Polynomials Of Arbitrary Degree Research Articles

Probabilistic analysis of the steady state of weakly perturbed linear oscillators subject to a class of Gaussian inputs

This paper aims to probabilistically study a class of nonlinear oscillator subject to weak perturbations and driven by stationary zero-mean Gaussian stochastic processes. For the sake of generality in the analysis, we assume that the perturbed term is a polynomial of arbitrary degree in the spatial position, that contains, as a particular case, the important case of the Duffing equation. We then take advantage of the so-called stochastic equivalent linearization technique to construct an equivalent linear model so that its behavior consistently approximates, in the mean-square sense, that of the nonlinear oscillator. This approximation allows us to take extensive advantage of the probabilistic properties of the solution of the linear model and its first mean-square derivative to construct reliable approximations of the main statistical moments of the steady state. From this key information, we then apply the principle of maximum entropy to construct approximations of the probability density function of the steady state. We illustrate the superiority of the equivalent linearization technique over the perturbation method through some examples.

Inf-sup stabilized Scott–Vogelius pairs on general shape-regular simplicial grids for Navier–Stokes equations

This paper considers the discretization of the time-dependent Navier–Stokes equations with the family of inf-sup stabilized Scott–Vogelius pairs recently introduced in [John/Li/Merdon/Rui, Math. Models Methods Appl. Sci., 2024] for the Stokes problem. Therein, the velocity space is obtained by enriching the H1-conforming Lagrange element space with some H(div)-conforming Raviart–Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used.In the present paper two alternatives for discretizing the convective terms are considered. One variant leads to a scheme that still only involves volume integrals, and the other variant employs upwinding known from DG schemes. Both variants ensure the conservation of linear momentum and angular momentum in some suitable sense. In addition, a pressure-robust and convection-robust velocity error estimate is derived, i.e., the velocity error bound does not depend on the pressure and the constant in the error bound for the kinetic energy does not blow up for small viscosity. After condensation of the enrichment unknowns and all non-constant pressure unknowns, the method can be reduced to a Pk−P0-like system for arbitrary velocity polynomial degree k. Numerical studies verify the theoretical findings.

Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves

We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations.

Residue sums of Dickson polynomials over finite fields

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

Bernstein inequalities for quaternionic polynomials in the setting of generalized polynomials

Recently, several authors proved a Bernstein type inequality for so called quaternionic unilateral polynomials. Although these polynomials are not generalized complex polynomials between normed spaces, it will be proved in this contribution that their restrictions to each complex plane satisfy a Bernstein inequality for the real Fréchet derivative. This result will imply Bernstein's inequality for quaternionic unilateral polynomials which in addition holds for all complex norms in which the unit closed ball contains the real unit quaternion. The second main contribution consists in providing Bernstein-type inequalities for so called quaternionic canonical generalized polynomials. These polynomials have a factorized form which, due to the lack of commutativity for quaternionic multiplication leads to a completely different type of investigation comparing to the other class of quaternionic unilateral polynomials. However, Bernstein's inequality is proved for some remarkable classes of such polynomials of degree less than or equal to 3 and for a class of polynomials of arbitrary degree having at most two distinct roots.

Square-free values of random polynomials

The question of whether or not a given integral polynomial takes infinitely many square-free values has only been addressed unconditionally for polynomials of degree at most 3. We address this question, on average, for polynomials of arbitrary degree.

The quadratic sum problem for symplectic pairs

Let (b,u) be a pair consisting of a symplectic form b on a finite-dimensional vector space V over a field F, and of a b-alternating endomorphism u of V (i.e. b(x,u(x))=0 for all x in V).Let p and q be arbitrary polynomials of degree 2 with coefficients in F. We characterize, in terms of the invariant factors of u, the condition that u splits into u1+u2 for some pair (u1,u2) of b-alternating endomorphisms such that p(u1)=q(u2)=0.

