## 澳洲代写｜STAT6039｜Principles of Mathematical Statistics数学统计原理 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University ）Principles of Mathematical Statistics数学统计原理澳洲代写代考辅导服务！

A first course in mathematical statistics with emphasis on applications; probability, random variables, moment generating functions and correlation, sampling distributions, estimation of parameters by the methods of moments and maximum likelihood, hypothesis testing, the central limit theorem, and Bayesian statistics.

## Principles of Mathematical Statistics数学统计原理问题集

Posterior inference: suppose you have a $\operatorname{Beta}(4,4)$ prior distribution on the probability $\theta$ that a coin will yield a ‘head’ when spun in a specified manner. The coin is independently spun ten times, and ‘heads’ appear fewer than 3 times. You are not told how many heads were seen, only that the number is less than 3 . Calculate your exact posterior density (up to a proportionality constant) for $\theta$ and sketch it.

Predictive distributions: consider two coins, $C_1$ and $C_2$, with the following characteristics: $\operatorname{Pr}\left(\right.$ heads $\left.\mid C_1\right)=0.6$ and $\operatorname{Pr}\left(\right.$ heads $\left.\mid C_2\right)=0.4$. Choose one of the coins at random and imagine spinning it repeatedly. Given that the first two spins from the chosen coin are tails, what is the expectation of the number of additional spins until a head shows up?

Posterior distribution as a compromise between prior information and data: let $y$ be the number of heads in $n$ spins of a coin, whose probability of heads is $\theta$.
(a) If your prior distribution for $\theta$ is uniform on the range $[0,1]$, derive your prior predictive distribution for $y$,
$$\operatorname{Pr}(y=k)=\int_0^1 \operatorname{Pr}(y=k \mid \theta) d \theta,$$
for each $k=0,1, \ldots, n$.
(b) Suppose you assign a $\operatorname{Beta}(\alpha, \beta)$ prior distribution for $\theta$, and then you observe $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies between your prior mean, $\frac{\alpha}{\alpha+\beta}$, and the observed relative frequency of heads, $\frac{y}{n}$.
(c) Show that, if the prior distribution on $\theta$ is uniform, the posterior variance of $\theta$ is always less than the prior variance.
(d) Give an example of a $\operatorname{Beta}(\alpha, \beta)$ prior distribution and data $y, n$, in which the posterior variance of $\theta$ is higher than the prior variance.

## 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。

statistics-labTM为您提供澳大利亚国立大学（The Australian National University ）Advanced Mathematical Statistics高级数学统计澳洲代写代考辅导服务！

This course introduces important algorithms and techniques of scientific computing, focussing on the area of numerical optimisation. The course will present both theoretical and practical aspects of the algorithms. The course is highly relevant to students from disciplines such as science, engineering or economics where skills in numerical computations are important.

A total of twelve hundred graduates of State Tech have gotten into medical school in the past several years. Of that number, one thousand earned scores of twenty-seven or higher on the MCAT and four hundred had GPAs that were 3.5 or higher. Moreover, three hundred had MCATs that were twenty-seven or higher and GPAs that were 3.5 or higher. What proportion of those twelve hundred graduates got into medical school with an MCAT lower than twenty-seven and a GPA below 3.5 ?

Let $A, B$, and $C$ be any three events defined on a sample space $S$. Let $N(A), N(B), N(C), N(A \cap$ $B), N(A \cap C), N(B \cap C)$, and $N(A \cap B \cap C)$ denote the numbers of outcomes in all the different intersections in which $A, B$, and $C$ are involved. Use a Venn diagram to suggest a formula for $N(A \cup B \cup C)$. [Hint: Start with the sum $N(A)+N(B)+N(C)$ and use the Venn diagram to identify the “adjustments” that need to be made to that sum before it can equal $N(A \cup B \cup C)$.] As a precedent, note that $N(A \cup B)=N(A)+N(B)-N(A \cap B)$. There, in the case of two events, subtracting $N(A \cap B)$ is the “adjustment.”

A poll conducted by a potential presidential candidate asked two questions: (1) Do you support the candidate’s position on taxes? and (2) Do you support the candidate’s position on homeland security? A total oftwelve hundred responses were received; six hundred said “yes” to the first question and four hundred said “yes” to the second. If three hundred respondents said “no” to the taxes question and “yes” to the homeland security question, how many said “yes” to the taxes question but “no” to the homeland security question?
For two events $A$ and $B$ defined on a sample space $S, N\left(A \cap B^C\right)=15, N\left(A^C \cap B\right)=50$, and $N(A \cap B)=2$. Given that $N(S)=120$, how many outcomes belong to neither $A$ nor $B$ ?

