Format: Hardback, 364 pages, height x width: 235x155 mm, weight: 797 g, 7 Illustrations,
color; 80 Illustrations, black and white; X, 364 p. 87 illus., 7 illus. in color
Series: Springer Proceedings in Mathematics & Statistics 386
Pub. Date: 09-Jun-2022
ISBN-13: 9783030983260
This book collects together original research and survey articles highlighting the fertile interdisciplinary applications of convex lattice polytopes in modern mathematics. Covering a diverse range of topics, including algebraic geometry, mirror symmetry, symplectic geometry, discrete geometry, and algebraic combinatorics, the common theme is the study of lattice polytopes. These fascinating combinatorial objects are a cornerstone of toric geometry and continue to find rich and unforeseen applications throughout mathematics. The workshop Interactions with Lattice Polytopes assembled many top researchers at the Otto-von-Guericke-Universitat Magdeburg in 2017 to discuss the role of lattice polytopes in their work, and many of their presented results are collected in this book. Intended to be accessible, these articles are suitable for researchers and graduate students interested in learning about some of the wide-ranging interactions of lattice polytopes in pure mathematics.
Gennadiy Averkov, Difference between families of weakly and strongly maximal integral lattice-free polytopes.- Victor Batyrev, Alexander Kasprzyk, and Karin Schaller, On the Fine interior of three-dimensional canonical Fano polytopes.- Monica Blanco, Lattice distances in 3-dimensional quantum jumps.- Amanda Cameron, Rodica Dinu, Mateusz Michalek, and Tim Seynnaeve, Flag matroids: algebra and geometry.- Daniel Cavey and Edwin Kutas, Classification of minimal polygons with specified singularity content.- Tom Coates, Alessio Corti, and Genival da Silva Jr, On the topology of Fano smoothings.- Sandra Di Rocco and Anders Lundman, Computing Seshadri constants on smooth toric surfaces.- Akihiro Higashitani, The characterisation problem of Ehrhart polynomials of lattice polytopes.- Johannes Hofscheier, The ring of conditions for horospherical homogeneous spaces.- Katharina Jochemko, Linear recursions for integer point transforms.- Valentina Kiritchenko and Maria Padalko, Schubert calculus on Newton-Okounkov polytopes.- Bach Le Tran, An Eisenbud-Goto-type upper bound for the Castelnuovo-Mumford regularity of fake weighted projective spaces.- Milena Pabiniak, Toric degenerations in symplectic geometry.- Andrea Petracci, On deformations of toric Fano varieties Thomas Prince, Polygons of finite mutation type.- Hendrik Suss, Orbit spaces of maximal torus actions on oriented Grassmannians of planes .- Akiyoshi Tsuchiya, The reflexive dimension of (0, 1)-polytopes
Format: Paperback / softback, 321 pages, height x width: 235x155 mm, weight: 510 g, 2 Illustrations,
color; 2 Illustrations, black and white; X, 321 p. 4 illus., 2 illus. in color.,
Series: Trends in Mathematics
Pub. Date: 24-Jul-2022
ISBN-13: 9783030709761
This proceedings volume features selected contributions from the conference Positivity X. The field of positivity deals with ordered mathematical structures and their applications. At the biannual series of Positivity conferences, the latest developments in this diverse field are presented. The 2019 edition was no different, with lectures covering a broad spectrum of topics, including vector and Banach lattices and operators on such spaces, abstract stochastic processes in an ordered setting, the theory and applications of positive semi-groups to partial differential equations, Hilbert geometries, positivity in Banach algebras and, in particular, operator algebras, as well as applications to mathematical economics and financial mathematics. The contributions in this book reflect the variety of topics discussed at the conference. They will be of interest to researchers in functional analysis, operator theory, measure and integration theory, operator algebras, and economics. Positivity X was dedicated to the memory of our late colleague and friend, Coenraad Labuschagne. His untimely death in 2018 came as an enormous shock to the Positivity community. He was a prominent figure in the Positivity community and was at the forefront of the recent development of abstract stochastic processes in a vector lattice context.
