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Issue 4 − C
Volume 3 (2015)

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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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Article Type |
: | Research Article |

Title |
: | Concept of Quadratic Equation of Rectangle to Relation all Mathematics Method |

Country |
: | India |

Authors |
: | Mr.Deshmukh Sachin Sandipan |

**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.

**Keywords: ** Rectangle, Sidemeasurement, Relation, Formula, Quadratic equation.

Mr.Deshmukh Sachin Sandipan

Corps of Signal (Indian Army), Maharashtra, India.

E-mail: rm.ram9414@gmail.com