## Issue 3 − AVolume 4 (2016)

ISSUE 3 − A ISSUE 3 − B
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 Article Type : Research Article Title : Analysis of Successive Occurrence of Digit 0 in Prime Numbers till 1 Trillion Country : India Authors : Neeraj Anant Pande

Abstract: All primes less than $1$ trillion are analyzed for successive occurrence of digit $0$ in them. Multiple successive occurrences of $0$'s are examined. The first and last instances of successive occurrence of all possible repetitions of $0$'s are determined within initial $12$ ranges of increasing powers of $10$.

Keywords: Prime numbers, digit $0$, successive occurrences.

Neeraj Anant Pande
Associate Professor, Department of Mathematics \& Statistics,Yeshwant Mahavidyalaya (College), Nanded, Maharashtra, India.
E-mail: napande@gmail.com

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 Article Type : Research Article Title : Spaces Over Non-Newtonian Numbers Country : India Authors : P.Sharma and S.Mishra

Abstract: We construct the spaces of non-Newtonian numbers using some new operations between them. Then we define generalized metric and generalized norm on the set of non-Newtonian numbers which attains the value as a non-negative non-Newtonian real number (depending on system of measurement) rather than non-negative real. Further, we have studied some order and convergence structures on these spaces.

Keywords: Non-Newtonian numbers, Ordered vector space, $*_{\gamma}$-Banach space.

P.Sharma
Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India.
E-mail: pranav15851@mgail.com

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 Article Type : Research Article Title : Fractional Order Model in Generalist Predator--Prey Dynamics Country : Saudi Arabia Authors : Moustafa El-Shahed, A. M. Ahmed and Ibrahim M.E.Abdelstar

Abstract: In this paper, we construct the fractional order generalist predator--prey dynamics model. The stability of free and positive fixed points is studied. The Adams-Bashforth-Moulton algorithm has been used to solve and simulate the system of differential equations.

Keywords: Generalist predator, Fractional order, Stability, Limit cycle, Numerical method.

Moustafa El-Shahed
Department of Mathematics, Faculty of Arts and Sceinces, Qassim University, Qassim, Unizah, Saudi Arabia.
E-mail: elshahedm@yahoo.com

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 Article Type : Research Article Title : Commutators of Fractional Integral with Variable Kernel on Variable Exponent Herz and Lebesgue Spaces Country : P.R. China Authors : Afif Abdalmonem, Omer Abdalrhman and Shuangping Tao

Abstract: In this paper, we study the boundedness for commutators of the fractional integral with variable kernel on Lebesgue spaecs with variable exponent. Also, we study the boundedness of the fractional integral operator and their commutator generated by BMO function is obtained on those Herz spaces with two variable exponent $p(\cdot), q(\cdot)$.

Keywords: Fractional integral, variable kernel, commutator, variable exponent, BMO function, Lebesgue spaecs, Herz spaces.

Afif Abdalmonem
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, P.R. China.
E-mail: afeefy86@gmail.com

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 Article Type : Research Article Title : Collatz Conjecture for Modulo an Integer Country : India Authors : T.Kannan and C.Ganesa Moorthy

Abstract: A function $T_m$ from a set \{1,2,3,...$m$\} into itself defined by \\$T_m(x)=\frac{x}{2},$ for even $x$ and by $T_m(x)=\frac{3x+1}{2}$ (mod $m$), for odd $x$ is considered in this article. The asymptotic behaviour of this function is studied in this article for some cases.

Keywords: Congruence, Collatz conjecture.

T.Kannan
Department of Mathematics, Sree Sevugan Annamalai college, Devakottai, TamilNadu, India.
E-mail: hardykannan@gmail.com

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 Article Type : Research Article Title : Time Dependent Solution of M/M/C Feedback Queue With Catastrophes Country : India Authors : S.Shanmugasundaram and S.Chitra

Abstract: In this paper we study the time dependent solution of M/M/C feedback queue under the catastrophic effect. To solve transient solution of this model, we use generating function technique. We derive stationary probability distribution of this model and we give numerical example for the feasibility of this model.

Keywords: Catastrophes, Departure Process, Feedback customer, Poisson process, Stationary probability distribution, Transient probability.

