Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Motivated by L\'{e}vy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator $f\mapsto{1/2}f''$. We then consider the special case where the state space is the self-similar set $\{\pm q^k:k\in \mathbb{Z}\}\cup\{0\}$ for some $q>1$. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan's ``lost'' notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, $q$-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding It\^{o} excursion measure and the Laplace exponent of the inverse local time -- both again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve $q$-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.Comment: Published in at http://dx.doi.org/10.1214/193940307000000383 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

A User''s Guide to Solving Dynamic Stochastic Games Using the Homotopy Method

This paper provides a step-by-step guide to solving dynamic stochastic games using the homotopy method. The homotopy method facilitates exploring the equilibrium correspondence in a systematic fashion; it is especially useful in games that have multiple equilibria. We discuss the theory of the homotopy method and its implementation and present two detailed examples of dynamic stochastic games that are solved using this method.

Storing images in warm atomic vapor

Reversible and coherent storage of light in atomic medium is a key-stone of future quantum information applications. In this work, arbitrary two-dimensional images are slowed and stored in warm atomic vapor for up to 30 $\mu$s, utilizing electromagnetically induced transparency. Both the intensity and the phase patterns of the optical field are maintained. The main limitation on the storage resolution and duration is found to be the diffusion of atoms. A techniqueanalogous to phase-shift lithography is employed to diminish the effect of diffusion on the visibility of the reconstructed image

Charge asymmetry in high-energy μ+μ−\mu^+\mu^- photoproduction in the electric field of a heavy atom

The charge asymmetry in the differential cross section of high-energy $\mu^+\mu^-$ photoproduction in the electric field of a heavy atom is obtained. This asymmetry arises due to the Coulomb corrections to the amplitude of the process (next-to-leading term with respect to the atomic field). The deviation of the nuclear electric field from the Coulomb field at small distances is crucially important for the charge asymmetry. Though the Coulomb corrections to the total cross section are negligibly small, the charge asymmetry is measurable for selected final states of $\mu^+$ and $\mu^-$. We further discuss the feasibility for experimental observation of this effect.Comment: 6 pages, 3 figure

The Influence of Design Features in the Biomechanical Performance of a Fixator for the Lumbar Spine

PhDSpinal fixation systems using pedicular screws have gained popularity in manging the damaged spine. However, the loading to which individual components of a fixator are exposed are largely unknown. This thesis describes the use of a Corpectomy injury model to investigate the mechanical response of a commercial internal spinal fixator and the resultant loads acting on its rods and screws, under four separatelo ading regimens. The fixator was instrumentedw ith strain gaugesa nd tested using specially designed jigs. The results were then compared to theoretical models and any differences highlighted. An evaluation was also performed on a range of transpedicular screw designs under tensile loads. An increase in the tightening torque of the fixator clamps, ranging from 5 to 15Nm, and the inclusion of transverse elements across its vertical rods produced a combined increase in overall torsional rigidity of 89%. However, no such changes were found under axial compression and both simulated flexion and extension tests. The relative ineffectivenesso f the transversee lementsu nder sagittal loads was probably due to their spatial relationship with the fixator. The results from the instrumented fixator indicated several load response pathways, as predicted by the theoretical analysis. These pathways were influenced by several factors including, the screw angulation, the boundary conditions of the test and the addition of the transverse elements. Clamp design was critical in minimising rotational slippage of both screws and transverse elements. The results from the instrumented fixator revealed that the transpedicular screws were exposed to complex loads under each of the tests. Under tensile loads, both the increasei n screw insertion depth and a decreasein screw pitch were found to be the important parameters which affect screw performance. Analysis showed the state of stress and strain along the thread was the overriding factor in the tensile performance of these screws. This work hase mphasisedth e importance of a full biornechanicale valuation of any future designs of spinal fixators

Measurement of Dicke Narrowing in Electromagnetically Induced Transparency

Dicke narrowing is a phenomena that dramatically reduces the Doppler width of spectral lines, due to frequent velocity-changing collisions. A similar phenomena occurs for electromagnetically induced transparency (EIT) resonances, and facilitates ultra-narrow spectral features in room-temperature vapor. We directly measure the Dicke-like narrowing by studying EIT line-shapes as a function of the angle between the pump and the probe beams. The measurements are in good agreement with an analytic theory with no fit parameters. The results show that Dicke narrowing can increase substantially the tolerance of hot-vapor EIT to angular deviations. We demonstrate the importance of this effect for applications such as imaging and spatial solitons using a single-shot imaging experiment, and discuss the implications on the feasibility of storing images in atomic vapor.Comment: Introduction revise