强K(a)hler-Finsler流形上(p,q)形式的中值Laplace算子

本文给出了强K(a)hler-Finsler流形上中值Laplace算子的一些性质,如自伴性质,散度形式等.与K(a)hler流形上利用逆变基本张量[11]及其在Finsler流形上的变形[5,10]作为密度函数定义流形上的逐点内积及整体内积不同,作者利用强K(a)hler-Finsler流形上的逆变密切Kahler度量作为密度函数定义了流形上的逐点内积和整体内积,并定义了强K(a)hler-Finsler流形上的Hodge-Laplace算子,它可看作函数情形中值Laplace算子的推广.     

Frames for vector spaces and affine spaces

A finite frame for a finite dimensional Hilbert space is simply a spanning sequence. We show that the linear functionals given by the dual frame vectors do not depend on the inner product, and thus it is possible to extend the frame expansion (and other elements of frame theory) to any finite spanning sequence for a vector space. The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact that the associated Gramian matrix is an orthogonal projection. Existing generalisations of the frame expansion to Banach spaces involve an analogue of the frame bounds and frame operator.The potential applications of our results are considerable. Whenever there is a natural spanning set for a vector space, computations can be done directly with it, in an efficient and stable way. We illustrate this with a diverse range of examples, including multivariate spline spaces, generalised barycentric coordinates, and vector spaces over the rationals, such as the cyclotomic fields.     

Nevanlinna Functions,Perturbation Formulas,and Triplets of Hilbert Spaces

Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space ?? and let A be a selfadjoint operator extension of S in ??. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, 5 is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q - function of S and A belongs to the subclass No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space ?? the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space ?? is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q - function of S and A belongs to the subclass N₁ of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.     

Scale-independent complementary bivariational principles in a complex Hilbert space,and their application to a scattering problem

Scale-independent complementary bivariational principles in a complex Hilbert space are derived from the stationary principle. These principles consist of two scale-independent functionals which yield upper bounds and lower bounds, respectively, to both the real and the imaginary part of a particular quantity associated with an inhomogeneous linear equation. They have the advantage that one need only guess the form of solutions of the equation and its auxiliary equation, not their size. Moreover, for a given pair of trial functions, they yield better bounds than the scale-dependent complementary bivariational principles obtained by Barnsley and Baker. Their application to a scattering problem yields scale-independent complementary bivariational principles for the scattering amplitude as well as those for the total scattering cross section.     

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

We continue the study of counting complexity begun in [13], [14], [15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. The reduction is based on a new formula for the coefficients of the Hilbert polynomial of a smooth variety. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound in Bach [1].     

Spectral approximation for compact integral operators by degenerate kernel methods

We establish a new improved error estimate for the solution of the integral equation eigenvalue problem by degenerate kernel methods. In [6] these estimates were proved under the assumption of normality of the original kernel as well as of the approximating degenerate kernel. Now we consider any compact integral operator and a general Banach space situation, in contrast to the Hilbert space setting in [6], This will be done by combining the techniques in [6] with the suitably transformed estimates of [5]. Our results show that degenerate kernel methods have, besides their overall property of furnishing easy approximations to eigenfunctions, for eigenvalues an order of convergence comparable to quadrature methods.     

Spectrum localization in Banach spaces II

A continuation of [6]. Gershgorin-type estimates for spectra in Banach spaces and Hilbert spaces are established when the set of perturbations of a given operator is a line segment, a linear image of the unit operator ball on a Hilbert space, and a ball of operators on a Banach space.     

The Iterated Aluthge Transform of an Operator

The Aluthge transform (defined below) of an operator T on Hilbert space has been studied extensively, most often in connection with p-hyponormal operators. In [6] the present authors initiated a study of various relations between an arbitrary operator T and its associated , and this study was continued in [7], in which relations between the spectral pictures of T and were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates { ⁽ⁿ⁾} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence { ⁽ⁿ⁾} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6). Submitted: December 5, 2000? Revised: August 30, 2001.     

Normal Extensions of Operators to Krein Spaces

In this paper, it is proved that every bounded linear operator on a Hilbert space has a normal extension to a Krein space. Two criteria for J-subnormality are given. In particular, in order that T be subnormal, it suffices that exp(-\bar \Lambda T^*)exp(\Lambda T) be a positive definite operator function on a bounded infinite subset of complex plane. This improves the condition of Bram [4]. Also it is proved that the local spectral subspaces are closed for J-subnormal operators.     

