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Zheyuan Wu
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# Beginning of the Summer Program
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## Schedule
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Presentation starts next week
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Start with examples, do exploratory work, it's just a summer.
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Final work: Find certain topic you are interested in, and do a expository paper.
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Find the motivation, background, definition, theorem, example, application for the theory you are interested in.
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At least 3 presentations is required.
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Collect the papers you interested in as you go the research, it is not linear.
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Symposium on November.
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Lightning talk (3 minutes) on end of July.
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# Schedule Presentation
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# Topic 1
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# Topic 1: Probability under language of measure theory
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## Probability Theory under Language of Measure Theory
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## Section 1: Uniform Random Numbers
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### Uniform random numbers
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### Basic Definitions
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Define picking a random number from the interval $[0,1]$ form the uniform probability distribution.
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#### Definition of Random Variables
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As a function $f:[0,1]\to S$, where $S$ is the space of potential outcomes of a random phenomenon. (Note, this definition inverts the axis of "probability" and "event" so that we can apply the measure theory to probability theory. Before, we define the probability of an event as a function $P:S\to [0,1]$, where $S\in A$ and $\int_A P(x)dx=1$.)
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A random variable is a function $f:[0,1]\to S$, where $[0,1]\subset \mathbb{R}$ and $S$ is a set of potential outcomes of a random phenomenon.
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$\ket{1}= \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ is a vector in a Hilbert space.
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#### Definition of Uniform Distribution
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The uniform distribution is defined by the length of function on subsets of $[0,1]$ as a measure of probability ([Lebesgue measure](https://notenextra.trance-0.com/Math4121/Math4121_L30#lebesgue-measure) by default).
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Let $X$ be a random number taken from $[0,1]$ and having the uniform distribution. The probability that $X$ should be the probability of the event that $X$ lies in $A$.
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$$
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\operatorname{Prob}(X\in A) =\lambda(A)=\text{length of }A
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$$
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#### Definition of Expectation
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Let $f:[0,1]\to \mathbb{R}$ be a random variable (with nice properties such that it is integrable). Then the expectation of $f$ is defined as
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$$
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\mathbb{E}[f]=\mathbb{E}[f(X)]=\int_0^1 f(x)dx
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$$
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#### Definition of Indicator Function
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The indicator function of an event $A$ is defined as
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$$
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\mathbb{I}_A(x)=\begin{cases}
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1 & \text{if } x\in A \\
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0 & \text{if } x\notin A
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\end{cases}
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$$
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#### Definition of Law of variable X
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The law of a random variable $X$ is the probability distribution of $X$.
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Let $Y$ be the outcome of $f(X)$. Then the law of $Y$ is the probability distribution of $Y$.
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$$
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\mu_Y(A)=\lambda(f^{-1}(A))=\lambda(\{x\in [0,1]: f(x)\in A\})
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$$
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### 1.1 Mathematical Coin Flip model
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A coin flip if a random experiment with two possible outcomes: $S=\{0,1\}$. with probability $p$ for $0$ and $1-p$ for $1$, where $p\in (0,1)\subset \mathbb{R}$.
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#### Definition of Independent Events
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Two events $A$ and $B$ are independent if
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$$
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\lambda(A\cap B)=\lambda(A)\lambda(B)
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$$
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or equivalently,
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$$
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\operatorname{Prob}(X\in A\cap B)=\operatorname{Prob}(X\in A)\operatorname{Prob}(X\in B)
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$$
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Generalization to $n$ events:
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$$
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\lambda(A_1\cap A_2\cap \cdots \cap A_n)=\lambda(A_1)\lambda(A_2)\cdots \lambda(A_n)
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$$
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#### Definition of Outcome selecting function
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Let the set of all possible outcomes represented by a Cartesian product $S=\{0,1\}^{\mathbb{N}}$. $(a_1,a_2,a_3,\cdots)\subset S$ is an infinite (or finite) sequence of coin flips.
