On Taking Square Roots without Quadratic Nonresidues over Finite Fields

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\tilde{O}(\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\tilde{O}(\log^3 N)$ for some integers $N$.Comment: 14 page

Micro-Scale, Meso-Scale, Macro-Scale, and Temporal Scale: Comparing the Relative Importance for Robbery Risk in New York City

We compare the relative importance of four dimensions for explaining the micro location of robberies: 1) the micro spatial scale of street segments; 2) the meso spatial scale surrounding the street segment; 3) the temporal pattern, and 4) the macro-scale of the surrounding 2.5 miles. This study uses crime, business, and land use data from New York City and aggregates it to street segments and hours of the day. Although the measures capturing the micro-scale of the street segment explained the largest amount of unique variance, the measures capturing temporal scale across hours of the day (and weekdays) explained the next largest amount of unique variance. The measures of the characteristics in the 2.5 miles macro scale explained the next largest amount of unique variance, and combined with the measures at the meso-scale explained nearly as much of the variance as the street segment measures

Fabrication and Characterization of Controllable Grain Boundary Arrays in Solution Processed Small Molecule Organic Semiconductor Films

We have produced solution-processed thin films of 6,13-bis(triisopropyl-silylethynyl) pentacene with grain sizes from a few micrometers up to millimeter scale by lateral crystallization from a rectangular stylus. Grains are oriented along the crystallization direction, and the grain size transverse to the crystallization direction depends inversely on the writing speed, hence forming a regular array of oriented grain boundaries with controllable spacing. We utilize these controllable arrays to systematically study the role of large-angle grain boundaries in carrier transport and charge trapping in thin film transistors. The effective mobility scales with the grain size, leading to an estimate of the potential drop at individual large-angle grain boundaries of more than one volt. This result indicates that the structure of grain boundaries is not molecularly abrupt, which may be a general feature of solution processed small molecule organic semiconductor thin films where relatively high energy grain boundaries are typically formed. This may be due to the crystal Transient measurements after switching from positive to negative gate bias or between large and small negative gate bias reveal reversible charge trapping with time constants on the order of 10 s, and trap densities that are correlated with grain boundary density. We suggest that charge diffusion along grain boundaries and other defects is the rate determining mechanism of the reversible trapping.Comment: 12 pages, 11 figure

A study of complement inactivation with reagents derived from human complement components

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