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澳洲大学｜MATH2701｜Abstract algebra and fundamental analysis抽象代数和基本分析新南威尔士大学

Posted on 2023年10月16日2024年1月7日 by statistics-lab

statistics-lab^TM为您提供新南威尔士大学（The University of New South Wales）Abstract algebra and fundamental analysis抽象代数和基本分析澳洲代写代考和辅导服务！

课程介绍：

Mathematics went through quite a revolution around the turn of the 20th century. In particular, an axiomatic approach infiltrated the mathematical paradigm, both as a tool to ensure mathematical rigour and to abstract common principles working in a variety of different settings.

First year mathematics emphasizes computation over abstraction and rigour. Later year courses (and Pure Mathematics in general) reverse this, so students need to learn some new skills and some new ways of thinking about mathematical objects.

This course is designed to help you develop the ability to write rigorous mathematical proofs in a setting where the level of abstraction is still quite modest. As such it will serve as an excellent preparation for the third year Pure Mathematics courses.

The course consists of two halves, algebra and analysis.

澳洲大学｜MATH2701｜Abstract algebra and fundamental analysis抽象代数和基本分析新南威尔士大学

Groups群论定义

Definition. Let $G$ be a set. A binary operation is a map of sets:
$$
: G \times G \rightarrow G \text {. } $$ For ease of notation we write $(a, b)=a * b \forall a, b \in G$. Any binary operation on $G$ gives a way of combining elements. As we have seen, if $G=\mathbb{Z}$ then + and $\times$ are natural examples of binary operations. When we are talking about a set $G$, together with a fixed binary operation $*$, we often write $(G, *)$.

Fundamental Definition. A group is a set $G$, together with a binary operation *, such that the following hold:

(Associativity): $(a * b) * c=a *(b * c) \forall a, b, c \in G$.
(Existence of identity): $\exists e \in G$ such that $a * e=e * a=a \forall a \in G$.
(Existence of inverses): Given $a \in G, \exists b \in G$ such that $a * b=b * a=e$.
Remarks. 1. We have seen five different examples thus far: $(\mathbb{Z},+),(\mathbb{Q},+),(\mathbb{Q} \backslash{0}, \times)$, $(\mathbb{Z} / m \mathbb{Z},+)$, and $(\mathbb{Z} / m \mathbb{Z} \backslash{[0]}, \times)$ if $m$ is prime. Another example is that of a real vector space under addition. Note that $(\mathbb{Z}, \times)$ is not a group. Also note that this gives examples of groups which are both finite and infinite. The more mathematics you learn the more you’ll see that groups are everywhere.
A set with a single element admits one possible binary operation. This makes it a group. We call this the trivial group.
A set with a binary operation is called a monoid if only the first two properties hold. From this point of view, a group is a monoid in which every element is invertible. $(\mathbb{Z}, \times)$ is a monoid but not a group.

定义设 $G$ 是一个集合。二元运算是集合的映射：
$$
: G （times） G （rightarrow） G （text） {. } $$ 为了便于记述，我们把 G$ 中的所有 a, b 写成 $(a, b)=a * b \。$G$ 上的任何二元运算都提供了一种组合元素的方法。正如我们所看到的，如果 $G=\mathbb{Z}$ 那么 + 和 $\times$ 就是二元运算的自然例子。当我们谈论一个集合 $G$ 以及一个固定的二进制运算 $$ 时，我们通常会写 $(G,)$。

基本定义。一个群是一个集合 $G$，加上一个二元运算 *，使得以下条件成立：

（关联性）： $(a * b) * c=a *(b * c) （对于 G$ 中的所有 a, b, c）。
2.（存在同一性）：对于 G$ 中的所有 a，在 G$ 中存在 e，使得 $a * e=e * a=a.
3.（倒数的存在）： Given $a \in G, \exists b \in G$ such that $a * b=b * a=e$.
备注. 1. 到目前为止，我们已经看到了五个不同的例子： $(\mathbb{Z},+),(\mathbb{Q},+),(\mathbb{Q} \backslash{0}, \times)$, $(\mathbb{Z} / m \mathbb{Z},+)$, 以及 $(\mathbb{Z} / m \mathbb{Z} \backslash{[0]}, \times)$ 如果 $m$ 是素数的话。另一个例子是实向量空间的加法。注意 $(\mathbb{Z}, \times)$ 不是一个群。还要注意，这给出了有限群和无限群的例子。数学学得越多，你就会发现群无处不在。
具有单元素的集合允许一种可能的二进制操作。这使它成为一个群。我们称之为三元组。
如果只有前两个性质成立，那么具有二元运算的集合就叫做单元组。从这个角度看，一个群是每个元素都可反演的单元。$(\mathbb{Z},\times)$是一个单元组，但不是一个群。

Cluster analysis聚类分析入门

Cluster: A collection of data objects

similar (or related) to one another within the same group
dissimilar (or unrelated) to the objects in other groups Cluster analysis (or clustering, data segmentation, …)
Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters
Unsupervised learning: no predefined classes (i.e., learning by observations vs. learning by examples: supervised)
Typical applications
As a stand-alone tool to get insight into data distribution
As a preprocessing step for other algorithms

数据集群数据对象的集合

在同一组中彼此相似（或相关
与其他组中的对象不相似（或不相关聚类分析（或聚类、数据分割……)
根据数据中发现的特征找出数据之间的相似性，并将相似的数据对象分组
无监督学习：没有预定义的类别（即通过观察进行学习，而不是通过示例进行学习：有监督学习）
典型应用
作为独立工具，深入了解数据分布情况
作为其他算法的预处理步骤

The Orbit-Stabiliser Theorem and Sylow’s Theorem轨道稳定器定理和西洛定理

Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. We can define an equivalence relation on $S$ by
$$
s \sim t \Longleftrightarrow \exists g \in G \text { such that } g(s)=t
$$
Remarks. This is an equivalence relation as a consequence of the group axioms, together with the definition of an action. I leave it as an exercise to check this.

Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. Under the above equivalence relation we call the equivalence classes orbits, and we write
$$
\operatorname{Orb}(s):={t \in S \mid \exists g \in G \text { such that } g(s)=t} \subset S
$$
for the equivalence class containing $s \in S$. We call it the orbit of $s$.
It is important to observe that $\operatorname{Orb}(s)$ is a subset of $S$ and hence is merely a set with no extra structure.

Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. We say that $G$ acts transitively on $S$ is there is only one orbit. Equivalently, $\varphi$ is transitive if given $s, t \in S$, $\exists g \in G$ such that $g(s)=t$.

An example of a transitive action is the natural action of $\Sigma(S)$ on $S$. This is clear because given any two points in a set $S$ there is always a bijection which maps one to the other. If $G$ is not the trivial group (the group with one element) then conjugation is never transitive. To see this observe that under this action $\operatorname{Orb}(e)={e}$.

Definition. Let $(G, *)$ be a group, together with an action $\varphi$ on a set $S$. Let $s \in S$. We define the stabiliser subgroup of $s$ to be all elements of $G$ which fix $s$ under the action. More precisely
$$
\operatorname{Stab}(s)={g \in G \mid g(s)=s} \subset G
$$
For this definition to make sense we must prove that $\operatorname{Stab}(s)$ is genuinely a subgroup.
Proposition. Stab(s) is a subgroup of $G$.
Proof. 1. $e(s)=s \Rightarrow e \in \operatorname{Stab}(s)$

$x, y \in \operatorname{Stab}(s) \Rightarrow(x * y)(s)=x(y(s))=x(s)=s \Rightarrow x * y \in \operatorname{Stab}(s)$.
$x \in \operatorname{Stab}(s) \Rightarrow x^{-1}(s)=x^{-1}(x(s))=\left(x^{-1} * x\right)(s)=e(s)=s \Rightarrow x^{-1} \in \operatorname{Stab}(s)$
Thus we may form the left cosets of $\operatorname{Stab}(s)$ in $G$ :
$$
G / \operatorname{Stab}(s):={x \operatorname{Stab}(s) \mid x \in G} .
$$
Recall that these subsets of $G$ are the equivalence classes for the equivalence relation:
$$
\text { Given } x, y \in G, x \sim y \Longleftrightarrow x^{-1} * y \in \operatorname{Stab}(s),
$$
hence they partition $G$ into disjoint subsets.
Proposition. Let $x, y \in G$ then $x \operatorname{Stab}(s)=y \operatorname{Stab}(s) \Longleftrightarrow x(s)=y(s)$.

