Volume 33, Issue 3

We study scatter properties of the modular addition operation for imprimitivity systems of the translation group of the binary vector space V n = {0, 1} n . We describe all the subspaces of the space V n that induce imprimitivity systems with worst possible scatter by the modular addition operation.

In a generalized allocation scheme of n particles over N cells we consider the random variable η n , N ( K ) which is the number of particles in a given set consisting of K cells. We prove that if n , K , N → ∞, then under some conditions random variables η n , N ( K ) are asymptotically normal, and under another conditions η n , N ( K ) converge in distribution to a Poisson random variable. For the case when N → ∞ and n is a fixed number, we find conditions under which η n , N ( K ) converge in distribution to a binomial random variable with parameters n and s = KN $\begin{array}{} \displaystyle \frac{K}{N} \end{array}$, 0 < K < N , multiplied by a integer coefficient. It is shown that if for a generalized allocation scheme of n particles over N cells with random variables having a power series distribution defined by the function B ( β ) = ln(1 − β ) the conditions n , N , K → ∞, KN $\begin{array}{} \displaystyle \frac{K}{N} \end{array}$ → s , N = γ ln( n ) + o (ln( n )), where 0 < s < 1, 0 < γ < ∞, are satisfied, then distributions of random variables η n,N(K)n $\begin{array}{} \displaystyle \frac{\eta_{n,N}(K)}{n} \end{array}$ converge to a beta-distribution with parameters sγ and (1 − s ) γ .

We consider the class EP ℕ of exponentially-polynomial functions which can be obtained by arbitrary superpositions of the constants 0, 1 and arithmetic operations of addition, multiplication, and powering. For this class, we solve the algorithmic equality problem of two functions that assume a finite number of values. Next, this class is restricted to the class PEP ℕ , in which the function x y is replaced by a sequence of functions { pix $\begin{array}{} \displaystyle p_i^x \end{array}$}, where p 0 , p 1 , … are all prime numbers. For the class PEP ℕ , the problem of membership of a function to a finitely generated class is effectively reduced to the equality problem of two functions. In turn, the last problem is effectively solved for the set of all one-place PEP ℕ -functions.

The fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set 𝕎 of vertices of a connected graph G in which the vector of distances to the vertices in 𝕎 uniquely determines all the vertices of the graph G . The minimum cardinality of a resolving set of G is known as the metric dimension of G . If 𝕎 ∖ ρ is also a resolving set for each ρ in 𝕎. In that case, 𝕎 is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set 𝕎. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures.

The functional system of Boolean vector functions with the naturally defined superposition operation is considered. It is shown that every closed class of this system admits a finite basis.