Grupper og gruppeteori

Målet med denne bog er at give læseren en grundig indføring i familieterapiens historie, grundlag og de vigtigste familieterapeutiske modeller i en aktuel og let forståelig fremstilling. Fra Systemteori til Familieteori, er oprindeligt skrevet af Borrik Schiødt og Thor Aage Egelund og er oversat til dansk med forord af en af Danmarks ledende specialister, Torben Marner som har skrevet bogens forord og forsynet den med et kapitel om familieterapien i Danmark i dag.Bogens forfattere har opdelt bogen i 4 overordnede dele hvor læseren indledningsvis får et indblik i Familieterapiens historie og fremvækst. Derefter præsenteres det bagvedliggende teoretiske grundlag, hvor bl.a. en indføring i kommunikationsbegreber, niveauer af logiske typer og definitioner af relationer gennemgås. Tredje del går i dybden med en nærmere beskrivelse af familien, og beskriver diagnoser, medicinske & psykiatriske modeller, familie typologier og et særskilt afsnit om familiens udviklingFjerde del bearbejder emner som strukturel familieterapi, strategisk familieterapi og systemisk familie terapi samt relaterede terapi modeller. Læseren har adgang til en uddybende ordliste, modeller og litteraturhenvisningerBogen er nu udgivet i 4 oplag og er bredt anerkendt indenfor sit felt.

Otte mænd samles hen over otte uger omkring otte temaer, der forbinder dem omkring en fælles livsudfordring. Denne bog er skrevet til dig, der søger inspiration til at facilitere livsstyrkende og meningsskabende gruppeprocesser for mænd, hvor samlivet slår knuder.I bogen deler Jørgen Juul Jensen ud af det gruppesamtaleformat, som han har udviklet, afprøvet og undervist i gennem 14 år. Dels i Mandecentret København og dels i et samarbejde med Frivilligcentre & Selvhjælp Danmark og Socialt Udviklingscenter.Bogen tilbyder læseren en "gør det selv guide", der indeholder et teoretisk perspektiv, en række metodiske greb til at facilitere komplekse gruppeprocesser og ideer til hvordan de enkelte gruppemøder kan gribes an. Alt er spækket med spændende spørgsmålstyper.Det er forfatterens erfaring, at når mænd mødes i et nærende fællesskab omkring nogle af de samme udfordringer, lærer kunsten at lytte og agere vidner til hinandens fortællinger, kan det gøre en verden til forskel. Det er denne proces, hvor der skabes et rum for personlig og fælles bevægelse, forfatteren ønsker at bidrage til.Bogen giver et indblik i mænds udfordrede position i samfundet og tilbyder et udsyn til den stille kulturrevolution, som foregår i det daglige samspil i familien og mand og mand imellem. Du er hermed inviteret til at være med i skabelsen af bedre sociale verdener."Otte mænd Otte" uger finder i dag bred anvendelse rundt i landet, og forfatteren bistår gerne, hvis du/ I vil guides på vejen.

This textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups.Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory ¿ whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the Drinfeld¿Lusztig double of a group algebra, appearing for the first time in a textbook at this level.Based on over twenty years of teaching various aspects of algebra, mainly at the École Normale Supérieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course ¿ with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level.

This book describes a novel approach to the study of Siegel modular forms of degree two with paramodular level. It introduces the family of stable Klingen congruence subgroups of GSp(4) and uses this family to obtain new relations between the Hecke eigenvalues and Fourier coefficients of paramodular newforms, revealing a fundamental dichotomy for paramodular representations. Among other important results, it includes a complete description of the vectors fixed by these congruence subgroups in all irreducible representations of GSp(4) over a nonarchimedean local field.Siegel paramodular forms have connections with the theory of automorphic representations and the Langlands program, Galois representations, the arithmetic of abelian surfaces, and algorithmic number theory. Providing a useful standard source on the subject, the book will be of interest to graduate students and researchers working in the above fields.

This book lays the foundation for a theory of coarse groups: namely, sets with operations that satisfy the group axioms ¿up to uniformly bounded error¿. These structures are the group objects in the category of coarse spaces, and arise naturally as approximate subgroups, or as coarse kernels.The first aim is to provide a standard entry-level introduction to coarse groups. Extra care has been taken to give a detailed, self-contained and accessible account of the theory. The second aim is to quickly bring the reader to the forefront of research. This is easily accomplished, as the subject is still young, and even basic questions remain unanswered.Reflecting its dual purpose, the book is divided into two parts. The first part covers the fundamentals of coarse groups and their actions. Here the theory of coarse homomorphisms, quotients and subgroups is developed, with proofs of coarse versions of the isomorphism theorems, and it is shown how coarse actions are related to fundamental aspects of geometric group theory. The second part, which is less self-contained, is an invitation to further research, where each thread leads to open questions of varying depth and difficulty. Among other topics, it explores coarse group structures on set-groups, groups of coarse automorphisms and spaces of controlled maps. The main focus is on connections between the theory of coarse groups and classical subjects, including: number theory; the study of bi-invariant metrics on groups; quasimorphisms and stable commutator length; groups of outer automorphisms; and topological groups and their actions.The book will primarily be of interest to researchers and graduate students in geometric group theory, topology, category theory and functional analysis, but some parts will also be accessible to advanced undergraduates.

