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Concepts158

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

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

Filtering by:

#competitive programming

#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

∑MathIntermediate

Multiplicative Functions

A multiplicative function is an arithmetic function f with f(mn) = f(m)f(n) whenever gcd(m, n) = 1.

#multiplicative function#dirichlet convolution#mobius function+12

∑MathIntermediate

Euler's Totient Function

Euler's Totient Function φ(n) counts how many integers from 1 to n are coprime with n.

#euler totient#phi function#coprime count+12

∑MathIntermediate

Modular Arithmetic Basics

Modular arithmetic is arithmetic with wrap-around at a fixed modulus m, like numbers on a clock.

#modular arithmetic#mod#modulo c+++12

∑MathIntermediate

Modular Inverse

A modular inverse of a modulo m is a number a_inv such that a × a_inv ≡ 1 (mod m).

#modular inverse#extended euclidean algorithm#fermats little theorem+12

∑MathIntermediate

Euler's Theorem

Euler’s Theorem says that if a and n are coprime, then a raised to the power φ(n) is congruent to 1 modulo n.

#euler totient#euler theorem#modular exponentiation+12

∑MathIntermediate

Fermat's Little Theorem

Fermat's Little Theorem says that for a prime p and integer a not divisible by p, a^{p-1} ≡ 1 (mod p).

#fermat's little theorem#modular inverse#binary exponentiation+11

∑MathIntermediate

Fast Exponentiation

Fast exponentiation (binary exponentiation) computes a^n using repeated squaring in O(log n) multiplications.

#binary exponentiation#fast power#modular exponentiation+11

∑MathIntermediate

Chinese Remainder Theorem

The Chinese Remainder Theorem (CRT) reconstructs an integer from its remainders modulo pairwise coprime moduli and guarantees a unique answer modulo the product.

#chinese remainder theorem#crt#modular arithmetic+12

⚙️AlgorithmAdvanced

Sqrt Decomposition on Queries

Sqrt decomposition on queries (time blocking) processes Q operations in blocks of size about \(\sqrt{Q}\) to balance per-query overhead and rebuild cost.

#sqrt decomposition#time blocking#query blocking+12