In this article, we study the short- and long-range perturbations of periodic Schrödinger operators. The asymptotic completeness is proved in the short-range case by referring to known results on the stationary approach and more explicitly with the time-dependent approach. In the long-range case, one is able to construct modified wave operators. In both cases, the asymptotic observables can be defined as elements of a commutative $C^{*}$-algebra of which the spectrum equals or is contained in the Bloch variety. Especially, the expression of the mean velocity as the gradient of the Bloch eigenvalues is completely justified in this framework, even when the Bloch variety presents singularities.

Given an $H$-map $i : Y \to X$, we say that $i$ is mod $p$ homotopy normal if the commutator map from $X_{(p)}\times Y_{(9)}$ to $X_{(p)}$ can be deformed into $Y_{(p)}$. In this paper, we study necessary conditions of mod $p$ homotopy normality for the cases that $X$ are exceptional Lie groups with odd torsion in the cohomology, by using the Morava K-theory.

Let $\rho : G \hookrightarrow \mathrm{GL}(n, \mathbf{F})$ be a representation of a finite group over the field $\mathbf{F}$ of characteristic $p$, and $h_{1}, \ldots ,h_{m} \in \mathbf{F}[V]^{G}$ invariant polynomials that form a regular sequence in $\mathbf{F}[V]$. In this note we introduce a tool to study the problem of whether they form a regular sequence in $\mathbf{F}[V]^{G}$. Examples show they need not. We define the cohomology of $G$ with coefficients in the Koszul complex \[ (\mathscr{K},\partial )=(\mathbf{F}[V]\bigotimes E(s^{-1}h_{1}, \ldots ,s^{-1}h_{n}), \partial (s^{-1}h_{i}:i=1,\ldots ,n), \] which we denote by $H^{*}(G;(\mathscr{K}, \partial ))$, and use it to study the homological codimension of rings of invariants of permutation representations of the cyclic group of order $p$, for $p \neq 0$, and to answer the above question in this case.