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.     

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.     

Carmichael Numbers of Order 3

In 2002, Agrawal, Kayal and Saxena presented a polynomial time primality testing algorithm(the AKS algorithm) in [1], which can determine whether an input number is prime or composite.Papers [2] and [3] play an important role in their primality testing algorithm. In [3], the authors     

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.     

Description of a Class of Solids in ℝ3

Let be a compact convex body such that for each parallel projection onto any plane no two opposite faces of Q are projected strictly inside the projection of the entire Q. Then Q is either a cone, or a frustum of a trihedral pyramid, or a prism (possibly with nonparallel bases). Bibliography: 1 title.     

Some Properties of an Annulus Sum of 3-manifolds

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The Graph of Binary Symmetric Matrices of Order 3

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The Hilbert function of generic plane sections of curves of ℙ3

Summary A characteristic condition is given on a zero-dimensional differentiable 0-sequenceH={h _i}_i0,h ₁3, in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of ³, hence of points of ² with the uniform position property. In this way an answer is given to some question stated by Harris in [Ha₂].The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in ³ whose generic plane section has an assigned Hilbert function satisfying that condition.A.M.S. (1980) Subject classification. Primary 14C20, 14H99. Secondary 14C05     

Computing the ℤ2-Cocharacter of 3 × 3 Matrices of Odd Degree

Let F be a field of characteristic 0 and A = M _{2, 1}(F) the algebra of 3 × 3 matrices over F endowed with the only non trivial ?₂-grading. Aver'yanov in [1 Aver'yanov , I. V. ( 2009 ). Basis of graded identities of the superalgebra M _{1, 2}(F) . Mathematical Notes 85 ( 4 ): 467 – 483 .[Crossref] , [Google Scholar]] determined a set of generators for the T ₂-ideal of graded identities of A. Here we study the identities in variables of homogeneous degree 1 via the representation theory of the symmetric group, and we determine the decomposition of the corresponding character into irreducibles.     

Meson Algebras of Order ≥3

Meson algebras of order 2 have already drawn much attention, and their study has brought plenty of interesting knowledge. This fact motivated the definition and the study of meson algebras of greater order. Unfortunately, these algebras finally proved to be disappointing; probably there is almost nothing to add to the information given in the present article. Almost all meson algebras of order ≥3 are trivial, and the only two cases that give nontrivial algebras, are completely described here.     

DCI-and CI-Groups of Order p~3

Let G be a finite group and S a subset of G.We define the Cayley digraphX=X(C,S)of G with respect to S bywhere V(X)and E(X)are the vertex-and edge-sets of X,respectively.S is saidto be a CI—subset of G if any graphisomorphism X(G,S)≌X(G,T),where TG,implies that there exists a group automorphism α∈ Aut G such that S~α=T.     

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.     

The computation of Belyi pairs of 6-edged dessins d’enfants of genus 3 with automorphism groups of order 12 and 3

In this article, we study dessins d’enfants of genus 3 with six edges and only one vertex, having groups of automorphisms of order at least 3. We find all such dessins and explain the method of the enumeration. For each of them the Belyi pair is computed. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 137–148, 2007.     

Cobordism of immersions of surfaces¶in non-orientable 3-manifolds

Some properties of non-orientable 3-manifolds are shown. In particular, for a connected, non-orientable 3-manifold M, the group of cobordism clases of immersions of surfaces in M is isomorphic to a group structure on the set H ₂(M,Z/2Z)×H ₁(M,Z/2Z)×Z/2Z. Received: 8 June 2000 / Revised version: 2 October 2000     

Decompositions of the 3-uniform hypergraphs K_v~(3) into hypergraphs of a certain type

In this paper,we investigate a generalization of graph decomposition,called hypergraph decomposition.We show that a decomposition of a 3-uniform hypergraph K_v~(3)into a special kind of hypergraph K_4~(3)-e exists if and only if v≡0,1,2(mod 9)and v≥9.     

A Note on 3-Divisibility of Class Number of Quadratic Field

In this paper, the authors show that there exists infinitely many family of pairs of quadratic fields Q(D^1/2) and Q((D+n)(1/2)) with D, n∈Z whose class numbers are both divisible by 3.