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Unity of Combinatorics [Pehme köide]

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  • Formaat: Paperback / softback, 353 pages
  • Sari: Carus Mathematical Monographs
  • Ilmumisaeg: 30-May-2020
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470465094
  • ISBN-13: 9781470465094
Teised raamatud teemal:
Unity of CombinatoricsSuurem pilt
  • Formaat: Paperback / softback, 353 pages
  • Sari: Carus Mathematical Monographs
  • Ilmumisaeg: 30-May-2020
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470465094
  • ISBN-13: 9781470465094
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470465094.html
Combinatorics, or the art and science of counting, is a vibrant and active area of pure mathematical research with many applications. The Unity of Combinatorics succeeds in showing that the many facets of combinatorics are not merely isolated instances of clever tricks but that they have numerous connections and threads weaving them together to form a beautifully patterned tapestry of ideas. Topics include combinatorial designs, combinatorial games, matroids, difference sets, Fibonacci numbers, finite geometries, Pascal's triangle, Penrose tilings, error-correcting codes, and many others. Anyone with an interest in mathematics, professional or recreational, will be sure to find this book both enlightening and enjoyable.

Few mathematicians have been as active in this area as Richard Guy, now in his eighth decade of mathematical productivity. Guy is the author of over 300 papers and twelve books in geometry, number theory, graph theory, and combinatorics. In addition to being a life-long number-theorist and combinatorialist, Guy's co-author, Ezra Brown, is a multi-award-winning expository writer. Together, Guy and Brown have produced a book that, in the spirit of the founding words of the Carus book series, is accessible ""not only to mathematicians but to scientific workers and others with a modest mathematical background.
Preface ix
Credits and Permissions xiii
Introduction 1(4)
Chapter 1 Blocks, sequences, bow ties, and worms
5(16)
1.1 Langford sequences
5(2)
1.2 Partitioning sets of integers
7(5)
1.3 Penrose tilings
12(9)
Chapter 2 Combinatorial games
21(18)
2.1 Wythoffs game
21(1)
2.2 Combinatorial games: rules and examples
22(3)
2.3 Developing strategies: P-positions and N-positions
25(2)
2.4 Nim, nimbers, and the Sprague-Grundy Theorem
27(8)
2.5 Nim arithmetic and Nim algebra
35(4)
Chapter 3 Fibonacci, Pascal, and Catalan
39(22)
3.1 Fibonacci numbers
39(5)
3.2 The triangle of Pingala, Al Karaji, Yang Hui, and Pascal
44(10)
3.3 The Catalan numbers and the central column of Pascal's triangle
54(7)
Chapter 4 Catwalks, Sandsteps, and Pascal pyramids
61(16)
Chapter 5 Unique rook circuits
77(12)
Appendix
86(3)
Chapter 6 Sums, colorings, squared squares, and packings
89(18)
6.1 Triples satisfying x + y = z
89(2)
6.2 Coil diagrams and the Ringel-Youngs Theorem
91(3)
6.3 Squaring the square
94(3)
6.4 Euler's polyhedral formula
97(2)
6.5 Packings and coverings of the complete graph
99(4)
6.6 Steiner triple systems
103(4)
Chapter 7 Difference sets and combinatorial designs
107(16)
7.1 Difference sets
107(6)
7.2 Multipliers
113(2)
7.3 Difference sets, de Bruijn cycles, and de Bruijn graphs
115(2)
7.4 Block designs
117(6)
Chapter 8 Geometric connections
