A Structural Equation Analysis of Weiner's Attribution-Affect Model of Helping Behavior

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【毕业设计毕业论文】物理专业毕业论文文献翻译中英文对照麦克斯韦方程组

外文资料翻译麦克斯韦方程组麦克斯韦方程组是英国物理学家麦克斯韦在19世纪建立的描述电场与磁场的四个基本方程。

方程组的微分形式，通常称为麦克斯韦方程。

在麦克斯韦方程组中，电场和磁场已经成为一个不可分割的整体。

该方程组系统而完整地概括了电磁场的基本规律，并预言了电磁波的存在。

麦克斯韦提出的涡旋电场和位移电流假说的核心思想是：变化的磁场可以激发涡旋电场，变化的电场可以激发涡旋磁场；电场和磁场不是彼此孤立的，它们相互联系、相互激发组成一个统一的电磁场。

麦克斯韦进一步将电场和磁场的所有规律综合起来，建立了完整的电磁场理论体系。

这个电磁场理论体系的核心就是麦克斯韦方程组。

麦克斯韦方程组在电磁学中的地位，如同牛顿运动定律在力学中的地位一样。

以麦克斯韦方程组为核心的电磁理论，是经典物理学最引以自豪的成就之一。

它所揭示出的电磁相互作用的完美统一，为物理学家树立了这样一种信念：物质的各种相互作用在更高层次上应该是统一的。

另外，这个理论被广泛地应用到技术领域。

历史背景1845年，关于电磁现象的三个最基本的实验定律：库仑定律（1785年），安培—毕奥—萨伐尔定律（1820年），法拉第定律（1831-1845年）已被总结出来，法拉第的“电力线”和“磁力线”概念已发展成“电磁场概念”。

场概念的产生，也有麦克斯韦的一份功劳，这是当时物理学中一个伟大的创举，因为正是场概念的出现，使当时许多物理学家得以从牛顿“超距观念”的束缚中摆脱出来，普遍地接受了电磁作用和引力作用都是“近距作用”的思想。

1855年至1865年，麦克斯韦在全面地审视了库仑定律、安培—毕奥—萨伐尔定律和法拉第定律的基础上，把数学分析方法带进了电磁学的研究领域，由此导致麦克斯韦电磁理论的诞生。

积分形式麦克斯韦方程组的积分形式：这是1873年前后，麦克斯韦提出的表述电磁场普遍规律的四个方程。

其中：（1）描述了电场的性质。

在一般情况下，电场可以是库仑电场也可以是变化磁场激发的感应电场，而感应电场是涡旋场，它的电位移线是闭合的，对封闭曲面的通量无贡献。

ON THE SOLUTIONS TO THE NORMAL FORM OF THE NAVIER–STOKES EQUATIONS

2 2 1 u(t) − qn (t)e−4π νnt/L−4π
2
ν (N +ε)t/L2
),
t → ∞,
for some ε > 0 depending on N . A normal form of the Navier–Stokes equations is obtained in an explicit way from the map u(0) → (qn (0))∞ n=1 . The associated normalization map W is deﬁned by W (u(0)) = (Rn qn (0))∞ , n=1 where Rn , n = 1, 2, 3, . . ., are speciﬁed projections (see the subsection 2.2). The following three related questions are still open: 1. When does the asymptotic expansion actually converge? 2. In what natural normed spaces does our normal form of the Navier–Stokes equations constitute a well-behaved inﬁnite-dimensional system of ordinary diﬀerential equations? 3. What is the range of the normalization map? This paper is devoted to the study of those questions to which we give some partial answers. For instance, we show that if the asymptotic expansion of a regular solution u(t) is absolutely convergent in the Sobolev space H 1 at time t = 0 then it converges in H 1 to u(t) for all time t large enough (see Theorem 5.3 and Proposition 5.4; see also Corollaries 5.11 and 5.12 relevant to Question 3); also we give examples of normed spaces in which the answer to Question 2 is positive (see Section 7). Although the questions 1, 2 and 3 are not yet settled, the asymptotic expansions of regular solutions of the Navier–Stokes equations were recently used as an essential tool in the study of the asymptotic behavior of the helicity [4] as well as in the statistical analysis of a decaying ﬂow [5]. The analytic method developed in [4] also turns out to be useful in the study of the asymptotic expansions. Moreover, the method can be applied to a new construction of the regular solutions to the Navier–Stokes equations that gives additional insight into the algebraic role played by the nonlinear terms in those equations. This construction is motivated by the 2 2 system of diﬀerential equations satisﬁed by the terms un (t) = qn (t)e−4π νnt/L in the above asymptotic expansion and is ﬁrst shown to yield the classical existence results for the regular solutions to the Navier–Stokes equations. Furthermore, the construction is well adapted to the study of the asymptotic expansions. With some new a priori estimates we show that the solutions obtained in this way are global solutions in suitable normed spaces to an extended system of the Navier–Stokes equations. These spaces turn out to provide a suitable setting for the study of the normalization map and the normal form and the extended Navier–Stokes equations. In this case the formal inverse of the normalization map is Lipschitz continuous in some of those normed spaces. At present, our estimates for the terms in the asymptotic expansions are not sharp. Thus, there is still room for improvements in our method. With more reﬁned estimates it may be possible to establish classical convergence of the asymptotic expansions for small values of the normalization map. Moreover, we still do not

麦克斯韦方程和规范理论的观念起源

麦克斯韦方程和规范理论的观念起源*2014－11－12收到†email：************************杨振宁1，2著汪忠1，† 译DOI：10.7693/wl20141201(1清华大学高等研究院北京100084)(2香港中文大学物理系香港沙田)早在法拉第的“电紧张态(electrotonic state)”和麦克斯韦的矢量势(vector potential) 概念中，规范自由度(gauge freedom)的存在就已经不可避免。

它如何演化成为一个支撑粒子物理标准模型的对称原理？这里有一段值得叙说的故事。

人们常说，继库仑(Charles Augustin de Cou- lomb)、高斯(Carl Friedrich Gauss)、安培(AndréMarie Ampère)、法拉第(Michael Faraday) 发现了电学和磁学的四条实验定律之后，麦克斯韦(James Clerk Maxwell)引入了位移电流，在他的麦克斯韦方程组中实现了电磁学的伟大综合。

