MagicInternetMath 4 months ago

📖 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:

Magic Internet Math

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

💡 Proposition I.41 If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Logistic Function $p(X) = \\frac{e^{\\beta_0 + \\beta_1 X}}{1 + e^{\\beta_0 + \\beta_1 X}}$ models the probability that $Y = 1$ given $X$, ensuring $0 \\leq p(X) \\leq 1$. From: Intro to Statistical Learning Learn more:

Introduction to Statistical Learning | Magic Internet Math

An interactive course based on James et al.

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📐 Polynomial Uniqueness A polynomial of degree $t-1$ is uniquely determined by $t$ points. Conversely, $t-1$ points leave infinitely many degree $t-1$ polynomials consistent. From: frost Learn more:

Magic Internet Math

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: Calculus I Learn more:

Calculus | Math Academy

An interactive course based on Tom M. Apostol

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Definition IV.3 A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure lies on the circumference of the circle. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Bayes Error Rate The Bayes error rate is $1 - E[\\max_j \\Pr(Y = j | X)]$, the lowest possible test error rate for a given problem. It is analogous to the irreducible error. From: Intro to Statistical Learning Learn more:

Introduction to Statistical Learning | Magic Internet Math

An interactive course based on James et al.

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Cyclic Code A linear code is cyclic if every cyclic shift of a codeword is also a codeword. Equivalently, an ideal in $\\mathbb{Z}_2[x]/(x^n - 1)$. From: intro-discrete Learn more:

Magic Internet Math

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📐 Insolvability of the Quintic The general polynomial of degree $\\geq 5$ is not solvable by radicals, since $S_n$ is not solvable for $n \\geq 5$. From: df-course Learn more:

Magic Internet Math

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Character The \\textbf{character} of a representation $\\rho: G \\to GL(V)$ is the function $\\chi: G \\to k$ defined by $\\chi(g) = \\text{tr}(\\rho(g))$. From: df-course Learn more:

Magic Internet Math

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📐 Radius of Convergence Every power series has a radius $R \\in [0, \\infty]$ such that it converges absolutely for $|x-a| < R$ and diverges for $|x-a| > R$. 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.

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Support Vector Classifier (Soft Margin) Allows some observations to be on wrong side of margin or hyperplane, controlled by tuning parameter $C$. Larger $C$ = wider margin, more violations allowed. From: Intro to Statistical Learning Learn more:

Introduction to Statistical Learning | Magic Internet Math

An interactive course based on James et al.

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📐 Chinese Remainder Theorem If $\\gcd(m, n) = 1$, the system $x \\equiv a \\pmod{m}$, $x \\equiv b \\pmod{n}$ has a unique solution modulo $mn$. Proof: Since $\\gcd(m,n) = 1$, write $mk + nt = 1$. Then $c = bmk + ant$ satisfies both congruences. Uniqueness: if $c, d$ both work, then $mn \\mid (c - d)$. From: intro-discrete Learn more:

Magic Internet Math

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📐 Polynomial GCD The GCD of polynomials can be computed using the Euclidean algorithm, yielding the unique monic divisor of highest degree. From: intro-discrete Learn more:

Magic Internet Math

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Sensitivity and Specificity Sensitivity = $\\frac{\\text{TP}}{\\text{TP} + \\text{FN}}$ (true positive rate). Specificity = $\\frac{\\text{TN}}{\\text{TN} + \\text{FP}}$ (true negative rate). From: Intro to Statistical Learning Learn more:

Introduction to Statistical Learning | Magic Internet Math

An interactive course based on James et al.

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

💡 Proposition V.2 If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Uniform Continuity $f$ is uniformly continuous on $S$ if $\\forall \\varepsilon > 0, \\exists \\delta > 0: |x-y| < \\delta \\Rightarrow |f(x) - f(y)| < \\varepsilon$ for all $x, y \\in S$. 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.

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

💡 Proposition I.28 If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

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

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

📖 Continuity at a Point $f$ is continuous at $c$ if $\\lim_{x \\to c} f(x) = f(c)$. 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.

Explore all courses:

Magic Internet Math

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

MagicInternetMath 4 months ago

💡 Proposition VII.33 Given as many numbers as we please, to find the least of those which have the same ratio with them. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

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

Explore all courses:

Magic Internet Math

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