## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|STAT4063

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|A Highly Absorptive Surface Whose Reflectivity is Strongly Specular

Aeroglaze ${ }^{\circledR} \mathrm{Z} 302[2]$ is a polyurethane-based paint whose absorptivity typically exceeds $90 \%$ in the visible part of the spectrum, depending on the coating thickness. It is unique in that the reflected component of radiation is mostly specular. Its special properties make it the coating of choice for many aerospace and optical applications where a surface must be an exceptionally efficient absorber but where diffuse reflection is undesirable. A typical application is the interior surface of a blackbody cavity used as a calibration target. In this case the cavity geometry would be such that several specular reflections would occur before an incident ray could escape, and any diffuse component of reflectivity present would diminish the effectiveness of the design because it would allow some power to escape the cavity with each reflection. Such diffuse “leaks” can

be significant when the effective emissivity of the cavity must be unity to better than three nines.

Prokhorov and Prokhorova [3] describe a three-component semiempirical model based on their own measurements of the BRDF of Z302 at a wavelength of $\lambda=10.6 \mu \mathrm{m}$. We have used the same data, represented by the symbols in Figure $4.4$, to derive a purely empirical four-component model, represented by the curves in the figure. Both the Prokhorov and Prokhorova model and our model are in excellent agreement with the measurements. Our four-component model [4] has the form
$$B R D F=\rho_{1}^{\prime \prime}+\rho_{2}^{\prime \prime}+\rho_{3}^{\prime \prime}+\rho_{4}^{\prime \prime},$$
where
$$\rho_{n}^{\prime \prime}=A_{n} \frac{1}{\sqrt{2 \pi} \sigma_{n}} e^{-\left(\vartheta_{v}-\vartheta_{i}\right)^{2} / 2 \sigma_{n}^{2}}+O_{n}, \quad n=1,2,3,4$$
In Eq. (4.11), $\vartheta_{i}$ and $\vartheta_{v}$ are the incidence and viewing angles shown in the inset in Figure $4.4$, and $A_{n}, \sigma_{n}$, and $O_{n}$ are empirical curve-fitling parameters. The form of Eq. (4.11) is recognizable as the normal probability distribution function multiplied by a scaling factor $A_{n}$ and shifted

in amplitude by an offset $O_{n}$. In practice the additive offsets for the four values of $n$ are gathered into a single constant. The fit illustrated in Figure $4.4$ was obtained by defining the standard deviation
$$\sigma_{n}=\frac{1}{\sqrt{2 \pi} b_{n} \vartheta_{i}}$$
and the multiplicative constant
$$A_{n}=\frac{B_{n}}{b_{n} \vartheta_{i}},$$

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|A Highly Reflective Surface Whose Reflectivity is Strongly Diffuse

We next consider a practical application whose accurate simulation requires a bidirectional spectral reflection model. The integrating sphere [5] often plays a central role in radiometric instrument calibration, surface optical behavior measurement, and radiant source characterization $[6,7]$. The purpose of an integrating sphere is to convert a collimated beam of monochromatic light, such as might be provided by a laser source, into a larger, weaker source of diffuse light at the same wavelength. The essential property of the integrating sphere is its ability to produce a Lambertian source of monochromatic radiation due to multiple scattering from its interior walls. This requirement will be satisfied exactly for a completely enclosed spherical cavity, even when the wall coating is not itself a perfectly diffuse reflector. However, a practical integrating sphere must be fitted with ports that allow the illuminating beam to enter and the instrument under calibration to

view the interior wall. The ports inevitably allow some of the entering radiation to escape before being completely diffused by reflections, thereby compromising the desired effect. In practice the ports are made as small as possible compared to the diameter of the sphere, and the interior walls are treated with a highly reflective, highly diffuse coating.
The author and his coworkers [8] have investigated the departure from ideal behavior of a practical integrating sphere, with emphasis on the influence of directionality. The results of that investigation are offered here as an example of an application in which the diffuse gray assumption may be inadequate. We consider an application in which a relatively small integrating sphere is to be used on-orbit to calibrate a radiometer against a reference radiometer by having both instruments observe the same sector of the interior wall through two separate ports. Because the calibrations will be repeated over a period of up to several years, it is important to know how the calibration factor might be expected to vary with aging of the wall coating. It is further interesting to know how the degree of directionality of reflections from the walls might influence its performance.

We use the MCRT method to simulate illumination of the interior by a quasi-monochromatic light source at a wavelength, $0.9 \mu \mathrm{m}$, for which the $B R F$ model developed below may be considered valid. For purposes of the current investigation the diameter of the port through which the narrow light beam is admitted may be considered sufficiently small compared to the diameters of the two viewing ports to neglect its presence in the ray trace. The hypothetical experimental arrangement is illustrated in Figure 4.12.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|The Band-Averaged Spectral Radiation

The case studies presented in Sections $4.3$ and $4.4$ exemplify direct application of the MCRT method without recourse to radiation distribution factors, which were not needed to accomplish the stated goals. Furthermore, they involve situations for which the wavelength interval of interest is sufficiently narrow that the surface models used are, to an acceptable approximation, independent of wavelength. However, in some cases radiation distribution factors are required, as explained in Chapter 3. The radiation distribution factor introduced and used in Chapter 3 for gray surfaces had two subscripts, $i$ and $j$, the indices of the emitting and absorbing surface. For the case of spectral radiation it is necessary to add a third subscript, $k$, representing the wavelength interval $\Delta \lambda_{k}$ in which the distribution factor applies. We define the band-averaged spectral radiation distribution factor $D_{i j k}$ as the fraction of power emitted in wavelength

interval $\Delta \lambda_{k}$ by surface element $i$ that is absorbed by surface element $j$, both directly and due to all possible reflections within the enclosure.
Estimation of the band-averaged spectral radiation distribution factor matrix assumes the availability of a dense data set ultimately based on extensive laboratory measurements. Imagine a bookshelf in a virtual thermophysical properties library bearing the label “Bidirectional Spectral Reflectivity.” Upon perusal of this bookshelf we might find books with titles “Z302,” “Spectralon,” “Gold Black,” and other optical coatings. When we take down one of these books and open to its table of contents; we find chapter titles such as “Wavelength Interval Between $0.01$ and $0.10 \mu \mathrm{m}$,” and “Wavelength Interval Between $0.10$ and $1.00 \mu \mathrm{m}$,” and so forth. Then when we flip through Chapter 1 we notice page headings “Angle of Incidence $=5^{\circ}$,” “Angle of Incidence $=10^{\circ}$,” and, on the last page, “Angle of Incidence $=85^{\circ}$.” Finally, when we scan one of these pages from top to bottom we find on the first line “Angle of Reflectance $=5^{\circ}$, $B R D F=44.21$,” and on the second line “Angle of Reflectance $=10^{\circ}$, $B R D F 38.45$,” and so forth. Upon plotting the data found in one of these books, we recognize that the spacing between successive wavelengths, angles of incidence, and angles of reflectance is sufficiently small to allow accurate linear interpolation between tabulated values.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|A Highly Absorptive Surface Whose Reflectivity is Strongly Specular

Prokhorov 和 Prokhorova [3] 描述了一个三分量半经验模型，该模型基于他们自己对 Z302 在波长为λ=10.6μ米. 我们使用了相同的数据，由图中的符号表示4.4，推导出一个纯经验的四分量模型，由图中的曲线表示。Prokhorov 和 Prokhorova 模型以及我们的模型都与测量结果非常吻合。我们的四分量模型 [4] 具有以下形式

