## Issue 4Volume 6 (2018)

 Authors : P. Maragatha Meenakshi and J. Sathya Title : On $\beta^\star$-closed Spaces in Terms of Nets Volume, Issue, Year : 6(4)(2018) Pages : 1−4

Abstract: The purpose of this paper is to obtain various characterizations of $\beta^\star$-closed spaces interms nets.

Keywords: Topological spaces, $\beta^\star$-open sets, $\beta^\star$-closed spaces.

 Authors : Fauzia Raza and A. K. Tiwari Title : Radiation Effect on MHD Flow of Nanofluids Over an Exponentially Stretching Sheet Volume, Issue, Year : 6(4)(2018) Pages : 5−12

Abstract: The problem of MHD radiation flow over an exponentially sheet through a porous medium is considered. The consequences of various parameters will be analyzed in Copper $(Cu)$ and Silver $(Ag)$ nanofluid. The non-linear governing equations of flow of thermal fields are converted to ordinary differential equation using similarity transformations and their numerical solution is obtained using MATLAB software bvp4c'' under the related boundary conditions. The interesting outcomes for variant physical parameters are exhibited through plots and numerical tables.

Keywords: Boundary layer, exponentially stretching sheet, MHD, nanofluid, porous medium, thermal radiation.

 Authors : Syama P. Bhattacharya and Asis Ghosh Title : An Observation on the Interrelationship Between the Gravitational Weight and the Area of a Flat Sheet: A New Concept of Rapid Measurement of Areas Bended by Irregular Outlines Volume, Issue, Year : 6(4)(2018) Pages : 13−20

Abstract: An accurate measurement of a bended flat area with many bends and deflections is practically impossible to perform using any of the existing methods of mensuration. Therefore, an alternative method has been investigated which insists on the measurement of gravitational weight of the portion of paper occupied by the area of a map. However, the map should be drawn on a thin flat sheet having the quality of uniform gravitational weight. A simple method for the selection of an uniform thin sheet has also been discussed. A theorem, $I = W10^{2}$ has been proposed for the rapid determination of index of any flat surface irrespective of sizes and geometric forms.

Keywords: Uniform gravitational weight, Transparency, Index, Spot-Photograph, Asteroids.

 Authors : A. Ibrahim and M. Indhumathi Title : Fuzzy PWI-Ideals of Lattice Pseudo-Wajsberg Algebras Volume, Issue, Year : 6(4)(2018) Pages : 21−31

Abstract: In this paper, we introduce the notions of fuzzy pseudo-Wajsberg implicative ideal (fuzzy PWI-ideal) and fuzzy pseudo lattice ideal in lattice pseudo-Wajsberg algebra. Also, we obtain some of their related properties. Moreover, we investigate the properties of fuzzy PWI-ideal related to homomorphism and kernel.

Keywords: Wajsberg algebra, Pseudo-Wajsberg algebra, Lattice pseudo-Wajsberg algebra, Fuzzy PWI-Ideal, Fuzzy Pseudo lattice ideal, Homomorphism, Kernel.

 Authors : A. Ibrahim and D. Saravanan Title : Strong Implicative and Fuzzy Strong Implicative Filters of Lattice Wajsberg Algebras Volume, Issue, Year : 6(4)(2018) Pages : 33−41

Abstract: In this paper, we consider and discuss some properties of strong implicative filter of lattice Wajsberg algebra. Also, we introduce the notion of fuzzy strong implicative filter and investigate some properties with interesting illustrations. Moreover, we obtain the relation between a strong implicative filter and fuzzy strong implicative filter in lattice Wajsberg algebra.

Keywords: Wajsberg algebra, Lattice Wajsberg algebra, Implicative filter, Strong implicative filter, Fuzzy Strong implicative filter.

