📖 Convolution When $X$ and $Y$ are independent with densities $f_X$ and $f_Y$, the density of $X + Y$ is the \\textbf{convolution}: $f_{X+Y}(u) = \\int_{-\\infty}^{\\infty} f_X(x) f_Y(u - x)\\,dx = (f_X * f_Y)(u)$. From: calc2-course Learn more:

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💡 Proposition I.16 (Exterior Angle Theorem) In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles. From: Euclid's Elements Learn more:

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📖 Axiomatic Definition of Determinant A function $d$, defined for each ordered $n$-tuple of vectors $A_1, \\ldots, A_n$ in $n$-space, is called a \\textbf{determinant function} of order $n$ if it satisfies: Axiom 1 (Homogeneity): $d(\\ldots, tA_k, \\ldots) = t \\cdot d(\\ldots, A_k, \\ldots)$, Axiom 2 (Additivity): $d(\\ldots, A_k + C, \\ldots) = d(\\ldots, A_k, \\ldots) + d(\\ldots, C, \\ldots)$, Axiom 3 (Vanishing): $d(A_1, \\ldo... From: calc2-course Learn more:

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📖 Definition of a Linear Space Let $V$ denote a nonempty set of objects, called elements. The set $V$ is called a \\textbf{linear space} if it satisfies ten axioms: (1) Closure under addition, (2) Closure under scalar multiplication, (3) Commutative law: $x + y = y + x$, (4) Associative law: $(x + y) + z = x + (y + z)$, (5) Existence of zero element: $x + 0 = x$, (6) Existence of negatives: $x + (-1)x = 0$, (7) Associative l... From: calc2-course Learn more:

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📖 Group Action A \\textbf{group action} of $G$ on a set $A$ is a map $G \\times A \\to A$, written $(g, a) \\mapsto g \\cdot a$, satisfying: (1) $e \\cdot a = a$ for all $a$; (2) $(gh) \\cdot a = g \\cdot (h \\cdot a)$ for all $g, h \\in G$, $a \\in A$. From: df-course Learn more:

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📐 Uniqueness of Zero Vector Every vector space has exactly one zero vector $\\mathbf{0}$. Proof: Suppose $\\mathbf{0}$ and $\\mathbf{0}'$ are both zero vectors. Then: $$\\mathbf{0} = \\mathbf{0} + \\mathbf{0}' = \\mathbf{0}'$$ where the first equality uses that $\\mathbf{0}'$ is a zero vector and the second uses that $\\mathbf{0}$ is a zero vector. From: Advanced Linear Algebra Learn more:

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💡 Proposition I.5 (Pons Asinorum) In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. From: Euclid's Elements Learn more:

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📖 Definition of Trace The \\textbf{trace} of an $n \\times n$ matrix $A$ is the sum of its diagonal elements: $\\text{tr} \\, A = \\sum_{i=1}^n a_{ii}$. From: calc2-course Learn more:

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📖 Definition VII.22 (Perfect Number) A perfect number is that which is equal to the sum of its own parts. From: Euclid's Elements Learn more:

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💡 Proposition II.7 If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to twice the rectangle contained by the whole and the said segment and the square on the remaining segment. From: Euclid's Elements Learn more:

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🎯 Groups of Prime Order are Cyclic If $|G| = p$ is prime, then $G$ is cyclic. Proof: Let $a \\in G$ with $a \\neq e$. Then $|a|$ divides $p$, so $|a| = p$ (since $|a| > 1$). Thus $G = \\langle a \\rangle$. From: df-course Learn more:

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📖 Definition 3.5.1 (Ordered Pair) If $x$ and $y$ are objects (possibly equal), we define the **ordered pair** $(x, y)$ to be a new object, defined in such a way that $(x, y) = (x\ From: tao-analysis-1 Learn more:

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📐 Subgroup Test A non-empty subset $H \\subseteq G$ is a subgroup if and only if for all $a, b \\in H$, we have $ab^{-1} \\in H$. Proof: ($\\Rightarrow$) If $H$ is a subgroup, then $a, b \\in H$ implies $b^{-1} \\in H$, so $ab^{-1} \\in H$ by closure. ($\\Leftarrow$) Suppose the condition holds. Since $H \\neq \\emptyset$, let $a \\in H$. Then $e = aa^{-1} \\in H$. For any $a \\in H$, $a^{-1} = ea^{-1} \\in H$. For $a, b \\in H$... From: df-course Learn more:

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💡 Proposition VII.5 If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one. From: Euclid's Elements Learn more:

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💡 Unintentional Short Bitcoin Funds holding banks, payment processors, fiat currencies, long-duration bonds, and gold miners may be structurally short Bitcoin. These assets compete with Bitcoin and will lose to it over time. From: bfi Learn more:

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💡 Proposition IV.1 Into a given circle to fit a straight line equal to a given straight line which is not greater than the diameter of the circle. 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: ln-bolts Learn more:

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💡 Proposition 5.4.7 (Trichotomy for Reals) Let $x, y$ be real numbers. Then exactly one of $x = y$, $x < y$, or $x > y$ is true. From: tao-analysis-1 Learn more:

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📖 Definition 9.9.2 (Uniform Continuity) Let $X \\subseteq \\mathbb{R}$ and $f : X \\to \\mathbb{R}$. We say $f$ is **uniformly continuous** iff for every $\\varepsilon > 0$ there exists $\\delta > 0$ such that $|f(x) - f(y)| \\leq \\varepsilon$ whenever $x, y \\in X$ and $|x - y| < \\delta$. From: tao-analysis-1 Learn more:

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📖 Forward Stepwise Selection Start with null model, then repeatedly add the predictor that provides the greatest additional improvement to the fit, until all predictors are in the model. From: Intro to Statistical Learning Learn more:

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