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$\pi_i:S\to \{0,1\}$ is the $i$-th projection function defined as $\pi_i((a_1,a_2,a_3,\cdots))=a_i$.
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> Note, this representation is isomorphic to the dyadic rationals (i.e., numbers that can be written as a fraction whose denominator is a power of 2) in the interval $[0,1]$.
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## Section 2: Formal definitions
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> Recall, the $\sigma$-algebra (denoted as $\mathcal{A}$ in Math4121) is the collection of all subsets of a set $S$ satisfying the following properties:
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>
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> 1. $\emptyset\in \mathcal{A}$ (empty set is in the $\sigma$-algebra)
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> 2. If $A\in \mathcal{A}$, then $A^c\in \mathcal{A}$ (if a set is in the $\sigma$-algebra, then its complement is in the $\sigma$-algebra)
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> 3. If $A_1,A_2,A_3,\cdots\in \mathcal{A}$, then $\bigcup_{i=1}^{\infty}A_i\in \mathcal{A}$ (if a countable sequence of sets is in the $\sigma$-algebra, then their union is in the $\sigma$-algebra)
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### Event, probability, and random variable
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Let $\Omega$ be a non-empty set.
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Let $\mathscr{F}$ be a $\sigma$-algebra on $\Omega$ (Note, $\mathscr{F}$ is a collection of subsets of $\Omega$ that satisfies the properties of a $\sigma$-algebra).
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#### Definition of Event
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An event is a element of $\mathscr{F}$.
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#### Definition of Probability Measure
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A probability measure $P$ is a function $P:\mathscr{F}\to [0,1]$ satisfying the following properties:
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1. $P(\Omega)=1$
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2. If $A_1,A_2,A_3,\cdots\in \mathscr{F}$ are pairwise disjoint ($\forall i\neq j, A_i\cap A_j=\emptyset$), then $P(\bigcup_{i=1}^{\infty}A_i)=\sum_{i=1}^{\infty}P(A_i)$
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#### Definition of Probability Space
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A probability space is a triple $(\Omega, \mathscr{F}, P)$ defined above.
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An event $A$ is said to occur almost surely (a.s.) if $P(A)=1$.
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#### Definition of Random Variable
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A random variable is a function $X:\Omega\to \mathbb{R}$ that is measurable with respect to the $\sigma$-algebra $\mathscr{F}$.
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That is, for any Borel set $B\subset \mathbb{R}$, the preimage $f^{-1}(B)\in \mathscr{F}$.
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$$
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f^{-1}(B)=\{x\in \Omega: f(x)\in B\}\in \mathscr{F}
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$$
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#### Definition of sigma-algebra generated by a random variable
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Let $\{f_\alpha:\Omega\to \mathbb{R},\alpha\in I\}$ be a family of functions where $I$ is an index set which is not necessarily finite or countable. The $\sigma$-algebra generated by the family of functions $\{f_\alpha:\alpha\in I\}$, denoted as $\sigma\{f_\alpha:\alpha\in I\}$, is the smallest $\sigma$-algebra containing all the subsets of $\Omega$ of the form
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$$
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f_\alpha^{-1}(B)=\{\omega\in \Omega: f_\alpha(\omega)\in B\}\in \mathscr{F}
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$$
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for all $\alpha\in I$ and $B\in \mathscr{B}(\mathbb{R})$.
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Equivalently,
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$$
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\sigma\{f_\alpha:\alpha\in I\}=\sigma\left(\bigcup_{\alpha\in I}f_\alpha^{-1}(B)\right)
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$$
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the sigma-algebra generated by a random variable $X$ is the intersection of all $\sigma$-algebras on $\Omega$ containing the sets $f_\alpha^{-1}(B)$ for all $\alpha\in I$ and $B\in \mathscr{B}(\mathbb{R})$.
