一阶理论

For the given endpoint interpolation conditions, a new approximation method of multidegree reduction is presented. Using Chebyshev polynomial approximation theory, the nearly best uniform approximation under the interpolation conditions of endpoints can be obtained.

对于给定的首末端点的各阶插值条件，给出了一种新的一次降多阶逼近算法，应用Chebyshev多项式逼近理论达到了满足端点插值条件下的近似最佳一致逼近。

In Chapter 2, starting from the basic fractional ordinary differential equations,weapply a high order approximation of fractional derivative advanced by Lubich to frac-tional differential equation, construct a high numerical difference scheme to solve thefractional differential equation, present error analysis of the algorithms theoretically,and prove the consistency ,convergency and stability.

接下来的第二章中，首先从基本的分数阶常微分方程出发，对Lubich提出的一个关于分数阶导数的高阶近似，将其应用于分数阶微分方程，构造高阶数值差分格式来进行分数阶微分方程的数值求解，并在理论上给出这一算法的误差分析，证明了它的相容性，收敛性和稳定性。

In such doing, this dissertation serves as a step stone for papers of its counterparts to come, and, more importantly, it proposes a strategic alternative to the realization of models for image processing. This dissertation consists of three major parts. In the first part, detailed discussions and delicate analyses of academic papers on Cellular Neural Network will be provided in the hope of helping us see the potentiality of Cellular Neural Network in the applications of image processing. I will focus on the aforementioned limitations on hardware compilation as well. In the second part, I will put forth "texture analysis" as one basic model of analysis when we apply Cellular Neural Network to image processing. In this so-called texture analysis, a useful "spatial feature" is especially drawn to help us overcome possible problems of more complicated Cellular Neural Network applications in image processing."Spatial feature" also serves as a well-functioning mechanism for technology of image identification. In the last part of this thesis, I will look into a case study, where Cellular Neural Network is applied to help de-screen document image. Using it as an example, we will see how algorithms of Cellular Neural Network may be of marvelous use in applications in document image processing, since it would reduce a great deal of calculation and computation when applied to software compilation, yet opens up unlimited possibilities for higher-speed hardware compilation of high-level image processing.

这篇论文主要可以分为三大部分：在第一部份里，我们会详细地说明并讨论在过去到现在大部分将分子类神经网路应用於影像处理的相关文献及未来所有可能的发展和技术，另外也将分子类神经网路作一完整的介绍，除此之外，我们也会特别著重於分子类神经网路在影像处理相关应用理论的讨论以及其硬体实现化的考量；在第二部分里，我们提出了一个将分子类神经网路应用於影像辨识处理的基础分析—纹路分析，这是由於纹路分析的复杂性和普遍性会使得分子类神经网路於高阶影像处理的应用不会只局限在单一的影像处理技术，其中我们也提出了一个相当有用的空间特徵，此一特徵不但可以使复杂地高阶影像处理能够应用分子类神经网路，也为影像辨识技术提供了一个很好的辨识机制；在最后一部分里，我们也将文件影像分析做了一个完整的剖析，并以文件影像的去网点为例来说明在实际情况下的分子类神经网路的应用，如此演算法的开发也为文件影像处理提供了更多实际的应用，更考量了文件影像处理若以软体实现时的计算量负荷，而对未来高阶数位影像处理能够以硬体实现来提高处理速度提供了无限的可能。

The second-order linear homogeneous differential equations in differential equation theory occupies an important position, but the second order differential equation has no general solution method, most of the solutions of these equations is difficult to obtain analytical expressions and even some impossible to obtain the solution of the form, and the second order differential equation solving is divided into homogeneous and inhomogeneous problems.

二阶线性齐次微分方程在微分方程理论中占有重要位置，但二阶变系数微分方程却没有一般的求解方法，大多数这类方程很难求得解的分析表达式甚至有的根本不可能求得其解的分析形式，并且二阶变系数微分方程的求解又分为齐次的和非齐次的问题。

In order to obtain more general solution of second order linear differential equation with constant coefficients, which is important in theory and practice, on the basis of knowing a special of the second order linear differential equation with constant coefficients and by using the method of variation of constant, the second order linear differential equation with constant coefficients is transferred to the reduced differential equation and a general formula of the second order linear differential equation with constant coefficients is derived.

为了更多地得到理论上和应用上占有重要地位的二阶常系数线性非齐次微分方程的通解，这里使用常数变易法，在先求得二阶常系数线性齐次微分方程一个特解的情况下，将二阶常系数线性非齐次微分方程转化为可降阶的微分方程，从而给出了一种运算量较小的二阶常系数线性非齐次微分方程通解的一般公式，并且将通解公式进行了推广，实例证明该方法是可行的。

Based on the analysis of some aspects, such as the approximative algorithm of FOC, the Tustin transform theory and its generating function formula's character, the convergence guarantee of binomial power function by Maclaurin expanding, and the consideration of the limitation of conventional methods, an improved method is proposed to compute the numerical evalution of FOC using PSE and Tustin transform and is further applied to solving the linear FOS.

