Three-order form invariance and conserved quantity

In this paper, the definition of three-order form invariance is given. Then the relation between the three-order form invariance and the three-order Lie symmetry is discussed and the sufficient and necessary condition of Lie symmetry, which comes from the three-order form invariance, is obtained. Finally a three-order Hojman conserved quantity is studied and an example is given to illustrate the application of the obtained results.

Three-order pseudo-Hamilton canonical equationsReceived

Based on the three-order Lagrangian equations, Hamilton＇s function of acceleration H^＊ and generalized acceleration momentum P^＊α are defined, and pseudo-Hamilton canonical equations corresponding to three-order Lagrangian equations are obtained. The equations are similar to Hamilton＇s canonical equations of analytical mechanics in form.

Sequence-lithofacies paleogeographic characteristics and petroleum geological significance of Lower Permian Qixia Stage in Sichuan Basin and its adjacent areas,SW China

Through the analysis of logging,field outcrops,cores and geochemical data,and based on the study of the relationships between sea level changes,sequence filling,paleo-geomorphy and lithofacies,the sequence lithofacies paleo-geography and evolution process of the Lower Permian Liangshan-Qixia Formation(Qixia Stage for short)in Sichuan Basin and its surrounding areas are restored.The Qixia Stage can be divided into three third-order sequences,in which SQ0,SQ1 and SQ2 are developed in the depression area,and SQ1 and SQ2 are only developed in other areas.The paleo-geomorphy reflected by the thickness of each sequence indicates that before the deposition of the Qixia Stage in the Early Permian,the areas surrounding the Sichuan Basin are characterized by“four uplifts and four depressions”,namely,four paleo-uplifts/paleo-lands of Kangdian,Hannan,Shennongjia and Xuefeng Mountain,and four depressions of Chengdu-Mianyang,Kangdian front,Jiangkou and Yichang;while the interior of the basin is characterized by“secondary uplifts,secondary depressions and alternating convex-concave”.SQ2 is the main shoal forming period of the Qixia Formation,and the high-energy mound shoal facies mainly developed in the highs of sedimentary paleo-geomorphy and the relative slope break zones.The distribution of dolomitic reservoirs(dolomite,limy dolomite and dolomitic limestone)has a good correlation with the sedimentary geomorphic highs and slope break zones.The favorable mound-shoal and dolomitic reservoirs are distributed around depressions at platform-margin and along highs and around sags in the basin.It is pointed out that the platform-margin area in western Sichuan Basin is still the key area for exploration at present;while areas around Chengdu-Mianyang depression and Guangwang secondary depression inside the platform and areas around sags in central Sichuan-southern Sichuan are favorable exploration areas for dolomitic reservoirs of the Qixia Formation in the next step.

[Y(IO3)3(H2O)2]n: a New Rare Earth Iodate Featuring the Circle-shaped I4O12 Polyiodate Anion and Three-order Nonlinear Optical Property

A new layered yttrium iodate[Y(IO3)3(H2O)2]n(1)has been prepared from the hydrothermal reaction of Y(NO3)3?6H2O with I2O5 at 170℃,and its structure was determined by X-ray single-crystal diffraction method.It belongs to the triclinic system,space group P1 with a=7.355(5),b=7.515(5),c=9.413(7)?,α=79.65(2)o,β=85.18(3)o,γ=71.870(19)o,Z=2,V=486.2(6)?3.1 was further characterized by FTIR,powder X-ray diffraction(PXRD)and UV-Vis spectra.In 1,the Y centers in a monocapped trigonal prism environment are bound by IO-3 anion and unique circle-shaped I4O12 polyiodate anion to generate a wave-like 2-D layer.The adjacent layers are further linked with each other by hydrogen bonds to form a quasi-3-D supramolecular network.1 exhibits a reverse saturation absorption and a self-defocusing effect with the nonlinear absorption coefficientβbeing–0.66×10-5 mW-1,which stems mainly from the electron transition from O-2p to I-5p orbital within iodates upon theoretical calculation.

Center Conditions and Bifurcation of Limit Cycles at Nilpotent Critical Point in a Quintic Lyapunov System

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of computer algebra system MATHEMATICA, the first 8 quasi Lyapunov constants are deduced. As a result, the necessary and sufficient conditions to have a center are obtained. The fact that there exist 8 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems.