Carleson measures for Harmonically Weighted Dirichlet Spaces are characterized. It is shown a version of a maximal inequality for these spaces. Also, Interpolating Sequences and Closed-Range Composition Operators are studied in this context.

The views expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. / The numerical experiments, carried out through the use of a pressure-velocity coupled method to solve the Favre Averaged Navier-Stokes equations, on steady and sinusoidally oscillating flows at five different Keluegan-Carpenter numbers, and three periodicity levels are described. A second-order in time, second-order in space, second-level predictor-corrector finite difference scheme has been used. The solutions were solved by the CFD-ACE program from the CFD Research Corporation. The analysis has produced in-line force coefficients comparable to those obtained experimentally for sinusoidally-oscillating flows.

Harmonically mode-locked semiconductor lasers with external ring cavities offer high repetition rate pulse trains while maintaining low optical linewidth via long cavity storage times. Single frequency injection locking generates widely-spaced and tunable frequency combs from these harmonically mode-locked lasers, while stabilizing the optical frequencies. The output is stabilized long-term with the help of a feedback loop utilizing either a novel technique based on Pound-Drever-Hall stabilization or by polarization spectroscopy. Error signals of both techniques are simulated and compared to experimentally obtained signals. Frequency combs spaced by 2.5 GHz and ~10 GHz are generated, with demonstrated optical sidemode suppression of unwanted modes of 36 dB, as well as RF supermode noise suppression of 14 dB for longer than 1 hour. In addition to the injection locking of actively harmonically mode-locked lasers, the injection locking technique for regeneratively mode-locked lasers, or Coupled OptoElectronic Oscillators (COEOs), is also demonstrated and characterized extensively.