According to a family-oriented lobbying group, there is too much crude language and violence on television. Forty-two percent of the programs they screened had language they found offensive, $27 \%$ were too violent, and $10 \%$ were considered excessive in both language and violence. What percentage of programs did comply with the group’s standards?

Let $A$ and $B$ be any two events defined on $S$. Suppose that $P(A)=0.4, P(B)=0.5$, and $P(A \cap B)=$ 0.1 . What is the probability that $A$ or $B$ but not both occur?

## 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。

## 澳洲代写｜MATH3514｜Numerical Optimisation数值优化 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University Numerical Optimisation数值优化澳洲代写代考辅导服务！

This course introduces important algorithms and techniques of scientific computing, focussing on the area of numerical optimisation. The course will present both theoretical and practical aspects of the algorithms. The course is highly relevant to students from disciplines such as science, engineering or economics where skills in numerical computations are important.

## Numerical Optimisation数值优化问题集

Compute the gradient $\nabla f(x)$ and Hessian $\nabla^2 f(x)$ of the Rosenbrock function
$$f(x)=100\left(x_2-x_1^2\right)^2+\left(1-x_1\right)^2 .$$Show that $x^*=(1,1)^T$ is the only local minimizer of this function, and that the Hessian matrix at that point is positive definite.Show that the function $f(x)=8 x_1+12 x_2+x_1^2-2 x_2^2$ has only one stationary point, and that it is neither a maximum or minimum, but a saddle point. Sketch the contour lines of $f$.

Let $a$ be a given $n$-vector, and $A$ be a given $n \times n$ symmetric matrix. Compute the gradient and Hessian of $f_1(x)=a^T x$ and $f_2(x)=x^T A x$.

Consider the function $f: \mathbf{R}^2 \rightarrow \mathbf{R}$ defined by $f(x)=|x|^2$. Show that the sequence of iterates $\left{x_k\right}$ defined by
$$x_k=\left(1+\frac{1}{2^k}\right)\left[\begin{array}{c} \cos k \ \sin k \end{array}\right]$$
satisfies $f\left(x_{k+1}\right)<f\left(x_k\right)$ for $k=0,1,2, \ldots$. Show that every point on the unit circle $\left{x \mid|x|^2=1\right}$ is a limit point for $\left{x_k\right}$. Hint: Every value $\theta \in[0,2 \pi]$ is a limit point of the subsequence $\left{\xi_k\right}$ defined by
$$\xi_k=k(\bmod 2 \pi)=k-2 \pi\left\lfloor\frac{k}{2 \pi}\right\rfloor,$$
where the operator $\lfloor\cdot\rfloor$ denotes rounding down to the next integer.

Prove that all isolated local minimizers are strict. (Hint: Take an isolated local minimizer $x^$ and a neighborhood $\mathcal{N}$. Show that for any $x \in \mathcal{N}, x \neq x^$ we must have $f(x)>f\left(x^*\right)$.)

Suppose that $f(x)=x^T Q x$, where $Q$ is an $n \times n$ symmetric positive semidefinite matrix. Show using the definition (1.4) that $f(x)$ is convex on the domain $\mathbb{R}^n$. Hint: It may be convenient to prove the following equivalent inequality:
$$f(y+\alpha(x-y))-\alpha f(x)-(1-\alpha) f(y) \leq 0,$$
for all $\alpha \in[0,1]$ and all $x, y \in \mathbb{R}^n$.

## 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。

statistics-labTM为您提供澳大利亚国立大学（The Australian National UniversityAdvanced Complex Analysis高级复分析澳洲代写代考辅导服务！

This course is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered include:

Complex differentiability, conformal mapping; complex integration, Cauchy integral theorems, Taylor series representation, isolated singularities, residue theorem and applications to real integration. Topics chosen from: argument principle, Riemann surfaces, theorems of Picard, Weierstrass and Mittag-Leffler.

Note: This is an HPC. It emphasises mathematical rigour and proof and develops the material from an abstract viewpoint.

