Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of $H$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_5-P_4$ and $K_5-Y_4$-graphic sequences where $Y_4$ is a tree on 5 vertices and 3 leaves.
A matrix $\Cal A$ whose entries come from the set $\{+,-,0\}$ is called a {\it sign pattern matrix}, or {\it sign pattern}. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by ${\cal DSSP}(m,2)$, is introduced. We determine all potentially nilpotent sign patterns in ${\cal DSSP}(3,2)$ and ${\cal DSSP}(5,2)$, and prove that one sign pattern in ${\cal DSSP}(3,2)$ is potentially stable.
In the framework of a stochastic optimization problem, it is assumed that the stochastic characteristics of optimized system are estimated from randomly right-censored data. Such a case is frequently encountered in time-to-event or lifetime studies. The analysis of precision of such a solution is based on corresponding theoretical properties of estimated stochastic characteristics. The main concern is to show consistency of optimal solution even in the random censoring case. Behavior of solutions for finite data sizes is studied with the aid of randomly generated example.
First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if $R$ is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime $R$-modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.
We characterize prime submodules of $R\times R$ for a principal ideal domain $R$ and investigate the primary decomposition of any submodule into primary submodules of $R\times R.$.
Let $\mathcal {P}\mathcal {B}_m$ be the category of all principal fibred bundles with $m$-dimensional bases and their principal bundle homomorphisms covering embeddings. We introduce the concept of the so called $(r,m)$-systems and describe all gauge bundle functors on $\mathcal {P}\mathcal {B}_m$ of order $r$ by means of the $(r,m)$-systems. Next we present several interesting examples of fiber product preserving gauge bundle functors on $\mathcal {P}\mathcal {B}_m$ of order $r$. Finally, we introduce the concept of product preserving $(r,m)$-systems and describe all fiber product preserving gauge bundle functors on $\mathcal {P}\mathcal {B}_m$ of order $r$ by means of the product preserving $(r,m)$-systems.
In this paper we apply the notion of the product $MV$-algebra in accordance with the definition given by B. Riečan. We investigate the convex embeddability of an $MV$-algebra into a product $MV$-algebra. We found sufficient conditions under which any two direct product decompositions of a product $MV$-algebra have isomorphic refinements.
If $G$ is a connected graph with distance function $d$, then by a step in $G$ is meant an ordered triple $(u, x, v)$ of vertices of $G$ such that $d(u, x) = 1$ and $d(u, v) = d(x, v) + 1$. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2.