Volume 59 , issue 1$-$2 ( 2007 ) | back |

Weak forms of open mappings and strong forms of sequence-covering mappings | 1$-$8 |

**Abstract**

In this paper, we discuss some weak forms of open mappings and some strong forms of sequence-covering mappings, and establish some relations among these mappings. As some applications of these results, we obtain that images of metric spaces under certain weak forms of open mappings can be characterized as images of metric spaces under certain strong forms of sequence-covering mappings.

**Keywords:** Open (weak-open, $sn$-open) mapping, almost open (almost
weak-open, almost $sn$-open) mapping, 1-sequence-covering mapping,
2-sequence-covering mapping.

**MSC:** 54C05, 54C10

Some properties of ordered hypergraphs | 9$-$13 |

**Abstract**

In this paper, all graphs and hypergraphs are finite. For any ordered hypergraph $H$, the associated graph $G_H$ of $H$ is defined. Some basic graph-theoretic properties of $H$ and $G_H$ are compared and studied in general and specially via the largest negative real root of the clique polynomial of $G_H$. It is also shown that any hypergraph $H$ contains an ordered subhypergraph whose associated graph reflects some graph-theoretic properties of $H$. Finally, we define the depth of a hypergraph $H$ and introduce a constructive algorithm for coloring of $H$.

**Keywords:** Hypergraph, Clique polynomial, Interval cycle.

**MSC:** 05C65, 05C99

On sequence-covering $msss$-maps | 15$-$21 |

**Abstract**

This paper gives characterizations of metric spaces under some sequence-covering $msss$-maps by means of certain kind of $\sigma$-locally countable networks.

**Keywords:** $msss$-maps; 1-sequence-covering maps; sequence-covering maps;
strong compact-covering maps; weak-bases; bases; $s$-networks; $cs$-networks.

**MSC:** 54E99, 54C10; 54D55

On co$LC$ topologies | 23$-$30 |

**Abstract**

In this paper we introduce the concept of co$LC$ topologies and discuss some of their basic properties. We relate this concept to classes of functions between topological spaces.

**Keywords:** Lindelöf, $LC$-space, coLindelöf topology, co$LC$ topology.

**MSC:** 54A05, 54D20, 54G10

Spline-wavelet solution of singularly perturbed boundary problem | 31$-$46 |

**Abstract**

Boundary or interior layers typically appear in singularly perturbed boundary problems. Solution gradients are very sharp in these regions, and it seems natural to use wavelets to obtain numerical solution. As layers are usually positioned on boundaries, wavelets have to be modified appropriately. In this paper the collocation method based on spline-wavelets is derived and tested on simple linear one-dimensional singularly perturbed boundary problem.

**Keywords:** Boundary layers, spline wavelets, collocation.

**MSC:** 65L10, 65L60, 65T60

Norm and lower bounds of operators on weighted sequence spaces | 47$-$56 |

**Abstract**

This paper is concerned with the problem of finding the upper and lower bounds of matrix operators from weighted sequence spaces $l_p(v,I)$ into $l_p(v,F)$. We consider certain matrix operators such as Ces\`aro, Copson and Hilbert which were recently considered on the usual weighted sequence spaces $l_p(v)$.

**Keywords:** Matrix operator, norm and lower bound, weighted sequence spaces,
Ces\`aro operator, Copson operator, Hilbert operator.

**MSC:** 46A45, 40H05

On pseudo-sequence-covering $\pi$ images of locally separable metric spaces | 57$-$64 |

**Abstract**

In this paper, pseudo-sequence-covering $\pi$ images and pseudo-sequence-covering, $s$, $\pi$ images of locally separable metric spaces are discussed, their internal characterizations are given, which extends and improves the study of images of locally separable metric spaces.

**Keywords:** Pseudo-sequence-covering mappings; $\pi$ mappings; $wcs$-covers;
point-star networks.

**MSC:** 54E20, 54E40, 54D55

Some subclasses of close-to-convex and quasi-convex functions | 65$-$73 |

**Abstract**

In the present paper, the author introduce two new subclasses ${\Cal S}_{sc}^{(k)}(\alpha,\beta,\gamma)$ of close-to-convex functions and ${\Cal C}_{sc}^{(k)}(\alpha,\beta,\gamma)$ of quasi-convex functions with respect to $2k$-symmetric conjugate points. The coefficient inequalities and integral representations for functions belonging to these classes are provided, the inclusion relationships and convolution conditions for these classes are also provided.

**Keywords:** Close-to-convex functions, quasi-convex functions, differential
subordination, Hadamard product, $2k$-symmetric conjugate points.

**MSC:** 30C45