Narrow Results

In this paper we consider coherent systems (E,V) on an elliptic curve which are α-stable with respect to some value of a parameter α. We show that the corresponding moduli spaces, if non-empty, are smooth and irreducible of the expected dimension. Moreover we give precise conditions for non-emptiness of the moduli spaces. Finally we study the variation of the moduli spaces with α.

In this paper, we continue the investigation of coherent systems of type (n,d,k) on the projective line which are stable with respect to some value of a parameter α. We work mainly with k < n and obtain existence results for arbitrary k in certain cases, together with complete results for k = 3. Our methods involve the use of the "flips" which occur at critical values of the parameter.

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

In this paper, we continue the investigation of coherent systems of type (n, d, k) on the projective line which are stable with respect to some value of a parameter α. We consider the case k = 1 and study the variation of the moduli spaces with α. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank.

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

Let $(E,V)$ be a general generated coherent system of type $(n,d,n+m)$ on a general non-singular irreducible complex projective curve. A conjecture of D. C. Butler relates the semistability of $E$ to the semistability of the kernel of the evaluation map $V\otimes {{𝒪}}_X\to E$. The aim of this paper is to obtain results on the existence of generated coherent systems and use them to prove Butler’s Conjecture in some cases. The strongest results are obtained for type $(2,d,4)$, which is the first previously unknown case.

Nondeterministic variables of certain distributions are employed to represent uncertain lifetimes and repair times in repairable systems, which are usually treated as stochastic variables and the probability distributions of the stochastic variables have crisp parameters. However, in practice, the parameters may not be precisely represented due to a lack of sufficient data. The main purpose of this paper is to apply the random fuzzy theory to the establishment of repairable coherent system models. Then the expressions of limiting availability and steady state failure frequency of the random fuzzy coherent systems are proposed, respectively. Finally, some numerical examples are given for illustration.

This paper is concerned with the problem of stochastic comparisons between the lifetimes of two coherent systems with active redundancy. For this purpose, we consider both the active redundancy at the system level and the redundancy at the component level. We assume that the original components are identically distributed and possibly dependent. It is also assumed that for each component, there are $m$ redundant components with possibly different lifetime distributions which follow the proportional hazards (reversed hazards) model. Under some conditions on the domination function of the system, we compare the lifetimes of the systems based on majorization orders between the parameter vectors of the proportionality of the component lifetimes. We also give sufficient conditions under which adding more redundant components imply the system improvement.

The performance (lifetime, failure rate, etc.) of a coherent system in iid components is completely determined by its “signature” and the common distribution of its components. A system's signature, defined as a vector whose i^th element is the probability that the system fails upon the i^th component failure, was introduced by Samaniego (1985) as a tool for indexing systems in iid components and studying properties of their lifetimes. In this paper, several new applications of the signature concept are developed for the broad class of mixed systems, that is, for stochastic mixtures of coherent systems in iid components. Kochar, Mukerjee and Samaniego (1999) established sufficient conditions on the signatures of two competing systems for the corresponding system lifetimes to be stochastically ordered, hazard-rate ordered or likelihood-ratio ordered, respectively. Partial results are obtained on the necessity of these conditions, but all are shown not to be necessary in general. Necessary and sufficient conditions (NASCs) on signature vectors for each of the three order relations above to hold are then discussed. Examples are given showing that the NASCs can also lead to information about the precise number and locations of crossings of the systems' survival functions or failure rates in (0, ∞) and about intervals over which the likelihood ratio is monotone. New results are established relating the asymptotic behavior of a system's failure rate, and the rate of convergence to zero of a system's survival function, to the signature of the system.

In this article we review recent work on generalizations of the total time on test transform, and on stochastic orders that are based on these generalizations. Applications in economics, statistics, and reliability theory, are described as well.