以分数阶算子近似方法的分析研究为基础，基于Tustin变换理论及其用于分数阶算子的离散生成函数公式特点，利用二项式幂函数的Maclaurin展开能够保证收敛的特性，考虑常用算法的局限性，提出了一种改进的基于幂级数展开和Tustin变换的分数阶运算方法，并应用于线性分数阶系统的求解，给出了递推算法的详细推导。

The theories of third order contact analysis have been successfully applied in theoretical research and practice in the manufacture of circular-arc bevel gears with good effect.

三阶接触分析理论在弧齿锥齿轮的理论研究和加工实践中都取得了很好的效果，而这种理论与锥齿轮的齿制是无关的，本文的研究工作就是围绕如何将三阶接触分析理论应用到延伸外摆线锥齿轮的加工中这一核心问题展开的。

In this project, we study the theory of higher order differential equations in Banach spaces and related topics. We solve an open problem put forward by two American Mathematicians and two Italian Mathematicians concerning wave equations with generalized Weztzell boundary conditions, introduce an existence family of operators from a Banach space $Y$ to $X$ for the Cauchy problem for higher order differential equations in a Banach space $X$, establish a sufficient and necessary condition ensuring $ACP_n$ possesses an exponentially bounded existence family, as well as some basic results in a quite general setting about the existence and continuous dependence on initial data of the solutions of $ACP_n$ and $IACP_n$. We set up quite a few multiplicative and additive perturbation theorems for existence families governing a wide class of higher order differential equations, regularized cosine operator families, regularized semigroups, and solution operators of Volterra integral equations, obtain classical and strict solutions having optimal regularity for the inhomogeneous nonautonomous heat equations with generalized Wentzell boundary conditions, gain novel existence and uniqueness theorems,which extend essentially the existing results, for mild and classical solutions of nonlocal Cauchy problems for semilinear evolution equations, present a new theorem with regard to the boundary feedback stabilization of a hybrid system composed of a viscoelastic thin plate with one part of its edge clamped and the rest-free part attached to a visocelastic rigid body. Also we obtain many other research results.

在本研究中，我们对Banach空间中的高阶算子微分方程的理论以及相关理论进行了深入研究，解决了由美国和意大利的四位数学家联合提出的一个关于广义Wentzell边界条件下的波动方程适定性的公开问题，恰当地定义了Banach空间中的高阶算子微分方程Cauchy问题的算子存在族及唯一族，建立了齐次和非齐次高阶算子微分方程Cauchy问题适定性的判别定理，获得了关于高阶退化算子微分方程的算子存在族、正则余弦算子族、正则算子半群、Volterra积分方程解算子族的乘积扰动和混合扰动定理，得到了关于以依赖于时间的二阶微分算子为系数的一大类非自治热方程非齐次情形下的时变广义Wentzell动力边值问题的古典解、严格解的最大正则性结果，获得了半线性发展方程非局部Cauchy问题广义解和经典解存在唯一的判别条件，从实质上推广了现有的相关结果；得到了一部分边缘固定而另一部分附在一粘弹性刚体上的薄板构成的混合粘弹性系统的边界反馈稳定化的新稳定化定理，还建立了一系列其他研究结果。

The spreading mechanism of the Bragg lines in shipborne HFSWR is analyzed and mathematical spreading model is given.

理论分析了舰载 HFSWR 一阶海杂波谱的展宽机理，给出其展宽数学模型。

In order to get the optimal solution of an inverse problem,we analyse systematically the double perturbation processing technique, the first order Bor...

为了得到反问题的最优解，文中系统地研究了层状介质中双摄动处理技术，以及层状介质中波场摄动的一阶 Born 近似解与理论重力异常摄动解。

We have no common name for a mime of Sophron or Xenarchus and a Socratic Conversation; and we should still be without one even if the imitation in the two instances were in trimeters or elegiacs or some other kind of verse--though it is the way with people to tack on 'poet' to the name of a metre, and talk of elegiac-poets and epic-poets, thinking that they call them poets not by reason of the imitative nature of their work, but indiscriminately by reason of the metre they write in.

索夫农 、森那库斯和苏格拉底式的对话采用的模仿没有一个公共的名称；三音步诗、挽歌体或其他类型的诗的模仿也没有——人们把&诗人&这一名词和格律名称结合到一起，称之为挽歌体诗人或者史诗诗人，他们被称为诗人，似乎只是因为遵守格律写作，而非他们作品的模仿本质。

The relationship between communicative competence and grammar teaching should be that of the ends and the means.

交际能力和语法的关系应该是目标与途径的关系。