Analytic Combinatorics

ISBN-10: 0521898064
ISBN-13: 9780521898065
Edition: 2009
List price: $113.00 Buy it from $109.04
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Description: Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and  More...

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Book details

List price: $113.00
Copyright year: 2009
Publisher: Cambridge University Press
Publication date: 1/15/2009
Binding: Hardcover
Pages: 826
Size: 7.00" wide x 9.75" long x 1.75" tall
Weight: 1.012
Language: English

Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Preface
An Invitation to Analytic Combinatorics
Symbolic Methods
Combinatorial Structures and Ordinary Generating Functions
Symbolic enumeration Methods
Admissible constructions and specifications
Integer compositions and partitions
Words and regular languages
Tree structures
Additional constructions
Perspective
Labelled Structures and Exponential Generating Functions
Labelled classes
Admissible labelled constructions
Surjections, set partitions, and words
Alignments, permutations, and related structures
Labelled trees, mapping, and graphs
Additional constructions
Perspective
Combinatorial Parameters and Multivariate Generating Functions
An introduction to bivariate generating functions (BGFs)
Bivariate generating functions and probability distributions
Inherited parameters and ordinary MGFs
Inherited parameters and exponential MGFs
Recursive parameters
Complete generating functions and discrete models
Additional constructions
Extremal parameters
Perspective
Complex Asymptotics
Complex Analysis, Rational and Meromorphic Asymptotics
Generating functions as analytic objects
Analytic functions and meromorphic functions
Singularities and exponential growth of coefficients
Closure properties and computable bounds
Rational and meromorphic functions
Localization of singularities
Singularities and functional equations
Perspective
Applications Of Rational and Meromorphic Asymptotics
A roadmap to rational and meromorphic asymptotics
The supercritical sequence schema
Regular specifications and languages
Nested sequences, lattice paths, and continued fractions
Paths in graphs and automata
Transfer matrix models
Perspective
Singularity Analysis Of Generating Functions
A glimpse of basic singularity analysis theory
Coefficient asymptotics for the standard scale
Transfers
The process of singularity analysis
Multiple singularities
Intermezzo: functions amenable to singularity analysis
Inverse functions
Polylogarithms
Functional composition
Closure properties
Tauberian theory and Darboux's method
Perspective
Applications of Singularity Analysis
A roadmap to singularity analysis asymptotics
Sets and the exp-log schema
Simple varieties of trees and inverse functions
Tree-like structures and implicit functions
Unlabelled non-plane trees and P�lya operators
Irreducible context-free structures
The general analysis of algebraic functions
Combinatorial applications of algebraic functions
Ordinary differential equations and systems
Singularity analysis and probability distributions
Perspective
Saddle-Point Asymptotics
Landscapes of analytic functions and saddle-points
Saddle-point bounds
Overview of the saddle-point method
Three combinatorial examples
Admissibility
Integer partitions
Saddle-points and linear differential equations
Large powers
Saddle-points and probability distributions
Multiple saddle-points
Perspective
Random Structures
Multivariate Asymptotics and Limit Laws
Limit laws and combinatorial structures
Discrete limit laws
Combinatorial instances of discrete laws
Continuous limit laws
Quasi-powers and Guassian limit laws
Perturbation of meromorphic asymptotics
Pertubation of singularity analysis asymptotics
Perturbation of saddle-point asymptotics
Local limit laws
Large deviations
Non-Gaussian continuous limits
Multivariate limit laws
Perspective
Appendices
Auxiliary Elementary Notions
Arithmetical functions
Asymptotic notations
Combinatorial probability
Cycle construction
Formal power series
Lagrange inversion
Regular languages
Stirling numbers.
Tree Concepts
Basic Complex Analysis
Algebraic elimination
Equivalent definitions of analyticity
Gamma function
Holonomic functions
Implicit Function Theorem
Laplace's method
Mellin transforms
Several complex variables
Concepts Of Probability Theory
Probability spaces and measure
Random variables
Transforms of distributions
Special distributions
Convergence in law
Bibliography
Index

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