📖 Chromatic Number
The chromatic number $\\chi(G)$ is the minimum number of colors needed to properly color the vertices of $G$ (adjacent vertices get different colors).
From: Introduction to Graph Theory
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📖 Definition VII.12 (Relatively Prime)
Numbers prime to one another are those which are measured by a unit alone as a common measure.
From: Euclid's Elements
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📐 Chromatic Polynomial Properties
For any graph $G$ with $n$ vertices and $m$ edges: $P(G, k) = k^n - mk^{n-1} + \\ldots$ is a polynomial. For edge $e$: $P(G, k) = P(G-e, k) - P(G/e, k)$.
From: Introduction to Graph Theory
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📖 Subspace
A subset $W$ of a vector space $V$ is a subspace if $W$ is itself a vector space under the operations inherited from $V$.
From: Advanced Linear Algebra
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💡 Proposition I.48 (Converse of Pythagorean Theorem)
If in a triangle the square on one of the sides be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right.
From: Euclid's Elements
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📖 Definition VII.15 (Multiply)
A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
From: Euclid's Elements
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📖 Definition VII.8 (Even-Times Even)
An even-times even number is that which is measured by an even number according to an even number.
From: Euclid's Elements
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📐 Fermat
If $p$ is prime and $\\gcd(a, p) = 1$, then $a^{p-1} \\equiv 1 \\pmod{p}$.
Proof: Consider the residues $a, 2a, 3a, \\ldots, (p-1)a \\pmod{p}$. These are a permutation of $1, 2, \\ldots, p-1$ since $\\gcd(a,p)=1$ prevents any two from being congruent.
Multiplying all together: $a \\cdot 2a \\cdot 3a \\cdots (p-1)a \\equiv 1 \\cdot 2 \\cdot 3 \\cdots (p-1) \\pmod{p}$
This giv...
From: Disquisitiones Arithmeticae
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📖 Planar Graph
A graph is planar if it can be drawn in the plane with no edge crossings. Such a drawing is called a plane embedding.
From: Introduction to Graph Theory
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📖 Definition V.2 (Multiple)
The greater is a multiple of the less when it is measured by the less.
From: Euclid's Elements
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📖 Gini Index
$G = \\sum_{k=1}^{K}\\hat{p}_{mk}(1 - \\hat{p}_{mk})$ measures node purity. $G$ is small when node is pure (contains mostly observations from one class).
From: Intro to Statistical Learning
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📖 Polynomial Regression
$y_i = \\beta_0 + \\beta_1 x_i + \\beta_2 x_i^2 + \\cdots + \\beta_d x_i^d + \\epsilon_i$ extends linear regression by including polynomial terms of the predictors.
From: Intro to Statistical Learning
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💡 Proposition VI.18
On a given straight line to describe a rectilinear figure similar and similarly situated to a given rectilinear figure.
From: Euclid's Elements
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📐 Dual Space Dimension
If $V$ is finite-dimensional, then $\\dim(V^*) = \\dim(V)$.
Proof: Let $\\{v_1, \\ldots, v_n\\}$ be a basis for $V$. Define $f_i: V \\to F$ by:
$$f_i(v_j) = \\delta_{ij} = \\begin{cases} 1 & i = j \\\\ 0 & i \\neq j \\end{cases}$$
**Claim:** $\\{f_1, \\ldots, f_n\\}$ is a basis for $V^*$.
**Spanning:** For any $f \\in V^*$, let $c_i = f(v_i)$. Then for any $v ...
From: Advanced Linear Algebra
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💡 Proposition V.11 (Transitivity)
Ratios which are the same with the same ratio are also the same with one another.
From: Euclid's Elements
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💡 Proposition IV.5 (Circumcircle of Triangle)
About a given triangle to circumscribe a circle.
From: Euclid's Elements
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📖 Graph Isomorphism
Two graphs G and H are isomorphic (G \= H) if there exists a bijection f: V(G) \-> V(H) such that uv in E(G) if and only if f(u)f(v) in E(H).
From: Introduction to Graph Theory
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📖 Definition III.7
An angle of a segment is that contained by a straight line and a circumference of a circle.
From: Euclid's Elements
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📖 Definition V.12 (Alternando)
Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.
From: Euclid's Elements
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💡 Proposition V.18 (Componendo)
If magnitudes be proportional separando, they will also be proportional componendo.
From: Euclid's Elements
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🏆 Magic Internet Math - Daily Top 10
🥇 Derek Ross - 10 XP | Level 1
🥈 Anonymous - 0 XP | Level 1
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