📐 Multivariate Gaussian Distribution The multivariate Gaussian with mean $\\boldsymbol{\\mu}$ and covariance $\\Sigma$ has density $f(\\mathbf{x}) = \\frac{1}{(2\\pi)^{n/2}|\\Sigma|^{1/2}} \\exp\\left(-\\frac{1}{2}(\\mathbf{x}-\\boldsymbol{\\mu})^T\\Sigma^{-1}(\\mathbf{x}-\\boldsymbol{\\mu})\\right)$. From: Linear Algebra Learn more:

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📐 Constructible Numbers A length is constructible with straightedge and compass from a unit segment iff it can be obtained from rational numbers using only the operations $+, -, \\times, \\div, \\sqrt{\\phantom{x}}$. Proof: Line-line intersections involve solving linear equations, giving rational results from rational inputs. Line-circle intersections require solving a quadratic, introducing at most one square root. Circle-circle intersections also reduce to solving a quadratic after eliminating one variable. Theref... From: Four Pillars of Geometry Learn more:

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📐 Archimedes: Area of Circle The area of a circle equals that of a triangle with base equal to circumference and height equal to radius: $A = \\pi r^2$. From: Thales to Euclid Learn more:

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📐 Orthogonality Criterion Vectors $\\mathbf{u}$ and $\\mathbf{v}$ are perpendicular (orthogonal) if and only if $\\mathbf{u} \\cdot \\mathbf{v} = 0$. Proof: $\\mathbf{u} \\cdot \\mathbf{v} = |\\mathbf{u}||\\mathbf{v}|\\cos\\theta = 0$ iff $\\cos\\theta = 0$ (assuming nonzero vectors), which occurs iff $\\theta = 90°$. From: Four Pillars of Geometry Learn more:

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📐 Euclidean Algorithm The GCD can be computed by repeated division. If $b = aq_1 + r_1$, then $\\gcd(b, a) = \\gcd(a, r_1)$. Continue until remainder is 0. From: intro-discrete Learn more:

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💡 Proposition I.21 If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle. From: Euclid's Elements Learn more:

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📖 Chromatic Index The chromatic index $\\chi\ From: Introduction to Graph Theory Learn more:

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📐 Theorem 6.21 (FTC Part II) If $f \\in \\mathcal{R}$ on $[a, b]$ and there is a differentiable function $F$ on $[a, b]$ with $F\ From: rudin Learn more:

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📐 Finite Integral Domains are Fields Every finite integral domain is a field. Proof: Let $R$ be finite and $a \\neq 0$. The map $\\phi_a: R \\to R$ by $x \\mapsto ax$ is injective (since $ax = ay$ implies $a(x-y) = 0$, and $R$ has no zero divisors). Since $R$ is finite, injective implies surjective. So $1 \\in \\text{Im}(\\phi_a)$, meaning $ab = 1$ for some $b$. From: df-course Learn more:

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📖 Definition of Subspace Given a linear space $V$, let $S$ be a nonempty subset of $V$. If $S$ is also a linear space, with the same operations of addition and multiplication by scalars, then $S$ is called a \\textbf{subspace} of $V$. From: calc2 Learn more:

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📐 The Halving Series The infinite series $\\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\frac{1}{16} + \\ldots$ converges to $1$. Proof: Let $S = \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\ldots$ Multiply by 2: $2S = 1 + \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\ldots = 1 + S$ Therefore $S = 1$. From: Beginner Calculus Learn more:

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📖 Identity Matrix The $n \\times n$ \\textbf{identity matrix} $I_n$ has 1s on the main diagonal and 0s elsewhere. From: calc2 Learn more:

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📖 Ends and Means Ends are goals the actor wishes to achieve. Means are things employed to attain ends. The ultimate end of all action is removal of felt uneasiness. From: Human Action Learn more:

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Ludwig von Mises

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📐 Newton For every action, there is an equal and opposite reaction: $F_{12} = -F_{21}$ From: Men of Mathematics Learn more:

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📐 Fermat Light travels between two points along the path that takes the least time From: Men of Mathematics Learn more:

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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: fermat-minkowski Learn more:

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📐 Axiom of Infinity There exists a set $\\Omega$ containing $\\emptyset$ and closed under the successor operation $X \\mapsto X \\cup \\{X\\}$. From: numbers-geometry Learn more:

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📖 Definition 7.1.1 (Finite Series) Let $m, n$ be integers, and let $(a_i)_{i=m}^n$ be a finite sequence. We define $\\sum_{i=m}^n a_i$ recursively by $\\sum_{i=m}^m a_i := a_m$ and $\\sum_{i=m}^{n+1} a_i := (\\sum_{i=m}^n a_i) + a_{n+1}$. From: tao-analysis-1 Learn more:

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📐 RSA Correctness For any message $m$ with $\\gcd(m, n) = 1$: $(m^e)^d \\equiv m \\pmod{n}$. Proof: Since $ed \\equiv 1 \\pmod{\\varphi(n)}$, write $ed = 1 + k\\varphi(n)$ for some integer $k$. By Euler's theorem, $m^{\\varphi(n)} \\equiv 1 \\pmod{n}$ when $\\gcd(m,n) = 1$. Therefore: $m^{ed} = m^{1 + k\\varphi(n)} = m \\cdot (m^{\\varphi(n)})^k \\equiv m \\cdot 1^k = m \\pmod{n}$. From: Algebraic Number Theory Learn more:

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📖 Definition 2.32 (Compact Set) A set $K$ is **compact** if every open cover of $K$ has a finite subcover. From: rudin Learn more:

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