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Some probabilistic restatements of the four color conjecture
 Journal of Graph Theory
, 2004
"... With every triangulation of sphere we associate in a natural way a probabilistic space and define several random events. The Four Color Conjecture turns out to be equivalent to different statements about positive correlation among some pairs of these events. c ○ (2003) John Wiley & Sons, Inc. 1. ..."
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With every triangulation of sphere we associate in a natural way a probabilistic space and define several random events. The Four Color Conjecture turns out to be equivalent to different statements about positive correlation among some pairs of these events. c ○ (2003) John Wiley & Sons, Inc. 1.
The Four Colour Theorem as a possible corollary of binomial summation
, 1998
"... The Four Colour Conjecture is reformulated as a statement about nondivisibility of certain binomial coefficients. This reformulation opens a (hypothetical) way of proving the Four Colour Theorem by taking advantage of recent progress in finding closed forms for binomial summations. 1 ..."
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The Four Colour Conjecture is reformulated as a statement about nondivisibility of certain binomial coefficients. This reformulation opens a (hypothetical) way of proving the Four Colour Theorem by taking advantage of recent progress in finding closed forms for binomial summations. 1
2003) "Financial Communities
"... ABSTRACT. Whereas many studies in finance have examined and established a strong link between stock returns and information, the physical mechanics of this link have been relatively unexplored. With the advent of stock message boards, it has become feasible to look more closely at the group process ..."
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ABSTRACT. Whereas many studies in finance have examined and established a strong link between stock returns and information, the physical mechanics of this link have been relatively unexplored. With the advent of stock message boards, it has become feasible to look more closely at the group process by which information impacts prices and vice versa. This paper utilizes a large universe of messages posted to stock market discussion forums to understand how opinions are linked across tickers during small investor discussion. We define a collective information unit, the financial community. These are clusters of tickers sharing and accessing the same information generators. Graph theoretic techniques are used to detect financial communities and to summarize their properties. Community stocks display connectedness, and we find that the greater the connectedness in a financial community, the greater the covariance of returns within the community as opposed to that amongst stocks that are not part of a major financial community. Highly connected stocks, on average, have lower return variance and higher mean returns. Using eigenvector techniques, we detect stocks that are hubs for information flow, using a measure known as centrality. We find that stocks with high centrality scores tend to have greater average covariance with other stocks than those with low scores. Our analysis of connectedness and centrality establishes a link between one arena of the information generation process and stock return correlations. 1.
One Probabilistic Equivalent of the Four Color Conjecture
, 2003
"... For every twoconnected planar threevalent graph we introduce in a natural way a probabilistic space and define two random events; the Four Color Conjecture turns out to be eqivalent to (positive) correlation of these events. ..."
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For every twoconnected planar threevalent graph we introduce in a natural way a probabilistic space and define two random events; the Four Color Conjecture turns out to be eqivalent to (positive) correlation of these events.
Graph Planarity and Related Topics
 GRAPH DRAWING (PROC. GD ’99)
, 1999
"... This compendium is the result of reformatting and minor editing of the author's transparencies for his talk delivered at the conference. The talk covered Euler's Formula, Kuratowski's Theorem, linear planarity tests, Schnyder's Theorem and drawing on the grid, the two paths problem, Pfaffian ori ..."
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This compendium is the result of reformatting and minor editing of the author's transparencies for his talk delivered at the conference. The talk covered Euler's Formula, Kuratowski's Theorem, linear planarity tests, Schnyder's Theorem and drawing on the grid, the two paths problem, Pfaffian orientations, linkless embeddings, and the Four Color Theorem.
TemperleyLieb Algebras And The FourColor Theorem
"... The TemperleyLieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1, e0, el,..., en, where the generators satisfy the rela 2 = 2el, eiejei = ei if [i j[ = 1 and eiej = ejei if [i j[ > 2. We tions i  use the Four Color Theorem to give a necessary and sufficient con ..."
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The TemperleyLieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1, e0, el,..., en, where the generators satisfy the rela 2 = 2el, eiejei = ei if [i j[ = 1 and eiej = ejei if [i j[ > 2. We tions i  use the Four Color Theorem to give a necessary and sufficient condition for certain elements of Tn to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.
Exercises
"... Topics: History, equivalent formulations and an outline of a proof. What are the prospects for finding a computerfree proof? Recommended reading: [1, 2, 3] ..."
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Topics: History, equivalent formulations and an outline of a proof. What are the prospects for finding a computerfree proof? Recommended reading: [1, 2, 3]
Graph Theory in Isabelle/Isar
, 2002
"... Contents 1 Formalization of graphs 3 1.1 De nition of graphs . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Properties of the representing set . . . . . . . . . . . . . . . . 3 1.3 Fundamental properties of graphs . . . . . . . . . . . . . . . . 4 1.4 Basis Graph Operations . . . . . . . . . ..."
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Contents 1 Formalization of graphs 3 1.1 De nition of graphs . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Properties of the representing set . . . . . . . . . . . . . . . . 3 1.3 Fundamental properties of graphs . . . . . . . . . . . . . . . . 4 1.4 Basis Graph Operations . . . . . . . . . . . . . . . . . . . . . 6 1.4.1 Size of a graph . . . . . . . . . . . . . . . . . . . . . . 6 1.4.2 Wellfounded induction on the size of a graph . . . . . 6 1.4.3 Reverse induction . . . . . . . . . . . . . . . . . . . . 7 1.4.4 Empty graph . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.5 Insert edge . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.6 Delete edge . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.7 Delete vertex . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.8 union and dierence of two graphs . . . . . . . . . . . 10 1.4.9 Induction theorem . . . . . . . . . . . . . . . . . . . . 10 1.4.10 Subgraph . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.1
Some Probabilistic Restatements of the Four Color Conjecture
, 2003
"... With every triangulation of sphere we associate in a natural way a probabilistic space and define several random events. The Four Color Conjecture turns out to be equivalent to di#erent statements about positive correlation among some pairs of these events. ..."
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With every triangulation of sphere we associate in a natural way a probabilistic space and define several random events. The Four Color Conjecture turns out to be equivalent to di#erent statements about positive correlation among some pairs of these events.