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Free actions of semiprime rings with involution induced a derivation

Vukman J.

Demonstratio Mathematica

2005

Vol. 38, nr 4

811--817

Let R be an associative ring. An element a is an element of R is said to be dependent of a mapping F : R -> R in case F (x) a = ax holds for all x is an element of R. A mapping F : R -> R is called a free action in case zero is the only dependent element of F. In this paper free actions of semiprime *- rings induced by a derivation are considered. We prove, for example, that in case we have a derivation D : R -> R, where R is a semiprime *-ring, then the mapping F defined by F(x) = D(x*) + D(x)*,x is an element of R, is a free action. It is also proved that any Jordan *-derivation on a 2-torsion free semiprime *-ring is a free action.

Badanie niezawodności systemów o elementach z zależnymi czasami zdatności.

Załęska-Fornal A.

Zagadnienia Eksploatacji Maszyn

2004

Vol. 39, nr 1

37-50

W pracy wykonano symulacyjne badanie niezawodności systemów o zależnych wykładniczych czasach zdatności w oparciu o metodę Monte Carlo. Rozważano zależności między czasami zdatności elementów wynikające z istniejących w środowisku wspólnych źródeł uszkodzeń. Wykonano analizę niezawodnościową przykładowego systemu o sześcioelementowej strukturze niezawodnościowej.

Assumption concerning independence of system components is very strong and in real life it is not often satisfied. Computing of system reliability requires additional information about the object. In this paper there is considered dependence between components life times arising from common environmental sources of failures. By means of the Monte Carlo method the simulative investigation of systems reliability with dependent exponential life lengths is presented here. An algorithm of generating the multivariate random variables is given. Some cases of generating bivariate exponential distribution, trivariate and six-variate exponential distribution are presented. Those algorithms enable to analyse the reliability of the systems with particular structures. The knowledge about the system structure and the built generators of random variables made it possible to obtain the independent realizations of the system life length. The obtained sample allows us to construct the estimators of density functions of life length. Histograms was accepted as the estimators of density function and also some estimators from the well known classes of densities of probability distribution were taken under consideration (gamma distribution). On the basis on the computer simulations and on the obtained results it was found that the mean life lengths of systems with given structures, under assumption of independence and dependence of the random variables describing their components life lengths, are similar. However their variances differ a lot. In all investigated cases the variance of dependent random variables is much more greater than the variance of the independent ones. Also the minimum and maximum values of realisations of discussed random variables are different for the independent and dependent variables. The range of the longest and the shortest system life length is greater for the dependent random variables. For all generated realizations of random variables the empirical densities of probability distributions are fitted.