Ordinary Polynomials Research Articles

Ritt-Wu's characteristic set method for ordinary difference polynomial systems with arbitrary ordering

In this paper, a Ritt-Wu's characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.

Decomposition of ordinary difference polynomials

In this paper, we present an algorithm to decompose ordinary non-linear difference polynomials with rational functions as coefficients. The algorithm provides an effective reduction of the decomposition of difference polynomials to the decomposition of linear difference polynomials over the same coefficient field. The algorithm is implemented in Maple for the constant coefficient case. Experimental results show that the algorithm is quite effective and can be used to decompose difference polynomials with thousands of terms.

TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.

Jacobi’s bound for independent systems of algebraic partial differential equations

In this paper we prove Jacobi’s bound for systems of n partial differential polynomials in n differential variables, which are independent over a prime differential ideal . This generalizes on the one hand our result (Kondratieva et al. in On Jacobi’s bound for systems of differential polynomials, algebra. Moscow University Press, Moscow, pp 79–85, 1982) about Jacobi’s bound for ordinary differential polynomials independent over a prime differential ideal and, on the other hand, the result by Tomasovic (A generalized Jacobi conjecture for arbitrary systems of algebraic differential equations. PhD thesis, Columbia University, 1976), who proposed a version of Jacobi’s bound for partial differential polynomials and proved it in the linear case. In Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, 1999) we gave another proof of this result by Tomasovic. The exposition in the present paper follows our proof in Kondratieva et al. (Differential and difference dimension polynomials, Kluwer, Dordrecht, pp 273–280, 1999).

Estimação de componentes de co-variância para pesos corporais do nascimento aos 365 dias de idade de bovinos Guzerá empregando-se modelos de regressão aleatória

Um total de 19.770 pesos corporais de bovinos Guzerá, do nascimento aos 365 dias de idade, pertencentes ao banco de dados da Associação Brasileira dos Criadores de Zebu (ABCZ) foi analisado com os objetivos de comparar diferentes estruturas de variâncias residuais, considerando 1, 18, 28 e 53 classes residuais e funções de variância de ordens quadrática a quíntica; e estimar funções de co-variância de diferentes ordens para os efeitos genético aditivo direto, genético materno, de ambiente permanente de animal e de mãe e parâmetros genéticos para os pesos corporais usando modelos de regressão aleatória. Os efeitos aleatórios foram modelados por regressões polinomiais em escala de Legendre com ordens variando de linear a quártica. Os modelos foram comparados pelo teste de razão de verossimilhança e pelos critérios de Informação de Akaike e de Informação Bayesiano de Schwarz. O modelo com 18 classes heterogêneas foi o que melhor se ajustou às variâncias residuais, de acordo com os testes estatísticos, porém, o modelo com função de variância de quinta ordem também mostrou-se apropriado. Os valores de herdabilidade direta estimados foram maiores que os encontrados na literatura, variando de 0,04 a 0,53, mas seguiram a mesma tendência dos estimados pelas análises unicaracterísticas. A seleção para peso em qualquer idade melhoraria o peso em todas as idades no intervalo estudado.

Homogeneidade e heterogeneidade de variância residual em modelos de regressão aleatória sobre o crescimento de caprinos Anglo-Nubianos

O objetivo deste trabalho foi comparar modelos de regressão aleatória com diferentes estruturas de variâncias residuais, na estimação dos componentes de covariância e parâmetros genéticos de características de crescimento de caprinos. A função de regressão por meio de polinômios ortogonais de Legendre de ordem quatro foi usada para modelar a trajetória média de crescimento animal, e de ordem três, para modelar os efeitos genéticos aditivos direto e materno e de ambiente permanente. Diferentes estruturas de variâncias residuais foram consideradas, por meio de classes de uma a sete variâncias residuais, e funções de variâncias com uso dos polinômios ordinários e de Legendre, com ordens que variaram de linear até a do quarto grau. Os modelos foram comparados pelo teste de razão de verossimilhança, o critério de informação de Akaike e o critério bayesiano de Schwarz. Os modelos com funções de variâncias residuais foram superiores aos de classes de variância. O polinômio ordinário de terceira ordem apresentou melhores resultados. Os parâmetros genéticos são influenciados pelo ajuste da variância residual, no entanto, os estimados pelos modelos que consideraram classes de variância, polinômio ordinário e polinômio de Legendre, de terceira ordem, são semelhantes.

