📖 The Lasso
The lasso minimizes $\\sum_{i=1}^{n}(y_i - \\beta_0 - \\sum_{j=1}^{p}\\beta_j x_{ij})^2 + \\lambda\\sum_{j=1}^{p}|\\beta_j|$. Unlike ridge, lasso can force coefficients exactly to zero, performing variable selection.
From: Intro to Statistical Learning
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📖 Kaplan-Meier Estimator
$\\hat{S}(t) = \\prod_{j: t_j \\leq t}\\left(1 - \\frac{d_j}{r_j}\\right)$ where $d_j$ is number of events at time $t_j$ and $r_j$ is number at risk just before $t_j$.
From: Intro to Statistical Learning
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💡 Proposition I.42
To construct, in a given rectilineal angle, a parallelogram equal to a given triangle.
From: Euclid's Elements
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📖 Sample Definition
A function $f: A \\to B$ is a mapping from set $A$ to set $B$.
From: Calculus: A Liberal Art
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📐 Cantor\
For any set $A$, $|A| < |\\mathcal{P}(A)|$. The power set has strictly greater cardinality.
From: Real Analysis
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💡 Proposition VII.37
If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.
From: Euclid's Elements
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💡 Proposition II.4 (Square of a Sum)
If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
From: Euclid's Elements
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📖 Sample Definition
A function $f: A \\to B$ is a mapping from set $A$ to set $B$.
From: orange-is-new-green
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💡 Proposition VII.8
If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole.
From: Euclid's Elements
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📐 Lagrange
If $H \\leq G$ and $G$ is finite, then $|H|$ divides $|G|$. The index is $[G:H] = |G|/|H|$.
Proof: Cosets partition $G$ into $[G:H]$ sets, each of size $|H|$. Thus $|G| = [G:H] \\cdot |H|$.
From: intro-discrete
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📐 Diagonalizability via Minimal Polynomial
$T$ is diagonalizable if and only if its minimal polynomial is a product of distinct linear factors.
Proof: **(⇒)** If $T$ is diagonalizable with eigenvalues $\\lambda_1, \\ldots, \\lambda_k$, then $(T - \\lambda_1 I) \\cdots (T - \\lambda_k I) = 0$ on each eigenbasis vector, hence on all of $V$. The minimal polynomial divides $(x - \\lambda_1) \\cdots (x - \\lambda_k)$.
**(⇐)** If $m(x) = (x - \\lamb...
From: Advanced Linear Algebra
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📖 Partition
A partition $P$ of $[a,b]$ is a finite set $\\{a = x_0 < x_1 < \\cdots < x_n = b\\}$.
From: Real Analysis
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💡 The Socialist Roots of Nazism
Nazism emerged not as a reaction against socialism but as its evolution. The Nazis were recruited from socialist ranks, adopted socialist methods, and fulfilled socialist predictions about capitalism\
From: The Road to Serfdom
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📐 GCD and LCM from Prime Factorisations
If $a = \\prod p_i^{\\alpha_i}$ and $b = \\prod p_i^{\\beta_i}$, then $\\gcd(a,b) = \\prod p_i^{\\min(\\alpha_i, \\beta_i)}$ and $\\text{lcm}(a,b) = \\prod p_i^{\\max(\\alpha_i, \\beta_i)}$.
From: intro-discrete
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📐 RSA Correctness
For properly chosen $e$ and $d$, decryption correctly recovers the original message: $m^{ed} \\equiv m \\pmod{n}$.
Proof: **Setup:** Let $n = pq$ for distinct primes $p, q$. Let $e$ be coprime to $\\varphi(n) = (p-1)(q-1)$, and let $d = e^{-1} \\pmod{\\varphi(n)}$.
This means $ed = 1 + k \\cdot \\varphi(n)$ for some integer $k$.
**Proof:**
We need to show $m^{ed} \\equiv m \\pmod{n}$ for any message $m$ with $0 \...
From: Cryptography Math
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📖 Rank and Nullity
The rank of $T$ is $\\text{rank}(T) = \\dim(\\text{im}(T))$. The nullity of $T$ is $\\text{nullity}(T) = \\dim(\\ker(T))$.
From: Advanced Linear Algebra
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📐 Handshaking Lemma
For any graph $G$, $\\sum_{v \\in V(G)} d(v) = 2|E(G)|$.
Proof: Each edge contributes exactly 2 to the sum of degrees (one for each endpoint).
From: Introduction to Graph Theory
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📖 Fermat Prime
A Fermat prime is a prime of the form $F_k = 2^{2^k} + 1$. The known Fermat primes are $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$, $F_4 = 65537$.
From: Disquisitiones Arithmeticae
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📖 Eigenvalue and Eigenvector
A scalar $\\lambda$ is an eigenvalue of $T: V \\to V$ if there exists a nonzero vector $v$ such that $T(v) = \\lambda v$. Such $v$ is called an eigenvector.
From: Advanced Linear Algebra
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💡 Proposition II.10
If a straight line be bisected, and a straight line be added to it in a straight line, the square on the whole with the added straight line and the square on the added straight line both together are double of the square on the half and of the square described on the straight line made up of the half and the added straight line as on one straight line.
From: Euclid's Elements
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📖 Minimum Distance
The minimum distance $d$ of a code is the smallest Hamming distance between distinct codewords.
From: intro-discrete
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📐 Jordan Canonical Form Theorem
Every linear operator on a finite-dimensional complex vector space has a unique Jordan canonical form (up to ordering of blocks).
Proof: **Existence:** By the primary decomposition theorem, $V = V_1 \\oplus \\cdots \\oplus V_k$ where $V_i$ is the generalized eigenspace for $\\lambda_i$.
On each $V_i$, $(T - \\lambda_i I)$ is nilpotent. For nilpotent operators, there exists a Jordan basis giving blocks of the form $J_m(0)$.
Combi...
From: Advanced Linear Algebra
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💡 Proposition I.10
To bisect a given finite straight line.
From: Euclid's Elements
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