数学代写|Ross数学夏令营2023选拔代写

Posted on 2024年1月30日2024年1月30日 by statistics-lab

This document is one part of the application to the Ross Mathematics Program,and will remain posted at https://rossprogram.org/students/to-apply from January through March. The deadline for applications is March 15, 2024. The Admissions Committee will start reading applications on March 16.

You are not expected to answer every question perfectly; rather, take this application as an opportunity to explore some beautiful mathematics! We are interested in seeing how you approach unfamiliar, open-ended math problems, and we encourage you to write up your discoveries and conjectures, even if you can’t prove them.
We believe that the most valuable part of a problem is the time spent thinking on it, and your application should reflect this: we are not looking for quick answers written in minimal space. Instead, we hope to see evidence of your explorations, conjectures, proofs, and generalizations written in a readable format. If you make progress on these four problems (even if you don’t solve them completely), we encourage you to submit
your Ross application.
Submit your own work on these problems. If you’ve seen one of the problems before (e.g. in a class or online), please include a reference along with your solution.
Admission factors include the quality of mathematical exposition and the questions you pose, as well as the completeness and correctness of your solutions to those questions.

Work on the 2024 Application Problems which you can find at https://raw.githubusercontent.com/rossprogram/rossprogram.github.io/master/students/application-problems.pdf. There are four problems. Upload your carefully written solutions to those four problems.

The solution to each problem must be uploaded as a separate PDF.
Use the PDF file format. If your solution file is in some other fomat, please transform it to a PDF, check that the converted file is readable, and then upload that PDF.
We welcome solutions that have been prepared with LaTeX or some similar typesetting program that produces PDFs. It is OK to type up your work using a word processor (like Microsoft Word), and then converting that file to a PDF. Please verify that the mathematical symbols you use are readable.
You are allowed to write up your solutions by hand, provided you use black ink or very dark pencil. Scan each page, combine the pages for one problem into a single PDF, check that the file is readable, and then upload that file. Please be sure to reduce the file size to be less than five megabytes. The Ross system cannot accept any files of larger size.

The Admissions Committee is not looking for quick answers written in minimal space, but rather for readable mathematical expositions that includes evidence of your explorations, conjectures, and proofs.

Problem 1

A polynomial is integral when it has integer coefficients. The square root of 2 is a solution to the integral polynomial equation $x^2-2=0$.
A number is rational when it can be expressed as $\frac{a}{b}$ for integers $a$ and $b$ (with $b \neq 0$ ).
A number is irrational when it is not rational.
(a) Suppose $c$ is a non-square integer. (That is, $c \neq n^2$ for any $n$.) Explain why $\sqrt{c}$ is not rational. Similarly, if $c$ is a non-cube integer, does it follow that $\sqrt[3]{c}$ is irrational?
(b) Find an integral polynomial equation that has $\alpha=\sqrt{3}+\sqrt{5}$ as a solution. Show that $\alpha$ is irrational.
(c) Let $a$ and $b$ be integers. Find an integral polynomial equation which has $\sqrt{a}+\sqrt{b}$ as a solution. Must $\sqrt{a}+\sqrt{b}$ be irrational? If $a \neq b$, must $\sqrt{a}-\sqrt{b}$ be irrational?
(d) Formulate some generalizations. As a starting point, is $\beta=\sqrt{3}+\sqrt{5}+\sqrt{7}$ irrational? What about numbers like $\gamma=3 \sqrt{2}-2 \sqrt{3}-3 \sqrt{5}+\sqrt{6}$ and $\delta=\sqrt[3]{5}-\sqrt{2}$ ?

