On allocation of redundant components for systems with dependent components

In this paper we consider the problem of optimal allocation of a redundant component for series, parallel and k-out-of-n systems of more than two components, when all the components are dependent. We show that for this problem is naturally to consider multivariate extensions of the joint bivariates stochastic orders. However, these extensions have not been defined or explicitly studied in the literature, except the joint likelihood ratio order, which was introduced by Shanthikumar and Yao (1991). Therefore we provide first multivariate extensions of the joint stochastic, hazard rate, reversed hazard rate order and next we provide sufficient conditions based on these multivariate extensions to select which component performs the redundancy.     

An Invariance in the Priority Queue with Generalized Server Vacations and Structured Batch Arrivals

Abstract

Shanthikumar (Shanthikumar, J.G. Level crossing analysis of priority queues and a conservation identity for vacation models. Nav. Res. Log. 1989, 36, 797–806) studied the priority M/G/1 queue with server vacations and found that the difference between the waiting time distribution under the non‐preemptive priority (NPP) and that under the preemptive‐resume priority (PRP) is independent of the vacation policy. We extend this interesting property: (i) to the generalized vacations which includes the two vacation policies considered by Shanthikumar; (ii) to the structured batch Poisson arrival process; and (iii) to the discrete‐time queues.     

Stochastic orders of the Marshall-Olkin extended distribution

A general method of introducing a parameter, called tilt parameter, has been discussed by Marshall and Olkin (1997) to give more flexibility in modelling. In this paper, we take the tilt parameter of the Marshall-Olkin extended family as a random variable. The closure of this model under different stochastic orders viz. ageing intensity order, likelihood ratio order, shifted likelihood ratio orders and shifted hazard rate orders is discussed.     

On the DFR property of the compound geometric distribution with applications in risk theory

In 1988, Shanthikumar proved that the sum of a geometrically distributed number of i.i.d. DFR random variables is also DFR. In this paper, motivated by the inverse problem, we study monotonicity properties related to defective renewal equations, and obtain that if a compound geometric distribution is DFR, then the random variables of the sums are NWU (a class that contains DFR). Furthermore, we investigate some applications of risk theory and give a characterization of the exponential distribution.     

On the DFR property of the compound geometric distribution with applications in risk theory

In 1988, Shanthikumar proved that the sum of a geometrically distributed number of i.i.d. DFR random variables is also DFR. In this paper, motivated by the inverse problem, we study monotonicity properties related to defective renewal equations, and obtain that if a compound geometric distribution is DFR, then the random variables of the sums are NWU (a class that contains DFR). Furthermore, we investigate some applications of risk theory and give a characterization of the exponential distribution.     

Multivariate dependence of spacings of generalized order statistics

Multivariate dependence of spacings of generalized order statistics is studied. It is shown that spacings of generalized order statistics from DFR (IFR) distributions have the CIS (CDS) property. By restricting the choice of the model parameters and strengthening the assumptions on the underlying distribution, stronger dependence relations are established. For instance, if the model parameters are decreasingly ordered and the underlying distribution has a log-convex decreasing (log-concave) hazard rate, then the spacings satisfy the MTP₂ (S- MRR₂) property. Some consequences of the results are given. In particular, conditions for non-negativity of the best linear unbiased estimator of the scale parameter in a location-scale family are obtained. By applying a result for dual generalized order statistics, we show that in the particular situation of usual order statistics the assumptions can be weakened.     

A New Characterization on g-frames in Hilbert C·-Modules

In this note, we establish a new characterization on g-frames in Hilbert C~*-modules from the operator-theoretic point of view, with which we provide a correction to one result recently obtained by Yao(Yao X Y. Some properties of g-frames in Hilbert C~*-modules(in Chinese). Acta Math. Sinica, 2011, 54(1): 1–8.).     

关于似然比序随机比较的一些结果

研究了两组随机变量列之间似然比序的随机比较，得到的结果推广了Shaked 和 Shanthikumar (1994) 中的 Theorem 1.C.19 和Theorem 1.C.20.      

Orientor field equations in Banach spaces

This paper focuses on the interplay between seminormality requirements and convergence hypotheses on trajectories in the lower closure theorems for orientor field equations. With the use of a weak form of seminormality, called the intermediate property (Q*), we obtain lower closure theorems (and thereby closure theorems) for orientor field equations in Banach spaces, under the assumption of strong convergence of some coordinates of the trajectories, while only weak convergence is assumed in the others. In the Euclidean case, this requirement of property (Q*) is further reduced to mere Kuratowski property (K), under the usual growth-type conditions. Finally, in the appendix, property (Q*) is investigated in detail.This research has been partially supported by Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan.     

