Continuous Polynomials Research Articles

On interpolation spaces of piecewise polynomials on mixed meshes

We consider fractional Sobolev spaces Hθ, θ ∈ (0,1), on 2D domains and H1-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shaperegular triangles and quadrilaterals. We prove that the norm obtained from interpolating between the discrete space equipped with the L2-norm on the one hand and the H1-norm on the other hand is equivalent to the corresponding continuous interpolation Sobolev norm, and the norm-equivalence constants are independent of meshsize and polynomial degree. This characterization of the Sobolev norm is then used to show an inverse inequality between H1 and Hθ.

Unravelling the speed-going relationship: A proof of concept study from British turf flat and jump race meetings

The maximum galloping speeds of racehorses during a race are influenced by the functional performance of the ground (‘going’) amongst other factors. For turf racecourses in Britain, the ground is descriptively classified and numerically quantified on the morning of a race meeting by the clerk of the course and subsequently published to assist decision making. Importantly, this includes deciding whether a horse should or should not run. The going is also assessed and classified during the meeting by racing analysts using the normalized winning times from each race result. Differences between going assessments are regularly reported, therefore this study aimed to evaluate whether an alternative method of measuring going could better predict going measured from performances. Measurement and performance data from 25 flat and 25 jump meetings were compared using linear and nonlinear regression models. A continuous two-phase polynomial model for cushioning was found to be the best predictor of performance going for all 50 meetings (adjusted r2=0.819, P<0.001). As cushioning can provide a going related indicator of the forces that the horse will experience at gallop, this measurement may be useful when evaluating racecourse going. This initial model suggests that there is little performance advantage at maximal galloping speeds above a cushioning value of approximately10 kN, possibly due to changes in limb contact timings to manage limb forces limits as the ground becomes firmer. An expansion of objective measurements of going that relate to performance across a wider geographic region, if not internationally are needed to confirm this limit.

A novel approach for the numerical solution of nonlinear Fredholm integral equations using Hosoya polynomial method

Abstract In this paper, we study the graph theoretical polynomial known as the Hosoya polynomial obtained from one of the standard classes of graphs called path. Using this polynomial applied for the numerical solution of the nonlinear Fredholm integral equation, which reduces in the algebraic system of equation with collocation points, then solving this system using Newton’s iterative with the help of MATLAB, we obtain the required approximate solution. The desired results in terms of a set of continuous polynomials over a closed interval [0, 1]. Illustrative applications show the efficiency, accuracy and validity of the proposed technique.

Deformations and q-Convolutions. Old and New Results

This paper is the survey of some of our results related to q-deformations of the Fock spaces and related to q-convolutions for probability measures on the real line R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}. The main idea is done by the combinatorics of moments of the measures and related q-cumulants of different types. The main and interesting q-convolutions are related to classical continuous (discrete) q-Hermite polynomial. Among them are classical (q=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=1$$\\end{document}) convolutions, the case q=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=0$$\\end{document}, gives the free and Boolean relations, and the new class of q-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of q-convolutions. The main result is the construction of Brownian motion related to q-Discrete Hermite polynomial of type I.

Long-term autonomous time-keeping of navigation constellations based on sparse sampling LSTM algorithm

The system time of the four major navigation satellite systems is mainly maintained by multiple high-performance atomic clocks at ground stations. This operational mode relies heavily on the support of ground stations. To enhance the high-precision autonomous timing capability of next-generation navigation satellites, it is necessary to autonomously generate a comprehensive space-based time scale on orbit and make long-term, high-precision predictions for the clock error of this time scale. In order to solve these two problems, this paper proposed a two-level satellite timing system, and used multiple time-keeping node satellites to generate a more stable space-based time scale. Then this paper used the sparse sampling Long Short-Term Memory (LSTM) algorithm to improve the accuracy of clock error long-term prediction on space-based time scale. After simulation, at sampling times of 300 s, 8.64 × 104 s, and 1 × 106 s, the frequency stabilities of the spaceborne timescale reach 1.35 × 10–15, 3.37 × 10–16, and 2.81 × 10–16, respectively. When applying the improved clock error prediction algorithm, the ten-day prediction error is 3.16 × 10–10 s. Compared with those of the continuous sampling LSTM, Kalman filter, polynomial and quadratic polynomial models, the corresponding prediction accuracies are 1.72, 1.56, 1.83 and 1.36 times greater, respectively.

Intra-leaf modeling of Cannabis leaflet shape produces leaf models that predict genetic and developmental identities.

