Inevitable randomness in discrete mathematics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2009]
|
| Series: | University lecture series
Volume 49 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xi, 250 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821847565 |
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MARC
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| 100 | 1 | |a Beck, József |d 1952- |e Verfasser |0 (DE-588)1105583325 |4 aut | |
| 245 | 1 | 0 | |a Inevitable randomness in discrete mathematics |c József Beck |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2009] | |
| 264 | 4 | |c © 2009 | |
| 300 | |a xi, 250 Seiten |b Illustrationen, Diagramme | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
ix
Part A. Reading the shadows on the wall and formulating a vague
conjecture
1
Chapter
1.
Complex systems
3
1.
Order and Disorder
3
2.
Ideal gases and the Equiprobability Postulate
4
3.
Apparent randomness of primes and the Riemann Hypothesis
6
4.
Zoo of zeta-functions
10
Chapter
2.
Collecting data: Apparent randomness of digit sequences
13
1.
Normal numbers
13
2.
Continued fraction
14
3.
Equidistribution and continued fraction
16
4.
More on continued fraction and diophantine approximation
17
Chapter
3.
Collecting data: More randomness in number theory
21
1.
The Twin Prime Conjecture and Independence
21
2.
Finite fields and the congruence Riemann Hypothesis
23
3.
Randomness in the two classical lattice point counting problems
24
4.
The 3n
+ 1
Conjecture
27
5.
Primes represented by individual quadratic forms
28
6.
Continued fraction: The length of the period for quadratic irrationals
34
Chapter
4.
Laplace and the Principle of Insufficient Reason
37
1.
Introduction
37
2.
Randomness and probability
40
3.
Complexity and randomness of individual sequences
43
4.
Formulating a vague probabilistic conjecture
44
5.
Limitations of the SLG Conjecture
47
Chapter
5.
Collecting proofs for the SLG Conjecture
49
1.
When independence is more or less plausible
49
2.
Another Central Limit Theorem: Randomness of the square root of
2 53
3.
Problems without apparent independence: Inevitable irregularities
—
an illustration of the Solid-Liquid-Gas Conjecture
58
vi
CONTENTS
Part B.
More evidence for the SLG Conjecture: Exact solutions in
real game theory
67
Chapter
6.
Ramsey Theory and Games
69
1.
The usual quick jump from easy to hard
69
2.
A typical hard problem: Ramsey Numbers. A case of Inaccessible
Data
71
3.
Another hard problem: Ramsey Games
74
4.
Weak Ramsey Games: Here we know the right order of magnitude!
76
5.
Proof of the lower bound in
(6.10) 77
6.
An interesting detour: Extremal Hypergraphs of the
Erdős-Selfridge
theorem and the Move Number
81
7.
Concluding note on off-diagonal Ramsey Numbers
86
Chapter
7.
Practice session (I): More on Ramsey Games and strategies
89
1.
Halving strategy
89
2.
Switching to the complete bipartite graph
_rrn,¡.
Completing the proof
of
(6.10) 92
3.
Understanding the threshold in
(6.10).
Random Play Intuition
93
4.
Move Number
94
5.
An interesting detour: Game vs. Ramsey
96
Chapter
8.
Practice session (II): Connectivity games and more strategies
99
1.
Lehman s theorem
99
2.
Erdős s
random graph intuition
101
3.
Forcing isolated points
102
4.
The
Chvátal-Erdős
proof: Quick greedy building
103
5.
Slow building via blocking: The Transversal Hypergraph Method
106
6.
Proof of Proposition
8.3 108
Chapter
9.
What kind of games? Ill
1.
Introduction 111
2.
The Tic-Tac-Toe family
113
3.
Where is the breaking point from draw to win? A humiliating gap in
our knowledge!
117
4.
First simplification: Replacing ordinary Win with Weak Win
118
Chapter
10.
Exact solutions of games: Understanding via the Equiprobability
Postulate
123
1.
Another simplification: Switching from Maker-Breaker games to
Cut-and-Choose games
123
2.
Sim and other Clique Games on graphs
125
3.
The concentration of random variables in general
126
4.
How does the Equiprobability Postulate enter real game theory?
129
5.
Rehabilitation of Laplace?
133
Chapter
11.
Equiprobability Postulate with Constraints (Endgame Policy)
135
1.
Introduction
135
2.
Modifying the Equiprobability Postulate with an Endgame Policy
136
3.
Going back to 1-dimensional goals
138
CONTENTS
vii
4.
Finding the correct form of the Biased Weak Win Conjecture when
Maker is the topdog
139
5.
