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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

## 数学代写|数值方法作业代写numerical methods代考|Model for the Evaluation

Recent discoveries in high-energy particle accelerators are connected to the possibility to reach higher level of energies in the experiments [1]. One of the main limitations to the involved energy, that is to say to the current of the beam, is the instability of the particle due to the electromagnetic interaction with the surrounding structures [2]. The synthetic design parameter commonly adopted in literature to describe the electromagnetic interaction between a traveling particle and a structure is the coupling impedance [3-5]. This parameter is proportional to the energy lost by the traveling charge due to the interaction with the scattered fields produced by the surrounding structures. Equivalently, it is proportional to the energy that has to be spent to keep its speed constant, neglecting the slowing effect of the surrounding structures. For structures invariant along the charge traveling direction, a per-unit-length coupling impedance has to be introduced [4], whose longitudinal and transverse components can be defined as
$$Z_{|}(r, \varphi, k)=-\frac{1}{q} \frac{1}{L} \int_{-L / 2}^{L / 2} E_z(r, \varphi, z, \omega) e^{j k z / \beta} d z \quad Z_{\perp}(r, \varphi, k)=\frac{1}{k} \nabla_{\perp} Z_{|}(r, \varphi, k),$$
where $L$ is an unitary length, $E_z(r, \varphi, z, \omega)$ the $x$-component of the electric field in the frequency domain, $k$ the wavenumber, and the charge $q$ is moving at constant velocity $v=\beta c$ along the $z$ axis. The second equation in (1) is known as the Panofski-Wenzel theorem [6].

The research of new shapes of cavities with proper coupling impedances is actually of high interest for the design of even more efficient particle accelerators [7-9]. Nowadays, powerful tools allow performing the electromagnetic numerical analysis of complex structures [10]. However, analytical or semi-analytical solutions still play a valuable role in this field, enabling to better understand the physics of some phenomena. Modal analysis is often adopted for close structures [11,12], diffractive methods for high-frequency solutions, and integral formulations for open geometries or in the presence of edges $[13,14]$.

Most of the studies related to the coupling impedance consider, in a cylindrical reference system, axially symmetric geometries. This choice is both because they represent most of the structures of interest and because the symmetry allows finding the solution with less effort or even in a semi-analytical form. In this paper, we want to analyze the interaction of a particle with a axially asymmetric structure, in particular an angular slot, as shown in Figure 1. This configuration is representative of particle accelerator components that break the axial symmetry. The proposed method is quite general and can be adopted for a wide class of scattering and diffraction problems [15-20]. It can be easily generalized and adopted to analyze similar geometries.

## 数学代写|数值方法作业代写numerical methods代考|Electromagnetic Fields in the Slot Frame

In order to complete the problem formulation, it is proper to express the electromagnetic fields in the slot frame, too. This can be realized by applying the Lorentz transforms to the fields computed in the previous section in particle frame.

Let us consider at first the $z$ component of the electric field. The contribution provided by the traveling charge in the frequency domain is well known and is
$$E_{z, q}=\frac{j q \kappa \zeta_0}{2 \pi \beta \gamma} e^{-j z k / \beta} K_0\left(\kappa \sqrt{r^2+r_q^2-2 r r_q \cos (\varphi)}\right),$$
where $\gamma=1 / \sqrt{1-\beta^2}$ is the Lorentz factor, $\kappa=k /(\beta \gamma)$, and $\zeta_0=\sqrt{\mu_0 / \varepsilon_0}$ is the characteristic impedance of free space.

