From fb1ffcd040f856e39b141c27e12cb2d4fa3a0174 Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Mon, 27 Oct 2025 11:56:32 -0500 Subject: [PATCH] updates --- app/layout.tsx | 2 +- content/CSE442T/CSE442T_L16.md | 5 +- content/CSE442T/CSE442T_L17.md | 10 +- content/CSE442T/CSE442T_L20.md | 10 +- content/CSE5313/Exam_reviews/CSE5313_E1.md | 3 + .../Math401/Extending_thesis/Math401_R2.md | 2 - content/Math401/Math401_N3.md | 13 +- content/Math4111/Math4111_L1.md | 5 +- content/Math4111/Math4111_L2.md | 9 +- content/Math4111/Math4111_L3.md | 5 +- content/Math4111/Math4111_L4.md | 10 +- content/Math4111/Math4111_L5.md | 10 +- content/Math4111/Math4111_L6.md | 5 +- content/Math4111/Math4111_L7.md | 5 +- content/Math4201/Math4201_L25.md | 166 ++++++++++++++++++ content/Math4201/_meta.js | 1 + content/layout.tsx | 92 ---------- 17 files changed, 219 insertions(+), 134 deletions(-) create mode 100644 content/Math4201/Math4201_L25.md delete mode 100644 content/layout.tsx diff --git a/app/layout.tsx b/app/layout.tsx index c5f5285..6f56dd0 100644 --- a/app/layout.tsx +++ b/app/layout.tsx @@ -82,7 +82,7 @@ export default async function RootLayout({ children }) { sidebar={{ defaultMenuCollapseLevel: 1 }} pageMap={pageMap} // TODO: fix algolia search - // search={} + search={} > {children} {/* SpeedInsights in vercel */} diff --git a/content/CSE442T/CSE442T_L16.md b/content/CSE442T/CSE442T_L16.md index ad4af94..ef6bf8a 100644 --- a/content/CSE442T/CSE442T_L16.md +++ b/content/CSE442T/CSE442T_L16.md @@ -13,7 +13,8 @@ Ouput $(r,m\oplus f_i(r))$ $Dec_i(r,c):$ Output $c\oplus f_i(r)$ -Proof of security: +
+Proof of security Suppose $D$ distinguishes, for infinitly many $n$. @@ -35,7 +36,7 @@ $(r_1,F(r_1)),\ldots, (r_q,F(r_q))$ So $D$ distinguishing output of $r_1,\ldots, r_q$ of PRF from the RF, this contradicts with definition of PRF. -QED +
Noe we have diff --git a/content/CSE442T/CSE442T_L17.md b/content/CSE442T/CSE442T_L17.md index 225767c..488241b 100644 --- a/content/CSE442T/CSE442T_L17.md +++ b/content/CSE442T/CSE442T_L17.md @@ -32,7 +32,8 @@ Proof of the validity of the decryption: Exercise. The encryption scheme is secure under this construction (Trapdoor permutation (TDP), Hardcore bit (HCB)). -Proof: +
+Proof We proceed by contradiction. (Constructing contradiction with definition of hardcore bit.) @@ -76,7 +77,7 @@ $$ This contradicts the definition of hardcore bit. -QED +
### Public key encryption scheme (multi-bit) @@ -144,7 +145,8 @@ Output: $m$ #### Security of El-Gamal encryption scheme -Proof: +
+Proof If not secure, then there exists a distinguisher $\mathcal{D}$ that can distinguish the encryption of $m_1,m_2\in G_q$ with non-negligible probability $\mu(n)$. @@ -155,5 +157,5 @@ $$ And proceed by contradiction. This contradicts the DDH assumption. -QED +
diff --git a/content/CSE442T/CSE442T_L20.md b/content/CSE442T/CSE442T_L20.md index 47c8197..8a53e01 100644 --- a/content/CSE442T/CSE442T_L20.md +++ b/content/CSE442T/CSE442T_L20.md @@ -26,7 +26,8 @@ Under the discrete log assumption, $H$ is a CRHF. - It is easy to compute - Compressing by 1 bit -Proof: +
