On the periodic solutions of a particular Lame equation

We consider the Hill's equation: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqOVdGhabaGaamizaiaadsha% daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWGTbGaai% ikaiaad2gacqGHRaWkcaaIXaGaaiykaaqaaiaaikdaaaGaam4qamaa% CaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaaiykaiabe67a4jabg2% da9iaaicdaaaa!4973!\[\frac{{d^2 \xi }}{{dt^2 }} + \frac{{m(m + 1)}}{2}C^2 (t)\xi = 0\]Where C(t) = Cn (t, {frbuilt|1/2}) is the elliptic function of Jacobi and m a given real number. It is a particular case of theame equation. By the change of variable from t to defined by: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaOWaaiqaaq% aabeqaamaalaaajaaybaGaamizaGGaaiab-z6agbqaaiaadsgacaWG% 0baaaiabg2da9OWaaOaaaKaaGfaacaGGOaqcKbaG-laaigdajaaycq% GHsislkmaaleaajeaybaGaaGymaaqaaiaaikdaaaqcaaMaaeiiaiaa% bohacaqGPbGaaeOBaOWaaWbaaKqaGfqabaGaaeOmaaaajaaycqWFMo% GrcqWFPaqkaKqaGfqaaaqcaawaaiab-z6agjab-HcaOiab-bdaWiab% -LcaPiab-1da9iab-bdaWaaakiaawUhaaaaa!51F5!\[\left\{ \begin{array}{l}\frac{{d\Phi }}{{dt}} = \sqrt {(1 - {\textstyle{1 \over 2}}{\rm{ sin}}^{\rm{2}} \Phi )} \\\Phi (0) = 0 \\\end{array} \right.\]it is transformed to the Ince equation: (1 + · cos(2)) y + b · sin(2) · y + (c + d · cos(2)) y = 0 where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaiaadggacq% GH9aqpcqGHsislcaWGIbGaeyypa0JcdaWcgaqaaiaaigdaaeaacaaI% ZaGaaiilaiaabccacaWGJbGaeyypa0Jaamizaiabg2da9aaacaqGGa% WaaSaaaKaaGfaacaWGTbGaaiikaiaad2gacqGHRaWkcaaIXaGaaiyk% aaqaaiaaiodaaaaaaa!4777!\[a = - b = {1 \mathord{\left/{\vphantom {1 {3,{\rm{ }}c = d = }}} \right.\kern-\nulldelimiterspace} {3,{\rm{ }}c = d = }}{\rm{ }}\frac{{m(m + 1)}}{3}\]In the neighbourhood of the poles, we give the expression of the solutions.The periodic solutions of the Equation (1) correspond to the periodic solutions of the Equation (3). Magnus and Winkler give us a theory of their existence. By comparing these results to those of our study in the case of the Hill's equation, we can find the development in Fourier series of periodic solutions in function of the variable and deduce the development of solutions of (1) in function of C(t).     

云南省金沙江流域水土流失基本特征分析

在应用新近建立的云南金沙江流域土壤流失方程测算该流域各县(市、区)年均土壤流失总量和各地类年均 土壤流失量基础上,分析了该流域水土流失的总体特征、各地类水土流失特征和不同坡度级坡耕地的水土流失特 征,为因地制宜地防治水土流失提供了依据。     

共线条件方程式教学中的几个问题

指出了共线条件方程式教学中应注意的一些问题：共线条件方程式是联立的两个平面方程式，存在双主距（fx,fy)时的几何概念，以及它的变换式与直接线性变换关系式的异同点。     

Formation of Concentric Rings Around Sources

Zones of increased concentration formed by a solvent flowing from a source are considered. A matehmatical model for forming such zones is proposed. It takes into account that such a zone is composed of a set of independent particles. Hence the distribution of a substance around the source can be explained by movement of an individual particle. In the model this movement is a continuous semi-Markov process with terminal stopping at some random point in space. Parameters of the process depend on the velocity field of the flow. Forward and backward partial differential equations for the distribution density of a random stopping point of the process are derived. The forward equation is investigated for the centrally symmetric case. Solutions of the equation demonstrate either a maximum or a local minimum at the source location. In the latter case a concentric ring around the source is formed. If different substances vary in their absorption rates, they can form separable concentration zones as a family of concentric rings.     

按不同气候背景制作单站MOS预报方程

在基层业务体制的框架基本形成以后，县局在参考上级指导预报的同时，初步开展对单站数值预报模式的研究，注意中、小尺度天气的变化规律，以提高订正预报的精度。     

Barotropic local instability and severe storm process

By means of barotropic model, the characteristic and initial value problems are investigated to reveal the local two-dimensional barotropic instability of the nonuniform current to the dynamic mechanism of the formation of the Yangtze-Huaihe River severe storm in July 1991. Analytical theory and numerical experiment show that (i) the unstable developing modes are chiefly the two periods of about 44 d and 10 d, which are fundamentally consistent with that of the precipitation change of the Yangtze-Huaihe River. (ii) The growth rate of the local perturbation is dominated by the meridional wave numbern = 1–5 and zonal wave numberk = 1–12, i.e. the severe storm over the Yangtze-Huaihe River results from the interaction of the systems at different latitudes and waves of different scales, (iii) The perturbation over the Yangtze-Huaihe River possesses the property of local intensification, which slowly migrates westward over the lower and middle reaches of the Yangtze-Huaihe River. (iv) The growth rate of the instability, especially the propagation velocity of the perturbation, is sensitive to the external parameters ū and α. Project supported by the National Natural Science Foundation of China.     

格子Boltzmann方法地震波场模拟

格子Ｂｏｌｔｚｍａｎｎ方法是细胞自动机在某些学科中的具体化和应用。它根据微观运动过程的某些基本特征建立简化的、时间和空间完全离散的动力学模型，这种模型的平行行为符合宏观的微分方程。     

坡地系统土壤侵蚀定量评价方法

区域性土壤侵蚀的定量评在涉及到泥沙输移的非连续性难题和众多的非确定性因素，目前应用较普遍的小区定量难以适用。所提出的定量评价方法是利用ＧＩＳ技术、模糊数学、将以分布参数为特征的区域坡地系统划分成若干类具有集中参数特征的基本侵蚀单元，结合ＵＳＬＥ方程来模拟空间上不同侵蚀背景条件下土壤侵蚀的强度，并可以确定影响土壤侵蚀的主要因子及其排序，该方法适用性强，可用于不同空间尺度的土壤侵蚀定量评价。     

结构内设备的摩擦摆隔震响应分析

基于系统耦和振动微分方程，分析了结构内摩擦摆隔震设备的响应规律和隔震效果。计算结果表明，经过合理设计的摩擦摆系统能够显著控制设备的地震绝对加速度响应从而有效提高设备的抗震可靠度。