## Volume 5 (2017)Article in Press

ISSUE 1 − A
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 Article Type : Research Article Title : The Laplace-Adomian Decomposition Method Applied to the Kundu-Eckhaus Equation Country : India Authors : O.Gonz\'alez-Gaxiola

Abstract: The Kundu-Eckhaus equation is a nonlinear partial differential equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. In spite of its importance, exact solution to this nonlinear equation are rarely found in literature. In this work, we solve this equation and present a new approach to obtain the solution by means of the combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). Besides, we compare the behaviour of the solutions obtained with the only exact solutions given in the literature through fractional calculus. Moreover, it is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

Keywords: Kundu-Eckhaus equation, Nonlinear Schr\"{o}dinger equation, Adomian decomposition method.

O.Gonz\'alez-Gaxiola
Departamento de Matem\'aticas Aplicadas y Sistemas, Universidad Aut\'onoma Metropolitana-Cuajimalpa, Cuajimalpa, M\'exico D.F.
E-mail: ogonzalez@correo.cua.uam.mx

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 Article Type : Research Article Title : A Note on Construction of Finite Field of Order $p$ and $p^2$ Country : India Authors : S.K.Pandey

Abstract: In this note we construct finite field of order $p$ (here $p$ is a positive prime) and $p^{2}$ for $p>2$ through even square elements of $Z_{2p}$. It has been already noticed that a finite field of order $p^{2}$, $p>2$ can be directly constructed without using the concept of quotient rings. We utilize the same technique to yield a finite field of order $p^{2}$ for $p>2$ however here we use the notion of even square elements of a ring $R$. It is noticed that for all the finite fields constructed in this article the reducing modulo $m$ is a composite integer.

Keywords: Finite field, matrix field, matrix, even square element.

S.K.Pandey
Department of Mathematics, Sardar Patel University of Police, Security and Criminal Justice, Daijar, Jodhpur, Rajasthan, India.
E-mail: skpandey12@gmail.com

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 Article Type : Research Article Title : $m(\alpha_{i})-$Series to circulate Functions Arising From Generalized Alpha Difference Equation Country : India Authors : M.M.S.Manuel, G.D.Babu, G.Britto Antony Xavier and M.Meganathan

Abstract: In this paper, authors investigate the summation and closed form relations to certain types of generalized higher order alpha difference equations for finding the values of alpha multi series to circular functions in the field of finite difference methods. We provide examples verified by MATLAB to illustrate the alpha multi series to circular functions.

Keywords: Alpha-difference equation, Alpha multi-series, Closed form solution, Circular functions.

G.Britto Antony Xavier
Department of Mathematics, Sacred Heart College, Tirupattur, Vellore, Tamil Nadu, India.
E-mail: brittoshc@gmail.com

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 Article Type : Research Article Title : Intuitionistic Fuzzy Cone Metric Spaces and Fixed Point Theorems Country : India Authors : A.M.Ali and G.R.Kanna

Abstract: In this paper, the notion of intuitionistic fuzzy cone metric space is introduced. Some common fixed point theorems for occasionally weakly compatible mapping in intuitionistic fuzzy cone metric space are stated and proved.

Keywords: Intuitionistic Fuzzy Cone Metric, Occasionally Weakly Compatible, Fixed Point.

G.R.Kanna
Department of Mathematics, G.T.N. Arts College, Dindigul, Tamilnadu, India.
E-mail: ranjithkanna91@gmail.com

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 Article Type : Research Article Title : $\alpha$, $k$-Laplace Transform with Poly Binomial Numbers Country : India Authors : G.Britto Antony Xavier, B.Govindan, S.John Borg and M.Meganathan

Abstract: In this paper, we introduce $\alpha, \ k$-Laplace Transform which is an extension of Generalized Laplace Transform (GLT) obtained by an inverse difference operator in the field of Digital Signal Processing (DSP). It provides a better platform to study the nature and functioning of relevant signal. Here we take $k$ as variable and $\alpha$ as parameter. Suitable examples with relevant diagrams which are generated and verified using MATLAB are inserted to validate our main results.

Keywords: Generalized Laplace Transform, Generalized difference operator, factorials and Polynomial-binomial.

G.Britto Antony Xavier
Department of Mathematics, Sacred Heart College, Tirupattur, Vellore, Taminadu, India.
E-mail: brittoshc@gmail.com

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 Article Type : Research Article Title : Numerical Solution of Intuitionistic Fuzzy Differential Equation by Milne's Predictor - Corrector Method Under Generalised Differentiability Country : India Authors : V.Parimala, P.Rajarajeswari and V.Nirmala

Abstract: Nowadays many real life problems are identified with Fuzzy set theory. The Fuzzy set theory is a useful tool to describe the situation in which data are imprecise or vague or uncertain. This set theory is completely described by its membership function. A membership function of a classical fuzzy set assigns to each element of the universe of discourse a number from the interval $[0, 1]$ to indicate the degree of belongingness to the set under consideration. The degree of non belongingness is just automatically the complement to \ "1" of the membership degree. But many times, a human being does not express the degree of non membership as the complement to \ "1".There may be some hesitation about the belongingness and non-belongingness. This missing data or hesitation is accomplished by a set known as intuitionistic fuzzy set. In this paper, Milne's Predictor - Corrector method is used for finding numerical solution of an intuitionistic fuzzy differential equation (IFDE). The proposed method is based on the concept of generalized differentiability. IFDE is transformed into four ordinary differential systems and then Milne's Predictor - Corrector method is applied.Also,the convergence and stability of the proposed method is given and its applicability is illustrated by solving a first order IFDE.

Keywords: Intuitionistic fuzzy differential equation, Numerical methods, Milne's Predictor - Corrector, Generalised differentiability.

V.Nirmala
Department of Mathematics, University College of Engineering Tindivanam, Tamil Nadu, India.
E-mail: nirmalaucet@mail.com

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 Article Type : Research Article Title : Another New Proof of the Butterfly Theorem Country : India Authors : Dasari Naga Vijay Krishna

Abstract: In this article we present a new proof of butterfly theorem.

Keywords: Cyclic Quadrilateral, Butterfly theorem, Ptolemy's theorem.

Dasari Naga Vijay Krishna
Department of Mathematics, Narayana Educational Instutions Machilipatnam, Bengalore, India.
E-mail: vijay9290009015@gmail.com

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 Article Type : Research Article Title : Efficiently Dominating ($\gamma$, ed)-Number of Graphs Country : India Authors : A.Mahalakshmi, K.Palani and S.Somasundaram

Abstract: In this paper, we introduce the new concept connected and independent edge detour domination number of a graph and obtain the connected and independent edge detour domination number for some well known graphs.

Keywords: Edge detour domination, efficient domination, efficient($\gamma$,ed)-number of graphs.

A.Mahalakshmi
Department of Mathematics, Sri Sarada College For Women, Tirunelveli, Tamil Nadu, India.
E-mail: lakshmi19maths@gmail.com

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 Article Type : Research Article Title : Positive Solutions of Quadratic Fractional Integral Equation With Generalized Mittag Leffler Function Country : India Authors : Mohammed Mazhar Ul Haque and Tarachand L.Holambe

Abstract: In this paper we consider the quadratic fractional integral equation with generalized mittag leffler function and will find the positive solution of this integral equation. In this paper we will find the existence as well as approximations of the positive solutions for a nonlinear quadratic fractional integral equation with generalized mittag leffler function given on a closed and bounded interval of the real line.

Keywords: Quadratic fractional integral equation, Fractional derivatives and Integrals, approximate solution.

Mohammed Mazhar Ul Haque
E-mail: mazhar-ul-haque@hotmail.com

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 Article Type : Research Article Title : Analysis of $M^{X}/G/1$ Queueing System With Setup and Feedback Under Multiple Adapted Vacation Policy Country : India Authors : S.Nithya Rubini and M.I.Afthab Begum

Abstract: $M^{X}/G/1$ queueing model under server breakdown and server vacation is considered. In this model, it is assumed that, whenever breakdowns occur the server is sent for repair immediately and the service interrupted customers join the head of the queue and repeat a new service as soon as the server is fixed. The server adopts both types of vacation policy. Whenever the server completes a service, he may continue to serve the next customer with probability $\textbf{(1-p)}$ (or) take a single vacation with probability p. At the end of the vacation, the server serves the next customer. The process continues until the system becomes empty. Once the system becomes empty, the server adopts Multiple Adapted Vacation (MAV) policy which is determined by a set of probabilities. It is also assumed that the system requires a setup operation to start a new busy period. Thus the idle vacation period, setup period, busy period, breakdown period and vacation time between two consecutive services during busy period constitute a cycle.

