We are concerned with the existence of quasi-periodic solutions for the following equation x" + Fx (x, t)x' + ω2x + φ(x,t) = 0,where F and φ are smooth functions and 2π-periodic in t, ω＞ 0 is a constant. Under some assumptions on the parities of F and φ, we show that the Dancer's function, which is used to study the existence of periodic solutions, also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e. all solutions are bounded).
In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 < σ < 1, and each given integer b > 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.
REN Xiu-Fang Department of Mathematics, Nanjing University, Nanjing 210093, China
The north-trending Liupan Shan (六盘山) is an important tectonic boundary between the Tibetan Plateau and the Ordos platform. The Late Cenozoic red earth deposits of the Liupan Shan record its tectonic history and environmental effects. In this article we report a new Late Cenozoic red earth section from an intermontane basin in the southern part of the Liupan Shan. Lithofacies analysis, paleomagnetic and fission-track chronologies, and paleocurrent analysis have been employed to identify the tectonic uplift events of the Liupan Shan. Based on the age constraints of mammal fossils, the pa-leomagnetic polarity zones of the Huating (华亭) Section can be approximately correlated with the standard polarity zones that lie between C3An.2n and C5n.1n of the Geomagnetic Polarity Timescale; the bottom age of this section is approximately 10 Ma. Based on this and the previous studies, we infer that a tectonic event commenced in the southern Liupan Shan in this interval between 8.3 and 8.7 Ma, accompanied by a remarkable increase in sediment accumulation rate. Field observations, fission-track dating, determinations of grain-size frequency distributions and the vertebrate fossils found there suggest that the red earth deposits were reworked by water and mainly transported by fluvial-alluvial processes from the adjacent area.
This paper provides evidence that the variation of boreal winter sea level pressure (SLP) over the North Pacific is out-of-phase with SLP fluctuation over the tropical Indian Ocean on both the interdecadal and interannual time scales.Subsequently,a SLP between tropical Indian Ocean and North Pacific (TIO-NP) oscillation index is defined to indicate the variation of such out-of-phase fluctuation.Moreover,the simultaneous surface air temperature and precipitation anomalies in China are closely related to TIO-NP oscillations.Below-normal surface air temperature anomalies in the northern and the eastern part of China,and less rainfall in southern China,correspond to positive TIO-NP oscillation phase with negative SLP anomalies in tropical Indian Ocean and positive anomalies in North Pacific.The TIO-NP oscillation affects China's winter climate anomalies,possibly through modulating the northeast East Asia winter monsoon.
We investigate the dynamics of a relativistic fireball which decelerates as it sweeps up ambient matter. Not only the radiative and adiabatic cases, but also the realistic intermediate cases are calculated. We perform numerical calculation for various ambient media and sizes of beaming expansion, and find that the deceleration radius R0 may play an important role for the hydrodynamic evolution of GRB afterglow.
Accumulating data suggest that FGFs/FGFR1 plays essential roles in the bone development and human skeletal diseases.Conditional inactivation of fgfr1 caused different phenotypes displaying in different cells or specific organs and revealed some novel functions of FGFR1 in bone development.Fgfr1 mutation mainly induced 2 types of human skeletal diseases,craniosynostosis syndrome and dysplasias. Similar mutation of fgfr1 in mouse model just mimicked the phenotype that happened in human.These fa- cilitate the investigation on the underlying mechanism of the diseases.Here we mainly focused on the ad- vance of FGFR1 function in the bone development and its mutation caused skeletal diseases.
The stress induced martensitic phase transformation of spherical ZrCu nanocrystals embedded in an amorphous matrix was studied in this paper. Microstructural observations revealed that the martensitic transformation of the nanocrystal was hindered by the surrounding amorphous coating. The existence of two-step transformation from the austenite phase(B2) to the base structure martensite(B19') and finally to the most stable superstructure martensite(Cm) was also demonstrated. The Cm martensite with(021) type I twinning symmetrically accommodation was surrounded by the B19' martensite with dislocation morphologies.
