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Issue 2
Volume 5 (2017)

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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.

Adluru Narasimha Murthy

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

Department of Mathematics, Adiyaman University, Adiyaman, Turkey.

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.

V.Chinnadurai

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

E-mail: kv.chinnadurai@yahoo.com

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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.

A.Victor Devadoss

Department of Mathematics, Loyola College, Chennai, Tamilnadu, India.

E-mail: hanivictor@ymail.com

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Article Type |
: | Research Article |

Title |
: | General Solution, Stability and Non-Stability of Quattuorvigintic Functional Equation in Multi-Banach Spaces |

Country |
: | India |

Authors |
: | John Michael Rassias, R.Murali, Matina John Rassias and A.Antony Raj |

**Abstract: ** In this current work, we estabilish the general solution and Hyers-Ulam stability for a new form of Quattuorvigintic functional equation in Multi-Banach Spaces.

**Keywords: ** Hyers-Ulam stability, Multi-Banach Spaces, Quattuorvigintic Functional Equations, Fixed Point Method.

R.Murali

Department of Mathematics, Sacred Heart College, Tirupattur, TamilNadu, India.

E-mail: shcrmurali@yahoo.co.in