## Issue 4 − CVolume 3 (2015)

ISSUE 4 − A ISSUE 4 − B ISSUE 4 − C ISSUE 4 − D ISSUE 4 − E ISSUE 4 − F
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 Article Type : Research Article Title : $\breve{g}$-closed and $\breve{g}$-open Maps in Topological Spaces Country : India Authors : O.Ravi || S.Padmasekaran || S.Usharani || I.Rajasekaran

Abstract: A set A in a topological space (X, $\tau$) is said to be $\breve{g}$-closed set if cl(A)$\subseteq$U whenever A$\subseteq$U and U is B-open in X. In this paper, we introduce $\breve{g}$-closed map from a topological space X to a topological space Y as the image of every closed set is $\breve{g}$-closed, and also we prove that the composition of two $\breve{g}$-closed maps need not be a $\breve{g}$-closed map. We also obtain some properties of $\breve{g}$-closed maps.

Keywords: Topological space, $\breve{g}$-closed map, $\breve{g}^\star$-closed map, $\breve{g}$-open map, $\breve{g}^\star$-open map.

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 Decompositions of Generalized $\mu$-$\alpha$-sets Country : India Authors : M.Jeyaraman || S.C.Vasthirani || O.Ravi || R.Muthuraj

Abstract: The aim of this paper is to introduce the new notions called $\mu\text{-}\alpha$-locally closed sets, $\mu_{\alpha\text{-}t}$-sets and $\mu_{\alpha\text{-}B}$-sets and investigate their properties. Using these concepts we have obtained some decompositions.

Keywords: $\mu$-$\alpha$-locally closed set, $\mu_\alpha$$g$ closed set, $\mu_{\alpha\text{-}t}$-set, $\mu_{\alpha\text{-}B}$-set.

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 : Irreducible elements in $(\mathcal{Z}^{+},\leq_{C})$ Country : Oman Authors : Sankar Sagi

Abstract: A convolution is a mapping $\mathcal{C}$ of the set $Z^{+}$ of positive integers into the set $\mathcal{P}(Z^{+})$ of all subsets of $Z^{+}$ such that, for any $n\in Z^{+}$ , each member of $C(n)$ is a divisor of $n$. If $D(n)$ is the set of all divisors of $n$, for any $n$, then $D$ is called the Dirichlet's convolution. If $U(n)$ is the set of all Unitary(square free) divisors of $n$ , for any $n$, then $U$ is called unitary(square free) convolution. Corresponding to any general convolution $C$, we can define a binary relation $\leq_{C}$ on $Z^{+}$ by  $m\leq_{C}n$ if and only if $m\in C(n)$ '. In this paper, we present irreducible elements in $(\mathcal{Z}^{+},\leq_{\mathcal{C}})$ , where $\leq_{\mathcal{C}}$ is the binary relation induced by the convolution $\mathcal{C}$.

Keywords: Semi Lattice,Convolution,Multiplicative convolution,Irreducible elements.

Sankar Sagi
Assistant Professor, College of Applied Sciences, Sohar, Sultanate of Oman.
E-mail: sagi$_{-}$sankar@yahoo.co.in

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 Article Type : Research Article Title : Co-maximal Filters in $(\mathcal{Z}^{+},\leq_{C})$ Country : Oman Authors : Sankar Sagi

Abstract: A convolution is a mapping $\mathcal{C}$ of the set $Z^{+}$ of positive integers into the set $\mathcal{P}(Z^{+})$ of all subsets of $Z^{+}$ such that, for any $n\in Z^{+}$ , each member of $C(n)$ is a divisor of $n$. If $D(n)$ is the set of all divisors of $n$, for any $n$, then $D$ is called the Dirichlet's convolution\cite{Narkiewicz}. If $U(n)$ is the set of all Unitary(square free) divisors of $n$ , for any $n$, then $U$ is called unitary(square free) convolution. Corresponding to any general convolution $C$, we can define a binary relation $\leq_{C}$ on $Z^{+}$ by  $m\leq_{C}n$ if and only if $m\in C(n)$ '. In this paper, we discuss co-maximal filters in $(\mathcal{Z}^{+},\leq_{\mathcal{C}})$ , where $\leq_{\mathcal{C}}$ is the binary relation induced by the convolution $\mathcal{C}$.

Keywords: Partial order,Semi Lattice,Convolution,Filter,Prime Filter,co-maximal.

Sankar Sagi
Assistant Professor, College of Applied Sciences, Sohar, Sultanate of Oman.
E-mail: sagi$_{-}$sankar@yahoo.co.in

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 Article Type : Research Article Title : Solutions of Some Integrals in Hypergeometric Functions and their Generalization in Double Series Identities Country : India Authors : Usha Gill || Ranjana Shrivastava || Kaleem A. Quraishi || Z. A. Taqvi

Abstract: In this paper, we obtain solutions of some integrals in the form of hypergeometric functions. Further, we generalize these integrals in the form of double series identities involving bounded sequences. We also derive hypergeometric forms of these identities involving Gaussian hypergeometric function, Srivastava-Daoust double hypergeometric function and Kamp\'{e} de F\'{e}riet double hypergeometric function.

