Recursive Polynomial Research Articles

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

Adaptive model predictive control scheme for partially internal thermally coupled air separation column

Partially Internal Thermally Coupled Air Separation Column (P-ITCASC) is a highly energy-efficient and cost-effective technology. However, its complex dynamics resulting from thermal coupling pose a challenge to the operating stability of this technology. This article, therefore, proposes an adaptive model predictive control (AMPC) scheme for the P-ITCASC process. The controller incorporates an auto-regressive and exogenous inputs (ARX) model, a linear time-varying Kalman filter, and a recursive polynomial model estimator (RPME) algorithm. Within RPME, ARX polynomial models are identified and utilized to estimate the time-varying parameters and update the prediction model of the process. The process states are observed through the Kalman filter, and the constrained receding horizon optimization problem is solved using quadratic programming. This control scheme ensures the closed-loop system's feasibility and stability in the presence of output/input constraints. An adaptive generic model control, model predictive control, and adaptive internal model control schemes were also designed for benchmarking study. Numerical simulations show that AMPC is more efficient in handling nonlinear dynamics and maintaining product concentration to desired set points compared to other control schemes.

On the Connection Between Irrationality Measures and Polynomial Continued Fractions

AbstractLinear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence $${F}_{n}={F}_{n-1}+{F}_{n-2}$$ F n = F n - 1 + F n - 2 , have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $$\zeta \left(3\right)$$ ζ 3 , which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as $$\pi$$ π and $$e$$ e . More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $$\pi$$ π , $$e$$ e , $$\zeta \left(3\right)$$ ζ 3 , and the Catalan constant. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $$\zeta \left(5\right)$$ ζ 5 ).

Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursions

Via Andersen-Borot-Orantin's geometric recursion, a twist of the topological recursion was proposed, and a recursion for the Masur-Veech polynomials was uncovered. The purpose of this article is to explore generalizations of Mirzakhani's recursion based on physical two-dimensional gravity models related to the Jackiw-Teitelboim gravity and to provide an introduction to various realizations of topological recursion. For generalized Mirzakhani's recursions involving a Masur-Veech type twist, we derive Virasoro constraints and cut-and-join equations, and also show some computations of generalized volumes for the physical two-dimensional gravity models.

The transcendence of growth constants associated with polynomial recursions

Let [Formula: see text], [Formula: see text], be a polynomial of degree [Formula: see text]. Let [Formula: see text] be a sequence of integers satisfying [Formula: see text] Set [Formula: see text]. Then, under the assumption [Formula: see text], in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either [Formula: see text] is transcendental or [Formula: see text] can be an integer or a quadratic Pisot unit with [Formula: see text] being its conjugate over [Formula: see text]. In this paper, we study the nature of such [Formula: see text] without the assumption that [Formula: see text] is in [Formula: see text], and we prove that either the number [Formula: see text] is transcendental, or [Formula: see text] is a Pisot number with [Formula: see text] being the order of the torsion subgroup of the Galois closure of the number field [Formula: see text]. Other results presented in this paper investigate the solutions of the inequality [Formula: see text] in [Formula: see text], considering whether [Formula: see text] is rational or irrational. Here, [Formula: see text] represents a number field, and [Formula: see text]. The notation [Formula: see text] denotes the distance between [Formula: see text] and its nearest integer in [Formula: see text].

Real stable polynomials and the alternatingly increasing property

In this paper, we first present a criterion for the real stability of recursive polynomials by applying the characterization of Borcea and Brändén concerning linear operators preserving stability. As applications, we derive some real stability results occurred in the literature in a unified manner. Moreover, we prove the real-rootedness of peak polynomials in bicolored dyck paths and ternary polynomials.Second, based on the h-polynomials from combinatorial geometry, we use real stability to get two criteria for interlacing property of zeros between a polynomial and its reciprocal polynomial, which in particular imply the alternatingly increasing property of the original polynomial or its reciprocal polynomial. These criteria extend a result of Brändén and Solus and unify such properties for many combinatorial polynomials, including ascent polynomials for k-ary words, descent polynomials on signed Stirling permutations and colored permutations, q-analog of descent polynomials on colored permutations and descent polynomials on color-signed permutations. Furthermore, we also obtain a recurrence relation and zero-interlacing property of q-analog of descent polynomials on colored permutations that extend some results of Brändén and Brenti. In addition, as an application of weak Hurwitz stability, we get the alternatingly increasing property and zero-interlacing property for two kinds of peak polynomials on the dual set of Stirling permutations.

