## Issue 4Volume 5 (2017)

ISSUE 4− A ISSUE 4− B ISSUE 4− C ISSUE 4 − D ISSUE 4 − E ISSUE 4 − F
 Authors : Stephen Dinagar and Kamalanathan Title : Finding Fuzzy Critical Path using Revised PILOT Method of Ranking Volume, Issue, Year : 5(4−D)(2017) Pages : 429−436

Abstract: In this paper a new ranking method called Revised Point of Intersection of Legs of Trapezium (PILOT)'' have been proposed used to find the fuzzy critical path, fuzzy earliest, fuzzy latest times and fuzzy floats in a project management problem. It is quiet interest to study in calculating critical path using generalized trapezoidal fuzzy number. A numerical example is given for justifying the proposed method.

Keywords: Defuzzification, Fuzzy Number, Ranking of Fuzzy Numbers, Fuzzy Critical Path.

 Authors : Chiranjilal Kujur, D.A.Xavier, S.A.Amirtha Raja and F.Xavier Title : L(2,1)-Labeling for Bloom Graph Volume, Issue, Year : 5(4−D)(2017) Pages : 437−447

Abstract: An $L(2,1)-$labeling of a graph is a function from the vertex set $V(G)$ to the set of all non-negative integers such that $\vert f(u)-f(v)\vert \ge 2$ if and are vertices and $\vert f(u)-f(v)\vert \ge 1$ if $d(u,v)=2$, where $d(u,v)$ denotes the distance between u and v in G. The $L(2,1)$-labeling number of G, denoted by $\lambda (G)$, is the smallest number k such that there is an $L(2,1)$-labeling with maximum label k. In this paper we determine $L(2,1)$-Labeling for Bloom Graph.

Keywords: $L(2,1)$-labeling, $L(2,1)$-numbering, Bloom Graph.

 Authors : N. Balamani and A. Parvathi Title : $\psi^{*}\alpha$-Closed Sets in Bitopological Spaces Volume, Issue, Year : 5(4−D)(2017) Pages : 449−456

Abstract: In this paper we introduce $\psi^{*}\alpha$-closed sets in bitopological spaces and obtain the relationship between the other existing closed sets. Also we study the notion of $(i, j)$-$\psi^{*}\alpha$-closure operator and some of its properties. As applications we introduce $(i, j)$-${}_{\psi^*\alpha}T_{c}$-space, $(i, j)$-${}_{\psi^*\alpha}T_{\alpha}$-space and study some of their properties.

Keywords: $\psi^{*}\alpha$-closed set, $\psi^{*}\alpha$-open set, $\psi$g-open set, $\tau_{i}$-open set.

 Authors : Mukund V. Bapat Title : A Note on L-cordial Labeling of Graphs Volume, Issue, Year : 5(4−D)(2017) Pages : 457−460

Abstract: We discuss L-cordial labeling of some families of graph.We show that Book graph $\theta(C_{4},n)$, Book graph $\theta(C_{5},n)$, One point union of $C_{3}$ i.e., $(C_{3})^{n}$, a double triangular snake $2S(C_{3},n)$ are L-cordial.

Keywords: L-cordial labeling, double snake, Book graph, union.

 Authors : S. Muthammai Title : Harmonic and Geometric-Arithmetic Indices of Boolean Function Graph $B(K_p, INC,\overline{K_q})$ Volume, Issue, Year : 5(4−D)(2017) Pages : 461−468

Abstract: For any graph G, let $V(G)$ and $E(G)$ denote the vertex set and edge set of G respectively. The Harmonic index $H(G)$ of a graph G is defined as the sum of the weights $\frac{2}{d(u)+d(v)}$ of all edges uv of G and the Geometric-Arithmetic index $GA(G)$ of G is defined as the sum of the weights $\frac{2\sqrt{d(u)d(v)} }{d(u)+d(v)}$, where $d(u)$ denotes the degree of a vertex u in G. The Boolean function graph $B(K_p, INC,\overline{K_ q})$ of G is a graph with vertex set $V(G)\cup E(G)$ and two vertices in $B(K_p, INC,\overline{K_ q})$ are adjacent if and only if they correspond to two adjacent vertices of G, two nonadjacent vertices of G or to a vertex and an edge incident to it in G, For brevity, this graph is denoted by $B_{4}(G)$. In this paper, lower and upper bounds of $H(B_{4}(G))$ and $GA(B_{4}(G))$ are obtained. These indices are found for some particular graphs.

Keywords: Harmonic index, geometric-arithmetic index, Boolean Function Graph.