这种说法不能说是错的，但它并没有道出微妙的几何和物理直觉之间的关联，而正是这种关联促使场论在19 世纪取代了超距作用的概念，也正是它带来了20 世纪粒子物理中非常成功的标准模型。

1 19 世纪的历史1820 年奥斯特(Hans Christian Oersted，1777—1851)发现电流能使其附近的小磁针偏转。

这一发现使整个欧洲科学界大为振奋，带来的结果之一是安培(1775—1836) 关于“超距作用(action at a* 原文已发表于Physics Today，2014 年11 月刊，第45—51 页distance)”的成功理论。

在英格兰，法拉第(1791—1867)也因为奥斯特的发现而激动不已，但他缺乏足够的数学训练，所以无法理解安培的工作。

在1822 年9 月3 日写给安培的一封信中，法拉第叹息道：“很不幸，我不具备足够的数学知识，也不具备自如地进行抽象推理的能力。

结构方程模型(SEM)PPT课件

• 例如：我们以学生父母教育程度、父母职业及其收入(共六个外显变量)，作为学生家庭社会经济地位(潜变量)的指标，我们又以学生中、英、数三科成绩(三个外显变量)，作为学业成就(潜变量) 的指标。
SEM的特点
• 理论先验性 • 同时处理测量与分析问题 • 以协方差的应用为核心 • 适用大样本分析
SEM的来源
• 心理计量学：
• Spearman认为，人类心智能力测验得分之间的相互关系，可以被视为是由这些分数背后所具有的一个潜的共同因素（common factor）的影响结果。
• Thurston认为，在复杂的智力测量背后，应该存在着不同且独立的一组共同因素，他称之为核心心智能力（primary mental abilities），由于这一组共同因素的存在，构成了智力测验得分的复杂关系。研究者必须找出这些因素，才能利用此一因素结构来对智力测验得分之间的共变（协方差）关系，得到最理想的解释，得出最大的解释力。
• 期刊与论文：
• 专门期刊：《结构方程模型》（Structural Equation Modeling ）
• 很多社会、心理等变量，均不能准确地及直接地量度，这包括智力、社会阶层、学习动机等，我们只好退而求其次，用一些外显指标(observable indicators)，去反映这些潜变量。
SEM基本模型
• 测量模型：对于指标与潜变量(例如六个社会经
济指标与社会经济地位)间的关系,通常写成如下测量方程:
x=Λxξ+δ y=Λyη+ε
• x，y是外源(如六项社经指标)及内生(如中、英、数成绩)指标。δ,ε是X，Y测量上的误差。
• Λx是x指标与ξ潜变量的关系(如六项社会经济地位指标与潜社会经济地位的关系)。Λy是y指标与η潜变量的关系(如中、英、数成绩与学业成就间关系)。

王娜量子体系相干态Wigner函数6月17日

量子体系相干态的Wigner函数王娜（陕理工学院物理系物理学063班，陕西汉中，723001）指导教师：王剑华教授【摘要】：Wigner函数作为相空间中的一个准概率分布函数，它包含了量子态在整个相空间演化过程中的全部信息，具有十分重要的物理意义。

本文首先介绍了Wigner函数的定义和性质，其次计算了一维谐振子的相干态,并推广到三维谐振子相干态，将相干态表达式代入三维坐标空间Wigner函数的一般表达形式中，得到了相应的Wigner函数。

最后,介绍了增、减光子奇偶相干态下的Wigner函数及其所表现出的特性。

【关键词】：谐振子；量子体系；相干态；Wigner函数引言Wigner函数最早是由著名的物理学家Wigner于1932年[1]引进的，Wigner函数的引进是为了对热力学体系做量子修正而引入相空间中的准几率分布函数。

在描述量子光学、核物理、量子计算、量子混沌以及量子信息的控制和传递中，Wigner函数也有着非常重要的作用，并且是一个很好的半经典近似。

在上世纪70年代以前，Wigner函数并没有引起人们更多的关注。

直到1975年，Moyal才从量子力学的内部逻辑出发，发现了这个引人入胜的乘法量子化方法[2]。

在这种量子化方法中，我们不需要选定一个特定的表象空间，比如坐标表象或动量表象，而且在现代量子测量中，量子态Wigner函数的重构[3]和测量对研究量子体系的演化过程有着重要的意义，这个定义于相空间的实函数具有准概率分布函数的性质。

一般说来，Wigner函数既可以取正值，也可以取负值，故不能像经典物理中那样，把Wigner 函数看成粒子在同一时刻的坐标、动量的概率密度[4]。

准经典态的Wigner函数始终是非负的，比如一维谐振子的较低的两个能量本征态[5]的Wigner函数，其中基态波函数相应的Wigner函数为非负[6]的，具有相空间中的旋转不变形。

但对于它的激发态，Wigner函数则可正可负，呈现出明显的非经典特征。

21世纪英语读写译。b4-u01-a幻灯片PPT课件

Language Points
4 Who is great Defining who is great depends on how one measures success. But there are some criteria. “Someone who has made a lasting contribution to human civilization is great,” said Dean Keith Simonton, a professor of psychology at the University of California at Davis and author of the 1994 book Greatness: Who Makes History and Why. But he added a word of caution: “Sometimes great people don’t make it into history books. A lot of women achieved great things or were influential but went unrecognized.”
Language Points
5 In writing his book, Simonton combined historical knowledge about great figures with recent findings in genetics, psychiatry and the social sciences. The great figures he focused on include men and women who have won Nobel Prizes, led great nations or won wars, composed symphonies that have endured for centuries, or revolutionized science, philosophy, politics or the arts. Though he doesn’t have a formula to define how or why certain people rise above (too many factors are involved), he has come up with a few common characteristics.

