Reference: DMA-07-01 (March 2007)
Source:pdfAbstract: We discuss scaling limits of large bipartite planar maps. If $p\geq 2$ is a fixed integer, we consider, for every integer $n\geq 2$, a random planar map $M_n$ which is uniformly distributed over the set of all rooted $2p$-angulations with $n$ faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of $M_n$, equipped with the graph distance rescaled by the factor $n^{-1/4}$, converges in distribution as $n\to\infty$ towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of $p$ and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to $4$.
AMS Classification :****
Keywords :****
Reference: DMA-07-02 (March 2007)
Source:pdfAbstract: We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted $2k$-angulations are a.s.~homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk.
AMS Classification :****
Keywords :****
Reference: DMA-07-03 (August 2008)
Source:pdfAbstract: Nous montrons que l'existence d'un rayon de Hall dans le spectre de Lagrange des constantes d'approximation d'un nombre r\'eel par des nombres rationnels se g\'en\'eralise \`a de nombreux probl\`emes d'approximation diophantienne, comme cons\'equence de la possibilit\'e de prescrire arbitrairement une hauteur de p\'en\'etration asymptotique suffisamment grande d'une g\'eod\'esique dans un voisinage d'une pointe d'une vari\'et\'e riemannienne de volume fini \`a courbure strictement n\'egative.
AMS Classification :****
Keywords :****
Reference: DMA-07-04 (April 2007)
Source:pdfAbstract: We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincar\'e bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime $p$ is ordinary, we give a new construction of the two-variable $p$-adic measure interpolating special values of Hecke $L$-functions of imaginary quadratic fields, originally constructed by Manin-Vishik and Katz. Our method via theta functions also gives insight for the case when $p$ is supersingular. The method of this paper will be used in subsequent papers in constructing certain two-variable $p$-adic distribution for supersingular $p$ interpolating Eisenstein-Kronecker numbers in two-varibales, as well as explicit calculation in two-variables of the $p$-adic elliptic polylogarithms for CM elliptic curves.
AMS Classification :****
Keywords :****
Reference: DMA-07-05 (May 2007)
Source:pdfAbstract: We provide a mathematical analysis of appearance of the concentrations (as Dirac masses) of the solution to a Fokker-Planck system with asymmetric potentials. This problem has been proposed as a model to describe motor proteins moving along molecular filaments. The components of the system describe the densities of the different conformations of the proteins. Our results are based on the study of a Hamilton-Jacobi equation arising, at the zero diffusion limit, after an exponential transformation change of the phase function that rises a Hamilton-Jacobi equation. We consider different classes of conformation transitions coefficients (bounded, unbounded and locally vanishing).
AMS Classification :35B25, 49L25, 92C05
Keywords :Hamilton-Jacobi equations, molecular motors, Fokker-Planck equations
Reference: DMA-07-06 (May 2007)
Source:pdfAbstract: The so-called 'nonlocal Fisher' model takes into account an influence neighborhood for inhibition in classical Fisher ecological invasion. In this area, it has been introduced to represent front propagation with redistributed resources. More recently it has also been proposed as the simpler model exhibiting Turing instability and the biological interpretation refers to adaptive evolution. One aspect of the present paper is to propose a nonlinear analysis of these Turing patterns. More precisely, we introduce a rescaled equation in order to take into account rare mutations (small diffusion). We analyze in which circumstances such a model exhibits stable patterns, among them the Dirac concentrations (that are interpreted as morphs in adaptive dynamics) are remarkable. We use a change of variables, similar to the phase in WKB method, that describes more accurately the phenomenon and leads to a constrained Hamilton-Jacobi equation. It allows us to interpret several features of the patterns, as the weights of the Dirac concentration points, the asymmetry variable regulating their velocities and other relevant quantities.
AMS Classification :35K57, 35B25, 49L25, 92C15, 92D15
Keywords :Adaptive evolution, Redistributed resources, Turing instability, Nonlocal Fisher equation, Dirac concentrations, Hamilton-Jacobi equation.
Reference: DMA-07-07 (June 2007)
Source:pdfAbstract: Employing Morse theory and the method of analytic discs but no $\overline{ \partial}$ techniques, we establish a version of the Hartogs extension theorem in a singular setting, namely: for every domain $\Omega$ of an $(n-1)$-complete normal complex space of pure dimension $n \geqslant 2$, and for every compact set $K \subset \Omega$ such that $\Omega \backslash K$ is connected, holomorphic or meromorphic functions in $\Omega \backslash K$ extend holomorphically or meromorphically to $\Omega$.
AMS Classification :Primary: 32F10; Secondary: 32C20, 32C55
Keywords :****
Reference: DMA-07-08 (October 2007)
Source:pdfAbstract: This technical report is the summary of the numerical experiments performed during the internship of Boris Mauricette at DMA and CEREA from March to August 2007. It will be referenced in subsequent journal papers.
AMS Classification :****
Keywords :****
Reference: DMA-07-09 (November 2007)
Source:pdfAbstract: In the space of $C^k$ piecewise expanding unimodal maps, $k\geq2$, we characterize the $C^2$ smooth families of maps where the topological dynamics does not change (the ``smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of $C^{k-1+Lip}$ deformations tangent to every given $C^k$ horizontal direction, for $k\ge 2$.
AMS Classification :****
Keywords :****
Reference: DMA-07-10 (November 2007)
Source:pdfAbstract: The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,\gamma}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a logarithmic term. Oracle inequalities allow to derive the maxiset of $\tilde{f}_{n,\gamma}$. Then, minimax properties of $\tilde{f}_{n,\gamma}$ are established. We first prove that the rate of this estimator on Besov spaces ${\cal B}^\al_{p,q}$ when $p\leq 2$ is $(\ln(n)/n)^{\al/(1+2\al)}$. This result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\cal B}^\al_{p,q}$ with $p\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. This result is new. Furthermore, $\tilde{f}_{n,\gamma}$ is adaptive minimax up to a logarithmic term. When $p>2$, the situation changes dramatically and the rate of $\tilde{f}_{n,\gamma}$ on Besov spaces ${\cal B}^\al_{p,q}$ is worse than $(\ln(n)/n)^{\al/(1+2\al)}$. Finally, the random threshold depends on a parameter $\gamma$ that has to be suitably chosen in practice. Some theoretical results provide upper and lower bounds of $\gamma$ to obtain satisfying oracle inequalities. Simulations reinforce these results.\\
AMS Classification :Mathematics Subject Classification (2000) 62G05 62G20
Keywords :Adaptive estimation, Model selection, Oracle inequalities, Poisson process, Thresholding rule