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## Symmetries and Dynamics of Star Clusters

by Jaroslav HaasIn this PhD thesis, which was nominated for publication in this series by the Astronomical Institute at Charles University, Prague, the author investigates the orbital evolution of an initially thin stellar disc around a supermassive black hole, considering various perturbative sources of gravity. His findings, obtained by both direct numerical N-body modelling and using standard perturbation methods, offer a viable theoretical explanation for the observed configuration of young stars in the Galactic Centre. This marks a significant contribution to a topic of great interest in contemporary astrophysics. The author also shows in his thesis that a secular instability (m = 1 mode) may occur in the embedding spherical cluster of old stars. This increases the richness of possible evolution scenarios of the embedding cluster and may lead to effective feeding of supermassive black holes through tidal disruption of stars on extremely eccentric orbits.

## Symmetries and Groups in Signal Processing

by Virendra P. SinhaSymmetries and Groups in Signal Processing: An Introduction deals with the subject of symmetry, and with its place and role in modern signal processing. In the sciences, symmetry considerations and related group theoretic techniques have had a place of central importance since the early twenties. In engineering, however, a matching recognition of their power is a relatively recent development. Despite that, the related literature, in the form of journal papers and research monographs, has grown enormously. A proper understanding of the concepts that have emerged in the process requires a mathematical background that goes beyond what is traditionally covered in an engineering undergraduate curriculum. Admittedly, there is a wide selection of excellent introductory textbooks on the subject of symmetry and group theory. But they are all primarily addressed to students of the sciences and mathematics, or to students of courses in mathematics. Addressed to students with an engineering background, this book is meant to help bridge the gap.

## Symmetries and Integrability of Difference Equations

by Decio Levi Raphaël Rebelo Pavel WinternitzDifference equations are playing an increasingly important role in the natural sciences. Indeed many phenomena are inherently discrete and are naturally described by difference equations. Phenomena described by differential equations are therefore approximations of more basic discrete ones. Moreover, in their study it is very often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference equations. This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference ones. Each of the eleven chapters is a self-contained treatment of a topic, containing introductory material as well as the latest research results. The book will be welcomed by graduate students and researchers seeking an introduction to the field. As a survey of the current state of the art it will also serve as a valuable reference.

## Symmetries and Semi-invariants in the Analysis of Nonlinear Systems

by Antonio Tornambè Laura MeniniThis book details the analysis of continuous- and discrete-time dynamical systems described by differential and difference equations respectively. Differential geometry provides the tools for this, such as first-integrals or orbital symmetries, together with normal forms of vector fields and of maps. A crucial point of the analysis is linearization by state immersion. The theory is developed for general nonlinear systems and specialized for the class of Hamiltonian systems. By using the strong geometric structure of Hamiltonian systems, the results proposed are stated in a different, less complex and more easily comprehensible manner. They are applied to physically motivated systems, to demonstrate how much insight into known properties is gained using these techniques. Various control systems applications of the techniques are characterized including: computation of the flow of nonlinear systems; computation of semi-invariants; computation of Lyapunov functions for stability analysis and observer design.

## Symmetries in Atomic Nuclei

by Pieter Van Isacker Jan Jolie Alejandro FrankSymmetries in Atomic Nuclei aims to present an overview of recent applications of symmetry to the description of atomic nuclei. Special care is given to a pedagogical introduction of symmetry concepts using simple examples. After a historical overview of the applications of symmetry in nuclear physics, progress in the field during the last decade is reviewed. Special emphasis is put on the introduction of neutron-proton and boson-fermion degrees of freedom. Their combination leads to a supersymmetric description of pairs and quartets of nuclei. Both theoretical aspects and experimental signatures of dynamical (super)symmetries are carefully discussed. Case studies show how these symmetries are displayed by real atomic nuclei which have been studied experimentally using state-of-the art spectroscopy. This book focuses on nuclear structure physics and has been written by active investigators in the field, but its scope is wider and is intended for final-year or post-graduate students and researchers interested in understanding the power and beauty of symmetry methods in physics.

