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
Let $G$ be a mixed graph. The eigenvalues and eigenvectors of $G$ are respectively defined to be those of its Laplacian matrix. If $G$ is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of $G$ corresponding to its second smallest eigenvalue (also called the algebraic connectivity of $G$). For $G$ being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of $G$ corresponding to its smallest eigenvalue.
We prove the complete convergence of Shannon's, paired, genetic and α-entropy for random partitions of the unit segment. We also derive exact expressions for expectations and variances of the above entropies using special functions.
In this paper we investigate holomorphically projective mappings of generalized Kählerian spaces. In the case of equitorsion holomorphically projective mappings of generalized Kählerian spaces we obtain five invariant geometric objects for these mappings.
A method of estimation of intrinsic volume densities for stationary random closed sets in Rd based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the discrete approximation, which differs from the standard techniques used for measuring parallel sets in image analysis. A method of reducing the bias is proposed and tested on simulated data.