Satellite-borne optical sensors provide a promising means for global air-traffic monitoring, yet their observations are often fragmented and limited in temporal resolution, making reliable trajectory forecasting highly challenging. Here we present HiFormer, a direct multi-step deep learning framework specifically designed for trajectory prediction under sparse space-based observations. The framework integrates convolutional, recurrent, and attention-based sequence modeling within a unified architecture, enabling the capture of short-term maneuvers, medium-range motion trends, and long-range dependencies in a single forward process. To address the lack of suitable datasets, we construct a large-scale synthetic benchmark of 12,000 trajectories representing four canonical motion patterns, and compile 1000 fragmented ADS-B flight segments covering diverse global routes. Extensive experiments demonstrate that HiFormer reduces multi-step prediction errors by up to 30% on synthetic data and 10% on real-world ADS-B tracks compared with representative baselines. These results establish HiFormer as a robust framework for space-based air-traffic monitoring and highlight its potential for forecasting tasks across other domains with sparse and irregular observations.

For almost a century, the decidability of the Skolem Problem - that is, the problem of finding whether a given linear recurrence sequence (LRS) has a zero term - has remained open. A breakthrough in the 1980s established that the Skolem Problem is indeed decidable for algebraic LRS of order at most 3, and real algebraic LRS of order at most 4. However, for general algebraic LRS of order 4 the question of decidability has remained open. Our main contribution in this paper is to prove decidability for this last case, i.e. we show that the Skolem Problem is decidable for all algebraic LRS of order at most 4. 11 pages. This is the TheoretiCS journal version

This paper introduces the Generalized Tribonacci Hybrid Split Quaternion (GTHSQ), a novel split quaternion with coefficients derived from generalized Tribonacci hybrid numbers. This form unifies various existing number types, such as generalized hybrid Tribonacci numbers and Tribonacci numbers, offering a fresh perspective on quaternion theory. To systematically characterize GTHSQ, we develop a comprehensive mathematical framework. This includes defining GTHSQ through related number expressions, specifying its operational rules, and proving that it retains the third-order linear recurrence relation of generalized Tribonacci sequences. We derive its Binet-type formula and generating functions while exploring its core properties. Additionally, we extend classic combinatorial identities (Vajda, Catalan, Cassini) to GTHSQ, define the GTHSQ matrix, and analyze its product with the S-matrix—a generalized third-order linear recurrence sequence representation matrix—to obtain matrix and determinant expressions for GTHSQ. These findings verify the closed-form solution of GTHSQ in terms of combinatorial identities and matrix representation. Furthermore, we discuss potential applications of GTHSQ, including advancements in quaternion algebra, support for encryption algorithms in cryptography, and simplification of spatial transformations in physics, thereby providing new tools and insights for future research in quaternion theory and interdisciplinary studies.

Prepositional phrase lexical bundles are a numeral recurrent sequence of words, differing fundamentally from other categories of lexical bundles. These particular categories of lexical bundles have frequent usages in academic English writings and play a crucial role in articulating relation between ideas and arranging discourse. This study investigates, examines, and explores the usage of the prepositional phrase within lexical bundles written by Kurdish EFL students enrolled in the Department of English at Salahaddin University-Erbil. A total of 60 student’s essays were collected, analysed and examined using a corpus-based methodological approach. The analysis revealed that Kurdish learners primarily utilized referential framing bundles (e.g., in terms of the, in the field of) to explain contexts or structure arguments, alongside and in tandem with temporal phrases (e.g., at the same time, in recent years) to link concepts sequentially. The findings indicate that learners heavily rely on prepositional phrases to establish relationships between concepts, clarify abstract concepts, organize discourse, necessitating pedagogical strategies to diversify usage while addressing cross-linguistic influences on bundle selection.

This paper investigates D-finite discrete generating series and their sections. The concept of D-finiteness is extended to multidimensional discrete generating series and its equivalence to p-recursive sequences is rigorously established. It is further shown that sections of the D-finite series preserve D-finiteness, and that their generating functions satisfy systems of linear difference equations with polynomial coefficients. In the two-dimensional case, explicit difference relations are derived that connect section values with boundary data, while in higher dimensions general constructive methods are developed for obtaining such relations, including cases with variable coefficients. Several worked examples illustrate how the theory applies to solving difference equations and analyzing multidimensional recurrent sequences. The results provide a unified framework linking generating functions and recurrence relations, with applications in combinatorics, number theory, symbolic summation, and the theory of discrete recursive filters in signal processing.

Abstract Let $\{u_n\}_n$ be a nondegenerate linear recurrence sequence of integers with Binet’s formula given by ${u_n= \sum _{i=1}^{m} P_i(n)\alpha _i^n.}$ Assume $\max _i \vert \alpha _i \vert>1$ . In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon>0$ , there is an effectively computable constant $C(\epsilon )$ such that if $ \vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$ , then $n<C(\epsilon )$ . Using results of Schmidt and Evertse, a complete noneffective (qualitative) proof of this conjecture was given by Fuchs and Heintze [‘On the growth of linear recurrences in function fields’, Bull. Aust. Math. Soc.104(1) (2021), 11–20] and, independently, by Karimov et al. [‘The power of positivity’, Proc. LICS23 (2023), 1–11]. In this paper, we give an effective upper bound for the number of solutions of the inequality $\vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$ , thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.

Abstract We consider the series $$\sum _{n=1}^{\infty } z^{n} (a_{n} + x)^{-s}$$ ∑ n = 1 ∞ z n ( a n + x ) - s where $$\{a_{n}\}$$ { a n } satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex s-plane. Thus we may associate a Lerch-type zeta function $$\varphi (z,s,x)$$ φ ( z , s , x ) to a general recurrence. This subsumes all previous results which dealt only with the ordinary zeta and Hurwitz cases and degrees 2 and 3. Our method generalizes a formula of Ramanujan for the classical Hurwitz–Riemann zeta functions. We determine the poles and residues of $$\varphi $$ φ , which turn out to be polynomials in x. In addition we study the dependence of $$\varphi (z,s,x)$$ φ ( z , s , x ) on x and z, and its properties as a function of three complex variables.

This paper presents new matrix identities involving powers and products of structured matrices whose entries are derived from real-valued sequences. These identities reveal deep algebraic relationships among matrix powers and products, leading to closed-form expressions for classical number-theoretic sequences. Applications to the Fibonacci and Lucas sequences are discussed, and a generalization involving power-sum matrices is introduced. The results highlight the effectiveness of matrix-based techniques in deriving elegant and compact representations of linear recurrence sequences.

The application of data-driven techniques in the field of electrical impedance tomography (EIT) has gained increasing attention in the past years. This is related to the possibilities of machine learning (ML) in addressing nonlinear, inverse, and even ill-posed problems, like the EIT image reconstruction. A currently used EIT application is monitoring intensive care patients. In clinical applications, the imaging reliability is crucial. This contribution demonstrates the potential of applying recurrent sequences of EIT measurements to a combination of a Long Short-Term Memory (LSTM) cells and a Variational Autoencoder (VAE) to predict the next time-instance impedance distribution. This could lead to a more accurate reconstruction or enable the detection of deviations from the established cycle before they actually appear.

In this study, we intend to bring together Padovan and Perrin number sequences, which are one of the most popular third-order recurrence sequences, and hyperbolic spinors, which are used in several disciplines from physics to mathematics, with the help of the split quaternions. This paper especially improves the relationship between hyperbolic spinors, both a physical and mathematical concept, and number theory. For this aim, we combine the hyperbolic spinors and Padovan and Perrin numbers concerning the split Padovan and Perrin quaternions, and we determine two new special recurrence sequences named Padovan and Perrin hyperbolic spinors. Then, we give Binet formulas, generating functions, exponential generating functions, Poisson generating functions, and summation formulas. Additionally, we present some matrix and determinant equations with respect to them. Besides, we construct some special equations that give relations between Padovan and Perrin hyperbolic spinors and Padovan and Perrin numbers. Further, we give a short introduction for (s, t)-Padovan and (s, t)-Perrin hyperbolic spinors in order to shed light on future studies.

