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Current Issue
Volume 3 Issue 2 (2015)

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Article Type |
: | Research Article |

Title |
: | Decompositions of $\tilde{g}$-continuity |

Country |
: | India |

Authors |
: | O. Ravi || S. Margaret Parimalam || S. Murugesan || A. Pandi |

**Abstract: **The aim of this paper is to give decompositions of a weaker form of continuity, namely $\tilde{g}$-continuity, by providing the concepts of $\tilde{g}_t$-sets, $\tilde{g}{_\alpha*}$-sets, $\tilde{g}_t$-continuity and $\tilde{g}{_\alpha*}$-continuity.

**Keywords:** $\tilde{g}$-closed set, $\tilde{g}{_\alpha}$-closed set, $\tilde{g}$$_t$-set, $\tilde{g}$$_\alpha*$-set, $\tilde{g}$$_t$-continuity, $\tilde{g}{_\alpha*}$-continuity.

Dr. O. Ravi

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | Associated Properties of $\gamma$-$\pi g$$\gamma$-closed Functions |

Country |
: | India |

Authors |
: | O. Ravi || S. Margaret Parimalam || S. Murugesan || A. Pandi |

**Abstract: **The concept of $\gamma$-open sets was introduced in [3, 7, 11]. The primary purpose of this paper is to introduce and study pre-$\pi g\gamma$-closed functions by using $\gamma$-open sets.

**Keywords:** $\gamma$-closed function, strongly $\gamma$-closed function, $\pi g\gamma$-closed function, pre-$\pi g\gamma$-closed function, almost $\pi g\gamma$-closed function, regular $\pi g\gamma$-closed function, pre-$\pi g\gamma$-continuous function, quasi-$\gamma$-regular space, quasi-$\gamma$-normal space.

Dr. O. Ravi

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | Oscillation of Second Order Quasi−linear Neutral Delay Differential Equations |

Country |
: | China |

Authors |
: | Hongyan Guan || Yuanhong Yu |

**Abstract: **In this paper, by employing generalized Riccati transformation and some specific analytical skills, we will establish some new oscillation criteria for the second order quasi-linear neutral delay equation of the form $$ (r(t)|z^{'}(t)|^{\alpha-1}z^{'}(t))^{'}+q(t)f(x(\sigma(t)))=0,\ \ t\geq t_0, $$ where z(t)=x(t)+p(t)x(τ(t)), −1 < p_{1} ≤ p(t) ≤ 0, q(t) > 0, α > 0. The results obtianed essentially improve and extent the results in the cited literature.

**Keywords:** Oscillation, second order, neutral delay di
erential equation, generalized Riccati transformation.

Dr. Hongyan Guan

School of Mathematics and Systems Science, Shenyang Normal University, Shenyang, People's Republic of China.

E-mail: guanhy8010@163.com

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Article Type |
: | Research Article |

Title |
: | On Weakly (1,2)$^\star$-$g^{\#}$-closed Sets |

Country |
: | India |

Authors |
: | O. Ravi || A. Pandi || I. Rajasekaran || S. Ramachandran |

**Abstract: **In this paper, we introduce weakly (1,2)$^\star$-$g^\#$-closed sets and investigate the relationships among the related (1,2)$^\star$-generalized closed sets.

**Keywords:** (1,2)$^\star$-$g^{\#}$-closed set, (1,2)$^\star$-$g$-closed set, (1,2)$^\star$-$\alpha g$-closed set, (1,2)$^\star$-$g^{\#}$-irresolute function.

Dr. O. Ravi

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | On $\beta$-normal Spaces |

Country |
: | India |

Authors |
: | O. Ravi || I. Rajasekaran || S. Murugesan || A. Pandi |

**Abstract: **The aim of this paper is to study the class of β−normal spaces. The relationships among s−normal spaces, p−normal spaces and β−normal spaces are investigated. Moreover, we study the forms of generalized β−closed functions. We obtain characterizations of β−normal spaces, properties of the forms of generalized β−closed functions and preservation theorems.

**Keywords: **p−normal space, s−normal space, β−normal space, gβ−closed function, β−gβ−closed function, separation axioms.

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | Decompositions of $rg$-continuity via Idealization |

Country |
: | India |

Authors |
: | K. Indirani || V. Rajendran || M. Meharin || O. Ravi |

**Abstract: **In this paper, we introduce the notions of $r\alpha g$-$\mathcal{I}$-open sets, $gpr$-$\mathcal{I}$-open sets, $C_r$-$\mathcal{I}$-sets and $C_r^*$-$\mathcal{I}$-sets in ideal topological spaces and investigate some of their properties and using these notions we obtain three decompositions of $rg$-continuity.

