Lifetime distribution and estimation problems of consecutive-k-out-of-n:F systems

Explicit formula is given for the lifetime distribution of a consecutive-k-out-of-n:F system. It is given as a linear combination of distributions of order statistics of the lifetimes of n components. We assume that the lifetimes are independent and identically distributed. The results should make it possible to treat the parametric estimation problems based on the observations of the lifetimes of the system. In fact, we take up, as some examples, the cases where the lifetimes of the components follow the exponential, the Weibull, and the Pareto distributions, and obtain feasible estimators by moment method. In particular, it is shown that the moment estimator is quite good for the exponential case in the sense that the asymptotic efficiency is close to one.This research was partially supported by the ISM Cooperative Research Program (94-ISM-CRP-5).     

On allocating redundancies to k‐out‐of‐ n reliability systems

For two components in series and one redundancy with their lifetimes following the proportional hazard models, we build the likelihood ratio order and the hazard rate order for lifetimes of the redundant systems. Also, for k ‐out‐of‐ n system with components’ lifetimes having the arrangement increasing joint density and the redundancies having identically distributed lifetimes, allocating more redundancies to weaker components is shown to help improve the system's reliability. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal redundancy allocation problems in engineering systems with dependent component lifetimes

This paper deals with the problem of allocating spare components for improving the system reliability when lifetimes of components are dependent. Two policies, called active and standby redundancies, are investigated in details. The optimal allocations are given without specific assumptions on the dependency structure among the component lifetimes. When there exist the (positive/negative) quadratic dependence and (upper/lower) orthant order among the component lifetimes, the derived results are simplified. Various illustrative examples are also given. Copyright © 2015 John Wiley & Sons, Ltd.     

Some results for repairable systems with minimal repairs

In this paper, we consider a repairable system with minimal repairs whose number of repairs is a positive random variable with a given probability vector. Some preservation theorems and aging properties of repairable systems are established. Under the condition that at time t the system is working, a new random variable for the residual lifetime of the system is proposed. Some stochastic ordering results among the lifetimes and residual lifetimes of two systems are obtained. Similar results for coherent systems with independent components and exchangeable components were obtained in the previous literature. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimality of singular vector perturbation under maximum norm

Fan, Wang, and Zhong estimate the difference between the singular vectors of a matrix and those of a perturbed matrix in terms of the maximum norm. Their estimations are used effectively to establish the asymptotic properties of robust covariance estimators (see Journal of Machine Learning Research, 2018;18:1-42). In this paper, we give the corresponding lower bound estimates, which show Fan-Wang-Zhong's estimations optimal.     

Approximate computation of the permeability tensor of a periodic porous medium

The permeability tensor K of an infinite periodic porous medium, obtained using the homogenization theory, is considered. The solutions of an optimal control problem for the Dirichlet or Neumann equation are used to obtain optimal upper bounds for K. The test functions used for the estimations are simpler than those obtained by other authors. Some possibilities are given to obtain also lower bounds.     

The distribution of the zeros of L-functions and the least prime in some arithmetic progression

This paper gives some new estimations to the distribution of the zeros of L-functions and proves that the least prime in an arithmetic progression with a prime differenceq is ≪q ^4.5.     

Estimations inférieures du noyau de la chaleur sur les variétés coniques et transformée de Riesz

Résumé Nous complétons les estimations supérieures du noyau de la chaleur sur les variétés coniques obtenues dans [13] par des estimations inférieures, et nous en déduisons quune variété à croissance polynomiale du volume peut vérifier des estimations supérieures et inférieures gaussiennes du noyau de la chaleur sans que la transformée de Riesz y soit bornée sur L^p pour tout p > 2.     

Conformal capacity and quasiconformal mappings in\bar R^n

Conformal capacity of some ring domains inn-spaces is computed and applied in estimations of the modulus of Groetzsch and Teichmuller’s domains. Upper bounds for modR are given here by means of the specific diameters of the ring domainR. Convergence theorems of Lehto and Virtanen are generalized here ton-spacesn>2; and are applied to discrete groups ofn-Moebius transformations. This paper is mainly based on the author’s Ph.D. Thesis done at the Technion, Haifa, under the guidance of Professor E. Netanyahu. The author wishes to thank Professor E. Netanyahu for his help in preparing this paper.     