Preface.- Dedication.- Coenraad Christoffel Andries Labuschagne.- Inverse monotonicity of elliptic operators invariational form.- On compact operators between lattice normed spaces.- How to be positive in natural sciences?.- Real positive maps and conditional expectations on operator algebras.- Free vector lattices and free vector lattice algebras.- On disjointness, bands and projections in partially ordered vector spaces.- 101 years of vector lattice theory - a vector lattice-valued Daniell integral.- Ergodicity in Riesz spaces.- Multiplicative representation of real-valuedbi-Riesz homomorphisms on partially ordered vector spaces.- Orthogonality: An antidote to Kadison's antilattice theorem.- Binary Relations in Mathematical Economics: On Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot.- On fixed point theory in partially ordered (quasi-)metric spaces and an application to complexity analysis of algorithms.- Inheritance properties of positive cones induced by subalgebras and quotients of ordered Banach algebras.- Universally complete spaces of continuous functions.- Applications of generalized B -algebras to quantum mechanics.
Format: Paperback / softback, 336 pages, height x width: 235x155 mm, weight: 539 g, 1 Tables,
color; 1 Illustrations, color; 40 Illustrations, black and white; XIII, 336 p. 41 illus., 1 illus. in color
Series: Lecture Notes in Mathematics 2311
Pub. Date: 13-Aug-2022
ISBN-13: 9783031051210
This book provides an introduction to deformation quantization and its relation to quantum field theory, with a focus on the constructions of Kontsevich and Cattaneo & Felder. This subject originated from an attempt to understand the mathematical structure when passing from a commutative classical algebra of observables to a non-commutative quantum algebra of observables. Developing deformation quantization as a semi-classical limit of the expectation value for a certain observable with respect to a special sigma model, the book carefully describes the relationship between the involved algebraic and field-theoretic methods. The connection to quantum field theory leads to the study of important new field theories and to insights in other parts of mathematics such as symplectic and Poisson geometry, and integrable systems.
Based on lectures given by the author at the University of Zurich, the book will be of interest to graduate students in mathematics or theoretical physics. Readers will be able to begin the first chapter after a basic course in Analysis, Linear Algebra and Topology, and references are provided for more advanced prerequisites.
1. Introduction. -
2. Foundations of Differential Geometry. -
3. Symplectic Geometry. -
4. Poisson Geometry. -
5. Deformation Quantization. -
6. Quantum Field Theoretic Approach to Deformation Quantization.
Format: Paperback / softback, 282 pages, height x width: 235x155 mm,
1 Illustrations, black and white; XII, 282 p. 1 illus.,
Series: Lecture Notes in Mathematics 2316
Pub. Date: 08-Dec-2022
ISBN-13: 9783031148682
This book develops a new theory in convex geometry, generalizing positive bases and related to Caratheordory's Theorem by combining convex geometry, the combinatorics of infinite subsets of lattice points, and the arithmetic of transfer Krull monoids (the latter broadly generalizing the ubiquitous class of Krull domains in commutative algebra)This new theory is developed in a self-contained way with the main motivation of its later applications regarding factorization. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. Among the most important is the elasticity, which measures the ratio between the maximum and minimum number of atoms in any factorization. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. Via the developed material in convex geometry, we characterize when finite elasticity holds for any Krull domain with finitely generated class group $G$, with the results extending more generally to transfer Krull monoids. This book is aimed at researchers in the field but is written to also be accessible for graduate students and general mathematicians.
1. Introduction. -
2. Preliminaries and General Notation. -
3. Asymptotically Filtered Sequences, Encasement and Boundedness. -
4. Elementary Atoms, Positive Bases and Reay Systems. -
5. Oriented Reay Systems. -
6. Virtual Reay Systems. -
7. Finitary Sets. -
8. Factorization Theory.
Format: Paperback / softback, 347 pages, height x width: 235x155 mm, weight: 551 g,
1 Illustrations, black and white; X, 347 p. 1 illus
Series: Lecture Notes in Mathematics 2319
Pub. Date: 22-Sep-2022
ISBN-13: 9783031151262
Goodreads reviews
This book provides the foundations for geometric applications of convex cones and presents selected examples from a wide range of topics, including polytope theory, stochastic geometry, and Brunn?Minkowski theory. Giving an introduction to convex cones, it describes their most important geometric functionals, such as conic intrinsic volumes and Grassmann angles, and develops general versions of the relevant formulas, namely the Steiner formula and kinematic formula.
In recent years questions related to convex cones have arisen in applied mathematics, involving, for example, properties of random cones and their non-trivial intersections. The prerequisites for this work, such as integral geometric formulas and results on conic intrinsic volumes, were previously scattered throughout the literature, but no coherent presentation was available. The present book closes this gap. It includes several pearls from the theory of convex cones, which should be better known.
Basic notions and facts.- Angle functions.- Relations to spherical geometry.- Steiner and kinematic formulas.- Central hyperplane arrangements and induced cones.- Miscellanea on random cones.- Convex hypersurfaces adapted to cones.
