OpenLLM-France/wikimedia huggingface dataset:
# Yucheng, Shandong

Yucheng (Chinese: 禹城; pinyin: Yǔchéng) is a county-level city located in the northwest of Shandong province, China. It is administered by the prefecture-level city of Dezhou.
The population was 495,387 in 1999.
History The name of "Yucheng" means "the city of Great Yu", who is believed to be the first king of the Xia dynasty, and an ancient hero specializing in fighting with flood. The county of Yucheng was set up during the Tang dynasty in 692 AD.

## Administrative divisions

As 2012, this city is divided to 1 subdistrict, 7 towns and 3 townships.

### Subdistricts

* Shizhong Subdistrict (市中街道)

### Towns

| Lun (伦镇) Fangsi (房寺镇) Zhangzhuang (张庄镇) Xindian (辛店镇) | Anren (安仁镇) Xinzhai (辛寨镇) Liangjia (梁家镇) |

### Townships

* Litun Township (李屯乡)
* Shiliwang Township (十里望乡)
* Juzhen Township (莒镇乡)

## Climate

Climate data for Yucheng, elevation 21 m (69 ft), (1991–2020 normals) :
| Month | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | Year |
| Mean daily maximum °C (°F) | 3.9 (39.0) | 7.9 (46.2) | 14.4 (57.9) | 21.1 (70.0) | 26.7 (80.1) | 31.8 (89.2) | 32.2 (90.0) | 30.6 (87.1) | 27.2 (81.0) | 21.2 (70.2) | 12.5 (54.5) | 5.5 (41.9) | 19.6 (67.3) |
| Daily mean °C (°F) | −1.4 (29.5) | 2.2 (36.0) | 8.4 (47.1) | 15.1 (59.2) | 20.9 (69.6) | 25.9 (78.6) | 27.5 (81.5) | 25.9 (78.6) | 21.3 (70.3) | 15.0 (59.0) | 7.0 (44.6) | 0.4 (32.7) | 14.0 (57.2) |
| Mean daily minimum °C (°F) | −5.6 (21.9) | −2.4 (27.7) | 3.4 (38.1) | 9.7 (49.5) | 15.4 (59.7) | 20.6 (69.1) | 23.4 (74.1) | 22.1 (71.8) | 16.6 (61.9) | 10 (50) | 2.6 (36.7) | −3.7 (25.3) | 9.3 (48.8) |
| Average precipitation mm (inches) | 3.9 (0.15) | 9.9 (0.39) | 9.9 (0.39) | 33.4 (1.31) | 48.7 (1.92) | 91.7 (3.61) | 149.7 (5.89) | 139.8 (5.50) | 45.1 (1.78) | 31.7 (1.25) | 19.5 (0.77) | 4.4 (0.17) | 587.7 (23.13) |
| Average precipitation days (≥ 0.1 mm) | 2.0 | 3.0 | 2.9 | 4.9 | 6.6 | 7.8 | 11.3 | 9.8 | 6.2 | 4.9 | 4.1 | 2.4 | 65.9 |
| Average snowy days | 2.7 | 2.8 | 0.9 | 0.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0.8 | 2.1 | 9.5 |
| Average relative humidity (%) | 60 | 56 | 52 | 58 | 62 | 61 | 76 | 80 | 74 | 67 | 66 | 63 | 65 |
| Mean monthly sunshine hours | 165.1 | 167.1 | 218.8 | 237.6 | 265.7 | 236.1 | 197.5 | 202.5 | 203.2 | 194.7 | 160.2 | 160.2 | 2,408.7 |
| Percent possible sunshine | 53 | 54 | 59 | 60 | 61 | 54 | 45 | 49 | 55 | 57 | 53 | 54 | 55 |
| Source: China Meteorological Administration ||||||||||||||
# Cooper, Minneapolis

The Cooper neighborhood (part of the larger Longfellow community) resides along the west shore of the Mississippi River in south Minneapolis. It is bound by 34th St. E. on the south, 38th Ave. S. on the west, 27th St. E. on the north, and the river gorge to the east. Bordering neighborhoods are Seward to the north, Longfellow to the west, and Howe to the south. St. Paul's Merriam Park neighborhood is just across the Mississippi River to the east.
Cooper was named after the 19th-century author James Fenimore Cooper, best known for writing The Last of the Mohicans (1832).