**Keywords: **$r\alpha g$-$\mathcal{I}$-continuity, $gpr$-$\mathcal{I}$-continuity, $C_r$-$\mathcal{I}$-continuity, $C_r^*$-$\mathcal{I}$-continuity and $rg$-continuity.

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | The First and Second Zagreb Indices of Generalized Complementary Prisms |

Country |
: | India |

Authors |
: | S. Arockiaraj || Vijaya Kumari |

**Abstract: **The first and second zagreb indices of a graph $G$ are defined as $M_1(G)=\sum\limits_{uv\in E(G)} [deg(u)+deg(v)]$ (or equivalently $\sum\limits_{u\in V(G)}[deg(u)^2]$ and $M_2(G)=\sum\limits_{uv\in E(G)}[deg (u)deg(v)]$ respectively. In this paper, we have obtained the first and second zagreb indices of the generalized complementary prisms $G_{m+n}, \mathcal{G}_{m,n}, \mathcal{G}_{m,m}^c$ and $G_{m,m}^P$.

**Keywords: **First Zagreb Index, Second Zagreb index, Complementary Prism and Generalized Complementary Prism.

Dr. S. Arockiaraj

Department of Mathematics, Mepco Schlenk Engineering College, Sivakasi, Tamilnadu, India.

E-mail: psarockiaraj@gmail.com

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Article Type |
: | Research Article |

Title |
: | On $\tilde{g} (1,2)^\star$-closed Sets |

Country |
: | India |

Authors |
: | O. Ravi || R. Bhavani || S. Murugesan || I. Rajasekaran |

**Abstract: **In this present paper, we introduce a new class of sets namely $\tilde{g} (1,2)^\star$-closed sets in bitopological spaces. The notion of $\tilde{g} (1,2)^\star$-interior is defined and some of its basic properties are studied. Also we introduce the concept of $\tilde{g} (1,2)^\star$-closure in bitopological spaces using the notion of $\tilde{g} (1,2)^\star$-closed sets.

**Keywords: **$\tilde{g} (1,2)^\star$-closed set, $(1,2)^\star$-$\hat{g}$-closed set, $(1,2)^\star$-sg-closed set, $(1,2)^\star$-gsp-closed set, $(1,2)^\star$-$\ddot{g}$-closed set, $(1,2)^\star$-$\psi$-closed set.

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | $\mathcal{I}$.$\vee$$_{m}$-sets and $\mathcal{I}$.$\wedge$$_m$-sets |

Country |
: | India |

Authors |
: | O. Ravi || K. Kalpana || S. Tharmar |

**Abstract: **We define $\wedge$$_m$-sets and $\vee$$_m$-sets in minimal spaces and discuss their properties. Also define $\mathcal{I}$.$\wedge$$_m$-sets and $\mathcal{I}$.$\vee$$_m$-sets in ideal minimal spaces and discuss their properties. At the end of the paper, we characterize m-T$_1$-spaces using these new sets.

**Keywords: **m-$\mathcal{I}_g$-closed set, $\mathcal{I}$.$\wedge$$_m$-set, $\mathcal{I}$.$\vee$$_m$-set, m-T$_1$-space.

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | Transformation Formulas of Lauricella's Function of the Third kind of Several Variables |

Country |
: | Yemen |

Authors |
: | Ahmed Ali Atash |

**Abstract: **In this research paper a transformation formulas for Lauricella's functionof the third kind of several variables are established with the help of the Generalization of the Kummer's theorem on the sum of the series _{2}F_{1}(-1) obtained by Lavoie et al. [4]. The presented results are generalizations of the will known result due to Srivastava [6].

**Keywords: **Transformation formulas, Lauricella's function, Kummer's theorem, Hypergeometric functions.

Dr. Ahmed Ali Atach

Department of Mathematics, Aden University, Aden, Yemen.

