Discrete logarithms in free groups

For the free group on n generators we prove that the discrete logarithm is distributed according to the standard Gaussian when the logarithm is renormalized appropriately.Comment: 9 pages, 1 figure, Corrects a mistake in the Introduction and Section

In vivo imaging reveals transient microglia recruitment and functional recovery of photoreceptor signaling after injury.

Microglia respond to damage and microenvironmental changes within the central nervous system by morphologically transforming and migrating to the lesion, but the real-time behavior of populations of these resident immune cells and the neurons they support have seldom been observed simultaneously. Here, we have used in vivo high-resolution optical coherence tomography (OCT) and scanning laser ophthalmoscopy with and without adaptive optics to quantify the 3D distribution and dynamics of microglia in the living retina before and after local damage to photoreceptors. Following photoreceptor injury, microglia migrated both laterally and vertically through the retina over many hours, forming a tight cluster within the area of visible damage that resolved over 2 wk. In vivo OCT optophysiological assessment revealed that the photoreceptors occupying the damaged region lost all light-driven signaling during the period of microglia recruitment. Remarkably, photoreceptors recovered function to near-baseline levels after the microglia had departed the injury locus. These results demonstrate the spatiotemporal dynamics of microglia engagement and restoration of neuronal function during tissue remodeling and highlight the need for mechanistic studies that consider the temporal and structural dynamics of neuron-microglia interactions in vivo

A wall interference assessment/correction system

A Wall Signature method, the Hackett method, has been selected to be adapted for the 12-ft Wind Tunnel wall interference assessment/correction (WIAC) system in the present phase. This method uses limited measurements of the static pressure at the wall, in conjunction with the solid wall boundary condition, to determine the strength and distribution of singularities representing the test article. The singularities are used in turn for estimating wall interferences at the model location. The Wall Signature method will be formulated for application to the unique geometry of the 12-ft Tunnel. The development and implementation of a working prototype will be completed, delivered and documented with a software manual. The WIAC code will be validated by conducting numerically simulated experiments rather than actual wind tunnel experiments. The simulations will be used to generate both free-air and confined wind-tunnel flow fields for each of the test articles over a range of test configurations. Specifically, the pressure signature at the test section wall will be computed for the tunnel case to provide the simulated 'measured' data. These data will serve as the input for the WIAC method-Wall Signature method. The performance of the WIAC method then may be evaluated by comparing the corrected parameters with those for the free-air simulation. Each set of wind tunnel/test article numerical simulations provides data to validate the WIAC method. A numerical wind tunnel test simulation is initiated to validate the WIAC methods developed in the project. In the present reported period, the blockage correction has been developed and implemented for a rectangular tunnel as well as the 12-ft Pressure Tunnel. An improved wall interference assessment and correction method for three-dimensional wind tunnel testing is presented in the appendix

Ice Shape Scaling for Aircraft in SLD Conditions

This paper has summarized recent NASA research into scaling of SLD conditions with data from both SLD and Appendix C tests. Scaling results obtained by applying existing scaling methods for size and test-condition scaling will be reviewed. Large feather growth issues, including scaling approaches, will be discussed briefly. The material included applies only to unprotected, unswept geometries. Within the limits of the conditions tested to date, the results show that the similarity parameters needed for Appendix C scaling also can be used for SLD scaling, and no additional parameters are required. These results were based on visual comparisons of reference and scale ice shapes. Nearly all of the experimental results presented have been obtained in sea-level tunnels. The currently recommended methods to scale model size, icing limit and test conditions are described