Quiver Varieties, Category O for Rational Cherednik Algebras, and Hecke Algebras

We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Z-algebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the c-function) used to define a highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that. We also interpret geometrically another ordering function (the a-function) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZ-functor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties

On the benefits of clustering approaches in digital soil mapping: an application example concerning soil texture regionalization

High-resolution soil maps are urgently needed by land managers and researchers for a variety of applications. Digital soil mapping (DSM) allows us to regionalize soil properties by relating them to environmental covariates with the help of an empirical model. In this study, a legacy soil dataset was used to train a machine learning algorithm in order to predict the particle size distribution within the catchment of the Bode River in Saxony-Anhalt (Germany). The random forest ensemble learning method was used to predict soil texture based on environmental covariates originating from a digital elevation model, land cover data and geologic maps. We studied the usefulness of clustering applications in addressing various aspects of the DSM procedure. To improve areal representativity of the legacy soil data in terms of spatial variability, the environmental covariates were used to cluster the landscape of the study area into spatial units for stratified random sampling. Different sampling strategies were used to create balanced training data and were evaluated on their ability to improve model performance. Clustering applications were also involved in feature selection and stratified cross-validation. Under the best-performing sampling strategy, the resulting models achieved an R2 of 0.29 to 0.50 in topsoils and 0.16-0.32 in deeper soil layers. Overall, clustering applications appear to be a versatile tool to be employed at various steps of the DSM procedure. Beyond their successful application, further application fields in DSM were identified. One of them is to find adequate means to include expert knowledge. © Copyright

Jack polynomials with prescribed symmetry and hole propagator of spin Calogero-Sutherland model

We study the hole propagator of the Calogero-Sutherland model with SU(2) internal symmetry. We obtain the exact expression for arbitrary non-negative integer coupling parameter $\beta$ and prove the conjecture proposed by one of the authors. Our method is based on the theory of the Jack polynomials with a prescribed symmetry.Comment: 12 pages, REVTEX, 1 eps figur

Process-based analysis of terrestrial carbon flux predictability

Despite efforts to decrease the discrepancy between simulated and observed terrestrial carbon fluxes, the uncertainty in trends and patterns of the land carbon fluxes remains high. This difficulty raises the question to what extent the terrestrial carbon cycle is predictable, and which processes explain the predictability. Here, the perfect model approach is used to assess the potential predictability of net primary production (NPPpred) and heterotrophic respiration (Rhpred) by using ensemble simulations conducted with the Max-Planck-Institute Earth System Model. In order to asses the role of local carbon flux predictability (CFpred) on the predictability of the global carbon cycle, we suggest a new predictability metric weighted by the amplitude of the flux anomalies. Regression analysis is used to determine the contribution of the predictability of different environmental drivers to NPPpred and Rhpred (soil moisture, air temperature and radiation for NPP and soil organic carbon, air temperature and precipitation for Rh). NPPpred is driven to 62 and 30 % by the predictability of soil moisture and temperature, respectively. Rhpred is driven to 52 and 27 % by the predictability of soil organic carbon temperature, respectively. The decomposition of predictability shows that the relatively high Rhpred compared to NPPpred is due to the generally high predictability of soil organic carbon. The seasonality in NPPpred and Rhpred patterns can be explained by the change in limiting factors over the wet and dry months. Consequently, CFpred is controlled by the predictability of the currently limiting environmental factor. Differences in CFpred between ensemble simulations can be attributed to the occurrence of wet and dry years, which influences the predictability of soil moisture and temperature. This variability of predictability is caused by the state dependency of ecosystem processes. Our results reveal the crucial regions and ecosystem processes to be considered when initializing a carbon prediction system

Upper Cretaceous Gosau deposits of the Apuseni Mountains (Romania) - similarities and differences to the Eastern Alps

The Apuseni Mountains were formed during Late Cretaceous convergence between the Tisia and the Dacia microplates as part of the Alpine orogen. The mountain range comprises a sedimentary succession similar to the Gosau Group of the Eastern Alps. This work focuses on the sedimentological and geodynamic evolution of the Gosau basin of the Apuseni Mts. and attempts a direct comparison to the relatively well studied Gosau Group deposits of the Eastern Alps. By analyzing the Upper Cretaceous Gosau sediments and the surrounding geological units, we were able to add critical evidence for reconstructing the Late Mesozoic to Paleogene geodynamic evolution of the Apuseni Mountains. Nannoplankton investigations show that Gosau sedimentation started diachronously after Late Turonian times. The burial history indicates low subsidence rates during deposition of the terrestrial and shallow marine Lower Gosau Subgroup and increased subsidence rates during the period of deep marine Upper Gosau Subgroup sedimentation.The Gosau Group of the Apuseni Mountains was deposited in a forearc basin supplied with sedimentary material from an obducted forearc region and the crystalline hinterland, as reflected by heavy mineral and paleocurrent analysis. Zircon fission track age populations show no fluctuation of exhumation rates in the surrounding geological units, which served as source areas for the detrital material, whereas increased exhumation at the K/Pg boundary can be proven by thermal modeling on apatite fission track data. Synchronously to the Gosau sedimentation, deep marine turbidites were deposited in the deep-sea trench basin formed by the subduction of the Transylvanian Ocean. The similarities to the Gosau occurrences of the Eastern Alps lead to direct correlation with the Alpine paleogeographic evolution and to the assumption that a continuous ocean basin (South Penninic - Transylvanian Ocean Basin) was consumed until Late Cretaceous times