Problem 2

Here’s a (paraphrased) conversation that took place between two Ross students.
A: Hey, want to see a magic trick?
B: Sure, how does it go?
A: Think of any number. Any nonnegative integer, I should say.
B: Okay.
A: Multiply it by 3 .
B: Okay….
A: Now divide it by 2, and if you get a decimal, then round down. Tell me if you rounded down or not.
B: I did have to round down.
A: Multiply it by 3 then divide it by 2 again. Again, tell me if you rounded down.
B: Okay, hang on… I did not have to round down this time.
A: Great, now just tell me: how many times does 9 go into this last number?
B: You want me to divide by 9 ?
A: Yes, just the quotient.
B: Um, the quotient is 4 .
A: So your original number was 19 .
B: That’s right! How did you do that? Let me think….
(a) Figure out how A’s magic trick works, and write up a clear, mathematical explanation of how to perform it.
(b) What variants of this trick can you come up with using the same principles? Can you change the numbers 2,3 , and 9 in the trick, or maybe the operations involved? What about the information you ask for?

问题 1

当一个多项式有整数系数时，它就是积分多项式。2 的平方根是积分多项式方程 $x^2-2=0$ 的解。
当一个数可以用 $ rac{a}{b}$ 表示整数 $a$ 和 $b$（其中 $b
等于 0$ ）。
当一个数不是有理数时，它就是无理数。
(a) 假设 $c$ 是一个非平方整数。即
eq n^2$ for any $n$）。请解释为什么 $\sqrt{c}$ 不是有理数。同样，如果 $c$ 是一个非立方整数，那么 $\sqrt[3]{c}$ 是不是无理数呢？
(b) 找出一个以 $lpha=\sqrt{3}+\sqrt{5}$ 为解的积分多项式方程。证明 $lpha$ 是无理数。
(c) 设 $a$ 和 $b$ 为整数。求一个以 $\sqrt{a}+\sqrt{b}$ 为解的积分多项式方程。$sqrt{a}+\sqrt{b}$ 一定是无理数吗？如果 $a
eq b$，那么 $\sqrt{a}-\sqrt{b}$ 一定是无理数吗？
(d) 提出一些概括。作为一个起点，$ea=\sqrt{3}+\sqrt{5}+\sqrt{7}$是无理数吗？那么像 $\gamma=3 \sqrt{2}-2 \sqrt{3}-3 \sqrt{5}+\sqrt{6}$ 和 $\delta=sqrt[3]{5}-\sqrt{2}$ 这样的数呢？

问题 2

下面是罗斯大学两个学生之间的一段对话（转述）。
A: 嘿，想看魔术吗？
乙：当然，怎么变？
甲：随便想一个数。应该说是任何一个非负整数。
乙：好的。
甲：乘以 3 。
乙：好的….。
甲：现在除以 2，如果是小数，就四舍五入。告诉我你有没有四舍五入。
乙：我必须四舍五入。
甲：乘以 3，然后再除以 2。同样，告诉我你是否四舍五入了。
乙：好的，等一下……这次我不用四舍五入。
甲：很好，现在告诉我：最后这个数是 9 的几倍？
乙：你要我除以 9？
甲：是的，只要商。
乙：嗯，商是 4。
甲：所以你原来的数字是 19 。
乙：没错！你是怎么算出来的？让我想想….
(a) 找出 A 的魔术是如何变出来的，并写出清晰的数学解释。
(b) 利用同样的原理，你还能变出什么魔术？你能改变魔术中的数字 2、3 和 9 吗？你要求的信息是什么？

ROSS数学夏令营2023选拔代写

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

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）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

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。

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。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

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。

回归分析代写

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

斯坦福大学数学夏令营保录取Sumac代写2024

Posted on 2024年1月30日2024年1月30日 by statistics-lab

准备申请 SUMaC 和其他暑期课程实际上是大学入学申请的绝佳实践……但规模较小。如果您在SUMaC入学考试方面需要帮助，请查看statistics的数学辅导和数学代写服务。您将在10min内联系到专业数学辅导老师得到联系。

当你参加斯坦福大学数学营时，你不仅会参与对数学的深入探索并发展成为一名数学家，而且你将沉浸在一个与你有相同数学天赋和好奇心的人组成的社区中。在三周的时间里，你将参加在线课程，结交新朋友，并接受智力上的挑战。许多参与者说这一经历改变了他们的生活。

