hotchpotch/wikipedia-multilingual-ir-related-lead-pairs huggingface dataset:
Plane-wave expansion
In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves:

where

i is the imaginary unit,

k is a real or complex wave vector of length k,

r is a position vector of length r,

jℓ are spherical Bessel functions,

Pℓ are Legendre polynomials, and

the hat ^ denotes the unit vector.

In the special case where k is aligned with the z axis,

where θ is the spherical polar angle of r.

For proof, expand  in Legendre polynomials , and evaluate the coeffient integrals.

Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in  (pronounced "v-hat"). The term normalized vector is sometimes used as a synonym for unit vector.

The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,

where ‖u‖ is the norm (or length) of u and .

The proof is the following:

A unit vector is often used to represent directions, such as normal directions.
Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors.
Clapham Junction (film)
Clapham Junction is a 2007 British television film, written by Kevin Elyot.  Directed by Adrian Shergold, the film centres on the experiences of several gay men during a 36-hour period in the Clapham area of London and the consequences when their lives collide. It was first broadcast on Channel 4 on 22 July 2007, repeated on 1 September 2009, and later released on DVD on Region 1.

Gay men
Gay men are male homosexuals. Historic terminology for gay men has included inverts and uranians.

Gay men continue to face significant discrimination in large parts of the world, particularly in parts of Asia and Africa. In the Western world, many gay men still experience discrimination in their daily lives, though some openly gay men have reached national success and prominence, including Apple CEO Tim Cook and heads of state or government such as Edgars Rinkēvičs (president of Latvia since 2023).

The word gay is recommended by LGBTQ groups and style guides to describe all people exclusively attracted to members of the same sex, while lesbian refers specifically to female homosexuals, and gay men to male homosexuals.
Kemba (rapper)
Matthew Jefferson, also known as Kemba, formerly known as YC the Cynic, (born November 21, 1990) is an American rapper from Hunts Point, Bronx, New York. He was named one of the "Top 5 Underground Hip-Hop Artists in New York" by Deli Magazine (2010) and was ordained "DXNext" by the hip hop website HipHopDX (2011). Kemba talks openly about his sense of responsibility to the art form, “It feels like there’s a void in the Bronx, and I don’t necessarily feel like I have to fill it, but I feel like I could."
He was also a feature on XXXTENTACION's final posthumous album, Bad Vibes Forever.

Hunts Point, Bronx
Hunts Point is a neighborhood located on a peninsula in the South Bronx of New York City. It is the location of one of the largest food distribution facilities in the world, the Hunts Point Cooperative Market. Its boundaries are the Bruckner Expressway to the west and north, the Bronx River to the east, and the East River to the south. Hunts Point Avenue is the primary street through Hunts Point.

The neighborhood is part of Bronx Community District 2, and its ZIP Code is 10474. The neighborhood is served by the New York City Police Department's 41st Precinct. NYCHA property in the area is patrolled by P.S.A. 7 at 737 Melrose Avenue located in the Melrose section of the Bronx.
Babu Owino
Paul Ongili Owino (born 10 October 1989), also known as Babu Owino, is a Kenyan politician and member of parliament for Embakasi East Constituency. He gained media recognition the chairperson of Students Organisation of Nairobi University (SONU). He is also an Advocate of the High Court of Kenya.

National Assembly (Kenya)
The National Assembly of the Republic of Kenya (formerly House of Representatives) is one of the two Houses of the Parliament of Kenya. Between 1966 and 2013, it served as a unicameral house. In 2013 (11th Parliament), it became the lower house when the Senate was reestablished.

It has a total of 349 seats: 290 elected from the constituencies, 47 women elected from the counties and 12 nominated representatives. The Speaker of the National Assembly of Kenya serves as an ex officio member.

The High Court of Kenya ordered lawmakers to introduce gender quotas, or face dissolution in the mid-2010s, following the implementation of the 2010 Constitution.
Gene Farrell
Gene Farrell is a Gaelic footballer from Camp, County Kerry in Ireland. He played with the Kerry intercounty team during the 1990s winning Munster Championships at Minor (1990), Under 21 (1992) and Senior level (1996).

