混合序列加权和的强收敛性

本文给出混合序列加权和的强收敛性的一些充分条件，这些结论推广和改进了文［１］定理３，文［２］定理３；文［３］定理４．１５以及文［４］定理４．     

再谈欧氏空间的变换和线性变换

文［３］推广了文［１］～［２］中的全部定理,文［４］又推广了文［３］的全部定理,本文推广了文［４］的全部定理,指出了文［４］中的一个错误论断,并给出了线性变换的一个等价条件．为简便,文中的Ｖ表示欧氏空间,ＶＶ表示Ｖ的所有变换的集合,Ｌ（Ｖ）表示Ｖ的所...     

谈欧氏空间中的变换和线性变换

谈欧氏空间中的变换和线性变换彭明海彭学梅（湖南省吉首大学４１６０００）文［３］推广了文［１］、文［２］的全部定理，本文不仅推广了文［３］的全部定理，而且得出新的结果．同时，论证也较简捷．今介绍给同行参考．定理１σ和τ是欧氏空间Ｖ的两个变换，φ是Ｖ的线...     

I.J.Matrix定理的更广泛推广

Ｉ．Ｊ．Ｍａｔｒｉｘ定理的更广泛推广张之正（河南洛阳师专数学系４７１０２２）近年来，数学通报连续讨论了１．Ｊ．Ｍａｔｒｉｘ定理的一些推广及其应用（文［ｌ］，［２］用复交函数中的留数定理，文［３Ｊ用初等方法，文［４］用线性代数中的Ｖａｎｄｅｒｍｏｎｄｅ...     

“向量极值问题的最优性条件”一文的注记

本文给出了文［１］中的几个主要结果（即［１］中的推论２，定理４，定理６及定理７）的反例，进而更正了有关结果     

有约束错误指定模型下的优化问题

文［１］ 利用文［3］的结果证明了一般有约束线性模型下参数的最优估计的方差协方差阵与有约束错误指定模型下最优估计的方差协方差阵间差是非负定的充分条件［２］也是必要的。更进一步，文［4］将［1］中的定理1推广到奇异线性模型上。该文的主要目的是证明了［２］的猜想在奇异线性模型中也是正确的，同时，推广了［１］中的定理2。     

欧氏空间中的变换和线性变换

欧氏空间中的变换和线性变换郝秀梅，杨子胥（山东财政学院数学系２５００１４）文［１］、［２］讨论了欧氏空间中几类与内积相关联的变换必为线性变换，本文不仅推广了该二文的全部定理，而且刻划了文［１］定理１的变换Ｔ．定理１设Ｔ是欧氏空间Ｖ的一个变换．如果存在...     

有心圆锥曲线的阿基米德定理的统一证法

有心圆锥曲线的阿基米德定理的统一证法孔繁秋（厦门市禾山中学３６１００９）过圆锥曲线弦的两端的切线与弦围成的三角形称为阿基米德三角形，弦叫做这三角形的底边．文［１］给出了抛物线的阿基米德定理，文［２］给出了圆锥曲线的阿基米德定理的统一表述，即定理圆锥曲...     

Vandermonde行列式的一个推广及其在初等数学中的应用

Ｖａｎｄｅｒｍｏｎｄｅ行列式的一个推广及其在初等数学中的应用肖振纲（湖南岳阳师专４１４０００）文［１］，［２］先后利用复变函数中的留数定理讨论了初等数学中的Ｉ．Ｊ．Ｍａｔｒｉｘ定理的两个推广，文［３］则用初等方法讨论了所述两个推广的关系，本文首先给出...     

两个立阵式对应元素之和的立阵式

文［１］介绍了立阵式并讨论了立阵式的性质，文［２］介绍了立阵式的Ｌａｐｌａｃｅ定理，本文在文此基础上，讨论了两个立阵式的对应元素之和的立阵式，并建立了它与牛顿二项式定理中，项之间的对应联系和计算形式。     

Givens' Transformation Applied to Quaternion Valued Vectors

Givens' transformation (1954) was originally applied to real matrices. We shall give an extension to quaternion valued matrices. The complex case will be treated in the introduction. We observe that the classical Givens' rotation in the real and in the complex case is itself a quaternion using an isomorphism between certain (2×2) matrices and R ⁴ equipped with the quaternion multiplication. In the real and complex case Givens' (2×2) matrix is determined uniquely up to an arbitrary (real or complex) factor with ||=1. However, because of the noncommutativity of quaternions, we shall show that in the quaternion case such a factor must obey certain additional restrictions. There are two numerical examples including a MATLAB program and some hints for implementation. Matrices with quaternion entries arise, e.g., in quantum mechanics.This revised version was published online in October 2005 with corrections to the Cover Date.     

