Theorem Let $(a_n)$ be a sequence of real numbers. If $(a_n)$ is nondecreasing and bounded above, then it converges to $\sup{a_n:n\in\mathbb{N}}$. If $(a_n)$ is nonincreasing and bounded bel...

Theorem for sequences Let $(x_n)$ be a bounded sequence in the finite-dimensional Euclidean space $\mathbb{R}^d$, where $d\geq 1$. Then $(x_n)$ has a convergent subsequence. The limit lies in $\ma...

Definitions 1. Rings, fields, and distributive laws A ring is a set $R$ equipped with two binary operations, addition $+$ and multiplication $\cdot$, such that: $(R,+)$ is an abelian group. ...

This article collects basic definitions used in set theory, algebra, and analysis. 1. Set A set is a well-defined collection of distinct objects, called its elements. A statement such as $x \in X...

Bézout’s identity for integers Let $a,b\in\mathbb Z$ be not both zero, and let [g=\gcd( a , b )>0.] Then there are integers $x,y$ such that [a...

Flow networks and feasible flows A flow network is a finite directed graph $G=(V,E)$ with a distinguished source $s$ and sink $t$, where $s\ne t$. Each edge $e$ has a finite, nonnegative capacity ...

The theorem Let $G=(V,E)$ be a finite directed network with distinct source $s$ and sink $t$. Each directed edge $e$ has a finite, nonnegative capacity $c_e$. A feasible flow assigns a value $f_e$...

https://www.acmicpc.net/problem/1006 Problem model There are two rows of enemy units arranged in a circle of N columns. One squad covers either one cell, or two adjacent cells whose enemy counts ...

This repository separates validation from publication. A feature-branch build proves the site is valid; only a production workflow publishes generated files to gh-pages. What triggers each workflo...

What the transform computes For a coefficient vector $a_0,\dots,a_{n-1}$, define the discrete Fourier transform (DFT) by evaluating its polynomial at the $n$th roots of unity: [X_k = \sum_{j=0}^{...