OpenLLM-France/wikimedia huggingface dataset:
# Bipolar theorem

In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.$^{:76–77}$

## Preliminaries

Suppose that $X$ is a topological vector space (TVS) with a continuous dual space $X^{\prime }$ and let $\left\langle x,x^{\prime }\right\rangle :=x^{\prime }(x)$ for all $x\in X$ and $x^{\prime }\in X^{\prime }.$ The convex hull of a set $A,$ denoted by $\operatorname {co} A,$ is the smallest convex set containing $A.$ The convex balanced hull of a set $A$ is the smallest convex balanced set containing $A.$
The polar of a subset $A\subseteq X$ is defined to be: $A^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{a\in A}\left|\left\langle a,x^{\prime }\right\rangle \right|\leq 1\right\}.$ while the prepolar of a subset $B\subseteq X^{\prime }$ is: $^{\circ }B:=\left\{x\in X:\sup _{x^{\prime }\in B}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.$ The bipolar of a subset $A\subseteq X,$ often denoted by $A^{\circ \circ }$ is the set $A^{\circ \circ }:=^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\sup _{x^{\prime }\in A^{\circ }}\left|\left\langle x,x^{\prime }\right\rangle \right|\leq 1\right\}.$

## Statement in functional analysis

Let $\sigma \left(X,X^{\prime }\right)$ denote the weak topology on $X$ (that is, the weakest TVS topology on $X$ making all linear functionals in $X^{\prime }$ continuous).
    The bipolar theorem: The bipolar of a subset $A\subseteq X$ is equal to the $\sigma \left(X,X^{\prime }\right)$-closure of the convex balanced hull of $A.$

## Statement in convex analysis

    The bipolar theorem:$^{:54}$ For any nonempty cone $A$ in some linear space $X,$ the bipolar set $A^{\circ \circ }$ is given by:
$A^{\circ \circ }=\operatorname {cl} (\operatorname {co} \{ra:r\geq 0,a\in A\}).$

### Special case

A subset $C\subseteq X$ is a nonempty closed convex cone if and only if $C^{++}=C^{\circ \circ }=C$ when $C^{++}=\left(C^{+}\right)^{+},$ where $A^{+}$ denotes the positive dual cone of a set $A.$ Or more generally, if $C$ is a nonempty convex cone then the bipolar cone is given by $C^{\circ \circ }=\operatorname {cl} C.$

## Relation to the Fenchel–Moreau theorem

Let $f(x):=\delta (x|C)={\begin{cases}0&x\in C\\\infty &{\text{otherwise}}\end{cases}}$ be the indicator function for a cone $C.$ Then the convex conjugate, $f^{*}(x^{*})=\delta \left(x^{*}|C^{\circ }\right)=\delta ^{*}\left(x^{*}|C\right)=\sup _{x\in C}\langle x^{*},x\rangle$ is the support function for $C,$ and $f^{**}(x)=\delta (x|C^{\circ \circ }).$ Therefore, $C=C^{\circ \circ }$ if and only if $f=f^{**}.$$^{:54}$
# Jarmosht

Jarmosht or Jarmasht (Persian: جرمشت), also rendered as Garmosht or Jarmast, may refer to:
* Garmosht, Khonj County
* Jarmosht-e Bala, Jahrom County
* Jarmosht-e Pain, Jahrom County
# Bolshoy Cheremshan

Bolshoy Cheremshan (Russian: Большой Черемшан, literally Greater Cheremshan, Tatar: Олы Чирмешән) is a river in Russia, a left tributary of the Volga between the Kama and Samara. It is 336 kilometres (209 mi) long, and has a drainage basin of 11,500 square kilometres (4,400 sq mi). It flows southwest to the Volga near Dimitrovgrad. The main inflows are the Bolshaya Sulcha and Maly Cheremshan. The maximal discharge is 1,660 cubic metres per second (59,000 cu ft/s) (1979), and the minimal mineralization is 600-800 mg/l. The riverbed is meandering and the meadows are wide. From around 1650 the Trans-Kama Line of forts ran along or near the Cheremshan.
# Philip Mårtensson

Philip Mårtensson (born 1 September 1993) is a Swedish football goalkeeper who last played for Varbergs BoIS.
# ACJC

ACJC may refer to:
# Asgardian

Asgardian may refer to:
* Asgard, a location associated with the gods in Norse cosmology
* Asgard (Archaea), a proposed superphylum of the Archaea domain of organisms
* Asgard (comics), the fictional realm in Marvel Comics
** Asgardians of the Galaxy, a team of superheroes
* Asgardia, a micronation in space
* Asgardian (Young Avengers), former name of Wiccan, a fictional character in Marvel Comics
# Tyszki-Gostery

