Matrix subspaces and determinantal hypersurfaces

Nonsingular matrix subspaces can be separated into two categories: by being either invertible, or merely possessing invertible elements. The former class was introduced for factoring matrices into the product of two matrices. With the latter, the problem of characterizing the inverses and related nonlinear matrix geometries arises. For the singular elements there is a natural concept of spectrum defined in terms of determinantal hypersurfaces, linking matrix analysis with algebraic geometry. With this, matrix subspaces and the respective Grassmannians are split into equivalence classes. Conditioning of matrix subspaces is addressed.     

非奇H-矩阵的实用判别准则

非奇H-矩阵在科学和工程实际中有着广泛地应用,但在实际中判定一个矩阵是否为非奇H-矩阵是比较困难的.通过构造不同的正对角阵,结合不等式的放缩技巧,给出了一些比较实用的新条件,改进和推广了现有的一些结论,并给出相应的一些数值算例来说明结果的有效性.     

Equivalence,reduction and minimization of finite fuzzy-automata

A new algebraic approach to the problem of equivalence, reduction and minimization of some kinds of fuzzy-automata is given. A system of necessary and sufficient conditions for equivalence of weakly initial fuzzy-automata is formulated. Some algorithms considering the equivalence for two fuzzy-automata are constructed.     

具有输出字符功能的模糊自动机的最小化问题

通过文献[8]中两类具有输出字符功能的Fuzzy自动机和Fuzzy有限状态自动机的强等价性,等价性和弱等价性的条件,在以往仅仅给出的Fuzzy有限状态自动机的最小化问题基础上,讨论了具有更广泛意义的具有输出字符功能的Fuzzy自动机的最小化问题,以及其最小化自动机与Fuzzy有限状态自动机的最小化自动机在不同条件下的关系。     

Real four-dimensional M-triquadrics

Nonsingular maximal intersections of three real six-dimensional quadrics are considered. Such algebraic varieties are referred to for brevity as real four-dimensionalM-triquadrics. The dimensions of their cohomology spaces with coefficients in the field of two elements are calculated.     

On the classification of nonsingular 2×2×2×2 hypercubes

As a first step in the classification of nonsingular 2×2×2×2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2×2×2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 ≤ i < j ≤ 4) and prove that a hypercube is the product of two 2×2×2 hypercubes if and only if its 12-rank is at most 2. We derive a ‘standard form’ for nonsingular 2×2×2×2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2×2×2×2 hypercube M of 12-rank 2 depends only on the value of an invariant δ ₀(M) which derives in a natural way from the Cayley hyperdeterminant det₀ M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.     

Indefinite inner-product spaces associated with hermitian polynomial matrices

Nonsingular hermitian polynomial matrices are completely characterized by bases and shift operators of suitably defined inner-product spaces.     

Lie symmetries and exact solutions of variable coefficient mKdV equations: An equivalence based approach

Group classification of classes of mKdV-like equations with time-dependent coefficients is carried out. The usage of equivalence transformations appears to be a crucial point for the exhaustive solution of the problem. We prove that all the classes under consideration are normalized. This allows us to formulate the classification results in three ways: up to two kinds of equivalence (which are generated by transformations from the corresponding equivalence groups and all admissible point transformations) and using no equivalence. A simple way for the construction of exact solutions of mKdV-like equations using equivalence transformations is described.     

Limit theorems for the multivariate binomial distribution

Nonsingular limit distributions are determined for sequences of affine transformations of random vectors whose distributions are multivariate binomial. Each of these limit distributions is that of an affine transformation of a random vector having a multivariate normal distribution or a multivariate Possion distribution or a joint distribution of two independent random vectors, one normal and the other Poisson.     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.     

有限Abel群的Fuzzy子群的等价分类及其类数

定义Fuzzy子群的两种等价关系,给出了有限Abel群的Fuzzy子群在这两种等价关系下的等价类数的求解公式。     

两类教育最优投资模型等价性及其相应最优投资准则

本研究了两类教育最优投资模型等价性，并且给出了相应的最优投资准则。     

Dual Representations of Convex Sets and Gateaux Differentiability Spaces

The duality of two kinds of representations of convex sets is studied: the tangential representation of a convex body and the representations of its polar or negative polar by means of their weak^* exposed points. The equivalence of the representations is proved and a condition for their validity is obtained for individual sets (the case of arbitrary locally convex space) and for classes of sets (the case of Gâteaux differentiability locally convex space). Properties of Gâteaux differentiability locally convex spaces are studied and some examples of such spaces are given.     

Real four-dimensional GM-triquadrics

Nonsingular intersections of three real six-dimensional quadrics are considered. Such algebraic varieties are referred to for brevity as real four-dimensional triquadrics. Necessary and sufficient conditions for a real four-dimensional triquadric to be a GM-variety are established.     

一类非奇异H-矩阵判定的新条件

非奇异H-矩阵是在许多领域具有广泛应用的重要矩阵类,但实际判定一个非奇异H-矩阵是十分困难的.在本文中,我们给出了一类关于非奇异H-矩阵新的判定条件,改进了近期的相关结果,并用数值例子说明了文中结果判定范围的更广泛性.     

图匹配多项式的因式分解及其应用

通过讨论几类图簇匹配多项式的因式分解,给出了两类图簇匹配等价图的结构性质,从而得到几类新的非匹配唯一图.     

Dual Representations of Convex Sets and Gâteaux Differentiability Spaces

The duality of two kinds of representations of convex sets is studied: the tangential representation of a convex body and the representations of its polar or negative polar by means of their weak^* exposed points. The equivalence of the representations is proved and a condition for their validity is obtained for individual sets (the case of arbitrary locally convex space) and for classes of sets (the case of Gâteaux differentiability locally convex space). Properties of Gâteaux differentiability locally convex spaces are studied and some examples of such spaces are given.

     

关于微分同胚的一个充分必要条件

确定两个流形是否C^r-微分同胚是微分流形研究中的重要课题,本文定义了反层的概念,给出了反层范畴,由此找到两个微分流形,C^r-微分同胚的特征刻画,于是给出了一种较以往更优的判定法．     

On the Hyperbolicity of Toral Endomorphisms

Mathematical Notes - Nonsingular endomorphisms of the m-torus $$\mathbb{T}$$ m, m ≥ 2, which are C1 perturbations of linear hyperbolic endomorphisms are considered. Sufficient conditions for...     

一类非奇H-矩阵的实用判据

非奇H-矩阵在数值分析和矩阵理论的研究中非常重要,但实际判定一个非奇异H-矩阵却非常困难.给出一类非奇异H-矩阵新的判定条件,改进了近期的相关结果,并用数值例子说明了结果判定范围的更广泛性.