 Authors : Sarfraz Ahmed and Amar Jyoti Goswami Title : Mathematical Modeling for Poiseuille Flow of Blood Fluid Model with an Axial Velocity and Variation of Apparent Viscosity Slip Along an Artery Wall in Presence of Magnetic Field Volume, Issue, Year : 5(4−D)(2017) Pages : 469−477

Abstract: The effects of non-Newtonian nature of blood and pulsatility on flow through stenosed artery have been investigated. It is of interest to note that the thickness of the viscous flow region with axial distances. Mathematical modeling for poiseuille flow of blood fluid model with an axial velocity slip along an artery wall in presence of magnetic field, is considered. It is observed that when Hartmann number increases the fluid velocity is greatly affected. Many standard result regarding Newtonian fluid flows, uniform and steady flow in an artery can be obtained in the present analysis as the special cases. Applications of this theoretical modelling to cardiovascular diseases and the role of slip in the better functioning of the diseased or occluded arteries are included in brief.

Keywords: Bingham plastic fluid model Reynolds Number, Hartmann number, Velocity profile, Non -Newtonian fluid.

 Authors : A. Manshath, V. Balaji and P. Sekar Title : On Relaxed Skolem Mean Labling For Five Star Volume, Issue, Year : 5(4−D)(2017) Pages : 479−484

Abstract: In this paper, we prove $\ell \le m < n$, the five star $K _{1,\ell } \cup K _{1,\ell } \cup K _{1,\ell } \cup K _{1,m} \cup K _{1,n}$ is a Relaxed skolem mean graph if $\left|m - n\right| \le 3\ell + 6$ for $\ell = 2,4,3,{\dots}$; $m = 2,3,4,{\dots}$ and $3\ell + m \le n \le 3\ell + m +6$.

Keywords: Relaxed Skolem mean graph and star.

 Authors : E. Krishnan and C. S. Preethi Title : Unit-Regular Semigroups and Rings Volume, Issue, Year : 5(4−D)(2017) Pages : 485−490

Abstract: Multiplicative semigroups of rings form an important class of semigroups and one theme in the study of semigroups is how the structure of this semigroup affects the structure of the ring. An important tool in analyzing the structure of a semigroup are the Green's relations. In this paper, we study some properties of these relations on the multiplicative semigroup of a unit regular ring with unity. This also gives easier proofs of some known results on ring theory.

Keywords: Sandwich set, Dedekind finite, I-square, Green's relations, unit-regular rings.

 Authors : Pablo I. Pucheta Title : A New Extended Riemann-Liouville Fractional Operator Volume, Issue, Year : 5(4−D)(2017) Pages : 491−498

Abstract: In this paper we will introduce a new and modified Riemann-Liouville fractional operator that resulted from modifying the extended fractional derivative due to M. Ozarslan (\cite{5}). We will study some familiar functions regarding this new operator, the transform Laplace and Mellin are calculate of the potential function and we will also define a new hypergeometric function in term of extended beta function due to Pucheta (\cite{11}).

Keywords: Extended beta function, Hypergeometric function, Fractional Calculus, Laplace and Mellin transform.

 Authors : Jitendra Awasthi Title : A Subclass of Starlike Functions Defined with a Differential Operator Volume, Issue, Year : 5(4−D)(2017) Pages : 499−504

Abstract: This paper deals with a new subclass of starlike functions $A_{n,\lambda }^{\ast }(p,\mu ,$ $\beta ,\gamma ,\delta )$. Coefficients inequality, distortion theorems and closure theorems have been obtained for this class. Further radii of starlikeness and convexity are also obtained for this class.

Keywords: Analytic functions, univalent, al-oboudi differential operator, extremal functions, Starlikeness, convexity.

 Authors : G.Somashekhara, N.Pavani and S.Girish Babu Title : A Note on Invariant Submanifolds of LP-Sasakian Manifolds Volume, Issue, Year : 5(4−D)(2017) Pages : 505−512

Abstract: The object of this paper is to obtain some necessary and sufficient conditions for an invariant submanifold of a LP-Sasakian manifold to be totally geodesic.We consider the pseudo projective and Quasi conformal invariant submanifolds of Lorentzian para-sasakian manifolds.

Keywords: Invariant submanifold, LP-sasakian manifold, totally geodesic.

 Authors : R.Krishnakumar and T.Mani Title : Common Fixed Point of Contractive Modulus on Complete Metric Space Volume, Issue, Year : 5(4−D)(2017) Pages : 513−520

Abstract: In this paper, we have proved the existence of unique common fixed point of contractive maps on complete metric space through a weakly compatible maps and contractive modulus.

Keywords: Complete Metric Space, Common Fixed Point, Contractive Modulus, Weakly Compatible.

