Volume 16, Issue 2

We suggest an algorithm to test numbers of the form N = 2 kp m − 1 for primality, where 2 k < p m , k is an odd positive integer, 2 k < p m , p is a prime number, and p = 3 (mod 4). The algorithm makes use of the Lucas functions. First we present an algorithm to test numbers of the form N = 2 k 3 m − 1. Then the same technique is used in the more general case where N = 2 kp m − 1. The algorithms suggested here are of complexity O ((log N ) 2 log log N log log log N ).

An incomplete market with European type contingent claim based on two underlying assets whose price evolutions are assumed to be binary is considered. We obtain explicit formulas for upper and lower bounds for the rational price interval as well as for upper and lower hedging strategies. The obtained strategies are semi-self-financing in the sense that the extraction of funds is assumed at each time step with non-negative probability. The results are extended to the case where the stock price jumps are distributed over a closed rectangle and the pay-off function is convex. No assumptions are imposed on the joint distribution of the price jumps.

A linear discrete time dynamic control system with quadratic cost function perturbed by a sequence of dependent random variables is investigated from the point of view of the so-called probabilistic optimality criteria. In problems of stochastic optimisation, these criteria are related to the study of the asymptotic behaviour (in some probabilistic sense) of an integral cost functional as the horizon of planning tends to infinity. We obtain estimates of the rate of increasing of the defect of the optimal control, that is, the positive part of the difference between values of the cost functional under the optimal control and an arbitrary control, it is shown that these estimates are connected with parameters of the perturbing process. The results are applied to a model of optimal pension funding as a model of dynamic control.

A branching process Z n with geometric distribution of descendants in a random environment represented by a sequence of independent identically distributed random variables (the Smith–Wilkinson model) is considered. The asymptotics of large deviation probabilities P (ln Z n > θn ), θ > 0, are found provided that the steps of the accompanying random walk S n satisfy the Cramér condition. In the cases of supercritical, critical, moderate, and intermediate subcritical processes the asymptotics follow that of the large deviations probabilities P ( S n ≤ θn ). In strongly subcritical case the same asymptotics hold for θ greater than some θ * (for θ ≤ θ * the asymptotics of large deviation probabilities are different).

Given a set S of n elements drawn from a linearly ordered set and a set K = { k 1 , k 2 , . . . , k r } of positive integers between 1 and n , the multiselection problem is to select the k i th smallest element for all values of i , 1 ≤ i ≤ r . We present an efficient randomised algorithm to solve this problem in time O ( n log r ) with probability at least 1− cn −1 , where c is a positive constant.

We consider the complexity of realisation of the monotone functions by straight-line programs with conditional stop. It is shown that the mean complexity of each monotone function of n variables does not exceed a 2 n / n 2 (1 + o (1)) as n → ∞, and the mean complexity of almost all monotone functions of n variables is at least b 2 n / n 2 (1 + o (1)) as n → ∞, where a and b are constants.

We consider realisation of Boolean functions over the basis { x ∨ y ∨ z , x & y & z , } by circuits of unreliable functional elements which are subject to single-type constant faults at inputs of the elements. Let γ be the probability of a fault at an input of an element. By the unreliability of a circuit is meant the greatest probability of error at its output. In this paper, we find the asymptotically best realisation of an arbitrary Boolean function f ( x 1 , . . . , x n ) such that the functions x i , i = 1, 2, . . . , n , are realised absolutely reliably, the constants 0 and 1 are realised as reliably as we wish, and the remaining functions are realised with unreliability asymptotically equal to γ 3 as γ → 0.