[en] In this survey we consider results and open problems related to two major ideas in the theory of optimal quadrature formulae: the ideas of Gauss and Kolmogorov

[en] This paper discusses generalizations of the model introduced by Kehr and Kunter of the random walk of a particle on a one-dimensional chain which in turn has been constructed by a random walk procedure. The superimposed random walk is randomised in time according to the occurrences of a stochastic point process. The probability of finding the particle in a particular position at a certain instant is obtained explicitly in the transform domain. It is found that the asymptotic behaviour for large time of the mean-square displacement of the particle depends critically on the assumed structure of the basic random walk, giving a diffusion-like term for an asymmetric walk or a square root law if the walk is symmetric. Many results are obtained in closed form for the Poisson process case, and these agree with those given previously by Kehr and Kunter. (author)

[en] The asymptotic behaviour of a stochastic non-linear nuclear reactor modelled by a master equation is analysed in two different limits: the thermodynamic limit and the zero-neutron-source limit. In the first limit a finite steady neutron density is obtained. The second limit predicts the neutron extinction. The interplay between these two limits is studied for different situations. (author)

[en] It is shown how Kolmogorov's ideas and results led to the establishment that some Newtonian systems are in essence identical to random processes

[en] If the marginal distributions and the moments of order 1 and 2 of a multivariate distribution are known, one can interpolate the total distribution while conserving all this information and keeping a large freedom on to be determined parameters of the synthesis. We can use it to conserve other characteristics of the distribution, or to enforce some criteria. This synthesis method is then applied to the calculation of distributions in the frame of probabilistic dynamics. We can obtain a system of hyperbolic PDEs for the marginal distributions of the solution of the Chapman-Kolmogorov equation. These PDEs depend only on the dynamics of the problem and on the formerly computed (Devooght, 1994) first and second moments in order. The solution method of these equations refers to the properties of the Lie algebras, and the calculation of the marginal distributions is reduced to the one of time quadratures. (author)

[en] Let π(i,x-bar,t) be the probability density for a physical system to be in a component state i with physical variables x-bar at time t. Its evolution is given by the Chapman-Kolmogorov equation, which is only analytically solvable in very simple cases. In this paper, we show how to obtain the first moments in order of the distributions. These moments are solutions of a large and coupled differential system that we have to close first. A specific algorithm is presented for this problem and is illustrated on different applications. (author)

[en] Short communication

[en] It is considered the problem of neutron absorption by a slab of absorbing and multiplying medium, within the framework of the theory of birth and death process. The number of transmitted neutrons is a random variable the probability of which is a solution of Chapman-Kolmogorof equations

[en] A given finite sequence of letters over a finite alphabet can always be algorithmically generated, in particular by a Turing machine. This fact is at the heart of complexity theory in the sense of Kolmogorov and Chaitin. A relevant question in this context is whether, given a statistically 'sufficiently long' sequence, there exists a deterministic finite automaton that generates it. In this paper we propose a simple criterion, based on measuring block entropies by lumping, which is satisfied by all automatic sequences. On the basis of this, one can determine that a given sequence is not automatic and obtain interesting information when the sequence is automatic. Following previous work on the Feigenbaum sequence, we give a necessary entropy-based condition valid for all automatic sequences read by lumping. Applications of these ideas to representative examples are discussed. In particular, we establish new entropic decimation schemes for the Thue-Morse, the Rudin-Shapiro and the paperfolding sequences read by lumping. (author)

No abstract available