## Geography

The Mississippi River Gorge is unique to this stretch of the entire Mississippi. The river cut an eighty-plus foot gorge as what is now St. Anthony Falls moved upstream over thousands of years, finally stopping at downtown Minneapolis due to engineering work during the city's flour milling boom times. Today, the Mississippi River Gorge is a protected natural landscape.

## Schools

2005 marked the closing of Cooper Elementary School, forcing Cooper's children to move to the neighboring Longfellow Elementary, which itself closed in 2010. Today, elementary students in Cooper are split between Dowling Elementary School and Hiawatha Elementary School.
Sanford Middle School and Andersen Middle School serve the Cooper neighborhood with students from Hiawatha/Howe tracking to Sanford and Dowling students tracking to Andersen.
South High School and Roosevelt High School serve the neighborhood with Sanford students tracking to Roosevelt and Andersen students tracking to South. Spanish Immersion magnet students at Andersen track to Roosevelt for high school.
Minnehaha Academy's (private) high school is located in Cooper along West River Road. Their middle school is further south on West River Road in the Howe neighborhood.

## Political representation

Cooper is part of City Council Ward 12 and Minnesota Legislative District 63A.
* City Council (Ward 12): Aurin Chowdhury
* Minneapolis Park Board (District 3): Becky Alper
* Minneapolis Mayor: Jacob Frey
* Hennepin County Commissioner (District 4): Angela Conley
* Minnesota House of Representatives (District 63A): Samantha Sencer-Mura
* Minnesota Senate (District 63): Zaynab Mohamed
* US House of Representatives (District 5): Ilhan Omar

## Notable residents

* Jesse Ventura, former Minnesota Governor, grew up in the house on the southwest corner of 32nd St. E. and 46th Ave. S.
# Li County, Hebei

Li County or Lixian (simplified Chinese: 蠡县; traditional Chinese: 蠡縣; pinyin: Lǐ xiàn) is a county under the jurisdiction of Baoding prefecture-level city, Hebei, China.

## Administrative divisions

* Towns : Liwu (蠡吾镇), Liushi (留史镇), Dabaichi (大百尺镇), Xinxing (辛兴镇), Beiguodan (北郭丹镇), Wan'an (万安镇), Sangyuan (桑园镇), Nanzhuang, Li County, Hebei (南庄镇)
* Townships : Xiaochen Township (小陈乡), Linbao Township (林堡乡), Beiniantou Township (北埝头乡), Baoxu Township (鲍墟乡), Daquti Township (大曲堤乡)

## Climate

Climate data for Lixian, elevation 19 m (62 ft), (1991–2020 normals, extremes 1981–2010) :
| Month | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | Year |
| Record high °C (°F) | 16.7 (62.1) | 23.0 (73.4) | 30.8 (87.4) | 34.0 (93.2) | 38.7 (101.7) | 40.5 (104.9) | 40.9 (105.6) | 37.4 (99.3) | 35.5 (95.9) | 32.0 (89.6) | 23.6 (74.5) | 19.3 (66.7) | 40.9 (105.6) |
| Mean daily maximum °C (°F) | 2.8 (37.0) | 7.1 (44.8) | 14.4 (57.9) | 21.8 (71.2) | 27.7 (81.9) | 32.1 (89.8) | 32.4 (90.3) | 30.6 (87.1) | 26.8 (80.2) | 20.3 (68.5) | 11.0 (51.8) | 4.2 (39.6) | 19.3 (66.7) |
| Daily mean °C (°F) | −3.5 (25.7) | 0.6 (33.1) | 7.6 (45.7) | 15.1 (59.2) | 21.1 (70.0) | 25.8 (78.4) | 27.3 (81.1) | 25.7 (78.3) | 20.7 (69.3) | 13.6 (56.5) | 4.9 (40.8) | −1.6 (29.1) | 13.1 (55.6) |
| Mean daily minimum °C (°F) | −8.5 (16.7) | −4.6 (23.7) | 1.6 (34.9) | 8.7 (47.7) | 14.6 (58.3) | 19.7 (67.5) | 22.7 (72.9) | 21.4 (70.5) | 15.6 (60.1) | 8.2 (46.8) | 0.2 (32.4) | −5.9 (21.4) | 7.8 (46.1) |
| Record low °C (°F) | −22.5 (−8.5) | −17.3 (0.9) | −10.6 (12.9) | −3.5 (25.7) | 4.7 (40.5) | 9.2 (48.6) | 15.6 (60.1) | 12.8 (55.0) | 4.5 (40.1) | −5.0 (23.0) | −15.1 (4.8) | −20.1 (−4.2) | −22.5 (−8.5) |
| Average precipitation mm (inches) | 1.9 (0.07) | 4.6 (0.18) | 8.0 (0.31) | 25.7 (1.01) | 34.6 (1.36) | 59.9 (2.36) | 132.8 (5.23) | 108.2 (4.26) | 47.2 (1.86) | 25.9 (1.02) | 13.5 (0.53) | 2.3 (0.09) | 464.6 (18.28) |
| Average precipitation days (≥ 0.1 mm) | 1.3 | 2.1 | 2.7 | 5.0 | 6.0 | 8.3 | 11.4 | 10.5 | 6.5 | 5.0 | 3.6 | 1.7 | 64.1 |
| Average snowy days | 2.2 | 2.1 | 0.9 | 0.2 | 0 | 0 | 0 | 0 | 0 | 0 | 1.2 | 2.4 | 9 |
| Average relative humidity (%) | 59 | 53 | 50 | 53 | 57 | 60 | 74 | 79 | 74 | 68 | 67 | 62 | 63 |
| Mean monthly sunshine hours | 150.5 | 161.2 | 217.2 | 244.2 | 270.3 | 235.4 | 205.0 | 202.5 | 191.7 | 180.7 | 157.0 | 146.7 | 2,362.4 |
| Percent possible sunshine | 49 | 52 | 58 | 61 | 61 | 53 | 46 | 49 | 52 | 53 | 52 | 50 | 53 |
| Source: China Meteorological Administration ||||||||||||||

