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Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
This article introduces a non-parametric information-theoretic approach to inference about the tail of a continuous or a discrete distribution. Leveraging a new concept named tail profile -- a set of information-theoretic quantities…
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
Detection of periodic patterns of interest within noisy time series data plays a critical role in various tasks, spanning from health monitoring to behavior analysis. Existing learning techniques often rely on labels or clean versions of…
Sequential pattern discovery is a well-studied field in data mining. Episodes are sequential patterns describing events that often occur in the vicinity of each other. Episodes can impose restrictions to the order of the events, which makes…
Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and…
Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter…
We propose an information criterion for determining an unknown number of periodic components in functional time series. Identifying the number of frequencies in large-scale time series has been a central focus. To achieve this goal, we…
Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case…
In this paper we introduce a new modification of the Jacobi-Perron algorithm in three dimensional case and prove its periodicity for the case of totally-real conjugate cubic vectors. This provides an answer in the totally-real case to the…
From Rauzy graph Rauzy Scheme can be obtaining by uniting sequence of vertices of ingoing and outgoing degree 1 by arches. This notion is a tool to describe Rauzy graph behavior. For morphic superword we prove periodicity of Rauzy schemes.…
The horizontal visibility algorithm has been recently introduced as a mapping between time series and networks. The challenge lies in characterizing the structure of time series (and the processes that generated those series) using the…
In this work we extend our study on a link between automaticity and certain algebraic power series over finite fields. Our starting point is a family of sequences in a finite field of characteristic $2$, recently introduced by the first…
A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the…
In order to find Hamiltonian cycle, algorithm should find edges that creates a Hamiltonian cycle. Higher number of edges creates more possibilities to check to solve the problem. Algorithm rests on analysis of original graph and opposite…
The linear complexity of a sequence has been used as an important measure of keystream strength, hence designing a sequence which possesses high linear complexity and $k$-error linear complexity is a hot topic in cryptography and…
Given an integer base $b>1$, a set of integers is represented in base $b$ by a language over $\{0,1,...,b-1\}$. The set is said to be $b$-recognisable if its representation is a regular language. It is known that eventually periodic sets…
The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute…
In directed graphs, a cycle can be seen as a structure that allows its vertices to loop back to themselves, or as a structure that allows pairs of vertices to reach each other through distinct paths. We extend these concepts to temporal…
The properties of continued fractions whose partial quotients belong to a quadratic number field K are distinct from those of classical continued fractions. Unlike classical continued fractions, it is currently impossible to identify…