The aim of this paper is to use the so-called Cayley transform to compute the LS category of Lie groups and homogeneous spaces by giving explicit categorical open coverings. When applied to U (n), U (2n)/Sp(n) and U (n)/O(n) this method is simpler than those formerly known. We also show that the Cayley transform is related to height functions in Lie groups, allowing to give a local linear model of the set of critical points. As an application we give an explicit covering of Sp (2) by categorical open sets. The obstacles to generalize these results to Sp(n) are discussed.

It is known that a 2 × 2 quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted by computing the topological degree of a characteristic map associated to the matrix and discussing the rank of the differential. The same techniques are extended to 3 × 3 matrices, which are still lacking a complete classification.

We characterize all possible relative positions between a circular hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. As an application, this provides a method to detect contact between the 2 surfaces by a simple calculation in many real world applications.

Abstract.A complete characterization is obtained of the 2 × 2 symplectic matrices that have an infinite number of left eigenvalues. Also, a new proof is given of a result of Huang and So on the number of eigenvalues of a quaternionic matrix. This is achieved by applying an algorithm for the resolution of equations due to De Leo et al.

No abstract