Arrhenius equation - Wikipedia

Where
k is the rate constant T is the absolute temperature (in kelvin) A is the pre-exponential factor, a constant for each chemical reaction that defines the rate due to frequency of
Arrhenius equation
Contents
1 2 3 4 Equation Arrhenius plot Modified Arrhenius' equation Theoretical interpretation of the equation 4.1 Arrhenius' Concept of Activation Energy 4.2 Collision theory 4.3 Transition state theory 4.4 Limitations of the idea of Arrhenius activation energy See also References Bibliography External links
frequency factor
attempt frequency
Given the small temperature range kinetic studies occur in, it is reasonable to approximate the activation energy as being independent of the temperature. Similarly, under a wide range of practical conditions, the weak temperature dependence of the pre-exponential factor is negligible compared to the temperature dependence of the factor; except in the case of "barrierless" diffusion-limited reactions, in which case the pre-exponential factor is dominant and is directly observable.

结构方程模型的理论与应用

2012/4/16
7
此种分析时利用协方差矩阵来进行模型的统和分析，比较研究者所提的假设模型隐含的协方差矩阵与实际搜集数据导出的协方差矩阵之间的差异。
2012/4/16
8
LISREL （ Linear Structural Relationship ）
LISREL由统计学者Karl G. Joreskog 和
（六）SEM的应用
理论模型构建
文献综述模型建构
数据分析
描述性统计信度分析
变量确定
研究假设
EFA
CFA SEM
研究设计
变量的测量问卷设计
假设检验
潜变量假设检验
数据收集
研究方法和研究工具
2012/4/16
中介变量假设检验
调节变量假设检验
SEM 是对一般线性模型（ general linear
model,GLM）的扩展。
• 一般线性模型即对平均数检验的方差分析,
• 适用于回归分析、方差和协方差分析、多水平模
型等具体的统计模型
2012/4/16
27
这些线性模式包括：路径分析、典型相关、
因素分析、判别分析、多元方差分析以及多
元回归分析。其中的每种分析都只是结构方
201241629六sem的应用理论模型构建?文献综述?模型建构?变量确定?研究假设研究设计?变量的测量?问卷设计?数据收集?研究方法和研究工具201241630数据分析?描述性统计?信度分析?efa?cfa?sem假设检验?潜变量假设检验?中介变量假设检验?调节变量假设检验第二章sem的组成sem的组成可以简单总结为
Exploratory Factor Analysis）可以求得

结构方程模型-CFA 部分

Structural Equation Model 21
若11 21
（六）不可识别与超识别
1
x1 x2 x3 x4 x5
2 3 4 5
11 21 1 31
42 52
如果一个模型含有一个不可识别的独立子模型，则其不可识别
2
Structural Equation Model
13个变量
2 11 11 11 2 21 11 11 2111 22 2 311111 312111 31 11 33 42 1121 42 2121 42 3121 52 2121 52 3121 52 11 21
固定负荷
固定方差
11 1
11 1
Structural Equation Model 17
（四）三指标模型的识别
1
x1 x2 x3
2 3
11 21 1 31
x1 111 1 x2 211 2 x3 311 3
Structural Equation Model
测量方程Measurement equation
基本假设：
x1 111 1 x2 211 2 x3 311 3 x4 42 2 4 x5 52 2 5 (9.1)
可放松
Cov i , j 0, i, j Cov i , j 0, i j
结论：当两个潜变量不相关时，模型是不可识别的。
Structural Equation Model
16
（三）指定测量单位
在CFA中，每个因子都需要指定其测量单位，否则任何模型都不可识别。因为因子负荷会随着潜变量的测量单位变化而变化。探索性因子分析中是将所有的观测变量和潜变量标准化，所以不存在识别问题，但存在因子的最优旋转问题。两种方法：

高中物理宇宙热历史概述粒子退耦原子复合过程微波背景辐射大爆炸核合成

Non-flat Universe
中微子仅贡献很少的一部分暗物质（HDM）
原子复合
宇宙原初元素 H 及 He 为主, 复合过程
H e H , He e He , He e He
这里我们只考虑H 的复合. 当温度 T ~ 104 K, 质子、电子和氢原子的热力学平衡统计分布满足非相对论 Boltzmann 分布
热平衡对于一个粒子系统，当粒子间的相互作用足够频繁时，系统处于热平衡状态。在膨胀的宇宙中，当粒子间相互作用的时间尺度短于宇宙膨胀的时间尺度时，热平衡状态可以维持
n1n2 两体相互作用的频率这里n1, n2 为两种粒子的数密度则对于一个“1”类的粒子，它与“2”类粒子相互作用的时间尺度 3
m 15.5 MeV
宇宙学限制 (e.g., Komatsu et al. 2010) WMAP7+BAO+H0 m 0.58 eV (95%CL)
WMAP7+LRG power spectrum+H0m
0.44 eV

Constraints from Planck
Flat
(without lensing)
gB
mT g 2
3/ 2
m exp T
g* n gB
2
30
T
4
mn
3 nT mn 2
g
*
2
30
T 4, 7 g 8 F

3

3
7 2 2 3 gF T 8 45
nT
g* gB
s
2 2 3 gB T 45

结构方程模型

YI=B0+B1Xi1+B2Xi2+…+BpXip+ εi εi为残差值，表示因变量无法被自变量解释的部
分，在测量模型即测量误差，在结构模型中为干扰变量或残差项，表示内生变量无法被外生变量及其他内生变量解释的部分。
ηη11== γ ξ + γ111ξ11+ ζ11 ζ1 η 1= γ11 ξ1+ γ12 ξ2 +ζ1
符号表示
潜在变量：被假定为因的外因变量，以ξ（xi/ksi）表示；假定果的内因变量以η（eta）表示。
外因变量ξ的观测指标称为X变量，内因变量η观测值表称为Y变量。
它们之间的关系是：①ξ与Y、η与X无关②ξ的协差阵以Φ（phi）表示③ξ与η的关系以γ表示，即内因被外因解释的归回矩阵④ξ与X之间的关系，以Λx表示，X的测量误差以δ表示，δ间的协方差阵以Θε表示⑥内因潜变量η与η之间以β表示。
观察变量
观察变量作为反映潜在变量的指标变量，可分为反映性指标与形成性指标两种。
反映性指标又称为果指标，是指一个以上的潜在变量是引起观察变量或显性变量的因，此种指标能反映其相对应的潜在变量，此时，指标变量为果，而潜在变量为因。
相对的，形成性指标是指指标变量是成因，而潜在变量被定义为指标变量的线性组合，因此潜在变量变成内生变量，指标变量变为没有误差项的外生变量。
SEM包含了许多不同的统计技术
SEM融合了因子分析和路径分析两种统计技术，可允许同时考虑许多内生变量、外生变量与内生变量的测量误差，及潜在变量的指标变量，可评估变量的信度、效度与误差值、整体模型的干扰因素等。
SEM重视多重统计指标的运用
SEM所处理的是整体模型契合度的程度，关注整体模型的比较，因而模型参考的指标是多元的，研究者必须参考多种不同的指标，才能对模型的是陪读做整体的判断，个别参数显著与否并不是SEM的重点。