On the multiple q-Genocchi and Euler numbers

The purpose of this paper is to present a systematic study of some families of multiple q-Genocchi and Euler numbers by using the multivariate q-Volkenborn integral (= p-adic q-integral) on ℤ p . The investigation of these q-Genocchi numbers and polynomials of higher order leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning ordinary q-Genocchi numbers and polynomials.

A characteristic set method for ordinary difference polynomial systems

We prove several basic properties for difference ascending chains, including a necessary and sufficient condition for an ascending chain to be the characteristic set of its saturation ideal and a necessary and sufficient condition for an ascending chain to be the characteristic set of a reflexive prime ideal. Based on these properties, we propose an algorithm to decompose the zero set of a finite set of difference polynomials into the union of zero sets of certain ascending chains. This decomposition algorithm is implemented and used to solve the perfect ideal membership problem, and to prove certain difference identities automatically.

Vertex-weighted Wiener polynomials for composite graphs

Recently introduced vertex-weighted Wiener polynomials are a common generalization of both vertex-weighted Wiener numbers and ordinary Wiener polynomials. We present here explicit formulae for vertex-weighted Wiener polynomials of the most frequently encountered classes of composite graphs.

Exponential fitted Gauss, Radau and Lobatto methods of low order

Several exponential fitting Runge-Kutta methods of collocation type are derived as a generalization of the Gauss, Radau and Lobatto traditional methods of two steps. The new methods are capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential and ordinary polynomials. Theorems of the truncation error reveal the good behavior of the new methods for stiff problems. Plots of their absolute stability regions that include the whole of the negative real axis are provided. A different procedure to find the parameter of the method is proposed. The variable step Radau method of two stages is derived. Finally, numerical examples underscore the efficiency of the proposed codes, especially when they are integrating stiff problems.

A connection between ordinary and Laplacian spectra of bipartite graphs

Let G be a bipartite graph with n vertices and m edges. Let S(G) be the subdivision of G, obtained by inserting a new vertex on each edge of G. The ordinary characteristic polynomial of S(G) and the Laplacian characteristic polynomial of G are related as φ(S(G),λ) = λm−nψ(G,λ2). If μi, i=1,2, … ,h, are the non-zero Laplacian eigenvalues of G, then the ordinary spectrum of S(G) consists of the numbers , and of n+m−2h zeros. As a corollary, we demonstrate that if ψ (G,λ) is written in the form , then for any tree T of order n and for any k, ck(Sn)≤ ck(T) ≤ ck(Pn), where Sn and Pn are, respectively, the star and the path of order n.

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 $${\mathcal K}$$ 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.

A bound for the Rosenfeld–Gröbner algorithm

We consider the Rosenfeld–Gröbner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F , let M ( F ) be the sum of maximal orders of differential indeterminates occurring in F . We propose a modification of the Rosenfeld–Gröbner algorithm, in which for every intermediate polynomial system F , the bound M ( F ) ⩽ ( n − 1 ) ! M ( F 0 ) holds, where F 0 is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal.

Properties of multiple Hermite and multiple Laguerre polynomials by the generating function

A generating function is obtained for multiple Hermite and multiple Laguerre polynomials from the Rodrigues formula, and then interesting recurrence relations are given. We also find a relation between multiple orthogonal polynomials and ordinary orthogonal polynomials such as multiple Hermite polynomials with classical Hermite polynomials and multiple Laguerre polynomials with classical Laguerre polynomials. Lastly, we find a differential equation for the multiple orthogonal polynomials by the generating function, which is a reconstruction of the works by Aptekarev et al. [Aptekarev, A.I., Branquinho, A., Van Assche, W., 2003, Multiple orthogonal polynomials for classical weights. Transaction of American Mathematical Society, 355, 3887–3914.].

