📐 Strong Perfect Graph Theorem
A graph is perfect if and only if it contains no odd hole and no odd antihole as an induced subgraph.
From: Introduction to Graph Theory
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📐 Ratio Test
If $L = \\lim |a_{n+1}/a_n|$, then: $L < 1 \\Rightarrow$ converges absolutely; $L > 1 \\Rightarrow$ diverges; $L = 1 \\Rightarrow$ inconclusive.
From: Real Analysis
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📐 Sample Theorem
If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$
Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset.
Therefore, every element of $A$ is in $B$.
Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition.
Therefore, every element of $B$ is in $A$.
Since $A \\subseteq B$ and $B \\subseteq A...
From: Calculus: A Liberal Art
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📖 Holm Step-Down Procedure
Sort p-values: $p_{(1)} \\leq \\cdots \\leq p_{(m)}$. Reject $H_{0(j)}$ if $p_{(j)} < \\alpha/(m-j+1)$ for all $j \\leq L$, where $L$ is first index where condition fails.
From: Intro to Statistical Learning
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📖 Definition VII.11 (Prime Number)
A prime number is that which is measured by a unit alone.
From: Euclid's Elements
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📖 Spectrum of a Graph
The spectrum of $G$ is the multiset of eigenvalues of its adjacency matrix $A(G)$, usually listed as $\\lambda_1 \\geq \\lambda_2 \\geq \\cdots \\geq \\lambda_n$.
From: Introduction to Graph Theory
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📖 Naive Bayes Classifier
Naive Bayes assumes features are independent within each class: $f_k(x) = \\prod_{j=1}^{p} f_{kj}(x_j)$, greatly simplifying estimation.
From: Intro to Statistical Learning
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📖 PCA Loadings
The loadings $\\phi_{1}, \\ldots, \\phi_p$ define the first PC direction: $Z_1 = \\phi_{11}X_1 + \\cdots + \\phi_{p1}X_p$ subject to $\\sum_{j=1}^p \\phi_{j1}^2 = 1$.
From: Intro to Statistical Learning
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💡 Proposition I.6
If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.
From: Euclid's Elements
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🎮 Interactive: Proof Structure Visualizer
See the logical structure of mathematical proofs. Direct proof, contradiction, and induction explained visually.
From: Why Math?
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📐 Fundamental Theorem of Arithmetic
Every integer greater than 1 can be expressed as a unique product of prime numbers (up to reordering).
Proof: **Existence:** We prove by strong induction. For $n = 2$, we're done (2 is prime). For $n > 2$, either $n$ is prime (done) or $n = ab$ for $1 < a, b < n$. By induction, both $a$ and $b$ have prime factorizations, so their product $n$ does too.
**Uniqueness:** Suppose $n = p_1 p_2 \\cdots p_k = q...
From: Cryptography Math
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📖 Sample Definition
A function $f: A \\to B$ is a mapping from set $A$ to set $B$.
From: Human Action
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🎮 Interactive: Number Line Explorer
Visualize integers, rationals, and operations on the number line. The foundation of all arithmetic!
From: Basic Algebra
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📖 Absolute Convergence
A series $\\sum a_n$ converges absolutely if $\\sum |a_n|$ converges.
From: Real Analysis
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📖 Definition VI.2 (Reciprocally Related Figures)
Two figures are reciprocally related when the sides about corresponding angles are reciprocally proportional.
From: Euclid's Elements
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📖 Definition V.1 (Part)
A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.
From: Euclid's Elements
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📖 Definition III.5
And that straight line is said to be at a greater distance on which the greater perpendicular falls.
From: Euclid's Elements
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💡 Proposition III.4
If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another.
From: Euclid's Elements
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📖 Generalized Eigenvector
A generalized eigenvector of $T$ for eigenvalue $\\lambda$ is a nonzero $v$ such that $(T - \\lambda I)^k v = \\mathbf{0}$ for some $k \\geq 1$.
From: Advanced Linear Algebra
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📐 Stone-Weierstrass Theorem
A subalgebra of $C(X)$ that separates points and vanishes nowhere is dense in $C(X)$.
From: Real Analysis
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📐 Ramsey
For all positive integers $s, t$, there exists a minimum $R(s,t)$ such that every red-blue edge-coloring of $K_n$ with $n \\geq R(s,t)$ contains a red $K_s$ or a blue $K_t$.
Proof: Induction: $R(s,1) = R(1,t) = 1$. For $s,t > 1$: take $n = R(s-1,t) + R(s,t-1)$. Pick vertex $v$; partition neighbors by edge color. Red neighbors form set of size $\\geq R(s-1,t)$ or blue neighbors $\\geq R(s,t-1)$. By induction, find red $K_{s-1}$ or blue $K_t$ in red neighborhood (add $v$ for ...
From: Introduction to Graph Theory
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