### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|AEM6061

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• 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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Rays and Ray Segments

A ray is defined here as the continuous sequence of straight-line paths connecting a point on one surface, from which an energy bundle is emitted, to a point on a second surface – or perhaps even on the same surface – where it is ultimately absorbed. One or several reflections from intervening surfaces may occur between emission and absorption of the energy bundle. The path followed by the energy bundle between reflections is referred to as a ray segment. Two situations are normally considered: either (i) the power of the emitted energy bundle does not change as it is reflected from one surface to the next until it reaches the surface where all its power is ultimately absorbed; or (ii) a fraction of the emitted power is left behind with each reflection until the remaining power is deemed to have dropped below a threshold value, at which point the ray trace is terminated. Both approaches have their adherents and are in common use, and both are developed in detail in this book.

The enclosure is an essential concept in all approaches to radiation heat transfer analysis. We define the enclosure as an ensemble of surfaces, both real and imaginary, arranged in such a manner that a ray emitted into the interior of the enclosure cannot escape. Energy is conserved within the enclosure under this definition. If a ray does leave the enclosure through an opening, represented by an imaginary surface, the energy it carries is deducted from the overall energy balance.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Mathematical Preliminaries

Consider two points, $P_{0}$ and $P_{1}$, in three-dimensional space, as illustrated in Figure 1.1. Let the Cartesian coordinates of points $P_{0}$ and $P_{1}$ be $\left(x_{0}, y_{0}, z_{0}\right)$ and $\left(x_{1}, y_{1}, z_{1}\right)$, respectively. Then the vector directed from $P_{0}$ to $P_{1}$ is
$$\boldsymbol{V}=\left(x_{1}-x_{0}\right) \boldsymbol{i}+\left(y_{1}-y_{0}\right) \boldsymbol{j}+\left(z_{1}-z_{0}\right) \boldsymbol{k},$$

and its magnitude is
$$t \equiv \sqrt{|\boldsymbol{V} \cdot \boldsymbol{V}|}=\sqrt{\left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}} .$$
In Eq. (1.1) $\boldsymbol{i}, \boldsymbol{j}$, and $\boldsymbol{k}$ are the unit vectors directed along the $x$-, $y$-, and $z$-axes, respectively. Note that the distance $t$ from $P_{0}$ to $P_{1}$ must always be real and positive.
The unit vector in the direction of $V$ is
$$v \equiv V / t=L i+M j+N k,$$
where $L, M$, and $N$ are the direction cosines illustrated in Figure 1.1. The direction cosines are defined
$$L \equiv v \cdot i=\cos \alpha, M \equiv v \cdot j=\cos \beta, \text { and } N \equiv v \cdot k=\cos \gamma,$$
where $\alpha, \beta$, and $\gamma$ are the angles between the unit vector $v$ and the $x$-, $y$-, and $z$-axes, respectively. Equations (1.1) and (1.3) can be combined to define the equations for the line segment connecting point $P_{0}$ to point $P_{1}$
$$\left(x_{1}-x_{0}\right) / L=\left(y_{1}-y_{0}\right) / M=\left(z_{1}-z_{0}\right) / N=t .$$
The three equations embodied in Eq. (1.5) are arguably the most important relationships in geometrical optics, because they form the basis for navigation of rays within an enclosure.
The general equation for a surface in Cartesian coordinates is
$$S(x, y, z)=0 .$$

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Ideal Models for Emission, Reflection, and Absorption of Rays

To this point we have treated the ray as a strictly mathematical concept without considering its physical nature. However, as we move on to the phenomena of emission, absorption, reflection, scattering, and refraction,

which occur when a ray intersects a surface, it will be convenient to introduce certain models borrowed from geometrical optics. In later chapters, we explore the principles of physical optics underlying these models. However, for the present it is convenient to exploit their relative simplicity as a tool for developing ray-tracing skills. This is not to say that the models introduced in this section are of pedagogical interest only; indeed, they have been the basis for traditional radiation heat transfer practice for the past century, during which time they have consistently yielded results whose accuracy is at least as good as that afforded by contemporary conduction and convection heat transfer epistemology.
We have been using the generally well understood term “surface” without formal definition. It is now appropriate to formally define a surface as the interface separating two regions of space having different optical properties. In fact, true surfaces do not exist, although approximations of surface behavior can be approached to an arbitrarily high degree of precision.

The optical behavior of a material substance is characterized by its index of refraction and its extinction coefficient. As a ray encounters the interface between two materials having different values of these optical properties, a portion of its power is redirected away from the interface. This portion of the incident power is said to be “reflected.” Of the power that crosses into the second medium, a portion is said to be “absorbed” while the remainder is said to be either “scattered” or “refracted.” The scattered and refracted power continues to propagate through the second medium while the absorbed power is locally converted into sensible heat. The two most prevalent models for describing reflection at a surface are the specular reflection model and the diffuse reflection model. These two models are important because they represent opposite extremes, both of which are often the desired behavior in engineering applications.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Mathematical Preliminaries

(X1−X0)/大号=(是1−是0)/米=(和1−和0)/ñ=吨.

## 有限元方法代写

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## 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。