The iconic, palmately compound leaves of Cannabis have attracted significant attention in the past. However, investigations into the genetic basis of leaf shape or its connections to phytochemical composition have yielded inconclusive results. This is partly due to prominent changes in leaflet number within a single plant during development, which has so far prevented the proper use of common morphometric techniques. Here, we present a new method that overcomes the challenge of nonhomologous landmarks in palmate, pinnate, and lobed leaves, using Cannabis as an example. We model corresponding pseudo-landmarks for each leaflet as angle-radius coordinates and model them as a function of leaflet to create continuous polynomial models, bypassing the problems associated with variable number of leaflets between leaves. We analyze 341 leaves from 24 individuals from nine Cannabis accessions. Using 3591 pseudo-landmarks in modeled leaves, we accurately predict accession identity, leaflet number, and relative node number. Intra-leaf modeling offers a rapid, cost-effective means of identifying Cannabis accessions, making it a valuable tool for future taxonomic studies, cultivar recognition, and possibly chemical content analysis and sex identification, in addition to permitting the morphometric analysis of leaves in any species with variable numbers of leaflets or lobes.

A counterexample to a conjecture of M. Ismail

In an earlier work [K. Castillo et al., J. Math. Anal. Appl. 514, 126358 (2022)], we give positive answer to the first, and apparently more easy, part of a conjecture of M. Ismail concerning the characterization of the continuous q-Jacobi polynomials, Al-Salam-Chihara polynomials or special or limiting cases of them. In this note we present an example that disproves the second part of such a conjecture, and so this issue is definitively closed.

On the convergence of 2D P4+$$ {P}_{4+} $$ triangular and 3D P6+$$ {P}_{6+} $$ tetrahedral divergence‐free finite elements

AbstractWe show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the ‐ mixed finite element method for on 2D triangular grids or on tetrahedral grids, even in the case the inf‐sup condition fails. By a simple ‐projection of the discrete pressure to the space of continuous polynomials, we show this post‐processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.

Error analysis for the finite element approximations of Dirichlet parabolic boundary control problem with measure data

This article investigates the convergence analysis of finite element method for Dirichlet boundary control problem governed by parabolic equation with measure data. The presence of measure data in the source term and Dirichlet boundary control lead the solution of the state equation to have low regularity, which creates a challenging task for the error analysis. We prove the existence, uniqueness and regularity results of the solution to the control problem. For the discretization of the state and adjoint variables, we employ continuous piecewise linear polynomials while the control variable is approximated by piecewise constant functions. The backward Euler technique provides the foundation for the temporal discretization. For the completely discrete optimal control problem, a priori error bounds for the state variable in the L2(0,T;L2(Ω))-norm and for the control variable in the L2(0,T;L2(∂Ω))-norm are derived. Numerical experiment is performed to validate our theoretical analysis.

A CutFEM method for phase change problems with natural convection

In this paper, we present a Nitsche-based CutFEM approach for solving phase change problems while taking natural convection into account in the liquid phase. In the proposed algorithm, the interface is implicitly tracked through a level-set approach, where the level-set function is updated by solving a transport equation. To stabilize the finite element formulation of this equation, we use the continuous interior penalty method (CIP). We use the Backward Euler scheme for the time discretization and the of continuous piecewise linear polynomials (P1) for the spatial discretization of the temperature and the level-set function. For flow velocity and pressure in the liquid phase, the finite element formulation is based on the inf-sup stable pair of Hood-Taylor finite element function spaces P2−P1. As we are dealing with unfitted finite element methods, stability issues may arise due to geometry discretization. To address this issue, we incorporate ghost penalty terms into the finite element formulations of the heat and Navier–Stokes equations. According to Stefan’s condition, the normal velocity of the interface is proportional to the jump in the interfacial flux. For an efficient approximation of this jump, we use a ghost penalty-based domain integral approach which has shown to be highly accurate. To validate the proposed method, we first provide two manufactured solutions that confirm optimal convergence in both temporal and spatial discretizations. We next consider challenging problems such as the melting of n-octadecane in a differentially heated cavity, the multi-cellular melting of gallium, and the freezing of water to further validate our algorithm. We compare our results to existing numerical and experimental data.

