Volume 12, Issue 5

We consider the class A n (k) of all (0, 1)-matrices A k of size n × n with exactly k ones in each row and each column, k = 1, ... , n. We prove an asymptotic formula for the permanent per A k , which holds true as n → ∞ and 0 < n - k = o(n/ln n) uniformly with respect to A k ∈ A n (k). We discuss the known upper and lower bounds for the numbers of m × n Latin rectangles and of n × n Latin squares and asymptotic expressions of these numbers as n → ∞ and m = m(n). We notice that the well-known O'Neil conjecture on the asymptotic behaviour of the number of Latin squares holds in a strong form. We formulate new conjectures of such kind and deduce from these conjectures asymptotic estimates of the numbers of Latin rectangles and Latin squares that sharpen the results known before. In conclusion, we give a short review of the literature devoted to the questions discussed in the paper with formulations of the main results.

Let each vertex of a finite directed graph be associated with a finite-dimensional linear space and each arc, with a linear transformation of the corresponding space. Such objects are referred to as linear representations of graphs. They naturally arise in some fields of algebra and are deeply studied in the past three decades. Replacing all arcs entering into a sink by oppositely oriented ones, we arrive at a new directed graph. These two directed graphs are close to each other in the sense that the problem of classification of their representation are equivalent, as shown by Bernstein, Gelfand, and Ponomarev. Two orientations are equivalent if one is derived from another by means of a sequence of the above transformations. In the directed graph representation theory, of most interest are circuit-free orientations. In this paper, we give a simple criterion for equivalence of circuit-free orientations. We prove that two orientations are equivalent if and only if some integrals of these orientations are equal to each other.

We study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii. We consider the classical question: how the properties of the system of subgroups of a group influence on the properties of the group? We show that almost layer-finiteness is transferred on the group G from the normalisers of non-trivial finite subgroups of the group G if G is periodic conjugately biprimitively finite group with a strongly embedded subgroup. We study the structure of an infinite Sylow 2-subgroup in a periodic conjugately biprimitively finite group in the case that the normaliser of any finite non-trivial subgroup is almost layer-finite. The results of the paper can be useful in the study of the class of periodic conjugately biprimitively finite groups (periodic Shunkov groups).

In the context of a new functional approach to the study of classes of groups, V. A. Vedernikov and M. M. Sorokina introduced Ω-foliated formations, which gave a possibility to systematise a wide class of formations of finite groups. In this paper, we study n-multiply Ω-foliated formations with r-direction φ such that φ 0 ≤ φ, φ(A) ⊆ G A′ G A for all A ∈ ℑ whose lattice of all n-multiply Ω-foliated subformations with direction φ is Boolean.

We describe the n-multiply Ω-bicanonical Fitting classes with Boolean lattice of Fitting subclasses. In particular, it is shown that in this case a Fitting class is directly decomposable with the use of the set of all atoms of its lattice. Here the notion of a direct decomposition plays the key role. Therefore we study direct decompositions separately and consider Ω-foliated Fitting classes with more general directions.

We consider the notion of a point, give examples of groups with points and prove a theorem on the location of points in subgroups of infinite groups.

The minimal polynomials of some decimations from geometric progressions and linear recurring sequences of maximal period over a finite field of characteristic 2 are studied. The step of these decimations increases exponentially.

We consider non-deterministic initial finite automata without final states and the ω-languages determined by such automata. For such ω-languages, we consider the so-called languages of obstructions. We define and analyse billiard ω-languages determined in a special way for each n ≥ 3 over an alphabet consisting of n letters. Each ω-word of such ω-language can be obtained with the use of infinite number of reflections of a point from the cushions of billiards having the form of a regular n-polygon. For such ω-languages, we consider the languages of obstructions and show that for any n a language of obstructions is not regular.

We give a definition of a quantum branching program. A well-known Boolean function MOD p is considered. We prove that any deterministic branching program and any probabilistic ordered stable branching program which (1/2 + ε)-compute the function MOD p are of width no less than p. We construct a stable quantum branching program of width O(log p) which computes the function MOD p with one-sided error ε > 0.