Keywords: Pochhammer Symbol, Gaussian Hypergeometric Function, Bounded Sequences, Multiple Series Identities, Srivastava-Daoust double Hypergeometric Function, Kamp\'{e} de F\'{e}riet double Hypergeometric Function.

Usha Gill
Department of Applied Sciences and Humanities, Al-Falah School of Engineering and Technology, Faridabad, Haryana, India.
E-mail: ushagill79@gmail.com

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 Article Type : Research Article Title : Alpha Cuts of Fuzzy Basis Country : India Authors : M.Muthukumari || A.Nagarajan || M.Murugalingam

Abstract: We introduce fuzzy basis, strong fuzzy basis and alpha cuts of fuzzy basis and strong fuzzy basis.

Keywords: Fuzzy basis, Strong fuzzy basis, Alpha cuts.

M.Murugalingam
Department of Mathematics, Thiruvalluvar College, Papanasam, Tamilnadu, India.
E-mail: mmmurugalingam@yahoo.com

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 Article Type : Research Article Title : $(1,2)^\star$-$r \omega$-Continuous and $(1,2)^\star$-$r \omega$-Irresolute Functions Country : India Authors : O.Ravi || M.Kamaraj || S.Murugambigai || I.Rajasekaran

Abstract: In this paper, we introduce two types of bitopological functions called $(1,2)^\star$-$r \omega$-continuous functions and $(1,2)^\star$-$r \omega$-irresolute functions and study their properties.

Keywords: $(1,2)^\star$-$\omega$-continuity, $(1,2)^\star$-$r \omega$-continuity, $(1,2)^\star$-$gpr$-continuity.

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 : Ulam Stabilities of $K$ - AC - Mixed Type Functional Equations in Three Variables Country : India Authors : M.Arunkumar || M.J.Rassias || S.Hema Latha || Yanhui Zhang

Abstract: In this paper, we obtain the general solution and generalized Ulam - Hyers stability of a 3 - variable $k-$ AC - mixed type functional equation \begin{align*} &f(kx+y, kz+w, ku+v)-f(kx-y, kz-w, ku-v)\\ &=k^2[f(x+y, z+w, u+v)-f(x-y, z-w, u-v)]-2(k^2-1)f(y, w,v) \end{align*} where $k \ge 2$, in Banach space using direct and fixed point methods.

Keywords: Additive functional equations, cubic functional equation, Mixed type AC functional equation, Ulam - Hyers stability, Ulam - TRassias stability, Ulam - Gavruta - Rassias stability, Ulam - JRassias stability, generalized Ulam - Hyers stability, fixed point.

M.Arunkumar
Department of Mathematics, Government Arts College, Tiruvannamalai, TamilNadu, India.
E-mail: annarun2002@yahoo.co.in

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 Article Type : Research Article Title : Product Measure Spaces and Theorems of Fubini and Tonelli Country : India Authors : Parvinder Singh

Abstract: The product X$\times Y$ of measure spaces has as its measurable sub sets, the $\sigma$-algebra generated by the products A$\times$ B measurable sub sets of X and Y. Fubini's Theorem introduced by Guido Fubini in 1907 is a result which gives conditions under which it is possible to commute a double integral. It implies that two repeated integrals of a function of two variables are equal if the function is integrable. Tonelli's Theorem is a successor of the Fubini's Theorem. The conclusion of Tonelli's theorem is identical to that of Fubini's theorem, but the assumption that $|f|$ has a finite integral is replaced by the assumption that f is non-negative.

Keywords: Measure Spaces, Product of Measure Spaces, Theorems of Fubini and Tonelli.

Parvinder Singh
P.G.Department of Mathematics, S.G.G.S. Khalsa College, Mahilpur, Hoshiarpur, Punjab, India.
E-mail: parvinder070@gmail.com

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Abstract: In this research paper, the equation of rectangle explained in the form of quadratic equation. In this research paper, the main quadratic equation of rectangle is $x^{2}-B(\square PQRS)x+A(\square PQRS)=0$, which is outcome of \lq Basic theorem of perimeter relation of square-rectangle'. If the value of \textbf{a} is not equal to 1 $(a\neq1)$, then the quadratic equation of rectangle is $ax^{2}-B(\square PQRS)x+a.A(\square PQ'R'S')=0\;\;[ax^2-bx+c=0]$ and if the value of \textbf{a} is 1 ($a=1$), then quadratic equation of rectangle is $x^{2}-B(\square PQRS)x+A(\square PQRS)=0\;\;[x^2-bx+d =0]$. In this Research Paper Three methods of quadratic equation of rectangle are explained i.e. (i) Factorization method of rectangle (ii) Completing square of method of rectangle (iii) Formula method of rectangle. We are trying to give a new concept \lq\lq Relation All Mathematics" to the world. I am sure that this concept will be helpful in Agricultural, Engineering, Mathematical world etc.