Prime factors

We use Voiculescu’s free probability theory to prove the existence of prime factors, hence answering a longstanding problem in the theory of von Neumann algebras.

A recipe for randomness

Despite many diverse theories that address closely related themes—e.g., probability theory, algorithmic complexity, cryptoanalysis, and pseudorandom number generation—a near-void remains in constructive methods certified to yield the desired “random” output. Herein, we provide explicit techniques to produce broad sets of both highly irregular finite and normal infinite sequences, based on constructions and properties derived from approximate entropy (ApEn), a computable formulation of sequential irregularity. Furthermore, for infinite sequences, we considerably refine normality, by providing methods for constructing diverse classes of normal numbers, classified by the extent to which initial segments deviate from maximal irregularity.

The optimal measure of allelic association

Allelic association between pairs of loci is derived in terms of the association probability ρ as a function of recombination θ, effective population size N, linear systematic pressure v, and time t, predicting both ρrt, the decrease of association from founders and ρct, the increase by genetic drift, with ρt = ρrt + ρct. These results conform to the Malecot equation, with time replaced by distance on the genetic map, or on the physical map if recombination in the region is uniform. Earlier evidence suggested that ρ is less sensitive to variations in marker allele frequencies than alternative metrics for which there is no probability theory. This robustness is confirmed for six alternatives in eight samples. In none of these 48 tests was the residual variance as small as for ρ. Overall, efficiency was less than 80% for all alternatives, and less than 30% for two of them. Efficiency of alternatives did not increase when information was estimated simultaneously. The swept radius within which substantial values of ρ are conserved lies between 385 and 893 kb, but deviation of parameters between measures is enormously significant. The large effort now being devoted to allelic association has little value unless the ρ metric with the strongest theoretical basis and least sensitivity to marker allele frequencies is used for mapping of marker association and localization of disease loci.

Randomness and degrees of irregularity.

The fundamental question "Are sequential data random?" arises in myriad contexts, often with severe data length constraints. Furthermore, there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness--e.g., to evaluate the efficacy of therapy in medicine. We address both these issues from a computable framework via a quantification of regularity. ApEn (approximate entropy), defining maximal randomness for sequences of arbitrary length, indicating the applicability to sequences as short as N = 5 points. An infinite sequence formulation of randomness is introduced that retains the operational (and computable) features of the finite case. In the infinite sequence setting, we indicate how the "foundational" definition of independence in probability theory, and the definition of normality in number theory, reduce to limit theorems without rates of convergence, from which we utilize ApEn to address rates of convergence (of a deficit from maximal randomness), refining the aforementioned concepts in a computationally essential manner. Representative applications among many are indicated to assess (i) random number generation output; (ii) well-shuffled arrangements; and (iii) (the quality of) bootstrap replicates.

Game theory and reciprocity in some extensive form experimental games

We examine decision making in two-person extensive form game trees using nine treatments that vary matching protocol, payoffs, and payoff information. Our objective is to establish replicable principles of cooperative versus noncooperative behavior that involve the use of signaling, reciprocity, and backward induction strategies, depending on the availability of dominated direct punishing strategies and the probability of repeated interaction with the same partner. Contrary to the predictions of game theory, we find substantial support for cooperation under complete information even in various single-play treatments.

An information theory model of hydrophobic interactions.

A molecular model of poorly understood hydrophobic effects is heuristically developed using the methods of information theory. Because primitive hydrophobic effects can be tied to the probability of observing a molecular-sized cavity in the solvent, the probability distribution of the number of solvent centers in a cavity volume is modeled on the basis of the two moments available from the density and radial distribution of oxygen atoms in liquid water. The modeled distribution then yields the probability that no solvent centers are found in the cavity volume. This model is shown to account quantitatively for the central hydrophobic phenomena of cavity formation and association of inert gas solutes. The connection of information theory to statistical thermodynamics provides a basis for clarification of hydrophobic effects. The simplicity and flexibility of the approach suggest that it should permit applications to conformational equilibria of nonpolar solutes and hydrophobic residues in biopolymers.