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Issue 4 − D
Volume 3 (2015)

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Article Type |
: | Research Article |

Title |
: | Tietze's Extension Theorem for Intuitionistic Fuzzy $\zeta-$Basically Disconnected Spaces in Katetov-Tong's Sense |

Country |
: | India |

Authors |
: | S.Sharmila || I.Arockiarani |

**Abstract: ** In this paper we initiate the concept of $\zeta-$basically disconnectedness in intuitionistic fuzzy topological spaces.
We also apply these notions of $\zeta-$basically disconnectedness to analyse and prove Tietze extension
theorem.

**Keywords: ** Intuitionistic fuzzy $\zeta$ space, Intuitionistic fuzzy $\zeta-$basically disconnected space, Intuitionistic fuzzy real line, Intuitionistic fuzzy unit interval, Lower (Upper) intuitionistic fuzzy $\zeta-$continuous function.

S.Sharmila

Department of Mathematics, Nirmala College for women, Coimbatore, Tamilnadu, India.

E-mail: sharmi.skumar@gmail.com

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Article Type |
: | Research Article |

Title |
: | Some Relations on Hikami's Mock Theta Functions |

Country |
: | India |

Authors |
: | Ananta kr. Bora |

**Abstract: ** In this paper we obtain relations connecting Hikami's mock
theta functions,partial mock theta functions and infinite
products analogous to the identities of Ramanujan.

**Keywords: ** Mock theta functions, partial mock theta functions.

Ananta kr. Bora

Concept Jr. College, Nagaon, Assam, Itachali, India.

E-mail: anantakrbora.bora610@gmail.com

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Article Type |
: | Research Article |

Title |
: | Solution and Ulam - Hyers Stability of an Additive - Quadratic Functional Equation in Banach Space: Hyers Direct and Fixed Point Methods |

Country |
: | India |

Authors |
: | John M. Rassias || M.Arunkumar || P.Agillan |

**Abstract: ** In this paper, the authors establish the general solution and generalized Ulam - Hyers stability of an additive quadratic functional equation
\begin{align*}
f(x+2y+3z)&+f(x-2y+3z)+f(x+2y-3z)+f(x-2y-3z)\\
&= 4f(x)+8[f(y)+f(-y)]+18[f(z)+f(-z)]
\end{align*}
in Banach spaces, using the Hyers direct and fixed point methods.

**Keywords: ** Additive functional equations, quadratic functional equation, mixed type functional equation, generalized Ulam - Hyers stability, fixed point.

M.Arunkumar

Department of Mathematics, Government Arts College, Tiruvannamalai, Tamil Nadu, India.

E-mail: annarun2002@yahoo.co.in

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Article Type |
: | Research Article |

Title |
: | Degree Sequence of Isomorphic Fuzzy Graphs |

Country |
: | India |

Authors |
: | K.Radha || A.Rosemine |

**Abstract: **In this paper degree sequence of isomorphic fuzzy graphs are considered and some of its properties are studied and also gave a sufficient condition for a fuzzy graph and its $\mu$-complement have an identical degree sequence.

**Keywords: ** Degree of a vertex, degree sequence of a fuzzy graph, degree sequence of isomorphic fuzzy graphs.

A.Rosemine

PG and Research Department of Mathematics, Periyar E.V.R. College, Tiruchirappalli, Tamilnadu, India.

E-mail: rosemi2006@gmail.com

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Article Type |
: | Research Article |

Title |
: | Closed Form Solution for the Time-Fractional Schr\"{o}dinger Equation via Laplace Transform |

Country |
: | India |

Authors |
: | S.C.Sharma || R.K.Bairwa |

**Abstract: **In this work, we apply an efficient technique based on coupling of iterative method and Laplace transform method to obtain the exact solution of the linear and nonlinear time- fractional Schr\"{o}dinger equations. This method is termed as iterative Laplace transform method. The time -fractional derivative is considered in the Caputo sense and the solutions are found in closed form, in terms of Mittag-Leffler functions. Several numerical examples have been considered to show the effectiveness of the proposed method and the results are compared with the exact solution depicting high accuracy and efficiency.

**Keywords: ** Laplace transform, Iterative Laplace transform method, Schr\"{o}dinger equations, Caputo fractional derivative, Mittag-Leffler function, fractional partial differential equation.

R.K.Bairwa

Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan, India.

