OpenLLM-France/wikimedia huggingface dataset:
# Rouché–Capelli theorem

Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions of a system of linear equations, given the ranks of its augmented matrix and coefficient matrix. The theorem is variously known as the:
* Rouché–Capelli theorem in English speaking countries, Italy and Brazil;
* Kronecker–Capelli theorem in Austria, Poland, Ukraine, Croatia, Romania, Serbia and Russia;
* Rouché–Fontené theorem in France;
* Rouché–Frobenius theorem in Spain and many countries in Latin America;
* Frobenius theorem in the Czech Republic and in Slovakia.

## Statement

A system of linear equations with n variables and coefficients in a field K has a solution if and only if its coefficient matrix A and its augmented matrix [A|b] have the same rank. If there are solutions, they form an affine subspace of $K^{n}$ of dimension n − rank(A). In particular:
* if n = rank(A), the solution is unique,
* if n > rank(A) and K is an infinite field, the system of linear equations admits infinitely many solutions,
* if K is a finite field, the number of solutions is finite, namely $|K|^{n-\mathrm {rank} (A)}$.

## Example

Consider the system of equations
    x + y + 2z = 3,
    x + y + z = 1,
    2x + 2y + 2z = 2.
The coefficient matrix is
    $A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}},$
and the augmented matrix is
    $(A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&2\end{array}}\right].$
Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are infinitely many solutions.
In contrast, consider the system
    x + y + 2z = 3,
    x + y + z = 1,
    2x + 2y + 2z = 5.
The coefficient matrix is
    $A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}},$
and the augmented matrix is
    $(A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&5\end{array}}\right].$
In this example the coefficient matrix has rank 2, while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent columns has made the system of equations inconsistent.

## Proof

There are several proofs of the theorem. One of them is the following one.
The use of Gaussian elimination for putting the augmented matrix in reduced row echelon form does not change the set of solutions and the ranks of the involved matrices. The theorem can be read almost directly on the reduced row echelon form as follows.
The rank of a matrix is the number of nonzero rows in its reduced row echelon form. If the ranks of the coefficient matrix and the augmented matrix are different, then the last non zero row has the form $[0\ldots 0\mid 1],$ corresponding to the equation 0 = 1. Otherwise, the ith row of the reduced row echelon form allows expressing the ith pivot variable as the sum of a constant and a linear combination of the non-pivot variables, showing that the dimension of the set of solutions is the number of non-pivot variables.
# Bristol Town Hall

Bristol Town Hall, at 45 Summer Street, is the town hall of Bristol, New Hampshire. It is a single story Greek Revival structure, built in 1849, and was the town's first purpose-built town hall. It continues to serve as a municipal meeting and polling place, although town offices are now in a modern building on Lake Street. The building was listed on the National Register of Historic Places in 2015.

## Description and history

Bristol Town Hall is located on the fringe of the town's central downtown area, on the north side of Summer Street just east of School Street. It is a single-story wood-frame structure, with a front-facing gabled roof and clapboarded exterior. The front facade is three bays wide, the bays articulated by wide paneled pilasters. The center bay houses the building entrance in the center of a wide recess, and the flanking bays house paired slender sash windows. The pilasters rise to an entablature beneath the fully pedimented gable.
The town of Bristol was incorporated in 1819 out of portions of neighboring towns. Its early town meetings were held in schools and the local Methodist church. Discussions began 1841 for the construction of a dedicated town meeting hall, which came to fruition with the construction of this hall in 1849. The building was enlarged in 1872, by adding 20 feet (6.1 m) to the rear, in order to add a stage. In addition to being used for town meetings, the hall hosted theatrical productions, social events, and the meetings of local fraternal societies. The building is distinctive in the state as a late example of Greek Revival architecture, and is a relatively modestly scaled town hall for the period when it was built.
# Jurģis Kalns

Jurģis Kalns (born 5 October 1982) is a Latvian football manager and former footballer who is the manager of Latvian club FK Auda

