For smooth compact oriented Riemannian manifolds$M$of dimension 4$k$+2,$k$≥0, with or without boundary, and a vector bundle$F$on$M$with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of$M$twisted by$F$. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when$k$=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.

Let $(R, \frak{m}, k_{R})$ be a regular local$k$-algebra satisfying the weak Jacobian criterion, and such that$k$_{$R$}/$k$is an algebraic field extension. Let $\mathcal{D}_{R}$ be the ring of$k$-linear differential operators of$R$. We give an explicit decomposition of the $\mathcal{D}_{R}$ -module $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}_{R}^{n+1}$ as a direct sum of simple modules, all isomorphic to $\mathcal{D}_{R}/\mathcal{D}_{R} \frak{m}$ , where certain “Pochhammer” differential operators are used to describe generators of the simple components.

We prove the existence of a resonance-free region in scattering by a strictly convex obstacle $\mathcal{O}$ with the Robin boundary condition $\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0$ . More precisely, we show that the scattering resonances lie below a cubic curve ℑ$ζ$=−$S$|$ζ$|^{1/3}+$C$. The constant$S$is the same as in the case of the Neumann boundary condition$γ$=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.

Suppose$μ$is a positive measure on $\mathbb{R}^{2}$ given by$μ$=$ν$×$λ$, where$ν$and$λ$are Radon measures on $\mathcal{S}^{1}$ and $\mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}$ , respectively, which do not vanish on any open interval. We prove that if for either$ν$or$λ$there exists a set of positive measure$A$in its domain for which the upper and lower$s$-densities, 0<$s$≤1, are positive and finite for every$x$∈$A$then the uncentered Hardy–Littlewood maximal operator$M$_{$μ$}is weak-type (1,1) if and only if$ν$is doubling and$λ$is doubling away from the origin. This generalizes results of Vargas concerning rotation-invariant measures on $\mathbb{R}^{n}$ when$n$=2.

We provide characterizations of $p\mspace {1mu}$ -nilpotency for fusion systems and $p\mspace {1mu}$ -local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.