Consider
\begin{aligned} & f(z)=\frac{x y^2(x+\mathrm{i} y)}{x^2+y^4} \quad(z=x+\mathrm{i} y \neq 0) \ & f(0)=0 \end{aligned}
Verify that $\lim _{z \rightarrow z_0}\left(f(z)-z_0\right) /\left(z-z_0\right)=0$ as $z \rightarrow 0$ along any straight line, $z=(a+\mathrm{i} b) t, t \in \mathbb{R}$. This does not prove that $f^{\prime}(0)=0$, however. By considering $z \rightarrow 0$ along the path $z(t)=t^2+\mathrm{i} t$, show that $f$ is not differentiable at 0 . (This shows that when computing $f^{\prime}$ it is not enough to consider a limit taken along certain specific paths or types of path. An entire neighbourhood of the point concerned must be considered.)

Prove that if each of the series $\sum a_n z^n, \sum b_n z^n$, and $\sum a_n b_n z^n$ has radius of convergence equal to 1 , then so have the series $\sum a_n b_n^2 z^n$ and $\sum a_n^2 b_n z^n$.

For $\left|a_n\right| \leq 1$, show that $\sum a_n z^n$ is absolute convergent for all $|z|<1$. If $\sum a_n z^n=$ $f(z)$ for $\left|a_n\right|<1,|z|<1$, show that
$$|f(z)| \leq \frac{1}{1-|z|}$$

Prove that, for $z \neq 1$,
$$\sum_{n=1}^k \frac{z^n}{n}=\frac{z}{1-z}\left(\sum_{n=1}^{k-1} \frac{1}{n(n+1)}-\sum_{n=1}^{k-1} \frac{z^n}{n(n+1)}+\frac{1-z^k}{k}\right)$$
Show that the series $\sum_{n=1}^{\infty} z^n / n$ and $\sum_{n=1}^{\infty} z^n /(n(n+1))$ have radius of convergence 1 ; that the latter series converges everywhere on $|z|=1$, while the former series converges everywhere on $|z|=1$ except $z=1$.

Suppose that the power series $\sum_{n=0}^{\infty} a_n z^n$ has a recurring sequence of coefficients; that is, $a_{n+k}=a_n$ for some fixed positive integer $k$ and all $n$. Prove that the series converges for $|z|<1$ to a rational function $p(z) / q(z)$ where $p, q$ are polynomials, and that the roots of $q$ are all on the unit circle.
What happens if $a_{n+k}=a_n / k$ instead?

## 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。

## 澳洲代写｜MATH1003｜Algebra and Calculus Methods代数与微积分方法 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Algebra and Calculus Methods代数与微积分方法澳洲代写代考辅导服务！

The course will discuss the three main classes of equations, elliptic, parabolic and hyperbolic. It is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered will include fundamental solutions, maximum principles, regularity (smoothness) of solutions, variational problems, Holder and Sobolev spaces.

## Algebra and Calculus Methods代数与微积分方法问题集

Consider the expression
$$\left(\cdots\left(\left((x-2)^2-2\right)^2-2\right)^2-\cdots-2\right)^2 \quad \text { (n squares) } .$$
Find the coefficient of $x^2$.

Solution. Let $a_n$ be the coefficient of $x^2$, and $b_n$ the coefficient of $x$. We observe that for any $n$, the term not containing $x$ in the above development is 4 . It follows that
$$a_n=4 a_{n-1}+b_{n-1}^2 \quad \text { and } b_n=4 b_{n-1},$$
where $a_1=1$ and $b_1=-4$. We first obtain that $b_n=-4^n$. Substituting in the recurrence relation corresponding to $a_n$, we obtain $a_n=4 a_{n-1}+4^{2 n-2}$, which implies $a_n=4^{n-1}\left(4^n-1\right) / 3$.

An interesting additive decomposition of positive integers in terms of Fibonacci numbers is presented in what follows.

Prove that any positive integer $N$ may be written as the sum of distinct and nonconsecutive terms of the Fibonacci sequence.

Solution. Let $\left(F_n\right){n \geq 1}$ be the Fibonacci sequence. Then $F_1=F_2=1$ and $F{n+2}=$ $F_{n+1}+F_n$, for all $n \geq 1$. Let us assume that $F_n \leq N<F_{n+1}$. So, $0 \leq N-F_n<$ $F_{n-1}$. It follows that there exists $s<n-1$ such that $F_s \leq N-F_n<F_{s+1}$. Hence $0 \leq N-F_n-F_s<F_{s-1}$ and $s-1<n-2$. We thus obtain that $N$ may be written
1.3 Recurrent Sequences
as $N=F_n+F_s+F_p+\cdots+F_r$, where the consecutive subscripts $n, s, p, \ldots, r$ are nonconsecutive numbers.