A recursive polynomial grey prediction model with adaptive structure and its application

As a sparse data analysis algorithm, ensuring a reasonable model structure is an important challenge for grey models to identify the control mechanism of the uncertain system from observational data. To improve the intelligence and adaptability of the model, this study presents a synchronized optimization strategy for data prioritization and model structure for discrete polynomial grey prediction model. The proposed polynomial grey model contains two hyper-parameters: memory factor parameter and structural parameter. The memory factor is introduced into the discrete model to reconstruct the objective function of structural parameter optimization, thereby avoiding the problem of information superposition. The structural parameter is used to enhance the adaptability of grey prediction model in uncertain data analysis tasks. By employing a recursive estimation approach, an adaptive strategy for estimating model hyper-parameters is proposed, which focuses on minimizing prediction errors within the in-sample data. Additionally, a comparison is made between the proposed improved polynomial grey model and existing polynomial grey models in terms of data information mining, estimation stability, and robustness against measurement noise. The proposed model is applied to the practical engineering application of wear prediction, further validating the effectiveness of the proposed approach in non-equidistant time series prediction tasks.

The Periodic Integer Orbits of Polynomial Recursions With Integer Coefficients

We show that polynomial recursions $x_{n+1}=x_{n}^{m}-k$ where $k,m$ are integers and $m$ is positive have no nontrivial periodic integral orbits for $m\geq3$. If $m=2$ then the recursion has integral two-cycles for infinitely many values of $k$ but no higher period orbits. We also show that these statements are true for all quadratic recursions.

Using easy coefficients conjecture for rotation symmetric Boolean functions

A Boolean function in n variables is rotation symmetric (RS) if it is invariant under powers of ρ(x1,…,xn)=(x2,…,xn,x1). An RS function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ to a single monomial. Completing earlier research on special cases, the author showed in 2018 that for any RS function fn in n variables, the sequence of Hamming weights wt(fn) for all values of n satisfies a linear recurrence relation. It was also proved that the associated recursion polynomial could be explicitly calculated as the minimal polynomial of a rules matrix and an algorithm for computing the rules matrix was explained. Examples showed that the usual formula which gives the values of wt(fn)−2n−1 as a linear combination of powers of the irrational roots of the minimal polynomial has simple coefficients which are all 1/2 if the multiset made up of the roots of the characteristic polynomial of the rules matrix is used instead, no matter what the degree of the Boolean function is. The conjecture that this is always true is called the Easy Coefficients Conjecture (ECC). The special case of MRS quadratics was proved in an earlier paper. The present paper gives some ECC applications valid for any function for which the ECC is true, even though there is a proof only for the quadratic MRS case so far.

Detection of Data Integrity Attack Using Model and Data-Driven-Based Approach in CPPS

The cyber-physical power system (CPPS) is a modern infrastructure utilising information and communication technology that has become more vulnerable to cyberattacks in recent years. The attack magnitude injected by the adversary is stealthier and it cannot be detected using conventional bad data detection techniques. Protecting sensitive data from data integrity attacks (DIA) is essential for ensuring system security and reliability. A tragic event will occur if the attack goes unreported. Therefore, DIA detection is highly vital for the operator in the control centre to make important decisions. This paper addresses the attack impact on WAC applications and attack detection using the model-based method and data-driven-based methods. On the basis of the validation of performance indicators, various detection approaches are simulated and compared to determine the best detection strategy. Simulation results show that in the model-based anomaly detection method, the recursive polynomial model estimator (RPME) has better detection performance than the recursive least square estimator (RLSE). The convolutional neural network- (CNN-) based data-driven anomaly detection technique outperforms other machine learning (ML) techniques such as support vector machine (SVM), K-nearest neighbour (KNN), and random forest (RF). On the WSCC 3 machine 9-bus system, the efficacy of the suggested methods is evaluated.

Some results related to Hurwitz stability of combinatorial polynomials

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to observe some results and applications for the Hurwitz stability of polynomials in combinatorics and study other related problems.We first present a criterion for the Hurwitz stability of the Turán expressions of recursive polynomials. In particular, it implies the q-log-convexity or q-log-concavity of the original polynomials. We also give a criterion for the Hurwitz stability of recursive polynomials and prove that the Hurwitz stability of any palindromic polynomial implies its semi-γ-positivity, which illustrates that the original polynomial with odd degree is unimodal. In particular, we get that the semi-γ-positivity of polynomials implies their parity-unimodality and the Hurwitz stability of polynomials implies their parity-log-concavity. Those results generalize the connections between real-rootedness, γ-positivity, log-concavity and unimodality to Hurwitz stability, semi-γ-positivity, parity-log-concavity and parity-unimodality (unimodality). As applications of these criteria, we derive some Hurwitz stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating run polynomials of types A and B and the Hurwitz stability for alternating run polynomials defined on a dual set of Stirling permutations.Finally, we study a class of recursive palindromic polynomials and derive many nice properties including Hurwitz stability, semi-γ-positivity, non-γ-positivity, unimodality, strong q-log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be implied in a unified approach.

Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures.

