Mathematics

This updated edition of this classic book is devoted to ordinary representation theory and is addressed to finite group theorists intending to study and apply character theory. It contains many exercises and examples, and the list of problems contains a number of open questions.

This updated edition of this classic book is devoted to ordinary representation theory and is addressed to finite group theorists intending to study and apply character theory. It contains many exercises and examples, and the list of problems contains a number of open questions.

This monograph covers in a comprehensive manner the current state of classification theory with respect to infinite abelian groups. A wide variety of ways to characterise different classes of abelian groups by invariants, isomorphisms and duality principles are discussed.

This is the sixth volume of a comprehensive and elementary treatment of finite group theory. This volume contains many hundreds of original exercises (including solutions for the more difficult ones) and an extended list of about 1000 open problems. The current book is based on Volumes 1–5 and it is suitable for researchers and graduate students working in group theory.

This is the fifth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume include theory of linear algebras and Lie algebras.

The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

Im ersten Teil des Buches werden sämtliche Kohärenzmaße sowie ihre jeweiligen Vor- und Nachteile vorgestellt. Dabei werden sowohl Adäquatheitsbedingungen als auch Testfälle für Kohärenzmaße berücksichtigt. Im zweiten Teil stehen dann mögliche Anwendungen von Kohärenzmaßen im Bereich der Kohärenztheorie der Rechtfertigung im Mittelpunkt. Insbesondere die Frage nach der Wahrheits- und Verlässlichkeitsförderlichkeit werden mit Bezug auf die vorgestellten Maße diskutiert. Zudem wird der Zusammenhang zwischen Kohärenz und Konsistenz eingehend beleuchtet.

Der letzte Teil des Buches widmet sich dann den Problemen und Einwänden sowohl gegen einzelne Klassen von Maßen als auch gegen das Projekt der probabilistischen Kohärenzmaße insgesamt. Hier werden unter anderem das Problem der Individuierung von Überzeugungen sowie der Zusammenhang zwischen dem Begriff der Kohärenz und dem der Erkärung betrachtet.

This authoritative book on periodic locally compact groups is divided into three parts: The first part covers the necessary background material on locally compact groups including the Chabauty topology on the space of closed subgroups of a locally compact group, its Sylow theory, and the introduction, classifi cation and use of inductively monothetic groups.

The second part develops a general structure theory of locally compact near abelian groups, pointing out some of its connections with number theory and graph theory and illustrating it by a large exhibit of examples.

Finally, the third part uses this theory for a complete, enlarged and novel presentation of Mukhin’s pioneering work generalizing to locally compact groups Iwasawa’s early investigations of the lattice of subgroups of abstract groups.

Contents
Part I: Background information on locally compact groups
Locally compact spaces and groups
Periodic locally compact groups and their Sylow theory
Abelian periodic groups
Scalar automorphisms and the mastergraph
Inductively monothetic groups
Part II: Near abelian groups
The definition of near abelian groups
Important consequences of the definitions
Trivial near abelian groups
The class of near abelian groups
The Sylow structure of periodic nontrivial near abelian groups and their prime graphs
A list of examples
Part III: Applications
Classifying topologically quasihamiltonian groups
Locally compact groups with a modular subgroup lattice
Strongly topologically quasihamiltonian groups

This book is intended as an introduction to harmonic analysis and generalized Gelfand pairs. Starting with the elementary theory of Fourier series and Fourier integrals, the author proceeds to abstract harmonic analysis on locally compact abelian groups and Gelfand pairs. Finally a more advanced theory of generalized Gelfand pairs is developed.

This book is aimed at advanced undergraduates or beginning graduate students. The scope of the book is limited, with the aim of enabling students to reach a level suitable for starting PhD research. The main prerequisites for the book are elementary real, complex and functional analysis. In the later chapters, familiarity with some more advanced functional analysis is assumed, in particular with the spectral theory of (unbounded) self-adjoint operators on a Hilbert space.

From the contents

• Fourier series
• Fourier integrals
• Locally compact groups
• Haar measures
• Harmonic analysis on locally compact abelian groups
• Theory and examples of Gelfand pairs
• Theory and examples of generalized Gelfand pairs

The study of finite groups factorised as a product of two or more subgroups has become a subject of great interest during the last years with applications not only in group theory, but also in other areas like cryptography and coding theory. It has experienced a big impulse with the introduction of some permutability conditions. The aim of this book is to gather, order, and examine part of this material, including the latest advances made, give some new approach to some topics, and present some new subjects of research in the theory of finite factorised groups.

