Format: Hardback, 257 pages, height x width: 235x155 mm, 3 Illustrations,
color; 24 Illustrations, black and white; X, 265 p. 27 illus., 3 illus. in color.
Series: Springer Graduate Texts in Philosophy 4
Pub. Date: 20-May-2024
ISBN-13: 9783031562143
The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by L?szlo Fejes Toth in his Notes to the 2nd edition.
The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of:
a corrected English translation of the original Lagerungen, the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail.
The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.
Part I. Lagerungen - Arrangements in the Plane, on the Sphere, and in Space.-
1. Some Theorems from Elementary Geometry.-
2. Theorems from the Theory of Convex Bodies.-
3. Problems on Packing and Covering in the Plane.-
4. Efficiency of Packings and Coverings with a Sequence of Convex Disks.-
5. Extremal Properties of Regular Polyhedra.-
6. Irregular Packing on the Sphere.-
7. Packing in Space.- Part II. Notes and Additional
Chapters to the English Edition.-
8. Notes.-
9. Finite Variations on the Isoperimetric Problem.-
10. Higher Dimensions.-
11. Ball Packings in Hyperbolic Space.-
12. Mutliple Arrangements.-
13. Neighbors.-
14. Packing and Covering Properties fo Sequences of Convex Bodies.-
15. Four Classic Problems.-
16. Miscellaneous Problems about Packing and Covering.- References for Part I.- References for Part II.- Name Index.- Subject Index.
Format: Hardback, 417 pages, height x width: 235x155 mm, 1 Illustrations, color;
29 Illustrations, black and white; X, 420 p. 30 illus.
Series: KIAS Springer Series in Mathematics 2
Pub. Date: 30-May-2024
ISBN-13: 9789819717972
A-infinity structure was introduced by Stasheff in the 1960s in his homotopy characterization of based loop space, which was the culmination of earlier works of Sugawara's homotopy characterization of H-spaces and loop spaces. At the beginning of the 1990s, a similar structure was introduced by Fukaya in his categorification of Floer homology in symplectic topology. This structure plays a fundamental role in the celebrated homological mirror symmetry proposal by Kontsevich and in more recent developments of symplectic topology.
A detailed construction of A-infinity algebra structure attached to a closed Lagrangian submanifold is given in Fukaya, Oh, Ohta, and Ono's two-volume monograph Lagrangian Intersection Floer Theory (AMS-IP series 46 I & II), using the theory of Kuranishi structures?a theory that has been regarded as being not easily accessible to researchers in general. The present lecture note is provided by one of the main contributors to the Lagrangian Floer theory and is intended to provide a quick, reader-friendly explanation of the geometric part of the construction. Discussion of the Kuranishi structures is minimized, with more focus on the calculations and applications emphasizing the relevant homological algebra in the filtered context.
The book starts with a quick explanation of Stasheff polytopes and their two realizations?one by the rooted metric ribbon trees and the other by the genus-zero moduli space of open Riemann surfaces?and an explanation of the A-infinity structure on the motivating example of the based loop space. It then provides a description of the moduli space of genus-zero bordered stable maps and continues with the construction of the (curved) A-infinity structure and its canonical models. Included in the explanation are the (Landau?Ginzburg) potential functions associated with compact Lagrangian submanifolds constructed by Fukaya, Oh, Ohta, and Ono. The book explains calculations of potential functions for toric fibers in detail and reviews several explicit calculations in the literature of potential functions with bulk as well as their applications to problems in symplectic topology via the critical point theory thereof. In the Appendix, the book also provides rapid summaries of various background materials such as the stable map topology, Kuranishi structures, and orbifold Lagrangian Floer theory.
Based Loop Space and A8 Space.- A8 Algebras and Modules: Unfiltered Case.- Obstruction-Deformation Theory of Filtered A8 Bimodules.- Symplectic Geometry and Hamiltonian Dynamics.- Analysis of Pseudoholomorphic Curves and Bordered Stable Maps.- Critical Points of Potential Functions and Floer Cohomology.- Filtered Fukaya Category and its Bulk Deformations.
Format: Paperback / softback, 490 pages, height x width: 235x155 mm, XVIII, 490 p., 1
Pub. Date: 16-Mar-2024
ISBN-13: 9783031192951
This book is an English translation of an entirely revised version of the 1958 edition of the eighth chapter of the book Algebra, the second Book of the Elements of Mathematics.
It is devoted to the study of certain classes of rings and of modules, in particular to the notions of Noetherian or Artinian modules and rings, as well as that of radical.
This chapter studies Morita equivalence of module and algebras, it describes the structure of semisimple rings. Various Grothendieck groups are defined that play a universal role for module invariants.
The chapter also presents two particular cases of algebras over a field. The theory of central simple algebras is discussed in detail; their classification involves the Brauer group, of which several
descriptions are given. Finally, the chapter considers group algebras and applies the general theory to representations of finite groups.
At the end of the volume, a historical note taken from the previous edition recounts the evolution of many of the developed notions.
Artinian Modules and Noetherian Modules.- The Structure of Modules of Finite Length.- Simple Modules.- Semisimple Modules.- Commutation.- Morita Equivalence of Modules and Algebras.- Simple Rings.- Semisimple Rings.- Radical.- Modules over an Artinian Ring.- Grothendieck Groups.- Tensor Products of Semisimple Modules.- Absolutely Semisimple Algebras.- Central Simple Algebras.- Brauer Groups.- Other Descriptions of the Brauer Group.- Reduced Norms and Traces.- Simple Algebras over a Finite Field.- Quaternion Algebras.- Linear Representations of Algebras.- Linear Representations of Finite Groups.- Algebras without Unit Element.- Determinants over a Noncommunitative Field.- Hilbert's Nullstellensatz.- Trace of an Endomorphism of Finite Rank.- Historical Note.- Bibliography.- Notation Index.- Terminology Index.
