📖 Minimum A minimum is a point where a function reaches a local lowest value. At a minimum, the derivative is zero and the second derivative is positive. From: Beginner Calculus Learn more:

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Calculus Made Easy - An interactive introduction to calculus

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📐 Theorem 1.21 (Existence of nth Roots) For every real $x > 0$ and every positive integer $n$, there is one and only one positive real $y$ such that $y^n = x$. From: rudin Learn more:

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📐 Cyclic Decomposition Theorem Every finitely generated module $M$ over a PID $R$ is isomorphic to $R^r \\oplus R/(a_1) \\oplus \\cdots \\oplus R/(a_k)$ where $a_1 | a_2 | \\cdots | a_k$. Proof: Apply the primary decomposition first, then decompose each primary component into cyclic modules. The divisibility chain follows from the uniqueness of elementary divisors. From: adv_linalg Learn more:

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🎮 Interactive: Constructible Polygon Demo See which regular polygons can be constructed with compass and straightedge. Gauss proved the 17-gon is constructible at age 19! From: Disquisitiones Arithmeticae Try it:

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Gauss

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📖 Point of Inflection A point of inflection is where a curve changes from concave up to concave down (or vice versa). At such points, $\\frac{d^2y}{dx^2} = 0$. From: Beginner Calculus Learn more:

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Calculus Made Easy - An interactive introduction to calculus

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📐 The Action Axiom Human beings engage in purposeful behavior toward chosen goals. This is self-evident and cannot be denied without self-refutation—the very act of denying it is itself purposeful action. Proof: The action axiom is apodictically certain—true by the nature of human thought itself. To argue against it, one must: 1. Have a goal (refuting the axiom) 2. Choose means (constructing arguments) 3. Believe the argument can achieve the goal But this is precisely purposeful action. The denial... From: Human Action Learn more:

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Ludwig von Mises

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📐 Area of Circle The area of a circle is proportional to the square of its diameter: $A \\propto d^2$ Proof: Proved using the Method of Exhaustion: inscribe and circumscribe polygons with increasing numbers of sides. As the number of sides approaches infinity, both polygon areas approach the circle's area. Since polygon areas scale with the square of linear dimensions, so does the circle's area. From: Men of Mathematics Learn more:

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An interactive journey through 2500 years of mathematical history with E.T. Bell

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📖 Ring A \\textbf{ring} $R$ is a set with two binary operations $+$ and $\\cdot$ such that: (1) $(R, +)$ is an abelian group with identity $0$; (2) Multiplication is associative; (3) Distributive laws hold: $a(b+c) = ab + ac$ and $(a+b)c = ac + bc$. From: df-course Learn more:

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📖 Perfect Special Honest Verifier Zero-Knowledge PSHVZK: A PPT simulator $\\mathcal{S}$ exists such that real and simulated transcript distributions are identical, given the verifier\ From: lcn Learn more:

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💡 The Salvation of Humanity Through the Christ Impulse Without the intervention of the Christ Being, humanity would have descended ever deeper into material existence without the possibility of return to the spirit. The Christ impulse, entering Earth evolution at the Baptism in the Jordan and culminating at Golgotha, planted a new spiritual force into the Earth\ From: steiner-GA90a Learn more:

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💡 Proposition 2.3.6 (Distributive Law) For any natural numbers $a$, $b$, $c$, we have $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$. From: tao-analysis-1 Learn more:

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📐 Whitney A graph $G$ with at least 3 vertices is 2-connected if and only if every pair of edges lies on a common cycle. Proof: If 2-connected, any two edges $e = xy$ and $f = uv$ can be put on a cycle: by Menger, there are 2 internally disjoint $x$-$u$ paths. Together with edges $e$ and $f$ and paths from $y$ and $v$, we can form a cycle through both edges. Conversely, if every pair of edges lies on a cycle, there\ From: Introduction to Graph Theory Learn more:

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Interactive course based on Douglas B. West

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📐 Schur (1) Any $G$-homomorphism between irreducible representations is either zero or an isomorphism. (2) If $k$ is algebraically closed, any $G$-endomorphism of an irreducible representation is a scalar multiple of the identity. From: df-course Learn more:

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📐 Division Algorithm For any integers $a$ and $b$ with $b > 0$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that $a = bq + r$ and $0 \\le r < b$. Proof: \\textbf{Existence:} Let $S = \\{a - bk : k \\in \\mathbb{Z}, a - bk \\geq 0\\}$. Since $S$ is non-empty (choose $k$ sufficiently negative) and bounded below by 0, by the Well-Ordering Principle, $S$ has a minimum element $r = a - bq$ for some $q$. If $r \\geq b$, then $a - b(q+1) = r - b \\geq 0... From: df-course Learn more:

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💡 Proposition I.20 (Triangle Inequality) In any triangle two sides taken together in any manner are greater than the remaining one. 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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💡 Proposition VII.31 Any composite number is measured by some prime number. From: Euclid's Elements 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: tontines Learn more:

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📖 Legendre Symbol For odd prime $p$: $(a/p) = 0$ if $p|a$; $(a/p) = 1$ if $a$ is QR mod $p$; $(a/p) = -1$ if $a$ is QNR mod $p$. From: Algebraic Number Theory Learn more:

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📐 Rationality of Action Action is necessarily always rational in the sense that it involves selecting means believed suitable for attaining ends. Irrational action does not exist—people can be mistaken about which means achieve their ends, but error is not irrationality. From: Human Action Learn more:

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Ludwig von Mises

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📐 Theorem 2.47 (Connected Subsets of ℝ) A subset $E$ of $\\mathbb{R}$ is connected if and only if: whenever $x, y \\in E$ and $x < z < y$, we have $z \\in E$. That is, the connected subsets of $\\mathbb{R}$ are precisely the intervals. From: rudin Learn more:

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