Abstract

Generalizations of openness, such as semi-open, preopen, semi-pre-open, $\alpha$-open, etc\. are important in topological spaces and in particular in topological spaces on which ideals are defined. $\alpha$-equivalent topologies and $*$-equivalent topologies with respect to an ideal have some common properties. Relations between these aforementioned notions of openness are investigated within the framework of $\alpha$-equivalence and $*$-equivalence.

Keywords: $\alpha$-equivalent topologies; $*$-equivalent topologies; semi-open sets, preopen sets, semi-pre-open sets, $\alpha$-open sets.

MSC: 54A05, 54A10, 54C08, 54C10

Abstract

In this paper, we relate composition operators with multiplication operators on the general Banach function spaces on a $\sigma$-finite measure space. We use this relation to study the invertibility and Fredholmness properties of composition operators on the Banach function spaces.

Keywords: Banach function spaces, closed range, composition operators, Fredholm operator, invertibility, measurable transformation.

MSC: 47B33, 46E30, 47B07, 46B70

Abstract

In this paper, we introduce and study the idea of a new type of compactness, defined in terms of a grill $\cal G$ in a topological space $X$. Calling it $\cal G$-compactness, we investigate its relation with compactness, among other things. Analogues of Alexender's subbase theorem and Tychonoff product theorem are also obtained for $\cal G$-compactness. Finally, we exhibit a new method, in terms of the deliberations here, for construction of the well known one-point compactification of a $T_2$, locally compact and noncompact topological space.

Keywords: Grill, $\cal G$-compactness, $\cal G$-regularity, one-point compactification.

MSC: 54D35, 54D99

Abstract

If each edge of a 3-partite graph is assigned a positive or a negative sign then it is called a signed 3-partite graph. Also, signed degree of a vertex x in a signed 3-partite graph is the number of positive edges incident with x less than the number of negative edges incident with x. The set of distinct signed degrees of the vertices of a signed 3-partite graph is called its signed degree set. In this paper, we prove that every set of n integers is the signed degree set of some connected signed 3-partite graph.

Keywords: Signed graph, signed tripartite graph, signed degree, signed set.

MSC: 05C22

Abstract

Let $\cal H$ be a separable infinite dimensional complex Hilbert space and let $B(\cal H)$ denote the algebra of bounded operators on $\cal H$ into itself. The generalized derivation $\delta_{A,B}$ is defined by $\delta_{A,B}(X)=AX-XB$. For pairs $C=(A_{1},A_{2})$ and $D=(B_{1},B_{2})$ of operators, we define the elementary operator $\Phi_{C,D}$ by $\Phi_{C,D}(X)=A_{1}XB_{1}-A_{2}XB_{2}$. If $A_{2}=B_{2}=I$, we get the elementary operator $\Delta_{A_{1},B_{1}}(X)=A_{1}XB_{1}-X$. Let $d_{A,B}=\delta_{A,B}$ or $\Delta_{A,B}$. We prove that if $A, B^{*}$ are $\log$-hyponormal, then $f(d_{A,B})$ satisfies (generalized) Weyl's Theorem for each analytic function $f$ on a neighborhood of $\sigma(d_{A,B})$, we also prove that $f(\Phi_{C,D})$ satisfies Browder's Theorem for each analytic function $f$ on a neighborhood of $\sigma(\Phi_{C,D})$.

Keywords: Elementary Operators, $p$-hyponormal, $\log$-hyponormal, Weyl's Theorem, single valued extension property.

MSC: 47B47, 47A30, 47B20

Abstract

Submanifolds of the Euclidean spaces satisfying equality in the basic Chen's inequality have, as is known, many interesting properties. In this paper, we discuss the curvature conditions of the form $E\cdot S=0$ on such submanifolds, where $E$ is any of the standard 4-covariant curvature operators, $S$ is the Ricci curvature operator, and $E$ acts on $S$ as a derivation.

Keywords: Submanifolds, curvature conditions, basic equality.

MSC: 53B25, 53C40

Abstract

Due to a technical error, Table 3 in the paper ``Spline-wavelet solution of singularly perturbed boundary problem'' by Desanka Radunović, published in the last issue of Matematčki Vesnik {\bf59}, 1--2 (2007), 31--46, appeared wrongly. The right Table 3 is printed here.

Keywords: Boundary layers, spline wavelets, collocation.

MSC: 65L10, 65L60, 65T60