Commutators in the Two-Weight Setting

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $\mu,\lambda \in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;\mu) \to L^p(\mathbb{R}^n;\lambda)$ is shown to be equivalent to the function $b$ being in a BMO space adapted to $\mu$ and $\lambda$. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both $\lambda$ and $\mu$ being Lebesgue measure, and Bloom in the case of dimension one.Comment: v3: suggestions from two referees incorporate

Multiparameter Riesz Commutators

It is shown that product BMO of Chang and Fefferman, defined on the product of Euclidean spaces can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of Coifman, Rochberg, and Weiss, and at the same time extends the work of Lacey and Ferguson and Lacey and Terwilleger on multiparameter commutators with Hilbert transforms. The method of proof requires the real-variable methods throughout, which is new in the multi-parameter context.Comment: 38 Pages. References updated. To appear in American J Mat

Multi-Parameter Div-Curl Lemmas

We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that arise. Extensions to differential forms are also given.Comment: v1: 8 page

Schatten Class Estimates for Paraproducts in Multi-parameter setting and applications

Let $\Pi _{b}$ be a bounded $n$ parameter paraproduct with symbol $b$. We demonstrate that this operator is in the Schatten class $\mathbb S_ p$, $0<p<\infty$, if the symbol is in the $n$ parameter Besov space $B_p$. Our result covers both the dyadic and continuous version of the paraproducts in the multiparameter setting. A direct method of proof is given and an application to the commutator of the double Hilbert transforms is provided

Prospecting for an HIV Vaccine

Human immunodeficiency virus (HIV) sets several challenges for the development of a preventative HIV vaccine. Predictable, protective natural immunity against HIV does not occur and so unlike most other diseases for which vaccines exist, there are few guideposts from natural infection. Nonetheless, six vaccine efficacy trials have occurred. One in particular, the Thai trial called RV144, showed partial protective efficacy and potential ways ahead to a better vaccine approach. This coupled with other lessons from studies of acute infections as well as an increasingly complex knowledge of HIV-related vaccine immunology bring hope that a vaccine solution might be reached for this pervasive and deadly pandemic

Light-cone QCD on the lattice

Ideas and recent results for light-front Hamiltonian quantisation of lattice gauge theories.Comment: LATTICE99(Plenary), 5 pages, 3 figure

Design of power electronics for TVC and EMA systems

The EMA systems proposed for future space transportation applications are high power systems operating at voltages up to 270 Vdc and at current levels on the order of hundreds of amperes. The position of the actuator is controlled by modulating the flow of energy from the source to an electric motor with an inverter. Hard-switching of the semiconductor devices in the inverter results in considerable device switching stresses and losses and in the generation of substantial amounts of EMI. Both of these can be reduced by employing zero-voltage-switching (ZVS) techniques in the inverter. This project has focused on the development of a ZVS inverter for the Marshall Space Center EMA prototypes, which utilize brushless dc motors to convert electrical energy to mechanical energy. An inverter which permitted zero-voltage switching and a quasi-PWM operation was selected for study and implementation. A waveshaping circuit is added to the front of a standard three-phase inverter to achieve the desired switching properties. This circuit causes the input voltage of the three-phase inverter to ring to zero where it is clamped for a short period of time. During this zero-voltage period, any of the semiconductor switches in the three-phase inverter are switched on or off at zero voltage resulting in a reduction in switching losses and EMI. The operation of this waveshaping circuit and its interaction with the three-phase inverter are described. The different circuit modes were analyzed using equivalent circuits. Based on this analysis, design relationships were developed for calculating component values for the circuit elements in the waveshaping circuit. Waveforms of various voltages and currents in the waveshaping circuit were plotted and used to determine the ratings of the semiconductors in the waveshaping circuit. The implementation of this inverter are described. Block diagrams for the overall control system and the waveshaping circuit control are presented and discussed. The current control scheme employed in the controller is also described