Format: Hardback, 650 pages, Worked examples or Exercises
Series: Encyclopedia of Mathematics and its Applications
Pub. Date: 17-Nov-2022
Publisher: Cambridge University Press
ISBN-13: 9781009243773
The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras. This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.
This text develops a new theory extending the classical theory of connected graded Hopf algebras to reflection arrangements.
ntroduction
1. Coxeter groups and reflection arrangements
Part I. Coxeter Species:
2. Coxeter species and Coxeter bimonoids
3. Basic theory of Coxeter bimonoids
4. Examples of Coxeter bimonoids
5. Coxeter operads
6. Coxeter Lie monoids
7. Structure theory of Coxeter bimonoids
Part II. Coxeter Spaces:
8. Coxeter spaces and Coxeter bialgebras
9. Basic theory of Coxeter bialgebras
10. Examples of Coxeter bialgebras
11. Coxeter operad algebras
12. Coxeter Lie algebras
13. Structure theory of Coxeter bialgebras
Part III. Fock Functors:
14. Fock functors
15. Coxeter bimonoids and Coxeter bialgebras
16. Adjoints of Fock functors
17. Structure theory under Fock functors
18. Examples of Fock spaces
Appendix A. Category theory
References
List of Notations
List of Tables
List of Figures
List of Summaries
Author Index
Subject Index.
Format: Paperback / softback, 143 pages, height x width: 235x155 mm,
37 Illustrations, black and white; X, 143 p. 37 illus.,
Pub. Date: 19-Dec-2022
ISBN-13: 9783031142086
Classically Semisimple Rings is a textbook on rings, modules and categories, aimed at advanced undergraduate and beginning graduate students. The book presents the classical theory of semisimple rings from a modern, category-theoretic point of view. Examples from algebra are used to motivate the abstract language of category theory, which then provides a framework for the study of rings and modules, culminating in the Wedderburn-Artin classification of semisimple rings. In the last part of the book, readers are gently introduced to related topics such as tensor products, exchange modules and C*-algebras. As a final flourish, Rickart's theorem on group rings ties a number of these topics together. Each chapter ends with a selection of exercises of varying difficulty, and readers interested in the history of mathematics will find biographical sketches of important figures scattered throughout the text.Assuming previous knowledge in linear and basic abstract algebra, this book can serve as a textbook for a course in algebra, providing students with valuable early exposure to category theory.
Introduction.
Chapter 1. Motivation from Ring Theory.
Chapter 2. Constructions with Modules.
Chapter 3. The Isomorphism Theorems.
Chapter 4. Noetherian Modules.
Chapter 5. Artinian Modules.
Chapter 6. Simple and Semisimple Modules.
Chapter 7. The Artin-Weddeburn Theorem.
Chapter 8. Tensor Products of Modules.
Chapter 9. Exchange Modules and Exchange Rings.-
Chapter 10. Semiprimitivity of Group Rings.- Bibliography.- Index of Symbols.- Index.
Format: Hardback, 413 pages, height x width: 235x155 mm, 13 Tables, color; 16 Illustrations,
color; 12 Illustrations, black and white; XIX, 413 p. 28 illus., 16 illus. in color
Series: Springer Series in Statistics
Pub. Date: 08-Dec-2022
ISBN-13: 9783031142741
The first half of the book is aimed at quantitative research workers in biology, medicine, ecology and genetics. The book as a whole is aimed at graduate students in statistics, biostatistics, and other quantitative disciplines. Ten detailed examples show how the author approaches real-world statistical problems in a principled way that allows for adequate compromise and flexibility. The need to accommodate correlations associated with space, time and other relationships is a recurring theme, so variance-components models feature prominently. Statistical pitfalls are illustrated via examples taken from the recent scientific literature. Chapter 11 sets the scene, not just for the second half of the book, but for the book as a whole. It begins by defining fundamental concepts such as baseline, observational unit, experimental unit, covariates and relationships, randomization, treatment assignment, and the role that these play in model formulation. Compatibility of the model with the randomization scheme is crucial. The effect of treatment is invariably modelled as a group action on probability distributions. Technical matters connected with space-time covariance functions, residual likelihood, likelihood ratios, and transformations are discussed in later chapters.
Ten Projects.- Basic Concepts.- Principles.- Initial values.- Probability distributions.- Gaussian distributions.- Space-time processes. Likelihood.- Residual Likelihood.- Response transformation.- Presentations and reports.