Meath GAA
The Meath County Board of the Gaelic Athletic Association (GAA) (Irish: Cumann Lúthchleas Gael Coiste na Mí) or Meath GAA is one of the 32 county boards of the GAA in Ireland, and is responsible for Gaelic games in County Meath, as well as for Meath county teams.
Irvine Valley (ward)
Irvine Valley  is one of the nine electoral wards of East Ayrshire Council. Created in 2007, the ward elects three councillors using the single transferable vote electoral system. Originally a four-member ward, Irvine Valley was reduced in size following a boundary review and has elected three councillors since the 2017 East Ayrshire Council election.

As a four-member ward, the area favoured the Scottish National Party (SNP) but since the boundary changes in 2017, the ward has been split between the SNP, Labour and The Rubbish Party.

Waterside, East Ayrshire
Waterside is a village in East Ayrshire, Scotland, situated about five miles north of Galston on the Craufurdland Water in the Parish of Fenwick. It lies a few miles north of Moscow on the A719 and had a population of 141 in the Census of 2001.
Jason Kelce
Jason Daniel Kelce (/ˈkɛlsi/ ⓘ KEL-see; born November 5, 1987) is an American former professional football center who spent his entire 13-year career with the Philadelphia Eagles of the National Football League (NFL). Kelce played college football for the Cincinnati Bearcats and was selected by the Eagles in the sixth round of the 2011 NFL draft. He won Super Bowl LII, was a seven-time Pro Bowl selection, and six-time first-team All-Pro selection. Kelce is often regarded as one of the greatest centers in NFL history.

Outside of football, Kelce and his brother, Travis, co-host the podcast New Heights, on which they discuss their playing careers and various other topics. In November 2024, Apple announced that New Heights was the 8th most popular podcast show on Apple Podcasts in 2024.

In 2024, Kelce joined ESPN's Monday Night Countdown pregame crew.

Apple Podcasts
Apple Podcasts (known as simply Podcasts in Apple operating systems) is an audio streaming service and media player application developed by Apple for playing podcasts. Apple began supporting podcasts with iTunes 4.9 released in June 2005 and launched its first standalone mobile app in 2012. The app was later pre-installed with iOS beginning October 2014. The Apple Podcasts directory features more than two million shows. Apple Podcasts is available on iOS, iPadOS, macOS, watchOS, tvOS, CarPlay, visionOS, Microsoft Windows operating systems, web browsers, and on Amazon Alexa devices.
Founder mode
Founder mode is a term used and popularized by Y Combinator co-founder Paul Graham in a September 2024 essay in response to a talk delivered by Airbnb co-founder Brian Chesky. It describes a specific kind of leadership in which a founder has a direct, hands-on approach to their company rather than breaking up and delegating responsibility through a top-down structure. Often cited examples of leaders embodying founder mode include Steve Jobs, Elon Musk, and Jensen Huang.

Immediately after its online publication, Graham's essay led to widespread debate and discourse about the different methodologies regarding how to run a company, the pros and cons of micromanagement, and the nature of founder personalities versus that of non-founders, among other relevant concerns. It also spurred many memes, jokes, and stereotypes about the culture surrounding Silicon Valley and big tech.

Y Combinator
Y Combinator, LLC (YC) is an American technology startup accelerator and venture capital firm launched in March 2005 which has been used to launch more than 5,000 companies. The accelerator program started in Boston and Mountain View, relocated to San Francisco in 2019, and was entirely online during the COVID-19 pandemic. Companies started via Y Combinator include Airbnb, Coinbase, Cruise, DoorDash, Dropbox, Instacart, Reddit, Stripe, Scale AI, Deel, Helion Energy, Starcloud, and Twitch.
Bettie Sue Masters
Bettie Sue Siler Masters is an American biochemist who is adjunct professor at Duke University known for her work on nitric oxide synthase and cytochrome P450 reductase. She was the 1992 recipient of the FASEB Excellence in Science Award, and has been elected as a member of the National Academy of Medicine and as a fellow of the American Association for the Advancement of Science.

Roanoke College
Roanoke College is a private liberal arts college in Salem, Virginia. It has approximately 2,000 students who represent approximately 40 states and 30 countries. The college offers 35 majors, 57 minors and concentrations, and pre-professional programs. Roanoke awards bachelor's degrees in arts, science, and business administration and is one of 280 colleges with a chapter of the Phi Beta Kappa honor society.

