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El. knyga: Design Theory

(Auburn University, Alabama, USA), (Auburn University, Alabama, USA)
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In order to teach students some of the most important techniques used for constructing combinatorial designs, Lindner and Rodger (both Auburn U.) focus on several basic designs: Steiner triple systems, latin squares, and finite projective and affine plane. Having set these out, they start to add additional interesting properties that may be required, such as resolvability, embeddings, orthogonality, and even some more complicated structures such as Steiner quadruple systems. They do not mention a date for the first edition, but have added extensive material introducing embeddings, directed designs, the universal algebraic representations of designs, and intersection properties of designs. Check out their pictures, they say, which make the constructions more comprehensible to students. Answers are provided for selected problems. Annotation ©2009 Book News, Inc., Portland, OR (booknews.com)

Design Theory, Second Edition presents some of the most important techniques used for constructing combinatorial designs. It augments the descriptions of the constructions with many figures to help students understand and enjoy this branch of mathematics.

This edition now offers a thorough development of the embedding of Latin squares and combinatorial designs. It also presents some pure mathematical ideas, including connections between universal algebra and graph designs.

The authors focus on several basic designs, including Steiner triple systems, Latin squares, and finite projective and affine planes. They produce these designs using flexible constructions and then add interesting properties that may be required, such as resolvability, embeddings, and orthogonality. The authors also construct more complicated structures, such as Steiner quadruple systems.

By providing both classical and state-of-the-art construction techniques, this book enables students to produce many other types of designs.

Recenzijos

it is remarkable how quickly the book propels the reader from the basics to the frontiers of design theory Combined, these features make the book an excellent candidate for a design theory text. At the same time, even the seasoned researcher of triple systems will find this a useful resource.

Peter James Dukes (3-VCTR-MS; Victoria, BC), Mathematical Reviews, 2010

1 Steiner Triple Systems
1
1.1 The existence problem
1
1.2 upsilon identical to 3 (mod 6): The Bose Construction
4
1.3 upsilon identical to 1 (mod 6): The Skolem Construction
9
1.4 upsilon identical to 5 (mod 6): The 6n + 5 Construction
14
1.5 Quasigroups with holes and Steiner triple systems
17
1.5.1 Constructing quasigroups with holes
17
1.5.2 Constructing Steiner triple systems using quasigroups with holes
22
1.6 The Wilson Construction
27
1.7 Cyclic Steiner triple systems
31
1.8 The 2n + 1 and 2n + 7 Constructions
35
2 A-Fold Triple Systems
45
2.1 Triple systems of index λ greater than 1
45
2.2 The existence of idempotent latin squares
47
2.3 2-Fold triple systems
50
2.3.1 Constructing 2-fold triple systems
50
2.4 Mendelsohn triple systems
55
2.5 λ = 3 and 6
59
2.6 λ-Fold triple systems in general
62
3 Quasigroup Identities and Graph Decompositions
65
3.1 Quasigroup identities
65
3.2 Mendelsohn triple systems revisited
70
3.3 Steiner triple systems revisited
72
4 Maximum Packings and Minimum Coverings
77
4.1 The general problem
77
4.2 Maximum packings
82
4.3 Minimum coverings
87
5 Kirkman Triple Systems
95
5.1 A recursive construction
95
5.2 Constructing pairwise balanced designs
103
6 Mutually Orthogonal Latin Squares
119
6.1 Introduction
119
6.2 The Euler and MacNeish Conjectures
123
6.3 Disproof of the MacNeish Conjecture
135
6.4 Disproof of the Euler Conjecture
138
6.5 Orthogonal latin squares of order n identical to 2 (mod 4)
141
7 Affine and Projective Planes
155
7.1 Affine planes
155
7.2 Projective planes
157
7.3 Connections between affine and projective planes
159
7.4 Connection between affine planes and complete sets of MOLS
161
7.5 Coordinatizing the affine plane
165
8 Intersections of Steiner Triple Systems
169
8.1 Teirlinck's Algorithm
169
8.2 The general intersection problem
175
9 Embeddings
185
9.1 Embedding latin rectangles — necessary conditions
185
9.2 Edge-coloring bipartite graphs
187
9.3 Embedding latin rectangles: Ryser's Sufficient Conditions
191
9.4 Embedding idempotent commutative latin squares: Cruse's Theorem
194
9.5 Embedding partial Steiner triple systems
198
10 Steiner Quadruple Systems 207
10.1 Introduction
207
10.2 Constructions of Steiner Quadruple Systems
214
10.3 The Stern and Lenz Lemma
220
10.4 The (3upsilon — 2u)-Construction
229
Appendices 247
A Cyclic Steiner Triple Systems
249
B Answers to Selected Exercises
251
References 259
Index 263
Charles C. Lindner, Christopher A. Rodger
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