OpenLLM-France/wikimedia huggingface dataset:
# Subgroup distortion

In geometric group theory, a discipline of mathematics, subgroup distortion measures the extent to which an overgroup can reduce the complexity of a group's word problem. Like much of geometric group theory, the concept is due to Misha Gromov, who introduced it in 1993.
Formally, let S generate group H, and let G be an overgroup for H generated by S ∪ T. Then each generating set defines a word metric on the corresponding group; the distortion of H in G is the asymptotic equivalence class of the function $R\mapsto {\frac {\operatorname {diam} _{H}(B_{G}(0,R)\cap H)}{\operatorname {diam} _{H}(B_{H}(0,R))}}{\text{,}}$ where B$_X$(x, r) is the ball of radius r about center x in X and diam(S) is the diameter of S.$^{:49}$
A subgroup with bounded distortion is called undistorted, and is the same thing as a quasi-isometrically embedded subgroup.

## Examples

For example, consider the infinite cyclic group ℤ = ⟨b⟩, embedded as a normal subgroup of the Baumslag–Solitar group BS(1, 2) = ⟨a, b⟩. With respect to the chosen generating sets, the element $b^{2^{n}}=a^{n}ba^{-n}$ is distance 2$^n$ from the origin in ℤ, but distance 2n + 1 from the origin in BS(1, 2). In particular, ℤ is at least exponentially distorted with base 2.
On the other hand, any embedded copy of ℤ in the free abelian group on two generators ℤ$^2$ is undistorted, as is any embedding of ℤ into itself.

## Elementary properties

In a tower of groups K ≤ H ≤ G, the distortion of K in G is at least the distortion of K in H.
A normal abelian subgroup has distortion determined by the eigenvalues of the conjugation overgroup representation; formally, if g ∈ G acts on V ≤ G with eigenvalue λ, then V is at least exponentially distorted with base λ. For many non-normal but still abelian subgroups, the distortion of the normal core gives a strong lower bound.

## Known values

Every computable function with at most exponential growth can be a subgroup distortion, but Lie subgroups of a nilpotent Lie group always have distortion n ↦ n$^r$ for some rational r.
The denominator in the definition is always 2R; for this reason, it is often omitted. In that case, a subgroup that is not locally finite has superadditive distortion; conversely every superadditive function (up to asymptotic equivalence) can be found this way.

## In cryptography

The simplification in a word problem induced by subgroup distortion suffices to construct a cryptosystem, algorithms for encoding and decoding secret messages. Formally, the plaintext message is any object (such as text, images, or numbers) that can be encoded as a number n. The transmitter then encodes n as an element g ∈ H with word length n. In a public overgroup G with that distorts H, the element g has a word of much smaller length, which is then transmitted to the receiver along with a number of "decoys" from G \ H, to obscure the secret subgroup H. The receiver then picks out the element of H, re-expresses the word in terms of generators of H, and recovers n.
# Arithmetical ring

In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions hold:
* The localization $R_{\mathfrak {m}}$ of R at ${\mathfrak {m}}$ is a uniserial ring for every maximal ideal ${\mathfrak {m}}$ of R.
* For all ideals ${\mathfrak {a}},{\mathfrak {b}}$, and ${\mathfrak {c}}$,
    ${\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})=({\mathfrak {a}}\cap {\mathfrak {b}})+({\mathfrak {a}}\cap {\mathfrak {c}})$
* For all ideals ${\mathfrak {a}},{\mathfrak {b}}$, and ${\mathfrak {c}}$,
    ${\mathfrak {a}}+({\mathfrak {b}}\cap {\mathfrak {c}})=({\mathfrak {a}}+{\mathfrak {b}})\cap ({\mathfrak {a}}+{\mathfrak {c}})$
The last two conditions both say that the lattice of all ideals of R is distributive.
An arithmetical domain is the same thing as a Prüfer domain.
# Lunchclub

Lunchclub is an American social platform that uses artificial intelligence and machine learning to connect users with common interests and objectives. Lunchclub operates remote-first, and lacks an official physical headquarters. The platform is accessible through a mobile app or a web browser.

