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$?

An asymptotic formula for the density of states of the polyharmonic periodic operator (−δ)^{$l$}+$V$in$R$^{$n$},$n$≥2,$l$>1/2 is obtained. Special consideration is given to the case of the Schrödinger equation$n$=3,$l$=1,$V$being a periodic potential, where the second term of the asymptotic is found.

We study $\mathbb{Z}_{2}$ -graded identities of simple Lie superalgebras over a field of characteristic zero. We prove the existence of the graded PI-exponent for such algebras.

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We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.