Issue 2Volume 5 (2017)

ISSUE 2− A ISSUE 2− B ISSUE 2− C
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 Article Type : Research Article Title : Generalized U - H Stability of New n - type of Additive Quartic Functional Equation in Non - Archimedean Orthogonally Space Country : India Authors : S.Murthy, V.Govindhan and M.Sree Shanmuga Velan

Abstract: Using direct method, we prove the stability of the orthogonally additive-quartic functional equation of the form $f\left(nx+n^{2}y+n^{3}z\right)+f\left(nx-n^{2}y+n^{3}z\right)+f\left(nx+n^{2}y-n^{3}z\right)+f\left(-nx+n^{2}y+n^{3}z\right) = 2\left[f\left(nx+n^{2}y\right)+f\left(nx-n^{2}y\right)\right] + 2\left[f\left(n^{2}y+n^{3}z\right) + f\left(n^{2}y-n^{3}z\right)\right]+ 2\left[f\left(nx+n^{3}z\right)+f\left(nx-n^{3}z\right)\right]-3n\left[f\left(x\right)-f\left(-x\right)\right]-3n^{2}\left[f\left(y\right)-f\left(-y\right)\right] -3n^{3}\left[f\left(z\right)-f\left(-z\right)\right]-2n^{4}\left[f\left(x\right)+f\left(-x\right)\right]-2n^{8}\left[f\left(y\right) +f\left(-y\right)\right]-2n^{12}\left[f\left(z\right)+f\left(-z\right)\right]$ for all $x,y,z$ in $X$ with $x\bot y$, $y\bot z$ and $z\bot x$ in Banach non-Archimedean Orthogonally spaces. Here $\bot$ is the orthogonally in the sense of Ratz.

Keywords: Additive functional equation, Quartic functional equation, Banach Non-Archimedean Orthogonally spaces.

V.Govindhan
Department of Mathematics, Sri Vidya Mandir Arts and Science College, Uthangarai, Tamil Nadu, India.
E-mail: govindoviya@gmail.com

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 Article Type : Research Article Title : General Solution and Stability of Quadratic Functional Equation Country : India Authors : G.Balasubramaniyan, V.Govindan and C.Muthamilarasi

Abstract: In this paper, the authors present the general solution and generalized Ulam-Hyers stability of quadratic functional equation of the form $f\left(nx+n^{2} y+n^{3} z+n^{4} t\right)+f\left(nx-n^{2} y+n^{3} z+n^{4} t\right)+f\left(nx+n^{2} y-n^{3} z+n^{4} t\right)+f\left(nx+n^{2} y+n^{3} z-n^{4} t\right)+f\left(-nx+n^{2} y+n^{3} z+n^{4} t\right)=f\left(nx+n^{2} y\right)+f\left(nx+n^{3} z\right)+f\left(nx+n^{4} t\right)++f\left(-nx+n^{2} y+n^{3} z+n^{4} t\right)=f\left(nx+n^{2} y\right)+f\left(nx+n^{3} z\right)+f\left(nx+n^{4} t\right)+f\left(n^{2} y+n^{3} z\right)+f\left(n^{2} y+n^{4} t\right)+f\left(n^{3} z+n^{4} t\right)+2n^{2} f(x)+2n^{4} f(y)+2n^{6} f(z)+2n^{8} f(t)$ where n is positive integer with $n\ne 0$ in Banach algebra using direct and fixed point methods.

Keywords: Quadratic functional equation, Banach algebra, generalized Ulam-Hyers stability, fixed point theory.

V.Govindhan
Department of Mathematics, Sri Vidya Mandir Arts and Science College, Uthangarai, Tamil Nadu, India.
E-mail: govindoviya@gmail.com

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 Article Type : Research Article Title : Partial Sums for Multivalent Harmonic Maps Defined by Generalized Hypergeometric Functions Country : India Authors : Vimlesh Kumar Gupta and Poonam Sharma

Abstract: In this paper,we study partial sums of the series of certain multivalent harmonic functions involving the generalized hypergeometric functions which are in the class $P_{H}(m,A,B).$ We establish some new results giving the sharp bounds of the real parts of the ratios of harmonic multivalent functions to its sequences of partial sums.

Keywords: Multivalent harmonic functions, Generalized hypergeometric functions, Partial sums.

