Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve

Neuro-electronic technology in medicine and beyond

This dissertation looks at the technology and social issues involved with interfacing electronics directly to the human nervous system, in particular the methods for both reading and stimulating nerves. The development and use of cochlea implants is discussed, and is compared with recent developments in artificial vision. The final sections consider a future for non-medicinal applications of neuro-electronic technology. Social attitudes towards use for both medicinal and non-medicinal purposes are discussed, and the viability of use in the latter case assessed

Keeping track of worm trackers

C. elegans is used extensively as a model system in the neurosciences due to its well defined nervous system. However, the seeming simplicity of this nervous system in anatomical structure and neuronal connectivity, at least compared to higher animals, underlies a rich diversity of behaviors. The usefulness of the worm in genome-wide mutagenesis or RNAi screens, where thousands of strains are assessed for phenotype, emphasizes the need for computational methods for automated parameterization of generated behaviors. In addition, behaviors can be modulated upon external cues like temperature, O2 and CO2 concentrations, mechanosensory and chemosensory inputs. Different machine vision tools have been developed to aid researchers in their efforts to inventory and characterize defined behavioral “outputs”. Here we aim at providing an overview of different worm-tracking packages or video analysis tools designed to quantify different aspects of locomotion such as the occurrence of directional changes (turns, omega bends), curvature of the sinusoidal shape (amplitude, body bend angles) and velocity (speed, backward or forward movement)

K-Rational D-Brane Crystals

In this paper the problem of constructing spacetime from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi-Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Neron-Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.Comment: 36 page