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Improvement of the Fast Clustering Algorithm Improved by *K*-Means in the Big Data

## Abstract

Clustering as a fundamental unsupervised learning is considered an important method of data analysis, and *K*-means is demonstrably the most popular clustering algorithm. In this paper, we consider clustering on feature space to solve the low efficiency caused in the Big Data clustering by *K*-means. Different from the traditional methods, the algorithm guaranteed the consistency of the clustering accuracy before and after descending dimension, accelerated *K*-means when the clustering centeres and distance functions satisfy certain conditions, completely matched in the preprocessing step and clustering step, and improved the efficiency and accuracy. Experimental results have demonstrated the effectiveness of the proposed algorithm.

###### A Mathematical Model to describe the herd behaviour considering group defense

## Abstract

A model for predator-prey interactions with herd behaviour is proposed. Novelty includes a smooth transition from individual behaviour (low number of prey) to herd behaviour (large number of prey). The model is analysed using standard stability and bifurcations techniques. We prove that the system undergoes a Hopf bifurcation as we vary the parameter that represents the efficiency of predators (dependent on the predation rate, for instance), giving rise to sustained oscillations in the system. The proposed model appears to possess more realistic features than the previous approaches while being also relatively easier to analyse and understand.

###### Entropy Generation in Couette Flow Through a Deformable Porous Channel

## Abstract

The present study examines the entropy generation on Couette flow of a viscous fluid in parallel plates filled with deformable porous medium. The fluid is injected into the porous channel perpendicular to the lower wall with a constant velocity and is sucked out of the upper wall with same velocity .The coupled phenomenon of the fluid flow and solid deformation in the porous medium is taken in to consideration. The exact expressions for the velocity of fluid, solid displacement and temperature distribution are found analytically. The effect of pertinent parameters on the fluid velocity, solid displacement and temperature profiles are discussed in detail. In the deformable porous layer, it is noticed that the velocity of fluid, solid displacement and temperature distribution are decreases with increasing the suction/injection velocity parameter. The results obtained for the present flow characteristic reveal several interesting behaviors that warrant further study on the deformable porous media. Furthermore, the significance of drag and the volume fraction on entropy generation number and Bejan number are discussed with the help of graphs.

###### Mathematical analysis of a B-cell chronic lymphocytic leukemia model with immune response

## Abstract

A B-cell chronic lymphocytic leukemia has been modeled via a highly nonlinear system of ordinary differential equations. We consider the rather important theoretical question of the equilibria existence. Under suitable assumptions all model populations are shown to coexist.

###### New iterative schemes for solving variational inequality and fixed points problems involving demicontractive and quasi-nonexpansive mappings in Banach spaces

## Abstract

In this paper, we suggest and analyze a new iterative method for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of fixed points of a demicontractive mapping which is the unique solution of some variational inequality problems involving accretive operators in a Banach space. We prove the strong convergence of the proposed iterative scheme without imposing any compactness condition on the mapping or the space. Finally, applications of our theorems to some constrained convex minimization problems are given.

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Note On Jakimovski-Leviatan Operators Preserving *e*
^{–x
}

## Abstract

In the present article, a modification of Jakimovski-Leviatan operators is presented which reproduce constant and *e*
^{–x} functions. We prove uniform convergence order of a quantitative estimate for the modified operators. We also give a quantitative Voronovskya type theorem.

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Optical solitons to the fractional Schr*ö*dinger-Hirota equation

## Abstract

This study reaches the dark, bright, mixed dark-bright, and singular optical solitons to the fractional Schr*ö*dinger-Hirota equation with a truncated *M*-fractional derivative via the extended sinh-Gordon equation expansion method. Dark soliton describes the solitary waves with lower intensity than the background, bright soliton describes the solitary waves whose peak intensity is larger than the background, and the singular soliton solutions is a solitary wave with discontinuous derivatives; examples of such solitary waves include compactions, which have finite (compact) support, and peakons, whose peaks have a discontinuous first derivative. The constraint conditions for the existence of valid solutions are given. We use some suitable values of the parameters in plotting 3-dimensional surfaces to some of the reported solutions.

###### The new extended rational SGEEM for construction of optical solitons to the (2+1)–dimensional Kundu–Mukherjee–Naskar model

## Abstract

This work proposes the new extended rational sinh-Gordon equation expansion technique (SGEEM). The computational approach is formulated based on the well-known sinh-Gordon equation. The proposed technique generalizes the sine-Gordon/sinh-Gordon expansion methods in a rational format. The efficiency of the suggested technique is tested on the (2+1)imensional Kunduukherjeeaskar (KMN) model. Various of optical soliton solutions have been obtained using this new method. The conditions which guarantee the existence of valid solitons are given.

###### On graphs with equal dominating and c-dominating energy

## Abstract

Graph energy and domination in graphs are most studied areas of graph theory. In this paper we try to connect these two areas of graph theory by introducing c-dominating energy of a graph *G*. First, we show the chemical applications of c-dominating energy with the help of well known statistical tools. Next, we obtain mathematical properties of c-dominating energy. Finally, we characterize trees, unicyclic graphs, cubic and block graphs with equal dominating and c-dominating energy.

###### Two Reliable Methods for The Solution of Fractional Coupled Burgers’ Equation Arising as a Model of Polydispersive Sedimentation

## Abstract

In this article, we attain new analytical solution sets for nonlinear time-fractional coupled Burgers’ equations which arise in polydispersive sedimentation in shallow water waves using exp-function method. Then we apply a semi-analytical method namely perturbation-iteration algorithm (PIA) to obtain some approximate solutions. These results are compared with obtained exact solutions by tables and surface plots. The fractional derivatives are evaluated in the conformable sense. The findings reveal that both methods are very effective and dependable for solving partial fractional differential equations.