fundamentals of vector network analysis -回复

fundamentals of vector network analysis -回复Fundamentals of Vector Network AnalysisIntroduction:Vector Network Analysis (VNA) is a powerful technique used in the field of electrical engineering for measuring and characterizing high-frequency electrical networks. It provides a comprehensive understanding of the behavior of networks, allowing engineers to design and optimize complex systems in various industries like telecommunications, aerospace, and electronics. In this article, we will delve into the fundamentals of Vector Network Analysis, explaining the underlying principles, measurement techniques, and applications.1. What is Vector Network Analysis?Vector Network Analysis is a method used to measure and analyze the electrical properties of complex networks at high frequencies. It involves the use of a specialized instrument called a Vector Network Analyzer. A VNA measures the amplitude and phase of electronic signals at the input and output ports of the device under test (DUT). These measurements are then used to determine the characteristics of the network, such as transmission and reflectioncoefficients, impedance, and scattering parameters.2. Basic Measurement Principles:Vector Network Analysis relies on the principle of superposition, where the measured signals can be treated as a sum of individual frequency components. The VNA generates a continuous wave signal at specific frequencies and measures the response of the DUT. By varying the frequency, the VNA can capture the behavior of the network across a wide range.3. Measurement Techniques:To perform vector network analysis, the VNA sends a stimulus signal to the DUT and measures the response at its input and output ports. There are two main measurement techniques used in VNA:a) Transmission Measurement: In this technique, the VNA measures the signal transmitted through the DUT. By comparing the transmitted signal with the reference signal, the VNA determines the transmission coefficient, providing information about the network's gain or loss.b) Reflection Measurement: This technique involves the measurement of the signal reflected at the input or output ports of the DUT. By comparing the reflected signal with the incident signal, the VNA calculates the reflection coefficient, which indicates the impedance match or mismatch between the network and the VNA.4. Calibration:Calibration is a critical step in VNA to remove the systematic errors introduced by the measurement setup. It involves the use of calibration standards and reference standards to establish accurate measurement references. Common calibration techniques include the Short-Open-Load-Thru (SOLT) and the Reflect-Match-Reflect (RMR) methods.5. Network Parameters:Vector Network Analysis provides several key parameters that help characterize the behavior of networks. These parameters include:a) S-parameters: S-parameters describe the scattering behavior of networks. They consist of two parts, magnitude, and phase, representing the amplitude and phase shift of signals.S-parameters provide information about signal reflections,transmission, and isolation between ports.b) Impedance: Impedance is a critical parameter that reflects how a network responds to the flow of AC current. It is expressed in terms of real (resistance) and imaginary (reactance) components.c) Transmission and Reflection Coefficients: These coefficients represent the amount of signal transmitted or reflected at the ports of the DUT. They determine the efficiency and impedance match of the network.d) Group Delay: Group delay indicates the time delay of the signal passing through the network. It is crucial in applications where phase coherence and timing are essential, such as in communications systems.6. Applications:Vector Network Analysis finds applications in various fields such as:a) Antenna Design and Testing: VNA helps characterize the performance of antennas by measuring the impedance match and radiation patterns.b) RF/Microwave Component Characterization: VNA is used to measure the performance of components like filters, amplifiers, and mixers, ensuring their proper functioning and efficiency.c) Material Characterization: By analyzing the reflection and transmission of electromagnetic waves through materials, VNA can determine the dielectric properties and material behavior, enabling applications in fields like material science and quality control.d) Circuit Design: VNA plays a significant role in designing and optimizing circuits by measuring their impedance and transmission characteristics. It aids in identifying issues like signal reflections and matching problems.Conclusion:Vector Network Analysis is a fundamental technique inhigh-frequency electrical engineering. With its ability to measure and analyze complex networks accurately, it enables engineers to design, troubleshoot, and optimize systems for various industries. By understanding the principles, measurement techniques,calibration, and network parameters, engineers can harness the power of VNA to ensure efficient, reliable, and well-designed networks.。

反问题参考书目-推荐下载

References - books1.R. Kress, Linear Integral Equations, Springer-Verlag, New York,1992.2. A. N. Tikhonov, V. Y. Arsenin, On the solution of ill-posed problems, JohnWiley and Sons, New York, 1977.3.H. W. Engl, E. Hanke and A. Neubauer, Regularization of inverse problem,Kluwer, Dordrecht, 1996.4. C. W. Groetsch, The Theory of Tikhonov Regularization for FredholmEquations of the First Kind, Pitman, Boston, 1984.5. C. W. Groetsch, Inverse problem in the Mathematical Sciences, Vieweg,Braunschweig, 1993.6.V. A. Morozov, Regularization Methods for Ill-Posed Problems, CRCPress,1993.7. A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear ill-posedproblems, London , New York: Chapman & Hall, 1998.8.O. M. Alifanov, Inverse Heat Transfer Problems, Springer Verlag, 1994.9. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problem,Springer, 1996.10.C. Susanne, L. Brenner, S. Ridgway, The mathematical Theory of FiniteElement Methods, Springer-Verlag, New York, 1994.11.苏超伟，《偏微分方程逆问题的数值方法及其应用》，12.M. A. Golberg, C. S. Chen, Discrete Projection Methods for IntegralEquations, Computational Mechanics Publications, Southampton, 199713.V. Isakov, Inverse Problems for Partial differential Equations, Springer-Verlag, New York, 1998.14.J. R. Cannon, The one-dimensional heat equation, Addison-Wesley PublishingCompany, 1984.15.R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65.Academic Press, New York-London,1975.16.D. V. Widder, The heat equation, Academic Press, 1975.17.J. V. Beck, K. D. Cole, A. Haji-Sheikh, B. Litouhi, Heat conduction usingGreen’s functions, Hemisphere Publishing Corporation, 1992.18.D. Colton, Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, 1992.19.D. L. Colton, Solution of boundary value problems by the methods of integraloperators, Pitman Publishing, 1976.20.V. Isakov, Inverse Source Problem, AMS Providence, R. I., 1990.21.A. L. Bukhgeim, Introduction to the theory of inverse problem, VSP, 2000.22.G. Wahba, Spline models for observational data, Society for Industrial andApplied Mathematics, 1990.23.C. S. Chen, Y. C. Hon and R. A. Schaback, Scientific Computing with RadialBasis Functions, preprint.24.J.W. Thomas, Numerical partical Differential equations (Finite differencemethods), Springer,1995.25.C. W. Groetsh, Inverse problem activities for undergraduates, 翻译版，程晋，谭永基，刘继军。