Coalition Games
142
6.
Vague Equiprobability Conjecture
143
7.
Philosophical speculations on a probabilistic paradigm
145
Chapter
12.
Constraints and Threshold Clustering
147
1.
What are the constraints of ordinary win? What are the constraints
of Ramsey Theory?
147
2.
Delicate win or delicate draw? A wonderful question!
151
3.
Threshold Clustering
152
Chapter
13.
Threshold Clustering and a few bold conjectures
155
1.
Examples
155
2.
What to do next? Searching for simpler problems
161
Part C. New evidence: Games and Graphs, the Surplus, and the
Square Root Law
163
Chapter
14.
Yet another simplification: Sparse hypergraphs and the Surplus
165
1.
Row-Column Games
165
2.
Exact solutions
169
3.
The Core-Density and the Surplus
171
4.
Remarks
173
5.
Regular graphs
—
local randomness
174
6.
How sharp is Theorem
1? 175
Chapter
15.
Is Surplus the right concept? (I)
177
1.
Socialism does work on graphs!
177
2.
Do-It-First Lead
179
3.
Monopoly
179
4.
Shutout
180
5.
Inevitable Shutout
183
Chapter
16.
Is Surplus the right concept? (II)
185
1.
The Move Number
185
2.
Discrepancy and variance
188
3.
Summary
189
Chapter
17.
Working with a game-theoretic Partition Function
193
1.
Introduction
193
2.
The lower bound
195
3.
Some underdog versions of Proposition
17.3 197
Chapter
18.
An attempt to save the Variance
203
1.
Introduction
203
2.
An alternative approach
204
Chapter
19.
Proof of Theorem
1:
Combining the variance with an exponential
sum
209
1.
Defining a complicated potential function
209
viii CONTENTS
2.
Global balancing
212
3.
Average argument
215
Chapter
20.
Proof of Theorem
2:
The upper bound
219
1.
Can we use the Local Lemma in games?
219
2.
Danger function: Big-Game
&
small-game decomposition
220
Chapter
21.
Conclusion (I): More on Theorem
1 227
1.
Threshold Clustering: Generalizations of Theorem
1 227
2.
When threshold clustering fails: Shutout games
230
3.
Last remark about Theorem
1 233
Chapter
22.
Conclusion (II): Beyond the SLG Conjecture
237
1.
Wild speculations: Is it true that most unrestricted do-it-first games
are unknowable?
237
2.
Weak Win and Infinite Draw
240
Dictionary of the Phrases and Concepts
245
References
247
|
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| ctrlnum | (OCoLC)317361752 (DE-599)BVBBV035791237 |
| dewey-full | 519.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.3 |
| dewey-search | 519.3 |
| dewey-sort | 3519.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| spelling | Beck, József 1952- Verfasser (DE-588)1105583325 aut Inevitable randomness in discrete mathematics József Beck Providence, Rhode Island American Mathematical Society [2009] © 2009 xi, 250 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier University lecture series Volume 49 Game theory Random measures Zufälliges Maß (DE-588)4068049-6 gnd rswk-swf Kombinatorisches Spiel (DE-588)4164753-1 gnd rswk-swf Kombinatorisches Spiel (DE-588)4164753-1 s Zufälliges Maß (DE-588)4068049-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1644-7 University lecture series Volume 49 (DE-604)BV004153846 49 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018650569&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Beck, József 1952- Inevitable randomness in discrete mathematics University lecture series Game theory Random measures Zufälliges Maß (DE-588)4068049-6 gnd Kombinatorisches Spiel (DE-588)4164753-1 gnd |
| subject_GND | (DE-588)4068049-6 (DE-588)4164753-1 |
| title | Inevitable randomness in discrete mathematics |
| title_auth | Inevitable randomness in discrete mathematics |
| title_exact_search | Inevitable randomness in discrete mathematics |
| title_full | Inevitable randomness in discrete mathematics József Beck |
| title_fullStr | Inevitable randomness in discrete mathematics József Beck |
| title_full_unstemmed | Inevitable randomness in discrete mathematics József Beck |
| title_short | Inevitable randomness in discrete mathematics |
| title_sort | inevitable randomness in discrete mathematics |
| topic | Game theory Random measures Zufälliges Maß (DE-588)4068049-6 gnd Kombinatorisches Spiel (DE-588)4164753-1 gnd |
| topic_facet | Game theory Random measures Zufälliges Maß Kombinatorisches Spiel |
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| work_keys_str_mv | AT beckjozsef inevitablerandomnessindiscretemathematics |