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🔗 Mathematics

The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single character able to indicate any integer from 1 to 9,999.

Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its place value (units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century.

🔗 New Zealand

Taumatawhakatangi­hangakoauauotamatea­turipukakapikimaunga­horonukupokaiwhen­uakitanatahu is a hill near Porangahau, south of Waipukurau in southern Hawke's Bay, New Zealand. The height of the hill is 305 metres (1,001 ft). The hill is notable primarily for its unusually long name, which is of Māori origin; it is often shortened to Taumata for brevity. It has gained a measure of fame as it is the longest place name found in any English-speaking country, and possibly the longest place name in the world; according to World Atlas. The name of the hill (with 85 characters) has also been listed in the Guinness World Records as the longest place name. Other versions of the name, including longer ones, are also sometimes used.

🔗 United States 🔗 Politics 🔗 Politics/American politics 🔗 Elections and Referendums 🔗 United States/U.S. presidential elections

The 1876 United States presidential election was the 23rd quadrennial presidential election, held on Tuesday, November 7, 1876, in which Republican nominee Rutherford B. Hayes faced Democrat Samuel J. Tilden. It was one of the most contentious and controversial presidential elections in American history, and gave rise to the Compromise of 1877 by which the Democrats conceded the election to Hayes in return for an end to Reconstruction and the withdrawal of federal troops from the South. After a controversial post-election process, Hayes was declared the winner.

After President Ulysses S. Grant declined to seek a third term despite previously being expected to do so, Congressman James G. Blaine emerged as the front-runner for the Republican nomination. However, Blaine was unable to win a majority at the 1876 Republican National Convention, which settled on Governor Hayes of Ohio as a compromise candidate. The 1876 Democratic National Convention nominated Governor Tilden of New York on the second ballot.

The results of the election remain among the most disputed ever. Although it is not disputed that Tilden outpolled Hayes in the popular vote, after a first count of votes, Tilden had won 184 electoral votes to Hayes's 165, with 20 votes from four states unresolved: in Florida, Louisiana, and South Carolina, each party reported its candidate had won the state, while in Oregon, one elector was replaced after being declared illegal for being an "elected or appointed official". The question of who should have been awarded these electoral votes is the source of the continued controversy.

An informal deal was struck to resolve the dispute: the Compromise of 1877, which awarded all 20 electoral votes to Hayes; in return for the Democrats' acquiescence to Hayes' election, the Republicans agreed to withdraw federal troops from the South, ending Reconstruction. The Compromise in effect ceded power in the Southern states to the Democratic Redeemers, who proceeded to disenfranchise black voters thereafter.

The 1876 election is the second of five presidential elections in which the person who won the most popular votes did not win the election, but the only such election in which the popular vote winner received a majority (rather than a plurality) of the popular vote. To date, it remains the election that recorded the smallest electoral vote victory (185–184), and the election that yielded the highest voter turnout of the eligible voting age population in American history, at 81.8%. Despite not becoming president, Tilden was the first Democratic presidential nominee since James Buchanan in 1856 to win the popular vote and the first since Franklin Pierce in 1852 to do so in an outright majority (In fact, Tilden received a slightly higher percentage than Pierce in 1852, despite the fact that Pierce won in a landslide).

🔗 Computer science 🔗 Statistics

The Jaccard index, also known as the Jaccard similarity coefficient, is a statistic used for gauging the similarity and diversity of sample sets. It was developed by Grove Karl Gilbert in 1884 as his ratio of verification (v) and now is frequently referred to as the Critical Success Index in meteorology. It was later developed independently by Paul Jaccard, originally giving the French name coefficient de communauté, and independently formulated again by T. Tanimoto. Thus, the Tanimoto index or Tanimoto coefficient are also used in some fields. However, they are identical in generally taking the ratio of Intersection over Union. The Jaccard coefficient measures similarity between finite sample sets, and is defined as the size of the intersection divided by the size of the union of the sample sets:

${\displaystyle J(A,B)={{|A\cap B|} \over {|A\cup B|}}={{|A\cap B|} \over {|A|+|B|-|A\cap B|}}.}$

Note that by design, ${\displaystyle 0\leq J(A,B)\leq 1.}$ If A intersection B is empty, then J(A,B) = 0. The Jaccard coefficient is widely used in computer science, ecology, genomics, and other sciences, where binary or binarized data are used. Both the exact solution and approximation methods are available for hypothesis testing with the Jaccard coefficient.

Jaccard similarity also applies to bags, i.e., Multisets. This has a similar formula, but the symbols mean bag intersection and bag sum (not union). The maximum value is 1/2.

${\displaystyle J(A,B)={{|A\cap B|} \over {|A\uplus B|}}={{|A\cap B|} \over {|A|+|B|}}.}$

The Jaccard distance, which measures dissimilarity between sample sets, is complementary to the Jaccard coefficient and is obtained by subtracting the Jaccard coefficient from 1, or, equivalently, by dividing the difference of the sizes of the union and the intersection of two sets by the size of the union:

${\displaystyle d_{J}(A,B)=1-J(A,B)={{|A\cup B|-|A\cap B|} \over |A\cup B|}.}$

An alternative interpretation of the Jaccard distance is as the ratio of the size of the symmetric difference ${\displaystyle A\triangle B=(A\cup B)-(A\cap B)}$ to the union. Jaccard distance is commonly used to calculate an n × n matrix for clustering and multidimensional scaling of n sample sets.

This distance is a metric on the collection of all finite sets.

