A subset Y of a space X is almost countably compact in X if for every countable cover U of Y by open subsets of X, there exists a finite subfamily V of U such that Y ⊆ U V . In this paper we investigate the relationship between almost countably compact spaces and relatively almost countably compact subsets, and also study various properties of relatively almost countably compact subsets.
A subspace Y of a space X is almost Lindelöf (strongly almost Lindelöf) in X if for every open cover U of X (of Y by open subsets V of X), there exists a countable subset V of U such that Y ⊆ S {V : V ∈ V }. In this paper we investigate the relationships between relatively almost Lindelöf subset and relatively strongly almost Lindelöf subset by giving some examples, and also study various properties of relatively almost Lindelöf subsets and relatively strongly almost Lindelöf subsets.
Simple modules for restricted Lie superalgebras are studied. The indecomposability of baby Kac modules and baby Verma modules is proved in some situation. In particular, for the classical Lie superalgebra of type $A(n|0)$, the baby Verma modules $Z_{\chi }(\lambda )$ are proved to be simple for any regular nilpotent $p$-character $\chi $ and typical weight $\lambda $. Moreover, we obtain the dimension formulas for projective covers of simple modules with $p$-characters of standard Levi form.
The main topic of the first section of this paper is the following theorem: let A be an Archimedean f-algebra with unit element e, and T : A → A a Riesz homomorphism such that T 2 (f) = T(fT(e)) for all f ∈ A. Then every Riesz homomorphism extension Te of T from the Dedekind completion A δ of A into itself satisfies Te2 (f) = Te(fT(e)) for all f ∈ A δ . In the second section this result is applied in several directions. As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application of the above result is a new approach to the Dedekind completion of commutative d-algebras.
In this paper, two robust consensus problems are considered for a multi-agent system with various disturbances. To achieve the robust consensus, two distributed control schemes for each agent, described by a second-order differential equation, are proposed. With the help of graph theory, the robust consensus stability of the multi-agent system with communication delays is obtained for both fixed and switching interconnection topologies. The results show the leaderless consensus can be achieved with some disturbances or time delays.
In this note we consider a linear-fractionai programming problem with equality linear constraints. Following Rohn, we define a generalized relative sensitivity coefficient measuring the sensitivity of the optimal value for a linear program and a linear-fractional minimization problem with respect to the perturbations in the problem data. By using an extension of Rohn's result for the linear programming case, we obtain, via Charnes-Cooper variable change, the relative sensitivity coefficient for the linear-fractional problem. This coefficient involves only the measure of data perturbation, the optimal solution for the initial linear-fractional problem and the optimal solution of the dual problem of linear programming equivalent to the initial fractional problem.
Properties of $n$-ary groups connected with the affine geometry are considered. Some conditions for an $n$-ary $rs$-group to be derived from a binary group are given. Necessary and sufficient conditions for an $n$-ary group $<\theta ,b>$-derived from an additive group of a field to be an $rs$-group are obtained. The existence of non-commutative $n$-ary $rs$-groups which are not derived from any group of arity $m<n$ for every $n\ge 3$, $r>2$ is proved.