High-degree Polynomial Research Articles

High-Degree Polynomial Integrals of a Natural System on the Two-Dimensional Torus

We study a natural mechanical system on the two-dimensional torus which admitsan additional first integral polynomial in momenta of an odd degree N andindependent of the energy integral. For N=5,7 , we obtain the estimateson the number of straight lines in the spectrum of the potential.

Exact subdomain and embedded interface polynomial integration in finite elements with planar cuts

The implementation of discontinuous functions occurs in many of today’s state-of-the-art partial differential equation solvers. However, in finite element methods, this poses an inherent difficulty: efficient quadrature rules available when integrating functions whose discontinuity falls in the element’s interior are for low order degree polynomials, not easily extended to higher order degree polynomials, and cover a restricted set of geometries. Many approaches to this issue have been developed in recent years. Among them, one of the most elegant and versatile is the equivalent polynomial technique. This method replaces the discontinuous function with a polynomial, allowing integration to occur over the entire domain rather than integrating over complex subdomains. Although eliminating the issues involved with discontinuous function integration, the equivalent polynomial tactic introduces its problems. The exact subdomain integration requires a machinery that quickly grows in complexity when increasing the polynomial degree and the geometry dimension, restricting its applicability to lower order degree finite element families. The current work eliminates this issue. We provide algebraic expressions to exactly evaluate the subdomain integral of any degree polynomial on parent finite element shapes cut by a planar interface. These formulas also apply to the exact evaluation of the embedded interface integral. We provide recursive algorithms that avoid overflow in computer arithmetic for standard finite element geometries: triangle, square, cube, tetrahedron, and prism, along with a hypercube of arbitrary dimensions.

Whitney edge elements and the Runge phenomenon

It is well known that Lagrange interpolation based on equispaced nodes can yield poor results. Oscillations may appear when using high degree polynomials. For functions of one variable, the most celebrated example has been provided by Carl Runge in 1901, who showed that higher degrees do not always improve interpolation accuracy. His example was then extended to multivariate calculus and in this work we show that it is meaningful, in an appropriate sense, also for Whitney edge elements, namely for differential 1-forms.

Hermitian and Unitary Almost-Companion Matrices of Polynomials on Demand

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees.

On Induced Paths, Holes, and Trees in Random Graphs

The concentration of the sizes of largest induced paths and cycles (holes) is studied in Erdős–Rényi random graphs. A 2-point concentration is proved for the size of the largest induced path and cycle for all satisfying and , where is any constant. No such tight concentration (within two consecutive values) was previously known for induced paths and cycles. As a corollary, a significant additive improvement is obtained over a 40-year-old result of Erdős and Palka [Discrete Math., 46 (1983), pp. 145–150] concerning the size of the largest induced tree in a dense random graph. Further, the induced path decomposition number and induced tree decomposition number, i.e., the smallest number of parts into which the vertex set of a graph can be partitioned such that every part induces a (i) path or (ii) tree, respectively, are studied for . The arguments involve the second moment method together with an adaptation of a martingale-based technique of Krivelevich et al. [Random Structures Algorithms, 22 (2003), pp. 1–14] for monotone high-degree polynomial random variables to the nonmonotone setting. A lower bound is proved showing the tightness of the application of the inequality up to logarithmic factors in the exponent. The modified inequality is then stated and proved in a general setting, which may be of independent interest.