## Religion

The Catholic minority is pastorally served by the Apostolic Prefecture of Lixian (Lizhou/ Lichow).
# Frank Freitas

Daniel Frank Errol Freitas (23 February 1901 – 10 April 1968) was a New Zealand rugby union player. A loose forward, Freitas represented West Coast at a provincial level, and was a member of the New Zealand national side, the All Blacks, in 1928. He played four games for the All Blacks but no test matches. He died in the Wahine disaster in 1968.
# Arthur Rees (cricketer)

Arthur Westland Rees (9 September 1866 – 1 January 1921) was a New Zealand cricketer. He played first-class cricket for Auckland and Hawke's Bay between 1889 and 1897. In his first two first-class matches he took 25 wickets.

## Life and career

Rees's father was William Lee Rees, who was a cousin of W. G. Grace and other members of the Grace and Gilbert families. Arthur Rees was born in Hokitika and educated at Auckland Grammar School before going to Caius College, Cambridge. He returned to New Zealand in 1889 and was admitted as a barrister and solicitor in 1890. He established a law practice in Gisborne in 1893.
A left-arm slow bowler, Rees took 32 wickets for 226 in the three matches of Auckland's southern tour in 1889–90. He began with 6 for 27 and 8 for 36 against Otago, bowling unchanged on his first-class debut, and a few days later he took 7 for 35 and 4 for 63 against Canterbury. He was the leading bowler in New Zealand that season, with 38 wickets at an average of 10.94.
Rees married Mabel Margaret Crawford in Gisborne in May 1893. He died at his home in Gisborne early on New Year's Day 1921 after a short illness. His wife, a son and three daughters survived him.
# George Keefe

George Washington Keefe (January 7, 1867 – August 24, 1935) was an American professional baseball left-handed starting pitcher. He played in Major League Baseball (MLB) for the Washington Nationals, Buffalo Bisons, and Washington Statesmen from 1886 to 1891.

