In this paper trans nite interpolation/approximation is discussed. This is a powerful approach to generation of curves, surfaces and solids (and even higher dimensional manifolds) by blending lower dimensional geometric objects. Trans nite blending, e.g. used in Gordon-Coons patches, is well known to mathematicians and CAD people. It is presented here in a very simple conceptual and computation...

We give a necessary and su cient condition for a graphical regu lar representation to be adjacency transitive and provide an in nite family of nite simple undirected vertex transitive graphs such that neither nor c is adjacency transitive

We use algebraic techniques to obtain superlinear lower bounds on the size of bounded-width branching programs to solve a number of problems. In particular, we show that any bounded-width branching program computing a nonconstant threshold function has length (n log log n); improving on the previous lower bounds known to apply to all such threshold functions. We also show that any program over ...

Suppose a group G acts on a Gromov-hyperbolic space X properly discontinuously. If the limit set L(G) of the action has at least three points, then the second bounded cohomology group of G, H 2 b (G;R) is in nite dimensional. For example, ifM is a complete, pinched negatively curved Riemannian manifold with nite volume, thenH 2 b ( 1 (M);R) is in nite dimensional. As an application, we show tha...

This list is a short overview of some of my results. One can nd a more detailed description in my Research Statement or at 1. Simple associative algebras with nite Z-grading. Here, R is a simple algebra with a nite Z-grading over a commutative ring. A description of all such algebras was given 18]. As a corollary, it was proved that every (not necessarily nite dimensional) simple Lie algebra of...

There is a large collection of e ective algorithms for computing information about nite soluble groups. The success in computation with these groups is primarily due to a computationally convenient representation of them by means of (special forms of) power conjugate presentations. A notable omission from this collection of algorithms is an e ective algorithm for computing the automorphism grou...

in this paper, we show that in each nite dimensional hilbert space, a frame of subspaces is an ultra bessel sequence of subspaces. we also show that every frame of subspaces in a nite dimensional hilbert space has frameness bound.

The area starts with Galois and Gauss Group theory and expo nential sums were the two application areas then That tradition continues Without Chevalley groups over nite elds there would have been no structure theory of nite simple groups Weil s Riemann Hypothesis for curves over nite elds is a singular event Many error estimates in combinatorics derive from it Research in nite elds requires com...

We reformulate the self-dual Einstein equation as a trio of di erential form equations for simple two-forms. Using them, we can quickly show the equivalence of the theory and 2D sigma models valued in in nite-dimensional group, which was shown by Park and Husain earlier. We also derive other eld theories including the 2D Higgs bundle equation. This formulation elucidates the relation among thos...

The adjoint action of a nite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of split semisimple orbits of a given genus. This conjecture is proved for type A, and partial results are obtained for other types. For type A a probabilistic interpretation is given in terms of Solomon's descent algebra and card shu ing. 3