3 research outputs found
Positiveness of the permanent of 4-dimensional polystochastic matrices of order 4
A nonnegative multidimensional matrix is called polystochastic if the sum of
its entries over each line is equal to . In this paper we overview known
results on positiveness of the permanent of polystochastic matrices and prove
that the permanent of every -dimensional polystochastic matrix of order
is greater than zero.Comment: This version is expanded by a historical surve
Transversals, plexes, and multiplexes in iterated quasigroups
A -ary quasigroup of order is a -ary operation over a set of
cardinality such that the Cayley table of the operation is a
-dimensional latin hypercube of the same order. Given a binary quasigroup
, the -iterated quasigroup is a -ary quasigroup
that is a -time composition of with itself. A -multiplex (a -plex)
in a -dimensional latin hypercube of order or in the
corresponding -ary quasigroup is a multiset (a set) of entries such
that each hyperplane and each symbol of is covered by exactly elements
of . A transversal is a 1-plex.
In this paper we prove that there exists a constant such that if a
-iterated quasigroup of order has a -multiplex then for large
the number of its -multiplexes is asymptotically equal to . As a corollary we obtain that if the
number of transversals in the Cayley table of a -iterated quasigroup of
order is nonzero then asymptotically it is .
In addition, we provide limit constants and recurrence formulas for the
numbers of transversals in two iterated quasigroups of order 5, characterize a
typical -multiplex and estimate numbers of partial -multiplexes and
transversals in -iterated quasigroups
Transversals, near transversals, and diagonals in iterated groups and quasigroups
Given a binary quasigroup of order , a -iterated quasigroup
is the -ary quasigroup equal to the -times composition of with
itself. The Cayley table of every -ary quasigroup is a -dimensional latin
hypercube. Transversals and diagonals in multiary quasigroups are defined so as
to coincide with those in the corresponding latin hypercube.
We prove that if a group of order satisfies the Hall--Paige
condition, then the number of transversals in is equal to for large , where is the
commutator subgroup of . For a general quasigroup , we obtain similar
estimations on the numbers of transversals and near transversals in and
develop a method for counting diagonals of other types in iterated quasigroups.Comment: More details are added to proofs and definitions, typos and errors
are correcte