Separating algebras and finite reflection groups

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.Comment: 12 pages, corrected yet another typ

Method of evaporation

Liquids, such as juices, milk, molten metal and the like are concentrated by forming uniformly-sized, small droplets in a precision droplet forming assembly and deploying the droplets in free fall downwardly as a central column within an evacuated column with cool walls. A portion of the solvent evaporates. The vapor flows to the wall, condenses, and usually flows down the wall as a film to condensate collector and drain. The vertical column of freely falling droplets enters the splash guard. The condensate can be collected, sent to other towers or recycled

The separating variety for the basic representations of the additive group

For a group $G$ acting on an affine variety $X$, the separating variety is the closed subvariety of $X\times X$ encoding which points of $X$ are separated by invariants. We concentrate on the indecomposable rational linear representations $V_n$ of dimension $n+1$ of the additive group of a field of characteristic zero, and decompose the separating variety into the union of irreducible components. We show that if $n$ is odd, divisible by four, or equal to two, the closure of the graph of the action, which has dimension $n+2$, is the only component of the separating variety. In the remaining cases, there is a second irreducible component of dimension $n+1$. We conclude that in these cases, there are no polynomial separating algebras.Comment: 14 page

Diethyl 2,5-bis­[(1E)-(1H-pyrrol-2-yl­methyl­idene)amino]­thio­phene-3,4-dicarboxyl­ate

In the crystal structure of the title compound, C20H20N4O4S, the azomethine group adopt E conformations. The pyrrole units are twisted by 10.31 (4) and 18.90 (5)° with respect to the central thio­phene ring. The three-dimensional network is close packed and involves N—H⋯O, N—H⋯N, C—H⋯N and C—H⋯O hydrogen bonding

Modeling the dynamical interaction between epidemics on overlay networks

Epidemics seldom occur as isolated phenomena. Typically, two or more viral agents spread within the same host population and may interact dynamically with each other. We present a general model where two viral agents interact via an immunity mechanism as they propagate simultaneously on two networks connecting the same set of nodes. Exploiting a correspondence between the propagation dynamics and a dynamical process performing progressive network generation, we develop an analytic approach that accurately captures the dynamical interaction between epidemics on overlay networks. The formalism allows for overlay networks with arbitrary joint degree distribution and overlap. To illustrate the versatility of our approach, we consider a hypothetical delayed intervention scenario in which an immunizing agent is disseminated in a host population to hinder the propagation of an undesirable agent (e.g. the spread of preventive information in the context of an emerging infectious disease).Comment: Accepted for publication in Phys. Rev. E. 15 pages, 7 figure

Modelling ion populations in astrophysical plasmas: carbon in the solar transition region

The aim of this work is to improve the modelling of ion populations in higher density, lower temperature astrophysical plasmas, of the type commonly found in lower solar and stellar atmospheres. Ion population models for these regions frequently employ the coronal approximation, which assumes conditions more suitable to the upper solar atmosphere, where high temperatures and lower densities prevail. Using the coronal approximation for modelling the solar transition region gives theoretical lines intensities for the Li-like and Na-like isoelectronic sequences which can be factors of 2-5 times lower than observed. The works of Burgess & Summers (1969) and Nussbaumer & Storey (1975) showed the important part ions in excited levels play when included in the modelling. Their models, however, used approximations for the atomic rates to determine the ion balance. Presented here is the first stage in updating these earlier models of carbon by using rates from up-to-date atomic calculations and more recent photo-ionising radiances for the quiet Sun. Where such atomic rates are not readily available, in the case of electron-impact direct ionisation and excitation--auto-ionisation, new calculations have been made and compared to theoretical and experimental studies. The effects each atomic process has on the ion populations as density changes is demonstrated, and final results from the modelling are compared to the earlier works. Lastly, the new results for ion populations are used to predict line intensities for the solar transition region in the quiet Sun, and these are compared with predictions from coronal-approximation modelling and with observations. Significant improvements in the predicted line intensities are seen in comparison to those obtained from zero-density modelling of carbon.Comment: Draft accepted by A&A, 13 pages, 15 figure