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#### Definition of distribution of random variable
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Let $f:\Omega\to \mathbb{R}$ be a random variable. The distribution of $f$ is the probability measure $P_f$ on $\mathbb{R}$ defined by
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$$
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P_f(B)=P(f^{-1}(B))=P(\{x\in \Omega: f(x)\in B\})
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$$
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also noted as $f_*P$.
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#### Definition of joint distribution of random variables
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Let $f_1,f_2,\cdots,f_n:\Omega\to \mathbb{R}$ be random variables. The joint distribution of $f_1,f_2,\cdots,f_n$ is the probability measure $P_{f_1,f_2,\cdots,f_n}$ on $\mathbb{R}^n$ defined by
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$$
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P_{f_1,f_2,\cdots,f_n}(B)=P(f_1^{-1}(B_1)\cap f_2^{-1}(B_2)\cap \cdots \cap f_n^{-1}(B_n))=P(\omega\in \Omega: (f_1(\omega),f_2(\omega),\cdots,f_n(\omega))\in B)
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$$
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### Expectation of a random variable
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Let $f:\Omega\to \mathbb{R}$ be a random variable. The expectation of $f$ is defined as
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$$
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\mathbb{E}[f]=\mathbb{E}[f(X)]=\int_\Omega f(x)dP
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$$
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Note, $P$ is the probability measure on $\Omega$.
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#### Definition of variance
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The variance of a random variable $f$ is defined as
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$$
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\operatorname{Var}(f)=\mathbb{E}[(f-\mathbb{E}[f])^2]=\mathbb{E}[f^2]-(\mathbb{E}[f])^2
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$$
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#### Definition of covariance
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The covariance of two random variables $f,g:\Omega\to \mathbb{R}$ is defined as
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$$
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\operatorname{Cov}(f,g)=\mathbb{E}[(f-\mathbb{E}[f])(g-\mathbb{E}[g])]
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$$
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### Point measures
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#### Definition of Dirac measure
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The Dirac measure is a probability measure on $\Omega$ defined as
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$$
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\delta_\omega(A)=\begin{cases}
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1 & \text{if } \omega\in A \\
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0 & \text{if } \omega\notin A
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\end{cases}
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$$
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Note that $\int_\Omega f(x)d\delta_\omega(x)=f(\omega)$.
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### Infinite sequence of independent coin flips
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> Side notes from basic topology:
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>
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> **Definition of product topology**:
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>
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> It is a set constructed by the Cartesian product of the sets. Suppose $X_i$ is a set for all $i\in I$. The element of the product set is a tuple $(x_i)_{i\in I}$ where $x_i\in X_i$ for all $i\in I$.
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>
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> For example, if $X_i=[0,1]$ for all $i\in \mathbb{N}$, then the product set is $[0,1]^{\mathbb{N}}$. An element of such product set is $(1,0.5,0.25,\cdots)$.
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The set of outcomes of such infinite sequence of coin flips is the product set of the set of outcomes of each coin flip.
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$$
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S=\{0,1\}^{\mathbb{N}}
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$$
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### Conditional probability
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#### Definition of conditional probability
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The conditional probability of an event $A$ given an event $B$ is defined as
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$$
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P(A|B)=\frac{P(A\cap B)}{P(B)}
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$$
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The law of total probability:
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$$
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P(A)=\sum_{i=1}^{\infty}P(A|B_i)P(B_i)
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$$
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Bayes' theorem:
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$$
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P(B_i|A)=\frac{P(A|B_i)P(B_i)}{\sum_{j=1}^{\infty}P(A|B_j)P(B_j)}
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$$
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#### Definition of independence of random variables
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Two random variables $f,g:\Omega\to \mathbb{R}$ are independent if for any Borel sets $A,B\subset \mathscr{B}(\mathbb{R})$ the events
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$$
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\{\omega\in \Omega: f(\omega)\in A\}\text{ and } \{\omega\in \Omega: g(\omega)\in B\}
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$$
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are independent.