We need in what follows elementary differentiability properties of polynomials.

For any real number $a$ and for any positive integer $n$ define the sequence $\left(x_k\right){k \geq 0}$ by $x_0=0, x_1=1$, and $$x{k+2}=\frac{c x_{k+1}-(n-k) x_k}{k+1}, \quad \text { for all } k \geq 0 .$$
Fix $n$ and let $c$ be the largest real number such that $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k, 1 \leq k \leq n$.

Solution. We first observe that $x_{n+1}$ is a polynomial of degree $n$ in $c$. Thus, it is enough to find $n$ values of $c$ such that $X_{n+1}=0$. We will prove that these values are $c=n-1-2 r$, for $r=0,1, \ldots, n-1$. In this case, $x_k$ is the coefficient of $t^{k-1}$ in the polynomial $f(t)=(1-t)^r(1+t)^{n-1-r}$. This property follows after observing that $f$ satisfies the identity
$$\frac{f^{\prime}(t)}{f(t)}=\frac{n-1-r}{1+t}-\frac{r}{1-t},$$
that is,
$$\left(1-t^2\right) f^{\prime}(t)=f(t)[(n-1-r)(1-t)-r(1+t)]=f(t)[(n-1-2 r)-(n-1) t] .$$
Identifying the coefficients of $t^k$ in both sides, we obtain
$$(k+1) x_{k+2}-(k-1) x_k=(n-1-2 r) x_{k+1}-(n-1) x_k .$$
In particular, the largest $c$ is $n-1$, and $x_k=C_{n-1}^{k-1}$, for $k=1,2, \ldots, n$.
The next problem is devoted to the study of the normalized logistic equation, which is a successful model of many phenomena arising in genetics and mathematical biology. In its simplest form, the logistic equation is a formula for approximating the evolution of an animal population over time. The unknown $a_n$ in the following recurrent sequence represents the number of animals after the $n t h$ year. It is easy to observe that the sequence $\left(a_n\right)_{n \geq 1}$ converges to zero. The interesting part of the problem is to deduce the first-and second-order decay terms of this sequence.

## 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。

## 澳洲代写｜MATH8202｜Theory of Partial Differential Equations偏微分方程理论 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Theory of Partial Differential Equations偏微分方程理论澳洲代写代考辅导服务！

The course will discuss the three main classes of equations, elliptic, parabolic and hyperbolic. It is intended both for mathematics students continuing to honours work and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

Topics to be covered will include fundamental solutions, maximum principles, regularity (smoothness) of solutions, variational problems, Holder and Sobolev spaces.

## Partial Differential Equations偏微分方程问题集

(a) On the bounded domain $\Omega$ with smooth boundary, let $u$ be a solution of the problem
$$\Delta u+a_i(\mathbf{x}) \frac{\partial u}{\partial x_i}=f(\mathbf{x}), \quad \frac{\partial u}{\partial n}=0 \text { on } \partial \Omega .$$
Assume that $f \geq 0$ in $\Omega$. Show that $u$ is a constant and $f=0$.
(b) Show that problem (4.19) can have a solution only if
$$\int_{\Omega} f(\mathbf{x}) v(\mathbf{x}) d \mathbf{x}=0$$
for every solution $v$ of the “adjoint” equation
$$\Delta v-\frac{\partial}{\partial x_i}\left(a_i(\mathbf{x}) v\right)=0, \quad \frac{\partial v}{\partial n}-a_i(\mathbf{x}) n_i v=0 \text { on } \partial \Omega .$$
(c) Using techniques to be developed in later chapters, one can show that the condition)is also sufficient and that the solution space of is one-dimensional. Taking these facts for granted, show that solutions of are either non-negative or non-positive. Equations of the form are called Fokker-Planck equations and arise in statistical physics. Only non-negative solutions are physically meaningful.

Let $\Omega$ be a regular hexagon with side $a$. Let $\lambda \in \mathbb{R}$ be such that the equation $\Delta u+\lambda u=0$ with boundary condition $u=0$ has a nontrivial solution in $\Omega$. Give a lower bound for $\lambda$.

Verify that the function given by (4.22) is harmonic.

