Math

Construction of real numbers

This is a note for me who forget again in the future. There are several way to construct real numbers. Dedekind cuts https://en.wikipedia.org/wiki/Dedekind_cut A Dedekind cut is a partition of the rationals $\mathbb{Q}$ into two subsets $A$ and $B$ such that: $A$ is non emply. $A\neq\mathbb{O}$ If $x,y\in\mathbb{Q}$, $x<y$, and $y\in A$, then $x\in A$. ($A$ is “closed downwards”.) If $x\in A$, then therre exists a $y\in A$ such that $x<y$ ($A$ does not contain a greatest element.

Topological space (draft)

This is my note of the video https://youtu.be/1wyOoLUjUeI. It’s not a full guide of topology. I wanted to know what is continuity the meaning of “topology” Topological space Let $M$ be a set. Then, a choice $\mathcal{O}\subseteq\mathcal{P}(M)$ is called a topology on $M$ if $\phi\in\mathcal{O}$ and $M\in\mathcal{O}$, $O_i, O_j \in\mathcal{O} \Rightarrow\bigcap\{O_i, O_j\}\in\mathcal{O}$, and $\displaystyle O_i\in\mathcal{O} \Rightarrow \bigcup_i O_i\in\mathcal{O}$. The pair $(M, \mathcal{O})$ is then called a topological space.

Classification and construction of sets

This is my note of the video https://youtu.be/6EIWRg-6ftQ. It’s not a full guide how to construct “numbers” from the empty set. Classification of sets In Mathematics, we usuallly investigate the “structure” of “spaces” in terms of “maps”. Space and map The terminology space is usually meant to be some set equipped with some “structure”. A map $\phi: A\rightarrow B$ is a relation such that for every $a\in A$ there exists exactly one $b\in B$ such that $\phi(a,b)$.

Axioms of set theory (ZFC)

This is my note of the video https://youtu.be/AAJB9l-HAZs. It’s not a full guide of axiomatic set theory. Relation Relation is a predicate of two variables. $\in$-relation Set theory is built on the postulate that there is a fundamental relation called $\in$ (epsilon relation). There will be no definition of what $\in$ is, or of what a set is. Instead, we’ll define 9 axioms, that spesk of $\in$ and sets. We call $x$ as an element of $y$ if $x\in y$ is a proposition.

Propositional and predicates logic

This is my note of the video https://youtu.be/V49i_LM8B0E. It’s not a full guide of propositional and predicates logic. Mathmatics as a language of Physics Physics “casts” concepts about the real world into rigorous mathematical forms. Mathematics is just a language for the Physics. Physics interprets the meaning of mathematical statements. But to describe Mathematics, we need logic (especially, standard logic). Psycologically, an equivalent but a different form will help us to understand an interpretation.

Lebesgue measure (updating)

My personal motivation(s): want to know when it is allowed to swap limit and sum. want to know when it is allowed to swap limit and integral. want to know what the Lebesgue integral is. want to make a base for probability theory. $\sigma$-algebra Suppose a set $X$ and its power set $\mathcal{P}(X)$. The “measure” will be defined for some elements of $\mathcal{P}(X)$ later. $\Sigma\subseteq\mathcal{P}(X)$ is $\sigma$-algebra if

Category Theory (updating)

Definition of a category Objects and arrows exist. There should be identity arrows for all objects. Arrows should be composited. Associativity holds. The arrows are also called morphisms. Terminology: domain and codomain $\mathrm{dom}(f)\xrightarrow{f} \mathrm{cod}(f)$ Directed graphs and categories If $f:A\rightarrow B$ and $g:B\rightarrow C$ exist but $h := g\cdot f$ doesn’t, it violates the rule of composition. Hence the graphs are not categories in general. Initial object An initial object of a category $C$ is an object $I$ in $C$ such that for every object $X$ in $C$, there exists precisely one morphism $I \rightarrow X$.

Galois Theory (almost done)

Before the lecture Motivation My motivation was to understand why there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients (Abel-Ruffini theorem). I knew the fact but never had a chance to learn it. But I was so curious about it these days, and decided to check. About learning material YouTube video “Visual Group Theory - Lecture 6” was easy to understand.

(Draft) Lattice-based crypto - introduction

During the Corona situation, I decided to learn something new, and it was lattice-based cryptography. Intro to LWE from knapsack problem From this slide 1D (modular) knapsack problem Given $a_1, a_2,\cdots, a_n$ and $t$, $q$, where all variables are integers. Find $\{x_i\}_{i=1}^{n}\in\{0, 1\}^{n}$ s.t. , $$ t=\sum_{i=1}^{n}x_1a_i \bmod q. $$ Vector modular knapsack problem Given $\pmb{a}_1, \pmb{a}_2,\cdots, \pmb{a}_n$ and $\pmb{t}$, $q$, where $\pmb{a},\pmb{t}\in\mathbb{Z}^{m\times1}$ and $q \in \mathbb{Z}$. Find $\pmb{x}=\{x_i\}_{i=1}^{n}\in\{0,1\}^{m\times1}$ s.t. , $$ \pmb{t}=\sum_{i=1}^{n}x_i\pmb{a}_i \bmod q.