P789 Inflammatory Bowel Disease subgroups identified by longitudinal faecal calprotectin profiles: A 10-year retrospective analysis of the Lothian IBD Registry

Abstract Background Faecal calprotectin (FCAL) is routinely used to monitor mucosal inflammation, and high FCAL values are associated with poor outcomes in IBD. Having previously found clusters characterised by distinct FCAL profiles in a Crohn’s disease (CD) cohort of 356 subjects, we aimed to extend this approach to all IBD patients treated by the Edinburgh IBD Unit. Methods Subjects were required to have a diagnosis of IBD, three FCAL measurements within ten years of diagnosis between January 2005 and September 2019, and be treated by the Edinburgh IBD Unit. All FCAL was measured on the same ELISA essay as part of routine clinical care. Latent class mixed models, an extension of linear mixed effects models with cluster specific fixed effects, were used to model and cluster FCAL trajectories. Fixed and random effects were modelled using natural cubic splines, which allows flexible models with relatively low computational cost when compared to high degree polynomials. Akaike information criterion, Bayesian information criterion and alluvial plots were used to decide on the appropriate number of clusters. CD subjects were modelled independently of UC and IBDU subjects, with similar clusters across disease subtypes given the same numbering for ease of comparison. Hereafter, we will describe the UC and IBDU patients collectively. Results 1,214 CD and 1,194 UC/IBDU subjects met the inclusion criteria. Across follow-up, 10,503 (median 7 per subject) FCAL measurements were available for patients with CD and 7,495 (median 5 per subject) for those with UC/IBDU. Figure 1. Log-transformed ten-year faecal calprotectin profiles for Crohn’s disease subjects. Red curves demonstrate overall profile for each cluster. The blue dotted lines indicate an FCAL of log (250 μg/g), the commonly accepted threshold for biochemical remission. For CD, six clusters were found to be appropriate, whereas five clusters were found to be the most appropriate for UC/ IBDU. The overall cluster prmofiles were remarkably similar between CD and UC/IBDU. Both CD cluster 2 and UC cluster 2, the largest clusters, were characterised by consistently high FCAL across follow-up: remaining above the 250 μg/g threshold commonly used for biochemical remission (blue dotted lines in the figures). Figure 2. Log-transformed ten-year faecal calprotectin profiles for ulcerative colitis subjects Conclusion After grouping CD and UC/IBDU subjects in a large cohort by their 10-year FCAL trajectories, we found remarkably similar trajectories across the two groups. In both cases, we found a large cluster of subjects with consistently elevated FCAL, which warrants further investigation.

High-order schemes for the fractional coupled nonlinear Schrödinger equation

<abstract><p>This paper considers the fractional coupled nonlinear Schrödinger equation with high degree polynomials in the energy functional that cannot be handled by using the quadratic auxiliary variable method. To this end, we develop the multiple quadratic auxiliary variable approach and then construct a family of structure-preserving schemes with the help of the symplectic Runge-Kutta method for solving the equation. The given schemes have high accuracy in time and can both inherit the mass and Hamiltonian energy of the system. Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.</p></abstract>

The Factorisation, Zeroes and Roots of Higher Degree (n ≥ 3 ) Polynomials and Equations by Factorisation Method ( by Splitting Method).

The Factorisation, Zeroes and Roots of Higher Degree (n ≥ 3 ) Polynomials and Equations by Factorisation Method ( by Splitting Method).

Bivariate multiquadric quasi-interpolation operators of Lidstone type

<abstract><p>In this paper, a kind of bivariate multiquadric quasi-interpolant with the derivatives of a approximated function is studied by combining the known multiquadric quasi-interpolant with the generalized Taylor polynomials that act as the bivariate Lidstone interpolation polynomials. For practical purposes, a kind of improved approximation operator without any derivative of the approximated function is given by using bivariate divided differences to approximate the derivatives. It has the property of high-degree polynomial reproducing. In addition, the improved bivariate quasi-interpolation operators only demand information of the location points rather than the derivatives of the function approximated. Some error bounds in terms of the modulus of continuity of high order and Peano representations for the error are given. Several numerical comparisons with other existing methods are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the algorithm is very simple and easy to implement.</p></abstract>

A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

This article considers the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> static output-feedback control for linear time-invariant uncertain systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. By applying polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system, which characterizes stochastic state uncertainty propagation. A closed-loop polynomial chaos transformation is proposed to derive the closed-loop expanded system. The approach explicitly accounts for the closed-loop dynamics and preserves the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> -induced gain, which results in smaller transformation errors compared to existing polynomial chaos transformations. The effect of using finite-degree polynomial chaos expansions is first captured by a norm-bounded linear differential inclusion, and then addressed by formulating a robust polynomial chaos based control synthesis problem. This proposed approach avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of truncation errors, which significantly reduces computational complexity. A numerical example illustrates the effectiveness of the proposed approach.