E-mail: ah-a-atash@hotmail.com

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Article Type |
: | Research Article |

Title |
: | $b$-Chromatic Number for the Graphs Obtained by Duplicating Edges |

Country |
: | India |

Authors |
: | S. Arockiaraj || V. Premalatha |

**Abstract: ** A $b$-colouring of a graph $G$ is a proper vertex colouring of $G$ such that each colour class contains a vertex that has atleast one neighbour in every other colour class and $b$-chromatic number of a graph $G$ is the largest integer $\phi(G)$ for which $G$ has a $b$-colouring with $\phi(G)$ colours. In this paper, we have obtained the $b$-chromatic number of the graphs $E_n, F_n$ and the graphs obtained by duplicating all the edges of path, cycle, complete graph, wheel graph, Ladder graph $L_n$ by vertices.

**Keywords: ** $b$-colouring, $b$-chromatic number.

Dr. S. Arockiaraj

Department of Mathematics, Mepco Schlenk Engineering College, Sivakasi, Tamilnadu, India.

E-mail: psarockiaraj@gmail.com

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Article Type |
: | Research Article |

Title |
: | The Automatic Continuity of Epimorphisms in certain classes of Topological Algebras |

Country |
: | Morroco |

Authors |
: | Youssef TIDLI |

**Abstract: ** In this papers, we are interested in the study of automatic continuity in *semi -simples Complete p-normed *-algebras. To do this, we extend the results known in the case of Banach spaces with p-normed complete spaces.

**Keywords: ** *-algebra, *simple algebra, *semi-simple algebra, automatic continuity, separting space.

Dr.Youssef TIDLI

Sidi Mohamed Ben Abdellah University , Faculty of Sciences Dhar El Marhaz, Laboratory of Functional Analysis, Fes, Morroco.

E-mail: youssef.tidli@gmail.com

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Article Type |
: | Research Article |

Title |
: | $\star$-$A_{\mathcal{I}}^\star$-sets and DECOMPOSITIONS OF $\star$-$A_{\mathcal{I}}^\star$-continuity |

Country |
: | India |

Authors |
: | O.Ravi || G.Selvi || S.Murugesan || S.Vijaya |

**Abstract: ** The aim of this paper is to introduce and study the notions of $\star$-$A_{\mathcal{I}}^\star$-sets and $\star$-C$_{\mathcal{I}}$-sets in ideal topological spaces. Properties of $\star$-$A_{\mathcal{I}}^\star$-sets and $\star$-C$_{\mathcal{I}}$-sets are investigated. Moreover, decompositions of $\star$-$A_{\mathcal{I}}^\star$-continuous functions via $\star$-$A_{\mathcal{I}}^\star$-sets and $\star$-C$_{\mathcal{I}}$-sets in ideal topological spaces are established.

**Keywords: ** $\star$-$A_{\mathcal{I}}^\star$-set, $\star$-C$_{\mathcal{I}}$-set, C$_{\mathcal{I}}^\star$-set, pre-$\mathcal{I}$-regular set, ideal topological space, decomposition.

Department of Mathematics, P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.

E-mail: siingam@yahoo.com

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Article Type |
: | Research Article |

Title |
: | Fractional Complex Variables: Strong Local Fractional Complex Derivatives (LFCDs) of Non-Integer Rational Order |

Country |
: | USA |

Authors |
: | Steve Anglin |

**Abstract: ** Fractional complex variables focus on the fractional or non-integer order differential calculus of a complex variable. In fractional calculus, locality can narrow down pieces of a function where there may be better behavior in order to model in an analytic sense, as well as obtain more meaningful physical and/or geometric information. That's where we introduce the concepts of Strong Local Fractional Complex Derivatives or LFCDs. Strong LFCDs can "maximize" the opportunity that the piece of the function in a localized or local enough area is "well-behaved" (enough). We prove a theorem that shows where Strong LFCDs exist. Applications include index of stability in Complex or Real Fractional Advection Dispersion Equation (FADE).

**Keywords: **

Dr.Steve Anglin

Lecturer of Case Western Reserve University.

E-mail: stevemanglin@gmail.com

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Article Type |
: | Research Article |

Title |
: | Integral Representation of Linear Functionals on Function Spaces: Reisz-Markov Theorem |

Country |
: | India |

Authors |
: | Parvinder Singh |

**Abstract: **

**Keywords:**

Parvinder Singh

PG Department of Mathematics, S.G.G.S. Khalsa College,
Mahilpur, Punjab, India.

E-mail: parvinder070@gmail.com

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Article Type |
: | Research Article |

Title |
: | Connected Edge Domination in Fuzzy Graphs |

Country |
: | India |

Authors |
: | C. Y. Ponnappan || S. Basheer Ahamed || P. Surulinathan |

**Abstract: **

**Keywords:**

S. Basheer Ahamed

Department of Mathematics, P.S.N.A. College of Engineering and Technology, Dindigul, Tamilnadu, India.