The contribution produced by the induced current density on the slot can be obtained with some manipulations as function of the representation coefficients $\sigma_n$.
In the particle frame, starting from Equation (3) it is possible to obtain
$$e_z^{\prime}\left(r^{\prime}, \varphi^{\prime}, z^{\prime}\right)=\frac{a}{4 \pi \varepsilon_0} \int_{\mathcal{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right)\left(z^{\prime}-z_0\right) d \varphi_0 d z_0}{\left[r^{\prime 2}+a^2-2 r^{\prime} a \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}-z_0\right)^2\right]^{3 / 2}} .$$
Lorentz transforms are now applied to obtain the electric field in the slot frame. In this specific case they are
$$e_z^{\prime}=e_z, \sigma^{\prime}=\sigma \gamma, r^{\prime}=r, \varphi^{\prime}=\varphi, z^{\prime}=\gamma(z-v t) .$$
Applying these transforms to Equation (19), it is found that
$$e_z(r, \varphi, z, t)=\frac{a \gamma}{4 \pi \varepsilon_0} \int_{\mathcal{S}} \frac{\sigma\left(\varphi_0, z_0\right)\left(\gamma(z-v t)-z_0\right) d \varphi_0 d z_0}{\left[r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)+\left(\gamma(z-v t)-z_0\right)^2\right]^{3 / 2}} .$$
By means of Equation (8) and applying a spatial Fourier transform according to Equation (9), it is found that
$$e_z(r, \varphi, z, t)=\frac{j a \gamma}{2 \pi \varepsilon_0} \int_{-\varphi_a}^{+\varphi_a} \int_{-\infty}^{+\infty} \sigma\left(\varphi_0, w\right) w e^{j w \gamma v t} K_0\left(w \sqrt{r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)}\right) e^{-j w \gamma z} d \varphi_0 d w .$$
Finally, by performing a time Fourier transform and then the integral on $w$, the required field is finally found as
$$E_z(r, \varphi, z, \omega)=\frac{j a k \zeta_0}{\beta^2} e^{-j z k / \beta} \int_{-\varphi_a}^{+\varphi_a} \tilde{\sigma}\left(\varphi_0, \kappa\right) K_0\left(\kappa \sqrt{r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)}\right) d \varphi_0 .$$

# 数值方法代考

## 数学代写|数值方法作业代写numerical methods代考|Model for the Evaluation

$$Z_{\mid}(r, \varphi, k)=-\frac{1}{q} \frac{1}{L} \int_{-L / 2}^{L / 2} E_z(r, \varphi, z, \omega) e^{j k z / \beta} d z \quad Z_{\perp}(r, \varphi, k)=\frac{1}{k} \nabla_{\perp} Z_{\mid}(r, \varphi, k),$$

## 数学代写|数值方法作业代写numerical methods代考|Electromagnetic Fields in the Slot Frame

$$E_{z, q}=\frac{j q \kappa \zeta_0}{2 \pi \beta \gamma} e^{-j z k / \beta} K_0\left(\kappa \sqrt{r^2+r_q^2-2 r r_q \cos (\varphi)}\right),$$

$$e_z^{\prime}\left(r^{\prime}, \varphi^{\prime}, z^{\prime}\right)=\frac{a}{4 \pi \varepsilon_0} \int_{\mathcal{S}} \frac{\sigma^{\prime}\left(\varphi_0, z_0\right)\left(z^{\prime}-z_0\right) d \varphi_0 d z_0}{\left[r^{\prime 2}+a^2-2 r^{\prime} a \cos \left(\varphi^{\prime}-\varphi_0\right)+\left(z^{\prime}-z_0\right)^2\right]^{3 / 2}} .$$

$$e_z^{\prime}=e_z, \sigma^{\prime}=\sigma \gamma, r^{\prime}=r, \varphi^{\prime}=\varphi, z^{\prime}=\gamma(z-v t) .$$

$$e_z(r, \varphi, z, t)=\frac{a \gamma}{4 \pi \varepsilon_0} \int_{\mathcal{S}} \frac{\sigma\left(\varphi_0, z_0\right)\left(\gamma(z-v t)-z_0\right) d \varphi_0 d z_0}{\left[r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)+\left(\gamma(z-v t)-z_0\right)^2\right]^{3 / 2}} .$$

$$e_z(r, \varphi, z, t)=\frac{j a \gamma}{2 \pi \varepsilon_0} \int_{-\varphi_a}^{+\varphi_a} \int_{-\infty}^{+\infty} \sigma\left(\varphi_0, w\right) w e^{j \omega \gamma v t} K_0\left(w \sqrt{r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)}\right) e^{-j w \gamma z} d \varphi_0 d w .$$

$$E_z(r, \varphi, z, \omega)=\frac{j a k \zeta_0}{\beta^2} e^{-j z k / \beta} \int_{-\varphi_a}^{+\varphi_a} \tilde{\sigma}\left(\varphi_0, \kappa\right) K_0\left(\kappa \sqrt{r^2+a^2-2 r a \cos \left(\varphi-\varphi_0\right)}\right) d \varphi_0 .$$

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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。

assignmentutor™您的专属作业导师
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