14 research outputs found
The Shortest Path Problem for the Distant Graph of the Projective Line Over the Ring of Integers
The distant graph of the projective line
over the ring of integers is considered. The shortest path problem in this
graph is solved by use of Klein's geometric interpretation of Euclidean
continued fractions. In case the minimal path is non-unique, all the possible
splitting are described which allows us to give necessary and sufficient
conditions for existence of a unique shortest path.Comment: 20 pages, 4 figures, Bulletin of the Malaysian Mathematical Sciences
Society, online 201
Neumann property in the extended modular group and maximal nonparabolic subgroups of the modular group
It is know that any Neumann subgroup of the modular group is maximal
non-parabolic. The question arises as to whether these are the only maximal
non-parabolic subgroups. The wild class of maximal, non-parabolic, not Neumann
subgroups of the modular group was constructed by Brenner and Lyndon. The new
construction of such a class is presented. Those groups are obtained as
subgroups of elements of positive determinant of any Neumann subgroup of the
extended modular group (the notion which we introduce in the paper) and in this
sens they are as "close" as possible to Neumann subgroups of the modular group.Comment: 1 figur
Counting Berg partitions
We call a Markov partition of a two dimensional hyperbolic toral automorphism
a Berg partition if it contains just two rectangles. We describe all Berg
partitions for a given hyperbolic toral automorphism. In particular there are
exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same
connectivity matrix (k, l, m, n)
Selected aspects of complex, hypercomplex and fuzzy neural networks
This short report reviews the current state of the research and methodology
on theoretical and practical aspects of Artificial Neural Networks (ANN). It
was prepared to gather state-of-the-art knowledge needed to construct complex,
hypercomplex and fuzzy neural networks.
The report reflects the individual interests of the authors and, by now
means, cannot be treated as a comprehensive review of the ANN discipline.
Considering the fast development of this field, it is currently impossible to
do a detailed review of a considerable number of pages.
The report is an outcome of the Project 'The Strategic Research Partnership
for the mathematical aspects of complex, hypercomplex and fuzzy neural
networks' meeting at the University of Warmia and Mazury in Olsztyn, Poland,
organized in September 2022.Comment: 46 page
On group extensions of 2-fold simple ergodic actions
Compact group extensions of 2-fold simple actions of locally compact second countable amenable groups are considered. It is shown what the elements of the centralizer of such a system look like. It is also proved that each factor of such a system is determined by a compact subgroup in the centralizer of a normal factor
A note on the centralizer of topological isometric extensions
summary:The centralizer of a semisimple isometric extension of a minimal flow is described
A natural family of factors for product -actions
It is shown that if and are natural
families of factors (in the sense of [E. Glasner, M. K. Mentzen and A. Siemaszko, A natural family of factors for
minimal flows , Contemp. Math. 215 (1998), 19–42]) for minimal flows
and , respectively, then is a natural family of
factors for the product -action on
generated by and .
An example is given showing the existence of topologically
disjoint minimal flows and for which the family
of factors of the flow is strictly
bigger than the family of factors of the product
-action on generated by and .
There is also an example of a minimal distal system with no
nontrivial compact subgroups in the group of its automorphisms
On exact topological flows
It is shown that group endomorphisms are exact flows
if and only if they are exact in the measure-theoretic sense and
that all flows which are exact with respect to an invariant measure
with full support are exact. It is also proved that all locally eventually
dense (led) flows have uniformly positive entropy (u.p.e.)