E-mail: sbasheerahameds@gmail.com

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Article Type |
: | Research Article |

Title |
: | Contra Regular Generalized Semipre Continuous Mappings in Intuitionistic Fuzzy Topological Spaces |

Country |
: | India |

Authors |
: | V. Vaishnavy || D. Jayanthi |

**Abstract: **In this paper we introduce the notion of intuitionistic fuzzy contra regular generalized semipre continuous mappings. Furthermore we provide some properties of intuitionistic fuzzy contra regular generalized semipre continuous mappings and discuss some interesting theorems.

**Keywords:** Intuitionistic fuzzy topology, Intuitionistic fuzzy closed sets, Intuitionistic fuzzy regular generalized semipre closed sets, Intuitionistic fuzzy regular generalized semipre open sets, Intuitionistic fuzzy contra regular generalized semipre continuous mappings.

V. Vaishnavy

Avinashilingam University, Coimbatore, Tamil Nadu, India.

E-mail: vaishnavyviswanathan92@gmail.com

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Article Type |
: | Research Article |

Title |
: | Special Pythagorean Triangles and 6 Digit Dhuruva Number |

Country |
: | India |

Authors |
: | M. A. Gopalan || S. Vidhyalakshmi || N. Thiruniraiselvi || R. Presenna |

**Abstract: **Pythagorean triangles, each with a leg represented by a 6-digit Dhuruva numbers are obtained. A few interesting results are given.

**Keywords:** Pythagorean triangles, 6-digit Dhuruva number.

N. Thiruniraiselvi

Research Scholar, Department of Mathematics, SIGC, Trichy, Tamilnadu, India.

E-mail: nts.maths.ig@gmail.com

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Article Type |
: | Research Article |

Title |
: | Separation Axioms via $q$-$\mathcal{I}$-open Sets |

Country |
: | India |

Authors |
: | P.Kalaiselvi || N.Rajesh |

**Abstract: **In this paper, $q$-$\mathcal{I}$-open sets are used to define and study some weak separation axioms in ideal topological spaces.

**Keywords:** Ideal topological spaces, $q$-$\mathcal{I}$-$T_0$ spaces, $q$-$\mathcal{I}$-$T_1$ spaces, $q$-$\mathcal{I}$-$T_2$ spaces.

N.Rajesh

Department of Mathematics, Rajah Serfoji Government College, Thanjavur, Tamilnadu, India.

E-mail: nrajesh_topology@yahoo.co.in

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Article Type |
: | Research Article |

Title |
: | On Construction of Even Vertex Odd Mean Graphs |

Country |
: | India |

Authors |
: | G. Pooranam || R. Vasuki || S. Suganthi |

**Abstract: **A graph $G$ with $p$ vertices and $q$ edges is said to have an \linebreak even vertex odd mean labeling if there exists an injective function $f:V(G)\rightarrow\{0,2,4,\dots,2q-2,2q\}$ such that the induced map \linebreak $f^*:E(G)\rightarrow\{1,3,5,\dots, 2q-1\}$ defined by $f^*(uv)=\frac{f(u)+f(v)}{2}$ is a bijection. A graph that admits an even vertex odd mean labeling is called an even vertex odd mean graph. In this paper we discuss the construction of two kinds of even vertex odd mean graphs. Here we prove that $(P_n;S_1)$ for $n\geq 1,$ $(P_{2n};S_m)$ for $m\geq 2, n\geq 1,$ $(P_m;C_n)$ for $n\equiv 0(mod \ 4),$ $m\geq 1,$ $[P_m;C_n]$ for $n\equiv 0(mod \ 4),$ $m\geq 1$ and $[P_m;C_n^{(2)}]$ for $n\equiv 0(mod \ 4),$ $m\geq 1$ are even vertex odd mean graphs.

**Keywords:** Labeling, even vertex odd mean labeling, even vertex odd mean graph.

R. Vasuki

Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur, Tamil Nadu, India.