Cognitive Structure Trajectory in Developing Limit Concept Learning in Senior High School

The existing comprehension of the concept of limit is marked by a deficiency in prioritising the enhancement of students' cognitive frameworks. The aim of this study is to examine the formation of cognitive structures while acquiring the concept of limitations. This study utilises qualitative approaches, namely descriptive analysis, to examine the development of cognitive processes in the process of acquiring the concept of boundaries. This study investigates the investigation of cognitive structures to ascertain the process of drawing conclusions, with the aim of creating instructional materials that facilitate comprehension of the concept of constraints. The survey included students who were currently enrolled in the 2021/2022 academic year at three public senior high schools in Pontianak. The study aims to analyse the progression of students' cognitive structures during the acquisition of the concept of limitations. The results revealed that all 14 students showed competence in determining the limit of a continuous polynomial function, specifically in the context of a linear function and a quadratic function. Substitution is commonly used to determine the limit of a linear function and the limit of a quadratic function. Only a single student employs the attributes of the limit function and graph trajectory to solve these two categories of problems. Researchers can conduct further investigation to examine the misconceptions that students have regarding the limit of functions. These misconceptions can be detected by analysing problems that involve substitution and have solutions that are already known. In senior high school, it is advisable to use graphs and tables to teach the concept of limit function since they aid in facilitating the learning process.

Some properties of 𝑝-limited sets

Karn and Sinha [Glasg. Math. J. 56 (2014), pp. 427–437] introduced the p p -limited ( 1 ≤ p &gt; ∞ 1 \leq p &gt; \infty ) sets (see the definition below). We show that p p -limited sets are preserved by continuous polynomials as well as by the projective tensor product and that scalar-valued polynomials are p p -summable on p p -limited sets. Considering the notion of p p -limited set from the ℓ p \ell _p -valued operators point of view, we introduce in Section \ref{sec:3} two weaker types of p p -limitedness in the setting of Banach lattices and study their basic properties.

Q-fractional integral operators with two parameters

We use the Poisson kernel of the continuous q-Hermite polynomials to introduce families of integral operators. One of them is semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The action of the semigroups of operators on the Askey–Wilson polynomials is shown to only change the parameters but preserves the degrees, hence we produce transmutation relations for the Askey–Wilson polynomials. The transmutation relations are then used to derive bilinear generating functions involving the Askey–Wilson polynomials.

Splines of the Second and Seventh Order Approximation and the Stability of the Solution of the Fredholm Integral Equations of the Second Kind

This work is a continuation of a series of works on the use of continuous local polynomial splines for solving interpolation problems and for solving the Fredholm integral equation of the second kind. Here the construction of a numerical solution to the Fredholm integral equation of the second kind using local spline approximations of the second order and the seventh order of approximation is considered. This paper is devoted to the investigation of the stability of the solution of the integral equation using these local splines. Approximation constants are given in the theorem about the error of approximation by the considered splines. Numerical examples of the application of spline approximations of the second and seventh order of approximation for solving integral equations are given.

Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used.

Algebras of weakly symmetric functions on spaces of Lebesgue measurable functions

In this work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the $p$th power of the absolute value is Lebesgue integrable, where $p\in[1,+\infty),$ and Lebesgue measurable essentially bounded functions on $[0,1].$ We construct generating systems of algebras of all weakly symmetric continuous complex-valued polynomials on these spaces. Also we establish conditions under which sets of weakly symmetric analytic functions are algebras.

Solutions of differential equations using linearly independent Hosoya polynomials of trees

Abstract We present an algorithm for the result of differential equations (DEs) by using linearly independent Hosoya polynomials of trees. With the newly adopted strategy, the desired outcome is expanded in the form of a collection of continuous polynomials over an interval. Nevertheless, compared to other methods for solving differential equations, this method’s precision and effectiveness relies on the size of the collection of Hosoya polynomials, and the process is easier. Excellent agreement between the exact and approximate solutions is obtained when the current scheme is used to crack linear and nonlinear equations. Potentially, this method could be used in more intricate systems for which there are no exact solutions.

On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations

Abstract In this paper, we study the number of limit cycles bifurcated from the periodic orbits of a cubic uniform isochronous center with continuous and discontinuous quartic polynomial perturbations. Using the averaging theory of first order for continuous and discontinuous differential systems and comparing the obtained results, we show that the discontinuous systems have at least 6 more limit cycles than the continuous ones. This study needs some computations that have been verified using Maple.

Stability and stabilization analysis of periodic switching stochastic systems based on Lyapunov function with continuous time-varying matrix polynomials

In this paper, a Lyapunov function with continuous time-varying Lyapunov matrix polynomials is constructed to ensure the exponential stability in mean square of a periodic switched linear stochastic system, where the Lyapunov matrix of this Lyapunov function is no longer a constant matrix, but a continuous time-varying matrix polynomial. Then, based on this stability study, we design an aperiodic intermittent control for unstable periodic switching stochastic systems to solve the influence of unstable subsystems and switching rules on stability, and the corresponding control gains are continuous time-varying matrix polynomial. Finally, an example is demonstrated to illustrate the effectiveness of the proposed control strategy.

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.