Recurrences and explicit formulae for the expansion and connection coefficients in series of ordinary Bessel polynomials

A formula expressing explicitly the derivatives of ordinary Bessel polynomials of any degree and for any order in terms of the ordinary Bessel polynomials themselves is proved. Another explicit formula, which expresses the ordinary Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original ordinary Bessel coefficients, is also given. A formula for the ordinary Bessel coefficients of the moments of one single ordinary Bessel polynomial of certain degree is proved. Formula for the ordinary Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its ordinary Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between generalized Bessel and ordinary Bessel polynomials is described. Explicit formula for these coefficients between Jacobi and ordinary Bessel polynomials is given, of which the ultraspherical polynomials and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre–ordinary Bessel and Hermite–ordinary Bessel are also developed.

Weighted lattice polynomials of independent random variables

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.

Exponential fitting BDF–Runge–Kutta algorithms

In other papers, the authors presented exponential fitting methods of BDF type. Now, these methods are used to derive some BDF–Runge–Kutta type formulas (of second-, third- and fourth-order), capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential with parameter A and ordinary polynomials. Theorems of the truncation error reveal the good behavior of the new methods for stiff problems. Plots of their absolute stability regions that include the whole of the negative real axis are provided. Different procedures to find the parameter of the method are proposed, using these techniques there will not be necessary to compute the exponential matrix at each step, even when nonlinear problems are integrated. Numerical examples underscore the efficiency of the proposed codes, especially when they are integrating stiff problems.

Adapted BDF Algorithms: Higher-order Methods and Their Stability

We present BDF type formulas of high-order (4, 5 and 6), capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential with parameter A and ordinary polynomials. For A = 0, the new formulas reduce to the classical BDF formulas. Theorems of the local truncation error reveal the good behavior of the new methods with stiff problems. Plots of their 0-stability regions in terms of the eigenvalues of the parameter A h are provided. Plots of their absolute stability regions that include the whole of the negative real axis are provided. The weights of the method usually require the evaluation of a matrix exponential. However, if the dimension of the matrix is large, we shall not perform this calculus and shall only approximate those coefficients once. Numerical examples underscore the efficiency of the proposed codes, especially when one is integrating stiff oscillatory problems.

Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries

In recent years, good algorithms have been developed for finding the zeros of trigonometric polynomials and of ordinary polynomials when written in the form of a truncated Chebyshev polynomial or Legendre polynomial series. In each case, the roots can be found from the eigenvalues of a generalized Frobenius companion matrix whose elements are trivial functions of the Fourier coefficients or Chebyshev coefficients. However, the QR method for computing the companion matrix eigenvalues has a cost that grows proportionally to N 3 where N is the polynomial degree. (By exploiting the special structure of the companion matrices, the cost can be reduced to O ( N 2 ) , but only for large N .) Here, we show that if the polynomial has definite parity, such as a trigonometric polynomial composed only of cosines or a polynomial that is a sum only of Chebyshev polynomials of odd degree, one can exploit these symmetries to halve the size of the problem. This reduces costs in the companion matrix method by a factor ranging between four and eight. For trigonometric polynomials, we give transformations that dramatically reduce costs even if the roots are found by an algorithm other than the companion matrix procedure. We further give reductions for trigonometric polynomials with double parity symmetries which save a factor of sixteen to a factor of sixty-four in the companion matrix algorithm. Special functions such as spherical harmonics, Mathieu functions, prolate spheroidal wavefunctions and Hough functions, all represented by truncated Fourier series with double parity, are a rich source of applications.

Exponentially accurate Runge-free approximation of non-periodic functions from samples on an evenly spaced grid

Approximating a function from its values f ( x i ) at a set of evenly spaced points x i through ( N + 1 ) -point polynomial interpolation often fails because of divergence near the endpoints, the “Runge Phenomenon”. This report shows how to achieve an error that decreases exponentially fast with N . Normalizing the span of the points to [ − 1 , 1 ] , the new strategy applies a filtered trigonometric interpolant on the subinterval x ∈ [ − 1 + D , 1 − D ] and ordinary polynomial interpolation in the two remaining subintervals. Convergence is guaranteed because the width D of the polynomial interpolation subintervals decreases as N → ∞ , being proportional to 1 / N . Applications to the Gibbs Phenomenon and hydrodynamic shocks are discussed.