定义让 $(G, *)$ 是一个群，以及一个集合 $S$ 上的作用 $\varphi$。在上述等价关系下，我们称等价类为轨道，并写为
$$
\操作符名称{Orb}(s):={t \in S \mid \exists g \in G \text { such that } g(s)=t} \子集 S
$$
为在 S$ 中包含 $s 的等价类。我们称之为 $s$ 的轨道。
需要注意的是，$operatorname{Orb}(s)$ 是 $S$ 的子集，因此只是一个没有额外结构的集合。

定义让 $(G, *)$ 是一个群，以及一个作用 $\varphi$ 在一个集合 $S$ 上。我们说 $G$ 在 $S$ 上的作用是唯一的轨道. 等价地，如果给定 S$中的 $s，t，$G$中存在 g，使得 $g(s)=t$，$\varphi$就是传递作用的。

传递作用的一个例子是 $S$ 上 $\Sigma(S)$ 的自然作用。这一点很清楚，因为给定集合 $S$ 中的任意两点，总有一个双射作用将其中一点映射到另一点。如果 $G$ 不是三元组（只有一个元素的组），那么共轭从来都不是传递的。要看到这一点，请观察在这个作用下 $\operatorname{Orb}(e)={e}$.

定义. 让 $(G, *)$ 是一个群，以及一个集合 $S$ 上的作用 $\varphi$. 让 $s （在 S$中）。我们将 $s$ 的稳定子群定义为在作用下固定 $s$ 的所有 $G$ 元素。更精确地说
$$
\operatorname{Stab}(s)={g\in G \mid g(s)=s} \子集 G
$$
为了使这个定义有意义，我们必须证明 $\operatorname{Stab}(s)$ 是一个真正的子群。
命题. Stab(s) 是 $G$ 的一个子群.
证明. 1. $e(s)=s （右箭头 e 在 operatorname{Stab}(s)$ 中
$x, y 在（操作符{Stab}(s)）中 \Rightarrow(x * y)(s)=x(y(s))=x(s)=s \Rightarrow x * y 在（操作符{Stab}(s)）$.
$x \in \operatorname{Stab}(s) \Rightarrow x^{-1}(s)=x^{-1}(x(s))=left(x^{-1} * x\right)(s)=e(s)=s \Rightarrow x^{-1} \in \operatorname{Stab}(s)$
因此，我们可以在 $G$ 中形成 $\operatorname{Stab}(s)$ 的左余集：
$$
G / \operatorname{Stab}(s):={x \operatorname{Stab}(s) \mid x \in G} .
$$
回想一下，$G$ 的这些子集就是等价关系的等价类：
$$
\text { Given } x, y in G, x \sim y \Longleftrightarrow x^{-1} * y \in \operatorname{Stab}(s)、
$$
因此它们将 $G$ 分割成互不相交的子集。
命题让 $x, y 在 G$ 中，那么 $x \operatorname{Stab}(s)=y \operatorname{Stab}(s) \Longleftrightarrow x(s)=y(s)$.

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

澳洲代写｜BFW3540 ｜Modelling in finance金融建模蒙纳士大学

Posted on 2023年10月16日2024年1月7日 by statistics-lab

statistics-lab^TM为您提供蒙纳士大学（Monash University）Modelling in finance金融建模澳洲代写代考和辅导服务！

课程介绍：

Topics include the development and application of financial spreadsheets, Excel and Visual Basic programming in financial modelling, modelling company financial statements, fixed income securities analysis, asset allocation and portfolio analysis, optimization using Solver, Interest rate models, option pricing models, numerical methods and risk management models.

澳洲代写｜BFW3540 ｜Modelling in finance金融建模蒙纳士大学

Fourier Transform傅立叶变换案例

In this section, we recall the Fourier transform definition, both for notational reasons and for the reader’s convenience.
The Fourier transform, for $f \in \mathcal{S}\left(\mathbb{R}^m\right)$, is denoted here as
$$
\widehat{f}(\xi):=\mathcal{F}f:=\int_{\mathbb{R}^m} \mathrm{e}^{i \xi x} f(x) \mathrm{d} x,
$$
where $\mathcal{S}\left(\mathbb{R}^m\right)$ is the Schwartz space of $\mathcal{C}^{\infty}\left(\mathbb{R}^m\right)$ functions of rapid decrease, see [RS75]. This is not the usual definition found in the mathematical literature. However, it is standard in probability, see [Chu01] and in the finance literature, see [CT04].

The Fourier transform is a linear bijection from $\mathcal{S}\left(\mathbb{R}^m\right)$ onto $\mathcal{S}\left(\mathbb{R}^m\right)$, whose inverse is given by the Fourier inversion formula

\begin{equation}
f(x)=\mathcal{F}^{-1}\widehat{f}=\frac{1}{(2 \pi)^m} \int_{\mathbb{R}^m} \mathrm{e}^{-i \xi x} \widehat{f}(\xi) \mathrm{d} \xi
\end{equation}

We also recall the Fourier transform for $f \in \mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$, where $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ is the space of tempered distributions, which is the dual of $\mathcal{S}\left(\mathbb{R}^m\right)$, the Fourier transform can be defined as
$$
(\mathcal{F}[f], \varphi)=(2 \pi)^m\left(f, \mathcal{F}^{-1}[\varphi]\right) \quad \varphi \in \mathcal{S}\left(\mathbb{R}^m\right),
$$
see [RR04]. This definition makes the Fourier transform in $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ an extension of the Fourier transform in $\mathcal{S}\left(\mathbb{R}^m\right)$. The Fourier transform for $L^1\left(\mathbb{R}^m\right)$ and $L^2\left(\mathbb{R}^m\right)$ are restrictions of the Fourier transform for $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$.

The Fourier transform has several useful properties. Some of them are reviewed below with the purpose of calling attention to the notation used here:

$\mathcal{F}f(x-a)=\mathrm{e}^{i a \xi} \widehat{f}(\xi)$
$D^\alpha \widehat{f}(\xi)=\mathcal{F}\left(i x)^\alpha f\right$
$(-i \xi)^\alpha \widehat{f}(\xi)=\mathcal{F}\leftD^\alpha f\right$
Some specific distributions are often used in this thesis. To present the notation, we give a brief overview of them. First, consider the Cauchy principal value
$$
\begin{aligned}
1 / x: \mathcal{S}(\mathbb{R}) & \rightarrow \mathbb{R} \
f & \rightarrow(1 / x, f):=f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x,
\end{aligned}
$$
where
$$
f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x:=\lim {\epsilon \downarrow 0}\left(\int\epsilon^{\infty} \frac{f(x)}{x} \mathrm{~d} x+\int_{-\infty}^{-\epsilon} \frac{f(x)}{x} \mathrm{~d} x\right) .
$$
This defines a distribution in $\mathcal{S}^{\prime}(\mathbb{R})$.

Probability and Stochastic Processes概率与随机过程案例

In this section, we present a brief overview of the topics on probability and stochastic processes used herein. References on the subject are [CW90] and [Sat99].

