Introduction to Combinatorial Designs  book cover
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2nd Edition

Introduction to Combinatorial Designs




ISBN 9781584888383
Published May 17, 2007 by Chapman and Hall/CRC
327 Pages - 42 B/W Illustrations

 
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Book Description

Combinatorial theory is one of the fastest growing areas of modern mathematics. Focusing on a major part of this subject, Introduction to Combinatorial Designs, Second Edition provides a solid foundation in the classical areas of design theory as well as in more contemporary designs based on applications in a variety of fields.

After an overview of basic concepts, the text introduces balanced designs and finite geometries. The author then delves into balanced incomplete block designs, covering difference methods, residual and derived designs, and resolvability. Following a chapter on the existence theorem of Bruck, Ryser, and Chowla, the book discusses Latin squares, one-factorizations, triple systems, Hadamard matrices, and Room squares. It concludes with a number of statistical applications of designs.

Reflecting recent results in design theory and outlining several applications, this new edition of a standard text presents a comprehensive look at the combinatorial theory of experimental design. Suitable for a one-semester course or for self-study, it will prepare readers for further exploration in the field.

To access supplemental materials for this volume, visit the author’s website at http://www.math.siu.edu/Wallis/designs

Table of Contents

Basic Concepts
Combinatorial Designs
Some Examples of Designs
Block Designs
Systems of Distinct Representatives
Balanced Designs
Pairwise Balanced Designs
Balanced Incomplete Block Designs
Another Proof of Fisher’s Inequality
t-Designs
Finite Geometries
Finite Affine Planes
Finite Fields
Construction of Finite Affine Geometries
Finite Projective Geometries
Some Properties of Finite Geometries
Ovals in Projective Planes
The Desargues Configuration
Difference Sets and Difference Methods
Difference Sets
Construction of Difference Sets
Properties of Difference Sets
General Difference Methods
Singer Difference Sets
More about Block Designs
Residual and Derived Designs
Resolvability
The Main Existence Theorem
Sums of Squares
The Bruck–Ryser–Chowla Theorem
Another Proof
Latin Squares
Latin Squares and Subsquares
Orthogonality
Idempotent Latin Squares
Transversal Designs
More about Orthogonality
Spouse-Avoiding Mixed Doubles Tournaments
Three Orthogonal Latin Squares
Bachelor Squares
One-Factorizations
Basic Ideas
The Variability of One-Factorizations
Starters
Applications of One-Factorizations
An Application to Finite Projective Planes
Tournament Applications of One-Factorizations
Tournaments Balanced for Carryover
Steiner Triple Systems
Construction of Triple Systems
Subsystems
Simple Triple Systems
Cyclic Triple Systems
Large Sets and Related Designs
Kirkman Triple Systems and Generalizations
Kirkman Triple Systems
Kirkman Packings and Coverings
Hadamard Matrices
Basic Ideas
Hadamard Matrices and Block Designs
Further Hadamard Matrix Constructions
Regular Hadamard Matrices
Equivalence
Room Squares
Definitions
Starter Constructions
Subsquare Constructions
The Existence Theorem
Howell Rotations
Further Applications of Design Theory
Statistical Applications
Information and Cryptography
Golf Designs
References
ANSWERS AND SOLUTIONS
INDEX

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