hotchpotch/wikipedia-multilingual-ir-related-lead-pairs huggingface dataset:
Fourier inversion theorem
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.

The theorem says that if we have a function  satisfying certain conditions, and we use the convention for the Fourier transform that

then

In other words, the theorem says that

This last equation is called the Fourier integral theorem.

Another way to state the theorem is that if  is the flip operator i.e. ⁠⁠, then

The theorem holds if both  and its Fourier transform are absolutely integrable (in the Lebesgue sense) and  is continuous at the point ⁠⁠. However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.

Absolutely integrable function
In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite.

For a real-valued function, since

where

both  and  must be finite. In Lebesgue integration, this is exactly the requirement for any measurable function f to be considered integrable, with the integral then equaling , so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions.

The same thing goes for a complex-valued function. Let us define




where  and  are the real and imaginary parts of . Then

so

This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function is Lebesgue integrable only if all the four integrals are finite. So having a finite integral of the absolute value is equivalent to the conditions for the function to be "Lebesgue integrable".
Rosa 'Seashell'
Rosa 'Seashell',  (aka KORshel),  is a salmon-pink hybrid tea rose cultivar, bred by Reimer Kordes in Germany before 1976. The rose was introduced into the United States by Jackson & Perkins in 1976. The cultivar was named an All-America Rose Selections winner in 1976.

W. Kordes' Söhne
W. Kordes' Söhne (en. W. Kordes' Sons) is a German rose breeding company in Klein Offenseth-Sparrieshoop in Schleswig-Holstein, Germany. The company is one of the world's leading rose breeders and producers for cut roses and garden roses, annually selling worldwide more than two million rose plants at retail and wholesale. Each year, more than 50,000 new crosses of garden roses and cut roses are tested, leading to four to six marketable varieties after a trial period of eight to ten years.
Callovosaurus
Callovosaurus (meaning "Callovian lizard") is a genus of iguanodontian dinosaur known from most of a left thigh bone discovered in Middle Jurassic-age rocks of England. At times, it has been considered dubious or a valid genus of basal iguanodontian, perhaps a dryosaurid.

Middle Jurassic
The Middle Jurassic is the second epoch of the Jurassic Period. It lasted from about 174.1 to 161.5 million years ago. Fossils of land-dwelling animals, such as dinosaurs, from the Middle Jurassic are relatively rare, but geological formations containing land animal fossils include the Forest Marble Formation in England, the Kilmaluag Formation in Scotland, the Calcaire de Caen of France, the Daohugou Beds in China, the Itat Formation in Russia, the Tiourarén Formationof Niger, and the Isalo III Formation of western Madagascar. Rocks of the Middle Jurassic were formerly (until about 1980s) in Europe called Dogger or Brown Jurassic.
Yaroslav Vazhynskyi
Yaroslav Volodymyrovych Vazhynskyi (Ukrainian: Ярослав Володимирович Важинський; born 23 March 1994) is a Ukrainian football goalkeeper who plays for FC Alians Lypova Dolyna in the Ukrainian Second League.

Ukrainian Second League
The Ukrainian Second League (Ukrainian: Друга ліга, Druha Liha) is a professional football league in Ukraine which is part of the Professional Football League of Ukraine, a collective member of the Ukrainian Association of Football. As the third tier it was established in 1992 as the Transitional League and changed its name the next season.

The league is lower than the Ukrainian First League (Persha Liha) and the lowest level of professional football competitions in the country. Since 1996 the league, after being merged with its lower tier (in 1992–1995 there was the Third League), consists of two main regions roughly north-west and south-east. The league's relegated teams lose their professional status and return to their regional associations.
Spermatic fascia
The spermatic fascia is a bilayered fascia covering the testis; both layers are derived from abdominal muscle or fascia.

The more superficial of these two layers, the external spermatic fascia, lies deep to the skin and dartos fascia of the testes, superficial to the cremaster muscle, and is a continuation of the aponeurosis of the external oblique muscle.

The deeper internal spermatic fascia is deep to the cremaster muscle, directly surrounds the spermatic cord and its contents, and is a continuation of the abdominal transversalis fascia.

