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

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

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

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

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

Inference Under Convex Cone Alternatives for Correlated Data

In this research, inferential theory for hypothesis testing under general convex cone alternatives for correlated data is developed. While there exists extensive theory for hypothesis testing under smooth cone alternatives with independent observations, extension to correlated data under general convex cone alternatives remains an open problem. This long-pending problem is addressed by (1) establishing that a "generalized quasi-score" statistic is asymptotically equivalent to the squared length of the projection of the standard Gaussian vector onto the convex cone and (2) showing that the asymptotic null distribution of the test statistic is a weighted chi-squared distribution, where the weights are "mixed volumes" of the convex cone and its polar cone. Explicit expressions for these weights are derived using the volume-of-tube formula around a convex manifold in the unit sphere. Furthermore, an asymptotic lower bound is constructed for the power of the generalized quasi-score test under a sequence of local alternatives in the convex cone. Applications to testing under order restricted alternatives for correlated data are illustrated.Comment: 31 page

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 radiation of sound from a propeller at angle of attack

The mechanism by which the noise generated at the blade passing frequency by a propeller is altered when the propeller axis is at an angle of attack to the freestream is examined. The measured noise field is distinctly non axially symmetric under such conditions with far field sound pressure levels both diminished and increased relative to the axially symmetric values produced with the propeller at zero angle of attack. Attempts have been made to explain this non axially symmetric sound field based on the unsteady (once per rev) loading experienced by the propeller blades when the propeller axis is at non zero angle of attack. A calculation based on this notion appears to greatly underestimate the measured azimuthal asymmetry of noise for high tip speed, highly loaded propellers. A new mechanism is proposed; namely, that at angle of attack, there is a non axially symmetric modulation of the radiative efficiency of the steady loading and thickness noise which is the primary cause of the non axially symmetric sound field at angle of attack for high tip speed, heavily loaded propellers with a large number of blades. A calculation of this effect to first order in the crossflow Mach number (component of freestream Mach number normal to the propeller axis) is carried out and shows much better agreement with measured noise data on the angle of attack effect