E-mail: rajendrabairwa1984@gmail.com

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Article Type |
: | Research Article |

Title |
: | Weakly $(\pi_\mathcal{H},\lambda)$-Continuity in Generalized Topologies |

Country |
: | India |

Authors |
: | R. Mariappan || M. Murugalingam || R. Ramesh |

**Abstract: ** In this paper, we introduce and study the notions of weakly $\pi$-$\mathcal{H}$-open sets and weakly $(\pi_\mathcal{H})$-continuity in hereditary generalized topological spaces. Also we prove that a function $f:(X,\mu,\mathcal{H})\rightarrow(Y,\lambda)$ is $(\mu,\lambda)$-continuous function if and only if it is weakly $(\pi_\mathcal{H},\lambda)$-continuous function and $(\delta_\mathcal{H},\lambda)$-continuous function where $\mathcal{H}$ is $\mu$-codense.

**Keywords: ** Generalized topology, hereditary class, weakly $\pi$-$\mathcal{H}$-open, weakly $(\pi_\mathcal{H})$-continuity.

R. Mariappan

Department of Science and Humanities, Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamil Nadu, India.

E-mail: mariappanbagavathi@gmail.com

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Article Type |
: | Research Article |

Title |
: | Bi-Univalent Coefficient Estimates for Certain Subclasses of Close-to-Convex Functions |

Country |
: | India |

Authors |
: | C.Selvaraj || T.R.K.Kumar |

**Abstract: ** In this paper,we introduce two new subclasses of the function class $\Sigma$ of
Bi-univalent functions defined in the open unit disk $\mathbb{U}=\left\{z:\left|z\right|<1\right\}$.
Besides, we find estimates on the coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ for functions in these new subclasses.

**Keywords: ** Bi-univalent functions, close-to-convex,quasi convex,coefficient bounds.

T.R.K.Kumar

Department of Mathematics, R.M.K.Engineering College, R.S.M.Nagar, Kavaraipettai, Tamilnadu, India.

E-mail: kumartrk@yahoo.com

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Article Type |
: | Research Article |

Title |
: | Semicircle : A Case Study on Concept of Trigonometrical Ratios of Acute Angles |

Country |
: | India |

Authors |
: | Manoj Kumar Srivastav |

**Abstract: ** Trigonometric ratios of an acute angle in a right angle triangle express the relationship between the angle and length of its sides. Angles in a semicircle is a right angle. The angle inscribed in a semicircle is always 90째 In the present paper ,author tried to present the properties of semicircle in which it is possible to explain the concept regarding trigonometric ratios of acute angles.

**Keywords: ** Semicircle, trigonometric ratios, acute angle.

Manoj Kumar Srivastav

Champdani Adarsh Sharmik Vidyamandir, 3, R. B. S. Road, Champdani, Baidyabati, Hooghly, W.B., India.

E-mail: mksrivastav2011@rediffmail.com

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Article Type |
: | Research Article |

Title |
: | On Middle Neighborhood Graphs |

Country |
: | India |

Authors |
: | V.R.Kulli |

**Abstract: ** The middle neighborhood graph $M_{nd}(G)$ of a graph $G = (V, E)$ is the graph with the vertex set $V\cup S$ where S is the set of all open neighborhood sets of G in which two vertices u and v are adjacent if u, $v\in S$ and $u\cap v\neq\phi$ or $u\in V$ and v is an open neighborhood set of G containing u. In this paper, some properties of this new graph are established. Also characterizations are given for graphs (i) whose middle neighborhood graphs are connected, (ii) whose middle neighborhood graphs are Eulerian.

**Keywords: ** Open neighborhood set, neighborhood graph, middle neighborhood graph, Eulerian.

V.R.Kulli

Department of Mathematics, Gulbarga University, Gulbarga, India.

E-mail: vrkulli@gmail.com

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Article Type |
: | Research Article |

Title |
: | Finding the Rank Orders of the Elementary Concepts of Arithmetic Used in I.C.S.E. Examination |

Country |
: | India |

Authors |
: | Dr. Sk. Samsul Alam |

**Abstract: ** In this paper, an attempt has been made to find the rank orders of the identified top fifty eight elementary concepts of arithmetic which were used to solve the arithmetical problems of two years question papers of mathematics of Indian Certificate of Secondary Education (I.C.S.E.) Examination (2005 & 2006) of The Council for the Indian School Certificate Examinations (C.I.S.C.E.) through task analysis technique. The teachers as well as learners will be benefited in several ways like to diagnose the weakness of the students, to find the difficult order of these concepts, to emphasize on the learning of these concepts at appropriate grade levels etc. after knowing these elementary concepts. This study will help the entire curriculum development of school mathematics.

**Keywords: ** Identifying, major concept, elementary concept, arithmetic, task analysis technique, mathematics question papers, frequency of use, rank orders.

Dr. Sk. Samsul Alam

Assistant Teacher, Silut Basantapur High School, Sahapur Basantapur, Burdwan, West Bengal, India.

E-mail: samsulalam_s@yahoo.in