Two-layer spatial beam with inter-layer slip in longitudinal and transverse direction

The paper presents a new mathematical and numerical model for analysis of two-layered spatial beams with inter-layer slip taken into account. Each of the two layers can be made of different material, such as timber, concrete or steel. These layers, connected together, form a composite beam, for example concrete–steel, concrete–timber or timber–timber composite beam. There are several novelties in the presented mathematical and numerical model. The first novelty is that the model enables the analysis of the influence of shear deformations on the stiffness of two-layered spatial beams. The second novelty of the model is that the inter-layer slip can only happen in longitudinal and transverse direction, while inter-layer slip in perpendicular direction is not possible. Yet another important novelty of the model is a consistent separation of the basic equations of the model into two unrelated groups and introduction of appropriate constraining equations. The negative influence of poorly conditioned equations due to the constraint equations, which would otherwise occur during the solving of the equations of the model, is thus avoided. The equations of the presented numerical model are solved by applying the finite elements method. Thus, a new family of deformation based finite elements has been developed, where all the deformation quantities of the beam model are interpolated with Lagrange polynomials of arbitrary degree. By introducing a new deformation based finite element we are able to avoid all types of locking that is typical for displacement-based finite elements. Convergence and parametric studies, presented in the paper, show: (i) that the deformation-based finite elements are very accurate and therefore suitable for the analysis of two-layer spatial beams, (ii) that the inter-layer slip in both considered directions significantly affects the values of physical quantities and therefore has to be considered in the analysis and (iii) that shear deformations have smaller effect on the stiffness of two-layer spatial beams than they have on the stiffness of homogeneous spatial beams.

Local parameter selection in the C0 interior penalty method for the biharmonic equation

Abstract The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages.

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

Abstract We consider a general nonsymmetric second-order linear elliptic partial differential equation in the framework of the Lax–Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive meshrefinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, for example, an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.

Orthogonal polynomials on a class of planar algebraic curves

AbstractWe construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form in where and ϕ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree are computed via the Lanczos algorithm in operations.

A pressure-robust HHO method for the solution of the incompressible Navier–Stokes equations on general meshes

Abstract In a recent work (Castanon Quiroz & Di Pietro (2020) A hybrid high-order method for the incompressible Navier–Stokes problem robust for large irrotational body forces. Comput. Math. Appl., 79, 2655–2677), we have introduced a pressure-robust hybrid high-order method for the numerical solution of the incompressible Navier–Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error estimates for the velocity that are fully independent of the pressure. A crucial question was left open in that work, namely whether the proposed construction could be extended to general polytopal meshes. In this paper, we provide a positive answer to this question. Specifically, we introduce a novel divergence-preserving velocity reconstruction that hinges on the solution inside each element of a mixed problem on a subtriangulation, then use it to design discretizations of the body force and convective terms that lead to pressure robustness. An in-depth theoretical study of the properties of this velocity reconstruction, and their reverberation on the scheme, is carried out for arbitrary polynomial degrees $k\geq 0$ and meshes composed of general polytopes. The theoretical convergence estimates and the pressure robustness of the method are confirmed by an extensive panel of numerical examples.

Topology of real multi-affine hypersurfaces and a homological stability property

Let R be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in Rn defined by a multi-affine polynomial of degree d is bounded by 2d−1. This bound is sharp and is independent of n (as opposed to the classical bound of d(2d−1)n−1 on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree d in Rn due to Petrovskiĭ and Oleĭnik, Thom and Milnor). Moreover, we show there exists c>1, such that given a sequence (Bn)n>0 where Bn is a closed ball in Rn of positive radius, there exist hypersurfaces (Vn⊂Rn)n>0 defined by symmetric multi-affine polynomials of degree 4, such that ∑i⩽5bi(Vn∩Bn)>cn, where bi(⋅) denotes the i-th Betti number with rational coefficients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.

