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

View from the Top: How Corporate Boards Can Engage on Sustainability Performance

Corporate boards are responsible for overseeing the interests of shareholders in the long term and have a critical role to play in championing sustainability across the enterprise. Over the years, Wall Street research, academic papers, corporate reports and trends from major investors have all underscored the same message: Companies that adopt sustainable practices deliver superior financial results and can face the future with more resilience.Based on interviews conducted with dozens of corporate directors, senior corporate leaders and governance experts, this Ceres report identifies key strategies for effective board engagement that can produce tangible environmental and social impacts. Specifically, the report recommends two inter-related approaches for weaving sustainability more deeply across board functions:Integrating sustainability into board governance systems, andIntegrating sustainability into board actions.By combining robust systems and meaningful actions, boards will have a far better chance of encouraging substantive performance improvements

A Bilinear Approach to Discrete Miura Transformations

We present a systematic approach to the construction of Miura transformations for discrete Painlev\'e equations. Our method is based on the bilinear formalism and we start with the expression of the nonlinear discrete equation in terms of $\tau$-functions. Elimination of $\tau$-functions from the resulting system leads to another nonlinear equation, which is a ``modified'' version of the original equation. The procedure therefore yields Miura transformations. In this letter, we illustrate this approach by reproducing previously known Miura transformations and constructing new ones.Comment: 7 pages in TeX, to appear in Phys. Letts.

Bilinear structure and Schlesinger transforms of the qq-PIII_{\rm III} and qq-PVI_{\rm VI} equations

We show that the recently derived ($q$-) discrete form of the Painlev\'e VI equation can be related to the discrete P$_{\rm III}$, in particular if one uses the full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$.Comment: 10 pages, Plain Te

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

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

Discrete Painleve equations: coalescences, limits and degeneracies

Starting from the standard form of the five discrete Painlev\'e equations we show how one can obtain (through appropriate limits) a host of new equations which are also the discrete analogues of the continuous Painlev\'e equations. A particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new $n$-dependence and introduces more new discrete Painlev\'e equations.Comment: PlainTe