## Career

Kalns was born on 5 October 1982 in Liepāja, Latvia. He is a native of Liepāja, Latvia. He mainly operated as a forward. He started his career with FK Liepājas Metalurgs. In 2005, he signed for FC Daugava Riga. In 2008, he signed for FK Ventspils. In 2009, he signed for FC Daugava Riga. In 2010, he signed for FK Liepājas Metalurgs. In 2014, he signed for FK Metta. In 2015, he signed for FK Auda. He was described as "one of the most productive players of the Latvian football superleague... outstanding for several years".
Kalns obtained a UEFA A License. He was regarded to prefer the 4-3-3 formation. In 2015, Kalns was appointed manager of Latvian side FK Auda. In 2018, he was appointed manager of Latvian side SK Super Nova. In 2019, he was appointed manager of Latvian side FK Tukums 2000. He also worked as their assistant manager. He suffered relegation while managing the club. In 2022, he was appointed manager of Latvian side Valmiera FC. He previously worked as their assistant manager. He helped the club win their first league title.
In 2024, he was appointed manager of Cambodian side Visakha FC, but left in January 2025 and returned to FK Auda.
# Vivaha Panchami

Vivaha Panchami (Sanskrit: विवाहपञ्चमी, romanized: Vivāhapañcamī) is a Hindu festival celebrating the wedding of Rama and Sita in Janakpurdham which was the capital city of Mithila. It is observed on the fifth day of the Shukla paksha or waxing phase of moon in the month of Agrahayana (November – December) in the Bikram Samvat, calendar also known as Mangsir. The day is observed as the Vivaha Utsava of Sita and Rama in temples and sacred places associated with Rama, such as the Mithila region of India and Nepal and Ayodhya in India.

## Observances

The day is of great importance at Janakpurdham and Vivah Mandap in Janakpur, Nepal, where thousands of pilgrims arrive many from India and from other part of the country, as it is mentioned in the Ramayana that Sita married Rama here.
* Vivah Mandap, Janakpur
* Idols of Lord Rama and Mata Sita in Vivah Mandap
* Vivah Mandap view with Janakpur Dham
# Bathurst High School (New Brunswick)

Bathurst High School is an English-language secondary school located in Bathurst, New Brunswick, Canada.

## School crest

The school crest is a black shield with red edging bearing the school's initials ("BHS") stylised across the central face of the shield, written in red, with a white line that follows the top contour above the initials, and a red maple leaf below the initials, bearing the Latin motto Pari Passu in white in a scroll-like space, in the base of the shield, and included within the unbroken red edging that marks the perimeter of the crest. The shield appears to get larger about halfway down, and the expansion is occupied by large, tapering red lines that occupy the expanded space.

## School mascot

The school mascot is The Phantom. Varsity sports teams are called the "Phantoms" or the "BHS Phantoms", and the typical image of the mascot is of a hooded person or ghost, usually faceless, bearing resemblance to the iconic "Grim Reaper" image, carrying a scythe. The origins of this mascot are not related to the "Grim Reaper", but rather the legend of the "Phantom Ship of the Bay of Chaleur", a longstanding legend in the Bathurst area that a ghost ship, or phantom ship, has been seen on fire floating in the local harbour throughout the city's history.

## School motto

The official school motto is the Latin phrase Pari Passu. Curiously, although this is not the "official" motto of the school, it is the most prominent and recognized motto by students and alumni and appears on the school crest. The origins and raison d'être of the motto appear to be lost, but it is probable that the context refers to the equality of all graduates, each with an equal share to the claim of a graduate of the school. "On equal footing" is the translation of the school crest.

## Bathurst van collision

Bathurst High School made headlines across Canada on January 12, 2008, following a motor vehicle crash that killed seven students from the school's basketball team, as well as the wife of their coach. The 1997 Ford Econoline E350 Club Wagon in which they were travelling spun across the centre line and struck a tractor trailer in freezing rain conditions. Four in the van survived, including the team's coach, who was driving the van, and his daughter. The vehicle was determined to have been in poor condition, with worn all-season tires, and a partially rusted frame.

## Notable graduates

* Scott Smith, president of Hockey Canada
* Maxwell family
# Bucharest Corporate Center

Bucharest Corporate Center also known as the Generali Tower is a class A office building in Bucharest. It has 15 floors and a surface of 18,000 m$^2$. The building is owned by the real estate branch of insurance giants Assicurazioni Generali.
# Busisiwe

