hotchpotch/wikipedia-multilingual-ir-related-lead-pairs huggingface dataset:
Binomial approximation
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x.  It states that

It is valid when  and  where  and  may be real or complex numbers.

The benefit of this approximation is that  is converted from an exponent to a multiplicative factor.  This can greatly simplify mathematical expressions (as in the example below) and is a common tool in physics.

The approximation can be proven several ways, and is closely related to the binomial theorem.  By Bernoulli's inequality, the left-hand side of the approximation is greater than or equal to the right-hand side whenever  and .

Binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ⁠⁠ expands into a polynomial with terms of the form ⁠⁠, where the exponents ⁠⁠ and ⁠⁠ are nonnegative integers satisfying ⁠⁠ and the coefficient ⁠⁠ of each term is a specific positive integer depending on ⁠⁠ and ⁠⁠. For example, for ⁠⁠,

The coefficient ⁠⁠ in each term ⁠⁠ is known as the binomial coefficient ⁠⁠ or ⁠⁠ (the two have the same value). These coefficients for varying ⁠⁠ and ⁠⁠ can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where ⁠⁠ gives the number of different combinations (i.e. subsets) of ⁠⁠ elements that can be chosen from an ⁠⁠-element set. Therefore ⁠⁠ is usually pronounced as "⁠⁠ choose ⁠⁠".
Elizabeth Clarke Wolstenholme-Elmy
Elizabeth Clarke Wolstenholme-Elmy (née Wolstenholme; 1833 – 12 March 1918) was a British teacher, campaigner and organiser, significant in the history of women's suffrage in the United Kingdom. She wrote essays and some poetry, using the pseudonyms E and Ignota.

Congleton
Congleton is a market town and civil parish in the borough of Cheshire East, in Cheshire, England. The town is on the River Dane, 21 miles (34 km) south of Manchester and 13 miles (21 km) north of Stoke on Trent. At the 2021 census, the town's built-up area had a population of 30,005, while the parish had a population of 28,497.
Ritual Entertainment
Ritual Entertainment was an American video game developer established in 1996 by Robert Atkins, Mark Dochtermann, Jim Dosé, Richard 'Levelord' Gray, Michael Hadwin, Harry Miller, and Tom Mustaine. Based in Dallas, Texas, Ritual Entertainment was formerly known as Hipnotic Interactive, during which period they began development of their signature video game SiN.

Bethesda Game Studios
Bethesda Game Studios is an American video game developer and a studio of ZeniMax Media based in Rockville, Maryland. It is best known for its action role-playing franchises, including The Elder Scrolls, Fallout, and Starfield. Bethesda Game Studios opened in 2001 as the development unit of Bethesda Softworks, separating from publishing operations. Todd Howard serves as the studio's executive producer, leading it with managing director Ashley Cheng and studio director Angela Browder. As of November 2023, Bethesda Game Studios had 450 employees.
Prastos
Prastos (Greek: Πραστός, Tsakonian: Πραστέ) is a settlement in Arcadia, Greece. Formerly, Prastos was the premier town of the Tsakonian region, but declined in importance after its devastation by Ibrahim Pasha of Egypt during the Greek War of Independence and a general economic migration to urban areas that occurred in the following decades. It is considered a traditional settlement.

Tsakonian Greek
Tsakonian or Tsaconian (also Tzakonian or Tsakonic, Greek: τσακωνικά, η τσακωνική διάλεκτος and Tsakonian: τσακώνικα, α τσακώνικα γρούσσα) is a highly divergent modern variety of Greek, spoken in the Tsakonian region of the Peloponnese, Greece. Unlike all other extant varieties of Greek, Tsakonian derives from Doric Greek rather than from the Attic–Ionic branch. Although it is conventionally treated as a dialect of Greek, some compendia treat it as a separate language.
Tsakonian is critically endangered, with only a few hundred or a few thousand, mostly elderly, fluent speakers left. Although Tsakonian and standard Modern Greek are related, they are not mutually intelligible.
Eden Valley Railway
The Eden Valley Railway (EVR) was a railway in Cumbria, England. It ran between Clifton Junction near Penrith and Kirkby Stephen via Appleby-in-Westmorland.

Passenger traffic ended in 1962 and the line was reduced to the track between the junction at Appleby station with the Settle-Carlisle Line and Kirkby Stephen which served a quarry. By 1976, all that was left was 6 miles (10 km) of track between Appleby in Westmorland and Flitholme. It was used by infrequent British Army services to Warcop Training Area until 1989.