ρn′′=一个n12圆周率σn和−(ϑ在−ϑ一世)2/2σn2+○n,n=1,2,3,4

σn=12圆周率bnϑ一世

## 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。

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|MAST90083

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Use of Radiation Distribution Factors When Some Surface Net Heat Fluxes Are Specified

We now address the situation where surface elements $1,2, \ldots, N$ have specified net heat fluxes and surfaces $N+1, N+2, \ldots, n$ have specified temperatures. To begin, let us consider the special case where $N=1$; that is, where only the first of $n$ surface elements has a specified net heat flux, with the remaining surfaces having specified temperatures. In this case Eq. (3.35) may be written
$$q_{1}=\varepsilon_{1}\left[\left(1-D_{11}\right) \sigma T_{1}^{4}-\sum_{j=2}^{n} \sigma T_{j}^{4} D_{i j}\right]$$
Equation (3.36) can be solved explicitly for the unknown surface temperature $T_{1}$ in terms of the known surface net heat flux $q_{1}$ and the known surface temperatures; that is,
$$\sigma T_{1}^{4}=\frac{1}{1-D_{11}}\left(\frac{q_{1}}{\varepsilon_{1}}+\sum_{j=2}^{n} \sigma T_{j}^{4} D_{i j}\right)$$
In the more general case where several surface elements have specified net heat fluxes we can rearrange Eq. (3.35) to obtain
$$q_{i}+\varepsilon_{i} \sum_{j=N+1}^{n} \sigma T_{j}^{4} D_{i j}=\varepsilon_{i} \sum_{j=1}^{N} \sigma T_{j}^{4}\left(\delta_{i j}-D_{i j}\right), \quad 1 \leq i \leq N_{\star}$$

Equation (3.38) represents $N$ equations in the $N$ unknown surface temperatures in terms of the $N$ known surface net heat fluxes and the $n-N$ known surface temperatures. It can be rewritten symbolically as
$$\Theta_{i}=\Psi_{i j} \Omega_{j},$$
where
$$\Theta_{i}=q_{i}+\varepsilon_{i} \sum_{j=N+1}^{n} \sigma T_{j}^{4} D_{i j}, \quad 1 \leq i \leq N,$$
is a known vector,
$$\Psi_{i j}=\varepsilon_{i}\left(\delta_{i j}-D_{i j}\right), \quad 1 \leq i \leq N, \quad 1 \leq j \leq N,$$
and
$$\Omega_{j}=\sigma T_{j}^{4}, \quad 1 \leq j \leq N,$$
is an unknown vector whose elements are sought. We obtain the unknown surface temperatures by inverting the matrix defined by Eq. (3.41) and then using it to operate on the vector defined by Eq. (3.40); that is,
$$\Omega_{i}=\left[\Psi_{i j}\right]^{-1} \Theta_{j}, \quad 1 \leq i \leq N .$$
The unknown surface net heat fluxes are then computed using Eq. (3.35) applied over the range $N+1 \leq i \leq n$.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Bidirectional Spectral Surfaces

Experience confirms that reflection from a surface is generally neither diffuse nor specular. Rather, at a given wavelength the distribution of reflected energy depends on the mechanical and chemical preparation

of the surface and on the direction of incidence. Diffuse and specular reflections represent the two extremes of bidirectional reflectivity. Both extremes may be approached but rarely achieved in practice. Various theories have been proposed for predicting bidirectional spectral reflection and directional spectral emission and absorption for generic surfaces. The interested reader is referred to Chapter 4 in Ref. [1], where this topic is pursued in more detail. However, metrology remains the only sure path to surface models that accurately capture the directional spectral behavior of real surfaces.

A simple two-component model for directional reflectivity was introduced in Chapter 1 (Figure 1.9), where it is suggested that a directional reflection pattern can be somewhat approximated as a suitably weighted combination of spectral and diffuse reflection. Different versions of this approximation would have to be applied for each wavelength of interest, using different weight factors for each wavelength. Any success this approach might have would be due in large measure to the fact that the distribution of radiant energy within an enclosure is governed by integral equations rather than by differential equations. Integration at least partially “averages out” positive and negative excursions from reality, as illustrated in Figure 4.1. Inspection of the figure reveals that, even though

$f_{1}(x)$ is a more detailed and presumably more accurate description of local behavior than $f_{2}(x)$, it may nonetheless be true, to an acceptable approximation, that
$$\int_{0}^{1} f_{1}(x) d x \cong \int_{0}^{1} f_{2}(x) d x$$

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Principles Underlying a Practical Bidirectional Reflection Model

Useful bidirectional reflection models are generally based on measurements, although they are frequently informed by theory. As we learned in Sections $2.12$ and $2.13$, the optical behaviors of electrically non-conducting (dielectric) and electrically conducting (metal) surfaces are fundamentally different. In general, metal surfaces are strong specular reflectors while dielectric surfaces tend to be weak diffuse reflectors. In both cases the directional distribution of reflected radiation is known to be strongly influenced by the topography, chemical state, and degree of contamination of the surface. With the exception of certain optical components (such as mirrors, lenses, and filters), it is unlikely that a bidirectional spectral reflectivity model based entirely on theory would accurately represent the optical behavior of a surface of practical engineering interest. Therefore, in cases where high accuracy is required, a successful surface optical model must be at least semiempirical if not based entirely on measurements of the optical behavior of the surface to be modeled.

In this chapter we first demonstrate the application of semiempirical approaches for two surface coatings engineered to exhibit specific $-$ and somewhat unique – optical behaviors. In the first example we consider a highly absorptive commercial coating whose small component of reflectivity is highly directional to the point of being almost specular, and in the second example we consider another commercial coating that is highly reflective but whose reflectivity is nearly diffuse. Both of these coatings are widely used in optical applications requiring an unusual combination of both metallic and dielectric behaviors. We then follow up by presenting a completely general approach suitable for applications where a full set of experimental data is available.

We begin by recalling the bidirectional spectral reflectivity from Chapter 2,\begin{aligned} \rho_{\lambda}^{\prime \prime} &=\rho\left(\lambda, \vartheta_{i}, \varphi_{i}, \vartheta_{r}, \varphi_{r}\right) \equiv \frac{d i_{\lambda, r}\left(\lambda, \vartheta_{i}, \varphi_{i}, \vartheta_{r}, \varphi_{r}\right)}{i_{\lambda, i}\left(\lambda, \vartheta_{i}, \varphi_{i}\right) \cos \vartheta_{i} d \Omega_{i}} \ & \equiv B R D F\left(\lambda, \vartheta_{i}, \varphi_{i}, \vartheta_{r}, \varphi_{r}\right) \end{aligned}

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Use of Radiation Distribution Factors When Some Surface Net Heat Fluxes Are Specified

q1=e1[(1−D11)σ吨14−∑j=2nσ吨j4D一世j]

σ吨14=11−D11(q1e1+∑j=2nσ吨j4D一世j)

q一世+e一世∑j=ñ+1nσ吨j4D一世j=e一世∑j=1ñσ吨j4(d一世j−D一世j),1≤一世≤ñ⋆

θ一世=Ψ一世jΩj,

θ一世=q一世+e一世∑j=ñ+1nσ吨j4D一世j,1≤一世≤ñ,

Ψ一世j=e一世(d一世j−D一世j),1≤一世≤ñ,1≤j≤ñ,

Ωj=σ吨j4,1≤j≤ñ,

Ω一世=[Ψ一世j]−1θj,1≤一世≤ñ.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Bidirectional Spectral Surfaces

F1(X)是对局部行为的更详细和可能更准确的描述F2(X)，但在可接受的近似值上，它可能是正确的，即

∫01F1(X)dX≅∫01F2(X)dX

## 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。

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|STAT5003

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Binning of Rays on a Surface Element; Illustrative Example

Meshing and indexing are treated in Section 1.6. A closely related idea is binning, in which indexing allows us to keep track of the spatial distribution of absorbed energy bundles. In the case of the instrument in Figure 3.2, it might be desirable to know how the power incident to the entrance aperture is distributed across the detector face. In this situation, after executing Step 11, control is transferred to a binning step not shown in Figure 3.1. The detector surface, subdivided into a number of bins, is illustrated in Figure 3.5. In a given application the actual number of bins and their spatial distribution would be dictated by the shape of the surface and the desired spatial resolution.

