Format: Hardback, 577 pages, height x width: 235x155 mm, XI, 577 p.
Series: Progress in Mathematics 349
Pub. Date: 31-Jan-2024
ISBN-13: 9783031496530
The purpose of this monograph is to provide a systematic account of the theory of noncommutative integration in semi-finite von Neumann algebras. It is designed to serve as an introductory graduate level text as well as a basic reference for more established mathematicians with interests in the continually expanding areas of noncommutative analysis and probability. Its origins lie in two apparently distinct areas of mathematical analysis: the theory of operator ideals going back to von Neumann and Schatten and the general theory of rearrangement invariant Banach lattices of measurable functions which has its roots in many areas of classical analysis related to the well-known Lp-spaces. A principal aim, therefore, is to present a general theory which contains each of these motivating areas as special cases.
A review of relevant operator theory.- Measurable operators.- Singular value functions.- Symmetric spaces of measurable operators.- Strongly symmetric spaces of measurable operators.- Examples.- Interpolation.
Format: Hardback, height x width: 235x155 mm, Approx. 315 p.
Pub. Date: 21-Feb-2024
ISBN-13: 9789819982844
Most of the existing monographs on generalized inverses are based on linear algebra tools and geometric methods of Banach (Hilbert) spaces to introduce generalized inverses of complex matrices and operators and their related applications, or focus on generalized inverses of matrices over special rings like division rings and integral domains, and does not include the results in general algebraic structures such as arbitrary rings, semigroups and categories, which are precisely the most general cases.
In this book, five important generalized inverses are introduced in these algebraic structures. Moreover, noting that the (pseudo) core inverse was introduced in the last decade and has attracted much attention, this book also covers the very rich research results on it, so as to be a necessary supplement to the existing monographs. This book starts with decompositions of matrices, introduces the basic properties of generalized inverses of matrices, and then discusses generalized inverses of elements in rings and semigroups, as well as morphisms in categories. The algebraic nature of generalized inverses is presented, and the behavior of generalized inverses are related to the properties of the algebraic system.
Scholars and graduate students working on the theory of rings, semigroups and generalized inverses of matrices and operators will find this book helpful.
Preface.- Algebraic Basic Knowledge.- Moore-Penrose Inverses.- Group Inverses.- Drazin Inverses.- Core Inverses.- Pseudo Core Inverses.- Bibliography.- Index.
Format: Hardback, height x width: 235x155 mm, 45 Illustrations, color; Approx. 330 p. 45 illus. in color.
Pub. Date: 04-Mar-2024
ISBN-13: 9783031484865
This book, the ninth of 15 related monographs, discusses a two product-cubic dynamical system possessing different product-cubic structures and the equilibrium and flow singularity and bifurcations for appearing and switching bifurcations. The appearing bifurcations herein are parabola-saddles, saddle-sources (sinks), hyperbolic-to-hyperbolic-secant flows, and inflection-source (sink) flows. The switching bifurcations for saddle-source (sink) with hyperbolic-to-hyperbolic-secant flows and parabola-saddles with inflection-source (sink) flows are based on the parabola-source (sink), parabola-saddles, inflection-saddles infinite-equilibriums. The switching bifurcations for the network of the simple equilibriums with hyperbolic flows are parabola-saddles and inflection-source (sink) on the inflection-source and sink infinite-equilibriums. Readers will learn new concepts, theory, phenomena, and analysis techniques.
・ Two-different product-cubic systems
・ Hybrid networks of higher-order equilibriums and flows
・ Hybrid series of simple equilibriums and hyperbolic flows
・ Higher-singular equilibrium appearing bifurcations
・ Higher-order singular flow appearing bifurcations
・ Parabola-source (sink) infinite-equilibriums
・ Parabola-saddle infinite-equilibriums
・ Inflection-saddle infinite-equilibriums
・ Inflection-source (sink) infinite-equilibriums
・ Infinite-equilibrium switching bifurcations.
Chapter 1 Cubic Systems with Two different Product Structures.
Chapter
2 Parabola-saddle and Saddle-source (sink) Singularity.-
Chapter 3
Inflection-source (sink) flows and parabola-saddles.
Chapter 4Saddle-source
(sink) with hyperbolic flow singularity.
Chapter 5 Equilibrium matrices with
hyperbolic flows.
Format: Hardback, height x width: 235x155 mm, 31 Illustrations, color; Approx. 300 p. 31 illus. in color.,
Pub. Date: 04-Mar-2024
ISBN-13: 9783031484902
This book, the tenth of 15 related monographs, discusses product-cubic nonlinear systems with two crossing-linear and self-quadratic products vector fields and the dynamic behaviors and singularity are presented through the first integral manifolds. The equilibrium and flow singularity and bifurcations discussed in this volume are for the appearing and switching bifurcations. The double-saddle equilibriums described are the appearing bifurcations for saddle source and saddle-sink, and for a network of saddles, sink and source. The infinite-equilibriums for the switching bifurcations are also presented, specifically:
・ Inflection-saddle infinite-equilibriums,
・ Hyperbolic (hyperbolic-secant)-sink and source infinite-equilibriums
・ Up-down and down-up saddle infinite-equilibriums,
・ Inflection-source (sink) infinite-equilibriums.
