Busemann functions and barrier functions

We show that Busemann functions on a smooth, non-compact, complete, boundaryless, connected Riemannian manifold are viscosity solutions with respect to the Hamilton-Jacobi equation determined by the Riemannian metric and consequently they are locally semi-concave with linear modulus. We also analysis the structure of singularity sets of Busemann functions. Moreover we study barrier functions, which are analogues to Mather's barrier functions in Mather theory, and provide some fundamental properties. Based on barrier functions, we could define some relations on the set of lines and thus classify them. We also discuss some initial relations with the ideal boundary of the Riemannian manifold.Comment: comments are welcome

Syntenin-1 is a promoter and prognostic marker of head and neck squamous cell carcinoma invasion and metastasis.

Metastasis represents a key factor associated with poor prognosis of head and neck squamous cell carcinoma (HNSC). However, the underlying molecular mechanisms remain largely unknown. In this study, our liquid chromatography with tandem mass spectrometry analysis revealed a number of significantly differentially expressed membrane/membrane-associated proteins between high invasive UM1 and low invasive UM2 cells. One of the identified membrane proteins, Syntenin-1, was remarkably up-regulated in HNSC tissues and cell lines when compared to the controls, and also over-expressed in recurrent HNSC and high invasive UM1 cells. Syntenin-1 over-expression was found to be significantly associated with lymph node metastasis and disease recurrence. HNSC patients with higher syntenin-1 expression had significantly poorer long term overall survival and similar results were found in many other types of cancers based on analysis of The Cancer Genome Atlas data. Finally, knockdown of syntenin-1 inhibited the proliferation, migration and invasion of HNSC cells, and opposite findings were observed when syntenin-1 was over-expressed. Collectively, our studies indicate that syntenin-1 promotes invasion and progression of HNSC. It may serve as a valuable biomarker for lymph node metastasis or a potential target for therapeutic intervention in HNSC

On invariant sets in Lagrangian graphs

In this exposition, we show that a Hamiltonian is always constant on a compact invariant connected subset which lies in a Lagrangian graph provided that the Hamiltonian and the graph are smooth enough. We also provide some counterexamples for the case that the Hamiltonians are not smooth enough.Comment: 4 page

Study of Peeling of Single Crystal Silicon by Intense Pulsed Ion Beam

The surface peeling process induced by intense pulsed ion beam (IPIB) irradiation was studied. Single crystal silicon specimens were treated by IPIB with accelerating voltage of 350 kV current density of 130 A/cm2. It is observed that under smaller numbers of IPIB shots, the surface may undergo obvious melting and evaporation..

General distance-like functions on the Wasserstein space

Viscosity solutions to the eikonal equations play a fundamental role to study the geometry, topology and geodesic flows. The classical definition of viscosity solution depends on the differential structure and can not extend directly to a general metric space. However, the distance-like functions, which are exactly viscosity solutions of the eikonal equation on a Riemannian manifold, is independent of the differential structure and well-defined on a non-compact, complete, separable metric space. In this paper, we study the viscosity solutions of the eikonal equation on Wasserstein space Pp(X) (p > 1), whose ambient space X is complete, separable, non-compact, locally compact length space. But the Wasserstein space is not locally compact, the co-rays (or calibrated curves) of the distance-like functions may not exist. Fortunately, we show that if the distance-like function is induced by a sequence of closed subset diverging to infinity consisting of Dirac probability measures in the Wasserstein space, the co-rays do indeed exist. The concrete conditions of the existence of the co-rays are also provided. We also show that a distance-like function on the ambient space can induce a distance-like function on the associated Wasserstein space.Comment: 19 page

Seasonal variations of C-1-C-4 alkyl nitrates at a coastal site in Hong Kong: Influence of photochemical formation and oceanic emissions

Five C-1-C-4 alkyl nitrates (RONO2) were measured at a coastal site in Hong Kong in four selected months of 2011 and 2012. The total mixing ratios of C-1-C-4 RONO2 (Sigma 5RONO2) ranged from 15.4 to 143.7 pptv with an average of 65.9 +/- 33.0 pptv. C-3-C-4 RONO2 (2-butyl nitrate and 2-propyl nitrate) were the most abundant RONO2 during the entire sampling period. The mixing ratios of C-3-C-4 RONO2 were higher in winter than those in summer, while the ones of methyl nitrate (MeONO2) were higher in summer than those in winter. Source analysis suggests that C-2-C-4 RONO2 were mainly derived from photochemical formation along with biomass burning (58.3-71.6%), while ocean was a major contributor to MeONO2 (53.8%) during the whole sampling period. The photochemical evolution of C-2-C-4 RONO2 was investigated, and found to be dominantly produced by the parent hydrocarbon oxidation. The notable enrichment of MeONO2 over C-3-C-4 RONO2 was observed in a summer episode when the air masses originating from the South China Sea (SCS) and MeONO2 was dominantly derived from oceanic emissions. In order to improve the accuracy of ozone (O-3) prediction in coastal environment, the relative contribution of RONO2 from oceanic emissions versus photochemical formation and their coupling effects on O-3 production should be taken into account in future studies. (C) 2017 Elsevier Ltd. All rights reserved