In 1995, the Eden Valley Railway Society was formed with the aim of reinstating services and restoring the line. Heritage railway services resumed operations in 2006 between Warcop and Sandford, with an extension of 1⁄2 mile (0.8 km) being opened in 2013, giving a current running line of almost 2+1⁄4 miles (3.6 km).

Kirkby Stephen East railway station
54°27′44″N 2°21′30″W﻿ / ﻿54.46227°N 2.35837°W﻿ / 54.46227; -2.35837
Kirkby Stephen East railway station was situated on the South Durham & Lancashire Union Railway (known as the Stainmore Line) between Barnard Castle and Tebay. It served the town of Kirkby Stephen in England and was a junction station for the Eden Valley Railway.

The station re-opened to traffic as the base of the Stainmore Railway Company in August 2011. The new layout of the station consists of two platforms, one of which is partly covered by the remaining train shed roof, and a bay platform between them, situated in the opposite direction to the original station layout.
Acadie (electoral district)
Acadie (pronounced [akadi]) is a provincial electoral district in Quebec, Canada, which elects a member to the National Assembly of Quebec. It is located in northern Montreal and consists of parts of the Saint-Laurent and Ahuntsic-Cartierville boroughs.

It was created as L'Acadie ([lakadi]) for the 1973 general election from parts of the Ahuntsic and Saint-Laurent electoral districts. It changed to its present name in 1989. In the change from the 2001 to the 2011 electoral map, its territory was unchanged.

Since 2022 its Member of the National Assembly (MNA) has been André Morin of the Quebec Liberal Party.

Ahuntsic-Cartierville
Ahuntsic-Cartierville (Canadian French pronunciation: [aɔntsɪk kaʁt͡sjevɪl]) is a borough (arrondissement) of the city of Montreal, Quebec, Canada. The borough was created following the 2002 municipal reorganization of Montreal. It comprises two main neighbourhoods, Ahuntsic, a former village annexed to Montreal in 1910 and Cartierville, a town annexed to Montreal in 1916.

Ahuntsic-Cartierville is located in the north end of Montreal, on the banks of the Rivière des Prairies. It traces its history to the fortified Sault-au-Récollet settlement, which was established by the Sulpicians in 1696. This in turn led to the colonization of the area.
Similarity (signal processing)
Similarity between two different signals is important in the field of signal processing. Below are some common methods for calculating similarity.

For instance, let's consider two signals represented as  and , where  and .

Signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing signals, such as sound, images, potential fields, seismic signals, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, improve subjective video quality, and to detect or pinpoint components of interest in a measured signal.
History of Fall River, Massachusetts
For much of its history, the city of Fall River, Massachusetts has been defined by the rise and fall of its cotton textile industry. From its beginnings as a rural outpost of the Plymouth Colony, the city grew to become the largest textile producing center in the United States during the 19th century, with over one hundred mills in operation by 1920. Even with the demise of local textile productions during the 20th century, there remains a lasting legacy of its impact on the city.

American Printing Company (Fall River Iron Works)
The American Printing Company, located in Fall River, Massachusetts grew to become the largest producer of printed cotton cloth in the United States by the early 20th Century. The company grew as an offshoot of the Fall River Iron Works, established in 1821 by Colonel Richard Borden and Major Bradford Durfee. The American Print Works was established in 1835 by Holder Borden. It employed several thousand workers at its peak during World War I.
Cleavable detergent
Cleavable detergents, also known as cleavable surfactants, are special surfactants (detergents) that are used in biochemistry and especially in proteomics to enhance protein denaturation and solubility. The detergent is rendered inactive by cleavage, usually under acidic conditions, in order to make the sample compatible with a following procedure or in order to selectively remove the cleavage products.

Applications for cleavable detergents include protease digestion of proteins such as in-gel digestion with trypsin after SDS PAGE and peptide extractions from electrophoresis gels. Cleavable detergents are mainly used in sample preparations for mass spectrometry.

Detergent
A detergent is a product for cleaning that contains surfactants plus other components. Detergents comprise surfactants as main functional components to remove hydrophobic grease or dirt by dispersing them in water. They often further comprise water (to facilitate application), builders (to soften water), enzymes (for breaking down proteins, fats, or starches), and dyes or fragrances (to improve the user's sensory experience).

Common surfactants used in detergents are alkylbenzene sulfonates, which are soap-like compounds that are more soluble than soap in hard water, because the polar sulfonate is less likely than the polar carboxylate of soap to bind to calcium and other ions found in hard water.
MS City of Rayville
SS City of Rayville, also referred to as the MV and/or MS City of Rayville was a 5883-ton American steamship. She was built in 1920 by Oscar Daniels & Co. of Tampa, Florida. It was the first American vessel sunk by enemy action in World War II.