Some of the topics covered by this book include groups whose subnormal subgroups are normal, permutable, or Sylow-permutable, products of nilpotent groups, and an exhaustive structural study of totally and mutually permutable products of finite groups and their relation with classes of groups.

This monograph is mainly addressed to graduate students and senior researchers interested in the study of products and permutability of finite groups. A background in finite group theory and a basic knowledge of representation theory and classes of groups is recommended to follow it.

This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters.

The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous.

In the second part, Chapters 6 through 9, the Stone-Cêch compactification βG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G, βG contains 2²|G| minimal right ideals.

In the third part, using the semigroup βG, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω-resolvable, and consequently, can be partitioned into ω subsets such that every coset modulo infinite subgroup meets each subset of the partition.

The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.

This is the first of three volumes of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular p-groups and regularity criteria, (c) p-groups of maximal class and their numerous characterizations, (d) characters of p-groups, (e) p-groups with large Schur multiplier and commutator subgroups, (f) (p‒1)-admissible Hall chains in normal subgroups, (g) powerful p-groups, (h) automorphisms of p-groups, (i) p-groups all of whose nonnormal subgroups are cyclic, (j) Alperin's problem on abelian subgroups of small index.

The book is suitable for researchers and graduate students of mathematics with a modest background on algebra. It also contains hundreds of original exercises (with difficult exercises being solved) and a comprehensive list of about 700 open problems.

Dieses Lehrbuch ist eine Einführung in die Techniken der Gruppentheorie und behandelt alle wichtigen Begriffe aus diesem Gebiet, wobei der Schwerpunkt im Bereich der endlichen Gruppen liegt. Es beginnt dort, wo die Gruppentheorie beginnt: bei den Permutationsgruppen. Danach werden wesentliche Strukturen und Methoden, wie das Arbeiten mit Kommutatoren und die Konstruktion von neuen aus gegebenen Gruppen behandelt. Nächstes Ziel sind die Fittinggruppe und ihre Verallgemeinerung, wozu nilpotente Gruppen studiert werden. Danach wendet sich der Text den einfachen Gruppen zu. Zu guter Letzt wird zunächst die Einfachheit der projektiven linearen Gruppen bewiesen und ein Überblick über orthogonale, symplektische und unitäre Gruppen gegeben. Weiter werden die sporadischen Mathieu-Gruppen und die Higman-Sims-Gruppe konstruiert.

Das Buch ist geschrieben für Studierende im Bachelor- und Masterstudium. Es setzt den Besuch der üblichen Algebra-Vorlesungen und somit nur allgemeine Kenntnisse über Gruppen voraus.

This is the third volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume:

• impact of minimal nonabelian subgroups on the structure of p-groups,
• classification of groups all of whose nonnormal subgroups have the same order,
• degrees of irreducible characters of p-groups associated with finite algebras,
• groups covered by few proper subgroups,
• p-groups of element breadth 2 and subgroup breadth 1,
• exact number of subgroups of given order in a metacyclic p-group,
• soft subgroups,
• p-groups with a maximal elementary abelian subgroup of order p²,
• p-groups generated by certain minimal nonabelian subgroups,
• p-groups in which certain nonabelian subgroups are 2-generator.

The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.

Starting with the Schur-Zassenhaus theorem, this monograph documents a wide variety of results concerning complementation of normal subgroups in finite groups. The contents cover a wide range of material from reduction theorems and subgroups in the derived and lower nilpotent series to abelian normal subgroups and formations.

Contents
Prerequisites
The Schur-Zassenhaus theorem: A bit of history and motivation
Abelian and minimal normal subgroups
Reduction theorems
Subgroups in the chief series, derived series, and lower nilpotent series
Normal subgroups with abelian sylow subgroups
The formation generation
Groups with specific classes of subgroups complemented

This is the second revised and extended edition of the successful book on the algebraic structure of the Stone-Čech compactification of a discrete semigroup and its combinatorial applications, primarily in the field known as Ramsey Theory. There has been very active research in the subject dealt with by the book in the 12 years which is now included in this edition.

