Modern Graph Theory

This research level monograph reflects the current state of the field and provides a reference for graduate students entering the field as well as for established researchers.

Set Theory: On the Structure of the Real Line

The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a sum- mary of recent invariant-measure characterizations. Then a new statisti- cal derivation of the law in the form of a CLT-like theorem for significant digits is presented. If distributions are selected at random (in any "unbi- ased" way) and random samples are then taken from each of these dis- tributions, the significant digits of the combined sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant-digit phenomenon in many different empir- ical contexts and helps justify its recent application to computer design, mathematical modelling and detection of fraud in accounting data.

/pdf/a-statistical-derivation-of-the-significant-digit-law-r7fqztvpcz.pdf

A Statistical Derivation of the Significant-Digit Law

Condorcet pensait que le but d'un vote est de faire un choix qui soit le meilleur pour la societe. Selon ce point de vue, il y a un choix qui est objectivement le meilleur, un autre vient en second et ainsi de suite. Pour l'etablissement d'une regle de vote, l'objectif devrait etre de choisir l'alternative qui est le plus probablement la meilleure. Condorcet resolut ce probleme en utilisant une forme d'estimation par le maximum de vraisemblance. La procedure qu'il en a tiree peut aussi etre justifiee a partir d'une perspective axiomatique moderne. C'est la seule fonction du choix social qui satisfasse une variante de l'axiome d'independance en meme temps que plusieurs autres proprietes classiques

Condorcet's theory of voting

Condcrcet's criterion states that an alternative that defeats every other by a simple majority is the socially optimal choice. Condorcet argued that if the object of voting is to determine the “best” decision for society but voters sometimes make mistakes in their judgments, then the majority alternative (if it exists) is statistically most likely to be the best choice. Strictly speaking, this claim is not true; in some situations Bordas rule gives a sharper estimate of the best alternative. Nevertheless, Condorcet did propose a novel and statistically correct rule for finding the most likely ranking of the alternatives. This procedure, which is sometimes known as “Kemeny's rule,” is the unique social welfare function that satisfies a variant of independence of irrelevant alternatives together with several other standard properties.

/pdf/condorcet-s-theory-of-voting-1r508yu416.pdf

Some combinatorial problems concerning uncountable cardinals

Boolean-valued models and independence proofs in set theory

Let us consider the following infinite game between two players, Empty and Nonempty. We are given a large set S. Empty opens the game by choosing a large subset S0 of S; then Nonempty chooses a large set S1 ⊆ S0; then Empty chooses large S2 ⊆ S, etc. The game is over after ω moves. If ⋂n=0xSn is empty then Empty wins, and if ⋂n=0∞Sn is nonempty then Nonempty wins.If “large” means “infinite”, then Empty can beat Nonempty rather easily: he chooses So countable, S0 = {a0, a1,…, an,…}, and then he chooses S2 such that a0 ∉ S2, S4 such that a1, ∉ S4 and so on.Next we assume that S is a set of uncountable cardinality, and that “large” means “of cardinality ∣S∣”. Then still Empty can win, but his winning strategy is somewhat more sophisticated: Let us identify S with a cardinal number κ. Thus each subset of S of size κ is a set of ordinals below κ. For each X ⊆ κ of size κ, let fx be the unique order-preserving mapping of X onto κ, and let F(X) = {x ϵ X: f(x) is a successor ordinal}. Empty's strategy is to play S0 = F(K), and when Nonempty plays S2k − 1, let S2k = F(S2k − 1).

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002248120004963X

An ideal game

Axiomatic set theory

(1983). The Ranking of Incomplete Tournaments: A Mathematician's Guide to Popular Sports. The American Mathematical Monthly: Vol. 90, No. 4, pp. 246-266.

Thomas Jech

Papers

Some combinatorial problems concerning uncountable cardinals

Boolean-valued models and independence proofs in set theory

An ideal game

Axiomatic set theory

The Ranking of Incomplete Tournaments: A Mathematician's Guide to Popular Sports