## Baseball career

Keefe was born in Washington, D.C., in 1867. In 1886, he joined the Washington Nationals of the National League (NL) and made his major league debut on July 30 at the age of 19. The sixth-youngest player in the league, he went 0–3 with a 5.17 earned run average (ERA) in four games during his first big league season.
In 1887, Keefe appeared in only one game. He allowed 16 hits, four walks, and 20 runs (eight of which were earned) and completed (and lost) his only game of the season. He was also the sixth-youngest player in 1887.
In 1888, Keefe went 6–7 with a 2.84 ERA in 13 starts for the Nationals. He spent most of the season with the Troy Trojans of the International Association. With Troy, he pitched 343 innings and went 10–28 with a 3.46 ERA and 184 strikeouts.
Back with the Nationals in 1889, Keefe went 8–18 with a 5.13 ERA, finishing fifth in the league in walks allowed (143) and seventh in losses. On May 1, he set the NL record for most walks in an inning, when he walked seven batters in the fifth inning. The record was subsequently tied by Bob Ewing and Tony Mullane, and Dolly Gray walked eight batters in an inning in 1909. Gray was an American League pitcher, however.
Keefe played for the Buffalo Bisons of the Players' League in 1890. That year, he went 6–16 with a 6.52 ERA. He was ninth in the league in home runs allowed (11) and 10th in the league in losses. In 196 innings, he walked 138 batters and struck out only 55.
Keefe played his final MLB season in 1891 for the Washington Statesmen of the American Association. With the Statesmen, he went 0–3 with a 2.68 ERA in five games (four starts). He played his final major league game on April 28.
During his MLB career, Keefe went 20–48 in 78 games, 71 of which he started. He completed 68 games and had one shutout. In 6161⁄3 innings, Keefe posted a 5.05 ERA, 360 walks, and 213 strikeouts. In 250 career at-bats, he hit .172.
Keefe then played in the minor leagues from 1892 to 1901.
Keefe died in Washington, D.C., in 1935. He was buried in Cedar Hill Cemetery in Suitland, Maryland.
# Larry Wogan

Lawrence William Wogan (18 September 1890 – August 1979) was a rugby union player who represented Australia.
Wogan, a centre, was born in Hokitika and claimed a total of 22 international rugby caps for Australia.
# All The Best Pandya

All The Best Pandya (Gujarati: ઓલ ધ બેસ્ટ પંડ્યા) is a 2025 Indian Gujarati family drama film directed and edited by Rahul Bhole and Vinit Kanojia. It Stars Malhar Thakar, Darshan Jariwala, Vandana Pathak, Yukti Randeria and others. The film is produced by Jeegar Chauhan, Malhar Thakar, Jigar Parmar and Jimmy Satish Asija and it will be released nationwide on 14 March 2025.

## Plot

Akshay Pandya and his father, Hasmukh Pandya, share a complicated bond, often avoiding direct confrontations. Their lives take an unexpected turn when Hasmukh, a principled and honest man, is accused of bribery. With the legal system failing to secure his release, Akshay steps into the courtroom to fight for his father's justice. Blending emotional depth with gripping courtroom drama, the film explores the complexities of their relationship while unraveling a case that challenges integrity, family, and the truth.

## Cast

* Malhar Thakar as Akshay Pandya
* Darshan Jariwala as Hasmukh Pandya
* Vandana Pathak as Indu
* Yukti Randeria as Bhoomi
* Smit Joshi as Fenku
* Prem Gadhavi as Mudresh
* Archan Trivedi as Kishor
* Dhara Shah as Manisha Bhabhi
* Satish Bhatt as Dr. Majumdar
* Nisarg Trivedi as Lawyer Mukul Desai
* Lipi Trivedi as Maitri
* Kartavya Shah as Manan Talati
* Firoz Irani as Judge
* Bhargav Joshi as Arvind
* Nikita Shah as News Anchor
* Urmila Solanki as Sneha
* Hiranya Zinzuwadia as Young Akshay
* Pratham Patel as Spiritual Guru
* Sandeep Prajapati as Nago

## Soundtrack
### Tracklist

All lyrics are written by Bhargav Purohit, Parth Tarpara, Dilip Rawal; all music is composed by Prashant Satose

## Marketing and Releases

The film was announced on 5 February 2025, along with its release date. The film's official poster was revealed on 19 February 2025, and the teaser was released on 21 February 2025. The film is scheduled for release on 14 March 2025.
# Mikael Borglund

Mikael Borglund (born 1956) is the managing director of Beyond International, one of Australia's leading film and television production companies.
# George Lyons (baseball)

George Tony Lyons (January 25, 1891 – August 12, 1981) was an American professional baseball pitcher who played a total of 33 games in Major League Baseball (MLB) for the 1920 St. Louis Cardinals and the 1924 St. Louis Browns. Listed at 5 feet 11 inches (1.80 m) and 180 pounds (82 kg), he threw and batted right-handed.