+Proof The hash function $h$ is a CRHF @@ -72,7 +73,7 @@ So $\mathcal{B}$ can break the discrete log assumption with non-negligible proba So $h$ is a CRHF. -QED +
To compress by more, say $h_k:{0,1}^n\to \{0,1\}^{n-k},k\geq 1$, then we can use $h: \{0,1\}^{n+1}\to \{0,1\}^n$ multiple times. @@ -106,7 +107,8 @@ One-time secure: Then ($Gen',Sign',Ver'$) is one-time secure. -Ideas of Proof: +
+Ideas of Proof If the digital signature scheme ($Gen',Sign',Ver'$) is not one-time secure, then there exists an adversary $\mathcal{A}$ which can ask oracle for one signature on $m_1$ and receive $\sigma_1=Sign'_{sk'}(m_1)=Sign_{sk}(h_i(m_1))$. @@ -119,7 +121,7 @@ Case 1: $h_i(m_1)=h_i(m_2)$, Then $\mathcal{A}$ finds a collision of $h$. Case 2: $h_i(m_1)\neq h_i(m_2)$, Then $\mathcal{A}$ produced valid signature on $h_i(m_2)$ after only seeing $Sign'_{sk'}(m_1)\neq Sign'_{sk'}(m_2)$. This contradicts the one-time secure of ($Gen,Sign,Ver$). -QED +
### Many-time Secure Digital Signature diff --git a/content/CSE5313/Exam_reviews/CSE5313_E1.md b/content/CSE5313/Exam_reviews/CSE5313_E1.md index 48abd57..f76ee00 100644 --- a/content/CSE5313/Exam_reviews/CSE5313_E1.md +++ b/content/CSE5313/Exam_reviews/CSE5313_E1.md @@ -240,6 +240,9 @@ class Polynomial(): def as_number(self) -> int: return sum([coefficient.value * self.p ** i for i, coefficient in enumerate(self.coefficients)]) ``` + +### Finite fields + ```python class FiniteField(): def __init__(self, p: int, n: int = 1, value: Polynomial = None, irreducible_polynomial: Polynomial = None): diff --git a/content/Math401/Extending_thesis/Math401_R2.md b/content/Math401/Extending_thesis/Math401_R2.md index a124128..fedc04a 100644 --- a/content/Math401/Extending_thesis/Math401_R2.md +++ b/content/Math401/Extending_thesis/Math401_R2.md @@ -202,8 +202,6 @@ $$ \end{aligned} $$ -QED - #### Proof of the Levy's concentration theorem via the Maxwell-Boltzmann distribution law diff --git a/content/Math401/Math401_N3.md b/content/Math401/Math401_N3.md index 9da26ea..c4712a7 100644 --- a/content/Math401/Math401_N3.md +++ b/content/Math401/Math401_N3.md @@ -187,7 +187,8 @@ $$ where $L(\mu)$ is the minimum mean code word length of all uniquely decipherable codes for $(A,\mu)$. -Proof: +
+Proof First, we show that @@ -278,7 +279,7 @@ $$ \end{aligned} $$ -QED +
### Entropy @@ -387,13 +388,9 @@ n−1 symbols. By the inductive hypothesis, the code on $A'$ is optimal. is optimal. -By Step 2 above, assigning the two merged symbols $a$ and $b$ codewords $w_0$ and $w_1$ (based on -1 -$w_1$ (based on $c$'s codeword $w$) results in the optimal solution for $A$. +By Step 2 above, assigning the two merged symbols $a$ and $b$ codewords $w_0$ and $w_1$ (based on 1.1.4) results in the optimal solution for $A$. Therefore, by induction, Huffman’s algorithm gives an optimal prefix code for any $n$. - -QED - + diff --git a/content/Math4111/Math4111_L1.md b/content/Math4111/Math4111_L1.md index fea3b34..5a82ddb 100644 --- a/content/Math4111/Math4111_L1.md +++ b/content/Math4111/Math4111_L1.md @@ -24,7 +24,8 @@ $\equiv\cancel{\exist} p\in \mathbb{Q}, p^2=2$ $\equiv p\in \mathbb{Q},p^2\neq 2$ -#### Proof +
+Proof Suppose for contradiction, $\exist p\in \mathbb{Q}$ such that $p^2=\mathbb{Q}$. @@ -36,7 +37,7 @@ So $m^2$ is divisible by 4, $2n^2$ is divisible by 4. So $n^2$ is even. but they are not both even. -QED +
