### 数学代写|离散数学作业代写discrete mathematics代考|Analysis

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|离散数学作业代写discrete mathematics代考|Analysis

We start by reminding the reader of the definition of a metric.
Definition $1.5$ (Metric, Triangle inequality). Let $X$ be a set. We say that $d: X \times X \rightarrow \mathbb{R}$ is a metric on $X$ if the following are satisfied for all $x, y, z \in X$ :
(i) $d(x, y) \geq 0$;
(ii) $d(x, y)=0$ if and only if $x=y$;
(iii) $d(x, y)=d(y, x)$;
(iv) $d(x, y) \leq d(x, z)+d(z, y)$ (this is referred to as the triangle inequality).
We will have the need to appeal to Fekete’s Lemma, which is quite useful for many combinatorial functions, not just in Ramsey theory.

Lemma 1.6 (Fekete’s Lemma). For any sequence of real numbers $\left{s_{i}\right}_{i=1}^{\infty}$, if either (i) $s_{i+j} \leq s_{i}+s_{j}$ for all $i, j \in \mathbb{Z}^{+}$or (ii) $s_{i+j} \geq s_{i}+s_{j}$ for all $i, j \in \mathbb{Z}^{+}$, then
$$\lim {n \rightarrow \infty} \frac{s{n}}{n}$$
exists and equals $\inf {n} \frac{s{n}}{n}$ if (i) is satisfied; it equals $\sup {n} \frac{s{n}}{n}$ if (ii) is satisfied.
An easy corollary of Fekete’s Lemma is also useful (and is often referred to as Fekete’s Lemma, too).

Corollary 1.7. For any sequence of real numbers $\left{s_{i}\right}_{i=1}^{\infty}$, if either (i) $s_{i+j} \leq$ $s_{i}, s_{j}$ for all $i, j \in \mathbb{Z}^{+}$or (ii) $s_{i+j} \geq s_{i} \cdot s_{j}$ for all $i, j \in \mathbb{Z}^{+}$, then
$$\lim {n \rightarrow \infty}\left(s{n}\right)^{\frac{1}{n}}$$
exists and equals $\inf {n}\left(s{n}\right)^{\frac{1}{n}}$ if (i) is satisfied; it equals $\sup {n}\left(s{n}\right)^{\frac{1}{n}}$ if (ii) is satisfied.

## 数学代写|离散数学作业代写discrete mathematics代考|Probability

The probability we use is basic. Recall that if $E$ and $F$ are independent events, then
$$\mathbb{P}(E \cap F)=\mathbb{P}(E) \cdot \mathbb{P}(F),$$
but that for general events, we have
$$\mathbb{P}(E \cap F)=\mathbb{P}(E) \cdot \mathbb{P}(F \mid E)$$
If $E$ and $F$ are mutually exclusive, i.e., $E \cap F=\emptyset$, then
$$\mathbb{P}(E \sqcup F)=\mathbb{P}(E)+\mathbb{P}(F)$$
while for general events, we have
$$\mathbb{P}(E \cup F)=\mathbb{P}(E)+\mathbb{P}(F)-\mathbb{P}(E \cap F)$$
We will also use expectation of a random variable. If $X$ is a random variable taking on possible values $x_{1}, x_{2}, \ldots$, then
$$\mathbb{E}(X)=\sum_{i} x_{i} \mathbb{P}\left(X=x_{i}\right)$$
We will often use indicator random variables (i.e., Bernoulli random variable, which take on values of 0 and 1 only). For any indicator random variable $X$, we have
$$\mathbb{E}(X)=\mathbb{P}(X=1),$$
since $\mathbb{E}(X)=0 \cdot \mathbb{P}(X=0)+1 \cdot \mathbb{P}(X=1)$.
We will almost exclusively be dealing with finite sample spaces that have equally likely outcomes so that when we randomly choose an element from a sample space with $n$ elements, the probability of choosing that element is $\frac{1}{n}$.

## 数学代写|离散数学作业代写discrete mathematics代考|Algebra

We will use some linear algebra but will remind the reader of the relevant facts as needed.

Our main reminder regarding abstract algebra is for what occurs in Sections $3.3 .3$ and $7.1$, where we use the coset decompositions of groups. For completeness, let $H$ be a subgroup of group $G$. Then a (left) coset of $H$ in $G$ has form
$$a H={a h: h \in H}$$
where $a \in G$. As far as cosets are concerned, we will only be using left cosets (and, mostly, our groups will be Abelian so that the left/right distinction is immaterial). By Lagrange’s Theorem, we know that every coset of $H$ has the

same number of elements, namely $|H|$, and that no two cosets of $H$ have non-empty intersection. It follows that the number of cosets of $H$ in $G$ is
$$|G: H|=\frac{|G|}{|H|} .$$
We will also be using group actions; that is, if $G$ is a group and $S$ is a set, we use $*: G \times S \rightarrow S$ (akin to a binary operation). Applying group actions, we will be using the concepts of orbits and stabilizers, defined next.

Definition $1.10$ (Orbit). Let $*$ be a group action on set $S$ by group $G$. For $s \in S$, the orbit of $s$ is
$$\mathcal{O}{s}={t \in S: g * s=t \text { for some } g \in G} .$$ Definition $1.11$ (Stabilizer). Let * be a group action on set $S$ by group $G$. For $s \in S$, the stabilizer of $s$ is $$G{s}={g \in G: g * s=s} .$$
In Exercise 1.17, the reader is asked to prove that $G_{s}$ is a subgroup of $G$.
In Section 3.3.3, we will be appealing to the Orbit-Stabilizer Theorem:
Theorem $1.12$ (Orbit-Stabilizer Theorem). Let $G$ be a finite group acting on a finite set $S$. Then
$$\left|\mathcal{O}{s}\right| \cdot\left|G{s}\right|=|G|$$
for any $s \in S$.

## 数学代写|离散数学作业代写discrete mathematics代考|Analysis

(二)d(X,是)=0当且仅当X=是;
㈢d(X,是)=d(是,X);
(四)d(X,是)≤d(X,和)+d(和,是)（这被称为三角不等式）。

Fekete 引理的一个简单推论也是有用的（通常也被称为 Fekete 引理）。

|G:H|=|G||H|.

|这s|⋅|Gs|=|G|

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