Format: Hardback, 351 pages, height x width: 254x178 mm, 3 Tables, color; 3 Illustrations, color;
37 Illustrations, black and white; XIII, 351 p. 40 illus., 3 illus. in color. With online files/update.,
Series: Springer Undergraduate Texts in Mathematics and Technology
Pub. Date: 31-Oct-2023
ISBN-13: 9783031395611
This textbook is designed for a first course in linear algebra for undergraduate students from a wide range of quantitative and data driven fields. By focusing on applications and implementation, students will be prepared to go on to apply the power of linear algebra in their own discipline. With an ever-increasing need to understand and solve real problems, this text aims to provide a growing and diverse group of students with an applied linear algebra toolkit they can use to successfully grapple with the complex world and the challenging problems that lie ahead. Applications such as least squares problems, information retrieval, linear regression, Markov processes, finding connections in networks, and more, are introduced on a small scale as early as possible and then explored in more generality as projects. Additionally, the book draws on the geometry of vectors and matrices as the basis for the mathematics, with the concept of orthogonality taking center stage. Important matrix factorizations as well as the concepts of eigenvalues and eigenvectors emerge organically from the interplay between matrix computations and geometry.
The R files are extra and freely available. They include basic code and templates for many of the in-text examples, most of the projects, and solutions to selected exercises. As much as possible, data sets and matrix entries are included in the files, thus reducing the amount of manual data entry required.
Introduction.-
1. Vectors.-
2. Matrices.-
3. Matrix Contexts.-
4. Linear
Systems.-
5. Least Squares and Matrix Geometry.
6. Orthogonal Systems.-
7.
Eigenvalues.-
8. Markov Processes.-
9. Symmetric Matrices.-
10. Singular
Value Decomposition.-
11. Function Spaces.-Bibliography.-Index.
Format: Hardback, 240 pages, height x width: 235x155 mm, 2 Tables, color; 4 Illustrations,
color; 6 Illustrations, black and white; X, 240 p. 10 illus., 4 illus. in color.,
Series: Probability and Its Applications
Pub. Date: 04-Nov-2023
ISBN-13: 9783031395451
This monograph highlights the connection between the theory of neutron transport and the theory of non-local branching processes. By detailing this frequently overlooked relationship, the authors provide readers an entry point into several active areas, particularly applications related to general radiation transport. Cutting-edge research published in recent years is collected here for convenient reference. Organized into two parts, the first offers a modern perspective on the relationship between the neutron branching process (NBP) and the neutron transport equation (NTE), as well as some of the core results concerning the growth and spread of mass of the NBP. The second part generalizes some of the theory put forward in the first, offering proofs in a broader context in order to show why NBPs are as malleable as they appear to be. Stochastic Neutron Transport will be a valuable resource for probabilists, and may also be of interest to numerical analysts and engineers in the field of nuclear research.
Part I Neutron Transport Theory.- Classical Neutron Transport Theory.- Some background Markov process theory.- Stochastic Representation of the Neutron Transport Equation.- Many-to-one, Perron-Frobenius and criticality.- Pal-Bell equation and moment growth.- Martingales and path decompositions.- Discrete evolution.- Part II General branching Markov processes.- A general family of branching Markov processes.- Moments.- Survival at criticality.- Spines and skeletons.- Martingale convergence and laws of large numbers.
Format: Hardback, 240 pages, height x width: 235x155 mm, 48 Illustrations, color; 5 Illustrations, black and white; X, 240 p. 53 illus., 48 illus. in color.
Pub. Date: 02-Nov-2023
ISBN-13: 9783031396144
Today, the theory of complex-valued functions finds widespread applications in various areas of mathematical research, as well as in electrical and mechanical engineering, aeronautics, and other disciplines. Complex analysis has become a basic course in mathematics, physics, and select engineering departments. This concise textbook provides a thorough introduction to the function theory of one complex variable. It presents the fundamental concepts with clarity and rigor, offering concise proofs that avoid lengthy and tedious arguments commonly found in mathematics textbooks. It goes beyond traditional texts by exploring less common topics, including the different approaches to constructing analytic functions, the conformal mapping criterion, integration of analytic functions along arbitrary curves, global analytic functions and their Riemann surfaces, the general inverse function theorem, the Lagrange-Burmann formula, and Puiseux series. Drawing from several decades of teaching experience, this book is ideally suited for one or two semester courses in complex analysis. It also serves as a valuable companion for courses in topology, approximation theory, asymptotic analysis, and functional analysis. Abundant examples and exercises make it suitable for self-study as well.