斯坦福大学数学营教给学生的不仅仅是当代数学的抽象概念和数学符号。参与者实际上这个项目中建立联系和友谊，许多人回想起来SUMaC的往事都说这是一次改变人生的经历。Mykel Kochenderfer是斯坦福大学航空和工程系副教授。Kochenderfer教授在高中时代参加了SUMaC，然后作为学生来到斯坦福大学，并最终加入了斯坦福大学成为了一名教授。

大量的高级数学课程有助于增加你的录取机会。如果一个对数学感兴趣的高中学生正在为未来的申请制定策略，我建议如果可能的话，尝试从学校的基础数学课程中考出来，而选择更多的高级数学课程。许多参加SUMaC的同学（10年级和11年级的学生）已经完成了微积分，有些人甚至对更高级的数学课题有经验，如三角学。

PSAT高分也有助于你的申请。与所有标准化考试一样，实践出真知。确保在你的PSAT考试日期之前进行几次模拟测试。

任何能证明你对数学的热情的额外课外活动也会帮助你脱颖而出；无论是参加数学竞赛，还是你以前参加过其他数学营，一定要强调你对数学活动感兴趣的所有证据。

申请还涉及到一个书面测试，主要是基于写证明。

一个有竞争力的SUMaC申请人应该有

高的GPA，包括但不限于数学课程的高成绩
高标准的分数，特别是数学部分的分数
通过数学竞赛等课外活动表现出对数学的热情
参加过以前的数学训练营
特别是：在SUMaC基于证明的入学考试中表现优异！

Sumac代写2024

斯坦福大学数学夏令营保录取Sumac代写2024|For use by SUMaC 2024 申请流程

In addition to the online application form, a complete application includes:

Academic Records
We require unofficial transcripts for each school or academic program from which you earned grades (such as letter or numerical grades) from Fall 2021 through Fall 2023. If you have multiple grade reports from one school, rather than one transcript from that school, please combine that into a single file for upload. Homeschooled applicants should submit a document similar to a transcript with their coursework from the years listed above. Homeschooled students should also submit transcripts from any graded courses they took with a school or program during this period.
Teacher Recommendation
We require one online recommendation form from a math teacher.
Optional Video Essay
This video essay is an opportunity to share more about your interest in our program. You can consider it a conversation between you and our Admissions Committee. If you attend a school in which the primary language of instruction is a language other than English, we strongly encourage you to complete the video essay.
$65 Application Fee
You may submit the $65 application fee online using a credit card. See instructions for payment at the end of the Online Application. Fee waivers can be provided if qualifications are met. Please contact [email protected].

斯坦福大学数学夏令营保录取Sumac代写2024|qualified probelm

A pair of distinct two-digit integers $(n, d)$ is misleading if the first digit of $d$ equals the second digit of $n$, and if you write the numbers as a fraction $\frac{n}{d}$ and “cancel” (i.e., cross out) the common digits, you get an equal fraction. For example, $(16,64)$ is a misleading pair since
$$
\frac{16}{64}=\frac{1}{4}=\frac{16}{64} .
$$
(a) Find all pairs of two-digit misleading integers. Explain how you know you have found all possible misleading two-digit pairs.
(b) Are there any misleading pairs of three-digit integers? For a three-digit pair to be misleading, when written as a fraction, “cancelling” the first digit of $d$ with either the second or third digit of $n$ (where the digits being canceled are equal) results in a fraction equal in value to the original fraction. Furthermore, the two integers cannot both end in 0 (otherwise, the pair could be obtained directly from a pair of misleading two-digit numbers).

