Trilinear Operators Research Articles

Conformally and Projective Covariant Differential Operators

Motivated by the structure of conformal anomalies in two-dimensional gravity and its generalizations, the projective and conformal covariance properties of linear, bilinear and trilinear differential operators are investigated in some detail and the triviality of the covariant trilinear operators is demonstrated.

Toward a definition of F freudenthal trilinear operation

Toward a definition of F freudenthal trilinear operation

Terminal trilinear operations

Terminal trilinear operations

The Akivis algebra of a homogeneous loop

A (local) Lie loop is a real analytic manifold M with a base point e and three analytic functions ( x , y ) → x ° y , x y , x / y : M × M → M (respectively, U × U → M for an open neighbourhood U of e in M ) such that the following conditions are satisfied: (i) x ° e = e ° x = e , (ii) x ° ( x y ) = y , and (iii) ( x / y )° y = x for all x , y e M (respectively, U ). If the multiplication ° is associative, then M is a (local) Lie group. The tangent vector space L ( M ) in e is equipped with an anticommutative bilinear operation ( X , Y ) →[ X , Y ] and a trilinear operation ( X , Y , Z ) →〈 X , Y , Z 〉. These are defined as follows: Let B be a convex symmetric open neighbourhood of 0 in L ( M ) such that the exponential function maps B diffeomorphically onto an open neighbourhood V of e in M and transport the operation ° into L ( M ) by defining X ° Y = (exp| B ) −1 ((exp X )° (exp Y )) for X and Y in a neighbourhood C of 0 in B such that (exp C ) ° (exp C ) ⊂ V . Similarly, we transport / and \.

Factorization of positive multilinear maps

For an excellent survey of these theorems and the many applications of them we refer to J. E. Gilbert’s paper [2]. In this same article Gilbert indicates that there are available versions of weak-type factorizations for maximal operators defined by multilinear operators, but it was also noticed that strong-type factorizations for multilinear operators had not yet been studied. In this paper we shall prove strong-type factorizations for positive multilinear operators. Our approach uses the positive projective tensor product of Banach lattices and we also use some of the linear operator results of Nikisin and Maurey. The results for bilinear operators are typical for the multilinear case, but we could not restrict ourselves to the bilinear case. To prove Theorem 3.2 and Theorem 3.5 for bilinear operators with values in Lr with r > 0, we need the result of the same theorems for trilinear operators. Therefore we consider the general multilinear case. For the same reason we shall consider tensor products of n Banach lattices. The organization of this paper is as follows. In Section 2 we develop the necessary machinery of the theory of tensor products of Banach lattices. In Section 3 we prove the factorization theorems for positive multilinear operators from Lpl x... x L,,---, Lq where q>_O.

Constraints from conformal covariance on the mixing of operators of lowest twist

In massless QCD the operators which diagonalize the anomalous dimension matrix to first order in α s form representations of the conformal group. This is proved by using anomalous Ward identities. Only a subgroup, the so-called collinear conformal group SU(1, 1) ≅ SO(2, 1), acts on the restricted class of operators of lowest twist. Towers of local operators bilinear and trilinear in the quark fields are constructed. They are considered as polynomials in the derivatives acting on the basic quark fields: bilinear operators are given by Jacobi polynomials, trilinear operators by Appell's polynomials. By using the explicit form of the anomalous dimension matrix we verify that the constructed bilinear collinear conformal operators are in fact eigenvectors. As the conformal trilinear operator basis is degenerate, the eigenvectors of the mixing matrix of trilinear operators are not determined by conformal covariance alone. However, by taking a conformal basis the number of independent operators is reduced considerably.

New Fast Algorithms for Matrix Operations

A new technique of trilinear operations of aggregating, uniting and canceling is introduced and applied to constructing fast linear noncommutative algorithms for matrix multiplication. The result is an asymptotic improvement of Strassen’s famous algorithms for matrix operations.