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 $2\pi$. 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 $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ 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 $(Y,\omega)$, where $Y$ is any compact oriented 3-manifold with toroidal boundary and $\omega$ 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 $\mathbb{C}\times\Sigma$ or $\mathbb{H}^2_+\times\Sigma$, where $\Sigma$ is any compact Riemann surface of genus $\geq 1$. Our first result states that any finite energy solution to the perturbed equations on $\mathbb{C}\times\Sigma$ is necessarily trivial. The second asserts that any small energy solutions on $\mathbb{H}^2_+\times\Sigma$ 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