Abstract Classical orthogonal polynomials are solutions of a second order differential, difference, or q -difference equation for continuous, discrete, or q -discrete variables, respectively. Furthermore, all such systems satisfy a three-term recurrence equation of the form: $$\begin{aligned} p_{n+1}(x) = (A_n x + B_n)p_n(x) - C_n p_{n-1}(x), \end{aligned}$$ p n + 1 ( x ) = ( A n x + B n ) p n ( x ) - C n p n - 1 ( x ) , for $$n\ge 0$$ n ≥ 0 with $$ p_{-1}=0,~p_0=1$$ p - 1 = 0 , p 0 = 1 . Given a holonomic three-term recurrence equation, we implement in and an algorithm which detects its classical orthogonal polynomial solutions for the continuous, discrete, and q -discrete variables when they exist. With our implementations, the results obtained using the Maple implementations by Koepf and Schmersau (Appl Math Comput 128:303–327, 2002) and Koorwinder and Swarttouw (Priv Commun, 1998) are easily recovered. In addition, we obtain new relations that extend beyond those previously established in the literature.

By means of the zero-curvature equation and two sets of Lenard recursion sequences, we construct a nonisospectral generalized 3 × 3 Ablowitz–Kaup–Newell–Segur (AKNS) integrable hierarchy. The general inhomogeneous Dirac–Manakov equations with temporally and spatially modulated coefficients are derived. Under some special reductions, the two-component nonlinear Schrödinger (NLS) or Manakov equations with helicoidal spin-orbit (SO) coupling, Rabi coupling, and mixed helicoidal SO and Rabi couplings are obtained. In particular, we propose several novel inhomogeneous two-component NLS equations which have variable helicoidal SO and Rabi couplings as well as external potentials. The N -fold Darboux-dressing transformation related to the nonisospectral generalized 3 × 3 AKNS matrix eigenvalue problem is established, and further the N -soliton solution represented in a compact determinant form is given. The classification of solitons for a two-component NLS equations with temporally modulated mixed helicoidal SO and Rabi couplings is discussed in detail. Specifically, three types of solitons including beating, bell-shaped and multi-peak solitons as well as their parabolic, periodic, exponential and kinked deformations, and some unconventional nonlinear superpositions are presented.

This paper investigates the qualitative behavior and explicit solutions of a system of rational difference equations of the form with arbitrary initial conditions u−2, u−1, u0, v−2, v−1, v0 ∈ \(\mathbb{R}\). The constants ai ∈ {−1, 1} for i = 1, 2, 3, 4. Four special cases of the system are examined based on the sign combinations in the denominators. For each case, we prove that all solutions are periodic with period twelve and derive explicit closed-form formulas for the solutions in terms of the initial values. The proofs are established using mathematical induction. Numerical examples are provided to verify and illustrate the theoretical results, demonstrating the periodic nature of the solutions. This work contributes to the understanding of higher-order rational difference equations and their periodic behavior.

This study concerns some properties of the generalized nonhomogeneous Horadam model and its application to four types of known sequences of numbers. Regarding the four sequences of this model, we provide results on combinatorial and linear expressions, and also on the analytical representation of Binet. Our approach is based on results we have developed regarding the general setting of nonhomogeneous linear recursive sequences, especially the fundamental sequence related to their homogeneous part.

A comprehensive soil pollution monitoring system based on convolutional recursive sequence network and terahertz spectroscopy.

This study enhances wind speed forecasting by implementing the second-order Generalized Least Deviation Method (GLDM), focusing on wind turbines in Turkey. The research aims to improve predictive accuracy and operational efficiency in renewable energy systems through advanced mathematical modeling in meteorology. The GLDM, utilizing a quasilinear recurrence equation, addresses the inherent non-linearity and variability of wind speed data. By applying the method to extensive SCADA data, this study minimizes residuals in nonlinear big data environments, integrating both linear and nonlinear components to refine predictions. A critical aspect of this research is the comparison between the second-order GLDM and traditional forecasting models, including statistical methods and machine learning approaches. The results demonstrate the superior performance of GLDM, as indicated by lower prediction errors and greater accuracy across key metrics. The study also underscores the importance of GLDM coefficients, ????????????????, in improving predictive capabilities. The findings advocate for the adoption of GLDM in wind speed forecasting, highlighting its potential to significantly enhance wind energy management through increased accuracy. This study also sets a precedent for broader applications of advanced mathematical models in environmental science, illustrating the effectiveness of GLDM in optimizing renewable energy resources.