Cape Otway
Cape Otway is a cape and a bounded locality of the Colac Otway Shire in southern Victoria, Australia on the Great Ocean Road; much of the area is enclosed in the Great Otway National Park. The cape marks the boundary between the Southern Ocean on the west and Bass Strait on the east.
Pediese, chief of the Ma
Pediese  was a Chief of the Ma and a High Priest of Ptah under the Twenty-second Dynasty of Egypt, who was involved in the replacement of an Apis bull, which had died in the Year 28 of Shoshenq III, and again in the replacement of the subsequent Apis, in the Year 2 of Pami. Both the steles were found in the Serapeum of Saqqara and both are now in The Louvre.

His son Peftjauawybast succeeded him as High Priest of Ptah.

Pediese in front of the dead Apis on the stele of the Year 28 of Shoshenq III (IM. 3749)

Pediese appears again on the stele of the Year 2 of Pami

High Priest of Ptah
The High Priest of Ptah was sometimes referred to as "the Greatest of the Directors of Craftsmanship" (wr-ḫrp-ḥmwt). This title refers to Ptah as the patron god of the craftsmen.

The office of the high priest of Ptah was located in Memphis in Lower Egypt. The temple of Ptah in Memphis was dedicated to Ptah, his consort Sekhmet and their son Nefertem.
Czech Corner
The Czech Corner (Czech: Český koutek, Polish: Czeski kątek,  German: Böhmischer Winkel) is a territory found in the western end of Klodzko land, close to the current Czech–Polish border. This area consisted of eleven villages which were inhabited by Bohemian Czechs.

County of Kladsko
The County of Kladsko (Czech: Kladské hrabství, German: Grafschaft Glatz, Polish: Hrabstwo kłodzkie) was a historical administrative unit within Bohemia as a part of the Kingdom of Bohemia and later in the Kingdom of Prussia with its capital at Kłodzko (Kladsko) on the Nysa river. The territory comprises the Kłodzko Land with the Kłodzko Valley in center within the Sudetes mountain range and roughly corresponds with the present-day Kłodzko County in the Polish Lower Silesian Voivodeship.
Germagno
Germagno is a comune (municipality) in the Province of Verbano-Cusio-Ossola in the Italian region Piedmont, located about 110 kilometres (68 mi) northeast of Turin and about 9 kilometres (6 mi) west of Verbania.

Germagno borders the following municipalities: Casale Corte Cerro, Loreglia, Omegna, Quarna Sopra.

Loreglia
Loreglia is a comune (municipality) in the Province of Verbano-Cusio-Ossola in the Italian region Piedmont, located about 110 kilometres (68 mi) northeast of Turin and about 9 kilometres (6 mi) southwest of Verbania. As of 31 December 2004, it had a population of 279 and an area of 9.2 square kilometres (3.6 sq mi).

Loreglia borders the following municipalities: Casale Corte Cerro, Germagno, Ornavasso, Quarna Sopra, Valstrona.
Japan National Route 302
National Route 302 is a national highway of Japan connecting Nakagawa-ku, Nagoya and Nakagawa-ku, Nagoya in Japan, with a total length of 54 km (33.55 mi). Route 302 serves as Nagoya's inner loop and runs concurrent with much of the Mei-Nikan Expressway.

Mei-Nikan Expressway
The Mei-Nikan Expressway (名二環, Mei-Nikan), or Nagoya Daini Kanjō Expressway (名古屋第二環状自動車道, Nagoya Daini-Kanjō Jidōsha-dō), is a partially completed tolled expressway in Japan. It is owned and operated by the Central Nippon Expressway Company. Upon completion, the expressway will form a second ring road around Nagoya in conjunction with the Isewangan Expressway. It is signed as C2 under the "2016 Proposal for Realization of Expressway Numbering."
Shōgi no Tatsujin: Master of Syougi
Syougi no Tatsujin is a board arcade video game developed by ADK and originally published by SNK on September 28, 1995, before being ported to the Neo Geo CD the same year. It was ported in a scaled-down version to the Neo Geo Pocket and later given improved graphics on the Neo Geo Pocket Color. The game was only released in Japan.

