📖 Zariski Topology The \\textbf{Zariski topology} on $k^n$ has closed sets exactly the affine algebraic sets $V(I)$. From: df-course Learn more:

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📖 Hamming Code The $[7, 4]$ Hamming code has minimum distance 3, correcting single-bit errors. From: intro-discrete Learn more:

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📖 Convergence of a Sequence A sequence $(a_n)$ converges to $L$ if $\\forall \\varepsilon > 0, \\exists N: n > N \\Rightarrow |a_n - L| < \\varepsilon$. 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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📖 Hamming Distance $d(x, y)$ is the number of positions where codewords $x$ and $y$ differ. From: intro-discrete Learn more:

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📐 Brooks For a connected graph $G$ that is neither complete nor an odd cycle, $\\chi(G) \\leq \\Delta(G)$. Proof: If $G$ is not 2-connected, apply induction to blocks. If 2-connected with $\\Delta \\geq 3$: find vertices $u, v$ both adjacent to some vertex $w$ but not to each other. Order vertices so $u, v$ come first and $w$ comes last (BFS from $w$). Give $u, v$ the same color. Greedily color the rest; eac... From: Introduction to Graph Theory Learn more:

Introduction to Graph Theory | Math Academy

Interactive course based on Douglas B. West

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🎮 Interactive: Vector Operations Visualizer See vector addition, scalar multiplication, and linear combinations in action. The building blocks of linear algebra! From: Linear Algebra Try it:

Introduction to Linear Algebra | Math Academy

Interactive course based on Gilbert Strang

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📐 Hyperbolic Distance Formula In the Poincare disk, the hyperbolic distance from $0$ to $r$ (where $0 < r < 1$) is $d(0, r) = \\ln\\frac{1+r}{1-r}$. The general formula uses the cross-ratio. 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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🎯 Orientation and Reflections An isometry preserves orientation iff it is a composition of an even number of reflections. Proof: A single reflection reverses orientation (it swaps clockwise/counterclockwise). Two reflections preserve orientation (reverse then reverse). Translations (2 reflections in parallel lines) and rotations (2 reflections in intersecting lines) preserve orientation. Reflections and glide reflections (... 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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📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: base-race Learn more:

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🎮 Interactive: Primitive Root Explorer Discover which integers generate all residues modulo a prime. Explore the cyclic structure of multiplicative groups that underlies Diffie-Hellman key exchange. From: Disquisitiones Arithmeticae Try it:

Disquisitiones Arithmeticae | Math Academy

Gauss

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📐 Multiplicative Group of Finite Field The non-zero elements of $GF(q)$ form a cyclic group of order $q - 1$. From: intro-discrete Learn more:

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📐 First Fundamental Theorem of Calculus If $f$ is continuous on $[a,b]$ and $F(x) = \\int_a^x f(t)\\,dt$, then $F\ 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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📖 Basis A basis of a vector space $V$ is a linearly independent set that spans $V$. 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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📖 Quadratic Discriminant Analysis (QDA) QDA is similar to LDA but assumes each class has its own covariance matrix $\\Sigma_k$, resulting in quadratic decision boundaries. 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 VII.39 To find the number which is the least that will have given parts. 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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🎮 Interactive: Riemann Sum Visualizer Approximate area under a curve with rectangles. Watch how the limit of Riemann sums defines the integral. From: Calculus I Try it:

Calculus | Math Academy

An interactive course based on Tom M. Apostol

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📖 Euclidean Domain A \\textbf{Euclidean domain} is an integral domain $R$ with a function $N: R \\setminus \\{0\\} \\to \\mathbb{Z}_{\\geq 0}$ such that for all $a, b$ with $b \\neq 0$, there exist $q, r$ with $a = bq + r$ and either $r = 0$ or $N(r) < N(b)$. From: df-course Learn more:

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📐 Recursive Binary Splitting At each step, select the predictor $X_j$ and cutpoint $s$ that minimize RSS: $\\sum_{i: x_i \\in R_1}(y_i - \\hat{y}_{R_1})^2 + \\sum_{i: x_i \\in R_2}(y_i - \\hat{y}_{R_2})^2$ 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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📐 Unique Reduced Representative Every positive definite binary quadratic form is properly equivalent to a unique reduced form. From: Disquisitiones Arithmeticae Learn more:

Disquisitiones Arithmeticae | Math Academy

Gauss

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📖 Quotient Group If $N \\trianglelefteq G$, the \\textbf{quotient group} $G/N$ is the set of cosets $\\{gN : g \\in G\\}$ with operation $(aN)(bN) = (ab)N$. From: df-course Learn more:

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