Multivalued solutions to the eikonal equation in stratified media

In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geometric solutions, using the notion of Legendrian immersions with an initial point source or an initial smooth front. · Then, we study the singularities of the solutions in the case of a smooth or piecewise Hamiltonian in a boundaryless stratified medium. Finally, we study the singularities of the solutions in a domain with a boundary that describes the propagating field in a waveguide

Managing the climate commons at the nexus of ecology, behaviour and economics

Sustainably managing coupled ecological–economic systems requires not only an understanding of the environmental factors that affect them, but also knowledge of the interactions and feedback cycles that operate between resource dynamics and activities attributable to human intervention. The socioeconomic dynamics, in turn, call for an investigation of the behavioural drivers behind human action. We argue that a multidisciplinary approach is needed in order to tackle the increasingly pressing and intertwined environmental challenges faced by modern societies. Academic contributions to climate change policy have been constrained by methodological and terminological differences, so we discuss how programmes aimed at cross-disciplinary education and involvement in governance may help to unlock scholars' potential to propose new solutions

Markov-Perfect Nash Equilibria in Models with a Single Capital Stock

Many economic problems can be formulated as dynamic games in which strategically interacting agents choose actions that determine the current and future levels of a single capital stock. We study necessary conditions that allow us to characterize Markov perfect Nash equilibria (MPNE) for these games. These conditions result in an auxiliary system of ordinary differential equations that helps us to explore stability, continuity and differentiability of MPNE. The techniques are used to derive detailed properties of MPNE for several games including the exploitation of a finite resource, the voluntary investment in a public capital stock, and the inter-temporal consumption of a reproductive asset

Multivalued solutions to the eikonal equation in stratified media

In the present paper we study the geometric properties of the multivalued solutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating high frequency waves in stratified media. We consider the case of a fixed Hamiltonian imposed by the medium, and we present the geometric framework that describes the geometric solutions, using the notion of Legendrian immersions with an initial point source or an initial smooth front. · Then, we study the singularities of the solutions in the case of a smooth or piecewise Hamiltonian in a boundaryless stratified medium. Finally, we study the singularities of the solutions in a domain with a boundary that describes the propagating field in a waveguide