This monograph adopts an operational and functional analytic approach to the following problem: given a short exact sequence (group extension) 1 N G H 1 of finite groups, describe the irreducible representations of G by means of the structure of the group extension. This problem has attracted many mathematicians, including I. Schur, A.H. Clifford, and G. Mackey and, more recently, M. Isaacs, B. Huppert, Y.G. Berkovich & E.M. Zhmud, and J.M.G. Fell & R.S. Doran.The main topics are, on the one hand, Clifford Theory and the Little Group Method (of Mackey and Wigner) for induced representations, and, on the other hand, Kirillov's Orbit Method (for step-2 nilpotent groups of odd order) which establishes a natural and powerful correspondence between Lie rings and nilpotent groups. As an application, a detailed description is given of the representation theory of the alternating groups, of metacyclic, quaternionic, dihedral groups, and of the (finite) Heisenberg group. The Little Group Method may be applied if and only if a suitable unitary 2-cocycle (the Mackey obstruction) is trivial. To overcome this obstacle, (unitary) projective representations are introduced and corresponding Mackey and Clifford theories are developed. The commutant of an induced representation and the relative Hecke algebra is also examined. Finally, there is a comprehensive exposition of the theory of projective representations for finite Abelian groups which is applied to obtain a complete description of the irreducible representations of finite metabelian groups of odd order.

This book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the ¿abelian¿ Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the ¿non-abelian¿ modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included.These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms.Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.

Bei klassischer Vorgehensweise werden die Identität des Erstellers und die Echtheit eines Dokuments durch manuelle Unterschrift auf Papier bezeugt. Dieses Vorgehen ist bei digitalen Texten und Nachrichten so nicht mehr möglich. Will man digitalisierte Daten unterschreiben und somit deren Echtheit und die eigene Identität bezeugen, so spricht man von einer digitalen Signatur. Dieses Buch beschreibt die drei wichtigsten Verfahren, nämlich die RSA-Signatur, die DSA-Signatur und die ECDSA-Signatur. Hierfür werden die jeweiligen algebraischen Grundlagen bereitgestellt.

This book focuses on symmetries in the analysis and synthesis of architectural designs. Crucial in the history of architecture, principles of symmetry provided the means to achieve balance and harmony of spatial composition in architecture. Less well known is the importance of symmetry principles in the analysis of the distinct constituents in a contemporary architectural design which may, at first glance, appear disorganized or even random. The revelation of different hierarchical levels wherein various types of symmetry or subsymmetry are superimposed provides a key for deciphering the underlying structure of spatial logic. The interaction between local and global subsymmetries is of particular interest. Operating with symmetry concepts in this manner offers architects, designers and students an explicit method for understanding the symmetrical logics of sophisticated designs and gaining insights into new designs.This book has two complementary objectives: to explore the fundamental principles of architectural composition founded on the algebraic structure of symmetry groups in mathematics and to apply the principles in the analysis and synthesis of architectural and urban designs. By viewing and decomposing architectural and urban designs in this manner, the hidden spatial logic and underlying order in a design become transparent.

This volume presents a completely self-contained introduction to the elaborate theory of locally compact quantum groups, bringing the reader to the frontiers of present-day research. The exposition includes a substantial amount of material on functional analysis and operator algebras, subjects which in themselves have become increasingly important with the advent of quantum information theory. In particular, the rather unfamiliar modular theory of weights plays a crucial role in the theory, due to the presence of 'Haar integrals' on locally compact quantum groups, and is thus treated quite extensivelyThe topics covered are developed independently, and each can serve either as a separate course in its own right or as part of a broader course on locally compact quantum groups. The second part of the book covers crossed products of coactions, their relation to subfactors and other types of natural products such as cocycle bicrossed products, quantum doubles and doublecrossed products. Induced corepresentations, Galois objects and deformations of coactions by cocycles are also treated. Each section is followed by a generous supply of exercises. To complete the book, an appendix is provided on topology, measure theory and complex function theory.

This volume collects papers based on lectures given at the XXXIX Workshop on Geometric Methods in Physics, held in Biäystok, Poland in June 2022. These chapters provide readers an overview of cutting-edge research in geometry, analysis, and a wide variety of other areas. Specific topics include:Classical and quantum field theoriesInfinite-dimensional groupsIntegrable systemsLie groupoids and Lie algebroidsRepresentation theoryGeometric Methods in Physics XXXIX will be a valuable resource for mathematicians and physicists interested in recent developments at the intersection of these areas.