In this paper, we introduce and investigate a new third order recurrence sequence so called generalized co-Narayana sequence and its two special subsequences which are related to generalized Narayana numbers and its two subsequences. There are close interrelations between recurrence equations of and roots of characteristic equations of generalized Narayana and generalized co-Narayana numbers. We present Binet's formulas, generating functions, some identities, Simson's formulas, recurrence properties, sum formulas and matrices related with these sequences.

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in Sagan and Tirrell (Adv Math 374:107387, 2020). The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the p-adic valuations of Lucas atoms for any prime p, answering to a problem left open in Sagan and Tirrell (2020), where the authors treated only some specific cases for p∈{2,3}. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two.

Lexical bundles (LBs) are recurrent continuous sequences of two or more words that frequently appear in discourse. Acquiring these bundles helps compensate for short-term memory limitations, supports efficient language processing, and promotes both comprehension and production in English. Lexical bundles can be learned through English language (EL) textbook input. Numerous researchers advocate for the inclusion and emphasis of native-like lexical bundles in EL textbooks; however, relatively few studies have evaluated lexical bundles in these textbooks. This study seeks to review the literature on lexical bundles in EL textbooks over the past two decades (2004-2024) to identify research trends and gaps. The review relied on three databases: Web of Science Core Collection, Scopus, and ERIC. After a rigorous screening process, 18 relevant studies were identified and coded based on study identification, language context, research aims, textbook types, methodologies, target lexical bundles, and main findings (cf. Appendix A). The findings recommend future research in this domain continue to employ the corpus-assisted method to investigate lexical bundles in EL textbooks and conduct a comprehensive evaluation using multiple analytical dimensions to explore three- and four-word bundles in foundational-level English for General Purposes (EGP) textbooks and English for Specific Purposes (ESP) textbooks.

Abstract For a wide class of integer linear recurrence sequences $(u(n))_{n=1}^\infty $ , we give an upper bound on the number of s-tuples $\left (n_1, \ldots , n_s\right ) \in \left ({\mathbb Z}\cap [M+1,M+ N]\right )^s$ such that the corresponding elements $u(n_1), \ldots , u(n_s)$ in the sequence are multiplicatively dependent.

С целью использования обобщенного быстрого преобразования Фурье для синхронизации многозначных (в общем случае p-ичных) линейных рекуррентных последовательностей максимального периода разработаны способы их преобразования к функциям Виленкина-Крестенсона на основе предлагаемых вариантов построения матриц-циркулянтов исходных последовательностей с использованием автоморфных мультипликативных групп расширенного поля Галуа по модулю неприводимого примитивного полинома. Данный подход позволяет не только преобразовать любой циклический сдвиг синхронизируемой последовательности к любой функции Виленкина-Крестенсона, но и в зависимости от условий преобразования указать, к какой функции этот сдвиг приведен при учете варианта ее расположения в соответствующей матрице, упорядоченной, например, по Кронекеру. Этот позволяет упростить алгоритм синхронизации многозначных псевдослучайных последовательностей при известном диапазоне их циклических сдвигов, особенно в случае больших периодов их повторения, а также снизить вычислительную сложность алгоритма их обработки при работе в усеченном базисе функций ВиленкинаКрестенсона. In order to use the generalised fast Fourier transform for the synchronisation of multivalued (in the general case p-ary) linear recurrent sequences of maximum period, methods of their transformation into Vilenkin-Krestenson functions have been developed on the basis of the proposed variants of the construction of matrix-circulants of initial sequences using automorphic multiplicative groups of the extended Galois field modulo an irreducible primitive polynomial. This approach allows not only to transform any cyclic shift of the synchronised sequence into any VilenkinKrestenson function, but also to specify, depending on the conditions of the transformation, to which function this shift is reduced, taking into account the variant of its arrangement in the corresponding matrix ordered, for example, by Kronecker. This makes it possible to simplify the algorithm of synchronisation of multivalued pseudo-random sequences with a known range of their cyclic shifts, especially in the case of large periods of their repetition, and also to reduce the computational complexity of the algorithm of their processing when working in the truncated basis of Vilenkin-Krestenson functions.

