Abstract Nonabelian 2-groups H containing a cyclic subgroup of index 2 are dihedral groups, generalized quaternion groups, quasidihedral groups and modular maximal-cyclic groups. Earlier the authors introduced the classes of piecewise quasiaffine transformations on an arbitrary nonabelian 2-group H with a cyclic subgroup of index 2. For the generalized group of quaternions of order 2 m we have obtained a complete classification of orthomorphisms, complete transformations and their left analogues in the class of piecewise quasiaffine transformations under consideration. This paper presents a similar classification for the remaining three groups (the dihedral group, the quasidihedral group and the modular maximal-cyclic group).

Abstract There exist well-known distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph. An example is given by the Johnson graph J (8, 3) with the intersection array {15, 8, 3;1, 4, 9}. The paper is concerned with the problem of the existence of distance-regular graphs Γ with the intersection arrays {78, 50, 9;1, 15, 60} and {174, 110, 18;1, 30, 132} for which Γ 3 is a triangle-free graph.

Abstract 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.

Abstract We show that the system of elementary conjunctions Ω n , 2 k = K 0 , … , K 2 k − 1 $ \Omega_{n,2^k} = {K_0,\ldots,K_{2^{k} -1}} $ such that each conjunction depends essentially on n variables and corresponds to some codeword of a linear (n, k)-code can be implemented by a separating contact circuit of complexity at most 2 k+1 +4k(n − k) − 2. We also show that if a contact (1, 2 k )-terminal network is separating and implements the system of elementary conjunctions Ω n , 2 k $ \Omega_{n,2^k} $ , then the number of contacts in it is at least 2 k+1 − 2.

Abstract A class of one-dimensional discrete power series distributions containing negative binomial distributions is considered. Properties of distributions of this class are investigated. Limit theorems generalizing similar theorems for the negative binomial distributions are proved. The proofs are based both on elementary asymptotic methods and on a modification of saddle-point method.

Abstract We study matrices over quotient rings modulo univariate polynomials over a two-element field. Lower bounds for the fraction of the invertible matrices among all such matrices of a given size are obtained. An efficient algorithm for calculating the determinant of matrices over these quotient rings and an algorithm for generating random invertible matrices (with uniform distribution on the set of all invertible matrices) are proposed and analyzed. An effective version of the latter algorithm for quotient rings modulo polynomials of form x r − 1 is considered and analyzed. These methods may find practical applications for generating keys of cryptographic schemes based on quasi-cyclic codes such as LEDAcrypt.

Abstract A hypergraph H = (V, E) has the property Bk if there exists an assignment of two colors to V such that each edge contains at least k vertices of each color. A hypergraph is called simple if every two edges of it have at most one common vertex. We obtain a new lower bound for the minimal number of edges of n-uniform simple hypergraph without the property Bk .

Abstract The enumeration closure operator (the Π-operator) is considered on the set Pk of functions of the k-valued logic. It is proved that, for any k ⩾ 2, any positively precomplete class in Pk is also Π-precomplete. It is also established that there are no other Π-precomplete classes in the three-valued logic.