Busisiwe (Sibusiso male) is a feminine given name derived from the Nguni word busisa, meaning "to bless". Shortened familiar versions include Busi/Bucie or Siwe. Notable people with the name include:
* Busisiwe Mavuso (born 1977 or 1978), South African businesswoman
* Busisiwe Mkhwebane (born 1970), South African advocate and prosecutor
* Busisiwe Mncube, South African politician
* Busisiwe Ndimeni (born 1991), South African footballer
* Busisiwe Nolubabalo Nqwiliso (born 1987), South African singer known as Bucie
* Busisiwe Shiba (born 1965), South African politician
* Busisiwe Thwala (born 1987), South African musician and actress known as Cici (singer)
* Busisiwe Tshwete (born 1981), South African politician
* Victoria Busisiwe Mhlongo (1947–2010), singer, dancer and composer known as Busi Mhlongo
# 611 Place

611 Place (displayed as AT&T CENTER) is a 42-story, 189 m (620 ft) skyscraper at 611 West 6th Street in Downtown Los Angeles, California, designed by William L. Pereira & Associates and completed in 1969. The building was commissioned by the now-defunct Crocker Citizen's Bank, and served as its Southern California headquarters until 1983, when it moved to Crocker Center, now Wells Fargo Center (Los Angeles). It was subsequently bought by AT&T. It was the tallest building in Los Angeles upon completion, and the first building to surpass Los Angeles City Hall in terms of structural height (many buildings had surpassed City Hall with decorative spires, the first being Richfield Tower). It consists of a cross-shaped tower clad in vertical aluminum beams, and supported on its west side by an immense, blank slab of concrete running the entire height of the building, which houses elevator and utility shafts and is used to display corporate logos. The building features a number of Pereira's design trademarks, including cleft vertical columns, grid patterned ceilings, and architectural lanterns fitted to the exterior.
The building was described in 2023 as "long-vacant." A 2007 plan to convert the building to condominiums did not move forward.

## In popular culture

The building has appeared in several movies and television series:
* The Elevator (1974), where it appeared as a skyscraper in which an elevator stalls and traps the occupants.
* Mr. Mom (1983), where it appeared as the location of the Richardson Advertising Agency.
* Uncommon Valor (1983), where it is featured as the Houston headquarters of an oil executive.
* Con Air (1997), the building be seen from an aerial view and street view as a dead body falls from an aircraft and lands on a car near the base of the building in the city of Fresno, California.
* Epicenter (2000), This building is destroyed by an earthquake in this movie.
* The Day After Tomorrow (2004), where it appeared in shots of Manhattan.
* Along Came Polly (2004), where it was the starting point of an ill-fated BASE jump.
* The Morning Show (2019 - 1st season, 2021 - 2nd season), where 611 Place is shown as UBA's headquarters.
# Carlos Cordero (runner)

Carlos Cordero Gómez (born January 7, 1977, in Irapuato, Guanajuato) is a Mexican athlete. He participated at the 2008 Summer Olympics finishing 32nd in the men's Marathon. and he also represented Mexico at the 2012 Summer Olympics.
He won a silver medal in the 1500 metres at the 2010 Central American and Caribbean Games

## Achievements

| Year | Competition | Venue | Position | Event | Notes |
| 2010 | 2010 Central American and Caribbean Games | Mayagüez, Puerto Rico | 2nd | Marathon | 2:22.06 |
| 2011 | 2011 Pan American Games | Guadalajara, Mexico | 6th | Marathon | 2:20:49 |
| 2008 | 2008 Summer Olympics | Beijing, China | 32nd | Marathon | 2:18:40 |
| 2009 | 2009 World Championships in Athletics | Berlin, Germany | 49th | Marathon | 2:22:16 |
| 2012 | 2012 Summer Olympics | London, United Kingdom | 60th | Marathon | 2:22:08 |
# Port Macquarie Airport

Port Macquarie Airport (IATA: PQQ, ICAO: YPMQ) is an airport in Port Macquarie, New South Wales, Australia. The airport is 2.25 nautical miles (4.17 km; 2.59 mi) west of the city centre and is owned and managed by the Port Macquarie-Hastings Council on behalf of its owners - the constituents of the Hastings. The ownership is not to be confused with the Port Macquarie Airport Proprietary Limited. The airport had 222,534 revenue passengers, the 32nd busiest in Australia, and 4,821 aircraft movements in 2017–2018.

