### 物理代写|电动力学代写electromagnetism代考|PHYS3040

statistics-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|电动力学代写electromagnetism代考|Complete Integration

Unlike algebra or calculus for instance, historians of the theory of probability often claim that, while it has a pre-history in gambling practice, this subject is a relative newcomer in mathematical terms.

Ideas of random variability and probability were put on a firmer mathematical basis in the course of the nineteenth century, and the modern form of the theory was well established by the mid-twentieth century.

An elementary link between statistics and probability is demonstrated in Sections $2.1$ and $2.2$ above, and the Riemann sum calculations of Example 4 indicate the central role of mathematical integration in analysis of random variation.

Twentieth century developments in probability and random variation are closely linked to developments in the theory of measure and integration culminating in Lebesgue’s theory of the integral [100]. A.N. Kolmogorov [93] made this the foundation of probability theory by identifying-

• the probability of an event as the measure of a set,
• a random variable as a measurable function, and
• the expected value of a random variable as the integral of a measurable function with respect to a probability measure.

One of the standard ways of defining the Lebesgue integral of a $\mu$-measurable function $f(\omega)(\omega \in \Omega)$ is to form finite sums
$$s_{j}=\sum_{r=1}^{n} \phi_{j}^{(r)}(\omega) \mu\left(A_{j}^{(r)}\right)$$
of simple functions $\phi_{j}(\omega)$ which converge to $f$ as $j \rightarrow \infty$, and then define the Lebesgue integral of $f$ on $\Omega$ by
$$\int_{\Omega} f(\omega) d \mu=\lim {j \rightarrow \infty} s{j} .$$

The dominated convergence theorem emerges from this: Suppose Lebesgue integrable functions $f_{j}(\omega)$ converge almost everywhere to $f(\omega)$, with $\left|f_{j}(\omega)\right| \leq g(\omega)$ almost everywhere, $g$ also being Lebesgue integrable. Then $f$ is Lebesgue int egrable, and $\int_{\Omega} f_{j}$ converges to $\int_{\Omega} f$ as $j \rightarrow \infty$.

Depending on the measurable integrand $f$, the measurable sets $A_{j}^{(r)}$ in Definition $3.1$ can be intervals, open sets, closed sets, isolated points, and/or various countable combinations of these and other even more complicated sets. One could say that the means used to access the integral of $f$ are themselves somewhat arcane and inaccessible. The “cure” (finding appropriate measurable sets) could be worse than the “disease” (finding the integral). ${ }^{1}$

The entrance to measure theory and Lebesgue integration is guarded by such fearsome sets and functions as the Cantor set, and the Devil’s Staircase or Cantor function [102]. But while it is unwise to enter the house of Lebesgue without keeping an eye out for monsters ${ }^{2}$, in the simple examples of essentially finite domains of preceding chapters we managed to negotiate our way fairly painlessly through the relevant measurable sets/functions. (Would we be so lucky if the domains were infinite, or the functions a bit more complicated?)
These monsters will never completely go away. But perhaps it would be better to not have to wrestle with them as a pre-condition of gaining entry to the house. It would be nice if the monsters were kept locked up in the cellar, not on guard at the front door. Is there any other way to deal with probability which provides full mathematical power and rigour? Is there another house, one that is more easily accessible, and closer to the “naive” or realistic view of random variability, as outlined in Chapter 2 above, and in [MTRV] pages $15-17 ?$

## 物理代写|电动力学代写electromagnetism代考|Burkill-complete Stochastic Integral

In contrast, the Burkill integral ([13], [14], [68]) involves integrator functions $h(I)$ which are not additive it is not required that $h(I)=h\left(I^{\prime}\right)+h\left(I^{\prime \prime}\right)$. (Of course, it is not forbidden either!)

It turns out that a version of the Burkill integral is very useful in a reformulated theory of stochastic integration, and in the Feynman integral theory of quantum mechanics. In [MTRV], in addition to dependence on cells $I$, an extended Burkill integrand $h(s, I)$ is allowed to depend also on tag points $s$ of cells $I$; and from this is developed a Burkill-complete form of integration. (A Burkill-complete integrand $h$ is not additive in respect of its dependence on cells $I$ – if it is additive it receives a different designation.)

Definition 6 below fits into the -complete structure of definitions. It deals with integrands $h(s, I)$ which are functions of tagged intervals $(\bar{s}, I)$ (or associated point-interval pairs $(\bar{s}, I))$; for instance, with $\left.I=] s^{\prime}, s^{\prime \prime}\right], \bar{s}=s^{\prime}$ or $s^{\prime \prime}$,
$$h(s, I)=\sqrt{\bar{s}\left(s^{\prime \prime}-s^{\prime}\right)}$$
and a partition $\mathcal{P}=\left{0=s_{0}, s_{1}, \ldots, s_{n-1}, s_{n}=1\right}$ of $\left.] 0,1\right]$ is a finite sample of points of the domain. Then $h(s, I)=\sqrt{\bar{s}{j}\left(s{j}-s_{j-1}\right)}$ with $\bar{s}{j}=s{j}$ or $s_{j-1}$; and a Riemann sum $(\mathcal{P}) \sum h(s, I)$ is a functional of samples of points:
$$(\mathcal{P}) \sum h(x, I)=\sum_{j=1}^{n} \sqrt{\bar{s}{j}\left(s{j}-s_{j-1}\right)}$$
This formulation changes the perspective of -complete integration from pointcell pairs to finite samples of points. Nevertheless, as in chapter 4 of [MTRV], the underlying structures can be readily conveyed in terms of relationships between cells or intervals $I$ of the domain.

