Blow-up Dynamics for L2-Critical Fractional Schrödinger Equations

In this paper, we consider the $L^2$-critical fractional Schrödinger equation $iu_t-|D|^{\beta }u+|u|^{2\beta }u=0$ with initial data $u_0\in H^{\beta /2}(\mathbb{R})$ and $\beta $ close to $2$. We show that if the initial data have negative energy and slightly supercritical mass, then the solution blows up in finite time. We also give a specific description for the blow-up dynamics. This is an extension of the works of F. Merle and P. Raphaël for $L^2$-critical Schrödinger equations. However, the nonlocal structure of this equation and the lack of some symmetries make the analysis more complicated, hence some new strategies are required.

Nonlocal Tensor Sparse Representation and Low-Rank Regularization for Hyperspectral Image Compressive Sensing Reconstruction

Hyperspectral image compressive sensing reconstruction (HSI-CSR) is an important issue in remote sensing, and has recently been investigated increasingly by the sparsity prior based approaches. However, most of the available HSI-CSR methods consider the sparsity prior in spatial and spectral vector domains via vectorizing hyperspectral cubes along a certain dimension. Besides, in most previous works, little attention has been paid to exploiting the underlying nonlocal structure in spatial domain of the HSI. In this paper, we propose a nonlocal tensor sparse and low-rank regularization (NTSRLR) approach, which can encode essential structured sparsity of an HSI and explore its advantages for HSI-CSR task. Specifically, we study how to utilize reasonably the l 1 -based sparsity of core tensor and tensor nuclear norm function as tensor sparse and low-rank regularization, respectively, to describe the nonlocal spatial-spectral correlation hidden in an HSI. To study the minimization problem of the proposed algorithm, we design a fast implementation strategy based on the alternative direction multiplier method (ADMM) technique. Experimental results on various HSI datasets verify that the proposed HSI-CSR algorithm can significantly outperform existing state-of-the-art CSR techniques for HSI recovery.

Global versus local — Mach’s principle versus the equivalence principle

The equivalence principle is the conceptual basis for general relativity. In contrast, Mach’s principle, although said to have been influential on Einstein in his formulation of general relativity, has not been shown to be central to the structure of general relativity. In this paper, we suggest that the quantum effects of Hawking and Unruh radiation are a manifestation of a thermal Mach’s principle, where the local thermodynamic properties of the system are determined by the nonlocal structure of the quantum fields which determine the vacuum of a given spacetime. By comparing Hawking and Unruh temperatures for the same local acceleration we find a violation of the Einstein elevator version of the equivalence principle, which vanishes in the limit that the horizon is approached.