steinharmonic analysis

steinharmonic analysisStein's harmonic analysis, developed by mathematician Elias M. Stein, is a powerful mathematical tool used to study functions in signal processing, image analysis, and various other areas of mathematics. In this article, we will explore the key concepts and techniques involved in Stein's harmonic analysis.Harmonic analysis is a branch of mathematics that deals with the representation and decomposition of functions as a sum of simpler periodic functions called harmonics. Stein's harmonic analysis focuses on the study of functions and their properties using harmonic analysis techniques.One of the fundamental concepts in Stein's harmonic analysis is the Fourier transform. The Fourier transform of a function is a mathematical operation that decomposes the function into a collection of sinusoidal functions with varying frequencies and amplitudes. This decomposition provides valuable information about different components of the function, which can be useful in understanding and analyzing signals.The Fourier transform can be visualized as a way to transform afunction from the time/space domain to the frequency domain. In the frequency domain, the function is represented as a sum of sinusoidal functions, each associated with a particular frequency. This transformation allows us to analyze the function in terms of its frequency content, enabling us to identify important features and patterns that may be hidden in the time/space domain representation.Stein's harmonic analysis builds upon the Fourier transform by introducing techniques such as wavelet analysis andtime-frequency analysis. Wavelet analysis is a mathematical tool used to analyze signals at different resolutions. Unlike the Fourier transform, which provides a global frequency analysis, wavelet analysis allows for a localized analysis of the frequency components of a signal. This can be particularly useful when dealing with signals that contain transient or localized events.Time-frequency analysis, on the other hand, focuses on analyzing how the frequency content of a signal changes over time. This can be achieved using techniques such as the short-time Fourier transform or the spectrogram. These methods provide a representation of the signal that captures both its time andfrequency properties, allowing for a detailed analysis of transient events and changes in frequency content.Stein's harmonic analysis also incorporates concepts from functional analysis, which deals with the study of vector spaces of functions and operators. Functional analysis provides a rigorous mathematical framework for studying the properties and behavior of functions and operators in a more general context.One of the key contributions of Stein's harmonic analysis is the development of theory and techniques for dealing with non-linear and time-varying systems. Traditional harmonic analysis techniques often assume linearity and time-invariance, which may not hold in many practical applications. Stein's approach allows for the analysis of more complex systems that exhibit non-linear and time-varying behavior.In conclusion, Stein's harmonic analysis is a valuable mathematical tool for studying functions and their properties in various fields, including signal processing and image analysis. The concepts and techniques involved, such as Fourier transforms, wavelet analysis,and time-frequency analysis, provide powerful tools for analyzing and understanding the frequency content and behavior of functions. By incorporating ideas from functional analysis, Stein's approach also allows for the analysis of more complex andnon-linear systems. Overall, Stein's harmonic analysis has proven to be a versatile and powerful tool in the field of mathematics.。

Levinson定理-中国科学院高能物理研究所电子邮件系

• 值时，不变，
减少
• 而经过值时，增加一，即
•
跳进。
•3.临界情况， • 对小的E值，
已经是负值。
•薛定谔方程的Levinson 定理
•当势能满足条件
时有
•半束缚态发生在S波的临界情况：
•势函数在无穷远存在尾巴的情况
满足Levinson定理，而 •满足修改的Levinson定理。
•Newton的两个反例
•时的相移为
•薛定谔方程相移的性质
•1. 由于因子，
很小，
•薛定谔方程相移的性质
•1. 由于因子，
很小，
•例外：和
•
，
时是半整数
•薛定谔方程相移的性质
•1. 由于因子，
很小，
•例外：和
•
，
时是半整数
•随跳跃变化，每次跳
•随跳跃变化，每次跳
•薛定谔方程相移的性质
•1. 很小时，
跳进
•薛定谔方程的Levinson 定理
•临界情况， •在区域有解
• 是束缚态， • 是半束缚态，
取负值。取无穷大。
•薛定谔方程相移的性质
•在区域径向方程依赖于势，设解为 •在区域径向方程可解，E>0时为
•可算得对数微商为
•薛定谔方程相移的性质
•由衔接条件 •解得
•薛定谔方程相移的性质
•一个相角随能量单调变化
•Professor C. N. Yang pointed out •In a talk on monopole (1981)
•“For the Sturm-Liouville problem, •the fundamental trick is the •definition of a phase angle which •is monotonic with respect to the •energy.”