There is also a version of the Jaccard distance for measures, including probability measures. If ${\displaystyle \mu }$ is a measure on a measurable space ${\displaystyle X}$, then we define the Jaccard coefficient by

${\displaystyle J_{\mu }(A,B)={{\mu (A\cap B)} \over {\mu (A\cup B)}},}$

and the Jaccard distance by

${\displaystyle d_{\mu }(A,B)=1-J_{\mu }(A,B)={{\mu (A\triangle B)} \over {\mu (A\cup B)}}.}$

Care must be taken if ${\displaystyle \mu (A\cup B)=0}$ or ${\displaystyle \infty }$, since these formulas are not well defined in these cases.

The MinHash min-wise independent permutations locality sensitive hashing scheme may be used to efficiently compute an accurate estimate of the Jaccard similarity coefficient of pairs of sets, where each set is represented by a constant-sized signature derived from the minimum values of a hash function.

🔗 Spaceflight

SNAP-10A (Systems for Nuclear, Auxiliary Power, aka Snapshot for Space Nuclear Auxiliary Power Shot, also known as OPS 4682, COSPAR 1965-027A) was a US experimental nuclear powered satellite launched into space in 1965 as part of the SNAPSHOT program. The test marked the world's first operation of a nuclear reactor in orbit, and also the first operation of an ion thruster system in orbit. It is the only fission reactor power system launched into space by the United States. The reactor stopped working after just 43 days due to a non-nuclear electrical component failure. The Systems Nuclear Auxiliary Power Program (SNAP) reactor was specifically developed for satellite use in the 1950s and early 1960s under the supervision of the U.S. Atomic Energy Commission.

🔗 Finance & Investment 🔗 Economics 🔗 Business 🔗 Rock music 🔗 Business/Accounting

A celebrity bond is commercial debt security issued by a holder of fame-based intellectual property rights to receive money upfront from investors on behalf of the bond issuer and their celebrity clients in exchange for assigning investors the right to collect future royalty monies to the works covered by the intellectual property rights listed in the bond. Typically backed by music properties, the investment vehicle was pioneered in 1997 by rock and roll investment banker David Pullman through his $55 million David Bowie bond deal.

🔗 Computer science

HAKMEM, alternatively known as AI Memo 239, is a February 1972 "memo" (technical report) of the MIT AI Lab containing a wide variety of hacks, including useful and clever algorithms for mathematical computation, some number theory and schematic diagrams for hardware — in Guy L. Steele's words, "a bizarre and eclectic potpourri of technical trivia". Contributors included about two dozen members and associates of the AI Lab. The title of the report is short for "hacks memo", abbreviated to six upper case characters that would fit in a single PDP-10 machine word (using a six-bit character set).

🔗 Military history 🔗 Military history/Early Modern warfare 🔗 Military history/Medieval warfare 🔗 Former countries 🔗 Czech Republic 🔗 Former countries/Holy Roman Empire

The Defenestrations of Prague (Czech: Pražská defenestrace, German: Prager Fenstersturz, Latin: Defenestratio Pragensis) were three incidents in the history of Bohemia in which people were defenestrated (thrown out of a window). Though already existing in Middle French, the word defenestrate ("out of the window") is believed to have first been used in English in reference to the episodes in Prague in 1618 when the disgruntled Protestant estates threw two royal governors and their secretary out of a window of the Hradčany Castle and wrote an extensive apologia explaining their action. In the Middle Ages and early modern times, defenestration was not uncommon—the act carried elements of lynching and mob violence in the form of murder committed together.

The first governmental defenestration occurred in 1419, the second in 1483 and the third in 1618, although the term "Defenestration of Prague" more commonly refers to the third. Often, however, the 1483 event is not recognized as a "significant defenestration", which leads to some ambiguity when the 1618 defenestration is referred to as the "second Prague defenestration". The first and third defenestrations helped to trigger a prolonged religious conflict inside Bohemia (the Hussite Wars, 1st defenestration) or beyond (Thirty Years' War, 3rd defenestration), while the second helped establish a religious peace in the country for 31 years (Peace of Kutná Hora, 2nd defenestration).

🔗 Computer science

TLA⁺ is a formal specification language developed by Leslie Lamport. It is used to design, model, document, and verify programs, especially concurrent systems and distributed systems. TLA⁺ has been described as exhaustively-testable pseudocode, and its use likened to drawing blueprints for software systems; TLA is an acronym for Temporal Logic of Actions.

For design and documentation, TLA⁺ fulfills the same purpose as informal technical specifications. However, TLA⁺ specifications are written in a formal language of logic and mathematics, and the precision of specifications written in this language is intended to uncover design flaws before system implementation is underway.

Since TLA⁺ specifications are written in a formal language, they are amenable to finite model checking. The model checker finds all possible system behaviours up to some number of execution steps, and examines them for violations of desired invariance properties such as safety and liveness. TLA⁺ specifications use basic set theory to define safety (bad things won't happen) and temporal logic to define liveness (good things eventually happen).

TLA⁺ is also used to write machine-checked proofs of correctness both for algorithms and mathematical theorems. The proofs are written in a declarative, hierarchical style independent of any single theorem prover backend. Both formal and informal structured mathematical proofs can be written in TLA⁺; the language is similar to LaTeX, and tools exist to translate TLA⁺ specifications to LaTeX documents.

TLA⁺ was introduced in 1999, following several decades of research into a verification method for concurrent systems. A toolchain has since developed, including an IDE and distributed model checker. The pseudocode-like language PlusCal was created in 2009; it transpiles to TLA⁺ and is useful for specifying sequential algorithms. TLA⁺² was announced in 2014, expanding language support for proof constructs. The current TLA⁺ reference is The TLA⁺ Hyperbook by Leslie Lamport.