Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials

A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and the existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class (neither potential nor kinetic term). These higher-order Abel equations are studied by means of their Darboux polynomials and Jacobi multipliers. In all the cases a family of constants of the motion is explicitly obtained. The general n-dimensional case is also studied

G´en´etique Clinique dans le Service de P´ediatrie et de G´en´etique M´edicale du Centre National Hospitalier et Universitaire de Cotonou : Etat des Lieux et Perspectives

Il s’agissait d’une ´etude r´etrospective descriptive portant sur les patients rec¸us en consultation de g´en´etique m´edicale de Septembre 2004 `a Aoˆut 2007. Les patients b´en´eficiaient des examens dysmorphologique et physique, des bilans cytog´en´etiques et/ou mol´eculaires, des interventions th´erapeutiques et un suivi `a long terme. Les variables ´etudi´ees ´etaient les donn´ees sociod´emographiques et cliniques. Soixante et seize patients ont ´et´e rec¸us durant la p´eriode avec une pr´edominance masculine (57,89%). Les motifs de consultation ´etaient domin´es par le retard psychomoteur (38,15%), la dysmorphie faciale (30,26%) et les malformations (19,73%). Les principales malformations portaient sur les extr´emit´es et la face. Les pathologies confirm´ees comprenaient des aberrations chromosomiques (46,05%) avec une pr´edominance de la trisomie 21 et des maladies monog´eniques (7,89%). Le rendement de nos recherches pourrait ˆetre am´elior´e par l’acc`es `a la technique FISH. C’est une exp´erience quasi unique en Afrique de l’ouest et permet d’apporter des r´eponses aux personnes souffrant d’affections h´er´editaires.Mots Cl´es g´en´etique clinique ; retard psychomoteur ; dysmorphie ; malformation ; aberration chromosomique ; maladie monog´eniqu

An extended scaling analysis of the S=1/2 Ising ferromagnet on the simple cubic lattice

It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the Renormalization Group Theory can only be applied over a narrow temperature range, the "critical region"; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions which is traditional but which is very inefficient for data covering wide temperature ranges. An alternative "extended caling" approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the 3d Ising model. After a discussion of the exact expressions for the historic 1d Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice 3d Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from Tc up to infinite temperature.Comment: 16 pages, 17 figure

Supersymmetry of the Nonstationary Schr\"odinger equation and Time-Dependent Exactly Solvable Quantum Models

New exactly solvable quantum models are obtained with the help of the supersymmetric extencion of the nonstationary Schr/"odinger equation.Comment: Talk at the 8th International Conference "Symmetry Methods in Physics". Dubna, Russia, 28 July - 2 August, 199

Connection between the Green functions of the supersymmetric pair of Dirac Hamiltonians

The Sukumar theorem about the connection between the Green functions of the supersymmetric pair of the Schr\"odinger Hamiltonians is generalized to the case of the supersymmetric pair of the Dirac Hamiltonians.Comment: 12 pages,Latex, no figure

Darboux transformations for a 6-point scheme

We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6-point difference scheme. The scheme is appropriate to solving a hyperbolic type initial-boundary value problem. We discuss several reductions and specifications of the transformations as well as construction of other Darboux covariant schemes by means of existing ones. In particular we introduce a 10-point scheme which can be regarded as the discretization of self-adjoint hyperbolic equation

Darboux transformations for quasi-exactly solvable Hamiltonians

We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of the first type give singular intermediate potentials and the ones of the second type give complex-valued intermediate potentials while final potentials are meaningful in all cases. These developments are illustrated on the so-called radial sextic oscillator.Comment: 11 pages, Late

Shape invariance through Crum transformation

We show in a rigorous way that Crum's result on equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. It can be shown that all neighbouring Darboux-transformed potentials of higher order, u_{k} and u_{k+1}, satisfy the condition of shape invariance provided the original potential u does. We use this result to proof that under the condition of shape invariance the n-th iteration of the original Sturm-Liouville problem defined through shape invariance is equal to the n-th Crum transformationComment: 26 pp, one more reference, J.-M. Sparenberg and D. Baye, J. Phys. A 28, 5079 (1995), has been added as Ref. 18 in the published version, which has 47 ref

Integrability and explicit solutions in some Bianchi cosmological dynamical systems

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G_2 cosmologies. By using Darboux's theory in order to study ordinary differential equations in the complex projective plane CP^2 we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases

Multivortex Solutions of the Weierstrass Representation

The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlev\'e test and analyse the possibility of deriving the B\"acklund transformation from the singularity analysis of the complex Sine-Gordon equation. We make use of the analysis using the known relations for the Painlev\'{e} equations to construct explicit formulae in terms of the Umemura polynomials which are $\tau$-functions for rational solutions of the third Painlev\'{e} equation. New classes of multivortex solutions of a Weierstrass system are obtained through the use of this proposed procedure. Some physical applications are mentioned in the area of the vortex Higgs model when the complex Sine-Gordon equation is reduced to coupled Riccati equations.Comment: 27 pages LaTeX2e, 1 encapsulated Postscript figur