Minimal invariant varieties and first integrals for algebraic foliations

Let $X$ be an irreducible algebraic variety over $\mathbb{C}$, endowed with an algebraic foliation ${\cal{F}}$. In this paper, we introduce the notion of minimal invariant variety $V({\cal{F}},Y)$ with respect to $({\cal{F}},Y)$, where $Y$ is a subvariety of $X$. If $Y=\{x\}$ is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through $x$. First we prove that for very generic $x$, the varieties $V({\cal{F}},x)$ have the same dimension $p$. Second we generalize a result due to X. Gomez-Mont. More precisely, we prove the existence of a dominant rational map $F:X\to Z$, where $Z$ has dimension $(n-p)$, such that for every very generic $x$, the Zariski closure of $F^{-1}(F(x))$ is one and only one minimal invariant variety of a point. We end up with an example illustrating both results.Comment: 15 page

Insights into the semiclassical Wigner treatment of bimolecular collisions

The semiclassical Wigner treatment of bimolecular collisions, proposed by Lee and Scully on a partly intuitive basis [J. Chem. Phys. 73, 2238 (1980)], is derived here from first principles. The derivation combines E. J. Heller's ideas [J. Chem. Phys. 62, 1544 (1975); 65, 1289 (1976); 75, 186 (1981)], the backward picture of molecular collisions [L. Bonnet, J. Chem. Phys. 133, 174108 (2010)] and the microreversibility principle

The Graph Motif problem parameterized by the structure of the input graph

The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. Though, in the many applications of Graph Motif, the input graph originates from real-life and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. For the FPT cases, we also give some kernelization lower bounds as well as some ETH-based lower bounds on the worst case running time. Interestingly, we establish that Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way.Comment: 24 pages, accepted in DAM, conference version in IPEC 201

The case for absolute ligand discrimination : modeling information processing and decision by immune T cells

Some cells have to take decision based on the quality of surroundings ligands, almost irrespective of their quantity, a problem we name "absolute discrimination". An example of absolute discrimination is recognition of not-self by immune T Cells. We show how the problem of absolute discrimination can be solved by a process called "adaptive sorting". We review several implementations of adaptive sorting, as well as its generic properties such as antagonism. We show how kinetic proofreading with negative feedback implements an approximate version of adaptive sorting in the immune context. Finally, we revisit the decision problem at the cell population level, showing how phenotypic variability and feedbacks between population and single cells are crucial for proper decision