On the rank of 3×3×3-tensors

Let U, V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U???V???W is the minimum dimension of a subspace of U???V???W containing τ and spanned by fundamental tensors, i.e. tensors of the form u???v???w for some u in U, v in V and w in W. We prove that if U, V and W have dimension three, then the rank of a tensor in U???V???W is at most six, and such a bound cannot be improved, in general. Moreover, we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U???V???W when the dimensions of U, V and W are higher.     

Units OF Z(G × C 2)

Abstract

In this paper, we are mainly interested in describing constructively u(Z(G × C ₂)), where u(ZG) has been described in some way. We first consider unitary units and prove that if u(ZG) is generated by unitary units, then U(Z(G × C₂)) is also generated by unitary units. Then we consider bicyclic units and ask the following question: If G has a normal complement generated by bicyclic units, does G × C ₂ also have a normal complement generated by bicyclic units? We give a negative answer to the above in general, showing that none of the normal complements of D ₈ × C ₂ × C ₂ is generated by bicyclic units, although a normal complement of D ₈ × C ₂ is indeed generated by bicyclic units.     

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.     

COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS ON T_d×Z

The author considers the contact process on a branching plane Td×Z, which is the product of a regular tree Td and the line Z. It is shown that above the second critical point, the complete convergence theory holds.     

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.     

COMPLETE CONVERGENCE THEORY OF THE CONTACT PROCESS ON Td × Z1

The author considers the contact process on a branching plane T_d × Z, which is the product of a regular tree T_d and the line Z. It is shown that above the second critical point, the complete convergence theory holds.     

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.