An SDP randomized approximation algorithm for max hypergraph cut with limited unbalance

We consider the design of semidefinite programming(SDP) based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance(MHC-LU): Find a partition of the vertices of a weighted hypergraph H =(V, E) into two subsets V1, V2 with ‖V2|- |V1‖ u for some given u and maximizing the total weight of the edges meeting both V1 and V2. The problem MHC-LU generalizes several other combinatorial optimization problems including Max Cut, Max Cut with Limited Unbalance(MC-LU), Max Set Splitting,Max Ek-Set Splitting and Max Hypergraph Bisection. By generalizing several earlier ideas, we present an SDP randomized approximation algorithm for MHC-LU with guaranteed worst-case performance ratios for various unbalance parameters τ = u/|V|. We also give the worst-case performance ratio of the SDP-algorithm for approximating MHC-LU regardless of the value of τ. Our strengthened SDP relaxation and rounding method improve a result of Ageev and Sviridenko(2000) on Max Hypergraph Bisection(MHC-LU with u = 0), and results of Andersson and Engebretsen(1999), Gaur and Krishnamurti(2001) and Zhang et al.(2004) on Max Set Splitting(MHC-LU with u = |V|). Furthermore, our new formula for the performance ratio by a tighter analysis compared with that in Galbiati and Maffioli(2007) is responsible for the improvement of a result of Galbiati and Maffioli(2007) on MC-LU for some range of τ.     

Spectral analysis for performance evaluation in a bus network

This paper deals with the performance evaluation of a public transportation system in terms of waiting times at various connection points. The behaviour of a bus network is studied in the framework of Discrete Event Systems (DES). Two possible operating modes of buses can be observed at each connection stop: periodic and non-periodic mode. Two complementary tools, Petri nets and (max, +) algebra, are used to describe the network by a non-stationary linear state model. This one can be solved after solving the structural conflicts associated to the graphical representation. From the characteristic matrix of the mathematical model, we determine eigenvalues and eigenvectors that we use to evaluate the connection times of passengers. This work is finally illustrated with a numerical example.     

Min–max and min–max regret versions of combinatorial optimization problems: A survey

Min–max and min–max regret criteria are commonly used to define robust solutions. After motivating the use of these criteria, we present general results. Then, we survey complexity results for the min–max and min–max regret versions of some combinatorial optimization problems: shortest path, spanning tree, assignment, min cut, min s–t cut, knapsack. Since most of these problems are NP-hard, we also investigate the approximability of these problems. Furthermore, we present algorithms to solve these problems to optimality.     

On the calculation of extended max and min operations between convex fuzzy sets of the real line

In this paper we will identify the sets of so-called sub- and pseudo-highest intersection points of convex fuzzy sets of the real line and will explore their properties. Based on the properties of these sets, an algorithm for calculating extended max and min operations between two or more convex fuzzy sets of the real line with general membership functions, not necessarily continuous, is proposed.     

MXenes as Emerging Materials: Synthesis,Properties, and Applications

Due to their unique layered microstructure, the presence of various functional groups at the surface, earth abundance, and attractive electrical, optical, and thermal properties, MXenes are considered promising candidates for the solution of energy- and environmental-related problems. It is seen that the energy conversion and storage capacity of MXenes can be enhanced by changing the material dimensions, chemical composition, structure, and surface chemistry. Hence, it is also essential to understand how one can easily improve the structure–property relationship from an applied point of view. In the current review, we reviewed the fabrication, properties, and potential applications of MXenes. In addition, various properties of MXenes such as structural, optical, electrical, thermal, chemical, and mechanical have been discussed. Furthermore, the potential applications of MXenes in the areas of photocatalysis, electrocatalysis, nitrogen fixation, gas sensing, cancer therapy, and supercapacitors have also been outlooked. Based on the reported works, it could easily be observed that the properties and applications of MXenes can be further enhanced by applying various modification and functionalization approaches. This review also emphasizes the recent developments and future perspectives of MXenes-based composite materials, which will greatly help scientists working in the fields of academia and material science.