Again, Linearizable Mappings

We examine a family of 3-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through the application of the singularity confinement criterion and are explicitly integrated.Comment: 14 pages, no figures, to be published in Physica

The Gambier Mapping

We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painleve property. In the continuous case, the solutions of the Gambier equation is obtained through a system of Riccati equations. The same holds true in the discrete case also. We use the singularity confinement criterion in order to study the integrability of this new mapping.Comment: PlainTe

Another integrable case in the Lorenz model

A scaling invariance in the Lorenz model allows one to consider the usually discarded case sigma=0. We integrate it with the third Painlev\'e function.Comment: 3 pages, no figure, to appear in J. Phys.

Discrete and Continuous Linearizable Equations

We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one and reduce the system to a single differential equation. This equation is of the form of those singled-out by Painlev\'e in his quest for integrable forms. In the discrete case, we extend previous results of ours showing that, again by elimination of variables, the general projective system can be written as a mapping for a single variable. We show that this mapping is a member of the family of multilinear systems (which is not integrable in general). The continuous limit of multilinear mappings is also discussed.Comment: Plain Tex file, 14 pages, no figur

Constructing Integrable Third Order Systems:The Gambier Approach

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlev\'e method for ODE's and the singularity confinement method for mappings).Comment: 14 pages, TEX FIL

Discrete systems related to some equations of the Painlev\'e-Gambier classification

We derive integrable discrete systems which are contiguity relations of two equations in the Painlev\'e-Gambier classification depending on some parameter. These studies extend earlier work where the contiguity relations for the six transcendental Painlev\'e equations were obtained. In the case of the Gambier equation we give the contiguity relations for both the continuous and the discrete system.Comment: 10 page

Integrable systems without the Painlev\'e property

We examine whether the Painlev\'e property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlev\'e continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities

Bilinear Discrete Painleve-II and its Particular Solutions

By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $\tau$ function for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te

Vestiges humains des niveaux de l’Aurignacien ancien du site de Brassempouy (Landes)

Le gisement de Brassempouy est localisé au sud du département des Landes (France), à deux kilomètres du village de Brassempouy et à quarante kilomètres au sud de Mont-de-Marsan, en Chalosse. Il comprend plusieurs cavités (grotte du Pape, grotte des Hyènes, galerie Dubalen, galerie du Mégacéros) qui appartiennent à un réseau karstique complexe creusé dans une formation calcaire éocène à quelques mètres sous le sol naturel. Les vestiges humains étudiés dans cet article ont été découverts de 1981 à 1996 par H. Delporte dans des niveaux de la grotte des Hyènes, de la galerie Dubalen et de la grotte du Pape attribués à l’Aurignacien ancien. Ces niveaux se placent dans l’intervalle 34 000 ans BP-30 000 ans BP. L’échantillon de vestiges humains comprend 13 dents, un fragment de mandibule, un fragment de crâne et deux phalanges distales de la main représentant des adultes et des enfants. Ces vestiges sont intéressants à un double titre. D’une part, les vestiges humains en contexte aurignacien ancien sont rares et, d’autre part, quatre des dents d’adulte ont une racine perforée ou rainurée intentionnellement et le fragment de crâne présente des cassures effectuées vraisemblablement sur os frais. L’objectif majeur de cet article est de présenter les caractéristiques morphométriques des différents vestiges et de discuter leur position taxinomique. L’analyse de ces vestiges, en particulier celle des dents isolées, montre qu’ils s’intègrent dans la variabilité des hommes d’anatomie moderne du Paléolithique moyen du Proche-Orient et du Gravettien d’Europe ainsi que dans celle des rares Aurignaciens d’Europe et des Néandertaliens du Würm. Il est impossible de conclure sur leurs affinités taxinomiques. Ailleurs en Europe, les vestiges humains découverts en contexte aurignacien sont aussi très fragmentaires et peu diagnostiques. Les mieux conservés sont en effet souvent mal datés. Aussi l’anatomie des artisans de l’Aurignacien en Europe reste très mal connue de telle sorte qu’il est actuellement impossible d’affirmer que les techno-complexes aurignaciens anciens relèvent uniquement de populations d’anatomie moderne. Bien que l’analyse détaillée des modifications artificielles soit présentée dans un autre article, on peut d’ores et déjà noter qu’en l’absence de sépultures primaires aurignaciennes, les dents percées de Brassempouy constituent un des rares indices d’intervention anthropique sur le corps, susceptible de relever d’un geste funéraire.The Brassempouy locality is situated in the Chalosse region in the southern part of the French department of Landes, two kilometers from the village of Brassempouy and forty kilometers south of Mont-de-Marsan. It includes several caves (grotte du Pape, grotte des Hyènes, galerie Dubalen, galerie du Mégacéros) which are part of a complex karstic system developed in an Eocene limestone formation just a few meters beneath the natural ground surface. The human remains described in the present article were discovered by H. Delporte between 1981 and 1996, in levels attributed to the early Aurignacian in the grotte des Hyènes, the galerie Dubalen and the grotte du Pape. These levels date to between 34,000 and 30,000 years BP. The sample of human remains, representing both adults and children, is comprised of 13 teeth, one mandibular fragment, one cranial fragment and two distal phalanges of the hand. The scientific interest of these remains is two-fold. First, human remains are rare in Aurignacian contexts. Second, four of the adult teeth show intentionally perforated or circum-incised roots, and the cranial fragment shows fracture patterns consistant with breakage of fresh bone. The primary goal of this article is to outline the morphometric characters of the different remains and to evaluate their taxonomic status. The analysis of these remains, especially the isolated teeth, shows that they fall within the range of variation, not only of anatomically modern humans of the Near Eastern Middle Paleolithic and the European Gravettian, but also of the few known European Aurignacians and of the Wurm II Neandertals. It is, therefore, impossible to draw conclusions concerning their precise taxonomic affiliation. Elsewhere in Europe, Aurignacian remains are also very fragmentary and equally undiagnostic. Those that are best preserved are often poorly dated. In sum, the anatomy of the European makers of the Aurignacian is so poorly known that it remains impossible to confirm that the early Aurignacian techno-complex is uniquely the product of anatomically modern populations. Although the intentional modifications of the Brassempouy human remains are presented in detail in a separate publication, we wish to emphasize here that, with primary Aurignacian burials being completely unknown, the modified human teeth from Brassempouy represent one of the rare traces of human intervention with dead bodies that might be interpreted as funerary behavior