Ore Principal Subresultant Coefficients in Solutions

The principal subresultant coefficients of polynomials play a fundamental role in elimination theory and computer algebra. Recently they have been extended to Ore polynomials. They are defined by an expression in the coefficients of Ore polynomials. In this paper, we provide another expression for them. This expression is written in terms of the “solutions” of Ore polynomials (in “generic” case). It can be viewed as a generalization of the well known expression for resultants of two commutative polynomials: the product of the pair-wise differences of their roots. Received: August 16, 1999; revised version: July 3, 2000     

Hydrodynamic coefficients for water-wave diffraction by spherical structures

Evaluation of hydrodynamic coefficients and loads on submerged or floating bodies is of great significance in designing these structures. Some special regular-shaped geometries such as those of cylindrical (circular, elliptic) and spherical (hemisphere, sphere, spheroid) structures are usually considered to obtain analytical solutions to wave diffraction and radiation problems. The work presented here is the result of water-wave interaction with submerged spheres. Analytical expressions for various hydrodynamic coefficients and loads due to the diffraction of water waves by a submerged sphere are obtained. The exciting force components due to surge and heave motions are derived by solving the diffraction problem. Theory of multipole expansions is used to express the velocity potentials in terms of an infinite series of associated Legendre polynomials with unknown coefficients and the orthogonality of the polynomials is utilized to simplify the expressions. Since the infinite series appearing in various expressions have excellent truncation properties, they are evaluated by considering only a finite number of terms. Gaussian quadrature is used to evaluate the integrals. Numerical estimates for the analytical expressions for the hydrodynamic coefficients and loads are presented for various depth to radius ratios. Consideration of more values for depth makes it easy to compare the results with those available. The results obtained match closely with those obtained earlier by Wang and Wu and their coworkers     

Eight-node conforming straight-side quadrilateral element with high-order completeness (QH8-C1)

A new eight-node conforming quadrilateral element with high-order completeness, denoted as QH8-C1, is proposed in this article. First, expressions for the interpolation displacement function satisfying the requirements for high-order completeness in the global coordinate system are constructed. Second, the displacement function expression in global coordinates is transformed into isoparametric coordinates, and the relationships between the two series of coefficients for the two kinds of displacement function expressions are found. Third, the displacement function expression is modified to satisfy the requirements of nodal freedom and interelement boundary continuity. The key to the new element construction is the derivation of the linear relationship expressions among 12 coefficients of element displacement interpolation polynomials in the global and isoparametric coordinate systems. As a result, the relationship between quadratic completeness and interelement continuity is explicitly given, and a proof of the completeness and the continuity was conducted to theoretically guarantee the validity of the derivation results. Furthermore, in order to verify the correctness of the theoretical work, nine numerical examples were performed. The computation results from these examples demonstrate that QH8-C1 exhibited excellent performance, including high simulation accuracy, fast convergence, insensitivity to mesh distortion, and monotonic convergence.     

On the relative power of polynomials with real, rational, and integer coefficients in proofs of termination of rewriting

In the seventies, Manna and Ness, Lankford, and Dershowitz pionneered the use of polynomial interpretations with integer and real coefficients in proofs of termination of rewriting. More than twenty five years after these works were published, however, the absence of true examples in the literature has given rise to some doubts about the possible benefits of using polynomials with real or rational coefficients. In this paper we prove that there are, in fact, rewriting systems that can be proved polynomially terminating by using polynomial interpretations with (algebraic) real coefficients; however, the proof cannot be achieved if polynomials only contain rational coefficients. We prove a similar statement with respect to the use of rational coefficients versus integer coefficients. Tel.: +34 96 387 7007 (ext. 73531)     

Moments of Second Order Polynomials with Simulation Applications

Practical solutions to complicated sampling problems can often be found through the use of polynomial approximators. The approximators can yield exact expressions for (approximate) statistical properties, such as moments, obviating the need for sampling; on the other hand, should sampling be used, the approximators can be applied as control variates to sharpen Monte Carlo results. In this paper we give exact expressions for the first three moments of second order polynomials of independent and identically distributed random variables. The utility of these polynomials is illustrated in three examples: The first concerns the evaluation of the moments of the sample variance, the second investigates properties of a present value when interest rates vary, and the third involves control variates in a complicated sampling problem with nonlinear regression.     

Stability of schur polynomials under coefficient perturbation

Abstract

In this paper, we are concerned with the stability robustness of characteristic polynomials with perturbed coefficients for linear discrete‐time systems. An upper bound on the allowable coefficient perturbation of a Schur polynomial with retaining stability is obtained. The proposed upper bound is directly formulated in terms of the polynomial under consideration and its value can be determined by numerical computation. We also provide a sufficient condition for the stability of interval polynomials.     

Various New Expressions for Subresultants and Their Applications

This article is devoted to presenting new expressions for Subresultant Polynomials, written in terms of some minors of matrices different from the Sylvester matrix. Moreover, via these expressions, we provide new proofs for formulas which associate the Subresultant polynomials and the roots of the two polynomials. By one hand, we present a new proof for the formula introduced by J. J. Sylvester in 1839, formula written in terms of a single sum over the roots. By other hand, we introduce a new expression in terms of the roots by considering the Newton basis.Partially supported by the European Union funded project RAAG (HPRN–CT–2001–00271) and by the spanish grant BFM2002-04402-C02-0     

On the complexity of skew arithmetic

In this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed almost linearly in terms of the cost of multiplication.     

