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Concepts38

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📐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

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AllBeginnerIntermediate
∑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#euclidean algorithm+12
∑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
1234
Advanced
Filtering by:
#competitive programming
#integer division trick
#block decomposition
+12
∑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

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
∑MathIntermediate

Bayes' Theorem

Bayes' Theorem tells you how to update the probability of a hypothesis after seeing new evidence.

#bayes' theorem#posterior probability#prior probability+11
∑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
∑MathIntermediate

Lucas' Theorem

Lucas' Theorem lets you compute C(n, k) modulo a prime p by working digit-by-digit in base p.

#lucas theorem#binomial coefficient modulo p#prime power modulus+12
∑MathIntermediate

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle (IEP) corrects overcounting by alternately adding and subtracting sizes of intersections of sets.

#inclusion-exclusion#derangements#surjections+12