 Authors : V. R. Kulli Title : On Augmented Revan Index and its Polynomial of Certain Families of Benzenoid Systems Volume, Issue, Year : 6(4)(2018) Pages : 43−50

Abstract: Chemical Graph Theory is a branch of Mathematical Chemistry whose focus of interest is to finding topological indices of molecular graph which correlate well with chemical properties of the chemical molecules. In this paper, we propose the augmented Revan index and augmented Revan polynomial of a graph. Also we determine the augmented Revan index and its polynomial of triangular benzenoids, benzenoid rhombus, benzenoid hourglass and jagged rectangle benzenoid systems.

Keywords: Augmented Revan index, augmented Revan polynomial, benzenoid.

 Authors : M. Amin Bahraoui Title : Diagonal Bundle Methods for Convex Minimization : Theoritical Aspect Volume, Issue, Year : 6(4)(2018) Pages : 51−57

Abstract: The bundle method can be seen as an approximate form of the proximal method. We study in this paper the convergence of general bundle method and we give a diagonal version of it.

Keywords: Convex minimization, bundle method, proximal method, Mosco-convergence, diagonalization.

 Authors : Stephanie A. Omega and Ina Marie P. Kintanar Title : Strictly Locating Sets in a Graph Volume, Issue, Year : 6(4)(2018) Pages : 59−63

Abstract: Let $G$ be a connected graph. A subset $S$ of $V(G)$ is a locating set in $G$ if for all $u,v\in V(G)\backslash S$, $N_{G}(u)\cap S\neq N_{G}(v)\cap S$. A subset $S$ of $V(G)$ is a strictly locating set in $G$ if $S$ is a locating set in $G$ and $N_{G}(w)\cap S\neq S$ $\forall w\in V(G)\backslash S$. The minimum cardinality of a strictly locating set in $G$, denoted by $sln(G)$, is called the strictly locating number of $G$. In this paper, the concept of strictly locating set in a graph is investigated. Moreover, the strictly locating sets in the join and corona of graphs are characterized and the strictly locating numbers of these graphs are determined.

Keywords: Locating set, strictly locating set, join, corona.

 Authors : S. Praveena and M. Kamaraj Title : Some New Results on Bipolar-Valued Fuzzy d-Algebras Volume, Issue, Year : 6(4)(2018) Pages : 65−73

Abstract: In this paper we introduce definitions of generalized bipolar-valued fuzzy $d$-algebra and investigate some associate results.

Keywords: Fuzzy $d$-algebra, bipolar-valued fuzzy $d$-algebra, bipolar-valued fuzzy $d$-subalgebra and upper bipolar-valued fuzzy sets, $t$-level-cut of bipolar-valued fuzzy sets and some results.

 Authors : Geetanjali Sharma, Akshay Sharma and Pankaj Tiwari Title : Common Fixed Point Theorems in Menger Space Volume, Issue, Year : 6(4)(2018) Pages : 75−80

Abstract: In this paper, the concept of occasionally weakly compatible maps in Menger space has been introduced to prove common fixed point theorems. Which generalize the well known results.

Keywords: Common fixed points, Metric space, Menger space, Compatible maps, Occasionally weakly compatible mappings and weak compatible mappings.

 Authors : S. N. Banasode and Y. M. Umathar Title : Topological Energies of Some Standard Graphs Volume, Issue, Year : 6(4)(2018) Pages : 81−90

Abstract: Let $G$ be simple, finite, undirected graph (molecular graph) with vertex set $V=\{v_1,v_2,v_3,\dots,v_n\}$ and edge set $E=\{e_1,e_2,e_3,\dots ,e_n\}$. The energy concept originated in chemistry and was introduced by I Gutman in 1978. Similar to the energies like dominating energy, minimum dominating energy, minimum total dominating energy, Zagreb energy is having its own importance in the study of physic-chemical properties of chemical compound that are often modeled by means of topological indices. In this paper, we find, first Zagreb energy [1] (introduced by Gutman), second Zagreb energy and hyper Zagreb energy of some standard graphs like Complete graph, complete bipartite graph, star graph and crown graph.

Keywords: Graph, Topological Indices, Topological Energy, Zagreb matrix, Zagreb energy.

