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Issue 4 − E
Volume 3 (2015)

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Article Type |
: | Research Article |

Title |
: | Homogenization of a Nonlinear Fibre-Reinforced Structure with Contact Conditions on the Interface Matrix-Fibres |

Country |
: | Morocco |

Authors |
: | H.Samadi || M.El Jarroudi |

**Abstract: ** We study the homogenization of a nonlinear problem posed in a
fibre-reinforced composite with matrix-fibres interfacial condition
depending on a parameter $\lambda=\lambda(\varepsilon)$, $\varepsilon$ being the size of the basic cell. Using $\Gamma$-convergence methods, three homogenized problems are determined according to the limit of the ratio$ \gamma = \frac{\lambda(\varepsilon)}{\varepsilon}$. The main result is that the effective constitutive relations reveal
non-local terms associated with the microscopic interactions between the
matrix and the fibers.

**Keywords: ** Composite material, periodic fibres, interfacial conditions, $\Gamma$-convergence, homogenized models.

H.Samadi

National School of Applied Science, LABTIC, UAE, Tangier, Morocco.

E-mail: ha.samadi@gmail.com

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Article Type |
: | Research Article |

Title |
: | A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Li\'{e}nard Type |

Country |
: | Argentina |

Authors |
: | Juan E. Napoles Valdes || Luciano M. Lugo Motta Bittencurt || Paulo M. Guzman |

**Abstract: ** In this note we study the continuability of the solutions of a Lienard
type equation with forcing term under suitable assumptions.

**Keywords: ** Continuability, second order, non-autonomous.

Juan E. Napoles Valdes

UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina and UTN, FRRE, French 414, (3500) Resistencia, Argentina.

E-mail: jnapoles@exa.unne.edu.ar

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Article Type |
: | Research Article |

Title |
: | On the Boundedness of Some Nonlinear Differential Equation of Second Order |

Country |
: | Argentina |

Authors |
: | Paulo M. Guzman || Juan E. Napoles Valdes || Luciano M.Lugo |

**Abstract: **In this paper we study the boundedness of the solutions of some nonlinear differential equation using as a key tool the Second Lyapunov method, i.e. find sufficient conditions under which the solutions of this equation are bounded. Various particular cases and methodological remarks are included at the end of paper.

**Keywords: ** Lyapunov's Second Method, boundedness, second order nonlinear equation.

Juan E. Napoles Valdes

UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina and UTN, FRRE, French 414, (3500) Resistencia, Argentina.

E-mail: jnapoles@exa.unne.edu.ar

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Article Type |
: | Research Article |

Title |
: | Approximating Fixed Point in CAT(0) Space by s-iteration Process for a Pair of Single Valued and Multivalued Mappings |

Country |
: | India |

Authors |
: | Manjula Tripathi || Anil Kumar Dubey |

**Abstract: **Suppose $K$ is a
closed convex subset of a complete CAT(0) space $X$. $T$ is
mapping from $K$ to $X$. $F(T)$ is set of fixed point of $T$ which
is nonempty. Sequence $\{x_n\}$ is defined by an element $x_1 \in
k$ such that
\begin{eqnarray*}
x_{n+1} &=& P((1-\alpha_n) Tx_n \oplus \alpha_n y_n)\\
y_n &=& P((1-\beta_n) x_n \oplus \beta_n T x_n) \ \ \ \forall \geq
1
\end{eqnarray*} where $P$ is the nearest point projection from $X$
onto $k$. $\{\alpha_n\}, \{\beta_n\}$ are real sequences in (0,1)
with the condition
\begin{eqnarray*}
\displaystyle \sum^{\infty}_{n=1}\alpha_n \beta_n (1-\beta_n)
=\infty
\end{eqnarray*} Then $\{x_n\}$ converges to some point $x^*$ in
$F(T)$. This result is extension of the result of
Abdul Rehman Razani and saeed Shabhani. [Approximating fixed
points for nonself mappings in CAT(0) spaces Springer 2011:65]

**Keywords: ** S-iteration, CAT(0) spaces, fixed point condition E, nonself mapping, condition C.

Anil Kumar Dubey

Assistant Professor, Bhilai Institute of Technology, Durg, India.

E-mail: anilkumardby70@gmail.com

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Article Type |
: | Research Article |

Title |
: | On Certain Subclasses of Multivalent Functions Involving A Differintegral Operator |

Country |
: | India |

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

**Abstract: ** In this paper, we introduce the class $S_{p}^{m,l,\lambda}\left(\alpha,\beta,\gamma,\sigma;A,B\right)$
of p-valent functions in the unit disc $\mathbb{U}=\left\{z :\left|z\right|<1\right\}$. We obtain coefficient estimate,
distortion and closure thoerems and radii of close-to-convexity, starlikeness and convexity for this class.

**Keywords: ** p-Valent analytic functions; Linear operator; Generalized neighborhoods; Fractional calculus
operators; Hypergeometric function; Differential subordination.

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 |
: | Fuzzy Quasi-maximal Spaces |

Country |
: | India |

Authors |
: | S.Anjalmose || G.Thangaraj |

**Abstract: ** In this paper the concept of Fuzzy Quasi-maximal spaces are introduced. Several properties and examples of fuzzy Quasi-maximal spaces are also studied. Relations between fuzzy Quasi-maximal space and fuzzy Baire space, fuzzy D-Baire space are studied.

