Recurrent Sequences Research Articles

On the p-Adic Valuation of Third Order Linear Recurrence Sequences

On the p-Adic Valuation of Third Order Linear Recurrence Sequences

Random balancing-like sequences

AbstractA balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.

Tiling of dominoes with ranked colors

Several articles deal with tilings with various colors and shapes. In this paper, we present a new type of tiling problem of a (1×n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1\ imes n)$$\\end{document}—board where the colors have a prescribed order of preference and the size of colored dominoes is bounded by (1×s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1\ imes s)$$\\end{document}. We show that the total number of tilings can be given as a linearly recurrent sequence of order ks, and at the same time by a higher order self-convolution of s-generalized Fibonacci sequences.

A methodological exploration of the P-frame approach: Implications for developing phrase lists

The widespread availability of corpora and corpus tools has led to considerable advances in our understanding of phraseology and how it can be investigated. An increasingly influential approach to investigating phraseology is the retrieval and analysis of phrase frames (p-frames), recurrent word sequences with a variable slot. Although the p-frame approach is quite well-established, pedagogically oriented research in this area is still quite new, in particular the application of phrase frames to generate useful lists of phrases. This paper aims to contribute to methodological efforts in this direction by presenting a case study to illustrate decision-making throughout the process from corpus compilation to the generation of a final pedagogical list of phrases. Using a corpus of research article (RA) introductions in Health Sciences, this study shows how key decisions including p-frame length, frequency and range thresholds were arrived at. It also discusses exclusion criteria focusing on variability and predictability of p-frame fillers, and the fine-tuning of the resulting list, focusing on semantic coherence and pedagogical usefulness. Finally, we present the results of an initial evaluation of the list by stakeholders. The main contributions of the study to p-frame research methodology lie in the clarification of threshold settings, the potential contribution of variability and predictability to decision-making and the proposal of more in-depth concordance analysis of co-text to aid in the identification of phrases.

Kepler sets of second-order linear recurrence sequences over Qp

Let [Formula: see text] be a second-order linear recurrence sequence with constant coefficients over the field of [Formula: see text]-adic numbers [Formula: see text]. We study the set of limit points of the sequence of consecutive ratios [Formula: see text] in [Formula: see text].

Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support {0,1,…,m+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\{0,1,\\ldots ,m+1\\}}$$\\end{document}, for some integer m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{m\\ge 1}$$\\end{document}. The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

The gaussian sequence 3th order in finite groups

Recent research examines Gaussian numbers in third-order recurrent sequences in connection with finite groups in the field of Abstract Algebra. Thus, some definitions were introduced aiming to transform infinite sequences into finite ones. In this regard, the Fourier transform was used as a visualization technique, explored in Google Colab. The mathematical theorems presented were established with the purpose of examining the patterns of these sequences and their corresponding cycles. As a future perspective, it is intended to investigate other mathematical theorems to generalize the sequences into finite groups.

Sequential Behavior of Broiler Chickens in Enriched Environments under Varying Thermal Conditions Using the Generalized Sequential Pattern Algorithm: A Proof of Concept.

Behavior analysis is a widely used non-invasive tool in the practical production routine, as the animal acts as a biosensor capable of reflecting its degree of adaptation and discomfort to some environmental challenge. Conventional statistics use occurrence data for behavioral evaluation and well-being estimation, disregarding the temporal sequence of events. The Generalized Sequential Pattern (GSP) algorithm is a data mining method that identifies recurrent sequences that exceed a user-specified support threshold, the potential of which has not yet been investigated for broiler chickens in enriched environments. Enrichment aims to increase environmental complexity with promising effects on animal welfare, stimulating priority behaviors and potentially reducing the deleterious effects of heat stress. The objective here was to validate the application of the GSP algorithm to identify temporal correlations between heat stress and the behavior of broiler chickens in enriched environments through a proof of concept. Video image collection was carried out automatically for 48 continuous hours, analyzing a continuous period of seven hours, from 12:00 PM to 6:00 PM, during two consecutive days of tests for chickens housed in enriched and non-enriched environments under comfort and stress temperatures. Chickens at the comfort temperature showed high motivation to perform the behaviors of preening (P), foraging (F), lying down (Ld), eating (E), and walking (W); the sequences <{Ld,P}>; <{Ld,F}>; <{P,F,P}>; <{Ld,P,F}>; and <{E,W,F}> were the only ones observed in both treatments. All other sequential patterns (comfort and stress) were distinct, suggesting that environmental enrichment alters the behavioral pattern of broiler chickens. Heat stress drastically reduced the sequential patterns found at the 20% threshold level in the tested environments. The behavior of lying laterally "Ll" is a strong indicator of heat stress in broilers and was only frequent in the non-enriched environment, which may suggest that environmental enrichment provides the animal with better opportunities to adapt to stress-inducing challenges, such as heat.