No attempt has been made in creating Figure $3.5$ to suggest a particular scheme for determining the spacing between consecutive radial rings. However, two schemes come to mind: (i) either the ring boundaries could be equally spaced or (ii) they could be spaced to achieve equal-area rings. The former scheme is, of course, easy to achieve but is probably

not as useful as the latter. A plot of power absorbed as a function of radial position would be easier to interpret if the rings were divided into equal areas. This may be achieved using the formula
$$r_{m}=(d / 2) \sqrt{m / M}$$
where $d$ is the diameter of the detector, $M$ is the total number of radial bands to be created, and $r_{m}$ is the radial position of the outer radius of the $m$ th band. It is both natural and convenient to base the circumferential divisions on equal angles,
$$\varphi_{n}=2 \pi n / N,$$
where $N$ is the total number of angular sectors, $n$ is the $n$th such sector, and $\varphi_{n}$ is the angle corresponding to the upper angular limit of the sector.

We arrive at the binning step with knowledge of $x_{1}, y_{1}, z_{1}$, and $P_{1}$, where $P_{1}$ is the power carried by the current ray. The Matlab function for determining the values of $m$ and $n$ and updating the power accumulated in bin $(m, n)$ appears in Figure 3.6. The “floor” operator on lines 8 and 12 of Figure $3.6$ returns the least integer in the argument.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Thermal and Optical Analysis

Figure $3.7$ is a detailed data sheet for the radiometric instrument concept illustrated in Figure 3.2. Our task is to provide a preliminary evaluation of its optical and thermal performance. The approach is to model the

optical behavior of the instrument under the assumption of specularly reflecting mirrors and diffuse-specular non-mirror surfaces in the visible wavelength range $(0.4-0.7 \mu \mathrm{m})$, and to model the thermal behavior of the instrument in the infrared. Optical characterization will require assessment of baffle and telescope performance, while thermal characterization will require estimation of the distribution factor matrix. The listing for the Matlab code developed to accomplish this task is available at the companion website listed on p. xix. Study of the code listing reveals that the component geometries and dimensions have been entered “by hand.” However, it is possible to import system geometries directly into Matlab from the CAD environment where they were created.

The first step is to determine the distance from the entrance aperture to the focal plane. This task is accomplished by performing a numerical experiment in which an on-axis collimated beam consisting of 100000 rays is introduced through the instrument aperture. The detector is then displaced back and forth along the optical axis until the waist diameter of the twice-reflected beam is minimized. The result of this process, obtained by iteration, is shown in Figure 3.8, in which the rays have converged to a circle of minimum diameter at $z=58.5823 \mathrm{~mm}$. The notation “TBD” in Figure $3.7$ can now be replaced with this value, and the preliminary design of the instrument is complete.

In Figures $3.9$ and 3.10, the intersections of rays with the primary and secondary mirrors are shown as individual dots. These and similar plots for the other surfaces of the instrument are invaluable for confirming the basic design assumptions, such as symmetry in this example.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Use of Radiation Distribution Factors for the Case of Specified Surface Temperatures

Consider the special case where all of the surface elements making up the generic enclosure shown schematically in Figure $3.17$ have specified temperatures, and where all of the surface net heat fluxes are unknown. Note that the view of some surface elements may be either partially or fully blocked when viewed from other surface elements. This does not present a problem in the following analysis because partial or full blockage of direct radiation has already been accounted for in the definition and estimation of the distribution factors, as has the effect of surface curvature.

Consistent with the definition of the diffuse-specular graybody radiation distribution factor, we can express the radiation heat flux $\left(\mathrm{W} \mathrm{m}^{-2}\right.$ )

absorbed by surface element $i$ due to emission from all of the surface elements making up the enclosure
$$q_{i, a}=\frac{Q_{i, a}}{A_{i}}=\frac{1}{A_{i}} \sum_{j=1}^{n} \varepsilon_{j} A_{j} \sigma T_{j}^{4} D_{j i}, \quad 1 \leq i \leq n,$$
which, with the introduction of reciprocity, Eq. (3.5), becomes
$$q_{i, a}=\varepsilon_{i} \sum_{j=1}^{n} \sigma T_{j}^{4} D_{i j} . \quad 1 \leq i \leq n .$$
The radiation heat flux emitted from surface element $i$ is
$$q_{i, e}=\frac{Q_{i, e}}{A_{i}}=\frac{\varepsilon_{i} A_{i} \sigma T_{i}^{4}}{A_{i}}=\varepsilon_{i} \sigma T_{i}^{4}, \quad 1 \leq i \leq n$$
Then the net heat flux from surface element $i$ is
$$q_{i}=q_{i, e}-q_{i, a}=\varepsilon_{i} \sigma T_{i}^{4}-\varepsilon_{i} \sum_{j=1}^{n} \sigma T_{j}^{4} D_{i j}, \quad 1 \leq i \leq n$$
or, with the introduction of the Kronecker delta
$$\delta_{i j} \equiv \begin{cases}1, & i=j \ 0, & i \neq j\end{cases}$$
Eq. (3.33) can be written
$$q_{i}=\varepsilon_{i} \sum_{j=1}^{n} \sigma T_{j}^{4}\left(\delta_{i j}-D_{i j}\right) \quad 1 \leq i \leq n$$

r米=(d/2)米/米

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Use of Radiation Distribution Factors for the Case of Specified Surface Temperatures

q一世,一个=问一世,一个一个一世=1一个一世∑j=1nej一个jσ吨j4Dj一世,1≤一世≤n,

q一世,一个=e一世∑j=1nσ吨j4D一世j.1≤一世≤n.

q一世,和=问一世,和一个一世=e一世一个一世σ吨一世4一个一世=e一世σ吨一世4,1≤一世≤n

q一世=q一世,和−q一世,一个=e一世σ吨一世4−e一世∑j=1nσ吨j4D一世j,1≤一世≤n

d一世j≡{1,一世=j 0,一世≠j

q一世=e一世∑j=1nσ吨j4(d一世j−D一世j)1≤一世≤n

## 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。

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|MATH 483

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|The Monte Carlo Ray-Trace Method and the Radiation Distribution Factor

The Monte Carlo ray-trace (MCRT) method is a two-step process. The first step involves estimation of the radiation distribution factor matrix

$D_{i j}$, and the second step involves multiplication of $D_{i j}$ by a vector whose components are the source strengths of the surfaces making up an enclosure. Individual elements of the distribution factor matrix may be interpreted as the sensitivity of the power absorbed by surface $j$ to the power emitted by surface $i$; that is,
$$D_{i j} \equiv \partial Q_{i j} / \partial Q_{i},$$
where $Q_{i j}$ is the total power in watts emitted from surface $i$ that is absorbed on surface $j$, and $Q_{i}$ is the total power emitted from surface $i$. If $Q_{i}$, the total power emitted from surface $i$, is known and the distribution factor matrix $D_{i j}$ is available for any combination of two surfaces $i$ and $j$, then the heat absorbed by surface $j$ is
$$Q_{j}=\sum_{\mathrm{i}=1}^{\mathrm{n}} Q_{i j}, \quad 1 \leq j \leq n$$
where $n$ is the total number of surfaces and
$$Q_{i j}=Q_{i} D_{i j} .$$
Calculation of $Q_{i}$ for a given surface condition has already been treated in Chapter 2. The current chapter deals with calculation of the distribution factor $D_{i j}$ and its subsequent use in determining the distribution of thermal radiation among surfaces of an enclosure.

