两端固定的奇异梁方程的多重正解

设 n 是一个任意的自然数. 证明了一个两端固定的奇异梁方程的 n 个正解的存在性, 其中非线性项是一个Carathéodory 函数. 主要工具是涉及非线性项的高度函数与锥压缩锥拉伸型的 Krasnoselskii不动点定理. 进一步的研究表明,如果非线性项在零点和无穷远处的增长极限均为无界函数, 该方程仍可能具有正解.     

一类二阶拟线性边值问题的可解性

当非线性项奇异和无穷远处的极限增长函数存在时,考察了一类二阶拟线性边值问题.通过引入非线性项在有界集合上的高度函数,并且考察高度函数的积分,证明了一个解的存在定理.该定理表明当极限增长函数的积分具有适当值时此问题有一个解.     

求解非定常对流扩散方程的流函数松弛方法

提出了一种求解线性和非线性对流扩散方程的流函数松弛方法.方法的主要思想是利用流函数松弛近似将原始的方程转化成等价的松弛方程组,新的松弛方程组是带源项的双曲系统.通过稳定性分析可以知道新系统的耗散系数可由松弛系数调整.数值实现亦证明这个方法可以快速有效地描述对流扩散方程的解.     

定常的热传导-对流问题的Galerkin/Petrov最小二乘混合元方法

1.引言 热传导-对流问题是大气动力学中的一个重要的方程,这个方程组也称为强迫耗散的非线性系统方程组,其较Navier-Stokes方程多了一个未知函数温度场,且温度与速度和压力之间存在着复杂的非线性关系.从热动力学可知,任何运动都会产生热量即有温度,而且温度与速度和压力之间必定互相转化,因此对该非线性系统的研究更具有实际意义.[1]先对     

具组合型非线性项与调和位势的非线性Schr(o)dinger方程

讨论了一类带有组合型非线性项与调和位势的非线性Schr(o)dinger方程.通过构造变分问题,引入位势井方法.给出了位势井的结构和位势井深度函数的性质.得到了问题的相关集合在流之下的不变性.揭示了只要问题的初值属于位势井内或位势井外,则问题在今后所有时间内的解都存在于位势井内或井外.结合凹性方法,给出了解的整体存在性的最佳条件.     

求二次比式和问题全局解的一个新的确定性算法

本文研究一类二次比式和规划问题.首先,利用等价转换的方法把原问题转化为一个非线性规划问题,并且这个非线性规划问题的目标函数通项的分子和分母都分别是两项线性函数乘积和再加上一个线性函数的形式,再根据两项线性函数乘积和的特性,对目标函数进行线性松弛,以确定原问题最优值的下界,从而提出一个求解线性规划问题的分支定界算法,并证明该算法的收敛性.最后,数值结果表明所提出的算法是可行有效的.     

相关免疫函数的结构与构造

一、引言 在流密码学中,人们通常采用一个线性移位寄存器或多个线性移位寄存器再加一个滤波函数来作为流密钥序列生成器。非线性滤波函数的作用是为了提高流密钥序列的线性复杂度。Siegenthaler指出:如果滤波函数选择不当,破译者在仅知密文的情况下可以使用相关攻击的方法来攻击该体制。为了抵抗相关攻击的方法,Siegenthaler提出了相关免疫函数的概念,并给出开关函数是相关免疫函数的必要条件。肖国镇和J.L.     

基于灵敏度函数的连续发酵混杂系统的最优控制

针对一类微生物连续发酵生产1,3-丙二醇(1,3-PD)问题,综合考虑胞内、胞外各物质浓度的变化,建立了一个新的非线性系统动力学模型.根据模型中指示函数的不同取值,将连续发酵过程拆分为16种切换模式.考虑到部分系统参数依赖于决策变量,讨论了系统状态关于参数与决策变量的灵敏度,得到了含有切换的梯度公式.基于灵敏度函数,建立了切换最优控制模型.应用序列二次规划算法对该模型进行了数值求解,通过大规模计算,得到了甘油流加速率和初始注入浓度的最优策略.数值结果表明,依该流加策略可以有效提高终端时刻1,3-PD的浓度.     

具非线性记忆的高阶阻尼双曲系统的一个爆破结果

该文研究了一类具非线性记忆项的高阶阻尼双曲系统的初值问题,利用试验函数方法,给出了弱解的一个爆破结果.     

具任意次非线性项的Camassa-Holm方程的双孤子新解

给出辅助方程、函数变换与变量分离解相结合的方法,构造了具任意次非线性项的Camassa-Holm方程的双孤子和双周期新解.首先,通过两个辅助方程、函数变换与变量分离解,将具任意次非线性项的Camassa-Holm方程的求解问题转化为非线性代数方程的求解问题.然后,借助符号计算系统Mathematica求出该方程组的解,并用辅助方程的相关结论,构造了双周期解和双孤子新解.     