Bounds on the number of squares in recurrence sequences

We investigate the number of squares in a very broad family of binary recurrence sequences with u0=1. We show that there are at most two distinct squares in such sequences (the best possible result), except under very special conditions where we prove there are at most three such squares.

Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁p(·)

Let C denote a convex subset within the vector space 𝓁p(·), and let T represent a mapping from C onto itself. Assume α=(α1,⋯,αn) is a multi-index in [0,1]n such that ∑i=1nαi=1, where α1&gt;0 and αn&gt;0. We define Tα:C→C as Tα=∑i=1nαiTi, known as the mean average of the mapping T. While every fixed point of T remains fixed for Tα, the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for Tα and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in R. Our focus then shifts to variable exponent modular vector spaces 𝓁p(·), where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework.

On multiplicative dependent vectors with coordinates from certain higher order recurrences

In this paper, we prove that there are finitely many multiplicative dependent vectors with coordinates from non-degenerate linear recurrence sequences [Formula: see text] of a fixed order [Formula: see text] These sequences satisfy the recurrence relation [Formula: see text] with initials [Formula: see text]. Here, the coefficients of the recurrence relation are positive integers satisfying [Formula: see text] or [Formula: see text] In both these conditions, [Formula: see text]

On Robustness for the Skolem, Positivity and Ultimate Positivity Problems

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

The ring of finite algebraic numbers and its application to the law of decomposition of primes

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.

On some $q$-Bessel type continuous wavelet transform

In this paper we continue as in \cite{Rezguietal} to exploit the modified variants of Bessel function in the framework of $q$-theory to construct wavelet operators. A generalized $q$-Bessel type function has been introduced leading to an associated mother wavelet which in turns has induced a continuous wavelet transform. Finally, Plancherel/Parceval type relations have been proved. Such variants of wavelets permit in some sense to approximate solutions of ODEs and PDEs by transforming them to recurrent sequences.

Abstract PO1-08-02: Population-based screening of Uruguayan Ashkenazi Jews for recurrent BRCA1 and BRCA2 pathogenic sequence variants

Abstract In Ashkenazi Jews (AJ) three recurring pathogenic sequence variants (PSVs) are detected in ~2.5% of the general population in the BRCA1 (c.68_69del = 185delAG, c.5266dup = 5382insC), and BRCA2 (c.5946del = 6174delT). Population-based screening for these PSVs in AJ women is part of the health basket in Israel. To assess the feasibility and outcome of BRCA genotyping in the Jewish population of Uruguay, AJ in the greater Montevideo area were recruited using ethically approved protocol and without pretest counseling were genotyped for the three predominant AJ PSVs in the BRCA genes. Independently confirmed PSV carriers were counseled, and genetic testing was offered to additional family members. Overall, 327 participants were enrolled: 312 (95%) female, 261 (80%) had all four grandparents AJ, and 14 (4%) women were breast cancer survivors with a mean age ± standard deviation (SD) 50 ± 11.5 years. The BRCA1 c.68_69del PSV was detected in three cancer free participants (0.92%, CI 95% 0.31-2.6), all with a suggestive family history. No carriers of the other two recurrent PSVs were detected. Online oncogenetic counseling was provided for all carriers. In conclusion, the rate of the BRCA1 c.68_69del PSV was similar with the rate in other AJ communities. AJ population BRCA genotyping screens in Uruguay seem feasible and should be promoted. Pedigrees of the carrier cases (a-c) Citation Format: Cecilia Castillo, Natalia Camejo, Nora Artagaveytia, Lucia Brignoni, Yael Laitman, Alfonso Cayota, Gabriel Krygier, Lucia Delgado, Eitan Friedman. Population-based screening of Uruguayan Ashkenazi Jews for recurrent BRCA1 and BRCA2 pathogenic sequence variants [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO1-08-02.

Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event

Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event

Вычисление тропических последовательностей, ассоциированных с последовательностями Сомоса, в пакете Gfan

This paper examines tropical recurrent sequences associated with Somos sequences. The classical Somos sequences have applications in the theory of elliptic curves. Due to the Laurent character of the classical sequences, a pattern can be inferred between the classical sequences and their tropical counterparts. The greatest interest is the increase in the dimension of the solution space of tropical sequences depending on the length of the final sequence. For a set of tropical sequences described by tropical recurrence relations, D.Yu.Grigoriev put forward a hypothesis about the stabilization of the maximum dimensions of the components of the corresponding tropical prevarieties. This hypothesis has been proven for tropical linear recurrent sequences. As part of this work, for tropical recurrent sequences associated with the sequences Somos-4 and Somos-5, the corresponding tropical prevarieties were investigated using the Gfan package in order to test Grigoriev's hypothesis.

Positivity-hardness results on Markov decision processes

This paper investigates a series of optimization problems for one-counter Markov decision processes (MDPs) and integer-weighted MDPs with finite state space. Specifically, it considers problems addressing termination probabilities and expected termination times for one-counter MDPs, as well as satisfaction probabilities of energy objectives, conditional and partial expectations, satisfaction probabilities of constraints on the total accumulated weight, the computation of quantiles for the accumulated weight, and the conditional value-at-risk for accumulated weights for integer-weighted MDPs. Although algorithmic results are available for some special instances, the decidability status of the decision versions of these problems is unknown in general. The paper demonstrates that these optimization problems are inherently mathematically difficult by providing polynomial-time reductions from the Positivity problem for linear recurrence sequences. This problem is a well-known number-theoretic problem whose decidability status has been open for decades and it is known that decidability of the Positivity problem would have far-reaching consequences in analytic number theory. So, the reductions presented in the paper show that an algorithmic solution to any of the investigated problems is not possible without a major breakthrough in analytic number theory. The reductions rely on the construction of MDP-gadgets that encode the initial values and linear recurrence relations of linear recurrence sequences. These gadgets can flexibly be adjusted to prove the various Positivity-hardness results.

Near-squares in binary recurrence sequences

We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. We show that for a given [Formula: see text], there is at most one [Formula: see text] such that [Formula: see text] is a near-square. With the exceptions of [Formula: see text] and [Formula: see text], any such [Formula: see text] can be a near-square only if [Formula: see text], [Formula: see text] is prime and [Formula: see text]. This is a part of a more general phenomenon regarding near-squares in nondegenerate recurrence sequences defined for the integers [Formula: see text] and [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. This arises from a novel Aurifeuillean-type factorization of [Formula: see text] we have found.

Combinatorial approach on the recurrence sequences: An evolutionary historical discussion about numerical sequences and the notion of the board

The tradition of studies involving the combinatorial approach to recurring numerical sequences has accumulated a few decades of tradition, and several problems continue to attract the interest of mathematicians in several countries. This work specifically discusses the Fibonacci, Pell, and Jacobsthal sequences, focusing on Mersenne sequences. The often-used definition of board involves considering how to fill a specific regular surface -the board- with a limited quantity of regularly shaped tiles. On the other hand, an analogous problem can be generalized and exemplifies current research developments. Finally, the examples covered constitute unexpected ways of exploring visualization and other skills in mathematics teachers’ learning, consequently inspiring them for their teaching context.