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In general, a finite or infinite family of random variables $f_1,f_2,\cdots,f_n:\Omega\to \mathbb{R}$ are independent if every finite collection of random variables from this family are independent.
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#### Definition of independence of sigma-algebras
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Let $\mathscr{G}$ and $\mathscr{H}$ be two $\sigma$-algebras on $\Omega$. They are independent if for any Borel sets $A\subset \mathscr{B}(\mathbb{R})$ and $B\subset \mathscr{B}(\mathbb{R})$, the finite collection of events are independent.
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## Section 3: Further definitions in measure theory and integration
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### $L^2$ space
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#### Definition of $L^2$ space
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Let $(\Omega, \mathscr{F}, P)$ be a measure space. The $L^2$ space is the space of all square integrable, complex-valued measurable functions on $\Omega$.
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Denoted by $L^2(\Omega, \mathscr{F}, P)$.
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The square integrable functions are the functions $f:\Omega\to \mathbb{C}$ such that
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$$
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\int_\Omega |f(\omega)|^2 dP(\omega)<\infty
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$$
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With inner product defined by
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$$
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\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
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$$
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The $L^2(\Omega, \mathscr{F}, P)$ space is a Hilbert space.
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337
pages/Math401/Math401_T2.md
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337
pages/Math401/Math401_T2.md
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# Topic 2: Finite-dimensional Hilbert spaces
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Recall the complex number is a tuple of two real numbers, $z=(a,b)$ with addition and multiplication defined by
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$$
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(a,b)+(c,d)=(a+c,b+d)
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$$
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$$
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(a,b)\cdot(c,d)=(ac-bd,ad+bc)
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$$
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or in polar form,
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$$
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z=re^{i\theta}=r(\cos\theta+i\sin\theta)
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$$
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where $r=\sqrt{a^2+b^2}=\sqrt{z\overline{z}}$ and $\theta=\tan^{-1}(b/a)$.
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The complex conjugate of $z$ is $\overline{z}=(a,-b)$.
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## Section 1: Finite-dimensional Complex Vector Spaces
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Here, we use the field $\mathbb{C}$ of complex numbers. or the field $\mathbb{R}$ of real numbers as the field $\mathbb{F}$ we are going to encounter.
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### Definition of vector space
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A vector space $\mathscr{V}$ over a field $\mathbb{F}$ is a set equipped with an **addition** and a **scalar multiplication**, satisfying the following axioms:
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1. Addition is associative and commutative. For all $u,v,w\in \mathscr{V}$,
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Associativity:
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$$
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(u+v)+w=u+(v+w)
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$$
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Commutativity:
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$$
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u+v=v+u
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$$
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2. Additive identity: There exists an element $0\in \mathscr{V}$ such that $v+0=v$ for all $v\in \mathscr{V}$.
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3. Additive inverse: For each $v\in \mathscr{V}$, there exists an element $-v\in \mathscr{V}$ such that $v+(-v)=0$.
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4. Multiplicative identity: There exists an element $1\in \mathbb{F}$ such that $v\cdot 1=v$ for all $v\in \mathscr{V}$.
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5. Multiplicative inverse: For each $v\in \mathscr{V}$ and $c\in \mathbb{F}$, there exists an element $c^{-1}\in \mathbb{F}$ such that $v\cdot c^{-1}=1$.
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6. Distributivity: For all $u,v\in \mathscr{V}$ and $c,d\in \mathbb{F}$,
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$$
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c(u+v)=cu+cv
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$$
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A vector is an ordered pair of elements over the field $\mathbb{F}$.
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If we consider $\mathbb{F}=\mathbb{C}^n$, $n\in \mathbb{N}$, then $u=(a_1,a_2,\cdots,a_n), v=(b_1,b_2,\cdots,b_n)\in \mathbb{C}^n$ are vectors.
