We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter $\varepsilon .$ The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.
In this article, we consider the operator $L$ defined by the differential expression \[ \ell (y)=-y^{\prime \prime }+q(x) y ,\quad - \infty < x < \infty \] in $L_2(-\infty ,\infty)$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition \[ \sup _{-\infty < x < \infty} \Big \lbrace \exp \bigl (\epsilon \sqrt{|x|}\bigr ) |q(x)|\Big \rbrace < \infty, \quad \epsilon > 0 \] holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.
A graph $G$ is stratified if its vertex set is partitioned into classes, called strata. If there are $k$ strata, then $G$ is $k$-stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color $X$ in a stratified graph $G$, the $X$-eccentricity $e_X(v)$ of a vertex $v$ of $G$ is the distance between $v$ and an $X$-colored vertex furthest from $v$. The minimum $X$-eccentricity among the vertices of $G$ is the $X$-radius $\mathop {\mathrm rad}\nolimits _XG$ of $G$ and the maximum $X$-eccentricity is the $X$-diameter $\mathop {\mathrm diam}\nolimits _XG$. It is shown that for every three positive integers $a, b$ and $k$ with $a \le b$, there exist a $k$-stratified graph $G$ with $\mathop {\mathrm rad}\nolimits _XG=a$ and $\mathop {\mathrm diam}\nolimits _XG=b$. The number $s_X$ denotes the minimum $X$-eccetricity among the $X$-colored vertices of $G$. It is shown that for every integer $t$ with $\mathop {\mathrm rad}\nolimits _XG \le t \le \mathop {\mathrm diam}\nolimits _XG$, there exist at least one vertex $v$ with $e_X(v)=t$; while if $\mathop {\mathrm rad}\nolimits _XG \le t \le s_X$, then there are at least two such vertices. The $X$-center $C_X(G)$ is the subgraph induced by those vertices $v$ with $e_X(v)=\mathop {\mathrm rad}\nolimits _XG$ and the $X$-periphery $P_X(G)$ is the subgraph induced by those vertices $v$ with $e_X(G)=\mathop {\mathrm diam}\nolimits _XG$. It is shown that for $k$-stratified graphs $H_1, H_2, \dots , H_k$ with colors $X_1, X_2, \dots , X_k$ and a positive integer $n$, there exists a $k$-stratified graph $G$ such that $C_{X_i}(G) \cong H_i \ (1 \le i \le k)$ and $d(C_{X_i}(G), C_{X_j}(G)) \ge n \text{for} i \ne j$. Those $k$-stratified graphs that are peripheries of $k$-stratified graphs are characterized. Other distance-related topics in stratified graphs are also discussed.
We give characterizations of the distributional derivatives $D^{1,1}$, $D^{1,0}$, $D^{0,1}$ of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.
This paper is a continuation of [6], where irreducibility in the sense of Duffus and Rival (DR-irreducibility) of monounary algebras was defined. The definition is analogous to that introduced by Duffus and Rival [1] for the case of posets. In [6] we found all connected monounary algebras A possessing a cycle and such that A is
DR-irreducible. The main result of the present paper is Thm. 4.1 which describes all connected monounary algebras A without a cycle and such that A is DR-irreducible.Other types of irreducibility of monounary algebras defined by means of the notion of a retract were studied in [2]–[5].