物理代写|理论力学代写theoretical mechanics代考|PHYC30022

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我们提供的理论力学theoretical mechanics及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|PHYC30022

物理代写|理论力学代写theoretical mechanics代考|Results of the Measurements

The amplitude characteristics are presented in the Table 2. The analysis of the obtained data shows that obvious filtration properties of first, second and third samples begin after the frequency $0.6 \mathrm{MHz}$. The increase of the distance between the rows, used in the third sample, has no effect on the through-transmitted amplitude in the latter case; however an obvious change of the impulse shape is quite clear, much more notable than for the first two samples. One may conclude that the increase in the distance between the rows complicated the diffraction field inside the sample. The fourth sample begins to demonstrate its filtration properties just at the frequency of $0.4 \mathrm{MHz}$, that is obviously connected with smaller ratio of the US wave length above the size of the obstacle.

The preliminary investigations $[1,2]$ show that after the first filtration strip there is a strip of almost perfect transmission. As can be seen from Fig. 5 and Table 2, for the first three samples such a frequency strip begins from $1.8 \mathrm{MHz}$. This effect is less pronounced for the fourth sample, though the amplitude of the through-transmitted signal is still higher than at the frequency $1.25 \mathrm{MHz}$.

Analyzing the table, one may conclude that the increase of the size of the holes (the fourth sample) results in the worst through-transmission in the meta-material, cutting off more than $90 \%$ of energy, beginning from the frequency $-1 \mathrm{MHz}$. The increase of the distance between the rows along the wave propagation also reduces the carrying capacity for higher frequencies, and the passage to the first filtration band becomes smoother (which is obvious for the frequency equal to $0.6 \mathrm{MHz}$, where the sample 3 demonstrates the best through-transmission). The shift of the rows in the second sample has not so strong effect at low frequencies, and in some cases even improves the through-transmission of the US signal, as can be seen for example, for the frequency $1.25 \mathrm{MHz}$. Nevertheless, for higher frequencies one can see a significant suppression of the transmission, which may be connected with a complex structure of the re-reflections inside the meta-material.

物理代写|理论力学代写theoretical mechanics代考|Simulation Method Using the Wang-Landau Algorithm

Monte-Carlo method use broad class of computational algorithms which are based on random walks. The typical problem in statistical physics that can be solved by these method is calculating mean values of macroscopic variables (energy, order parameter,

etc.) at different temperatures for systems which follows Boltzmann statistics. There are some techniques for Monte-Carlo method: Metropolis [18], Wolff [19], Lee [20], Wang-Landau algorithms [21], parallel tempering [22]. In this section, Metropolis and Wang-Landau algorithms are described and illustrated on the example of twodimensional Ising model.

The Ising model consists of spins which have two possible orientations. Originally developed for simulation of ferromagnetic materials, now, this model has many applications including the simulation of ferroelectrics [23], spin glasses [24], image data processing [25], neuroscience, etc. In 1944 , the two-dimensional Ising model on a square lattice was analytically solved by Onsager [26]. The Hamiltonian of this model is determined by the formula:
E=-J \sum_{\langle i, j\rangle} \overrightarrow{S_{i}} \overrightarrow{S_{j}}-\vec{H} \sum_{i} \overrightarrow{S_{i}}
where $\overrightarrow{S_{i}}$ is the value of spin located in site $i$, the symbol $\langle i, j\rangle$ denotes the pairs of nearest-neighbor segments, $J$ is a parameter of spin interactions, $\vec{H}$ is the external magnetic field strength.

The Metropolis algorithm generates the sequence of states at a predetermined temperature using the probability distribution for the system. For the Ising model, the Metropolis algorithm should be applied as follows:

  1. A random spin is chosen and rotated.
  2. The new system configuration is accepted with probability:
    P=\min \left(-\frac{\Delta E}{k_{\mathbb{B}} T}, 1\right),
    where $\Delta E$ is energy change due to the spin rotation, $k_{B}$ is the Boltzmann constant, $T$ is the temperature.
  3. Steps 1 and 2 are repeated.
    The results of simulation for the two-dimensional Ising model with periodic boundary conditions obtained by means of the Metropolis algorithm are presented in Fig. 1. The heat capacity was determined by the formula:
    C=\frac{\left\langle E^{2}\right\rangle-\langle E\rangle^{2}}{k_{B} T^{2}} .

物理代写|理论力学代写theoretical mechanics代考|Investigation of the Influence of Bulk Properties

The surface properties of layers are determined not only by chemical composition of the substance, but also by their physical structure and the orientational order of polymer chains [29]. Intermolecular orientation interactions are much weaker than valence interactions; therefore, the self-organization of the system with the given chemical structure is determined by intermolecular interactions. In this chapter, we consider the equilibrium properties and phase transitions on the surface of ferroelectric polymer system, in which orientational interactions both between the surface molecules and molecules located in the bulk are taken into account.

Model. Usually, polymer chains have predominantly planar orientation relatively to the interphase boundary [30]. Therefore, in this paper, to describe the surface of ferroelectric polymer systems, we use a two-dimensional model, which consist of $M$ freely-jointed chains, each of which is a sequence of $N$ connected rigid segments, located in parallel to the surface (Fig. 4).