## History

Lunchclub was established in November 2017. The parent company of Lunchclub is Elliot Technologies, Inc. The platform started its services in cities across the United States and United Kingdom and, as of 2021, it is available worldwide.
Lunchclub received $4 million in a seed funding round from a16z to connect professionals offline. In 2020, Lunchclub raised a $24 million series A round led by Coatue and Lightspeed Venture Partners. After the series A funding, the company's estimated valuation is above $100 million. Investors including Michael Ovitz, Max Levchin, Adam D’Angelo, Vlad Tenev, and Ruchi Sanghvi have also helped fund the company.
In March 2020, in response to the COVID-19 pandemic, Lunchclub added video chat to the platform, using the same technology.

## Features

Lunchclub uses artificial intelligence and machine learning to connect people that share similar interests, values, and goals. The user defines their interests including career, funding, partnerships, new business ideas, and more. Then, its machine learning and AI process create profiles from user information and their public data sources. They then match people up, using this data, based on their interests and objectives.
# Igok station

Igok Station is a station of the Daegu Metro Line 2 in Igok-dong, Dalseo District, Daegu.
# Michael Hardy

Michael Hardy may refer to:
* Hardy (singer) (Michael William Hardy, born 1990), American country musician
* Michael Hardy (attorney) (1955–2024), American civil rights lawyer
* Michael C. Hardy (born 1972), American historian
# Dhaa

Music of Nepal :
General topics :
Dhimay
Bhusyah
Madal
Sarangi
Damphu
Dholak
Chyabrung
Bansuri :
Genres :
Dapa
Dohori
Selo
Bhajan
Rodhi
Adhunik Geet
Nepali Rock
Nepali Hip hop :
Specific forms :
Tamang Music
Newa music
Gurung music
Khas
Maithili music
Sunuwar music
Magar music
Bhojpuri music
Sherpa music
Thakali music
Chepang music
Rai music
Tibetan music
Lepcha music
Limbu music
Kusunda music :
Media and performance :
Nationalistic and patriotic songs :
vte :
The Dhaa (or Dhah) is a two-headed drum, "slightly smaller than the Dhimay." It belongs to the membranophone group of Newar traditional musical instruments. It is a kind of drum specially played during the month of Gunlaa, the ninth month of Newar calendar. Dhaa is also known as "Gunlaa Baajan".
It is made of a hollow wooden trunk covered at both sides with animal skin. The left side is covered with a thicker skin producing flat sound whereas the right side is covered with thinner skin producing sharper sound.
The Dhaa is played by a group of even number of people standing. The instrument is suspended with a belt over the right shoulder. Dhaa is always accompanied with Taa and Bhushyaa along with the tunes from some aerophones.
# Gray Line

Gray Line or Grey Line may refer to:

## Transportation

* Gray Line Worldwide, an international sightseeing bus operator
* Grey Line (Bangkok), a planned monorail line in Bangkok, Thailand
* Grey Line (Delhi Metro), Delhi, India
* Gray Coach, a former Canadian inter-city bus line based in Toronto
* E Embarcadero, San Francisco, California
* Jubilee line, London, England
* Line 6 Finch West, Toronto, Canada
* Line 9 (Shenzhen Metro), Shenzhen, China
* Metro (Minnesota), lettered lines are color-coded "METRO Gray", Minneapolis—Saint Paul, Minnesota
* Olympic Park railway line, Sydney, Australia
* Pujiang line, Shanghai, China
* Serpukhovsko-Timiryazevskaya line, known as the Grey Line, Moscow, Russia
* Silver Line (DART), a commuter rail line in Texas
* Silver Line (Grand Rapids), Grand Rapids, Michigan
* Lynx Silver Line, a proposed transit extension in Charlotte, North Carolina
* Silver Line (MBTA), a system of bus lines in Boston, Massachusetts
* METRORapid Silver Line, Houston, Texas
* Silver Line (Mexico City), Mexico City
* Silver Line (San Diego Trolley), a heritage streetcar line operated by the San Diego Trolley
* Silver Line (Washington Metro), a rapid transit line of the Washington Metro system
* HealthLine, Cleveland, Ohio
* J Line (Los Angeles Metro), formerly the Silver Line, Los Angeles, California
* Silver Line (Pittsburgh), a light rail line of the Pittsburgh Light Rail system

