The effect of linear perturbations on the Yamabe problem

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold $\left( M,g\right) $ is compact. Established in the locally conformally flat case by Schoen (Lecture Notes in Mathematics, vol. 1365, pp. 120–154. Springer, Berlin 1989, Surveys Pure Application and Mathematics, 52 Longman Science, Technology, pp. 311–320. Harlow 1991) and for $n\le 24$ by Khuri–Marques–Schoen (J Differ Geom 81(1):143–196, 2009), it has revealed to be generally false for $n\ge 25$ as shown by Brendle (J Am Math Soc 21(4):951–979, 2008) and Brendle–Marques (J Differ Geom 81(2):225–250, 2009). A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential $\frac{n-2}{4(n-1)} {{\mathrm{Scal}}}_g,\, {{\mathrm{Scal}}}_g$ being the Scalar curvature of $\left( M,g\right) $ . We show that a-priori $L^\infty $ –bounds fail for linear perturbations on all manifolds with $n\ge 4$ as well as a-priori gradient $L^2$ –bounds fail for non-locally conformally flat manifolds with $n\ge 6$ and for locally conformally flat manifolds with $n\ge 7$ . In several situations, the results are optimal. Our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of $g$ .     

Assessing views of coaching via a video-based tool

Mathematics coaches represent a unique group of didacticians, or individuals who work with practicing teachers. Twenty-eight mathematics coaches participated in this exploratory study, which used video viewing to examine the coach–teacher dynamic. To gather data about participants’ views of effective coaching practices, we developed the Video Assessment of Coaching instrument, which provided coaches with opportunities to express their views of effective practice and implementation. The participants expressed views of effective coaching that often did not align with those of coaching authors. The significance of this research lies in its efforts to document the views that mathematics coaches develop as practitioners, as an early step in the examination of the relationships between the views of coaches and coaches’ effectiveness in improving teacher practice, knowledge, and attitudes.     

Dissipative operators and additive perturbations in locally convex spaces

Let be a densely defined operator on a Banach space X. Characterizations of when generates a C₀‐semigroup on X are known. The famous result of Lumer and Phillips states that it is so if and only if is dissipative and is dense in X for some . There exists also a rich amount of Banach space results concerning perturbations of dissipative operators. In a recent paper Tyran–Kamińska provides perturbation criteria of dissipative operators in terms of ergodic properties. These results, and others, are shown to remain valid in the setting of general non–normable locally convex spaces. Applications of the results to concrete examples of operators on function spaces are also presented.     

Anatomy of a laminar starting thermal plume at high Prandtl number

We present an experimental study of the dynamics of a plume generated from a small heat source in a high Prandtl number fluid with a strongly temperature-dependent viscosity. The velocity field was determined with particle image velocimetry, while the temperature field was measured using differential interferometry and thermochromic liquid crystals. The combination of these different techniques run simultaneously allows us to identify the different stages of plume development, and to compare the positions of key-features of the velocity field (centers of rotation, maximum vorticity locations, stagnation points) respective to the plume thermal anomaly, for Prandtl numbers greater than 10³. We further show that the thermal structure of the plume stem is well predicted by the constant viscosity model of Batchelor (Q J R Met Soc 80: 339–358, 1954) for viscosity ratios up to 50.     

Reaction mechanisms with exotic nuclei

This article examines a number of reaction mechanisms for scattering initiated by an exotic projectile. Comparisons are made with recent experimental data, in order to extract information on the peculiarity of the nuclear structure under extreme conditions and to test the accuracy of the available theoretical methods. Predictions for future experiments are also made.     

The Cadmium Oxide Route to Alkali Metal Oxometallates with Uncommon Structural Features

The aim of this research report is to give an introduction to a specific solid state reaction applied in the field of alkali metal rich transition metal compounds. The approach is an explorative investigation of the oxidation of transition metals with CdO in the presence of alkali metal oxides and / or compounds with complex anions such as Na₂SO₄. Thereby, a range of unusual compounds have been obtained and structurally characterized. In particular, low valences and uncommon coordination numbers (C.N.) of the late 3d transition metals, as well as interesting varieties in the anionic part of the structures are accessible. The presented examples were selected mainly in order of their inherent aspects concerning the reactivity, structural features or electronic structures.     

A new synthesis of flavonoids via Heck reaction

Several naturally occurring flavonoids have been synthesised following a new proposed method based on the use of the Heck reaction. The key step involves the coupling of an aryl vinyl ketone with an aryl iodide. This procedure affords the flavonoid moiety in a single step.     

Surface and interface electronic properties of group III-nitride heterostructures

For group III-nitrides with wurtzite structure the presence of fixed polarization interface charges yields new challenges in order to understand and control Schottky barrier heights, band offsets and 2D confinement in heterostructure FETs. In this short review experimental results obtained by in situ photoemission spectroscopy on MBE AlGaN/GaN heterostructures grown on 6H–SiC are discussed, with emphasis on the presence and interplay of surface electronic states. Schrödinger–Poisson calculations are performed to get the complete band scheme at the selected heterojunctions. Results on the polarity dependence of Pt/GaN Schottky barrier values from the literature are also discussed.