Abstract Stanley Milgram's Obedience to authority experiments are widely known to have demonstrated the human proclivity to follow violent orders coming from a legitimate authority. The present paper examines the extent to which Milgram's participants, during the obedient phase of their sessions, did in fact follow the full set of instructions that was given by the experimenter: a recursive sequence of procedures in a fake laboratory experiment on “Memory and learning.” Results show that while participants did not fail to administer electric shocks, (1) no entirely “fully obedient” participant fully obeyed the procedures Milgram's experimenter instructed them to do; (2) violations of the procedures occurred on average 48.4% of the time in “fully obedient” sessions; and (3) violations occurred significantly more frequently in “fully obedient” than in the obedient phase of “disobedient” sessions. In sum, in sessions traditionally regarded as fully obedient legitimate violence got to a substantial extent transformed into illegitimate violence. The paper concludes with the possibility that, the context of illegitimate violence that has been constructed implicitly by the experimenter constituted a coercive environment and thus made a causal contribution to obedient participants' inability to resist the experimenter to the point of disobedience.

Industrial intelligent diagnosis is a core task for ensuring the reliability and safety of mechanical equipment operation. However, traditional Transformer models have problems such as weak ability to model local time-frequency features, scattered focus on key features, and insufficient frequency sensitivity when dealing with non-stationary industrial signals, which pose many challenges in fault detection and classification. In addition, existing methods that combine discrete wavelet transform (DWT) with deep learning models often adopt a "preprocessing-classification" sequential approach. This involves first performing wavelet decomposition on the original signal and then inputting the results into a classification network for training. Such methods exhibit a disconnection between feature extraction and classification, making it challenging to achieve end-to-end optimization. This paper introduces a novel WET-BiLSTM model, whose primary innovation lies in designing DWT as a differentiable embedded module that is seamlessly integrated with Transformer and BiLSTM architectures. This design facilitates the construction of a truly end-to-end intelligent diagnostic framework.The design not only leverages self-attention for global dependency modeling but also integrates the time-frequency multi-scale decomposition capabilities of DWT with the local smoothing and recursive sequence modeling abilities of BiLSTM. This combination enables a more effective capture of critical fault information. The entire process facilitates an end-to-end mapping from the original vibration signal to the fault type, thereby eliminating the disconnect between preprocessing and model training commonly found in traditional methods. To verify the effectiveness of the proposed method, experiments are conducted on two typical industrial fault datasets, namely rolling bearing and reciprocating compressors. The results showed that WET-BiLSTM outperformed other models in terms of fault identification accuracy, stability and noise resistance. Further attention visualization analysis shows that the introduction of the DWT module enables the model to allocate more concentrated and discriminative attention weights at critical timing positions, improving the interpretability and fault awareness of the model. The study provides an efficient, reliable and physically explainable solution for the intelligent diagnosis of complex industrial equipment.

ABSTRACT Video anomaly detection (VAD) remains a challenging task due to the computational burden of modelling long video sequences and the difficulty in capturing subtle anomalies in small spatial regions. In order to address these issues, a novel framework is proposed, namely HSM‐MAE, which integrates scalable spatio‐temporal modelling with discrepancy‐aware memory mechanisms. In the field of temporal modelling, HSM‐MAE employs a state‐space model‐based encoder, facilitating recursive sequence representation with linear computational complexity with respect to sequence length. This property improves scalability when longer temporal contexts are required, rather than guaranteeing an unconditional end‐to‐end speedup under all settings. In order to preserve local spatial continuity, a Hilbert scanning strategy is introduced, which serialises spatial regions within video frames while better preserving their neighbourhood structure, thereby enhancing the model's sensitivity to fine‐grained anomalies. Furthermore, a state memory module is employed to retain latent states across frames, and a state discrepancy metric is designed to measure temporal variation amplitudes, guiding the model to focus on abnormal transitions. A student–teacher reconstruction architecture has been shown to amplify prediction errors in anomalous regions via a distillation strategy, while a random masking mechanism has been demonstrated to strengthen robustness against occlusion and background noise. Extensive experimentation on standard benchmarks demonstrates that HSM‐MAE achieves competitive real‐time throughput and a favourable accuracy–efficiency trade‐off under our evaluation setting, while delivering consistent improvements in detection performance.

We derive a Binet-type formula for operator-valued sequences satisfying linear recurrence relations, extending the classical scalar case to the setting of bounded operators on Hilbert spaces. In this framework, we analyze the operator moment problem as an application, establishing new connections between recursive operator sequences and moment sequences.