Prove the second claim in the subsection on subharmonic functions.
. Show that if $u$ is of class $C^2$ and subharmonic in the sense defined in this section, then $\Delta u \geq 0$.

Let the sequence $f_m$ be equicontinuous at each point of a compact set $S$. Show that it is uniformly equicontinuous on $S$.

Let $\Omega=\left{(x, y) \in \mathbb{R}^2 \mid 0<x^2+y^2<1\right}$. Prove that there is no solution to the Dirichlet problem $\Delta u=0$ in $\Omega, u(\mathbf{x})=1$ for $x^2+y^2=1$, $u(\mathbf{0})=0$. Hint: Show first that if there is a solution, then there is also a radially symmetric solution.

## 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。

## 澳洲代写｜MATH6406｜Partial Differential Equations, Fourier Analysis and Complex Analysis偏微分方程、傅立叶分析和复分析 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Partial Differential Equations, Fourier Analysis and Complex Analysis偏微分方程、傅立叶分析和复分析澳洲代写代考辅导服务！

Many physical processes such as vibrating strings, diffusion of heat and fluid flows are well modelled by partial differential equations and/or integral equations. This course provides an introduction to methods for solving and analysing standard partial differential equations and integral equations, including an introduction to complex analytic techniques.

## Partial Differential Equations偏微分方程问题集

Let $\Omega$ be bounded and assume $c \leq 0$ in $\Omega$. Let $L u \geq 0$ (or, respectively, $L u \leq 0$ ). Then
$$\max {\bar{\Omega}} u \leq \max {\partial \Omega} u^{+} \text {(or, resp., } \min {\bar{\Omega}} u \geq \min {\partial \Omega} u^{-} \text {). }$$
Here, $u^{+}=\max (u, 0), u^{-}=\min (u, 0)$. In particular, if $L u=0$ in $\Omega$, then
$$\max {\bar{\Omega}}|u|=\max {\partial \Omega}|u| \text {. }$$
Proof. If $u \leq 0$ throughout $\Omega$, the corollary is trivially true. Hence we may assume that $\Omega^{+}=\Omega \cap{u>0} \neq \emptyset$. On $\Omega^{+}$, we have $-c u \geq 0$, and hence
$$a_{i j} \frac{\partial^2 u}{\partial x_i \partial x_j}+b_i \frac{\partial u}{\partial x_i} \geq 0 .$$
Hence the previous theorem implies that the maximum of $u$ on the closure of $\Omega^{+}$is equal to its maximum on $\partial \Omega^{+}$. Since $u=0$ on $\partial \Omega^{+} \cap \Omega$, this maximum must be achieved on $\partial \Omega$.

The following corollary is typically used in applications. It yields a uniqueness result as well as a comparison principle.

Assume $L u \geq 0$ ( $L u \leq 0$ ) in $\Omega$ (not necessarily bounded) and assume that $u$ is not constant. If $c=0$, then $u$ does not achieve its maximum (minimum) in the interior of $\Omega$. If $c \leq 0, u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c, u$ cannot be zero at an interior maximum (minimum).
Proof. Assume that $u$ assumes its maximum $M$ at an interior point and let $\Omega^{-}=\Omega \cap{u<M}$. If $\Omega^{-}$is not empty, then $\partial \Omega^{-} \cap \Omega$ is not empty. Let $\mathbf{y}$ be a point in $\Omega^{-}$that is closer to $\partial \Omega^{-}$than to $\partial \Omega$ and let $B$ be the largest ball contained in $\Omega^{-}$and centered at $\mathbf{y}$. Let $\mathbf{x}_0$ be a point on $\partial B \cap \partial \Omega^{-}$. Then we can apply the previous lemma to $B$. We conclude that $\nabla u$ is nonzero at $\mathbf{x}_0$, contradicting the assumption that $u$ assumes its maximum there.

Let $u$ be a solution of $\Delta u=u^3-u$ on a bounded domain $\Omega$. Assume that $u=0$ on $\partial \Omega$. Show that $u \in[-1,1]$ throughout $\Omega$. Can the values \pm 1 be achieved?
Assume that the $n \times n$ matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive semidefinite. Show that $\operatorname{tr}(\mathbf{A B}) \geq 0$. Hint: $\mathbf{B}=\sum_k \lambda_k \mathbf{q}_k \mathbf{q}_k^T$, where the $\lambda_k$ and $\mathbf{q}_k$ are the eigenvalues and eigenvectors of $\mathbf{B}$.