Format: Paperback / softback, 370 pages, height x width: 235x155 mm,
7 Illustrations, color; 113 Illustrations, black and white; X, 370 p. 119 illus., 6 illus. in color.,
Series: Springer Undergraduate Mathematics Series
Pub. Date: 04-Jul-2024
ISBN-13: 9783031585128
This textbook offers a hands-on introduction to general topology, a fundamental tool in mathematics and its applications. It provides solid foundations for further study in mathematics in general, and topology in particular.
Aimed at undergraduate students in mathematics with no previous exposure to topology, the book presents key concepts in a mathematically rigorous yet accessible manner, illustrated by numerous examples. The essential feature of the book is the large sets of worked exercises at the end of each chapter. All of the basic topics are covered, namely, metric spaces, continuous maps, homeomorphisms, connectedness, and compactness. The book also explains the main constructions of new topological spaces such as product spaces and quotient spaces. The final chapter makes a foray into algebraic topology with the introduction of the fundamental group.
Thanks to nearly 300 solved exercises and abundant examples, Point-Set Topology is especially suitable for supplementing a first lecture course on topology for undergraduates, and it can also be utilized for independent study. The only prerequisites for reading the book are familiarity with mathematical proofs, some elements of set theory, and a good grasp of calculus.
Introduction.- Topological spaces.- Proximity on a topological
space.- Metric spaces.- Continuity.- Homeomorphisms and topological
invariants.- Product topology.- Connectedness.- Compactness.- Quotient
topology.- The fundamental group.
Format: Hardback, 310 pages, height x width: 235x155 mm, 30 Illustrations,
color; 57 Illustrations, black and white; XIV, 314 p. 65 illus. With online files/update.,
Series: Undergraduate Texts in Mathematics
Pub. Date: 04-Jun-2024
ISBN-13: 9783031559631
Monte Carlo Methods are among the most used, and useful, computational tools available today. They provide efficient and practical algorithms to solve a wide range of scientific and engineering problems in dozens of areas many of which are covered in this text. These include simulation, optimization, finance, statistical mechanics, birth and death processes, Bayesian inference, quadrature, gambling systems and more.
This text is for students of engineering, science, economics and mathematics who want to learn about Monte Carlo methods but have only a passing acquaintance with probability theory. The probability needed to understand the material is developed within the text itself in a direct manner using Monte Carlo experiments for reinforcement. There is a prerequisite of at least one year of calculus and a semester of matrix algebra.
Each new idea is carefully motivated by a realistic problem, thus leading to insights into probability theory via examples and numerical simulations. Programming exercises are integrated throughout the text as the primary vehicle for learning the material. All examples in the text are coded in Python as a representative language; the logic is sufficiently clear so as to be easily translated into any other language. Further, Python scripts for each worked example are freely accessible for each chapter. Along the way, most of the basic theory of probability is developed in order to illuminate the solutions to the questions posed. One of the strongest features of the book is the wealth of completely solved example problems. These provide the reader with a sourcebook to follow towards the solution of their own computational problems. Each chapter ends with a large collection of homework problems illustrating and directing the material.
This book is suitable as a textbook for students of engineering, finance, and the sciences as well as mathematics. The problem-oriented approach makes it ideal for an applied course in basic probability as well as for a more specialized course in Monte Carlo Methods. Topics include probability distributions, probability calculations, sampling, counting combinatorial objects, Markov chains, random walks, simulated annealing, genetic algorithms, option pricing, gamblers ruin, statistical mechanics, random number generation, Bayesian Inference, Gibbs Sampling and Monte Carlo integration.
1. Introduction to Monte Carlo Methods.-
2. Some Probability Distributions and Their Uses.-
3. Markov Chain Monte Carlo.-
4. Random Walks.-
5. Optimization by Monte Carlo Methods.-
6. More on Markov Chain Monte Carlo.- A. Generating Uniform Random Numbers.- B. Perron Frobenius Theorem.- C. Kelly Allocation for Correlated Investments.- D. Donsker's Theorem.- E. Projects.- References.- List of Notation.- Code Index.
Format: Hardback, 202 pages, height x width: 235x155 mm, 8 Illustrations, color;
22 Illustrations, black and white; XII, 200 p. 11 illus., 7 illus. in color.,
Series: Abel Symposia 17
Pub. Date: 02-Jul-2024
ISBN-13: 9783031577888
In recent years, triangulated categories have proved very successful as a common mathematical framework for formulating important advances in various fields, and at the same time for the interaction between different subject areas. The purpose of the symposium was therefore not only the study of triangulated categories in itself, but rather fruitful exchanges between disciplines. The symposium brought together established researchers who have made important contributions involving triangulated categories. Many participants came from representation theory, but there were also participants with backgrounds in commutative algebra, geometry and algebraic topology.
- Descent in Tensor Triangular Geometry.- An Introduction to Mackey and
Green 2-Functors.- Local Dualisable Objects in Local Algebra.- The
Donovan--Wemyss Conjecture via the Derived Auslander--Iyama Correspondence.-
A Brief Introduction to the Q-Shaped Derived Category.- Completions of
Triangulated Categories.- Obstructions to the Existence of Bounded
TStructures.- Some Applications of Extriangulated Categories.- Homological
Residue Fields as Comodules over Coalgebras.