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Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche", reprinted in (Hilbert 2013, p.730), and was popularized through George Gamow's 1947 book One Two Three... Infinity.

A whale fall occurs when the carcass of a whale has fallen onto the ocean floor at a depth greater than 1,000 m (3,300 ft), in the bathyal or abyssal zones. On the sea floor, these carcasses can create complex localized ecosystems that supply sustenance to deep-sea organisms for decades. This is unlike in shallower waters, where a whale carcass will be consumed by scavengers over a relatively short period of time. Whale falls were first observed in the late 1970s with the development of deep-sea robotic exploration. Since then, several natural and experimental whale falls have been monitored through the use of observations from submersibles and remotely operated underwater vehicles (ROVs) in order to understand patterns of ecological succession on the deep seafloor.

Deep sea whale falls are thought to be hotspots of adaptive radiation for specialized fauna. Organisms that have been observed at deep-sea whale fall sites include giant isopods, squat lobsters, bristleworms, prawns, shrimp, lobsters, hagfish, Osedax, crabs, sea cucumbers, and sleeper sharks. In the past three years whale fall sites have come under scrutiny, and new species have been discovered, including potential whale fall specialists. It has been postulated that whale falls generate biodiversity by providing evolutionary stepping stones for multiple lineages to move and adapt to new environmentally-challenging habitats. Researchers estimate that 690,000 carcasses/skeletons of the nine largest whale species are in one of the four stages of succession at any one time. This estimate implies an average spacing of 12 km (7.5 mi) and as little as 5 km (3.1 mi) along migration routes. They hypothesize that this distance is short enough to allow larvae to disperse/migrate from one to another.

Whale falls are able to occur in the deep open ocean due to cold temperatures and high hydrostatic pressures. In the coastal ocean, a higher incidence of predators as well as warmer waters hasten the decomposition of whale carcasses. Carcasses may also float due to decompositional gases, keeping the carcass at the surface. The bodies of most great whales (baleen and sperm whales) are slightly denser than the surrounding seawater, and only become positively buoyant when the lungs are filled with air. When the lungs deflate, the whale carcasses can reach the seafloor quickly and relatively intact due to a lack of significant whale fall scavengers in the water column. Once in the deep-sea, cold temperatures slow decomposition rates, and high hydrostatic pressures increase gas solubility, allowing whale falls to remain intact and sink to even greater depths.

A fossil natural nuclear fission reactor is a uranium deposit where self-sustaining nuclear chain reactions have occurred. This can be examined by analysis of isotope ratios. The conditions under which a natural nuclear reactor could exist had been predicted in 1956 by Paul Kazuo Kuroda. The phenomenon was discovered in 1972 in Oklo, Gabon by French physicist Francis Perrin under conditions very similar to what was predicted.

Oklo is the only known location for this in the world and consists of 16 sites at which self-sustaining nuclear fission reactions are thought to have taken place approximately 1.7 billion years ago, and ran for a few hundred thousand years, averaging probably less than 100 kW of thermal power during that time.

The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.

A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".

The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a volume, which happens to be different from the volume at the start.

Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.

It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.

In programming jargon, Yoda conditions (also called Yoda notation) is a programming style where the two parts of an expression are reversed from the typical order in a conditional statement. A Yoda condition places the constant portion of the expression on the left side of the conditional statement. The name for this programming style is derived from the Star Wars character named Yoda, who speaks English with a non-standard syntax.

Yoda conditions are part of the Symfony, and the WordPress coding standards.

The "PayPal Mafia" is a group of former PayPal employees and founders who have since founded and developed additional technology companies such as Tesla Motors, LinkedIn, Palantir Technologies, SpaceX, YouTube, Yelp, and Yammer. Most of the members attended Stanford University or University of Illinois at Urbana–Champaign at some point in their studies.

Eigengrau (German: "intrinsic gray", lit. "own gray"; pronounced [ˈʔaɪ̯gn̩ˌgʁaʊ̯]), also called Eigenlicht (Dutch and German: "own light"), dark light, or brain gray, is the uniform dark gray background that many people report seeing in the absence of light. The term Eigenlicht dates back to the nineteenth century, but has rarely been used in recent scientific publications. Common scientific terms for the phenomenon include "visual noise" or "background adaptation". These terms arise due to the perception of an ever-changing field of tiny black and white dots seen in the phenomenon.

Eigengrau is perceived as lighter than a black object in normal lighting conditions, because contrast is more important to the visual system than absolute brightness. For example, the night sky looks darker than Eigengrau because of the contrast provided by the stars.

The Buddhist games list is a list of games that Gautama Buddha is reputed to have said that he would not play and that his disciples should likewise not play, because he believed them to be a 'cause for negligence'. This list dates from the 6th or 5th century BCE and is the earliest known list of games.

There is some debate about the translation of some of the games mentioned, and the list given here is based on the translation by T. W. Rhys Davids of the Brahmajāla Sutta and is in the same order given in the original. The list is duplicated in a number of other early Buddhist texts, including the Vinaya Pitaka.

1. Games on boards with 8 or 10 rows. This is thought to refer to ashtapada and dasapada respectively, but later Sinhala commentaries refer to these boards also being used with games involving dice.
2. The same games played on imaginary boards. Akasam astapadam was an ashtapada variant played with no board, literally "astapadam played in the sky". A correspondent in the American Chess Bulletin identifies this as likely the earliest literary mention of a blindfold chess variant.
3. Games of marking diagrams on the floor such that the player can only walk on certain places. This is described in the Vinaya Pitaka as "having drawn a circle with various lines on the ground, there they play avoiding the line to be avoided". Rhys Davids suggests that it may refer to parihāra-patham, a form of hop-scotch.
4. Games where players either remove pieces from a pile or add pieces to it, with the loser being the one who causes the heap to shake (similar to the modern game pick-up sticks).
5. Games of throwing dice.
6. "Dipping the hand with the fingers stretched out in lac, or red dye, or flour-water, and striking the wet hand on the ground or on a wall, calling out 'What shall it be?' and showing the form required—elephants, horses, &c."
7. Ball games.
8. Blowing through a pat-kulal, a toy pipe made of leaves.
9. Ploughing with a toy plough.
10. Playing with toy windmills made from palm leaves.
11. Playing with toy measures made from palm leaves.
12. Playing with toy carts.
13. Playing with toy bows.
14. Guessing at letters traced with the finger in the air or on a friend's back.
15. Guessing a friend's thoughts.
16. Imitating deformities.

Although the modern game of chess had not been invented at the time the list was made, earlier chess-like games such as chaturaji may have existed. H.J.R. Murray refers to Rhys Davids' 1899 translation, noting that the 8×8 board game is most likely ashtapada while the 10×10 game is dasapada. He states that both are race games.