Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic

Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property

The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable equations have the Painlev\'e property, that is, all solutions are single-valued around all movable singularities. In this expository article, we review methods for analysing such singularity structure. In particular, we describe well known techniques of nonlinear regular-singular-type analysis, i.e. the Painlev\'e tests for ordinary and partial differential equations. Then we discuss methods of obtaining sufficiency conditions for the Painlev\'e property. Recently, extensions of \textit{irregular} singularity analysis to nonlinear equations have been achieved. Also, new asymptotic limits of differential equations preserving the Painlev\'e property have been found. We discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B. Grammaticos and K. Tamizhman

Concours annuel de 1984. Groupe I/a. Rapports sur le mémoire de M. Patrick Esser Rapports des premier et deuxième commissaires

Bureau F.J., Garnir Henri Georges, Gillis Paul P. Concours annuel de 1984. Groupe I/a. Rapports sur le mémoire de M. Patrick Esser Rapports des premier et deuxième commissaires. In: Bulletin de la Classe des sciences, tome 70, 1984. pp. 557-558

Concours annuel de 1984. Groupe I/a. Rapports sur le mémoire de M. Patrick Esser Rapports des premier et deuxième commissaires

Bureau F.J., Garnir Henri Georges, Gillis Paul P. Concours annuel de 1984. Groupe I/a. Rapports sur le mémoire de M. Patrick Esser Rapports des premier et deuxième commissaires. In: Bulletin de la Classe des sciences, tome 70, 1984. pp. 557-558

The hazards of starting and quitting smoking: some Australian evidence

The empirical analysis employs individual level data from the Australian Health Survey combined with retrospective data on tobacco price matched to the age at which the individual started and quit smoking. Split-population hazard models are estimated for both starting and quitting smoking. The analysis suggests price plays a significant role in the decision to start smoking but not in the decision to quit. Further sensitivity analysis of different age groups and an alternative data source, questions the robustness of the significant role of price in the smoking initiation decision. From a policy perspective, the results indicate that increases in tobacco taxation can be an important instrument in reducing the incidence of smoking, but should be combined with other mechanisms such as mandating smoke-free environments and antismoking education. Our results strongly support the targeting of antismoking campaigns towards teenagers.<br /