Trivial improvements of predictive skill due to direct reconstruction of global carbon cycle

State-of-the-art carbon cycle prediction systems are initialized from reconstruction simulations in which state variables of the climate system are assimilated. While currently only the physical state variables are assimilated, biogeochemical state variables adjust to the state acquired through this assimilation indirectly instead of being assimilated themselves. In the absence of comprehensive biogeochemical reanalysis products, such approach is pragmatic. Here we evaluate a potential advantage of having perfect carbon cycle observational products to be used for direct carbon cycle reconstruction. Within an idealized perfect-model framework, we define 50 years of a control simulation under pre-industrial CO2 levels as our target representing observations. We nudge variables from this target onto arbitrary initial conditions 150 years later mimicking an assimilation simulation generating initial conditions for hindcast experiments of prediction systems. We investigate the tracking performance, i.e. bias, correlation and root-mean-square-error between the reconstruction and the target, when nudging an increasing set of atmospheric, oceanic and terrestrial variables with a focus on the global carbon cycle explaining variations in atmospheric CO2. We compare indirect versus direct carbon cycle reconstruction against a resampled threshold representing internal variability. Afterwards, we use these reconstructions to initialize ensembles to assess how well the target can be predicted after reconstruction. Interested in the ability to reconstruct global atmospheric CO2, we focus on the global carbon cycle reconstruction and predictive skill. We find that indirect carbon cycle reconstruction through physical fields reproduces the target variations on a global and regional scale much better than the resampled threshold. While reproducing the large scale variations, nudging introduces systematic regional biases in the physical state variables, on which biogeochemical cycles react very sensitively. Global annual surface oceanic pCO2 initial conditions are indirectly reconstructed with an anomaly correlation coefficient (ACC) of 0.8 and debiased root mean square error (RMSE) of 0.3 ppm. Direct reconstruction slightly improves initial conditions in ACC by +0.1 and debiased RMSE by −0.1 ppm. Indirect reconstruction of global terrestrial carbon cycle initial conditions for vegetation carbon pools track the target by ACC of 0.5 and debiased RMSE of 0.5 PgC. Direct reconstruction brings negligible improvements for air-land CO2 flux. Global atmospheric CO2 is indirectly tracked by ACC of 0.8 and debiased RMSE of 0.4 ppm. Direct reconstruction of the marine and terrestrial carbon cycles improves ACC by 0.1 and debiased RMSE by −0.1 ppm. We find improvements in global carbon cycle predictive skill from direct reconstruction compared to indirect reconstruction. After correcting for mean bias, indirect and direct reconstruction both predict the target similarly well and only moderately worse than perfect initialization after the first lead year. Our perfect-model study shows that indirect carbon cycle reconstruction yields satisfying initial conditions for global CO2 flux and atmospheric CO2. Direct carbon cycle reconstruction adds little improvements in the global carbon cycle, because imperfect reconstruction of the physical climate state impedes better biogeochemical reconstruction. These minor improvements in initial conditions yield little improvement in initialized perfect-model predictive skill. We label these minor improvements due to direct carbon cycle reconstruction trivial, as mean bias reduction yields similar improvements. As reconstruction biases in real-world prediction systems are even stronger, our results add confidence to the current practice of indirect reconstruction in carbon cycle prediction systems

Generalized Calogero-Moser systems from rational Cherednik algebras

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups we also obtain generalized Calogero-Moser systems with added quadratic potential.Comment: 36 pages; the main change is an improvement of section 7 so that it now deals with an arbitrary complex reflection group; Selecta Math, 201

Depositional age and exhumation of Tethyan Sedimentary rocks intruded by Oligo-Miocene granite

Abstract HKT-ISTP 2013 A

Survival of ancient landforms in a collisional setting as revealed by combined fission track and (U-Th)/He thermochronometry: A case study from Corsica (France)

The age of high-elevation planation surfaces in Corsica is constrained using new apatite (U-Th)/He data, field observations, and published work (zircon fission track, apatite fission track [AFT] data and landform/stratigraphical analysis). Thermal modeling results based on AFT and (U-Th)/He data, and the Eocene sediments uncomformably overlapping the Variscan crystalline basement indicate that present-day elevated planation surfaces in Corsica are the remnants of an erosion surface formed on the basement between ∼120 and ∼60 Ma. During the Alpine collision in the Paleocene-Eocene, the Variscan crystalline basement was buried beneath a westward-thinning wedge of flysch, and the eastern portion was overridden by the Alpine nappes. Resetting of the apatite fission track thermochronometer suggests an overburden thickness of >4 km covering Variscan Corsica. Protected by soft sediment, the planation surface was preserved. In the latest Oligocene to Miocene times, the surface was re-exposed and offset by reactivated faults, with individual basement blocks differentially uplifted in several phases to elevations of, in some cases, >2 km.Currently the planation surface remnants occur at different altitudes and with variable tilt. This Corsican example demonstrates that under favorable conditions, paleolandforms typical of tectonically inactive areas can survive in tectonically active settings such as at collisional plate margins. The results of some samples also reveal some discrepancies in thermal histories modeled from combined AFT and (U-Th)/He data. In some cases, models could not find a cooling path that fit both data sets, while in other instances, the modeled cooling paths suggest isothermal holding at temperature levels just below the apatite partial annealing zone followed by final late Neogene cooling. This result appears to be an artifact of the modeling algorithm as it is in conflict with independent geological constraints. Caution should be used when cross-validating the AFT and (U-Th)/He systems both in the case extremely old terrains and in the case of rocks with a relatively simple, young cooling history