Sur les identités polynomiales vérifiées par les algèbres de rétrocroisement

We study the ideal of polynomial identities of a single indeterminate satisfied by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and su?cient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities satified by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8.     

Two-Orthogonal Polynomial Sequences as Eigenfunctions of a Third-Order Differential Operator

This paper presents functional identities fulfilled by the forms of the dual sequence of polynomial eigenfunctions of certain differential operators, belonging to the class of the two-orthogonal polynomial sequences. For a specific third-order lowering operator, the correspondent matrix differential identity is deduced, proving that the resultant polynomial sequence is a classical polynomial sequence in the Hahn’s sense. As an example, the vectorial relation fulfilled by the tuple of functionals (u ₀, u ₁) of a two-orthogonal polynomial sequences analogous to the classical Laguerre polynomials is given, treated in a work of Ben Cheikh and Douak.     

Matrices of Formal Power Series Associated to Binomial Posets

We introduce an operation that assigns to each binomial poset a partially ordered set for which the number of saturated chains in any interval is a function of two parameters. We develop a corresponding theory of generating functions involving noncommutative formal power series modulo the closure of a principal ideal, which may be faithfully represented by the limit of an infinite sequence of lower triangular matrix representations. The framework allows us to construct matrices of formal power series whose inverse may be easily calculated using the relation between the Möbius and zeta functions, and to find a unified model for the Tchebyshev polynomials of the first kind and for the derivative polynomials used to express the derivatives of the secant function as a polynomial of the tangent function.On leave from the Rényi Mathematical Institute of the Hungarian Academy of Sciences.     

Polynomial identities for the ternary cyclic sum

We simplify the results of Bremner and Hentzel [J. Algebra 231 (2000) 387–405] on polynomial identities of degree 9 in two variables satisfied by the ternary cyclic sum [a, b, c] = abc + bca + cab in every totally associative ternary algebra. We also obtain new identities of degree 9 in three variables which do not follow from the identities in two variables. Our results depend on (i) the LLL algorithm for lattice basis reduction, and (ii) linearization operators in the group algebra of the symmetric group which permit efficient computation of the representation matrices for a non-linear identity. Our computational methods can be applied to polynomial identities for other algebraic structures.     

Ryser's permanent identity in the symmetric algebra

The polynomial algebra ofver a field is canonically isomorphic to the symmetric algebra over a vector space. Several identities expressing homogeneous polynomials in terms of sums of powers of linear polynomials are exploited to obtain Ryser'ś permanent identity along with extensions of some recent identities due to Bebiano.     

The Perturbation ϕ2, 1 of the M(p,p + 1) Models of Conformal Field Theory and Related Polynomial-Character Identities

Using q-trinomial coefficients of Andrews and Baxter along with the technique of telescopic expansions, we propose and prove a complete set of polynomial identities of Rogers-Ramanujan type for M(p, p + 1) models of conformal field theory perturbed by the operator _{2, 1}. The bosonic form of our polynomials is closely related to corner transfer matrix sums which arise in the computation of the order parameter in the regime 1⁺ of A _p–1 dilute models. In the limit where the degree of the polynomials tends to infinity our identities provide new companion fermionic representations for all Virasoro characters of unitary minimal series.     

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L²[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered. These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.     

关于分圆多项式的Schinzel等式

对一无平方因子的奇数ｎ＞１，　分圆多项式φｎ（ｘ）　　满足Ｓｃｈｉｎｚｅｌ等式，　φｎ（ｘ）＝Ｐ２ｎ，ｍ（ｘ）－（－１／ｍ）ｍｘＱ２ｎ，ｍ（ｘ），　　这里Ｐｎ，ｍ（ｘ）和　Ｑｎ，ｍ（ｘ）是整系数多项式且　ｍ｜ｎ．本文给出两个简明的公式来计算　Ｐｎ，ｍ（ｘ）　和　Ｑｎ，ｍ（ｘ）　　．     

Polynomial interpolation of operators

Necessary and sufficient conditions for the solvability of the polynomial operator interpolation problem in an arbitrary vector space are obtained (for the existence of a Hermite-type operator polynomial, conditions are obtained in a Hilbert space). Interpolational operator formulas describing the whole set of interpolants in these spaces as well as a subset of those polynomials preserving operator polynomials of the corresponding degree are constructed. In the metric of a measure space of operators, an accuracy estimate is obtained and a theorem on the convergence of interpolational operator processes is proved for polynomial operators. Applications of the operator interpolation to the solution of some problems are described. Bibliography: 134 titles. This paper is a continuation of the work published inObchyslyuval'na ta Prykladna Maternatyka, No. 78 (1994). The numeration of chapters, assertions, and formulas is continued. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp 10–116.     

Shape preserving polynomial curves

We introduce particular systems of functions and study the properties of the associated Bézier-type curve for families of data points in the real affine space. The systems of functions are defined with the help of some linear and positive operators, which have specific properties: total positivity, nullity diminishing property and which are similar to the Bernstein polynomial operator. When the operators are polynomial, the curves are polynomial and their degrees are independent of the number of data points. Examples built with classical polynomial operators give algebraic curves written with the Jacobi polynomials, and trigonometric curves if the first and the last data points are identical.     

基于吴方法的几何定理证明的恒等式方法

多年来通常认为以吴方法为代表的几何定理机器证明的坐标法给出的证明不可读,或不是图灵意义下的类人解答.其实,只要对吴氏的算法做不多的改进,即将命题的结论多项式表示为其条件多项式的线性组合,就能获得不依赖于理论、算法和大量计算过程的恒等式明证.这样的恒等式可以转化为其他更简明且更有直观几何意义的点几何形式或向量及其他形式,从而获得多种证明方法.这也证明了点几何恒等式明证方法对等式型几何命题的普遍有效性.     

