Sheaves on the alcoves and modular representations II

We relate the category of sheaves on alcoves that was constructed in "Sheaves on the alcoves and modular representations I" to the representation theory of reductive algebraic groups. In particular, we show that its indecomposable projective objects encode the simple rational characters of a reductive algebraic group in all characteristics above the Coxeter number.Comment: 38 pages. Some corrections in version 3. This is the companion article to "Sheaves on the alcoves and modular representations I". Both articles replace our earlier submissions arXiv:1508.05579 and arXiv:1504.0169

Exploiting correlation in the construction of D-optimal response surface designs.

Cost considerations and difficulties in performing completely randomized experiments often dictate the necessity to run response surface experiments in a bi-randomization format. The resulting compound symmetric error structure not only affects estimation and inference procedures but it also has severe consequences for the optimality of the designs used. Fir this reason, it should be taken into account explicitly when constructing the design. In this paper, an exchange algorithm for constructing D-optimal bi-randomization designs is developed and the resulting designs are analyzed. Finally, the concept of bi-randomization experiments is refined, yielding very efficient designs, which, in many cases, outperform D-optimal completely randomized experiments.Structure;

Semi-bayesian D-optimal designs and estimation procedures for mean and variance functions.

Semi-Bayesian D-optimal designs for fitting mean and variance functions are derived for some prior distributions on the variance function parameters. The impact of the mean of the prior and of the uncertainty about this mean is analyzed. Simulation studies are performed to investigate whether the choice of design has a substantial impact on the efficiency of the mean and the variance function parameter estimation and whether the D-optimality criterion is appropriate irrespective of the method applied to estimate the variance function parameters.Functions;

Outperforming completely randomized designs.

Bi-randomization designs have become increasingly popular in industry because some of the factors under investigation are often hard-to-change. It is well-known that the resulting compound symmetric error structure not only affects estimation and inference procedures but also the efficiency of the experimental designs used. In this paper, the use of bi-randomization designs is shown to outperform completely randomized designs in terms of D-efficiency. This result suggests that bi-randomization designs should be considered as an alternative to completely randomized designs even if all experimental factors are easy-to-change.Optimal;

Estimating the intercept in an orthogonally blocked experiment when the block effects are random.

Abstract: For an orthogonally blocked experiment, Khuri (1992) has shown that the ordinary least squares estimator and the generalized least squares estimator of the factor effects in a response surface model with random block effects coincide. However, the equivalence does not hold for the estimation of the intercept when the block sizes are heterogeneous. When the block sizes are homogeneous, ordinary and generalized least squares provide an identical estimate for the intercept.Effects;

The optimal design of an experiment with blocks of size two for quadratic regression on one variable.

Exact D-optimal designs are derived for an optometry experiment for the estimation of a quadratic polynomial in one explanatory variable. Two observations are made for each subject participating in the experiment, such that each subject serves as a block of two possibly correlated observations. The exact D-optimal designs are compared to the best possible three-level designs and to the continuous D-optimal designs.Optimal;

Degenerate flag varieties and Schubert varieties: a characteristic free approach

We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.Comment: 23 pages; A few typos corrected; Authors affiliation adde