伪对称点集与正则单形

要证明了E~n中的有序向量集是伪对称点集的充要条件.利用这一充分必要条件,得到了有关正则单形的几个等价描述,给出了伪对称点集与正则单形的关系的一个结论:设■={A_1,A_2,…,A_(n+1)}是E~n中的点集,则■是n维对称点集的充要条件是以(?)为顶点的单形是正则单形.     

矩阵可对角化的零化多项式的判定

给定数域F上的方阵A,借助等价标准形和数学归纳法证明了如果存在数域F上互素的一次因式乘积的多项式是A的零化多项式,则A可对角化.     

关于切点单形的两个不等式

Definition Let A_i(i=1,2,…,n+1) be the vertex of a simplex Σ_A in n-dimen-sional Euclidean space Eⁿ and A_i′ be the tangent points which the inscribed sphere of Σ_A is tangent to the side face of Σ_A, then the simplex with the tangent points as vertexes is called the tangent points simplex.     

一类n-阶m-点奇异边值问题的正解

研究n-阶m-点奇异边值问题其中h(t)允许在t=0,t=1处奇异,f(t,v_0,v_1,…,v_(n-2))允许在v_i=0(i=0,1,…,n-2)处奇异.利用锥拉伸与压缩不动点定理得到了上述奇异边值问题正解的存在性.     

幂等阵幂零阵多项式可逆的充要条件及一般方阵多项式可逆性的判定

1　幂等阵多项式设Ａ∈Ｆｎ×ｎ 为一幂等矩阵 ,由于Ａ2 =Ａ ,所以任取ｆ(ｘ) ∈Ｆ[ｘ].ｆ(Ａ)总可以化为ｋＡ+ｌＩｎ的形式 (ｋ,ｌ∈Ｆ ,Ｉｎ 为ｎ阶单位阵 ) .对此我们有定理 1　若Ａ为ｎ阶非零幂等矩阵 ,ｋ≠ 0 ,ｌ≠ 0 ,则ｋＡ +ｌＩｎ 可逆 ｋ≠-ｌ.证　充分性因为ｌ≠ 0 ,ｋ≠-ｌ,所以ｌ(ｋ +ｌ)Ｉｎ 可逆又　 (ｋＡ +ｌＩｎ) [-ｋＡ+(ｋ+ｌ)Ｉｎ]=-ｋ2 Ａ2 +ｋ2 Ａ+ｋｌＡ-ｋｌＡ +ｌ(ｋ+ｌ)Ｉｎ=ｌ(ｋ+ｌ)Ｉｎ所以ｋＡ+ｌＩｎ 可逆 .必要性若ｋＡ+ｌＩｎ 可逆 ,而ｋ=-ｌ,则由Ａ(ｋＡ+ｌＩｎ) =ｋＡ2 +ｌＡ =0 ｎｎ,可知Ｒ(Ａ) +Ｒ(…     

零维多项式系统保持重数的零点分解

对零维多项式系统,基于经典的吴方法给出了一个保持重数的零点分解定理及其算法.在一定条件下,该算法计算出的分解是三角化的.     

因子中套代数的极大的n-幂零理想

本文研究因子中套代数的极大的n-幂零理想.利用vN代数中的投影的比较理论,获得了因子中套代数的一个理想是极大的n-幂零理想的一个充要条件,该结果推广了陆芳言等人的一些结果.     

矩阵的秩与零化多项式

从一道考研数学试题出发,深入探讨了矩阵的秩与零化多项式之间的内在联系,推广了已知的相关结果,给出了该类问题的一般处理技巧.     

矩阵的秩与零化多项式

从一道考研数学试题出发,深入探讨了矩阵的秩与零化多项式之间的内在联系,推广了已知的相关结果,给出了该类问题的一般处理技巧.     