 Authors : Premkumar T. Lalchandani Title : Exact Zero-Divisor Graph of a Commutative Ring Volume, Issue, Year : 6(4)(2018) Pages : 91−98

Abstract: The aim of this article is to continue the study of exact zero-divisor graph of a commutative ring with nonzero identity. We discuss the properties and nature of exact zero-divisor graph and compare some of its properties with zero-divisor graph.

Keywords: Zero Divisor, Exact Zero Divisor, Exact Zero-Divisor Graph.

 Authors : A. Albina and U. Mary Title : A Study on Dominator Coloring of Friendship and Barbell Graphsg Volume, Issue, Year : 6(4)(2018) Pages : 99−105

Abstract: A dominator coloring of a graph $G$ is a proper coloring of $G$ in which every vertex should dominate every vertex of at least one color class. The dominator chromatic number $\chi_d(G)$ is the minimum number of color classes in a dominator coloring of $G$. In this paper we study the dominator chromatic number of the barbell graph $B_n$ and friendship graph $F_n$.

Keywords: Coloring, Domination, Dominator coloring.

 Authors : K. Divya Joseph and P. A. Dinesh Title : The Magnetic, Joule's and Ohmic Effect on Incompressible Viscous Flow Over a Hyperbolic Stretching Circular Cylinder Volume, Issue, Year : 6(4)(2018) Pages : 107−116

Abstract: In we study the heat transfer of boundary layer flow of an incompressible viscous fluid over hyperbolic stretching cylinder. The governing nonlinear partial differential equations are converted into ordinary differential equations by using suitable transformations, which are then tackled using the homotopy method. The homotopy method gives us solutions in the form of series. The influence of the Magnetic, Joule's and Ohmic effect on velocity as well as temperature profiles are investigated and results can be seen visually in graphs. The computational results without these effects agree excellently with the previous results by \cite{1}.

Keywords: Joule's effect, Ohmic effect, Magnetohydrodynamic, incompressible, boundary layer.

 Authors : B. Basavanagoud and Chetana S. Gali Title : On the General Product-connectivity Index of Generalized xyz-point-line Transformation Graphs Volume, Issue, Year : 6(4)(2018) Pages : 117−132

Abstract: The general product-connectivity index is a molecular descriptor defined as $R_{\alpha}(G)=\sum\limits_{uv\in E(G)}(d_{G}(u)d_{G}(v))^\alpha$, where $d_{G}(u)$ denotes the degree of vertex $u$ in a graph $G$ and $\alpha$ is a real number. In this paper, we compute the expressions for general product-connectivity index of generalized \textit{xyz}--point-line transformation graphs.

Keywords: Subdivision graph, Semitotal-point graph, Semitotal-line graph, General product-connectivity index, Generalized \textit{xyz}--point-line transformation graphs.

 Authors : B. C. Dhage, N. S. Jadhav and S. D. Sarkate Title : Differential Inequalities and Comparison Principles for Linearly Perturbed Differential Equations of First Type Volume, Issue, Year : 6(4)(2018) Pages : 133−141

Abstract: In this paper, some results concerning the global existence as well as comparison theorems for an initial value problem of first order hybrid differential equations with a linear perturbation of first type have been proved. The main results rely on the hybrid fixed point technique of Dhage involving the sum of two operators in a Banach space. Our results include several basic results for unperturbed nonlinear differential equations as special cases.

Keywords: First order hybrid differential equation, Hybrid fixed point principle, Existence theorem, Maximal and minimal solutions.

 Authors : Rani Rajeevan and T. K. Mathew Varkey Title : SFS and SFS-2 Domination Volume, Issue, Year : 6(4)(2018) Pages : 143−147

Abstract: Let $G_{A,V}$ be a fuzzy soft graph and $S\subseteq V$ is a fuzzy soft dominating set in $G_{A,V}$, then $S$ is said to be a secure fuzzy soft dominating set if for each vertex $x_i\in V-S$ is adjacent to a vertex $x_j\in S$ such that $\left(S-\left\{x_j\right\}\right)\cup \left\{x_i\right\}$ is a dominating set for all $e\in A$ and the minimum fuzzy soft cardinality taken over all minimal secure fuzzy soft dominating set is called secure fuzzy soft domination number and is denoted by $\gamma_{sefs}(G_{A,V})$.