## Symmetries in Fundamental Physics

by Kurt SundermeyerOver the course of the last century it has become clear that both elementary particle physics and relativity theories are based on the notion of symmetries. These symmetries become manifest in that the "laws of nature" are invariant under spacetime transformations and/or gauge transformations. The consequences of these symmetries were analyzed as early as in 1918 by Emmy Noether on the level of action functionals. Her work did not receive due recognition for nearly half a century, but can today be understood as a recurring theme in classical mechanics, electrodynamics and special relativity, Yang-Mills type quantum field theories, and in general relativity. As a matter of fact, as shown in this monograph, many aspects of physics can be derived solely from symmetry considerations. This substantiates the statement of E. P. Wigner ". . . if we knew all the laws of nature, or the ultimate Law of nature, the invariance properties of these laws would not furnish us new information. " Thanks to Wigner we now also understand the implications of quantum physics and symmetry considerations: Poincare invariance dictates both the characteristic properties of particles (mass, spin, . . . ) and the wave equations of spin 0, 1/2, 1, . . . objects. Further, the work of C. N. Yang and R. Mills reveals the consequences of internal symmetries as exemplified in the symmetry group of elementary particle physics. Given this pivotal role of symmetries it is thus not surprising that current research in fundamental physics is to a great degree motivated and inspired by considerations of symmetry. The treatment of symmetries in this monograph ranges from classical physics to now well-established theories of fundamental interactions, to the latest research on unified theories and quantum gravity.

## Symmetries in Graphs, Maps, and Polytopes

by Jozef Širáň Robert JajcayThis volume contains seventeen of the best papers delivered at the SIGMAP Workshop 2014, representing the most recent advances in the field of symmetries of discrete objects and structures, with a particular emphasis on connections between maps, Riemann surfaces and dessins d'enfant. Providing the global community of researchers in the field with the opportunity to gather, converse and present their newest findings and advances, the Symmetries In Graphs, Maps, and Polytopes Workshop 2014 was the fifth in a series of workshops. The initial workshop, organized by Steve Wilson in Flagstaff, Arizona, in 1998, was followed in 2002 and 2006 by two meetings held in Aveiro, Portugal, organized by Antonio Breda d'Azevedo, and a fourth workshop held in Oaxaca, Mexico, organized by Isabel Hubard in 2010. This book should appeal to both specialists and those seeking a broad overview of what is happening in the area of symmetries of discrete objects and structures. iv>

## Symmetries, Integrable Systems and Representations

by Sophie Morier-Genoud Kenji Iohara Bertrand RémyThis volume is the result of two international workshops; Infinite Analysis 11 - Frontier of Integrability - held at University of Tokyo, Japan in July 25th to 29th, 2011, and Symmetries, Integrable Systems and Representations held at Université Claude Bernard Lyon 1, France in December 13th to 16th, 2011. Included are research articles based on the talks presented at the workshops, latest results obtained thereafter, and some review articles. The subjects discussed range across diverse areas such as algebraic geometry, combinatorics, differential equations, integrable systems, representation theory, solvable lattice models and special functions. Through these topics, the reader will find some recent developments in the field of mathematical physics and their interactions with several other domains.

## Symmetries of Integro-Differential Equations: With Applications in Mechanics and Plasma Physics

by N. Kh. Ibragimov Sergey V. Meleshko Yurii N. Grigoriev Vladimir F. KovalevThis book aims to coherently present applications of group analysis to integro-differential equations in an accessible way. The book will be useful to both physicists and mathematicians interested in general methods to investigate nonlinear problems using symmetries. Differential and integro-differential equations, especially nonlinear, present the most effective way for describing complex processes. Therefore, methods to obtain exact solutions of differential equations play an important role in physics, applied mathematics and mechanics. This book provides an easy to follow, but comprehensive, description of the application of group analysis to integro-differential equations. The book is primarily designed to present both fundamental theoretical and algorithmic aspects of these methods. It introduces new applications and extensions of the group analysis method. The authors have designed a flexible text for postgraduate courses spanning a variety of topics.

