International Journal of Industrial Mathematics
Volume:8 Issue: 2, Spring 2016

Clustering is a widespread data analysis and data mining technique in many fields of study such as engineering, medicine, biology and the like. The aim of clustering is to collect data points. In this paper, a Cultural Algorithm (CA) is presented to optimize partition with N objects into K clusters. The CA is one of the effective methods for searching into the problem space in order to find a near optimal solution. This algorithm has been tested on different scale datasets and has been compared with other well-known algorithms in clustering, such as K-means, Genetic Algorithm (GA), Simulated Annealing (SA), Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO) algorithm. The results illustrate that the proposed algorithm has a good proficiency in obtaining the desired results.

ýIn this paper, based on [A. Razani, V. Rako$\check{c}$evi$\acute{c}$ and Z. Goodarzi, Nonself mappings in modular spaces and common fixed point theorems, Cent. Eur. J. Math. 2 (2010) 357-366.] a fixed point theorem for non-self contraction mapping $T$ in the modular space $X_\rho$ is presented. Moreover, we study a new version of Krasnoseleskii's fixed point theorem for $S$, where $T$ is a continuous non-self contraction mapping and $S$ is continuous mapping such that $S(C)$ resides in a compact subset of $X_\rho$, where $C$ is a nonempty and complete subset of $X_\rho$, also $C$ is not bounded. Our result extends and improves the result announced by Hajji and Hanebally [A. Hajji and E. Hanebaly, Fixed point theorem and its application to perturbed integral equations in modular function spaces, Electron. J. Differ. Equ. 2005 (2005) 1-11]. As an application, the existence of a solution of a nonlinear integral equation on $C(I, L^\varphi) $ is presented, where $C(I, L^\varphi)$ denotes the space of all continuous function from $I$ to $L^\varphi$, $L^\varphi$ is the Musielak-Orlicz space and $I=[0,b] \subset \mathbb{R}$. In addition, the concept of quasi contraction non-self mapping in modular space is introduced. Then the existence of a fixed point of these kinds of mapping without $\Delta_2$-condition is proved. Finally, a three step iterative sequence for non-self mapping is introduced and the strong convergence of this iterative sequence is studied. Our theorem improves and generalized recent know results in the ýliterature.ý

ýHepatitis B virus (HBV) infection is a major public health problem in the world todayý. ýA mathematical model is formulated to describe the spread of hepatitis Bý, ýwhich can be controlled by vaccination as well as treatmentý. ýWe study the dynamical behavior of the system with fixed control for both vaccination and treatmentý. ýThe results shows that the dynamics of the model is completely determined by the basic reproductive number R_0. ýif R_01, ýthe disease-free equilibrium is unstable and the disease is uniformly persistentý. ýFurthermoreý, ýIf R_0>1, ýthe unique endemic equilibrium is globally asymptotically stable by using a generalization of the Poincar e-Bendixson ýcriterion.ý

This article investigates the nonlinear, steady boundary layer flow and heat transfer of an incompressible Eyring-Powell non-Newtonian fluid from an isothermal sphere with Biot number effects. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The influence of a number of emerging dimensionless parameters, namely the Eyring-Powell rheological fluid parameter $\left( \varepsilon \right) $, the local non-Newtonian parameter based on length scale $\left( \delta \right) $, Prandtl number (Pr), Biot number $\left( \gamma\right) $ and dimensionless tangential coordinate $\left(\xi \right) $ on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is found that the velocity and heat transfer rate (Nusselt number) decrease with increasing $\left( \varepsilon \right) $, whereas temperature and skin friction increase. An increasing $\left(\delta\right) $ is observed to enhance velocity, local skin friction and heat transfer rate but reduces the temperature. An increase $\left( \gamma \right) $ is seen to increase velocity, temperature, local skin friction and Nusselt number. The study is relevant to chemical materials processing ýapplications.ý

ýIn this paperý, ýwe have investigated the chaotic behavior of thermal convection in couple stress liquid saturated porous layer subject to gravityý, ýheated from below and cooled from aboveý, ýbased on theory of dynamical systemý. ýA low dimensional Lorenzý- ýlike model is obtained by using Galerkin-truncation approximationý. ýWe found that there is proportional relation between scaled couple stress parameter and rescaled Rayleigh numberý. We analyzed that increase in level of couple stress parameter increases the level of chaosý.

Recently, the block-pulse functions (BPFs) are used in solving electromagnetic scattering problem, which are modeled as linear Fredholm integral equations (FIEs) of the second kind. But the theoretical aspect of this method has not fully investigated yet. In this article, in addition to presenting a new approach for solving FIE of the second kind, the theory of both methods is investigated as a main part. By providing a new method based on BPFs for solving FIEs of the second kind, the least squares and non-least squares solutions are defined for this problem. First, the convergence of the non-least squares solution is proved by the Nystr$\ddot{o}$m method. ýTýhen, considering the fact that the set of all invertible matrices is an open set, the convergence of the least squares solution is investigated. The convergence of Nystr$\ddot{o}$m method has the main role in proving the basic results. Because the presented convergence trend is independent of the orthogonality of the basis functions, the given method can be applied for any arbitrary ýmethod.ý

ýThere are a few finite groups that are determined up to isomorphism solely by their order, such as $\mathbb{Z}_{2}$ or $\mathbb{Z}_{15}$. Still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of Sylow $p$-subgroups for each prime $p$, etc. In this paper, we investigate the possibility of characterizing the projective special linear groups $L_{n}(2)$ by simple conditions when $2^{n}-1$ is a prime number. Our result states that: $G\cong L_{n}(2)$ if and only if $|G|=|L_{n}(2)|$ and $G$ has one conjugacy class length $\frac{|L_{n}(2)|% }{2^{n}-1}$, where $2^{n}-1=p$ is a prime number. Furthermore, we will show that Thompson's conjecture holds for the simple groups $L_{n}(2)$, where $2^{n}-1$ prime is a prime number. By Thompson's conjecture if $L$ is a finite non-Abelian simple group, $G$ is a finite group with a trivial center, and the set of the conjugacy classes size of $L$ is equal to $G$, then $L\cong G$ý.ý