The Geometry of the Newton Method on Non-Compact Lie Groups

An important class of optimization problems involve minimizing a cost function on a Lie group. In the case where the Lie group is non-compact there is no natural choice of a Riemannian metric and it is not possible to apply recent results on the optimization of functions on Riemannian manifolds. In this paper the invariant structure of a Lie group is exploited to provide a strong interpretation of a Newton iteration on a general Lie group. The paper unifies several previous algorithms proposed in the literature in a single theoretical framework. Local asymptotic quadratic convergence is proved for the algorithms considered.     

Doppler cooling a single Ca+ ion with a violet extended-cavity diode laser

We present a scheme for employing a violet extended-cavity diode laser in experiments with single, trapped ions. For this the grating-stabilised laser is spatially and spectrally filtered and referenced to a Fabry–Pérot cavity. We measure an upper limit to the line width by observing a 305-kHz FWHM beat note with the second harmonic of a titanium sapphire laser. The laser is subsequently used to optically cool a single ⁴⁰Ca⁺ ion close to the Doppler limit. PACS 03.67.Lx; 32.80.Pj; 42.55.Px     

Mapping Properties of a Projection Related to the Helmholtz Equation

We study the continuity properties of a projection derived from a recent characterization of Herglotz Wave Functions in the plane. Herglotz Wave Functions are the entire solutions of the Helmholtz equation which have L²-Far-Field-Pattern. The behavior of this projection is reminiscent of the Disc Multiplier Operator on both L^p and mixed L^p-normed spaces. In particular a Kakeya type set plays a central role in our analysis. As a consequence we answer a question posed by Alvarez, Folch-Gabayet and Pérez-Esteva.     

Proximality and Chebyshev sets

This paper is a companion to a lecture given at the Prague Spring School in Analysis in April 2006. It highlights four distinct variational methods of proving that a finite dimensional Chebyshev set is convex and hopes to inspire renewed work on the open question of whether every Chebyshev set in Hilbert space is convex.     

Kinetic and thermodynamic considerations of the bracketing method: Entropy-driven proton-transfer reactions in a Fourier transform mass spectrometer

Several reports of experimentally derived proton affinity values and gas-phase basicity values for amino acids and peptides have recently appeared in the literature. Here, we show that the thermodynamic quantity that is measured by the Fourier transform mass spectrometry proton transfer bracketing of amino acids and peptides is gas-phase basicity and not proton affinity. Both experimental and theoretical evidence supports this conclusion. The difference between the values determined by proton transfer bracketing measurements for lysine versus leucine is consistent with a difference in gas-phase basicity rather than proton affinity. The rate of proton transfer from protonated lysine to a series of reference compounds have been measured. Entropy-driven, endothermic proton transfer is found to occur at the collision rate. Recent ab initio and semi-empirical calculations of the proton affinity of lysine are found to agree with the value that is derived from bracketing studies when one assumes that gas-phase basicity is measured. While entropy-driven reactions have been observed previously in high-pressure mass spectrometers, this is the first evidence for such reactions at low pressure in a Fourier transform mass spectrometer.     

The probability of generating a soluble subgroup of a finite group

It is proved that for finite groups G, the probability thattwo randomly chosen elements of G generate a soluble subgrouptends to zero as the index of the largest soluble normal subgroupof G tends to infinity.