Recurrence Research Articles

On the Representation of Non-Holonomic Power Series

Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like tan(z), z/(exp(z)-1), sec(z), etc. are not holonomic, therefore their power series are inaccessible by non-pattern matching implementations like the current Maple convert/FormalPowerSeries up to Maple 2021. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of tan(z) and sec(z) the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete representations when linear summations for power series coefficients of non-holonomic functions appear, might be seen as a step forward towards the representation of non-holonomic power series.
 By adapting the method of ansatz with undetermined coefficients, we build an algorithm that computes least-order quadratic differential equations with polynomial coefficients for a large class of non-holonomic functions. A differential equation resulting from this procedure is converted into a recurrence equation by applying the Cauchy product formula and rewriting powers into polynomials and derivatives into shifts. Finally, using enough initial values we are able to give normal form representations (Geddes et al. 1992) to characterize several non-holonomic power series. As a consequence of the defined normal relation, it turns out that our algorithm is able to detect identities between non-holonomic functions that were not accessible in the past. We discuss this algorithm and its implementation for Maple 2022.
 Our Maple and Maxima implementations are available under the FPS software which can be downloaded at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm.

Applicability of standard forward column/row recurrence equations for ALFs

Applicability of standard forward column/row recurrence equations for ALFs

A Note on Neo Balancing Sequence, Generalized Recursive Sequences and Pythagorean Triples

We proved theorems about creating Pythagorean Triples from generalized recursive sequences in this study. Pythagorean triples are a very old concept that has made significant progress in recent mathematical studies. Pythagorean triples can be generated using Neo balancing sequences and generalized recursive sequences, and vice-versa. For greater understanding, a general procedure is presented with appropriate illustrations.

Common values of a class of linear recurrences

Let (an),(bn) be linear recursive sequences of integers with characteristic polynomials A(X),B(X)∈Z[X] respectively. Assume that A(X) has a dominating and simple real root α, while B(X) has a pair of conjugate complex dominating and simple roots β,β̄. Assume further that α,β,α/β and β̄/β are not roots of unity and δ=log|β|/log|α|∈Q. Then there are effectively computable constants c0,c1>0 such that the inequality |an−bm|>|an|1−(c0log2n)/nholds for all n,m∈Z≥02 with max{n,m}>c1. We present c0 explicitly.

Algebro-Geometric Solutions of the Sine-Gordon Hierarchy

On the basis of two sets of Lenard recursion sequences and zero-curvature equation associated with a matrix spectral problem, we derive the entire sine-Gordon hierarchy, which is composed of all the positive and negative flows. Using the theory of hyperelliptic curves, the Abel-Jacobi coordinates are introduced, from which the corresponding positive and negative flows are linearized. The algebro-geometric solutions of the entire sine-Gordon hierarchy are constructed by using the asymptotic properties of the meromorphic function.

Virtual cyclic cellular automata, finite group actions and recursive properties

The aim of these notes is three fold. First we introduce virtual cyclic cellular automata and show that the inverse of a reversible (2R+1)-cyclic cellular automaton with periodic boundary conditions is a virtual cyclic cellular automaton. These virtual automata have two special characteristics: they have active and non active cells at specific steps of times and they reflect certain periodicity. We will relate these particularities with a finite cyclic group action on the cellular automaton, and prove that the inverse transition dipolynomial is an invariant dipolynomial under this action. Secondly, we use a recursive estimation of neighbours (REN) algorithm to produce direct examples of virtual cyclic cellular automata, which moreover generalize some of the cellular automata used in applications like collective control or traffic patterns. We also propose a new REN algorithm which allows us to reinterprete a (2R+1)-cyclic cellular automata as a recursive sequence originated from the elementary cellular automaton with base rule 150, and which motivate us to introduce a new notion of a recursive Wolfram number for a (2R+1)-cyclic cellular automaton. Finally we show that this recursive Wolfram number can be computed by the new REN algorithm applied to the base rule 150 and its complementary 105 rule.

