580 research outputs found
Mathematical approaches to differentiation and gene regulation
We consider some mathematical issues raised by the modelling of gene
networks. The expression of genes is governed by a complex set of regulations,
which is often described symbolically by interaction graphs. Once such a graph
has been established, there remains the difficult task to decide which
dynamical properties of the gene network can be inferred from it, in the
absence of precise quantitative data about their regulation. In this paper we
discuss a rule proposed by R.Thomas according to which the possibility for the
network to have several stationary states implies the existence of a positive
circuit in the corresponding interaction graph. We prove that, when properly
formulated in rigorous terms, this rule becomes a theorem valid for several
different types of formal models of gene networks. This result is already known
for models of differential or boolean type. We show here that a stronger
version of it holds in the differential setup when the decay of protein
concentrations is taken into account. This allows us to verify also the
validity of Thomas' rule in the context of piecewise-linear models and the
corresponding discrete models. We discuss open problems as well.Comment: To appear in Notes Comptes-Rendus Acad. Sc. Paris, Biologi
Upper bounds for regularized determinants
Let be a holomorphic vector bundle on a compact K\"ahler manifold . If
we fix a metric on , we get a Laplace operator acting upon
smooth sections of over . Using the zeta function of , one
defines its regularized determinant . We conjectured elsewhere
that, when varies, this determinant remains bounded from
above.
In this paper we prove this in two special cases. The first case is when
is a Riemann surface, is a line bundle and , and the second case is when is the projective line, is a line
bundle, and all metrics under consideration are invariant under rotation around
a fixed axis.Comment: 22 pages, plain Te
Congruence schemes
A new category of algebro-geometric objects is defined. This construction is
a vast generalization of existing F1-theories, as it contains the the theory of
monoid schemes on the one hand and classical algebraic theory, e.g.
Grothendieck schemes, on the the other. It also gives a handy description of
Berkovich subdomains and thus contains Berkovich's approach to abstract
skeletons. Further it complements the theory of monoid schemes in view of
number theoretic applications as congruence schemes encode number theoretical
information as opposed to combinatorial data which are seen by monoid schemes
On the slopes of the lattice of sections of hermitian line bundles
In this paper we apply Arakelov theory to study the distribution of the Petersson norms of classical cusp forms as well as the distribution of the sup norms of rational functions on adelic subsets of curves. The method in both cases is to study the limiting distribution of the successive minima of norms of global sections of powers of a metrized ample line bundle as one takes increasing powers of the bundle. We develop a general method for computing the measure associated to this distribution. We also study measures associated to the zeros of sections which have small norm
An arithmetic Riemann-Roch theorem in higher degrees
We prove an analogue in Arakelov geometry of the Grothendieck-Riemann-Roch
theorem
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