In this thesis, the triple $(\Omega, \mathcal{F}, \mathbb{P})$ denotes a complete probability space, where $\Omega$ is a set of points $\omega, \mathcal{F}$ is a $\sigma$-algebra containing all $\mathbb{P}$-null sets, and $\mathbb{P}$ is a probability measure. When we say that $X$ is a random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, we mean that $X$ is real-valued function on $\Omega$, measurable with respect to $\mathcal{F}$.
The characteristic function of a random variable is defined as
$$
\varphi(z)=\mathbb{E}\left[\mathrm{e}^{i z X}\right] .
$$
For properties of the characteristic function and a review of probability theory we refer to [CW90].

The filtered complete probability space is denoted by $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$, where, as in [Pro04], we write $\mathbb{F}$ for the filtration $\left(\mathcal{F}t\right){0 \leq t \leq \infty}$ and we assume that $\mathcal{F}_0$ contains all the $\mathbb{P}$-null sets. We use $\stackrel{\mathbb{P}}{\longrightarrow}$ to denote convergence in probability, see [CW90] and the French acronyms càdlàg (continu à droite, limité à gauche) is used to define the right continuous, left limited process, see [Pro04].

The main class of stochastic processes we are interested in this work are the Levy processes, see [Sat99] for a comprehensive treatment of the subject. We briefly review the definition of a Levy process
Definition: A Levy process is a càdlàg stochastic process, $\left(X_t\right)_{t \geq 0}$, on $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ taking values in $\mathbb{R}$ and with the following properties:

Independent increments. That is, given $t_0 \leq \ldots \leq t_N$, and defined $Y_n:=X_{t_n}-X_{t_{n-1}}$ we have $\left{Y_n\right}_{n=1}^N$ independent;
Stationary increments. That is, the distribution of $X_{t+s}-X_t$ does not depend on $t$;
Stochastic continuity. That is,
$$
X_{t+h} \underset{h \downarrow 0}{\stackrel{\mathbb{P}}{\longrightarrow}} X_t
$$
An important stochastic process is the Brownian motion.

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

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｜PHYC30018｜Quantum Physics量子计算墨尔本大学

Posted on 2023年10月16日2024年1月7日 by statistics-lab

statistics-lab^TM为您提供墨尔本大学（The University of Melbourne，简称UniMelb，中文简称“墨大”）Quantum Physics量子计算澳洲代写代考和辅导服务！

课程介绍：

Quantum mechanics plays a central role in our understanding of fundamental phenomena, primarily in the microscopic domain. It lays the foundation for an understanding of atomic, molecular, condensed matter, nuclear and particle physics.

Quantum Physics量子计算问题集

问题 1. Consider a two-dimensional strip of material of length $L$ and width $W$, placed in a magnetic field perpendicular to the strip and with an electric field established in the direction of the length $L$.
(a) Suppose that the resistivity matrix is given by the classical result
$$
\rho=\left(\begin{array}{cc}
\rho_0 & -\rho_H \
\rho_H & \rho_0
\end{array}\right)
$$
where $\rho_H=B /$ nec is the Hall resistivity and $\rho_0$ is the usual Ohmic resistivity. Find the conductivity matrix, $\sigma=\rho^{-1}$. Write it in the form:
$$
\sigma=\left(\begin{array}{cc}
\sigma_0 & \sigma_H \
-\sigma_H & \sigma_0
\end{array}\right) .
$$
What are $\sigma_0$ and $\sigma_H$ ?
(b) Suppose $B=0$, so the Hall resistivity is zero. Notice that the Ohmic conductivity, $\sigma_0$, is just $1 / \rho_0$. In particular, note that $\sigma_0 \rightarrow \infty$ as $\rho_0 \rightarrow 0$. Now suppose $\rho_H \neq 0$. Show that $\sigma_0 \rightarrow 0$ as $\rho_0 \rightarrow 0$, so it is possible to have both $\sigma_0$ and $\rho_0$ equal to zero

问题 2.

This problem asks you to give a complete presentation of a calculation that is almost the same as one you saw in lecture.

Consider a constant electric field, $\vec{E}=\left(0, E_0, 0\right)$ and a constant magnetic field, $\vec{B}=\left(0,0, B_0\right)$.
(a) Choose an electrostatic potential $\phi$ and a vector potential $\vec{A}$ which describe the $\vec{E}$ and $\vec{B}$ fields, and write the Hamiltonian for a charged particle of mass $m$ and charge $q$ in these fields. Assume that the particle is restricted to move in the $x y$-plane.
(b) What are the allowed energies as a function of $B_0$ and $E_0$ ? Draw a figure to show how the Landau levels (energy levels when $E_0=0$ ) change as $E_0$ increases.

问题 3.

You will see the “standard presentation” of the Aharonov-Bohm effect in lecture, on the day that this problem set is due. The standard presentation has its advantages, and in particular is more general than the presentation you will work through in this problem. However, students often come away from the standard presentation of the Aharonov-Bohm effect thinking that the only way to detect this effect is to do an interference experiment. This is not true, and the purpose of this problem is to disabuse you of this misimpression before you form it.

As Griffiths explains on pages 385-387 (344-345 in 1st Ed.), the Aharonov-Bohm effect modifies the energy eigenvalues of suitably chosen quantum mechanical systems. In this problem, I ask you to work through the same physical example that Griffiths uses, but in a different fashion which makes more use of gauge invariance.

Imagine a particle constrained to move on a circle of radius $b$ (a bead on a wire ring, if you like.) Along the axis of the circle runs a solenoid of radius $a<b$, carrying a magnetic field $\vec{B}=\left(0,0, B_0\right)$. The field inside the solenoid is uniform

and the field outside the solenoid is zero. The setup is depicted in Griffiths’ Fig. 10.10. (10.12 in 1st Ed.)
(a) Construct a vector potential $\vec{A}$ which describes the magnetic field (both inside and outside the solenoid) and which has the form $A_r=A_z=0$ and $A_\phi=\alpha(r)$ for some function $\alpha(r)$. I am using cylindrical coordinates $z, r$, $\phi$.
(b) Since $\vec{\nabla} \times \vec{A}=0$ for $r>a$, it must be possible to write $\vec{A}=\vec{\nabla} f$ in any simply connected region in $r>a$. [This is a theorem in vector calculus.] Show that if we find such an $f$ in the region
$$
r>a \text { and }-\pi+\epsilon<\phi<\pi-\epsilon,
$$
then
$$
f(r, \pi-\epsilon)-f(r,-\pi+\epsilon) \rightarrow \Phi \text { as } \epsilon \rightarrow 0 .
$$
Here, the total magnetic flux is $\Phi=\pi a^2 B_0$. Now find an explicit form for $f$, which is a function only of $\phi$.
(c) Now consider the motion of a “bead on a ring”: write the Schrödinger equation for the particle constrained to move on the circle $r=b$, using the $\vec{A}$ you found in (a). Hint: the answer is given in Griffiths.
(d) Use the $f(\phi)$ found in (b) to gauge transform the Schrödinger equation for $\psi(\phi)$ within the angular region $-\pi+\epsilon<\phi<\pi-\epsilon$ to a Schrödinger equation for a free particle within this angular region. Call the original wave function $\psi(\phi)$ and the gauge-transformed wave function $\psi^{\prime}(\phi)$.
(e) The original wave function $\psi$ must be single-valued for all $\phi$, in particular at $\phi=\pi$. That is, $\psi(\pi-\epsilon)-\psi(-\pi+\epsilon) \rightarrow 0$ and $\frac{\partial \psi}{\partial \phi}(\pi-\epsilon)-\frac{\partial \psi}{\partial \phi}(-\pi+\epsilon) \rightarrow 0$ as $\epsilon \rightarrow 0$. What does this say about the gauge-transformed wave function? I.e., how must $\psi^{\prime}(\pi-\epsilon)$ and $\psi^{\prime}(-\pi+\epsilon)$ be related as $\epsilon \rightarrow 0$ ?
[Hint: because the $f(\phi)$ is not single valued at $\phi=\pi$, the gauge transformed wave function $\psi^{\prime}(\phi)$ is not single valued there either.]
(f) Solve the Schrödinger equation for $\psi^{\prime}$, and find energy eigenstates which satisfy the boundary conditions you derived in (e). What are the allowed energy eigenvalues?
(g) Undo the gauge transformation, and find the energy eigenstates $\psi(\phi)$ in the original gauge. Do the energy eigenvalues in the two gauges differ?
(h) Plot the energy eigenvalues as a function of the enclosed flux, $\Phi$. Show that the energy eigenvalues are periodic functions of $\Phi$ with period $\Phi_0$, where you must determine $\Phi_0$. For what values of $\Phi$ does the enclosed magnetic field have no effect on the spectrum of a particle on a ring? Show that the

