In this paper two Denjoy type extensions of the Pettis integral are defined and studied. These integrals are shown to extend the Pettis integral in a natural way analogous to that in which the Denjoy integrals extend the Lebesgue integral for real-valued functions. The connection between some Denjoy type extensions of the Pettis integral is examined.
We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than $c$ has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight $c$ which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of $\pi$-weight less than $\mathfrak p$ has a dense completely Hausdorff (and hence Urysohn) subspace. We show that there exists a Tychonoff space without dense normal subspaces and give other examples of spaces without “good” dense subsets.
Alice Jedličková (ed.) ; translation Melvyn Clarke, Martina Kurtyová ... [et al.]., Obsahuje seznam citovaných děl, and Obsahuje český text, částečně přeloženo
Let G be a connected graph of order n ≥ 3 and let c : E(G) → {1, 2, . . . , k} be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v with respect to c is the k-tuple c(v) = (a1, a2, . . . , ak), where ai is the number of edges incident with v that are colored i (1 ≤ i ≤ k). The coloring c is detectable if distinct vertices have distinct color codes. The detection number det(G) of G is the minimum positive integer k for which G has a detectable k-coloring. We establish a formula for the detection number of a path in terms of its order. For each integer n ≥ 3, let Du(n) be the maximum detection number among all unicyclic graphs of order n and du(n) the minimum detection number among all unicyclic graphs of order n. The numbers Du(n) and du(n) are determined for all integers n > 3. Furthermore, it is shown that for integers k ≥ 2 and n ≥ 3, there exists a unicyclic graph G of order n having det(G) = k if and only if du(n) ≤ k ≤ Du(n).
In this contribution, we present the final outcome of the program initiated in [23], aimed at the determination of a periodic unit cell for plain weave composites with reinforcement imperfections. The emphasis is put on a realistic geometrical description of these material systems utilizing the information provided by in-situ two-dimensional micrographs. Complex geometry of an analyzed composite is approximated using a two-layer periodic unit cell allowing for a mutual shift as well as nesting of individual layers. The parameters of the idealized unit cell are derived via matching appropriate statistical descriptors related to the real material and the idealized geometrical model. Once the optimal geometry of the unit cell is determined, it can be converted to a CAD model and used ot generate the periodic finite element mesh applicable in the subsequent numerical treatment. The individual steps of this procedure are demonstrated in detail for a real world carbon-carbon composite system. and Obsahuje seznam literatury
The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume-finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume-finite element schemes). However, it is of the first order only. (b) Pure discontinuous Galerkin finite element method of higher order combined with a technique avoiding spurious oscillations in the vicinity of shock waves.
We study conditions of discreteness of spectrum of the functional-differential operator Lu = −u ′′ + p(x)u(x) + ∫ ∞ −∞ (u(x) − u(s)) dsr(x, s) on (−∞,∞). In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.
Performance of coherent reliability systems is strongly connected with distributions of order statistics of failure times of components. A crucial assumption here is that the distributions of possibly mutually dependent lifetimes of components are exchangeable and jointly absolutely continuous. Assuming absolute continuity of marginals, we focus on properties of respective copulas and characterize the marginal distribution functions of order statistics that may correspond to absolute continuous and possibly exchangeable copulas. One characterization is based on the vector of distribution functions of all order statistics, and the other concerns the distribution of a single order statistic.
Let $G$ be a simple graph. A subset $S \subseteq V$ is a dominating set of $G$, if for any vertex $v \in V~- S$ there exists a vertex $u \in S$ such that $uv \in E (G)$. The domination number, denoted by $\gamma (G)$, is the minimum cardinality of a dominating set. In this paper we prove that if $G$ is a 4-regular graph with order $n$, then $\gamma (G) \le \frac{4}{11}n$.
We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc., Miloš Arsenović, Miroslav Pavlović., and Seznam literatury