Keywords: Secure Fuzzy soft domination, Secure Fuzzy soft domination number, total secure fuzzy soft domination, fuzzy soft 2-dominating set, total fuzzy soft 2-dominating set, secure fuzzy soft 2-dominating set, total secure fuzzy soft 2-dominating set.

 Authors : B. Osoba and T. Y. Oyebo Title : On Multiplication Groups of Middle Bol Loop Related to Left Bol Loop Volume, Issue, Year : 6(4)(2018) Pages : 149−155

Abstract: In this paper, we show that an element of Bryant-Schneider group of middle Bol loop $(Q, \circ)$ is an automorphism and pseudo-automorphism. We proved that multiplication groups of middle autotopy nuclei of middle Bol loop $(Q, \circ)$ coincided with the corresponding left Bol loop $(Q, \circ)$ and their multiplication groups were show to be normal subgroups. It was found that the corresponding left Bol loop $(Q, \circ)$ is an indexed two.

Keywords: Quasigroup, Loop, Bryant-Schneider group, middle Bol loop.

 Authors : Mitropam Chakraborty and S. K. Samanta Title : A Fixed Point Theorem for Kannan-type Maps in Metric Spaces Volume, Issue, Year : 6(4)(2018) Pages : 157−161

Abstract: We prove a generalization of Kannan's fixed point theorem, based on a recent result of Vittorino Pata.

Keywords: Kannan map, fixed point, convergence rate.

 Authors : L. Vanitha, K. Dhanalakshmi and C. Ramachandran Title : On Third Order Hankel Determinant for Some Special Class of Analytic Functions Related with Generalized Sakaguchi Functions Volume, Issue, Year : 6(4)(2018) Pages : 163−173

Abstract: In this paper, we investigate the third order Hankel determinant for some special class of analytic functions related with generalized Sakaguchi functions in the open unit disk using subordination.

Keywords: Univalent functions, Starlike functions, Sakaguchi functions, Subordination, Hankel determinant.

 Authors : Prabhat Bansal and Gajendra K. Saraswat Title : An M/M/1 Queuing System with Three Types Input Sources Volume, Issue, Year : 6(4)(2018) Pages : 175−186

Abstract: This paper considers an M/M/1 queuing system with three types input sources, where the rate of arrival and service capacity follow Poisson distribution. The arrival process consists of three stages said to be active state, sick state and passive state. The system remains in three state for a random time which is exponentially distributed. The queue discipline is first-come-first-serve (FCFS). Laplace transforms of the various probability generating functions are obtained and the steady state results are derived. The probability that the arrival process (input) will be in active state, sick state and passive state, is also analyzed.

Keywords: Queuing Theory, Markovian Process, Laplace Transform, Exponential Distribution, Poisson distribution and Probability Generating Function.

 Authors : Sreelekshmi C. Warrier Title : On Soft Multi Games Volume, Issue, Year : 6(4)(2018) Pages : 187−194

Abstract: Molodstov introduced the concept of soft set theory for modelling uncertainty and vagueness.In this paper, after introduce soft multi game and their basic operations. Then we extended some theories based on Nash equilibrium using soft multiset approach. Finally we give an application of decision making problem between three persons using soft multi Nash equilibrium method.

Keywords: Soft set, Soft multiset, Game theory, Soft game, Soft multi game.

