The Shortest Path Problem for the Distant Graph of the Projective Line Over the Ring of Integers

The distant graph $G = G(\mathbb{P}(Z),\triangle)$ 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

Integrability and geometry of the Wynn recurrence

We show that the Wynn recurrence (the missing identity of Frobenius of the Pad\'{e} approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev-Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Pad\'{e} theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions.Comment: 17 pages, 7 figures; v2 presentation improved +2 page

Hermite-Pad\'{e} approximation and integrability

We show that solution to the Hermite-Pad\'{e} type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Pad\'{e} approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Pad\'{e} problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.Comment: 17 pages, 4 figures; v2 introduction expande

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 natural family of factors for product Z2\mathbb{Z}^2-actions

It is shown that if ${\mathcal N}$ and ${\mathcal N}'$ 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 $(X,T)$ and $(X',T')$, respectively, then $\{R\otimes R'\colon R\in{\mathcal N},R'\in{\mathcal N}'\}$ is a natural family of factors for the product $\mathbb{Z}^2$-action on $X\times X'$ generated by $T$ and $T'$. An example is given showing the existence of topologically disjoint minimal flows $(X,T)$ and $(X',T')$ for which the family of factors of the flow $(X\times X',T\times T')$ is strictly bigger than the family of factors of the product $\mathbb{Z}^2$-action on $X\times X'$ generated by $T$ and $T'$. There is also an example of a minimal distal system with no nontrivial compact subgroups in the group of its automorphisms

A note on the centralizer of topological isometric extensions

summary:The centralizer of a semisimple isometric extension of a minimal flow is described