In this paper we use cohomology of Lie algebras to study the variety of laws associated with filiform Lie algebras of a given dimension. As the main result, we describe a constructive way to find a small set of polynomials which define this variety. It allows to improve previous results related with the cardinal of this set. We have also computed explicitly these polynomials in the case of dimensions 11 and 12.
In this paper, we introduce and study a new class of completely generalized nonlinear variational inclusions for fuzzy mappings and construct some new iterative algorithms. We prove the existence of solutions for this kind of completely generalized nonlinear variational inclusions and the convergence of iterative sequences generated by the algorithms.
In this paper we first study what changes occur in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then we study the set of all convex geometries which have the same poset of join-irreducible elements. We show that this set---ordered by set inclusion---is a ranked join-semilattice and we characterize its cover relation. We prove that the lattice of all ideals of a given poset $P$ is the only convex geometry having a poset of join-irreducible elements isomorphic to $P$ if and only if the width of $P$ is less than 3. Finally, we give an algorithm for computing all convex geometries having the same poset of join-irreducible elements.
In the Shapley-Scarf economy each agent is endowed with one unit of an indivisible good (house) and wants to exchange it for another, possibly the most preferred one among the houses in the market. In this economy, core is always nonempty and a core allocation can be found by the famous Top Trading Cycles algorithm. Recently, a modification of this economy, containing Q >= 2 types of goods (say, houses and cars for Q=2) has been introduced. We show that if the number of agents is 2, a complete description of the core can be found efficiently. However, when the number of agents is not restricted, the problem to decide the nonemptyness of the core becomes NP-hard already in the case of two types of goods. We also show that even the problem to decide whether an allocation exists in which each agent strictly improves compared to his endowment, is NP-complete.
In the minimization of the number of subtours made by the insertion head of an SMD placement machine a variant of the network flow problem arose. In a network with <span class="tex">n</span> vertices and <span class="tex">m</span> arcs a set <span class="tex">F</span> of arcs (parametrized arcs) is given. The task is to find a flow of a given size such that the maximum of flow values along the arcs from <span class="tex">F</span> is minimized. This problem can be solved by a sequence of maximum flow computations in modified networks where the capacities of the parametrized arcs are successively set to an increasing sequence of target parameter values. We show that it suffices to consider at most <span class="tex">O(|F|)</span> different target values and so this approach leads to a strongly polynomial algorithm consisting of at most <span class="tex">O(|F|)</span> maximum flow computations.
The surface topography persists for a long time in the forefront interest of many research and development organisations due to the perpetual quest for more precise and perfect measurement methods. So far used contact methods cannot be applied in many cases where their application can damage the measured surface and therefore waste the measured element, for instance lens surface or silicon plates etc. The great effort is therefore devoted to the development of contactless methods that employ various physical principles. In this article the brief description of light interference is presented including some algorithms for evaluation of the interference field and potentials of actual measuring instruments manufactured by prestigious producers and examples of model which are available at the today market. and Problematika topografie ploch je již dlouhou dobu v popředí zájmu řady výzkumných a vývojových pracovišť, neboť oblast vědy a techniky vyžaduje stale přesnější a dokonalejší metody měření. Dosud využívané kontaktní metody jsou v řadě případů zcela nepoužitelné vzhledem k tomu, že při jejich aplikaci dochází k poškození měřeného povrchu a tím ke znehodnocení měřeného prvku (např. plochy čočky, křemíkových desek apod.). Velké úsilí je proto věnováno právě vývoji bezkontaktních metod, pracujících na různých fyzikálních principech. V článku je stručně popsána interference světla a některé algoritmy pro vyhodnocování interferenčního pole se současným uvedením aktuálních možností měřicích přístrojů renomovaných světových výrobců a příklady modelů, které jsou v dnešní době na trhu k dispozici.