The median stabilization degree (msd, for short) of a median algebra measures the largest possible number of steps needed to generate a subalgebra with an arbitrary set of generators. With computer assistance, we found that msd of the lattice {−1, 0, 1}4 equals 2. This value is of critical importance to determine msd of {−1, 0, 1}n for all n ≥ 5 and to determine msd of the free median algebra λ...

In this paper we consider the problem of finding a vector that can be written as a nonnegative, integer and linear combination of given 0-1 vectors, the generators, such that the l1-distance between this vector and a given target vector is minimized. We prove that this closest vector problem is NP-hard to approximate within an additive error of (ln 2− ǫ)d ≈ (0, 693 − ǫ) d for all ǫ > 0 where d ...

In this paper we consider the problem of finding a vector that can be written as a nonnegative, integer and linear combination of given 0-1 vectors, the generators, such that the l1-distance between this vector and a given target vector is minimized. We prove that this closest vector problem is NP-hard to approximate within an additive error of (ln 2− ǫ)d ≈ (0, 693 − ǫ) d for all ǫ > 0 where d ...

We consider column-sparse positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. We develop a new rounding scheme based on the Partial Resampling variant of the Lovász Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + ln(∆1+1) amin +O( √ log(∆...

We present an approximate analytic study of our previously introduced model of evolution including the effects of genetic exchange. This model is motivated by the process of bacterial transformation. We solve for the velocity, the rate of increase of fitness, as a function of the fixed population size, N. We find the velocity increases with ln N, eventually saturating at an N which depends on t...

chronic kidney disease is one of the most common complication of systemic lupus erythematosus, which if untreated can lead to  the end-stage renal disease (esrd). early diagnosis and adequate treatment of lupus nephritis (ln) is critical to prevent the chronic kidney disease incidence and to reduce the development of esrd. the treatment of ln has changed significantly over the past decade. in p...