Aponeurosis
An aponeurosis (/ˌæpənjʊəˈroʊsɪs/; pl.: aponeuroses) is a flattened tendon by which muscle attaches to bone or fascia. Aponeuroses exhibit an ordered arrangement of collagen fibres, thus attaining high tensile strength in a particular direction while being vulnerable to tensional or shear forces in other directions. They have a shiny, whitish-silvery color, are histologically similar to tendons, and are very sparingly supplied with blood vessels and nerves. When dissected, aponeuroses are papery and peel off by sections. The primary regions with thick aponeuroses are in the ventral abdominal region, the dorsal lumbar region, the ventriculus in birds, and the palmar (palms) and plantar (soles) regions.
Rachel Lawson
Rachel Lawson (born June 11, 1972) is an American former collegiate softball first baseman and current head coach at the University of Kentucky. Lawson played college softball for the UMass Minutewomen in the Atlantic 10 Conference from 1991 to 1994, helping them to the 1992 Women's College World Series.

1992 NCAA Division I softball tournament
The 1992 NCAA Division I softball tournament was the eleventh annual tournament to determine the national champion of NCAA women's collegiate softball. Held during May 1992, twenty Division I college softball teams contested the championship. The tournament featured eight regionals of either two or three teams, each in a double elimination format. The 1992 Women's College World Series was held in Oklahoma City, Oklahoma from May 23 through May 26 and marked the conclusion of the 1992 NCAA Division I softball season.  UCLA won their seventh championship by defeating defending champions Arizona 2–0 in the final game.
2024 Macau Guia Race
The 2024 Macau Guia Race - Kumho FIA TCR World Tour Event of Macau was sixty-second edition of the Macau Guia Race and the ninth edition under the TCR regulations held at Guia Circuit in Macau on 14–17 November 2024. The race was contested with TCR touring cars and run in support of the 2024 edition of the Macau Grand Prix. It was part of the TCR World Tour and TCR China Touring Car Championship, as the final round for both series.

Race one was won by Thed Björk, who started from pole position, with Norbert Michelisz and Mikel Azcona joining him on the podium. Dušan Borković won race two ahead of Esteban Guerrieri and Marco Butti, making it a full sweep of the podium for their team GOAT Racing.

Mikel Azcona
Mikel Azcona Troyas (born 25 June 1996) is a Spanish auto racing driver who competes in the TCR World Tour for BRC Hyundai N Squadra Corse and the Porsche Supercup and Porsche Carrera Cup Germany for Hadeca Racing. He won the TCR Europe Touring Car Series championship in 2018 and 2021 and the WTCR championship in 2022.
Rahima Orchient Yayah
Rahima Orchient Yayah (born 1974) is a Malaysian businesswoman, former model and beauty pageant titleholder who was crowned as Miss Malaysia World 1994. She represented Malaysia in Miss World 1994 pageant in Sun City, South Africa where she placed in the top 10 semi-finalists.

Sun City (South Africa)
Sun City is a luxury resort and casino, situated in the North West Province of South Africa. It is located between the Elands River and the Pilanesberg, built on the land of Bakubung Village (known as Ledig) about 140 km (90 miles) northwest of Johannesburg, near the city of Rustenburg. The complex borders the Pilanesberg National Park. It is made up of a number of themed sub-resorts with hotels on each, including the original Sun City Resort, The Cabanas, The Cascades and the Lost City (The Palace).
Rager
Rager is a surname. Notable people with the surname include:

Chris Rager, American voice actor

Jordan Rager (born 1994), American country music singer

Mose Rager (1911–1986), American guitarist

Roger Rager (1948–2022), American racing driver

Yitzhak Rager (1932–1997), Israeli journalist, diplomat and a Likud mayor of Beersheba

Chris Rager
Chris Rager is an American voice actor who works on a number of English versions of Japanese anime series. His major roles include Mr. Satan in Dragon Ball, Will Powers in Ace Attorney, and Arlong in the Funimation dub of One Piece. In video games, he voices Mr. Torgue in the Borderlands series.
Aberdeen, South Africa
Aberdeen, officially Xamdeboo after a name change in February 2026, is a small town in the Sarah Baartman District Municipality of the Eastern Cape province of South Africa. With its numerous examples of Victorian architecture, it is one of the architectural conservation areas of the Karoo.

Aberdeen lies some 55 km south-west of Graaff-Reinet, 155 km east-south-east of Beaufort West and 32 km south of the Camdeboo Mountains. Laid out on the farm Brakkefontein as a settlement of the Dutch Reformed Church in 1856 (now the NGK), it became a municipality in 1858. It is named after the Scottish city of Aberdeen.