A characterization of the multivariate excess wealth ordering

In this paper, some new properties of the upper-corrected orthant of a random vector are proved. The univariate right-spread or excess wealth function, introduced by Fernández-Ponce et al. (1996), is extended to multivariate random vectors, and some properties of this multivariate function are studied. Later, this function was used to define the excess wealth ordering by Shaked and Shanthikumar (1998) and Fernández-Ponce et al. (1998). The multivariate excess wealth function enable us to define a new stochastic comparison which is weaker than the multivariate dispersion orderings. Also, some properties relating the multivariate excess wealth order with stochastic dependence are described.     

Condition for inducing topology in a vector lattice by the order completion

Properties of an order topology in vector lattices and Boolean algebras are studied. The main result is the following: in a vector lattice or a Boolean algebra with the condition of “closure by one step” (a generalization of the well-known “regularity” property of Boolean algebras and K-spaces) the order topology is induced by the topology of its Dedekind completion. Bibliography: 4 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16, 1997, pp. 204–207.     

On the solution existence of implicit quasivariational inequalities with discontinuous multifunctions

In this article, we established some solution existence theorems for implicit quasivariational inequalities. We first established some results in finite-dimensional spaces and then a solution existence result in infinite-dimensional spaces was derived. Our theorems are proved for discontinuous mappings and sets which may be unbounded. The results presented in this article are improvements of results in Cubiotti, and Yao (Cubiotti, P. and Yao, J.C., 1997, Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings. Mathematical Methods of Operations Research, 46, 213–328; Cubiotti, P. and Yao, J.C., 2007, Discontinuous implicit generalized quasi-variational inequalities in Banach spaces. Journal of Global Optimization (To appear))     

Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii). The second author's work was partially supported by the fund for the promotion of research at the Technion-Israel Institute of Technology. The third author's work was partially supported by the Swedish Natural Science Research Council, and by the 1992 Wallenberg Prize from the Swedish Mathematical Society.     

On the recognition complexity of some graph properties

By applying a topological approach due to Kahn, Saks and Sturtevant, we prove that all decreasing graph properties consisting of bipartite graphs only are elusive. This is an analogue to a well-known result of Yao.     

Prüfer's ascent result via Inc

Let R be an integral domain whose integral closure is a Pr¨fer domain. It is proved that R ? T has the incomparability property for each integral domain T which contains R and is algebraic over R. As a corollary, one has a new proof of Pr¨fer's ascent result, which states that if R is as above and T is the integral closure of R in some field containing R, then T is a Pr¨fer domain.     

Equivariant closure operators and trisp closure maps

A trisp closure map ? is a special map on the vertices of a trisp (triangulated space) T with the property that T collapses onto the subtrisp induced by the image of ?. We study the interaction between trisp closure maps and group operations on the trisp, and give conditions such that the quotient map is again a trisp closure map. Special attention is on the case that the trisp is the nerve of an acyclic category, and the relationship between trisp closure maps and closure operators on posets is studied.     

Existentially Incomplete Tame Models and a Conjecture of Ellentuck

We construct a recursive ultrapower F/U such that F/U is a tame 1-model in the sense of [6, §3] and FU is existentially incomplete in the models of II₂ arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some do not.     

Multivariate dispersive ordering of spacings of generalized order statistics

The purpose of this paper is to investigate conditions on the underlying distributions and the parameters, on which generalized order statistics are based, to establish the Shaked–Shanthikumar multivariate dispersive ordering of normalized and unnormalized spacings of generalized order statistics from one sample and two samples. Several probability inequalities concerning spacings of generalized order statistics are also given.     

A Characterization Theorem for the Canonical Basis of a Closure Operator

The purpose of this paper is to provide a characterization result for the canonical basis of an arbitrary closure operator. This theorem strengthens the result of Burosch, Demetrovics and Katona, who propose a characterization of the generating system of a closure operator defined by the quasi-closed sets of the closure operator.     

Representation of convex geometries by circles on the plane

Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an $n$-dimensional space equipped with a convex hull operator, by the result of Kashiwabara et al. (2005). Allowing circles rather than points, as was suggested by Czédli (2014), may presumably reduce the dimension for representation. This paper introduces a property, the Weak 2 × 3-Carousel rule, which is satisfied by all convex geometries of circles on the plane, and we show that it does not hold in all finite convex geometries. This raises a number of representation problems for convex geometries, which may allow us to better understand the properties of Euclidean space related to its dimension.