### Theorem (No closest rational for a irrational number) diff --git a/content/Math4111/Math4111_L2.md b/content/Math4111/Math4111_L2.md index 68127d7..c659c7b 100644 --- a/content/Math4111/Math4111_L2.md +++ b/content/Math4111/Math4111_L2.md @@ -47,19 +47,18 @@ Let $S$ be an ordered set and $E\subset S$. We say $\alpha\in S$ is the LUB of $ 1. $\alpha$ is the UB of $E$. ($\forall x\in E,x\leq \alpha$) 2. if $\gamma<\alpha$, then $\gamma$ is not UB of $E$. ($\forall \gamma <\alpha, \exist x\in E$ such that $x>\gamma$ ) -#### Lemma - -Uniqueness of upper bounds. +#### Lemma (Uniqueness of upper bounds) If $\alpha$ and $\beta$ are LUBs of $E$, then $\alpha=\beta$. -Proof: +
+Proof Suppose for contradiction $\alpha$ and $\beta$ are both LUB of $E$, then $\alpha\neq\beta$ WLOG $\alpha>\beta$ and $\beta>\alpha$. -QED +
We write $\sup E$ to denote the LUB of $E$. diff --git a/content/Math4111/Math4111_L3.md b/content/Math4111/Math4111_L3.md index 1538986..4944059 100644 --- a/content/Math4111/Math4111_L3.md +++ b/content/Math4111/Math4111_L3.md @@ -26,7 +26,8 @@ Let $S=\mathbb{Z}$. Proof that $LUBP\implies GLBP$. -Proof: +
+Proof Let $S$ be an ordered set with LUBP. Let $B ### Field diff --git a/content/Math4111/Math4111_L4.md b/content/Math4111/Math4111_L4.md index 4b33fb7..b321f90 100644 --- a/content/Math4111/Math4111_L4.md +++ b/content/Math4111/Math4111_L4.md @@ -27,7 +27,8 @@ (Archimedean property) If $x,y\in \mathbb{R}$ and $x>0$, then $\exists n\in \mathbb{N}$ such that $nx>y$. -Proof +
+Proof Suppose the property is false, then $\exist x,y\in \mathbb{R}$ with $x>0$ such that $\forall v\in \mathbb{N}$, nx\leq y$ @@ -39,7 +40,7 @@ This implies $(m+1)x>\alpha$ Since $(m+1)x\in \alpha$, this contradicts the fact that $\alpha$ is an upper bound of $A$. -QED +
### $\mathbb{Q}$ is dense in $\mathbb{R}$ @@ -51,7 +52,8 @@ $$ x<\frac{m}{n} +Proof Let $x,y\in\mathbb{R}$, with $x1$, and $\ex So $-m_21+nx\geq 1+(m-1)=m$ -QED +
### $\sqrt{2}\in \mathbb{R}$, $(\sqrt[n]{x}\in\mathbb{R})$ diff --git a/content/Math4111/Math4111_L5.md b/content/Math4111/Math4111_L5.md index 0b12af7..05e62aa 100644 --- a/content/Math4111/Math4111_L5.md +++ b/content/Math4111/Math4111_L5.md @@ -30,7 +30,8 @@ $\forall x\in \mathbb{R}_{>0},\forall n\in \mathbb{N},\exist$ unique $y\in \math (Because of this Theorem we can define $x^{1/x}=y$ and $\sqrt{x}=y$) -Proof: +
+Proof We cna assume $n\geq 2$ (For $n=1,y=x$) @@ -94,7 +95,7 @@ So want $k\leq \frac{y^n-x}{ny^{n-1}}$ [For actual proof, see the text.] -QED +
### Complex numbers @@ -151,7 +152,8 @@ $$ (\sum a_j b_j)^2=(\sum a_j^2)(\sum b_j^2) $$ -Proof: +
+Proof For real numbers: @@ -169,7 +171,7 @@ let $t=C/B$ to get $0\leq A-2(C/B)C+(C/B)^2B=A-\frac{C^2}{B}$ to generalize this to $\mathbb{C}$, $A=\sum |a_j|^2,B=\sum |b_j|^2,C=\sum |a_j \bar{b_j}|$. -QED +
