Diophantine property in the group of affine transformations of the line

We investigate the Diophantine property of a pair of elements in the group of affine transformations of the line. We say that a pair of elements g_1,g_2 in this group is Diophantine if there is a number A such that a product of length l of elements of the set {g_1,g_2,g_1^{-1},g_2^{-1}} is either the unit element or of distance at least A^{-l} from the unit element. We prove that the set of non-Diophantine pairs in a certain one parameter family is of Hausdorff dimension 0.Comment: 12 pages, no figures, reference to [ABRS] update

Random walks in compact groups

Let X_1,X_2,... be independent identically distributed random elements of a compact group G. We discuss the speed of convergence of the law of the product X_l*...*X_1 to the Haar measure. We give poly-log estimates for certain finite groups and for compact semi-simple Lie groups. We improve earlier results of Solovay, Kitaev, Gamburd, Shahshahani and Dinai.Comment: 35 pages, no figures, revision based on referee's report, results and proofs unchange

Field strength scaling in quasi-phase-matching of high-order harmonic generation by low-intensity assisting fields

High-order harmonic generation in gas targets is a widespread scheme used to produce extreme ultraviolet radiation, however, it has a limited microscopic efficiency. Macroscopic enhancement of the produced radiation relies on phase-matching, often only achievable in quasi-phase-matching arrangements. In the present work we numerically study quasi-phase-matching induced by low-intensity assisting fields. We investigate the required assisting field strength dependence on the wavelength and intensity of the driving field, harmonic order, trajectory class and period of the assisting field. We comment on the optimal spatial beam profile of the assisting field

Local Limit Theorem for the Lorentz Process and Its Recurrence in the Plane

For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Lai-Sang Young's axioms (which imply exponential decay of correlation and the CLT) a local CLT is proven. In fact, a unified version of the local CLT is found, covering among others the absolutely contionuous and the arithmetic cases. For the planar Lorentz process with a finite horizon this result implies a.) the local CLT and b.) the recurrence. For the latter case ($d=2$, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt, and J.-P. Conze, could already establish recurrence

Photoreactions in Phycomyces. Responses to the stimulation of narrow test areas with ultraviolet light

Equipment has been developed for ultraviolet illumination of sharply bounded test areas of the growing zone of sporangiophores of Phycomyces. The growing zone is opaque for this light and the tropic responses are negative. Periodic short narrow stimuli on alternating sides produce periodic tropic responses when applied at x > 0.5 mm, but none for x 0.1 mm. The lag between stimulus and response is 3.3 min. for any x > 0.5 mm. For smaller x the lag increases progressively. In all cases the tropic bend occurs at values of x > 0.5 mm. Sustained tropic stimuli, applied at constant height relative to ground, produce relatively sharp tropic bends. The center of the bend is at all times close to the simultaneous position of the stimulated area. The boundaries of a light-adapted zone move less than 0.1 mm in 10 min. relative to the sporangium. It is concluded that the receiving and adapting structures do not move relative to the sporangium, and that the responding system does not move relative to ground. The two systems move relative to each other with the speed of growth. The responding system does not extend above x = 0.5 mm

Random walks in Euclidean space

Consider a sequence of independent random isometries of Euclidean space with a previously fixed probability law. Apply these isometries successively to the origin and consider the sequence of random points that we obtain this way. We prove a local limit theorem under a suitable moment condition and a necessary non-degeneracy condition. Under stronger hypothesis, we prove a limit theorem on a wide range of scales: between e^(-cl^(1/4)) and l^(1/2), where l is the number of steps.Comment: 62 pages, 1 figure, revision based on referee's report, proofs and results unchange