hotchpotch/wikipedia-multilingual-ir-related-lead-pairs huggingface dataset:
Lang's theorem
In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field , then, writing  for the Frobenius, the morphism of varieties

is surjective. Note that the kernel of this map (i.e., ) is precisely .

The theorem implies that   
vanishes,  and, consequently, any G-bundle on  is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type.

It is not necessary that G is affine. Thus, the theorem also applies to abelian varieties (e.g., elliptic curves.) In fact, this application was Lang's initial motivation. If G is affine, the Frobenius  may be replaced by any surjective map with finitely many fixed points (see below for the precise statement.)

The proof (given below) actually goes through for any  that induces a nilpotent operator on the Lie algebra of G.

Serge Lang
Serge Lang (French: [lɑ̃ɡ]; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He received the Frank Nelson Cole Prize in 1960 and was a member of the Bourbaki group.

As an activist, Lang campaigned against the Vietnam War, and also successfully fought against the nomination of the political scientist Samuel P. Huntington to the National Academies of Science. Later in his life, Lang was an HIV/AIDS denialist. He claimed that HIV had not been proven to cause AIDS and protested Yale's research into HIV/AIDS.
Ministry of Local Government (Sweden)
The Ministry of Local Government (Swedish: Kommundepartementet, Kn) was a ministry in Sweden established in 1974. The ministry was responsible for the county administrative boards, the municipalities, the tax charge offices, the local enforcement offices, etc., and later also questions regarding the Church of Sweden and other religious communities. The ministry was headed by the minister for physical planning and local government. The ministry ceased in 1982 and the Ministry for Civil Service Affairs took over its tasks.

Minister for Physical Planning and Local Government
The Minister for Physical Planning and Local Government (Swedish: Kommunminister) was a member of the government of Sweden. The minister for physical planning and local government was the head of the Ministry of Local Government from 1974 to 1982 which was responsible for the county administrative boards, the municipalities, the tax charge offices, the local enforcement offices, etc., and later also questions regarding the Church of Sweden and other religious communities.
OK-N
OK-National (OK-N) is an international kart racing class sanctioned by the Commission Internationale de Karting (CIK-FIA) for drivers aged 14 and over. Debuting in international competition in 2023, OK-N is the secondary senior class in FIA championships to OK.

Approved by the FIA World Motor Sport Council in 2022, OK-N was designed as a simplified version of OK regulations to reduce costs for competitors. The class debuted in international competition the following year, being contested in several championships across Europe and the Middle East.

Its inaugural World Cup was held in 2024—with qualifying decided via national championships held in 12 countries—and won by Kyuho Lee. The Arrive & Drive World Cup hosted its first edition in 2025, using a modified OK-N engine in a single-design format.

Kart racing
Kart racing or karting is a motorsport discipline using open-wheel, four-wheeled vehicles known as go-karts or shifter karts. They are usually raced on scaled-down circuits, although some professional kart races are also held on full-size motorsport circuits. Karting is commonly perceived as the stepping stone to the higher ranks of motorsports. Most modern Formula One drivers, including Ayrton Senna, Michael Schumacher, Fernando Alonso, Kimi Räikkönen, Lewis Hamilton, Sebastian Vettel, Nico Rosberg, and Max Verstappen, have begun their racing careers in karting. NASCAR Cup driver Connor Zilisch also got his start in karting.

Karts vary widely in speed, and some (known as superkarts) can reach speeds exceeding 160 kilometres per hour (100 mph), while recreational go-karts intended for the general public may be limited to lower speeds.
Ruhlsdorf (Strausberg)
Ruhlsdorf is a village and a civil parish (Ortsteil) of the German town of Strausberg, located in the district of Märkisch-Oderland in Brandenburg. With a population of 44 (as of 2007), it is the littlest one of its municipality.

Strausberg
Strausberg (German: [ˈʃtʁaʊsˌbɛʁk] ⓘ) is a town in Brandenburg, in eastern Germany, located 30 km (19 mi) east of Berlin. With a population of about 27,000 it is the largest town in the district of Märkisch-Oderland.
Exchange Irrigation Scheme
18°48′03″S 29°01′48″E﻿ / ﻿18.800917°S 29.029886°E﻿ / -18.800917; 29.029886

Exchange Irrigation Scheme is a 165 hectares (410 acres) irrigated arable land in Zhombe Communal Land but in Silobela constituency in Kwekwe District of the Midlands Province of Zimbabwe. It is 37 km (23 mi) southwest of Zhombe Joel, 83 km (52 mi) northwest of Kwekwe and 25 km (16 mi) north of Crossroads DSC.