### New York City Subway

* BMT Canarsie Line
* L (New York City Subway service)
* S (New York City Subway service), three separate rapid transit subway shuttles

## Other uses

* Grayline lascar
* Terminator (solar) or grey line, delimiting the illuminated and dark sides of a planetary body
# Double Happiness (calligraphy)

Double Happiness (simplified Chinese: 双喜; traditional Chinese: 雙喜; pinyin: shuāngxǐ) sometimes translated as Double Happy, is a Chinese traditional ornament design, commonly used as a decoration symbol of marriage. Outside of China, it is also used in the United States, Europe, East Asia and Southeast Asia by members of the Overseas Chinese community.

## Characteristics

Double Happiness is a ligature, "囍" composed of 喜喜 – two copies of the Chinese character 喜 (xǐ) literally meaning joy, compressed to assume the square shape of a standard Chinese character (much as a real character may consist of two parts), and is pronounced simply as xǐ or as a polysyllabic Chinese character, being read as 双喜 (shuāngxǐ).
Typically the character "囍" is written in Chinese calligraphy, and frequently appears on traditional decorative items, associated with marriage. The double happiness symbol is often found all over wedding ceremonies, as well as on gift items given to the bride and groom. The color of the character is usually red, occasionally black.
Since 2017, the version 10 of the Unicode Standard features a rounded version of the character in the "Enclosed Ideographic Supplement" block, at code point U+1F264 🉤 ROUNDED SYMBOL FOR SHUANGXI.

## In popular culture

Nowadays shuāngxǐ (alternatively transcribed shuang hsi) is used as a brand names for things like fashion, jewelry, cigarettes, matches, soy sauce, etc. It is also featured as decoration on many items by Chinese luxury brand Shanghai Tang.
Hong Kong lifestyle retail store G.O.D. designs many products themed with the double happiness symbol, including scented candles, accessories and Ming-inspired tableware and tea sets.

## Gallery

* A double happiness character on the door ring of Soong Ching-ling's ancestral home in Wenchang, Hainan
* A porcelain vase from the Qing dynasty with double happiness characters
* Rice bowl cover decorated with a medallion of the double happiness and longevity (Shou) symbol in the center, from Joseon dynasty Korea
* A Vietnamese wedding decoration, with a double happiness character
* Palace of Earthly Tranquility (坤宁宫) of the Forbidden City, with traditional Chinese wedding decorations and a double happiness character in the foreground
* Chinese wedding invitation cards with double happiness characters
* Chinese wedding invitation card with a double happiness character in the center
* Double happiness decorations in Vietnam
* Chinese wedding ceremony teaware, with double happiness characters on them
* Chinese wedding ceremony teaware
* Gold jewelry with double happiness character, Hong Kong
* A traditional Chinese wedding reception, with double happiness decoration in the middle
* Double Happiness gate at Tomb of Le Van Duyet, Ho Chi Minh City, Vietnam, taken during the Tet Holiday of 2024.
# FA Sápmi

FA Sápmi is a football association which is the organizer of the Sápmi football team representing the indigenous Sámi people. It is a member of the Confederation of Independent Football Associations (ConIFA), but not a member of either FIFA or UEFA. President of the FA Sápmi is Håkan Kuorak from Jokkmokk, Sweden.
FA Sápmi was established in March 2014 as the successor of the 2003 founded Sámi Football Association (SSL, Northern Sami: Sámi Spábbáčiekčanlihttu), which had not been active in few years and had some financial trouble. FA Sápmi is the host of 2014 ConIFA World Football Cup.
# Al Burnett

Al Burnett (1906–1973) was a British impresario, known as the "nightclub king". He ran the Stork Club in Swallow Street in London's West End.

