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Concepts140

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

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🔷All∑Math⚙️Algo🗂️DS📚Theory

Level

AllBeginnerIntermediateAdvanced
∑MathIntermediate

Vectors & Vector Spaces

A vector is an element you can add and scale, and a vector space is any collection of such elements closed under these operations.

#vector space#basis#span+12
∑MathAdvanced

Floor Sum Formula

The floor sum computes S(n,m,a,b) = sum_{i=0}^{n-1} floor((a i + b)/m) efficiently in O(log(min(a,m))) time.

#floor sum#atcoder library
678910
#euclidean algorithm
+12
∑MathIntermediate

Legendre's Formula

Legendre's formula gives the exponent of a prime p in n! by summing how many multiples of p, p^2, p^3, ... are ≤ n.

#legendre's formula#p-adic valuation#binomial divisibility+10
∑MathIntermediate

Harmonic Lemma

The Harmonic Lemma says that the values of \lfloor n/i \rfloor only change about 2\sqrt{n} times, so you can iterate those value blocks in O(\sqrt{n}) instead of O(n).

#harmonic lemma#integer division trick#block decomposition+12
∑MathIntermediate

Sprague-Grundy Theorem

Sprague–Grundy theory turns every finite impartial game (normal play) into an equivalent Nim heap with a size called the Grundy number.

#sprague-grundy#grundy number#mex+11
∑MathIntermediate

Game Theory - Nim

Nim is a two-player impartial game with several piles where a move removes any positive number of stones from exactly one pile.

#nim#game theory#xor+11
∑MathIntermediate

Game Theory - Calculation Techniques

Sprague–Grundy theory converts any impartial, normal-play game into an equivalent Nim heap using a Grundy number.

#sprague-grundy#grundy numbers#nim-sum+12
∑MathIntermediate

Variance and Covariance

Variance measures how spread out a random variable is around its mean, while covariance measures how two variables move together.

#variance#covariance#standard deviation+12
∑MathAdvanced

Generating Functions - EGF

Exponential generating functions (EGFs) encode a sequence (a_n) as A(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!}, which naturally models labeled combinatorial objects.

#exponential generating function#egf#binomial convolution+11
∑MathIntermediate

Linearity of Expectation Applications

Linearity of expectation says the expected value of a sum equals the sum of expected values, even if the variables are dependent.

#linearity of expectation#indicator variables#expected inversions+12
∑MathIntermediate

Expected Value

Expected value is the long-run average outcome of a random variable if you could repeat the experiment many times.

#expected value#linearity of expectation#indicator variables+12
∑MathAdvanced

Pólya Enumeration

Pólya Enumeration Theorem generalizes Burnside’s Lemma by turning counting under symmetry into a polynomial substitution problem.

#pólya enumeration#cycle index#burnside lemma+12