47 research outputs found
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems
We show that unless P=NP, there exists no polynomial time (or even
pseudo-polynomial time) algorithm that can decide whether a multivariate
polynomial of degree four (or higher even degree) is globally convex. This
solves a problem that has been open since 1992 when N. Z. Shor asked for the
complexity of deciding convexity for quartic polynomials. We also prove that
deciding strict convexity, strong convexity, quasiconvexity, and
pseudoconvexity of polynomials of even degree four or higher is strongly
NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd
degree polynomials can be decided in polynomial time.Comment: 20 page
The role of overproduction in the development of the primate visual cortex
Biomedical Research10SUPPL. 277-79BRES
Functional morphology in the inferior colliculus of the marmoset
Hearing Research381-267-79HERE
Pyramidal neurons are immunopositive for peptides, but not GABA, in the temporal cortex of the macaque monkey (Macaca fascicularis)
Brain Research533124-41BRRE