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

Neuronal architecture of the human temporal cortex

Anatomy and Embryology1814351-364ANEM