1 Complex Numbers and Complex Plane.- 2 Analytic Functions.- 3
Elementary Analytic Functions.- 4 Integration of Functions of a Single
Complex Variable.- 5 Complex Power Series.- 6 Laurent Series. Isolated
Singularities of Analytic Functions.- 7 Residue Calculus.- 8 Analytic
Continuations.- 9 Qualitative Properties of Analytic Functions.
Format: Hardback, 201 pages, height x width: 235x155 mm, 12 Illustrations, color; 1 Illustrations, black and white; X, 201 p. 13 illus., 12 illus. in color.
Series: CMS/CAIMS Books in Mathematics 11
Pub. Date: 26-Oct-2023
ISBN-13: 9783031395239
This book provides a short introduction to partial differential equations (PDEs). It is primarily addressed to graduate students and researchers, who are new to PDEs. The book offers a user-friendly approach to the analysis of PDEs, by combining elementary techniques and fundamental modern methods.
The author focuses the analysis on four prototypes of PDEs, and presents two approaches for each of them. The first approach consists of the method of analytical and classical solutions, and the second approach consists of the method of weak (variational) solutions.
1 Notations and review.- 2 Partial differential equations.- 3 First
order PDEs. Classical and weak solutions.- 4 Second-order linear elliptic
PDEs. maximum principle and classical solutions - 5 Distributions.- 6 Sobolev
spaces.- 7 Second order linear elliptic PDEs. Weak solutions.- 8 Second order
parabolic and hyperbolic PDEs.- 9 Annex.
Format: Paperback / softback, 87 pages, height x width: 235x155 mm, 10 Illustrations, black and white; III, 87 p. 10 illus.,
Series: SpringerBriefs in Mathematics
Pub. Date: 18-Oct-2023
ISBN-13: 9783031391613
This book serves as an introductory asset for learning metric geometry by delivering an in-depth examination of key constructions and providing an analysis of universal spaces, injective spaces, the Gromov-Hausdorff convergence, and ultralimits. This book illustrates basic examples of domestic affairs of metric spaces, this includes Alexandrov geometry, geometric group theory, metric-measure spaces and optimal transport.
Researchers in metric geometry will find this book appealing and helpful, in addition to graduate students in mathematics, and advanced undergraduate students in need of an introduction to metric geometry. Any previous knowledge of classical geometry, differential geometry, topology, and real analysis will be useful in understanding the presented topics.
1 Definitions. - A. Metric spaces. -B. Topology. - C. Variations. -D.
Maximal metric and gluing. - E. Completeness. -F. Compact spaces. - G. Proper
spaces. -H. Geodesics. - I. Metric trees. - J. Length. - K. Length spaces. -
2 Universal spaces. - A. Embedding in a normed space. - B. Extension
property. - C. Universality. - D. Uniqueness and homogeneity. - E. Remarks. -
3 Injective spaces. - A. Definition. - B. Admissible and extremal functions.
- C. Equivalent conditions. - D. Space of extremal functions. - E. Injective
envelope. - F. Remarks. - 4 Space of subsets. - A. Hausdorff distance. -B.
Hausdorff convergence. - C. An application. -D. Remarks. - 5 Space of spaces.
- A. Gromov-Hausdorff metric. - B. Approximations and almost isometries 49;
C. Optimal realization 50; D. Convergence 51; E. Uniformly totally bonded
families 52; F. Gromov selection theorem. - G. Universal ambient space. - H.
Remarks. - 6 Ultralimits. - A. Faces of ultrafilters. - B. Ultralimits of
points. - C. An illustration. - D. Ultralimits of spaces. - E. Ultrapower. -
F. Tangent and asymptotic spaces. - G. Remarks. - Semisolutions. -Index. -
Bibliography
Format: Hardback, height x width: 235x155 mm, Approx. 335 p.