A new form of QCD coherence for multiple soft emissions using Glauber-SCET

Amplitude-level factorization for a soft gluon emission has long been understood in terms of a product of loop-expanded soft-gluon currents and hard scattering matrix elements, both of which are infrared (IR) divergent. Thus, the amplitude for multiple soft gluon emissions, ordered in their relative softness, can be written as a product of IR divergent soft gluon currents and the matrix elements. In a more recent work, Angeles-Martinez, Forshaw and Seymour [1] (AMFS) showed that the result for this amplitude can in fact be re-expressed in an ordered evolution approach, involving IR finite one-loop insertions where the virtual loop momentum is constrained in a highly non-trivial way by the kT of the adjacent real emissions. The result thus exhibits a novel amplitude level QCD coherence where the IR divergences originating only from the very last, softest, gluon emission remain, and the rest cancel. The proof of the AMFS result at one-loop in QCD, however, involves many diagrams, and only after carefully grouping and summing over all the diagrams does the correct ordering variable emerge, making the higher order extension a challenging task. Moreover, the compact, Markovian nature of the final AMFS result is suggestive of a deeper underlying physics that is obscured in the derivation using traditional diagrammatic QCD. By considering a (recursive) sequence of effective field theories (EFTs) with Glauber-SCET operators, we present an elegant derivation of this result involving only a handful of diagrams. The SCET derivation offers clean physical insights, and makes a higher order extension of the AMFS result tractable. We also show that the grouping of QCD graphs necessary to derive the AMFS result in full theory is already implicit in the Feynman rules of Glauber-SCET operators such that the same result can alternatively be derived with significantly less effort in a single EFT with multiple ordered soft gluon emissions.

FPS in action

Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is "simple" enough. Simplicity is related to the compactness of the formula due to the presence of algebraic numbers: "the smaller, the simpler". This poster showcases the capacity of recent updates on the Formal Power Series (FPS) algorithm, implemented in Maxima and Maple (convert/FormalPowerSeries), to find simple formulas for sequences like those from https://oeis.org/A307717, https://oeis.org/A226782, or https://oeis.org/A226784 by computing power series representations of their correctly guessed generating functions. We designed the algorithm for the more general context of univariate P -recursive sequences. Our implementations are available at http://www.mathematik.uni-kassel.de/~bteguia/FPS_webpage/FPS.htm

An Implementation of the Generalized Differential Transform Scheme for Simulating Impulsive Fractional Differential Equations

In this research study, the generalized differential transform scheme has been applied to simulate impulsive differential equations with the noninteger order. One specific tool of the implemented scheme is that it converts the problems into a recurrence equation that finally leads easily to the solution of the considered problem. The validity and reliability of this method have successfully been accomplished by applying it to simulate the solution of some equations. It is shown that the considered method is very suitable and efficient for solving classes of fractional-order initial value problems for impulsive differential equations and might find wide applications.

Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory

By embedding a Markov-modulated random recurrence equation in continuous time, we derive the Markov-modulated generalized Ornstein-Uhlenbeck process. This process turns out to be the unique solution of a stochastic differential equation driven by a bivariate Markov-additive process. We present this stochastic differential equation as well as its solution explicitely in terms of the driving Markov-additive process. Moreover, we give necessary and sufficient conditions for strict stationarity of the Markov-modulated generalized Ornstein-Uhlenbeck process, and prove that its stationary distribution is given by the distribution of a certain exponential functional of Markov-additive processes. Finally, we propose a Markov-modulated risk model with investment that generalizes Paulsen’s risk process to a Markov-switching environment, and derive a formula for the ruin probability in this risk model.

Tails of bivariate stochastic recurrence equation with triangular matrices

We study bivariate stochastic recurrence equations with triangular matrix coefficients and we characterize the tail behavior of their stationary solutions W=(W1,W2). Recently it has been observed that W1,W2 may exhibit regularly varying tails with different indices, which is in contrast to well-known Kesten-type results. However, only partial results have been derived. Under typical “Kesten–Goldie” and “Grey” conditions, we completely characterize tail behavior of W1,W2. The tail asymptotics we obtain has not been observed in previous settings of stochastic recurrence equations.

