title: "$\LaTeX$ 語法筆記" path: "LaTeX 語法筆記" :::info 此頁歡迎共筆。 ::: <img src="https://rintaroutw.github.io/fsg/test/Flag_of_the_Republic_of_China.svg" width="300" height="200"></img> HackMD 是透過 MathJax 支援 $\LaTeX$ ，雖未支援所有語法，對多數人應已足夠，一起把 Web 上醜死的公式換成美美的吧～

HackMD 筆記 Book Graphviz digraph { graph [bgcolor=transparent]; node[fontcolor="#888888";color="#888888"]; edge [color="#888800"]; a->b; a->c;

學海無涯，思境無維；數乃六藝，理之始也。 或有一得足矣 [name=愚千慮] $$ \require{newcommand} \newcommand{\bmatrix}[1] {\begin{bmatrix}#1\end{bmatrix}} \newcommand{\vmatrix}[1] {\begin{vmatrix}#1\end{vmatrix}} $$

--- title: Dark Theme --- <style> body { background-color: #131a21 !important; color: #888 !important; } .topbar { background-color: #131a21 !important; color: #bbb !important; } .btn:hover { background-color: #333 !important; color: #fff !important; } nav, .toolbar, .summary { background-color: #131a21 !important; color: #999 !important; } h1, h2, h3, h4, h5, h6, p { font-family: serif; font-weight: 500; font-size: 100%; } h1.part:not(:first-of-type), h2.part { ma

{%hackmd @RintarouTW/About %} $$ \cases{ b^2=h^2+d^2\ a^2=h^2+(c-d)^2 }\ a^2-b^2=c^2-2cd\ d=\frac{b^2+c^2-a^2}{2c} $$

{%hackmd @RintarouTW/About %} https://open.umn.edu/opentextbooks/textbooks/abstract-algebra-theory-and-applications Preliminaries Equivalence Relation Rational Number Equivalence Function Equivalence Equivalence Classes (form a partition) of a Set

{%hackmd @RintarouTW/DarkTheme %} 愚千慮の筆記本 數學 (Math) 複數 𝒾 的應用 複數與旋轉 (Complex Number ＆ Rotation) 二項式定理 (Binomial Theorem) 研究 代數與命名 座標、向量與矩陣

{%hackmd @RintarouTW/About %} https://open.umn.edu/opentextbooks/textbooks/applied-discrete-structures Set Theory Combinatorics Logic Number Theory Relation

{%hackmd @RintarouTW/About %} :::info A field is a commutative ring with unity such that each nonzero element has a multiplicative inverse. ::: The most common infinite fields $$ \cases{ [\mathbb{Q};+,\cdot]\

{%hackmd @RintarouTW/About %} digraph { graph [bgcolor=transparent]; node[fontcolor="#888888";color="#888888"]; edge [fontcolor="#AAAAAA";fontsize="9";color="#888800"]; "Magama"->"Quasigroup" [xlabel="divisibility"]; "Magama"->"Unital magma" [xlabel="identity"]; "Magama"->"Semigroup"[xlabel="associativity"]; "Quasigroup"->"Loop"[xlabel="identity"];

{%hackmd @RintarouTW/About %} The structures similar to the set of integers are called rings, and those similar to the set of real numbers are called fields. Ring https://math.stackexchange.com/questions/61497/why-are-rings-called-rings In German, "Ring" can also mean a (close) group of people with shared interests, an association. Maybe it is because of that: you've got some closely interacting elements but for all their interaction, they never leave the group. – Raphael Sep 2 '11 at 23:05 :::info

{%hackmd @RintarouTW/About %} 1.1 Set Notation and Relations Set : A collections of elements. Set A, if $x\in A$, $x$ is an element of A. Or $x\notin A$, $x$ is not an element of A. $$ \cases{ A={1,2,3}\ \mathbb{N}:the\ natrual\ numbers, {1, 2, 3, \ldots}\ \mathbb{Z}:the\ integers, {\ldots,-3, -2, -1, 0, 1, 2, 3, \ldots}\ as\ a\ set\

{%hackmd @RintarouTW/About %} $$ [V;*_1, *_2,\ldots, *_n]\cases{ V : \text{a set called domain of elements}\ *_n : operations } $$ Operations

{%hackmd @RintarouTW/About %} https://medium.com/@AlainGalvan/raw-webgpu-58522e638b17 Spec: https://gpuweb.github.io/gpuweb/ WGSL: https://gpuweb.github.io/gpuweb/wgsl.html (Rust based) SPIR: https://www.khronos.org/spir/ (Satndard Portable Intermediate Representation) Samples: https://austineng.github.io/webgpu-samples/ Chrome: Get Started with GPU Compute on the Web

{%hackmd @RintarouTW/About %} Monoid :::info A Monoid is a set $M$ with a binary operation $\cdot$ with the properties: $\cdot$ is associative : $\forall a,b,c \in M, a(bc)=(ab)c$ $\cdot$ has an identity : $\forall a\in M,\exists e\in M, ae=ea=a$ :::

{%hackmd @RintarouTW/About %} :::info Cycle : A cycle is a circuit whose edge list contains no duplicates. It is customary to use $C_n$ to denote a cycle with $n$ edges. A Tree contains no cycle or self-loops. A forest is an undirected graph whose components are all trees. :::

{%hackmd @RintarouTW/About %} Vector Space :::info $$ vector\ addition\ [V;+] \cases{ \forall x,y\in V, x+y=y+x\ (communitive)\ \forall x,y,z\in V, x+(y+z)=(x+y)+z\ (associative)\

{%hackmd @RintarouTW/About %} 3.1 Propositions and Logical Operators :::info Proposition(主張): A proposition is a sentence to which one and only one of the terms true or false can be meaningfully applied. ::: Example: “Four is even,”, “4 ∈ {1, 3, 5}” and “43 > 21” are propositions

{%hackmd @RintarouTW/About %} Many Faces of Recusion Binomial Coefficients with Recursive Definition :::info $$ \cases{n,k\in \mathbb{P}, n\geq k\geq 0\ \binom{n}{0}=1\ \binom{n}{n}=1\ }\

{%hackmd @RintarouTW/About %} :::info Function/mapping/map/transformation $$ f:\underbrace{A}{domain}\rightarrow \underbrace{B}{codomain} $$ Function. A function from a set A into a set B is a relation from A into B such that each element of A is related to exactly one element of the set B. The set A is called the domain of the function and the set B is called the codomain.