1. Since the origins of the birational geometry of algebraic varieties cai be traced back to Riemann's theory of algebraic functions, it is not surprising that topological considerations have played a considerable role in the theory of algebraic varieties defined over the field of complex numbers. For instance in the theory of algebraic functions of two complex variables, due mainly tc Picard, ...

The Connes formula giving the dual description for the distance between points of a Riemannian manifold is extended to the Lorentzian case. It resulted that its validity essentially depends on the global structure of spacetime. The duality principle classifying spacetimes is introduced. The algebraic account of the theory is suggested as a framework for quantization along the lines proposed by ...

We propose a new computational approach based on descriptor state space algorithms for computing normalized coprime factorizations of arbitrary rational matrices. The proposed approach applies to both continuousand discrete-time rational transfer-function matrices and shows that each rational matrix possesses a normalized coprime factorization with proper factors. The new method is conceptually...

Most of the research, to date, on optimizing program transformations for declarative languages has focused on sequential execution strategies. In this paper, we consider a class of commonly encountered computations whose \natural" speciica-tion is essentially sequential, and show how algebraic properties of the operators involved can be used to transform them into divide-and-conquer programs th...

We discuss the relationships among Jacobi matrices, orthogonal polynomials, spectral measure, moments, minors, Gaussian quadrature, resolvents and continued fractions in the simplest setting, namely the finite-dimensional one. The formal structure is essentially the same as that in the infinite-dimensional setting, where it leads into the rich analytic world of orthogonal polynomials on the rea...

We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions of these results. Finally, we obtain the first nontrivial upper bounds for the fundamental problem of the maximal size of independent systems. These bounds d...