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🔗 Mathematics

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.

🔗 Astronomy 🔗 Astronomy/Astronomical objects

Wolf 359 is a red dwarf star located in the constellation Leo, near the ecliptic. At a distance of approximately 7.9 light years from Earth, it has an apparent magnitude of 13.54 and can only be seen with a large telescope. Wolf 359 is one of the nearest stars to the Sun; only the Alpha Centauri system (including Proxima Centauri), Barnard's Star and the brown dwarfs Luhman 16 and WISE 0855−0714 are known to be closer. Its proximity to Earth has led to its mention in several works of fiction.

Wolf 359 is one of the faintest and lowest-mass stars known. At the light-emitting layer called the photosphere, it has a temperature of about 2,800 K, which is low enough for chemical compounds to form and survive. The absorption lines of compounds such as water and titanium(II) oxide have been observed in the spectrum. The surface has a magnetic field that is stronger than the average magnetic field on the Sun. As a result of magnetic activity caused by convection, Wolf 359 is a flare star that can undergo sudden increases in luminosity for several minutes. These flares emit strong bursts of X-ray and gamma ray radiation that have been observed by space telescopes. Wolf 359 is a relatively young star with an age of less than a billion years. Two planetary companions are suspected but as yet no debris disks have been unmasked.

🔗 Economics 🔗 Lists

This is a list of some of the major unsolved problems, puzzles, or questions in economics. Some of these are theoretical in origin and some of them concern the inability of orthodox economic theory to explain an empirical observation.

🔗 Linguistics 🔗 Linguistics/Applied Linguistics 🔗 Languages

The word mamihlapinatapai is derived from the Yaghan language of Tierra del Fuego, listed in The Guinness Book of World Records as the "most succinct word", and is considered one of the hardest words to translate. It has been translated as "a look that without words is shared by two people who want to initiate something, but that neither will start" or "looking at each other hoping that the other will offer to do something which both parties desire but are unwilling to do".

A romantic interpretation of the meaning has also been given, as "that look across the table when two people are sharing an unspoken but private moment. When each knows the other understands and is in agreement with what is being expressed. An expressive and meaningful silence."

🔗 Heraldry and vexillology 🔗 Visual arts

In Western art history, Mise en abyme (French pronunciation: ​[miz ɑ̃n‿abim]; also mise en abîme) is a formal technique of placing a copy of an image within itself, often in a way that suggests an infinitely recurring sequence. In film theory and literary theory, it refers to the technique of inserting a story within a story. The term is derived from heraldry and literally means "placed into abyss". It was first appropriated for modern criticism by the French author André Gide.

A common sense of the phrase is the visual experience of standing between two mirrors, seeing as a result an infinite reproduction of one's image. Another is the Droste effect, in which a picture appears within itself, in a place where a similar picture would realistically be expected to appear. That is named after the 1904 Droste cocoa package, which depicts a woman holding a tray bearing a Droste cocoa package, which bears a smaller version of her image.

Avram Noam Chomsky (born December 7, 1928) is an American professor and public intellectual known for his work in linguistics, political activism, and social criticism. Sometimes called "the father of modern linguistics", Chomsky is also a major figure in analytic philosophy and one of the founders of the field of cognitive science. He is a laureate professor of linguistics at the University of Arizona and an institute professor emeritus at the Massachusetts Institute of Technology (MIT). Among the most cited living authors, Chomsky has written more than 150 books on topics such as linguistics, war, and politics. Ideologically, he aligns with anarcho-syndicalism and libertarian socialism.

Born to Ashkenazi Jewish immigrants in Philadelphia, Chomsky developed an early interest in anarchism from alternative bookstores in New York City. He studied at the University of Pennsylvania. During his postgraduate work in the Harvard Society of Fellows, Chomsky developed the theory of transformational grammar for which he earned his doctorate in 1955. That year he began teaching at MIT, and in 1957 emerged as a significant figure in linguistics with his landmark work Syntactic Structures, which played a major role in remodeling the study of language. From 1958 to 1959 Chomsky was a National Science Foundation fellow at the Institute for Advanced Study. He created or co-created the universal grammar theory, the generative grammar theory, the Chomsky hierarchy, and the minimalist program. Chomsky also played a pivotal role in the decline of linguistic behaviorism, and was particularly critical of the work of B. F. Skinner.

An outspoken opponent of U.S. involvement in the Vietnam War, which he saw as an act of American imperialism, in 1967 Chomsky rose to national attention for his anti-war essay "The Responsibility of Intellectuals". Becoming associated with the New Left, he was arrested multiple times for his activism and placed on President Richard Nixon's list of political opponents. While expanding his work in linguistics over subsequent decades, he also became involved in the linguistics wars. In collaboration with Edward S. Herman, Chomsky later articulated the propaganda model of media criticism in Manufacturing Consent, and worked to expose the Indonesian occupation of East Timor. His defense of unconditional freedom of speech, including that of Holocaust denial, generated significant controversy in the Faurisson affair of the 1980s. Chomsky's commentary on the Cambodian genocide and the Bosnian genocide also generated controversy. Since retiring from active teaching at MIT, he has continued his vocal political activism, including opposing the 2003 invasion of Iraq and supporting the Occupy movement. An anti-Zionist, Chomsky considers Israel's treatment of Palestinians to be worse than South African-style apartheid, and criticizes U.S. support for Israel.

