Narrow Results

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differs by at most 1, and its size is the number of edges that go across the two parts. In this paper, motivated by a well-known result of Edwards about Max-Cut, we study maximum bisections of graphs. Let ${𝒢}=\{P_{\ell }⊠P_k,P_{\ell }⊠C_k,C_{\ell }⊠C_k\}$. For each $G\in {𝒢}$, we show that $G$ admits a bisection of size larger than Edwards’ bound. We also study the maximum bisection of graphs that seem closely related to the strong product of two paths.

The d–dimensional k-ary array is the d–fold Cartesian product graph of the path graph P_k with k vertices. We show that the (edge) isoperimetric number of is given by and identify the cardinalities and the structure of the isoperimetric sets. For odd k, the cardinalities of isoperimetric sets in are , whereas every isoperimetric set for k even has cardinality .

Designing energy-efficient scheduling algorithms on heterogeneous distributed systems is increasingly becoming the focus of research. State-of-the-art works have studied scheduling by combining dynamic voltage and frequency scaling (DVFS) technology and turning off the appropriate processors to reduce dynamic and static energy consumptions. However, the methods for turning off processors are ineffective. In this study, we propose a novel method to assign priorities to processors for facilitating effective selection of turned-on processors to decrease static energy consumption. An energy-efficient scheduling algorithm based on bisection (ESAB) is proposed on this basis, and this algorithm directly turns on the most energy-efficient processors depending on the idea of bisection to reduce static energy consumption while dynamic energy consumption is decreased by using DVFS technology. Experiments are performed on fast Fourier transform, Gaussian elimination, and randomly generated parallel applications. Results show that our ESAB algorithm makes a better trade-off between reducing energy consumption and low computation time of task assignment (CTTA) than existing algorithms under different scale conditions, deadline constraints, and degrees of parallelism and heterogeneity.

Bitubular structural configurations, where the outer tube is circular, square and curvy square in shape while the inner-tube section is curvy triangular, square and hexagonal in different proposed configurations, are numerically crushed under dynamic axial loading. The crashworthiness effectiveness for changing inner-tube polygonal cross-section for each of the outer tube sections is studied and compared with changing outer tube shape. The deformation plots and energy absorption (EA) parameters such as peak crushing force (PCF) mean crushing force (MCF), energy absorption and crush force efficiency for each case are evaluated. Most of the configurations showed ovalization with low PCF and MCF and moderate crush force efficiency. Afterwards, effects of $L/D$ and $t/R$ on deformation modes and EA are demonstrated by selecting one of the configurations from each group using published experimental results.

A short and novel MATLAB implementation of the local mesh refinement using newest bisection or longest bisection is presented in this paper. This short implementation is helpful for the teaching of adaptive finite element methods and programming in more advanced languages.