It is in region 3; the climatic conditions are semi-arid with an average yearly rainfall of 632 mm (24.9 in). Estimated elevation is 1,200 metres (3,900 ft) above sea level.

Zhombe
Zhombe, originally known as Jombe, is a rural communal area in Kwekwe District, Midlands Province, Zimbabwe. It is an area of mixed Shona and Northern Ndebele People. It lies along the Mnyathi border line between Midlands and Mashonaland Provinces. There are a few commercial farms within its borders and a handful of resettlement areas. Most of it is within the Zhombe Constituency. Its administrative centre is the Zhombe Joel Growth Point, and it is under the Zibagwe Rural District Council.

There are fourteen business centres in Zhombe. Ten of the business centres are electrified. There are also two rural service centres: Empress and Zhombe Joel. As of 2011, there were 53 primary schools and 18 secondary schools. There are ten clinics (and one mission hospital) and several health centers in Zhombe.

The area is primarily rangeland supporting over 48,000 cattle, with 13 animal health centers and 26 dip tanks.
Henry Markram
Henry John Markram (Hebrew: הנרי מרקרם; born 28 March 1962) is a South African-born Israeli neuroscientist, professor at the École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland and director of the Blue Brain Project and founder of the Human Brain Project.

École Polytechnique Fédérale de Lausanne
EPFL (officially no longer an acronym; originally short for French: École Polytechnique Fédérale de Lausanne, lit. 'Federal School of Technology in Lausanne') is a public research university in Lausanne, Switzerland. Founded in 1969, the university primarily teaches and conducts research in science, technology, engineering, and mathematics (STEM) fields.

Like its sister institution ETH Zurich, EPFL is part of the Swiss Federal Institutes of Technology Domain, a consortium of universities and research institutes under the Swiss Federal Department of Economic Affairs, Education and Research. As of 2024, EPFL enrolled 14,012 students from over 130 countries.

Students, faculty, and researchers affiliated with EPFL include a Nobel laureate, two Fields Medalists, two Pritzker Prize winners, and one Turing Award recipient. EPFL has an urban campus that extends alongside Lake Geneva, and includes the EPFL Innovation Park as well as university research centers and affiliated laboratories.
Marilyn McLeod
Marilyn McLeod (May 27, 1939 – November 24, 2021) was an American songwriter and occasional singer. McLeod began her career as a songwriter for Motown. Together with Pam Sawyer, she wrote the 1976 Diana Ross hit "Love Hangover".

Pam Sawyer
Pamela Joan Sawyer (born 1938) is an English songwriter/lyricist, who started writing songs in the mid-1960s and whose credits as a co-writer at Motown included "Love Child", "If I Were Your Woman", "My Whole World Ended (The Moment You Left Me)", and "Love Hangover".
Emil Georg Bührle
Emil Georg Bührle (German pronunciation: [ˈeːmɪl ɡeˈɔʁk ˈbyːʁlə]; 31 August 1890 - 26 November 1956) was a German-born Swiss industrialist, controversial armament manufacturer and art collector. Bührle was long-term managing owner of Oerlikon-Bührle and the founding patron of Foundation E.G. Bührle. By the end of World War II, Bührle had become Switzerland's richest man after having been told by the Swiss authorities to not only supply weapons to the Allies but also to Nazi Germany. He was the patriarch of the Bührle family.

Oerlikon-Bührle
Oerlikon-Bührle (full name Werkzeugmaschinenfabrik Oerlikon, Bührle & Co., initially Werkzeugmaschinenfabrik Oerlikon) was a Swiss armaments company based in Oerlikon (Zürich) from 1906 to 1999.

After restructurings and the sale of various core divisions, in particular the armaments division Oerlikon Contraves Defence to the German Rheinmetall DeTec, the holding company was merged into the present OC Oerlikon in January 2000.
EPH (company)
Energetický a průmyslový holding, a.s. (EPH) is a Czech Republic (Prague) based company currently investing mainly in the energy sector in Europe, founded in 2009. The group's business lines covers the complete value chain ranging from highly efficient cogeneration, power generation, and natural gas transmission, gas storage, gas and electricity distribution and supply. The majority owner is Czech entrepreneur and lawyer Daniel Křetínský (94%), who is ranked among the richest Czechs in the long term and according to Forbes he became a dollar billionaire in 2017. The remaining 6% is distributed among selected managers of EPH. EPH employs almost 25.000 people and comprises over 50 companies.