给定零点的最近复系数多项式计算问题

Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f(z) and a zero α, in this paper we explore the problem of computing a complex polynomial f(z) such that f(α) = 0 and the distance ∥f-f ∥ is minimal. Considering most of the existing works focus on either certain polynomial basis or certain vector norm, we propose a common computation framework based on both general polynomial basis and general vector norm, and summarize the computing process into a four-step algorithm. Further, to find the explicit expression of f(z), we focus on two specific norms which generalize the familiar lp-norm and mixed norm studied in the existing works, and then compute f(z) explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.     

The Least-Perimeter Partition of a Sphere into Four Equal Areas

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.     

The Zero Set of Fractional Brownian Motion Is a Salem Set

The existence of sets supporting a Borel measure such that its Fourier transform tends to zero at infinity can be traced back to the problem of uniqueness of trigonometric series, studied extensively by Cantor. Given \(\alpha \in (0, 1)\), Beurling asked if there exists a subset of the real line of Hausdorff dimension \(\alpha \) supporting a Borel measure whose Fourier transform converges to zero at infinity with rate \(\alpha /2\). Salem answered the question in the affirmative and such sets are now called Salem sets or rounded sets. Kahane showed that images of compact sets by fractional Brownian motion are Salem sets and this was recently extended to Gaussian random fields with stationary increments and to multi-parameter Brownian sheets. He asked if the level sets of fractional Brownian motion are also Salem sets and the problem has remained open since. This paper answers Kahane’s question in the affirmative. The argument is based on the study of oscillatory integrals with non-smooth amplitudes and new properties of the generalised Euler spiral which have independent interest.     

The Zero Set of a Real Analytic Function

Mathematical Notes -     

The Case Where the Sum of Three Partial Reflections is Equal to Zero

Up to unitary equivalence, we describe all irreducible triples of self-adjoint operators A ₁, A ₂, A ₃ such that (A _i) |–1, 0, 1}, i = 1, 2, 3, and A ₁ + A ₂ + A ₃ = 0.     

General Existence Theorem of Zero Points

Let X be a nonempty, compact, convex set in and let be an upper semicontinuous mapping from X to the collection of nonempty, compact, convex subsets of . It is well known that such a mapping has a stationary point on X; i.e., there exists a point X such that its image under has a nonempty intersection with the normal cone of X at the point. In the case where, for every point in X, it holds that the intersection of the image under with the normal cone of X at the point is either empty or contains the origin 0 ⁿ , then must have a zero point on X; i.e., there exists a point in X such that 0 ⁿ lies in the image of the point. Another well-known condition for the existence of a zero point follows from the Ky Fan coincidence theorem, which says that, if for every point the intersection of the image with the tangent cone of X at the point is nonempty, the mapping must have a zero point. In this paper, we extend all these existence results by giving a general zero-point existence theorem, of which the previous two results are obtained as special cases. We discuss also what kind of solutions may exist when no further conditions are stated on the mapping . Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.     

多目标对策的弱Pareto-Nash平衡点集的稳定性研究

通过定义多目标对策的加权Nash平衡点集,得出它和对应对策的弱Pareto-Nash平衡点集之间的关系,证明了在一定条件下的多目标对策的弱Pareto-Nash平衡点集的稳定性.     

范德蒙德行列式与多项式的不动点

以范德蒙德行列式为工具,得到关于多项式不动点的两个新结论,以及线性代数中某已知结论的一个新证明,另外提供实例说明多项式不动点的应用．     

粗糙集理论在属性约简及知识分类中的应用

本针对不完备信息系统属性约简的两种定义，证明了两的等价性。在此基础上结合粗糙集理论提出了相似矩阵、相似区间的概念，并将其应用于不完备信息系统知识分类的问题中。     

On Sets of Unbounded Divergence at Each Point of Multiple Fourier Series of a Function Equal Zero on a Closed Set

The problem investigated is to characterize sets E, the sets of unbounded divergence (at each point) of single and multiple Fourier series under condition of convergence of these series to zero at each point of the complement of E.For any nonempty open set B T ^N = [0, 2] ^N , N 1, a Lebesgue integrable function f ₀ is constructed which equals zero on the set U = T ^N \ B whose multiple trigonometric Fourier series diverges unboundedly (in the case of summation over squares) at each point of the set