## 澳洲代写｜ECOS3007｜ International Macroeconomics国际宏观经济学 悉尼大学

statistics-labTM为您悉尼大学（英语：The University of Sydney），简称悉大、USYD，简称“NCL”Stochastic Processes随机过程代写代考辅导服务！

This unit studies macroeconomic theory and policy in a global trading world. The microfoundations of the various sectors are examined in the context of an open economy. The evolution of international money and capital markets is described, the operation of the foreign exchange market is examined, showing how its microstructure affects its macro performance. Theories and tests of the efficiency of international capital markets are surveyed, as well as core theories and tests of exchange rate and asset price determination. The unit develops the macroeconomic implications of monetary and fiscal policies for small and large open economies for different regimes.

## International Macroeconomics国际宏观经济学相关

Assume that there exists free international capital mobility and that the world interest rate, $r^$, is $10 \%$ per period (i.e., $r^=0.1$ ). Finally,

assume that the economy’s initial net foreign asset position is zero $\left(B_0^*=0\right)$.
(a) Compute the firm’s optimal levels of period-1 investment and period-2 profits.
(b) State the maximization problem of the representative household and solve for the optimal levels of consumption in periods 1 and 2.
(c) Find the country’s net foreign asset position at the end of period 1 , the trade balance in periods 1 and 2 , and the current account in periods 1 and 2 .
(d) Now consider an investment surge. Specifically, assume that as a result of a technological improvement, the production technology becomes $Q_2=2 \sqrt{T_1}$. Find the equilibrium levels of savings, investment, the trade balance, the current account, and the country’s net foreign asset position in period 1. Compare your results with those obtained in items (a)-(c) providing interpretation and intuition.

(a) 计算公司第一期投资和第二期利润的最佳水平。
(b) 陈述代表性家庭的最大化问题并求解第 1 期和第 2 期的最优消费水平。
(c) 求该国在第 1 期末的外国净资产头寸、第 1 期和第 2 期的贸易差额以及第 1 期和第 2 期的经常账户。
(d) 现在考虑投资激增。 具体来说，假设由于技术改进，生产技术变为$Q_2=2 \sqrt{T_1}$。 求第 1 期储蓄、投资、贸易平衡、经常账户和国家净外国资产头寸的均衡水平。将您的结果与 (a)-(c) 项中获得的结果进行比较，以提供解释和直觉。

An investment surge
Suppose that in period 1 agents learn that in period 2 the productivity of capital will increase. For example, suppose that the production function in period 2 was initially given by $F\left(K_2\right)=\sqrt{K_2}$ and that due to a technological advancement it changes to $\tilde{F}\left(K_2\right)=2 \sqrt{K_2}$. Another example of an investment surge is given by an expected increase in the price of exports. In Norway, for instance, the oil price increase of 1973 unleashed an investment boom of around $10 \%$ of GDP. In response to this news, firms will choose to increase investment in period 1 for any given level of the interest rate. This scenario is illustrated in figure 6.4. Initially, the investment schedule is $I^0\left(r_1\right)$ and the saving schedule is $S^0\left(r_1, Q_1\right)$. Given the world interest rate $r^*$, investment is $I_1^0$ and savings is $S_1^0$. As shown in panel (b), the current account schedule is $C A^0\left(r_1, Q_1\right)$, and the equilibrium current account balance is $C A_1^0$. The news of the future productivity increase shifts the investment schedule to the right to $I^1\left(r_1\right)$, and the new equilibrium level of investment is $I_1^1$, which is higher than $I_1^0$. The expected increase in productivity might also affect current saving through its effect on expected future income. Specifically, in period 2, firms will generate higher profits which represent a positive income effect for households who are the owners of such firms. Households will take advantage of the expected increase in profits by increasing consumption in period 1 , thus cutting savings. Therefore, the savings schedule shifts to the left to $S^1\left(r_1, Q_1\right)$ and the equilibrium level of savings falls from $S_1^0$ to $S_1^1$. With this shifts in the investment and savings schedules it follows that, for any given interest rate, the current account is lower. That is, the current account schedule shifts to the left to $C A^1\left(r_1, Q_1\right)$. Given the world interest rate $r^*$, the current account deteriorates from $C A_1^0$ to $C A_1^1$. Note that if the economy was closed, the investment surge would trigger a rise in the domestic interest rate from $r_c^0$ to $r_c^1$ and thus investment would increase by less than in the open economy.