Aharonov-Bohm effect can only be used to determine the fractional part of $\Phi / \Phi_0$.
[Note: you have shown that even though the bead on a ring is everywhere in a region in which $\vec{B}=0$, the presence of a nonzero $\vec{A}$ affects the energy eigenvalue spectrum. However, the effect on the energy eigenvalues is determined by $\Phi$, and is therefore gauge invariant. To confirm the gauge invariance of your result, you can compare your answer for the energy eigenvalues to Griffiths’ result, obtained using a different gauge.]

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

澳洲大学｜MATH5855｜Multivariate Analysis多变量分析新南威尔士大学

Posted on 2023年10月16日2024年1月7日 by statistics-lab

statistics-lab^TM为您提供新南威尔士大学（The University of New South Wales）Quantum Physics of Solids and Devices固体和器件的量子物理澳洲代写代考和辅导服务！

课程介绍：

Quantum mechanics plays an important role in the properties of solids, and will be central to new generations of
electronic devices across the coming decades, e.g., quantum computers. Existing devices, such as laser diodes and superconducting quantum interference devices (SQUIDs), also exploit quantum phenomena for their operation. This course covers three main areas. The first is ‘The Quantum Physics of Solids’, with topics including crystal structure, phonons as quantum oscillations, electrons as quantum particles in solids, band structure and unconventional materials. The second is ‘Interactions in Quantum Systems’, with topics including paramagnetism, diamagnetism and ferromagnetism, electron-electron interactions and their role in screening and plasmonic effects, and superconductivity. The third is ‘From Semiconductors to Quantum Devices’, with topics including charge carriers in semiconductors, p-n junctions and diodes, finite solids and heterojunctions, quantum confinement and low-dimensional devices, nanoelectronics. The course will appeal to those seeking a better contextual understanding of quantum mechanics and to learn about its real world applications: past, present and future.

澳洲大学｜MATH5855｜Multivariate Analysis多变量分析新南威尔士大学

Detail	Information
Course Code	MATH5855
Prerequisite	UNSW UG students – MATH2801, MATH2901 or admitted to postgraduate programs of the School of Mathematics and Statistics and as an elective in some approved programs.
Academic Unit	School of Mathematics and Statistics
Course Convener	Not mentioned in the provided text
Units	6

Discriminant analysis判别分析入门

Introduction
There are two prototypical situations in multivariate analysis that are, in a sense, different sides of the same coin. Suppose we have identifiable groups, and they may (or may not) differ in their means (and possibly in their covariance structure) on one or more response measures.

How can we test whether the groups are significantly different?
If the groups are different, how can we construct a rule that allows us to accurately assign an individual to one of several groups, depending on their scores on the response measures?
In this module, we will deal with the second problem, examining, in detail, a method known as discriminant analysis.
However, the first problem, related to a technique known as MANOVA (Multivariate Analysis of Variance) is closely related to the first.

多元分析中有两种典型情况，从某种意义上说，它们是一枚硬币的两面。假设我们有可识别的群体，他们在一个或多个反应测量上的均值（可能还有协方差结构）可能（也可能）不同。

我们如何检验这些组别是否存在显著差异？
如果组别不同，我们怎样才能构建一个规则，使我们能够根据个体在反应测量指标上的得分，准确地将其分配到几个组别中的一个？
在本模块中，我们将讨论第二个问题，详细研究一种称为判别分析的方法。
然而，第一个问题与一种称为 MANOVA（多变量方差分析）的技术密切相关。

Cluster analysis聚类分析入门

Cluster: A collection of data objects

similar (or related) to one another within the same group
dissimilar (or unrelated) to the objects in other groups Cluster analysis (or clustering, data segmentation, …)
Finding similarities between data according to the characteristics found in the data and grouping similar data objects into clusters
Unsupervised learning: no predefined classes (i.e., learning by observations vs. learning by examples: supervised)
Typical applications
As a stand-alone tool to get insight into data distribution
As a preprocessing step for other algorithms

数据集群数据对象的集合

在同一组中彼此相似（或相关
与其他组中的对象不相似（或不相关聚类分析（或聚类、数据分割……)
根据数据中发现的特征找出数据之间的相似性，并将相似的数据对象分组
无监督学习：没有预定义的类别（即通过观察进行学习，而不是通过示例进行学习：有监督学习）
典型应用
作为独立工具，深入了解数据分布情况
作为其他算法的预处理步骤

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｜MTH3130 ｜Topology: The mathematics of shape拓扑学形状数学蒙纳士大学

Posted on 2023年10月16日2023年10月16日 by statistics-lab

statistics-lab^TM为您提供蒙纳士大学（Monash University）Topology: The mathematics of shape拓扑学形状数学澳洲代写代考和辅导服务！

课程介绍：

From point-set topology to manifolds: sets, topological spaces, basis of topology, and properties of spaces such as compact, connected, and Hausdorff. Maps between spaces and their properties, including continuity, homeomorphism, and homotopy.

Constructing spaces via subspace, product, identification, and cell complexes. Manifolds. Additional topics from algebraic and low-dimensional topology may include fundamental group and Seifert-van Kampen theorem, classification of surfaces, and topics in knot theory. Throughout, examples of spaces will include Euclidean spaces, surfaces (real projective plane, Klein bottle, Mobius strip), complexes, function spaces, and others.

澳洲代写｜MTH3320｜Computational linear algebra计算线性代数蒙纳士大学

Topology拓扑学问题集

问题 1.

Show that if $f: X \rightarrow Y$ induces an isomorphism in homology with coefficients in the prime fields $\mathbb{F}_p$ (for all primes $p$ ) and $\mathbb{Q}$, then it induces an isomorphism in homology with coefficients in $\mathbb{Z}$. (Hint: 18 (c).)

问题 2.

Let $A \subseteq X$ and $B \subseteq Y$ be subsets. Construct a natural chain map
$$
S_(X, A) \otimes S_(Y, B) \rightarrow S_(X \times Y, A \times Y \cup X \times B) $$ that is a homology isomorphism if $A$ and $B$ are open. (Hint: Problem 26., or its proof, might be useful.) So there is a natural “relative cross product” map $$ H_(X, A ; R) \otimes_R H_(Y, B ; R) \rightarrow H_(X \times Y, A \times Y \cup X \times B ; R)
$$
that is an isomorphism if $A$ and $B$ are open, $R$ is a PID, and either $H_(X, A ; R)$ or $H_(Y, B ; R)$ is free over $R$.

问题 3.

(a) What is the $k$ th Betti number of $\left(S^1\right)^n$ ?
(b) Define an equivalence relation on $\mathbb{R}^n$ by saying that two vectors are equivalent if they differ by a vector with entries in $\mathbb{Z}$. Identify the quotient space of $\mathbb{R}^n$ by this equivalence relation with the product space $\left(S^1\right)^n$. Let $M$ be an $n \times n$ matrix with entries in $\mathbb{Z}$. It defines a linear map $\mathbb{R}^n \rightarrow \mathbb{R}^n$ in the usual way. Show that this map descends to a self-map of $\left(S^1\right)^n$. Compute the effect of this map on $H_n\left(\left(S^1\right)^n\right)$.