Roanoke is an NCAA Division III school competing in the Old Dominion Athletic Conference. The college fields varsity teams in eleven men's and ten women's sports. Roanoke's athletic nickname is Maroons and the mascot is Rooney, a maroon-tailed hawk.
The Wondrous Tale of Alroy
The Wondrous Tale of Alroy is the sixth novel written by Benjamin Disraeli, who would later become a Prime Minister of Britain. Originally published in 1833, a "new edition” was published in 1834, a heavily revised edition in 1846 entitled Alroy: A Romance and another in 1871 based on that. It is a fictionalised account of the life of David Alroy. Its significance lies in its portrayal of Disraeli's "ideal ambition" and for its being his only novel with a distinctive Jewish subject. Cecil Roth described it as perhaps the earliest Jewish historical novel and Adam Kirsch as "a significant proto-Zionist text". Philip Rieff described an answer by Alroy as "perhaps one of the earliest Zionist perorations given in Western literature".

Philip Rieff
Philip Rieff (December 15, 1922 – July 1, 2006) was an American sociologist and cultural critic, who taught sociology at the University of Pennsylvania from 1961 until 1992, and also, during the 1950s, at the University of Chicago.

He was the author of a number of books on Sigmund Freud and his legacy, including Freud: The Mind of the Moralist (1959) and The Triumph of the Therapeutic: Uses of Faith after Freud (1966).

At the University of Chicago, he married his 17-year-old student Susan Sontag after 10 days of courtship in the 1950s. The marriage lasted eight years. Sontag and Rieff had a son together, David Rieff, a writer and the editor of his mother's personal journals. Rieff's second wife and widow, Alison Douglas Knox, died December 12, 2011.
Hayward Regional Shoreline
Hayward Regional Shoreline is a regional park located on the shores of the San Francisco Bay in Hayward, California. It is part of the East Bay Regional Parks system. The 1,713-acre (693 ha) park extends to the shores of San Lorenzo. Part of the park is former commercial salt flats purchased in 1996. A former landfill, now capped with soil and plants, is located in the park. The park includes the 250-acre (100 ha) tidal wetland, Cogswell Marsh, and the 364-acre (147 ha) Oro Loma Marsh (constructed in 1997). Located to the south of the park is the Hayward Shoreline Interpretive Center, which provides information on the Bay shore habitats. The San Francisco Bay Trail runs through the park, which connects the park with San Lorenzo Creek.

San Lorenzo, California
San Lorenzo (Spanish for "Saint Laurence") is a census-designated place (CDP) located in the East Bay of the San Francisco Bay Area in Alameda County, California, United States. The population was 29,581 at the 2020 census. It is an unincorporated community, located at the banks of San Lorenzo Creek. It was originally named Squattersville in 1851, but later renamed to San Lorenzo.

In 1944, under contract to the U.S. Navy, The David D. Bohannon Company began construction of San Lorenzo Village, which was one of the nation's first planned communities, with parcels designated for schools, churches, parks, and several retail centers. Bohannon's pioneering pre-cutting techniques, referred to as the "California method," were used in later developments, such as the more famous Levittown, Pennsylvania.
Portugal at the Rugby World Cup
Portugal has appeared at the Rugby World Cup twice, in 2007 and 2023.

At the 2007 Rugby World Cup the team lost all four matches but succeeded in scoring points and one try in every match, and was able to gain a bonus point on the final game against Romania.

At the 2023 Rugby World Cup the team achieved its first draw against Georgia and also its first ever win beating Fiji 24 points to 23.

Portugal has never made it past the pool stage of the tournament.

David Penalva
David Penalva (born 26 January 1980 in France) is a Portuguese rugby union footballer. He plays as a lock.

Penalva played for Nice, in the French Fédérale 1, for the 2010–11 season.

He had 35 caps for Portugal, from 2002 to 2012, scoring 1 try, 5 points on aggregate. Penalva made his debut for the national side on 3 March 2002, in a 13–10 win over Spain, in Caldas da Rainha for the Six Nations B. 
He was called for the 2007 Rugby World Cup, playing in all the four games and scoring a try in the third game against Italy in a 31–5 loss. That would be the single points of his international career. Penalva most recent game was at 10 March 2012, in a 23–17 win over Spain, in Coimbra, for the Six Nations B. He has been absent from the National Team since then.
Andrea Gonsebate
Andrea Ruth Gonsebate  (born 7 May 1977) is an Argentine women's international footballer who plays as a defender. She is a member of the Argentina women's national football team. She was part of the team at the 2003 FIFA Women's World Cup.