strictly true for the small investor, given the opportunity to hold commingled funds
and mutual funds it is not an unrealistic assumption.
The third assumption is the absence of personal income tax. This means, for
example, that the individual is indifferent to the form (dividends or capital gains) in
which the return on the investment is received. This is a realistic assumption for
pension and other tax-exempt funds.
The fourth assumption is that an individual’s or institution’s purchases or sales
cannot affect the price of a stock. This is analogous to the assumption of perfect

C. 最简单的均衡概念–标准资本资产定价模型

10

1. 基本假设
CAPM 模型描述了投资者应如何行为，以及市场清算的价格和收益如何设定。
回报率。在对该模型进行更详细的讨论之前，有必要说明一下 CAPM 的基本假设。
CAPM 的基本假设。
CAPM 的第一个假设是不存在交易成本。
买卖任何资产都没有成本（摩擦）。如果存在交易成本
存在，那么拥有任何资产的预期收益以及拥有该资产的可取性
将是投资者是否已经拥有该资产的函数。因此
交易成本会增加模型的复杂性。虽然交易成本
虽然交易成本可能足够大，足以在小投资者的决策过程中发挥作用
或影响作为近似替代品的替代投资（如本附录前后部分所述）。
对足够多的（大）投资者来说，交易成本已经小到足以影响他们的决策。
(大）投资者而言，它们的影响足够小，因此对均衡价格的影响应该很小。
CAPM 背后的第二个假设是资产是可分割的。
这意味着投资者可以在一项投资中持有任何头寸，无论其财富大小。
他们的财富大小。例如，他们可以购买 1 美元的 IBM 股票。虽然这
虽然这对小投资者来说并非完全正确，但如果有机会持有混合基金和共同基金
和共同基金的机会，这并非不切实际的假设。
第三个假设是没有个人所得税。这意味着
例如，个人对投资收益的形式（股息或资本利得）并不关心。
投资收益的形式。对于
对于养老金和其他免税基金来说，这是一个现实的假设。
第四个假设是个人或机构的购买或出售
不会影响股票价格。这类似于完美投资假设。

## 知识点： financial markets in discrete time离散时间金融市场

Proposition 2.1. Constant repayments. Assuming that the loan is completely repaid at the beginning of year $N+1$, the amount $m$ refunded every year is given by
$$m=\frac{r(1+r)^N A}{(1+r)^N-1}=\frac{r}{1-(1+r)^{-N}} A .$$
Proof. Denoting by $A_k$ the amount owed by the borrower at the beginning of year $\mathrm{n}^o k=1,2, \ldots, N$ with $A_1=A$, the amount $m$ refunded at the end of the first year can be decomposed as
$$m=r A_1+\left(m-r A_1\right),$$
into $r A_1$ paid in interest and $m-r A_1$ in principal repayment, i.e. there remains
\begin{aligned} A_2 & =A_1-\left(m-r A_1\right) \ & =(1+r) A_1-m, \end{aligned}
to be refunded. Similarly, the amount $m$ refunded at the end of the second year can be decomposed as
$$m=r A_2+\left(m-r A_2\right),$$

into $r A_2$ paid in interest and $m-r A_2$ in principal repayment, i.e. there remains
\begin{aligned} A_3 & =A_2-\left(m-r A_2\right) \ & =(1+r) A_2-m \ & =(1+r)\left((1+r) A_1-m\right)-m \ & =(1+r)^2 A_1-m-(1+r) m \end{aligned}
to be refunded. After repeating the argument we find that at the beginning of year $k$ there remains
\begin{aligned} A_k & =(1+r)^{k-1} A_1-m-(1+r) m-\cdots-(1+r)^{k-2} m \ & =(1+r)^{k-1} A_1-m \sum_{i=0}^{k-2}(1+r)^i \ & =(1+r)^{k-1} A_1+m \frac{1-(1+r)^{k-1}}{r} \end{aligned}
to be refunded, i.e.

