紧支撑正交对称和反对称小波的构造

１．引言 近年来,人们分别从数学和信号的观点对正交小波进行了广泛的研究．尤其是２尺度小波,它克服了短时 Ｆｏｕｒｉｅｒ变换的一些缺陷．目前最常用的 ２尺度小波是 Ｄａｕｂｅｃｈｉｅｓ 小波,但 ２尺度小波也存在一些问题：如 Ｄａｕｂｅｃｈｉｅｓ［２］已证明了除 Ｈａａｒ小波外不存在既正交又对称的紧支撑 ２尺度小波．因此人们提出了 ａ尺度小波理论［３］－［６］,文献［４］－［６］对 ４尺度小波迸行研究．本文的目的是研究４尺度因子时紧支撑正交对称和反对称小波的构造方法．并指出对同一紧支撑正交对称尺度函数而言,…     

周期多尺度分析的特征及它的一个应用

在这篇文章里，我们研究了周期多尺度分析的性质，给出了尺度函数序列的一个特征．这个特征能够使我们从一个尺度函数序列得到另一个尺度函数序列．最后，我们给出了主要结果的一个应用．     

多-尺度加细函数及其性质的研究

引入伸缩因子为a的多-尺度加细函数和平移不变子空间的概念．研究了多-尺度加细方程解存在的条件．特别地,给出这种方程的解是正交的充分必要条件．建立了多-尺度加细函数与两尺度加细函数之间的关系．并讨论了它们的一些性质．最后给出相应的构造算例．     

湍流不同尺度间的近程和共振作用规律及应用

从Navier-Stokes方程出发,研究了湍流不同尺度间的相互作用规律,给出相近尺度间近程粘性应力的积分和微分表达式．引入极相近尺度之间共振相互作用的概念,得到共振粘性应力的微分表达式．利用共振粘性应力张量获得不含经验关系和常数、近似封闭的大涡模拟（LES）方程组．利用近程和共振粘性应力张量获得不含经验关系和常数、近似封闭的湍流多尺度方程组．讨论了湍流多尺度方程的性质及用于湍流计算的优点,尺度间相互作用的近程特性说明:多尺度模拟是湍流计算很有价值的方法,并列举了算例．     

区间[-1，1]上的α尺度双正交多小波的构造

本文研究了一元α尺度紧支撑、双正交多小波的构造．在区间[-1，1]，给出了利用α尺度双正交尺度向量构造α尺度双正交多小波的推导过程得到了一种有效的小波构造算法，并给出了数值算例．     

周期结构区域中压电问题的双尺度有限元方法

近年来纤维压电复合材料的力电性能预测已发展为一个重要的研究领域．对力电耦合周期结构的复合材料问题,通过引入匹配的边界层得到了电势与位移解的新型双尺度有限元计算方法,建立了电势与位移的双尺度耦合关系,分析了双尺度有限元解的误差.数值算例验证了方法的有效性．     

用局部平均速度结构函数检测湍流边界层多尺度相干结构

测量风洞中平板湍流边界层不同法向位置的流向速度时间序列信号,提出了基于局部多尺度平均意义的湍流多尺度局部平均结构函数的概念,以描述湍流多尺度结构的伸缩变形和相对运动;基于多尺度局部平均结构函数的概念与Harr子波变换的一致性,提出了用湍流多尺度局部平均结构函数的平坦因子检测湍流边界层多尺度相干结构及其间歇性的检测方法,提取近壁区域不同法向位置处多尺度相干结构的条件相位平均波形,以研究多尺度相干结构猝发的动力学过程及其对湍流统计性质的影响．     

水滴清除气溶胶过程的随机算法和数值模拟

气溶胶尺度分布的时间演变可量化气溶胶的湿沉降过程,它在数学上可由考虑湿沉降的通用动力学方程来描述．该方程为一典型的部分积分微分方程,与气溶胶尺度分布和雨滴尺度分布均相关,且由于需要考虑Brown扩散、拦截和惯性碰撞等湿沉降机制而使得清除系数模型非常复杂,普通的数值方法难以求解．为此发展了一种新的多重Monte Carlo算法,以求解考虑湿沉降的通用动力学方程,并用于模拟实际环境中气溶胶的湿沉降．对于对数正态分布的气溶胶尺度分布和雨滴尺度分布, 多重Monte Carlo算法进行的数值模拟表明, 雨滴几何平均尺度越小, 雨滴几何标准偏差越小,越有利于小尺度和中等尺度气溶胶的湿去除,而稍微不利于大尺度气溶胶的湿去除．     