Daniel Křetínský
Daniel Křetínský (Czech: [ˈdanɪjɛl ˈkr̝ɛtiːnskiː]; born 9 July 1975) is a Czech billionaire businessman and lawyer, who is the chief executive officer (CEO) and 94% owner of Energetický a průmyslový holding (EPH), co-owner and president of football club Sparta Prague, and director and major shareholder of English football club West Ham United.
Wiener Neustädter Lokomotivfabrik
The Wiener Neustädter Lokomotivfabrik (Wiener Neustadt locomotive factory) was the largest locomotive and engineering factory in the Austro-Hungarian Empire. During World War II the company produced armaments as part of Rax-Werk Ges.mbH which was associated with the Mauthausen concentration camp.

Raxwerke
Raxwerke or Rax-Werke was a facility of the Wiener Neustädter Lokomotivfabrik at Wiener Neustadt in Lower Austria. During World War II, the company also produced lamps for Panzer tanks and anti-aircraft guns.  Two Raxwerke plants employed several thousand forced laborers from the Mauthausen-Gusen concentration camp (on 20 June 1943 Mauthausen delivered ~500 prisoners to the Rax-Werke).: 189
Installment sales method
The installment sales method is one of several approaches used to recognize revenue under the US GAAP, specifically when revenue and expense are recognized at the time of cash collection rather than at the time of sale. Under the US GAAP, it is the principal method of revenue recognition when the recognition occurs subsequently to the sale.

Revenue recognition
In accounting, the revenue recognition principle states that revenues are earned and recognized when they are realized or realizable, no matter when cash is received.

It is a cornerstone of accrual accounting together with the matching principle. Together, they determine the accounting period in which revenues and expenses are recognized. In contrast, the cash accounting recognizes revenues when cash is received, no matter when goods or services are sold.

Cash can be received in an earlier or later period than when obligations are met, resulting in the following two types of accounts:

Accrued revenue: Revenue is recognized before cash is received.

Deferred revenue: Revenue is recognized when cash is received.
Georgia State Route 20
State Route 20 (SR 20) is a 165.345-mile-long (266.097 km) state highway roughly in the shape of a capital J rotated ninety degrees to the left, which travels through portions of Floyd, Bartow, Cherokee, Forsyth, Gwinnett, Walton, Rockdale, Newton, and Henry counties in the northwestern and north-central parts of the U.S. state of Georgia. Its counterclockwise, or western terminus is at the Alabama state line in Floyd County, and its clockwise, or eastern terminus occurs at its interchange with Lower Woolsey Road southwest of Hampton in Henry County south-southeast of the Atlanta Motor Speedway.

Floyd County, Georgia
Floyd County is a county in the Northwest region of the U.S. state of Georgia. As of the 2020 United States census, the population was 98,584. The county seat is Rome. Floyd County comprises the Rome, Georgia metropolitan statistical area.
Puszcza Niepołomice
MKS Puszcza Niepołomice (Polish pronunciation: [ˈpuʂt͡ʂa ɲɛpɔwɔˈmit͡sɛ]) is a professional football club located in Niepołomice, Poland. The team competes in the I liga, the second level of the Polish football league system. The team's colors are yellow, white and green.

In 2023, the club secured promotion to the Ekstraklasa for the first time in its history, remaining in the top flight until 2025.

Niepołomice
Niepołomice (pronounced Niepowomitse [ɲɛpɔwɔˈmʲit͡sɛ]) is a town in southern Poland, seat of Gmina Niepołomice in the Wieliczka County in the Lesser Poland Voivodeship.

It is situated on the Vistula River, on the verge of the large virgin Niepołomice Forest.

There is a 14th-century hunting castle in town initially built by Casimir III, as well as a conservation center for European bison (Polish: Żubry) nearby.

The town is also home to professional football club Puszcza Niepołomice.

Hunting castle, exterior

Town hall

Church of 10,000 Martyrs
Turbo C
Turbo C is a discontinued integrated development environment (IDE) and compiler for the C programming language from Borland. First introduced in 1987, it was noted for its integrated development environment, small size, fast compile speed, comprehensive manuals and low price.

