Trisecant Lemma for Non Equidimensional Varieties

The classic trisecant lemma states that if $X$ is an integral curve of \PP^3 then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. In this paper, our purpose is first to present another derivation of this result and then to introduce a generalization to non-equidimensional varities. For the sake of clarity, we shall reformulate our first problem as follows. Let $Z$ be an equidimensional variety (maybe singular and/or reducible) of dimension $n$, other than a linear space, embedded into \PP^r, $r \geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $\dim(V_{1,3}(Z)) = 2n$. Then we inquire the more general case, where $Z$ is not required to be equidimensional. In that case, let $Z$ be a possibly singular variety of dimension $n$, that may be neither irreducible nor equidimensional, embedded into \PP^r, where $r \geq n+1$, and $Y$ a proper subvariety of dimension $k \geq 1$. Consider now $S$ being a component of maximal dimension of the closure of \{l \in \G(1,r) \vtl \exists p \in Y, q_1, q_2 \in Z \backslash Y, q_1,q_2,p \in l\}. We show that $S$ has dimension strictly less than $n+k$, unless the union of lines in $S$ has dimension $n+1$, in which case $dim(S) = n+k$. In the latter case, if the dimension of the space is stricly greater then $n+1$, the union of lines in $S$ cannot cover the whole space. This is the main result of our work. We also introduce some examples showing than our bound is strict

Modelling the vertical distribution of eggs of anchovy (Engraulis encrasicolus) and sardine (Sardina pilchardus)

Measurements were made of the density and settling velocity of eggs of sardine (Sardina pilchardus) and anchovy (Engraulis encrasicolus), using a density-gradient column. These results were related to observed vertical distributions of eggs obtained from stratified vertical distribution sampling in the Bay of Biscay. Eggs of both species had slightly positive buoyancy in local seawater throughout most of their development until near hatching, when there was a marked increase in density and they became negatively buoyant. The settling velocity of anchovy eggs, which are shaped as prolate ellipsoids, was close to predictions for spherical particles of equivalent volume. An improved model was developed for prediction of the settling velocity of sardine eggs, which are spherical with a relatively large perivitelline volume; this incorporated permeability of the chorion and adjustment of the density of the perivitelline fluid to ambient seawater. Eggs of both species were located mostly in the top 20 m of the water column, in increasing abundance towards the surface. A sub-surface peak of egg abundance was sometimes observed at the pycnocline, particularly where this was pronounced and associated with a low-salinity surface layer. There was a progressive deepening of the depth distributions for successive stages of egg development. Results from this study can be applied for improved plankton sampling of sardine and anchovy eggs and in modelling studies of their vertical distribution

Skill Comparisons between Neural Networks and Canonical Correlation Analysis in Predicting the Equatorial Pacific Sea Surface Temperatures.

Among the statistical methods used for seasonal climate prediction, canonical correlation analysis (CCA), a more sophisticated version of the linear regression (LR) method, is well established. Recently, neural networks (NN) have been applied to seasonal climate prediction. Unlike CCA and LR, NN is a nonlinear method, which leads to the question whether the nonlinearity of NN brings any extra prediction skill. In this study, an objective comparison between the three methods (CCA, LR, and NN) in predicting the equatorial Pacific sea surface temperatures (in regions Niño1+2, Niño3, Niño3.4, and Niño4) was made. The skill of NN was found to be comparable to that of LR and CCA. A cross-validated t test showed that the difference between NN and LR and the difference between NN and CCA were not significant at the 5% level. The lack of significant skill difference between the nonlinear NN method and the linear methods suggests that at the seasonal timescale the equatorial Pacific dynamics is basically linear. Copyright 2000 American Meteorological Society (AMS). Permission to use figures, tables, and brief excerpts from this work in scientific and educational works is hereby granted provided that the source is acknowledged. Any use of material in this work that is determined to be “fair use” under Section 107 of the U.S. Copyright Act or that satisfies the conditions specified in Section 108 of the U.S. Copyright Act (17 USC §108, as revised by P.L. 94-553) does not require the AMS’s permission. Republication, systematic reproduction, posting in electronic form, such as on a web site or in a searchable database, or other uses of this material, except as exempted by the above statement, requires written permission or a license from the AMS. Additional details are provided in the AMS Copyright Policy, available on the AMS Web site located at (http://www.ametsoc.org/) or from the AMS at 617-227-2425 or [email protected], Faculty ofEarth and Ocean Sciences, Department ofReviewedFacult