有线性阻尼和变长度的单摆

应用多尺度法和势能逼近法研究有线性阻尼和变长度的单摆．通过摩擦因数与摆长慢变参数阶的比较,详细讨论了3种不同的情况并得到振幅、频率和解的渐近分析表达式．势能逼近法的应用使结果对大幅振动也有效．当摩擦因数不是很小时,修正的多尺度法用于得到更精确的首次近似解．与数值结果的比较表明本文的方法是有效的．     

线张力作用下微纳米尺度液滴的非线性粘附

三相接触线上的线张力对微纳米尺度液滴的粘附行为有至关重要的影响．首次在由表面张力、线张力和液滴尺度组成的全参数空间中研究了液滴粘附的非线性行为．研究表明:液滴粘附解空间可以被归纳为4个特征区;在每个特征区内,液滴的粘附行为是相同的;在具有高度非线性的特征区中,给定材料体系,存在着多重粘附态;液滴粘附解空间中存在着两个公共不动点．这些新结果,与数值计算和实验观测相符合．     

Lump solutions to a generalized Hietarinta-type equation via symbolic computation

Lump solutions are one of important solutions to partial differential equations, both linear and nonlinear. This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms. The key is a Hirota bilinear form. Lump solutions are constructed via symbolic computations with Maple, and specific reductions of the resulting lump solutions are made. Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented, together with three-dimensional plots and density plots of the lump solutions.     

Decomposition method for the Camassa–Holm equation

The Adomian decomposition method is applied to the Camassa–Holm equation. Approximate solutions are obtained for three smooth initial values. These solutions are weak solutions with some peaks. We plot those approximate solutions and find that they are very similar to the peaked soliton solutions. Also, one single and two anti-peakon approximate solutions are presented. Compared with the existing method, our procedure just works with the polynomial and algebraic computations for the CH equation.     

A CLASS OF SINGULARLY PERTURBED EQUATIONS FOR LARGE PARAMETER WITH THREE TURNING POINTS

A class of singularly perturbed problem of third order equation with two para-meters is studied. Using singular perturbation method, the structure of solutions to the problem is discussed in three different cases about two small parameters. The asymptotic solutions to the problem are given. The structure of solutions and the different limit behaviors are revealed. And the solutions are compared with the exact solutions to the equation in which the coefficients are constants and a relatively more perfect res...     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

扩展的(G'/G)展开法求量子Zakharov-Kuznetsov方程的精确解

为得到量子Zakharov-Kuznetsov方程的一些新精确解,借助行波解的思想,结合齐次平衡原理和一类非线性常微分方程解的结构,利用扩展的(G'/G)展开方法,研究了其相应的更加丰富的精确解表达形式.新精确解的表达式主要由双曲函数、三角函数和有理数函数构成,出现了某些怪波解的情形.通过对比不同情况下解的形式,利用M...     

含任意次正幂项的广义五阶KdV方程的精确解

利用齐次平衡原理,通过引进含非线性辅助微分方程（sub-ODE）,获得了含任意次正幂项的广义五阶KdV方程的精确解,包括钟状孤波解,扭状孤波解和三角函数表示的周期波解.所得精确解与前人用其它方法所获得一致,并包含了以往文献未提供的部分解,扩充并完善了以往文献的相关结果.     

Absolute and relative choreographies in rigid body dynamics

For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev-Chaplygin case, and the Steklov solution. The “genealogy” of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).      

Nonlocal Symmetries and Explicit Solutions of the Boussinesq Equation

The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev′e equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.     

A multiple-scales solution of the undular hydraulic jump problem

A non-linear third-order ODE governing the free surface of an undular hydraulic jump is solved by means of a multiple-scales analysis. The resulting solutions are compared with direct numerical solutions of the ODE, and good agreement is found. Furthermore, new flow features, which are not evident in the numerical solutions, are revealed by the multiple-scales solutions. The results of the analysis are also compared with numerical solutions of the full equations of motion and with experimental data, and satisfactory agreement is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类推广的KdV方程的新行波解

本文研究了一类推广的Kd V方程的行波解求解的问题.利用新的G展开法,并借助Mathematica计算软件,获得了该方程的含有多个任意参数的新的行波解,分别为三角函数解、双曲函数解、有理函数解和指数函数解,扩大了该类方程的解的范围.