Analysis of the model-based corrector approach for explicit cosimulation

In many industries, the focus of testing is currently shifting away from classical hardware tests to the virtual verification and validation of products. To this end, cosimulation has become a common tool for the simulation and analysis of complex systems that span multiple engineering domains and usually involve multiple, heterogeneous and application-specific simulation environments. In particular, the so-called explicit cosimulation allows a widespread application since it has minimal requirements regarding the capabilities of the tool interfaces. However, explicit cosimulation also poses a numerical challenge, especially when the system includes stiff coupling loops. The model-based corrector approach presented in Haid et al. (The 5th Joint International Conference on Multibody System Dynamics, 2018) provides a method for the efficient cosimulation of such systems. In this article, this model-based corrector approach is extended to additional extrapolation methods. By modeling the cosimulation process through a linear recurrence equation and applying it to the two-mass oscillator test model, the influence of model-based correction on the underlying extrapolation methods in terms of stability, accuracy, and error convergence is analyzed. It is shown that adding model-based correction can significantly improve the overall cosimulation, allowing >10 times larger macrostep sizes or reducing the cosimulation error by a factor of 10 or more in some cases.

Symbolic Conversion of Holonomic Functions to Hypergeometric Type Power Series

A term $a_n$ is $m$-fold hypergeometric, for a given positive integer $m$, if the ratio $a_{n+m}/a_n$ is a rational function over a field $K$ of characteristic zero. We establish the structure of holonomic recurrence equation, i.e. linear and homogeneous recurrence equations having polynomial coefficients, that have $m$-fold hypergeometric term solutions over $K$, for any positive integer $m$. Consequently, we describe an algorithm, say $mfoldHyper$, that extends van Hoeij's algorithm (1998) which computes a basis of the subspace of hypergeometric $(m=1)$ term solutions of holonomic recurrence equations to the more general case of $m$-fold hypergeometric terms. We generalize the concept of hypergeometric type power series introduced by Koepf (1992), by considering linear combinations of Laurent-Puiseux series whose coefficients are $m$-fold hypergeometric terms. Thus thanks to $mfoldHyper$, we deduce a complete procedure to compute these power series; indeed, it turns out that every linear combination of power series with $m$-fold hypergeometric term coefficients, for finitely many values of $m$, is detected. On the other hand, we investigate an algorithm to represent power series of non-holonomic functions. The algorithm follows the same steps of Koepf's algorithm, but instead of seeking holonomic differential equations, quadratic differential equations are computed and the Cauchy product rule is used to deduce recurrence equations for the power series coefficients. This algorithm defines a normal function that yields together with enough initial values normal forms for many power series of non-holonomic functions. Therefore, non-trivial identities are automatically proved using this approach. This paper is accompanied by implementations in the Computer Algebra Systems (CAS) Maxima 5.44.0 and Maple 2019.

Recursive sequences attached to modular representations of finite groups

The core of a finite-dimensional modular representation M of a finite group G is its largest non-projective summand. We prove that the dimensions of the cores of M⊗n have algebraic Hilbert series when M is Omega-algebraic, in the sense that the non-projective summands of M⊗n fall into finitely many orbits under the action of the syzygy operator Ω. Similarly, we prove that these dimension sequences are eventually linearly recursive when M is what we term Ω+-algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences.