This book is a self-contained exposition of the theory of compact right semigroups for discrete semigroups and the algebraic properties of these objects. The methods applied in the book constitute a mosaic of infinite combinatorics, algebra, and topology. The reader will find numerous combinatorial applications of the theory, including the central sets theorem, partition regularity of matrices, multidimensional Ramsey theory, and many more.

This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p-groups play an important role. Topics covered in this volume include:

• subgroup structure of metacyclic p-groups
• Ishikawa’s theorem on p-groups with two sizes of conjugate classes
• p-central p-groups
• theorem of Kegel on nilpotence of H p-groups
• partitions of p-groups
• characterizations of Dedekindian groups
• norm of p-groups
• p-groups with 2-uniserial subgroups of small order

The book also contains hundreds of original exercises and solutions and a comprehensive list of more than 500 open problems. This work is suitable for researchers and graduate students with a modest background in algebra.

This is the second of three volumes devoted to elementary finite p-group theory. Similar to the first volume, hundreds of important results are analyzed and, in many cases, simplified. Important topics presented in this monograph include: (a) classification of p-groups all of whose cyclic subgroups of composite orders are normal, (b) classification of 2-groups with exactly three involutions, (c) two proofs of Ward's theorem on quaternion-free groups, (d) 2-groups with small centralizers of an involution, (e) classification of 2-groups with exactly four cyclic subgroups of order 2ⁿ > 2, (f) two new proofs of Blackburn's theorem on minimal nonmetacyclic groups, (g) classification of p-groups all of whose subgroups of index p² are abelian, (h) classification of 2-groups all of whose minimal nonabelian subgroups have order 8, (i) p-groups with cyclic subgroups of index p² are classified.

This volume contains hundreds of original exercises (with all difficult exercises being solved) and an extended list of about 700 open problems. The book is based on Volume 1, and it is suitable for researchers and graduate students of mathematics with a modest background on algebra.

This book is an introduction to financial mathematics.

The first part of the book studies a simple one-period model which serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of risk.

In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Such models are typically incomplete: They involve intrinsic risks which cannot be hedged away completely. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk.

In addition to many corrections and improvements, this second edition contains several new sections, including a systematic discussion of law-invariant risk measures and of the connections between American options, superhedging, and dynamic risk measures.

These Proceedings contain 22 refereed research and survey articles based on lectures given at the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, held in Turku, Finland, from May 31 to June 4, 1999. The subject of the symposium was number theory in a broad sense with an emphasis on recent advances and modern methods. The topics covered in this volume include various questions in elementary number theory, new developments in classical Diophantine problems - in particular of the Fermat and Catalan type, the ABC-conjecture, arithmetic algebraic geometry, elliptic curves, Diophantine approximations, Abelian fields, exponential sums, sieve methods, box splines, the Riemann zeta-function and other Dirichlet series, and the spectral theory of automorphic functions with its arithmetical applications.

Observability in Mathematics were developed by authors based on denial of infinity idea. We introduce Observers into arithmetic, and arithmetic becomes dependent on Observers. And after that the basic mathematical parts also become dependent on Observers. One of such parts is geometry. Geometry plays important role not only in pure Mathematics but in contemporary Physics, for example, in Relativity theory, Quantum Yang-Mills theory. We call New Geometry both Observers in arithmetics and in geometry. We reconsider the basis of classic geometry (points, straight lines, planes and space) from this Mathematics point of view. The relations of connection, order, parallels (Euclid, Gauss-Bolyai-Lobachevsky, Riemann), congruence, continuity are discovered and have new properties. We show that almost all classic geometry theorems are satisfied in Mathematics with Observers geometry with probabilities less than 1.

That means classic geometries are not a limiting cases of the Observer’s geometry, but are only particular cases. And new geometry opens the road to reconsider differential geometry, algebraic geometry, geometric algebra, topology, and also to reconsider geometrical applications to various parts of contemporary physics.

We proved that Mathematics with Observers gives a birth a new geometry.

The goal of this book is to cover the active developments of arithmetically Cohen-Macaulay and Ulrich bundles and related topics in the last 30 years, and to present relevant techniques and multiple applications of the theory of Ulrich bundles to a wide range of problems in algebraic geometry as well as in commutative algebra.