Accuracy of high-order, discrete approximations to the lifting-line equation

AbstractThe accuracy of several numerical schemes for solving the lifting-line equation is investigated. Circulation is represented on discrete elements using polynomials of varying degree, and a novel scheme is introduced based on a discontinuous representation that permits arbitrary polynomial degrees to be used. Satisfying the Helmholtz theorems at inter-element boundaries penalises the discontinuities in the circulation distribution, which helps ensure the solution converges towards the correct, continuous behaviour as the number of elements increases. It is found that the singular vorticity at the wing tips drives the leading-order error of the solution. With constant panel widths, numerical schemes exhibit suboptimal accuracy irrespective of the basis degree; however, driving the width of the tip panel to zero at a rate faster than the domain average enables improved accuracy to be recovered for the quadratic-strength elements. In all cases considered, higher-order circulation elements exhibit higher accuracy than their lower-order counterparts for the same total degrees of freedom in the solution. It is also found that the discontinuous quadratic elements are more accurate than their continuous counterparts while also being more flexible for geometric representation.

Envelopes are Solving Machines for Quadratics and Cubics and Certain Polynomials of Arbitrary Degree

Summary Everybody knows from school how to solve a quadratic equation of the form x 2 − p x + q = 0 graphically. To solve more than one equation this method can become tedious, as for each pair (p, q) a new parabola has to be drawn. Stunningly, there is one single curve that can be used to solve every quadratic equation by drawing tangent lines through a given point (p, q) to this curve. In this article we derive this method in an elementary way and generalize it to equations of the form x n − p x + q = 0 for arbitrary n ≥ 2 . Moreover, the number of solutions of a specific equation of this form can be seen immediately with this technique. In concluding the article, we point out connections to the duality of points and lines in the plane and to the concept of Legendre transformation.

Images of graded polynomials on matrix algebras

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra Mn(K) over a field K endowed with its canonical Zn-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field Q of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on n×n matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over Mn(K) when K is a quadratically closed field of characteristic zero or greater than n and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on M2(K).

Integral Representations and Quadrature Schemes for the Modified Hilbert Transformation

Abstract We present quadrature schemes to calculate matrices where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when the modified Hilbert transformation is used for the variational setting. This work provides the calculation of these matrices to machine precision for arbitrary polynomial degrees and non-uniform meshes. The proposed quadrature schemes are based on weakly singular integral representations of the modified Hilbert transformation. First, these weakly singular integral representations of the modified Hilbert transformation are proven. Second, using these integral representations, we derive quadrature schemes, which treat the occurring singularities appropriately. Thus, exponential convergence with respect to the number of quadrature nodes for the proposed quadrature schemes is achieved. Numerical results, where this exponential convergence is observed, conclude this work.

Explicitly solvable systems of first-order ordinary differential equations with homogeneous right-hand sides, and their periodic variants

In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous--which feature periodic solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data.

The Bubble Transform and the de Rham Complex

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh $${{\mathscr {T}}}$$ of the domain $$\varOmega $$ , we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in $$L^2$$ and also on the appropriate subspace consisting of k-forms with exterior derivatives in $$L^2$$ .

A general vertical decomposition of Euler equations: Multilayer-moment models

In this work, we present a general framework for vertical discretizations of Euler equations. It generalizes the usual moment and multilayer models and allows to obtain a family of multilayer-moment models. It considers a multilayer-type discretization where the layerwise velocity is a polynomial of arbitrary degree N on the vertical variable. The contribution of this work is twofold. First, we compare the multilayer and moment models in their usual formulation, pointing out some advantages/disadvantages of each approach. Second, a family of multilayer-moment models is proposed. As particular interesting case we shall consider a multilayer-moment model with layerwise linear horizontal velocity. Several numerical tests are presented, devoted to the comparison of multilayer and moment methods, and also showing that the proposed method with layerwise linear velocity allows us to obtain second order accuracy in the vertical direction. We show as well that the proposed approach allows to correctly represent the vertical structure of the solutions of the hydrostatic Euler equations. Moreover, the measured efficiency shows that in many situations, the proposed multilayer-moment model needs just a few layers to improve the results of the usual multilayer model with a high number of vertical layers.