Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group

Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of this result, we show that if $\alpha_{1}I+S_{f}$, where $S_{f}$ is the operator given by convolution with $f$, $f\in L^{1}_{v}(\mathbb{H})$, is invertible in \B(L^{p}(\mathbb{H})), then (\alpha_{1}I+S_{f})^{-1}=\alpha_{2}I+S_{g}$, and$g\in L^{1}_{v}(\mathbb{H})$. We prove analogous results for twisted convolution operators and apply the latter results to a class of Weyl pseudodifferential operators. We briefly discuss relevance to mobile communications.Comment: This version corrects two mistakes and recognizes the work of other authors related to a corollary of our main theore

On negatively curved bundles with hyperbolic fibers outside the Igusa stable range

We prove that the Teichm\"{u}ller space $\mathcal{T}^{<0}(M)$ of negatively curved metrics on a hyperbolic manifold $M$ has nontrivial $i$-th rational homotopy groups for some $i> \dim M$. Moreover, some elements of infinite order in \pi_i B\mbox{Diff}(M) can be represented by bundles over $S^i$ with fiberwise negatively curved metrics.Comment: Referrences added;other minor change

Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

Critical incident analysis through narrative reflective practice: A case study

Teachers can reflect on their practices by articulating and exploring incidents they consider critical to themselves or others. By talking about these critical incidents, teachers can make better sense of seemingly random experiences that occur in their teaching because they hold the real inside knowledge, especially personal intuitive knowledge, expertise and experience that is based on their accumulated years as language educators teaching in schools and classrooms. This paper is about one such critical incident analysis that an ESL teacher in Canada revealed to her critical friend and how both used McCabe’s (2002) narrative framework for analyzing an important critical incident that occurred in the teacher’s class