### Euclidean spaces diff --git a/content/Math4111/Math4111_L6.md b/content/Math4111/Math4111_L6.md index ae90125..8afe89e 100644 --- a/content/Math4111/Math4111_L6.md +++ b/content/Math4111/Math4111_L6.md @@ -126,7 +126,8 @@ $A$ is countable, $n\in \mathbb{N}$, $\implies A^n=\{(a_{1},...,a_{n}):a_1\in A, a_n\in A\}$, is countable. -Proof: +
+Proof Induct on $n$, @@ -138,7 +139,7 @@ Induction step: suppose $A^{n-1}$ is countable. Note $A^n=\{(b,a):b\in A^{n-1},a Since $b$ is fixed, so this is in 1-1 correspondence with $A$, so it's countable by Theorem 2.12. -QED +
#### Theorem 2.14 diff --git a/content/Math4111/Math4111_L7.md b/content/Math4111/Math4111_L7.md index a9be63d..067bf6d 100644 --- a/content/Math4111/Math4111_L7.md +++ b/content/Math4111/Math4111_L7.md @@ -80,7 +80,8 @@ Let $(X,d)$ be a metric space, $\forall p\in X,\forall r>0$, $B_r(p)$ is an ope *every ball is an open set* -Proof: +
+Proof Let $q\in B_r(p)$. @@ -88,7 +89,7 @@ Let $h=r-d(p,q)$. Since $q\in B_r(p),h>0$. We claim that $B_h(q)$. Then $d(q,s) ### Closed sets diff --git a/content/Math4201/Math4201_L25.md b/content/Math4201/Math4201_L25.md new file mode 100644 index 0000000..3991778 --- /dev/null +++ b/content/Math4201/Math4201_L25.md @@ -0,0 +1,166 @@ +# Math4201 Topology I (Lecture 25) + +## Continue on compact spaces + +### Compact spaces + +#### Definition of compact spaces + +A compact space $X$ is a topological space such that any open covering of $X$ has a finite subcovering. + +$$ +X=\bigcup_{\alpha\in A} U_\alpha\implies \exists \alpha_1, ..., \alpha_n\in A \text{ such that } X=\bigcup_{i=1}^n U_{\alpha_i} +$$ + +
+Example of compact spaces + +$(0,1)$ is not compact, consider the open cover $\{(0,1/n):n\in \mathbb{N}\}$ which does not have a finite subcover. + +--- + +$\mathbb{R}$ is not compact, consider the open cover $\{(-n,n):n\in \mathbb{N}\}$ which does not have a finite subcover. + +--- + +Later we will see that $[0,1]$ is compact. (more generally, any closed and bounded interval is compact) + +
+ +> [!TIP] +> +> A property (or definition) is good for topologists if it is preserved by homeomorphism, or even better, by continuous maps. + +#### Proposition of compact spaces preserved by continuous maps + +Let $X$ be a compact space and $f:X\to Y$ be a continuous map. Then $f(X)$ is compact. + +
+Proof + +Consider an open covering of $f(X)$, So, there are open sets $\{f(x)\cap U_{\alpha}\}_{x\in X}$ such that $f(X)=\bigcup_{\alpha\in I} (f(x)\cap U_{\alpha})$. + +This implies that $\{f^{-1}(f(x)\cap U_{\alpha})\}_{x\in X, \alpha\in I}$ consists of: + +1. $f^{-1}(U_{\alpha})$ is open because $f$ is continuous. +2. $f^{-1}(f(x)\cap U_{\alpha})$ covers $X$ because $\forall x\in X, f(x)\in f(X)\subseteq \bigcup_{\alpha\in I} (f(x)\cap U_{\alpha})$ so $x\in f^{-1}( U_{\alpha})$. + +Since $X$ is compact, there are finitely many $x_1, ..., x_n\in X$ such that $X=\bigcup_{i=1}^n f^{-1}(U_{\alpha_i})$. + +So, $f(X)=\bigcup_{i=1}^n f(f^{-1}(U_{\alpha_i}))=\bigcup_{i=1}^n U_{\alpha_i}$. + +This implies that $f(X)$ is compact. + +