S.Chitra
Department of Mathematics, Sengunthar College of Engineering, Thiruchengode, Namakkal, Tamilnadu, India.
E-mail: chitrasmaths@gmail.com

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 Article Type : Research Article Title : A Time Truncated Two Stage Group Acceptance Sampling Plan For Compound Rayleigh Distribution Country : India Authors : A.R.Sudamani Ramaswamy and N.Sowmya

Abstract: In this paper, two stage group sampling plan is developed assuming that life time of the test units follow Compound Rayleigh Distribution and the life tests is terminated at a prefixed time. The minimum number of groups required for ensuring the specified mean life at specified consumer's confidence level have been determined. The Operating characteristic values for various quality levels are obtained and the results are discussed with the help of tables and examples. The minimum mean ratios are also obtained for a specified level of producer's risk.

Keywords: Acceptance sampling, Compound Rayleigh distribution, Producer's risk, Consumer's risk, Operating Characteristic curve, Truncated life test, Two stage Group Sampling Plan.

N.Sowmya
Department of Mathematics, Avinashilingam Institute For Home Science and Higher Education for Women, Coimbatore, India.
E-mail: sowmyanagarajan92@gmail.com

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 Article Type : Research Article Title : Interpretation of Planar Ternary Ring in Desargues Plane Like Usual Ring Country : Macedonia Authors : Flamure Sadiki, Alit Ibraimi, Krutan Rasimi and Agim Rushiti

Abstract: Use of ring'' in the determine of a planar ternary ring seems unjustified at first sight. In this paper we show that in Desargues affine plane, in certain condition, planar ternary ring (S, t) turns into usually associative ring $(S,\ +,\ \cdot)$. So, in an affine plane $\mathpzc{A}=(\mathcal{P},\ \mathcal{L},\ \mathcal{I})$ coordinatized from a coordinative system (O, I, OX, OY, OI) and bijection $\sigma :\ (OI)\to S$, determine a ternary operation $t :\ S^{3} \to S$, which we call planar ternary operation. We prove that every coordinatizing affine plane $\mathpzc{A}=(\mathcal{P},\ \mathcal{L},\ \mathcal{I})$ determine a planar ternary ring (S, t), where S is the set of coordinates of affine plane and t its planar ternary operation, and vise versa. Also we introduce the binary operation of addition and multiplication in S and underline relations $a+b=t(a,\ 1,\ b)$, $a\cdot b=t(a,\ b,\ 0)$. Since affine plane is fulfilled with the first Desargues axiom D1, related to the structure $(S,\ +,\ \cdot)$ in that plane, considering some isomorphism imply that $(S,\ +)$ is abelian group. In the following, we show that when in an affine plane except first Desargues axiom D1 also hold the second Desargues axiom D2, then its planar ternary ring $(S,\ t)$ is the usual associative ring $(S,\ +,\ \cdot)$.

Keywords: Ternary operation, planar ternary operation, planar ternary ring, coordinatized affine plane, first D1 and the second D2 Desargues axiom.

Department of Mathematics, University of Tetovo, Tetovo, Macedonia.
E-mail: sflamure@gmail.com

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 Article Type : Research Article Title : A Fixed Point Theorem for Meir Keeler Type Contraction Using Result of Wong, Chi Song Country : India Authors : Sujata Goyal

Abstract: In this paper, we establish a new fixed point theorem for Meir Keeler type contraction using result of Wong [2]. The presented theorem is an extension of the result of Wong [2].

Keywords: Complete metric space, Fixed point, Meir-Keeler type contraction.

Sujata Goyal
Department of Mathematics, A.S.College, Khanna, India.
E-mail: sujatagoyal636@yahoo.com

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 Article Type : Research Article Title : Deriving And Deducing The Equation Of The Curve Of Quickest Descent Country : India Authors : Priyanka Priyadarshini Mishra

Abstract: In Mathematics, we repeatedly come across the concept that a straight line is the shortest distance between two points. However hardly we find discussions on the curve that takes minimum time. My essay consider this issue not limited to 2-Dimensional but extended to spheres and other complex geometrical shapes and explores the research question “How do we find the equation to the curve of quickest descent between two places using calculus and justify the same comparatively with different plane curves?” The investigation aims to derive the equation of the curve of quickest descent. Since time is a crucial element in everyone’s life, achieving the aim can expunge some of the biggest problems.\\ The extended essay explores the research subject step wise with consideration of certain assumptions, however most of the paramount factors are included in the essay. It investigates different curves such as straight line, parabolic arc, elliptical arc and cycloid on 2D plane taking the example of a ball (negligible mass) allowed to travel along slides of different shapes. Then the result of part 1 is applied to a real life situation where it is proposed that if a road or an underground tunnel were built between the cities of Bhubaneswar, Orissa, India and New Delhi, India, which shape should it take to optimize travel time. Second part of essay examines a larger scope of problem. I employ important identities and formulae such as Beltrami Identity and Great Circle Distance Formula as well as calculus of Variations to derive the equations of the curves especially cycloid and hypocycloid.\\ The essay concludes in the first part that given a small-scale example, the curve of quickest descent is a cycloid. With this result, it further goes on to conclude that Hypocycloid is the Brachistochrone between Bhubaneswar and New Delhi.