## 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。

## PX281-15｜Computational Physics计算物理学 沃里克大学

statistics-labTM为您沃里克大学（英语：University of Warwick）Computational Physics计算物理学英国代写代考辅导服务！

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Vectorized programming in Python using Numpy
2. Handling, processing and analysing physics data: plotting distributions, data fitting, hypothesis testing
3. Monte Carlo simulation for physics modelling: Different types of random numbers, generation of random numbers according to specific distributions. Brownian motion and diffusion
4. Digital Signal Processing: the Fourier transform and convolution method, digital filters
5. Numerical solutions of ordinary differential equations: the Verlet algorithm for many coupled ODE’s
6. Speeding up Python: why, when and what again is a compiler

## Computational Physics计算物理学计案例

(a) If $u$ has $p-1$ continuous derivatives in $L^2(\mathbb{R})$, and a pth derivative of bounded value, $(p \geq 1)$ then
$$|\hat{v}(k)-\hat{u}(k)|=O\left(h^{p+1}\right) \text { as } h \rightarrow 0 .$$
(b) If $u$ has $\infty$ many continuous derivatives in $L^2(\mathbb{R})$, then
$$|\hat{v}(k)-\hat{u}(k)|=O\left(h^m\right) \text { as } h \rightarrow 0 \text { for every } m>0,$$
i.e., the expansion converges super-algebraically.

(a) If $u$ has $p-1$ continuous derivatives in $L^2(\mathbb{R})$ for some $p \geq v+1$, and a pth derivative of bounded value, $(p \geq 1)$ then
$$\left|w_j-u^{(v)}\left(x_j\right)\right|=O\left(h^{p-v}\right) \text { as } h \rightarrow 0 .$$
(b) If $u$ has $\infty$ many continuous derivatives in $L^2(\mathbb{R})$, then
$$\left|w_j-u^{(v)}\left(x_j\right)\right|=O\left(h^m\right) \text { as } h \rightarrow 0 \text { for every } m>0$$

## 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。

## MATH0039｜Differential and Integral Calculus微分和积分微积分 伦敦大学学院

statistics-labTM为您提供伦敦大学学院（University College London，简称：UCL）Differential and Integral Calculus微分和积分微积分英国代写代考辅导服务！

This course provides a fairly rapid introduction to calculus. Calculus underlies almost all areas
of mathematics and a great deal of science and engineering. The aim of the course will be to
provide a solid grounding in this fundamental branch of mathematics for students who have a
limited mathematical background.

## Differential and Integral Calculus微分和积分微积分案例

On a toll road a driver takes a time stamped toll-card from the starting booth and drives directly to the end of the toll section. After paying the required toll, the driver is surprised to receive a speeding ticket along with the toll receipt. Which of the following best describes the situation?

(a) The booth attendant does not have enough information to prove that the driver was speeding.
(b) The booth attendant can prove that the driver was speeding during his trip.
(c) The driver will get a ticket for a lower speed than his actual maximum speed.
(d) Both (b) and (c).
Be prepared to justify your answer.

Answer: (d). First answer this question in the easiest form, discussing instantaneous and average velocity – then go back and try to answer again in a way that allows the application of the mean value theorem easily. The fact that (c) is also true, can be shown by using a graph. This is probably the hardest part of the problem.

Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.

Let variables $x$ and $y$ represent two nonnegative numbers. The sum of the two numbers is given to be
$$9=x+y,$$
so that
$$y=9-x$$
We wish to MAXIMIZE the PRODUCT
$$P=x y^2$$
However, before we differentiate the right-hand side, we will write it as a function of $x$ only. Substitute for $y$ getting
$$\begin{gathered} P=x y^2 \ =x(9-x)^2 . \end{gathered}$$
Now differentiate this equation using the product rule and chain rule, getting
$$\begin{gathered} P^{\prime}=x(2)(9-x)(-1)+(1)(9-x)^2 \ =(9-x)[-2 x+(9-x)] \ =(9-x)[9-3 x] \ =(9-x)(3)[3-x] \ =0 \end{gathered}$$
for
$$x=9 \text { or } x=3 \text {. }$$

2A-5 Making reasonable assumptions, if a person 5 feet tall weighs on the average $120 \mathrm{lbs}$., approximately how much does a person $5^{\prime} 1^{\prime \prime}$ tall weigh?