## Symmetry

by Marcus Du SautoySymmetry is all around us. Of fundamental significance to the way we interpret the world, this unique, pervasive phenomenon indicates a dynamic relationship between objects. Combining a rich historical narrative with his own personal journey as a mathematician, Marcus du Sautoy takes a unique look into the mathematical mind as he explores deep conjectures about symmetry and brings us face-to-face with the oddball mathematicians, both past and present, who have battled to understand symmetry's elusive qualities.

## Symmetry

by Kristopher TappThis textbook is perfect for a math course for non-math majors, with the goal of encouraging effective analytical thinking and exposing students to elegant mathematical ideas. It includes many topics commonly found in sampler courses, like Platonic solids, Euler's formula, irrational numbers, countable sets, permutations, and a proof of the Pythagorean Theorem. All of these topics serve a single compelling goal: understanding the mathematical patterns underlying the symmetry that we observe in the physical world around us. The exposition is engaging, precise and rigorous. The theorems are visually motivated with intuitive proofs appropriate for the intended audience. Students from all majors will enjoy the many beautiful topics herein, and will come to better appreciate the powerful cumulative nature of mathematics as these topics are woven together into a single fascinating story about the ways in which objects can be symmetric.

## Symmetry Analysis of Differential Equations

by Daniel J. ArrigoA self-contained introduction to the methods and techniques of symmetry analysis used to solve ODEs and PDEsSymmetry Analysis of Differential Equations: An Introduction presents an accessible approach to the uses of symmetry methods in solving both ordinary differential equations (ODEs) and partial differential equations (PDEs). Providing comprehensive coverage, the book fills a gap in the literature by discussing elementary symmetry concepts and invariance, including methods for reducing the complexity of ODEs and PDEs in an effort to solve the associated problems. Thoroughly class-tested, the author presents classical methods in a systematic, logical, and well-balanced manner. As the book progresses, the chapters graduate from elementary symmetries and the invariance of algebraic equations, to ODEs and PDEs, followed by coverage of the nonclassical method and compatibility. Symmetry Analysis of Differential Equations: An Introduction also features:Detailed, step-by-step examples to guide readers through the methods of symmetry analysisEnd-of-chapter exercises, varying from elementary to advanced, with select solutions to aid in the calculation of the presented algorithmic methodsSymmetry Analysis of Differential Equations: An Introduction is an ideal textbook for upper-undergraduate and graduate-level courses in symmetry methods and applied mathematics. The book is also a useful reference for professionals in science, physics, and engineering, as well as anyone wishing to learn about the use of symmetry methods in solving differential equations.

## Symmetry and Economic Invariance

by Ryuzo Sato Rama V. RamachandranSymmetry and Economic Invariance (second enhanced edition) explores how the symmetry and invariance of economic models can provide insights into their properties. Although the professional economist of today is adept at many of the mathematical techniques used in static and dynamic optimization models, group theory is still not among his or her repertoire of tools. The authors aim to show that group theoretic methods form a natural extension of the techniques commonly used in economics and that they can be easily mastered. Part I provides an introduction that minimizes prerequisites including prior knowledge of group theory. Part II discusses recent developments in the field.

## Symmetry and Pattern in Projective Geometry

by Eric LordSymmetry and Pattern in Projective Geometry is a self-contained study of projective geometry which compares and contrasts the analytic and axiomatic methods. The analytic approach is based on homogeneous coordinates, and brief introductions to Plücker coordinates and Grassmann coordinates are presented. This book looks carefully at linear, quadratic, cubic and quartic figures in two, three and higher dimensions. It deals at length with the extensions and consequences of basic theorems such as those of Pappus and Desargues. The emphasis throughout is on special configurations that have particularly interesting symmetry properties. The intricate and novel ideas of 'Donald' Coxeter, who is considered one of the great geometers of the twentieth century, are also discussed throughout the text. The book concludes with a useful analysis of finite geometries and a description of some of the remarkable configurations discovered by Coxeter. This book will be appreciated by mathematics students and those wishing to learn more about the subject of geometry. It makes accessible subjects and theorems which are often considered quite complicated and presents them in an easy-to-read and enjoyable manner.

