📐 Economic Calculation Problem Without market prices, central planners cannot calculate whether resources are being used efficiently. This is the fundamental problem that makes rational socialist planning impossible. Proof: To allocate resources rationally, one must know the relative values of different uses. Prices emerge only from voluntary exchange. Socialism abolishes private ownership of means of production. Without private ownership, there is no exchange of capital goods. Without exchange, there are no... From: Human Action Learn more:

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Ludwig von Mises

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📖 Prometheus Unchained John Galt is Prometheus who changed his mind—after centuries of being torn by vultures for bringing fire to men, he withdrew his fire until men withdraw their vultures. From: Atlas Shrugged Learn more:

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📐 Euclid Given any finite list of primes $p_1, p_2, \\ldots, p_k$, there exists a prime not on this list. Proof: Consider $n = p_1 p_2 \\cdots p_k + 1$. By the Lemma, $n$ has some prime divisor $p$. But $n \\equiv 1 \\pmod{p_i}$ for each $i$, so $p$ cannot equal any $p_i$. Therefore there exists a prime not on the original list. Since any finite list can be extended, there are infinitely many primes. From: Thales to Euclid Learn more:

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Journey through 2,500 years of mathematical history, from ancient Egypt through modern foundations

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📖 Vertex and Edge Connectivity The vertex connectivity $\\kappa(G)$ is the minimum size of a vertex cut (or $n-1$ if $G = K_n$). The edge connectivity $\\kappa\ From: Introduction to Graph Theory Learn more:

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Interactive course based on Douglas B. West

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📖 Binomial Coefficient For $0 \\leq k \\leq n$, $\\binom{n}{k} = \\frac{n!}{k!(n-k)!}$ where $n! = n(n-1)\\cdots 1$ and $0! = 1$. From: intro-discrete Learn more:

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📐 Rank-Nullity Theorem (Dimension Theorem) If $V$ is finite-dimensional, then $T(V)$ is also finite-dimensional, and: $\\dim N(T) + \\dim T(V) = \\dim V$. In other words, the nullity plus the rank of a linear transformation equals the dimension of its domain. Proof: Let $n = \\dim V$ and let $e_1, \\ldots, e_k$ be a basis for $N(T)$ where $k = \\dim N(T)$. By Theorem 1.7, these are part of some basis for $V$: $e_1, \\ldots, e_k, e_{k+1}, \\ldots, e_n$ where $k + r = n$. We show that $T(e_{k+1}), \\ldots, T(e_n)$ form a basis for $T(V)$, proving $\\dim T(V) =... From: calc2 Learn more:

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📖 Rent The factor price of land services, determined by the discounted marginal value product of land in its most valuable use. From: Man, Economy, and State Learn more:

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Murray Rothbard

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📐 Fundamental Theorem of Calculus (Part 1) If $f$ is continuous on $[a, b]$ and $F(x) = \\int_a^x f(t)\\,dt$, then $F$ is differentiable and $F\ Proof: The derivative of an accumulation function is the original function. If you're accumulating something at rate $f(x)$, then the rate of change of your total is exactly $f(x)$. Formally: $F'(x) = \\lim_{h \\to 0} \\frac{F(x+h) - F(x)}{h} = \\lim_{h \\to 0} \\frac{1}{h}\\int_x^{x+h} f(t)\\,dt = ... From: Calculus: A Liberal Art Learn more:

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📐 Cayley-Hamilton Theorem Every linear operator satisfies its characteristic polynomial: if $p(t) = \\det(tI - A)$, then $p(A) = 0$. Proof: The minimal polynomial divides the characteristic polynomial by structure theory. Since both are monic of the same degree when considering the module $F[t]/(m_\\tau)$, they are equal up to units. From: adv_linalg Learn more:

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📐 Uncertainty Action occurs in time, and the future is always uncertain. If the future were known with certainty, there would be no need to choose. From: Man, Economy, and State Learn more:

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Murray Rothbard

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📖 Leibniz Notation for Derivatives The derivative is written as $\\frac{dy}{dx}$, representing the ratio of infinitesimal changes. This notation suggests correctly that derivatives behave like fractions. From: Calculus: A Liberal Art Learn more:

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📐 Theorem 6.8 (Continuous ⟹ Integrable) If $f$ is continuous on $[a, b]$, then $f \\in \\mathcal{R}$ on $[a, b]$. From: rudin 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: foundation Learn more:

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📐 Fundamental Theorem of Arithmetic Every positive integer greater than $1$ has a unique factorization into prime numbers (up to order). From: Thales to Euclid Learn more:

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Journey through 2,500 years of mathematical history, from ancient Egypt through modern foundations

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📐 Newton Force equals mass times acceleration: $F = ma = m\\frac{d^2x}{dt^2}$. This is fundamentally a differential equation. From: Calculus: A Liberal Art Learn more:

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📖 Field A field $F$ is a set with addition and multiplication where $(F, +)$ is an abelian group, $(F \\setminus \\{0\\}, \\cdot)$ is an abelian group, and distributivity holds. From: Algebraic Number Theory Learn more:

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📐 Hyperbolic Isometries The isometries of the Poincare disk are the Mobius transformations that preserve the unit disk. These have the form $z \\mapsto e^{i\\theta}\\frac{z - a}{1 - \\bar{a}z}$ for $|a| < 1$. From: Four Pillars of Geometry Learn more:

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Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📐 Theorem 7.12 (Uniform Limit of Continuous) If $f_n \\to f$ uniformly on $E$ and each $f_n$ is continuous on $E$, then $f$ is continuous on $E$. Proof: Fix $x \\in E$ and $\\varepsilon > 0$. By uniform convergence, there exists $N$ with $|f_n(t) - f(t)| < \\varepsilon/3$ for all $t \\in E$ when $n \\geq N$. Since $f_N$ is continuous at $x$, there exists $\\delta > 0$ with $|f_N(t) - f_N(x)| < \\varepsilon/3$ when $|t - x| < \\delta$. Then for... From: rudin Learn more:

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📖 Riemann Integrable $f$ is Riemann integrable if $\\sup_P L(f,P) = \\inf_P U(f,P)$. The common value is $\\int_a^b f$. From: Real Analysis Learn more:

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A rigorous introduction to real analysis covering limits, continuity, differentiation, and integration through formal mathematical proofs.

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💡 Tocqueville\ "Democracy extends the sphere of individual freedom, socialism restricts it. Democracy attaches all possible value to each man; socialism makes each man a mere agent. Democracy and socialism have nothing in common but one word: equality. But notice the difference: while democracy seeks equality in liberty, socialism seeks equality in restraint and servitude." From: The Road to Serfdom Learn more:

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