Local models of Shimura varieties and a conjecture of Kottwitz

We give a group theoretic definition of "local models" as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a $p$-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.Comment: 88 pages, several corrections and change

A Computational Study Of The Role Of Spatial Receptive Field Structure In Processing Natural And Non-Natural Scenes

The center-surround receptive field structure, ubiquitous in the visual system, is hypothesized to be evolutionarily advantageous in image processing tasks. We address the potential functional benefits and shortcomings of spatial localization and center-surround antagonism in the context of an integrate-and-fire neuronal network model with image-based forcing. Utilizing the sparsity of natural scenes, we derive a compressive-sensing framework for input image reconstruction utilizing evoked neuronal firing rates. We investigate how the accuracy of input encoding depends on the receptive field architecture, and demonstrate that spatial localization in visual stimulus sampling facilitates marked improvements in natural scene processing beyond uniformly-random excitatory connectivity. However, for specific classes of images, we show that spatial localization inherent in physiological receptive fields combined with information loss through nonlinear neuronal network dynamics may underlie common optical illusions, giving a novel explanation for their manifestation. In the context of signal processing, we expect this work may suggest new sampling protocols useful for extending conventional compressive sensing theory

Gravitational lensing statistical properties in general FRW cosmologies with dark energy component(s): analytic results

Various astronomical observations have been consistently making a strong case for the existence of a component of dark energy with negative pressure in the universe. It is now necessary to take the dark energy component(s) into account in gravitational lensing statistics and other cosmological tests. By using the comoving distance we derive analytic but simple expressions for the optical depth of multiple image, the expected value of image separation and the probability distribution of image separation caused by an assemble of singular isothermal spheres in general FRW cosmological models with dark energy component(s). We also present the kinematical and dynamical properties of these kinds of cosmological models and calculate the age of the universe and the distance measures, which are often used in classical cosmological tests. In some cases we are able to give formulae that are simpler than those found elsewhere in the literature, which could make the cosmological tests for dark energy component(s) more convenient.Comment: 14 pages, no figure, Latex fil

NAM: Non-Adversarial Unsupervised Domain Mapping

Several methods were recently proposed for the task of translating images between domains without prior knowledge in the form of correspondences. The existing methods apply adversarial learning to ensure that the distribution of the mapped source domain is indistinguishable from the target domain, which suffers from known stability issues. In addition, most methods rely heavily on `cycle' relationships between the domains, which enforce a one-to-one mapping. In this work, we introduce an alternative method: Non-Adversarial Mapping (NAM), which separates the task of target domain generative modeling from the cross-domain mapping task. NAM relies on a pre-trained generative model of the target domain, and aligns each source image with an image synthesized from the target domain, while jointly optimizing the domain mapping function. It has several key advantages: higher quality and resolution image translations, simpler and more stable training and reusable target models. Extensive experiments are presented validating the advantages of our method.Comment: ECCV 201

On the gravitational wave background from compact binary coalescences in the band of ground-based interferometers

This paper reports a comprehensive study on the gravitational wave (GW) background from compact binary coalescences. We consider in our calculations newly available observation-based neutron star and black hole mass distributions and complete analytical waveforms that include post-Newtonian amplitude corrections. Our results show that: (i) post-Newtonian effects cause a small reduction in the GW background signal; (ii) below 100 Hz the background depends primarily on the local coalescence rate $r_0$ and the average chirp mass and is independent of the chirp mass distribution; (iii) the effects of cosmic star formation rates and delay times between the formation and merger of binaries are linear below 100 Hz and can be represented by a single parameter within a factor of ~ 2; (iv) a simple power law model of the energy density parameter $\Omega_{GW}(f) ~ f^{2/3}$ up to 50-100 Hz is sufficient to be used as a search template for ground-based interferometers. In terms of the detection prospects of the background signal, we show that: (i) detection (a signal-to-noise ratio of 3) within one year of observation by the Advanced LIGO detectors (H1-L1) requires a coalescence rate of $r_0 = 3 (0.2) Mpc^{-3} Myr^{-1}$ for binary neutron stars (binary black holes); (ii) this limit on $r_0$ could be reduced 3-fold for two co-located detectors, whereas the currently proposed worldwide network of advanced instruments gives only ~ 30% improvement in detectability; (iii) the improved sensitivity of the planned Einstein Telescope allows not only confident detection of the background but also the high frequency components of the spectrum to be measured. Finally we show that sub-threshold binary neutron star merger events produce a strong foreground, which could be an issue for future terrestrial stochastic searches of primordial GWs.Comment: A few typos corrected to match the published version in MNRA

Numerical simulations of negative-index refraction in wedge-shaped metamaterials

A wedge-shaped structure made of split-ring resonators (SRR) and wires is numerically simulated to evaluate its refraction behavior. Four frequency bands, namely, the stop band, left-handed band, ultralow-index band, and positive-index band, are distinguished according to the refracted field distributions. Negative phase velocity inside the wedge is demonstrated in the left-handed band and the Snell's law is conformed in terms of its refraction behaviors in different frequency bands. Our results confirmed that negative index of refraction indeed exists in such a composite metamaterial and also provided a convincing support to the results of previous Snell's law experiments.Comment: 18 pages, 6 figure

Correcting for selection bias via cross-validation in the classification of microarray data

There is increasing interest in the use of diagnostic rules based on microarray data. These rules are formed by considering the expression levels of thousands of genes in tissue samples taken on patients of known classification with respect to a number of classes, representing, say, disease status or treatment strategy. As the final versions of these rules are usually based on a small subset of the available genes, there is a selection bias that has to be corrected for in the estimation of the associated error rates. We consider the problem using cross-validation. In particular, we present explicit formulae that are useful in explaining the layers of validation that have to be performed in order to avoid improperly cross-validated estimates.Comment: Published in at http://dx.doi.org/10.1214/193940307000000284 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org