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The addition and scalar multiplication are defined by
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$$
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u+v=(a_1+b_1,a_2+b_2,\cdots,a_n+b_n)
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$$
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||||
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||||
$$
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cu=(ca_1,ca_2,\cdots,ca_n)
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$$
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$c\in \mathbb{C}$.
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The matrix transpose is defined by
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$$
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u^T=(a_1,a_2,\cdots,a_n)^T=\begin{pmatrix}
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a_1 \\
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a_2 \\
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\vdots \\
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a_n
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\end{pmatrix}
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$$
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The complex conjugate transpose is defined by
|
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|
||||
$$
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||||
u^*=(a_1,a_2,\cdots,a_n)^*=\begin{pmatrix}
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||||
\overline{a_1} \\
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\overline{a_2} \\
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\vdots \\
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\overline{a_n}
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\end{pmatrix}
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$$
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> In physics, the complex conjugate is sometimes denoted by $z^*$ instead of $\overline{z}$.
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> The complex conjugate transpose is sometimes denoted by $u^\dagger$ instead of $u^*$.
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### Hermitian inner product and norms
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||||
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On $\mathbb{C}^n$, the Hermitian inner product is defined by
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||||
$$
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\langle u,v\rangle=\sum_{i=1}^n \overline{u_i}v_i
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$$
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||||
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The norm is defined by
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$$
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\|u\|=\sqrt{\langle u,u\rangle}
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$$
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#### Definition of Inner product
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Let $\mathscr{H}$ be a complex vector space. An inner product on $\mathscr{H}$ is a function $\langle \cdot, \cdot \rangle: \mathscr{H}\times \mathscr{H}\to \mathbb{C}$ satisfying the following axioms:
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||||
|
||||
1. For each $u\in \mathscr{H}$, $v\mapsto \langle u,v\rangle$ is a linear map.
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||||
$$
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||||
\langle u,av+bw\rangle=a\langle u,v\rangle+b\langle u,w\rangle
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||||
$$
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||||
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||||
For all $u,v,w\in \mathscr{H}$ and $a,b\in \mathbb{C}$.
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||||
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||||
2. For all $u,v\in \mathscr{H}$, $\langle u,v\rangle=\overline{\langle v,u\rangle}$.
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||||
$u\mapsto \langle u,v\rangle$ is a conjugate linear map.
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3. $\langle u,u\rangle\geq 0$ and $\langle u,u\rangle=0$ if and only if $u=0$.
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||||
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||||
#### Definition of norm
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||||
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||||
Let $\mathscr{H}$ be a complex vector space. A norm on $\mathscr{H}$ is a function $\|\cdot\|: \mathscr{H}\to \mathbb{R}$ satisfying the following axioms:
|
||||
|
||||
1. For all $u\in \mathscr{H}$, $\|u\|\geq 0$ and $\|u\|=0$ if and only if $u=0$.
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||||
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||||
2. For all $u\in \mathscr{H}$ and $c\in \mathbb{C}$, $\|cu\|=|c|\|u\|$.
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||||
|
||||
3. Triangle inequality: For all $u,v\in \mathscr{H}$, $\|u+v\|\leq \|u\|+\|v\|$.
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||||
|
||||
#### Definition of inner product space
|
||||
|
||||
A complex vector space $\mathscr{H}$ with an inner product is called a **Hilbert space**.
|
||||
|
||||
#### Cauchy-Schwarz inequality
|
||||
|
||||
For all $u,v\in \mathscr{H}$,
|
||||
|
||||
$$
|
||||
|\langle u,v\rangle|\leq \|u\|\|v\|
|
||||
$$
|
||||
|
||||
#### Parallelogram law
|
||||
|
||||
For all $u,v\in \mathscr{H}$,
|
||||
|
||||
$$
|
||||
\|u+v\|^2+\|u-v\|^2=2(\|u\|^2+\|v\|^2)
|
||||
$$
|
||||
|
||||
#### Polarization identity
|
||||
|
||||
For all $u,v\in \mathscr{H}$,
|
||||
|
||||
$$
|
||||
\langle u,v\rangle=\frac{1}{4}(\|u+v\|^2-\|u-v\|^2+i\|u+iv\|^2-i\|u-iv\|^2)
|
||||
$$
|
||||
|
||||
#### Additional definitions
|
||||
|
||||
Let $u,v\in \mathscr{H}$.
