The influence of the cardinality of the alphabet on the quality of reconstruction of a symbolic periodic sequence from a sequence with noise

В статье рассмотрена задача восстановления символьных периодических последовательностей, искаженных шумами вставки, а также замены и удаления символов. Поскольку степень детализации символьного описания процесса определяется мощностью алфавита, представляет интерес исследование влияния степени детализации символьного описания на возможность восстановления полной информации об исходной периодической последовательности. Представлено экспериментальное исследование зависимости характеристик качества предложенного авторами метода восстановления периода от мощности алфавита. Для алфавитов разной мощности приводятся доля последовательностей с удовлетворительно восстановленным периодом и относительная погрешность определения длины периода. Качество восстановления оценивается отношением редакционного расстояния от восстановленной периодической последовательности до исходной строго периодической последовательности The relevance of this study is associated with the presence of a wide range of applied problems in real-world data processing and analysis. It is sensible to encode information using symbols from a finite alphabet in such problems. By varying the cardinality of the alphabet, in the description of the process, the symbolic representation provides a level of detail sufficient for real-world data analysis. However, for a number of subject areas in which it is possible to use symbolic coding of trajectories of the examined processes researchers face the presence of distortions, noise, and fragmentation of information. This occurs in bioinformatics, medicine, digital economy, time series forecasting and analysis of business processes. Periodic processes are widely represented in these subject areas. Without noise, these processes correspond to periodic symbolic sequences, i.e. words over a finite alphabet. A researcher often receives a sequence distorted by noises of various origins as the experimental data, instead of the expected periodic symbolic sequence. Under these conditions, when solving the problem of identifying the periodicity, which includes both the determination of a periodically repeating symbolic fragment and its length, hereinafter called the period, the problem requires reducing the effect of noise on the experimental results. The article deals with the problem of recovering periodic sequences, distorted by presence of noise along the replaced and deleted symbols. Since the level of detail in the description of the process depends on the cardinality of the alphabet, it is of interest to study the influence of the level of detail in the symbolic description on the possibility of recovering complete information about the initially periodic sequences. The article experimentally examines the dependence of the cardinality of the alphabet on the quality characteristics of the period recovery method proposed by the authors. For alphabets of different cardinalities, the proportion of sequences with a satisfactorily reconstructed period and the relative error in determining the length of the period are given. The quality of reconstruction of a periodically repeating fragment is estimated by the ratio of the editing distance from the reconstructed periodic sequence to the original sequence distorted by noise

Divisibility of integers obtained from truncated periodic sequences

A finite set of prime numbers [Formula: see text] is called unavoidable with respect to [Formula: see text] if for each [Formula: see text] the sequence of integer parts [Formula: see text], [Formula: see text] contains infinitely many elements divisible by at least one prime number [Formula: see text] from the set [Formula: see text]. It is known that an unavoidable set exists with respect to [Formula: see text] and that it does not exist if [Formula: see text] is an integer such that [Formula: see text] is not square free. In this paper, we show that no finite unavoidable sets exist with respect to [Formula: see text] if [Formula: see text] is a prime number or [Formula: see text] belongs to some explicitly given arithmetic progressions, for instance, [Formula: see text] and [Formula: see text], [Formula: see text]

Generation of Periodic Sequences in the Sieve of Eratosthenes

The arithmetic properties of prime numbers are hard to predict and it is one of the fundamental problem in number theory. In this paper, with Sieve of Eratosthenes as basis a periodic sequence of numbers is generated. This periodic pattern is used to understand the prime numbers in a better way.

Clustering of periodic orbits of a hyperbolic automorphism on the torus T2

This paper deals with clustering of periodic orbits of the hyperbolic toral automorphism induced by matrix A. We prove that Ta satisfies the Axiom A. The clustering of periodic orbits of Ta is ivestigated via the notion of 'p-closeness' of periodic sequences of the respective symbolic dynamical system. We also provide the number of clusters of periodic sequences with given periods in the case of 2-closeness.

Sublacunary sets and interpolation sets for nilsequences

<p style='text-indent:20px;'>A set <inline-formula><tex-math id="M1">\begin{document}$ E \subset \mathbb{N} $\end{document}</tex-math></inline-formula> is an interpolation set for nilsequences if every bounded function on <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula> can be extended to a nilsequence on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{N} $\end{document}</tex-math></inline-formula>. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here <inline-formula><tex-math id="M4">\begin{document}$ \{r_n: n \in \mathbb{N}\} \subset \mathbb{N} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ r_1 &lt; r_2 &lt; \ldots $\end{document}</tex-math></inline-formula> is <i>sublacunary</i> if <inline-formula><tex-math id="M6">\begin{document}$ \lim_{n \to \infty} (\log r_n)/n = 0 $\end{document}</tex-math></inline-formula>. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for <inline-formula><tex-math id="M7">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.</p>

Convergence of Generalized Collatz Problem in k-Adic Field

In this article, we define a new series transformation called transformation and probe into its fixed point and periodicity. We extend the number field of the transform period problem to a wider field. Different constraints are imposed on then different periodic columns are formed after finite transformations. We obtain that their periodic sequences are and respectively after derivation. As an application, it can provide a reference for C problems in more complex algebraic systems.