The transitivity of the Hardy-Weinberg law

The reduction of multi-allelic polymorphisms to variants with fewer alleles, two in the limit, is addressed. The Hardy-Weinberg law is shown to be transitive in the sense that a multi-allelic polymorphism that is in equilibrium will retain its equilibrium status if any allele together with its corresponding genotypes is deleted from the population. Similarly, the transitivity principle also applies if alleles are joined, which leads to the summation of allele frequencies and their corresponding genotype frequencies. These basic polymorphism properties are intuitive, but they have apparently not been formalized or investigated. This article provides a straightforward proof of the transitivity principle, and its usefulness in practical genetic data analysis with multi-allelic markers is explored. In general, results of statistical tests for Hardy-Weinberg equilibrium obtained with polymorphisms that are reduced by deletion or joining of alleles are seen to be consistent with the formulated transitivity principle. We also show how the transitivity principle allows one to identify equilibrium-offending alleles, and how it can provide clues to genotyping problems and evolutionary changes. For microsatellites, which are widely used in forensics, the transitivity principle implies one expects similar results for statistical tests that use length-based and sequence-based alleles. High-quality autosomal microsatellite databases of the US National Institute of Standards and Technology are used to illustrate the use of the transitivity principle in testing both length-based and sequence-based microsatellites for Hardy-Weinberg proportions. Test results for Hardy-Weinberg proportions for the two types of microsatellites are seen to be largely consistent and can detect allele imbalance.

On Uniform Nonintegrability and Weak Uniform Nonintegrability of a Sequence of Random Variables with Respect to a Nonnegative Array

Chandra, Hu and Rosalsky [1] introduced the notion of a sequence of random variables being uniformly nonintegrable and they established a de La Vallée Poussin type criterion for this notion. Inspired by the Chandra, Hu and Rosalsky [1] article, Hu and Peng [2] introduced the weaker notion of a sequence of random variables being weakly uniformly nonintegrable and they also established a de La Vallée Poussin type criterion for this notion using a modification of the Chandra, Hu and Rosalsky [1] argument. In this correspondence, we introduce the more general notion of uniform nonintegrability and weak uniform nonintegrability with respect to an array of nonnegative real numbers together with a de La Vallée Poussin type criterion for this notion. This criterion immediately yields as particular cases the criteria of Chandra, Hu and Rosalsky [1] and Hu and Peng [2] , and it has a substantially simpler and more straightforward proof.

Presentations for Temperley–Lieb Algebras

We give a new and conceptually straightforward proof of the well-known presentation for the Temperley–Lieb algebra, via an alternative new presentation. Our method involves twisted semigroup algebras, and we make use of two apparently new submonoids of the Temperley–Lieb monoid.

On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

We consider the inverse function $z=g(w)$ of a (normalized) starlike function $w=f(z)$ of order $\alpha$ on the unit disk of the complex plane with $0<\alpha<1.$ Krzy{\. z}, Libera and Z\l otkiewicz obtained sharp estimates of the second and the third coefficients of $g(w)$ in their 1979 paper. Prokhorov and Szynal gave sharp estimates of the fourth coefficient of $g(w)$ as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of $g(w)$ together with explicit forms of the extremal functions.

On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition on the spectrum of A, coincides with the C 0 -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.

Virtual tangles and fiber functors

We define a category [Formula: see text] of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to [Formula: see text] which induces an equivalence of categories. On the other hand, we show that [Formula: see text] is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

On the Single-Parity Locally Repairable Codes with Multiple Repairable Groups

Locally repairable codes (LRCs) are a new family of erasure codes used in distributed storage systems which have attracted a great deal of interest in recent years. For an [ n , k , d ] linear code, if a code symbol can be repaired by t disjoint groups of other code symbols, where each group contains at most r code symbols, it is said to have availability- ( r , t ) . Single-parity LRCs are LRCs with a constraint that each repairable group contains exactly one parity symbol. For an [ n , k , d ] single-parity LRC with availability- ( r , t ) for the information symbols (single-parity LRCs), the minimum distance satisfies d ≤ n - k - ⌈ k t / r ⌉ + t + 1 . In this paper, we focus on the study of single-parity LRCs with availability- ( r , t ) for information symbols. Based on the standard form of generator matrices, we present a novel characterization of single-parity LRCs with availability t ≥ 1 . Then, a simple and straightforward proof for the Singleton-type bound is given based on the new characterization. Some necessary conditions for optimal single-parity LRCs with availability t ≥ 1 are obtained, which might provide some guidelines for optimal coding constructions.

A straightforward proof of Carleman estimate for second-order elliptic operator and a three-sphere inequality

In this paper we provide a simple proof of a Carleman estimate for a second-order elliptic operator P with Lipschitz leading coefficients. We apply such a Carleman estimate to derive a three-sphere inequality for solutions to equation {Pu=0}.

DISPERSIVE ORDERING FOR THE MULTIVARIATE NORMAL DISTRIBUTION

Let (X1, …, Xn) be a multivariate normal random vector with any mean vector, variances equal to 1 and covariances equal and positive. Turner and Whitehead [9] established that the largest order statistic max{X1, …, Xn} is less than the standard normal random variable in the dispersive order. In this paper, we give a new and straightforward proof for this result. Several possible extensions of this result are also discussed.