The fixation probability and time for a doubly beneficial mutant

For a highly beneficial mutant $A$ entering a randomly reproducing population of constant size, we study the situation when a second beneficial mutant $B$ arises before $A$ has fixed. If the selection coefficient of $B$ is greater than the selection coefficient of $A$, and if $A$ and $B$ can recombine at some rate $\rho$, there is a chance that the double beneficial mutant $AB$ forms and eventually fixes. We give a convergence result for the fixation probability of $AB$ and its fixation time for large selection coefficients.     

A randomized algorithm for the decomposition of matrices

Given an $m\times n$ matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying $A^{\mathrm{T}}$ to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and $A^{\mathrm{T}}$ can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as $\sqrt{lm}$ times the $(k+1)$st greatest singular value ${\sigma }_{k+1}$ of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when $l?k$ is a fixed small nonnegative integer. For example, according to one of our estimates for $l?k=20$, the probability that the spectral norm $6A?Z6$ is greater than $10\sqrt{(k+20)m}{\sigma }_{k+1}$ is less than ${10}^{?17}$. The paper contains a number of estimates for $6A?Z6$, including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and $A^{\mathrm{T}}$ can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples.     

Copositive and semidefinite relaxations of the quadratic assignment problem

Semidefinite relaxations of the quadratic assignment problem ($QAP$) have recently turned out to provide good approximations to the optimal value of $QAP$. We take a systematic look at various conic relaxations of $QAP$. We first show that $QAP$ can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the well-studied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size.     

Sequences of integers with missing quotients and dense points without neighbors

Let $A$ be a pre-defined set of rational numbers. We say that a set of natural numbers $S$ is an $A$-quotient-free set if no ratio of two elements in $S$ belongs to $A$. We find the maximal asymptotic density and the maximal upper asymptotic density of $A$-quotient-free sets when $A$ belongs to a particular class.It is known that in the case $A=\{p,q\}$, where $p$, $q$ are coprime integers greater than 1, the latter problem is reduced to the evaluation of the largest number of non-adjacent lattice points in a triangle whose legs lie on the coordinate axes. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.     

On a problem of partitions of the set of nonnegative integers with the same representation functions

Dombi has shown that the set $\mathbb{N}$ of all non-negative integers can be partitioned into two subsets with identical representation functions. In this paper, we prove that one cannot partition $\mathbb{N}$ into more than two subsets with identical representation functions, while for any integer $k\ge 3$ there is a partition $\mathbb{N}=A_1\cup ?\cup A_k$ such that $A_i$ and $A_{k+1?i}$ have the same representation function for any integer $1\le i\le k$.     

Ideal amenability of module Lau Banach algebra

Given Banach algebras $A$ and $B$, where $A$ be a Banach $B$-bimodule. In this paper we study the ideal amenability, approximate ideal and cyclic ideal amenability of module Lau Banach algebra $A{\times }_{\alpha }B$.