The existence of a perfect -factorization of the complete graph with nodes, namely, , for arbitrary even number , is a 40-year-old open problem in graph theory. So far, two infinite families of perfect -factorizations have been shown to exist, namely, the factorizations of and , where is an arbitrary prime number . It was shown in previous work that finding a perfect -factorization of is relate...

Integer factorization is an important problem in modern cryptography as it is the basis of RSA encryption. I have implemented two integer factorization algorithms: Pollard’s rho algorithm and Dixon’s factorization method. While the results are not revolutionary, they illustrate the software design difficulties inherent to integer factorization. The code for this project is available at https://...

The high dimensionality of the data, the expressions of thousands of features in a much smaller number of samples, presents challenges that affect applicability of the analytical results. In principle, it would be better to describe the data in terms of a small number of meta-features, derived as a result of matrix factorization, which could reduce noise while still capturing the essential feat...

New perturbation analyses are presented for the block Cholesky downdating problem U T U = R T R ? X T X. These show how changes in R and X alter the Cholesky factor U. There are two main cases for the perturbation matrix R in R: (1) R is a general matrix; (2))R is an upper triangular matrix. For both cases, rst order perturbation bounds for the downdated Cholesky factor U are given using two ap...

The Elliptic Curve Method (ECM) of factorization can be used in the relation collection phase of the Number Field Sieve (NFS) to help identify smooth integers. This requires rapidly nding small prime factors for a large number of composites, each of a few machine words in size. We present a software implementation of ECM that is optimized for high throughput operation and compare it with recent...

The success of high-throughput sequencing has lead to an increasing number of projects which sequence large populations of a species. Storage and analysis of sequence data is a key challenge in these projects, because of the sheer size of the datasets. Compression is one simple technology to deal with this challenge. Referential factorization and compression schemes, which store only the differ...

A list of technical reports, including some abstracts and copies of some full reports may be found at: Abstract We compare implementations of two integer factorization algorithms, the elliptic curve method (ECM) and a variant of the Pollard \rho" method, on three machines with parallel and/or vector architectures. ECM is scalable and well suited for both vector and parallel architectures. The \...

We compare implementations of two integer factorisation algorithms, the elliptic curve method (ECM) and a variant of the Pollard “rho” method, on three machines (the Fujitsu AP1000, VP2200 and VPP500) with parallel and/or vector architectures. ECM is scalable and well suited for both vector and parallel architectures.

in this paper, we consider an arbitrary binary polynomial sequence {a_n} and then give a lower triangular matrix representation of this sequence. as main result, we obtain a factorization of the innite generalized pascal matrix in terms of this new matrix, using a riordan group approach. further some interesting results and applications are derived.

in this paper‎, ‎we determine the simple groups $g=ab$‎, ‎where $b$ is isomorphic to $l_{3}(4)$ and $a$ isomorphic to an alternating or a symmetric group on $ngeq5$‎, ‎letters‎.