If we can somehow obtain the $D_{i j}$ matrix, we already have the answer to one of the most pressing questions in optical and thermal design: “How sensitive is the heat absorbed by a specific surface $j$ to the heat emitted from a specific surface $i$ ?’ Consideration of Eqs. (3.2) and (3.3) reveals that calculation of the power distribution among the surfaces of an enclosure is a straightforward vector multiplication once the distribution factor matrix is known. Knowledge of the distribution factor matrix greatly facilitates thermal or optical design because it permits targeted analysis of heat transfer among a limited number of surfaces of particular interest.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Properties of the Total Radiation Distribution Factor

It can be demonstrated (see Problems 3.1-3.3) that, subject to the graybody assumption defined in Chapter 2 , the total radiation distribution factor has the following useful properties:

1. Conservation of energy:
$$\sum_{j=1}^{n} D_{i j}=1, \quad 1 \leq i \leq n$$
2. Reciprocity:
$$\varepsilon_{i} A_{i} D_{i j}=\varepsilon_{j} A_{j} D_{j i}, \quad 1 \leq i \leq n, \quad 1 \leq j \leq n$$
3. Combination of 1 and 2 :
$$\sum_{i=1}^{n} \varepsilon_{i} A_{i} D_{i j}=\varepsilon_{j} A_{j}, \quad 1 \leq j \leq n$$
In Eqs. (3.4)-(3.6), $n$ is the number of surface elements making up the enclosure, $\varepsilon$ is the emissivity, and $A$ is the surface area.

Equation (3.6), which is obtained by summing both sides of Eq. (3.5) over the index $i$ and then substituting Eq. (3.4) into the result, is useful for detecting and eliminating errors made during calculation of the distrihution factors for an enclosure. It can also he used to provide a statistically meaningful measure of the accuracy with which the distribution factor matrix for a given enclosure has been computed. The conservation of energy relationship, Eq. (3.4), and the reciprocity relationship, Eq. (3.5), are also useful for detecting errors or for finding unknown distribution factors from known distribution factors using distribution factor algebra. However, note that these relationships cannot be used both for error detection and for finding unknown distribution factors in the same enclosure.

Finally, we note that distribution factors can also be defined for radiation entering the enclosure through an opening $o$ with a specified directional distribution; e.g., collimated in a specific direction. In this case, the appropriate relation for defining the distribution of radiation on the surface elements making up the enclosure is
$$Q_{v j}=Q_{s} D_{v j,} \quad 1 \leq j \leq n,$$
where $Q_{o}$ is the power (W) entering the enclosure through opening $o$ and $D_{o j}$ is the fraction of this power absorbed by surface element $j$.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Estimation of the Distribution Factor Matrix Using

The Monte Carlo ray-trace method is a statistical approach in which the analytical solution of a problem is avoided in favor of a numerical

simulation whose outcome is expected to be the same as that of the analytical solution but which is easier to obtain. In practice, this often means that the numerical simulation is obtainable while the equivalent analytical solution is for all practical purposes unobtainable.

In the case of a thermal radiation problem, a given quantity of energy is uniformly divided into a large number $N_{i}$ of discrete energy bundles. Here, we feel no obligation to distinguish between energy and power, as in the steady state the former is the latter multiplied by an appropriate time interval. The $N_{i}$ energy bundles are followed from their emission by surface element $i$ (or from their entry into the enclosure through opening $o$ ), through a series of reflections from other surface elements, to their absorption on surface element $j$ of the enclosure. The optical models of the enclosure walls and the laws of probability are used to determine the number of energy bundles $N_{i j}$ absorbed by a given surface element $j$, where $j=i$ is a possibility.

A consequence of the definition of the radiation distribution factor is that its value approaches the ratio of $N_{i j}$ to $N_{i}$ as $N_{i}$ increases; that is,
$$D_{i j} \approx N_{i j} / N_{i} .$$
The accuracy with which $N_{i j} / N_{i}$ estimates $D_{i j}$ depends on the number of energy bundles traced from surface $i$. Of course, as in any model-based analysis, it also depends on the accuracy with which the enclosure geometry and the surface optical models are known. Furthermore, as shown in Chapter 7, the uncertainty associated with the estimate corresponding to the value of $N_{i}$ can be stated to within a specified confidence interval. Moreover, the ultimate accuracy of the solution to a radiation heat transfer problem using the MCRT method depends in a statistically meaningful way on the product of the number of surfaces $n$ making up the enclosure and the number of rays $N$ traced per surface. The ability of the MCRT method to attribute a confidence level to the uncertainty of the results obtained is a compelling argument for its use.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|The Monte Carlo Ray-Trace Method and the Radiation Distribution Factor

D一世j，第二步涉及乘法D一世j由一个向量组成，其分量是构成外壳的表面的源强度。分布因子矩阵的各个元素可以解释为表面吸收功率的灵敏度j到表面发出的功率一世; 那是，

D一世j≡∂问一世j/∂问一世,

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Properties of the Total Radiation Distribution Factor

1. 能量守恒：
∑j=1nD一世j=1,1≤一世≤n
2. 互惠：
e一世一个一世D一世j=ej一个jDj一世,1≤一世≤n,1≤j≤n
3. 1 和 2 的组合：
∑一世=1ne一世一个一世D一世j=ej一个j,1≤j≤n
在方程式中。(3.4)-(3.6),n是组成外壳的表面元素的数量，e是发射率，并且一个是表面积。

D一世j≈ñ一世j/ñ一世.

## 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。

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|MAT 359

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

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

A surface is formally defined in Section $1.4$ as the interface between two regions of space having different optical properties, where the optical properties in question are the refractive and absorptive indices $n$ and $k$. We distinguish between these two properties and the models used to characterize the interaction between thermal radiation and surfaces. We are now in a position to elaborate on the definition and use of the surface

interaction models for emission, reflection, and absorption introduced in Chapter $1 .$

We define the directional spectral emissivity $\varepsilon(\lambda, T, \vartheta, \varphi)$ as the ratio of the spectral intensity of emission from a real body in direction $(\vartheta, \varphi)$ to the spectral intensity of a blackbody at the same temperature,
$$\varepsilon(\lambda, T, \vartheta, \varphi) \equiv \frac{i_{\lambda, e}(\lambda, T, \vartheta, \varphi)}{i_{b \lambda}(\lambda, T)}$$
Note that the symbol for the directional spectral emissivity can also be written $\varepsilon_{\lambda}^{\prime}(T)$, where the prime $\left({ }^{\prime}\right)$ indicates directionality and the subscript $\lambda$ identifies the model as spectral.

