Harmonic Labeling of Graphs

Which graphs admit an integer value harmonic function which is injective and surjective onto $\Z$? Such a function, which we call harmonic labeling, is constructed when the graph is the $\Z^2$ square grid. It is shown that for any finite graph $G$ containing at least one edge, there is no harmonic labeling of $G \times \Z$

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

A deep photometric survey of the eta Chamaeleontis cluster down to the brown dwarf - planet boundary

We report the outcome of the deep optical/infrared photometric survey of the central region (33 X 33 arcmin or 0.9 pc^2) of the eta Chamaeleontis pre-main sequence star cluster. The completeness limits of the photometry are I = 19.1, J = 18.2 and H = 17.6; faint enough to reveal low mass members down to the brown dwarf and planet boundary of ~ 13 M_Jup. We found no such low mass members in this region. Our result combined with a previous shallower (I = 17) but larger area survey indicates that low mass objects (0.013 < M/M(solar mass) < 0.075) either were not created in the eta Cha cluster or were lost due to the early dynamical history of the cluster and ejected to outside the surveyed areas.Comment: 5 pages with 4 figures, accepted by MNRA