Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application

Stability properties of the solution set of generalized inequality systems with locally Lipschitz functions are obtained under a regularity condition on the generalized Jacobian and the Clarke tangent cone. From these results, we derive sufficient conditions for the optimal value function in a nonsmooth optimization problem to be continuous or locally Lipschitz at a given parameter.     

下半连续函数的Proximal—次微分与广义中值定理

该文对定义在Hilbert空间E上的一般下半连续函数证明了如［9］中形式的逼近中值定理在Proximal-次微分意义下也成立．若E＝［a，b］R，则得到了不等式形式的中值定理．作为应用给出了函数凸性、Lipschitz性质及常数性质的Proximal-次微分刻划．     

定义在区间值上的Fuzzy关系方程

Ｆｕｚｚｙ关系方程Ａ。Ｘ＝Ｂ中的矩阵Ａ、Ｂ的元素均为区间值，称为定义在区间值上的Ｆｕｚｚｙ关系方程。它是目前关于Ｆｕｚｚｙ关系方程理论研究中的主要问题之一￣［１］。本文讨论了区间值上Ｆｕｚｚｙ关系方程解的定义，并用符号定值法求出其全体解集。     

Some properties of bivariate Gumbel Type A distributions with proportional hazard rates

We call a set of univariate distributions with the same mathematical form but different parameter values a family $\text{J}$. Consider a bivariate Gumbel Type A survival distribution, S₁₂(x₁, x₂), defined in (2.1), for which both marginal distributions, S₁(x₁), S₂(x₂), belong to the same family, $\text{J}$ of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h₁(t) and h₂(t), are proportional if and only if the marginal hazard rates, λ₁(t) and λ₂(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X₁, X₂), and of the identified minimum, W_i = X_i, for X_i < X_j, j ≠ i, belong to the same family $\text{J}$ as do S₁(x₁), S₂(x₂) (Corollary 1). Counter-examples of distributions other than Gumbel Type A, for which these properties do not hold, are given. Some applications to the analysis of competing risks, using a family of Gompertz distributions, are discussed.     

Periodic precipitation of cobalt(II) oxinate in agar gel: factors influencing the flocculation

The influence of concentration, ageing and pH of the gel medium on the periodic precipitation of cobalt oxinate in agar gel is reported. The results are explained on the basis of Shinohara’s revised flocculation theory. The flocculation value (F) increases with increase in the gel concentration whereas it decreases with increase in ageing and pH of the gel. The raise in temperature of the gel increases the solubility of the sparingly soluble substance and hence theF value. The effect of additives on the periodic precipitation of cobalt (II) oxinate is reported.     

Boundary value problems in queueing theory

Recently complex function techniques have been developed for the analysis of queueing systems which need for their modelling a two dimensional state space. A variety of computer- and communication networks gives rise to such two-dimensional queueing systems and their analysis is needed for the performance evaluation of these aggregates. The present study reviews these developments     

When is the pseudo-best estimator blue?

Summary Linear unbiased estimation of the mean of a random variable in Hilbert space is treated in the typical case where the mean belongs to a known subspace. The best linear estimate depends on the underlying covariance operatorB ₀ of the random variable. This operatorB ₀, however, is rarely completely known, so that an auxiliary operatorB is used to compute a “pseudo-best” estimate. It is shown that the best and the pseudo-best estimates coincide, if and only ifB ₀ B ⁻¹ leavesM invariant. Applications to linear regression are to be found in the references.     

Some comments on fuzzy variables

Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X₁, X₂,…,X_n are mutually unrelated fuzzy variables with common membership function μ and α₁, α₂,…,α_n are real numbers satisfying α_i ? o for every i and Σ_i=1ⁿα_i=1, when does does Z = Σ_{i = 1}ⁿα_iX_i have the same membership function μ?