|
||||
|
||||
$||v||$ is the length of $v$.
|
||||
|
||||
$v$ is a unit vector if $\|v\|=1$.
|
||||
|
||||
$u,v$ are orthogonal if $\langle u,v\rangle=0$.
|
||||
|
||||
#### Definition of orthonormal basis
|
||||
|
||||
A set of vectors $\{e_1,e_2,\cdots,e_n\}$ in a Hilbert space $\mathscr{H}$ is called an orthonormal basis if
|
||||
|
||||
1. $\langle e_i,e_j\rangle=\delta_{ij}$ for all $i,j\in \{1,2,\cdots,n\}$.
|
||||
|
||||
$$
|
||||
\delta_{ij}=\begin{cases}
|
||||
1 & \text{if } i=j \\
|
||||
0 & \text{if } i\neq j
|
||||
\end{cases}
|
||||
$$
|
||||
|
||||
2. $n=\dim \mathscr{H}$.
|
||||
|
||||
### Subspaces and orthonormal bases
|
||||
|
||||
#### Definition of subspace
|
||||
|
||||
A subset $\mathscr{W}$ of a vector space $\mathscr{V}$ is a subspace if it is closed under addition and scalar multiplication.
|
||||
|
||||
#### Definition of orthogonal complement
|
||||
|
||||
Let $E$ be a subset of a Hilbert space $\mathscr{H}$. The orthogonal complement of $E$ is the set of all vectors in $\mathscr{H}$ that are orthogonal to every vector in $E$.
|
||||
|
||||
$$
|
||||
E^\perp=\{v\in \mathscr{H}: \langle v,w\rangle=0 \text{ for all } w\in E\}
|
||||
$$
|
||||
|
||||
#### Definition of orthogonal projection
|
||||
|
||||
Let $E$ be a $m$-dimensional subspace of a Hilbert space $\mathscr{H}$. An orthogonal projection of $E$ is a linear map $P_E: \mathscr{H}\to E$
|
||||
|
||||
$$
|
||||
P_E(v)=\sum_{i=1}^m \langle v,e_i\rangle e_i
|
||||
$$
|
||||
|
||||
#### Definition of orthonormal direct sum
|
||||
|
||||
A inner product space $\mathscr{H}$ is the direct sum of $E_1,E_2,\cdots,E_n$ if
|
||||
|
||||
$$
|
||||
\mathscr{H}=E_1\oplus E_2\oplus \cdots \oplus E_n
|
||||
$$
|
||||
|
||||
and $E_i\cap E_j=\{0\}$ for all $i\neq j$.
|
||||
|
||||
That is, $\forall v\in \mathscr{H}$, there exists a unique $v_i\in E_i$ such that $v=v_1+v_2+\cdots+v_n$.
|
||||
|
||||
#### Definition of meet and join of subspaces
|
||||
|
||||
Let $E$ and $F$ be two subspaces of a Hilbert space $\mathscr{H}$. The meet of $E$ and $F$ is the subspace $\mathscr{H}$ such that
|
||||
|
||||
$$
|
||||
E\land F=E\cap F
|
||||
$$
|
||||
|
||||
The join of $E$ and $F$ is the subspace $\mathscr{H}$ such that
|
||||
|
||||
$$
|
||||
E\lor F=\{u+v: u\in E, v\in F\}
|
||||
$$
|
||||
|
||||
### Null space and range
|
||||
|
||||
#### Definition of null space
|
||||
|
||||
Let $A$ be a linear map from a vector space $\mathscr{V}$ to a vector space $\mathscr{W}$. The null space of $A$ is the set of all vectors in $\mathscr{V}$ that are mapped to the zero vector in $\mathscr{W}$.