澳洲代写｜ETC3250｜Introduction to machine learning机器学习入门蒙纳士大学

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

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｜CITS5508｜Machine Learning机器学习西澳大学

Posted on 2023年10月13日2023年10月13日 by statistics-lab

statistics-lab^TM为您提供西澳大学（University of Western Australia，简称:UWA）Machine Learning机器学习澳洲代写代考和辅导服务！

课程介绍：

There is an explosion in data generation and data collection due to improvements in sensing technologies and business processes. Extracting meaningful knowledge from large amounts of data has become a priority for businesses as well as scientific domains. Machine learning provides core underlying theory and techniques to data analytics, where algorithms iteratively learn from data to uncover hidden insights. In this unit, students will develop in-depth understanding of machine learning techniques that are applicable to both scientific and business data. The topics covered by the unit include supervised classification, unsupervised classification, regression, support vector machines, decision trees, random forests, dimensionality reduction, artificial neural networks, deep neural networks, autoencoders, and reinforcement learning.

Computational linear algebra计算线性代数问题集

问题 1.

We will explore here the use of the Bayesian Information Criterion (BIC) for model selection, specifically to select features for a Naive Bayes model. The input consists of $d$-dimensional binary feature vectors $\mathbf{x}=\left[x_1, \ldots, x_d\right]^T$ where $x_i \in{-1,1}$ and binary class labels $y \in{-1,1}$.
A Naive Bayes model may be used, for example, to classify emails as spam $(y=1)$ or not-spam $(y=-1)$. The body of each email can be represented as a bag of words (i.e., we ignore frequency, placement, and grammar etc.). We could restrict ourselves to a dictionary of $d$ words $\left{w_1, \ldots, w_d\right}$. and coordinate $x_i$ could serve as an indicator of word $w_i: x_i=1$ if $w_i$ is present in the email and $x_i=-1$ if it is absent.
Recall that the Naive Bayes model assumes that the features are conditionally independent given the label so that
$$
P(\mathbf{x}, y)=\left[\prod_{i=1}^d P\left(x_i \mid y\right)\right] P(y)
$$
We parameterize our model by introducing separate parameters for each component:
$$
\begin{aligned}
P(\mathbf{x} \mid y, \theta) & =\left[\prod_{i=1}^d P\left(x_i \mid y, \theta_i\right)\right] \text { where } \
P\left(x_i \mid y, \theta_i\right) & =\theta_{i \mid y}^{\frac{x_i+1}{2}}\left(1-\theta_{i \mid y}^{\frac{1-x_i}{2}}\right)
\end{aligned}
$$
Here $\theta=\left[\theta_1, \ldots, \theta_d\right]^T$ and $\theta_i=\left[\theta_{i \mid 1}, \theta_{i \mid-1}\right]^T$ so that
$$
\begin{aligned}
\theta_{i \mid 1} & =P\left(x_i=1 \mid y=1\right) \
\theta_{i \mid-1} & =P\left(x_i=1 \mid y=-1\right)
\end{aligned}
$$
In class, we have already discussed how to compute the Maximum Likelihood estimates of these parameters. We will take a Bayesian approach, however, and introduce a prior probability over the parameters $\theta$ as:
$$
\begin{aligned}
P(\theta) & =\prod_{i=1}^d \prod_{y \in{1,-1}} P\left(\theta_{i \mid y}\right) \
P\left(\theta_{i \mid y}\right) & =\frac{\Gamma\left(r^{+}+r^{-}+2\right)}{\Gamma\left(r^{+}+1\right) \Gamma\left(r^{-}+1\right)} \cdot \theta_{i \mid y}^{r^{+}}\left(1-\theta_{i \mid y}\right)^{r^{-}}
\end{aligned}
$$

where $r^{+}$and $r^{-}$are hyper-parameters, common across all the $\theta_{i \mid y}$ ‘s, and, for integer $k, \Gamma(k+1)=k$ !. The hyper-parameters characterize our beliefs about the parameters prior to seeing any data. You may assume here that $r^{+}$and $r^{-}$are non-negative integers.
(a) Eqn 5 specifies the prior $P\left(\theta_{i \mid y}\right)$ as a Beta distribution. This choice of the prior is particularly convenient, as we will see now. Show that the posterior distribution $P(\theta \mid \mathcal{D})$, given $n$ training examples $\left{\left(\mathbf{x}j, y_j\right) \mid j=1, \ldots, n\right}$, has the following form: $$ P(\theta \mid \mathcal{D}) \propto L(\mathcal{D} ; \theta) P(\theta)=\prod{i=1}^d \prod_{y \in{-1,1}} \theta_{i \mid y}^{m_{i \mid y}^{+}}\left(1-\theta_{i \mid y}\right)^{m_{i \mid y}^{-}}
$$
where $L(\mathcal{D} ; \theta)$ is the likelihood function. What are $m_{i \mid y}^{+}$and $m_{i \mid y}^{-}$? Your answer should use the $\hat{n}y$ and $\hat{n}{i y}\left(x_i, y\right)$ notation used in the lectures; e.g., $\hat{n}_{k y}(1,-1)$ is the number of examples where $y=-1$ and $x_k=1$.

You have just shown that the Beta distribution is a conjugate prior to the Binomial distribution. In other words, if the prior probability $P(\theta)$ has the form of a Beta distribution, and the likelihood $P(\mathcal{D} \mid \theta)$ has the form of a Binomial distribution, then the posterior probability $P(\theta \mid \mathcal{D})$ will be a Beta distribution. When the class labels and features are not binary but take on $k$ values, the same relationship holds between the Dirichlet and Multinomial distributions.

问题 2.

(b) Recall that, in the Bayesian scheme for model selection, our aim is to find the model that maximizes the marginal likelihood, $P\left(\mathcal{D} \mid \mathcal{F}_d\right)$, which is the normalization constant for the posterior:
$$
P\left(\mathcal{D} \mid \mathcal{F}_d\right)=\int L(\mathcal{D} ; \theta) P(\theta) d \theta
$$
where $\mathcal{F}_d$ denotes the model. Compute a closed-form expression for $P\left(\mathcal{D} \mid \mathcal{F}_d\right)$. (Hint: use the form of the normalization constant for the beta distribution).

问题 3.

(c) Let us now use our results to choose between two models $\mathcal{F}1$ and $\mathcal{F}_2$. The only difference between them is in how they use feature $i: \mathcal{F}_2$ involves $P\left(x_i \mid y\right)$ term that connects the feature to the label $y$ whereas $\mathcal{F}_1$ only has $P\left(x_i\right)$, assuming that feature $x_i$ is independent of the label. In $\mathcal{F}_2$ the distribution of $x_i$ ‘s is parameterized by two parameters $\left[\theta{i \mid 1}, \theta_{i \mid-1}\right]^T$ as described before. In contrast, $\mathcal{F}_1$ only requires a single parameter $\theta_i^{\prime}$,
$$
P\left(x_i \mid \theta_i^{\prime}\right)=\theta_i^{\prime \frac{x_i+1}{2}}\left(1-\theta_i^{\prime}\right)^{\frac{1-x_i}{2}}
$$
The prior probability over $\theta_i^{\prime}$ is the same as Eqn 6 :
$$
P\left(\theta_i^{\prime}\right) \propto \theta_i^{r^{+}}\left(1-\theta_i^{\prime}\right)^{r^{-}}
$$
The expressions of the marginal likelihood from the two models will differ only in terms of feature $i$. To compare the two models we can ignore all the other terms. From your results in part (b), extract from $P\left(\mathcal{D} \mid \mathcal{F}_2\right)$ the factors involving feature $i$. Use a similar analysis to evaluate the terms involving feature $i$ in $P\left(\mathcal{D} \mid \mathcal{F}_1\right)$. Using these, evaluate the “decision rule”, $\log \left[P\left(\mathcal{D} \mid \mathcal{F}_2\right) / P\left(\mathcal{D} \mid \mathcal{F}_1\right)\right]>0$ to include feature $x_i$.