 Authors : Asma Ali and Ambreen Bano Title : Identities with Multiplicative (Generalized) $(\alpha,\beta)$-derivations in Semiprime Rings Volume, Issue, Year : 6(4)(2018) Pages : 195−202

Abstract: Let $R$ be an associative ring and $\alpha,\beta$ be automorphisms on $R$. A mapping $F:R\rightarrow R$ (not necessarily additive) is said to be a multiplicative (generalized)-$(\alpha,\beta)$-derivation if $F(xy)=F(x)\alpha(y)+\beta(x)d(y)$ holds for all $x,y\in R$, where $d$ is any mapping on $R$. Suppose that $G$ and $F$ are multiplicative (generalized)-$(\alpha,\beta)$-derivations associated with the mappings $g$ and $d$ on $R$ respectively. The main objective of this article is to study the following situations: (i) $G(xy)+F[x,y]\pm\alpha[x,y]=0$; (ii) $G(xy)+F(x\circ y)\pm\alpha(x\circ y)=0$; (iii) $G(xy)+F(x)F(y)\pm\alpha(xy)=0$; (iv) $G(xy)+F(x)F(y)\pm\alpha[x,y]=0$; (v) $G(xy)+F(x)F(y)\pm\alpha(x\circ y)=0$; for all $x,y$ in some non-zero subsets of a semiprime ring.

Keywords: Prime rings, Semiprime rings, Multiplicative (generalized)-$(\alpha,\beta)$-derivations.

 Authors : Snehlata Rathore and Gajendra K. Saraswat Title : Homotopy Perturbation Approach to the Solution of Non-linear Burger's Equation Volume, Issue, Year : 6(4)(2018) Pages : 203−211

Abstract: The aim of this paper is to build another effective intermittent connection to explain nonlinear Burgers' equation. The homotopy perturbation technique is utilized to explain this equation. Burger's equation is a popular reaction diffusion equation in the biomathematics on the grounds that Burgers equations emerge in numerous applications, it is worth trying new solution methods. In this method, the solution is considered as an infinite series expansion where it converges rapidly to the exact solution. Numerical experimentation demonstrates the precision of a minimum error of order third for different space steps and coefficient of kinematic viscosity. The technique is viewed as high in accuracy. All calculations has been completed utilizing MATLAB.

Keywords: Homotopy perturbation method, Burger's equation, kinematic viscosity, MATLAB.

 Authors : V. R. Kulli Title : Computing Square Revan Index and its Polynomial of Certain Benzenoid Systems Volume, Issue, Year : 6(4)(2018) Pages : 213−219

Abstract: The aim of this paper is to build another effective intermittent connection to explain nonlinear Burgers' equation. The homotopy perturbation technique is utilized to explain this equation. Burger's equation is a popular reaction diffusion equation in the biomathematics on the grounds that Burgers equations emerge in numerous applications, it is worth trying new solution methods. In this method, the solution is considered as an infinite series expansion where it converges rapidly to the exact solution. Numerical experimentation demonstrates the precision of a minimum error of order third for different space steps and coefficient of kinematic viscosity. The technique is viewed as high in accuracy. All calculations has been completed utilizing MATLAB.

Keywords: Homotopy perturbation method, Burger's equation, kinematic viscosity, MATLAB.

 Authors : Sk. Samsul Alam Title : Algebraic Expression: Identification, Sequencing \& Application of its Elementary Concepts Volume, Issue, Year : 6(4)(2018) Pages : 221−230

Abstract: In this paper, total 312 elementary concepts of `algebraic expression' have been identified and sequenced. Out of these only 181 is found to be considered by National Council of Educational Research and Training (N.C.E.R.T.) and 110 by West Bengal Board of Secondary Education (W.B.B.S.E.) in their prescribed text books. The number of use of these elementary concepts together with their frequency of use in the West Bengal, The Council for the Indian School Certificate Examinations (C.I.S.C.E.) and Visva-Bharati final examinations have been identified. Two lists of elementary concepts of algebraic expression, one of the three Boards combined and the other only WB.B.S.E. respectively have been done. Afterwards, a test was developed on the identified top 12 elementary concepts of algebraic expression according to the frequency of their use in Madhyamik Pariksha (M.P.) under W.B.B.S.E. The test was then applied on $X^{th}$-Standard 194 students (boys-107, girls-87). The sample was selected randomly from Bengali medium schools of Burdwan district and affiliated to W.B.B.S.E. Several materials and methods have been adopted for the study which are mentioned above.

