Format: Paperback / softback, 425 pages, height x width: 235x155 mm, weight: 664 g,
1 Illustrations, black and white; X, 425 p. 1 illus.,
Pub. Date: 03-Jan-2024
ISBN-13: 9783031238192
This book introduces the reader to quantum groups, focusing on the simplest ones, namely the closed subgroups of the free unitary group.
Although such quantum groups are quite easy to understand mathematically, interesting examples abound, including all classical Lie groups, their free versions, half-liberations, other intermediate liberations, anticommutation twists, the duals of finitely generated discrete groups, quantum permutation groups, quantum reflection groups, quantum symmetry groups of finite graphs, and more.
The book is written in textbook style, with its contents roughly covering a one-year graduate course. Besides exercises, the author has included many remarks, comments and pieces of advice with the lone reader in mind. The prerequisites are basic algebra, analysis and probability, and a certain familiarity with complex analysis and measure theory. Organized in four parts, the book begins with the foundations of the theory, due to Woronowicz, comprising axioms, Haar measure, PeterWeyl theory, Tannakian duality and basic Brauer theorems. The core of the book, its second and third parts, focus on the main examples, first in the continuous case, and then in the discrete case. The fourth and last part is an introduction to selected research topics, such as toral subgroups, homogeneous spaces and matrix models.
Introduction to Quantum Groups offers a compelling introduction to quantum groups, from the simplest examples to research level topics.
Part I. Quantum groups.
Chapter 1. Quantum spaces.
Chapter 2. Quantum groups.
Chapter 3. Representation theory.
Chapter 4. Tannakian duality.-
Part II. Quantum rotations.
Chapter 5. Free rotations.
Chapter 6. Unitary groups.
Chapter 7. Easiness, twisting.
Chapter 8. Probabilistic aspects.-
Part III. Quantum permutations.
Chapter 9. Quantum permutations.
Chapter 10. Quantum reflections.
Chapter 11. Classification results.
Chapter 12. The standard cube.- Part IV. Advanced topics.
Chapter 13. Toral subgroups.-
Chapter 14. Amenability, growth.
Chapter 15. Homogeneous spaces.
Chapter 16. Modelling questions.- Bibliography.- Index.
Format: Paperback / softback, 574 pages, height x width: 235x155 mm, weight: 1070 g,
100 Illustrations, black and white; XXVII, 574 p. 100 illus.,
Pub. Date: 17-Jan-2024
ISBN-13: 9783031224249
This textbook demonstrates the strong interconnections between linear algebra and group theory by presenting them simultaneously, a pedagogical strategy ideal for an interdisciplinary audience. Being approached together at the same time, these two topics complete one another, allowing students to attain a deeper understanding of both subjects. The opening chapters introduce linear algebra with applications to mechanics and statistics, followed by group theory with applications to projective geometry. Then, high-order finite elements are presented to design a regular mesh and assemble the stiffness and mass matrices in advanced applications in quantum chemistry and general relativity.
This text is ideal for undergraduates majoring in engineering, physics, chemistry, computer science, or applied mathematics. It is mostly self-contained?readers should only be familiar with elementary calculus. There are numerous exercises, with hints or full solutions provided. A series of roadmaps are also provided to help instructors choose the optimal teaching approach for their discipline.
The second edition has been revised and updated throughout and includes new material on the Jordan form, the Hermitian matrix and its eigenbasis, and applications in numerical relativity and electromagnetics.
Part I: Introduction to Linear Algebra.- Vectors and Matrices.- Determinant
and Vector Product in Physics.- Markov Matrix and its Spectrum: Towards
Search Engines.- Special Relativity: Algebraic Point of View.- Part II:
Introduction to Group Theory.- Groups and Isomorphism Theorems.- Projective
Geometry in Computer Graphics.- Quantum Mechanics: Algebraic Point of View.-
Part III: Polynomials and Basis Functions.- Polynomials and Their Gradient.-
Basis Functions: Barycentric Coordinates in 3D.- Part IV: Finite Elements
in 3-D. - Automatic Mesh Generation.- Mesh Regularity.- Numerical Integration.-
Spline: Variational Model in 3D.- Part V: Permuation Group in Quantum Chemistry.-
Determinant and Electronic Structure.- Part VI: The Jordan Form.- The Jordan
Form.- Jordan Decomposition.- Algebras and their Derivation.- Part VII:
Linearization in Numerical Relativity.- Einstein Equations and their Linearization.
Format: Paperback / softback, 285 pages, height x width: 235x155 mm, weight: 456 g, 8 Illustrations, color;
5 Illustrations, black and white; X, 285 p. 13 illus., 8 illus. in color.,
Pub. Date: 03-Jan-2024
ISBN-13: 9783031107986
This proceedings volume, the fifth in a series from the Combinatorial and Additive Number Theory (CANT) conferences, is based on talks from the 19th annual workshop, held online due to the COVID-19 pandemic. Organized every year since 2003 by the New York Number Theory Seminar at the CUNY Graduate Center, the workshops survey state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. The CANT 2021 meeting featured over a hundred speakers from North and South America, Europe, Asia, Australia, and New Zealand, and was the largest CANT conference in terms of the number of both lectures and participants.
These proceedings contain peer-reviewed and edited papers on current topics in number theory. Topics featured in this volume include sumsets, minimal bases, Sidon sets, analytic and prime number theory, combinatorial and discrete geometry, numerical semigroups, and a survey of expansion, divisibility, and parity. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.
