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🔗 Mathematics

Graham's number is an immense number that arises as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is named after mathematician Ronald Graham, who used the number in conversations with popular science writer Martin Gardner as a simplified explanation of the upper bounds of the problem he was working on. In 1977, Gardner described the number in Scientific American, introducing it to the general public. At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof. The number was published in the 1980 Guinness Book of World Records, adding to its popular interest. Other specific integers (such as TREE(3)) known to be far larger than Graham's number have since appeared in many serious mathematical proofs, for example in connection with Harvey Friedman's various finite forms of Kruskal's theorem. Additionally, smaller upper bounds on the Ramsey theory problem from which Graham's number derived have since been proven to be valid.

Graham's number is much larger than many other large numbers such as Skewes' number and Moser's number, both of which are in turn much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Thus Graham's number cannot be expressed even by power towers of the form ${\displaystyle a^{b^{c^{\cdot ^{\cdot ^{\cdot }}}}}}$.

However, Graham's number can be explicitly given by computable recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Graham. As there is a recursive formula to define it, it is much smaller than typical busy beaver numbers. Though too large to be computed in full, the sequence of digits of Graham's number can be computed explicitly through simple algorithms. The last 12 digits are ...262464195387. With Knuth's up-arrow notation, Graham's number is ${\displaystyle g_{64}}$, where

🔗 Law 🔗 Time

"Justice delayed is justice denied" is a legal maxim. It means that if legal redress or equitable relief to an injured party is available, but is not forthcoming in a timely fashion, it is effectively the same as having no remedy at all.

This principle is the basis for the right to a speedy trial and similar rights which are meant to expedite the legal system, because of the unfairness for the injured party who sustained the injury having little hope for timely and effective remedy and resolution. The phrase has become a rallying cry for legal reformers who view courts, tribunals, judges, arbitrators, administrative law judges, commissions or governments as acting too slowly in resolving legal issues — either because the case is too complex, the existing system is too complex or overburdened, or because the issue or party in question lacks political favour. Individual cases may be affected by judicial hesitancy to make a decision. Statutes and court rules have tried to control the tendency; and judges may be subject to oversight and even discipline for persistent failures to decide matters timely, or accurately report their backlog. When a court takes a matter "under advisement" – awaiting the issue of a judicial opinion, order or judgement and forestalls final adjudication of a lawsuit or resolution of a motion – the issue of timeliness of the decision(s) comes into play.

🔗 Greece 🔗 Writing systems

Boustrophedon (Ancient Greek: βουστροφηδόν, boustrophēdón "ox-turning" from βοῦς, bous, "ox", στροφή, strophē, "turn" and the adverbial suffix -δόν, "like, in the manner of"; that is, turning like oxen in ploughing) is a type of bi-directional text, mostly seen in ancient manuscripts and other inscriptions. Alternate lines of writing are flipped, or reversed, with reversed letters. Rather than going left-to-right as in modern European languages, or right-to-left as in Arabic and Hebrew, alternate lines in boustrophedon must be read in opposite directions. Also, the individual characters are reversed, or mirrored. It was a common way of writing in stone in Ancient Greece.

🔗 Law 🔗 Statistics

The Howland will forgery trial was a U.S. court case in 1868 to decide Henrietta Howland Robinson's contest of the will of Sylvia Ann Howland. It is famous for the forensic use of mathematics by Benjamin Peirce as an expert witness.

🔗 China/Chinese history 🔗 China 🔗 Toys

The bamboo-copter, also known as the bamboo dragonfly or Chinese top (Chinese zhuqingting (竹蜻蜓), Japanese taketonbo 竹蜻蛉), is a toy helicopter rotor that flies up when its shaft is rapidly spun. This helicopter-like top originated in Jin dynasty China around 320 AD, and was the object of early experiments by English engineer George Cayley, the inventor of modern aeronautics.

In China, the earliest known flying toys consisted of feathers at the end of a stick, which was rapidly spun between the hands and released into flight. "While the Chinese top was no more than a toy, it is perhaps the first tangible device of what we may understand as a helicopter."

The Jin dynasty Daoist philosopher Ge Hong's (c. 317) book Baopuzi (抱樸子 "Master Who Embraces Simplicity") mentioned a flying vehicle in what Joseph Needham calls "truly an astonishing passage".

