关于多项式映射自同构恒等集的一个注记

本文给出了判定一个仿射代数集是否是一个自同构恒等集的充分条件.作为一个推论,我们给出了Mckay-Wang的一个问题的一个新证明.我们也给出了一些具体的例子来说明主要的定理.     

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.     

ON AN IDENTITY OF RAMANUJIAN

In the present paper the authors will give a new proof of the classical Ra-manujian identity by the theory of distributions.     

单位积决定的若当矩阵代数(英文)

本文研究了单位积决定的若当矩阵代数M=Mn(R)的条件及分类问题.利用基矩阵及巧妙对对称双线性映射{·,·}进行构造和扩充,用初等矩阵的方法,获得了一系列新的同样重要的定义,结论与证明(与参考文献[1]相比较),推广了参考文献[1]的结论,作为其应用可以进一步证明了Mn(R)上的任意可逆线性映射都是保单位积的.     

关于高维Dedekind和的恒等式

以Ｄｏｎｚａｇｉｅｒ中高维Ｄｅｄｅｋｉｎｄ和为基础，研究了其算术性质，得到了一个有趣的恒等式。     

Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification

The spectral properties of Hermitian matrix polynomials with real eigenvalues have been extensively studied, through classes such as the definite or definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix polynomials, and overdamped or gyroscopically stabilized quadratics. We give a unified treatment of these and related classes that uses the eigenvalue type (or sign characteristic) as a common thread. Equivalent conditions are given for each class in a consistent format. We show that these classes form a hierarchy, all of which are contained in the new class of quasidefinite matrix polynomials. As well as collecting and unifying existing results, we make several new contributions. We propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line. By analyzing their effect on eigenvalue type, we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient, which is important for treating definite and quasidefinite polynomials. We also give a sufficient and necessary condition for a quasihyperbolic matrix polynomial to be strictly isospectral to a real diagonal quasihyperbolic matrix polynomial of the same degree, and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial, thereby identifying an important new class of diagonalizable matrix polynomials.     

Critère d’existence d’idempotent basé sur les algèbres de Rétrocroisement

We study the relationship of backcrossing algebras with mutation algebras and algebras satisfying ω-polynomial identities: we show that in a backcrossing algebra every element of weight 1 generates a mutation algebra and that for any polynomial identity f there is a backcrossing algebra satisfying f. We give a criterion for the existence of idempotent in the case of baric algebras satisfying a nonhomogeneous polynomial identity and containing a backcrossing subalgebra. We give numerous genetic interpretations of the algebraic results.     

Group identities on units of locally finite algebras and twisted group algebras

We study locally finite algebras and twisted group algebras with units satisfying a group identity. As a preliminary result, we obtain a necessary condition for twisted group algebras to satisfy a generalized polynomial identity.     

Banach空间中线性离散时间系统的一致多项式膨胀性

给出了Banach 空间中线性离散时间系统一致多项式膨胀性的概念,并讨论了其离散特征。借助Lyapunov函数给出了线性离散时间系统满足一致多项式膨胀的充要条件。所得结论将一致指数稳定性、指数膨胀性及多项式稳定性中的若干经典结论推广到了一致多项式膨胀性的情形。     

新的Q整图类K_n~(-tk2)

在他人研究整图,Laplace整图和Seidel-整图的基础上,刻画了Q整图新类.对图类K-tk2n的无符号拉普拉斯特征多项式进行研究分析,应用矩阵的初等变换,给出了图类K-tk2n是Q整图的充分必要条件,得到了新的Q整图类K-tk2n及其Q谱.     

A closed-form expression for the Drinfeld modular polynomial Φ<Subscript><Emphasis Type="Italic">T</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>, <Emphasis Type="Italic">Y</Emphasis>)

In this paper we give a closed-form expression for the Drinfeld modular polynomial \({\Phi_T(X,Y) \in \mathbb{F}_q(T)[X,Y]}\) for arbitrary q and prove a conjecture of Schweizer. A new identity involving the Catalan numbers plays a central role.     

