SOS-Hankel Tensors: Theory and Application

Hankel tensors arise from signal processing and some other applications. SOS (sum-of-squares) tensors are positive semi-definite symmetric tensors, but not vice versa. The problem for determining an even order symmetric tensor is an SOS tensor or not is equivalent to solving a semi-infinite linear programming problem, which can be done in polynomial time. On the other hand, the problem for determining an even order symmetric tensor is positive semi-definite or not is NP-hard. In this paper, we study SOS-Hankel tensors. Currently, there are two known positive semi-definite Hankel tensor classes: even order complete Hankel tensors and even order strong Hankel tensors. We show complete Hankel tensors are strong Hankel tensors, and even order strong Hankel tensors are SOS-Hankel tensors. We give several examples of positive semi-definite Hankel tensors, which are not strong Hankel tensors. However, all of them are still SOS-Hankel tensors. Does there exist a positive semi-definite non-SOS-Hankel tensor? The answer to this question remains open. If the answer to this question is no, then the problem for determining an even order Hankel tensor is positive semi-definite or not is solvable in polynomial-time. An application of SOS-Hankel tensors to the positive semi-definite tensor completion problem is discussed. We present an ADMM algorithm for solving this problem. Some preliminary numerical results on this algorithm are reported

Constraints on Neutrino Velocities Revisited

With a minimally modified dispersion relation for neutrinos, we reconsider the constraints on superluminal neutrino velocities from bremsstrahlung effects in the laboratory frame. Employing both the direct calculation approach and the virtual Z-boson approach, we obtain the generic decay width and energy loss rate of a superluminal neutrino with general energy. The Cohen-Glashow's analytical results for neutrinos with a relatively low energy are confirmed in both approaches. We employ the survival probability instead of the terminal energy to assess whether a neutrino with a given energy is observable or not in the OPERA experiment. Moreover, using our general results we perform systematical analyses on the constraints arising from the Super-Kamiokande and IceCube experiments.Comment: RevTex4, 14 pages, 5 figures, minor corrections, version to appear in Phys. Rev.

Enhancing Hydrogen Generation Through Nanoconfinement of Sensitizers and Catalysts in a Homogeneous Supramolecular Organic Framework.

Enrichment of molecular photosensitizers and catalysts in a confined nanospace is conducive for photocatalytic reactions due to improved photoexcited electron transfer from photosensitizers to catalysts. Herein, the self-assembly of a highly stable 3D supramolecular organic framework from a rigid bipyridine-derived tetrahedral monomer and cucurbit[8]uril in water, and its efficient and simultaneous intake of both [Ru(bpy)3 ]2+ -based photosensitizers and various polyoxometalates, that can take place at very low loading, are reported. The enrichment substantially increases the apparent concentration of both photosensitizer and catalyst in the interior of the framework, which leads to a recyclable, homogeneous, visible light-driven photocatalytic system with 110-fold increase of the turnover number for the hydrogen evolution reaction

The Arabidopsis NLP7 gene regulates nitrate signaling via NRT1.1-dependent pathway in the presence of ammonium.

Nitrate is not only an important nutrient but also a signaling molecule for plants. A few of key molecular components involved in primary nitrate responses have been identified mainly by forward and reverse genetics as well as systems biology, however, many underlining mechanisms of nitrate regulation remain unclear. In this study, we show that the expression of NRT1.1, which encodes a nitrate sensor and transporter (also known as CHL1 and NPF6.3), is modulated by NIN-like protein 7 (NLP7). Genetic and molecular analyses indicate that NLP7 works upstream of NRT1.1 in nitrate regulation when NH4+ is present, while in absence of NH4+, it functions in nitrate signaling independently of NRT1.1. Ectopic expression of NRT1.1 in nlp7 resulted in partial or complete restoration of nitrate signaling (expression from nitrate-regulated promoter NRP), nitrate content and nitrate reductase activity in the transgenic lines. Transcriptome analysis revealed that four nitrogen-related clusters including amino acid synthesis-related genes and members of NRT1/PTR family were modulated by both NLP7 and NRT1.1. In addition, ChIP and EMSA assays results indicated that NLP7 may bind to specific regions of the NRT1.1 promoter. Thus, NLP7 acts as an important factor in nitrate signaling via regulating NRT1.1 under NH4+ conditions