Cluster structures on quantum coordinate rings

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C_w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.Comment: v2: minor corrections. v3: references updated, final version to appear in Selecta Mathematic

Categorification of skew-symmetrizable cluster algebras

We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with the action of a finite group G, we construct a G-equivariant mutation on the set of maximal rigid G-invariant objects of C. Using an appropriate cluster character, we can then attach to these data an explicit skew-symmetrizable cluster algebra. As an application we prove the linear independence of the cluster monomials in this setting. Finally, we illustrate our construction with examples associated with partial flag varieties and unipotent subgroups of Kac-Moody groups, generalizing to the non simply-laced case several results of Gei\ss-Leclerc-Schr\"oer.Comment: 64 page

Variation of ionic conductivity in a plastic-crystalline mixture

Ionically-conducting plastic crystals are possible candidates for solid-state electrolytes in energy-storage devices. Interestingly, the admixture of larger molecules to the most prominent molecular PC electrolyte, succinonitrile, was shown to drastically enhance its ionic conductivity. Therefore, binary mixtures seem to be a promising way to tune the conductivity of such solid-state electrolytes. However, to elucidate the general mechanisms of ionic charge transport in plastic crystals and the influence of mixing, a much broader data base is needed. In the present work, we investigate mixtures of two well-known plastic-crystalline systems, cyclohexanol and cyclooctanol, to which 1 mol% of Li ions were added. Applying differential scanning calorimetry and dielectric spectroscopy, we present a thorough investigation of the phase behavior and the ionic and dipolar dynamics of this system. All mixtures reveal plastic-crystalline phases with corresponding orientational glass-transitions. Moreover, their conductivity seems to be dominated by the "revolving-door" mechanism, implying a close coupling between the ionic translational and the molecular reorientational dynamics of the surrounding plastic-crystalline matrix. In contrast to succinonitrile-based mixtures, there is no strong variation of this coupling with the mixing ratio.Comment: 8 pages, 6 figures, final version as accepted for publicatio

From pairs of most similar sequences to phylogenetic best matches

Background Many of the commonly used methods for orthology detection start from mutually most similar pairs of genes (reciprocal best hits) as an approximation for evolutionary most closely related pairs of genes (reciprocal best matches). This approximation of best matches by best hits becomes exact for ultrametric dissimilarities, i.e., under the Molecular Clock Hypothesis. It fails, however, whenever there are large lineage specific rate variations among paralogous genes. In practice, this introduces a high level of noise into the input data for best-hit-based orthology detection methods. Results If additive distances between genes are known, then evolutionary most closely related pairs can be identified by considering certain quartets of genes provided that in each quartet the outgroup relative to the remaining three genes is known. A priori knowledge of underlying species phylogeny greatly facilitates the identification of the required outgroup. Although the workflow remains a heuristic since the correct outgroup cannot be determined reliably in all cases, simulations with lineage specific biases and rate asymmetries show that nearly perfect results can be achieved. In a realistic setting, where distances data have to be estimated from sequence data and hence are noisy, it is still possible to obtain highly accurate sets of best matches. Conclusion Improvements of tree-free orthology assessment methods can be expected from a combination of the accurate inference of best matches reported here and recent mathematical advances in the understanding of (reciprocal) best match graphs and orthology relations. Availability Accompanying software is available at https://github.com/david-schaller/AsymmeTree

Mixing layer height as an indicator for urban air quality?

The mixing layer height (MLH) is a measure for the vertical turbulent exchange within the boundary layer, which is one of the controlling factors for the dilution of pollutants emitted near the ground. Based on continuous MLH measurements with a Vaisala CL51 ceilometer and measurements from an air quality network, the relationship between MLH and near-surface pollutant concentrations has been investigated. In this context the uncertainty of the MLH retrievals and the representativeness of ground-based in situ measurements are crucial. We have investigated this topic by using data from the BAERLIN2014 campaign in Berlin, Germany, conducted from June to August 2014. To derive the MLH, three versions of the proprietary software BL-VIEW and a novel approach COBOLT were compared. It was found that the overall agreement is reasonable if mean diurnal cycles are considered. The main advantage of COBOLT is the continuous detection of the MLH with a temporal resolution of 10 min and a lower number of cases when the residual layer is misinterpreted as mixing layer. We have calculated correlations between MLH as derived from the different retrievals and concentrations of pollutants (PM10, O-3 and NOx) for different locations in the metropolitan area of Berlin. It was found that the correlations with PM10 are quite different for different sites without showing a clear pattern, whereas the correlation with NOx seems to depend on the vicinity of emission sources in main roads. In the case of ozone as a secondary pollutant, a clear correlation was found. We conclude that the effects of the heterogeneity of the emission sources, chemical processing and mixing during transport exceed the differences due to different MLH retrievals. Moreover, it seems to be unrealistic to find correlations between MLH and near-surface pollutant concentrations representative for a city like Berlin (flat terrain), in particular when traffic emissions are dominant. Nevertheless it is worthwhile to use advanced MLH retrievals for ceilometer data, for example as input to dispersion models and for the validation of chemical transport models

