## Impagliazzo’s Hardcore Lemma and Computational Hardness

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 …

## Searching and Replacing with grep and sed

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 …

## Turn Off Voice Assistant on Apple Mac

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 …

## A Double Sum of Binomials

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 $${i+j \choose j} = {i+j-1 \choose j} + {i+j-1 \choose j-1}$$, …