The directional total emissivity of a surface is then related to the directional spectral emissivity according to
$$\varepsilon^{\prime}(T)=\varepsilon(T, \vartheta, \varphi)=\frac{\int_{\lambda=0}^{\omega} \varepsilon(\lambda, T, \vartheta, \varphi) i_{b \lambda}(\lambda, T) d \lambda}{\int_{\lambda=0}^{\infty} i_{b \lambda}(\lambda, T) d \lambda} .$$
We know that the denominator is $\sigma T^{4} / \pi$, so Eq. (2.31) can be rewritten
$$\varepsilon^{\prime}(T)=\varepsilon(T, \vartheta, \varphi)=\frac{\pi}{\sigma T^{4}} \int_{\lambda=0}^{\infty} \varepsilon(\lambda, T, \vartheta, \varphi) i_{b \lambda}(\lambda, T) d \lambda .$$
A surface is said to be gray in a given direction $\left(\vartheta_{1}, \varphi_{1}\right)$ if the directional spectral emissivity is independent of wavelength in that direction. Equation (2.32) then defines a gray equivalent directional emissivity for spectral surfaces. The spectral intensities of a blackbody, a graybody, and a hypothetical real surface, all at $6000 \mathrm{~K}$, are compared in Figure $2.11$.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Introduction to the Radiation Behavior of Surfaces

The primary surface conditions that influence the radiation behavior of a solid surfacé ăré its bulk êlectricall propertiês (èléctrical conductor or non-conductor), its topography (smooth, polished, sanded, sand blasted, turned, lapped, honed, ground, peened, etc.), its chemical condition (reduced, oxidized, anodized, galvanized, etc.), its degree of contamination (clean or dirty, dusty, dry or oily, etc.), and its surface grain structure (annealed, cold rolled, hot rolled, etc.). Surfaces may also be painted, sputter coated, or evaporation coated to enhance or diminish emission, absorption, or reflection or to bias directionality and/or wavelength dependence. In addition, all surface preparations are subject to damage and aging. The result is a bewilderingly subjective array of adjectives, often open to interpretation, which renders effective communication between designers and modelers difficult if not impossible. Still, it is imperative, once a project moves out of the preliminary design phase, that engineers charged with performance modeling have access to accurate models for surface radiation behavior. In the extreme this often requires a surface characterization campaign, typically based on measurement of the bidirectional spectral reflectivity of key surfaces. The following brief review is intended as a guide to the reader tasked with formulating surface radiation behavior models, a topic treated in more detail in Chapter $4 .$
Maxwell’s Equations [10],
$$\nabla \times \boldsymbol{H}=\varepsilon_{0} \frac{\partial \boldsymbol{E}}{\partial t}+\frac{\boldsymbol{E}}{r_{e}}$$

$$\begin{gathered} \nabla \times \boldsymbol{E}=-\mu_{0} \frac{\partial \boldsymbol{H}}{\partial t} \ \nabla \cdot \boldsymbol{E}=\frac{\rho_{e}}{\varepsilon} \ \nabla \cdot \boldsymbol{H}=0 \end{gathered}$$
are the point of departure for understanding the interaction of EM radiation with a surface. In Eqs. (2.62) to (2.65), $\boldsymbol{H}\left(\mathrm{Am}^{-1}\right)$ is the magnetic field strength, $\mathrm{E}\left(\mathrm{V} \mathrm{m}^{-1}\right)$ is the electric field strength, $\varepsilon_{0}=8.854 \mathrm{C}^{2} / \mathrm{N} \cdot \mathrm{m}^{2}$ is the permittivity of free space, $r_{e}(\Omega-\mathrm{m})$ is the electrical resistivity, $\rho_{e}\left(\mathrm{C} \mathrm{m}^{-3}\right)$ is the electric charge density, and $\mu_{0}=4 \pi \times 10^{-7} \mathrm{~N} \mathrm{~A}^{-2}$ is the magnetic permeability. These celebrated equations were formulated in 1864 by the British physicist and mathematician James Clerk Maxwell, who synthesized them from already known relationships between electricity and magnetism. Their solution permitted for the first time the theoretical calculation (see Problem 2.7) of the already known speed of light in a vacuum, thereby removing any doubt as to their validity.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Radiation Behavior of Surfaces Composed

In the following paragraphs we consider the two limiting cases in which a monochromatic EM wave is incident to the plane interface separating two ideal regions. In the first extreme, the wave passes from a dielectric whose optical properties are $n_{1}$ and $k_{1}=0\left(r_{e} \rightarrow \infty\right)$ into another dielectric whose optical properties are $n_{2}$ and $k_{2}=0$; in the second extreme, the wave passes from a dielectric whose optical properties are $n_{1}$ and $k_{1}=0$ into an electrical conductor, or metal, whose optical properties are $n_{2}$ and $k_{2} \neq 0$

The case of a smooth plane interface between two dielectrics with $n_{2}>n_{1}$ is illustrated in Figure 2.19. In the figure $\boldsymbol{E}_{p, i}$ represents the

eléctric component of a transverse-magnetic (TM), p-polarized, monochromatic electromagnetic wave incident to the interface, or surface. The arrows labeled “Incident,” “Reflected,” and “Transmitted” can be thought of as “rays,” in which case the lines passing normal to the rays indicate wavefronts separated by a distance $\lambda$. Following convention, the subscript ” $p$ ” is used to remind us that the electric field vector in this case lies in the plane of incidence, the plane containing both the incident ray and the unit normal vector $\boldsymbol{n}=-i$. Without loss of generality we consider only the real part of the incident electric field,
$$\operatorname{Re}\left[E_{p, i}\right]=\left|E_{p, i}\right| \cos \left[\omega\left(\frac{n_{1} x^{\prime}}{c_{0}}-t\right)\right],$$
where $\omega=2 \pi c_{0} / \lambda$ and $x^{\prime}=y / \sin \vartheta_{i}$. This is equivalent to assuming that the phase angle of the incident wave, $\phi_{i}=\tan ^{-1}\left[\operatorname{Im}\left(E_{p, i}\right) / \operatorname{Re}\left(E_{p, i}\right)\right]$, is zero.

Careful consideration of Figure $2.19$ reveals that the $y$-component (parallel to the interface) of the electric field above the interface (region 1) is
\begin{aligned} \left|E_{y, 1}\right|=&\left|E_{p, i}\right| \cos \vartheta_{i} \cos \left[\omega\left(\frac{n_{1} y / \sin \vartheta_{i}}{c_{0}}-t\right)\right] \ &-\left|E_{p, r}\right| \cos \vartheta_{r} \cos \left[\omega\left(\frac{n_{1} y / \sin \vartheta_{r}}{c_{0}}-t\right)\right] \end{aligned}

and the $y$-component below the interface (region 2) is
$$\left|E_{y, 2}\right|=\left|E_{p, t}\right| \cos \vartheta_{t} \cos \left[\omega\left(\frac{n_{2} y / \sin \vartheta_{t}}{c_{0}}-t\right)\right] .$$

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

e(λ,吨,ϑ,披)≡一世λ,和(λ,吨,ϑ,披)一世bλ(λ,吨)

e′(吨)=e(吨,ϑ,披)=∫λ=0ωe(λ,吨,ϑ,披)一世bλ(λ,吨)dλ∫λ=0∞一世bλ(λ,吨)dλ.

e′(吨)=e(吨,ϑ,披)=圆周率σ吨4∫λ=0∞e(λ,吨,ϑ,披)一世bλ(λ,吨)dλ.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Introduction to the Radiation Behavior of Surfaces

∇×H=e0∂和∂吨+和r和

∇×和=−μ0∂H∂吨 ∇⋅和=ρ和e ∇⋅H=0

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Radiation Behavior of Surfaces Composed

|和是,1|=|和p,一世|因⁡ϑ一世因⁡[ω(n1是/罪⁡ϑ一世C0−吨)] −|和p,r|因⁡ϑr因⁡[ω(n1是/罪⁡ϑrC0−吨)]

|和是,2|=|和p,吨|因⁡ϑ吨因⁡[ω(n2是/罪⁡ϑ吨C0−吨)].