Keywords: Brachistochrone, curve of quickest descent, hypocycloid.

D-62 Utkal Royal Residency, Gautam Nagar, Kalpana Square, Bhubaneswar, India.
E-mail: mishra.priyanka99@gmail.com

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 Article Type : Research Article Title : A Generalisation of Quasi-Convex Sequence Spaces Country : India Authors : Ajaya Kumar Singh

Abstract: The object of the paper is to generalise the concept of quasi-convex sequence spaces by means of the generalised the fractional difference.

Keywords: Banach limit, Banach spaces, Binomial co-efficient, Cauchy sequence, fractional difference, Generalised almost convergence, Generalised binomial coefficient, Linear space, Quasi-convex sequence spaces, Sub-linear functional, Topological properties.

Ajaya Kumar Singh
Department of Mathematics, P.N. College (Autonomous), Khordha, India.
E-mail: ajayakumarsingh1966@gmail.com

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 Article Type : Research Article Title : Effects of Thermal Radiation and Variable Fluid Viscosity on Stagnation Point Flow Past a Porous Stretching Sheet Embedded in Porous Medium With Partial Slip Condition Country : India Authors : Manish Raj, Abhay Jha and Anil Sharma

Abstract: The effects of thermal radiation and variable fluid viscosity on stagnation point flow past a porous stretching sheet embedded in a porous medium with partial slip condition are studied. The governing non-linear partial differential equation are transformed to a system of non-linear ordinary differential equation by using suitable similarity transformation. The transformed governing equation are solved numerically using Runge-Kutta method along with shoating technique the result are presented in the graphical form. The effects of various physical parameters on the velocity and temperature profiles are discussed.

Keywords: Temperature-dependent fluid viscosity, function stagnation point flow, thermal radiation, porous medium, partial slip condition.

Abhay Jha
Department of Mathematics, JECRC University, Jaipur, India.
E-mail: itsabhay@rediffmail.com

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 Article Type : Research Article Title : New Class of $\alpha$g*p-Continuous Functions in Topological Spaces Country : India Authors : J.Raja Kumari and C.Sekar

Abstract: The aim of this paper is to introduce and study the new class of functions namely $\alpha$g*p-totally continuous functions and totally $\alpha$g*p-continuous functions. Further the relationship between this new class with other classes of existing functions are established.Also strongly ($\alpha$g*p)*-continuous functions in topological spaces are introduced and studied.

Keywords: $\alpha$g*p-totally continuous function, totally $\alpha$g*p-continuous function, strongly ($\alpha$g*p)*-continuous function, $\alpha$g*p-co-Hausdorff space.

J.Raja Kumari
Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur, TamilNadu, India.
E-mail: rajakumarieben@gmail.com

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 Article Type : Research Article Title : Snakes Related Sum Perfect Square Graphs Country : India Authors : S.G.Sonchhatra and G.V.Ghodasara

Abstract: Let $G = (V, E)$ be a simple $(p,q)$ graph and $f:V(G)\rightarrow\{0,1,2,\ldots, p-1\}$ be a bijection. We define $f^{\ast}:E(G)\rightarrow\ \mathbb{N}$ by $f^{\ast}(uv)=(f(u))^{2} + (f(v))^{2} + 2 f(u) \cdot f(v)$, $\forall uv \in E(G)$. If $f^{\ast}$ is injective, then $f$ is called sum perfect square labeling. A graph which admits sum perfect square labeling is called sum perfect square graph. In this paper we prove that several snakes related graphs are sum perfect square.

Keywords: Sum perfect square graphs, triangular snakes, quadrilateral snakes.

S.G.Sonchhatra
Department of Mathematics, Government Engineering College, Rajkot, India.
E-mail: sonchhabdasunil.20@mail.com

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 Article Type : Research Article Title : Lattice Labelling of a Graph Country : India Authors : S.Lakshminarayana and M.Vishukumar

Abstract: Let $G(u,v)$ be a connected graph. Let $L = (L, \wedge,\vee)$ be a lattice under partial order $\tau$. Let $f:v\longrightarrow{L}$ be a injective function. Let $f^\star:E\longrightarrow L'$ be the function induced by f such that $f^\star(xy)= [f(x)\vee f(y)]-[f(x)\wedge f(y)]$ then we say f is a $\tau$ labeling if L is injective and L' is a subset of L. A $\tau$ labeling is called lattice labeling if L' is a sub lattice of L. In this paper we give lattice labeling of certain graphs.

