Ree geometries

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines

REE From EOF

It is well-known that entanglement of formation (EOF) and relative entropy of entanglement (REE) are exactly identical for all two-qubit pure states even though their definitions are completely different. We think this fact implies that there is a veiled connection between EOF and REE. In this context, we suggest a procedure, which enables us to compute REE from EOF without relying on the converse procedure. It is shown that the procedure yields correct REE for many symmetric mixed states such as Bell-diagonal, generalized Vedral-Plenino, and generalized Horodecki states. It also gives a correct REE for less symmetric Vedral-Plenio-type state. However, it is shown that the procedure does not provide correct REE for arbitrary mixed states.Comment: 17 pages, 1 figure, several typos corrected, final version to appear in Quantum Information Processin

Origin and fluxes of atmospheric REE entering an ombrotrophic peat bog in Black Forest (SW Germany): Evidence from snow, lichens and mosses

The fate of the Rare Earth Elements (REE) were investigated in different types of archives of atmospheric deposition in the Black Forest, Southern Germany: (1) a 70 cm snow pack collected on the domed part of a raised bog and representing 2 months of snow accumulation, (2) a snow sample collected close to the road about 500 m from the peat bog, (3) two species of lichens and (4) a peat profile representing 400 years of peat accumulation as well as a “preanthropogenic” sample and the living moss layer from the top of the core. REE concentrations in peat are significantly correlated to Ti which is a lithogenic conservative element suggesting that REE are immobile in peat bog environments. Snow, lichens and peat samples show similar PAAS (Post Archean Australian Shale) normalized REE distributions suggesting that the complete atmospheric REE signal is preserved in the peat profile. However, the annual flux of REE accumulated by the peat is ca. 10 times greater than that of the bulk winter flux of REE. This difference probably indicates that the REE concentrations in the snowpack are not representative of the average REE flux over the whole year. Despite the pronounced geological differences between this site (granite host-rock) and a previously studied peat bog in Switzerland (limestone host-rock) similar REE distribution patterns and accumulation rates were found at both sites. Given that both sites confirm an Upper Continental Crust signature, the data suggests both sites are influenced by regional and not local, soil-derived lithogenic aerosols

Introduction to a Resources Special Issue on Criticality of the Rare Earth Elements: Current and Future Sources and Recycling

The rare earth elements (REE) are vital to modern technologies and society and are amongst the most important of the critical elements. This special issue of Resources examines a number of facets of these critical elements, current and future sources of the REE, the mineralogy of the REE, and the economics of the REE sector. These papers not only provide insights into a wide variety of aspects of the REE, but also highlight the number of different areas of research that need to be undertaken to ensure sustainable and secure supplies of these critical metals into the future

Conformal bootstrap to R\'enyi entropy in 2D Liouville and super-Liouville CFTs

The R\'enyi entanglement entropy (REE) of the states excited by local operators in two-dimensional irrational conformal field theories (CFTs), especially in Liouville field theory (LFT) and $\mathcal{N}=1$ super-Liouville field theory (SLFT), has been investigated. In particular, the excited states obtained by acting on the vacuum with primary operators were considered. {We start from evaluating the second REE in a compact $c=1$ free boson field theory at generic radius, which is an irrational CFT. Then we focus on the two special irrational CFTs, e.g., LFT and SLFT. In these theories, the second REE of such local excited states becomes divergent in early and late time limits. For simplicity, we study the memory effect of REE for the two classes of the local excited states in LFT and SLFT. In order to restore the quasiparticles picture, we define the difference of REE between target and reference states, which belong to the same class. The variation of the difference of REE between early and late time limits always coincides with the log of the ratio of the fusion matrix elements between target and reference states. Furthermore, the locally excited states by acting generic descendent operators on the vacuum have been also investigated. The variation of the difference of REE is the summation of the log of the ratio of the fusion matrix elements between the target and reference states, and an additional normalization factor. Since the identity operator (or vacuum state) does not live in the Hilbert space of LFT and SLFT and no discrete terms contribute to REE in the intermediate channel, the variation of the difference of REE between target and reference states is no longer the log of the quantum dimension which is shown in the 1+1-dimensional rational CFTs (RCFTs).Comment: 53 pages,add a new section (Section 2.4

A new perspective to rational expectations: maximin rational expectations equilibrium

We introduce a new notion of rational expectations equilibrium (REE) called maximin rational expectations equilibrium (MREE), which is based on the maximin expected utility (MEU) formulation. In particular, agents maximize maximin expected utility conditioned on their own private information and the information that the equilibrium prices generate. Maximin equilibrium allocations need not to be measurable with respect to the private information of each individual and with respect to the information that the equilibrium prices generate, as it is in the case of the Bayesian REE. We prove that a maximin REE exists universally (and not generically as in Radner (1979) and Allen (1981)), it is effcient and incentive compatible. These results are false for the Bayesian REE