Volume 27, Issue 4

A general approach is proposed for obtaining estimates of the number of solutions of systems of nonlinear equations. Final estimates are established in the case when the arguments of functions in the system are the signs of linear recurrent sequences over Galois rings.

We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with parameter τ > 0. There are two critical values of this parameter: τ = 1 and τ = 2. The properties of a graph change significantly when τ = τ ( N ) passes these points as N → ∞. Let G N , n be the subset of random graphs under the condition that sum of degrees of its vertices is equal to n . The limit theorem for the maximum vertex degree in G N , n as N , n → ∞ and τ → 1 or τ → 2 is proved.

The subset A of the group G is called ( k , l )- sumset if there exists subset B ⊆ G such that A = kB - lB , where kB - lB = { x 1 + ... + x k - x k +1 ... - x k + l | x 1 ,..., x k + l ∈ B }. Upper and lower bounds of the number of ( k , l )-sumsets in the Abelian group are obtained.

The paper is concerned with the asymptotic behaviour of the independence number for the binomial model of a random k -regular hypergraph H ( n , k , p ) in a sparse case, when p=c/n− 1k− 1 $p=c/{n-1\choose k-1}$ with positive constant c > 0. The independence number α ( H ( n , k , p )) is shown to satisfy the law of large numbers α (H(n,k,p))n→ Pγ (c) as n→ +∞ $$\frac{\alpha(H(n,k,p))}{n}\stackrel{{\sf P}}{\to}\gamma(c)\;\;\mbox{ as }n\to+\infty$$ with some constant γ ( c ) > 0. We also shows that γ ( c ) > 0 is a solution of some transcendental equation for small values of c ⩽ ( k - 1) -1 .

For shift-invariant probability measures on the set of infinite two-sided binary sequences (one-dimensional covers) we introduce the notion of capacity as a maximum portion of embedded into the cover uniformly distributed (purely random) binary sequence (message) that admits special correction of the cover restoring its distribution up to distribution of n -tuples (subwords of some fixed length n ). “Special correction” is carried out using the proposed new algorithm that changes some of the cover’s symbols not occupied by embedded message. The features of the introduced capacity are examined for the Markov cover. In particular, we show how capacity may be significantly increased by weakening of the standard constraint that positions for message embedding have to be chosen by independent unfair coin tosses. Experimental results are presented for correction of real steganographic covers after LSB-embedding.