In effect, adjacent pairs of points $\left(s_{j}, s_{j-1}\right)$ from the finite sample $\mathcal{P}$ must satisfy conditions corresponding to Axioms DS1 to DS8 in chapter 4 (pages 111-113 of [MTRV]). Of course, in simple domains such as $] 0,1]$ it is natural to visualize pairs of points $\left(s_{j}, s_{j-1}\right)$ as intervals $I_{j}$. But in the more complicated and structured domains used in quantum field theory (Chapters 8 and 9 below), the alternative “samples of points” perspective may be helpful.

Lebesgue integration uses functions $\mu(A)$ of measurable subsets of a domain. In contrast, -complete integration uses functions $\mu(I)$ of subintervals of the domain. The latter can be replaced by $\mu\left(s^{\prime}, s^{\prime \prime}\right)$ (where $\left.\left.I=\right] s^{\prime}, s^{\prime \prime}\right]$ ). But measurable sets $A$ can consist of infinitely many intervals and discrete points, ruling out the “finite sample of points” approach.

## 物理代写|电动力学代写electromagnetism代考|The Henstock Integral

The origins of the ideas in chapter 4 of [MTRV] are as follows. Starting with his $1948 \mathrm{PhD}$ thesis, Ralph Henstock (1923-2007) worked in non-absolute integration, including the Riemann-complete or gauge integral which, independently, Jaroslav Kurzweil also discovered in the 1950 s. As a Cambridge undergraduate (1941-1943) Henstock took a course of lectures, given by J.C. Burkill, on the integration of non-additive interval functions. Later, under the supervision of Paul Dienes in Birkbeck College, London, he undertook research into the ideas of Burkill (interval function integrands) and of Dienes (Stieltjes integrands); and he presented this thesis in December $1948 .$

In terms of overall approach and methods of proof, the thesis contains the germ of Henstock’s later work as summarized in chapter 4 of [MTRV]. For example, a notable innovation is a set of axioms for constructing any particular system of integration. This approach highlights the features held in common by various systems, so that a particular property or theorem can, by a single, common proof, be shown to hold for various kinds of integration. These ideas are the basis of the theory in chapter 4 of [MTRV].

Within this approach, Henstock’s thesis places particular emphasis on various alternative ways of selecting Riemann sums, as constituting the primary distinguishing feature of different systems of integration. This was central to his subsequent work and achievement. Accordingly, the theory in chapter 4 of [MTRV] is designated there as the Henstock integral, from which almost all systems of integration can be deduced.

Robert Bartle’s book ([5], page 15) has a discussion of titles for this kind of

integral-variously called Kurzweil-Henstock, gauge, or generalized Riemann. Bartle suggests that it could equally be called “the Denjoy-Perron-KurzweilHenstock integral”. Evading this litany, Bartle settles for “generalized Riemann”, or simply “the integral”.

The first worked-out version of this kind of integration was in Henstock’s Theory of Integration [70], published in 1962 and re-published in 1963 , in which the integral was designated “Riemann-complete”. In support of the “-complete” appendage, Henstock’s presentation has theorems which justify the integration of limits of integrable functions, differentiation under the integral sign, Fubini’s theorem, along with a theory of variation corresponding to measure theory.

## 物理代写|电动力学代写electromagnetism代考|Complete Integration

• 事件的概率作为集合的度量，
• 作为可测量函数的随机变量，以及
• 随机变量的期望值，作为可测量函数相对于概率测度的积分。

sj=∑r=1nφj(r)(ω)μ(一个j(r))

∫ΩF(ω)dμ=林j→∞sj.

## 物理代写|电动力学代写electromagnetism代考|Burkill-complete Stochastic Integral

H(s,我)=s¯(s′′−s′)

(磷)∑H(X,我)=∑j=1ns¯j(sj−sj−1)

Lebesgue 积分使用函数μ(一个)域的可测量子集。相比之下，-complete 积分使用函数μ(我)域的子区间。后者可以替换为μ(s′,s′′)（在哪里我=]s′,s′′]）。但可测集一个可以由无限多的间隔和离散点组成，排除了“点的有限样本”方法。

## 物理代写|电动力学代写electromagnetism代考|The Henstock Integral

[MTRV]第4章的思想起源如下。从他的开始1948磷HD在论文中，Ralph Henstock (1923-2007) 研究了非绝对积分，包括 Jaroslav Kurzweil 在 1950 年代独立发现的黎曼完全积分或规范积分。作为剑桥大学本科生 (1941-1943)，Henstock 参加了 JC Burkill 教授的关于非加性区间函数积分的课程。后来，在伦敦伯克贝克学院的 Paul Dienes 的指导下，他对 Burkill（区间函数被积函数）和 Dienes（Stieltjes 被积函数）的思想进行了研究；他在 12 月提交了这篇论文1948.

Robert Bartle 的书（[5]，第 15 页）讨论了这种类型的标题

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。