One Controller at a Time (1-CAT): A mimo design methodology

The One Controller at a Time (1-CAT) methodology for designing digital controllers for Large Space Structures (LSS's) is introduced and illustrated. The flexible mode problem is first discussed. Next, desirable features of a LSS control system design methodology are delineated. The 1-CAT approach is presented, along with an analytical technique for carrying out the 1-CAT process. Next, 1-CAT is used to design digital controllers for the proposed Space Based Laser (SBL). Finally, the SBL design is evaluated for dynamical performance, noise rejection, and robustness

Dirac--Lie systems and Schwarzian equations

A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the Dirac's description of constrained systems, we introduce and analyse a particular class of Lie systems on Dirac manifolds, called Dirac--Lie systems, which are associated with `Dirac--Lie Hamiltonians'. Our results enable us to investigate constants of the motion, superposition rules, and other general properties of such systems in a more effective way. Several concepts of the theory of Lie systems are adapted to this `Dirac setting' and new applications of Dirac geometry in differential equations are presented. As an application, we analyze traveling wave solutions of Schwarzian equations, but our methods can be applied also to other classes of differential equations important for Physics.Comment: 41 page

Injection locking of two frequency-doubled lasers with 3.2 GHz offset for driving Raman transitions with low photon scattering in 43^{43}Ca+^+

We describe the injection locking of two infrared (794 nm) laser diodes which are each part of a frequency-doubled laser system. An acousto-optic modulator (AOM) in the injection path gives an offset of 1.6 GHz between the lasers for driving Raman transitions between states in the hyperfine split (by 3.2 GHz) ground level of $^{43}$Ca$^+$. The offset can be disabled for use in $^{40}$Ca$^+$. We measure the relative linewidth of the frequency-doubled beams to be 42 mHz in an optical heterodyne measurement. The use of both injection locking and frequency doubling combines spectral purity with high optical power. Our scheme is applicable for providing Raman beams across other ion species and neutral atoms where coherent optical manipulation is required.Comment: 3 pages, 3 figure

Exact Relations for a Strongly-interacting Fermi Gas from the Operator Product Expansion

The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k^4 at large momentum k, as pointed out by Shina Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.Comment: 4 pages, 2 figure