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Artur Siemaszko

Publications and source records attributed to Artur Siemaszko.

14 recordsLinked to original sources
arXiv2026

An incremental approximation to the prime-counting function via polygonal numbers

We construct an incremental prime-counting approximation from corrected right-endpoint sums on polygonal-number partitions. For each fixed polygonal family, retaining completed contributions gives $\mathcal O(1)$ numerical operations per consecutive node and $\mathcal O(\sqrt X)$ operations for the full traversal through $X$, in the unit-cost model. The scheme uses a fixed number of working numerical words rather than repeated summation. Asymptotic formulas for the discretization error and terminal scale yield an unconditional relation with the classical normalized prime-counting error. We prove that the logarithmic mean of the normalized estimator error converges to $1/\sqrt2$ if and only if the Riemann hypothesis holds, and give an independent Mellin proof of the converse. Under the hypothesis, this constant uniquely removes the limiting signed mean bias. Computations at powers of ten through $10^{19}$ with coefficient $0.7071$ show accuracy comparable to the Riemann approximation at the sampled arguments

math.NT↗
arXiv2026

A note on the diameter of graphs of two-dimensional simplex codes

Let $Γ^s(2,q)$ be the graph induced in the Grassmann graph by the $q$-ary simplex codes of dimension 2. For $q=4$, this graph is known to have diameter 3. We prove that the same diameter occurs for every prime power $q\geq5$. Together with the elementary cases $q=2,3$, this gives diam$Γ^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\geq4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\geq5$ an explicit diagonal pair of simplex lines gives the matching lower bound; we give two proofs, one using moments and one using products. For $q\geq7 we also retain an independent counting proof. The case $q=4$ is handled separately.

math.CO↗
arXiv2026

On aggregation-quantization permutability problem for discrete-time Markov chains

Given random walk on a graph, the corresponding discrete-time quantum walk can be constructed using the method proposed by Szegedy. On the other hand, given a partition of the set of states of a Markov chain, one can study the corresponding aggregated process. We extend the aggregation technique to the level of quantum Markov chains. We provide conditions under which application of these two operations - Szegedy's quantization and aggregation - give the same result. These consist of the strong lumpability of the underlying Markov chain supplemented by nonlinear relations, generalizing Kolmogorov's cycle condition, that account for the coefficients of both the original and the aggregated transition matrices. In particular, we show that the conditions are satisfied in the case of the random walk on graphs equipped with equitable partitions. We present several examples, which include the classical/quantum walks on Platonic solids. We discuss also relation of discrete-time classical/quantum walks on $N$-dimensional hypercube and the Ehrenfests urn model with $N$ particles. We apply our technique for of discrete-time walks on Cayley graphs of free groups. We also compare our results with those obtained using Cantero-Moral-Velazquez uniformization of unitary matrices.

quant-ph↗
arXiv2026

Elastodynamical properties of Sturmian structured media

In this paper, wave propagation in structured media with quasiperiodic patterns is investigated. We propose a methodology based on Sturmian sequences for the generation of structured mechanical systems from a given parameter. The approach is presented in a general form so that it can be applied to waveguides of different nature, as long as they can be modeled with the transfer matrix method. The bulk spectrum is obtained and its fractal nature analyzed. For validation of the theoretical results, three numerical examples are presented. The obtained bulk spectra show different shapes for the studied examples, but they share features which can be explained from the proposed theoretical setting.

math-ph↗
arXiv2024

Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle

We define quantization scheme for discrete-time random walks on the half-line consistent with Szegedy's quantization of finite Markov chains. Motivated by the Karlin and McGregor description of discrete-time random walks in terms of polynomials orthogonal with respect to a measure with support in the segment $[-1,1]$, we represent the unitary evolution operator of the quantum walk in terms of orthogonal polynomials on the unit circle. We find the relation between transition probabilities of the random walk with the Verblunsky coefficients of the corresponding polynomials of the quantum walk. We show that the both polynomials systems and their measures are connected by the classical Szegő map. Our scheme can be applied to arbitrary Karlin and McGregor random walks and generalizes the so called Cantero-Grünbaum-Moral-Velázquez method. We illustrate our approach on example of random walks related to the Jacobi polynomials. Then we study quantization of random walks with constant transition probabilities where the corresponding polynomials on the unit circle have two-periodic real Verblunsky coefficients. We present geometric construction of the spectrum of such polynomials (in the general complex case) which generalizes the known construction for the Geronimus polynomials. In the Appendix we present the explicit form, in terms of Chebyshev polynomials of the second kind, of polynomials orthogonal on the unit circle and polynomials orthogonal on the real line with coefficients of arbitrary period.

math-ph↗
arXiv2023

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.

cs.LG↗
arXiv2023

Hermite-Padé approximation and integrability

We show that solution to the Hermite-Padé 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-Padé 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-Padé 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.

nlin.SI↗
arXiv2023

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.

math.GR↗
arXiv2022

Integrability and geometry of the Wynn recurrence

We show that the Wynn recurrence (the missing identity of Frobenius of the Padé 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 Padé 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.

nlin.SI↗
arXiv2020

The Cayley Graph of Neumann Subgroups

All Cayley representations of the distant graph $Γ_Z$ over integers are characterized as Neumann subgroups of the extended modular group. Possible structures of Neumann subgroups are revealed and it is shown that every such a structure can be realized.

math.GR↗
arXiv2015

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.

math.CO↗
arXiv2012

Counting Berg partitions via Sturmian words and substitution tilings

We develop the connection of Berg partitions with special substitution tilings of two tiles. We obtain a new proof that the number of Berg partitions with a fixed connectivity matrix is equal to half of the sum of its entries, \cite{S-W}. This approach together with the formula of Séébold \cite{Seb}, for the number of substitutions preserving a given Sturmian sequence, shows that all of the combinatorial substitutions can be realized geometrically as Berg partitions. We treat Sturmian tilings as intersection tilings of bi-partitions. Using the symmetries of bi-partitions we obtain geometrically the palindromic properties of Sturmian sequences (Theorem 3) established combinatorially by de Luca and Mignosi, \cite{L-M}.

math.DS↗
arXiv2010

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).

math.DS↗