ADK Corporation
ADK Corporation (Japanese: 株式会社エーディーケイ, Hepburn: Kabushiki Kaisha ADK), formerly known as Alpha Denshi Corporation (アルファ電子株式会社), was a Japanese video game developer founded in 1980. ADK began as a developer of arcade games and is best known for their library of SNK Neo Geo titles, including for its home consoles, produced in partnership with SNK. Most notable among these are their fighting games and, in particular, the World Heroes series and Aggressors of Dark Kombat. The company closed with properties sold to SNK Playmore in 2003.
San Gabriel High School
San Gabriel High School (SGHS) is a public high school located in Los Angeles County, California and operated by the Alhambra Unified School District. It is almost entirely in the city limits of Alhambra, with a small portion and the school's address in the city limits of San Gabriel.

It traditionally served students residing in San Gabriel although the balance was shifted in 1994 when San Gabriel Unified School District separated itself from Alhambra Unified. Upon separation, SGUSD opened Gabrielino High School to serve its students. San Gabriel High School now serves students from portions of Alhambra, San Gabriel, and Rosemead.

Alhambra, California
Alhambra (/ælˈhæmbrə/ ⓘ, /ɑːlˈhɑːmbrə/ ⓘ, Spanish pronunciation: [aˈlambɾa] ⓘ; from "Alhambra")  is a city located in the western San Gabriel Valley region of Los Angeles County, California, United States, approximately 8 miles (13 km) east from the downtown Los Angeles civic center. It was incorporated on July 11, 1903. As of the 2020 census, the population was 82,868. The city's ZIP Codes are 91801 and 91803 (plus 91802 for P.O. boxes).
Aaro Pajari
Aaro Olavi Pajari (17 July 1897 – 14 October 1949) was a major general in the Finnish Army. During World War II, he became one of the four double recipients of the Mannerheim Cross 2nd Class.

His greatest achievement was the Finnish victory at the Battle of Tolvajärvi in the Winter War, where his small Finnish force smashed a far larger Soviet army. Pajari went on to serve throughout the Winter, Continuation, and Lapland wars, becoming famous for his success in leading small unit, and guerrilla style operations.

Mannerheim Cross
The Mannerheim Cross (Finnish: Mannerheim-risti, Swedish: Mannerheimkorset), officially Mannerheim Cross of the Cross of Liberty (Finnish: Vapaudenristin Mannerheim-risti, Swedish: Frihetskorsets Mannerheimkors) is the most distinguished Finnish military honour. A total of 191 people received the cross between 22 July 1941 and 7 May 1945, with six of the recipients receiving a cross twice. Available in two classes, the 1st class medal has only been awarded twice, with both recipients also having received the medal in the 2nd class. Although still active de jure, no crosses have been awarded since 1945. Tuomas Gerdt, the last living Knight of the Mannerheim Cross, died on 1 November 2020.
Roger Redgate
Roger Redgate (born 1958) is a British composer, conductor and improvisor. Born in Bolton, he began his musical training at Chetham's School of Music in Manchester, then attended the Royal College of Music where he studied with Edwin Roxburgh and Lawrence Casserley. Under a DAAD (German Academic Exchange) scholarship he also studied with Brian Ferneyhough and Klaus Huber in Freiburg. In 1984, with the new complexity composers Richard Barrett and Michael Finnissy, he co-founded Ensemble Exposé.

Redgate has worked in the fields of jazz, improvised music, film and television. His piano works have been recorded by Nicolas Hodges.

In September 1996 Redgate took up the position of Professor in Composition and Director, Contemporary Music Research at Goldsmiths, University of London. His compositions are published by Editions Henry Lemoine, Paris, United Music Publishers, London and the Associated Board of the Royal Schools of Music.

Brian Ferneyhough
Brian John Peter Ferneyhough (/ˈfɜːrnihoʊ/ ⓘ; born 16 January 1943) is an English composer. Ferneyhough is typically considered the central figure of the New Complexity movement. Ferneyhough has taught composition at the Hochschule für Musik Freiburg and the University of California, San Diego; he teaches at Stanford University and is a regular lecturer in the summer courses at Darmstädter Ferienkurse. He has resided in California since 1987.
Randolph County School System (North Carolina)
The Randolph County School System (RCSS) is a school district headquartered in Asheboro, North Carolina, United States. It serves Randolph County, except for the City of Asheboro, which is served by Asheboro City Schools.

Randolph County, North Carolina
Randolph County is a county located in the U.S. state of North Carolina. As of the 2020 census, the population was 144,171. Its county seat is Asheboro.

Randolph County is included in the Greensboro-High Point, NC Metropolitan Statistical Area, which is also included in the Greensboro–Winston-Salem–High Point, NC Combined Statistical Area.