## 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。

## 澳洲代写｜MATH6216｜Advanced Topics in Algebra 代数高级专题 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Advanced Topics in Algebra 代数高级专题澳洲代写代考辅导服务！

This course introduces students to key concepts and techniques in advanced algebra. Topics will be taken from contemporary research areas in Algebra.

Possible topics include:

Algebraic number theory, Analytic number theory, Algebraic geometry and scheme theory, Sheaf theory, Derived and Triangulated categories, Algebraic curves and Riemann surfaces

## Advanced Topics in Algebra 代数高级专题问题集

$\sharp$ EXERCISE. Let $K$ be a field and let $a_1, \ldots, a_n \in K$. Show that the ideal
$$\left(X_1-a_1, \ldots, X_n-a_n\right)$$
of the ring $K\left[X_1, \ldots, X_n\right]$ (of polynomials with coefficients in $K$ in indeterminates $\left.X_1, \ldots, X_n\right)$ is maximal.

The concept of maximal ideal in a commutative ring immediately leads to the very important idea of the Jacobson radical of such a ring.

Definition. Let $R$ be a commutative ring. We define the Jacobson radical of $R$, sometimes denoted by $\operatorname{Jac}(R)$, to be the intersection of all the maximal ideals of $R$.

Thus $\operatorname{Jac}(R)$ is an ideal of $R$ : even in the case when $R$ is trivial, our convention concerning the intersection of the empty family of ideals of a commutative ring means that $\operatorname{Jac}(R)=R$.

Note that when $R$ is quasi-local, $\operatorname{Jac}(R)$ is the unique maximal ideal of $R$.

We can provide a characterization of the Jacobson radical of a commutative ring.

Let $R$ be a commutative ring, and let $r \in R$. Then $r \in$ $\mathrm{Jac}(R)$ if and only if, for every $a \in R$, the element $1-r a$ is a unit of $R$.
Proof. ( $\Rightarrow$ ) Suppose that $r \in \operatorname{Jac}(R)$. Suppose that, for some $a \in R$, it is the case that $1-r a$ is not a unit of $R$. Then, by 3.11, there exists a maximal ideal $M$ of $R$ such that $1-r a \in M$. But $r \in M$ by definition of $\operatorname{Jac}(R)$, and so
$$1=(1-r a)+r a \in M$$
$(\Leftarrow)$ Suppose that, for each $a \in R$, it is the case that $1-r a$ is a unit of $R$. Let $M$ be a maximal ideal of $R$ : we shall show that $r \in M$. If this were not the case, then we should have
$$M \subset M+R r \subseteq R$$
Hence, by the maximality of $M$, we deduce that $M+R r=R$, so that there exist $b \in M$ and $a \in R$ with $b+a r=1$. Hence $1-r a \in M$, and so cannot be a unit of $R$. This contradiction shows that $r \in M$, as claimed. As this is true for each maximal ideal of $R$, we have $r \in \mathrm{Jac}(R)$.

## 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。

## 澳洲代写｜MATH6213｜Complex Analysis复分析 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Complex Analysis复分析澳洲代写代考辅导服务！

This course is intended both for continuing mathematics students and for other students using mathematics at a high level in theoretical physics, engineering and information technology, and mathematical economics.

## Complex Analysis复分析问题集

Let $\gamma_1=[\mathrm{i}, 0]$ and $\gamma_2=[0,1]$. Show that $\gamma_1+\gamma_2$ is defined, but $\gamma_2+\gamma_1$ is not.
Let $\gamma_1=[0,1]$ and $\gamma_2=[1,0]$. Show that $\gamma_1+\gamma_2$ and $\gamma_2+\gamma_1$ are both defined, but they are different paths.

Let $S$ be a subset of $\mathbb{C}$. If $z, w \in S$, define $z \sim w$ if and only if there is a path from $z$ to $w$. Show that $\sim$ is an equivalence relation. The equivalence classes are components of $S$. If $S$ is open and non-empty, show that each component is a domain.

Let $S$ be a path-connected subset of $\mathbb{C}$, and let $f: S \rightarrow \mathbb{C}$ be a continuous function. Prove that $f(S)$ is path-connected (even though it may not be open).

Give explicit functions for paths that describe the curves of Figure 2.28 in the direction indicated by the arrows. (All subpaths are parts of circles or line segments; $0<\varepsilon<R$ and $x_1, x_2, y_1, y_2$ are positive reals.)

