We obtain a simple construction for particular subclasses of several varieties of lattice expansions. The construction allows a unified approach to the characterization of the subdirectly irreducible algebras.
A graph G is a k-tree if either G is the complete graph on k + 1 vertices, or G has a vertex v whose neighborhood is a clique of order k and the graph obtained by removing v from G is also a k-tree. Clearly, a k-tree has at least k + 1 vertices, and G is a 1-tree (usual tree) if and only if it is a 1-connected graph and has no K_{3} -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of k-trees as follows: if G is a graph with at least k + 1 vertices, then G is a k-tree if and only if G has no K_{k+2} -minor, G does not contain any chordless cycle of length at least 4 and G is k-connected., De-Yan Zeng, Jian-Hua Yin., and Obsahuje seznam literatury
This work deals with a multivariate random coefficient autoregressive model (RCA) of the first order. A class of modified least-squares estimators of the parameters of the model, originally proposed by Schick for univariate first-order RCA models, is studied under more general conditions. Asymptotic behavior of such estimators is explored, and a lower bound for the asymptotic variance matrix of the estimator of the mean of random coefficient is established. Finite sample properties are demonstrated in a small simulation study.
This paper provides a theoretical proof illustrating that for a certain class of functions having the property that the partial derivatives have the same equation with respect to all variables, the optimum value (minimum or maximum) takes place at a point where all the variables have the same value. This information will help the researchers working with high dimensional functions to minimize the computational burden due to the fact that the search has to be performed only with respect to one variable.
We investigate the existence of positive solutions for a nonlinear second-order differential system subject to some m-point boundary conditions. The nonexistence of positive solutions is also studied.
We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)<q(x)< \frac{N\cdot m(x)}{(N-m(x))}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda $ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}} _2$-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.
We say that a real normed lattice is quasi-Baire if the intersection of each sequence of monotonic open dense sets is dense. An example of a Baire-convex space, due to M. Valdivia, which is not quasi-Baire is given. We obtain that $E$ is a quasi-Baire space iff $(E, T({\mathcal U}),T({\mathcal U}^{-1}))$, is a pairwise Baire bitopological space, where $\mathcal U$, is a quasi-uniformity that determines, in $L$. Nachbin’s sense, the topological ordered space $E$.
The actual construction of the Szász-Mirakyan operators and its various modifications require estimations of infinite series which in a certain sense restrict their usefulness from the computational point of view. Thus the question arises whether the Szász-Mirakyan operators and their generalizations cannot be replaced by a finite sum. In connection with this question we propose a new family of linear positive operators.
We assign to each positive integer $n$ a digraph whose set of vertices is $H=\lbrace 0,1,\dots ,n-1\rbrace $ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\hspace{4.44443pt}(\@mod \; n)$. We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.
Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see \cite{Combs_Andrews_1998,Jayaram2008} and \cite{Baczynski_Jayaram_2008}). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see \cite{Baczynski_Jayaram_2009})
f(min(x+y,a))=min(f(x)+f(y),b),
where a,b>0 and f:[0,a]→[0,b] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation
f(m1(x+y))=m2(f(x)+f(y)),
where m1,m2 are functions defined on some intervals of \R satisfying additional assumptions. We analyze the cases when m2 is injective and when m2 is not injective.