Approximation on Nash sets with monomial singularities

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.Comment: 39 page

A Prospective Study of Long-term Outcomes in Female Patients with Nonalcoholic Steatohepatitis Using Age- and Body Mass Index-matched Cohorts

In patients with nonalcoholic steatohepatitis (NASH), the prevalence of cirrhosis is higher among women than men, and hepatocellular carcinoma (HCC) develops mainly in the cirrhotic stage among women. However, the long-term outcomes in female patients with NASH have not been fully elucidated, and age, gender and BMI were not simultaneously adjusted in previous studies on the prognosis of NASH. To elucidate the outcomes in female patients with NASH, we prospectively compared NASH patients with advanced fibrosis (advanced NASH) with hepatitis C virus-related advanced fibrosis (advanced CHC) patients and NASH patients with mild fibrosis (mild NASH) using study cohorts that were adjusted for body mass index (BMI) in addition to age. The median follow-up period was 92.5 months. Liver-related complication-free survival was significantly reduced in the advanced NASH group compared to the mild NASH group. No liver-related complications developed in the mild NASH group. The overall survival, liver-related complication- and cardiovascular/cerebrovascular disease-free survival were not significantly different between the advanced NASH and CHC groups. Female patients with NASH and advanced fibrosis may have a less favorable prognosis for liver-related complications than the matched cohorts with NASH and mild fibrosis, but may have a similar prognosis to the matched cohorts with CHC

From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem

The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of $N$ player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article we emphasize the role of optimal transport theory in: 1) the passage from Nash to Cournot-Nash equilibria as the number of players tends to infinity, 2) the analysis of Cournot-Nash equilibria

Implementation in mixed Nash equilibrium

A mechanism implements a social choice correspondence f in mixed Nash equilibrium if at any preference profile, the set of all pure and mixed Nash equilibrium outcomes coincides with the set of f-optimal alternatives at that preference profile. This definition generalizes Maskin’s definition of Nash implementation in that it does not require each optimal alternative to be the outcome of a pure Nash equilibrium. We show that the condition of weak set-monotonicity, a weakening of Maskin’s monotonicity, is necessary for implementation. We provide sufficient conditions for implementation and show that important social choice correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium

Blow-Nash types of simple singularities

We address the question of the classification under blow-Nash equivalence of simple Nash function germs. We state that this classification coincides with the real analytic classification. We prove moreover that a simple germ can not be blow-Nash equivalent to a nonsimple one. The method is based on the computation of relevant coefficients of the real zeta functions associated to a Nash germ via motivic integration.Comment: 16 page