Dynamics of nearly unstable axisymmetric liquid bridges

The dynamics of a noncylindrical, axisymmetric, marginally unstable liquid bridge between two equal disks is analyzed in the inviscid limit. The resulting model allows for the weakly nonlinear description of both the (first stage of) breakage for unstable configurations and the (slow) dynamics for stable configurations. The analysis is made for both slender and short liquid brides. In the former range, the dynamics breaks reflection symmetry on the midplane between the supporting disks and can be described by a standard Duffing equation, while for short bridges reflection symmetry is preserved and the equation is still Duffing-like but exhibiting a quadratic nonlinearity. The asymptotic results compare well with existing experiments

New efficient constructive heuristics for the hybrid flowshop to minimise makespan: A computational evaluation of heuristics

This paper addresses the hybrid flow shop scheduling problem to minimise makespan, a well-known scheduling problem for which many constructive heuristics have been proposed in the literature. Nevertheless, the state of the art is not clear due to partial or non homogeneous comparisons. In this paper, we review these heuristics and perform a comprehensive computational evaluation to determine which are the most efficient ones. A total of 20 heuristics are implemented and compared in this study. In addition, we propose four new heuristics for the problem. Firstly, two memory-based constructive heuristics are proposed, where a sequence is constructed by inserting jobs one by one in a partial sequence. The most promising insertions tested are kept in a list. However, in contrast to the Tabu search, these insertions are repeated in future iterations instead of forbidding them. Secondly, we propose two constructive heuristics based on Johnson’s algorithm for the permutation flowshop scheduling problem. The computational results carried out on an extensive testbed show that the new proposals outperform the existing heuristics.Ministerio de Ciencia e Innovación DPI2016-80750-

On prime numbers of the form 2n±k2^n \pm k

Consider the set $\mathcal{K}$ of integers $k$ for which there are infinitely many primes $p$ such that $p+k$ is a power of $2$. The aim of this paper is to show a relationship between $\mathcal{K}$ and the limits points of some set rational numbers related to a sequence of polynomials $C_n(q)$ introduced by Kassel and Reutenauer [KasselReutenauer]

A relationship between the ideals of Fq[x,y,x−1,y−1]\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right] and the Fibonacci numbers

Let $C_n(q)$ be the number of ideals of codimension $n$ of $\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$, where $\mathbb{F}_q$ is the finite field with $q$ elements. Kassel and Reutenauer [KasselReutenauer2015A] proved that $C_n(q)$ is a polynomial in $q$ for any fixed value of $n \geq 1$. For $q = \frac{3+\sqrt{5}}{2}$, this combinatorial interpretation of $C_n(q)$ is lost. Nevertheless, an unexpected connexion with Fibonacci numbers appears. Let $f_n$ be the $n$-th Fibonacci number (following the convention $f_0 = 0$, $f_1 = 1$). Define the series $$F(t) = \sum_{n \geq 1} f_{2n}\,t^n.$$ We will prove that for each $n \geq 1$, $$C_n\left( \frac{3+\sqrt{5}}{2}\right) = \lambda_n \, \left(f_{2n} \frac{3+\sqrt{5}}{2} - f_{2n-2} \right) ,$$ where the integers $\lambda_n \geq 0$ are given by the following generating function $$ \prod_{m \geq 1} \left(1+F\left( t^m\right)\right) = 1 + \sum_{n \geq 1} \lambda_n\,t^n. $

Non-Axisymmetric Effects on Long Liquid Bridges

The stability of long liquid bridges under non-axisymmetric disturbances like a microgravitational force acting perpendicular to the liquid bridge axis or a non-coaxiality of the disks is analyzed through an asymptotic method based on bifurcation techniques. Results obtained indicate that such non-axisymmetric effects are of higher order than those produced by axisymmetric perturbations

On a function introduced by Erd\"{o}s and Nicolas

Erd\"os and Nicolas [erdos1976methodes] introduced an arithmetical function $F(n)$ related to divisors of $n$ in short intervals $\left] \frac{t}{2}, t\right]$. The aim of this note is to prove that $F(n)$ is the largest coefficient of polynomial $P_n(q)$ introduced by Kassel and Reutenauer [kassel2015counting]. We deduce that $P_n(q)$ has a coefficient larger than $1$ if and only if $2n$ is the perimeter of a Pythagorean triangle. We improve a result due to Vatne [vatne2017sequence] concerning the coefficients of $P_n(q)$