Keywords: Algebraic expression, major concept, elementary concept, identification, sequencing, frequently used elementary concepts, text book scanning process, task analysis technique, achievement test.

 Authors : N. V. Srinivasa Rao, M. N. R. Chowdary and D. Bala Krishna Title : A Note on Study of Game Problem Using Simplex Method Volume, Issue, Year : 6(4)(2018) Pages : 231−233

Abstract: In this paper, optimal strategies of game problem which does not have a saddle point using simplex (Big-M) method calculated.

Keywords: Linear programming problem, Game problem, Optimal strategies, Big-M method.

 Authors : S. Zion Chella Ruth Title : Entire Double Sequence Space of Interval Numbers Volume, Issue, Year : 6(4)(2018) Pages : 235−239

Abstract: In the past decades, modal analysis has become a major technology in the quest for determining, improving and optimizing dynamic characteristics of engineering structures. Not only has it been recognized in mechanical and aeronautical engineering, but modal analysis has also been discovered in profound applications for civil and building structures, space structures, transportation and nuclear problems [4]. In this paper we introduced the new concept of double interval sequence spaces $\Gamma (gI)$ and $\Lambda (gI)$. We present the different properties like completeness, solidness etc. Also, we have given some new definitions and theorems about the sequence space of double interval numbers.

Keywords: Complete metric space, double interval numbers, dual sequence space, solid.

 Authors : W. Stanly and R. Vasanthakumari Title : A Study on Dusty Couple Stress Fluid Heated From Below in the Presence of Horizontal Magnetic Field with Hall Currents Volume, Issue, Year : 6(4)(2018) Pages : 241−250

Abstract: An investigation is made on the effect of Hall currents and uniform horizontal Magnetic field on the thermal stability of dusty couple-stress fluid is considered. The analysis is carried out within the limitation of framework of linear stability theory and normal mode technique. A dispersion relation governing the effect of dust particles, Hall currents, magnetic field and couple stress are derived. For the case of stationary convection, dust particles and Hall currents are found destabilizing effect whereas couple-stress has stabilizing effect on the system. Magnetic field has a stabilizing or destabilizing effect on the thermal convection under the restrictions. It has been observed that oscillatory modes are introduced due to the presence of magnetic field and Hall currents which were non-existent in their absence. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics.

Keywords: Thermal convection, couple-stress fluid, magnetic field, Hall currents, Dust particles.

 Authors : Tripti Sharma Title : An Algorithm to Balance an Unbalanced Transportation Problem to Achieve Improved Initial Solution Volume, Issue, Year : 6(4)(2018) Pages : 251−256

Abstract: This paper suggests an algorithm to balance an unbalanced transportation problem in order to obtain improved initial feasible solution of unbalanced transportation problem proceed with Vogel's Approximation Method, in comparison to other existing algorithms to balance the transportation problem. The algorithm is supported by numerical examples.

Keywords: Unbalanced Transportation Problem, dummy row/column, Transportation Problem, Vogel's Approximation Method.

 Authors : S. Aditya Krishnan Title : New Method For Optimal For Transportation Problem (Aditya's Method) Volume, Issue, Year : 6(4)(2018) Pages : 257−262

Abstract: The main aim of this paper is to provide a new method (Aditya's Method) to obtain the optimal solution for transportation by skipping the modi method and obtain the correct result.

Keywords: Vogel's Approximation Method, MODI method, Initial Basic Feasible Solution, Optimality Test, Transportation Problem.

 Authors : K. Selvam Title : New Approach to Find the Maxima and Minima of a Function Volume, Issue, Year : 6(4)(2018) Pages : 263−266

Abstract: In this paper we have introduced a new and effective method to find the maxima and minima of a function. The proposed method is very short compared with existing methods. The proposed method is illustrated by the examples.

Keywords: Differentiation, Homogenous functions, and variables.