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

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｜MTH3320｜Computational linear algebra计算线性代数蒙纳士大学

Posted on 2023年10月13日2023年10月13日 by statistics-lab

statistics-lab^TM为您提供蒙纳士大学（Monash University）Computational linear algebra计算线性代数澳洲代写代考和辅导服务！

课程介绍：

The overall aim of this unit is to study the numerical methods for matrix computations that lie at the core of a wide variety of large-scale computations and innovations in the sciences, engineering, technology and data science. You will receive an introduction to the mathematical theory of numerical methods for linear algebra (with derivations of the methods and some proofs). This will broadly include methods for solving linear systems of equations, least-squares problems, eigenvalue problems, and other matrix decompositions. Special attention will be paid to conditioning and stability, dense versus sparse problems, and direct versus iterative solution techniques. You will learn to implement the computational methods efficiently, and will learn how to thoroughly test their implementations for accuracy and performance. You will work on realistic matrix models for applications in a variety of fields. Applications may include, for example: computation of electrostatic potentials and heat conduction problems; eigenvalue problems for electronic structure calculation; ranking algorithms for webpages; algorithms for movie recommendation, classification of handwritten digits, and document clustering; and principal component analysis in data science.

Computational linear algebra计算线性代数问题集

问题 1.

Let $\mathbf{A}=\left(\begin{array}{ll}1 & 2 \ 3 & 4\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{ll}5 & 6 \ 7 & 8\end{array}\right)$. Determine (a) $(\mathbf{A}-\mathbf{B})(\mathbf{A}+\mathbf{B})(\mathrm{b}) \mathbf{A}^2-\mathbf{B}^2$ Explain why $\mathbf{A}^2-\mathbf{B}^2 \neq(\mathbf{A}-\mathbf{B})(\mathbf{A}+\mathbf{B})$.
University of Hertfordshire, UK
Let $\mathbf{A}$ and $\mathbf{B}$ be invertible (non-singular) $n$ by $n$ matrices. Find the errors, if any, in the following derivation:
$$
\begin{aligned}
\mathbf{A B}(\mathbf{A B})^{-1} & =\mathbf{A B A}^{-1} \mathbf{B}^{-1} \
& =\mathbf{A A}^{-1} \mathbf{B B}^{-1} \
& =\mathbf{I} \times \mathbf{I}=\mathbf{I}
\end{aligned}
$$
You need to explain why you think there is an error.
University of Hertfordshire, UK
Given the matrix
$$
\mathbf{A}=\frac{1}{7}\left(\begin{array}{rrr}
3 & -2 & -6 \
-2 & 6 & -3 \
-6 & -3 & -2
\end{array}\right)
$$
(a) Compute $\mathbf{A}^2$ and $\mathbf{A}^3$.
(b) Based on these results, determine the matrices $\mathbf{A}^{-1}$ and $\mathbf{A}^{2004}$.

问题 2.

Give an example of the following, or state that no such example exists: $2 \times 2$ matrix $\mathbf{A}$ and $2 \times 1$ non-zero vectors $\mathbf{u}$ and $\mathbf{v}$ such that $\mathbf{A u}=\mathbf{A v}$ yet $\mathbf{u} \neq \mathbf{v}$.

Illinois State University, USA (part question)
(a) If $\mathbf{A}=\left(\begin{array}{ll}1 & 2 \ 3 & 4\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{rr}0 & 1 \ -1 & 0\end{array}\right)$, compute $\mathbf{A}^2, \mathbf{B}^2, \mathbf{A B}$ and $\mathbf{B A}$.
(b) If $\mathbf{A}=\left(\begin{array}{ll}a & b \ c & d\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{ll}e & f \ g & h\end{array}\right)$, compute $\mathbf{A B}-\mathbf{B A}$.
Queen Mary, University of London, UK
Let $\mathbf{M}=\left(\begin{array}{ll}1 & 1 \ 1 & 1\end{array}\right)$. Compute $\mathbf{M}^n$ for $n=2,3,4$. Find a function $c(n)$ such that $\mathbf{M}^n=c(n) \mathbf{M}$ for all $n \in \mathbb{Z}, n \geq 1$. (You are not required to prove any of your results.) Queen Mary, University of London, UK (part question)
Let $\mathbf{A}=\left(\begin{array}{cc}\frac{1}{3} & \frac{1}{3} \ \frac{1}{3} & \frac{1}{3}\end{array}\right)$. Determine (i) $\mathbf{A}^2$ (ii) $\mathbf{A}^3$ Prove that $\mathbf{A}^n=\frac{1}{2}\left(\frac{2}{3}\right)^n \mathbf{A}$.
University of Hertfordshire, UK How many rows does $\mathbf{B}$ have if $\mathbf{B C}$ is a $4 \times 6$ matrix? Explain.

问题 3.

Prove (give a clear reason): If $\mathbf{A}$ is a symmetric invertible matrix then $\mathbf{A}^{-1}$ is also symmetric.

Massachusetts Institute of Technology USA
If $\mathbf{A}$ is a matrix such that $\mathbf{A}^2-\mathbf{A}+\mathbf{I}=\mathbf{O}$ show that $\mathbf{A}$ is invertible with inverse I – A.

McGill University Canada 2007
(part question)
(a) Define what is meant by a square matrix $\mathbf{A}$ being invertible. Show that the inverse of $\mathbf{A}$, if it exists, is unique.
(b) Show that the product of any finite number of invertible matrices is invertible.
(c) Find the inverse of the matrix
$$
\mathbf{A}=\left[\begin{array}{rrr}
1 & 0 & 1 \
-1 & 1 & 1 \
0 & 1 & 0
\end{array}\right]
$$
University of Sussex, UK
Let $\mathbf{A}$ and $\mathbf{B}$ be $n \times n$ invertible matrices, with $\mathbf{A X A}^{-1}=\mathbf{B}$. Explain why $\mathbf{X}$ is invertible and calculate $\mathbf{X}^{-1}$ in terms of $\mathbf{A}$ and $\mathbf{B}$.

\begin{prob}

Show that, for any non-zero vector $\mathbf{u}$ in $\mathbb{R}^n$, we have $\left|\frac{1}{|\mathbf{u}|} \mathbf{u}\right|=1$.
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\mathbb{R}^n$. Disprove the following propositions:
(a) If $\mathbf{u} \cdot \mathbf{v}=0$ then $\mathbf{u}=\mathbf{O}$ or $\mathbf{v}=\mathbf{O}$.
(b) $|\mathbf{u}+\mathbf{v}|=|\mathbf{u}|+|\mathbf{v}|$
Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \ldots, \mathbf{u}_n$ be orthogonal vectors in $\mathbb{R}^n$. Prove
(i) $\left|\mathbf{u}_1+\mathbf{u}_2\right|^2=\left|\mathbf{u}_1\right|^2+\left|\mathbf{u}_2\right|^2$
(ii) $\left|\mathbf{u}_1+\mathbf{u}_2+\cdots+\mathbf{u}_n\right|^2=\left|\mathbf{u}_1\right|^2+\left|\mathbf{u}_2\right|^2+\cdots+\left|\mathbf{u}_n\right|^2$
For part (ii) use mathematical induction.

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

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｜COMP30027｜Machine Learning机器学习墨尔本大学

Posted on 2023年10月12日2023年10月12日 by statistics-lab

statistics-lab^TM为您提供墨尔本大学（The University of Melbourne，简称UniMelb，中文简称“墨大”）Complex Analysis复杂分析澳洲代写代考和辅导服务！

课程介绍：

Machine Learning, a core discipline in data science, is prevalent across Science, Technology, the Social Sciences, and Medicine; it drives many of the products we use daily such as banner ad selection, email spam filtering, and social media newsfeeds. Machine Learning is concerned with making accurate, computationally efficient, interpretable and robust inferences from data. Originally borne out of Artificial Intelligence, Machine Learning has historically been the first to explore more complex prediction models and to emphasise computation, while in the past two decades Machine Learning has grown closer to Statistics gaining firm theoretical footing.