## Symmetry and Symmetry-Breaking in Semiconductors: Fine Structure of Exciton States (Springer Tracts in Modern Physics #279)

by Bernd Hönerlage Ivan PelantThis book discusses group theory investigations of zincblende and wurtzite semiconductors under symmetry-breaking conditions. The text presents the group theory elements required to develop a multitude of symmetry-breaking problems, giving scientists a fast track to bypass the need for recalculating electronic states. The text is not only a valuable resource for speeding up calculations but also illustrates the construction of effective Hamiltonians for a chosen set of electronic states in crystalline semiconductors. Since Hamiltonians have to be invariant under the transformations of the point group, the crystal symmetry determines the multiplet structure of these states in the presence of spin-orbit, crystal-field, or exchange interactions. Symmetry-breaking leads to additional coupling of the states, resulting in shifts and/or splittings of the multiplets. Such interactions may be intrinsic, as in the case of the quasi-particle dispersion, or extrinsic, induced by magnetic, electric, or strain fields. Using a power expansion of the perturbations these interaction terms can be determined in their parameterized form in a unique way. The hierarchic structure of this invariant development allows to estimate the importance of particular symmetry-breaking effects in the Hamiltonian. A number of selected experimental curves are included to illustrate the symmetry-based discussions, which are especially important in optical spectroscopy. This text is written for graduate students and researchers who want to understand and simulate experimental findings reflecting the fine structure of electronic or excitonic states in crystalline semiconductors.

## Symmetry and the Beautiful Universe

by Christopher T. Hill Leon M. LedermanWhen scientists peer through a telescope at the distant stars in outer space or use a particle-accelerator to analyze the smallest components of matter, they discover that the same laws of physics govern the whole universe at all times and all places. Physicists call the eternal, ubiquitous constancy of the laws of physics symmetry. Symmetry is the basic underlying principle that defines the laws of nature and hence controls the universe. This all-important insight is one of the great conceptual breakthroughs in modern physics and is the basis of contemporary efforts to discover a grand unified theory to explain all the laws of physics.Nobel Laureate Leon M. Lederman and physicist Christopher T. Hill explain the supremely elegant concept of symmetry and all its profound ramifications to life on Earth and the universe at large in this eloquent, accessible popular science book. They not only clearly describe concepts normally reserved only for physicists and mathematicians, but they also instill an appreciation for the profound beauty of the universe's inherent design.Central to the story of symmetry is an obscure, unpretentious, but extremely gifted German mathematician named Emmy Noether. Though still little known to the world, she impressed no less a scientist than Albert Einstein, who praised her "penetrating mathematical thinking." In some of her earliest work she proved that the law of the conservation of energy was connected to the idea of symmetry and thus laid the mathematical groundwork for what may be the most important concept of modern physics.Lederman and Hill reveal concepts about the universe, based on Noether's work, that are largely unknown to the public and have wide-reaching implications in connection with the Big Bang, Einstein's theory of relativity, quantum mechanics, and many other areas of physics. Through ingenious analogies and illustrations, they bring these astounding notions to life. This book will open your eyes to a universe you never knew existed.From the Trade Paperback edition.

