Let$f:V$→$R$be a function defined on a subset$V$of$R$^{$n$}×$R$^{$d$}let$⃜:x→$inf{$f(x t)$;$t$such that$(x t)∈V}$denote the$shadow$of$f$and let$Φ$=${(x t)∈V; f(x t)=⃜(x)}$This paper deals with the characterization of some properties of ⃜ in terms of the infinitesimal behavior of$f$near points ζ∈$Φ$proving in particular a conjecture of J M Trépreau concerning the case$d$=1 Characterizations of this type are provided for the convexity the subharmonicity or the$C$^{1 1}regularity of ⃜ in the interior of$I={x∈$$R$^{n};ε$R$^{d}$(x t)∈V}$and in the$C$^{1 1}case an expression for$D$^{2}⃜ is given To some extent an answer is given to the following question: which convex function ⃜:$I$→$R$$I$interval ϒ$R$(resp which function √:$I$→$R$of class$C$^{1 1}) is the shadow of a$C$^{2}function$f:I$×$R→R$?

The continuity of weak solutions of elliptic partial differential equations $$div \mathcal{A}(x,\nabla u) = 0$$ is considered under minimal structure assumptions. The main result guarantees the continuity at the point$x$_{0}for weakly monotone weak solutions if the structure of$A$is controlled in a sequence of annuli $$B(x_0 ,R_j )\backslash \bar B(x_0 ,r_j )$$ with uniformly bounded ratio$R$_{$j$}$/r$_{$j$}such that lim_{$j→∞$}$R$_{$j$}=0. As a consequence, we obtain a sufficient condition for the continuity of mappings of finite distortion.

The spectral order on$R$^{$n$}induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree$n$. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^{∞}of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$ -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coef. cient functions can easily be estimated via a simple local regression. This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental data set.

This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to an arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (ie,| 2 g 2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their