## Airlines and destinations

| Airlines | Destinations |
| Eastern Air Services | Lord Howe Island |
| QantasLink | Brisbane, Sydney |
| Rex Airlines | Sydney |
Historically a number of airlines have flown to the airfield. These include: -Virgin Australia -JetGo -Bonza -Ansett Australia -Hazelton Airlines -Kendell Airlines -East-West Airlines -FlyPelican -Brindabella Airlines

## Virgin Australia

On 4 February 2008, Virgin Blue (now Virgin Australia) commenced direct scheduled services between Sydney and Port Macquarie. The flights operated daily in the morning. The service used to be operated by an Embraer E-170 aircraft, and was the first scheduled jet service for the airport. Connections were offered to all major Australian cities via Sydney. An increase to twice-daily service commenced 4 August 2008.
Competition by Virgin resulted in a significant drop in fare prices for the Port Macquarie - Sydney route, where Qantas previously operated a monopoly after Hazelton Airlines ceased services in 2001.
On 28 July 2011, Virgin Australia (in partnership with Skywest Airlines) announced its intention to commence a daily Brisbane to Port Macquarie service from October 2011. On 19 October 2011, Virgin Australia's ATR 72 departed on its first flight to Brisbane. The airline also announced that, as a result of the removal of Embraer 170 aircraft from its fleet, Sydney to Port Macquarie services would also be operated by ATR72 aircraft.
Both routes have since been withdrawn due to the removal of ATR72 aircraft from the Virgin Australia fleet in 2020.

## Aircraft operations

Busiest domestic routes into and out of Port Macquarie (year ending December 2016) :
| Rank | Airport | Passengers | % change | Carriers |
| 1 | Sydney | no data yet | no data yet | Rex, QantasLink |
| 2 | Brisbane | no data yet | no data yet | QantasLink |
| 3 | Canberra | no data yet | no data yet | FlyPelican |
| 4 | Lord Howe Island | no data yet | no data yet | Eastern Air Services |
Connections to Brisbane (via Coffs Harbour) by Brindabella Airlines ceased 31 December 2010, citing poor loads on the route. Despite this, Virgin Australia entered the route on 19 October 2011 as a daily service with their ATR72 aircraft.
# Xiaomi Mi 8 EE

The Xiaomi Mi 8 Explorer Edition is a flagship Android smartphone developed by Xiaomi Inc. It was launched at an event held in Shenzhen, China.

## Specifications
### Hardware

The Xiaomi Mi 8 EE is powered by the Qualcomm Snapdragon 845 processor, with 8 GB LPDDR4X RAM and Adreno 630 GPU. It has a 6.21-inch (158 mm) FullHD plus AMOLED display and 128 GB of storage. The handset features a fingerprint scanner integrated into the display and a 3,000 mAh battery with a USB-C reversible connector which supports Quick Charge 4.0+. It does not feature a 3.5mm headphone jack. The Mi 8 EE includes a dual camera setup with a 12 MP sensor with wide angle lens and a 12 MP sensor with telephoto lens. The front camera has a 20 MP sensor with aperture f/2.0. The Mi 8 EE camera has an overall score of 99 and a photo score of 105 on DxOMark. It also introduces a 3D facial recognition, IR facial unlock and a dual band GPS which allows reception of L1 and L5 signals simultaneously.
The Xiaomi Mi 8 EE features a transparent back which appears to show the working internal components of the phone. A controversy occurred over allegations that the visible internals are not the actual components of the phone but merely a sticker.

## Xiaomi Mi 8 Pro

Pro has more color options than standard or EE. It also has an under–screen fingerprint sensor, but 3D facial recognition was cancelled.
# Bonetiella

Bonetiella is a monotypic genus of shrubs in the subfamily Anacardioideae of the cashew and sumac family Anacardiaceae. It contains the single species Bonetiella anomala, which is endemic to northern and central Mexico.
# East Metro Integration District 6067

EMID East Metro Integration District 6067 was a special school district in the U.S. state of Minnesota created as a collaboration between Saint Paul Public Schools and nine suburban school districts in the eastern Minneapolis–Saint Paul metro area. EMID is designed to foster voluntary cultural integration among schools in the Twin Cities.
EMID has four components:

## Harambee Community Cultures/ Environmental Science School

Harambee is a K-5 year-round elementary school. It gets its name from the harambee tradition of Kenyan community self-help events.