|
||||
|
||||
$$
|
||||
\text{Null}(A)=\{v\in \mathscr{V}: Av=0\}
|
||||
$$
|
||||
|
||||
#### Definition of range
|
||||
|
||||
Let $A$ be a linear map from a vector space $\mathscr{V}$ to a vector space $\mathscr{W}$. The range of $A$ is the set of all vectors in $\mathscr{W}$ that are mapped from $\mathscr{V}$.
|
||||
|
||||
$$
|
||||
\text{Range}(A)=\{w\in \mathscr{W}: \exists v\in \mathscr{V}, Av=w\}
|
||||
$$
|
||||
|
||||
### Dual spaces and adjoints of linear maps\
|
||||
|
||||
#### Definition of linear map
|
||||
|
||||
A linear map $T: \mathscr{V}\to \mathscr{W}$ is a function that satisfies the following axioms:
|
||||
|
||||
1. Additivity: For all $u,v\in \mathscr{V}$ and $a,b\in \mathbb{F}$,
|
||||
|
||||
$$
|
||||
T(au+bv)=aT(u)+bT(v)
|
||||
$$
|
||||
|
||||
2. Homogeneity: For all $u\in \mathscr{V}$ and $a\in \mathbb{F}$,
|
||||
|
||||
$$
|
||||
T(au)=aT(u)
|
||||
$$
|
||||
|
||||
#### Definition of linear functionals
|
||||
|
||||
A linear functional $f: \mathscr{V}\to \mathbb{F}$ is a linear map from $\mathscr{V}$ to $\mathbb{F}$.
|
||||
|
||||
Here, $\mathbb{F}$ is the field of complex numbers.
|
||||
|
||||
#### Definition of dual space
|
||||
|
||||
Let $\mathscr{V}$ be a vector space over a field $\mathbb{F}$. The dual space of $\mathscr{V}$ is the set of all linear functionals on $\mathscr{V}$.
|
||||
|
||||
$$
|
||||
\mathscr{V}^*=\{f:\mathscr{V}\to \mathbb{F}: f\text{ is linear}\}
|
||||
$$
|
||||
|
||||
If $\mathscr{H}$ is a finite-dimensional Hilbert space, then $\mathscr{H}^*$ is isomorphic to $\mathscr{H}$.
|
||||
|
||||
Note $v\in \mathscr{H}\mapsto \langle v,\cdot\rangle\in \mathscr{H}^*$ is a conjugate linear isomorphism.
|
||||
|
||||
#### Definition of adjoint of a linear map
|
||||
|
||||
Let $T: \mathscr{V}\to \mathscr{W}$ be a linear map. The adjoint of $T$ is the linear map $T^*: \mathscr{W}\to \mathscr{V}$ such that
|
||||
|
||||
$$
|
||||
\langle Tv,w\rangle=\langle v,T^*w\rangle
|
||||
$$
|
||||
|
||||
for all $v\in \mathscr{V}$ and $w\in \mathscr{W}$.
|
||||
|
||||
#### Definition of self-adjoint operators
|
||||
|
||||
A linear operator $T: \mathscr{V}\to \mathscr{V}$ is self-adjoint if $T^*=T$.
|
||||
|
||||
#### Definition of unitary operators
|
||||
|
||||
A linear map $T: \mathscr{V}\to \mathscr{V}$ is unitary if $T^*T=TT^*=I$.
|
||||
|
||||
### Dirac's bra-ket notation
|
||||
|
||||
#### Definition of bra and ket
|
||||
|
||||
Let $\mathscr{H}$ be a Hilbert space. The bra-ket notation is a notation for vectors in $\mathscr{H}$.
|
||||
|
||||
$$
|
||||
\langle v|w\rangle
|
||||
$$
|
||||
|
||||
is the inner product of $v$ and $w$.