澳洲代写｜COMP30027｜Machine Learning机器学习墨尔本大学

Machine Learning机器学习问题集

问题 1.

The data and scripts for this problem are available in hw2/prob1. You can load the data using the MATLAB script load_al_data. This script should load the matrices y_noisy, y_true, X_in. The $y$ vectors are $n \times 1$ while $\mathrm{X}{-}$in is a $n \times 3$ matrix with each row corresponding to a point in $\mathcal{R}^3$. The $y{\text {true }}$ vectors correspond to the ideal $y$ values, generated directly from the “true” model (whatever it may be) without any noise. In contrast, the $y_{\text {noisy }}$ vectors are the actual, noisy observations, generated by adding Gaussian noise to the $y_{\text {true }}$ vectors. You should use $y_{n o i s y}$ for any estimation. $y_{\text {true }}$ is provided only to make it easier to evaluate the error in your predictions (simulate an infinite test data). You would not have $y_{\text {true }}$ in any real task.
(a) Write MATLAB functions theta = linear_regress $(\mathrm{y}, \mathrm{X})$ and $\mathrm{y}$ hat $=$ linear_pred(theta, X_test). Note that we are not explicitly including the offset parameter but instead rely on the feature vectors to provide a constant component. See part (b).
(b) The feature mapping can substantially affect the regression results. We will consider two possible feature mappings:
$$
\begin{aligned}
& \phi_1\left(x_1, x_2, x_3\right)=\left[1, x_1, x_2, x_3\right]^T \
& \phi_2\left(x_1, x_2, x_3\right)=\left[1, \log x_1^2, \log x_2^2, \log x_3^2\right]^T
\end{aligned}
$$
Use the provided MATLAB function feature mapping to transform the input data matrix into a matrix
$$
X=\left[\begin{array}{c}
\phi\left(\mathbf{x}1\right)^T \ \phi\left(\mathbf{x}_2\right)^T \ \cdots \ \phi\left(\mathbf{x}{\mathbf{n}}\right)^T
\end{array}\right]
$$
For example, $\mathrm{X}$ = feature_mapping ( $\mathrm{X}_{-}$in, 1 ) would get you the first feature representation. Using your completed linear regression functions, compute the mean squared prediction error for each feature mapping (2 numbers).

问题 2.

(c) The selection of points to query in an active learning framework might depend on the feature representation. We will use the same selection criterion as in the lectures, the expected squared error in the parameters, proportional to $\operatorname{Tr}\left[\left(X^T X\right)^{-1}\right]$. Write a MATLAB function $\mathrm{idx}=\operatorname{active_ learn}(\mathrm{X}, \mathrm{k} 1, \mathrm{k} 2)$. Your function should assume that the top $k_1$ rows in $X$ have been queried and your goal is to sequentially find the indices of the next $k_2$ points to query. The final set of $k_1+k_2$ indices should be returned in idx. The latter may contain repeated entries. For each feature mapping, and $k_1=5$ and $k_2=10$, compute the set of points that should be queried (i.e., $\mathrm{X}(:, \mathrm{idx})$ ). For each set of points, use the feature mapping $\phi_2$ to perform regression and compute the resulting mean squared prediction errors (MSE) over the entire data set (again, using $\phi_2$ ).

问题 3.

(d) Let us repeat the steps of part (c) with randomly selected additional points to query. We have provided a MATLAB function $i d x=\operatorname{randomly}(\operatorname{select}(\mathrm{X}, \mathrm{k} 1, \mathrm{k} 2)$ which is essentially the same as active_learn except that it selects the $k_2$ points uniformly at random from $X$. Repeat the regression steps as in previous part, and compute the resulting mean squared prediction error again. To get a reasonable comparison you should repeat this process 50 times, and use the median MSE. Compare the resulting errors with the active learning strategies. What conclusions can you draw?

问题 4.

(e) Let us now compare the two sets of points chosen by active learning due to the different feature representations. We have provided a function plot_points(X,idx_r,idx_b) which will plot each row of $X$ as a point in $\mathbf{R}^3$. The points indexed by $i d x _r$ will be circled in red and those marked by idx_b will be circled (larger) in blue (some of the points indexed by idx_r and idx_b might be common). Plot the original data points using the indexes of the actively selected points based on the two feature representations. Also plot the same indexes using $\mathrm{X}$ from the second feature representation with its first constant column removed. In class, we saw an example where the active learning strategy chose points at the extrema of the available space. Can you see evidence of this in the two plots?

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｜MAST30021｜Complex Analysis复杂分析墨尔本大学

Posted on 2023年10月12日2023年10月12日 by statistics-lab

statistics-lab^TM为您提供墨尔本大学（The University of Melbourne，简称UniMelb，中文简称“墨大”）Complex Analysis复杂分析澳洲代写代考和辅导服务！

课程介绍：

Complex analysis is a core subject in pure and applied mathematics, as well as the physical and engineering sciences. While it is true that physical phenomena are given in terms of real numbers and real variables, it is often too difficult and sometimes not possible, to solve the algebraic and differential equations used to model these phenomena without introducing complex numbers and complex variables and applying the powerful techniques of complex analysis.

Topics include:the topology of the complex plane; convergence of complex sequences and series; holomorphic functions, the Cauchy-Riemann equations, harmonic functions and applications; contour integrals and the Cauchy Integral Theorem; singularities, Laurent series, the Residue Theorem, evaluation of integrals using contour integration, conformal mapping; and aspects of the gamma function.

澳洲代写｜MAST30021｜Complex Analysis复杂分析墨尔本大学

Complex Analysis复杂分析问题集

问题 1.

By writing $z$ in the form $z=a+b \mathrm{i}$, find all solutions $z$ of the following equations:
(i) $z^2=-5+12 \mathrm{i}$
(ii) $z^2=2+\mathrm{i}$
(iii) $(7+24 \mathrm{i}) z=375$
(iv) $z^2-(3+\mathrm{i}) z+(2+2 \mathrm{i})=0$
(v) $z^2-3 z+1+\mathrm{i}=0$

问题 2.

If $\lambda$ is a positive real number, show that
$$
{z \in \mathbb{C}:|z|=\lambda|z-1|}
$$
is a circle, unless $\lambda$ takes one particular value (which?)

问题 3.

Draw the set of points
$$
{z \in \mathbb{C}: \operatorname{re}(z+1)=|z-1|}
$$
by substituting $z=x+\mathrm{i} y$ and computing the real equation relating $x$ and $y$.
Now note that re $(z+1)$ is the distance from $z$ to the line $y=-1$, and $|z-1|$ is the distance between $z$ and 1. Compare with the classical ‘focus-directrix’ definition of a parabola: the locus of a point equidistant from a fixed line (here $y=-1$ ) and a fixed point (here $(x, y)=(1,0)$ ).

问题 4.

Let $r, s, \theta, \phi$ be real. Let
$$
\begin{aligned}
z & =r(\cos \theta+\mathrm{i} \sin \theta) \
w & =s(\cos \phi+\mathrm{i} \sin \phi)
\end{aligned}
$$
Form the product $z w$ and use the standard formulas for $\cos (\theta+\phi), \sin (\theta+\phi)$ to show that $\arg (z w)=\arg (z)+\arg (w)$ (for any values of $\arg$ on the right, and some value of arg on the left).
By induction on $n$, derive De Moivre’s Theorem
$$
(\cos \theta+\mathrm{i} \sin \theta)^n=\cos n \theta+\mathrm{i} \sin n \theta
$$
for all natural numbers $n$.

