Format: Paperback / softback, 140 pages, height x width: 235x155 mm, X, 140 p.
Series: Universitext
Pub. Date: 02-May-2023
ISBN-13: 9783031263057
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.
1. Groups and Hopf algebras.-
2. Constructions of algebras, coalgebras, bialgebras, and Hopf algebras.-
3. The restricted dual of an algebra.-
4. The restricted dual of Hopf algebras: examples of calculations.-
5. The quantum double.-
6. Applications in knot theory.
Format: Hardback, 199 pages, height x width: 235x155 mm, X, 199 p.
Series: CMS/CAIMS Books in Mathematics 6
Pub. Date: 26-Apr-2023
ISBN-13: 9783031253256
The main subject of the book is stochastic analysis and its various applications to mathematical finance and statistics of random processes. The main purpose of the book is to present, in a short and sufficiently self-contained form, the methods and results of the contemporary theory of stochastic analysis and to show how these methods and results work in mathematical finance and statistics of random processes. The book can be considered as a textbook for both senior undergraduate and graduate courses on this subject. The book can be helpful for undergraduate and graduate students, instructors and specialists on stochastic analysis and its applications.
1 Probabilistic Foundations.- 2 Random variables and their quantitative characteristics.- 3 Expectations and convergence of sequences of random variables.- 4 Weak convergence of sequences of random variables.- 5 Absolute continuity of probability measures and conditional expectations.- 6 Discrete time stochastic analysis: basic results.- 7 Discrete time stochastic analysis: further results and applications.- 8 Elements of classical theory of stochastic processes.- 9 Stochastic differential equations, diffusion processes and their applications.- 10 General theory of stochastic processes under "usual conditions".- 11 General theory of stochastic processes in applications.- 12 Supplementary problems.- References.- Index.
Format: Paperback / softback, height x width: 235x155 mm, Approx. 110 p.
Series: SpringerBriefs on PDEs and Data Science
Pub. Date: 16-Apr-2023
ISBN-13: 9789819906840
This book provides a general treatment of a class of functionals modelled on convolution energies with kernel having finite p-moments. A general asymptotic analysis of such non-local functionals is performed, via Gamma-convergence, in order to show that the limit may be a local functional representable as an integral. Energies of this form are encountered in many different contexts and the interest in building up a general theory is also motivated by the multiple interests in applications (e.g. peridynamics theory, population dynamics phenomena and data science). The results obtained are applied to periodic and stochastic homogenization, perforated domains, gradient flows, and point-clouds models. This book is mainly intended for mathematical analysts and applied mathematicians who are also interested in exploring further applications of the theory to pass from a non-local to a local description, both in static problems and in dynamic problems.
Preface Introduction Convolution-type energies The -limit of a class of reference energies Asymptotic embedding and compactness results A compactness and integral-representation result Periodic homogenization A generalization and applications to point clouds Stochastic homogenization Application to convex gradient flows References Index
Format: Paperback / softback, 180 pages, height x width: 235x155 mm, 4 Tables, color;
4 Illustrations, color; 1 Illustrations, black and white; X, 180 p. 5 illus., 4 illus. in color
Pub. Date: 23-Apr-2023
ISBN-13: 9783031264276
This book provides a comprehensive introduction to complex analysis in several variables. One major focus of the book is extension phenomena alien to the one-dimensional theory (Hartog's Kugelsatz, theorem of Cartan-Thullen, Bochner's theorem). The book primarily aims at students starting to work in the field of complex analysis in several variables and teachers who want to prepare a university lecture. Therefore, the book contains more than 50 examples and more than 100 supporting exercises.
- Elementary theory of several complex variables. - Continuation on circular and polycircular domains. - Biholomorphic maps. - Analytic Sets. - Hartogs' Kugelsatz.- Continuation on Tubular Domains. - Cartan-Thullen Theory. - Local Properties of holomorphic functions. - Hints to the exercises.
Format: Hardback, 229 pages, height x width: 235x155 mm, 8 Illustrations, black and white; XVI, 229 p. 8 illus.
Series: Undergraduate Texts in Mathematics
Pub. Date: 11-May-2023
ISBN-13: 9783031250019
The principle aim of this unique text is to illuminate the beauty of the subject both with abstractions like proofs and mathematical text, and with visuals, such as abundant illustrations and diagrams. With few mathematical prerequisites, geometry is presented through the lens of linear fractional transformations. The exposition is motivational and the well-placed examples and exercises give students ample opportunity to pause and digest the material. The subject builds from the fundamentals of Euclidean geometry, to inversive geometry, and, finally, to hyperbolic geometry at the end. Throughout, the author aims to express the underlying philosophy behind the definitions and mathematical reasoning. This text may be used as primary for an undergraduate geometry course or a freshman seminar in geometry, or as supplemental to instructors in their undergraduate courses in complex analysis, algebra, and number theory. There are elective courses that bring together seemingly disparate topics and this text would be a welcome accompanimen
Motivation.- I Euclidean and Inversive Geometry.- Euclidean Isometries and Similarities.- Inversive Geometry.- Applications of Inversive Geometry.- II Non-Euclidean Geometry.- Spherical Geometry.- Appendix: Set Theory.