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废液缸
Academic Waste
Drafts WIP and shelved writings.
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A Table of Lie Groups and Lie Algebras
math
algebra
Type
2025/10/16
zh
非结合代数括号制造艺术
TAOCP: The Art of Creating Parentheses
math
algebra
combinatorics
二元运算接受两个输入，输出一个结果．如果我们有多个输入，就需要使用括号来明确运算顺序．例如，给定四个元素
\(a, b, c, d\)
，可以有以下五种不同的括号方式：
\[ ((ab)c)d \quad (a(bc))d \quad (ab)(cd) \quad a((bc)d) \quad a(b(cd)) \]
2025/10/07
zh
short
分析力学乱炖
math
analysis
physics
Newton 力学的基本假设是质点的运动服从
Newton 第二定律
\[ m \ddot{\vb* q} = -\pdv{U}{\vb* q} \]
这里
\(U(\vb* q)\)
是质点在位置
\(\vb* q\)
处的势能．物理学实践似乎表明这种形式下的力学方程长相往往比较丑陋，不易于守恒量的发现．考虑到光学中光路传播的最短路径原理具有优…
2025/09/21
zh
零点定理乱炖
Suspicious Nullstellensatz Soup
math
algebra
On Hilbert’s Nullstellensatz and the ideal–variety correspondence.
2025/08/12
zh
Artin–Rees–Krull 乱炖
Artin–Rees Lemma and Krull Intersection Theorem
math
algebra
本文所指环均为交换幺环．Krull 交定理刻画了 Noether 环中任意理想幂次组成的降链的极限形态：
2025/08/04
zh
从一般线性群
\(\operatorname{GL}_n(\mathbb F_q)\)
的 Sylow
\(p\)
-子群谈起
Rambling from the Sylow
\(p\)
-subgroups of the General Linear Group
\(\operatorname{GL}_n(\mathbb F_q)\)
math
algebra
设
\(p\)
为素数，
\(q = p^\alpha\)
．我们引入一点先进的
\(q\)
-analog 记号：
\[ \begin{aligned}[] [n]_q &:= \frac{q^n - 1}{q - 1} = q^{n-1} + q^{n-2} + \cdots + q + 1 \\ [n]_q! &:=…
2025/07/12
zh
神奇流形在哪里
Fantastic Manifolds and Where to Find Them
math
geometry
\[ \gamma(t) := (e^{2\pi i t}, e^{2 \pi i \alpha t}) \]
is an
\(\mathbb R \to \mathbb T^2 \subset \mathbb C^2\)
immersion (Note that it’s injective!). The corresponding…
2025/06/02
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