Hyperbolic systems of conservation laws with Lipschitz continuous flux-functions: The Riemann problem

For strictly hyperbolic systems of conservation laws with Lipschitz continuous flux-functions we generalize Lax's genuine nonlinearity condition and shock admissibility inequalities and we solve the Riemann problem when the left- and right-hand initial data are sufficiently close. Our approach is based on the concept of multivalued representatives ofL functions and a generalized calculus for Lipschitz continuous mappings. Several interesting features arising with Lipschitz continuous flux-functions come to light from our analysis.Dedicated to Constantine Dafermos on his 60^th birthday     

Asymptotics of the solutions of the one-dimensional nonlinear system of equations of shallow water with degenerate velocity

A system of one-dimensional nonlinear equations of shallow water with degenerate velocity is considered. The change of variables taking the given system to a nonlinear system with small nonlinearity is proposed. Formal asymptotic solutions near the point of degeneracy are obtained.     

Construction of type II blow-up solutions for a semilinear parabolic system with higher dimension

This paper is concerned with blow-up solutions for a semilinear parabolic system with a power type nonlinearity. Non self-similar blow-up solution is constructed by the matched asymptotic expansions. One component of this solution converges to the singular steady state, and another component converges to zero in self-similar variables.     

投影寻踪模型的改进及其在城市水资源承载能力预测中的应用

在城市水资源承载能力研究中,偏最小二乘回归方法能有效地处理自变量间多重线性相关性问题,但不能较好地处理因变量与自变量间复杂的非线性问题.投影寻踪神经网络耦合模型是处理非线性问题的有力工具,而且神经网络投影寻踪耦合模型稳健性高,但不能较好地处理自变量间多重线性相关性问题.本文把这两种方法结合在一起,建立了基于偏最小二乘回归的神经网络投影寻踪耦合模型,对城市水资源承载能力进行了预测,并取得了满意效果.     

Normal correlation coefficient of non-normal variables using piece-wise linear approximation

The correlation coefficient of non-normal variables is expressed as a function of the correlation coefficient of normal variables using piece-wise linear approximation of each univariate transform of normal to anything, and the second order moments of a multiply truncated bivariate normal distribution. For the inverse problem, an algorithm iterates this analytic function in order to assign a normal correlation coefficient to two non-normal variables. The algorithm is applied for the generation of randomized bivariate samples with given correlation coefficient and marginal distributions and used in a randomization test for bivariate nonlinearity. The test correctly does not reject the null hypothesis of linear correlation if the nonlinearity is plausible and due to the sample transform alone.     

Exact solutions of semilinear radial wave equations in n dimensions

Exact solutions are derived for an n-dimensional radial wave equation with a general power nonlinearity. The method, which is applicable more generally to other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE system of group-invariant variables given by group foliations of the wave equation, using the one-dimensional admitted point symmetry groups. (These groups comprise scalings and time translations, admitted for any nonlinearity power, in addition to space-time inversions admitted for a particular conformal nonlinearity power.) This is shown to yield not only group-invariant solutions as derived by standard symmetry reduction, but also other exact solutions of a more general form. In particular, solutions with interesting analytical behavior connected with blow-ups as well as static monopoles are obtained.     

A route to chaos in a conservative system of three interacting Langmuir waves

This paper presents the results of numerical calculations of a route to chaos in a conservative Hamiltonian system of three Langmuir waves interacting with each other through three-wave couplings. The route is investigated by studying time series, power spectra, phase space portraits and Lyapnov exponents of wave variables for several combinations of wave vectors. The results show that the system follows a route which is very similar to the Ruelle–Takens–Newhouse scenario observed in dissipative systems, and widths and shifts of peaks in power spectra appeared due to the three moderate strength wave interactions. The breaks of tori in the system are also numerically investigated by studying the dependency of Maximum Lyapnov exponents for wave-variables on a parameter which represents the nonlinearity of the system.     

Optimal Controller for Dithered Systems with Backlash or Hysteresis

Dither is a high-frequency signal introduced into a nonlinear system to improve the system performance. A nonlinearity with memory (backlash, hysteresis) is considered in this paper; a dither of a given amplitude is being injected at the input of the nonlinearity. The dither can effectively eliminate memory of the nonlinearity. The significance of the dither frequency lies in its effect on the deviation of the dithered system from its corresponding model, the smoothed system, in which the nonlinearity is a smooth function. The deviation amount can be improved as the dither frequency increases. If the dither has a sufficiently high frequency, the outputs of the smoothed system and the dithered system can be made as close as desired. This enables us to predict the stability of the dithered system by establishing the stability of the smoothed system.     

Patterson-Wiedemann construction revisited

In 1983, Patterson and Wiedemann constructed Boolean functions on n=15 input variables having nonlinearity strictly greater than 2^n-1-2^(n-1)/2. Construction of Boolean functions on odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions are known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995 and subsequently these functions can be described as repetitions of a particular binary string. As example we elaborate the cases for n=15,21. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating autocorrelation and generalized nonlinearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all GF(2)-linear transformations of GF(2^ab) which acts on PG(2,2^a).     

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

We consider the nonlinear Schrödinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrödinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.