The enumeration of simultaneous higher-order contacts between plane curves

Using the Semple bundle construction, we derive an intersection-theoretic formula for the number of simultaneous contacts of specified orders between members of a generic family of degree $d$ plane curves and finitely many fixed curves. The contacts counted by the formula occur at nonsingular points of both the members of the family and the fixed curves.Comment: 32 pages, AmS-TeX v2.1 (Revised statement and proof of one lemma; other minor changes.

Cartan Prolongation of a Family of Curves Acquiring a Node

Using the monster/Semple tower construction, we study the structure of the Cartan prolongation of the family $x_1x_2 = t$ of plane curves with nodal central member

The structural invariants of Goursat distributions

This is the first of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. Here we study those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In the subsequent paper we will relate these structural invariants to the small-growth invariants.Comment: 35 pages, 6 figure

The Reproductive Seasonality and Gametogenic Cycle of Acropora cervicornis off Broward County, Florida, USA

Reproductive characters of the Caribbean reef-building coral Acropora cervicornis were investigated based on histological samples collected from April 2001 through October 2002. Oogenesis commenced in early to mid-October through November and spermatogenesis was initiated from January to March. The onset of gametogenesis was staggered, exhibiting up to approximately a 1-month delay within colonies. In the hermaphroditic polyps, the observed male-to-female gonad ratio was nearly 1:1 and ripe oocytes represented over 70% of the total gonadal volume. Fecundity estimates based on Stage IV ova ranged between 10.4 and 21.8 mm3 per square centimeter per year, comparable to A. cervicornis in Puerto Rico and other broadcasting Indo-Pacific Acropora. Fecundity estimates based on Stage III vitellogenic oocytes indicated statistically significant differences among study sites. Spawning in field conditions was observed in 2001, 2003, and 2004 from 2300 to 2330 h. Gamete release generally occurred synchronously between nights two and seven after the full moon of July or August. However in 2003, multiple, small-scale gamete release episodes occurred over more than one lunar cycle. This coincided with the full moon occurring early in the month of July. While prolific gamete production is reported in this study, low levels of recruitment have been reported for this species. Thus, the highly fragmenting A. cervicornis may rely heavily on asexual reproduction for population maintenance and expansion, and recovery after disturbance may be greatly protracted