We characterize those Tychonoff quasi-uniform spaces $(X,\mathcal {U})$ for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family $\mathcal {K}_{0}(X)$ of nonempty compact subsets of $X$. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space $X$ is uniformly locally compact on $\mathcal {K}_{0}(X)$ if and only if $X$ is paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is $\sigma $-compact if and only if its (lower) semicontinuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces $(X,\mathcal {U})$ for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on $\mathcal {K}_{0}(X)$ is obtained.
The aim of this paper is to present sufficient conditions for all bounded solutions of the second order neutral differential equation \[ \big (x(t)-px(t-\tau )\big )^{\prime \prime }- q(t)x\big (\sigma (t)\big )=0 \] to be oscillatory and to improve some existing results. The main results are based on the comparison principles.
Let X = (X,Y) be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let Xt = [(X-t,Y-t) | X>t, Y>t] denotes the corresponding pair of residual lifetimes after time t, with t >= 0. This note deals with stochastic comparisons between X and Xt: we provide sufficient conditions for their comparison in usual stochastic and lower orthant orders. Some of the results and examples presented here are quite unexpected, since they show that there is not a direct correspondence between univariate and bivariate aging. This work is mainly based on, and related to, recent papers by Bassan and Spizzichino ([4] and [5]), Averous and Dortet-Bernadet [2], Charpentier ([6] and [7]) and Oakes [16].
For a connected graph G of order n > 2 and a linear ordering s : v1, v2, . . . , vn of vertices of G, d(s) = ∑ nP−1 i=1 d(vi , vi+1), where d(vi , vi+1) is the distance between vi and vi+1. The upper traceable number t +(G) of G is t +(G) = max{d(s)}, where the maximum is taken over all linear orderings s of vertices of G. It is known that if T is a tree of order n ≥ 3, then 2n−3 ≤ t +(T) ≤ ⌊n 2 /2⌋−1 and t +(T) ≤ ⌊n 2 /2⌋−3 if T ≠ Pn. All pairs n, k for which there exists a tree T of order n and t +(T) = k are determined and a characterization of all those trees of order n ≥ 4 with upper traceable number ⌊n 2 /2⌋ − 3 is established. For a connected graph G of order n ≥ 3, it is known that n − 1 ≤ t +(G) ≤ ⌊n 2 /2⌋ − 1. We investigate the problem of determining possible pairs n, k of positive integers that are realizable as the order and upper traceable number of some connected graph.
We first investigate factorizations of elements of the semigroup $S$ of upper triangular matrices with nonnegative entries and nonzero determinant, provide a formula for $\rho (S)$, and, given $A\in S$, also provide formulas for $l(A)$, $L(A)$ and $\rho (A)$. As a consequence, open problem 2 and problem 4 presented in N. Baeth et al. (2011), are partly answered. Secondly, we study the semigroup of upper triangular matrices with only positive integral entries, compute some invariants of such semigroup, and also partly answer open Problem 1 and Problem 3 in N. Baeth et al. (2011).
Eric van Douwen produced in 1993 a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of βN. We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of βN expanding on earlier work of Balcar and Błaszczyk (1990) and Simon (1987).
In this paper we investigate the relation between the lattice of varieties of pseudo $MV$-algebras and the lattice of varieties of lattice ordered groups.
Let X be a normed linear space. We investigate properties of vector functions F : [a, b] → X of bounded convexity. In particular, we prove that such functions coincide with the delta-convex mappings admitting a Lipschitz control function, and that convexity Kb a F is equal to the variation of F ′ + on [a, b). As an application, we give a simple alternative proof of an unpublished result of the first author, containing an estimate of convexity of a composed mapping.