In 2023, it was proposed for Aberdeen to be renamed to Camdeboo. In 2024, this was rejected by the Minister of Sport, Arts and Culture, Gayton McKenzie. In 2026, a name with a slight difference in spelling (Xamdeboo) was officially approved by the minister in January, with the name change taking place in February.

Dutch Reformed Church, Aberdeen
The Dutch Reformed Church in Aberdeen is a congregation of the Dutch Reformed Church in the Synod of Eastern Cape. It is one of six congregations established in 1855 in what was then the Cape Colony, and the penultimate of the year. Jansenville, also in the Eastern Province, was first on February 4, Ceres was second on March 21, Sutherland third, then Aberdeen, Heidelberg fifth and Simon's Town sixth. The church is the tallest congregation in the country.
Final Battle 2009
Final Battle 2009 was a professional wrestling pay-per-view (PPV) event produced by Ring of Honor (ROH), which was only available online. It took place on December 19, 2009 from the Manhattan Center in New York City. Nine matches were aired live from the show, and a bonus match aired during the intermission.

It was the seventh annual event in the Final Battle chronology, with the first taking place in 2002. It was the first ROH event to be shown as an online pay-per-view, and the first to be aired live.

Manhattan Center
The Manhattan Center is a building in Midtown Manhattan, New York City. Built in 1906 and located at 311 West 34th Street, it houses Manhattan Center Studios, the location of two recording studios; its Grand Ballroom; and the Hammerstein Ballroom, a performance venue. In 1976, the building was purchased by its current owner, the Unification Church, for $3 million. Some of the enterprises residing in the building include: Telemundo, Macy's, CFDA, WeWork, Facebook, iHeart Media, Samsung, American Heart Association, Robin Hood, FX Network, Endeavor, MAC Cosmetics, Viacom, SiriusXM, NBA, NBC Universal, Masterbeat, Formerly Al Jazeera America HQ, Broadway Cares Equity Fights AIDS.
Amy Parmenter
Amy Parmenter (born 1 August 1997) is an Australian netball player in the Suncorp Super Netball league, playing as the Captain for Melbourne Mavericks, for which she also serves as the club's Australian Netball Players’ Association delegate.

Parmenter was elevated to the Giants senior ahead of the 2019 season, having been a training partner with Canberra Giants in the Australian Netball League for the previous two years. Before her time in the ANL, Parmenter represented Netball New South Wales and Australia at under 17/19 levels. Parmenter had an outstanding debut season at the Giants, recording the most intercepts of any midcourter in the competition and ending the year by being awarded the Rising Star award.

Netball New South Wales
Netball New South Wales is the governing body for netball in New South Wales. It is affiliated to Netball Australia. It is responsible for organizing and managing two elite level teams, New South Wales Swifts and Giants Netball, who compete in Suncorp Super Netball. It is also responsible for organizing and managing the Netball NSW Premier League as well as numerous other leagues and competitions for junior and youth teams. Its headquarters are based at Netball Central, Sydney Olympic Park.
Sefwi Bodi
Sefwi Bodi is a town in the Western North Region of Ghana. The people in this community are mainly Sefwi people. In June 2012, Sefwi Bodi wa elevated as the capital of the then newly created Bodi (district). Bodi is 60 km west of Sefwi Wiawso the regional capital. Bodi is considered one of the breadbaskets of Ghana.
Greater percentage of foodstuffs sent to major cities like Kumasi and Accra come from Sefwi Bodi.
Sefwi Bodi is noted for the Bodi Senior High School which is the only second cycle institution in that area of the country.

Western North Region
The Western North region is one of the six new regions of Ghana created in 2019. The region is bounded by the Ivory Coast (Comoé District) on the west, the Central region in the southeast, and the Ashanti, Ahafo, Bono East and Bono regions in the north. The Western North region has the highest rainfall in Ghana, lush green hills, and fertile soils. There are numerous small and large-scale gold mines companies in the region. The ethnic culture of the region is dominated by the Sefwis. The main languages spoken are Sefwi, Akan, French and English.
Edel Randem
Edel Signe Randem (married name: Uvland; born 11 September 1910 in Oslo – died 26 November 2001 in Stabekk) was a Norwegian figure skater.