Compact Hyperhermitian–Weyl and Quaternion Hermitian–Weyl Manifolds

Let (M, g, H) be a quaternion Hermitian manifold. The additional datum of a torsion-free connection D preserving both the quaternionic structure H and the conformal class of g defines on M the structure of quaternion Hermitian–Weyl manifold. Under the compactness assumption of both M and the leaves of a canonical foliation, M is here shown to project on a locally 3-Sasakian orbifold P. Then M is proved to admit both a compatible global complex structure and a finite covering M carrying a hyperhermitian–Weyl structure. The uniqueness of the Weyl structure compatible with a given quaternion Hermitian metric and some restrictions on the Betti numbers are also obtained.     

Time–frequency analysis of bivariate signals

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. We show that an adequate quaternion Fourier transform permits to build relevant time–frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, a bivariate counterpart of the usual analytic signal of real signals is introduced, called the quaternion embedding of bivariate signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time–frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.     

标准算子代数上的Jordan映射

Let A be a standard operator algebra on a Banach space of dimension 〉 1 and B be an arbitrary algebra over Q the field of rational numbers. Suppose that M ： A → B and M^＊ ： B → A are surjective maps such that {M（r（aM^＊（x）＋M^＊（x）a））=r（M（a）x＋xM（a））, M^＊（r（M（a）x＋xM（a）））=r（aM^＊（x）＋M^＊（x）a） for all a ∈ A, x ∈ B, where r is a fixed nonzero rational number. Then both M and M^＊ are additive.     

四元数矩阵方程AX=B的实部半正定解(英文)

一个ｎ×ｎ实四元数矩阵称为实部半正定（或正定）矩阵，如果对于任意的非零ｎ维四元数列向量ｘ，有Ｒｅ［ｘＡｘ］≥０（或＞０）．本文给出了四元数矩阵方程ＡＸ＝Ｂ有实部半正定（或正定）矩阵解的充要条件及其通解的表达式，并给出了四元数分块阵为实部半正定（或正定）矩阵的一个判别法则     

四元数矩阵方程AXAH=B的最小二乘解

引入了四元数矩阵范数的概念，通过使用四无数矩阵的奇异值分解，给出了四元数矩阵方程AXA^H＝B在最小二乘意义下的Hermitian解以及Skew-Hermitian解．     

The Least Square Solutions to the Quaternion Matrix Equation AX=B

61.IntroductionTheconsiderableprogresseshavebeenmadeinthetheoryofmatricesoverskewfields-However,sofar,wehavenotseenanyofstudyresultsofthe1eastsquareproblemofmatricesoverskewfields.Thematrixequation'AX=B,(l)whichisveryimportant,wereinvestigateddeeplyin[1~3J.lnthispaper,wedefineanormofarealquaternionmatrix,giveexpressionsoftheleastsquaresolutionsofthequaternionma-trixequation(l)andtheequationwiththeconstraintconditionDX=E.Throughoutthispaper,wedenotetherealquaternionfieldbyH,thesetofallmXnma…     

ORDER PROPERTIES AND CONSTRUCTION OF SYMPLECTIC RUNGE-KUTTA METHODS

1. IntroductionFOr a given s stage Runge-Kutta methodwith A = [ail], p = [pl, PZt... 5 P.]T and ac = [afl, ry23... ) %]T / 0, we introduce thefollowing simplifying conditions as in Butcher [1]and make the notational convensionwhere 1 5 m? pi(x), i ~ 1, 2, 3,' ? are arbitrarily given i--th polynomials with the property that pi(0) = 0,Note that B(P), C(P) and D(P) are equivalent to BI,. = 0, CI,P = 0 and DI,. = 0respectively. We shall always denote BI,., CI,., DI,. and VI,. by B, …     

A proof of the Pfister Factor Conjecture

It is shown that any split product of quaternion algebras with orthogonal involution is adjoint to a Pfister form. This settles the Pfister Factor Conjecture formulated by D.B. Shapiro. A more general problem on decomposability for algebras with involution is posed and solved in the case where the algebra is equivalent to a quaternion algebra.     

Yet Another Description of the Connes—Higson Functor

Mathematical Notes - Suppose that A and B are C*-algebras, A is separable, and B is stable. The elements of the group E1(A, B) in Connes—Higson E-theory are represented by *-homomorphisms...