Generic 2×2 matrices with involution

Procesi has given a linear basis for the ring ofm generic 2×2 matrices. We do the same for the ring ofm generic 2×2 matrices with transpose involution.     

Generic properties in Euclidean kinematics

The motion of a rigid body in a Euclidean space E ⁿ is represented by a path in the Euclidean isometry group E(n). A normal form for elements of the Lie algebra of this group leads to a stratification of the algebra which is shown to be Whitney regular. Translating this along invariant vector fields give rise to a stratification of the jet bundles J ^k (R, E(n)) for k=1, 2 and, hence, via the transversality theorem, to generic properties of rigid body motions. The relation of these to the classical centrodes and axodes of motions is described, together with applications to planar 4-bar mechanisms and the dynamics of a rigid body.     

Generic properties of the complementarity problem

Givenf: R ₊ ⁿ R ⁿ , the complementarity problem is to find a solution tox 0,f(x) 0, and x, f(x) = 0. Under the condition thatf is continuously differentiable, we prove that for a generic set of such anf, the problem has a discrete solution set. Also, under a set of generic nondegeneracy conditions and a condition that implies existence, we prove that the problem has an odd number of solutions.This work was partially supported by N.S.F. Grants GP-8007 and 010185.     

Generic properties of compact metric spaces

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose elements enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.     

Structural properties of Riordan matrices and extending the matrices

We consider an infinite lower triangular matrix L=[?_n,k]_n,k∈N0 and a sequence Ω=(ω_n)_n∈N0 called the (a,b)-sequence such that every element ?_n+1,k+1 except lying in column 0 can be expressed as

     

Topological properties of orthostochastic matrices

The set of n×n orthostochastic matrices with the topology induced by the Euclidean matric is shown to be compact and path-connected. For n<3, the set of orthostochastic matrices is identical to the set of doubly stochastic matrices. In this paper, it is shown that for n3 the orthostochastic matrices are not everywhere dense in the set of doubly stochastic matrices, thus answering a question of L. Mirsky in his survey article on doubly stochastic matrices [2].     

On properties of cell matrices

In this paper properties of cell matrices are studied. A determinant of such a matrix is given in a closed form. In the proof a general method for determining a determinant of a symbolic matrix with polynomial entries, based on multivariate polynomial Lagrange interpolation, is outlined. It is shown that a cell matrix of size n>1 has exactly one positive eigenvalue. Using this result it is proven that cell matrices are (Circum-)Euclidean Distance Matrices ((C)EDM), and their generalization, k-cell matrices, are CEDM under certain natural restrictions. A characterization of k-cell matrices is outlined.     

Microlocal properties of scattering matrices

We consider the scattering theory for a pair of operators H₀ and H = H₀ + V on L²(M, m), where M is a Riemannian manifold, H₀ is a multiplication operator on M, and V is a pseudodifferential operator of order ? μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.