Keywords: Graph Labeling, Lattice labelling and $\tau$ labelling.

S.Lakshminarayana
School of Physical Sciences and Computer Applications, REVA University, Bangalore, India.
E-mail: narayan\_sl@reva.edu.in

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 Article Type : Research Article Title : Best Proximity Points for Generalized $\alpha-\psi$-Geraghty Contractive Mappings Country : India Authors : S.Arul Ravi and A.Anthony Eldred

Abstract: In this paper we consider a generalization of $\alpha-\psi$-Geraghty contractions and investigate the existence and uniqueness of best proximity point for the mappings satisfying this condition using the P-property.

Keywords: Best proximity point, P-property, $\alpha-\psi$-Geraghty contraction, $\alpha$-admissible, triangular-$\alpha$-admissible, $\alpha$-regular.

S.Arul Ravi
P.G. and Research Department of Mathematics, St.Joseph's College (Autonomous), Tiruchirappalli, India.
E-mail: ammaarulravi@gmail.com

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 Article Type : Research Article Title : Balanced Mean Cordial Labeling and Graph Operations Country : India Authors : V J Kaneria, M J Khoda and H M Karavadiya

Abstract: Balanced mean cordial labeling is a mean cordial labeling $f$ with \linebreak $|v_{f}(i)- v_{f}(j)|=0$, $|e_{f}(i)- e_{f}(j)|=0$, $\forall$ $i,j\in \{0,1,2\}$. In this paper, we investigate mean cordial labeling for P($t\cdot H$), where H be any graph and $t\equiv 0$(mod $3$). We also investigate balanced mean cordial labeling for TP($t\cdot H$), G$^{*}$, P$_{t}$$\times G, C_{t}$$\times G$, where H and G both are balanced cordial graphs.

Keywords: Path Union, Mean Cordial, Balanced mean cordial.

M J Khoda
Department of Mathematics, VVP Engineering College, Virda Vajadi, Rajkot, Gujarat, India.
E-mail: mjkmaths@gmail.com

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 Article Type : Research Article Title : Generalization of Common Coupled Fixed Point Theorems in Complex Valued b-Metric Spaces Country : India Authors : A.K.Dubey

Abstract: Recently, Azam et al. [1] introduced the complex valued metric space and obtained sufficient conditions for the existence of common fixed points. Rao et al. [20] introduce the notion of complex valued b-metric spaces. In this paper, some common coupled fixed point theorems have been established for a pair of mappings in a complete complex valued b-metric space in view of diverse contractive conditions. Our results extend and improve several fixed point theorems in the literature.

Keywords: Common fixed point, Coupled fixed point, Complete complex valued b-metric space, Complex-valued metric spaces.

A.K.Dubey
Department of Mathematics, Institute of Technology, Bhilai House, Chhattisgarh, India.
E-mail: anilkumardby70@gmail.com

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 Article Type : Research Article Title : Higher Order Backlund Transformation for $u_t+uu_x+\alpha u-u_{xx}+\gamma u_{xxx}+u_{yy} = 0$ Country : India Authors : S.Padmasekaran, S.Lavanya Devi and G.Sivagami

Abstract: The homogeneous balance (HB) method has drawn lots of interests in seeking the solitary wave solution and other kinds of solutions. Wang showed the HB method is powerful for finding analytic solitary wave solutions of PDE. In this paper, the solitary wave solution of nonlinear PDE has been obtained by this method.

Keywords: Homogeneous Balance method, solitary wave solutions, auto-Backlund transformation.

G.Sivagami
Department of Mathematics, SBM College of Engineering and Technology, Dindigul, Tamilnadu, India.
E-mail: drgsivagami@gmail.com

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Abstract: In this paper we establish again that the nonclassical method accounts for more general results than those obtained by direct method and Lie's classical method with the help of a nonlinear parabolic equation with a variable coefficient $u_t= u_{xx} + V(t,x) u^p, p>1$. A perturbation solution for the reduced equation $z^2 ff^{\prime \prime}+l_5 f^2+({1+p})/({1-p})z^2 f^{\prime^2}+\epsilon z^{n_1+2}=0$ is obtained.