25,179 research outputs found
Microlocal KZ functors and rational Cherednik algebras
Following the work of Kashiwara-Rouquier and Gan-Ginzburg, we define a family
of exact functors from category for the rational Cherednik algebra
in type to representations of certain "coloured braid groups" and calculate
the dimensions of the representations thus obtained from standard modules. To
show that our constructions also make sense in a more general context, we also
briefly study the case of the rational Cherednik algebra corresponding to
complex reflection group .Comment: Revised to improve exposition, giving more details on the
construction of the microlocal local systems and providing background
information on twisted D-modules in an appendi
On the Combinatorics of Locally Repairable Codes via Matroid Theory
This paper provides a link between matroid theory and locally repairable
codes (LRCs) that are either linear or more generally almost affine. Using this
link, new results on both LRCs and matroid theory are derived. The parameters
of LRCs are generalized to matroids, and the matroid
analogue of the generalized Singleton bound in [P. Gopalan et al., "On the
locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is
given for matroids. It is shown that the given bound is not tight for certain
classes of parameters, implying a nonexistence result for the corresponding
locally repairable almost affine codes, that are coined perfect in this paper.
Constructions of classes of matroids with a large span of the parameters
and the corresponding local repair sets are given. Using
these matroid constructions, new LRCs are constructed with prescribed
parameters. The existence results on linear LRCs and the nonexistence results
on almost affine LRCs given in this paper strengthen the nonexistence and
existence results on perfect linear LRCs given in [W. Song et al., "Optimal
locally repairable codes," IEEE J. Sel. Areas Comm.].Comment: 48 pages. Submitted for publication. In this version: The text has
been edited to improve the readability. Parameter d for matroids is now
defined by the use of the rank function instead of the dual matroid. Typos
are corrected. Section III is divided into two parts, and some numberings of
theorems etc. have been change
Fitting stochastic predator-prey models using both population density and kill rate data
Most mechanistic predator-prey modelling has involved either parameterization
from process rate data or inverse modelling. Here, we take a median road: we
aim at identifying the potential benefits of combining datasets, when both
population growth and predation processes are viewed as stochastic. We fit a
discrete-time, stochastic predator-prey model of the Leslie type to simulated
time series of densities and kill rate data. Our model has both environmental
stochasticity in the growth rates and interaction stochasticity, i.e., a
stochastic functional response. We examine what the kill rate data brings to
the quality of the estimates, and whether estimation is possible (for various
time series lengths) solely with time series of population counts or biomass
data. Both Bayesian and frequentist estimation are performed, providing
multiple ways to check model identifiability. The Fisher Information Matrix
suggests that models with and without kill rate data are all identifiable,
although correlations remain between parameters that belong to the same
functional form. However, our results show that if the attractor is a fixed
point in the absence of stochasticity, identifying parameters in practice
requires kill rate data as a complement to the time series of population
densities, due to the relatively flat likelihood. Only noisy limit cycle
attractors can be identified directly from population count data (as in inverse
modelling), although even in this case, adding kill rate data - including in
small amounts - can make the estimates much more precise. Overall, we show that
under process stochasticity in interaction rates, interaction data might be
essential to obtain identifiable dynamical models for multiple species. These
results may extend to other biotic interactions than predation, for which
similar models combining interaction rates and population counts could be
developed
- …