## 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。

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|STAT 40820

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|The Blackbody Radiation Distribution Function

By definition, a blackbody is a perfect absorber of thermal radiation; that is, it absorbs all incident radiation from all directions and at all wavelengths. It follows from this definition and from the Second Law of Thermodynamics that no body at a given temperature can emit more thermal radiation than a blackbody at the same temperature. Therefore, we say that a blackbody is an ideal emitter. Furthermore, it can be demonstrated through the use of “thought experiments” (see Ref. [1], pp. 32-33) that an isothermal enclosure is filled with blackbody radiation, which is both uniform and isotropic. According to the Stefan-Boltzmann law, the emissive power escaping from an isothermal enclosure through a vanishingly small hole in the wall is
$$e_{b}=\sigma T^{4}\left(\mathrm{~W} / \mathrm{m}^{2}\right),$$
where $T(\mathrm{~K})$ is the absolute temperature and $\sigma=5.6696 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \cdot \mathrm{K}^{4}$ is the Stefan-Boltzmann constant. The German physicist Josef Stefan (1835-1893) first suggested the form of Eq. (2.15) in 1879 on the basis of data already in the literature (see Problem 2.4) [2]. Stefan discovered that a straight line results when the initial cooling rate of a body suspended in a vacuum is plotted against the difference between its absolute temperature to the fourth power and that of its surroundings to the fourth power. Five years later Stefan’s student, Austrian physicist Ludwig Boltzmann (1844-1906), derived the form of Eq. (2.15) on the basis of classical thermodynamics [3]. Boltzmann’s derivation is also available in Ref. [1] (pp. $38-42$ ).

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

We learn in the first paragraph in the previous section that blackbody radiation is isotropic. We conclude that the intensity of a blackbody must be independent of direction; that is, for a blackbody
$$i_{\lambda}(\lambda, \vartheta, \varphi)=i_{b \lambda}(\lambda, T) .$$

It then follows from Eq. (2.11) that the directional spectral emissive power of a blackbody is
$$e_{b \lambda}(\lambda, T, \vartheta)=i_{b \lambda}(\lambda, T) \cos \vartheta .$$
Thus, the directional spectral emissive power of a blackbody varies as the cosine of the angle with respect to the local surface normal. This is sometimes referred to as Lambert’s cosine law, and surfaces that conform to this law are frequently referred to as Lambertian. While all blackbodies are Lambertian, not all Lambertian surfaces are blackbodies. The blackbody hemispherical spectral emissive power, $e_{b \lambda}(\lambda, T, \vartheta)$, is
$$e_{b \lambda}(\lambda, T)=i_{b \lambda}(\lambda, T) \int_{2 \pi} \cos \vartheta d \Omega=\pi i_{b \lambda}(\lambda, T),$$
and the blackbody total intensity, $i_{b}(T)$, is
$$i_{b}(T)=\int_{\lambda=0}^{\infty} i_{b \lambda}(\lambda, T) d \lambda .$$
Evaluation of the integral in Eq. (2.23) is complicated by the form of the integrand, given by Eq. (2.19). The approach is to introduce a change of variables, $\eta=C_{2} / \lambda T$, after which
$$i_{b}(T)=\frac{C_{1} T^{4}}{C_{2}^{4}} \int_{\eta=0}^{\infty} \frac{\eta^{3}}{e^{\eta}-1} d \eta .$$

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Emission and Absorption Mechanisms

To this point we have characterized thermal radiation as a wave phenomenon. However, in 1905 Albert Einstein introduced the idea of the photon as an alternative view of EM radiation. In Einstein’s photoelectric theory the photon is a particle whose energy $e$ (not to be confused with emissive power) is proportional to the frequency of a corresponding EM wave,
$$e=h v(\mathrm{~J}),$$
where $h$ is Planck’s constant. The dual wave-particle description of EM radiation is now firmly established, with one being more convenient to use than the other depending on the situation. Another important conclusion of the photoelectric theory is that, at the most fundamental level, radiation heat transfer always involves interactions between photons and electrons. Modern physics recognizes two categories of atomic particle: the fermions, which are the building blocks of matter, and bosons, which moderate interactions among the fermions. In the field of quantum electrodynamics (QED), photons and electrons form a boson-fermion pair whose interactions account for all electrical and magnetic phenomena. The reader interested in pursuing this fascinating topic further is referred to Richard P. Feynman’s highly readable classic QED: the Strange Theory of Light and Matter [8].

For the purposes of the following discussion, an atom may be viewed as a positively charged nucleus surrounded by a swarm of negatively charged electrons. In order for an atom to be electrically neutral, the number of electrons must exactly balance the positive charge of the nucleus. The rules of quantum mechanics require that the electrons organize themselves into layers, or “shells.” surrounding the nucleus. A discrete energy

level is identified with an electron depending on the shell it occupies, with electrons occupying the inner shells having less energy than those occupying outer shells. Electrons can migrate between shells only by gaining or giving up the amount of energy associated with the difference between their fixed energy in the two shells. The mechanism for gaining or giving up this energy is interaction with a photon, as required by Einstein’s photoelectric theory embodied in Eq. (2.27). Thus, when an electron moves from one energy level to another within an atom, the conservation of energy principle requires that a corresponding amount, or quantum, of energy be absorbed by or emitted from the atom. If the atomic transition occurs from energy level $E_{a}$ to a lower energy level $E_{b}$, then, according to Eq. (2.27), the frequency of the light emitted by the atom for this bound-bound transition is
$$v=\frac{E_{a}-E_{b}}{h} .$$

## 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。

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|ME 777

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

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

Further discussion of the nature of thermal radiation requires careful definition of certain concepts and terms of art. Chief among these is the concept of the solid angle $\Omega$ (sr), whose thorough understanding is critical to the study of both radiation heat transfer and applied optics. Consider Figure $2.3$, which shows a differential surface element $d S$ located a distance $r$ from a second differential area $d A$. The line of length $r$ connecting $d A$ and $d S$ intersects the normal to $d A$ at an angle $\beta_{A}$, and it intersects the normal to $d S$ at an angle $\beta_{S}$. As a concession to clarity, the surface elements $d A$ and $d S$ are necessarily drawn as finite in size but, in fact, both are arbitrarily small compared to the finite distance $r$. The surface element $d S \cos \beta_{S}$, which is hinged to surface element $d S$ along their common lower edge, is tilted toward $d A$, so that the line $r$ is normal to $d S \cos \beta_{S}$. Because both $d S$ and $d S \cos \beta_{S}$ are vanishingly small, the distance between the points where $r$ intersects them is negligible compared to the finite length $r$. Then the differential solid angle $d \Omega_{S}$ subtended by $d S \cos \beta_{S}$ at $d A$ is defined
$$d \Omega_{S} \equiv \frac{d S \cos \beta_{S}}{r^{2}} .$$
Note that the solid angle in steradians (sr) is actually a dimensionless ratio of area over length squared, just as a one-dimensional angle in radians (r) is a dimensionless ratio of lengths.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Intensity of Radiation

The term spectral and its synonym monochromatic (“mono” = one, “chrome” = color) refer to radiation confined to a vanishingly small wavelength interval $d \lambda$ centered about a specified wavelength $\lambda$. Thus, the polarized ray in Figure $2.1$ represents spectral radiation. The spectral intensity $i_{\lambda}(\lambda, \vartheta, \varphi)$ of a plane source is the power per unit wavelength in the wavelength interval $d \lambda$ centered about wavelength $\lambda$, per unit projected area of the source, per unit solid angle, passing in direction $(\vartheta, \varphi)$. Note that the symbol $\lambda$ appears twice in the notation. This is not

redundant usage; the subscript $\lambda$ signals that the spectral intensity is a per-unit-wavelength quantity, and the $\lambda$ in the argument list signals that the value of the spectral intensity depends on wavelength. While it is traditional to call this quantity “intensity” in the radiation heat transfer community, it is frequently referred to as “radiance” in the applied optics, astronomy, and earth sciences literature.