A very high order discontinuous Galerkin method for the numerical solution of stiff DDEs

We present a high order discontinuous Galerkin (DG) method for the numerical solution of systems of delay differential equations (DDEs). The method is based on Legendre orthogonal polynomials of high degree k (typically k=10) in each subinterval and is a generalisation to DDEs of a similar method which proved to be very efficient for ordinary differential equations (ODEs). We show how the error can be estimated allowing to control the size of the time step. We also propose a particularly efficient quasi Newton method for the solution of the resulting non linear systems based on a very accurate approximation of the Jacobian matrix that is very easy to implement and with excellent convergence properties. The method is then applied to very stiff systems of DDEs.

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balanced property, we adopt the similar idea of WENO-XS scheme (Xing and Shu (2005) [30]) to balance the flux gradients and the source terms. The fluxes in the original equations are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balanced property. Instead of using a five-point stencil in the fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction. Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.

The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory

This paper presents planning of smooth trajectories using the asymmetric profiles of displacement, velocity, acceleration and jerk of described higher degree polynomials. The presented way of determination of polynomial and asymmetry coefficients as well as motion time includes the procedure at both constraints of single kinematic quantities, i.e., velocity, acceleration, jerk and introducing a few constraints simultaneously. Moreover, the paper presents a method of planning trajectories with intermediate points consisting in concatenation of the analyzed polynomials. The proposed method of trajectory planning was compared to that of using the S-curve. The results in the form of kinematic quantity courses are presented graphically. The discussed method guarantees continuity of displacement, velocity, acceleration and jerk. It can be used for planning trajectories of various technical objects (e.g., manipulators, mobile robots, CNC machine tools, optical disk drives, autonomous vehicles, etc.).

Prediction of Primary Deformation Modulus Based on Bearing Capacity: A Case on Forest Road with a Light Falling Weight Deflectometer Zorn ZFG 3000 GPS

Bearing capacity and compaction are among the most important and frequently used geotechnical parameters in road construction. The aim of this study was to determine the possibility of predicting the value of the primary deformation modulus E1 (obtained from measurements using a static plate load test—PLT) based on measurements with a Zorn light falling weight deflectometer (LFWD), type ZFG 3000 GPS, with a drop weight of 10 kg. A regression analysis was performed on 245 bearing capacity measurements that were taken on 46 forest road sections with various road surfaces. Different regression models were tested, from linear to logarithmic, polynomial, exponential and power models, but excluding polynomials of fourth and higher degree. The results showed that the prediction of E1 values (PLT) from the dynamic deformation modulus values Evd (LFWD) was possible. However, the reported unsatisfactory strength of the relationship between the two parameters was associated with a high risk of error (r = 0.64, R2 = 0.41, Se = 49.78). Neither the use of more complex non-linear regression models, nor the use of multiple regression by introducing an additional estimator in the form of the s/v ratio, significantly improved estimation results. The quality of the prediction of the E1 value was not constant. It varied, depending on the type of forest road, the use of geosynthetic reinforcement and the type of road subgrade. During the study, it was also found that the quality of the prediction of the E1 value could be improved by limiting the range of Evd values tested from above. It is advisable to continue this type of research, as the obtained results could form the basis for future development of national standards for the use of LFWDs to control the bearing capacity and compaction of forest road pavements.

Stable factorization for phase factors of quantum signal processing

This paper proposes a new factorization algorithm for computing the phase factors of quantum signal processing. The proposed algorithm avoids root finding of high degree polynomials by using a key step of Prony&amp;apos;s method and is numerically stable in the double precision arithmetics. Experimental results are reported for Hamiltonian simulation, eigenstate filtering, matrix inversion, and Fermi-Dirac operator.

