1,602 research outputs found
CR eigenvalue estimate and Kohn-Rossi cohomology
Let be a compact connected CR manifold with a transversal CR -action
of dimension , which is only assumed to be weakly pseudoconvex. Let
be the -Laplacian. Eigenvalue estimate of
is a fundamental issue both in CR geometry and analysis. In this
paper, we are able to obtain a sharp estimate of the number of eigenvalues
smaller than or equal to of acting on the -th Fourier
components of smooth -forms on , where and
. Here the sharp means the growth order with respect to
is sharp. In particular, when , we obtain the asymptotic estimate of
the growth for -th Fourier components of
as . Furthermore, we establish a Serre
type duality theorem for Fourier components of Kohn-Rossi cohomology which is
of independent interest. As a byproduct, the asymptotic growth of the
dimensions of the Fourier components for is established. Compared with previous results in this field, the
estimate for already improves very much the corresponding estimate
of Hsiao and Li . We also give appilcations of our main results, including
Morse type inequalities, asymptotic Riemann-Roch type theorem,
Grauert-Riemenscheider type criterion, and an orbifold version of our main
results which answers an open problem.Comment: 39 pages, submitted on January 17, 2018. Comments welcome! arXiv
admin note: text overlap with arXiv:1506.06459, arXiv:1502.02365 by other
author
Learning Structured Inference Neural Networks with Label Relations
Images of scenes have various objects as well as abundant attributes, and
diverse levels of visual categorization are possible. A natural image could be
assigned with fine-grained labels that describe major components,
coarse-grained labels that depict high level abstraction or a set of labels
that reveal attributes. Such categorization at different concept layers can be
modeled with label graphs encoding label information. In this paper, we exploit
this rich information with a state-of-art deep learning framework, and propose
a generic structured model that leverages diverse label relations to improve
image classification performance. Our approach employs a novel stacked label
prediction neural network, capturing both inter-level and intra-level label
semantics. We evaluate our method on benchmark image datasets, and empirical
results illustrate the efficacy of our model.Comment: Conference on Computer Vision and Pattern Recognition(CVPR) 201
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