Some have made flying cars [feiche 飛車] with wood from the inner part of the jujube tree, using ox-leather (straps) fastened to returning blades so as to set the machine in motion [huan jian yi yin chiji 環劍以引其機]. Others have had the idea of making five snakes, six dragons and three oxen, to meet the "hard wind" [gangfeng 罡風] and ride on it, not stopping until they have risen to a height of forty li. That region is called [Taiqing 太清] (the purest of empty space). There the [qi] is extremely hard, so much so that it can overcome (the strength of) human beings. As the Teacher says: "The kite (bird) flies higher and higher spirally, and then only needs to stretch its two wings, beating the air no more, in order to go forward by itself. This is because it starts gliding (lit. riding) on the 'hard wind' [gangqi 罡炁]. Take dragons, for example; when they first rise they go up using the clouds as steps, and after they have attained a height of forty li then they rush forward effortlessly (lit. automatically) (gliding)." This account comes from the adepts [xianren 仙人], and is handed down to ordinary people, but they are not likely to understand it.

Needham concludes that Ge Hong was describing helicopter tops because "'returning (or revolving) blades' can hardly mean anything else, especially in close association with a belt or strap"; and suggests that "snakes", "dragons", and "oxen" refer to shapes of man-lifting kites. Other scholars interpret this Baopuzi passage mythologically instead of literally, based on its context's mentioning fantastic flights through chengqiao (乘蹻 "riding on tiptoe/stilts") and xian (仙 "immortal; adept") techniques. For instance, "If you can ride the arches of your feet, you will be able to wander anywhere in the world without hindrance from mountains or rivers … Whoever takes the correct amulet and gives serious thought to the process may travel a thousand miles by concentrating his thoughts for one double hour." Compare this translation.

Some build a flying vehicle from the pith of the jujube tree and have it drawn by a sword with a thong of buffalo hide at the end of its grip. Others let their thoughts dwell on the preparation of a joint rectangle from five serpents, six dragons, and three buffaloes, and mount in this for forty miles to the region known as Paradise.

This Chinese helicopter toy was introduced into Europe and "made its earliest appearances in Renaissance European paintings and in the drawings of Leonardo da Vinci." The toy helicopter appears in a c. 1460 French picture of the Madonna and Child at the Musée du Palais de Tesse’ in Mans depicting the Child holding a toy copter sitting in Mary’s lap next to St Benôit (unknown artist), and in a 16th-century stained glass panel at the Victoria and Albert Museum in London. A picture from c. 1560 by Pieter Breughel the Elder at the Kunsthistorisches Museum in Vienna, Children's Games, depicts a helicopter top with three airscrews.

"The helicopter top in China led to nothing but amusement and pleasure, but fourteen hundred years later it was to be one of the key elements in the birth of modern aeronautics in the West." Early Western scientists developed flying machines based upon the original Chinese model. The Russian polymath Mikhail Lomonosov developed a spring-driven coaxial rotor in 1743, and the French naturalist Christian de Launoy created a bow drill device with contra-rotating feather propellers.

In 1792, George Cayley began experimenting with helicopter tops, which he later called "rotary wafts" or "elevating fliers". His landmark (1809) article "On Aerial Navigation" pictured and described a flying model with two propellers (constructed from corks and feathers) powered by a whalebone bow drill. "In 1835 Cayley remarked that while the original toy would rise no more than about 20 or 25 feet (6 or 7.5 metres), his improved models would 'mount upward of 90 ft (27 metres) into the air'. This then was the direct ancestor of the helicopter rotor and the aircraft propeller."

Discussing the history of Chinese inventiveness, the British scientist, sinologist, and historian Joseph Needham wrote, "Some inventions seem to have arisen merely from a whimsical curiosity, such as the 'hot air balloons' made from eggshells which did not lead to any aeronautical use or aerodynamic discoveries, or the zoetrope which did not lead onto the kinematograph, or the helicopter top which did not lead to the helicopter."

🔗 Companies 🔗 Brands 🔗 Retailing 🔗 Spain

El Corte Inglés S.A. (Spanish pronunciation: [el ˈkoɾte iŋˈɡles]), headquartered in Madrid, is the biggest department store group in Europe and ranks third worldwide. El Corte Inglés is Spain's only remaining department store chain. El Corte Inglés has been a member of the International Association of department stores since 1998.

