Volume 14, Issue 1

Let be a finite set, where X t , t = 1, 2, . . . , T , are pairwise nonoverlapping sets, N t = | X t | be the cardinality of the set X t , t = 0, 1, . . . , T . Let ℱ 1 be the class of all mappings f of the set X′ = X \ X 0 into X such that the image y = f ( x ) ∈ X t −1 ∪ X t for any x ∈ X t , t = 1, . . . , T . The cardinality of the set of all mappings of the class ℱ 1 is . With the use of non-homogeneous branching processes, we study some asymptotical properties of the uniformly distributed on ℱ 1 random mapping f as N t → ∞, t = 1, 2, . . . , T . Similar results are obtained for some other classes of random mappings f of the set X .

The method of boundary functionals suggested by A. A. Sapozhenko who applied it to a series of enumeration problems is successfully employed in the case of partially ordered sets with regular structure of the strata, for example, on the unit n -dimensional cube B n . But partially ordered sets occurring in some problems, say, the three-valued n -dimensional lattice , do not possess a regular structure. In this paper, the method of boundary functionals is extended to the case of such sets and is used to find the asymptotic behaviour of the number of antichains in the three central strata of .

For a vector combinatorial optimisation problem with bottleneck sub-criteria, we give a formula for limiting perturbations of parameters of sub-criteria which do not yield new Pareto-optimal solutions.

A Galois–Eisenstein ring or a GE-ring is a finite commutative chain ring. We consider two methods of enumeration of all solutions of some system of polynomial equations over a GE-ring R . The first method is the general method of coordinate-wise linearisation. This method reduces to solving the initial polynomial system over the quotient field = R / Rad R and then to solving a series of linear equations systems over the same field. For an arbitrary ideal of the ring R [ x 1 , . . . , x k ] a standard base called the canonical generating system (CGS) is constructed. The second method consists of finding a CGS of the ideal generated by the polynomials forming the left-hand side of the initial system of equations and solving instead of the initial system the system with polynomials of the CGS in the left-hand side. For systems of such type a modification of the coordinate-wise linearisation method is presented.

We construct a standard basis of an ideal of the polynomial ring R [ X ] = R [ x 1 , . . . , x k ] over commutative Artinian chain ring R , which generalises a Gröbner base of a polynomial ideal over fields. We adopt the notion of the leading term of a polynomial suggested by D. A. Mikhailov and A. A. Nechaev, but using the simplification schemes introduced by V. N. Latyshev. We prove that any canonical generating system constructed by D. A. Mikhailov and A. A. Nechaev is a standard basis of the special form. We give an algorithm (based on the notion of S -polynomial) which constructs standard bases and canonical generating systems of an ideal. We define minimal and reduced standard bases and give their characterisations. We prove that a Gröbner base χ of a polynomial ideal over the field = R / rad( R ) can be lifted to a standard basis of the same cardinality over R with respect to the natural epimorphism ν : R [ X ] → [ X ] if and only if there is an ideal I R [ X ] such that I is a free R -module and Ī = (χ).

We give systems of generators of symmetric and alternating permutation groups whose software implementation relies upon simple arithmetical operations. For some given systems, we estimate the group depth (the generator system index).