# Mathematica Moravica, Vol. 14-1 (2010)

A Class of Polynomials in Two Variables
Mathematica Moravica, Vol. 14-1 (2010), 1–14.

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Abstract. In this paper, we present some families of polynomials in two variables. Some further results of these polynomials as generating function, Rodriguez formula and recurrence relations are discussed. We derive various families of bilinear and bilateral generating functions. We also give some particular cases reduced to Hermite-Hermite and Laguerre-Laguerre polynomials.
Keywords. Rodriguez formula, recurrence relation, generating function, bilinear and bilateral generating function.

Warped Product Semi-Invariant Submanifolds in Almost Paracontact Metric Manifolds
Mathematica Moravica, Vol. 14-1 (2010), 15–21.

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Abstract. In this paper we have investigated the existence of warped product semi-invariant submanifolds in almost paracontact metric manifolds. Finally, we see that there exists no any warped product semi invariant submanifold in almost paracontact metric manifolds such that the contra variant vector field tangent to submanifold.
Keywords. Paracontact metric manifold, warped product, warped product semi-invariant submanifold.

Geometry of Semi-Invariant Submanifolds of a Riemannian Product Manifold
Mathematica Moravica, Vol. 14-1 (2010), 23–34.

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Abstract. In this paper, we show new results on semi-invariant submanifolds of a Riemannian product manifold and introduce equations related with geometry of semi-invariant submanifold in real product space forms. We characterize and study the geometry of the semiinvariant submanifolds in a Riemannian product manifold.
Keywords. Riemannian product manifold, real space form; semiinvariant submanifold, Semi-Riemannian product, partially integrable and Ricci tensor and Scalar curvature.

Some Remarks on the Notion of Contraction of Lie Group Representations
Mathematica Moravica, Vol. 14-1 (2010), 35–46.

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Abstract. In the series of papers [1-4], L. Barker developed a general notion of convergence for sequences of Hilbert spaces and related objects (vectors, operators...). In this paper, we remark that Barker's convergence for sequences of operators provides a notion of contraction of Lie group (unitary) representations and we compare it to the usual one introduced by J. Mickelsson and J. Niederle. This allows us to illustrate Barker's convergence of operators by various examples taken from contraction theory.
Keywords. Contractions, Lie groups, unitary representations, sequences of Hilbert spaces.

Some Equalities which Hold in the $(n,m)$-Group $(Q,A)$ for $n\geq 2m$
Mathematica Moravica, Vol. 14-1 (2010), 47–51.

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Abstract. In this paper, we have proved two equalities which hold in an $(n,m)$-group $(Q,A)$ for $n\geq 2m$. The first of them is a generalization of the equality $(a\cdot b)^{-1} = b^{-1}\cdot a^{-1}$, which holds in the binary group $(Q,\cdot)$. The second of them is equality $A(x_{1}^{m}, b_{1}^{n-2m}, y_{1}^{m}) = A\bigl(A(x_{1}^{m}, a_{1}^{n-2m}, (a_{1}^{n-2m}, e(b_{1}^{n-2m}))^{-1}), a_{1}^{n-2m}, y_{1}^{m}\bigr).$ Keywords. $(n,m)$-group, $\{1, n-m+1\}$-neutral operation, inverse operation of the $(n,m)$-groupoid.

Central Operation of the $(n,m)$-Group
Mathematica Moravica, Vol. 14-1 (2010), 53–59.

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Abstract. In this paper we have defined a central operation of the $(n,m)$-group, as a mapping $\alpha$ of the set $Q^{n-2m}$ into the set $Q^{m}$, such that for every $a_{1}^{n-2m}, b_{1}^{n-2m}\in Q$ and for every $x_{1}^{m}\in Q^{m}$ the following equality holds: $A(\alpha(a_{1}^{n-2m}), a_{1}^{n-2m}, x_{1}^{m}) = A(x_{1}^{m}, \alpha(b_{1}^{n-2m}), b_{1}^{n-2m}).$ This is a generalization of the notion of a central operation of the $n$-group, i.e. of the central element of a binary group. The notion of the central operation of the $n$-group was defined by Janez Ušan in [4]. Furthermore, in this paper we have proved some claims which hold for the central operation of the $(n, m)$-group.
Keywords. $(n, m)$-group, $\{1, n-m+1\}$-neutral operation, inverse operation of the $(n, m)$-groupoid, central operation.

The γ-open Open Topology for Function Spaces
Mathematica Moravica, Vol. 14-1 (2010), 61–67.

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Abstract. In this paper we have introduced the notion of $\gamma$-open open topology and proved some properties which the topology does possess. We have also introduced the concept of convergence of nets in $\gamma H(X)$ (where $\gamma H(X)$ is the set of all self $\gamma$-homeomorphisms on a topological space $X$) and showed when $\gamma H(X)$ is complete.
Keywords. $\gamma$-open sets, $\gamma$-open open topology, $\gamma H(X)$-the set of all self $\gamma$-homeomorphisms on $X$, $\gamma$-convergence, $\gamma$-regular.

