The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet–Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord R...

We present a new algorithm for finding maximum a-posteriori (MAP) assignments of values to belief networks. The belief network is compiled into a network consisting only of nodes with boolean (i.e. only 0 or 1) conditional probabilities. The MAP assignment is then found using a best-first search on the resulting network. We argue that, as one would anticipate, the algorithm is exponential for t...

This paper analyses the reliability of a cold standby system consisting of two repairable units, a switch and a repairman. At any time, one of the two units is operating while the other is on cold standby. The repairman may not always at the job site, or take vacation. We assume that shocks can attack the operating unit. The arrival times of the shocks follow a homogeneous Poisson process and t...

be the first-passage time of B over g , and let F denote the distribution function of . The first-passage problem seeks to determine F when g is given. The inverse first-passage problem seeks to determine g when F is given. Both the process B and the boundary g in these formulations may be more general, and our choice of Brownian motion is primarily motivated by the tractability of the expositi...

Two cumulative damage models are considered, the inverse gamma process and a composed gamma process. They can be seen as ‘continuous’ analogues of Poisson and compound Poisson processes, respectively. For these models the first passage time distribution functions are derived. Inhomogeneous versions of these processes lead to models closely related to the Weibull failure model. All models show i...

A classical assumption in the theory of degradation models is that the quality of a device is described by a non-negative diffusion process {X(t), t ≥ 0} starting at X(0) = x0 > 0. Assuming that large values of X(t) correspond to a better quality, the level X(t) = 0 stands for zero quality, i.e. for the lack of functionality of the device. Consequently, the first-passage time T of {X(t)} throug...

Abstract. We present exact results for the spectrum of the fractional Laplacian in a bounded domain and apply them to First Passage Time (FPT) Statistics of Lévy flights. We specifically show that the average is insufficient to describe the distribution of FPT, although it is the only quantity available in the existing literature. In particular, we show that the FPT distribution is not peaked a...

We describe a method for inverting Gentzen’s cut-elimination in classical first-order logic. Our algorithm is based on first computing a compressed representation of the terms present in the cut-free proof and then cut-formulas that realize such a compression. Finally, a proof using these cut-formulas is constructed. This method allows an exponential compression of proof length. It can be appli...

We show that the vertex visited last by a LexBFS has eccentricity at least diam(G) − 2 for house-hole-free graphs, at least diam(G) − 1 for house-hole-domino-free graphs, and equal to diam(G) for house-hole-domino-free and AT-free graphs. To prove these results we use special metric properties of house-hole-free graphs with respect to LexBFS. ? 1999 Elsevier Science B.V. All rights reserved.