Volume 25 in the series De Gruyter Studies in Mathematics
The subject matter of compact groups is frequently cited in fi elds like algebra, topology, functional analysis, and theoretical physics. This book serves the dual purpose of providing a text for upper level graduate students, and of being a source book for researchers who need the structure and representation theory of compact groups. After a gentle introduction to compact groups and their representation theory, the book presents self-contained courses on linear Lie groups and on locally compact abelian groups.Appended chapters contain the material for self-contained courses on abelian groups and on category theory.Using the Lie algebras and the exponential function of arbitrary compact groups, the book avoids unnecessary restrictions to finite dimensional or abelian compact groups. Earlier editions of 1998, 2006, 2013, and 2020 have been quoted for instruction and research. The present edition conceptually sharpens, polishes, and improves the earlier material. For instance, it includes a treatment of the Bohr compactifi cation of topological groups which fi ts perfectly into the general treatment of adjoint functors that the book treats in an appendix of its own, and which, in the abelian environment, connects neatly with the Pontryagin--van Kampen duality of compact abelian groups having been discussed in the book in great detail. The link between arbitrary compact groups and their weakly complete group algebras is as extensively discussed as is now the theory of weakly complete universal enveloping algebras of the Lie algebras of compact groups. All of this is based on the category of weakly complete real and complex vector spaces and its precise duality to the category of ordinary real, respectively, complex vector spaces, is treated in an appendix systematically.
Thorough introduction to compact groups, which have applications in algebra, topology, functional analysis, and theoretical physics.
Thrust of the book points to structure theory of infinite dimensional, not necessarily commutative compact groups.
Karl H. Hofmann, TU Darmstadt, Germany; Sydney A. Morris, Federation University, Australia.
Algebra and Number Theory
Combinatorics and Graph Theory
Mathematics
Volume 74 in the series De Gruyter Expositions in Mathematics
This book presents a wide-ranging approach to operator-valued measures and integrals of both vector-valued and set-valued functions. It covers convergence theorems and an integral representation for linear operators on spaces of continuous vector-valued functions on a locally compact space. These are used to extend Choquet theory, which was originally formulated for linear functionals on spaces of real-valued functions, to operators of this type.
A new and comprehensive approach to integrals of vector- and set-valued functions with respect to operator-valued measures,
Integral representations for linear operators on function spaces,
An extension of Choquet theory to linear operators
Walter Roth, Emeritus Professor, Philippines.
Analysis
Mathemat
Volume 96 in the series De Gruyter Studies in Mathematics
Polynomials are useful tools, are simply defined, easy to calculate, and integrate and can be pieced together to form spline functions. Classes of polynomials are used in many fields of science. There is a trove of material available on existing polynomials, specific classes and applications, but it is not presented coherently. This volume solves this issue by introducing a systematic study of sequences of polynomials and related applications.
Presents the latest developments in polynomial sequences
Provides a simple and systematic treatment, that will appeal to readers .
Can be used as a tool for reseachers or in advanced graduate courses.
Francesco Aldo Costabile, Maria Italia Gualtieri and Anna Napoli, University of Calabria, Italy.
Analysis
Differential Equations and Dynamical Systems
Mathematics
Volume 75 in the series De Gruyter Expositions in Mathematics
The second edition presents schemes, simplicial sets, higher categories, model categories, derived algebraic geometry, and spectral algebraic geometry in a self-contained manner. It discusses Motives, Goodwillie Calculus, Higher Galois, Supersymmetry, and topics in physical mathematics. A new chapter on Derived Motivic Spectrais now included as is an extended introduction to Infinity Category as well as a revised chapter on Stacks.
Presents a concise treatment of derived algebraic geometry.
Covers the the construction of a mathematical model of physical phenomena.
Describes clear connections between algebra and geometry and topics in physics and mathematics.
Renaud Gauthier, University of Mary, USA.
Algebra and Number Theory
Geometry and Topology
Mathematics
Format: 115 pages, height x width: 235x155 mm, weight: 213 g, 123 Illustrations, color;
45 Illustrations, black and white; XIV, 115 p. 168 illus., 123 illus. in color.
Pub. Date: 03-Sep-2023
ISBN-13: 9783031074448
This book draws on elements from everyday life, architecture, and the arts to provide the reader with elementary notions of geometric topology. Pac Man, subway maps, and architectural blueprints are the starting point for exploring how knowledge about geometry and, more specifically, topology has been consolidated over time, offering a learning journey that is both dense and enjoyable.
The text begins with a discussion of mathematical models, moving on to Platonic and Keplerian theories that explain the Cosmos. Geometry from Felix Klein's point of view is then presented, paving the way to an introduction to topology. The final chapters present the concepts of closed, orientable, and non-orientable surfaces, as well as hypersurface models. Adopting a style that is both rigorous and accessible, this book will appeal to a broad audience, from curious students and researchers in various areas of knowledge to everyone who feels instigated by the power of mathematics in representing our world - and beyond.
Reviews
This book is an excellent introduction to geometry and topology, for people who do not know much about these subjects. This book constitutes a very interesting reading for any advanced high-school pupil and for non-mathematicians. It should be also useful for university undergraduates. Geometers and topologists will also find it pleasant to skim, because of the historical remarks, the questions and relations that the author establishes between geometry and topology and the things of everyday life. (Athanase Papadopoulos, zbMATH 07555479, 2023)
Preface.- Mathematical Models.- The Big Bang Theory of Ancient Greece.-
Geometry: From disorder to order.- Topology.- Fourth dimension.-
Non-orientable surfaces.- Hypersurfaces.
Format: 434 pages, height x width: 235x155 mm, weight: 688 g, 2 Illustrations, black and white; XVI, 434 p. 2 illus
Series: Developments in Mathematics 77
Pub. Date: 25-Sep-2023
ISBN-13: 9783031116216
This monograph is devoted to developing a theory of combined measure and shift invariance of time scales with the related applications to shift functions and dynamic equations. The study of shift closeness of time scales is significant to investigate the shift functions such as the periodic functions, the almost periodic functions, the almost automorphic functions, and their generalizations with many relevant applications in dynamic equations on arbitrary time scales. First proposed by S. Hilger, the time scale theorya unified view of continuous and discrete analysishas been widely used to study various classes of dynamic equations and models in real-world applications. Measure theory based on time scales, in its turn, is of great power in analyzing functions on time scales or hybrid domains. As a new and exciting type of mathematicsand more comprehensive and versatile than the traditional theories of differential and difference equations, the time scale theory can precisely depict the continuous-discrete hybrid processes and is an optimal way forward for accurate mathematical modeling in applied sciences such as physics, chemical technology, population dynamics, biotechnology, and economics and social sciences. Graduate students and researchers specializing in general dynamic equations on time scales can benefit from this work, fostering interest and further research in the field. It can also serve as reference material for undergraduates interested in dynamic equations on time scales. Prerequisites include familiarity with functional analysis, measure theory, and ordinary differential equations.
Riemann Integration, Stochastic Calculus and Shift Operators on Time
Scales.- -Measurability and Combined Measure Theory on Time Scales .- Shift
Invariance and Matched Spaces of Time Scales.- Almost Periodic Functions
under Matched Spaces of Time Scales.- Almost Automorphic Functions under
Matched Spaces of Time Scales.- C0-Semigroup and Stepanov-like Almost
Automorphic Functions on Hybrid Time Scales.- Almost Periodic Dynamic
Equations under Matched Spaces.- Almost Automorphic Dynamic Equations under
Matched Spaces.- Applications on Dynamics Models under Matched Spaces.