+ +#### Corollary of compact spaces preserved by homeomorphism + +If $f:X\to Y$ is homeomorphism and $X$ is compact, then $Y$ is compact. + +#### Lemma of compact subspaces + +Let $X$ be a topological space and $Y\subseteq X$ be a subspace with subspace topology from $X$. + +Then $Y$ is compact if and only if for any open cover $\{U_\alpha\}_{\alpha\in I}$ of $Y$, there exists a finite subcover $\{U_{\alpha_1}, ..., U_{\alpha_n}\}$ of $Y$. + +#### Proposition of closed compact sets + +Every closed subspace $Y$ of a compact space $Y\subseteq X$ is compact. + +
+Proof + +Let $\{U_\alpha\}_{\alpha\in I}$ be an open cover of $Y$. Since $Y$ is closed, $X-Y$ is open. So, $(X-Y)\cup \bigcup_{\alpha\in I} U_\alpha$ is an open cover of $X$. + +Since $X$ is compact, there are finitely many $\alpha_1, ..., \alpha_n\in I$ such that $X=\bigcup_{i=1}^n U_{\alpha_i}$ and possibly $X-Y\subseteq U_{\alpha_m}$. + +So, $Y=\bigcup_{i=1}^n (U_{\alpha_i}\cap Y)=\bigcup_{i=1}^n U_{\alpha_i}$. + +This implies that $Y$ is compact. + +
+ +> [!WARNING] +> +> The converse of the proposition is almost true. + +#### Proposition of compact subspaces with Hausdorff property + +If $Y$ is compact subspace of a **Hausdorff space** $X$, then $Y$ is closed in $X$. + +
+Proof + +To show the claim, we need to show $x$ outside $y$, there is an open neighborhood of $x$ that is disjoint from $Y$. + +For any $y\in Y$, there are disjoint open neighborhoods $U_y$ and $V_y$ of $x$ and $y$ respectively (by the Hausdorff property of $X$). + +So $\bigcup_{y\in Y} V_y\supseteq Y$ and $Y$ is a compact subspace of $X$, so there are finitely many $y_1, ..., y_n\in Y$ such that $Y\subseteq \bigcup_{i=1}^n V_{y_i}$. + +Since for each $y_i\in V_{y_i}$, there exists an open neighborhood $U_{y_i}$ of $x$ such that $U_{y_i}\cap V_{y_i}=\emptyset$, we have $U_{y_i}\cap Y=\emptyset$. + +So $\bigcap_{i=1}^n U_{y_i}$ is disjoint from $\bigcup_{i=1}^n V_{y_i}\supseteq Y$, so disjoint from $Y$. + +Furthermore, $x\in \bigcap_{i=1}^n U_{y_i}$, so $\bigcap_{i=1}^n U_{y_i}$ is open in $X$ because it is an finite intersection of open sets. + +This holds for any $x\in X-Y$, so $X-Y$ is open in $X$, so $Y$ is closed in $X$. + +
+ +This the course of proving this proposition, we showed the following: + +#### Proposition + +If $X$ is Hausdorff and $Y\subseteq X$ is compact, and $x\in X-Y$, then there are disjoint open neighborhoods $U,V\subseteq X$ such that $x\in U$ and $Y\subseteq V$. + +
+Proof + +Use the proof from last proposition, take $U=\bigcap_{i=1}^n U_{y_i}$ and $V=\bigcup_{i=1}^n V_{y_i}$. + +
+ +#### Theorem of closed maps from compact and Hausdorff spaces + +If $f:X\to Y$ is continuous and $X$ is compact, $Y$ is Hausdorff, then $f$ is a closed map. + +In particular, if $f:X\to Y$ is continuous and bijection with $X$ compact and $Y$ Hausdorff, then $f$ is a homeomorphism. + +