Argentina women's national football team
The Argentina women's national football team represents Argentina in international women's football. Like their men's counterpart, the women's team has been known or nicknamed "La Albiceleste" (The White and Sky Blue).

Women's football in Argentina remains largely in the shadow of the men's game in terms of play development and fan support; in women's sports in Argentina, field hockey and volleyball are also more popular. Almost all its members were amateur players until 1991, when the Campeonato de Fútbol Femenino was founded to increase football popularity among women in Argentina.

The Argentina–Brazil football rivalry in women's football cannot be compared to that of men, given the big differences between both countries; Brazil has the clear advantage in matches between them, and has been hosting a competitive professional women's league for many years, while Argentina only introduced it in 2019.
Zeppelinfjellet
Zeppelinfjellet is a mountain in Oscar II Land at Spitsbergen, Svalbard. It reaches a height of 556 m.a.s.l. (1824 ft) and is located on the peninsula of Brøggerhalvøya, south of the former mining society of Ny-Ålesund and near Zeppelinhamna. The mountain is named after German military officer and airship designer Ferdinand von Zeppelin.

Zeppelinhamna
Zeppelinhamna is a cove at the northern side of the peninsula Brøggerhalvøya in Oscar II Land at Spitsbergen, Svalbard. It extends between the points Mainzodden and Gluudneset, and the island Prins Heinrichøya is located in the bay. The cove is named after German airships designer Ferdinand von Zeppelin.
Lawang Sewu: Dendam Kuntilanak
Lawang Sewu: Dendam Kuntilanak (Lawang Sewu: Kuntilanak's Vengeance) is a 2007 Indonesian Horror film directed by Arie Azis. The film tells about a group of seven teenagers from Jakarta is in Semarang to celebrate their high school graduation. Along the road home after partying, they stopped by Lawang Sewu, a supposedly haunted building in Semarang, to relieve themselves. One of the girls went inside the building to avoid the boys and didn't come back. When they start to find their missing friend, the night turns into a nightmare for other six, who find themselves haunted and terrorized by a Kuntilanak.

Bunga Jelitha
Bunga Jelitha Ibrani (born 6 September 1991) is an Indonesian actress, model and beauty pageant titleholder who won Supermodel International 2011 and also Puteri Indonesia 2017. She represented Indonesia at the Miss Universe 2017, where she didn't make into the Top 16, but she won "Miss Photogenic" award.
Auchincruive
Auchincruive is a former country house and estate in South Ayrshire, Scotland. It is located 4 kilometres (2.5 mi) east of Ayr, on the north bank of the River Ayr. Auchincruive House was built in the 18th century on the site of an earlier mansion. In 1927, the estate became the West of Scotland College of Agriculture, and the house was renamed Oswald Hall. The college became the Scottish Agricultural College in 1990. In 2007 the college announced that the site would be disposed of for redevelopment, and masterplan proposals were approved by South Ayrshire Council in January 2011. The house is protected as a category A listed building, along with other buildings on the estate. The estate is included on the Inventory of Gardens and Designed Landscapes in Scotland, the national listing of significant gardens. The house was returned to its original name of Auchincruive House in May of 2020, the listed building records have also been updated to reflect this.

River Ayr
The River Ayr (/ɛər/ AIR; Scottish Gaelic: Uisge Àir) is a river in Ayrshire, Scotland. At 65 km (40 mi) it is the longest river in the county.

The river was held as sacred by pre-Christian cultures. The remains of several prehistoric sacrificial horse burials have been found along its banks, mainly concentrated around the town of Ayr.
Extended natural numbers
In mathematics, the extended natural numbers is a set which contains the values  and  (infinity). That is, it is the result of adding a maximum element  to the natural numbers. Addition and multiplication work as normal for finite values, and are extended by the rules  (),  and  for .

With addition and multiplication,  is a semiring but not a ring, as  lacks an additive inverse. The set can be denoted by ,  or . It is a subset of the extended real number line, which extends the real numbers by adding  and .

Additive inverse
In mathematics, the additive inverse of an element x, denoted −x, is the element that when added to x, yields the additive identity. This additive identity is often the number 0 (zero), but it can also refer to a more generalized zero element.