Pub. Date: 18-Sep-2023
ISBN-13: 9783031394881
This book provides a comprehensive algebraic treatment of hypergroups, as defined by F. Marty in 1934. It starts with structural results, which are developed along the lines of the structure theory of groups. The focus then turns to a number of concrete classes of hypergroups with small parameters, and continues with a closer look at the role of involutions (modeled after the definition of group-theoretic involutions) within the theory of hypergroups. Hypergroups generated by involutions lead to the exchange condition (a genuine generalization of the group-theoretic exchange condition), and this condition defines the so-called Coxeter hypergroups. Coxeter hypergroups can be treated in a similar way to Coxeter groups. On the other hand, their regular actions are mathematically equivalent to buildings (in the sense of Jacques Tits). A similar equivalence is discussed for twin buildings. The primary audience for the monograph will be researchers working in Algebra and/or Algebraic Combinatorics, in particular on association schemes.
1 Basic Facts : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1
1.1 Neutral Elements and Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Complex Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Thin Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Groups and Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Actions of Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Hypergroups Admitting Regular Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 Association Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Closed Subsets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27
2.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Dedekind Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Generating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 The Thin Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.7 Foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Elementary Structure Theory: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 47
3.1 Centralizers and Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Su cient Conditions for Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Strong Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Computations in Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6 Homomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 The Homomorphism Theorem and the Isomorphism Theorems . . . . . . . . . . 71
4 Subnormality and Thin Residues : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 79
4.1 Subnormal Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 The Thin Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Thin Residues of Thin Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5 Residually Thin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6 Finite Residually Thin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.7 Solvable Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Tight Hypergroups : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 107
5.1 Tight Hypergroup Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 The Set S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 The Sets a b \ Fc and Sa;b(Fc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4 The Sets bf1b \ Fa and Sb;(f1;:::;fn)(Fa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.5 Structure Constants of Finite Tight Hypergroups . . . . . . . . . . . . . . . . . . . . . 122
5.6 Rings Arising from Certain Finite Tight Hypergroups . . . . . . . . . . . . . . . . . 126
5.7 Finite Metathin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.8 Finite Metathin Hypergroups with Restricted Thin Residue . . . . . . . . . . . . 132
6 Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 137
6.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 Cosets of Closed Subsets Generated by an Involution, I . . . . . . . . . . . . . . . . 142
.3 Cosets of Closed Subsets Generated by an Involution, II . . . . . . . . . . . . . . . 145
6.4 Cosets of Closed Subsets Generated by an Involution, III . . . . . . . . . . . . . . . 147
6.5 Length Functions De ned by Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . 152
6.6 Hypergroups Generated by Two Distinct Involutions . . . . . . . . . . . . . . . . . . 156
6.7 Dichotomy and the Exchange Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.8 Projective Hypergroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7 Hypergroups with a Small Number of Elements : : : : : : : : : : : : : : : : : : : : : : 171
7.1 Hypergroups of Cardinality at Most 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.2 Non-Symmetric Hypergroups of Cardinality 4 . . . . . . . . . . . . . . . . . . . . . . . . 179
7.3 Hypergroups of Cardinality 6 with a Non-Normal Closed Subset, I . . . . . . 190
7.4 Hypergroups of Cardinality 6 with a Non-Normal Closed Subset, II . . . . . . 202
7.5 Non-Normal Closed Subsets Missing Four Elements . . . . . . . . . . . . . . . . . . . 215
7.6 Non-Normal Closed Subsets Missing Four Elements and Thin Elements . . 221
8 Constrained Sets of Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 223
8.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.2 Constrained Sets of Involutions and Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.3 Constrained Sets of Involutions and the Thin Radical . . . . . . . . . . . . . . . . . . 230
8.4 Constrained Sets of Involutions and Dichotomy . . . . . . . . . . . . . . . . . . . . . . . 233
8.5 Constrained Sets of Non-Thin Involutions and Dichotomy . . . . . . . . . . . . . . 239
8.6 Constrained Sets of Involutions and Foldings . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.7 Dichotomic Constrained Sets of Involutions and Foldings . . . . . . . . . . . . . . . 248
9 Coxeter Sets of Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 251
9.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.2 The Sets V1(U) for Subsets U of Coxeter Sets V of Involutions . . . . . . . . . . 256
9.3 The Sets V????1(U) for Subsets U of Coxeter Sets V of Involutions . . . . . . . . . 263
9.4 Sets of Subsets of Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.5 Spherical Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
9.6 Subsets of Spherical Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . 273
9.7 Coxeter Sets of Involutions and Foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
9.8 Coxeter Sets of Involutions and Their Coxeter Numbers . . . . . . . . . . . . . . . . 280
9.9 Coxeter Sets of Involutions and Type Preserving Bijections . . . . . . . . . . . . . 286
10 Regular Actions of (Twin) Coxeter Hypergroups: : : : : : : : : : : : : : : : : : : : : 293
10.1 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.2 Twin Buildings, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
10.3 Twin Buildings, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
10.4 Regular Actions of Coxeter Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
10.5 Regular Actions of Twin Coxeter Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . 315
References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 333
Format: Hardback, 463 pages, height x width: 235x155 mm, 6 Illustrations, black and white; IV, 463 p. 6 illus.
