We provide a new family of elliptic curves that results in a one to two percent performance improvement of the elliptic curve integer factorization method. The speedup is confirmed by extensive tests for factors ranging from 15 to 63 bits.

Heavy nuclei introduce a new scale into the pQCD description of hard processes on nuclei, the saturation scale, and the familiar linear k ⊥ –factorization breaks down. It is replaced by a new concept, the nonlinear k ⊥ –factorization. Here we give a brief overview how topological cross sections for hard processes on nuclei are obtained from nonlinear k ⊥ –factorization.

The generalized Cholesky factorization and the Cholesky-like factorization are two generalizations of the classic Cholesky factorization. In this paper, the rigorous multiplicative perturbation bounds for the two factorizations are derived using the matrix equation and the refined matrix equation approaches. The corresponding first-order multiplicative perturbation bounds, as special cases, are...

We provide a construction of free factorization algebras in algebraic geometry and link factorization homology of a scheme with coefficients in a free factorization algebra to the homology of its (unordered) configuration spaces. As an application, this construction allows for a purely algebro-geometric proof of homological stability of configuration spaces. CONTENTS

We compare results on diffractive W-boson production at the Tevatron with predictions based on the diffractive structure function measured in deep inelastic scattering at HERA assuming (a) conventional factorization or (b) hard factorization combined with a rapidity gap distribution scaled to the total gap probability. We find that conventional factorization fails, while the scaling prediction ...

We discuss factorization in heavy quarkonium production in high energy collisions using NRQCD. Infrared divergences at NNLO are not matched by conventional NRQCD matrix elements. However, we show that gauge invariance and factorization require that conventional NRQCD production matrix elements be modified to include Wilson lines or non-abelian gauge links. With this modification NRQCD factoriza...

We show that existing proofs of factorization imply the cancellation of certain multiladder contributions that Gotsman, Levin, and Maor had suggested would invalidate the basic factorization theorem in QCD. No modifications of the original argument are necessary, although the details of the example offer useful insight into the mechanisms of factorization.

Familiarly, in Z, we have unique factorization. We investigate the general ring and what conditions we can impose on it to necessitate analogs of unique factorization. The trivial ideal structure of a field, the extent to which primary decomposition is unique, that a Noetherian ring necessarily has one, that a principal ideal domain is a unique factorization domain, and that a Dedekind domain h...

This article aimed at a general audience of computational scientists, surveys the Cholesky factorization for symmetric positive definite matrices, covering algorithms for computing it, the numerical stability of the algorithms, and updating and downdating of the factorization. Cholesky factorization with pivoting for semidefinite matrices is also treated.  2009 John Wiley & Sons, Inc. WIREs Co...