Vimlesh Kumar Gupta
Department of Mathematics \& Astronomy, University of Lucknow, Lucknow, UP, India.
E-mail: vim987@gmail.com

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 Article Type : Research Article Title : Paired Domination of Certain Nanotubes Country : Kwara State Authors : A.S.Shanthi and A.Sajiya Merlin Mahizl

Abstract: A vertex subset \textit{D} of a graph \textit{G} is a dominating set if every vertex of \textit{G} is either in \textit{D} or is adjacent to a vertex in \textit{D}. The paired-domination problem on \textit{G} asks for a minimum-cardinality dominating set \textit{S} of \textit{G} such that the subgraph induced by \textit{S} contains a perfect matching. A set \textit{S} of vertices in \textit{G} is a total dominating set of \textit{G} if every vertex of \textit{V} is adjacent to some vertex in \textit{S}. In this paper we construct a minimum paired dominating set for \textit{H}-Naphtalenic Nanotorus, \textit{TUC}${}_{4}$\textit{C}${}_{8}$ (\textit{S}) nanotube and \textit{V}-phenylenic nanotube and hence we determine the total dominating set.

Keywords: Nanotubes, Perfect Matching, Paired Domination.

A.S.Shanthi
Department of Mathematics, Stella Maris College, Chennai, Tamilnadu, India.
E-mail: shanthu.a.s@gmail.com

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 Article Type : Research Article Title : Generalized Discrete $\alpha$, $k$-Fourier Transform Country : India Authors : G.Britto Antony Xavier and B.Govindan

Abstract: In this paper, we obtain $\alpha$, $k$-discrete Fourier transform and the closed form solutions of various functions by using inverse difference operators $\alpha$ and $k$. Appropriate examples are inserted to exemplify the main results.

Keywords: Discrete Fourier Transform, Polynomial factorials and Generalized difference operator.

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 : On Certain Class of Meromorphically Starlike Functions with Alternating Coefficients Country : India Authors : Adluru Narasimha Murthy and P.Thirupathi Reddy

Abstract: In this paper, we introduce the subclass $\sigma\left(n, \alpha, \beta \right)$ of meromorphically starlike functions in the punctured unit disk. Also we investigate coefficient inequality, distortion properties, and closure properties. Further preserving integral operators are obtained.

Keywords: Regular, Meromorphic, starlike, coefficient inequality.

Department of Mathematics, Government Aided A.V.V.Junior College, Warangal, Telangana State, India.
E-mail: reddypt2@gmail.com

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 Article Type : Research Article Title : On Some Properties of Metric F-Structure Satisfying $F^{2k+1}+F=0$ Country : India Authors : Lakhan Singh and Shailendra Kumar Gautam

Abstract: In this paper, we have studied various properties of the \textit{F}-structure satisfying $F^{2k+1}+F=0$. Where $k$ is positive integer The metric F- structure, $f$ induced on each integral manifold of tangent bundle $l^{*}$ have also been discussed.

Keywords: Differentiable manifold, projection operators, tangent bundles and metric.

Shailendra Kumar Gautam
Department of Mathematics, Eshan College of Engineering, Mathura(UP), India.
E-mail: skgautamrbs@yahoo.com

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 Article Type : Research Article Title : Superior Julia Sets and Superior Mandelbrot Sets in SP Orbit Country : India Authors : Mandeep Kumari, Sudesh Kumari and Renu Chugh

Abstract: The aim of this paper is to generate new Superior Julia and Superior Mandelbrot sets for complex-valued polynomials such as quadratic, cubic and higher degree polynomials using SP orbit, which is an example of four-step iterative procedure. In this paper, we visualize the graphical images of Superior Julia sets and Superior Mandelbrot sets and analyze their patterns. It is surprising to see that a few Superior Mandelbrot sets take the shape of decorated coupled Urns (Kalash in Hindi language), a decorated coupled Lightening lamp (Diya in Hindi language) and beautiful scared patterns (Rangolies in Hindi language). We believe that our results generalize and extend the existing results in the literature of Fractal theory.

Keywords: Superior Julia set, Superior Mandelbrot set, Superior escape criterion, SP orbit, complex polynomials.

Mandeep Kumari
Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana, India.
E-mail: kumarimandeep28@gmail.com

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 Article Type : Research Article Title : Solution and Stability of a,b,c,d Mixed Type Functional Equation in BS (Banach Space) and BA (Banach Algebra) Using Two Different Methods Country : India Authors : S.Murthy and J.Satish Kumar

Abstract: In this article, the authors introduce the general solution and generalized Ulam-Hyers stability of a generalized a,b,c,d mixed type functional equation of the form $g\left(\frac{a}{b} x+\frac{b}{c} y+\frac{c}{d} z\right)+g\left(\frac{a}{b} x-\frac{b}{c} y+\frac{c}{d} z\right)+g\left(\frac{a}{b} x+\frac{b}{c} y-\frac{c}{d} z\right)+g\left(-\frac{a}{b} x+\frac{b}{c} y+\frac{c}{d} z\right)=\frac{a}{b} [g(x)-g(-x)]+\frac{b}{c} [g(y)-g(-y)]+\frac{c}{d} [g(z)-g(-z)] +2\left(\frac{c}{d} \right)^{2} [g(z)+g(-z)]+2\left(\frac{a}{b} \right)^{2} [g(x)+g(-x)]+2\left(\frac{b}{c} \right)^{2} [g(y)+g(-y)]$, where a,b,c,d are positive integers with $a\ne b\ne c\ne d\ne 0$, in BS( Banach Space) and BA (Banach Algebras )using two different methods.