## Symmetry Breaking in Syntax

by Hubert HaiderIn this illuminating new theory of grammar, Hubert Haider demonstrates that there is a basic asymmetry in the phrase structure of any language, whatever sentence structure it takes. Moreover, he argues that understanding this asymmetry is the key to understanding the grammatical causality underlying a broad range of core syntactic phenomena. Until now, Germanic languages have been seen to fall into two distinct classes: those which take an object-verb sentence structure (OV) or a verb-object one (VO). However, by examining the nature of this universal underlying asymmetry, Hubert Haider reveals a third syntactic type: 'Type III'. In particular, he employs the third type to explore the cognitive evolution of grammar which gave rise to the structural asymmetry and its typological implications. Symmetry Breaking in Syntax will appeal to academic researchers and graduate students involved in comparative and theoretical syntax and the cognitive evolution of grammar.

## Symmetry Discovered: Concepts and Applications in Nature and Science

by Joe RosenSymmetry provides an insight into the way nature works and is often used by scientists and technologists to help solve problems. Symmetry has numerous other applications as well -- with more being discovered all the time in science, the arts and other fields of human endeavor.This classic work provides an excellent introduction to the basic concepts and terminology (including, optionally, group theory), as well as lucid discussions of geometric symmetry, other symmetries and appropriate symmetry, symmetry in nature, uses of symmetry in science and much more.Readers wishing to pursue specific topics will find many references that reflect the author's wide reading in the subject and his own obvious enthusiasm. For this edition, Dr. Rosen has provided a new preface, solutions to the problems, and an addendum to the bibliography.

## Symmetry, Group Theory, and the Physical Properties of Crystals

by Richard C PowellThis book demonstrates the importance of symmetry in determining the properties of solids and the power of using group theory and tensor algebra to elucidate these properties. It provides the fundamentals necessary for the reader to understand how to utilize these techniques in many different applications without becoming lost in a heavy formal treatment of the subject matter. The book begins by discussing the concepts of symmetry relevant to crystal structures. This is followed by a summary of the basics of group theory and how it applies to quantum mechanics. Next is a discussion of the description of the macroscopic properties of crystals by tensors and how symmetry determines the form of these tensors. The basic concepts covered in these early chapters are then applied to a series of different examples including crystal field theory treatment of point defects in solids, molecular orbitals, two-photon processes, the optical properties of solids, the nonlinear optical properties of solids, lattice vibrations, the Jahn-Teller effect, and the effects of translational symmetry on electronic energy bands in solids.. Emphasis is placed on showing how group theory and tensor algebra can provide important information about the properties of a system without resorting to first principal quantum mechanical calculations. The book also features a comprehensive set of relevant tables, including crystal symmetries, point group character tables, matter tensors of different rank, and other tensor properties.

## Symmetry in Chemistry

by Hans H. Jaffé Milton OrchinBecause symmetry arguments are a powerful tool in teaching such concepts as hybridization, group and molecular orbitals, selection rules in absorption spectroscopy, crystal structure, and other topics, a book devoted exclusively to symmetry in chemistry, developed in an essentially nonmathematical way, is a must for students and research workers interested in these subjects. This is such a book. After an introduction to symmetry, the authors offer lucid discussions of symmetry elements and operations, multiple symmetry operations, multiplication tables and point groups, group theory, applications, and crystal symmetry. Three appendices provide complete character tables, tables of the number of normal vibrations in various symmetry species, and tables showing the direct sums of excited states and combination states of degenerate vibrations. Intended as a supplementary text for both undergraduate and graduate students who seek a broad background for understanding structural problems, this concise treatment will also appeal to the large group of practicing chemists who wish to review the applications of symmetry and group theory.

## Symmetry in Syntax

by Barbara CitkoWhile much has been written on asymmetric aspects of sentence structure, symmetric aspects have been largely ignored, or claimed to be non-existent. Does symmetry in syntax exist, and if it does, how do we account for it? Barbara Citko sets out to tackle these questions and offers a unified approach to a number of phenomena that have so far been studied only in isolation. Focusing on three core minimalist mechanisms: merge, move and labeling, she advances a new theory of these mechanisms, by showing that under certain well-defined circumstances merge can create symmetric structures, move can target either of two potentially moveable objects, and labels can be constructed symmetrically from the features of two objects. This book is aimed at researchers and graduate students interested in minimalist syntax, the structure of questions, relative clauses, coordination, double object constructions and copular sentences.

