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Concepts39

Groups

📐Linear Algebra15 📈Calculus & Differentiation10 🎯Optimization14 🎲Probability Theory12 📊Statistics for ML9 📡Information Theory10 🔺Convex Optimization7 🔢Numerical Methods6 🕸Graph Theory for Deep Learning6 🔵Topology for ML5 🌐Differential Geometry6 ∞Measure Theory & Functional Analysis6 🎰Random Matrix Theory5 🌊Fourier Analysis & Signal Processing9 🎰Sampling & Monte Carlo Methods10 🧠Deep Learning Theory12 🛡️Regularization Theory11 👁️Attention & Transformer Theory10 🎨Generative Model Theory11 🔮Representation Learning10 🎮Reinforcement Learning Mathematics9 🔄Variational Methods8 📉Loss Functions & Objectives10 ⏱️Sequence & Temporal Models8 💎Geometric Deep Learning8

Category

🔷All ∑Math ⚙️Algo 🗂️DS 📚Theory

Level

All Beginner Intermediate

∑MathAdvanced

Burnside's Lemma

Burnside's Lemma says the number of distinct objects up to a symmetry group equals the average number of objects fixed by each symmetry.

#burnside's lemma#cauchy-frobenius#polya enumeration+12

∑MathAdvanced

Partition Function

The partition function p(n) counts the number of ways to write n as a sum of positive integers where order does not matter.

#partition function

#integer partitions

#euler pentagonal theorem

+11

∑MathAdvanced

Generating Functions - OGF

An ordinary generating function (OGF) encodes a sequence (a_n) as a formal power series A(x) = \sum_{n \ge 0} a_n x^n.

#ordinary generating function#ogf#coefficient extraction+12

∑MathAdvanced

Stirling Numbers of First Kind

Stirling numbers of the first kind count permutations by their number of cycles and connect power polynomials to rising/falling factorials.

#stirling numbers of the first kind#unsigned cycle numbers#signed stirling numbers+12

∑MathAdvanced

Stirling Numbers of Second Kind

Stirling numbers of the second kind S(n,k) count how many ways to split n labeled items into k non-empty, unlabeled groups.

#stirling numbers of the second kind#set partitions#bell numbers+12

∑MathAdvanced

Primitive Roots

A primitive root modulo n is a number g that cycles through all units modulo n when you repeatedly multiply by g, so its multiplicative order equals \(\varphi(n)\).

#primitive root#multiplicative order#euler totient+10

∑MathAdvanced

Discrete Logarithm

The discrete logarithm problem asks for x such that g^x ≡ h (mod p) in a multiplicative group modulo a prime p.

#discrete logarithm#baby-step giant-step#pollard rho dlp+12

∑MathAdvanced

Pollard's Rho Factorization

Pollard's Rho is a randomized algorithm that finds a non-trivial factor of a composite integer by walking a pseudorandom sequence modulo n and extracting a factor with a gcd.

#pollard's rho#integer factorization#cycle detection+10

∑MathAdvanced

Quadratic Residues

A quadratic residue modulo an odd prime p is any a for which x^2 ≡ a (mod p) has a solution; exactly half of the nonzero classes are residues.

#quadratic residues#legendre symbol#euler criterion+12

∑MathAdvanced

Möbius Function and Inversion

The Möbius function μ(n) is 0 if n has a squared prime factor, otherwise it is (-1)^k where k is the number of distinct prime factors.

#mobius function#mobius inversion#dirichlet convolution+12

∑MathAdvanced

Divisor Function Sums

Summing the divisor function d(i) up to n equals counting lattice points under the hyperbola xy ≤ n, which can be done in O(√n) using floor-division blocks.

#divisor function#euler totient#mobius function+11

∑MathAdvanced

Game Theory - Advanced Games

Sprague–Grundy (SG) theory solves impartial, normal-play, terminating games by assigning each position a nonnegative integer called its Grundy value.

#sprague-grundy#grundy number#nim-sum+12