The Helton-Nie Conjecture (HNC) is the proposition that every convex semialgebraic set is a spectrahedral shadow. Here we prove that HNC is equivalent to another proposition related to quadratically constrained quadratic programming. Namely, that the convex hull of the rank-one elements of any spectrahedron is a spectrahedral shadow. In the case of compact convex semialgebraic sets, the spectra...

We give an output-sensitive algorithm to compute the first k convex or maximal layers in 0( n log I&) -time where Hk is the number of points participating in the first k layers. Computing only the first k layers is interesting in various problems that arise in computational geometry (k-sets and dually k-levels, k-hulls and dually k-belts), pattern recognition, statistics, operations research, etc.

this experiment was carried out using 200 ross 308 broiler chickens in a completely randomized design with 5 treatments and 4 replicates. the true metabolisable energy of the barley hulls were determined using cecectomized roosters.the dry matter, true metabolisable energy, ash, ether extracts, crude protein, crude fiber, ndf, adf, ca and total p content of barely hulls were: 92%, 978 kcal/kg, ...

Divide-and-conquer is a central paradigm for the design of algorithms, through which some fundamental computational problems, such as sorting arrays and computing convex hulls, are solved in optimal time within Θ(n logn) in the worst case over instances of size n. A finer analysis of those problems yields complexities within O(n(1 + H(n1, . . . , nk))) ⊆ O(n(1+ log k)) ⊆ O(n logn) in the worst ...

Let k, d, λ ≥ 1 be integers with d ≥ λ. What is the maximum positive integer n such that every set of n points in R has the property that the convex hulls of all k-sets have a transversal (d−λ)-plane? What is the minimum positive integer n such that every set of n points in general position in R has the property that the convex hulls of all k-sets do not have a transversal (d− λ)-plane? In this...

In this paper, the behavior of the pseudopolynomial numerical hull of a square complex matrix with respect to structured perturbations and its radius is investigated.

The modified Gram–Schmidt algorithm is a well-known and widely used procedure to orthogonalize the column vectors of a given matrix. When applied to ill-conditioned matrices in floating point arithmetic, the orthogonality among the computed vectors may be lost. In this work, we propose an a posteriori reorthogonalization technique based on a rank-k update of the computed vectors. The level of o...

The modified Gram–Schmidt algorithm is a well–known and widely used procedure to orthogonalize the column vectors of a given matrix. When applied to ill–conditioned matrices in floating point arithmetic, the orthogonality among the computed vectors may be lost. In this work, we propose an a posteriori reorthogonalization technique based on a rank–k update of the computed vectors. The level of o...

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...