 Authors : Vivek S.Sharma and Milan Joshi Title : A Comparison Between Two Approaches to Solve Functional Differential Equations: DTM and DJM Volume, Issue, Year : 5(4−D)(2017) Pages : 521−527

Abstract: Differential Transform Method (DTM) is a Taylor series method for finding Adomian Polynomials along with the method of steps which is used to solve many linear and non-linear differential equations or systems of physical significance. Whereas, the Daftardar-Gejji andJafari Method (DJM) is a method developed by Varsha Daftardar-Gejji and H. Jafari \cite{17} to solve non-linear Integral equations, Fractional differential equations, system of ODEs and so on. Montri Thongmoon and Sasitorn Pusjuso \cite{13} have compared DTM and Laplace transform method to solve system of differential equations. Josef Rebenda et al. \cite{2,10,11} have recently shown that DTM can be applied to functional differential equations with delay, whereas, Sachin Bhalekar and Jayvant Patade \cite{16} have recently obtained solutions of system of delay differential equations with proportional delay using DJM in the form of a special function and studied it's properties. We compare these two iterative methods to solve differential equations. We further try to explore the possibility of these two numerical methods on delayed, hybrid dynamical and delayed hybrid models.

Keywords: Functional Differential Equations, Differential Transform Method (DTM), delayed hybrid models, Daftardar-Gejji and Jafari Method(DJM).

 Authors : G. Balaji, K. Geetha and N. Rajesh Title : On $\omega$-compact Spaces Volume, Issue, Year : 5(4−D)(2017) Pages : 529−534

Abstract: In \cite{J} the author, introduced the notion of $\omega$-compactness and investigated its fundamental properties. In this paper, we investigate some more properties of this type of compact space.

Keywords: Topological spaces, $\omega$-open set, $\omega$-compact spaces.

 Authors : S. Venkatesan and P. Sekar Title : Exceptionality of Edge Graceful Labeling of Tricycle Graphs Volume, Issue, Year : 5(4−D)(2017) Pages : 535−538

Abstract: Graceful labeling is assignment of distinct labels from $f\{0, |E|\}$ to vertices of graph, where edges are labeled by difference of absolute values of adjacent vertices and every label from $f\{1, 2, \dots, |E|\}$ is used exactly once as an edge label. The edge - gracefulness is reversal of gracefulness. It involves labeling edges, rather than vertices, with an induced numbering consisting of sums of edge- labels, rather than differences of vertex labels. In this paper, we are discussing about exceptionality of edge graceful labeling of tricycle graphs by the verification by Lo's theorem.

Keywords: Graph labeling, graceful labeling, edge - graceful labeling, tricycle graph.

 Authors : K. Bhavya, G. Somashekhara and G.S. Shivaprasanna Title : Ricci Solitons on ($\varepsilon,\delta$) Trans-Sasakian Manifolds Volume, Issue, Year : 5(4−D)(2017) Pages : 539−545

Abstract: The object of this paper is to study the characterisation of Ricci Solitons in generalised ricci recurrent and $\phi$ recurrent ($\varepsilon,\delta$) trans-sasakian manifolds based on the 1-form.

Keywords: Ricci, $\phi$-recurrent, pseudo-projective, ($\varepsilon,\delta$) Trans sasakian, Ricci recurrent, shrinking, expanding, steady.

 Authors : Ritika and Savita Rathee Title : Strong and Delta-Convergence Theorems for Multivalued Mappings in CAT(0) Spaces Volume, Issue, Year : 5(4−D)(2017) Pages : 547−553

Abstract: In this paper we suggest two new three-step iterative schemes for the certain class of multi-valued mappings in CAT(0) spaces and also prove the strong and delta-convergence results for such iterative processes.

Keywords: Strong convergence, $\Delta$-convergence, Quasi-nonexpansive mappings, Condition (E), CAT(0) spaces.

 Authors : Ritika Title : CR-Iterative Process for Asymptotically Quasi- Nonexpansive Mappings Volume, Issue, Year : 5(4−D)(2017) Pages : 555−560

Abstract: In this paper strong convergence theorems for CR-iterative process for asymptotically quasi-nonexpansive mappings are established in the framework of CAT(0) spaces.

Keywords: CAT(0) spaces, Asymptotically Quasi-Nonexpansive mapping, Strong convergence, CR-iterative process.

 Authors : Vishal Vincent Henry, Swapnil Srivastava and Ajit Paul Title : Group Extension Through c-left Groupoid Volume, Issue, Year : 5(4−D)(2017) Pages : 561−564

Abstract: In this paper the structure of c-left groupoid has been defined and on the basis of that group extension of G of a group H has been solved for which S will be a left transversal to H in G such that the corresponding c-left groupoid is the given one.

Keywords: Groupoid, left transversals, group extension.

