Searcharxiv⌕ Search

arXiv subjects

Maciej P. Wojtkowski

Publications and source records attributed to Maciej P. Wojtkowski.

10 recordsLinked to original sources
arXiv2025

On Sequence Groups

Linear second order recursive sequences with arbitrary initial conditions are studied. For sequences with the same parameters a ring and a group is attached, and isomorphisms and homomorphisms are established for related parameters. In the group, called the {\it sequence group}, sequences are identified if they differ by a scalar factor, but not if they differ by a shift, which is the case for the Laxton group. Prime divisors of sequences are studied with the help of the sequence group $\mod p$, which is always cyclic of order $p\pm 1$. Even and odd numbered subsequences are given independent status through the introduction of one rational parameter in place of two integer parameters. This step brings significant simplifications in the algebra. All elements of finite order in Laxton groups and sequence groups are described effectively. A necessary condition is established for a prime $p$ to be a divisor of a sequence: {\it the norm (determinant) of the respective element of the ring must be a quadratic residue $\mod p$}. This leads to an uppers estimate of the set of divisors by a set of prime density $1/2$. Numerical experiments show that the actual density is typically close to $0.35$. A conjecture is formulated that the sets of prime divisors of the even and odd numbered elements are independent for a large family of parameters.

math.NT↗
arXiv2025

The index and its prime divisors

We propose a new interpretation of the classical index of appearance for second order linear recursive sequences. It stems from the formula \[ C_{n}(t)-2 =\fracΔ{Q^{n}}\ L_n^2,\ \ \ \text{where} \ \ t= (T^2-2Q)/Q, \ Δ= T^2-4Q, \] connecting the Chebyshev polynomials of the first kind $C_n(x)$ with the Lucas sequence defined for integer $T,Q\neq 0$ by the recursion $L_{n+1}= TL_n-QL_{n-1}, L_0=0, L_1 = 1$. We build on the results of \cite{L-W}. We prove that for any prime $r\geq 2$ the sets $Π_j(t,r), j=1,2,\dots$, of primes $p$ such that $j$ is the highest power of $r$ dividing the index of appearance, have prime density equal to $\frac{1}{(r+1)r^{j-1}}$, for $r$-generic values of $t$. We give also complete enumeration of non-generic cases and the appropriate density formulas. It improves on the work of Lagarias, \cite{L}, and Ballot, \cite{B1},\cite{B2},\cite{B3}, on the sets of prime divisors of sequences of "finite order". Our methods are sufficient to prove that for any linear recursive sequence of second order (with some trivial exceptions) the set of primes not dividing any element contains a subset of positive density. We consider also some applications in arithmetic dynamics.

math.NT↗
arXiv2020

Gaussian vectors with Markov property

We demonstrate the parallel between the properties of Gaussian vectors and the Euclidean geometry. In particular we study the Markov property and give various equivalent Euclidean and probabilistic characterizations. We also give a simple Euclidean proof of the conditional maximality of the differential entropy for the Markov Gaussian vector (related to the Burg's Theorem).

math.PR↗
arXiv2020

Partitions of primes by Chebyshev polynomials

Partitions of the set of primes are introduced based on the Chebyshev polynomials at rationals. The prime densities of all such partitions are established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is the bridge between the algebra of Chebyshev polynomials and number-theoretic properties of the partitions. It is shown how to obtain in this way some of the classical theory of Lucas sequences. A hidden symmetry of the problem is revealed by the new language. As an application number-theoretic properties of simple dynamical systems (rotations and certain interval maps) are discussed.

math.NT↗
arXiv2015

Integrability via Reversibility

A class of left-invariant second order reversible systems with functional parameter is introduced which exhibits the phenomenon of robust integrability: an open and dense subset of the phase space is filled with invariant tori carrying quasi-periodic motions, and this behavior persists under perturbations within the class. Real-analytic volume preserving systems are found in this class which have positive Lyapunov exponents on an open subset, and the complement filled with invariant tori.

math.DS↗
arXiv2015

Homogeneous Weyl connections of non-positive curvature

We study homogenous Weyl connections with non-positive sectional curvatures. The Cartesian product $\mathbb S^1 \times M$ carries canonical families of Weyl connections with such a property, for any Riemmanian manifold $M$. We prove that if a homogenous Weyl connection on a manifold, modeled on a unimodular Lie group, is non-positive in a stronger sense (streched non-positive), then it must be locally of the product type.

math.DG↗
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↗