123(32)
8.1 A quick tour of projective geometry
123(9)
8.2 Finite projective geometries and Singer designs
132(5)
8.3 Examples: n - 2, q = 2 and 3
137(3)
8.4 Affine planes and magic squares
140(2)
8.5 Heawood's map on the torus revisited
142(2)
8.6 (7,3,1) and Nim
144(4)
8.7 The automorphism group of the Fano plane
148(7)
Chapter 9 The groups PSL(2,7) and GL(3,2) and why they are isomorphic
155(10)
9.1 The group GL(3,2)
157(2)
9.2 The group P5L(2,7)
159(2)
9.3 Constructing an isomorphism of PSL(2,7) onto GL(3,2)
161(4)
Chapter 10 Incidence matrices, codes, and sphere packings
165(22)
10.1 Introducing incidence matrices
165(2)
10.2 Error-correcting codes
167(7)
10.3 Sphere packing
174(7)
10.4 Hadamard matrices and Hadamard difference sets
181(2)
10.5 Hadamard matrices and projective geometries
183(4)
Chapter 11 Kirkman's schoolgirls, fields, spreads, and hats
187(22)
11.1 Kirkman's Schoolgirls Problem
187(1)
11.2 Fifteen young ladies at school
188(1)
11.3 Resolvable block designs and Kirkman triple systems
189(2)
11.4 Kirkman's schoolgirls and difference sets
191(3)
11.5 K = Q (√2, √3, √5, √7) and the designs it contains
194(5)
11.6 Spreads in PG(3,F2) and the geometry of Kirkman
199(3)
11.7 Fifteen schoolgirls, fifteen hats, and coding theory
202(3)
11.8 Questions
205(4)
Chapter 12 (7,3,1) and combinatorics
209(8)
12.1 (7,3,1) and the Heawood graph
210(1)
12.2 (7,3,1) and Latin squares
211(1)
12.3 (7,3,1) and round-robin tournaments
212(5)
Chapter 13 (7,3,1) and normed algebras
217(12)
13.1 Sums of squares
217(3)
13.2 The quaternions and the octonions
220(5)
13.3 Beyond the octonions
225(4)
Chapter 14 (7,3,1) and matroids
229(14)
14.1 Why matroids?
230(1)
14.2 Declaration of (in)dependence
230(4)
14.3 Thus, matroids
234(5)
14.4 Matroids and greed
239(4)
Chapter 15 Coin-turning games and Mock Turtles
243(14)
15.1 A review of some combinatorial game theory
243(2)
15.2 Turning Turtles
245(2)
15.3 Turning Corners: coins on a grid
247(3)
15.4 Mock Turtles: more turtles in a line
250(2)
15.5 More about turtle-turning games
252(5)
Chapter 16 The (11,5,2) biplane, codes, designs, and groups
257(22)
16.1 "How do you make math exciting for students?"
257(2)
16.2 Difference sets, block designs, and biplanes
259(2)
16.3 The automorphism group of the biplane
261(4)
16.4 Incidence matrices, revisited
265(1)
16.5 Error-correcting codes
266(4)
16.6 Steiner systems
270(4)
16.7 Automorphisms, transitivity, simplicity, and the Mathieu groups
274(5)
Chapter 17 Rick's Tricky Six Puzzle: More than meets the eye
279(32)
17.1 Sliding-block puzzles
279(3)
17.2 What is the exception?
282(1)
17.3 Not much of a puzzle?
283(3)
17.4 What is the automorphism group of the Tricky Six Puzzle?
286(1)
17.5 Two different group actions
287(7)
17.6 The projective plane of order 4
294(5)
17.7 Buy one, get several free!
299(4)
17.8 The Hoffman-Singleton graph
303(2)
17.9 The Steiner system S(5,6,12)
305(2)
17.10 A (12,132,4) binary code and Golay's ternary code Su
307(2)
17.11 Conclusions
309(2)
Chapter 18 S(5,8,24)
311(6)
Chapter 19 The Miracle Octad Generator
317(12)
19.1 The Miracle Octad Generator (MOG)
318(1)
19.2 An elementary approach
319(5)
19.3 A more mathematical approach
324(5)
Bibliography 329(10)
Index 339
Ezra Brown, Virginia Polytechnic Institute and State University, Blacksburg, VA.

Richard K. Guy, University of Calgary, AB, Canada.