For an arbitrary set of numbers $S$, define $S+S$ to be the set of all sums $x+y$ where $x$ and $y$ are in $S$. Let $A, B, C, D$ be sets of integers such that $(A \cup B)+(A \cup B)=(C \cup D)+(C \cup D)$.
For example, if $A=B=C=D={1,2}$ then
$$
(A \cup B)+(A \cup B)=(C \cup D)+(C \cup D)={1+1,1+2,2+1,2+2}={2,3,4}
$$Can you find four sets of integers $A, B, C, D$ so that$$(A \cup B)+(A \cup B)=(C \cup D)+(C \cup D)=(A \cup C)+(A \cup C) \neq(B \cup D)+(B \cup D) ?$$

(a) On a $3 \times 3$ grid, draw a path that starts in the center of a square in the corner of the grid, and repeatedly moves from the center of one square to the center of an adjacent square, without visiting any squares twice, and such that every square in the grid is included. For example, the following is a path that meets these conditions:
What are the possible ending positions for a path drawn according to these rules? What if you started from an edge square or the center square of the grid? Explain.
(b) Repeat the above problem for a $4 \times 4$ grid.
(c) Can you describe what happens in general for an $n \times n$ grid where $n$ is a positive integer?

Let $P_1=\left(x_1, y_1\right)$ and $P_2=\left(x_2, y_2\right)$ be two points in the $(x, y)-$ plane such that $y>0$ and $x_1<x_2$.
We call a circle $C$ a Yeppeun circle for $P_1$ and $P_2$ if $C$ contains $P_1$ and $P_2, C$ is tangent to the $x$-axis at a point $P_3=\left(x_3, 0\right)$, and the $x$-coordinate of the point of tangency lies between the $x$-coordinates of $P_1$ and $P_2$; that is, $x_1<x_3<x_2$. For example, the circle $C$ in the following diagram is a Yeppeun circle for the given $P_1$ and $P_2$
Given any two such points $P_1$ and $P_2$ above the $x$-axis, does a Yeppeun circle always exist? Explain.

数学夏令营

斯坦福大学数学夏令营保录取Sumac代写2024|For use by SUMaC 2024 applicants only

尽可能多地解决以下问题。您在这些问题上的工作以及您在学校的成绩、老师的推荐以及对申请表上问题的回答，将用于评估您的 SUMaC 申请。尽管 SUMaC 非常挑剔，申请者群体竞争激烈，但入学并不需要对每个问题的正确答案。
$\therefore$ 除申请截止日期外，此考试没有时间限制。
$\therefore$ 请包括对所有解决方案的清晰、详细的解释；没有解释的数字答案或公式对评估您的申请没有用。

如果您无法完全解决问题，我们鼓励您写下任何您认为抓住您的想法导致解决方案的部分进展。
$\therefore$ 您需要创建一个包含您的解决方案和解释的单独文档。本文件可以是打字或手写的，只要您上传的最终文件清晰可辨以供我们审核。
太这些问题都不需要计算器或计算机，我们鼓励您不要使用计算工具或计算机程序来解决您的问题。
您应该在不使用任何外部资源（书籍、老师或家长帮助、互联网搜索等）的情况下完成自己的工作。如果您从其他来源认识到其中一个问题，或者如果您得到任何帮助，请在您的报告中注明。

斯坦福大学数学夏令营保录取Sumac代写2024|qualified probelm

第一题部分解答

(a) Let $(n, d)=(10 m+a, 10 a+b)$ be a pair of misleading 2-digit integers.
We have $1 \leq m, a \leq 9, \quad 0 \leq b \leq 9, \quad m, a, b \in \mathbb{Z}$.
Consider $\quad \frac{n}{d}=\frac{10 m+a}{10 a+b}=\frac{m}{b} \quad(30 \quad b \neq 0)$.
$$
\begin{aligned}
& \Leftrightarrow(10 m+a) b=(10 a+b) m \quad 3 m+1=2 m \
& \Leftrightarrow 9 m b+a b=10 m a \quad \text { (*). } \
&
\end{aligned}
$$

For (*), we want to search for all solutions.
$$
\begin{aligned}
\bmod 2: & b m+a b=b(a+m) \equiv 0 \
& \Rightarrow 2 \mid b \text { or } 2 \mid a+m .
\end{aligned}
$$
$\bmod 5: \quad a b-m b=b(a-m)=0$
$\Rightarrow 5 \mid b$ or $5 \mid a-m$.
$\bmod$ q: $\quad a b-m a=a(b-m)=0$
$\Rightarrow$ either $91 a$
or $3|a \& 3|(b-m)$
(because $q f(b-m)$ when $1 \leq b, m \leq q$.)
$$
\Rightarrow \quad a \in{3,6,9} \text {. }
$$

Note that $a, b, m$ cannot be all equal.
(i) $a=3$.
If $5 \mid b$, then $b=5 \Rightarrow m=-1$.
So $5+b \Rightarrow 513-m \Rightarrow m=3$ or 8 .