In May 1990, Borland replaced Turbo C with Turbo C++. In 2006, Borland reintroduced the Turbo moniker.

Turbo C++
Turbo C++ is a discontinued C++ compiler and integrated development environment originally from Borland. It was designed as a home and hobbyist counterpart for Borland C++. As the developer focused more on professional programming tools, later Turbo C++ products were made as scaled down versions of its professional compilers.
TACAM T-60
The TACAM T-60 (Tun Anticar pe Afet Mobil T-60 – "Anti-tank gun on T-60 mobile gun carriage") was a Romanian tank destroyer used during World War II. It was built by removing the turret of captured Soviet T-60 light tanks and building a pedestal to mount a captured Soviet 76.2 mm (3.00 in) M-1936 F-22 field gun in its place. A three-sided fighting compartment was built to protect the gun and its crew. Thirty-four were built in 1943, and they served in the Jassy-Kishniev Offensive, and the Budapest Offensive.

T-60 tank
The T-60 scout tank was a light tank produced by the Soviet Union from 1941 to 1942. During this period, 6,292 units were built. The tank was designed to replace the obsolete T-38 amphibious scout tank and saw action during World War II.

The Kingdom of Romania used the T-60 chassis to build some locally-designed tank destroyers.
Nabeela (actress)
Shireen Gul, also known as Nabeela (Urdu:نبیلہ) was a Pakistani actress. She was known as Malka-e-Jazbat (The Queen of Emotions) because of the tragic roles she often portrayed in films. She worked in both Urdu and Punjabi films and is known for her roles in films Ashiana (1964), Badnaam (1966), Babul Da Wehra (1968), Jagg Beeti (1968), Pagri Sanbhal Jatta (1968), Saiqa (1968), Veryam (1969), Ishq Na Puchhay Zaat (1969), Sheran Day Puttar Sher (1969), Sajna Door Deya (1970), Maa Puttar (1970), Dunya Matlab Di (1970), Babul (1971), Charagh Kahan Roshni Kahan (1971).

Ashiana (1964 film)
Ashiana is a 1964 Pakistani film directed by S. M. Yusuf, who also co-produced it.

Ashiana is the third collaboration of S. M. Yusuf with Fayyaz Hashmi and A.Hameed, after Saheli (1960)  and Aulad (1962). S. M. Yusuf received his third Nigar Award for best director at the annual award ceremony.
Al-Okhdood Club
Al-Okhdood Club (Arabic: نادي الأخدود, romanized: nādī ʿukhdūdi, lit. 'Ditch') is a Saudi professional football club based in Najran, in the southern region of Saudi Arabia. Founded in 1976, the club competes in the Saudi Professional League, the first tier of Saudi football. The club have won the Saudi Second Division twice in 1992 and most recently in 2021. Besides football, the club also consists of various other departments including karate, handball, taekwondo, volleyball, water polo, basketball and cycling. The club's name alludes to the People of the Ditch mentioned in the Qur'an.

2020–21 Saudi Second Division
The 2020–21 Saudi Second Division was the 45th season of the Saudi Second Division since its establishment in 1976. The season started on 30 October 2020 as a consequence of the postponement of the previous season's conclusion due to the COVID-19 pandemic. The group stage draw was held on 22 September 2020. The final was played on 19 April 2021 between Al-Akhdoud and Al-Orobah. Al-Akhdoud defeated Al-Orobah 4–2 on penalties (3–3 after extra time) to win their second title and first since 1992.
Miami River (Florida)
The Miami River is a river in the U.S. state of Florida that drains out of the Everglades and runs through the city of Miami, including the downtown area. The 5.5-mi-long (8.9 km) river flows from the terminus of the Miami Canal at Miami International Airport to Biscayne Bay. It was originally a natural river inhabited at its mouth by the Tequesta Native Americans, but it was dredged and is now polluted throughout its route through Miami-Dade County. The mouth of the river is home to the Port of Miami and many other businesses, whose pressure to maintain it has helped to improve the river's condition.

Tequesta
The Tequesta, also Tekesta, Tegesta, Chequesta, Vizcaynos, were a Native American tribe on the Southeastern Atlantic coast of Florida. They had infrequent contact with Europeans and had largely migrated by the middle of the 18th century.
End of Basque home rule in France
The end of Basque home rule or foruak/fors, the native institutional and legal system, took place during the French revolutionary period (1789-1795). The final violent dissolution of the semi-autonomous Basque institutional and legal system was coupled with the arrival of French troops to the Basque Country within the War of the Pyrenees and a deliberate terror on the Basque population centred in Labourd.

