Volume 30, Issue 3

Integer random walk { S n , n ≥ 0} with zero drift and finite variance σ 2 stopped at the moment T of the first visit to the half axis (-∞, 0] is considered. For the random process which associates the variable u ≥ 0 with the number of visits the state ⌊ uσ n $\begin{array}{} \displaystyle \sqrt{n} \end{array}$⌋ by this walk conditioned on T > n , the functional limit theorem on the convergence to the local time of stopped Brownian meander is proved.

The portfolio theory is used to formulate a multicriteria investment Boolean escaped gain minimization problem for searching all extreme portfolios. Stability aspects of this set against perturbed parameters of minimax Savage criteria are studied. We give lower and upper estimates for the stability radius for arbitrary Hölder norms on the three-dimensional space of initial data.

The complexity of the decision of polynomial (functional) completeness of a finite quasigroup is investigated. It is shown that the polynomial completeness of a finite quasigroup may be checked in time polynomially dependent on the order of the quasigroup.

Orthomorphisms of Abelian groups with the minimum possible Cayley distance between them are studied. We describe the class of transformations mapping the given orthomorphism into the set of all orthomorphisms such that their Cayley distance from this orthomorphism takes the minimal possible value 2. Algorithms are suggested that construct all orthomorphisms separated from the given one by the minimum possible distance, and the complexity of these algorithms is estimated. Examples of Abelian groups are presented such that the set of all their orthomorphisms contains elements which are separated from all other orthomorphisms by a distance larger than the minimum possible.

The previous paper was concerned with systems of equations over a certain family 𝓢 of quasigroups. In that work a method of elimination of an outermost variable from the system of equations was suggested and it was shown that further elimination of variables requires that the family 𝓢 of quasigroups satisfy the generalized distributive law (GDL). In this paper we describe families 𝓢 that satisfy GDL. The results are applied to construct classes of easily solvable systems of equations.

Predicates that are preserved by a semi-lattice function are considered. These predicates are called weak positive. A representation of these predicates are proposed in the form of generalized conjunctive normal forms (GCNFs). Properties of GCNFs of these predicates are obtained. Based on the properties obtained, more efficient polynomial-time algorithms are proposed for solving the generalized satisfiability problem in the case when all initial predicates are preserved by a certain semi-lattice function.