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Second-order Differential Equation

0 comments 学习笔记 Math
Second-order Linear Differential Homogeneous EquationsConsider operator$$ \gdef\mydef{\gdef}\mydef\s#1{\left(#1\right)}\mydef\m#1{\left[#1\right]}\mydef\b#1{\left\{#1\right\}}\mydef\abs#1{\left\lve...

Discrete Fourier Transform and Fast Fourier Transform

0 comments 学习笔记 Polynomial FFT
ConvolutionConsider two polynomials $\gdef\mydef{\gdef}\mydef\s#1{\left(#1\right)}\mydef\m#1{\left[#1\right]}\mydef\b#1{\left\{#1\right\}}\mydef\abs#1{\left\lvert#1\right\rvert} F\s x = \sum_kf_kx^...

Riemann Integral

0 comments 学习笔记 Math
When I was taught integration, my teacher told me that when we divide the area into strips with sufficiently small widths, the error will decrease to zero. However, there was still some confusion i...

Schröder Bernstein Theorem

0 comments 学习笔记 Math Set Theory
StatementIf there exists an injective function $f: \mathcal S \to \mathcal T$ and an injective function $g: \mathcal T \to \mathcal S$, then there exists a bijective function $h: \mathcal S \to \ma...

IMO 2010 P5

0 comments 做题笔记 Math
StatementEach of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowedType 1) Choose a non-empty box $B_j$, $1 \leq j \leq 5$ remo...

Dantzig's Simplex Method

1 comments 学习笔记 Matrix Math Linear Programming
OverviewDantzig's simplex method is used to solve the following linear programming problem.Minimize $z = \mathbf c^{\mathrm T} \mathbf x$Subject to $A\mathbf x \leq \mathbf b$ and $\mathbf x \geq 0...

CF1519F Chests and Keys

0 comments 做题笔记 Dynamic Programming Flows
Problem StatementAlice and Bob play a game. Alice has got $n$ treasure chests (the $i$-th of which contains $a_i$ coins) and $m$ keys (the $j$-th of which she can sell Bob for $b_j$ coins).Firstly,...

CF2147E Maximum OR Popcount

0 comments 做题笔记 Greedy
Problem StatementYou are given an array of $n$ non-negative integers.You want to answer given $q$ independent scenarios. In the $i$-th scenario, you are allowed to perform the following operation a...

Boids Algorithm

0 comments 杂事 No tags
OverviewLet us watch a demo first. Go to https://boids.rbtr.ee/ to see further.As you can see, the Boids algorithm is a method to simulate the movements of social creatures.AlgorithmThe Boids algo...

CEOI 2023 Grading Server

0 comments 做题笔记 Dynamic Programming
Grading ServerDespite running low on oil, the opening ceremony was a great success. Now, the committee is looking forward to the first competition day. But what’s this? The chairman of the Technica...