Preface.-
1. On the Number of Dot Product Chains in Finite Fields and Rings (V. Blevins, D. Crosby, E. Lynch, S. Senger).-
2. Completeness of Positive Linear Recurrence Sequences (E. Boldyriew, J. Haviland, P. Lam, J.Lentfer, S.J. Miller, F.T. Suarez).-
3. Length Density and Numerical Semigroups (C. Brower, S. Chapman, T. Kulhanek, J. McDonough, C. O'Neill, V.Pavlyuk, V. Ponomarenko).-
4. On a Problem of Cilleruelo-Nathanson, II (Y.-G.Chen, J. Fang).- 5. Linked Partition Ideals and a Schur-type Identity of Andrews (S. Chern).-
6. Semi-magic Matrices for Dihedral Groups (R. Donley).-
7. Is the Syracuse Falling Time Bounded by 12? (S. Eliahou, J. Fromentin, R Simonetto).-
8. Genera of Numerical Semigroups and Polynomial Identities for Degrees of Syzygies (L. Fel).-
9. Expansion, Divisibility, and Parity (H.A.Helfgott).-
10. Sums of Squares (R.J. Hendel).-
11. Bilinear Generalized Radon Transforms in the Plane (A. Greenleaf, A. Iosevich, B. Krause, A.Liu).-
12. GeneralizedBernoulli Numbers, Cotangent Power Sums, and Higher-order Arctangent Numbers (B. Isaacson).-
13. A New Class of Minimal Asymptotic Bases (M.B. Nathanson).-
14. An Inverse Problem for Infinite Sidon Sets (M.B. Nathanson).
Format: Hardback, 363 pages, height x width: 235x155 mm, 9 Illustrations, color;
23 Illustrations, black and white; XVI, 313 p. 63 illus., 10 illus. in color.
Series: Springer Theses
Pub. Date: 01-Jun-2024
ISBN-13: 9783031544453
This book offers a systematic introduction to the Hopf algebra of renormalization in quantum field theory, with a special focus on physical motivation, the role of Dyson?Schwinger equations, and the renormalization group. All necessary physical and mathematical constructions are reviewed and motivated in a self-contained introduction. The main part of the book concerns the interplay between Dyson?Schwinger equations (DSEs) and renormalization conditions. The book is explicit and consistent about whether a statement is true in general or only in particular renormalization schemes or approximations and about the dependence of quantities on regularization parameters or coupling constants. With over 600 references, the original literature is cited whenever possible and the book contains numerous references to other works discussing further details, generalizations, or alternative approaches. There are explicit examples and remarks to make the connection from the scalar fields at hand to QED and QCD. The book is primarily targeted at the mathematically oriented physicist who seeks a systematic conceptual overview of renormalization, Hopf algebra, and DSEs. These may be graduate students entering the field as well as practitioners seeking a self-contained account of the Hopf algebra construction. Conversely, the book also benefits the mathematician who is interested in the physical background of the exciting interplay between Hopf algebra, combinatorics and physics that is renormalization theory today.
Introduction to perturbative quantum field theory.- Hopf algebra theory of renormalization.- Renormalized Green functions in kinematic renormalization.- Renormalization group and Dyson-Schwinger equations in non-kinematic renormalization.- Field diffeomorphisms and symmetries.- Conclusion and outlook
Format: Hardback, 230 pages, height x width: 235x155 mm, 43 Illustrations, color;
4 Illustrations, black and white; XII, 188 p. 46 illus., 43 illus. in color.,
Series: Undergraduate Lecture Notes in Physics
Pub. Date: 21-May-2024
ISBN-13: 9783031552670
This textbook provides a didactic introduction to the topic of group theory in physics, with a special focus on solid state physics issues. The book is useful for students who encounter such problems in their first scientific work (in theory or experiment). In addition to the basic introduction to group theory and representation theory, the book deals with point groups, double point groups, and space groups, which are essential in solid state physics. As an example for systems with space group symmetry, electrons in periodic potentials are discussed.
Furthermore, there are chapters on material tensors and the Wigner Eckart theorem for the evaluation of matrix elements. The latter is especially interesting for students dealing with spectroscopic problems. The content is accompanied by a series of exercises and examples. A set of solutions can be found in the appendix.
Introduction.- Groups: Definitions and Properties.- Point Groups.- Representations and Characters.- Orthogonality Theorems.- Quantum Mechanics and Group Theory.- Irreducible Representations of the Point Groups in Solids.- Group Theory in Stationary Perturbation Theory Calculations.- Material Tensors and Tensor Operators.- Matrix Elements of Tensor Operators: The Wigner-Eckart Theorem.- Double Groups and their Representations.- Space Groups.- Representations of Space Groups.- Particles in Periodic Potentials.
Format: Hardback, 106 pages, height x width: 240x168 mm, 17 Illustrations, color;
2 Illustrations, black and white; XV, 106 p. 19 illus., 17 illus. in color.
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 05-Mar-2024
ISBN-13: 9783031545047
This book provides a comprehensive review on tensor algebra, including tensor products, tensor unfolding, tensor eigenvalues, and tensor decompositions. Tensors are multidimensional arrays generalized from vectors and matrices, which can capture higher-order interactions within multiway data. In addition, tensors have wide applications in many domains such as signal processing, machine learning, and data analysis, and the author explores the role of tensors/tensor algebra in tensor-based dynamical systems where system evolutions are captured through various tensor products. The author provides an overview of existing literature on the topic and aims to inspire readers to learn, develop, and apply the framework of tensor-based dynamical systems.
Tensors and Tensor Algebra.- Tucker Product Representation.- Einstein
Product Representation.- CP/tensor Train Decomposition
Representation.- Tensor Vector Product Representation.- Contract Product
Representation.- T-product Representation.