She competed at the 1931 World Figure Skating Championships, where she finished sixth. She also finished sixth at the 1930 European Figure Skating Championships, and she participated at the 1928 Winter Olympics, where she finished 13th.

Randem won the Norwegian Figure Skating Championships in 1930 and 1931.

1931 World Figure Skating Championships
The World Figure Skating Championships is an annual figure skating competition sanctioned by the International Skating Union in which figure skaters compete for the title of World Champion.

The competitions took place from February 28 to March 1 in Berlin, Germany. It was the second year when all competitions were held at the same location and the same time, but the first time in Europe.
Santa María del Lago
Santa María del Lago (Spanish: Humedal Santa María del Lago) is a wetland, part of the Wetlands of Bogotá, located in the locality Engativá, Bogotá, Colombia. The wetland on the Bogotá savanna covers an area of 12 hectares (30 acres), of which 4 hectares (9.9 acres) water.

Engativá
Engativá is the 10th locality (localidad) of Bogotá, the capital city of Colombia. located in the city’s west‑northwest sector, Originally an independent municipality of the Department of Cundinamarca, Engativá was annexed to Bogotá in 1954, forming part of the city’s expansion.

Its territory is a mix of predominantly residential neighborhoods, commercial areas, industrial zones, and a lot of rural-like spaces used for agriculture and green zones.

It lies at an average elevation of around 2,600 meters (about 8,500 ft) above sea level.
1971 Kew state by-election
A by-election for the seat of Kew was held on 17 April 1971 following the resignation of then Deputy Premier, Arthur Rylah. The by-election was won by Liberal candidate, Rupert Hamer, he won with a 64.8% primary vote and a swing of 23.1% in his primary vote.

Deputy Premier of Victoria
The deputy premier of Victoria is the second-most senior officer in the Government of Victoria. The deputy premier position was created in May 1932, with Robert Menzies being the first person to hold the position. The deputy premier is appointed by the Governor on the advice of the premier. The deputy premier is usually also a minister in the government.

When the Labor Party forms government, the deputy leader of the Labor parliamentary party typically becomes the deputy premier. The same was the case when the Liberal Party formed government on its own. When the Liberal-National coalition is in government, the deputy premier is usually the leader of the junior coalition partner, the Nationals (or its predecessor, the Country Party). The current deputy premier is Ben Carroll of the Labor Party, who has held the position since 2 October 2023.
Labyrinth of Dreams (film)
Labyrinth of Dreams (ユメノ銀河, Yume no ginga) is a 1997 Japanese mystery film directed by Gakuryū Ishii. It was screened in the Contemporary World Cinema section of the 1997 Toronto International Film Festival. The movie is based on a novel by Kyuusaku Yumeno.

Yumeno Kyūsaku
Yumeno Kyūsaku (夢野 久作; 4 January 1889 – 11 March 1936) was the pen name of Sugiyama Yasumichi (杉山 泰道), an early Shōwa period Japanese author, Zen priest, post office director and sub-lieutenant. The pen name roughly means "a person who always dreams". His Dharma name was Goshin-in Gin'en Taidō-koji (悟真院吟園泰道居士). He wrote detective novels and is known for his avant-gardism and his surrealistic, wildly imaginative and fantastic, even bizarre narratives.
Bassano Politi
Bassano Politi was a 16th-century Italian mathematician.

He published Questio de modalibus, a book where he collected several medieval treatises by Thomas Bradwardine, Nicole Oresme, Biagio Pelacani, and Giovanni de Casali.

Blasius of Parma
Biagio Pelacani da Parma; c. 1350 – 1416), known in English as Blasius of Parma, was an Italian philosopher, mathematician and astrologer.
He popularised English and French philosophical work in Italy, where he associated both with scholastics and with early Renaissance humanists.

He was professor of mathematics at the University of Padua, where he taught from 1382 to 1388; he taught also at the University of Pavia (1374? to 1378, and again 1389 to 1407), and the University of Bologna (1389 to 1382). His students included Vittorino da Feltre.
Wrestling at the 2011 Pan American Games – Men's freestyle 60 kg
The men's freestyle 60 kg competition of the wrestling events at the 2011 Pan American Games in Guadalajara, Mexico, was held on October 24 at the CODE II Gymanasium.  The defending champion was Yandro Quintana from Cuba.

