Volume 17, Issue 5

We study a random graph which can be treated as a model of large-scale data transmission systems, including the Internet. We give conditions for existence of a giant component in such a graph and prove that the limit distribution of the size of this component is normal.

We consider the set of random forests consisting of N rooted trees ordered in one of N ! possible ways and of n nonroot unlabelled vertices. As N, n → ∞, we find the limit distributions of the ( N – p )/th term of the set of order statistics obtained by arranging the sizes of the trees of a random unlabelled forest in nondescending order for fixed p = 1, 2, … We find that a giant tree (that is, a tree of size n + o ( n )) emerges in the only case where N, n → ∞ so that

We consider in detail the phenomenon of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables related to the generalised allocation scheme from one lattice to another.

We consider multiple optimal stopping rules for a finite (with horizon N ) sequence of independent random variables. We are interested in finding a stopping rule which maximises the expected sum of k , 1 < k < N , observations. The optimal stopping rule and the value of the game are obtained. This result can be applied in the house-selling problem and in behavioural ecology problems.

We consider a random nonhomogeneous hypergraph on n vertices with M = M(n) edges, M i = M i (n) edges consist of i vertices, For each edge, vertices are chosen by random and equiprobable sampling with replacement out of n vertices. Under the condition that N → ∞ and we show that the probability that the random hypergraph consists of hypertrees and components with one cycle tends to one. Similar results for random graphs and random homogeneous hypergraphs have been obtained earlier.

We investigate the structure of the set of reversible cellular automata. In the class of two-dimensional binary linear cellular automata with variable structure, the classes with decidable and undecidable reversibility property are separated, which contain almost all such cellular automata. With the use of this result, we prove undecidability of reversibility in the class of cellular automata with Γ-pattern and sixteen states of a cell and in the class of two-dimensional binary cellular automata with self-dual local transition functions. The complete description is given for the structure of the set of reversible cellular automata of the class of binary cellular automata with local transition functions from the Post classes.

The problem of packing rectangles into several strips is considered. It is shown that for this problem there exist on-line algorithms with multiplicative error asymptotically close to 2 e , where e is the base of the natural logarithm. It is proved that none of the on-line algorithms can have the asymptotic multiplicative error less than e .

A Boolean function g is said to be an annihilator of a Boolean function f if fg = 0. In some problems concerning finite automata, it is required to find non-zero annihilators of low algebraic degree for a function f . In this paper we suggest Algorithm M2 which, given the Zhegalkin polynomial for a function f , yields a basis of the space of its annihilators of degree not exceeding d . Algorithm M2 is an enhancement of a previously known algorithm and allows in a series of cases to decrease calculations. The total complexity of Algorithm M2 is the same as for the previous algorithm.