Adjoint Methods in Data Assimilation for Estimating Model Error

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure. This revised version was published online in July 2006 with corrections to the Cover Date.     

The oxidative mannich reaction catalyzed by dirhodium caprolactamate

Dirhodium caprolactamate [Rh2(cap)4] is a highly effective catalyst for the oxidative Mannich reaction. The reaction proceeds via C-H oxidation of a tertiary amine followed by nucleophilic capture. This green transformation is conducted in protic solvent using inexpensive T-HYDRO (70% t-BuOOH in water). Synthetically valuable gamma-aminoalkyl butenolides are obtained.     

Recirculating, passive micromixer with a novel sawtooth structure

A microfluidic device capable of recirculating nano to microlitre volumes in order to efficiently mix solutions is described. The device consists of molded polydimethyl siloxane (PDMS) channels with pressure inlet and outlet holes sealed by a glass lid. Recirculation is accomplished by a repeatedly reciprocated flow over an iterated sawtooth structure. The sawtooth structure serves to change the fluid velocity of individual streamlines differently depending on whether the fluid is flowing backwards or forward over the structure. Thus, individual streamlines can be accelerated or decelerated relative to the other streamlines to allow sections of the fluid to interact that would normally be linearly separated. Low Reynolds numbers imply that the process is reversible, neglecting diffusion. Computer simulations were carried out using FLUENT. Subsequently, fluorescent indicators were employed to experimentally verify these numerical simulations of the recirculation principal. Finally, mixing of a carboxyfluorescein labeled DMSO plug with an unlabeled DMSO plug across an immiscible hydrocarbon plug was investigated. At cycling rates of 1 Hz across five sawtooth units, the time was recorded to reach steady state in the channels, i.e. until both DMSO plugs had the same fluorescence intensity. In the case of the sawtooth structures, steady state was reached five times faster than in channels without sawtooth structures, which verified what would be expected based on numerical simulations. The microfluidic mixer is unique due to its versatility with respect to scaling, its potential to also mix solutions containing small particles such as beads and cells, and its ease of fabrication and use.     

A duality theorem for hope module algebras over dedekind rings

Let L be a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p≧3, which has been proved to be generalized restricted in the sense of [Shul, Shu2]. For a generalized restricted L-module M, the homological support variety ‖L‖_M is defined to be that of the primitive p-envelope P{L). A realization L of P(L) is given in Der(&;(m : n)). Furthermore, a class of generalized restricted highest weight L-modules lift to Dist(Tx)V(p)-module structures and their support varieties can be computed by using algebraic group techniques developed in [LN].