|
||||
|
||||
$$
|
||||
|v\rangle
|
||||
$$
|
||||
|
||||
is the vector (or linear map) $v$.
|
||||
|
||||
$$
|
||||
|u\rangle\langle v|
|
||||
$$
|
||||
|
||||
is a linear map from $\mathscr{H}$ to $\mathscr{H}$.
|
||||
|
||||
11
pages/Math401/_meta.js
Normal file
11
pages/Math401/_meta.js
Normal file
@@ -0,0 +1,11 @@
|
||||
export default {
|
||||
index: "Course Description",
|
||||
"---":{
|
||||
type: 'separator'
|
||||
},
|
||||
Math401_N1: "Math 401, Notes 1",
|
||||
Math401_N2: "Math 401, Notes 2",
|
||||
Math401_N3: "Math 401, Notes 3",
|
||||
Math401_T1: "Math 401, Topic 1: Probability under language of measure theory",
|
||||
Math401_T2: "Math 401, Topic 2: Finite-dimensional Hilbert spaces",
|
||||
}
|
||||
@@ -3,3 +3,67 @@
|
||||
This is a course about symmetrical group and bunch of applications in other fields of math.
|
||||
|
||||
Prof. Renado Fere is teaching this course.
|
||||
|
||||
The course is split into two parts:
|
||||
|
||||
1. Symmetrical group
|
||||
2. Summer research project
|
||||
|
||||
## Symmetrical group (Spring 2025 Course)
|
||||
|
||||
Notes from N1-N3.
|
||||
|
||||
Basically are overview of some interesting topics related to symmetrical group or other related math topics.
|
||||
|
||||
I don't record them carefully, but I will try to update them if they are necessary for my future reference.
|
||||
|
||||
## Summer research project
|
||||
|
||||
### Schedule
|
||||
|
||||
Presentation starts next week
|
||||
|
||||
Start with examples, do exploratory work, it's just a summer.
|
||||
|
||||
Final work: Find certain topic you are interested in, and do a expository paper.
|
||||
|
||||
Find the motivation, background, definition, theorem, example, application for the theory you are interested in.
|
||||
|
||||
At least 3 presentations is required.
|
||||
|
||||
Collect the papers you interested in as you go the research, it is not linear.
|
||||
|
||||
Symposium on November.
|
||||
|
||||
Lightning talk (3 minutes) on end of July.
|
||||
|
||||
### Topic of interest
|
||||
|
||||
I am interested in the following topics:
|
||||
|
||||
1. Quantum error correction
|
||||
2. Von Neumann algebra and other operator algebras which are related to quantum algorithms
|
||||
|
||||
### Notes
|
||||
|
||||
T1-T7 should be the notes for the spring course Math 444. Taught by Prof. Renado Fere on Spring 2025 but I don't know that and they are helpful for understanding the material for the book that might contains my subject of interest.
|
||||
|
||||
[The Functional Analysis of Quantum Information Theory](https://arxiv.org/abs/1410.7188)
|
||||
|
||||
The original lecture notes, by Prof. Renado Fere, are here, may move to other places. Last updated on 2025-06-14.
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 1](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes01.pdf)
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 2](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes02.pdf)
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 3](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes03.pdf)
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 4](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes04.pdf)
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 5](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes05.pdf)
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 6](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes06.pdf)
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 7](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes07.pdf)
|
||||
|
||||
[Math 444 Spring 2025 Notes, Lecture 9](https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Notes09.pdf)
|
||||
|
||||
@@ -35,6 +35,9 @@ export default {
|
||||
Math416: {
|
||||
type: 'page',
|
||||
},
|
||||
Math401: {
|
||||
type: 'page',
|
||||
},
|
||||
CSE332S: {
|
||||
type: 'page',
|
||||
},
|
||||
|
||||
Reference in New Issue
Block a user