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).     

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

A new method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is solved exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g., when the Courant-Friedrichs-Lewy (CFL) condition failed).     

Development of site-specific design ground motions in Western and Eastern North America

This paper presents results recently obtained for generating site-specific ground motions needed for design of critical facilities. The general approach followed in developing these ground motions using either deterministic or probabilistic criteria is specification of motions for rock outcrop or very firm soil conditions followed by adjustments for site-specific conditions. Central issues in this process include development of appropriate attenuation relations and their uncertainties, differences in expected motions between Western and Eastern North America, and incorporation of site-specific adjustments that maintain the same hazard level as the control motions, while incorporating uncertainties in local dynamic material properties. For tectonically active regions, such as the Western United States (WUS), sufficient strong motion data exist to constrain empirical attenuation relations for M up to about 7 and for distances greater than about 10–15 km. Motions for larger magnitudes and closer distances are largely driven by extrapolations of empirical relations and uncertainties need to be substantially increased for these cases.

For the Eastern United States (CEUS), due to the paucity of strong motion data for cratonic regions worldwide, estimation of strong ground motions for engineering design is based entirely on calibrated models. The models are usually calibrated and validated in the WUS where sufficient strong motion data are available and then recalibrated for applications to the CEUS. Recalibration generally entails revising parameters based on available CEUS ground motion data as well as indirect inferences through intensity observations. Known differences in model parameters such as crustal structure between WUS and CEUS are generally accommodated as well. These procedures are examined and discussed.     

河南及邻区地震前中小地震活跃-平静的定量指标研究

以累计频度定量计算方法，分区讨论了河南及邻区1970年以来地震活动非线性度ZL值的时间进程曲线，系统计算了M≥5．0级地震前的ZL值，结果表明：开始出现活跃(平静)异常的时间集中在地震前1—2年，结束异常的时间在地震前一年内，集中于0—4个月。同时提出了异常的定量指标以及发震类型。     

Nonlinear Rate Dependent Model of High Damping Rubber Bearing

The accurate analysis of the response of isolated structures requires the application of appropriate models of isolation devices. The purpose of this paper is to analyse a nonlinear strain rate dependent model of a high damping rubber bearing which simulates the horizontal behaviour of the device under specified vertical load using a nonlinear elastic spring-dashpot element. The effectiveness of the model is checked by fitting the experimental data concerning three different rubber bearings. The results of the study show that the model can simulate the bearing behaviour over a wide shear strain range with small simulation errors. This revised version was published online in July 2006 with corrections to the Cover Date.     

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。     

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。