Dieses Buch befasst sich mit dem Wissen, das zum Arbeiten mit realen Geodaten benötigt wird. In allen Geowissenschaften sind umfangreiche Datensätze zu verarbeiten und zu visualisieren. Dafür sind fundierte numerische und geometrische, an zufälligen Objekten und Strukturen auch stochastisch-geometrische, Grundlagenkenntnisse nötig. Existierende Lehrwerke behandeln dies nur sehr oberflächlich und entsprechen nicht hinreichend dem Bedarf. Dieses Werk füllt diese Lücke.

This volume is the proceedings of a conference on Finite Geometries, Groups, and Computation that took place on September 4-9, 2004, at Pingree Park, Colorado (a campus of Colorado State University). Not accidentally, the conference coincided with the 60^th birthday of William Kantor, and the topics relate to his major research areas.

Participants were encouraged to explore the deeper interplay between these fields. The survey papers by Kantor, O'Brien, and Penttila should serve to introduce both students and the broader mathematical community to these important topics and some of their connections while the volume as a whole gives an overview of current developments in these fields.

Dieser Titel aus dem De Gruyter-Verlagsarchiv ist digitalisiert worden, um ihn der wissenschaftlichen Forschung zugänglich zu machen. Da der Titel erstmals im Nationalsozialismus publiziert wurde, ist er in besonderem Maße in seinem historischen Kontext zu betrachten. Mehr erfahren Sie hier.

Hermitian symmetric spaces are an important class of manifolds that can be studied with methods from Kähler geometry and Lie theory. This work gives an introduction to Hermitian symmetric spaces and their submanifolds, and presents classifi cation results for real hypersurfaces in these spaces, focusing on results obtained by Jürgen Berndt and Young Jin Suh in the last 20 years.

This unique book’s subject is meanders (connected, oriented, non-self-intersecting planar curves intersecting the horizontal line transversely) in the context of dynamical systems. By interpreting the transverse intersection points as vertices and the arches arising from these curves as directed edges, meanders are introduced from the graphtheoretical perspective. Supplementing the rigorous results, mathematical methods, constructions, and examples of meanders with a large number of insightful figures, issues such as connectivity and the number of connected components of meanders are studied in detail with the aid of collapse and multiple collapse, forks, and chambers. Moreover, the author introduces a large class of Morse meanders by utilizing the right and left one-shift maps, and presents connections to Sturm global attractors, seaweed and Frobenius Lie algebras, and the classical Yang-Baxter equation.

Contents

Seaweed Meanders

Meanders

Morse Meanders and Sturm Global Attractors

Right and Left One-Shifts

Connection Graphs of Type I, II, III and IV

Meanders and the Temperley-Lieb Algebra

Representations of Seaweed Lie Algebras

CYBE and Seaweed Meanders

This monograph describes some of the most interesting results obtained by the mathematicians and physicists collaborating in the CRC 647 "Space – Time – Matter", in the years 2005 - 2016. The work presented concerns the mathematical and physical foundations of string and quantum field theory as well as cosmology. Important topics are the spaces and metrics modelling the geometry of matter, and the evolution of these geometries. The partial differential equations governing such structures and their singularities, special solutions and stability properties are discussed in detail.

Contents
Introduction
Algebraic K-theory, assembly maps, controlled algebra, and trace methods
Lorentzian manifolds with special holonomy – Constructions and global properties
Contributions to the spectral geometry of locally homogeneous spaces
On conformally covariant differential operators and spectral theory of the holographic Laplacian
Moduli and deformations
Vector bundles in algebraic geometry and mathematical physics
Dyson–Schwinger equations: Fix-point equations for quantum fields
Hidden structure in the form factors ofN = 4 SYM
On regulating the AdS superstring
Constraints on CFT observables from the bootstrap program
Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities
Yangian symmetry in maximally supersymmetric Yang-Mills theory
Wave and Dirac equations on manifolds
Geometric analysis on singular spaces
Singularities and long-time behavior in nonlinear evolution equations and general relativity

Proceedings of the 8th International Conference of Topological Algebras and Their Applications (ICTAA-2014), held on May 26-30, 2014 in Playa de Villas de Mar Beach, dedicated to the memory of Anastasios Mallios (Athens, Greece). This series of conferences started in 1999 in Tartu, Estonia and were subsequently held in Rabat, Moroco (2000), Oulu, Finland (2001), Oaxaca, Mexico (2002), Bedlewo, Poland (2003), Athens, Greece (2005) and Tartu, Estonia (2008 and 2013).