Figure $2.5$ represents a beam of monochromatic light of spectral power $d^{3} P(\lambda, \vartheta, \varphi)$ (W) leaving the plane surface element $d A$ in direction $(\vartheta, \varphi)$ at an angle $\vartheta$ with respect to the surface normal and contained in a beam whose solid angle is $d \Omega_{s}$. Then the spectral intensity of this beam is
$$i_{\lambda}(\lambda, \vartheta, \varphi)=\frac{d^{3} P(\lambda, \vartheta, \varphi)}{d A \cos \vartheta d \Omega_{S} d \lambda}\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{sr} \cdot \mu \mathrm{m}\right) .$$
The superscript ” 3 ” on the differential operator in the numerator of Eq. (2.7) is required for notational consistency; in order for the intensity to be a finite quantity, the number of differential symbols $d$ must be the same in both the numerator and the denominator.

Another useful concept is the total intensity of a beam, which is obtained by integrating the spectral intensity over all wavelengths, i.e.,
$$i(\vartheta, \varphi)=\int_{\lambda=0}^{\infty} i_{\lambda}(\lambda, \vartheta, \varphi) d \lambda .$$
The word “total” is used exclusively in this context in the radiation heat transfer literature.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Directional Spectral Emissive Power

The directional spectral emissive power $e_{\lambda}(\lambda, \vartheta, \varphi)$ of a plane source is the power per unit wavelength in a specified wavelength interval $d \lambda$ about wavelength $\lambda$, per unit source surface area $d A$, emitted in direction $(\vartheta, \varphi)$ per unit solid angle into the space above the source. Then referring once again to Figure $2.5$, the differential directional spectral emissive power contained in the solid angle $d \Omega_{s}$ is
$$d E_{\lambda} \equiv \frac{d^{3} P(\lambda, \vartheta, \varphi)}{d A d \lambda}\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\right) .$$
Invoking Eq. (2.7) we have
$$d E_{\lambda}=i_{\lambda}(\lambda, \vartheta, \phi) \cos \vartheta d \Omega_{s}\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\right)$$
Finally, the directional spectral emissive power is
$$e_{\lambda}(\lambda, \vartheta, \varphi) \equiv \frac{d E_{\lambda}}{d \Omega_{\mathrm{S}}}=i_{\lambda}(\lambda, \vartheta, \phi) \cos \vartheta\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{sr} \cdot \mu \mathrm{m}\right) .$$

dΩ小号≡d小号因⁡b小号r2.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Directional Spectral Emissive Power

d和λ≡d3磷(λ,ϑ,披)d一个dλ(在/米2⋅μ米).

d和λ=一世λ(λ,ϑ,φ)因⁡ϑdΩs( 在/米2⋅μ米)

## 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。

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|STAT31511

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Scattering and Refraction

The power incident to a surface that is neither absorbed locally nor reflected is then either scattered in a process analogous to reflection, or it is refracted. In the MCRT description of radiation heat transfer, scattering is modeled by subdividing the incident ray into many equal-power rays, with each scattered ray continuing in a direction determined by an appropriate scattering model. The complex phenomenon of scattering is treated in detail in Chapter 5 , which deals with radiation propagating through a participating medium. The simplest and most basic model for scattering, which is used in the early chapters of this book, is the assumption that scattering can be neglected, as is often the case in radiation heat transfer.

In the ray-trace description of geometrical optics, refraction refers to the abrupt change in direction of the ransmitted ray as it passes through an interface. The Snell-Descartes law, illustrated in Figure 1.11, represents reality very well, especially for interfaces between air and common materials used in the fabrication of lenses, filters, retarder plates, and windows. According to the Snell-Descartes law
$$\sin \left(\vartheta_{1}\right) / \sin \left(\vartheta_{2}\right)=n_{2} / n_{1},$$
where $n_{1}$ and $n_{2}$ are the refractive indices of the two materials whose interface provokes refraction. Problem $1.14$ is an important application of this principle in applied optics.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Meshing and Indexing

The MCRT method used throughout this book requires that the modeling space be subdivided into surface and volume elements, i.e., that it be appropriately meshed. While entire books have been written on this topic alone, the limited treatment offered here is adequate for the needs of most MCRT analyses. The meshes used in the MCRT method must be amenable to indexing. Indexing refers to the process of systematically numbering the surface and volume elements in such a way that the numbers, called indices, can be determined algorithmically from the coordinates of a point lying on a surface element or within a volume element.
Pedagogical considerations favor limitation of the discussion presented here to rectilinear spaces, i.e., to spaces that can be represented by rectangular solid blocks. As used here, the word “solid”‘ implies only that the spaces are three-dimensional. Many, if not most, enclosures of practical engineering interest can be accurately represented using a rectilinear mesh if care is taken to ensure that the surface element unit normal vectors represent the actual local curvature. The methods presented in this section can be extended to spaces consisting of trapezoidal, cylindrical (both circular and noncircular), and spheroidal solids.

Consider the hollow, three-dimensional rectilinear space illustrated in Figure 1.12. Use of the MCRT method often requires that the space be divided into $N$ surface elements and $n-N$ volume elements, with a unique number, or “index”, algorithmically assigned to each element. Furthermore, square surface elements and cubic volume elements are highly desirable. Finally, the resulting mesh must be sufficiently dense to assure adequate spatial resolution of the results obtained using an MCRT analysis. How do we go about satisfying all of these requirements? Consider the following numerical examples.

Electromagnetic (EM) waves, whose properties are explored in this chapter, carry energy from one location to another, even – indeed, especially – in a vacuum. The mathematical form of the magnitude of the electric field component of an EM wave propagating along the $x$-axis is
$$E(x, t)=E_{0} e^{i(k x-\omega t)},$$
where $E\left(\mathrm{~V} \mathrm{~m}^{-1}\right)$ is the instantaneous electric field strength at position $x(\mathrm{~m})$ and time $t(\mathrm{~s}), E_{0}$ is the amplitude of its oscillation, $k\left(\mathrm{~m}^{-1}\right)$ is

the wave number, and $\omega\left(\mathrm{r} \mathrm{s}^{-1}\right)$ is the angular frequency. An analogous equation can be written for the magnetic field component, $H\left(\mathrm{Am}^{-1}\right)$. Figure $2.1$ illustrates a $y$-polarized EM wave propagating along the $x$-axis. The power carried by this EM wave is given by the Poynting vector,
$$\boldsymbol{P}=\boldsymbol{E} \times \boldsymbol{H}=E_{y} \boldsymbol{j} \times H_{z} \boldsymbol{k}=E_{y} H_{z} \boldsymbol{i}\left(\mathrm{W} / \mathrm{m}^{2}\right) .$$
This is the basic mechanism of radiation heat transfer.
The frequency $v=\omega / 2 \pi\left(\mathrm{s}^{-1}\right)$ of an electromagnetic wave is determined at its origin and does not change as it propagates. However, its wavelength $\lambda=2 \pi / k(\mathrm{~m})$ varies according to the speed of light $c\left(\mathrm{~m} \mathrm{~s}^{-1}\right)$ in the medium through which it propagates according to
$$\lambda=c / v=c_{0} / n v,$$
where $n \equiv c_{0} / c$ is the index of refraction and $c_{0}$ is the speed of light in a vacuum $\left(\sim 2.9979 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}\right)$.

