THE LOCAL CONTINUITY MODULI FOR TWO CLASSES OF GAUSSIAN PROCESSES

§ 1　IntroductionLet{ W(t) } be the standard Wiener process,i.e.a Gaussian process with stationaryindependentincrements;EW(t) =0 and Cov(W(t) ,W(t s) ) =tfor all,t,s≥ 0 .CsorgoandRévész[1 ] gave the following local continuity modulus theorem for the Wiener process;lim supt→ 0 | W(t) | / (2 tlog log(1 / t) ) 1 / 2 =1 ,a.s. (1 )lim supt→ 0 sup0≤ s≤ t| W(s) | / (2 s log log(1 / s) ) 1 / 2 =1 .a.s. (2 )(“Local continuity modulus” was also called“another version of continuity m…     

上期课外练习参考解答

初一年级1.解原方程化为{x}=11-2[x]/5. ∴.0<11+2[x]<5. ∴-5.5<[x]<-3. ∴[x]=-4,或[x]=-5, 当[x]=-4时,{x}=0.6. 当[x]=-5时.{x}=0.2. ∴x=-3.4或x=-4.8. 2.解将三个式子相加,得a4+b4+c4-a2b2-b2c2-c2a2=0. 配方得(a2-b2)2+(b2-c2)2+(c2-a2)2=0.     

也谈慎用图象解法

文 [1 ]告诫人们 :代数问题应慎用图象解法 .下面再举三个例子说明为什么要“慎用图象解法” .例 1　求函数ｙ =2ｓｉｎｘ-13ｃｏｓｘ +2 的最大值和最小值 (文 [2 ]例 2 ) .关于例 1 ,文 [2 ]的解答是解法 1　引进参数ｕ =3ｃｏｓｘ,ｖ =2·ｓｉｎｘ,则点 (ｕ ,Ｖ)在椭圆ｕ23 +ｖ22 =1上 ,易知 (-2 ,1 )到椭圆的切线斜率是ｙ的最大值与最小值 ,设切线方程ｖ =ｋ(ｕ+2 ) +1 ,将其代入椭圆方程并化简为(3ｋ2 +2 )ｕ2 +(1 2ｋ2 +6ｋ)ｕ +(1 2ｋ2 +1 2ｋ -3 ) =0 ,由Δ =0得 ,ｋ2 +4ｋ -1 =0 ,所以ｋ =-2 ± 5 ,所以ｙｍａｘ =-2 +5 ,ｙｍｉｎ …     

方程组(E′_3):■出现五个极限环的例子

<正> 本文利用[1]的方法,证明数字系数的方程组(dx)/(dt)=λx-y-(5+δ)x~3+(12-C)x~2y+(25+γ)xy~2-(4+β)y~3,(dy)/(dt)=x+λy+4x~3+(65+3δ)x~2y-(12-C)xy~2-25y~3,(1)其中λ=10~(-2,830),γ=-10~(-1,407),β=10~(-698),δ=-10~(-226),C=10~(-46),出现五个围绕原点的极限环.     

对一道“通讯杯”竞赛题的看法

20 0 3年“通讯杯”高中数学综合应用能力竞赛已落下帷幕 .这次竞赛 ,无疑对提高参赛者的数学综合应用能力有很大帮助 .美中不足的是命题组提供的最后一道解答题的解法 1有一点问题 .现将解法1抄录如下 (见文 [1]) .解法 1　 1) ,2 )略 .3)不妨设n为k位整数 ,则n xn≤n[1+1 10 +(1 10 ) 2 +… +(1 10 ) k -2 ]=n·10 9·[1- (1 10 ) k -1].9.下略 .解法 1,3)中证明xn>.9时用到的不等式n xn ≤n[1+1 10 +(1 10 ) 2 +… +(1 10 ) k-2 ](1)有误 ,举反例如下 :当n =190时 ,由题中所给信息知n xn=2 10 .(1)式右边 =190× (1+1 10…     

TESTING FOR VARYING DISPERSION IN DISCRETE EXPONENTIAL FAMILY NONLINEAR MODELS

§ 1　 Introduction and modelsThe general form of exponential family nonlinear models isg(μi) =f(xi,﹀) , (1 )where,g(· ) is a monotonic link function,f is a known differentiable nonlinear functionand﹀ is a p-vectoroffixed population parameters;μi=E(yi) and the density of response yiisp(yi) =exp{[yiθi -b(θi) -c(yi) ] -12 a(yi,) } ,(2 )whereθi is the natural parameter, is the dispersion parameter.From [1 1 ] ,μi=b(θi) ,Vi=Var(yi) =- 1 b(θi) .If f(xi,β) =x Ti ﹀,then mod…     

《常规的分离参数解法为何半途而废》一文的瑕疵与改进

近日拜读了文[1]——《常规的分离参数解法为何半途而废》,发现有几处瑕疵,笔者进行了一些改进.一、几处瑕疵1.方程et=21+(t-1)2(021+(t-1)2,0相似文献   

有界区域上非线性 Schrodinger 方程解的全局性质

本文解决了 H.Bréis 在文[1]中提出的一个问题,对于非线性 Schr(?)dinger 方程(?) (1)H.Brézis 在文[1]中证明了,如果 Ω=R~2,p=3,K≤0或k>0,同时K integral from R~2|u_0|~2dx<4,(1)式有解 u(x,t),u∈C~1([0,∞),L~2(R~2))∩C([0,∞),H~2(R~2)).并且指出,在 Ω=R~2的情形,如果 p≥3,并且初值 u_0满足     

