216 research outputs found
The Fenchel-type inequality in the 3-dimensional Lorentz space and a Crofton formula
We generalize the Fenchel theorem to strong spacelike (which means that the
tangent vector and the curvature vector span a spacelike 2-plane at each point)
closed curves with index 1 in the 3-dimensional Lorentz space, showing that the
total curvatures must be less than or equal to . A similar generalization
of the Fary-Milnor theorem is also obtained. We establish the Crofton formula
on the de Sitter 2-sphere which implies the above results.Comment: 9 pages, 4 figures. Comments are welcom
Monopoles and Landau-Ginzburg Models II: Floer Homology
This is the second paper of this series. We define the monopole Floer
homology for any pair , where is a compact oriented 3-manifold
with toroidal boundary and is a suitable closed 2-form. This
generalizes the work of Kronheimer-Mrowka for closed oriented 3-manifolds. The
Euler characteristic of this Floer homology recovers the Milnor torsion
invariant of the 3-manifold by a theorem of Meng-Taubes.Comment: 147 pages. We add a finiteness resul
Monopoles and Landau-Ginzburg Models I
The end point of this series of papers is to construct the monopole Floer
homology for any pair , where is any compact oriented
3-manifold with toroidal boundary and is a suitable closed 2-form. In
the first paper, we exploit the framework of the gauged Landau-Ginzburg models
to address two model problems for the (perturbed) Seiberg-Witten moduli spaces
on either or , where
is any compact Riemann surface of genus . Our first result
states that any finite energy solution to the perturbed equations on
is necessarily trivial. The second asserts that any
small energy solutions on necessarily have energy
decay exponentially in the spatial direction. These results will lead
eventually to the compactness theorem in the second paper.Comment: 56 pages. v2. We add an appendix explaining the case for higher genus
surfaces. v3. The statement of the main results are revised to incorporate
higher genus surfaces as wel
Cross-LKTCN: Modern Convolution Utilizing Cross-Variable Dependency for Multivariate Time Series Forecasting Dependency for Multivariate Time Series Forecasting
The past few years have witnessed the rapid development in multivariate time
series forecasting. The key to accurate forecasting results is capturing the
long-term dependency between each time step (cross-time dependency) and
modeling the complex dependency between each variable (cross-variable
dependency) in multivariate time series. However, recent methods mainly focus
on the cross-time dependency but seldom consider the cross-variable dependency.
To fill this gap, we find that convolution, a traditional technique but
recently losing steam in time series forecasting, meets the needs of
respectively capturing the cross-time and cross-variable dependency. Based on
this finding, we propose a modern pure convolution structure, namely
Cross-LKTCN, to better utilize both cross-time and cross-variable dependency
for time series forecasting. Specifically in each Cross-LKTCN block, a
depth-wise large kernel convolution with large receptive field is proposed to
capture cross-time dependency, and then two successive point-wise group
convolution feed forward networks are proposed to capture cross-variable
dependency. Experimental results on real-world benchmarks show that Cross-LKTCN
achieves state-of-the-art forecasting performance and improves the forecasting
accuracy significantly compared with existing convolutional-based models and
cross-variable methods
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