## 数学代写|应用数学代写applied mathematics代考|MTH103

2023年3月22日
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• Statistical Computing 统计计算
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|应用数学代写applied mathematics代考|Dimensional considerations

From (3.17), the material function $J$ has the dimension of energy length. It is often written as $J=E I$ where $E$ is Young’s modulus for the elastic material making up the rod, and $I$ is the moment of inertia of a cross-section.

Young’s modulus gives the ratio of tensile stress to tensile strain in an elastic solid. Strain, which measures a deformed length to an undeformed length, is dimensionless, so $E$ has the dimension of stress, force/area, meaning that
$$[E]=\frac{M}{L T^2} \text {. }$$
For example, the Young’s modulus of steel is approximately $200 \mathrm{kN} / \mathrm{mm}^2$.
The moment of inertia, in this context, is a second area moment of the rod cross-section, and has the dimension of $L^4$. The term ‘moment of inertia’ is also used to describe the relationship between angular velocity and angular momentum for a rotating rigid body; the moment of inertia here corresponds to this notion with mass replaced by area.

Explicitly, we define the components of a second-order, area-moment tensor of a region $\Omega \subset \mathbb{R}^2$ in the plane, with Cartesian coordinates $x_i, i=1,2$, by
$$I_{i j}=\int_{\Omega} x_i x_j d A$$
In general, this symmetric, positive-definite tensor has two positive real eigenvalues, corresponding to the moments of inertia about the principal axes defined by the corresponding eigenvectors. If these eienvalues coincide, then we get the the isotropic case with $I_{i j}=I \delta_{i j}$ where $I$ is the moment of inertia. For example, if $\Omega$ is a disc of radius $a$, then $I=\pi a^4 / 4$.

## 数学代写|应用数学代写applied mathematics代考|The persistence length of DNA

An interesting application of rod theories is to the modeling of polymers whose molecular chains resist bending, such as DNA. A statistical mechanics of flexible polymers may be derived by supposing that the polymer chain undergoes a random walk due to thermal fluctuations. Such polymers typically coil up because there are more coiled configurations that straight ones, so coiling is entropically favored.
If a polymer has elastic rigidity, then the increase in entropy that favors its coiling is opposed by the bending energy required to coil. As a result, the tangent vector of the polymer chain is highly correlated over distances short enough that significant bending energies are required to change its direction, while it is decorrelated over much longer distances. A typical lengthscale over which the tangent vector is correlated is called the persistence length of the polymer.

According to statistical mechanics, the probability that a system at absolute temperature $T$ has a specific configuration with energy $E$ is proportional to
$$e^{-E / k T}$$
where $k$ is Boltzmann’s constant. Boltzmann’s constant has the approximate value $k=1.38 \times 10^{-23} \mathrm{~J} K^{-1}$. The quantity $k T$ is an order of magnitude for the random thermal energy of a single microscopic degree of freedom at temperature $T$.

The bending energy of an elastic rod is set by the coefficent $J$ in (3.17), with dimension energy · length. Thus, the quantity
$$A=\frac{J}{k T}$$
is a lengthscale over which thermal and bending energies are comparable, and it provides a measure of the persistence length. For DNA, a typical value of this length at standard conditions is $A \approx 50 \mathrm{~nm}$, or about 150 base pairs of the double helix.

The statistical mechanics of an elastica, or ‘worm-like chain,’ may be described, formally at least, in terms of path integrals (integrals over an infinite-dimensional space of functions). The expected value $\mathbf{E}[\mathcal{F}(\vec{r})]$ of some functional $\mathcal{F}(\vec{r})$ of the elastica configuration is given by
$$\mathbf{E}[\mathcal{F}(\vec{r})]=\frac{1}{Z} \int \mathcal{F}(\vec{r}) e^{-\mathcal{E}(\vec{r}) / k T} D \vec{r},$$
where the right-hand side is a path integral over a path space of configurations $\vec{r}(s)$ using the Boltzmann factor (3.21) and the elastica energy (3.17).

# 应用数学代考

## 数学代写|应用数学代写applied mathematics代考|Dimensional considerations

$$[E]=\frac{M}{L T^2} \text {. }$$

$$I_{i j}=\int_{\Omega} x_i x_j d A$$

## 数学代写|应用数学代写applied mathematics代考|The persistence length of DNA

$$e^{-E / k T}$$

$$A=\frac{J}{k T}$$

elastica 或“蠕虫状链条”的统计力学至少可以根据路径积 分 (无限维函数空间上的积分) 进行形式化描述。期望值 $\mathbf{E}[\mathcal{F}(\vec{r})]$ 一些功能性的 $\mathcal{F}(\vec{r})$ elastica 配置的由下式给出
$$\mathbf{E}[\mathcal{F}(\vec{r})]=\frac{1}{Z} \int \mathcal{F}(\vec{r}) e^{-\mathcal{E}(\vec{r}) / k T} D \vec{r}$$

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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。