## Biography

Nicknamed "Smooth", Lyons played minor league baseball from 1914 to 1928, except for 1916 when there is no record of him playing professionally. Lyons played only six games during the 1918 season, due to military service from May 1918 to January 1919. In 13 minor league seasons, he accrued a 124–163 win–loss record for seasons that records are available for.
Lyons' first major league stint came in September 1920 with the St. Louis Cardinals. His best outing was a complete game 4–2 win over the Philadelphia Phillies on September 22. With the Cardinals, Lyons pitched in seven games (two starts) registering a 2–1 record with a 3.09 earned run average (ERA) while striking out five batters in 23+1⁄3 innings pitched.
Lyons returned to the major leagues in 1924 with the St. Louis Browns, when he pitched in 26 games (six starts). He had two complete game wins in June; one an 11–4 win over the Boston Red Sox, the other a 10–2 win over the Philadelphia Athletics. His record with the Browns was 3–2 with a 5.21 ERA and 25 strikeouts in 77+2⁄3 innings pitched.
Lyons' major league pitching appearances yielded an overall 5–3 record with a 4.72 ERA in 33 games pitched. As a batter, he accrued a .222 batting average (6-for-27) with four RBIs. Defensively, he committed no errors in 39 total chances for a 1.000 fielding average.
Born in Bible Grove, Illinois, in 1891, Lyons served in the United States Army during World War I. He died at age 90 in Nevada, Missouri, and was interred there. The ballpark in the city of Nevada—used by the local high school and a M.I.N.K. Collegiate Baseball League team—is named Lyons Stadium in his honor.
# Mike Burgoyne (rugby union)

Michael Martin Burgoyne (27 March 1951 – 22 November 2016) was a New Zealand rugby union player. A flanker, Burgoyne represented North Auckland at a provincial level, and was a member of the New Zealand national side, the All Blacks, in 1979. He played six matches for the All Blacks including games against Argentina and Italy, but neither was recognised as a full international by the New Zealand Rugby Union. Of Ngāti Kahu descent, Burgoyne made eight appearances for New Zealand Māori between 1975 and 1979.
Burgoyne died suddenly in 2016 while on the Fijian island of Yanuca.
# Craig Smith (rugby league, born 1971)

Craig Smith (born 31 October 1971) is a New Zealand former professional rugby league footballer who played as a prop in the 1990s and 2000s. He played for the South Sydney Rabbitohs, Illawarra Steelers, St. George Illawarra Dragons and the Newcastle Knights as well as representing New Zealand, New Zealand Māori and Queensland.

## Background

Smith was born in Kaitaia, New Zealand and originally was a rugby union player in Northland. In 1988, Smith moved to Queensland to complete his final years of schooling at Ipswich Grammar School. He later took up rugby league.

## Playing career
### Club career
#### South Sydney Rabbitohs

Smith commenced his top level career with the South Sydney Rabbitohs in the 1995 ARL season.

#### Illawarra Steelers

He then moved to the Illawarra Steelers.

#### St. George Illawarra Dragons

Smith played at prop forward for the St. George Illawarra Dragons in their 1999 NRL Grand Final loss to the Melbourne Storm.

#### Wigan Warriors

Following a series of suspensions in 2001 for contrary conduct and striking, Smith left the NRL and Australia in 2002 to play for the Super League club, the Wigan Warriors. With Wigan, he won the 2002 Challenge Cup Final. In 2003 he was named at prop in the season's Super League Dream Team. Smith played for the Wigan Warriors at prop in the 2003 Super League Grand Final, which was lost to the Bradford Bulls.

#### Newcastle Knights

In 2005, Smith agreed to a one-year deal and a return to the NRL with the Newcastle Knights.
Although originally intending to retire at the end of the 2005 season, Smith was re-signed by Newcastle for the 2006 season. A broken jaw in August 2006 sidelined him for several rounds but he was fit to return for Newcastle's finals appearances. Smith played his last NRL game in Newcastle's semi-finals defeat by the Brisbane Broncos.

### Representative career
#### Queensland

Smith was selected to represent Queensland as a prop for games I and II of the 1997 State of Origin series and on the bench for game III.

#### New Zealand

Smith was selected to go on the 1998 New Zealand tour of Great Britain and played for the Kiwis twelve times between 1998 and 2001.
Smith was selected for the New Zealand team to compete in the end of season 1999 Rugby League Tri-Nations tournament. In the final against Australia he played at prop forward in the Kiwis' 20-22 loss.