A fifth-order finite difference HWENO scheme combined with limiter for hyperbolic conservation laws

In this paper, a simple fifth-order finite difference Hermite WENO (HWENO) scheme combined with limiter is proposed for one- and two-dimensional hyperbolic conservation laws. The fluxes in the governing equation are approximated by the nonlinear HWENO reconstruction which is the combination of a quintic polynomial with two quadratic polynomials, where the linear weights can be artificial positive numbers only if the sum equals one. And other fluxes in the derivative equations are approximated by high-degree polynomials directly. For the purpose of controlling spurious oscillations, an HWENO limiter is applied to modify the derivatives. Instead of using the modified derivatives both in fluxes reconstruction and time discretization as in the modified HWENO scheme (Z. Zhao et al. (2020) [27]), we only apply the modified derivatives in time discretization while remaining the original derivatives in fluxes reconstruction. Comparing with the modified HWENO scheme, the proposed HWENO scheme is simpler, more accurate, efficient, and has better resolution. In addition, the HWENO scheme has a more compact spatial reconstructed stencil and is more efficient than the same order finite difference WENO scheme of Jiang and Shu (1996) [11] and WENO scheme with artificial linear weights of Zhu and Qiu (2016) [29]. Various benchmark numerical examples are presented to show the fifth-order accuracy, efficiency, high resolution, and robustness of the proposed HWENO scheme.

Trend Analysis on Water Quality Index Using the Least Squares Regression Models

River water pollution requires continuous water quality monitoring that promotes the improvement of water resources. Therefore, the trend analysis on water quality data using mathematical model is an important task to determine whether the measured data increase or decrease during the time period. This paper is intended to highlight the applicability of the least squares regression models to fit the WQI data of the Skudai River, Tebrau River and Segget River located in Johor, Malaysia. As per the 12 years of trend analysis, the data of WQI are collected from the Environmental Quality Reports 2009-2020. The least squares method is utilized to estimate the unknown constants of the linear, quadratic, cubic, polynomial of degree four and degree five regression models. The advantage of using proposed models is that it can be implemented easily even on relatively low computational power systems. The results show that the higher degree polynomial model fits the data reasonably well, in which the polynomials of degree 4 and 5 have lowest average error. Assessment of actual and predictable values of WQI shows that the trends in WQI for all study areas are downward year after year.

A Scalable and Robust Vertex-Star Relaxation for High-Order FEM

Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a $p$-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate.

A New Fuzzy Vault based Biometric System robust to Brute-Force Attack

Fuzzy vault based biometric systems use fuzzy vaults during the coding that occurs within the enrollment stage inside a biometric system. In that stage the biometric system creates a vault from the biometric data, user’s key and chaff points. In the verification stage, a new biometric data is introduced and user’s key can be recovered through the use of the Lagrange polynomial interpolation method. As a consequence, in this kind of systems brute force attack would be successful because fuzzy vaults are finite. Most of the related works focus on designing secure fuzzy vault systems through the use of a password or using hybrid systems. This leads to security falling on the same user or increasing the number of security elements such as chaff points, the degree of the polynomial, or multiple biometric samples. This paper proposes a new system that considers cryptography to achieve a fuzzy vault biometric system robust against brute-force attacks. It is important to say that our system does not need a higher number of chaff points or even a higher polynomial degree. Obtained results show that this new fuzzy vault biometric system not only would be secure for current times but also would be for the future time.

Confluent Vandermonde with Arnoldi

In this note, we extend the Vandermonde with Arnoldi method recently advocated by Brubeck et al. (2021) to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples. We note that one of the approaches orthogonalizes only the function values and is equivalent to the formula given by Brubeck and Trefethen (2022). The other approach orthogonalizes the Hermite data. About which approach is preferable to another, we made the comparison, and the result is problem dependent.