Didier Arquès

Functional relations and numbering of rooted genus one maps

Abstract There was no known closed formula enumerating rooted genus one maps by their numbers of edges or vertices and faces. We solve this problem here. We first construct functional relations whose solutions are the generating functions for rooted genus one maps and for doubly rooted planar maps. We then obtain parametric systems and explicit formulaes giving the number of rooted genus one maps with m edges or with m 1 faces and m 2 vertices.

A code in the protein coding genes.

A statistical analysis with 12,288 autocorrelation functions applied in protein (coding) genes of prokaryotes and eukaryotes identifies three subsets of trinucleotides in their three frames: T0 = X0 [symbol: see text] {AAA, TTT} with X0 = {AAC, AAT, ACC, ATC, ATT, CAG, CTC, CTG, GAA, GAC, GAG, GAT, GCC, GGC, GGT, GTA, GTC, GTT, TAC, TTC} in frame 0 (the reading frame established by the ATG start trinucleotide), T1 = X1 [symbol: see text] {CCC} in frame 1 and T2 = X2 [symbol: see text] {GGG} in frame 2 (the frames 1 and 2 being the frame 0 shifted by one and two nucleotides, respectively, to the right). These three subsets are identical in these two gene populations and have five important properties: (i) the property of maximal (20 trinucleotides) circular code for X0 (resp. X1, X2) allowing to retrieve automatically the frame 0 (resp. 1, 2) in any region of the gene without start codon; (ii) the DNA complementarity property C (e.g. C(AAC) = GTT): C(T0) = T0, C(T1) = T2 and C(T2) = T1 allowing the two paired reading frames of a DNA double helix simultaneously to code for amino acids; (iii) the circular permutation property P (e.g. P(AAC) = ACA): P(X0) = X1 and P(X1) = X2 implying that the two subsets X1 and X2 can be deduced from X0; (iv) the rarity property with an occurrence probability of X0 = 6 x 10(-8); and (v) the concatenation properties in favour of an evolutionary code: a high frequency (27.5%) of misplaced trinucleotides in the shifted frames, a maximum (13 nucleotides) length of the minimal window to retrieve automatically the frame and an occurrence of the four types of nucleotides in the three trinucleotide sites. In Discussion, a simulation based on an independent mixing of the trinucleotides of T0 allows to retrieve the two subsets T1 and T2. Then, the identified subsets T0, T1 and T2 replaced in the 2-letter genetic alphabet {R, Y} (R = purine = A or G, Y = pyrimidine = C or T) allow to retrieve the RNY model (N = R or Y) and to explain previous works in the alphabet {R, Y}. Then, these three subsets are related to the genetic code. The trinucleotides of T0 code for 13 amino acids: Ala, Asn, Asp, Gln, Glu, Gly, Ile, Leu, Lys, Phe, Thr, Tyr and Val. Finally, a strong correlation between the usage of the trinucleotides of T0 in protein genes and the amino acid frequencies in proteins is observed as six among seven amino acids not coded by T0, have as expected the lowest frequencies in proteins of both prokaryotes and eukaryotes.

A Physically-Based BRDF Model for Multilayer Systems with Uncorrelated Rough Boundaries

This paper presents a new BRDF model allowing the simulation of the optical behaviour of multilayer systems formed of homogeneous and isotropic thin films with random rough boundaries. The boundaries are supposed to be locally smooth and generated by a stationary and isotropic Gaussian process. Moreover, it is assumed that they are mutually independent from the statistical point of view. The BRDF is composed of three terms: specular, directional diffuse and uniform diffuse terms, and accounts for interference, diffraction and polarization effects. The expressions for the specular and directional diffuse components are derived analytically, by means of the Abeles formalism, within the framework of the Kirchhoff theory of diffraction. We present pictures of composite multilayer materials obtained by incorporating this model in a spectral ray-tracing algorithm.

Rooted maps on orientable surfaces, Riccati's equation and continued fractions

We present a new approach in the study of rooted maps without regard to genus. We prove the existence of a new type of equation for the generating series of these maps enumerated with respect to edges and vertices. This is Riccatis equation. It seems to be the first time that such a differential equation appears in the enumeration of rooted maps. Solving this equation leads to different closed forms of the studied generating series. The most interesting consequence is a development of this generating function in a very nice continued fraction leading to a new equation generalizing the well-known Dyck equation for rooted planar trees. In a second part, we also obtain a differential equation for the generating series of rooted trees regardless of the genus, with respect to edges. This also leads to a continued fraction for the generating series of rooted genus independent trees and to an unexpected relation between both previous generating series of trees and rooted maps.

An approach to geometrical and optical simulation of soap froth

Abstract This paper deals with a complete computer simulation of structures that are of great interest to physicists and mathematicians: foams, and, peculiarly, soap froth. We first present a minimization algorithm, based on Plateaus rules, which allows to model the geometry of irregular patterns of bubbles. The resulting model accounts for the thickness variation of the films, and their tapering under gravity. It involves non-parallel plane films, in the shape of wedges. Then we derive the equations describing the interaction between a plane monochromatic electromagnetic wave and this kind of film. Assuming the material – soap – that constitutes these films is a linear, homogeneous isotropic dielectric medium, the expressions of the intensity reflection and transmission coefficients, also called reflectivity and transmissivity are deduced and simplified in the case of soap films. The expressions given for wedge films can be applied more generally to the case of surfaces with gentle slope variations. The optical simulation of the interaction between light waves and soap froth is performed using a ray-tracing algorithm. We show in this paper how this algorithm had to be adapted, in order to integrate these new expressions of reflectivity and transmissivity. Finally, the consistency of the results is analysed, studying the effect of a variation of the parameters, and making a comparison with reality.