On the concept of normal shift in non-metric geometry

1. What is normal shift ? Brief historical overview. Phenomenon of normal shift is very simple by its nature. Let’s consider it in three-dimensional Euclidean space R3 . Suppose that σ is some smooth orientable surface in R3 . At each point p ∈ σ one can draw unit normal vector n such that n = n(p) would be a smooth vector-valued function on σ . Let’s move each point p of σ in the direction of vector n(p) to the distance t which is the same for all points p ∈ σ . Then moved points pt would form another surface σt as shown on Fig. 1.1. Changing parameter t we would obtain one-parametric family of surfaces. This construction is known as Bonnet transformation. In Bonnet construction initial surface σ is transformed by moving each point of σ . Trajectories of motion in this case are straight lines directed along normal vectors and points of σ move along them with a constant speed |v| = 1. Therefore parameter t, which is the distance of displacement, can also be interpreted as time variable. Bonnet noted that all surfaces σt in his construction are perpendicular to the trajectories of moving points. For this reason his construction is also known as normal displacement or normal shift. Basic observation by Bonnet, i. e. orthogonality of surfaces σt and shift trajectories, gave an impetus for generalization of his construction. This was done by me

具左右分数阶导数的时滞微分方程的正解存在性及迭代求解法

文章编号：1007 − 6735(2020)05 − 0417 − 07DOI: 10.13255/ki.jusst.20191008001具左右分数阶导数的时滞微分方程的正解存在性及迭代求解法魏春艳，刘锡平（上海理工大学理学院，上海 200093）摘要：研究了带有左右Riemann-Liouville分数阶导数的非线性时滞泛函微分方程积分边值问题。

运用上下解方法，得到了边值问题正解的存在性和唯一性的新结论，给出了求边值问题近似解的迭代方法，并对近似解进行了误差估计。

最后给出了具体实例用于说明本文所得结论与方法具有广泛的适用性。

关键词：左右分数阶导数；时滞；边值问题；正解；迭代方法中图分类号：O 175.8 文献标志码：AExistence and iteration for the delay differential equations involving left and right fractional derivativesWEI Chunyan， LIU Xiping(College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China)Abstract: The integral boundary value problems of nonlinear delay functional differential equations with left and right Riemann-Liouville fractional derivatives were studied by using the method of lower and upper solutions. Some new results on the existence and uniqueness of solutions were established by using the method of upper and lower solutions, iteration method for solving differential equations and the error estimations were presented. Finally, an example was given out to illustrate the wide applicability of the results and methods.Keywords:left and right fractional derivatives;delay;boundary value problem;positive solution;iteration method1 问题的提出近年来，分数阶微分方程受到了人们的广泛关注[1-12]，在化学工程、粘弹力学以及人口动态等问题中得到了广泛应用[13-15]。

建筑物震动

1.1
General Features of Building Vibrations
A single-story building may be thought of as a single mass (the roof) supported by elastic walls, which also have some damping. When the mass of this structural system
1
Introduction to Building Vibrations
In this web-based module for analysis, visualization and experimentation you will learn about structural dynamics and the vibrations of buildings. You will learn how one mathematically describes the dynamic behavior of a three-story building in terms of its oscillatory response to earthquakes. You will ﬁnd that for a three-story building, one can mathematically describe the vibrations of the building using a set of three second-order diﬀerential equations, similar to the spring-mass-damper equation that you studied in your ﬁrst-year physics course. The key diﬀerence between the single-mass system you studied in physics, and the three-mass system you will learn about here, is that the three equations used in the three-mass system are inter-related, or coupled. Physically, this means that the response and oscillations of one ﬂoor mass depends on the oscillations of all the other ﬂoors. In coupled systems it is often convenient to write the diﬀerential equations in terms of matrices. Matrix notation helps to simplify and unify how one mathematically describe dynamic systems, like this three-story building. To get a general idea of what building vibrations are about, please use the navigation bar on the left to view some videos of our small-scale building model oscillating at various frequencies. Right now this unfortunately requires Internet Explorer and Windows Media Player. Browser-independent versions of the video will be up soon. The ﬁrst part of this tutorial presents an introduction to the response of buildings to earthquakes. The second part describes a small-scale web-based experiment of a three-story building model, how earthquake-like motions are applied to its base, and how the model is instrumented to measure the vibratory response. The third part of the tutorial provides instructions on how to run web-based computer simulations and actual earthquake-response experiments, through your web-browser. In the last part we ask you to answer some evaluations about this web-based lab. Before continuing, please note the time. When you complete the evaluation questions, you will be asked how much time you spent working on this web-based lab. The educational objectives of this module are:

结构方程模型 ppt课件

CONTENTS
01 概念介绍 02 基本原理
03 案例分析
04 实际操作
ppt课件
2
01 概念介绍
1.基本概念
结构方程模型（Structural Equation Modeling, SEM）是一种验证性多元统计分析技术，是应用线性方程表示观测变量与潜变量之间，以及潜变量之间关系的一种多元统计方法，其实质是一种广义的一般线性模型。
ppt课件
19
02 基本原理
3.模型拟合——主要拟合度指标
（3）整体模型拟合度
a) χ2卡方拟合指数检验选定的模型协方差矩阵与观察数据协方差矩阵相匹配的假设。原假设是模型协方差阵等于样本协方差阵。如果模型拟合的好，卡方值应该不显著。在这种情况下，数据拟合不好的模型被拒绝。
b) RMR 是残差均方根。RMR 是样本方差和协方差减去对应估计的方差和协方差的平方和，再取平均值的平方根。 RMR应该小于0.08，RMR越小，拟合越好。
2.模型评价——参数估计（1）假设条件 ① 测量模型误差项δ，ε的均值为零 ② 结构模型的残差项ζ的均值为零 ③ 误差项ε，δ与因子η，ξ之间不相关，误差项ε与δ不相关 ④ 残差项ζ与ξ ，η ，δ之间不相关（2）参数估计策略 ① 加权最小平方策略（WLS） ② 最大概似法(ML) ③ 无加权最小平方法(ULS) ④ 一般化最小平方法（GLS） ⑤ 渐进分布自由法（ADF）