It resulted in the abrupt suppression of all native governmental and jurisdictional organs and the establishment of the departement of the Lower Pyrenees (Basses-Pyrénées), as well as the departement administrative system, as everywhere in France.

End of Basque home rule in Spain
The end of Basque home rule or fueros in Spain was a process coming to a head in the period extending from the First Carlist War (1833–1840) to the aftermath of the Third Carlist War (1876–1878). It brought to an end the status of sovereignty coupled with allegiance to the Crown of Castile held by the different Basque territories for centuries. In the French Basque Country, home rule was abruptly suppressed during the French Revolution (starting 1790).

The loss of home rule (fueros) was followed by the Basque Economic Agreement (1878), a period of shaky peace with occasional popular uprisings, like the Gamazada centred in Navarre, and the emergence of Basque nationalism.
Georgia State Route 7
State Route 7 (SR 7) is a 216.0-mile-long (347.6 km) state highway that travels in a southeast-to-northwest orientation through portions of Lowndes, Cook, Tift, Turner, Crisp, Dooly, Houston, Peach, Crawford, Monroe, Lamar, Pike, and Spalding counties in the southern and central parts of the U.S. state of Georgia. The highway connects the Florida state line southeast of Lake Park to the Griffin area, via Valdosta, Tifton, Cordele, Perry, and Barnesville. The highway is concurrent with either US 41 or US 341 for its entire length, and closely parallels I-75 for much of its length.

SR 7 was established at least as early as 1919 along nearly the same path it travels today. US 41/SR 7 was designated on a concurrency with I-75 northwest of Valdosta to Hahira in 1982. US 41/SR 7 were re-routed onto an eastern bypass of Valdosta in 2006.

Turner County, Georgia
Turner County is a county located in the south central portion of the U.S. state of Georgia. As of the 2020 census, the population was 9,006. The county seat is Ashburn. The county was created on August 18, 1905, and named for Henry G. Turner, U.S. representative and Georgia state Supreme Court justice.
Speyergau
Speyergau was a medieval county in the East Frankish (German) stem duchy of Franconia. It was centred around the administrative centre of Speyer and roughly covered the former Roman administrative area of Civitas Nemetum, which is today the south-eastern portion of the Palatinate region between the Rhine river, the Palatinate Forest range, and some smaller parts of northern Alsace. The Speyergau, together with the neighbouring Wormsgau and Nahegau, was part of the major possessions held by the Salian dynasty of German kings and Holy Roman Emperors.

Wormsgau
The Wormsgau (Latin: pagus wormatiensis) was a medieval county in the East Frankish (German) stem duchy of Franconia, comprising the surroundings of the city of Worms and further territories on the left bank of the Upper Rhine river. Together with the neighbouring Nahegau and Speyergau, it belonged to the central Rhenish Franconian possessions of the Imperial Salian dynasty.
Vidas robadas (Mexican TV series)
Vidas robadas is a Spanish-language telenovela produced by the Mexican television network TV Azteca. It is internationally known as Infamia. It stars Carla Hernandez, Andres Palacios, and marks the return of the grand actress, Cristian Bach to the television screen.

Carla Hernández
Carla Hernández (born February 18, 1987) is a Mexican actress, who attended TV Azteca's acting class, Centro de Formación Actoral. She made her official acting debut in Azteca's Vidas Robadas, as Luz and Camila, the protagonist and antagonist, alongside Christian Bach and Andrés Palacios, in 2010.
Sunda collared dove
The Sunda collared dove (Streptopelia bitorquata), sometimes referred to as the island collared dove, Javanese turtle dove, Javanese collared dove, or Indonesian collared dove, is a species of bird in the family Columbidae. Sometimes confused with the Eurasian collared dove (Streptopelia decaocto) which is very similar in appearance, the Sunda collared dove is a small to medium-sized bird, native to the tropical and subtropical islands of Indonesia.

Eurasian collared dove
The Eurasian collared dove (Streptopelia decaocto), often simply just collared dove, is a dove species native to Europe, Asia, and northern Africa. It has also been introduced to Japan, North and Central America, and the islands in the Caribbean.