λ=C/在=C0/n在,

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法Monte Carlo method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。我们的专家在代写蒙特卡洛方法Monte Carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法Monte Carlo method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

## 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

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。

## 统计代写|蒙特卡洛方法代写monte carlo method代考| Markov Jump Processes

statistics-lab™ 为您的留学生涯保驾护航 在代写蒙特卡洛方法学monte carlo method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。我们的专家在代写蒙特卡洛方法学monte carlo method代写方面经验极为丰富，各种代写蒙特卡洛方法学monte carlo method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

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Markov Jump Processes

A Markov jump process $X=\left{X_{t}, t \geqslant 0\right}$ can be viewed as a continuous-time generalization of a Markov chain and also of a Poisson process. The Markov property $(1.30)$ now reads
$$\mathbb{P}\left(X_{t+s}=x_{t+s} \mid X_{u}=x_{u}, u \leqslant t\right)=\mathbb{P}\left(X_{t+s}=x_{t+s} \mid X_{t}=x_{t}\right) .$$
As in the Markov chain case, one usually assumes that the process is timehomogeneous, that is, $\mathbb{P}\left(X_{t+s}=j \mid X_{t}=i\right)$ does not depend on $t$. Denote this probability by $P_{s}(i, j)$. An important quantity is the transition rate $q_{i j}$ from state $i$ to $j$, defined for $i \neq j$ as
$$q_{i j}=\lim {t \downarrow 0} \frac{P{t}(i, j)}{t} .$$
The sum of the rates out of state $i$ is denoted by $q_{i}$. A typical sample path of $X$ is shown in Figure 1.6. The process jumps at times $T_{1}, T_{2}, \ldots$ to states $Y_{1}, Y_{2}, \ldots$, staying some length of time in each state.

More precisely, a Markov jump process $X$ behaves (under suitable regularity conditions; see [3]) as follows:

1. Given its past, the probability that $X$ jumps from its current state $i$ to state $j$ is $K_{i j}=q_{i j} / q_{i}$.
2. The amount of time that $X$ spends in state $j$ has an exponential distribution with mean $1 / q_{j}$, independent of its past history.

The first statement implies that the process $\left{Y_{n}\right}$ is in fact a Markov chain, with transition matrix $K=\left(K_{i j}\right)$.

A convenient way to describe a Markov jump process is through its transition rate graph. This is similar to a transition graph for Markov chains. The states are represented by the nodes of the graph, and a transition rate from state $i$ to $j$ is indicated by an arrow from $i$ to $j$ with weight $q_{i j}$.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Birth-and-Death Process

A birth-and-death process is a Markov jump process with a transition rate graph of the form given in Figure 1.7. Imagine that $X_{t}$ represents the total number of individuals in a population at time $t$. Jumps to the right correspond to births, and jumps to the left to deaths. The birth rates $\left{b_{i}\right}$ and the death rates $\left{d_{i}\right}$ may differ from state to state. Many applications of Markov chains involve processes of this kind. Note that the process jumps from one state to

the next according to a Markov chain with transition probabilities $K_{0,1}=1$, $K_{i, i+1}=b_{i} /\left(b_{i}+d_{i}\right)$, and $K_{i, i-1}=d_{i} /\left(b_{i}+d_{i}\right), i=1,2, \ldots$. Moreover, it spends an $\operatorname{Exp}\left(b_{0}\right)$ amount of time in state 0 and $\operatorname{Exp}\left(b_{i}+d_{i}\right)$ in the other states.
Limiting Behavior We now formulate the continuous-time analogues of (1.34) and Theorem 1.13.2. Irreducibility and recurrence for Markov jump processes are defined in the same way as for Markov chains. For simplicity, we assume that $\mathscr{E}={1,2, \ldots}$. If $X$ is a recurrent and irreducible Markov jump process, then regardless of $i$,
$$\lim {t \rightarrow \infty} \mathbb{P}\left(X{t}=j \mid X_{0}=i\right)=\pi_{j}$$
for some number $\pi_{j} \geqslant 0$. Moreover, $\pi=\left(\pi_{1}, \pi_{2}, \ldots\right)$ is the solution to
$$\sum_{j \neq i} \pi_{i} q_{i j}=\sum_{j \neq i} \pi_{j} q_{j i}, \quad \text { for all } i=1, \ldots, m$$
with $\sum_{j} \pi_{j}=1$, if such a solution exists, in which case all states are positive recurrent. If such a solution does not exist, all $\pi_{j}$ are 0 .

As in the Markov chain case, $\left{\pi_{j}\right}$ is called the limiting distribution of $X$ and is usually identified with the row vector $\pi$. Any solution $\pi$ of (1.42) with $\sum_{j} \pi_{j}=1$ is called a stationary distribution, since taking it as the initial distribution of the Markov jump process renders the process stationary.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|GAUSSIAN PROCESSES

The normal distribution is also called the Gaussian distribution. Gaussian processes are generalizations of multivariate normal random vectors (discussed in Section 1.10). Specifically, a stochastic process $\left{X_{t}, t \in \mathscr{T}\right}$ is said to be Gaussian if all its finite-dimensional distributions are Gaussian. That is, if for any choice of $n$ and $t_{1}, \ldots, t_{n} \in \mathscr{T}$, it holds that
$$\left(X_{t_{1}}, \ldots, X_{t_{n}}\right)^{\top} \sim \mathrm{N}(\boldsymbol{\mu}, \Sigma)$$
for some expectation vector $\boldsymbol{\mu}$ and covariance matrix $\Sigma$ (both of which depend on the choice of $\left.t_{1}, \ldots, t_{n}\right)$. Equivalently, $\left{X_{t}, t \in \mathscr{T}\right}$ is Gaussian if any linear combination $\sum_{i=1}^{n} b_{i} X_{t_{i}}$ has a normal distribution. Note that a Gaussian process is determined completely by its expectation function $\mu_{t}=\mathbb{E}\left[X_{t}\right], t \in \mathscr{T}$, and covariance function $\Sigma_{s, t}=\operatorname{Cov}\left(X_{s}, X_{t}\right), s, t \in \mathscr{T}$.

The Wiener process can be defined as a Gaussian process $\left{X_{t}, t \geqslant 0\right}$ with expectation function $\mu_{t}=0$ for all $t$ and covariance function $\Sigma_{s, t}=s$ for $0 \leqslant s \leqslant t$. The Wiener process has many fascinating properties (e.g., [11]). For example, it is a Markov process (i.e., it satisfies the Markov property $(1.30)$ ) with continuous sample paths that are nowhere differentiable. Moreover, the increments $X_{t}-X_{s}$ over intervals $[s, t]$ are independent and normally distributed. Specifically, for any $t_{1}<t_{2} \leqslant t_{3}<t_{4}$,
$$X_{t_{4}}-X_{t_{3}} \quad \text { and } \quad X_{t_{2}}-X_{t_{1}}$$
are independent random variables, and for all $t \geqslant s \geqslant 0$,
$$X_{t}-X_{s} \sim \mathrm{N}(0, t-s) .$$
This leads to a simple simulation procedure for Wiener processes, which is discussed in Section 2.8.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Markov Jump Processes

q一世j=林吨↓0磷吨(一世,j)吨.

1. 鉴于它的过去，概率X从当前状态跳转一世陈述j是ķ一世j=q一世j/q一世.
2. 的时间量X在州花费j具有均值的指数分布1/qj，独立于其过去的历史。

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Birth-and-Death Process

∑j≠一世圆周率一世q一世j=∑j≠一世圆周率jqj一世, 对全部 一世=1,…,米

## 统计代写|蒙特卡洛方法代写monte carlo method代考|GAUSSIAN PROCESSES

(X吨1,…,X吨n)⊤∼ñ(μ,Σ)

X吨4−X吨3 和 X吨2−X吨1

X吨−Xs∼ñ(0,吨−s).

## 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。