Limit analysis of the statistics of quasi-steady non-linear aerodynamic forces for small turbulence intensities

This paper focuses on the estimation of statistical characteristics of a specific quasi-steady wind loading used in buffeting analyses. In this loading, the intrinsic non-linearity of aerodynamic coefficients is considered and approached by a polynomial expression of any a priori chosen degree. As rigorous developments of the statistical moments would result in impractical formulations, we suggest to consider the smallness of the turbulence intensities to construct, by means of a limit analysis, approximate expressions of the raw moments of aerodynamic forces. From these expressions, approximate cumulants and associated dimensionless characteristics, such as skewness and excess coefficients, are derived. The accuracy of the proposed analytical relations is assessed by comparison with Monte Carlo simulations, and the relevance of the sophisticated non-linear loading under consideration is compared to more traditional models.     

Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations

We develop a basic theory of Gröbner bases for ideals in the algebra of Laurent polynomials (and, more generally, in its monomial subalgebras). For this we have to generalize the notion of term order. The theory is applied to systems of linear partial difference equations (with constant coefficients) on Êⁿ. Furthermore, we present a method to compute the intersection of an ideal in the algebra of Laurent polynomials with the subalgebra of all polynomials.     

Decomposition of ordinary differential polynomials

In this paper, we present a complete algorithm to decompose nonlinear differential polynomials in one variable and with coefficients in a computable differential field of characteristic zero. The algorithm provides an efficient reduction of the problem to the factorization of LODOs over the same coefficient field. Besides arithmetic operations, the algorithm needs decomposition of algebraic polynomials, factorization of multi-variable polynomials, and solution of algebraic linear equation systems. The algorithm is implemented in Maple for the constant field case. The program can be used to decompose differential polynomials with thousands of terms effectively. This article was partially supported by a National Key Basic Research Project of China (NO. G1998030600) and by a USA NSF grant CCR-0201253.     

Orthonormal polynomials in wavefront analysis: analytical solution

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.     

On the computation of the nijboer-zernike aberration integrals at arbitrary defocus

Abstract

We present a new computation scheme for the integral expressions describing the contributions of single aberrations to the diffraction integral in the context of an extended Nijboer-Zernike approach. Such a scheme, in the form of a power series involving the defocus parameter with coefficients given explicitly in terms of Bessel functions and binomial coefficients, was presented recently by the authors with satisfactory results for small-to-medium-large defocus values. The new scheme amounts to systemizing the procedure proposed by Nijboer in which the appropriate linearization of products of Zernike polynomials is achieved by using certain results of the modern theory of orthogonal polynomials. It can be used to compute point-spread functions of general optical systems in the presence of arbitrary lens transmission and lens aberration functions and the scheme provides accurate data for any, small or large, defocus value and at any spatial point in one and the same format. The cases with high numerical aperture, requiring a vectorial approach, are equally well handled. The resulting infinite series expressions for these point-spread functions, involving products of Bessel functions, can be shown to be practically immune to loss of digits. In this respect, because of its virtually unlimited defocus range, the scheme is particularly valuable in replacing numerical Fourier transform methods when the defocused pupil functions require intolerably high sampling densities.     

基于加权正交多项式的M进制类小波基的构造

本文运用加权正交多项式和频带有限小波函数构造了L^2（[c,d]）空间的M进制类小波基,使该类小波基具有解析表达式和消失距性质,并给出了它的小波分解与重构算法和L^2（[c,d]）空间的多分辨分解.     

基于第二类切比雪夫多项式的语音线性谱对参数的计算

对线性谱对(LSP)参数的计算方法提出改进算法，该算法利用第二类切比雪夫多项式的迭代性质对初始函数降阶。理论分析表明，改进算法可以获得更简洁的数学表达式。实验结果显示，改进算法中基本消除了乘法运算，同时随着线性预测分析阶数的增加可以进一步降低算法复杂度。     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.     

A Riemann-Hilbert approach to the Akhiezer polynomials

In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the union of disjoint intervals, first introduced by N. I. Akhiezer, via a reformulation as a matrix factorization or Riemann-Hilbert problem. This approach complements the method proposed in a previous paper, which involves the construction of a certain meromorphic function on a hyperelliptic Riemann surface. The method described here is based on the general Riemann-Hilbert scheme of the theory of integrable systems and will enable us to derive, in a very straightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantities involving the corresponding recurrence coefficients. Both of these equations were obtained earlier by A. Magnus. In our approach, however, we are able to go beyond Magnus' results by actually solving the equations in terms of the Riemanni Theta-functions. We also show that the related Hankel determinant can be interpreted as the relevant tau-function.     

General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes

Zernike polynomials have been widely used to describe the aberrations in wavefront sensing of the eye. The Zernike coefficients are often computed under different aperture sizes. For the sake of comparison, the same aperture diameter is required. Since no standard aperture size is available for reporting the results, it is important to develop a technique for converting the Zernike coefficients obtained from one aperture size to another size. By investigating the properties of Zernike polynomials, we propose a general method for establishing the relationship between two sets of Zernike coefficients computed with different aperture sizes.     

Polynomial stress functions of anisotropic plane problems and their applications in hybrid finite elements

In this paper, systematic approaches to determine the polynomial stress functions for anisotropic plane problems are presented based on the Lekhnitskii’s theory of anisotropic elasticity. It is demonstrated that, for plane problems, there are at most four independent polynomials for arbitrary n-th order homogeneous polynomial stress functions: three independent polynomials for n equal to two and four for n greater than or equal to three. General expressions for such polynomial stress functions are derived in explicit forms. Unlike the isotropic case, the polynomials for anisotropic problems are functions of material constants, because the elastic constants cannot be eliminated in the governing equation for general anisotropic cases. The polynomials can be used as analytical trial functions to develop the new 8-node hybrid element (ATF-Q8) for anisotropic problems. This ATF-Q8 element demonstrated excellent performance in comparison with traditional numerical methods through several testing examples.