**Keywords: ** Fuzzy dense, fuzzy nowhere dense, fuzzy first category, fuzzy Quasi-maximal space, fuzzy Baire space and fuzzy D-Baire space.

S.Anjalmose

Department of Mathematics, St. Joseph's College of Arts $\&$ Science (Autonomous), Manjakuppam, Cuddalore, Tamil Nadu, India.

E-mail: ansalmose@gmail.com

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Article Type |
: | Research Article |

Title |
: | Continuous and Contra Continuous Functions in Bi-topological Spacese |

Country |
: | India |

Authors |
: | Parvinder Singh |

**Abstract: ** The concept of bi-topological spaces was first
introduced by J.C. Kelly \cite{2} in 1963. Many authors such as Levine
\cite{3} contributed as he defined the semi-open sets and semi-continuity
in bi-topological spaces. Maheshwari and Prasad \cite{5} contributed
semi-open sets and semi-continuity to bi-topological spaces. The notion
of $\beta$-open sets contributed by Mashhour et. al. \cite{6} and
Andrijevic \cite{1} define Semi pre-open sets. In this paper we discuss
pre-continuity and semi pre-continuity in bi-topological spaces.
LellisThivager et.al. \cite{4} introduces $g^{\ast}$-closed sets
topological spaces and initiated the concepts of ultra space by using $(1,2)\alpha$-open sets in bi-topological spaces and proved that
each $(1,2)\alpha$-open sets is (1, 2) semi-open and (1, 2) pre-open
but the converse of each is not true. R-Devi and S.Sampath Kumar and M.
Caldas \cite{7} introduced and studied a class of sets and maps between
bi-topological spaces Called supra $\alpha-$open sets and supra $\alpha$-continuous maps respectively.

**Keywords: ** Bi-topological Spaces, Continuous Functions, Contra Continuous Functions.

Parvinder Singh

P.G. Department of Mathematics, S.G.G.S. Khalsa College, Mahilpur (Hoshiarpur), India.

E-mail: parvinder070@gmail.com

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Article Type |
: | Research Article |

Title |
: | The New Proof of Euler's Inequality Using Spieker Center |

Country |
: | India |

Authors |
: | Dasari Naga Vijay Krishna |

**Abstract: ** If R is the Circumradius and r is the Inradius of a non-degenerate triangle then due to EULER we have an inequality referred as \lq\lq Euler's Inequality\rq\rq which states that $R\geq 2r$, and the equality holds when the triangle is Equilateral. In this article let us prove this famous inequality using the idea of \lq Spieker Center \rq.

**Keywords: ** Euler's Inequality, Circumcenter, Incenter, Circumradius, Inradius, Cleaver, Spieker Center, Medial Triangle, Stewart's Theorem.

Dasari Naga Vijay Krishna

Department of Mathematics, Keshava Reddy Educational Instutions, Machiliaptnam, Kurnool, India.

E-mail: vijay9290009015@gmail.com

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Article Type |
: | Research Article |

Title |
: | The Global Maximal Domination Number of a Graph |

Country |
: | India |

Authors |
: | V.R.Kulli |

**Abstract: ** A maximal dominating set D of a graph G is a global maximal dominating set if D is also a maximal dominating set of $\bar{G}$. The global maximal domination number $\gamma_{gm}(G)$ of G is the minimum cardinality of a global maximal dominating set. In this paper, bounds for $\gamma_{gm}(G)$ and exact values of $\gamma_{gm}(G)$ for some standard graphs are obtained. We characterize maximal dominating sets of G which are global maximal dominating sets. Also Nordhaus-Gaddum type results are obtained.

**Keywords: ** Dominating set, maximal dominating set, global maximal dominating set, global maximal domination number.

V.R.Kulli

Department of Mathematics, Gulbarga University, Gulbarga, India.

E-mail: vrkulli@gmail.com

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Article Type |
: | Research Article |

Title |
: | Sequencing Of The Elementary Concepts Of Unitary Method & Identifying These Are Used In M.P., I.C.S.E. Examination And S.C.E. |

Country |
: | India |

Authors |
: | Dr. Sk. Samsul Alam |

**Abstract: ** In this paper, total identified 27 elementary concepts of the major concept unitary method have been sequenced and presented here. This work was done by ¡®text book scanning process¡¯, ¡®task analysis technique¡¯ and incorporating ¡®experts¡¯ opinions¡¯. Here, It is also identified the elementary concepts which were used to solve the arithmetical problems of mathematics question papers of 10th standard of Madhyamik Pariksha (M.P.) (1999, 2000) of West Bengal Board of Secondary Education (W.B.B.S.E.), 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.), School Certificate Examination (S.C.E.) (1999, 2000) of Visva-Bharati (a Central University). This study will help the entire curriculum development of school mathematics.

**Keywords: ** Major concepts, elementary concepts, Identifying, sequencing, unitary method, text book scanning process, task analysis technique.

Dr. Sk. Samsul Alam

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

E-mail: samsulalam_s@yahoo.in