The main quantitative characteristic of the polymer chain flexibility is the persistent length $a$, which is related with the energetic constant of intrachain orientation interaction $K_{1}$ by the ratio:
K_{1}=\frac{a \cdot k_{B} T}{2}

Similar to the persistent length $a$, we introduce the interchain interaction parameter of $b$. The orientation interaction of neighboring polymer chain elements is described by the energy constant $K_{2}$,
K_{2}=\frac{b \cdot k_{B} T}{2} .
To take into account the interaction of surface molecules with molecules located in the bulk of the film, we use the mean field constant $V$ and the dimensionless mean field parameter $q$ :
q=\frac{V}{k_{B} T}
The internal energy in the low-temperature approximation can be represented as:
H=& \frac{1}{2} K_{1} \sum_{n, m=1}^{N, M}\left(\varphi_{n, m}-\varphi_{n-1, m}\right)^{2}+\frac{1}{2} K_{2} \sum_{n, m=1}^{N, M}\left(\varphi_{n, m}-\varphi_{n, m-1}\right)^{2} \
&-\mu V \sum_{n, m=1}^{N, M} \cos \left(\varphi_{n, m}\right)
where $\mu$ is the long-range orientation order parameter, which is defined as the average cosine of the angle between the directions of chain rigid element and the director, i.e. $\mu=\left\langle\cos \varphi_{\vec{n}}\right\rangle$.

物理代写|理论力学代写theoretical mechanics代考|PHYC30022


物理代写|理论力学代写theoretical mechanics代考|Results of the Measurements

幅度特性如表2所示。对所得数据的分析表明,第一、第二和第三样品的明显过滤特性在频率之后开始0.6米H和. 在第三个样本中使用的行间距的增加对后一种情况下的穿透幅度没有影响;然而,脉冲形状的明显变化非常明显,比前两个样本要明显得多。可以得出结论,行之间距离的增加使样品内部的衍射场复杂化。第四个样品仅在频率为0.4米H和,这显然与美国波长在障碍物大小之上的比例较小有关。

初步调查[1,2]表明在第一个过滤条之后有一条几乎完美的传输。从图 5 和表 2 可以看出,对于前三个样本,这样的频率带从1.8米H和. 对于第四个样本,这种影响不太明显,尽管通过传输信号的幅度仍然高于频率1.25米H和.

分析表格,可以得出结论,孔尺寸的增加(第四个样品)导致超材料中最差的穿透率,切断超过90%能量,从频率开始−1米H和. 沿波传播的行间距的增加也降低了对较高频率的承载能力,到第一个过滤带的通道变得更加平滑(这对于频率等于0.6米H和,其中样品 3 展示了最佳的穿透式传输)。第二个样本中的行移位在低频处没有那么强的影响,在某些情况下甚至改善了美国信号的直通传输,例如,对于频率1.25米H和. 然而,对于更高的频率,人们可以看到传输的显着抑制,这可能与超材料内部的再反射的复杂结构有关。

物理代写|理论力学代写theoretical mechanics代考|Simulation Method Using the Wang-Landau Algorithm


等)在不同温度下遵循玻尔兹曼统计的系统。Monte-Carlo 方法有一些技术:Metropolis [18]、Wolff [19]、Lee [20]、Wang-Landau 算法 [21]、并行回火 [22]。在本节中,Metropolis 和 Wang-Landau 算法以二维 Ising 模型为例进行描述和说明。

Ising 模型由具有两个可能方向的自旋组成。该模型最初是为模拟铁磁材料而开发的,现在,该模型具有许多应用,包括模拟铁电体 [23]、自旋玻璃 [24]、图像数据处理 [25]、神经科学等。 1944 年,二维伊辛模型Onsager [26] 对正方形晶格进行了解析求解。该模型的哈密顿量由以下公式确定:

在哪里小号一世→是位于现场的自旋值一世, 符号⟨一世,j⟩表示最近邻段对,Ĵ是自旋相互作用的参数,H→是外部磁场强度。

Metropolis 算法使用系统的概率分布在预定温度下生成状态序列。对于 Ising 模型,Metropolis 算法应用如下:

  1. 选择并旋转随机旋转。
  2. 新的系统配置很可能被接受:
  3. 重复步骤 1 和 2。
    通过 Metropolis 算法获得的具有周期性边界条件的二维 Ising 模型的模拟结果如图 1 所示。热容量由以下公式确定:

物理代写|理论力学代写theoretical mechanics代考|Investigation of the Influence of Bulk Properties


模型。通常,聚合物链相对于相界面具有主要的平面取向 [30]。因此,在本文中,为了描述铁电聚合物系统的表面,我们使用了一个二维模型,该模型由米自由连接的链,每个链都是一个序列ñ连接的刚性段,平行于表面(图 4)。



类似于持久长度一个,我们引入链间交互参数b. 相邻聚合物链元素的取向相互作用由能量常数描述ķ2,

为了考虑表面分子与位于薄膜主体中的分子的相互作用,我们使用平均场常数在和无量纲平均场参数q :


H=12ķ1∑n,米=1ñ,米(披n,米−披n−1,米)2+12ķ2∑n,米=1ñ,米(披n,米−披n,米−1)2 −μ在∑n,米=1ñ,米因⁡(披n,米)

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。



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