π Linear Congruence Theorem
$ax \\equiv b \\pmod{n}$ has solutions iff $\\gcd(a,n) \\mid b$. If solvable, there are exactly $\\gcd(a,n)$ solutions mod $n$.
From: intro-discrete
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π Leave-One-Out Cross-Validation (LOOCV)
LOOCV uses a single observation as the validation set and the remaining $n-1$ observations as training set, repeated for each observation. $\\text{CV}_{(n)} = \\frac{1}{n}\\sum_{i=1}^{n}\\text{MSE}_i$
From: Intro to Statistical Learning
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π Weierstrass M-Test
If $|f_n(x)| \\leq M_n$ for all $x$ and $\\sum M_n < \\infty$, then $\\sum f_n$ converges uniformly and absolutely.
From: Real Analysis
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π‘ Proposition V.12
If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
From: Euclid's Elements
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π‘ Proposition III.28
In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.
From: Euclid's Elements
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π Unique Representation
If $\\{v_1, \\ldots, v_n\\}$ is a basis for $V$, then every vector $v \\in V$ can be written uniquely as $v = c_1 v_1 + \\cdots + c_n v_n$.
Proof: **Existence:** Since the basis spans $V$, every $v$ is a linear combination.
**Uniqueness:** Suppose $v = \\sum c_i v_i = \\sum d_i v_i$. Then:
$$\\sum (c_i - d_i) v_i = \\mathbf{0}$$
By linear independence, $c_i - d_i = 0$ for all $i$, so $c_i = d_i$.
From: Advanced Linear Algebra
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π Differentiability Implies Continuity
If $f$ is differentiable at $c$, then $f$ is continuous at $c$.
From: Real Analysis
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π Jordan Block
A Jordan block $J_k(\\lambda)$ is a $k \\times k$ matrix with $\\lambda$ on the diagonal, $1$s on the superdiagonal, and $0$s elsewhere.
From: Advanced Linear Algebra
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π Two Squares Theorem
A positive integer $n$ is a sum of two squares if and only if every prime factor $p \\equiv 3 \\pmod{4}$ appears to an even power in the factorization of $n$.
From: Disquisitiones Arithmeticae
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π Random Forest
Like bagging, but at each split only a random subset of $m \\approx \\sqrt{p}$ predictors is considered. This decorrelates trees, reducing variance further.
From: Intro to Statistical Learning
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π‘ The German Path
Germany did not abandon liberalism because it failed, but because Germany never fully embraced it. What failed was socialismβbut socialists blamed liberalism for problems socialism created.
From: The Road to Serfdom
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π Natural Isomorphism to Double Dual
The map $\\phi: V \\to V^{**}$ defined by $\\phi(v)(f) = f(v)$ is a natural isomorphism for finite-dimensional $V$.
Proof: **Linearity:** For $u, v \\in V$ and $f \\in V^*$:
$$\\phi(u + v)(f) = f(u + v) = f(u) + f(v) = \\phi(u)(f) + \\phi(v)(f)$$
**Injectivity:** If $\\phi(v) = 0$, then $f(v) = 0$ for all $f \\in V^*$. Choosing $f$ from the dual basis shows $v = \\mathbf{0}$.
**Dimension:** $\\dim(V) = \\dim(V^*) =...
From: Advanced Linear Algebra
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π Intersection of Subspaces
The intersection of any collection of subspaces of $V$ is a subspace of $V$.
Proof: Let $\\{W_i\\}_{i \\in I}$ be subspaces of $V$ and let $W = \\bigcap_{i \\in I} W_i$.
- **Non-empty:** Each $W_i$ contains $\\mathbf{0}$, so $\\mathbf{0} \\in W$.
- **Closure under addition:** If $u, v \\in W$, then $u, v \\in W_i$ for all $i$. Since each $W_i$ is a subspace, $u + v \\in W_i$ fo...
From: Advanced Linear Algebra
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π Primitive Root of Unity
A primitive $n$-th root of unity is a root $\\zeta$ such that $\\zeta^n = 1$ but $\\zeta^k \\ne 1$ for $0 < k < n$. Equivalently, $\\zeta = e^{2\\pi i k/n}$ where $\\gcd(k, n) = 1$.
From: Disquisitiones Arithmeticae
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π ErdΕs-Gallai Theorem
A non-increasing sequence $(d_1, \\ldots, d_n)$ of non-negative integers is graphic if and only if $\\sum d_i$ is even and for each $k \\in [n]$: $\\sum_{i=1}^k d_i \\leq k(k-1) + \\sum_{i=k+1}^n \\min(d_i, k)$.
From: Introduction to Graph Theory
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π Euler
If $\\gcd(a, n) = 1$, then $a^{\\phi(n)} \\equiv 1 \\pmod{n}$.
Proof: Generalizes Fermat's proof. Multiplication by $[a]$ permutes $G_n$. Product of $[a]G_n$ is $[a]^{\\phi(n)}$ times product of $G_n$. Cancel to get $[a]^{\\phi(n)} = [1]$.
From: intro-discrete
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π Common Notion 1
Things which are equal to the same thing are also equal to one another.
From: Euclid's Elements
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π‘ Proposition VII.17
If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.
From: Euclid's Elements
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π Power Set
The power set $\\mathcal{P}(A)$ is the set of all subsets of $A$. If $|A| = n$, then $|\\mathcal{P}(A)| = 2^n$.
From: intro-discrete
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π Step Function
Step functions break the range of $X$ into bins and fit a constant in each bin: $y_i = \\beta_0 + \\beta_1 C_1(x_i) + \\cdots + \\beta_K C_K(x_i) + \\epsilon_i$ where $C_k(x)$ are indicator functions.
From: Intro to Statistical Learning
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