We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for \(T_n\). As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

This article revisits the Mankiw et al.’s specification of the Solow growth model by addressing the endogeneity concern raised by Acemoglu. To do so, we re-estimate the Mankiw et al.’s equation of long-run cross-country income difference via generalized method of moments (GMM) by using the future-time reference of language as an instrument for physical capital and the Protestant missionary activities in 1923 and primary school enrolment in 1900 as instruments for human capital for 1970–2019. Additionally, the GMM estimates allow us to analyse the role of fundamental causes of economic prosperity, determined by cultural and institutional factors, in explaining cross-country income differences. Although the numerical similarity of the ordinary least squares (OLS) and GMM estimates shows that the benchmark Mankiw et al.’s results are robust to endogeneity criticisms, their interpretations differ. The OLS estimates primarily capture the impact of proximate causes on cross-country income differences, while the GMM estimates reveal the role of capital accumulation, inferred by institutions and cultural factors, in explaining long-term cross-country income differences. JEL: O24, C26, E24

ABSTRACT In this work, we present a recursive solution of the discrete phase retrieval problem for Fourier measurements. We show that a solution of this problem may be obtained, if we are able to compute a rank one pre‐image of a certain low‐rank Hermitian matrix, under a known linear operator. Based on this observation, we construct a nonlinear recursive sequence of low‐rank Hermitian matrices, and we prove that under certain conditions, there exists a subsequence tending to a solution.

BackgroundFinger spasticity, which worsens hand function, is common in stroke patients. Botulinum toxin A (BoNT-A) treatment is effective in relieving spasticity but has little effect on arm-hand capacity. The adding of a rehabilitation program following BoNT-A injection has been suggested to enhance spasticity treatment outcomes. This pilot study aimed to compare the effects of BoNT-A combing with robot-assisted bimanual therapy (RBT) versus BoNT-A combing with RBT integrating mirror therapy (RBMT) in patients with chronic stroke and spastic fingers.MethodsPatients with chronic stroke and finger spasticity were recruited. They were randomly allocated to receive 24 training sessions of RBT or RBMT. Assessments were performed before BoNT-A injection, after 24 training sessions, and 3 months follow-up. The outcomes included Fugl-Meyer assessment for the upper extremity (FMA-UE-total, proximal and distal component scores), modified Ashworth scale (MAS), action research arm test (ARAT), box and block test (BBT), and motor activity log (MAL). We applied Friedman’s test to analysis intragroup, generalized estimating equations for intergroup differences of the treatment effects.ResultsThirty-one patients were randomly allocated to the RBMT (n = 16) and RBT (n = 15) groups. After BoNT-A injection and training, both groups showed statistically significant improvements in FMA-UE total and FMA-proximal, MAS, and ARAT scores. Only the RBMT group showed significant improvements in the FMA-UE distal, and MAL scores. Neither group showed a significant improvement in the BBT. In the RBMT group the injection and post-training gains were maintained at the 3-month follow-up, except for MAS. The RBT group did not show a maintenance effect, except for the amount of use the affected UE, which showed significant improvement from baseline to follow-up. Comparing the treating effects between the two groups, there were significant differences in the FMA distal scores at post-training, favoring RBMT.ConclusionsOur pilot data demonstrated that an adjunct to BoNT-A for post-stroke spastic fingers, RBMT may provide more benefit in distal UE function than did RBT. However, a larger sample size and adding a proper control group are required to validate our findings in future studies.Trial registration This study was registered at ClinicalTrials.gov (ID no. NCT04826900, registered retrospectively on 03/28/2021).

UDC 517.5 We study the relationship between certain Fibonacci polynomials and the Schwarz lemma within the framework of the complex analysis and control engineering. The Fibonacci polynomials are traditionally associated with combinatorial mathematics and recursive sequences. At the same time, they also play a significant role in the power series expansions of analytic functions. We present a version of the Schwarz lemma based on the first Fibonacci polynomial. By considering the boundary version of the Schwarz lemma, we deduce new inequalities that provide deeper insights into the relationship between these two mathematical concepts. Beyond their theoretical significance, we also present practical implications of our results. Specifically, we show that the deduced results can be applied to get marginally stable discrete-time transfer functions in digital control systems.