Rooted planar maps are well labeled trees

Abstract R. Cori and B. Vauquelin have constructed (cf[1]) a one to one correspondence from rooted planar maps onto rooted well-labeled trees (trees whose vertices are labeled with natural numbers that differ by at most one on adjacent vertices). This correspondence does not associate other families of planar maps (e.g. planar hypermaps,...) and easily definable families of trees. The main result of this paper (Theorem 1, Section II) is to construct a new one to one correspondence from rooted planar maps onto rooted well-labeled trees which also associates rooted planar hypermaps with n edge-ends (called ‘brin’ in French) and rooted very well-labeled trees (well labeled trees whose adjacent vertices have not the same label) with n edges. This last result is given in Section 3, Theorem 2. The coding of rooting very well-labeled trees by words extending Dycks words (or parenthesis systems), allows their enumeration, hence the enumeration of rooted planar hypermaps. This side is the subject of a work in progress under B. Vauquelin.

Thinning grayscale well-composed images

Usual approaches for constructing topological maps on discrete structures are based on cellular complexes topology. This paper aims to construct a coherent topological map defined on a square grid from a watershed transformation. The main idea behind the proposed approach is to impose some constraints on the original image in order to obtain good properties of the resulting watershed. We propose a definition of well-composed grayscale images based on the well-composed set theory and the cross-section topology. Properties of two different thinning algorithms are then studied and we show how to obtain a thin crest network. We derive an efficient algorithm that permits the construction of a meaningful topological map, resulting in a topological segmentation, i.e. a segmentation that describes in a coherent framework faces and contours. Finally, we demonstrate the usefulness of this algorithm for multilevel image segmentation.

An illumination model for a system of isotropic substrate- isotropic thin film with identical rough boundaries

A new physically-based illumination model describing the interaction of light with a system composed of an isotropic substrate coated by an isotropic film with geometrically identical statistical rough boundaries (ITF) is presented. This model divides the intensity reflected from the system into three components: specular, directional-diffuse and uniform diffuse intensity. The formulas for the intensity reflected coherently (specular) and incoherently (directionaldiffuse) from the system are derived within the framework of the scalar diffraction theory. Assuming that the slopes on the boundaries of the film are small, a first-order expansion of the reflection coefficient is used in the evaluation of the Helmholtz-Kirchhoff integral which allows to calculate the previous intensities. The consistency of the model is evaluated numerically and appraised visually by comparison with classic approximations.

Counting rooted maps on a surface

Several enumeration results are known about rooted maps on orientable surfaces, whereas rooted maps on non-orientable surfaces have seldom been studied. First, we unify both kind of maps, giving general functional equations for the generating series which counts rooted maps on any locally orientable surface, by number of vertices and faces. Then, we formally solve these equations, in order to establish a detailed common formula for all these generating series. All of them appear to be algebraic functions of the variables counting the number of vertices and faces. Explicit expressions and numerical tables for the series counting rooted maps on the non-orientable surfaces of genus 3 and 4 are given.

A possible code in the genetic code

In order to analyse the genetic code, the distribution of the 64 trinucleotides w (words of 3 letters on the gene alphabet {A,C,G,T}, w∈τ={AAA,⋯,TTT}) in the prokaryotic protein coding genes (words of large sizes) is studied with autocorrelation functions. The trinucleotides wp can be read in 3 frames p (p=0: reference frame, p=1: reference frame shifted by 1 letter, p=2: reference frame shifted by 2 letters) in coding genes. Then, the autocorrelation function wp(N)iw′ analyses the occurrence probability of the i-motif wp(N)iw′, i.e. 2 trinucleotides wp in frame p and w′ in any frame (w,w′∈ τ) which are separated by any i bases N (N=A, C, G or T). The 642×3=12288 autocorrelation functions applied to the prokaryotic protein coding genes are almost all non-random and have a modulo 3 periodicity among the 3 following types: 0 modulo 3, 1 modulo 3 and 2 modulo 3. The classification of 12288 i-motifs wp(N)iw′ according to the type of periodicity implies a constant preferential occurrence frame for w′ independent of w and p. Three sub-sets of trinucleotides are identified: 22 trinucleotides in frame 0 forming the subset τ0={AAA, AAC, AAT, ACC, ATC, ATT, CAG, CTC, CTG, GAA, GAC, GAG, GAT, GCC, GGC, GGT, GTA, GTC, GTT, TAC, TTC, TTT} and 21 trinucleotides in each of the frames 1 and 2 forming the sub-sets τ1 and τ2 respectively. Except for AAA, CCC, GGG and TTT, the sub-sets τ1 and τ2 are generated by a circular permutation P of τ0: P(τ0)=τ1 and P(τ1)=τ2. Furthermore, the complementarity property ∁ of the DNA double helix (i.e. ∁(A)=T, ∁(C)=G, ∁(G)=C, ∁(T)=A and if w=l1l2l3 then ∁(w)=∁(l3)∁(l2)∁(l1) with l1, l2, l3∈{A,C,G,T}) is observed in these 3 sub-sets: ∁(τ0)=τ0, ∁(τ1)=τ2 and ∁(τ2)=τ1.