2A-5 A reasonable assumption is that $w$ is propotional to volume $v$, which is in turn proportional to the cube of a linear dimension, i.e., a given person remains similar to him/herself, for small weight changes.) Thus $w=C h^3$; since 5 feet $=60$ inches, we get
$$\frac{w(60+\epsilon)}{w(60)}=\frac{C(60+\epsilon)^3}{C(60)^3}=\left(1+\frac{\epsilon}{60}\right)^3 \Rightarrow w(60+\epsilon) \approx w(60) \cdot\left(1+\frac{3 \epsilon}{60}\right) \approx 120 \cdot\left(1+\frac{1}{20}\right) \approx 126 .$$
[Or you can calculate the linearization of $w(h)$ arround $h=60$ using derivatives, and using the value $w(60)$ to determine $C$. getting $w(h) \approx 120+6(h-60)$

## 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。

## MATH20201｜Algebraic Structures 1代数结构 1 曼切斯特大学

statistics-labTM为您提供曼彻斯特大学（英语：The University of Manchester，缩写：UoM）Differential Geometry (Advanced)英国代写代考辅导服务！

This course unit provides an introduction to the main algebraic structures: groups and rings giving the main definitions, some
basic results and a wide range of examples. This builds on the study of topics such as properties of the integers, modular
arithmetic, and permutations included in MATH10101/MATH10111. These structures are fundamental concepts in
mathematics, particularly in the study of symmetry and of number theory.

## Algebraic Structures 1代数结构 1 案例

Problem 1. Show that
a) if $p(t), q(t) \in K[t]$ are two polynomials such that the corresponding principal ideals $(p(t))$ and $(q(t))$ coincide then there exists $c \in K-{0}$ such that $p(t)=c q(t)$,

(a) To compute $[\mathbb{C}: \mathbb{R}]$, note that $\mathbb{C}=\mathbb{R}(i)$ and $i$ is a solution to the polynomial $x^2+1 \in \mathbb{R}[x]$. This polynomial is irreducible, otherwise it would have a linear factor hence a solution in $\mathbb{R}$, which is impossible. So by problem 5 we see that $[\mathbb{C}: \mathbb{R}]=[\mathbb{R}(i): \mathbb{R}]=2$.

b) for any two non-zero polynomials $p(t), q(t) \in K[t]$ we have $\operatorname{deg}(p(t) q(t))=\operatorname{deg} p(t)+\operatorname{deg} q(t)$

(b) The field $\mathbb{Q}$ is countable. Hence any simple extension $\mathbb{Q}(\alpha)$ is also countable, but $\mathbb{R}$ is not countable.

To show that a simple extension of a countable field is also countable, note that such an extension is either isomorphic to a homomorphic image of the polynomial ring $F[x]$ (in the case of algebraic extension), or isomorphic to the field or rational functions $F(x)$ (in the case of transcendental extension). Since $F[x] \subset F(x)$ and $F(x)$ is countable, the claim follows.

c) if $p(t)$ is a non-zero polynomial of degree $n$ then it has no more then $n$ distinct roots.

(c) We use the definition of $K(\alpha)$ as the set of all elements of the form $f(\alpha) / g(\alpha)$ where $f, g \in K[x]$ and $g(\alpha) \neq 0$. Since this set is closed under additions, multiplications and taking inverses, it is a subfield of $L$. In fact, is it the minimal subfield of $L$ containing $K$ and $\alpha$.

## 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。

## MATH3968｜Differential Geometry (Advanced)微分几何学（高级） 悉尼大学

statistics-labTM为您提供悉尼大学（英语：The University of Sydney）Differential Geometry (Advanced)澳洲代写代考辅导服务！

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface.

## Differential Geometry (Advanced)微分几何学（高级）案例

Let $\kappa: I \rightarrow \mathbb{R}$ be a smooth function defined on an open interval I. Then there is a regular curve $\alpha: I \rightarrow \mathbb{R}^2$ parametrised by an length $s$ with curvature $\kappa$. Moreover, $\alpha$ is unique up to rigid motions of $\mathbb{R}^2$.

