Volume 23 - Issue 1 - June 1998

Single species dopant diffusion in silicon

W. Merz, K. Pulverer and E. Wilczok


In this article we consider a model for dopant diffusion in silicon. First, we describe the diffusion mechanisms and formulate a low concentration reaction-diffusion model which involves five equations, resulting from the interactions between the dopant and the different defect species always present in a silicon crystal. Assuming equilibrium conditions, the complete model may be reduced to a single linear diffusion equation, which is easy to handle.

Under extrinsic, i.e. high concentration doping conditions the situation is more complicated, since the concentration of charged defects is then influenced by the concentration of dopants and the effect of an internal electric field has to be taken into consideration as well. Assuming equilibrium, a nonlinear diffusion equation results for which we formulate an existence result in Sobolev spaces and perform numerical computations in two- and three-dimensional spaces.

Eight valuation methods in financial mathematics: the Black–Scholes formula as an example

Jesper Andreasen, Bjarke Jensen and Rolf Poulsen


This paper describes a large number of valuation techniques used in modern financial mathematics. Although the approaches differ in generality and rigour, they are consistent in a very noteworthy sense. Each model has the celebrated Black–Scholes formula for the price of a call-option as a special case.

Quit when you are ahead

David K. Neal


We consider a random walk boundary problem applied to gambling. Suppose one starts with $c, wins bets with probability p, receives $e for winning, and pays $d for losing. A maximum of k bets are to be made. We give numerical solutions for finding the probability of going ahead within these k bets, and for finding the average fortune when quitting upon going ahead or upon making k bets.

Ordinary and inverse sampling in the trinomial

J. C. W. Rayner and P. J. Davy


By using inverse rather than ordinary sampling on just one of the categories in a multinomial model, distributions for which both marginals are geometric can be constructed. The results, which are not widely known, are accessible by elementary techniques. In a companion paper, Davy and Rayner (1996) have shown that this approach leads to an alternative derivation of the bivariate negative binomial distribution of Edwards and Gurland (1961).