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.

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.

Free entropy and property T factors

We show that a large class of finite factors has free entropy dimension less than or equal to one. This class includes all prime factors and many property T factors.

Complex mtDNA constitutes an approximate 620-kb insertion on Arabidopsis thaliana chromosome 2: Implication of potential sequencing errors caused by large-unit repeats

Previously conducted sequence analysis of Arabidopsis thaliana (ecotype Columbia-0) reported an insertion of 270-kb mtDNA into the pericentric region on the short arm of chromosome 2. DNA fiber-based fluorescence in situ hybridization analyses reveal that the mtDNA insert is 618 ± 42 kb, ≈2.3 times greater than that determined by contig assembly and sequencing analysis. Portions of the mitochondrial genome previously believed to be absent were identified within the insert. Sections of the mtDNA are repeated throughout the insert. The cytological data illustrate that DNA contig assembly by using bacterial artificial chromosomes tends to produce a minimal clone path by skipping over duplicated regions, thereby resulting in sequencing errors. We demonstrate that fiber-fluorescence in situ hybridization is a powerful technique to analyze large repetitive regions in the higher eukaryotic genomes and is a valuable complement to ongoing large genome sequencing projects.

Persistent confusion of total entropy and chemical system entropy in chemical thermodynamics.

The change in free energy with temperature at constant pressure of a chemical reaction is determined by the sum (dS) of changes in entropy of the system of reagents, dS(i), and the additional entropy change of the surroundings, dS(H), that results from the enthalpy change, W. A faulty identification of the total entropy change on reaction with dS(i) has been responsible for the attribution of general validity to the expressions (d deltaG/dT)p = -deltaS(i) and d(deltaG/T)/d(1/T)= deltaH, which are found in most textbooks and in innumerable papers.