Formulas for zonal polynomials

New integral and differential formulas for zonal polynomials are proved. As illustrations, zonal polynomials corresponding to partitions of two parts are computed. A method is presented, based on a certain partial differential operator, for expressing an orthogonally invariant polynomial as a linear combination of zonals. Zonal polynomials are expressed as linear combinations of well-known symmetric polynomials.     

An efficient algorithm for factoring polynomials over algebraic extension field

An efficient algorithm is proposed for factoring polynomials over an algebraic extension field defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Gröbner basis, no extra Gröbner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a characteristic polynomial of a generic linear map. Then this polynomial is factorized over the ground field. From its factors, the factorization of the polynomial over the extension field is obtained. The algorithm has been implemented in Magma and computer experiments indicate that it is very efficient, particularly for complicated examples.     

Monogenic Differential Operators

In this paper we consider operators acting on Clifford algebra valued polynomials and, in particular, differential operators with polynomial coefficients. The decomposition of polynomials into homogeneous pieces leads to the classical homogeneous decomposition of operators and the further decomposition of homogeneous polynomials into monogenic polynomials leads to the concept of monogenic operator. Monogenic operators are characterized in terms of commutation relations and the monogenic decomposition of differential operators is studied in detail.     

On conservative approximation by linear polynomial operators an extension of the Bernstein's operator

In this work we study linear polynomial operators preserving some consecutive i-convexities and leaving invariant the polynomials up to a certain degree. First, we study the existence of an incom patibility between the conservation of certain i-convexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DeVore about the Bernstein's operator is extended. Finally, from these results a generalized Bernstein's operator is obtained. This work was supported by Junta de Andalucia. Grupo de investigación: Matemática Aplicada. Código: 1107     

On almost summing polynomials and multilinear mappings

In this article we explore the notion of everywhere almost summing polynomials and define a natural norm which makes this class a Banach polynomial ideal which is a holomorphy type and also coherent and compatible with the notion of almost summing linear operators. Similar results are not valid for the original concept of almost summing polynomials, due to G. Botelho.     

Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces

We show that a Banach space E has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on E can be uniformly approximated on compact sets by homogeneous polynomials which are members of the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established also for the compact approximation property.     

Solving evolutionary-type differential equations and physical problems using the operator method

We present a general operator method based on the advanced technique of the inverse derivative operator for solving a wide range of problems described by some classes of differential equations. We construct and use inverse differential operators to solve several differential equations. We obtain operator identities involving an inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. We present examples of using the operator method to construct solutions of equations containing linear and quadratic forms of a pair of operators satisfying Heisenberg-type relations and solutions of various modifications of partial differential equations of the Fourier heat conduction type, Fokker–Planck type, Black–Scholes type, etc. We demonstrate using the operator technique to solve several physical problems related to the charge motion in quantum mechanics, heat propagation, and the dynamics of the beams in accelerators.     

An algebraic identity and the Jacobi–Trudi formula

The first Jacobi–Trudi identity expresses Schur polynomials as determinants of matrices, the entries of which are complete homogeneous polynomials. The Schur polynomials were defined by Cauchy in 1815 as the quotients of determinants constructed from certain partitions. The Schur polynomials have become very important because of their close relationship with the irreducible characters of the symmetric groups and the general linear groups, as well as due to their numerous applications in combinatorics. The Jacobi–Trudi identity was first formulated by Jacobi in 1841 and proved by Nicola Trudi in 1864. Since then, this identity and its numerous generalizations have been the focus of much attention due to the important role which they play in various areas of mathematics, including mathematical physics, representation theory, and algebraic geometry. Various proofs of the Jacobi–Trudi identity, which are based on different ideas (in particular, a natural combinatorial proof using Young tableaux), have been found. The paper contains a short simple proof of the first Jacobi–Trudi identity and discusses its relationship with other well-known polynomial identities.     

Efficient application of nonlinear stationary operators in adaptive wavelet methods—the isotropic case

This paper deals with the efficient application of nonlinear operators in wavelet coordinates using a representation based on local polynomials. In the framework of adaptive wavelet methods for solving, e.g., PDEs or eigenvalue problems, one has to apply the operator to a vector on a target wavelet index set. The central task is to apply the operator as fast as possible in order to obtain an efficient overall scheme. This work presents a new approach of dealing with this problem. The basic ideas together with an implementation for a specific PDE on an L-shaped domain were presented firstly in [38]. Considering the approximation of a function based on wavelets consisting of piecewise polynomials, e.g., spline wavelets, one can represent each wavelet using local polynomials on cells of the underlying domain. Because of the multilevel structure of the wavelet spaces, the generated polynomial usually consists of many overlapping pieces living on different spatial levels. Since nonlinear operators, by definition, cannot generally be applied to a linear decomposition exactly, a locally unique representation is sought. The application of the operator to these polynomials now has a simple structure due to the locality of the polynomials and many operators can be applied exactly to the local polynomials. From these results, the values of the target wavelet index set can be reconstructed. It is shown that all these steps can be applied in optimal linear complexity. The purpose of the presented paper is to provide a self-consistent development of this operator application independent of the particular PDE, operator, underlying domain, types of wavelets, or space dimension, thereby extending and modifying the previous ideas from [38].