23 research outputs found
Random Planar Lattices and Integrated SuperBrownian Excursion
In this paper, a surprising connection is described between a specific brand
of random lattices, namely planar quadrangulations, and Aldous' Integrated
SuperBrownian Excursion (ISE). As a consequence, the radius r_n of a random
quadrangulation with n faces is shown to converge, up to scaling, to the width
r=R-L of the support of the one-dimensional ISE. More generally the
distribution of distances to a random vertex in a random quadrangulation is
described in its scaled limit by the random measure ISE shifted to set the
minimum of its support in zero.
The first combinatorial ingredient is an encoding of quadrangulations by
trees embedded in the positive half-line, reminiscent of Cori and Vauquelin's
well labelled trees. The second step relates these trees to embedded (discrete)
trees in the sense of Aldous, via the conjugation of tree principle, an
analogue for trees of Vervaat's construction of the Brownian excursion from the
bridge.
From probability theory, we need a new result of independent interest: the
weak convergence of the encoding of a random embedded plane tree by two contour
walks to the Brownian snake description of ISE.
Our results suggest the existence of a Continuum Random Map describing in
term of ISE the scaled limit of the dynamical triangulations considered in
two-dimensional pure quantum gravity.Comment: 44 pages, 22 figures. Slides and extended abstract version are
available at http://www.loria.fr/~schaeffe/Pub/Diameter/ and
http://www.iecn.u-nancy.fr/~chassain
Solution of coupled vertex and propagator Dyson-Schwinger equations in the scalar Munczek-Nemirovsky model
In a scalar model, we exactly solve the vertex and
propagator Dyson-Schwinger equations under the assumption of a spatially
constant (Munczek-Nemirovsky) propagator for the field. Various
truncation schemes are also considered.Comment: 7 pages,4 figures, minor changes, reference added for published
versio
On the Evolution of the Standard Genetic Code: Vestiges of Critical Scale Invariance from the RNA World in Current Prokaryote Genomes
Herein two genetic codes from which the primeval RNA code could have originated the standard genetic code (SGC) are derived. One of them, called extended RNA code type I, consists of all codons of the type RNY (purine-any base-pyrimidine) plus codons obtained by considering the RNA code but in the second (NYR type) and third (YRN type) reading frames. The extended RNA code type II, comprises all codons of the type RNY plus codons that arise from transversions of the RNA code in the first (YNY type) and third (RNR) nucleotide bases. In order to test if putative nucleotide sequences in the RNA World and in both extended RNA codes, share the same scaling and statistical properties to those encountered in current prokaryotes, we used the genomes of four Eubacteria and three Archaeas. For each prokaryote, we obtained their respective genomes obeying the RNA code or the extended RNA codes types I and II. In each case, we estimated the scaling properties of triplet sequences via a renormalization group approach, and we calculated the frequency distributions of distances for each codon. Remarkably, the scaling properties of the distance series of some codons from the RNA code and most codons from both extended RNA codes turned out to be identical or very close to the scaling properties of codons of the SGC. To test for the robustness of these results, we show, via computer simulation experiments, that random mutations of current genomes, at the rates of 10−10 per site per year during three billions of years, were not enough for destroying the observed patterns. Therefore, we conclude that most current prokaryotes may still contain relics of the primeval RNA World and that both extended RNA codes may well represent two plausible evolutionary paths between the RNA code and the current SGC
A new formulation of differential radiosity and a rendering application
Among all the existing realistic lighting methods, the radiosity method is the only one that gives precisely an
analytic solution of diffuse light exchanges. The gradient of this solution have been studied but not often used in
a context of pure rendering. We present in this article a method to render a surface using the radiosity contour
levels. First, we define a differential formulation of the radiosity equation which leads us to a new expression of
the gradient of radiosity. We deduce from this general equation a simpler equation of this gradient in the case of
a planar surface lighted by a light source reduced to a point. Then we present our method to render planar
surfaces using a radial mesh that follows the contour levels of the radiosity. This method is shown to improve
the quality of the rendering and decrease the number of vertices used for rendering
Mathematical Approximation for Real-time Lighting Rendering
Many shading models are able to provide realistic rendering of lighting effects under various atmospheric conditions but computational times are often expensive. This paper proposes a method to render lighting effects within participating media in real-time on a graphics workstation. It consists of mathematical approximations based on a re-formulation of the light transport equation considering atmospheric scattering with light sources described by their luminous intensity distribution. Hardware capabilities of graphics computer boards are used to accelerate parts of the rendering process