We wish to compute boolean functions by circuits, or more generally, compositions of simple boolean functions (gates). Suppose we are given a function which is hard to approximate by any small-size circuit. That is, any small-size circuit fails to compute…
Unix commands grep and sed can be used to efficiently search and replace strings in a file, a directory, or files acrose multiple directories. Specifically, grep is used to search for a certain string, and sed is used to replace…
VoiceOver, as a voice assistant, provides spoken description of items on the screen. The following are two ways of turning on/off VoiceOver. Toggle VoiceOver Use ⌘ Cmd + F5 to toggle on/off VoiceOver. Enable/disable VoiceOver from the menu Open System…
Prove the following summation of binomial coefficients. \[ \sum_{i=0}^n \sum_{j=0}^m {i+j \choose j} = {n + m +2 \choose n+1} - 1. \] First, by Pascal’s rule that \({i+j \choose j} = {i+j-1 \choose j} + {i+j-1 \choose j-1} \),…