In elementary mathematics, the additive inverse is often referred to as the opposite number, or the negative of a number. The unary operation of arithmetic negation is closely related to subtraction and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers.
McCurtain County Wilderness Area
The McCurtain County Wilderness Area is a 14,087 acres (5,701 ha) wilderness nature preserve 25 miles (40 km) north of Broken Bow, Oklahoma.  It has been owned by the Oklahoma Department of Wildlife Conservation.  It was designated a National Natural Landmark in December 1974 for its excellent example of a xeric upland oak-pine forest.

Broken Bow, Oklahoma
Broken Bow is a city in McCurtain County, Oklahoma, United States. The population was 4,228 at the 2020 census. It is named after Broken Bow, Nebraska, the former hometown of the city's founders, the Dierks brothers.
Constantine, Mabe and Mawnan (electoral division)
Constantine, Mabe and Mawnan is an electoral division of Cornwall in the United Kingdom which returns one member to sit on Cornwall Council. It was created at the 2021 local elections, being created from the former divisions of Constantine, Mawnan and Budock, and Mabe, Perranarworthal and St Gluvias. The current councillor is Anna Thomason-Kenyon, who won the election as a member of Reform UK but has now changed party to the Cornish independent non aligned group.

Cornwall Council
Cornwall Council (Cornish: Konsel Kernow [ˈkɔn.sɛl ˈkɛrnɔʊ]), known between 1889 and 2009 as Cornwall County Council (Cornish: Konteth Konsel Kernow), is the local authority for the non-metropolitan county of Cornwall in South West England. Since 2009 it has been a unitary authority, having taken over district-level functions when the county’s districts were abolished. The non-metropolitan county of Cornwall is slightly smaller than the ceremonial county, which additionally includes the Isles of Scilly. The council's headquarters is Lys Kernow (also known as New County Hall) in Truro.

The council has been under no overall control since July 2024. Following the May 2025 election an administration of the Liberal Democrats and independents formed to run the council.
Hårgalåten
"Hårgalåten", or the "Harga song", is a Swedish folk song. The song is connected to a 1785 story published by Johan Gabriel Lindstrom. The song describes how a mysterious fiddler came to the community near Hårgaberget or "Harga mountain", and played the fiddle. The youth danced to the tune that the fiddler played and they were unable to stop. Some of the youth noticed a cloven hoof, leading them to believe the fiddler was demonic. Even as morning came, the adults noticed they were all still dancing, and the fiddler still fiddling. Later, the dancers started dropping dead from exhaustion.

Cloven hoof
A cloven hoof, cleft hoof, divided hoof, or split hoof is a hoof split into two toes. Members of the mammalian order Artiodactyla that possess this type of hoof include pigs
(suborder Suina), as well as cattle, deer, antelopes, gazelles, goats, and sheep (suborder Ruminantia).

The two digits of cloven-hooved animals are homologous to the third and fourth fingers of the hand. They are called claws and are named for their relative location on the foot: the outer, or lateral, claw and the inner, or medial claw. The space between the two claws is called the interdigital cleft; the area of skin is called the interdigital skin. The hard outer covering of the hoof is called the hoof wall or horn. It is a hard surface, similar to the human fingernail.

The almost finger-like dexterity available to cloven-hooved mammals such as mountain goats and wild sheep combined with a hard outer shell and soft and flexible inner pads provides excellent traction in their precarious habitats.
Oost-Watergraafsmeer
Oost-Watergraafsmeer was a stadsdeel (borough) of Amsterdam until 2010. It bordered Diemen, Duivendrecht, and the boroughs Amsterdam-Centrum, Oud-Zuid, Zeeburg, and Zuideramstel. The Watergraafsmeer part of the borough is a polder.

Located in Oost/Watergraafsmeer:

Oosterpark

Science Park Amsterdam

Buitenplaats Frankendael

Buitenplaats
A buitenplaats (Dutch pronunciation: [ˈbœytə(n)ˌplaːts], lit. 'outside place') was a summer residence for rich townspeople in the Netherlands. During the Dutch Golden Age of the 17th century, many traders and city administrators in Dutch towns became very wealthy. Many of them bought country estates, at first mainly to collect rents, however soon mansions started to be built there, which were used only during the summer.