We first prove existence. Let $\theta: I \rightarrow \mathbb{R}$ be an indefinite integral of $\kappa$ (so that $\theta$ is a smooth function with $\theta^{\prime}=\kappa$ ), and let $x_1, x_2$ be indefinite integrals of $\cos \theta, \sin \theta$ respectively. If we let
$$\alpha(s)=\left(x_1(s), x_2(s)\right), \quad s \in I,$$
then $\alpha$ is a smooth curve, and
$$\alpha^{\prime}=(\cos \theta, \sin \theta) .$$
Hence
$$\alpha^{\prime \prime}=\theta^{\prime}(-\sin \theta, \cos \theta),$$
so that $\alpha$ is parametrised by arc length and has curvature $\theta^{\prime}=\kappa$.
We now prove the statement concerning uniqueness. So, let $\alpha_1(s)$ and $\alpha_2(s)$ be parametrised by arc length, both having the same curvature $\kappa$. We let $t_1=\left(\cos \theta_1, \sin \theta_1\right)$ and $\boldsymbol{t}_2=\left(\cos \theta_2, \sin \theta_2\right)$ be the unit tangent vectors to $\alpha_1$ and $\alpha_2$ respectively. Picking a base-point $s_0 \in I$ we may assume, by applying a suitable rigid motion of $\mathbb{R}^2$, that
$$\boldsymbol{\alpha}_1\left(s_0\right)=\alpha_2\left(s_0\right), \quad \boldsymbol{t}_1\left(s_0\right)=\boldsymbol{t}_2\left(s_0\right) .$$
Using Lemma 4, we see that $d \theta_1 / d s=d \theta_2 / d s$, so that $\theta_1-\theta_2$ is constant and hence, since $\theta_1\left(s_0\right)=\theta_2\left(s_0\right)$ by assumption, we see that $\theta_1=\theta_2$, so that $\boldsymbol{t}_1=\boldsymbol{t}_2$. But then, $d \boldsymbol{\alpha}_1 / d s=d \boldsymbol{\alpha}_2 / d s$, so a similar argument shows that $\boldsymbol{\alpha}_1=\boldsymbol{\alpha}_2$, and the uniqueness statement is proved.

We illustrate the existence part of the above proof by constructing directly all plane curves with constant non-zero curvature.

Let $\kappa: I \rightarrow \mathbb{R}$, $\tau: I \rightarrow \mathbb{R}$ be smooth functions defined on an open interval $I$, and assume that $\kappa>0$. Then there is a regular curve $\alpha: I \rightarrow \mathbb{R}^3$ parametrised by arc length $s$ with curvature $\kappa$ and torsion $\tau$. Moreover, $\alpha$ is unique up to rigid motions of $\mathbb{R}^3$.

For a given $\kappa$ and $\tau$, the Serret-Frenet formulae form a linear system of three first order ordinary differential equations for the $\mathbb{R}^3$-valued functions $\boldsymbol{t}, \boldsymbol{n}$ and $\boldsymbol{b}$, and the ODE theorem tells us that such a system has a unique solution ${\boldsymbol{t}, \boldsymbol{n}, \boldsymbol{b}}$ on $I$ for any set of initial conditions $\left{\boldsymbol{t}\left(s_0\right), \boldsymbol{n}\left(s_0\right), \boldsymbol{b}\left(s_0\right)\right}$. Then the six quantities $\boldsymbol{t} . \boldsymbol{t}, \boldsymbol{n} . \boldsymbol{n}, \boldsymbol{b} . \boldsymbol{b}, \boldsymbol{t} . \boldsymbol{n}, \boldsymbol{t} . \boldsymbol{b}, \boldsymbol{n} . \boldsymbol{b}$ satisfy a linear system of six first order ordinary differential equations (one of which, for instance, is $\left.\frac{d}{d s}(t . b)=\kappa n \cdot b+\tau t . n\right)$ for which $t \cdot t=n \cdot n=b \cdot b=1, t \cdot n=t \cdot b=n \cdot b=0$ is easily seen to be a solution. Thus, using the ODE theorem again, we see that any solution

of the Serret-Frenet formulae with initial trihedron being right-handed orthonormal will stay right-handed orthonormal.

We now prove the existence part of the theorem. For given functions $\kappa$ and $\tau$, let ${\boldsymbol{t}, \boldsymbol{n}, \boldsymbol{b}}$ be a right-handed orthonormal solution of the Serret-Frenet formulae, and let $\alpha: I \rightarrow \mathbb{R}^3$ be an indefinite integral of $t$. Then $\alpha$ is a smooth curve with $d \boldsymbol{\alpha} / d s=t$ and $d^2 \alpha / d s^2=$ $d \boldsymbol{t} / d s=\kappa \boldsymbol{n}$. It follows that $\alpha$ is parametrised by arc length, that $\boldsymbol{t}$ is the unit tangent vector, that $\boldsymbol{n}$ is the principal normal vector and $\kappa$ is the curvature. Thus $\boldsymbol{b}$ is the binormal, from which it follows that $\alpha$ has torsion $\tau$.