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｜ETC3250｜Introduction to machine learning机器学习入门蒙纳士大学

Posted on 2023年10月12日2023年10月12日 by statistics-lab

statistics-lab^TM为您提供蒙纳士大学（Monash University）Introduction to machine learning机器学习入门澳洲代写代考和辅导服务！

课程介绍：

Business analytics involves uncovering the hidden information in masses of business data using statistical graphics, models and algorithms. The most widely used prediction and classification models will be covered. Practical skills in applying techniques to different problems will be developed using a suitable software environment that involves doing reproducible analyses. Topics to be covered include dimension reduction with methods such as principal component analysis, supervised learning with methods such as linear models, discriminant analysis, decision trees and forests, support vector machines, neural networks, and unsupervised methods such as k-means clustering. Techniques for numerical optimisation, Monte Carlo simulation, and resampling methods including bootstrap, cross-validation, and bagging will be discussed. Modelling will include nonlinear relationships and nonparametric methods.

Introduction to machine learning机器学习入门案例

问题 1.

Problem 1. Rademacher Complexities and beyond
Let $\mathcal{F}$ be a class of functions from $\mathcal{X}$ to $\mathbb{R}$ and let $X_1, \ldots, X_n$ be iid copies of a random variable $X \in \mathcal{X}$. Moreover, let $\sigma_1, \ldots, \sigma_n$ be $n$ i.i.d. $\operatorname{Rad}(1 / 2)$ random variables and let $g_1, \ldots, g_n$ be $n$ i.i.d. $N(0,1)$. Assume that all these random variables are mutually independent.

Prove the desymmetrization inequality:
$$
\mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n \sigma_i\left[f\left(X_i\right)-\mathbb{E}[f(X)]\right]\right|\right] \leq 2 \mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n\left[f\left(X_i\right)-\mathbb{E}[f(X)]\right]\right|\right]
$$

(1) Let $Y_1, \ldots, Y_n$ be ghost copies of $X_1, \ldots, X_n$. Then we have
$$
\begin{aligned}
\mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n \sigma_i\left[f\left(X_i\right)-\mathbb{E}[f(X)]\right]\right|\right] & =\mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n \sigma_i\left[f\left(X_i\right)-\mathbb{E}\left[f\left(Y_i\right)\right]\right]\right|\right] \
& \leq \mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n \sigma_i\left[f\left(X_i\right)-f\left(Y_i\right)\right]\right|\right]
\end{aligned}
$$
by Jensen’s inequality,
$$
=\mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n\left[f\left(X_i\right)-f\left(Y_i\right)\right]\right|\right],
$$
as $\sigma_i\left[f\left(X_i\right)-f\left(Y_i\right)\right]$ and $f\left(X_i\right)-f\left(Y_i\right)$ have the same distribution,
$$
\begin{aligned}
& =\mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n\left[\left(f\left(X_i\right)-\mathbb{E}\left[f\left(X_i\right)\right]\right)-\left(f\left(Y_i\right)-\mathbb{E}\left[f\left(Y_i\right)\right]\right)\right]\right|\right] \
& \leq 2 \mathbb{E}\left[\sup {f \in \mathcal{F}}\left|\frac{1}{n} \sum{i=1}^n\left[f\left(X_i\right)-\mathbb{E}\left[f\left(X_i\right)\right]\right]\right|\right]
\end{aligned}
$$
by the triangle inequality.

问题 2.

Prove the Rademacher/Gaussian process comparison inequality
$$
\mathbb{E}\left[\sup {f \in \mathcal{F}} \sum{i=1}^n \sigma_i f\left(X_i\right)\right] \leq \sqrt{\frac{\pi}{2}} \mathbb{E}\left[\sup {f \in \mathcal{F}} \sum{i=1}^n g_i f\left(X_i\right)\right]
$$
Define $R_n(\mathcal{F})=\mathbb{E}\left[\sup {f \in \mathcal{F}} \frac{1}{n}\left|\sum{i=1}^n \sigma_i f\left(X_i\right)\right|\right]$. Let $\mathcal{F}$ and $\mathcal{G}$ be two set of functions from $\mathcal{X}$ to $\mathbb{R}$ and recall that $\mathcal{F}+\mathcal{G}={f+g: f \in \mathcal{F}, g \in \mathcal{G}}$.

(2) The distribution of $g_i$ is the same as that of $\left|g_i\right| \sigma_i$, so we can write
$$
\begin{aligned}
\mathbb{E}\left[\sup {f \in \mathcal{F}} \sum{i=1}^n g_i f\left(X_i\right)\right] & =\mathbb{E}{X_i, \sigma_i}\left[\mathbb{E}{g_i}\left[\sup {f \in \mathcal{F}} \sum{i=1}^n\left|g_i\right| \sigma_i f\left(X_i\right) \mid X_i, \sigma_i\right]\right] \
& \geq \mathbb{E}{X_i, \sigma_i}\left[\sup {f \in \mathcal{F}} \sum_{i=1}^n \mathbb{E}\left[\left|g_i\right|\right] \sigma_i f\left(X_i\right)\right],
\end{aligned}
$$
by Jensen’s inequality,
$$
=\sqrt{\frac{2}{\pi}} \mathbb{E}\left[\sup {f \in \mathcal{F}} \sum{i=1}^n \sigma_i f\left(X_i\right)\right],
$$
using the first absolute moment of the standard Gaussian.

问题 3.

Let $h \in \mathbb{R}^{\mathcal{X}}$ be a given function and define $\mathcal{F}+h={f+h: f \in \mathcal{F}}$. Show that
$$
R_n(\mathcal{F}+{h}) \leq R_n(\mathcal{F})+\frac{|h|_{\infty}}{\sqrt{n}},
$$
where $|h|_{\infty}=\sup _{x \in \mathcal{X}}|h(x)|$.

(3) We compute:
$$
\begin{aligned}
R_n(\mathcal{F}+h) & =\mathbb{E}\left[\sup {f \in \mathcal{F}} \frac{1}{n}\left|\sum{i=1}^n \sigma_i\left(f\left(X_i\right)+h\left(X_i\right)\right)\right|\right] \
& \leq \mathbb{E}\left[\sup {f \in \mathcal{F}} \frac{1}{n}\left|\sum{i=1}^n \sigma_i f\left(X_i\right)\right|\right]+\mathbb{E}\left[\frac{1}{n}\left|\sum_{i=1}^n h\left(X_i\right)\right|\right],
\end{aligned}
$$
by the triangle inequality,
$$
\begin{aligned}
& =R_n(\mathcal{F})+\mathbb{E}\left[\frac{1}{n}\left|\sum_{i=1}^n h\left(X_i\right)\right|\right], \
& \leq R_n(\mathcal{F})+\frac{1}{n} \sqrt{\mathbb{E}\left[\left(\sum_{i=1}^n \sigma_i h\left(X_i\right)\right)^2\right]},
\end{aligned}
$$
by Jensen’s inequality,
$$
\begin{aligned}
& =R_n(\mathcal{F})+\frac{1}{n} \sqrt{\sum_{i=1}^n \mathbb{E}\left[h\left(X_i\right)^2\right]+2 \sum_{i<j} \mathbb{E}\left[\sigma_i \sigma_j h\left(X_i\right) h\left(X_j\right)\right]} \
& =R_n(\mathcal{F})+\frac{1}{n} \sqrt{\sum_{i=1}^n \mathbb{E}\left[h\left(X_i\right)^2\right]}
\end{aligned}
$$
by symmetry,
$$
\begin{aligned}
& \leq R_n(\mathcal{F})+\frac{1}{n} \sqrt{n|h|_{\infty}^2} \
& =R_n(\mathcal{F})+\frac{|h|_{\infty}}{n}
\end{aligned}
$$

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

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