In the construction of a space-filling curve in Section 2.9, why can we not just take $\gamma_n$ to be a path that zigzags $n$ times across $\mathbb{U}^2$, as in Figure 2.29? Justify your answer.

Let $\mathbb{U}^3$ be the unit cube in $\mathbb{R}^3$. Show that there exists a continuous map from $[0,1]$ onto $\mathbb{U}^3$. (Hint: the hard way is to construct paths with similar properties to those in Section 2.9. The easy way is to observe that $\mathbb{U}^3=\mathbb{U}^2 \times[0,1]$. Map $[0,1] \times[0,1]$ onto $\mathbb{U}^2 \times[0,1]$; then compose with $\gamma:[0,1] \rightarrow[0,1] \times[0,1]$.)

Let $\mathbb{U}^n$ be the unit hypercube in $\mathbb{R}^n$. Show that there exists a continuous map from $[0,1]$ onto $\mathbb{U}^n$.

Let $m, n>0$ be integers. Prove that there exists a continuous map from $\mathbb{U}^m$ onto $\mathbb{U}^n$, even when $m<n$.

If $\gamma_n$ is a sequence of step paths in $\mathbb{U}^2$, can the union of their tracks equal $\mathbb{U}$ (rather than just being dense in $\mathbb{U}^2$ )? Prove your answer correct.

Does there exist a continuous map $\gamma:[0,1] \rightarrow \mathbb{U}^2$ that is one-one as well as onto? Justify your answer.

## 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。

## 澳洲代写｜MATH6005｜Discrete Mathematical Models离散数学模型 澳洲国立大学

statistics-labTM为您提供澳大利亚国立大学（The Australian National University）Foundations of Mathematics数学基础澳洲代写代考辅导服务！

This course is a critical approach to the foundations of mathematics. In other mathematics classes, the philosophical concepts at the most basic foundations are usually treated naively. The question of what exactly a number is, or what a set or a proof or an algorithm are, is completely ignored. Some evidence that these matters are not insubstantial is that in the early twentieth century, naive attempts to address them by the great logicians of the time led to famous paradoxes and a period known as the Crisis in Foundations of Mathematics.

## Discrete Mathematical Models离散数学模型问题集

Show by examples that neither the assertion in lemma 6.5 .2 nor Fermat’s “Little” Theorem remains valid if we drop the assumption that $p$ is a prime.
Consider a regular $p$-gon, and for a fixed $k(1 \leq k \leq p-1)$, consider all $k$-subsets of the set of its vertices. Put all these $k$-subsets into a number of boxes: We put two $k$-subsets into the same box if they can be rotated into each other. For example, all $k$-subsets consisting of $k$ consecutive vertices will belong to one and the same box.
(a) Prove that if $p$ is a prime, then each box will contain exactly $p$ of these rotated copies.
(b) Show by an example that (a) does not remain true if we drop the assumption that $p$ is a prime.
6.6 The Euclidean Algorithm
99
(c) Use (a) to give a new proof of Lemma

Imagine numbers written in base $a$, with at most $p$ digits. Put two numbers in the same box if they arike by a cyclic shift from each other. How many will be in each class? Give a new proof of Fermat’s Theorem this way.
Give a third proof of Fermat’s “Little” Theorem based on Exercise 6.3.5.
[Hint: Consider the product $a(2 a)(3 a) \cdots((p-1) a)$.]

Show that if $a$ and $b$ are positive integers with $a \mid b$, then $\operatorname{gcd}(a, b)=a$.
(a) Prove that $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, b-a)$.
(b) Let $r$ be the remainder if we divide $b$ by $a$. Then $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, r)$.

(a) If $a$ is even and $b$ is odd, then $\operatorname{gcd}(a, b)=\operatorname{gcd}(a / 2, b)$.
(b) If both $a$ and $b$ are even, then $\operatorname{gcd}(a, b)=2 \operatorname{gcd}(a / 2, b / 2)$.

How can you express the least common multiple of two integers if you know the prime factorization of each?

Suppose that you are given two integers, and you know the prime factorization of one of them. Describe a way of computing the greatest common divisor of these numbers.

Prove that for any two integers $a$ and $b$,
$$\operatorname{gcd}(a, b) \operatorname{lcm}(a, b)=a b .$$

## 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。