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

Zeta functions and Blow-Nash equivalence

We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which is an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo. The new definition is more natural and geometric. Moreover, this equivalence relation still does not admit moduli for a Nash family of isolated singularities. Some previous invariants are no longer invariants for this new relation, however, thanks to a Denef & Loeser formula coming from motivic integration in a Nash setting, we managed to derive new invariants for this equivalence relation.Comment: 12 page

Analytic and Nash equivalence relations of Nash maps

Let $M$ and $N$ be Nash manifolds, and $f$ and $g$ Nash maps from $M$ to $N$. If $M$ and $N$ are compact and if $f$ and $g$ are analytically R-L equivalent, then they are Nash R-L equivalent. In the local case, $C^infty$ R-L equivalence of two Nash map germs implies Nash R-L equivalence. This shows a difference of Nash map germs and analytic map germs. Indeed, there are two analytic map germs from $(R^2,0)$ to $(R^4,0)$ which are $C^infty$ R-L equivalent but not analytically R-L equivalent

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