Algebra-Based Reasoning for Loop Synthesis

Provably correct software is one of the key challenges of our software-driven society. Program synthesis—the task of constructing a program satisfying a given specification—is one strategy for achieving this. The result of this task is then a program that is correct by design. As in the domain of program verification, handling loops is one of the main ingredients to a successful synthesis procedure. We present an algorithm for synthesizing loops satisfying a given polynomial loop invariant. The class of loops we are considering can be modeled by a system of algebraic recurrence equations with constant coefficients, thus encoding program loops with affine operations among program variables. We turn the task of loop synthesis into a polynomial constraint problem by precisely characterizing the set of all loops satisfying the given invariant. We prove soundness of our approach, as well as its completeness with respect to an a priori fixed upper bound on the number of program variables. Our work has applications toward synthesizing loops satisfying a given polynomial loop invariant—program verification—as well as generating number sequences from algebraic relations. To understand viability of the methodology and heuristics for synthesizing loops, we implement and evaluate the method using the Absynth tool.

The interchange in art-psychotherapy

This paper addresses theories pertaining to what could be said of the state of mind of the creator in the art-making phase of art therapy, by reference to the interchange with the looking/talking phase. I pursue this exploration of the interchange within the concept of the triangular relationship: creator-artwork-viewer, as well as the three ‘waves’ of art therapy theory and practice. I conclude by making use of mathematical abstraction to describe the recursive sequence of art-making and looking/talking about the art, to pose questions about what might be occurring during these processes and which it is hoped could serve as a framework for further investigations from various theoretical standpoints.

The Behavior and Structures of Solution of Fifth-Order Rational Recursive Sequence

In this work, we aim to study some qualitative properties of higher order nonlinear difference equations. Specifically, we investigate local as well as global stability and boundedness of solutions of this equation. In addition, we will provide solutions to a number of special cases of the studied equation. Also, we present many numerical examples that support the results obtained. The importance of the results lies in completing the results in the literature, which aims to develop the theoretical side of the qualitative theory of difference equations.

Generalization of a DC–DC Modular Converter Topology for Fuel Cell Applications

This article presents a new modular dc–dc converter topology. As a modular topology, the converter can be used for several energy sources. Each converter module is based on a three-level dc–dc converter, which contains two output capacitors. The first module shares the second capacitor in common with the first capacitor of the next module. In this way, the N-module can be utilized for an N-energy source. The output capacitor voltages are balanced due to the proposed topology and the controller. The controller is based on the indirect sliding mode controller, where the input current of each energy source and the capacitor voltage balancing are taken into account. One of the aims of this article was to obtain the general system relationships for the N-module system. The process of obtaining recurrence equations for the duty cycles of the system is provided for the desired number of modules. Simulation and experimental results are provided to validate the topology.

Modified recurrent equation-based cubic spline interpolation for missing data recovery in phasor measurement unit (PMU)

Background: Smart grid systems require high-quality phasor measurement unit (PMU) data for proper operation, control, and decision-making. Missing PMU data may lead to improper actions or even blackouts. While the conventional cubic interpolation methods based on the solution of a set of linear equations to solve for the cubic spline coefficients have been applied by many researchers for interpolation of missing data, the computational complexity increases non-linearly with increasing data size. Methods: In this work, a modified recurrent equation-based cubic spline interpolation procedure for recovering missing PMU data is proposed. The recurrent equation-based method makes the computations of spline constants simpler. Using PMU data from the State Load Despatch Center (SLDC) in Madhya Pradesh, India, a comparison of the root mean square error (RMSE) values and time of calculation (ToC) is calculated for both methods. Results: The modified recurrent relation method could retrieve missing values 10 times faster when compared to the conventional cubic interpolation method based on the solution of a set of linear equations. The RMSE values have shown the proposed method is effective even for special cases of missing values (edges, continuous missing values). Conclusions: The proposed method can retrieve any number of missing values at any location using observed data with a minimal number of calculations.

Ultimate periodicity problem for linear numeration systems

We address the following decision problem. Given a numeration system U and a U-recognizable set [Formula: see text], i.e. the set of its greedy U-representations is recognized by a finite automaton, decide whether or not X is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linear recurrence sequences. Based on arithmetical considerations about the recurrence equation and on p-adic methods, the DFA given as input provides a bound on the admissible periods to test.