We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves Ek : x 3 + y = k of ranks 8, 9, 10, and 11 over Q. As a corollary we produce examples of elliptic curves over Q with a rational 3-torsion point and rank as high as 11. We also discuss the problem of finding the minimal curve Ek of a give...

In various applications, data in multi-dimensional space are normalized to unit length. This paper considers the problem of best fitting given points on the m-dimensional unit sphere S by k-dimensional great circles with k much less than m. The task is cast as an algebraically constrained low-rank matrix approximation problem. Using the fidelity of the low-rank approximation to the original dat...

We propose a new algorithm called Equipoise for the k-server problem, and we prove that it is 2-competitive for two servers and 11-competitive for three servers. For k = 3, this is a tremendous improvement over previously known constants. The algorithm uses several techniques for designing on-line algorithms { convex hulls, work functions and forgiveness.

We consider the Modified Gram-Schmidt orthogonalization applied to a matrix A ∈ Rm×n. This corresponds to a QR factorization : A = QR. We study this algorithm in finite precision computation when the matrix A has a numerical rank deficiency k. This subject has already been dealt with success by Björck and Paige in 1992 [1]. They give useful bounds in term of norms. We extend their results to pr...

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh size, applied to a finite-rank degenerate-kernel...

Tomato (Lycopersicum esculentum ‘Early Girl’) seedling growth was evaluated in substrates containing varying proportions of ground rice hulls. Substrates were formulated containing 0, 30, 60, and 90% ground rice hulls with one-half of the treatments also treated with a surfactant. Seedling growth in two of the ground rice hull-containing substrates was generally similar to the two controls of 9...

Given an m × n matrix A and a positive integer k, we introduce a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying AT to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient wheneverA andAT can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of th...

We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, n_{h}(A,t)dA , with enclosed area in the interval (A,A+dA) , is described, for a disordered initia...