Chomsky is widely recognized as having helped to spark the cognitive revolution in the human sciences, contributing to the development of a new cognitivistic framework for the study of language and the mind. Chomsky remains a leading critic of U.S. foreign policy, contemporary capitalism, U.S. involvement and Israel's role in the Israeli–Palestinian conflict, and mass media. Chomsky and his ideas are highly influential in the anti-capitalist and anti-imperialist movements. Since 2017, he has been Agnese Helms Haury Chair in the Agnese Nelms Haury Program in Environment and Social Justice at the University of Arizona.

🔗 Military history 🔗 Military history/North American military history 🔗 Military history/United States military history 🔗 Military history/Military science, technology, and theory 🔗 Physics 🔗 Military history/World War II 🔗 Physics/History

The Einstein–Szilárd letter was a letter written by Leó Szilárd and signed by Albert Einstein that was sent to the United States President Franklin D. Roosevelt on August 2, 1939. Written by Szilárd in consultation with fellow Hungarian physicists Edward Teller and Eugene Wigner, the letter warned that Germany might develop atomic bombs and suggested that the United States should start its own nuclear program. It prompted action by Roosevelt, which eventually resulted in the Manhattan Project developing the first atomic bombs.

🔗 Biography 🔗 Oregon 🔗 Perl

Randal L. Schwartz (born November 22, 1961), also known as merlyn, is an American author, system administrator and programming consultant.

He is known for his expertise in the Perl programming language, his promotional role within the Perl community, as a co-host of FLOSS Weekly, and for a controversial felony conviction resulting from State of Oregon vs. Randal Schwartz, later officially expunged.

🔗 Medicine 🔗 Psychology 🔗 Disability

Clanging (or clang associations) is a symptom of mental disorders, primarily found in patients with schizophrenia and bipolar disorder. This symptom is also referred to as association chaining, and sometimes, glossomania.

Steuber defines it as "repeating chains of words that are associated semantically or phonetically with no relevant context". This may include compulsive rhyming or alliteration without apparent logical connection between words.

Clanging refers specifically to behavior that is situationally inappropriate. While a poet rhyming is not evidence of mental illness, disorganized speech that impedes the patient's ability to communicate is a disorder in itself, often seen in schizophrenia.

🔗 Military history 🔗 Africa 🔗 United Kingdom 🔗 Military history/African military history 🔗 Africa/Tanzania 🔗 British Empire 🔗 Military history/European military history 🔗 Military history/British military history

The Anglo-Zanzibar War was a military conflict fought between the United Kingdom and the Zanzibar Sultanate on 27 August 1896. The conflict lasted between 38 and 45 minutes, marking it as the shortest recorded war in history. The immediate cause of the war was the death of the pro-British Sultan Hamad bin Thuwaini on 25 August 1896 and the subsequent succession of Sultan Khalid bin Barghash. The British authorities preferred Hamud bin Muhammed, who was more favourable to British interests, as sultan. In accordance with a treaty signed in 1886, a condition for accession to the sultanate was that the candidate obtain the permission of the British consul, and Khalid had not fulfilled this requirement. The British considered this a casus belli and sent an ultimatum to Khalid demanding that he order his forces to stand down and leave the palace. In response, Khalid called up his palace guard and barricaded himself inside the palace.

The ultimatum expired at 09:00 East Africa Time (EAT) on 27 August, by which time the British had gathered three cruisers, two gunboats, 150 marines and sailors, and 900 Zanzibaris in the harbour area. The Royal Navy contingent were under the command of Rear-Admiral Harry Rawson and the pro-Anglo Zanzibaris were commanded by Brigadier-General Lloyd Mathews of the Zanzibar army (who was also the First Minister of Zanzibar). Around 2,800 Zanzibaris defended the palace; most were recruited from the civilian population, but they also included the sultan's palace guard and several hundred of his servants and slaves. The defenders had several artillery pieces and machine guns, which were set in front of the palace sighted at the British ships. A bombardment, opened at 09:02, set the palace on fire and disabled the defending artillery. A small naval action took place, with the British sinking the Zanzibari royal yacht HHS Glasgow and two smaller vessels. Some shots were also fired ineffectually at the pro-British Zanzibari troops as they approached the palace. The flag at the palace was shot down and fire ceased at 09:40.

The sultan's forces sustained roughly 500 casualties, while only one British sailor was injured. Sultan Khalid received asylum in the German consulate before escaping to German East Africa (in the mainland part of present Tanzania). The British quickly placed Sultan Hamud in power at the head of a puppet government. The war marked the end of the Zanzibar Sultanate as a sovereign state and the start of a period of heavy British influence.