Series: Universitext
2014, XX, 170 p. 8 illus. in color.
ISBN 978-3-319-07964-6
Due: July 31, 2014
Treats problems from four different mathematical disciplines, under the common theme of asymptotic limits and generous use of generating function techniques
Presented rigorously and with enough detail to allow the advanced undergraduate student to use it for independent study
Excellent for a seminar course in which the students present the lectures
The primary intent of the book is to introduce an array of beautiful problems in a variety of subjects quickly, pithily and completely rigorously to graduate students and advanced undergraduates. The book takes a number of specific problems and solves them, the needed tools developed along the way in the context of the particular problems. It treats a melange of topics from combinatorial probability theory, number theory, random graph theory and combinatorics. The problems in this book involve the asymptotic analysis of a discrete construct as some natural parameter of the system tends to infinity. Besides bridging discrete mathematics and mathematical analysis, the book makes a modest attempt at bridging disciplines. The problems were selected with an eye toward accessibility to a wide audience, including advanced undergraduate students. The book could be used for a seminar course in which students present the lectures.
Content Level â Upper undergraduate
Keywords â Asymptotic problems - Combinatorics - Graph theory - Number theory - Probability
Related subjects â Number Theory and Discrete Mathematics - Probability Theory and Stochastic Processes
Partitions With Restricted Summands or "The Money Changing Problem".- The Asymptotic Density of Relatively Prime Pairs and of Square-Free Numbers.- A One-Dimensional Probabilistic Packing Problem.- The Arcsine Laws for the One-Dimensional Simple Symmetric Random Walk.- The Distribution of Cycles in Random Permutations.- Chebyshev's Theorem on the Asymptotic Density of the Primes.- Mertens' Theorems on the Asymptotic Behavior of the Primes.- The Hardy-Ramanujan Theorem on the Number of Distinct Prime Divisors.- The Largest Clique in a Random Graph and Applications to Tampering Detection and Ramsey Theory.- The Phase Transition Concerning the Giant Component in a Sparse Random Graph?a Theorem of Erd?s and Renyi.
Series: Springer Monographs in Mathematics
2014, X, 308 p. 66 illus.
Hardcover
ISBN 978-3-319-08030-7
Due: August 31, 2014
Contains more than 350 exercises
Emphasizes ideas rather than technique, leaving technical details for the reader as exercises
Uses each of the three types of algebraic properties (Burnside, finite basis, growth) to study each of the other two
Combinatorial Algebra: Syntax and Semantics explores the foundations of the subject and many important applications. With over 350 exercises at various levels of difficulty and with hints for the more difficult problems, this textbook aims to reach a wide and diversified audience; no prerequisites beyond standard courses in linear and abstract algebra are required, although a certain mathematical maturity (the ability to understand and produce complicated proofs) will help the reader.
The five chapters cover classical results that formed the core of modern combinatorial algebra, including Burnside-type problems, growth and identities of groups, semigroups and associative algebras as well as relatively recent breakthroughs, including the road coloring problem. In this self-contained exposition, the author delves into the deep connections between syntactic and semantic methods in combinatorial algebra. Many proofs emphasize several ``universal" tools, such as trees, subshifts, uniformly recurrent words, and so on.
The broad appeal of this textbook extends to a variety of student levels: from advanced high-schoolers to undergraduates and graduate students, including those in search of a Ph.D. thesis who will benefit from the gFurther reading and open problemsh sections at the end of Chapters 2?5. Additionally, this text opens a path to self-study, engaging those beyond the classroom setting: researchers, instructors, students, virtually anyone who wishes to learn and better understand this important area of mathematics.
Content Level â Graduate
Keywords â Burnside-type problems - Novikov?Adian theorem - amenable groups - combinatorics on words - semigroups - symbolic dynamics
Related subjects â Algebra - Mathematics
Introduction.- 1. Main definitions and basic fact.- 2. Words that can be avoided.- 3. Semigroups.- 4. Rings.- 5. Groups.- Bibliography.- Index.
Series: Algebra and Applications, Vol. 19
2014, XX, 707 p. 59 illus.
Hardcover
ISBN 978-3-319-07967-7
Due: July 31, 2014
Provides full proofs of key statements in the modular representation theory of groups
Contains a coherent treatment and full proofs of the main results on equivalences between derived categories
Introduces stable categories and different types of equivalences between them as well as their respective invariants
Is completely self-contained and only assumes a basic knowledge of algebra
Introducing the representation theory of groups and finite dimensional algebras, this book first studies basic non-commutative ring theory, covering the necessary background of elementary homological algebra and representations of groups to block theory.
It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field.
Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given ? such as the structure of blocks of cyclic defect groups ? whenever appropriate. Overall, many methods from the representation theory of algebras are introduced.
Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields, and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.
Rings, Algebras and Modules.- Modular Representations of Finite Groups.- Abelian and Triangulated Categories.- Morita theory.- Stable Module Categories.- Derived Equivalences.
Series: Lecture Notes in Mathematics, Vol. 2113
2014, X, 138 p.
Softcover
ISBN 978-3-319-08152-6
Due: August 31, 2014
Contains an elementary classification of the arithmetic classes of three-dimensional crystallographic groups
Gives a clear construction, for a geometrically important class of groups, of the classifying spaces that are used in applications of the Farrell-Jones
isomorphism conjecture
Shows how the Farrell-Jones isomorphism theorem is used in computations, assembling all of the required methods
The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.
Content Level â Research
Keywords â 20H15,19B28,19A31,19D35 - Algebraic K-theory - Classifying spaces - Crystallographic groups - Farrell-Jones isomorphism conjecture
Related subjects â Algebra - Geometry & Topology
Series: Springer Monographs in Mathematics
2014, XVIII, 138 p. 51 illus., 29 illus. in color.
Hardcover
ISBN 978-1-4939-1374-9
Due: August 31, 2014
Dynamical networks is a hot topic
The first book to highlight the isospectral matrix reduction of networks and put into context of traditional networks literature
Static and dynamical properties of networks are covered
This book presents a new approach to the analysis of networks, which emphasizes how one can compress a network while preserving all information relative to the network's spectrum. This approach can be applied to any network irrespective of the network's structure or whether the network is directed, undirected, weighted, unweighted, etc. Besides these compression techniques, the authors introduce a number of other isospectral transformations and demonstrate how, together, these methods can be applied to gain new results in a number of areas. This includes the stability of time-delayed and non time-delayed dynamical networks, eigenvalue estimation, pseudospectra analysis, and the estimation of survival probabilities in open dynamical systems.
The theory of isospectral transformations, developed in this text, can be readily applied in any area that involves the analysis of multidimensional systems and is especially applicable to the analysis of network dynamics. This book will be of interest to mathematicians, physicists, biologists, engineers and to anyone who has an interest in the dynamics of networks.
Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks.- Isospectral Matrix Reductions.- Dynamical Networks and Isospectral Graph Reductions.- Stability of Dynamical Networks.- Improved Eigenvalue Estimates.- Pseudospectra and Inverse Pseudospectra.- Improved Estimates of Survival Probabilities