This completes the proof of existence, and we now prove uniqueness. Let $\alpha_1$ and $\alpha_2$ be smooth curves parametrised by arc length, both having the same curvature $\kappa$ and torsion $\tau$, and let $\left{\boldsymbol{t}_1, \boldsymbol{n}_1, \boldsymbol{b}_1\right},\left{\boldsymbol{t}_2, \boldsymbol{n}_2, \boldsymbol{b}_2\right}$ be the corresponding unit tangent vectors, principal normals and binormals. Picking a base point $s_0 \in I$ we may assume, by applying a suitable rigid motion of $\mathbb{R}^3$, that
$$\boldsymbol{\alpha}_1\left(s_0\right)=\boldsymbol{\alpha}_2\left(s_0\right), \quad \boldsymbol{t}_1\left(s_0\right)=\boldsymbol{t}_2\left(s_0\right), \quad \boldsymbol{n}_1\left(s_0\right)=\boldsymbol{n}_2\left(s_0\right), \quad \boldsymbol{b}_1\left(s_0\right)=\boldsymbol{b}_2\left(s_0\right),$$
and the uniqueness part of the ODE theorem now shows that $\left{\boldsymbol{t}_1, \boldsymbol{n}_1, \boldsymbol{b}_1\right}=\left{\boldsymbol{t}_2, \boldsymbol{n}_2, \boldsymbol{b}_2\right}$. In particular, $t_1=t_2$ so that $d \alpha_1 / d s=d \alpha_2 / d s$, and it follows that $\alpha_1-\alpha_2$ is constant. Since we applied a rigid motion so that $\alpha_1\left(s_0\right)=\alpha_2\left(s_0\right)$ we see that $\alpha_1=\alpha_2$, and uniqueness is proved.

## 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。

## MATH0030｜Mathematical Ecology数学生态学 伦敦大学学院

statistics-labTM为您提供伦敦大学学院（University College London，简称：UCL）Mathematical Ecology数学生态学英国代写代考辅导服务！

Mathematical models are used extensively in many areas of the Biological Sciences. This course
aims to give a sample of the construction and mathematical analysis of such models in Population Ecology. The fundamental question to be addressed is: what natural (or human) factors
control the abundance and distribution of the various populations of animals and plants that
we see in Nature?
No special knowledge of Ecology is required or assumed. However, an interest in, and willingness
to learn about, concepts and problems in this area are essential. Mathematical techniques used
include calculus, mathematical methods and linear algebra, and those developed include the
important qualitative technique of phase plane analysis which the course uses extensively.

## Mathematical Ecology数学生态学案例

The general deterministic model for the dynamics of a single, unstructured population reads:
$$\dot{N}(t)=f(N, t)=f(N(t), t)$$
$N(t) \in \mathbf{R}$ is usually interpreted as population density rather than the total size.

The most basic model is exponential growth with
$$\dot{N}(t)=r N(t)$$
and explicit solution $N(t)=N_0 \exp (r t)$. The net growth rate $r$ is also called the Malthusian parameter (Thomas Malthus, 1798: Essay on the Principle of Population). Exponential growth with $r>0$ leads to population explosion and is unsustainable in Nature. With $r<0$, there is a trivial stable equilibrium at $N=0$.

The simplest model with a stable equilibrium population derives from a dynamics with immigration and death (negative growth),
$$\dot{N}(t)=c-d N(t)$$
with $c, d>0$ and explicit solution
$$N(t)=\frac{c}{d}+\left(N_0-\frac{c}{d}\right) \exp [-d t] .$$
The model describes the dynamics in a population sink. It also describes pure migration if immigration is constant and emigration occurs with a constant per-capita rate.

An alternative approach assumes that larvae (or juveniles) do not compete primarily among themselves, but with their adult conspecifics. This will be true, in particular, if adult fish eat eggs and larvae of their own species (which is indeed true for many species, where predation is purely size-dependent).

\end{prob}

We then get

$$\frac{\partial L_t(\tau)}{\partial \tau}=-m_1 L_t(\tau)-m_2 N_t L_t(\tau)$$
which simply results in
$$L_t(\Delta)=b N_t \exp \left[-\left(m_1+m_2 N_t\right) \Delta\right] .$$
We then get a so-called Ricker model for stock recruitment
$$N_{t+1}=s N_t+N_t \exp \left[r\left(1-N_t / K\right)\right]$$
with constants
$$r=\log [b]-m_1 \Delta \quad, \quad K=\frac{\log [b]-m_1 \Delta}{m_2 \Delta}$$

## 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。