Combinatorial covers and vanishing of cohomology

We use a Mayer–Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen–Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients and obtain new generalizations.     

Attractors for Damped Quintic Wave Equations in Bounded Domains

The dissipative wave equation with a critical quintic non-linearity in smooth bounded three-dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.     

Unidirectional amplification and shaping of optical pulses by three-wave mixing with negative phonons

A possibility to greatly enhance frequency-conversion efficiency of stimulated Raman scattering is shown by making use of extraordinary properties of three-wave mixing of ordinary and backward waves. Such processes are commonly attributed to negative-index plasmonic metamaterials. This work demonstrates the possibility to replace such metamaterials that are very challenging to engineer by readily available crystals which support elastic waves with contra-directed phase and group velocities. The main goal of this work was to investigate specific properties of indicated nonlinear optical process in short-pulse regime and to show that it enables elimination of fundamental detrimental effect of fast damping of optical phonons on the process concerned. Among the applications is the possibility of creation of a family of unique photonic devices such as unidirectional Raman amplifiers and femtosecond pulse shapers with greatly improved operational properties.     

Efficient synthesis of 3,5‐functionalized isoxazoles and isoxazolines via 1,3‐dipolar cycloaddition reactions of 1‐propargyl‐ and 1‐allylbenzotriazoles

3‐Aryl‐5‐(benzotriazol‐1‐ylmethyl)‐ 10a‐f and 3‐p‐methoxyphenyl‐5‐(α‐benzotriazol‐1‐yl‐α‐ethoxymethyl)‐isoxazole (13) were prepared in high yields by 1,3‐dipolar cycloadditions of 1‐propargyl‐benzotriazole (5) and (α‐ethoxypropargyl)benzotriazole (8), respectively, with nitrite oxides 3a‐f (prepared in situ from benzohydroximoyl chlorides 2a‐f). The benzotriazol‐1‐ylmethyl moiety was further elaborated by sequential lithiation and reaction with aldehydes, alkyl halides and Michael acceptors. Similar 1,3‐cycloadditions using 1‐allylbenzotriazole (6) and 1‐(α‐ethoxyallyl)benzotriazole (7) afforded 3,5‐substituted isoxazolines 11b, f and 12 in excellent yields.     

Molecular clouds,colliding flows and HEDLA experiments: Star formation with the AstroBEAR AMR code

In this contribution we present new simulations of colliding flows relevant to star formation studies which may also be relevant to future HEDLA experiments. We first discuss AstroBEAR a multi-physics MHD AMR code whose functionality and parallel performance make it a highly useful tool for star formation studies. We then present the results of two simulations of colliding flows that only differed with regards to the degree of inhomogeneity. One flow was completely uniform (Smooth), while the other contained many small over-densities in a lighter background (Clumpy) though with the same mean density, mass flux, & ram pressure. Despite these similarities, the two runs differed significantly with regards to their overall evolution, mixing, and core formation. The Smooth run formed cores early on throughout the cloud – while the inhomogeneities present in the Clumpy run drove a higher degree of small scale turbulence which seemed able to prevent local collapse. The Clumpy run did form cores but only after global collapse had ensued at around 20 Myr. The Smooth run managed to avoid global collapse for the duration of the simulation due to the increased splashing of material radially outwards as well as the higher pressure of the incoming material due to the details of the heating/cooling curve. In the final section we comment on possible applications of these studies to future HEDLA experiments. We conclude that key aspects of the flow evolution before gravitational collapse sets in may be amenable to laboratory studies.