An example of the organization of intelligent systems is presented as a further generalization of multi-level network projects implementing modes, measurement, calibration, monitoring, and correction of the control signal. The structure of executive means of intelligent control based on a vector-indicator and recurrent sequences in the form of a transformed expansion into a Taylor series is proposed. It is demonstrated that the exponential representation of the Euler sine with the recurrent expansion and the Laplace transform reduce the nonlinear model to an analytical one for the hybrid stepper motor. Angular movement of the rotor, currents in the phase windings and the transfer function are presented in the image of voltage drops. Five samples of existing hybrid stepper motor models were obtained and statistically investigated in the Matlab Simulink environment. Two experimental research methods of non-contact rotation angle measurement by a mirror-reflected laser beam and a laser rigidly connecting with a rotating shaft are presented. Research of the experimental models for full-step and half-step settings of the motor driver demonstrates the need for intelligent correction of the model for an adequate response of the executive means of the intelligent systems.

Let [Formula: see text] be a prime power and [Formula: see text]. In this paper we complete the classification of good polynomials of degree [Formula: see text] that achieve the best possible asymptotics (with an explicit error term) for the number of totally split places. Moreover, for degrees up to [Formula: see text], we provide an explicit lower bound and an asymptotic estimate for the number of totally split places of [Formula: see text]. Finally, we prove the general fact that the number [Formula: see text] of [Formula: see text] for which [Formula: see text] splits obeys a linear recurring sequence. For any [Formula: see text], this allows for the computation of [Formula: see text] for large [Formula: see text] by only computing a recurrence sequence over [Formula: see text].

This study investigates the blast-induced damage distribution in inclined shafts and surrounding rock masses at the Tianchi Pumped Storage Power Station, focusing on high geostress conditions during blasting excavations. By employing the numerical simulation capabilities of LS-DYNA, the study methodically assesses the influence of geostress and blasting parameters on the resultant damage, including the progressive impact of recurrent blasting sequences. The results elucidate that judicious modification of blasting parameters can efficaciously mitigate damage to the surrounding rock of both the excavation site and adjacent inclined shafts. In high geostress environments, an escalation in geostress markedly intensifies the extent of damage incurred during subterranean blasting excavations. Furthermore, successive blasting operations incrementally extend the zone of rock damage, albeit with a diminishing rate of expansion upon each subsequent detonation. This research contributes critical insights towards augmenting the safety protocols for blasting excavations within the inclined shafts of pumped storage power stations.

Abstract We introduce the notion of a twisted rational zero of a nondegenerate linear recurrence sequence (LRS). We show that any nondegenerate LRS has only finitely many such twisted rational zeros. In the particular case of the Tribonacci sequence, we show that and are the only twisted rational zeros that are not integral zeros.

A class of shell models is proposed where the shell variables are defined on a recurrent sequence of integer wave numbers such as the Fibonacci or the Padovan series or their variations, including a sequence made of square roots of Fibonacci numbers rounded to the nearest integer. Considering the simplest model, which involves only local interactions, the interaction coefficients can be generalized in such a way that the inviscid invariants, such as energy and helicity, can be conserved even though there is no exact self-similarity. It is shown that these models basically have identical features with standard shell models and produce the same power-law spectra, similar spectral fluxes, and analogous deviation from self-similar scaling of the structure functions, implying comparable levels of turbulent intermittency. Such a formulation potentially opens up the possibility of using shell models, or their generalizations along with discretized regular grids such as those found in direct numerical simulations, as either diagnostic tools or subgrid models. It also allows us to develop models where the wave-number shells can be interpreted as sparsely decimated sets of wave numbers over an initially regular grid. In addition to conventional shell models with local interactions that result in forward cascade, a particular a helical shell model with long-range interactions is considered on a similarly recurrent sequence of wave numbers, corresponding to the Fibonacci series, and found to result in the usual inverse cascade.