#### New Zealand Maori

Smith was again coaxed out of retirement to take part in the New Zealand Māori team that met the Indigenous Dreamtime team in a commemorative match at the 2008 Rugby League World Cup.

## Coaching career

Smith has been on the coaching staff for the under-20s side at Newcastle.
# Blair Urlich

Blair Wolfgang Urlich (born 26 January 1975) is a New Zealand former professional rugby union player.

## Rugby career

A loose forward from Kaitaia, Urlich played for Northland under-19s and got his start in first-class rugby in 1998 with North Harbour, going on to make 50 provincial appearances. He featured once for the Blues in each of the 2000 and 2001 Super 12 seasons, then found more regular selection during their 2002 campaign, playing eight games. After being overlooked by the Blues in 2003, Urlich continued his career in Japan, playing for the Mitsubishi DynaBoars. He had a season with Northland in 2008 and the following year was appointed coach of Takapuna.

### Representative

Urlich represented Croatia in rugby sevens and toured Australia in 2002 with New Zealand Māori.
# Gluing schemes

In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps.

## Statement

Suppose there is a (possibly infinite) family of schemes $\{X_{i}\}_{i\in I}$ and for pairs $i,j$, there are open subsets $U_{ij}$ and isomorphisms $\varphi _{ij}:U_{ij}{\overset {\sim }{\to }}U_{ji}$. Now, if the isomorphisms are compatible in the sense: for each $i,j,k$,
* $\varphi _{ij}=\varphi _{ji}^{-1}$,
* $\varphi _{ij}(U_{ij}\cap U_{ik})=U_{ji}\cap U_{jk}$,
* $\varphi _{jk}\circ \varphi _{ij}=\varphi _{ik}$ on $U_{ij}\cap U_{ik}$,
then there exists a scheme X, together with the morphisms $\psi _{i}:X_{i}\to X$ such that
* $\psi _{i}$ is an isomorphism onto an open subset of X,
* $X=\cup _{i}\psi _{i}(X_{i}),$
* $\psi _{i}(U_{ij})=\psi _{i}(X_{i})\cap \psi _{j}(X_{j}),$
* $\psi _{i}=\psi _{j}\circ \varphi _{ij}$ on $U_{ij}$.

## Examples
### Projective line

Let $X=\operatorname {Spec} (k[t])\simeq \mathbb {A} ^{1},Y=\operatorname {Spec} (k[u])\simeq \mathbb {A} ^{1}$ be two copies of the affine line over a field k. Let $X_{t}=\{t\neq 0\}=\operatorname {Spec} (k[t,t^{-1}])$ be the complement of the origin and $Y_{u}=\{u\neq 0\}$ defined similarly. Let Z denote the scheme obtained by gluing $X,Y$ along the isomorphism $X_{t}\simeq Y_{u}$ given by $t^{-1}\leftrightarrow u$; we identify $X,Y$ with the open subsets of Z. Now, the affine rings $\Gamma (X,{\mathcal {O}}_{Z}),\Gamma (Y,{\mathcal {O}}_{Z})$ are both polynomial rings in one variable in such a way
    $\Gamma (X,{\mathcal {O}}_{Z})=k[s]$ and $\Gamma (Y,{\mathcal {O}}_{Z})=k[s^{-1}]$
where the two rings are viewed as subrings of the function field $k(Z)=k(s)$. But this means that $Z=\mathbb {P} ^{1}$; because, by definition, $\mathbb {P} ^{1}$ is covered by the two open affine charts whose affine rings are of the above form.

### Affine line with doubled origin

Let $X,Y,X_{t},Y_{u}$ be as in the above example. But this time let $Z$ denote the scheme obtained by gluing $X,Y$ along the isomorphism $X_{t}\simeq Y_{u}$ given by $t\leftrightarrow u$. So, geometrically, $Z$ is obtained by identifying two parallel lines except the origin; i.e., it is an affine line with the doubled origin. (It can be shown that Z is not a separated scheme.) In contrast, if two lines are glued so that origin on the one line corresponds to the (illusionary) point at infinity for the other line; i.e, use the isomorphism $t^{-1}\leftrightarrow u$, then the resulting scheme is, at least visually, the projective line $\mathbb {P} ^{1}$.

## Fiber products and pushouts of schemes

The category of schemes admits finite pullbacks and in some cases finite pushouts; they both are constructed by gluing affine schemes. For affine schemes, fiber products and pushouts correspond to tensor products and fiber squares of algebras.