有界区域上非线性Schrdinger方程解的全局性质

本文解决了 H.Bréis 在文[1]中提出的一个问题,对于非线性 Schr(?)dinger 方程(?) (1)H.Brézis 在文[1]中证明了,如果 Ω=R~2,p=3,K≤0或k>0,同时K integral from R~2|u_0|~2dx<4,(1)式有解 u(x,t),u∈C~1([0,∞),L~2(R~2))∩C([0,∞),H~2(R~2)).并且指出,在 Ω=R~2的情形,如果 p≥3,并且初值 u_0满足     

一道联赛题的两种解法

1999年全国高中数学联赛第三题是一道三角不等式问题 ,难度适中 ,能充分考查学生的基本素质 .题目　已知当x∈ [0 ,1]时 ,不等式x2 cosθ -x( 1-x) +( 1-x) 2 sinθ >0恒成立 ,试求θ的取值范围 .命题组提供的解答构思巧妙 ,方法独特 ,但技巧性较强 ,学生不易想到 .下面介绍两种学生容易接受和掌握的常规解法 .方法一　 (判别式法 )设　f(x) =x2 cosθ-x( 1-x) +( 1-x) 2 sinθ=( 1+sinθ+cosθ)x2 -( 2sinθ +1)x+sinθ ,易知二次函数 f(x)的对称轴x =2sinθ +1( 2sinθ+1) +( 2cosθ +1) .由x∈ [0 ,1] ,f(x)恒正可知f( 0 ) =sinθ>0 ,　f…     

Symmetry reductions and soliton-like solutions for the variable coefficient MKdV equations

Two types of symmetry reductions are derived for the variable coefficient MKdV equation, which contain well-known Painleve II type equation and Jacobian elliptic equation. In addition, soliton-like solutions of the variable coefficient MKdV equation are also obtained. Finally, a transformation between the variable coefficient MKdV equation and the MKdV equation are also found.     

Comments on “New exact periodic solitary-wave solution of MKdV equation”

By means of a so-called extended homoclinic test method, Guo et al. (2009) [1] obtained exact periodic solitary-wave solutions of the modified Korteweg-de Vries (MKdV) equation. We point out that the bilinear form of MKdV equation given in above paper is not true, so that the obtained solutions in the above paper are not actual solutions of the equation. In this paper, we apply an extended ansatz to obtain some new exact periodic solitary-wave solutions to MKdV equation.     

Higher order solution of the dust ion acoustic solitons in a warm dusty plasma with vortex-like electron distribution

The ratios of dust to free electron and free to trapped electron temperatures are examined in warm dusty plasmas with vortex-like electron distribution through the derivation of a modified Korteweg–de Vries (MKdV) equation using a reductive perturbation theory. As the wave amplitude increases, the width and velocity of the soliton deviate from the prediction of the MKdV equation, i.e., the breakdown of the MKdV approximation. To describe the soliton of larger amplitude, the MKdV equation with the fifth-order dispersion term is employed and its higher-order solutions are obtained.     

Lie symmetry analysis to the time fractional generalized fifth-order KdV equation

In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional generalized fifth-order KdV equation. It shows that this equation can be reduced to an equation which is related to the Erdélyi–Kober fractional derivative. Of course, this method can also be applied to other nonlinear fractional partial differential equations.     

A note on “The integrable KdV6 equation: Multiple soliton solutions and multiple singular soliton solutions”

The goal of this short note is to provide another kind soliton solutions with Hirota form, which is different from what Wazwaz obtained in [A.M. Wazwaz, The integrable KdV6 equations: Multiple soliton solutions and multiple singular soliton solutions, Appl. Math. Comput. 204 (2008) 963-972]. Meanwhile we newly construct the MKdV6 equation and derive a Miura transformation between KdV6 equation and MKdV6 equation.     

耗散MKdV方程的指数吸引子

讨论了无界区域R~1上的MKdV方程,运用带权空间构造一类紧算子和算子分解的方法,得到该方程在H~2（R~1）上指数吸引子的存在性.     

N-soliton solution of a lattice equation related to the discrete MKdV equation

We present an N-soliton solution of a lattice equation related to the discrete MKdV equation under an arbitrary boundary value at infinity.     

Decoupling of the N-soliton solution for a new discrete MKdV equation

We prove that the N-soliton solution for a new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete MKdV equation, and approach single soliton solutions as t→∞. We present concrete expressions of soliton components in the case of N = 2, and illustrate the results.     

可积的与Hamilton形式的NLS-MKdV方程族

本文基于ｌｏｏｐ代数Ａ２的一个特殊子代数，设计了一个等谱问题，应用屠规彰格式计算出一族具有Ｈａｍｉｌｔｏｎ结构的可积系．此族含有非线性Ｓｃｈｒｄｉｎｇｅｒ方程与修正的ＫｄＶ方程，称之为ＮＬＳ ＭＫｄＶ方程族     

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method,the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations.New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple.The method is also valid for other (1+1)-dimensional and higher dimensional systems.