Factorization of J-expansive meromorphic operator-valued functions

The factorization theorems are a generalization for J-biexpansive meromorphic operator-valued functions on an infinite-dimensional Hilbert space of the theorems on decomposition of J-expansive matrix functions on a finite-dimensional Hilbert space due to A. V. Efimov and V. P. Potapov [Uspekhi Mat. Nauk28 (1973), 65–130; Trudy Moskov. Mat. Ob??.4 (1955), 125–236]. They also generalize theorems on factorization of J-expansive meromorphic operator functions due to Ju. P. Ginzburg [Izv. Vys?. U?ebn. Zaved. Matematika32 (1963), 45–53]. Within the framework of generalized network theory, the results can be applied to the J-biexpansive real operators that characterize a Hilbert port. Application of the extraction procedure to a given real operator leads to its splitting into a product of real factors, corresponding to Hilbert ports of a simpler structure. This can be interpreted as an extension of the classical method of synthesis of passive n-ports by factor decomposition.     

强Khler-Finsler流形上(p,q)形式的中值Laplace算子(英文)

本文给出了强Khler-Finsler流形上中值Laplace算子的一些性质,如自伴性质,散度形式等。与Khler流形上利用逆变基本张量及其在Finsler流形上的变形作为密度函数定义流形上的逐点内积及整体内积不同,作者利用强Khler-Finsler流形上的逆变密切Khler度量作为密度函数定义了流形上的逐点内积和整体内积,并定义了强Khler-Finsler流形上的Hodge-Laplace算子,它可看作函数情形中值Laplace算子的推广。     

On the iterates of Jackson type operator GS,N in Hilbert space

In this paper we study the limit of the iterates of Jackson type operator. Our results continue the works of Badea [2] and Nagler et al. [9, 10]. The proofs are based on spectral theory of linear operators and are performed at first for Hilbert space and then are extended for some Banach spaces.     

Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

Concerning the semilocal convergence of Newton’s method and convex majorants

We approximate a locally unique solution of an equation on a Banach space setting using Newton’smethod.Motivated by the work by Ferreira and Svaiter [5] but using more precise majorization sequences, and under the same computational cost we provide: a larger convergence region; finer error bounds on the distances involved, and an at least as precise information on the location of the solution than in [5]. The results can also compare favorably to the corresponding ones given byWang in [10]. Finally we complete the study with two concrete applications.      

Bohr Inequality for Multiple Op erators

An absolute value equation is established for linear combinations of two operators. When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained. Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2] and [3].     

LINEAR TRANSPORT EQUATION WITH INDEFINITE COLLISION OPERATOR

The operator theory on indefinite inner product spaces is used to discuss the halfrange problem of linear transport equation with indefinite collision operator. A counterexample to [1] is given and a relation between measures of nonuniqueness and noncompleteness is established.     

On the Convergence of an Interpolation Process to an Entire Operator in a Hilbert Space

For an entire operator in a Hilbert space, we consider an interpolation operator-valued polynomial that has a fairly simple structure. A theorem on pointwise convergence of the iteration process to the entire operator is proved.     

关于数值数学的一个典型问题

<正> Collatz L.在综述性文章[1]和[2]中就数值数学的典型问题归纳为五类,第一类是方程Tu=φ或Tu=u的解.关于这类问题主要是寻找解的存在性定理和解的存在区间以及唯一性定理等等. 如所周知,由初始元u_o出发,经过迭代     

Jensen’s inequality for operator monotone functions

In this paper, we shall show that some classical inequalities for monotone functions also hold for operator monotone functions on an arbitrary Hilbert spaceH. Such results can be found in the classical book [1, p. 83] or in a new book [2].     

带时滞机器人模型解的展开

研究一个带有时滞机器人模型解的性质, 其中机器人的动力学行为由一组含有时滞的微分方程描述.通过引入Hilbert状态空间将其写成一个发展方程, 利用半群理论得到抽象发展方程的适定性. 通过对系统算子的谱作细致分析,得到谱的渐近表达式, 并证明系统的本征函数并不构成空间的基.但我们证明了对时间大于5倍时滞时, 方程的解按照本征向量的展开式.