The Golden Partition Conjecture is motivated by conjectures whose root is a famous sorting problem. Imagine that a finite set S with cardinality n has a hidden total order ≺ that we would like to uncover by making comparisons between pairs of elements from S. Comparing a pair u, v ∈ S means uncovering either u ≺ v or v ≺ u. How many comparisons are needed in the worst case? The classical answer...

Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location p...

We propose a polynomial-time algorithm which takes as input a finite set of points of R3 and compute, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii. Almost three decades ago, an elegant...

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to e−|t| p and symmetric p-stable random variables, where p ∈ (0, 2]. We obtain various sharp moment an...

ball mills are common grinding equipment in mineral processing industries. ball wear results from three mechanisms namely impact, abrasion and corrosion. of these, the corrosion mechanism is the least investigated, due to its chemical-electrochemical nature. therefore, the aims of this research were to investigate the grinding chemistry influence (slurry ph, solid percentage, water chemistry an...

The exterior sphere condition is compared to proximal smoothness, and examples are provided, which show that the two properties are not necessarily equivalent. Then conditions are given under which equivalence holds, and an open question involving the union of uniform closed balls property is stated in the form of a conjecture.

Let M be the collection of all intersections of balls, considered as a subset of the hyperspace H of all closed, convex and bounded sets of a Banach space, furnished with the Hausdorff metric. We prove that M is uniformly very porous if and only if the space fails the Mazur intersection property.

In 1988 Simpson extended the Donaldson–Uhlenbeck–Yau theorem to context of Higgs bundles, and as an application he proved a uniformization which characterizes complex projective manifolds quasi-projective curves whose universal coverings are unit balls. this paper we give necessary sufficient condition for be uniformized by This generalizes Simpson. Several byproducts also obtained in paper.

We studied the following two geometric optimization problems with cardinality constraints. • Capacitated covering with unit balls: Given a set of n points in R and a positive integer L, partition them into a minimum number of groups such that each group contains at most L points, and can be covered by a unit ball in R. • Capacitated connected dominating set in unit disk graph: Given a positive ...