 Authors : Deepak Kumar Title : A Study On Standard Approach Applied to the Motion of a Celestial Body in Our Universe Volume, Issue, Year : 5(4−D)(2017) Pages : 565−566

Abstract: In this article, we describe that in standard mechanics,the energy of a celestial body circulating in their axis is constant. As we know, that in their axis is constant. As we know, that in Newton gravitational law, this energy is defined as the product of masses (one larger and other small) entering in the gravitional attraction as well as the size of the major semiaxis as we study in kepler laes. The objective of this paper is that the energy of moving body is decreasing when the quantum mechanic of the body is considered.

Keywords: Quantum Mechanics, Keplar Laws, Newtonian Mechanics.

 Authors : C.Sunitha Title : On Special Types of Numbers Volume, Issue, Year : 5(4−D)(2017) Pages : 567−570

Abstract: A positive integer n is said to be multiply perfect number if there is a k such that $\sigma\left(n\right)= kn$, where $k\geq1$. In this paper we survey some results of interest on perfect numbers, multiply perfect numbers, k-hyperperfect numbers, superperfect numbers and k-hyper super perfect numbers. To state some results established earlier we have (1). If $n=3^{k-1}\left(3^k-2\right)$ where $3^k-2$ is prime, then $n$ is a 2-hyperperfect number. (2). If $n=3^{p-1}$ where p and $\frac{3^p-1}{2}$ are primes, then n is a super-hyperperfect number.

Keywords: k-perfect numbers, hyperperfect numbers.

 Authors : Sangeeta Baruah and M. Borah Title : Assam's Inter-District Disparity in Education Sector Volume, Issue, Year : 5(4−D)(2017) Pages : 571−578

Abstract: This paper documents the extent of inequality in educational opportunity amongst the districts of Assam spanning the data of the year 2013-2014. The level of educational development is estimated with the help of composite index based on optimum combination of different developmental indicators. The main objective of the study is to classify the districts into different stages of development such as high level, medium level, developing and low level developed. The composite indicators are obtained with the help of two different methods. The district-wise data in respect of twenty five indicators are used f or twenty seven districts of the State. Irrespective of the indicators used, the district of Dima Hasao stands out as the least unequal in terms of educational opportunities. However, the district of Tinisukia stands at the bottom of the list in this regard. Ranks of the districts in the level of educational development obtained from both the methods are compared. For bringing out uniform regional development, potential targets have been estimated for low developed districts.

Keywords: Composite index, Model districts, Development indicators, Potential targets.

 Authors : Krishna Prasad and Bani Mukherjee Title : Modelling of an Inventory-Transportation Problem with Non Instantaneous Deteriorating Items Incorporating Partial Backlogging Volume, Issue, Year : 5(4−D)(2017) Pages : 579−587

Abstract: An optimal replacement policy of non-instantaneous deteriorating inventory has been developed to determine the transportation policy. The model has been studied considering exponentially increasing demand. The partial backlogging has been taken in to consideration which is depended on waiting time for next replacement. The total average cost of the system has been derived for various cases. We have also carried out the sensitivity analysis to identify the sensitiveness of the various parameter used in this model. Some interesting numerical examples have been presented to illustrate the model.

Keywords: Partial backlogging, non-instantaneous, transportation, deterioration.

 Authors : G. Uthra, K. Thangavelu and S. Shunmugapriya Title : Ranking Generalized Intuitionistic Pentagonal Fuzzy Number by Centroidal Approach Volume, Issue, Year : 5(4−D)(2017) Pages : 589−593

Abstract: In this paper we define a Generalized Intuitionistic Pentagonal fuzzy number and propose a new ranking formula which includes the area of both membership and non membership parts of the fuzzy number. The membership and the non membership area of the fuzzy number is splitted into three plane figures and centroid of the centroids of these plane figures are calculated. The ranking formula is calculated by finding the area of this centroid from the origin. The advantage of this paper is that the ranking GIPFN by this approach yields better solution when compared with ranking by Accuracy function. This approach is illustrated with numerical examples.

Keywords: Generalized Pentagonal fuzzy number, Intuitionistic fuzzy number, centroid.

 Authors : Tariq Ahmad Koka, V.H.Badshah and Riyaz Ahmad Shah Title : Single and Multi Server Queuing Models: A Study Volume, Issue, Year : 5(4−D)(2017) Pages : 595−603

Abstract: This paper describes queuing system and queuing networks which are successfully used for performance analysis of different systems such as computer, communications, transportation networks and manufacturing. Queues or waiting lines are a common phenomenon in everyday life. The essence of this phenomenon is the low efficiency of queuing system. In this paper we treat elementary queuing models. Attention is paid to methods for analysis of these models and also to application of queuing models. The queuing number, the service windows number, and the optimal service rate are investigated by means of queuing theory. The time of customer in a waiting line is reduced. The customer satisfaction is increased. By illustration, it is shown that the results are effective and practical.

Keywords: Queuing system, single server model, arrival rate, service rate, infinite and finite models.