Invariant measures of the 2D Euler and Vlasov equations

We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures, and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials. Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.Comment: 40 page

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization

Organizing risk: organization and management theory for the risk society

Risk has become a crucial part of organizing, affecting a wide range of organizations in all sectors. We identify, review and integrate diverse literatures relevant to organizing risk, building on an existing framework that describes how risk is organized in three ‘modes’ – prospectively, in real-time, and retrospectively. We then identify three critical issues in the existing literature: its fragmented nature; its neglect of the tensions associated with each of the modes; and its tendency to assume that the meaning of an object in relation to risk is singular and stable. We provide a series of new insights with regard to each of these issues. First, we develop the concept of a risk cycle that shows how organizations engage with all three modes and transition between them over time. Second, we explain why the tensions have been largely ignored and show how studies using a risk work perspective can provide further insights into them. Third, we develop the concept of risk translation to highlight the ways in the meanings of risks can be transformed and to identify the political consequences of such translations. We conclude the paper with a research agenda to elaborate these insights and ideas further

Towards clean material cycles: Is there a policy conflict between circular economy and non-toxic environment?

Lately, the idea of ‘non-toxic’ (or ‘clean’) material cycles has become increasingly popular as a starting point for the consideration of circular economy strategies among leading regulating actors, such as the European Commission (2017). However, it should be apparent that, by default, circular economy aspirations awaken a potential policy conflict between increased circulation of resources and reduction of exposure to hazardous substances (human- and eco-toxic substances of concern). Here, we build upon the great tradition of Waste Management & Research editorials on the topic debated, and not least on the recent opinion piece by Stanisavljevic and Brunner (2019). We argue that non-toxic material cycles are indeed a desirable vision, but that they will not come naturally in a world of circulated material resources – at least not without major supply chain disruptions and persistent conscious efforts. Let us elaborate