E-mail: vasukisehar@gmail.com

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Article Type |
: | Research Article |

Title |
: | Soft Expert Generalized Closed Sets With Respect to Soft Expert Ideals |

Country |
: | India |

Authors |
: | S. Tharmar |

**Abstract: **The soft expert models are richer than soft set models since the soft set models are created with the help of one expert whereas but the soft expert models are made with the opinions of all experts. In this paper, we introduce soft expert generalized closed sets and soft expert generalized open sets with respect to soft expert ideals in soft expert topological spaces and study their basic properties.

**Keywords:** Soft expert sets, Soft expert topological space, Soft expert g-closed sets, Soft expert $\mathcal{I}_g$-closed sets and Soft expert $\mathcal{I}_g$-open sets.

S. Tharmar

Department of Mathematics, Mohamed Sathak Engineering College, Ramanathapuram, Tamilnadu, India.

E-mail: tharmar11585@gmail.com

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Article Type |
: | Research Article |

Title |
: | Super pair sum labeling of $H$-graphs |

Country |
: | India |

Authors |
: | J. Venkateswari || R. Vasuki || P. Sugirtha |

**Abstract: **Let $G$ be a graph with $p$ vertices and $q$ edges.The graph $G$ is said to be a super pair sum labeling if there exists a bijection $f$ from $V(G)\cup E(G)$ to $\left\{0,\pm1,\pm2,\dots,\pm\left(\frac{p+q-1}{2}\right)\right\}$ when $p+q$ is odd and from $V(G)\cup E(G)$ to $\left\{\pm1,\pm2,\dots,\pm\left(\frac{p+q}{2}\right)\right\}$ when $p+q$ is even such that $f(uv)=f(u)+f(v).$ A graph that admits a super pair sum labeling is called a super pair sum graph. In this paper, we investigate super pair sum labeling of some $H$-graphs.

**Keywords:** Labeling, super pair sum labeling, super pair sum graph.

R. Vasuki

Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering, Tiruchendur, Tamil Nadu, India.

E-mail: vasukisehar@gmail.com

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Article Type |
: | Research Article |

Title |
: | A Nonagonal Fuzzy Number and Its Arithmetic Operation |

Country |
: | India |

Authors |
: | A. Felix || S. Christopher || A. Victor Devadoss |

**Abstract: ** In many real life cases, the decision data of human judgments with preferences are often vague so that the usual ways of using crisp values are inadequate also using fuzzy numbers such as triangular, trapezoidal are not suitable in few cases where the uncertainties arises in nine different points. Therefore, Nanogonal Fuzzy Number (NFN) and its arithmetic operations based on extension principle of fuzzy sets and $\alpha -$cut are introduced.

**Keywords:** Fuzzy Arithmetic, Linguistic values, Nonagonal Fuzzy Number (NFN), $\alpha$-cut.

A. Felix

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

E-mail: afelixphd@gmail.com

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Article Type |
: | Research Article |

Title |
: | A General Family of Generating Relations Involving Multivariable Hypergeometric Polynomials |

Country |
: | India |

Authors |
: | Mohd. Idris Qureshi || Deepak Kumar Kabra || Kaleem A. Quraishi |

**Abstract: ** The aim of this paper is to derive two general theorems on generating relations for a certain sequence of functions. The generating functions for multivariable generalized hypergeometric polynomials are shown here as special cases of a general class of generating relations. A number of results associated with Jacobi, Khandekar polynomials of several variables and other results of multiple Gaussian hypergeometric functions scattered in the literature of special functions follow as applications of main results.

**Keywords:** Generalized Rice polynomials and classical Jacobi polynomials, Orthogonal polynomials, Srivastava-Daoust multiple hypergeometric functions, Bounded multiple sequences, Generating functions and generating relations.

Deepak Kumar Kabra

Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Sangam University, Rajasthan, India.

E-mail: dkabra20@gmail.com

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Article Type |
: | Research Article |

Title |
: | Some properties of $r$-$T_{2}$ spaces |

Country |
: | Egypt |

Authors |
: | A. A. Ramadan || M. El Dardery || Y. C. Kim |

**Abstract: **We introduce $r$-$T_{2}$ spaces in fuzzy topological spaces in view of Sostak and investigate some properties of $r$-$T_{2}$ spaces. Moreover, we study properties of subspaces and products of $r$-$T_{2}$ spaces.

**Keywords:** $r$-$T_{2}$ spaces, subspaces and products of $r$-$T_{2}$ spaces.

Dr.A.A.Ramadan

Department of Mathematics, Faculty of Science, Beni-Suef
University, Beni-Suef, Egypt.

E-mail: aramadan58@gmail.com