In 2020, the center of population of North Carolina was located in Randolph County, east of Seagrove.
Slavko Brezoski
Slavko Brezoski (Macedonian: Славко Брезоски; 10 June 1922 – 7 March 2017) was a Macedonian architect, urban planner, painter, writer and educator known for his works in the genre of modern architecture realised during the middle of the 20th century in modern-day North Macedonia, Serbia, Croatia, Brazil and Libya. He was a professor and Dean at the Faculty of Architecture at the Ss. Cyril and Methodius University of Skopje.

Architecture of North Macedonia
The groups of people who have settled or controlled the territory of modern-day North Macedonia have influenced the country in many ways, one of the most visible being architecture. These groups of people include the Paionians, Illyrians, Ancient Macedonians, Romans, Byzantines, Bulgarians, Serbs, Ottomans, Yugoslavs, and ethnic Macedonians.
John Hensleigh Allen
John Hensleigh Allen DL (29 August 1769 – 12 April 1843) was the Whig member of parliament for Pembroke elected at the 1818 United Kingdom general election until the 1826 United Kingdom general election.

He was the son of John Bartlett Allen (1733–1792) a local landowner and colliery owner and his first wife Elizabeth Hensleigh.  He had 9 sisters, and his brothers-in-law included Josiah Wedgwood II, Sir James Mackintosh (both Whig MPs), John Wedgwood the horticulturist and Jean Charles Léonard de Sismondi the historian.

He was educated at Westminster School and Trinity College, Cambridge, following his elder brother Lancelot Baugh Allen.

He was High Sheriff of Pembrokeshire in 1809.

He married Gertrude, daughter of Lord Robert Seymour, on 9 November 1812.  They had five children:

Gertrude Elizabeth Allen (died 1824)

Seymour Phillips Allen (1814–1861)

Henry George Allen (1815–1908)

John Hensleigh Allen (1818–1868)

Isabel Georgina Allen (1820–1914), married George Lort Phillips

Lord Robert Seymour
Lord Robert Seymour JP (20 January 1748 – 23 November 1831) was a British politician who sat in the Irish House of Commons from 1771 to 1776 and in the British House of Commons from 1771 to 1807. He was known as Hon. Robert Seymour-Conway until 1793, when his father was created a marquess; he then became Lord Robert Seymour-Conway, but dropped the surname of Conway after his father's death in 1794.
Arviat (electoral district)
Arviat was a territorial electoral district (riding) for the Legislative Assembly of Nunavut, Canada. The riding consisted of the community of Arviat.

In 2011 the Nunavut Electoral Boundaries Commission recommended the district be split into two districts, Arviat North-Whale Cove and Arviat South. With the passage of Bill 22 in October 2011 the two new districts were created.

Arviat North-Whale Cove
Arviat North-Whale Cove (Inuktitut: ᐊᕐᕕᐊᑦ
ᐅᐊᖕᓇᖓ−ᑎᑭᕋᕐᔪᐊᖅ, French: Arviat Nord-Whale Cove, Inuinnaqtun: Arviat Tununga-Tikiraryuaq) is a territorial electoral district (riding) for the Legislative Assembly of Nunavut, Canada. The riding consists of part of the community of Arviat and Whale Cove. The district was created prior to the 28 October 2013 general election. The communities were previously in Arviat and Rankin Inlet South/Whale Cove.
Second Avenue (Manhattan)
Second Avenue is located on the East Side of the New York City borough of Manhattan extending from Houston Street at its south end to the Harlem River Drive at 128th Street at its north end. A one-way street, vehicular traffic on Second Avenue runs southbound (downtown) only, except for a one-block segment of the avenue in Harlem. South of Houston Street, the roadway continues as Chrystie Street south to Canal Street.

A bicycle lane runs in the leftmost lane of Second Avenue from 125th to Houston Streets. The section from 55th to 34th Streets closes a gap in the Manhattan Waterfront Greenway.

Second Avenue passes through a number of Manhattan neighborhoods including (from south to north) the Lower East Side, the East Village, Stuyvesant Square, Kips Bay, Tudor City, Turtle Bay, East Midtown, Lenox Hill, Yorkville and Spanish Harlem.

Yorkville, Manhattan
Yorkville is a neighborhood on the Upper East Side of Manhattan, New York City, United States. Its southern boundary is East 79th Street, its northern East 96th Street, its western Third Avenue, and its eastern the East River. Yorkville is one of the most densely populated city subdivisions in the world, and the most dense of such in the U.S.

Yorkville is part of Manhattan Community District 8, and its primary ZIP Codes are 10028, 10075, and 10128. It is patrolled by the 19th Precinct of the New York City Police Department.