## Symmetry: An Introduction to Group Theory and Its Applications

by Roy McweenyThe crucial significance of symmetry to the development of group theory and in the fields of physics and chemistry cannot be overstated, and this well-organized volume provides an excellent introduction to the topic.The text develops the elementary ideas of both group theory and representation theory in a progressive and thorough fashion, leading students to a point from which they can proceed easily to more elaborate applications. The finite groups describing the symmetry of regular polyhedral and of repeating patterns are emphasized, and geometric illustrations of all main processes appear here -- including more than 100 fully worked examples.Designed to be read at a variety of levels and to allow students to focus on any of the main fields of application, this volume is geared toward advanced undergraduate and graduate physics and chemistry students with the requisite mathematical background.

## Symmetry Properties in Transmission Lines Loaded with Electrically Small Resonators

by Jordi NaquiThis book discusses the analysis, circuit modeling, and applications of transmission lines loaded with electrically small resonators (mostly resonators inspired by metamaterials), focusing on the study of the symmetry-related electromagnetic properties of these loaded lines. It shows that the stopband functionality (resonance) that these lines exhibit can be controlled by the relative orientation between the line and the resonator, which determines their mutual coupling. Such resonance controllability, closely related to symmetry, is essential for the design of several microwave components, such as common-mode suppressed differential lines, novel microwave sensors based on symmetry disruption, and spectral signature radio-frequency barcodes. Other interesting aspects, such as stopband bandwidth enhancement (due to inter-resonator coupling, and related to complex modes) and magnetoelectric coupling between the transmission lines and split-ring resonators, are also included in the book.

## Symmetry: Representation Theory and Its Applications

by Roger Howe Markus Hunziker Jeb F. WillenbringNolan Wallach's mathematical research is remarkable in both its breadth and depth. His contributions to many fields include representation theory, harmonic analysis, algebraic geometry, combinatorics, number theory, differential equations, Riemannian geometry, ring theory, and quantum information theory. The touchstone and unifying thread running through all his work is the idea of symmetry. This volume is a collection of invited articles that pay tribute to Wallach's ideas, and show symmetry at work in a large variety of areas. The articles, predominantly expository, are written by distinguished mathematicians and contain sufficient preliminary material to reach the widest possible audiences. Graduate students, mathematicians, and physicists interested in representation theory and its applications will find many gems in this volume that have not appeared in print elsewhere. Contributors:D. Barbasch, K. Baur, O. Bucicovschi, B. Casselman, D. Ciubotaru, M. Colarusso, P. Delorme, T. Enright, W. T. Gan, A Garsia, G. Gour, B. Gross, J. Haglund, G. Han, P. Harris, J. Hong, R. Howe, M. Hunziker, B. Kostant, H. Kraft, D. Meyer, R. Miatello, L. Ni, G. Schwarz, L. Small, D. Vogan, N. Wallach, J. Wolf, G. Xin, O. Yacobi.

## Symmetry, Spectroscopy, and Crystallography

by Robert GlaserWritten in a clear and understandable manner, this book provides a comprehensive, yet non-mathematical, treatment of the topic, covering the basic principles of symmetry and the important spectroscopic techniques used to probe molecular structure. The chapters are extensively illustrated and deal with such topics as symmetry elements, operations and descriptors, symmetry guidelines, high-fidelity pseudosymmetry, crystallographic symmetry, molecular gears, and experimental techniques, including X-ray crystallography and NMR spectroscopy. As an additional feature, 3D animations of most of the structures and molecules covered are available online at wiley.com. As a result, chemists learn how to understand and predict molecular structures and reactivity. Authored by a renowned expert with numerous publications and an excellent track record in research and teaching, this is a useful source for graduate students and researchers working in the field of organic synthesis, physical chemistry, biochemistry, and crystallography, while equally serving as supplementary reading for courses on stereochemistry, organic synthesis, or crystallography.