3×3实矩阵的Volterra乘子

本文应用代数知识,详细讨论3×3实矩阵A存在Volterra乘子D的充分必要条件,以及A的Volterra乘子D唯一的充分必要条件,并且确定A的Volterra乘子中各元素的最小存在范围及具体表达式(当D存在唯一时)。     

On the hyperdeterminant for 2×2×3 arrays

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2?×?2?×?3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x _ijk where i, j?=?1,?2 and k?=?1,?2,?3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2?×?2?×?2 arrays.     

Hopf bifurcation withD 3 ×D 3-symmetry

A generic Hopf bifurcation involving an eight dimensional center eigenspace is considered for systems possessing aD ₃ ×D ₃-symmetry. This kind of Hopf bifurcation can occur in systems of three interacting groups of oscillators, where each group itself is composed of three individual oscillators. The terminology micro and Macro is introduced here to denote symmetry operations acting on individual oscillators and on the whole groups, respectively. The normal form for the Hopf bifurcation admits 11 distinct periodic solutions with maximal isotropy subgroups. These are classified and their branching-types and stabilities are determined in terms of the cubic and relevant quintic coefficients of the normal form. The symmetry properties of these solutions when only certain Macro variables in the oscillator groups are observed are discussed in the context of the remaining symmetry. Furthermore, the relation of the normal form to the corresponding one for a singleD ₃-symmetry is established by restricting the system to four dimensional fixed point subspaces associated with submaximal isotropy subgroups. Based on this information the possibility of quasiperiodic solutions and of a particular class of heteroclinic cycles is discussed.Dedicated to Prof. Klaus Kirchgässner on the occasion of his 60th birthday     

Indefinite numerical range of 3 × 3 matrices

The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler’s approach for definite inner product spaces. The classification of the associated curve is presented in the 3 × 3 indefinite case, using Newton’s classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range. The work of this author was partially supported by the Portuguese foundation FCT, in the scope of program POCI 2010.     

Solution of 3 × 3 games using graphical method

While graphical solution of 2×n games is described in all OR text books, solution of games of size 3×3 and higher sizes is obtained using simplex method only. This paper describes a method of solving games of size 3×3 graphically. The basic principle is to consider the problem as a three-dimensional model, and to convert it into plan and elevation, and thereby obtaining the solution. Each strategy is represented by a plane, and the common point where these planes intersect, is the solution point. The location of the plan of the solution point in relation to the strategies decides the probabilities with which they are to be played, and the height of the solution point gives the value of the game. Extension of this method for solving games of size 3×n is also discussed.     

A new central polynomial for 3 × 3 matrices

Abstract

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations.     

Eigenvalue problem for symmetric 3×3 octonionic matrix

The eigenvalue problem of symmetric 3×3 octonionic matrix has been analyzed. We have especially proved explicitly first that octonionic eigenfunctions have six independent solutions in general with four degeneracy each, and second that for different eigenvalues they satisfy a cubic orthogonality relation under some conditions, which has been previously discovered by Dray and Manogue by computer use. For these, the close relationship between the octonion algebra and a Clifford algebra plays a significant role.     

On the Yamabe constants of S2×R3 and S3×R2

We compare the isoperimetric profiles of $S^2\times {\mathbb{R}}^3$ and of $S^3\times {\mathbb{R}}^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2\times {\mathbb{R}}^3$ and $S^3\times {\mathbb{R}}^2$. Explicitly we show that $Y(S^3\times {\mathbb{R}}^2,[g_0^3+d{x}^2])>(3/4)Y(S^5)$ and $Y(S^2\times {\mathbb{R}}^3,[g_0^2+d{x}^2])>0.63Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in Petean and Ruiz (2011) [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.     

奇怪的算术《3×3=2》

3+3+3=2;6+3=2,这是笑話嗎?不,这是完全真实的一个真理。确实有这样一些算术:6+3=2;3×4=5;3÷4=6,这些奇怪的等式,是在所谓“残数算术”中出現的。应用这个算术我們能够很快地根据已知数算出过200年后或是300年前的某一天是星期几。在这个算术中的残数可以看成是一个特别的数列。在普通算术中,自然数1,2,3,……有无限多个,它們是用来数有順序的东西的。例如,数书的頁数;铁路的公里牌;紀元年次(1960,1961,……);排队的报数等等。但是要数站成一个圓圈的七个人却不是用的     

Mutation Classes of Skew-Symmetric 3 × 3-Matrices

In this article, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3 × 3-matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.     

Possible Spectrums of 3×3 Upper Triangular Operator Matrices

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M(D,E,F) a 3×3 upper triangular operator matrix acting on H1⊕H2⊕H3 of the form M(D,E,F)=(A D E 0 B F 0 0 C). For given A ∈ B(H1), B ∈ B(H2) and C ∈ B(H3), the sets UD,E,F σp(M(D,E,F)), ∪D,E,F σr(M(D,E,F)), ∪D,E,F σc(M(D,E,F)) and ∪D,E,F σ(M(D,E,F)) are characterized, where D ∈ B(H2,H1), E ∈ B(H3, H1), F ∈ B(H3, H2) and σ(·), σp(·), σr(·),σc(·) denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

A Bang-Bang representation for 3×3 embeddable stochastic matrices

Summary It is proved that a 3×3 embeddable stochastic matrix has a representation as a product of a finite number of elementary stochastic matrices, with only one off-diagonal element positive. In particular if the determinant is 1/2 then only 6 matrices are needed and a necessary and sufficient condition for embeddability in this case is given.     

一维3×3非线性系统脉冲波的干扰

利用非线性几何光学的方法,讨论一维情形下3×3半线性偏微分方程具有两个初始脉冲波的干扰问题.证明方程组近似解的存在性,得到脉冲波的干扰出现在一阶项中.通过对近似解进行误差分析,得到精确解的一个好的近似.     

Four Classes of Surfaces with Constant Mean Curvature in S 3 × R and H 3 × R

Deforming rotation surfaces with constant mean curvature in S ³ and H ³ to S ³ × R and H ³ × R respectvely, we give four classes of surfaces with mean curvature vector of constant length in S ³ × R and H ³ × R. We have complete minimal surfaces in S ³ × R and H ³ × R. Also we obtain minimal 2-tori in S ³ × S ¹, some of which are embedded.     

The Minimum Rank of a 3 × 3 Partial Block Matrix

After recalling the definition and some basic properties of finite hypergroups—a notion introduced in a recent paper by one of the authors—several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: ‘let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a ₁…a _n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then A_k =P_k (A) is a non-negative matrix, where p_k denotes the characteristic polynomial of the top k× k principal submatrix of A, for k=1,…,n. The matrices A_k as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) a_i =0 for all ior (ii) a_i =0 for i<n and a_n =1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups.     

3×3上三角算子矩阵值域的闭性研究

令H_1,H_2,H_3是可分的复Hilbert空间,记M=(AEF0BD00C)为H_1⊕H_2⊕H_3上的3×3上三角算子矩阵.设A∈B(H_1),B∈B(H_2),C∈B(H_3)是给定的算子,利用对角元算子A,B,C的值域和零空间性质描述了算子矩阵M值域R(M)的闭性.