On reliability analysis of consecutive k-out-of-n systems with arbitrarily dependent components

In this paper, we consider the linear and circular consecutive k-out-of-n systems consisting of arbitrarily dependent components. Under the condition that at least n?r+1 components (r ≤ n) of the system are working at time t, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k-out-of-n systems. In the following, we investigate the inactivity time of the component with lifetime T_r:n at the system level for the consecutive k-out-of-n systems under the condition that the system is not working at time t > 0, and obtain some stochastic properties of this conditional random variable.     

Numerical range of operators acting on Banach spaces

The aim of the paper is to propose a definition of numerical range of an operator on reflexive Banach spaces. Under this definition the numerical range will possess the basic properties of a canonical numerical range. We will determine necessary and sufficient conditions under which the numerical range of a composition operator on a weighted Hardy space is closed. We will also give some necessary conditions to show that when the closure of the numerical range of a composition operator on a small weighted Hardy space has zero.     

On the Schröder–Bernstein property for modules

The well known Schröder–Bernstein Theorem states that any two sets with one to one maps into each other are isomorphic. The question of whether any two (subisomorphic or) direct summand subisomorphic algebraic structures are isomorphic, has long been of interest. Kaplansky asked whether direct summands subisomorphic abelian groups are always isomorphic? The question generated a great deal of interest. The study of this question for the general class of modules has been somewhat limited. We extend the study of this question for modules in this paper. We say that a module Msatisfies the Schröder–Bernstein property (S-B property) if any two direct summands of M which are subisomorphic to direct summands of each other, are isomorphic. We show that a large number of classes of modules satisfy the S-B property. These include the classes of quasi-continuous, directly finite, quasi-discrete and modules with ACC on direct summands. It is also shown that over a Noetherian ring R, every extending module satisfies the S-B property. Among applications, it is proved that the class of rings R for which every R-module satisfies the S-B property is precisely that of pure-semisimple rings. We show that over a commutative domain R, any two quasi-continuous subisomorphic R-modules are isomorphic if and only if R is a PID. We study other conditions related to the S-B property and obtain characterizations of certain classes of rings via those conditions. Examples which delimit and illustrate our results are provided.     

Mixed mode axisymmetric annular cracks in a finite layer: Off angle crack initiation

The solutions of axisymmetric Volterra type climb and glide edge dislocations are obtained in a layer by means of the Hankel transforms. Utilizing the same procedure, Green’s function solution is obtained for a layer under self-equilibration normal ring traction. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks where the layer is under axisymmetric normal loads. These equations are solved numerically to obtain dislocation density on the cracks surfaces. The results are employed to determine stress intensity factors for annular and penny-shaped cracks and the interaction between two co-axial penny-shaped cracks is studied. Moreover, the stress intensity factors of the interacting cracks are determined such that they can be further used in conjunction with strain energy density (SED) failure criterion to obtain the possible direction of crack initiation that may not be apparent under mixed mode conditions.     

Practical reduction of imines by NaBH4/alumina under solvent-free conditions： An efficient route to secondary amine

Sodium borohydride supported on alumina reduces imines to the corresponding secondary amines in high to excellent isolated yields under solvent-free conditions. Noteworthy is that highly chemoselective reactions were achieved in the presence of other functional groups such as halogen, nitro, and cyano groups.