📐 Infinitude of Primes
There are infinitely many prime numbers.
Proof: Given primes $p_1, \\ldots, p_n$, consider $N = p_1 \\cdots p_n + 1$. No $p_i$ divides $N$, so $N$ has a prime factor not in the list.
From: intro-discrete
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📖 Direct Proof
To prove $P \\Rightarrow Q$, assume $P$ and derive $Q$ through logical steps.
From: intro-discrete
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📖 Spline
A spline of degree $d$ with knots at $\\xi_1, \\ldots, \\xi_K$ is a piecewise polynomial of degree $d$ that is continuous and has continuous derivatives up to order $d-1$ at each knot.
From: Intro to Statistical Learning
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📖 Definition V.7 (Greater Ratio)
When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.
From: Euclid's Elements
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📖 Polynomial Ring
$F[x]$ is the ring of polynomials over field $F$ with standard addition and multiplication.
From: intro-discrete
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📐 Backpropagation
Backpropagation computes gradients of the loss with respect to weights by applying the chain rule layer by layer from output to input, enabling gradient descent optimization.
From: Intro to Statistical Learning
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📐 König-Egerváry Theorem
In a bipartite graph, the maximum size of a matching equals the minimum size of a vertex cover.
Proof: A vertex cover must include at least one endpoint of each matched edge, so cover size $\\geq$ matching size. For equality: let $M$ be maximum and $U \\subseteq X$ be vertices reachable from unmatched $X$-vertices via $M$-alternating paths. Set $C = (X \\setminus U) \\cup (Y \\cap U)$. Then $|C| =...
From: Introduction to Graph Theory
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📐 Petersen
Every bridgeless cubic graph has a perfect matching.
Proof: Apply Tutte\
From: Introduction to Graph Theory
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💡 Proposition III.16
The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed.
From: Euclid's Elements
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💡 Proposition VI.17
If three straight lines be proportional, the rectangle contained by the extremes is equal to the square on the mean; and, if the rectangle contained by the extremes be equal to the square on the mean, the three straight lines will be proportional.
From: Euclid's Elements
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📐 Inverse Function Theorem (1D)
If $f$ is $C^1$ and $f\
From: Real Analysis
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📐 Weak Perfect Graph Theorem
A graph $G$ is perfect if and only if its complement $\\overline{G}$ is perfect.
Proof: A replication lemma shows that if $G$ is perfect, so is any graph obtained by "replicating" vertices. Using this, one shows $G$ perfect implies $\\chi(G) \\cdot \\alpha(G) \\geq n$ for all induced subgraphs. Applying this to $\\overline{G}$ (where $\\chi$ and $\\alpha$ swap roles with $\\omega$ a...
From: Introduction to Graph Theory
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📖 Matrix Representation
Let $T: V \\to W$ be linear with ordered bases $\\beta$ of $V$ and $\\gamma$ of $W$. The matrix $[T]_\\beta^\\gamma$ is defined by $[T(v_j)]_\\gamma = $ column $j$ of $[T]_\\beta^\\gamma$.
From: Advanced Linear Algebra
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📐 Generator Polynomial
Every cyclic code of length $n$ is generated by a unique monic polynomial $g(x)$ dividing $x^n - 1$. Dimension $k = n - \\deg g$.
From: intro-discrete
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💡 Proposition VII.35
If two numbers measure any number, the least number measured by them will also measure the same.
From: Euclid's Elements
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📐 p-Series Test
The series $\\sum \\frac{1}{n^p}$ converges if and only if $p > 1$.
From: Real Analysis
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📖 Complete Residue System
A complete residue system modulo $m$ is a set of $m$ integers containing exactly one representative from each congruence class modulo $m$. The standard choice is $\\{0, 1, 2, \\ldots, m-1\\}$.
From: Disquisitiones Arithmeticae
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📖 Matroid
A matroid $M = (E, \\mathcal{I})$ consists of a ground set $E$ and a collection $\\mathcal{I}$ of independent sets satisfying: (1) $\\emptyset \\in \\mathcal{I}$, (2) if $I \\in \\mathcal{I}$ and $J \\subseteq I$, then $J \\in \\mathcal{I}$, (3) if $I, J \\in \\mathcal{I}$ with $|I| < |J|$, then $\\exists e \\in J \\setminus I$ with $I \\cup \\{e\\} \\in \\mathcal{I}$.
From: Introduction to Graph Theory
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📐 Wilson
An integer $p > 1$ is prime if and only if $(p-1)! \\equiv -1 \\pmod{p}$.
Proof: For prime $p$, each $a \\in \\{1, 2, \\ldots, p-1\\}$ has a unique inverse $a^{-1}$ in that set. The elements that equal their own inverse satisfy $a^2 \\equiv 1$, i.e., $a \\equiv \\pm 1$. So only $1$ and $p-1$ are self-inverse.
Thus in $(p-1)! = 1 \\cdot 2 \\cdots (p-1)$, all elements except $...
From: Disquisitiones Arithmeticae
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📖 Mean Squared Error (MSE)
The mean squared error for regression is $\\text{MSE} = \\frac{1}{n}\\sum_{i=1}^{n}(y_i - \\hat{f}(x_i))^2$ where $\\hat{f}(x_i)$ is the prediction for the $i$th observation.
From: Intro to Statistical Learning
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