Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we don’t know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = 1, 2. It is known that a two dimensional Q-algebra is just a two dimensional commutative operator algebra on a two dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a three dimensional Q-algebra of the disc algebra on a three dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra.     

Linear operators in finite dimensional probabilistic normed spaces

In this paper, we consider strongly bounded linear operators on a finite dimensional probabilistic normed space and define the topological isomorphism between probabilistic normed spaces. Then we prove that every finite dimensional probabilistic normed space which is a topological vector space is complete.     

关于线性变换的像空间与核空间的直和

讨论数域P上有限维线性空间V上线性变换A的方幂A~k的像空间ImA~k与核空间KerA~k的直和,并将结论推广到无限维线性空间.证明了:V=ImA~k+KerA~k当且仅当ImA~k=ImA~(k+1),以及ImA~k∩KerA~k=0当且仅当KerA~k=KerA~(k+1).     

Solvable Jordan algebras of compact operators

It is proved that a Jordan algebra of compact operators which is closed is either an Engel Jordan algebra, or contains a nonzero finite rank operator; Moreover, it is showed that any solvable Jordan algebra of compact operators on an infinite dimensional Banach space is triangularizable.     

一类微分方程特解的逆矩阵求法

根据函数求导运算与函数的不定积分互为逆运算的思想,以及有限维函数向量空间在求导运算下不变的条件,利用逆矩阵方法得到向量空间中基函数的不定积分,从而给出一定条件下用逆矩阵法求一类常系数非齐次线性微分方程特解的新方法.     

Finite Element Approximations of Symmetric Tensors on Simplicial Grids in $\mathbb{R}^n$: The Higher Order Case

The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k$ ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k$ ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in $\mathbb{R}^n$ for any space dimension. On the contrary, the discrete stress space here is the space of $H(div, Ω\;\mathbb{S}) — P_k$ tensors, and the discrete displacement space here is the space of $L²(Ω ; \mathbb{R}^n) — P_{k-1}$ vectors for $k ≥ n$+1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang.     

Partitions lineaires arguesiennes d'un espace vectoriel

In this work, we study linear and desarguesian partitions of a finite dimensional vector space over a skew-field K. When K is finite, we describe the set of all these partitions as a homogeneous space of the general linear group and we give an enumeration formula.     

Ensembles de classe (0, 1, q,q + 1) de PG (d,q)

We determine all sets Q of points of any finite dimensional protective space P such that each line intersecting Q in more than one point, either is contained in Q or contains exactly one point not on Q. If P is a finite protective space of order q, these sets are the so called sets of class (0, 1, q, q + 1).     

Spreads and Ovoids Translation with Respect to Disjoint Flags

It is shown that if a spread of a finite split Cayley hexagon is translation with respect to two disjoint flags then it is either a hermitian spread or a Ree–Tits spread. Analogously, if an ovoid of a classical generalized quadrangle Q(4, q) is translation with respect to two disjoint flags then it is either an elliptic quadric or a Suzuki–Tits ovoid. In the course of obtaining these results, we introduce the notion of local polarity for ovoid-spread pairings and show that if an ovoid-spread pairing is locally polar at each of its elements then it arises from a polarity.     

Determining functionals for random partial differential equations

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional random dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results.We apply our results to the 2D Navier-Stokes equations forced by a white noise.     

Two-stage stochastic lot-sizing problem under cost uncertainty

For production planning problems, cost parameters can be uncertain due to marketing activities and interest rate fluctuation. In this paper, we consider a single-item two-stage stochastic lot-sizing problem under cost parameter uncertainty. Assuming cost parameters will increase or decrease after time period p each with certain probability, we minimize the total expected cost for a finite horizon problem. We develop an extended linear programming formulation in a higher dimensional space that can provide integral solutions by showing that its constraint matrix is totally unimodular. We also project this extended formulation to a lower dimensional space and obtain a corresponding extended formulation in the lower dimensional space. Final computational experiments demonstrate that the extended formulation is more efficient and performs more stable than the two-stage stochastic mixed-integer programming formulation.     

Three dimensional linear motion transformation in a higher order dimensional space

The objective of the work presented in this paper is an attempt at solving and transforming of the known from the classical mechanics three dimensional – single mass mathematical and mechanical vibration models in a higher order dimensional space with any virtual sectional curvature – positive or negative, constant or variable. The object of the investigation is a class of three dimensional surfaces. The aims of the work presented in this paper are to illustrate the performance of the common algorithm in three dimensional linear motion transformation, that means to transform 3D space in a higher order dimensional space and a comparison is derived on the behavior of the common algorithm depending on the surface properties. A characterization of the Riemannian Manifolds is performed by means of curvature operators in the three dimensional solution. The computer codes Mathematica and MATLAB are used in the numerical simulation. The system motion is investigated in a 3-D qualitative aspect in time and frequency domain. The application can be in topology when geodesists make snap shots of the surface profile, then the curved lines can be analyzed and transformed in the desired space dimension. Any kind of a trajectory of motion can be transformed successfully in a higher order dimensional space and vice verse by means of applying of the common algorithm.     