## Crosswinds East Metro Arts and Science School

Crosswinds is a 6-10 school in Woodbury, Minnesota. Crosswinds is an accredited International Baccalaureate Middle Years Program School
# 633 17th Street

633 17th Street, formerly known as the First Interstate Tower North, is a high-rise building in Denver, Colorado. The building was completed in 1974, and rises 32 floors and 434 feet (132 m) in height. The building stands as the twelfth-tallest building in Denver and Colorado. It also stood as the tallest building in the city at the time of its 1974 completion, and held that distinction for four years until it was surpassed by the 507-foot (155 m) 555 17th Street in 1978.
At the time of its completion, 633 17th Street was known as "First Interstate Tower North" after First Interstate Bancorp, its primary tenant. After First Interstate merged with Wells Fargo in 1996, the building's address became its official name. First Interstate Tower North was built adjacent to the pre-existing First Interstate Tower South, now known as 621 17th Street, which rises 386 feet (118 m) in height. The two buildings together form the First of Denver Plaza.
Exterior shots of the building were used in the opening credits of the television series Dynasty
# Edmond J. Safra Synagogue (Brooklyn)

The Edmond J. Safra Synagogue is an Orthodox Jewish synagogue, under construction, located at 2085 Ocean Parkway, Brooklyn, in New York City, New York, United States. The synagogue is one of several that are eponymous with Edmond J. Safra, a former banker and philanthropist, partially or fully funded by the Edmond J. Safra Foundation.
Founded in 2005, the congregation practices in the Sephardic rite and is composed of participants of Sephardic and Middle Eastern descent, including immigrants from: Syria, Lebanon, Egypt, Morocco, Israel, Yemen, and Turkey, along with their descendants.
The congregation is led by Rabbi Eli Mansour and Rabbi Nathan Escava.
The synagogue building was designed over a period of ten years, by Building Studio Architects, with the interior design completed by Ovadia Design Group. as of December 2023, Safra Synagogue was under construction. The limestone and granite building recalls Brooklyn's grand civic buildings from the 1920's-30's, with Art Deco and neo-Classical proportions and details completed in the Moderne style. The synagogue is expected to be completed in 2024.
# Niyi Adeolokun

Niyi Adeolokun (born 3 November 1990) is an Irish rugby union player. He primarily plays as a winger. Currently unattached he most recently played for Bristol Bears and previously for Connacht. Adeolokun joined Connacht in 2014 from Trinity College. During his time with Dublin University, Adeolokun also played rugby sevens for the club.

## Early life

Born in Ibadan, Nigeria, Adeolokun came to Ireland with his family in 2001 aged 11. He played a wide array of sports in his youth, playing Gaelic football with Templeogue Synge Street and also played soccer for League of Ireland side Shelbourne's under-20 team. Adeolokun first played rugby union as a student at De La Salle College in Churchtown.

## Club career
### Youth and amateur level

Adeolokun was tied with Leinster as a teenager, but after being dropped from their under-19 development squad, Adeolokun played for De La Salle senior school's cup team. In 2009, he was invited All-Ireland League side Dublin University by Director of Rugby, Tony Smeeth. Adeolokun played for the team for four seasons. During his time with the squad also played for the Dublin University Sevens team that won back-to-back All-Ireland Championships in 2011 and 2012.

### Connacht

In June 2014, it was announced that Adeolokun had signed for Irish provincial team, Connacht on a one-year deal. Connacht's academy coach Nigel Carolan had recommended him to the province, with Adeolokun having had a trial with the second tier side Connacht Eagles, in April 2014.
Adeolokun made his debut for Connacht on 6 September 2014, starting against Newport Gwent Dragons on the opening weekend of the 2014–15 Pro12, providing an assist to teammate Eoin McKeon for Connacht's first try of the new season. He scored his first try for the province against La Rochelle in the 2014–15 Rugby Challenge Cup. Adeolokun played a total of nine games in the league and four in the Challenge Cup in his first season, scoring a try in each. In November 2014, it was announced that Adeolokun had signed a new contract with Connacht to keep him with the team until the end of 2016–17 season.
On 28 May 2016, Connacht won their first ever major trophy, the 2015–16 Pro12 after a 20-10 win against Leinster in the final, Adeolokun scored the second try of the match.

### Bristol Bears

In August 2020 it was confirmed Adeolokun had joined Premiership Rugby side Bristol Bears on a short-term contract for the remainder of the 2019–20 season. He signed a one-year extension ahead of the 2020–21 season.

## International career

In November 2016 Adeolokun received his first call-up to the senior Ireland team. He made his debut for Ireland on 12 November 2016, coming as a 67th minute replacement for Craig Gilroy against Canada.