On the Hyper Order of Solutions of Linear Differential Equations with Entire Coefficients
Mathematica Moravica, Vol. 14-1 (2010), 69–82.

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Abstract. In this paper, we investigate higher order homogeneous linear differential equations with entire coefficients of finite order. We improve and extend the results due to the second author and Hamouda by introducing the concept of hyper-order. We also consider non homogeneous linear differential equations.
Keywords. Differential equations, meromorphic function, hyper-order.

Convergence to Common Fixed Points of Two Asymptotically Nonexpansive Type Mappings
Mathematica Moravica, Vol. 14-1 (2010), 83–88.

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Abstract. Common fixed points of two asymptotically non expansive type mappings have been approximated by strong convergence of an iteration scheme in a uniformly convex Banach space.
Keywords. Asymptotically nonexpansive type mappings, strong convergence, iteration scheme; common fixed points.

Common Random Fixed Point and Random Best Approximation in Non-Starshaped Domain of $q$-Normed Spaces
Mathematica Moravica, Vol. 14-1 (2010), 89–99.

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Abstract. The aim of this paper is to obtain some common random fixed point by extending the concept of uniformly $\mathcal{R}$-subweakly commuting mappings to the case of non-star shaped domain in $q$-normed space. Random best approximation results have also been obtained as its application. This work provides extension as well as substantial improvement of some results in the existing literature.
Keywords. Asymptotically non-expansive maps, uniformly asymptotically regular map, random best approximation, random fixed point, random operator, non-expansivemapping, uniformly $\mathcal{R}$-subweakly commuting maps, property $(N)$, property $(C)$, $q$-normed space.

Convergence of Common Fixed Point for Asymptotically Quasi-Nonexpansive Mappings in Convex Metric Spaces
Mathematica Moravica, Vol. 14-1 (2010), 101–111.

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Abstract. In this paper, the necessary and sufficient conditions for three-step iterative sequences with errors to converge to a common fixed point for three asymptotically quasi-nonexpansive mappings is established in convex metric spaces. The results of this paper are generalizations and improvements of the corresponding results of Chang [1] - [3], Kim et al. [8], Liu [9] - [11], Ghosh and Debnath [4], Xu and Noor [15], Shahzad and Udomene [13], Khan and Takahashi [6] and Khan and Uddin [7].
Keywords. Convex metric space, asymptotically quasi-nonexpansive mapping, asymptotically nonexpansive mapping, three-step iterative sequence with errors, common fixed point.

On the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces
Mathematica Moravica, Vol. 14-1 (2010), 113–119.

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Abstract. In the present paper we prove a common fixed point theorem for Ishikawa iterates of a pair of multivalued mappings on a Banach space, satisfying nonexpansive type condition which extend and generalize the results of Rhoades [16, 17] and others.
Keywords. Ishikawa-type iteration, Banach spaces, multivalued nonexpansive mappings, common fixed point.

The Matrix Transformations on Double Sequence Space of χπ2
Mathematica Moravica, Vol. 14-1 (2010), 121–127.

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Abstract. Let $\xi$ denote the space of all prime sense double gai sequences and $\Lambda^{2}$ the space of all prime sense double analytic sequences. First we show that the set $E = \{s^{(mn)}: m,n = 1, 2, 3, \dots\}$ is a determining set for $\chi_{\pi}^{2}$. The set of all finite matrices transforming $\chi_{\pi}^{2}$ into FK-space $Y$ denoted by $(\chi_{\pi}^{2}: Y)$. We characterize the classes $(\chi_{\pi}^{2}: Y)$ when $Y = c_{0}^{2}, c^{2}, \chi^{2}, l^{2}, \Lambda^{2}$. But the approach to obtain these result in the present paper is by determining set for $\chi_{\pi}^{2}$. First, we investigate a determining set for $\chi_{\pi}^{2}$ and then we characterize the classes of matrix transformations involving $\chi_{\pi}^{2}$ and other known sequence spaces.
Keywords. Determining set, gai sequence, analytic sequence, doublesequence.

CR-Warped Product Submanifolds of Lorentzian Manifolds
Mathematica Moravica, Vol. 14-1 (2010), 129–136.

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Abstract. In this paper, we study warped product $CR$-submanifolds of a Lorentzian Sasakian manifold. We show that the warped product of the type $M = N_{\perp}\times_{f} N_{T}$ in a Lorentzian Sasakian manifold is simply $CR$-product and obtain a characterization of $CR$-warped product submanifolds.
Keywords. Warped product, $CR$-submanifold, contact $CR$-warped product, Lorentzian Sasakian manifold.

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