## Early life

He was born Aaron Isaacs in Mile End, London.

## Career

Burnett was master of ceremonies (MC) at a nightclub in London's Regent Street called The Nut House, which was popular with jazz musicians during the war.
In the late 1950s, notable guests at the Stork Club included Harold Macmillan, John Profumo, Peter Sellers, Frank Sinatra, Lana Turner, Bette Davis, Ava Gardner, Elizabeth Taylor, King Hussein of Jordan, and Jean Simmons. In about 1969, he banned Barbara Windsor, whose agent was Burnett's son, Barry Burnett, for refusing to sit with a customer, even though she was not working as a hostess at the time. It was later owned by Oscar Owide, eventually becoming a "sleazy hostess joint", before Marco Pierre White and Piers Adam tried (and failed) to revive it as a glamorous destination.
Burnett owned The Astor, which was later acquired by Bertie Green.
He owned the Pigalle Club in Piccadilly, and won the 1958 English Greyhound Derby with his dog Pigalle Wonder, who set new records on seven British tracks.
Burnett was also an actor, and appeared in King Arthur Was a Gentleman (1942), Café Continental (1947), and Sweet Beat (1959).
# Susan Bennis/Warren Edwards

Susan Bennis/Warren Edwards was a successful New York-based shoe company founded in 1972 by Susan Bennis and Warren Edwards. It dissolved in 1997.

## Career

Bennis is a graduate of New York's Fashion Institute of Technology. In 1972 Bennis and Edwards had the opportunity to purchase the New York outlet of The Chelsea Cobbler, a British shoe brand founded in 1967 by Richard Smith and Mandy Wilkins. At the time, Bennis and Edwards hoped to combine their clothing designs (marketed under the name Kiss) with Chelsea Cobbler's footwear, but quickly decided to focus exclusively on their own shoe designs, which they had made in Italy. By 1980, the Susan Bennis/Warren Edwards brand was known for expensive shoes with "outrageous" design elements such as unexpected material combinations, loud fabric prints, and elaborate decoration.
Susan Bennis/Warren Edwards clients included Bruce Springsteen, Jacqueline Kennedy Onassis, and Cher, who wore their shoes in Moonstruck. They maintained exclusivity by refusing to wholesale their shoes, selling only through their own shops at first. In 1993, they granted the department store Henri Bendel permission to retail their shoes and handbags, including a large number of store exclusives, although it had been reported a couple of years earlier by Women's Wear Daily that Neiman Marcus would market their line.
In 1990 Bennis and Edwards moved their flagship store to West 57th Street between Fifth and Sixth Avenues in New York City. It closed down in 1997, the same year the company disbanded.
Following the closure, Warren Edwards went solo the following year, opening a shop on Park Avenue, New York, where he was still based in 2010. In 2000, he was specialising in slightly modified classic styles of business and business-casual footwear for New York professionals.

## Awards

The firm won a Coty Award in 1982, and the Cutty Sark Award in 1984.

## Exhibitions

Examples of Susan Bennis/Warren Edwards shoes are housed in the Metropolitan Museum of Art's Costume Institute, the Victoria and Albert Museum, London, and the Philadelphia Museum of Art, who also hold some of their handbags.
# Oreste Puliti

Oreste Puliti (18 February 1891 – 5 February 1958) was an Italian fencer. He won four gold medals and a silver at three Olympic Games.
His teammates were accused of attempting to inflate his score by losing to him in the final of the Men's Sabre in 1924. He was disqualified for threatening to assault a Hungarian judge, Gyorgy Kovacs.
# Orlando Polmonari

Orlando Polmonari (11 March 1924 – 27 August 2014) was an Italian gymnast. He won a bronze medal in the all-around with the Italian national team at the 1960 Summer Olympics. He was born and died in Ferrara, Italy.
# John Sorensen

John Sorensen may refer to:
* John Sørensen, Danish sprint canoeist
* John L. Sorenson, American anthropologist, scholar and author
# Paolo Ponzo

Paolo Ponzo (11 March 1972 – 24 March 2013) was an Italian footballer who last played as a midfielder for Liguria club Imperia.