Series: Fields Institute Communications 87
Pub. Date: 25-Oct-2023
ISBN-13: 9783031392696
The focus program on Analytic Function Spaces and their Applications took place at Fields Institute from July 1st to December 31st, 2021. Hilbert spaces of analytic functions form one of the pillars of complex analysis. These spaces have a rich structure and for more than a century have been studied by many prominent mathematicians. They also have several essential applications in other fields of mathematics and engineering, e.g., robust control engineering, signal and image processing, and theory of communication. The most important Hilbert space of analytic functions is the Hardy class H2. However, its close cousins, e.g. the Bergman space A2, the Dirichlet space D, the model subspaces Kt, and the de Branges-Rovnyak spaces H(b), have also been the center of attention in the past two decades. Studying the Hilbert spaces of analytic functions and the operators acting on them, as well as their applications in other parts of mathematics or engineering were the main subjects of this program. During the program, the world leading experts on function spaces gathered and discussed the new achievements and future venues of research on analytic function spaces, their operators, and their applications in other domains.
With more than 250 hours of lectures by prominent mathematicians, a wide variety of topics were covered. More explicitly, there were mini-courses and workshops on Hardy Spaces, Dirichlet Spaces, Bergman Spaces, Model Spaces, Interpolation and Sampling, Riesz Bases, Frames and Signal Processing, Bounded Mean Oscillation, de Branges-Rovnyak Spaces, Operators on Function Spaces, Truncated Toeplitz Operators, Blaschke Products and Inner Functions, Discrete and Continuous Semigroups of Composition Operators, The Corona Problem, Non-commutative Function Theory, Drury-Arveson Space, and Convergence of Scattering Data and Non-linear Fourier Transform. At the end of each week, there was a high profile colloquium talk on the current topic. The program also contained two semester-long advanced courses on Schramm Loewner Evolution and Lattice Models and Reproducing Kernel Hilbert Space of Analytic Functions. The current volume features a more detailed version of some of the talks presented during the program.
Absolute continuity in higher dimensions.- An indefinite analog of
Sarason's generalized interpolation theorem.- An Operator theoretical
approach of some inverse problems.- Applications of the automatic additivity
of positive homogenous order isomorphisms between positive definite cones in
C*-algebras.- Direct and Inverse Spectral Theorems for a Class of Canonical
Systems with two Singular Endpoints.- Nevanlinna domains and uniform
approximation by polyanalytic polynomial modules.- On meromorphic inner
functions in the upper half-plane.- On the norm of the Hilbert matrix.-
Radial limits of functions holomorphic in C or the polydisc.- Recent
developments in the interplay between function theory and operator theory for
block Toeplitz, Hankel, and model operators.- Sarason's Ha-plitz product
problem.- Sub-Hardy Hilbert spaces in the non-commutative unit row-ball.- The
relationship of the Gaussian curvature with the curvature of a Cowen-Douglas
operator.- Weighted Polynomial Approximation on the Cubes of the non-zero
Integers.- Index.