Extension of the Morris-Shore transformation to multilevel ladders

We describe situations in which chains of N degenerate quantum energy levels, coupled by time-dependent external fields, can be replaced by independent sets of chains of length N, N-1,...,2 and sets of uncoupled single states. The transformation is a generalization of the two-level Morris-Shore transformation [J.R. Morris and B.W. Shore, Phys. Rev. A 27, 906 (1983)]. We illustrate the procedure with examples of three-level chains

Non-universal coarsening and universal distributions in far-from equilibrium systems

Anomalous coarsening in far-from equilibrium one-dimensional systems is investigated by simulation and analytic techniques. The minimal hard core particle (exclusion) models contain mechanisms of aggregated particle diffusion, with rates epsilon<<1, particle deposition into cluster gaps, but suppressed for the smallest gaps, and breakup of clusters which are adjacent to large gaps. Cluster breakup rates vary with the cluster length x as kx^alpha. The domain growth law x ~ (epsilon t)^z, with z=1/(2+alpha) for alpha>0, is explained by a scaling picture, as well as the scaling of the density of double vacancies (at which deposition and cluster breakup are allowed) as 1/[t(epsilon t)^z]. Numerical simulations for several values of alpha and epsilon confirm these results. An approximate factorization of the cluster configuration probability is performed within the master equation resulting from the mapping to a column picture. The equation for a one-variable scaling function explains the above results. The probability distributions of cluster lengths scale as P(x)= 1/(epsilon t)^z g(y), with y=x/(epsilon t)^z. However, those distributions show a universal tail with the form g(y) ~ exp(-y^{3/2}), which disagrees with the prediction of the independent cluster approximation. This result is explained by the connection of the vacancy dynamics with the problem of particle trapping in an infinite sea of traps and is confirmed by simulation.Comment: 30 pages (10 figures included), to appear in Phys. Rev.

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Reflector antennas with low sidelobes, low cross polarization, and high aperture efficiency

Techniques are presented for computing the horn near field patterns on the subreflectors and for correcting the phase center errors of the horn pattern by shaping the subreflector surface. The diffraction pattern computations for scanned beams are described. The effects of dish aperture diffraction on pattern bandwidth are investigated. A model antenna consisting of a reflector, shaped subreflector, and corrugated feed horn is described

Sub-surface damage issues for effective fabrication of large optics

A new ultra precision large optics grinding machine, BoX®has been developed at Cranfield University. BoX®islocated at the UK's Ultra Precision Surfaces laboratory at the OpTIC Technium. This machine offers a rapidand economic solution for grinding large off-axis aspherical and free-form optical components.This paper presents an analysis of subsurface damage assessments of optical ground materials produced usingdiamond resin bonded grinding wheels. The specific materials used, Zerodur®and ULE®are currently understudy for making extremely large telescope (ELT) segmented mirrors such as in the E-ELT project.The grinding experiments have been conducted on the BoX®grinding machine using wheels with grits sizes of76 μm, 46 μm and 25 μm. Grinding process data was collected using a Kistler dynamometer platform. Thehighest material removal rate (187.5 mm3/s) used ensures that a 1 metre diameter optic can be ground in lessthan 10 hours. The surface roughness and surface profile were measured using a Form Talysurf. The subsurfacedamage was revealed using a sub aperture polishing process in combination with an etching technique.These results are compared with the targeted form accuracy of 1 μm p-v over a 1 metre part, surface roughnessof 50-150 nm RMS and subsurface damage in the range of 2-5 μm. This process stage was validated on a 400mm ULE®blank and a 1 metre hexagonal Z

Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field

In this paper we consider the minimum time population transfer problem for the $z$-component of the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let $(-E,E)$ be the two energy levels, and $|\Omega(t)|\leq M$ the bound on the field amplitude. For each couple of values $E$ and $M$, we determine the time optimal synthesis starting from the level $-E$ and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For $M/E<<1$, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency $\omega_R=2E$. On the other side, for $M/E>1$, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed $E$ we also prove that for $M\to\infty$ the time needed to reach the state two tends to zero. In the case $M/E>1$ there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the $x$ and $y$ directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as $M/E\to0$, giving a partial proof of a conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte

Three Dimensional Mapping of Texture in Dental Enamel

We have used synchrotron x-ray diffraction to study the crystal orientation in human dental enamel as a function of position within intact tooth sections. Keeping tooth sections intact has allowed us to construct 2D and 3D spatial distribution maps of the magnitude and orientation of texture in dental enamel. We have found that the enamel crystallites are most highly aligned at the expected occlusal points for a maxillary first premolar, and that the texture direction varies spatially in a three dimensional curling arrangement. Our results provide a model for texture in enamel which can aid researchers in developing dental composite materials for fillings and crowns with optimal characteristics for longevity, and will guide clinicians to the best method for drilling into enamel, in order to minimize weakening of remaining tooth structure, during dental restoration procedure