International Journal of Mathematics for Industry

$L(r,s,t)$-labeling problem ($Lrst$-$LP$) is an important topic in discrete mathematics due to its various applications, like in frequency assignment in mobile communication systems, signal processing, circuit design, etc. $L211$-$LP$ is a special case of $Lrst$-$LP$. An $L211$-labeling ($L211L$) of a graph $G=(V,E)$ is a mapping $F:V\to \{0,1,2,\dots \}$ such that $|F(\xi )-F(\eta )|\ge 2$ if and only if $d(\xi ,\eta )=1$, $|F(\xi )-F(\eta )|\ge 1$ if $d(\xi ,\eta )=2$ or 3, where $d(\xi ,\eta )$ is the distance between the nodes $\xi$ and $\eta$. In this work, we have determined the upper bound of $L211L$ for interval graph (IG) and obtained a tighter upper bound which is $4\mathrm{\Delta }-2$. Also, we proposed an efficient algorithm to label any IG by $L211L$ and also computed the time complexity of the proposed algorithm.

We construct a method for fairing a given discrete planar curve by using the integrable discrete analog of Euler’s elastica, which is a discrete version of the approximation algorithm presented by Brander et al. We first give a brief review of the integrable discrete analog of Euler’s elastica proposed by Bobenko and Suris, then we present a detailed account of the fairing algorithm, and we apply this method to an architectural problem of characterizing the keylines of Japanese handmade pantiles.

In this paper, we present differential geometric formulation of hanging membrane forms based on the equilibrium equations in the shell membrane theory and the geometric variational problems. We also present the equations of hanging membranes in isothermic coordinates which will be an effective representation for numerical analysis in our forthcoming paper.

In industrial problems linear elasticity simulation has been known to be successfully solved by the Finite Element Methods (FEMs). However, a mathematical model as well as a numerical solution program that can be applied to the deformation simulation of bimetallic materials are still the challenges in this research area. In this paper, we investigate a mathematical model for the linear elastic deformation of bimetallic structures. By reducing the linear elasticity problem to its weak form, we apply the continuous Galerkin approximation to obtain the discrete problem. The existence and the uniqueness of the solution of the weak problem are clearly demonstrated, ensuring the well-posedness of the problem. In addition, the numerical simulations with bimetallic materials have been built on Freefem$++$ software that illustrate the efficiency and reliability of the numerical method.

We consider Bernoulli first-passage percolation on the  $d$-dimensional hypercubic lattice with  $d\ge 2$. The passage time of edge  $e$ is 0 with probability  $p$ and 1 with probability  $1-p$, independently of each other. Let  $p_c$ be the critical probability for percolation of edges with passage time 0. When  $0\le p<p_c$, there exists a nonrandom, nonempty compact convex set  ${{ℬ}}_p$ such that the set of vertices to which the first-passage time from the origin is within  $t$ is well approximated by  $t{{ℬ}}_p$ for all large  $t$, with probability one. The aim of this paper is to prove that for  $0\le p<q<p_c$, the Hausdorff distance between  ${{ℬ}}_p$ and  ${{ℬ}}_q$ grows linearly in  $q-p$. Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the critical case.