+Example distinguishing these two properties + +Consider the map $f:[0,2\pi)\to \mathbb{S}^1$ defined by $f(x)=(\cos x, \sin x)$. This is a continuous bijection. + +$f$ is continuous bijection and $Y$ is Hausdorff, But $X$ is not compact. + +Then $f$ is not a homeomorphism because $f^{-1}$ is not continuous. + +
+ +
+Proof + +Consider $Z\subseteq X$ is closed and $X$ is compact, so $Z$ is compact. + +So $f(Z)$ is compact since $f$ is continuous. Note that $f(Z)\subseteq Y$ is Hausdorff, so $f(Z)$ is closed in $Y$. + +So $f$ is a closed map. + +
+ +#### Theorem of products of compact spaces + +If $X,Y$ are compact spaces, then $X\times Y$ is compact. (More generalized version: Tychonoff's theorem) + +
+Incomplete Proof + +Let $\{U_\alpha\}_{\alpha\in I}$ be an open cover of $X\times Y$. + +Step 1: For any $x\in X$, there are finitely many $\alpha_1, ..., \alpha_n\in I$ and open neighborhoods $x\in V\subseteq X$ such that $V\times Y\subseteq \bigcup_{i=1}^n U_{\alpha_i}\times Y$. + +For any $y\in Y$, there is $U_\alpha$ and $x\in U_y\subseteq X$ and $y\in V_y\subseteq Y$ such that $(x,y)\in U_y\times V_y\subseteq U_\alpha$. + +Continue next time... + +
\ No newline at end of file diff --git a/content/Math4201/_meta.js b/content/Math4201/_meta.js index b51774f..9526fe6 100644 --- a/content/Math4201/_meta.js +++ b/content/Math4201/_meta.js @@ -28,4 +28,5 @@ export default { Math4201_L22: "Topology I (Lecture 22)", Math4201_L23: "Topology I (Lecture 23)", Math4201_L24: "Topology I (Lecture 24)", + Math4201_L25: "Topology I (Lecture 25)", } diff --git a/content/layout.tsx b/content/layout.tsx deleted file mode 100644 index 504aad1..0000000 --- a/content/layout.tsx +++ /dev/null @@ -1,92 +0,0 @@ -/* eslint-env node */ -import { Footer, Layout} from 'nextra-theme-docs' -import { Banner, Head } from 'nextra/components' -import { getPageMap } from 'nextra/page-map' -import 'nextra-theme-docs/style.css' -import { SpeedInsights } from "@vercel/speed-insights/next" -import { Analytics } from "@vercel/analytics/react" -import { Navbar } from '../components/navbar' - -export const metadata = { - metadataBase: new URL('https://notenextra.trance-0.com'), - title: { - template: '%s - NoteNextra' - }, - description: 'A static note sharing site for minimum care', - applicationName: 'NoteNextra', - generator: 'Next.js', - appleWebApp: { - title: 'NoteNextra' - }, - other: { - 'msapplication-TileImage': '/ms-icon-144x144.png', - 'msapplication-TileColor': '#fff' - }, - twitter: { - site: 'https://notenextra.trance-0.com' - } -} - -export default async function RootLayout({ children }) { - const pageMap = await getPageMap() - const navbar = ( - - - - - - NoteNextra - - - } - projectLink="https://github.com/Trance-0/NoteNextra" - /> - ) - return ( - - - - - - MIT {new Date().getFullYear()} ©{' '} - - Trance-0 - - . - - - } - editLink="Edit this page on GitHub" - docsRepositoryBase="https://github.com/Trance-0/NoteNextra/tree/main" - sidebar={{ defaultMenuCollapseLevel: 1 }} - pageMap={pageMap} - > - {children} - {/* SpeedInsights in vercel */} - - {/* Analytics in vercel */} - - - - - ) -} \ No newline at end of file