FFmpeg is a powerful open-source command-line tool for processing video and audio files. It can be used to perform a wide range of tasks such as converting video and audio files from one format to another, extracting audio from a video file, splitting a video into smaller parts, and much more. In this guide, we’ll cover the basics of using FFmpeg for common video and audio processing tasks. Installation Before we can start using FFmpeg, we need to install it.

Goldbach’s Conjecture, first stated by German mathematician Christian Goldbach in a letter to Leibniz in 1742, is one of the oldest unsolved problems in mathematics. The conjecture states that: Every even integer greater than 2 can be expressed as the sum of two prime numbers. $$ \forall 2m \in \mathbb{N}, \exists p, q \in \mathbb{N} \text{ such that } p + q = 2m $$ Despite being over three centuries old, the conjecture remains unsolved to this day.

The Collatz Conjecture, also known as the $3n + 1$ conjecture, is a mathematical problem that has puzzled mathematicians for decades. It was first proposed by the German mathematician Lothar Collatz in 1937. The conjecture states that for any positive integer $n$, the sequence generated by repeatedly applying the function: will always reach 1 regardless of the value of $n$. Another way to think about it is that starting with 1, all positive numbers will be reachable by repeatedly applying the function in reverse:

Julia sets are a type of fractal that are closely related to the Mandelbrot set . Both sets are generated using complex numbers and iterating over a mathematical function. The Mandelbrot set is defined as the set of complex numbers c for which the function $f(z) = z^2 + c$, with z starting at 0, does not diverge when iterated. In other words, if the absolute value of f(z) remains less than or equal to 2 for all iterations, then c is considered to be a member of the Mandelbrot set.

The Mandelbrot set is a famous fractal named after the mathematician Benoit Mandelbrot, who introduced it in a paper in 1980. It is known for its intricate and detailed structure, which is made up of an infinite number of smaller copies of itself. The set has many interesting properties, such as self-similarity and non-differentiability. It is defined by the complex equation $z(n+1) = z(n)^2 + c$, where $z(0) = 0$ and $c$ is a complex number in the form of $a + bi$, where $a$ and $b$ are real numbers.