5

6
结构模型：反映潜在变量之间因果关系
方程式： 1 11 1 1 2 21 1 21 1 2
0 0
B

21
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一维非线性奇异问题有限元解的两种存在唯一性证明

内蒙古工业大学学报JOURNAL OF INNER M ONGOLIA第24卷　第4期UNIVERSITY OF T ECHNOLOGY Vol.24No.42005 文章编号:1001-5167(2005)04-0241-05一维非线性奇异问题有限元解的两种存在唯一性证明周凤玲1,于小平2,刘雪英1(1.内蒙古工业大学理学院数学系,呼和浩特010051;2.呼和浩特土默特中学,呼和浩特010051)摘要:讨论一维非线性奇异问题的有限元方法,用两种方法证明了弱解的存在唯一性,并给出有限元解的误差估计.关键词:加权So bolev空间;强单调半连续;Banach不动点定理中图分类号:O242.21 文献标识码:A0　引　言一维奇异稳态问题的有限元方法,已经有不少作者研究,并有一些文章给出了相应较理想的估计结果.比如,Eriksso n和T hom ee在文〔1〕中研究了奇异方程L u=-1x(x u′)′+qu=f(x)给出了L∞模估计,Jesper son D在文〔2〕中给出了一维奇异问题的加权L2-模估计和L2-模估计,李宏〔3〕和李美凤〔4〕以及李德茂,魏建强〔5〕研究了奇异非线性问题.本文旨在上述基础上,对系数奇异且右端非线性的问题,利用两种方法,给出有限元解的存在唯一性证明,并给出有限元解的误差估计.本文讨论如下一维奇异稳态问题L u=-1p(x)(p(x)a(x,u)u′)′=f(x,u),x∈(0,1)u′(0)=u(1)=0(1)其中p(0)=0,且当x∈(0,1)时,p′(x)>0对a(x,u)及f(x,u)作如下假设:1)a(x,u)∈C(I×R)且存在常数c1>0,使得0<a(x,u) c1(2)2)a u・u′∈C(I-)且存在常数c2,使得 a u・u′ c2(3)3)a(u)满足整体Lipschitz条件,即对 u,v∈R1,有a(u)-a(v) u-v (4) 4)a u・u′ 0 1+a(u) 21 c3 21, =( 0, 1)∈R2(5) 在(5)中取 =(0,1),则有　a(u) c3(6)5)f(x,u)∈L2p(I×R1)且关于u满足整体Lipschitz条件,即对 u,v∈R1,有f(x,u)-f(x,v) L u-v (7)收稿日期:2005-01-04基金项目:内蒙古自然科学基金项目(200208020104)和内蒙古工业大学校基金(X200517)资助作者简介:周凤玲(1972～),女,内蒙古巴彦淖尔盟人,内蒙古工业大学理学院讲师,硕士. 6)f (x ,t )关于t 可导,且f tc 3-1(8) 其中c 3见(5)式.1　变分问题定义加权Sobolev 空间L 2p =H 0p (I )={v ∫10p (x )v 2d x <+∞}H m p (I )={v D v ∈H 0p (I ), m }H o m p (I )={v v ∈H m p (I ),v (1)=0} 相应的范数与半范数定义为:‖v ‖0,p =(∫10p (x )v 2d x )12, v ∈L 2p (I )‖v ‖m ,p =( m ‖D v ‖20,p )12, v ∈H m p (I )v m ,p =(‖ =m ‖D v ‖20,p )12 定义V ={v v ∈H 1p (I ),v (1)=0},对 v ∈V ,用p (x )v 乘以(1)两边且在I 上积分,使用分部积分得∫10p (x )a (u )u ′v ′d x =∫10p (x )f (x ,u )v d x 令B (u ,v ,w )=∫10p (x )a (u )v ′w ′d x ,则得与(1)相应的变分问题:求u ∈V ,使得B (u ,u ,v )=(p (x )f (x ,u ),v ), v ∈V (9)对区间I -=[0,1]进行剖分:0=x 0<x 1<…<x n -1<x n =1,记I i =(x i -1,x i ),h i =x i -x i -1,i =1,2,…,n ,并设h =max 1 i n h i ,设剖分是拟一致的,即存在常数 >0,使得min 1 i n h i h,定义有限维空间V h 为V h ={v h /v h ∈C (I ),v /I i ∈P 1(x ),1 i n ,v h (1)=0}则(1)对应的近似变分问题为:求u h ∈V h ,使得B (u h ,v h ,w h )=(p (x )f (x ,u h ),v h ), v h ∈V h(10)为方便起见,以下将与u ,h 无关的常数均记为c ,仅u 与有关的常数记为c (u ).2　弱解的存在唯一性引理2.1　对 v ∈V ,有‖v ‖0,p c v 1,p引理2.2　设a (u )满足假设(1)(2)(3)(4),则对 u ,v ,w ∈V ,B (u ,v ,w )有如下性质:1) B (u ,v ,w ) c ‖v ‖1,p ‖w ‖1,p(2.1)2)B (u ,v ,v ) c v 21,p(2.2)3) B (u ,u ,w )-B (v ,v ,w ) c ‖u -v ‖1,p ‖w ‖1,p(2.3)4)B (u ,u ,u -v )-B (v ,v ,u -v ) c u -v 21,p (2.4)证明引文〔5〕中引理2.1.引理2.3　设X 为自反的Banach 空间,T :X →X ′半连续且满足强单调条件,即对 x ,y ∈X ,有(T x -Ty ,x -y )> ‖x -y ‖2成立,其中 (x )满足 (0)=0, (t )>0,(t >0)lim t →∞(t )=+∞,则T 为满射且为1-1的,详见文〔6〕.242内蒙古工业大学学报2005年定理2.4　设a (u )满足假设(1),(2),(3),(4),f (x ,u )满足假设(5),(6),则(1)存在唯一解.证明一　令A (u ,u ,v )=B (u ,u ,v )-(p (x )f (x ,u ),v ), v ∈V则由(5)得 f (x ,u ) f (x ,0) +L u从而有‖f ‖0,p c 4+c ‖u ‖0,p(2.5)对固定的u ,由(2.1),(2.5)及引理2.2,得A (u ,u ,v )B (u ,u ,v ) + (p (x )f (x ,u ),v )c ‖u ‖1,p ‖v ‖1,p +(c 4+c ‖u ‖0,p )‖v ‖1,p (c ‖u ‖1,p +c 4)‖v ‖1,p 即对固定的u ∈V ,A (u ,u ,・),是V 上的有界线性泛函,由Riesz 表示定理知,存在唯一的w ∈V ,使得A (u ,u ,v )=(w ,v )1, v ∈V 其中 (w ,v )1=∫I p (x )(w v +w ′v ′)d x令Tu =w ,则T :V →V 且A (u ,u ,v )=(T u ,v )1, u ,v ∈V 现证T u = 在V 存在唯一解,由引理2.2,只需证T 半连续且强单调即可.先证T 连续,从而半连续.