For an $\ell $-cyclically ordered set $M$ with the $\ell $-cyclic order $C$ let $P(M)$ be the set of all monotone permutations on $M$. We define a ternary relation $\overline{C}$ on the set $P(M)$. Further, we define in a natural way a group operation (denoted by $\cdot $) on $P(M)$. We prove that if the $\ell $-cyclic order $C$ is complete and $\overline{C}\ne \emptyset $, then $(P(M), \cdot ,\overline{C})$ is a half cyclically ordered group.
We present an existence theorem for monotonic solutions of a quadratic integral equation of Abel type in C[0, 1]. The famous Chandrasekhar’s integral equation is considered as a special case. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.
The set of points of upper semicontinuity of multi-valued mappings with a closed graph is studied. A topology on the space of multi-valued mappings with a closed graph is introduced.
If $Q$ is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group $\mathop {\mathrm Mlt}Q$ is a Frobenius group. Conversely, if $\mathop {\mathrm Mlt}Q$ is a Frobenius group, $Q$ a quasigroup, then $Q$ has to be isotopic to an Abelian group. If $Q$ is, in addition, finite, then it must be a central quasigroup (a $T$-quasigroup).
We reduce the problem on multiplicities of simple subquotients in an $\alpha $-stratified generalized Verma module to the analogous problem for classical Verma modules.
A vertex k-coloring of a graph G is a multiset k-coloring if M(u) ≠ M(v) for every edge uv ∈ E(G), where M(u) and M(v) denote the multisets of colors of the neighbors of u and v, respectively. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For an integer l ≥ 0, the l-corona of a graph G, corl (G), is the graph obtained from G by adding, for each vertex v in G, l new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for l-coronas of all complete graphs, the regular complete multipartite graphs and the Cartesian product Kr K2 of Kr and K2. In addition, we show that the minimum l such that χm(corl (G)) = 2 never exceeds χ(G) − 2, where G is a regular graph and χ(G) is the chromatic number of G.
We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations u ′′(x) +∑ i pi(x)u ′ (hi(x)) +∑ i qi(x)u(gi(x)) = 0 without the delay conditions hi(x), gi(x) ≤ x, i = 1, 2, . . ., and u ′′(x) + ∫ ∞ 0 u ′ (s)dsr1(x, s) + ∫ ∞ 0 u(s)dsr0(x, s) = 0.
The current article considers some infinite groups whose finitely generated subgroups are either permutable or pronormal. A group G is called a generalized radical, if G has an ascending series whose factors are locally nilpotent or locally finite. The class of locally generalized radical groups is quite wide. For instance, it includes all locally finite, locally soluble, and almost locally soluble groups. The main result of this paper is the following Theorem. Let G be a locally generalized radical group whose finitely generated subgroups are either pronormal or permutable. If G is non-periodic then every subgroup of G is permutable.
Let $(X,\mathbb I)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R\subseteq T^2\times X$ it is possible to find a set $A\subseteq T$ such that $R(A^2)$ is completely $\mathbb I $-nonmeasurable, i.e, it is $\mathbb I$-nonmeasurable in every positive Borel set. We also obtain such a set $A\subseteq T$ simultaneously for continuum many relations $(R_\alpha )_{\alpha <2^\omega }.$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.
Qualitative comparison of the nonoscillatory behavior of the equations \[ L_ny(t) + H(t,y(t)) = 0 \] and \[ L_ny(t) + H(t,y(g(t))) = 0 \] is sought by way of finding different nonoscillation criteria for the above equations. $L_n$ is a disconjugate operator of the form
\[ L_n = \frac{1}{p_n(t)} \frac{\mathrm{d}{}}{\mathrm{d}t} \frac{1}{p_{n-1}(t)} \frac{\mathrm{d}{}}{\mathrm{d}t} \ldots \frac{\mathrm{d}{}}{\mathrm{d}t} \frac{\cdot }{p_0(t)}. \] Both canonical and noncanonical forms of $L_n$ have been studied.