## Career
### Early career

Born in Cairo Montenotte, Liguria, Ponzo started his senior career with Serie D club Vado. In 1991–92 Serie A season, he was signed by one of the largest club in Liguria region, Genoa C.F.C. He played a season with its youth team and left for Montevarchi in 1992.

### Serie B teams

After 3 Serie C2 seasons with the Tuscany team, he was signed by Serie B club Cesena. After the club relegated, he left for Serie B club Ravenna in 1997.
He played 4 times for the Emilia–Romagna club and moved to regional and league rival Reggiana in January 1998, in exchange with Sebastiano Vecchiola.
In January 2000, he was signed by Serie B club Savoia from Reggiana, at that time at Serie C1. He played 15 times for the Campania club. Savoia finished as the bottom team and relegated.

### Modena

In 2000, he was signed by Modena, his second Serie C1 club. He followed the team promoted to Serie B as Serie C1 champion and promoted to Serie A as Serie B runner-up. From 2002 to 2004, he played 43 Serie A matches. He made his Serie A debut on 14 September 2002, losing to AC Milan 0–3. He followed the team relegated back to Serie B in 2004.

### Spezia

In 2005, he left for Serie C1 club Spezia, which is from his home region. He won promotion to Serie B again as champion. in 2006–07 Serie B season, he played 27 matches in all competitions, started 20 times. He was on the bench in the relegation play-out.

### Later career

In 2007, Ponzo was re-signed by Reggiana. He played 2 seasons for the club in Serie C2/Seconda Divisione. In July 2009, he left to play for Savona from his home province, the Province of Savona. He made 32 league appearances for the club and won promotion back to professional league as Group A champion.

## Death

On 24 March 2013 Ponzo died of a heart attack during the Maremontana running race.

## Career statistics

* Serie A Apps 43 : 2002–04 (Modena)
* Serie B Apps 331 : 1995–97 (Cesena), 1997 (Ravenna), 1999-00 (Savoia), 1998–99 (Reggiana), 2001–02 2004–05 (Modena), 2006–07 (Spezia)
* Serie C1 Apps 114 : 1999-00 2008–09 (Reggiana), 2000–01 (Modena), 2005–06 (Spezia)
* Serie C2 Apps 142 : 1992–95 (Montevarchi), 2007–08 (Reggiana), 2010–11 (Savona)
* Serie D Apps 32 : 2009–10 (Savona)
* Eccellenza Apps 76 : 1988–91 (Vado), 2011–12 (Imperia)
# Robin Brooks

Robin Brooks (born 1961) is a British radio dramatist, actor, and author.

## Selected credits
### Adaptations

* 2000 – The Art of Love, a comedy, emphasizing Ovid's role as lover, with Bill Nighy and Anne-Marie Duff
* 2004 – Mort by Terry Pratchett
* 2006 – Small Gods by Terry Pratchett
* 2008 – An Expert in Murder by Nicola Upson
* 2008 – Night Watch by Terry Pratchett
* 2009 – Armadale by Wilkie Collins
* 2010 – I, Claudius by Robert Graves
* 2012 – Ulysses by James Joyce
* 2012 – Mary Stuart by Friedrich Schiller
* 2013 – Eric by Terry Pratchett
* 2013 – "Jill" by Philip Larkin
* 2019 — One Day in the Life of Ivan Denisovich by Aleksandr Solzhenitsyn

