Fourier Series

Mathematicians of the eighteenth century, including Daniel Bernoulli and Leonhard Euler, expressed the problem of the vibratory motion of a stretched string through partial differential equations that had no solutions in terms of “elementary functions.” Their resolution of this difficulty was to introduce infinite series of sine and cosine functions that satisfied the equations. In the early nineteenth century, Joseph Fourier, while studying the problem of heat flow, developed a cohesive theory of such series. Consequently, they were named after him.

What Is a Torrent?

A torrent file stores data about files for use with BitTorrent, enables users to send and receive files in small pieces to and from computers across the Internet as of January 2012. Because small pieces of the files are downloaded and then reassembled by the software, BitTorrent uses less resources than direct downloading of files, resulting in faster speeds for the downloader and lower bandwidth costs for the file provider. 

At any given instant of time BitTorrent has, on average, more active users than YouTube and Facebook combined.

 

 In this animation, the colored bars beneath all of the 7 clients in the animation above represent the file, with each color representing an individual piece of the file. After the initial pieces transfer from the seed (large system at the bottom), the pieces are individually transferred from client to client. The original seeder only needs to send out one copy of the file for all the clients to receive a copy.