This freestyle wrestling competition followed a single-elimination format, with a repechage to award two bronze medals. The two finalists faced off for gold and silver medals. Wrestlers who lost to the finalists advanced to the repechage, leading to two bronze medal matches where semifinal losers faced the last remaining repechage opponent from their bracket.

Each bout consisted of up to three rounds, lasting two minutes apiece. The wrestler who scored more points in each round was the winner of that rounds; the bout finished when one wrestler had won two rounds (and thus the match).

Yandro Quintana
Yandro Miguel Quintana Rivalta (born January 30, 1980, in Ciego de Ávila) is a Cuban wrestler who competed in the Men's Freestyle 60 kg at the 2004 Summer Olympics which were also his first Olympics. He won the gold medal by defeating Masuod Jokar in the final with the help of his coach Filberto Delgado. He stands 5'2 and 132 lbs. In 2003 he came in first at the Pan American Games. He now resides in Havana, Cuba.
Isma'il Pasha Abu Jabal
'Amir al-Liwa' Ismail Pasha Abu Jabal (Egyptian Arabic: إسماعيل باشا أبو جبل) (1819-1818 / 1234 AH – 24 April 1883 / 17 Jumada al-Thani 1300 AH) was an Egyptian military and political leader who lived in the nineteenth century. He participated as a brigade commander in the Crimean War, which the Egyptian army fought alongside the armies of the Ottoman Empire, and then the wāli of Egypt, Sa'id Pasha, assigned him - on Muharram 15, 1272 AH, corresponding to September 27, 1855 AD - to lead the Egyptian armies in the Crimea after Ahmad Pasha al-Munkali was excused due to ill health.

Jumada al-Thani
Jumada al-Thani, also known as Jumada al-Akhirah, Jumada al-Akhir, or Jumada II, is the sixth month of the Islamic calendar. The word Jumda (جمد), from which the name of the month is derived, is used to denote dry, parched land, a land devoid of rain. Jumādā (جمادى) may also be related to a verb meaning 'to freeze', and another account relates that water would freeze in pre-Islamic Arabia during this time of year.
Madame l'archiduc
Madame l’archiduc (French pronunciation: [madam laʁʃidyk]) is an opéra bouffe, or operetta in three acts, by Jacques Offenbach, with a French libretto by Albert Millaud first performed at the Bouffes-Parisiens (Salle Choiseul) in Paris in 1874.

After a slow start Madame l’Archiduc had an opening run of 100 performances. It was seen in Vienna in 1875 and London in 1876. Highlights of the score include the quartet in cod-English for the count, countess and young couple in Act 1, an ‘alphabet’ sextet for Marietta, Giletti and the conspirators in Act 2, and a polka for the arrival of the dragoons.

A 1963 radio recording conducted by Jean-Claude Hartemann, with Lina Dachary as Marieta, Raymond Amade as Fortunato, and Gaston Rey and Janette Levasseur as the count and countess, was issued in 1990 in the Gaîeté-Lyrique/Musidisc series of French light opera.

Théâtre des Bouffes-Parisiens
The Théâtre des Bouffes-Parisiens (French: [teɑtʁ de buf paʁizjɛ̃]) is a Parisian theatre founded in 1855 by the composer Jacques Offenbach for the performance of opéra bouffe and operetta. The current theatre is located in the 2nd arrondissement at 4 rue Monsigny with an entrance at the back at 65 Passage Choiseul. In the 19th century the theatre was often referred to as the Salle Choiseul. With the decline in popularity of operetta after 1870, the theatre expanded its repertory to include comedies. The theatre is still active with regular productions of stage plays.
Basildon Meadows
Basildon Meadows is a 6.5-hectare (16-acre) biological Site of Special Scientific Interest on the southern outskirts of Basildon in Essex.

The site is composed of three unimproved meadows which have a wide variety of herbs. The principal grasses are creeping bent, rye-grass and crested dog's tail. Flowers include the green-winged orchid and yellow rattle, which are rare in Essex. There is also a small pond and scattered scrub.

The fields are private land with no public access.

Lolium perenne
Lolium perenne, common name perennial ryegrass, English ryegrass, winter ryegrass, or ray grass, is a grass from the family Poaceae. It is native to Europe, Asia and northern Africa, but is widely cultivated and naturalised around the world.