Best match graphs

Best match graphs arise naturally as the first processing intermediate in algorithms for orthology detection. Let T be a phylogenetic (gene) tree T and σ an assignment of leaves of T to species. The best match graph (G,σ) is a digraph that contains an arc from x to y if the genes x and y reside in different species and y is one of possibly many (evolutionary) closest relatives of x compared to all other genes contained in the species σ(y). Here, we characterize best match graphs and show that it can be decided in cubic time and quadratic space whether (G,σ) derived from a tree in this manner. If the answer is affirmative, there is a unique least resolved tree that explains (G,σ), which can also be constructed in cubic time

A remark on Leclerc's Frobenius categories

Leclerc recently studied certain Frobenius categories in connection with cluster algebra structures on coordinate rings of intersections of opposite Schubert cells. We show that these categories admit a description as Gorenstein projective modules over an Iwanaga-Gorenstein ring of virtual dimension at most two. This is based on a Morita type result for Frobenius categories.Comment: 5 pages, extended abstract for a talk at the Workshop Homological Bonds between Commutative Algebra and Representation Theory at CRM Barcelona, February 2015, comments welcom

Validation of the Aeolus L2B wind product with airborne wind lidar measurements in the polar North Atlantic region and in the tropics

During the first 3 years of the European Space Agency's Aeolus mission, the German Aerospace Center (Deutsches Zentrum für Luft- und Raumfahrt, DLR) performed four airborne campaigns deploying two different Doppler wind lidars (DWL) on board the DLR Falcon aircraft, aiming to validate the quality of the recent Aeolus Level 2B (L2B) wind data product (processor baseline 11 and 12). The first two campaigns, WindVal III (November–December 2018) and AVATAR-E (Aeolus Validation Through Airborne Lidars in Europe, May and June 2019), were conducted in Europe and provided first insights into the data quality at the beginning of the mission phase. The two later campaigns, AVATAR-I (Aeolus Validation Through Airborne Lidars in Iceland) and AVATAR-T (Aeolus Validation Through Airborne Lidars in the Tropics), were performed in regions of particular interest for the Aeolus validation: AVATAR-I was conducted from Keflavik, Iceland, between 9 September and 1 October 2019 to sample the high wind speeds in the vicinity of the polar jet stream; AVATAR-T was carried out from Sal, Cape Verde, between 6 and 28 September 2021 to measure winds in the Saharan dust-laden African easterly jet. Altogether, 10 Aeolus underflights were performed during AVATAR-I and 11 underflights during AVATAR-T, covering about 8000 and 11 000 km along the Aeolus measurement track, respectively. Based on these collocated measurements, statistical comparisons of Aeolus data with the reference lidar (2 µm DWL) as well as with in situ measurements by the Falcon were performed to determine the systematic and random errors of Rayleigh-clear and Mie-cloudy winds that are contained in the Aeolus L2B product. It is demonstrated that the systematic error almost fulfills the mission requirement of being below 0.7 m s−1 for both Rayleigh-clear and Mie-cloudy winds. The random error is shown to vary between 5.5 and 7.1 m s−1 for Rayleigh-clear winds and is thus larger than specified (2.5 m s−1), whereas it is close to the specifications for Mie-cloudy winds (2.7 to 2.9 m s−1). In addition, the dependency of the systematic and random errors on the actual wind speed, the geolocation, the scattering ratio, and the time difference between 2 µm DWL observation and satellite overflight is investigated and discussed. Thus, this work contributes to the characterization of the Aeolus data quality in different meteorological situations and allows one to investigate wind retrieval algorithm improvements for reprocessed Aeolus data sets.</p

Reconstruction of time-consistent species trees

Background The history of gene families—which are equivalent to event-labeled gene trees—can to some extent be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, paralogous or xenologous genes. The question then arises as whether inferred event-labeled gene trees are “biologically feasible” which is the case if one can find a species tree with which the gene tree can be reconciled in a time-consistent way. Results In this contribution, we consider event-labeled gene trees that contain speciations, duplications as well as horizontal gene transfer (HGT) and we assume that the species tree is unknown. Although many problems become NP-hard as soon as HGT and time-consistency are involved, we show, in contrast, that the problem of finding a time-consistent species tree for a given event-labeled gene can be solved in polynomial-time. We provide a cubic-time algorithm to decide whether a “time-consistent” species tree for a given event-labeled gene tree exists and, in the affirmative case, to construct the species tree within the same time-complexity