The mechanism of three important aldol reactions and a biomimetic transamination is investigated using a combination of experimental kinetic isotope effects (KIEs), standard theoretical calculations and dynamics trajectory simulations. This powerful mechanistic probe is found to be invaluable in understanding intricate details of the mechanism of these reactions. The successful application of variational transition state theory including multidimensional tunneling to theoretically predict isotope effects, described in this dissertation, represents a significant advance in our research methodology. The role of dynamic effects in aldol reactions is examined in great detail. The study of the proline catalyzed aldol reaction has revealed an intriguing new dynamic effect - quasiclassical corner cutting - where reactive trajectories cut the corner between reactant and product valleys and avoid the saddle point. This phenomenon affects the KIEs observed in this reaction in a way that is not predictable by transition state theory. The study of the Roush allylboration of aldehydes presents an example where recrossing affects experimental observations. The comparative study of the allylboration of two electronically different aldehydes, which are predicted to have different amounts of recrossing, suggests a complex interplay of tunneling and recrossing affecting the observed KIEs. The Mukaiyama aldol reaction has been investigated and the results unequivocally rule out the key carbon-carbon bond forming step as rate-limiting. This raises several interesting mechanistic scenarios - an electron transfer mechanism with two different rate-limiting steps for the two components, emerges as the most probable possibility. Finally, labeling studies of the base catalyzed 1,3- proton transfer reaction of fluorinated imines point to a stepwise process involving an azomethine ylide intermediate. It is found that dynamic effects play a role in determining the product ratio in this reaction.

In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.