🔗 Soviet Union 🔗 Russia 🔗 Russia/technology and engineering in Russia 🔗 Spaceflight 🔗 Dogs 🔗 Russia/science and education in Russia 🔗 Russia/history of Russia

Laika (Russian: Лайка; c. 1954 – 3 November 1957) was a Soviet space dog who was one of the first animals in space and the first to orbit the Earth. A stray mongrel from the streets of Moscow, she flew aboard the Sputnik 2 spacecraft, launched into low orbit on 3 November 1957. As the technology to de-orbit had not yet been developed, Laika's survival was never expected. She died of overheating hours into the flight, on the craft's fourth orbit.

Little was known about the impact of spaceflight on living creatures at the time of Laika's mission, and animal flights were viewed by engineers as a necessary precursor to human missions. The experiment, which monitored Laika's vital signs, aimed to prove that a living organism could survive being launched into orbit and continue to function under conditions of weakened gravity and increased radiation, providing scientists with some of the first data on the biological effects of spaceflight.

Laika died within hours from overheating, possibly caused by a failure of the central R‑7 sustainer to separate from the payload. The true cause and time of her death were not made public until 2002; instead, it was widely reported that she died when her oxygen ran out on day six or, as the Soviet government initially claimed, she was euthanised prior to oxygen depletion. In 2008, a small monument to Laika depicting her standing atop a rocket was unveiled near the military research facility in Moscow that prepared her flight. She also appears on the Monument to the Conquerors of Space in Moscow.

🔗 Statistics

The bean machine, also known as the Galton Board or quincunx, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximates a normal distribution. Among its applications, it afforded insight into regression to the mean or "regression to mediocrity".

🔗 Volcanoes 🔗 Islands 🔗 British Overseas Territories 🔗 Argentina 🔗 South America/South Georgia and the South Sandwich Islands 🔗 South America

South Georgia and the South Sandwich Islands (SGSSI) is a British Overseas Territory in the southern Atlantic Ocean. It is a remote and inhospitable collection of islands, consisting of South Georgia Island and a chain of smaller islands known as the South Sandwich Islands. South Georgia is 165 kilometres (103 mi) long and 35 kilometres (22 mi) wide and is by far the largest island in the territory. The South Sandwich Islands lie about 700 kilometres (430 mi) southeast of South Georgia. The territory's total land area is 3,903 km² (1,507 sq mi). The Falkland Islands are about 1,300 kilometres (810 mi) west from its nearest point.

No permanent native population lives in the South Sandwich Islands, and a very small non-permanent population resides on South Georgia. There are no scheduled passenger flights or ferries to or from the territory, although visits by cruise liners to South Georgia are increasingly popular, with several thousand visitors each summer.

The United Kingdom claimed sovereignty over South Georgia in 1775 and the South Sandwich Islands in 1908. The territory of "South Georgia and the South Sandwich Islands" was formed in 1985; previously, it had been governed as part of the Falkland Islands Dependencies. Argentina claimed South Georgia in 1927 and claimed the South Sandwich Islands in 1938.

Argentina maintained a naval station, Corbeta Uruguay, on Thule Island in the South Sandwich Islands from 1976 until 1982 when it was closed by the Royal Navy. The Argentine claim over South Georgia contributed to the 1982 Falklands War, during which Argentine forces briefly occupied the island. Argentina continues to claim sovereignty over South Georgia and the South Sandwich Islands.

Toothfish are vital to the islands' economy; as a result, Toothfish Day is celebrated on 4 September as a bank holiday in the territory.

🔗 United States 🔗 Companies 🔗 Video games 🔗 United States/Washington - Seattle 🔗 United States/Washington

Valve Corporation, also known as Valve Software, is an American video game developer, publisher, and digital distribution company headquartered in Bellevue, Washington. It is the developer of the software distribution platform Steam and the Half-Life, Counter-Strike, Portal, Day of Defeat, Team Fortress, Left 4 Dead, and Dota series.

Valve was founded in 1996 by former Microsoft employees Gabe Newell and Mike Harrington. Their debut product, the PC first-person shooter Half-Life, was released in 1998 to critical acclaim and commercial success, after which Harrington left the company. In 2003, Valve launched Steam, which accounted for around half of digital PC game sales by 2011. By 2012, Valve employed around 250 people and was reportedly worth over US$3 billion, making it the most profitable company per employee in the United States. In the 2010s, Valve began developing hardware, such as the Steam Machine, a brand of gaming PCs, as well as the HTC Vive and Valve Index virtual reality headsets.