The topics of the conference include all areas of mathematics, connected with (preferably general) topological algebras and their applications, including all kinds of topological-algebraic structures as topological linear spaces, topological rings, topological modules, topological groups and semigroups; bornological-algebraic structures such as bornological linear spaces, bornological algebras, bornological groups, bornological rings and modules; algebraic and topological K-theory; topological module bundles, sheaves and others.

Contents
Some results on spectral properties of unital algebras and on the algebra of linear operators on a unital algebra
Descriptions of all closed maximal one-sided ideals in topological algebras
On non self-adjoint operators defined by Riesz bases in Hilbert and rigged Hilbert spaces
Functional calculus on algebras of operators generated by a self-adjoint operator in Pontryagin space Π1
On Gelfand-Naimark type Theorems for unital abelian complex and real locally C*-, and locally JB-algebras
Multipliers and strictly real topological algebras
Multipliers in some perfect locally m-pseudo-convex algebras
Wedderburn structure theorems for two-sided locally m-convex H*-algebras
Homologically best modules in classical and quantized functional analysis
Operator Grüss inequality
Main embedding theorems for symmetric spaces of measurable functions
Mapping class groups are linear
Subnormable A-convex algebras
Commutative BP*-algebras and Gelfand-Naimark’s theorem
Discrete nonclosed subsets in maximally nondiscrete topological groups
Faithfully representable topological *-algebras: some spectral properties
On continuity of complementors in topological algebras
Dominated ergodic theorem for isometries of non-commutative Lp-spaces, 1 p p ≠ 2
Ranks and the approximate n-th root property of C*-algebras
Dense ideals in topological algebras: some results and open problems

This book is an introduction to classical knot theory. Topics covered include: different constructions of knots, knot diagrams, knot groups, fibred knots, characterisation of torus knots, prime decomposition of knots, cyclic coverings and Alexander polynomials and modules together with the free differential calculus, braids, branched coverings and knots, Montesinos links, representations of knot groups, surgery of 3-manifolds and knots, Jones and HOMFLYPT polynomials.

Knot theory has expanded enormously since the first edition of this book published in 1985. In this third completely revised and extended edition a chapter about bridge number and companionship of knots has been added.

The book contains many figures and some tables of invariants of knots. This comprehensive account is an indispensable reference source for anyone interested in both classical and modern knot theory.

Most of the topics considered in the book are developed in detail; only the main properties of fundamental groups, covering spaces and some basic results of combinatorial group theory are assumed to be known. The text is accessible to advanced undergraduate and graduate students in mathematics.

As in the field of "Invariant Distances and Metrics in Complex Analysis" there was and is a continuous progress this is now the second extended edition of the corresponding monograph. This comprehensive book is about the study of invariant pseudodistances (non-negative functions on pairs of points) and pseudometrics (non-negative functions on the tangent bundle) in several complex variables. It is an overview over a highly active research area at the borderline between complex analysis, functional analysis and differential geometry. New chapters are covering the Wu, Bergman and several other metrics.

The book considers only domains in Cⁿ and assumes a basic knowledge of several complex variables. It is a valuable reference work for the expert but is also accessible to readers who are knowledgeable about several complex variables. Each chapter starts with a brief summary of its contents and continues with a short introduction. It ends with an "Exercises" and a "List of problems" section that gathers all the problems from the chapter. The authors have been highly successful in giving a rigorous but readable account of the main lines of development in this area.

This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry – as differential geometry in general – has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces.
The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.

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In the early 1920s M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology.

Circle-valued Morse theory originated from a problem in hydrodynamics studied by S. P. Novikov in the early 1980s. Nowadays, it is a constantly growing field of contemporary mathematics with applications and connections to many geometrical problems such as Arnold's conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions, and the theory of knots and links in the three-dimensional sphere.

The aim of the book is to give a systematic treatment of geometric foundations of the subject and recent research results. The book is accessible to first year graduate students specializing in geometry and topology.

Using the full power of countable alphabet thermodynamic formalism and modern functional analysis, the book presents a systematic treatment of the perturbative approach to open dynamical systems, covering those generated by a large class of naturally occurring holes through which the dynamical medium escapes. It highlights escape rates, conditionally invariant measures, surviving equilibrium states, and Hausdorff dimension of surviving sets.