📖 Exponential Distribution A random variable $X$ has an \\textbf{exponential distribution} with parameter $\\lambda > 0$ if its distribution function is $F(t) = 1 - e^{-\\lambda t}$ for $t \\geq 0$, and $F(t) = 0$ for $t < 0$. The density function is $f(t) = \\lambda e^{-\\lambda t}$ for $t \\geq 0$. From: calc2 Learn more:

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💡 Proposition III.12 If two circles touch one another externally, the straight line joining their centres will pass through the point of contact. From: Euclid's Elements Learn more:

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The foundational text of Western mathematics. Study all 13 books of Euclid

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💡 The Principle of Correlation Between Planes Every being, force, and form on the physical plane has its counterpart on the astral and devachanic planes. Physical matter is the condensed expression of astral forces, which are in turn the densified expression of devachanic archetypes. What appears as a physical object to the senses appears as a living, luminous form on the astral plane and as a sounding, creative thought-being on the devach... From: steiner-GA90a Learn more:

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📐 Theorem 4.14 (Image of Compact) If $f: X \\to Y$ is continuous and $K \\subset X$ is compact, then $f(K)$ is compact. Proof: Let $\\{V_\\alpha\\}$ be an open cover of $f(K)$. Since $f$ is continuous, each $f^{-1}(V_\\alpha)$ is open. These sets cover $K$: $K \\subset \\bigcup_\\alpha f^{-1}(V_\\alpha)$. Since $K$ is compact, finitely many suffice: $K \\subset f^{-1}(V_{\\alpha_1}) \\cup \\cdots \\cup f^{-1}(V_{\\alp... From: rudin Learn more:

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📖 Ends and Means Every action aims at an end (goal) and employs means to achieve it. Means are always scarce relative to ends; otherwise no action would be needed. From: Man, Economy, and State Learn more:

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Murray Rothbard

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💡 Hidden Purpose and Sacrifice The apparent abandonment of great potential by Francisco and other strikers is actually purposeful sacrifice in service of a larger cause. From: Atlas Shrugged Learn more:

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📖 Invertible Matrix An $n \\times n$ matrix $A$ is \\textbf{invertible} (or \\textbf{nonsingular}) if there exists an $n \\times n$ matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I_n$. A matrix that is not invertible is called \\textbf{singular}. From: calc2 Learn more:

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📐 Theorem 11.41 (Riesz-Fischer) $L^2(\\mu)$ is complete with respect to the norm $\\|f\\|_2 = \\left(\\int |f|^2\\right)^{1/2}$. From: rudin Learn more:

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📖 Linear Transformation A linear transformation $T: \\mathbb{R}^n \\to \\mathbb{R}^m$ satisfies $T(\\mathbf{u} + \\mathbf{v}) = T(\\mathbf{u}) + T(\\mathbf{v})$ and $T(c\\mathbf{v}) = cT(\\mathbf{v})$. From: Four Pillars of Geometry Learn more:

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💡 Equilateral Triangle Construction (Euclid I.1) Given a line segment $AB$, construct circles centered at $A$ and $B$ with radius $|AB|$. If they intersect at point $C$, then $\\triangle ABC$ is equilateral. Proof: Since $C$ lies on the circle centered at $A$, we have $|AC| = |AB|$ by definition of circle. Since $C$ lies on the circle centered at $B$, we have $|BC| = |AB|$ by definition of circle. Therefore $|AB| = |AC| = |BC|$, making $\\triangle ABC$ equilateral. From: Four Pillars of Geometry Learn more:

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📐 Euclidean Algorithm $\\gcd(a,b) = \\gcd(b, a \\mod b)$. Repeated application computes $\\gcd(a,b)$ in $O((\\log b)^2)$ bit operations. Proof: If $a = bq + r$, then any common divisor of $a$ and $b$ also divides $r = a - bq$. Conversely, any common divisor of $b$ and $r$ also divides $a = bq + r$. Therefore $\\gcd(a,b) = \\gcd(b,r)$. The algorithm terminates because remainders strictly decrease. Each step halves the larger numbe... From: Algebraic Number Theory Learn more:

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📖 Injective Module An $R$-module $Q$ is \\textbf{injective} if for every injection $f: M \\to N$ and map $g: M \\to Q$, there exists an extension $\\tilde{g}: N \\to Q$ with $\\tilde{g} \\circ f = g$. From: df-course Learn more:

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📐 Absolute Convergence Implies Convergence If $\\sum |a_n|$ converges, then $\\sum a_n$ converges. From: Real Analysis Learn more:

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📐 Pappus Let $A, B, C$ be collinear and $D, E, F$ be collinear on a parallel line. If $AB \\parallel ED$ and $FE \\parallel BC$, then $AF \\parallel CD$. Proof: By similar triangles formed by the parallel lines, we establish proportions that force $AF \\parallel CD$. From: Four Pillars of Geometry Learn more:

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📐 Derived Demand Demand for factors is derived from demand for the consumers\ From: Man, Economy, and State Learn more:

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Murray Rothbard

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📐 Theorem 2.36 (Nested Compact Sets) If $\\{K_\\alpha\\}$ is a collection of compact subsets of a metric space such that every finite subcollection has nonempty intersection, then $\\bigcap_\\alpha K_\\alpha \\neq \\emptyset$. From: rudin Learn more:

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📐 Vertical Angles Theorem When two lines intersect, vertically opposite angles are equal. Proof: Let the lines intersect at $O$, forming angles $\\alpha, \\beta, \\alpha', \\beta'$ in order. Then $\\alpha + \\beta = 180°$ (angles on a straight line). Also $\\beta + \\alpha' = 180°$. Therefore $\\alpha = \\alpha'$. Similarly, $\\beta = \\beta'$. From: numbers-geometry Learn more:

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📐 Cross-Ratio is Projectively Invariant The cross-ratio of four collinear points is preserved by any projectivity (and hence by any central projection). Proof: A Mobius transformation $f(x) = \\frac{ax + b}{cx + d}$ preserves cross-ratio: $(f(a), f(b); f(c), f(d)) = (a, b; c, d)$. Direct calculation shows the cross-ratio is unchanged after substituting the linear fractional expressions. From: Four Pillars of Geometry Learn more:

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💡 World Wisdom and Human Wisdom There exists a cosmic wisdom (world wisdom) that permeates and orders the entire universe, and a human wisdom that is the portion of this cosmic wisdom accessible to human consciousness at its present stage of development. The task of spiritual science is to progressively expand human wisdom until it encompasses ever greater portions of world wisdom. This expansion occurs not through abstract s... From: steiner-GA90a Learn more:

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📖 Function A function $f: A \\to B$ assigns to each element of $A$ exactly one element of $B$. From: intro-discrete Learn more:

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