物理代写|光学工程代写Optical Engineering代考请认准statistics-lab™

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非参数统计代写

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广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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.

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

Field	Information
Course Code	STAT6039
Prerequisite Courses	Not mentioned in the provided text
Majors	Statistics
Teachers	Not mentioned in the provided text
Units	6 units

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

问题 1.

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.

问题 2.

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?

问题 3.

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.

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非参数统计代写

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广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜STAT3056｜Advanced Mathematical Statistics高级数学统计澳洲国立大学

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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.

澳洲代写｜STAT3056｜Advanced Mathematical Statistics高级数学统计澳洲国立大学

Field	Information
Course Code	STAT3056
Prerequisite Courses	STAT3013
Majors	Statistics
Teachers	Prof Alan Welsh
Units	6 units

Advanced Mathematical Statistics高级数学统计问题集

问题 1.

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.”

问题 2.

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 $B$ ?

问题 3.

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?

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广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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

Field	Information
Course Code	MATH3514
Prerequisite Courses	Not explicitly mentioned
Majors	Mathematics
Teachers	Not mentioned
Units	6 units

Numerical Optimisation数值优化问题集

问题 1.

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 $f_2(x)=x^T A x$.

问题 2.

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.

问题 3.

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 $x, y \in \mathbb{R}^n$.

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜MATH3228｜Advanced Complex Analysis高级复分析澳洲国立大学

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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.

澳洲代写｜MATH3228｜Advanced Complex Analysis高级复分析澳洲国立大学

Field	Information
Course Code	MATH3228
Prerequisite Courses	Not explicitly mentioned
Majors	Mathematics
Teachers	Dr James Tener
Units	6 units

Advanced Complex Analysis高级复分析问题集

问题 1.

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.)

问题 2.

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 $z=1$.

问题 3.

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?

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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.

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

Field	Information
Course Code	MATH1003
Prerequisite Courses	Secondary School: ACT Mathematical Methods or NSW HSC Mathematics Advanced or equivalent
Majors	Mathematics
Teachers	Not explicitly mentioned in the provided text
Units	6 units

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

问题 1.

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.

问题 2.

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.

问题 3.

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.

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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

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

Field	Information
Course Code	MATH8202
Prerequisite Courses	Not explicitly mentioned in the provided text
Majors	Mathematics
Teachers	Prof Xu-Jia Wang
Units	6 units

Partial Differential Equations偏微分方程问题集

问题 1.

(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.

问题 2.

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 $\lambda$.

问题 3.

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.

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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.

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

Field	Information
Course Code	MATH6406
Prerequisite Courses	Not explicitly mentioned in the provided text
Majors	Mathematics
Teachers	Noa Kraitzman
Units	6 units

Partial Differential Equations偏微分方程问题集

问题 1.

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.

问题 2.

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.

问题 3.

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 $\mathbf{B}$.

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年10月17日2024年1月7日 by statistics-lab

statistics-lab^TM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和辅导服务！

课程介绍：

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

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

Field	Information
Course Code	MATH6216
Prerequisite Courses	Not explicitly mentioned in the provided text
Majors	Mathematics
Teachers	AsPr James Borger, AsPr Uri Onn
Units	6 units

Advanced Topics in Algebra 代数高级专题问题集

问题 1.

$\sharp$ 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.

问题 2.

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.

问题 3.

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
$$
1=(1-r a)+r a \in M
$$
a contradiction.
$(\Leftarrow)$ 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 $r \in \mathrm{Jac}(R)$.

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写