Basic probabilistic and statistical concepts for maintenance of parts and systems

Mathematical models are presented which are useful for determiningwhen replacement or maintenance is needed. In addition, techniquesfor assessing the efficacy of maintenance and/or overhaul arediscussed. Since the underlying concepts and techniques fornonrepairable items are relatively well known, attention isfocused on repairable items. Moreover, great emphasis is placedon the major differences between the concepts, probabilisticmodels, and statistical analysis techniques appropriate fornonrepairable and repairable items respectively. Such emphasisis still required because the superficial similarities betweennonrepairable and repairable items have contributed to the widespreaduse of poor terminology and notation which, in turn, make thesimilarities appear to be substantive, rather than just superficial.This vicious circle—-which is still evident in most currentreliability texts and standards—-must be broken, and thispaper is intended to contribute to this campaign. It is alsostressed that, even to the very limited extent that repairablesystems concepts and techniques are discussed in the literature,excessive emphasis is placed on reliability growth or improvement.This has resulted in even less understanding of basic notionsof repairable-systems deterioration, i.e. of basic conceptsassociated with systems maintenance. This paper focuses on conceptsconnected with systems maintenance to help rectify this imbalance.Nonetheless, it is also stressed that the same models (withdifferent parameters) can often be used for both situations.     

A Basic Analogue of Hermite's Equation

Using basic number and the analogues of differentiation andintegration, a q-analogue of Hermite's equation is introduced.Series solutions are given, and it is shown that polynomialforms of these solutions are orthogonal with respect to basicintegration. By reversing the series representation of thesesolutions, a basic analogue of the Hermite polynomial is obtainedfor which a generating function and a three-term recurrencerelation are deduced. Finally, an orthogonality relation isgiven.     

Kondo effect and mesoscopic fluctuations

Two important themes in nanoscale physics in the last two decades are correlations between electrons and mesoscopic fluctuations. Here we review our recent work on the intersection of these two themes. The setting is the Kondo effect, a paradigmatic example of correlated electron physics, in a nanoscale system with mesoscopic fluctuations; in particular, we consider a small quantum dot coupled to a finite reservoir (which itself may be a large quantum dot). We discuss three aspects of this problem. First, in the high-temperature regime, we argue that a Kondo temperature T _k which takes into account the mesoscopic fluctuations is a relevant concept: for instance, physical properties are universal functions of T/T _k. Secondly, when the temperature is much less than the mean level spacing due to confinement, we characterize a natural cross-over from weak to strong coupling. This strong coupling regime is itself characterized by well-defined single-particle levels, as one can see from a Nozières Fermi-liquid theory argument. Finally, using a mean-field technique, we connect the mesoscopic fluctuations of the quasiparticles in the weak coupling regime to those at strong coupling.     

The Laplace Transform of the Macdonald Function of Argument x3/2 and the Airy Function Ai (x)

The Laplace transform of the Airy function of the first kindAi(x) is obtained conveniently as a special case of the Laplacetransform of the Macdonald function of argument 2x^3/2.