Recognition of plane-to-plane map-germs

We present a complete set of criteria for determining A-types of plane-to-plane map-germs of corank one with A-codimension <7, which provides a new insight into the A-classification theory from the viewpoint of recognition problem. As an application to generic differential geometry, we discuss about projections of smooth surfaces in 3-space.Comment: 22 page

The US Agriculture Greenhouse Emissions and Environmental Performance

Replaced with revised version of paper 09/23/11.Productivity, Technical Efficiency, environmental performance, Production Economics, Productivity Analysis, Resource /Energy Economics and Policy,

Binary differential equations at parabolic and umbilical points for 22-parameter families of surfaces

We determine local topological types of binary differential equations of asymptotic curves at parabolic and flat umbilical points for generic $2$-parameter families of surfaces in $\mathbb P^3$ by comparing our projective classification of Monge forms and classification of general BDE obtained by Tari and Oliver. In particular, generic bifurcations of the parabolic curve are classified. The flecnodal curve is also examined by direct computations, and we present new bifurcation diagrams in typical examples.Comment: 20 page