Global Weinstein Type Theorem on Multiple Rotating Periodic Solutions for Hamiltonian Systems

This paper concerns the existence of multiple rotating periodic solutions for $2n$ dimensional convex Hamiltonian systems. For the symplectic orthogonal matrix $Q$, the rotating periodic solution has the form of $z(t+T)=Qz(t)$, which might be periodic, anti-periodic, subharmonic or quasi-periodic according to the structure of $Q$. It is proved that there exist at least $n$ geometrically distinct rotating periodic solutions on a given convex energy surface under a pinched condition, so our result corresponds to the well known Ekeland and Lasry's theorem on periodic solutions. It seems that this is the first attempt to solve the symmetric quasi-periodic problem on the global energy surface. In order to prove the result, we introduce a new index on rotating periodic orbits.Comment: arXiv admin note: text overlap with arXiv:1812.0583

Long wavelength local density of states oscillations near graphene step edges

Using scanning tunneling microscopy and spectroscopy, we have studied the local density of states (LDOS) of graphene over step edges in boron nitride. Long wavelength oscillations in the LDOS are observed with maxima parallel to the step edge. Their wavelength and amplitude are controlled by the energy of the quasiparticles allowing a direct probe of the graphene dispersion relation. We also observe a faster decay of the LDOS oscillations away from the step edge than in conventional metals. This is due to the chiral nature of the Dirac fermions in graphene.Comment: 5 pages, 4 figures, to appear in Phys. Rev. Let

Endoscopic characteristics of different grades of melanosis coli complicated with polyps

Objective To investigate the endoscopic characteristics of different grades of melanosis coli （MC） complicated with polyps. Methods Clinical and endoscopic data of 597 patients diagnosed with MC who underwent colonoscopy were retrospectively analyzed. The correlation of different grades of MC with age， gender and intestinal diseases was analyzed. The gender， age， polyp size and pathological type of polyps in patients with different grades of MC were analyzed statistically. Results Polyps were more commonly detected in gradeⅠMC than in gradeⅡand gradeⅢMC. The size of polyps significantly differed among different grades of MC. The detection rate of < 0.5 cm polyps in gradeⅢMC was 37.50%， significantly lower than 67.11% in gradeⅠand 60.18% in gradeⅡMC. The detection rate of 0.6-1.0 cm polyps in gradeⅢMC was 57.50%， significantly higher than 23.49% in grade Ⅰand 25.66% in gradeⅡMC. The detection rates of hyperplastic polyps and tubular adenoma significantly differed among different grades of MC complicated with polyp （all P < 0.05）. No statistical significance was observed in the distribution of inflammatory polyp， retention polyp， serrated adenoma and tubular-serrated adenoma among different grades of MC complicated with polyp （all P > 0.05）. Conclusions Polyps are more commonly detected in gradeⅠMC. Polyps of 0.6-1.0 cm in size are more likely to be detected in grade ⅢMC. Hyperplastic polyps are more commonly detected in gradeⅠMC， whereas tubular adenoma in gradeⅡandⅢMC