Preface.- Crossing-linear and Self-quadratic Product Systems.- Double-saddles and switching dynamics.- Vertically Paralleled Saddle-source and Saddle-sink.- Horizontally Paralleled Saddle-source and Saddle-sink.- Simple Equilibrium Networks and Switching Dynamics
Format: Hardback, 243 pages, height x width: 235x155 mm, 113 Tables, color; 116 Illustrations, color;
1 Illustrations, black and white; XIV, 243 p. 117 illus., 116 illus. in color.
Pub. Date: 07-Mar-2024
ISBN-13: 9783031475108
The focus of this monograph is convertingreshapingone 3D convex polyhedron to another via an operation the authors call tailoring. A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a vertex) of a polyhedron and sutures closed the hole. This is akin to Johannes Keplers vertex truncation, but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful gluing theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoring allows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences.
Part II carries out a systematic study of vertex-merging, a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and pasted inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht Durer in the 15th century: whether every convex polyhedron can be unfolded to a planar net. Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studied in the literature but with considerable promise for future development.
This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrovs Gluing Theorem, shortest paths and cut loci, Cauchys Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the journey worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the conce
I. Tailoring for Every Body.-
1. Introduction to Part I.-
2. Preliminaries.-
3. Domes and Pyramids.-
4. Tailoring via Sculpting.-
5. Pyramid Seal Graph.-
6. Algorithms for Tailoring via Sculpting.-
7. Crests.-
8. Tailoring via Flattening.-
9. Applications of Tailoring.- II. Vertex-Merging and Convexity.-
10. Introduction to Part II.-
11. Vertex-Merging Reductions and Slit Graphs.-
12. Planar Spiral Slit Tree.-
13. Convexity on Convex Polyhedra.-
14. Minimal-length Enclosing Polygon.-
15. Spiral Tree on Polyhedron.-
16. Unfolding via Slit Trees.-
17. Vertices on Quasigeodesics.-
18. Conclusions.- Bibliography.- References.- Index.
Format: Paperback / softback, 287 pages, height x width: 235x155 mm, V, 287 p.
Series: UNITEXT 157
Pub. Date: 01-Mar-2024
ISBN-13: 9783031492457
This book aims at introducing students into the modern analytical foundations to treat problems and situations in the Calculus of Variations solidly and rigorously. Since no background is taken for granted or assumed, as the textbook pretends to be self-contained, areas like basic Functional Analysis and Sobolev spaces are studied to the point that chapters devoted to these topics can be utilized by themselves as an introduction to these important parts of Analysis. The material in this regard has been selected to serve the needs of classical variational problems, leaving broader treatments for more advanced and specialized courses in those areas. It should not be forgotten that problems in the Calculus of Variations historically played a crucial role in pushing Functional Analysis as a discipline on its own right. The style is intentionally didactic. After a first general chapter to place optimization problems in infinite-dimensional spaces in perspective, the first part of the book focuses on the initial important concepts in Functional Analysis and introduces Sobolev spaces in dimension one as a preliminary, simpler case (much in the same way as in the successful book of H. Brezis). Once the analytical framework is covered, one-dimensional variational problems are examined in detail including numerous examples and exercises. The second part dwells, again as a first-round, on another important chapter of Functional Analysis that students should be exposed to, and that eventually will find some applications in subsequent chapters. The first chapter of this part examines continuous operators and the important principles associated with mappings between functional spaces; and another one focuses on compact operators and their fundamental and remarkable properties for Analysis. Finally, the third part advances to multi-dimensional Sobolev spaces and the corresponding problems in the Calculus of Variations. In this setting, problems become much more involved and, for this same reason, much more interesting and appealing. In particular, the final chapter dives into a number of advanced topics, some of which reflect a personal taste. Other possibilities stressing other kinds of problems are possible. In summary, the text pretends to help students with their first exposure to the modern calculus of variations and the analytical foundation associated with it. In particular, it covers an extended introduction to basic functional analysis and to Sobolev spaces. The tone of the text and the set of proposed exercises will facilitate progressive understanding until the need for further challenges beyond the topics addressed here will push students to more advanced horizons.
1 Motivation and perspective.- Part I: Basic Functional Analysis and
Calculus of Variations.- 2 A first exposure to Functional Analysis.-
3 Introduction to convex analysis. The Hahn-Banach and Lax-Milgram theorems.-
4 The Calculus of Variations for one-dimensional problems.- Part II: Basic
Operator Theory.- 5 Continuous operators.- 6 Compact operators.- Part III:
Multidimensional Sobolev Spaces and Scalar Variational Problems.- 7
Multidimensional Sobolev spaces.- 8 Variational problems.- 9 Finer results in
Sobolev spaces and the Calculus of Variations.- Appendix A: Hints and
solutions to exercises.- Appendix B: So much to learn.