In [2], the author showed that a square matrix with nonnegative determinant can always be written as the product of five or fewer positive semi-definite matrices. This is an extension to the result in [1] asserting that every matrix with positive determinant is the product of five or fewer positive definite matrices. Analogous to the analysis in [1], the author of [2] studied those matrices whi...

We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controling the deformation theory of associative algebras and prove that there is a canonical solution within the class of graphs without circuits, without assuming the Jacobi identity. The proof is based on the unique factorization property of graph insertions.

Let Fq be a finite field with q elements and denote by θ : Fq → Fq an automorphism of Fq. In this paper, we deal with linear codes of Fnq invariant under a semi-linear map T : F q → F q for some n ≥ 2. In particular, we study three kind of their dual codes, some relations between them and we focus on codes which are products of module skew codes in the non-commutative polynomial ring as a subca...

The topic of this lecture is differential geometry on quantum groups. Several lecturers at this conference have talked about this subject. For a review and a fairly extensive list of references I would like to point the reader to the contributions of B. Jurco and S. L. Woronowicz in this proceedings. Here we shall propose a new rigid framework for the so-called Cartan calculus of Lie derivative...

Associated to every subset J of nodes of a Dynkin diagram is a standard parabolic subalgebra pJ of the corresponding Lie algebra g, generated by the positive Borel subalgebra of g together with the negative simple generators fj, j ∈ J . Furthermore one has a decomposition of g as a semi-direct product of pJ and the subalgebra n − J generated by the remaining negative simple generators. In the s...

We consider the string, like point particles and branes, to be an irreducible representation of semi-direct product Cartan involution invariant subalgebra [Formula: see text] its vector representation. show that preserves string charges, little algebra, is essentially Borel text]. also known physical states carry a parts this algebra.