设{u h } V 且满足‖u h -u ‖1,p →0,u ∈V 则由(7),(2.3)及Riesz 表示定理,得‖Tu h -T u ‖1,p =sup ‖v ‖1,p =1(T u h -T u ,v )1=sup ‖v ‖1,p =1〔A (u h ,u h ,v )-A (u ,u ,v )〕=sup ‖v ‖1,p =1〔B (u h ,u h ,v )-B (u ,u ,v )-(p (x )f (x ,u h ),v )+(p (x )f (x ,u ),v )〕sup ‖v ‖1,p=1〔 B (u ,u ,v )-B (u ,u ,v ) + p (x )(f (x ,u )-f (x ,u ),v ) sup ‖v ‖1,p=1〔c ‖u h -u ‖1,p ‖v ‖1,p +L ‖u h -u ‖0,p ‖v ‖0,p 〕c ‖u h -u ‖1,p →0因此T 连续.再证T 强单调.对 u ,v ∈V ,由(8),(2.1),(2.3),(2.4)及引理2.1,得(T u -T v ,u -v )1=A (u ,u ,u -v )-A (v ,v ,u -v )=B (u ,u ,u -v )-B (v ,v ,u -v )-(p (x )f (x ,u )-p (x )f (x ,v ),u -v )c u -v 21,p =c 5 u -v 21,pc ‖u -v ‖20,p ,其中c 5=max (c ,1)即T 强单调.由引理2.3知T u = 在V 中存在唯一解,即(9)在V 中存在唯一解.类似可证明问题(10)存在唯一解.在此基础上,可得误差估计为‖u -u h ‖0,p +h ‖(u -u h )′‖0,p c (u )h 2‖u ‖2,p 证明二　对任意固定的u ∈V ,定义映射F :V →R 为F (v )=B (u ,u ,v )显然F 是V 上的线性泛函,且由引理2.2的(2.1)可得F (v ) = B (u ,u ,v ) c ‖u ‖1,p ‖v ‖1,p即F 是V 上的有界线性泛函,而知V 是Hilbert 空间,由Riesz 表示定理,对 u ∈V , T (u )∈V ,使得(T (u ),v )V =B (u ,u ,v ), v ∈V 其中(・,・)V 表示V 中内积,其形式为243第4期周凤玲等　一维非线性奇异问题有限元解的两种存在唯一性证明(u ,v )V =∫Ip (x )(u ′v ′+uv )d x , u ,v ∈V 另一方面,显然(p (x )f ,v )是V 上有界线性泛函,再根据Riesz 表示定理,得(p (x )f ,v )=(f *,v )V从而可得B (u ,u ,v )-(p (x )f ,v )=(T (u )-f *,v )V v ∈V(2.6) 对 u ,w ∈V ,有(T (u )-T (w ),u -w )V =(T (u ),u -w )V -(T (w ),u -w )V=B (u ,u ,u -w )-B (w ,w ,u -w )c u -w 21,p ‖u -w ‖21,p(2.7) (T (u )-T (w ),v )V = B (u ,u ,v )-B (w ,w ,v ) k ‖u -w ‖1,p ‖v ‖1,p(2.8) 由于T (u ),T (w )∈V ,因此在(2.8)中令v =T (u )-T (w ),则由(2.8)可得T (u )-T (w ) 1,p k ‖u -w ‖1,p(2.9) 定义算子F :V →V 为F (u )=u - (T (u )-f *), u ∈V(2.10) 其中=k 21, <k 1 (k 1 k ) 由(2.7),(2.9)及(2.10)得‖F (u )-F (w )‖21,p =‖u -w - (T (u )-T (w ))‖21,p=(u -w - (T (u )-T (w )),u -w - (T (u )-T (w )))V=‖u -w ‖21,p -2 (T (u )-T (w ),u -w )V + 2‖T (u )-T (w )‖21,p(1-2 + 2k 21)‖u -w ‖21,p (1- 2k 21)‖u -w ‖21,p ‖u -w ‖21,p 则由Banach 不动点定理,对于算子F :F (u )=u 存在唯一解,记该解为u ,则由(2.10)得Tu =f* 再由(2.6)知,对 v ∈V ,有B (u ,u ,v )=(p (x )f ,v )即u 是(9)的唯一解.类似可证明问题(10)在中V h 存在唯一解.在此基础上,可得出误差估计为‖u -u h ‖0,p +h u -u h 1,p c (u )h 2‖u ‖2,p 参考文献:[1]　Eriksson K ,T hom ee V .G aler kin M ethods for Sing ular Bo undar y V alue Pr oblem in O ne Space Dimensio n [J ].M ath.Co mput ,1984,42(166):345～367.[2]　Jeper so n.R itz-G aler kin M ethods for Sing ular y Bo undar y V alue Pr o blem [J].SIA M J.N umer Anal,1978,15(4):813～834.[3]　李宏.一维非线性奇异问题的有限元方法[J ].内蒙古大学学报(自然科学版),1997,28(4):463～471.[4]　李美凤.一类非线性奇异问题的有限元方法[J].内蒙古大学学报(自然科学版),2000,31(1):7～11.[5]　李德茂,魏建强.非对称有限元方法[J].内蒙古大学学报(自然科学版),1994,25(1):30～34.[6]　王元明.非线性偏微分方程[M ].南京:东南大学出版社,1991,266～268.244内蒙古工业大学学报2005年T WO M ET HO DS FOR PRO VING T HE EXIST ENCEA N D U N IQ U ENESS O F N ON L IN EA R SING U LA R PROBLEM SIN ON E DIM ENSIO NZHOU Feng -ling 1,YU Xiao -ping 2,LIU Xue -y ing1(1.S chool of Science ,Inner M ong olia Univ ersity of T echnology ,H ohhot ,010051,P RC ;2.Tumote M iddle School ,H ohhot 010051,PR C ) Abstract :A class of no nlinear sing ular problems is discussed in this paper.T he existence and u-niqueness o f solutions for these pr oblem s ar e pro ved by tw o m ethods and the error estimates are giv-en .Keywords :w eighted Sobolev space;str ong ly m onotonic property and half co ntinuity;Banach ′sfix ed-po int theo rems 245第4期周凤玲等　一维非线性奇异问题有限元解的两种存在唯一性证明。