Quantum mechanical calculations for material modeling using Kohn-Sham density functional theory (DFT) involve the solution of a nonlinear eigenvalue problem for N smallest eigenvector-eigenvalue pairs, with N proportional to the number of electrons in the material system. These calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons. Large-scale matrix eigenvalue problems arising from the discretization of the Kohn-Sham DFT equations employing a systematically convergent basis traditionally rely on iterative orthogonal projection methods, which are shown to be computationally efficient and scalable on massively parallel computing architectures. However, as the size of the material system increases, these methods are known to incur dominant computational costs through the Rayleigh-Ritz projection step of the discretized Kohn-Sham Hamiltonian matrix and the subsequent subspace diagonalization of the projected matrix. This work explores the potential of polynomial expansion approaches based on recursive Fermi-operator expansion as an alternative to the subspace diagonalization of the projected Hamiltonian matrix to reduce the computational cost. Subsequently, we perform a detailed comparison of various recursive polynomial expansion approaches to the traditional approach of explicit diagonalization on both multi-node central processing unit and graphics processing unit architectures and assess their relative performance in terms of accuracy, computational efficiency, scaling behavior, and energy efficiency.

Gap labels for zeros of the partition function of the 1D Ising model via the Schwartzman homomorphism

Inspired by the 1995 paper of Baake–Grimm–Pisani, we aim to explain the empirical observation that the distribution of Lee–Yang zeros corresponding to a one-dimensional Ising model appears to follow the gap labelling theorem. This follows by combining two main ingredients: first, the relation between the transfer matrix formalism for the 1D Ising model and an ostensibly unrelated matrix formalism generating the Szegő recursion for orthogonal polynomials on the unit circle, and second, the gap labelling theorem for CMV matrices.

Carleman Linearization and Systems of Arbitrary Depth Polynomial Recursions

New approach to systems of polynomial recursions is developed based on the Carleman linearization procedure. The article is divided into two main sections: firstly, we focus on the case of uni-variable depth-one polynomial recurrences. Subsequently, the systems of depth-one polynomial recurrence relations are discussed. The corresponding transition matrix is constructed and upper triangularized. Furthermore, the powers of the transition matrix are calculated using the back substitution procedure. The explicit expression for a solution to a broad family of recurrence relations is obtained. We investigate to which recurrences the framework can be applied and construct sufficient conditions for the method to work. It is shown how introduction of auxiliary variables can be used to reduce arbitrary depth systems to the depth-one system of recurrences dealt with earlier. Finally, the limitations of the method are discussed, outlining possible directions for future research.

Recursive Prediction for Lossless Image Compression

This paper introduced an algorithm for lossless image compression to compress natural and medical images. It is based on utilizing various casual fixed predictors of one or two dimension to get rid of the correlation or spatial redundancy embedded between image pixel values then a recursive polynomial model of a linear base is used.
 The experimental results of the proposed compression method are promising in terms of preserving the details and the quality of the reconstructed images as well improving the compression ratio as compared with the extracted results of a traditional linear predicting coding system.

On the Galois Groups of Some Recursive Polynomials

On the Galois Groups of Some Recursive Polynomials

Arbitrary positive powers of semicirculant and r-circulant matrices

We provide a novel recursive method, which does not require any assumption, to compute the entries of the kth power of a semicirculant matrix by using a closed-form expression for the mth term of a certain recursive polynomial sequence. As an application, a method for computing the entries of the kth power of r-circulant matrices is also presented.

A Neural Network Monte Carlo Approximation for Expected Utility Theory

This paper proposes an approximation method to create an optimal continuous-time portfolio strategy based on a combination of neural networks and Monte Carlo, named NNMC. This work is motivated by the increasing complexity of continuous-time models and stylized facts reported in the literature. We work within expected utility theory for portfolio selection with constant relative risk aversion utility. The method extends a recursive polynomial exponential approximation framework by adopting neural networks to fit the portfolio value function. We developed two network architectures and explored several activation functions. The methodology was applied on four settings: a 4/2 stochastic volatility (SV) model with two types of market price of risk, a 4/2 model with jumps, and an Ornstein–Uhlenbeck 4/2 model. In only one case, the closed-form solution was available, which helps for comparisons. We report the accuracy of the various settings in terms of optimal strategy, portfolio performance and computational efficiency, highlighting the potential of NNMC to tackle complex dynamic models.

Some remarks on generalized recursive polynomials

Some remarks on generalized recursive polynomials

Identities for linear recursive sequences of order <inline-formula><tex-math id="M1">$ 2 $</tex-math></inline-formula>

<p style='text-indent:20px;'>We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [<xref ref-type="bibr" rid="b16">16</xref>]. Numerous identities are constructed, and many well known identities can be proved readily by using this unified rule. Various Catalan-like and Cassini-like identities are given for recursive number sequences and recursive polynomial sequences. Sets of identities for Diophantine quadruple are shown.</p>