The Brun–Hooley Sieve

Sieve Methods Lecture Notes, Part I the Brun-hooley Sieve

A sieve is a technique for bounding the size of a set after the elements with “undesirable properties” (usually of a number theoretic nature) have been removed. The undesirable properties could be divisibility by a prime from a given set, other multiplicative constraints (divisibility by a perfect square for example) or inclusion in a set of residue classes. The methods usually involve some kin...

The Brun-hooley Sieve

The Information Sieve

We introduce a new framework for unsupervised learning of representations based on a novel hierarchical decomposition of information. Intuitively, data is passed through a series of progressively fine-grained sieves. Each layer of the sieve recovers a single latent factor that is maximally informative about multivariate dependence in the data. The data is transformed after each pass so that the...

The number field sieve

We describe the main ideas underlying integer factorization using the number field sieve.

The Bernoulli Sieve Revisited

We consider an occupancy scheme in which “balls” are identified with n points sampled from the standard exponential distribution, while the role of “boxes” is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index K n of the last occupied box, the number Kn of occupied boxes, the number Kn...