidly developing areas of modern nonlinear physics and mathematics - theoretical, analytical andnumerical, studyofthestructureanddynamicsofone-dimensionalaswell as two- and three-dimensional solitons and nonlinear wave packets described by the Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), nonlinear Schr] odinger (NLS) and derivative nonlinear Schr] odinger (DNLS) classes of equations. Special attention is paid to generalizations (relevant to various complex physical media) of these equations, accounting for higher-order d- persion corrections, in?uence of dissipation, instabilities, and stochastic ?- tuations of the wave ?elds. We present here a coordinated approach to the theory, simulations, and applications of the nonlinear one-, two-, and three-dimensional solitary wave solutions. Overall, the content of the book is a systematic account of results notonlyalreadyknownintheliterature, butalsothoseofneworiginalstudies related to the theory of models allowing soliton solutions, and analyses of the stability and asymptotics of these solutions. We give signi?cant consideration to numerical methods and results of numerical simulations of the structure and dynamics of solitons and nonlinear wave packets. Together with deep insights into the theory, applications to various branches of modern physics are considered, especially to plasma physics (such as space plasmas including ionospheric and magnetospheric processes), hydrodynamics, and atmosphere dynamics. Presently, thetheoryofone-dimensionalnonlinearequationsoftheclasses consideredbytheauthorsiswelldeveloped, andtheprogressinstudiesofthe structure and evolution of one-dimensional solitons and wave packets is ob- ous. This progress was especially fast after the discovery of hidden algebraic symmetries of the KdV, NLS, and other (integrable by the inverse scatt- ing transform (IST) method) classes of one-dimensional evolution equations

The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schroedinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators, Hamburger moment problem) . Among others, a number of advanced special topics are treated on a text book level accompanied by numerous illustrating examples and exercises. The main themes of the book are the following: - Spectral integrals and spectral decompositions of self-adjoint and normal operators - Perturbations of self-adjointness and of spectra of self-adjoint operators - Forms and operators - Self-adjoint extension theory :boundary triplets, Krein-Birman-Vishik theory of positive self-adjoint extension

This book is a collection of multidisciplinary papers presented at the Department of Physics of Milan University's congress on 28 and 29 June 2017, which was also intended as a kick-off meeting for the design of a novel science campus at the Expo site in Milan. The congress presented a snapshot of the department's research to the academic community, the media, policymakers and authorities as well as the public at large, and also provided an opportunity to strengthen interdisciplinary collaborations between the members of the department and other communities. This book is a valuable resource for scientists looking for synergetic projects, policymakers wanting to grasp scientists' points of view and for prospective graduate students seeking expanding areas of research.