Keywords: Mixed type functional equations, Banach Algebra, generalized Ulam-Hyers Stability, Fixed point.

S.Murthy
Principal, Government Arts College (For Men), Krishnagiri, Tamilnadu, India.
E-mail: smurthy07@yahoo.co.in

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 Article Type : Research Article Title : The Triple $\chi^{3}$ Sequence Spaces Country : Turkey Authors : Ayhan Esi and N.Subramanian

Abstract: Let $\chi^{3}$ denote the space of all triple gai sequences and $\Lambda^{3}$ the space of all triple analytic sequences. This paper some theorems on general properties of $\chi^{3}$ sequences have been established.

Keywords: Gai sequence, analytic sequence, triple sequence, dual space, AK space, solid, $\delta-$ dual.

Ayhan Esi
E-mail: jessielawrence2703@gmail.com

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 Article Type : Research Article Title : Improvement On Operator Axioms And Fundamental Operator Functions Country : P.R. China Authors : Pith Xie

Abstract: The Operator axioms have been contructed to deduce number systems. In this paper, we slightly improve on the syntax of the Operator axioms and construct a semantics of the Operator axioms. Then on the basis of the improved Operator axioms, we define two fundamental operator functions to study the analytic properties of the Operator axioms. Finally, we prove two theorems about the fundamental operator functions and pose some conjectures. Real operators can give new equations and inequalities so as to precisely describe the relation of mathematical objects or scientific objects.

Keywords: Real Functions, Operator, Number System.

Pith Xie
Axiom Studio, PO Box \#3620, Jiangdongmen Postoffice, Gulou District, Nanjing, 210036, P.R. China.
E-mail: pith.xie@outlook.com

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 Article Type : Research Article Title : Some New Combination Graphs Country : India Authors : G.V.Ghodasara and Mitesh J.Patel

Abstract: A graph $G=(V,E)$ with $p$ vertices and $q$ edges is said to be combination graph, if there exists an injection $f:V(G)\rightarrow{}\{1,2,\ldots,p\}$ such that the induced edge function $g_{f}:E(G)\rightarrow \mathbb{N}$ defined by $g_{f}(uv)=\frac{(f(u))!}{| f(u)-f(v)|! (f(v))!} ( f(u)>f(v))$, for every $uv$ $\in$ $E(G)$ is injective. In this paper we prove that some cycle and wheel related graphs obtained with the use of graph operations are combination graph.

Keywords: Combination graph, prism graph, umbrella graph, wheel graph.

Mitesh J.Patel
Department of Mathematics, Tolani College of Arts and Science, Adipur-Kachchh, Gujarat, India.
E-mail: miteshmaths1984@gmail.com

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 Article Type : Research Article Title : $Q$-Cubic Ideals of Semigroups Country : India Authors : V.Chinnadurai, A.Swaminathan and K.Bharathivelan

Abstract: In this paper, for a set Q, the notion of Q-cubic ideals of semigroups is introduced and some properties of such ideals are investigated.

Keywords: Q-fuzzy set, Semigroup, An ideal, Q-Cubic semigroup, Q-Cubic ideal and Homomorphism.

Associate Professor, Department of Mathematics, Annamalai University, Annamalainagar, Tamilnadu, India.

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 Article Type : Research Article Title : Analyzing the Expectations of the Old Aged People Using Fuzzy Model Country : India Authors : A.Victor Devadoss and A.Lincy Amala Celine

Abstract: The advancement of technology and the necessities of the day to day life force the children to leave their parents to the old aged homes. Due to this process, the mushroom growths of old aged homes exist. And how far this home for aged satisfies the expectations of the parents and the responsibilities of the children are of major concern. The expectations of the elderly are very simple but not all children realize their importance or value their presence in their lives. The varying degree of acceptance of their ageing parents according to the human values is a valid criterion to study the expectations of the old aged people. The fuzzy logic serves as an appropriate tool to study the expectations as it is an unsupervised data. The problem of study is analyzed using the appropriate fuzzy model in this paper. The section one is of introductory nature. The second section gives the introduction to triangular fuzzy cognitive maps. The description of the problem is given in section three; and in the fourth section the model is adapted to the problem of study. The conclusion is derived in the final section.

Keywords: Triangular fuzzy number, linguistic variables, weightage and maximum weight.

Department of Mathematics, Loyola College, Chennai, Tamilnadu, India.
E-mail: hanivictor@ymail.com

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