Let α be an infinite cardinal. Let Tα be the class of all lattices which are conditionally α-complete and infinitely distributive. We denote by T'α the class of all lattices X such that X is infinitely distributive, α-complete and has the least element. In this paper we deal with direct factors of lattices belonging to T α - As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class T'α.
In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon B-spline functions. The properties of B-spline functions are presented. The operational matrix of derivative (Dϕ) and integration matrix (P) are introduced. These matrices are utilized to reduce the solution of nonlinear constrained quadratic optimal control to the solution of nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.
In the present paper we deal with generalized $MV$-algebras ($GMV$-algebras, in short) in the sense of Galatos and Tsinakis. According to a result of the mentioned authors, $GMV$-algebras can be obtained by a truncation construction from lattice ordered groups. We investigate direct summands and retract mappings of $GMV$-algebras. The relations between $GMV$-algebras and lattice ordered groups are essential for this investigation.
The main goal of this paper is to show an application of Graph Theory to classifying Lie algebras over finite fields. It is rooted in the representation of each Lie algebra by a certain pseudo-graph. As partial results, it is deduced that there exist, up to isomorphism, four, six, fourteen and thirty-four $2$-, $3$-, $4$-, and $5$-dimensional algebras of the studied family, respectively, over the field $\mathbb {Z}/2\mathbb {Z}$. Over $\mathbb {Z}/3\mathbb {Z}$, eight and twenty-two $2$- and $3$-dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented.
Recently, the eminently popular standard quantile regression has been generalized to the multiple-output regression setup by means of directional regression quantiles in two rather interrelated ways. Unfortunately, they lead to complicated optimization problems involving parametric programming, and this may be the main obstacle standing in the way of their wide dissemination. The presented R package modQR is intended to address this issue. It originates as a quite faithful translation of the authors' moQuantile toolbox for Octave and MATLAB, and provides all the necessary computational support for both the directional multiple-output quantile regression methods to the wide statistical public. The article offers a concise summary of the statistical theory behind modQR, overviews the package in brief, points out its departures from moQuantile, comments on its use and performance, and demonstrates its application.
We continue the study of directoid groups, directed abelian groups equipped with an extra binary operation which assigns an upper bound to each ordered pair subject to some natural restrictions. The class of all such structures can to some extent be viewed as an equationally defined substitute for the class of (2-torsion-free) directed abelian groups. We explore the relationship between the two associated categories, and some aspects of ideals of directoid groups.
It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.