On exhibiting inventory systems with Erlangian lifetimes under renewal demands

A continuous revies (s, S) inventory system with renewal demand in which one item is put into operation as an exhibiting piece is analyzed. The lifetime of any operating unit has Erlangian distribution, and on failure is replaced by another one from the stock and the failed item is disposed of. Replenishment of stock is instantaneous. The transient and stationary values of inventory level distribution and the mean reorder rate are obtained using the techniques of semi-regenerative processes. Decision rules for optimums andS that minimize the long-run expected cost rate are derived. The solution for a dual model with the distribution of lifetimes and inter-demand times interchanged is also given.     

On roots of exponential terms

In the present paper some tools are given to state the exact number of roots for some simple classes of exponential terms (with one variable). The result were obtained by generalizing Sturm's technique for real closed fields. Moreover for arbitrary non-zero terms t(x) certain estimations concerning the location of roots of t(x) are given. MSC: 03C65, 03C60, 12L12.     

Estimations optimales du noyau de la chaleur sur les variétés cuspidales

Résumé Dans cet article, on se propose d’étudier les estimations optimales du noyau de la chaleur, p _t (t > 0), sur les variétés cuspidales à base compacte. Une conséquence immédiate de notre estimations optimales de p _t est que la fonction maximale associée au noyau de la chaleur est de type faible (1, 1).      

Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains

Summary In this paper, methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied. In order to verify solutions using computer, it is necessary to determine some constants which appear in a priori error estimations. We propose some methods for determination of these constants. In numerical examples, calculating these constants for anL-shaped domain, we verify the solution of a nonlinear elliptic equation.     

On the value distribution of a certain type of differential polynomials

Some quantitative estimations on the value distribution of the functionafg ⁿ(n2) are obtained, whereg is a linear differential polynomial inf anda is a small function off.The research was partially supported by a U.G.C grant of Hong Kong     

On a Finite Group Having a Normal Series Whose Factors Have Bicyclic Sylow Subgroups

We consider the structure of a finite group having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigate groups of odd order and A ₄-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.     

Stokes型积分-微分方程Q_2-P_1元的超收敛分析

本文研究Q2-P1混合元对Stokes型积分-微分方程的有限元方法.利用积分恒等式技巧给出了关于流体速度u和压力p的误差估计,特别是在压力p的误差中去掉了影响解的稳定性的1因子t-2,改善了以往文献的结果.同时,通过构造适当的插值后处理算子得到了整体超收敛结果.     

Saturated graphs with minimal number of edges

Let F = {F₁,…} be a given class of forbidden graphs. A graph G is called F-saturated if no F_i ∈ F is a subgraph of G but the addition of an arbitrary new edge gives a forbidden subgraph. In this paper the minimal number of edges in F-saturated graphs is examined. General estimations are given and the structure of minimal graphs is described for some special forbidden graphs (stars, paths, m pairwise disjoint edges).     

Estimation of the acceleration coefficient of sequential system tests

This paper deals with some methods of constructing the relationship between lifetimes in normal and accelerated tests. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez pp. 120–131, Perm, 1991.     

Functionally perturbed stochastic differential equations

This paper is devoted to the large class of stochastic differential equations of the Ito type whose coefficients are functionally perturbed and depend on a small parameter. The solution of a such equation is compared with the solution of the corresponding unperturbed equation, in the (2m)‐th moment sense, on finite intervals or on intervals whose length tends to infinity as small parameter tends to zero. Some numerical estimations of these intervals with respect to the parameter are also given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and analysis of system reliability using phase‐type distribution closure properties

Phase‐type distribution closure properties are utilized to devise algorithms for generating reliability functions of systems with basic structures. These structures include series, parallel, K‐out‐of‐N, and standby structures with perfect/imperfect switch. The algorithms form a method for system reliability modeling and analysis based on the relationship between the system lifetime and component lifetimes for general structures. The proposed method is suitable for functional system reliability analysis, which can produce reliability functions of systems with independent components instead of only system reliability values. Once the system reliability function is obtained, other reliability measures such as the system's hazard function and mean time to failure can be obtained efficiently using only matrix algebra. Dimensional and numerical comparisons with computerized symbolic processing are also presented to show the superiority of the proposed method.