Notes for a course, held in the Biebrza Marshes, Poland, May 6–10, 2010, on Frobenius categories of finite groups of Lie type.

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In case of S3-symmetry, the Lie algebras are coordinatized by some nonassociative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These s...

We propose an efficient protocol for digital quantum simulation of quantum chemistry problems and enhanced digital-analog quantum simulation of transport phenomena in biomolecules with superconducting circuits. Along these lines, we optimally digitize fermionic models of molecular structure with single-qubit and two-qubit gates, by means of Trotter-Suzuki decomposition and Jordan-Wigner transfo...

This paper presents a parallel algorithm for matrix inversion on a torus interconnected MIMD-MC multi-processor. This method is faster than the parallel implementations of other widely used methods namely Gauss-Jordan, Gauss-Seidal or LU decomposition based inversion. This new algorithm also introduces a novel technique, called adaptive pivoting, for solving the zero pivot problem at no cost. O...

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element s in a dual group such that s = 1. We prove that there is a bijection between, on the one hand the set of quadratic unipotent characters of GL(n, q) or U(n, q) for all n ≥ 0 and on the other hand, the set of quadratic unipotent characte...

We compare the performance of the modular architecture, composed of competing expert networks, suggested by Jacobs, Jordan, Nowlan and Hinton (1991) to the performance of a single back-propagation network on a complex, but low-dimensional, vowel recognition task. Simulations reveal that this system is capable of uncovering interesting decompositions in a complex task. The type of decomposition ...

We introduce the notion of intrinsic subspaces of linear and affine pair geometries, which generalizes the one of projective subspaces of projective spaces. We prove that, when the affine pair geometry is the projective geometry of a Lie algebra introduced in [BeNe04], such intrinsic subspaces correspond to inner ideals in the associated Jordan pair, and we investigate the case of intrinsic sub...

We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected group Bia lynicki-Birula’s results of 1966–67. We prove “fusion theorems” for n-dimensional tori in the a...

Abstract We make some considerations and remarks on D = 11 supergravity its integral form. start from the geometrical formulation of by means form technique we provide a superspace action that reproduces (at quadratic level) recent in pure spinor framework. also Chevalley-Eilenberg cocycles their Hodge duals.