Solutions of the Knizhnik - Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models

In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral $n$-point functions, as well as the equations governing them, of the $A_1^{(1)}$ WZNW conformal theory and the corresponding Virasoro minimal models. The WZNW correlators are described as solutions of the Knizhnik - Zamolodchikov equations with rational levels and isospins. The technical tool exploited are certain relations in twisted cohomology. The results extend to arbitrary level $k+2 \neq 0$ and isospin values of the type $J=j-j'(k+2)$, $ \ 2j, 2j' \in Z\!\!\!Z_+$.Comment: 40 page

On Cayley's factorization with an application to the orthonormalization of noisy rotation matrices

The final publication is available at link.springer.comA real orthogonal matrix representing a rotation in four dimensions can be decomposed into the commutative product of a left- and a right-isoclinic rotation matrix. This operation, known as Cayley's factorization, directly provides the double quaternion representation of rotations in four dimensions. This factorization can be performed without divisions, thus avoiding the common numerical issues attributed to the computation of quaternions from rotation matrices. In this paper, it is shown how this result is particularly useful, when particularized to three dimensions, to re-orthonormalize a noisy rotation matrix by converting it to quaternion form and then obtaining back the corresponding proper rotation matrix. This re-orthonormalization method is commonly implemented using the Shepperd-Markley method, but the method derived here is shown to outperform it by returning results closer to those obtained using the Singular Value Decomposition which are known to be optimal in terms of the Frobenius norm.Peer ReviewedPostprint (author's final draft

Sylvester's Double Sums: the general case

In 1853 Sylvester introduced a family of double sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to charactise all members of the family, providing a uniform method.Comment: 16 pages, uses academic.cls and yjsco.sty. Revised version accepted for publication in the special issue of the Journal of Symbolic Computation on the occasion of the MEGA 2007 Conferenc

Computational issues in fault detection filter design

We discuss computational issues encountered in the design of residual generators for dynamic inversion based fault detection filters. The two main computational problems in determining a proper and stable residual generator are the computation of an appropriate leftinverse of the fault-system and the computation of coprime factorizations with proper and stable factors. We discuss numerically reliable approaches for both of these computations relying on matrix pencil approaches and recursive pole assignment techniques for descriptor systems. The proposed computational approach to design fault detection filters is completely general and can easily handle even unstable and/or improper systems

Efficient approximation of functions of some large matrices by partial fraction expansions

Some important applicative problems require the evaluation of functions $\Psi$ of large and sparse and/or \emph{localized} matrices $A$. Popular and interesting techniques for computing $\Psi(A)$ and $\Psi(A)\mathbf{v}$, where $\mathbf{v}$ is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from $A$ by a complex multiple of the identity matrix $I$ for computing $\Psi(A)\mathbf{v}$ or require inverting sequences of matrices with the same characteristics for computing $\Psi(A)$. Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that $A^{-1}$ shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests