### 物理代写|理论力学代写theoretical mechanics代考|PHYSICS 2532

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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

## 物理代写|理论力学代写theoretical mechanics代考|Effect of Thickness on the Magnitude of Spontaneous

To describe properties of the ferroelectric films and to study of ordering effects we use a three-dimensional lattice model (Fig. 10), consisting of $N_{1}, N_{2}$ and $N_{3}$ nodes along the respective axes of the Cartesian coordinate system. The position of the lattice node is characterized by the set of three numbers $\vec{n}=\left(n_{1}, n_{2}, n_{3}\right)$.

In this paper, the interaction energy of dipoles is described by a potential that takes into account the energy of orientation interactions (as in the classical Ising model) and the additional term representing the Lennard-Jones potential:
$$H=H_{o r}+\sum_{\vec{n}, \vec{m}} \varepsilon\left(\frac{r_{0}^{12}}{r_{\vec{n}, \vec{m}}^{12}}-\frac{2 r_{0}^{6}}{r_{\vec{n}, \vec{m}}^{6}}\right),$$
where $\varepsilon$ is the potential well depth of the Lennard-Jones potential, $r_{i, j}, j$ is the distance between the dipoles, $r_{0}$ is average distance in the absence of orientation interactions.

The second term of Eq. (14) does not depend on the temperature and the polarization, in contrast to the first term.

When the polarization decreases, therefore, we must take into account that the distance between the dipoles changes in transverse dimensions $N_{2}$ and $N_{3}$ of film. The potential of orientation interactions $H_{o r}$ is represented by the formula:
\begin{aligned} H_{o r}=&-\sum_{\vec{n}} K_{1} S_{n_{1}, n_{2}, n_{3}} S_{n_{1}-1, n_{2}, n_{3}}-\sum_{\vec{n}} K_{2} \frac{r_{0}^{3}}{r^{3}} S_{n_{1}, n_{2}, n_{3}} S_{n_{1}, n_{2}-1, n_{3}} \ &-\sum_{\vec{n}} K_{2} \frac{r_{0}^{3}}{r^{3}} S_{n_{1}, n_{2}, n_{3}} S_{n_{1}, n_{2}, n_{3}-1}+p \sum_{\vec{n}} S_{\vec{n}} E_{d} \end{aligned}
where the quantity $S_{-n}$ takes only two values $+1$ and $-1, K_{1}$ is the coefficient of exchange interactions in the longitudinal direction, $p$ is the dipole moment, $K_{2}$ is the constant of exchange interactions between the dipoles in the transverse direction, $E_{d}$ is the projection of the vector of the depolarizing field strength on the direction $N_{1}$.

## 物理代写|理论力学代写theoretical mechanics代考|Modeling of Geometric and Optical Properties

The solution of the problem of creating surfaces with certain properties is necessary both for stable functioning of products and technological control of the surface quality of such products [33]. The use of the fractal approach to describe structural in homogeneities, as well as the justification of general regularities, is one of the modern scientific trends in the surface physics and the chemistry of solids. At present, various mathematical models of fractals (Sierpinski rug, Mandelbrot set), describe well the real imperfections (Brownian) surfaces of metal layers, dielectric layers [34], semiconductor surfaces [35] those have defects of a symmetric type [36, 37]. However, when examining the surface of polymer coatings of metal sheet, the detected defects are anisotropic (Fig. 16a); therefore, these models cannot be used to describe their structure. In this paper, the three-dimensional anisotropic model based on the Julia set will be used to construct a fractal model of the surface.
Algorithm of creating of the fractals
To construct fractal surfaces of the extured polymer coating of sheet metal (Fig. 16a, b), the following algorithm was used:

1. The area in which the fractal is created is divided into $1000 \times 1000$ rectangles. Each rectangle is characterized by the coordinates $\left(X_{r, s}, Y_{r, s}\right)$ of its center.
2. A sequence is defined by the recurrence formula [38].
$$Z_{r, s}^{(n)}=\left(Z_{r, s}^{(n-1)}\right)^{2}+p+i q,$$
where values $p$ and $q$ are parameters of the fractal function (22). The first term of the sequence is defined as
$$Z_{r, s}^{(1)}=X_{r, s}+i Y_{r, s}$$
3. The value of $H$ is select inversely to the rate of increase of the modulus of the sequence term (1). $H$ is equal to the smallest number of the sequence term, when $\left|z_{i}\right|>Q$. In our calculations, we assumed that the value is $Q=10^{6}$.
The examples of fractal functions obtained are shown in Fig. 16c, d.

## 物理代写|理论力学代写theoretical mechanics代考|Problem Formulation

In an infinite two-dimensional elastic medium there is an array of obstacles. The obstacles can be of two types: absolutely solid and voids. In the array of obstacles, a pulse is introduced with a tonal filling by several periods of a planar high-frequency, monochromatic longitudinal or transverse elastic wave, and in a certain region of the elastic medium, a transmitted wave with any possible reflections (longitudinal wave to longitudinal one, transverse wave to transverse one) and transformations (longitudinal wave to transverse one, transverse wave to longitudinal one).

The aim of the study is to obtain analytical expressions for displacements in the transmitted longitudinal or transverse wave.

The structure of the input pulse makes it possible to investigate the problem in the regime of harmonic oscillations. The incident plane elastic wave is replaced by a set of point sources of cylindrical waves. Each cylindrical wave propagating in an angle with a vertex in the source directed toward the obstacles and a contracted semi-circle is replaced by a system of corresponding radial propagation rays of the elastic wave. Thus, the problem is reduced to a problem of short-wave diffraction of elastic waves in a local formulation. The total field in the region of reception of propagating elastic waves is composed of rays transmitted through a system of obstacles, which can be of the three types: rays transmitted through the obstacle system without diffraction; rays reflected from the system once or a finite number of times.

## 物理代写|理论力学代写theoretical mechanics代考|Effect of Thickness on the Magnitude of Spontaneous

H=H○r+∑n→,米→e(r012rn→,米→12−2r06rn→,米→6),

H○r=−∑n→ķ1小号n1,n2,n3小号n1−1,n2,n3−∑n→ķ2r03r3小号n1,n2,n3小号n1,n2−1,n3 −∑n→ķ2r03r3小号n1,n2,n3小号n1,n2,n3−1+p∑n→小号n→和d

## 物理代写|理论力学代写theoretical mechanics代考|Modeling of Geometric and Optical Properties

1. 创建分形的区域分为1000×1000矩形。每个矩形的特征是坐标(Xr,s,是r,s)的中心。
2. 序列由递归公式[38]定义。
从r,s(n)=(从r,s(n−1))2+p+一世q,
值在哪里p和q是分形函数 (22) 的参数。序列的第一项定义为
从r,s(1)=Xr,s+一世是r,s
3. 的价值H与序列项 (1) 的模数的增加率成反比。H等于序列项的最小数，当|和一世|>问. 在我们的计算中，我们假设该值为问=106.
得到的分形函数的例子如图 16c、d 所示。

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