The Assignment Problem
Given $n$ workers and $n$ jobs, and the cost $c_{ij}$ to train the $i$-th worker for the $j$-th job, find an assignment of one worker to each job which minimized the total training cost. ...
Given $n$ workers and $n$ jobs, and the cost $c_{ij}$ to train the $i$-th worker for the $j$-th job, find an assignment of one worker to each job which minimized the total training cost. ...
Find the smallest integer $n$ that there must be either three mutual friends or three mutual strangers among $n$ people, suppose each pair of people are either friends or strangers? ...
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. ...
Various ways of distributing balls into cells. ...
It is extremely difficult, and often impossible, to prevent errors when data are stored, retrieved, operated on, or transmitted. ...
Any mathematical system containing all the theorems of arithmetic is an incomplete system. ...
问题: 证明 $1+\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+\cdots=\dfrac{\pi^2}{6}$ 这一问题实际上有多种代数方法证明,但由于公式中出现了 $\pi$,则必定在几何学中可以找到圆来对应。 想象一位观察者站在数轴的原点,并在所有的正整数的位置放置一座灯塔。假设观察者接收到的第一座灯塔的亮度为 $1$。由于亮度和距离的平方成反比,则第 $n$ 座灯塔的亮度为 $\dfrac{1}{n^2}$。这样我们就将原问题转化为了物理的表达形式,我们只需要证明原点所接收到的总亮度为 $\dfrac{\pi^2}{6}$ 即可。 ...
问题: 在圆上任取 $n$ 个点,将每对点用直线连接,并规定三条线不能交于一点,这些直线会将圆分割成多少份? 首先我们列出一些简单情况来寻找规律: $2$ 个点将圆分为 $2$ 份 $3$ 个点将圆分为 $4$ 份 $4$ 个点将圆分为 $8$ 份 $5$ 个点将圆分为 $16$ 份 Simple Cases ...
想象你的麦克风在录制一段由四个纯音同时播放的音频,由于其只能捕捉气压——时间图像,因此最后的结果看起来相当复杂: Pressure-Time 如果给定这样一段音频,该如何将其分解为不同纯音的叠加? ...
问题: 在球面上随机选择四点组成四面体,问球心落在该四面体内部的概率? 首先将问题简化,考虑二维情形:在圆上随机选择三点 $P_1, P_2, P_3$ 组成三角形,求圆心落在该三角形内部的概率。 ...