### Plays

* 1998 – The Golden Triangle – a trilogy on the lives of the Pre-Raphaelite Brotherhood, consisting of:
** The Awakening Conscience (on William Holman Hunt and his model Annie Miller, taking its title from Hunt's painting of the same name)
** The Order of Release (on John Everett Millais, John Ruskin and Effie Gray, named after Millais's painting of the same name)
** Love Among The Ruins (on Edward Burne-Jones and Maria Zambaco, named after Burne-Jones's painting of the same name)
* 2003 – The Smallest Man in Christendom
* 2006 – Duce's Bonce
* 2007 – A Warning to the Furious
* 2018 – 4/4: Introduction and Allegro
* 2018 – 4/4: Rondo Mysterioso
* 2020 – Elizabeth and Essex

## References and notes

* ↑ Robin Brooks - allegra
* ↑ ROBIN BROOKS RADIO PLAYS
* ↑ One Day in the Life of Ivan Denisovich
* ↑ Radio: Pick of the day, Phil Daoust, The Guardian, 30 May 2006
* ↑ Radio – Martin Hoyle, Financial Times, 30 May 2006
* ↑ BBC – Afternoon Play – A Warning to the Furious
* ↑ BBC – Afternoon Drama – 4/4: Introduction and Allegro
* ↑ BBC – Afternoon Drama – 4/4: Rondo Mysterioso
* ↑ "BBC Radio 3 - Drama on 3, Elizabeth and Essex".

## External Sources

* Robin Brooks
# Piero Brumana

Piero Brumana (30 July 1901 – 17 December 1975) was an Italian racing cyclist. He rode in the 1926 Tour de France.
# Piero Gardoni

Piero Gardoni (12 February 1934 – 15 December 1994) was an Italian professional footballer who played as a defender. He played for various Italian clubs, most notably Atalanta, where he spent nine seasons and captained the team to its Coppa Italia victory in 1963.

## Career

Gardoni was born in Borgo Palazzo, Bergamo, in 1934, and grew up playing in the youth sector of his hometown club Atalanta. In 1953, Gardoni joined local Serie D club Caravaggio on loan for one season to gain playing experience. He then spent three seasons on loan at Palazzolo, also in Serie D, and one year on loan at Reggiana in Serie C, where he helped the club earn promotion to Serie B in 1958.
Gardoni then returned to Atalanta, where he would spend the next nine seasons. In his first season with the Atalanta senior team, Gardoni won the 1958–59 Serie B and promotion to Serie A. He was then named captain in 1960. In the 1962–63 season, Gardoni captained Atalanta to its first (and so far only) Coppa Italia triumph, forming part of a defensive line that allowed only three shots on target in the final against Torino; Atalanta won the match 3–1. The following season, Gardoni scored his first and only goal for Atalanta, in a 2–1 Serie A victory over Vicenza. In total, he recorded 213 appearances with Atalanta (209 in national league competitions and 4 in cup competitions) and played in various defensive roles: he began his career as a full-back and evolved into a libero.
In 1967, Gardoni left Atalanta for Serie B club Reggina, where he spent one season. He retired as a player after a short spell with Serie D club Pergolettese in 1969.

## Later life, death, and legacy

After retiring as a footballer, Gardoni worked in insurance. In the 1990s, he co-owned an insurance agency in Zogno with his son Marco Gardoni. Following a cancer diagnosis, Gardoni disappeared on 12 December 1994 and committed suicide three days later; his body was found in the Oglio river.
Gardoni is remembered for his grit and aerial strength as a defender and for his honest and humble demeanor as a person, as well as for his dedication to Atalanta. As the captain of the team that won the Coppa Italia in 1963, Gardoni is considered an icon by Atalanta ultras and is commemorated in their choreographic displays. Gardoni is also the subject of the fan song Ol Piero, written by Luciano Ravasio [it].

## Honors

Palazzolo
* IV Serie: 1955
Reggiana
* Serie C: 1958
Atalanta
* Serie B: 1959
* Coppa Italia: 1963
# Piervittorio Pampuro

Piervittorio Pampuro (22 April 1917 – 6 August 2007) was an Italian field hockey player. He competed in the men's tournament at the 1952 Summer Olympics.