Random generation of tournaments and asymmetric graphs with given out-degrees

In this paper, we first present a polynomial algorithm which computes a random tournament with given out-degrees; any tournament having these out-degrees has a nonzero probability to be computed. Then we give a necessary and sufficient condition for a sequence of numbers to be the out-degrees (or similarly the in-degrees) of an asymmetric graph. Lastly, using the above algorithm and this characterization, we design a second polynomial algorithm to compute a random asymmetric graph with given out-degrees, and any asymmetric graph with these out-degrees has a nonzero probability to be found.     

Improvement of the formal and numerical estimation of the constant in some Markov-Bernstein inequalities

Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant in Markov-Bernstein inequalities in L ² for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new sufficient condition is given in order that a polynomial has some complex zeros. This revised version was published online in June 2006 with corrections to the Cover Date.     

A polynomial identity for the bilinear operation in Lie–Yamaguti algebras

We use computer algebra to demonstrate the existence of a multilinear polynomial identity of degree 8 satisfied by the bilinear operation in every Lie–Yamaguti algebra. This identity is a consequence of the defining identities for Lie–Yamaguti algebras, but is not a consequence of anticommutativity. We give an explicit form of this identity as an alternating sum over all permutations of the variables in a nonassociative polynomial with 8 terms. Our computations show that no such identities exist in degrees less than 8.     

A continuous-state polynomial branching process

A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The process can also be obtained from a spectrally positive Lévy process through Lamperti type transformations. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly. Some of those are also new for the classical linear branching process. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branching process arises naturally as the rescaled limit of a sequence of discrete-state processes.     

Zero location by Hermitian forms: The singular case

The Schur-Cohn criterion for the number of zeros of a polynomial inside and outside the unit disc fails if the polynomial has a pair of conjugate zeros or a zero on the unit circle: the corresponding quadratic form is singular. Recently the authors have shown [5] that the classical Schur-Cohn criterion may be deduced from a simple algebraic identity; this yields not only a very simple proof but also a substantial generalization. The method produces a whole family of quadratic forms which may be used for testing the zeros. In the present paper the same algebraic identity is used to show that singularity of these quadratic forms is always due to the presence of pairs of conjugate zeros or zeros on the unit circle. There is a method for ascertaining zero distribution in the singular case by differentiation; we give a derivation of this test on the basis of our matrix-theoritic treatment. The second section deals with the same problem in the case of a general circle or half plane.     

齐次群上二阶半线性偏微分方程的一类Picone型恒等式和Sturmian比较定理

本文给出了齐次群上的一类广义Picone型恒等式,由此证明了以下半线性方程组(其中 表示齐次群上的广义梯度)的Sturmian比较定理及一类振荡定理,并用于Heisenberg群上一类半线性方程.然后利用这里的广义Picone型恒等式证明了Heisenberg群上一类更一般的Hardv型不等式     

The Neumann problem on ellipsoids

The Neumann problem on an ellipsoid in \(\mathbf {R}^n\) asks for a function harmonic inside the ellipsoid whose normal derivative is some specified function on the ellipsoid. We solve this problem when the specified function on the ellipsoid is a normalized polynomial (a polynomial divided by the norm of the normal vector arising from the definition of the ellipsoid). Specifically, we give a necessary and sufficient condition for a solution to exist, and we show that if a solution exists then it is a polynomial whose degree is at most the degree of the polynomial giving the specified function. Furthermore, we give an algorithm for computing this solution. We also solve the corresponding generalized Neumann problem and give an algorithm for computing its solution.     

完全六部图是S-整图的一个充要条件

在他人研究完全多部图的邻接谱的基础上,对整完全多部图的Seidel多项式进行研究分析,以期得到完全六部图G是S-整图的充要条件．从讨论完全六部图的Seidel多项式入手,应用矩阵行初等变换的方法给出完全六部图G是S-整图的充要条件．