This study aims to unravel the mathematical structure of the Flower of Life (FOL) by deriving closed-form algebraic expressions that govern its recursive progression. The research investigates how circles, hexagons, and six-petaled motifs evolve across successive growth classes and explores their connections with harmonic and recursive principles in mathematics, nature, and art. The study employed algebraic modeling, finite difference analysis, and computational validation using tabulated datasets. Patterns were derived by observing recursive growth sequences and confirmed through quadratic formulations. Comparative analyses of polygonal counts and areas were conducted to highlight harmonic progressions and proportional scaling across classes of the FOL. Results revealed that both circles and hexagons follow the same quadratic expression, , demonstrating their structural equivalence. Six-petaled motifs, however, showed an accelerated progression due to recursive intersections, reflecting a higher-order harmonic amplification. The computation of polygonal areas confirmed quadratic scaling, consistent with recursive growth laws found in natural and cultural systems. These findings validate the FOL as both a mathematically rigorous construct and a universal model of recursive growth. The study concludes that the FOL embodies algebraic harmony and recursive progression, with its structural relationships governed by quadratic formulations. This positions the FOL as a nexus of mathematics, biology, art, and cultural symbolism, bridging abstract geometry with observable phenomena in nature and design. By formalizing the recursive principles of the FOL, the study provides a mathematical framework applicable to computational modeling, educational pedagogy, and design innovation. Its insights may enhance interdisciplinary learning in STEM and STEAM education, support algorithmic design in architecture and art, and reinforce the cultural significance of sacred geometry through a rigorous mathematical lens.

The manuscript deals with solving a mathematical model of criminal activities (MCA model), in order to facilitate the understanding of the dynamics of crime and implement certain measures in its prevention and deterrence. For the MCA model, consists of two coupled ordinary differential equations of the diffusion type, the stability and existence of a unique Cauchy solution are first examined. Thereafter, coupled equations of the MCA model are transformed into a dimensionless system on the unit interval and solved using the homotopy perturbation method (HPM). In this way, recursive sequences of approximate HPM solutions are obtained and, under certain conditions, their convergence has been proven. In the following, using different initial conditions, several numerical simulations of the proposed HPM technique in solving MCA model were conducted, as well as a practical application of the proposed procedure in modeling real-world crime rate data.

We introduce REPTILE, a compiler that performs tiling optimizations for programs expressed as mathematical recurrence equations. REPTILE recursively decomposes a recurrence program into a set of unique tiles and then simplifies each into a different set of recurrences. Given declarative user specifications of recurrence equations, optimizations, and optional mappings of recurrence subexpressions to external libraries calls, REPTILE generates C code that composes compiler-generated loops with calls to external hand-optimized libraries. We show that for direct linear solvers expressible as recurrence equations, the generated C code matches and often exceed the performance of standard hand-optimized libraries. We evaluate REPTILE's generated C code against hand-optimized implementations of linear solvers in Intel MKL, as well as two non-solver recurrences from bioinformatics: Needleman-Wunsch and Smith-Waterman. When the user provides good tiling specifications, REPTILE achieves parity with MKL, achieving between 0.79--1.27x speedup for the LU decomposition, 0.97--1.21x speedup for the Cholesky decomposition, 1.61x--2.72x for lower triangular matrix inversion, 1.01--1.14x speedup for triangular solve with multiple right-hand sides, and 1.14--1.73x speedup over handwritten implementations of the bioinformatics recurrences.

ABSTRACTWe consider a borderline case: The central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the i.i.d. case, a well‐known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index . In the dependent case, we assume the stronger condition of sequential regular variation of the time series with tail index . We assume that a sample of size from this time series can be split into blocks of size such that as and that the block sums are asymptotically independent. Then we apply classical central limit theory for row‐wise i.i.d. triangular arrays. The necessary and sufficient conditions for such independent block sums will be verified by using large deviation results for the time series. We derive the central limit theorem for ‐dependent sequences, linear processes, stochastic volatility processes and solutions to affine stochastic recurrence equations whose marginal distributions have infinite variance and are regularly varying with tail index .

Fractional-order differential equations are fundamental in diverse scientific and engineering fields, including population dynamics, optimal control, and physics. This paper presents a non-polynomial spline method for their numerical solution, establishing a linear system of algebraic equations in the representation of three-term recurrence equations, which is solved using an elimination algorithm. The maximum error estimations and the order of convergence for each example demonstrate the success and core contribution of the method and its performance is demonstrated through numerical and graphical examples. There is a thorough explanation of a mathematical procedure provided, as well as graphical and numerical examples for solving several examples. Results confirm that the proposed approach achieves superior accuracy and reliability compared to existing techniques.