Keywords: Batch arrival, server breakdown, feedback, MAV policy.

S.Nithya Rubini
Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for Women, Tamilnadu, India.
E-mail: @mail.com

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 Article Type : Research Article Title : Some Properties of Semi ${}^{\# }$generalized Open Sets in Topological Spaces Country : India Authors : S.Saranya and K.Bageerathi

Abstract: In this paper a new class of generalized open sets in topological spaces, namely semi ${}^{\# }$generalized open(briefly, s${}^{\#}$g-open) sets is introduced. We prove that this class lies between the class of sg-open sets and the class of gs-open sets and we study some basic properties and characterizations of s${}^{\#}$g-open sets. Also we introduce s${}^{\#}$g-neighbourhood (shortly, s${}^{\#}$-nbhd), s${}^{\#}$g-interior, s${}^{\#}$g-closure and s${}^{\#}$g-derived set of a set in a topological spaces and investigate some basic properties of these sets.

Keywords: s${}^{\#}$g-open set, s${}^{\#}$g-neighbourhood, s${}^{\#}$g-closure, s${}^{\#}$g-interior, s${}^{\#}$g-derived set.

S.Saranya
Department of mathematics, Aditanar college of Arts and science, Tiruchendur, Tamil Nadu, India.
E-mail: saranya7873@gmail.com

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 Article Type : Research Article Title : Some Euler Spaces of Difference Sequences and Matrix Mappings Country : India Authors : Shadab Ahmad Khan and Ashfaque A.Ansari

Abstract: Altay and Ba\c{s}ar [2] and Altay, Ba\c{s}ar and Mursaleen [3] introduced the Euler sequence spaces $e^r_{0\ ,}{\ e}^r_{c\ ,\ \ }\ and\ e^r_{\infty \ ,}$ respectively. Polat and Ba\c{s}ar [25] introduced the spaces $e^r_0$(${\Delta }^{(m)}$), $e^r_{c\ \ }$(${\Delta }^{(m)}$), and $e^r_{\infty \ }$(${\Delta }^{(m)}$) consisting of all sequences whose $m^{th}$ order differences are in the Euler spaces $e^r_{0,}\ e^r_{c\ ,\ \ }and\ e^r_{\infty \ ,}$ respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the $\alpha$-, $\beta$-, $\gamma$-, and continuous duals of the spaces $e^r_0\ ,$ $e^r_{c\ \ },e^r_p,$ $e^r_{\infty \ },$ $e^r_0$(${\Delta }^{(m)}$), $e^r_{c\ \ }$(${\Delta }^{(m)}$) and $e^r_{\infty \ }$(${\Delta }^{(m)}$), for $1\le p<\infty$ and their basis have been constructed. The last section of the article is devoted to the characterization of some matrix mappings on the sequence spaces $e^r_{c\ \ }\$and $e^r_{c\ \ }\mathrm{(}{\Delta }^{(m)}\mathrm{)\ }$.

Keywords: The $\alpha$-, $\beta$-, $\gamma$-, and continuous duals, basis, matrix mappings and difference sequence spaces of order m.

Department of Mathematics & Statistics, D. D. U. Gorakhpur University, Gorakhpur, India.
E-mail: sakhanan15@gmail.com

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 Article Type : Research Article Title : Heronian Mean Labeling of Double Triangular and Double Quadrilateral Snake Graphs Country : India Authors : S.S.Sandhya, E.Ebin Raja Merly and S.D.Deepa

Abstract: In this paper, We contribute some new results for Heronian Mean labeling of graphs. We prove that Double Triangular snake, Alternate Double Triangular snake, Double Quadrilateral Snake and Alternate Double Quadrilateral Snake graphs are Heronian mean graphs.

Keywords: Heronian Mean labeling, Triangular snake, Double Triangular snake, Quadrilateral snake, Double Quadrilateral Snake.

S.D.Deepa
Department of Mathematics, Nesamony Memorial Christian College, Marthandam, Tamilnadu, India.

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