Solution of nonlinear wave equation of elastic rod

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia are studied. A nonlinear wave equation is derived. The equation is solved by the method of full approximation.     

A CONSTRUCTIVE PROOF OF THE INVERSION FORMULA FOR ZONAL FUNCTIONS ON SL(2,R）

1.IntroductionLeSL(2,R)denotethemultiplicativegroupofall2x2realm8triceswithdet-nat1.InthispaPer,weuseGtodenotebothSL(2,R)andthellnearLiegroupbecausetheyareisomorphictoeachother.Forj={o,1/2},s=1 iA(whereAER,andRisthesetofallrealnumbers),letVi,8betheprincipalcolltinuousseriesofunitaryrepresentationsofG(cf.[4]).SetBytheIwasawadecomposition,anygEGcanbeuniquelywhttenasg=usatnr,u8ESK,afESA,nrESN.Al8oanyginGhasaCartandecompositionasfollows:Afunctionf0nGissaidtobeazonalfunctionifitsatisf…     

An approximation method to estimate the Hausdorff measure of the Sierpinski gasket

In this paper, we firstly define a decreasing sequence {P^n(S)} by the generation of the Sierpinski gasket where each P^n(S) can be obtained in finite steps. Then we prove that the Hausdorff measure H^8(S) of the Sierpinski gasket S can be approximated by {P^n(S)} with P^n(S)/(1 1/2^n-3)s ≤ H^8(S)≤ Pb(S).An algorithm is presented to get P^n(S) for n≤ 5. As an application, we obtain the best lower bound of H^8(S) till now: H^8(S) ≥ 0.5631.     

<Emphasis Type="Italic">L</Emphasis><Superscript>p</Superscript> approximation by general Bernstein-Durrmeyer operator defined on simplex

As a generalization of the Bernstein-Durrmeyer operators defined on the simplex, a class of general Bernstein-Durrmeyer operators is introduced. With the weighted moduli of smoothness as a metric, we prove a strong direct theorem and an inverse theorem of weak type for these operators by using a decomposition way. From the theorems the characterization of L^p approximation behavior is derived     

Strong Approximations of Martingale Vectors and Their Applications in Markov-Chain Adaptive Designs

The strong approximations of a class of R~d-valued martingales are considered.The conditions usedin this paper are easier to check than those used in [3] and [9].As an application,the strong approximation ofa class of non-homogenous Markov chains is established,and the asymptotic properties are established for themulti-treatment Markov chain adaptive designs in clinical trials.     

On Approximation by Reciprocals of Spherical Harmonics in L p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.     

具有非连续性系数的两相不可压缩可混溶问题的有限元法

Two phase ,miscible,incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Many numerical methods have been given by didierent authors to this system,hut these methods need very high regularity conditions. Actualty,in most practical applications these regularity conditions couldn‘t be satisfied. In this paper,the problem of discontinuous coefficients with lower regularity conditions is considered and the error estimates are demonstrated.     

用“定积分法”探索数列不等式放缩、裂项的目标

我们在教学定积分时,主要着眼于用它解决相关的曲线围成的面积问题,忽略了它在处理其它数学问题中的独特功效.由文[1]我们知道：求曲边梯形的面积（定积分）是通过“分割、以直代曲、作和、逼近”来处理的,其中重要的步骤是以小曲边梯形的面积近似值作为数列的项（相应小矩形面积）,再求数列和,为数列和与定积分之间架起了桥梁.     

剖析一道中考试题的困难成因

一道看似熟悉而又简单的中考题,却成了学生们烫手的山芋.简单而又熟悉的背景,让学生们难舍难弃;千分析万琢磨,题设与结论却难成因果.是什么原因使这道题让考生感到如此困惑,我们不妨共作探讨与分析.     

一个极其驰名的数“圆周率”的不同时期

圆周率是一个极其驰名的数.从有文字记载的历史开始,这个数就引起了古今中外学者们的兴趣.圆周率最早是出于解决有关圆的计算问题.仅凭这一点,求出它的尽量准确的近似值,就是一个极其迫切的问题了.事实也是如此,几千年来作为数学家们的奋斗目标,古今中外一代一代的数学家为此献出了自己的智慧和劳动.