Random sieves, II

Sieves of Dimension 1 +

with fκ(s) as large as possible (resp. Fκ(s) as small as possible) given that the above inequality holds for all choices of A satisfying (1). Selberg [2] has shown (in a much more general context) that the functions fκ(s), Fκ(s) are continuous, monotone, and computable for s > 1, that they do not change if we replace (1) with (2), and that they tend to 1 exponentially as s goes to infinity. Mor...

Factoring Integers Using SIMD Sieves

We describe our single-instruction multiple data (SIMD) implementation of the multiple polynomial quadratic sieve integer factoring algorithm. On a 16K MasPar massively parallel computer, our implementation can factor 100 digit integers in a few days. Its most notable success was the factorization of the 110-digit RSA-challenge number, which took about a month.

Enantiomerically enriched, polycrystalline molecular sieves.

Zeolite and zeolite-like molecular sieves are being used in a large number of applications such as adsorption and catalysis. Achievement of the long-standing goal of creating a chiral, polycrystalline molecular sieve with bulk enantioenrichment would enable these materials to perform enantioselective functions. Here, we report the synthesis of enantiomerically enriched samples of a molecular si...

Random Triangles II

Let S denote the unit sphere in Euclidean 3-space. A spherical triangle T is a region enclosed by three great circles on S; a great circle is a circle whose center is at the origin. The sides of T are arcs of great circles and have length a, b, c. Each of these is ≤ π. The angle α opposite side a is the dihedral angle between the two planes passing through the origin and determined by arcs b, c...

On Random Matrices Ii