This text presents the basic theory of random walks on infinite, finitely generated groups, along with certain background material in measure-theoretic probability. The main objective is to show how structural features of a group, such as amenability/nonamenability, affect qualitative aspects of symmetric random walks on the group, such as transience/recurrence, speed, entropy, and existence or nonexistence of nonconstant, bounded harmonic functions. The book will be suitable as a textbook for beginning graduate-level courses or independent study by graduate students and advanced undergraduate students in mathematics with a solid grounding in measure theory and a basic familiarity with the elements of group theory. The first seven chapters could also be used as the basis for a short course covering the main results regarding transience/recurrence, decay of return probabilities, and speed. The book has been organized and written so as to be accessible not only to students in probability theory, but also to students whose primary interests are in geometry, ergodic theory, or geometric group theory.

This monograph studies the heat kernel for the spin-tensor Laplacians on Lie groups and maximally symmetric spaces. It introduces many original ideas, methods, and tools developed by the author and provides a list of all known exact results in explicit form ¿ and derives them ¿ for the heat kernel on spheres and hyperbolic spaces. Part I considers the geometry of simple Lie groups and maximally symmetric spaces in detail, and Part II discusses the calculation of the heat kernel for scalar, spinor, and generic Laplacians on spheres and hyperbolic spaces in various dimensions. This text will be a valuable resource for researchers and graduate students working in various areas of mathematics ¿ such as global analysis, spectral geometry, stochastic processes, and financial mathematics ¿ as well in areas of mathematical and theoretical physics ¿ including quantum field theory, quantum gravity, string theory, and statistical physics.

This textbook provides a concise, visual introduction to Hopf algebras and their application to knot theory, most notably the construction of solutions of the Yang¿Baxter equations.Starting with a reformulation of the definition of a group in terms of structural maps as motivation for the definition of a Hopf algebra, the book introduces the related algebraic notions: algebras, coalgebras, bialgebras, convolution algebras, modules, comodules. Next, Drinfel¿d¿s quantum double construction is achieved through the important notion of the restricted (or finite) dual of a Hopf algebra, which allows one to work purely algebraically, without completions. As a result, in applications to knot theory, to any Hopf algebra with invertible antipode one can associate a universal invariant of long knots. These constructions are elucidated in detailed analyses of a few examples of Hopf algebras. The presentation of the material is mostly based on multilinear algebra, with all definitions carefully formulated and proofs self-contained. The general theory is illustrated with concrete examples, and many technicalities are handled with the help of visual aids, namely string diagrams. As a result, most of this text is accessible with minimal prerequisites and can serve as the basis of introductory courses to beginning graduate students.

This monograph thoroughly explores the development of the theory of varieties of semigroups and of two related algebras: involution semigroups and monoids. Through this in-depth analysis, readers will attain a deeper understanding of the differences between these three types of varieties, which may otherwise seem counterintuitive. New results with detailed proofs are also presented that answer previously unsolved fundamental problems. Featuring both a comprehensive overview as well as highlighting the author¿s own significant contributions to the area, this book will help establish this subfield as a matter of timely interest. Advances in the Theory of Varieties of Semigroups will appeal to researchers in universal algebra and will be particularly valuable for specialists in semigroups.

The congruences of a lattice form the congruence lattice. Over the last several decades, the study of congruence lattices has established itself as a large and important field with a great number of interesting and deep results, as well as many open problems. Written by one of the leading experts in lattice theory, this text provides a self-contained introduction to congruences of finite lattices and presents the major results of the last 90 years. It features the author¿s signature ¿Proof-by-Picture¿ method, which is used to convey the ideas behind formal proofs in a visual, more intuitive manner. Key features include:an insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensionscomplete proofs, an extensive bibliography and index, and over 180 illustrationsadditional chapters covering new results of the lastseven years, increasing the size of this edition to 430 pages, 360 statements, and 262 referencesThis text is appropriate for a one-semester graduate course in lattice theory, and it will also serve as a valuable reference for researchers studying lattices. Reviews of previous editions:¿[This] monograph¿is an exceptional work in lattice theory, like all the contributions by this author. The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. ¿ Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica LII (1), 2007 "The book is self-contained, with many detailed proofs presented that can be followed step-by-step. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasis on the more 'geometric' aspects." ¿ Mathematical Reviews

Effective leadership and organizational performance are concepts that continue to receive widespread attention in the business world. This book explores the importance of strategic leadership and the value it adds to organizations.

First published in 1991, this book was unique in that it not only explores the role of communication in divorce mediation, but it also presents original research to support its claims. A series of empirical studies, it points readers to a more focused set of recommendations about communication than the typical practitioner's "How-to" books.