Study of a finite element method for the time-dependent generalized Stokes system associated with viscoelastic flow

A three-field finite element scheme designed for solving systems of partial differential equations governing time-dependent viscoelastic flows is studied. Once a classical backward Euler time discretization is performed, the resulting three-field system of equations allows for a stable approximation of velocity, pressure and extra stress tensor, by means of continuous piecewise linear finite elements, in both two- and three- dimensional space. This is proved to hold for the linearized form of the system. An advantage of the new formulation is the fact that it provides an algorithm for the explicit iterative resolution of system nonlinearities. Convergence in an appropriate sense applying to these three flow fields is demonstrated.     

Pathological asymptotic behavior of control systems in Banach space

New phenomena arising when a linear dynamical system is defined on an infinite dimensional Banach space, although negligible from an engineering standpoint when only a finite time-interval is considered, become crucial when the asymptotic (feedback) behavior of the system is of interest. Pathologies with respect to the correspondent finite dimensional case are displayed even when the operator acting on the state is bounded.In particular, although in such case, the classical controllability and observability theory admits a natural generalization to infinite dimensions, the finite dimensional relationships between controllability and stabilizability fails. A few examples are given of systems that are approximately controllable and yet are not stabilizable: Moreover, such examples are drawn from a class of systems that can never be exactly controllable. The analysis is carried out using the perturbation theory of the spectrum. Another new feature of the infinite dimensionality of the state space is that even if the spectrum of an operator has the max of its real part equal to 0, yet the associated homogeneous differential equation may be globally asymptotically stable: Its consequence on stabilizability is also examined.     

W(2)到Kac模的权导子空间

本文研究了Witt型模李超代数W(2)到Kac模K(λ)的权导子空间问题.利用分类讨论及线性方程组求解的方法,获得了W(2)到K(λ)的权导子空间要么是零维要么是一维的结果,推广了李代数到其模的权导子空间的相应结果.     

Abel群的一些分解定理的推广(Ⅲ)

从主理想整环上有界模分解的Prüfer-Baer定理出发,研究(无限维)向量空间的代数的线性变换的几个基本问题,得到了如下结果:设V是域F上的(无限维)向量空间,A是V上的一个代数的线性变换,则有(1)若任何与A可交换的线性变换均与线性变换B可交换,则B=f(A),其中f是F上的多项式.进而线性变换B也是代数的.(2) V中存在一组基,使A在这组基下的矩阵是有理标准型(经典标准型)矩阵.当F是代数闭域时,经典标准型矩阵即为若当标准型矩阵.(3)当F是代数闭域时,A存在相应的Jordan-Chevalley分解.进一步,该结论在完全域上仍成立.这些研究推广了有限维向量空间上线性变换的相关结果.     

A distance function for computing on finite subsets of Euclidean spaces

In practical purposes for some geometrical problems, specially the fields in common with computer science, we deal with information of some finite number of points. The problem often arises here is: “How are we able to define a plausible distance function on a finite three dimensional space?” In this paper, we define such a distance function in order to apply it to further purposes, e.g. in the field settings of transportation theory and geometry. More precisely, we present a new model for traveling salesman problem and vehicle routing problem for two dimensional manifolds in three dimensional Euclidean space, the second problem on which we focus on this line is, three dimensional triangulation.     

Strong unicity in normed linear spaces

We consider best approximations in a real or complex normed linear space by elements of a finite dimensional subspace. It is the purpose of this paper to characterize, when a best approximation to a given element is strongly unique.     

线性变换的值域与核的维数关系定理的证法探析

本文首先证明了有限维线性空间上的线性变换是单射当且仅当它是满射这一结论,其次分析了通常高等代数教材中关于线性变换的值域与核的维数关系定理的两种证明方法,并给出了该定理的一种新的证明方法.     

关于Hilbert空间上无限维线性定常系统的典型分解

有限维线性定常系统的典型分解定理是控制理论的重要成果之一 [1] ,也是计算机控制的重要理论基础之一 ,本文将这一结果推广到 Hilbert空间上的无限维线性定常系统 .