A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source

We consider the following geometric optics problem: Construct a system of two reflectors which transforms a spherical wavefront generated by a point source into a beam of parallel rays. This beam has a prescribed intensity distribution. We give a rigorous analysis of this problem. The reflectors we construct are (parts of) the boundaries of convex sets. We prove existence of solutions for a large class of input data and give a uniqueness result. To the author's knowledge, this is the first time that a rigorous mathematical analysis of this problem is given. The approach is based on optimal transportation theory. It yields a practical algorithm for finding the reflectors. Namely, the problem is equivalent to a constrained linear optimization problem.Comment: 5 Figures - pdf files attached to submission, but not shown in manuscrip

Two problems related to prescribed curvature measures

Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.Comment: 12 pages, Corrected typo

Writing Russia's future: paradigms, drivers, and scenarios

The development of prediction and forecasting in the social sciences over the past century and more is closely linked with developments in Russia. The Soviet collapse undermined confidence in predictive capabilities, and scenario planning emerged as the dominant future-oriented methodology in area studies, including the study of Russia. Scenarists anticipate multiple futures rather than predicting one. The approach is too rarely critiqued. Building on an account of Russia-related forecasting in the twentieth century, analysis of two decades of scenarios reveals uniform accounts which downplay the insights of experts and of social science theory alike