Siegel's theorem on integral points is a significant result in number theory, particularly in the study of Diophantine equations and the distribution of rational and integral solutions to these equations. The theorem essentially states that for a certain class of algebraic varieties, known as "affine" or "projective" varieties of general type, there are only finitely many integral (or rational) points on these varieties.

Awesome tool to view quick stuff quickly without generating files. Unfortunately it doesn't support all options that the ffmpeg CLI supports, e.g. ffplay multiple input files. One day, one day.

The Sum of Four Cubes Problem refers to the mathematical question of whether every integer can be expressed as the sum of four integer cubes.

The "sums of three cubes" problem refers to the mathematical challenge of expressing certain integers as the sum of three integer cubes. Specifically, the equation can be stated as: \[ n = x^3 + y^3 + z^3 \] where \( n \) is the integer we want to express, and \( x \), \( y \), and \( z \) are also integers.

"The Monkey and the Coconuts" is a traditional folk tale that often appears in various cultures, with different versions and details. The story typically involves a group of monkeys and a supply of coconuts that they find. The narrative usually revolves around themes such as intelligence, teamwork, problem-solving, and sometimes morality. In one common version of the tale, a group of monkeys discovers a coconut tree and figures out how to gather the coconuts.

Tijdeman's theorem is a result in number theory concerning the equation \( x^k - y^m = 1 \), where \( x \), \( y \) are positive integers, and \( k \), \( m \) are integers greater than or equal to 2. The theorem states that the only solutions in positive integers \( (x, y, k, m) \) to this equation occur for certain specific values.

The AMNH Exhibitions Lab, part of the American Museum of Natural History (AMNH) in New York City, is an innovative space dedicated to the design, development, and testing of new museum exhibitions. It serves as a collaborative environment where curators, educators, designers, and other professionals can come together to explore and create engaging and educational exhibits that align with the museum's mission to inspire understanding of the natural world and the universe.

Continuous version of the Fourier series.

Can be used to represent functions that are not periodic: math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the Fourier series is only for periodic functions.

Of course, every function defined on a finite line segment (i.e. a compact space).

Therefore, the Fourier transform can be seen as a generalization of the Fourier series that can also decompose functions defined on the entire real line.

As a more concrete example, just like the Fourier series is how you solve the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.

The Holyland Model of Jerusalem is a highly detailed scale model of the city of Jerusalem, representing its landscape and architecture at a specific point in history. Typically, the model depicts Jerusalem as it was during the Second Temple period, around 66 AD. This period is significant in Jewish history, as it was during this time that the Second Temple stood before its destruction by the Romans in 70 AD.

Illés Relief is a type of relief sculpture characterized by its intricate details and craftsmanship. It refers to a specific artwork that depicts the biblical prophet Elijah (Illés in Hungarian) in a dramatic context, including scenes from his life and miracles. The relief captures the essence of the narrative and emotions associated with the prophet, often showcasing his encounters with nature and divine intervention.

What happens when the underdogs get together and try to factor out their efforts to beat some evil dominant power, sometimes victoriously.

Or when startups use the cheapest stuff available and randomly become the next big thing, and decide to keep maintaining the open stuff to get features for free from other companies, or because they are forced by the Holy GPL.

Open source frees employees. When you change jobs, a large part of the specific knowledge you acquired about closed source a project with your blood and tears goes to the trash. When companies get bought, projects get shut down, and closed source code goes to the trash. What sane non desperate person would sell their life energy into such closed source projects that could die at any moment? Working on open source is the single most important non money perk a company can have to attract the best employees.

Open source is worth more than the mere pragmatic financial value of not having to pay for software or the ability to freely add new features.

Its greatest value is perhaps the fact that it allows people study it, to appreciate the beauty of the code, and feel empowered by being able to add the features that they want.

That is why Ciro Santilli thought:

But quoting Ciro's colleague S.:

And "can reverse engineer the undocumented GPU hardware APIs", Ciro would add.

While software is the most developed open source technology available in the 2010's, due to the "zero cost" of copying it over the Internet, Ciro also believes that the world would benefit enormously from open source knowledge in all areas on science and engineering, for the same reasons as open source.

"Lion Attacking a Dromedary" refers to a famous painting by the French artist Antoine-Louis Barye, created in the 19th century. Barye was known for his animal sculptures and paintings, and this particular work depicts the dramatic moment of a lion attacking a dromedary (a one-humped camel). The painting is noted for its dynamic composition and the vivid depiction of the struggle between the powerful predator and its prey.

"Man in the Mud" typically refers to a concept or metaphor that illustrates human struggle, resilience, or the complexities of life. It could represent individuals who find themselves in difficult circumstances or "stuck" situations, much like being trapped in mud.

The Museum of the Gorge is a local museum located in Ironbridge, Shropshire, England. It is part of the Ironbridge Gorge World Heritage Site, which is known for its historical significance in the development of the iron and coal industries during the Industrial Revolution. The museum is dedicated to showcasing the history and heritage of the Ironbridge Gorge area, particularly its industrial past.

A Nativity scene, also known as a creche or mangers, is a depiction of the birth of Jesus Christ, primarily used as a visual representation of the Nativity story from the Christian tradition. These scenes typically include figures representing key characters such as: - **Baby Jesus:** Often depicted in a manger or cradle. - **Mary:** The mother of Jesus, usually shown near her child. - **Joseph:** Mary's husband, often portrayed standing protectively by them.

The "Nutshell Studies of Unexplained Death" is a collection of dioramas created by Frances Glessner Lee in the 1940s. Glessner Lee was a pioneer in forensic science who aimed to improve the training of homicide investigators. The dioramas are incredibly detailed miniature scenes that depict various murder mysteries and unexplained deaths. Each diorama is designed to present a different set of circumstances surrounding a fictional death, complete with realistic props and meticulous attention to detail.

The Coriolis-Stokes force refers to the combined effects of the Coriolis force and the Stokes drag force in fluid dynamics. This force is particularly relevant in the study of geophysical flows, such as ocean currents and atmospheric movements, where both Earth's rotation and viscous forces play significant roles. 1. **Coriolis Force**: This is an inertial force described mathematically by the Coriolis effect, which arises from the rotation of the Earth.

Stronger version of the big O notation, basically means that ratio goes to zero. In big O notation, the ratio does not need to go to zero.

So in informal terms, big O notation means $\le$, and little-o notation means $<$.

E.g.: