💡 100% Bitcoin Allocation for Pensions Given medium-term problem horizon and ability to survive two Bitcoin halving cycles (eight years), 100% allocation is justified for pensions. Limited bailout availability argues for maximum volatility bet. From: bfi Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

💡 Proposition 4.4.1 (Interspersing of Integers) Let $x$ be a rational. Then there exist integers $N$ and $M$ such that $N \\leq x < M$. From: tao-analysis-1 Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Equivalence Class The equivalence class of $a$ is $[a] = \\{b \\in A : a \\sim b\\}$. From: intro-discrete Learn more:

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📐 Density of Rationals Between any two distinct real numbers, there exists a rational number. From: Real Analysis Learn more:

Real Analysis | Math Academy

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

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📖 Definition A.4 (Existential Quantifier) If $P(x)$ is a property, then "$\\exists x, P(x)$" means "there exists at least one $x$ for which $P(x)$ is true." From: tao-analysis-1 Learn more:

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📐 Injectivity Criterion A linear transformation $T: V \\to W$ is injective (one-to-one) if and only if $\\ker(T) = \\{\\mathbf{0}\\}$. Proof: **(⇒)** If $T$ is injective and $v \\in \\ker(T)$, then $T(v) = \\mathbf{0} = T(\\mathbf{0})$. By injectivity, $v = \\mathbf{0}$. **(⇐)** If $\\ker(T) = \\{\\mathbf{0}\\}$ and $T(u) = T(v)$, then $T(u - v) = \\mathbf{0}$, so $u - v \\in \\ker(T) = \\{\\mathbf{0}\\}$. Thus $u = v$. From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Math Academy

Interactive course for learning Advanced Linear Algebra based on Steven Roman

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📖 Hierarchical Clustering Hierarchical clustering produces a tree-like dendrogram showing nested clusters. Bottom-up (agglomerative): start with $n$ clusters and merge. Top-down (divisive): start with 1 cluster and split. From: Intro to Statistical Learning Learn more:

Introduction to Statistical Learning | Magic Internet Math

An interactive course based on James et al.

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💡 Proposition III.22 The opposite angles of quadrilaterals in circles are equal to two right angles. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

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

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📐 Menger The maximum number of internally disjoint $u$-$v$ paths equals the minimum size of a $u$-$v$ separating set. Proof: The number of disjoint paths cannot exceed the separator size (each path needs a distinct separator vertex). For the reverse: induct on $|E|$. If every $u$-$v$ separator contains a neighbor of $u$ and a neighbor of $v$, delete an edge $e$ on a shortest path; the separator bound decreases by at mo... From: Introduction to Graph Theory Learn more:

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Interactive course based on Douglas B. West

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📖 Axiom 3.5 (Axiom of Specification) Let $A$ be a set, and for each $x \\in A$, let $P(x)$ be a property pertaining to $x$. Then there exists a set $\\{x \\in A : P(x)\\}$ whose elements are exactly those elements $x \\in A$ for which $P(x)$ is true. From: tao-analysis-1 Learn more:

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💡 Proposition 5.1.15 (Cauchy Sequences are Bounded) Every Cauchy sequence $(a_n)_{n=1}^{\\infty}$ is bounded. From: tao-analysis-1 Learn more:

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💡 Proposition II.9 If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double of the square on the half and of the square on the straight line between the points of section. 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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📐 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: Numbers and Geometry Learn more:

Numbers and Geometry | Math Academy

Interactive course exploring the deep connections between numbers and geometry

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📖 Principal Ideal An ideal $I$ is \\textbf{principal} if $I = (a) = Ra = \\{ra : r \\in R\\}$ for some $a \\in R$. From: df-course Learn more:

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📖 Isometry An isometry of the plane is a transformation $T: \\mathbb{R}^2 \\to \\mathbb{R}^2$ that preserves distances: $|T(P)T(Q)| = |PQ|$ for all points $P, Q$. 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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📐 Slope as Tangent If a line makes angle $\\theta$ with the positive $x$-axis, its slope is $m = \\tan\\theta$. Proof: Consider a line through the origin making angle $\\theta$ with the $x$-axis. A point on this line has coordinates $(r\\cos\\theta, r\\sin\\theta)$ for some $r > 0$. The slope is $m = \\frac{r\\sin\\theta - 0}{r\\cos\\theta - 0} = \\frac{\\sin\\theta}{\\cos\\theta} = \\tan\\theta$. From: Four Pillars of Geometry Learn more:

The Four Pillars of Geometry | Magic Internet Math

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

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📖 Survival Time The survival time $T$ is the time until an event of interest (death, failure, etc.) occurs. We observe $(Y_i, \\delta_i)$ where $Y_i = \\min(T_i, C_i)$ and $\\delta_i$ indicates if event was observed. From: Intro to Statistical Learning Learn more:

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📖 Definition of Inner Product Let $V$ be a real linear space. A function that assigns to each pair of elements $x$ and $y$ in $V$ a real number $(x, y)$, called the \\textbf{inner product}, satisfies: (1) $(x, y) = (y, x)$ (symmetry), (2) $(x + y, z) = (x, z) + (y, z)$ (additivity), (3) $(cx, y) = c(x, y)$ (homogeneity), (4) $(x, x) \\geq 0$, and $(x, x) = 0$ iff $x = 0$ (positivity). From: calc2-course Learn more:

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📐 Order Divides Exponent If $a^n \\equiv 1 \\pmod{m}$, then $\\text{ord}_m(a)$ divides $n$. In particular, for prime $p$, $\\text{ord}_p(a)$ divides $p-1$. From: Disquisitiones Arithmeticae Learn more:

Disquisitiones Arithmeticae | Math Academy

Gauss

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📖 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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