Tyszki-Gostery is a village in the administrative district of Gmina Czerwin, within Ostrołęka County, Masovian Voivodeship, in east-central Poland. It lies approximately 12 kilometres (7 mi) east of Czerwin, 25 km (16 mi) east of Ostrołęka, and 106 km (66 mi) north-east of Warsaw.
# Bespalov

Bespalov (masculine, Russian: Беспалов) or Bespalova (feminine, Russian: Беспалова) is a Russian surname. Notable people with the surname include:
* Anastasiya Bespalova, Russian composer
* Irina Bespalova (born 1981), Russian swimmer
* Mariya Bespalova (born 1986), Russian hammer thrower
* Nikita Bespalov (born 1987), Russian ice hockey player
# List of LGBTQ theatre companies

This is a list of theatre companies specializing in LGBTQ theatre.

## Australia

* ActNow Theatre, theatre company based in Adelaide, South Australia

## Canada

* Buddies in Bad Times, Canadian theatre company based in Toronto, Ontario
* Frank Theatre Company, Canadian theater company, based in Vancouver, British Columbia

## United Kingdom

* Above the Stag Theatre, off-West End theatre company in London
* Aputheatre, theare company in Brighton, England
* Milk Presents, English LGBTQIA+ theatre company based in Derby

## United States

* Celebration Theatre, theatre company in Los Angeles
* The Five Lesbian Brothers, American theater company based in New York City
* Pomo Afro Homos, African-American gay theater troupe founded in San Francisco
* Split Britches, American performance troupe based in New York City
* StageQ, theatre company based in Madison, Wisconsin.
* The Theater Offensive, Boston-based theatrical organization
* Theatre Rhinoceros, gay and lesbian theatre based in San Francisco
* Triangle Theater Company, defunct LGBT+ theatrical organization in Boston
# Abajaluy

Abajaluy (Persian: اباجالوي), also rendered as Abajalu, may refer to:
* Abajaluy-e Olya
* Abajaluy-e Sofla
# Sydney Morse

Sydney Morse (1854–1929) was a rugby union international who represented England from 1873 to 1875.

## Early life

Sydney Morse was born on 1 June 1854 in Birmingham. He was the son of Rev Francis Morse, MA, and his wife Clarissa Catharine Morse. Francis, at the time of Sydney's birth, was the incumbent of the parish of St John's Church, Ladywood, Birmingham. Francis was the son of Thomas Morse, of Flixton, near Lowestoft, and was born in 1819, and educated at Shrewsbury School and St John's College, Cambridge. Sydney had ten siblings among whom were Catharine Elizabeth (b. 1850), Clara (b. 1851), Edward St John Morse (b. 1852), Harold (b. 1860), Harriet Emily (b. 1864), Winifred Mary, (b. 1868) and Margaret Ellinor, (b. 1870).

## Rugby union career

Morse played rugby at Marlborough College and went on to play for the school's old boys club in London, the Marlborough Nomads. However, he also played for the Law Club, a club open only to members of the legal profession, and it was whilst registered as a Law Club player that he won his first cap on 3 March 1873 at Hamilton Crescent, Glasgow in the Scotland vs England match. In total, he played three matches for his national side, the final two being registered as a Marlborough Nomads player. He played his final match for England on 8 March 1875 at Edinburgh against Scotland. He was described as "a dashing runner and good drop with either foot." Sydney's older brother, Edward St John Morse, played in the first meeting of Oxford and Cambridge match on 10 February 1872, representing Cambridge University, a game won by Oxford by a goal (from a try by Isherwood) to nil.

## Career and later life

Sydney became a very successful solicitor, setting up the firm of Sydney Morse & Co, based in the City of London. His firm was associated with many institutions themselves involved in new technologies, including forty or more tramway and electric lighting companies, through to gramophone manufacturers. Sydney married Juliet in 1878 with whom he had a number of children, including Leopold George Esmond Morse.
Sydney also became known as a collector of art, including the portrait of Sir John Everett Millais, 1st Bt by William Holman Hunt. This particular portrait remained in the family until the death of his wife Juliet Morse. At her sale, Christie's, 19 March 1937, it was bought by their son Leopold George Esmond Morse for presentation to the National Portrait Gallery in memory of his father. Amongst other works collected by Sydney Morse were important pieces by Blake, Whistler and a number of Pre-Raphaelite Brotherhood pieces. Holman Hunt also did a drawing of Sydney Morse himself, c.1897–8.
Sydney Morse died on 27 January 1929 at 14 Airlie Gardens, Campden Hill, "in his 75th year and the 51st year of his marriage".
Many years after his death Sydney Morse & Co continued to operate, becoming in the mid-twentieth century part of Waltons & Morse LLP, who continued into the twentieth-first century as one of the City of London's leading specialists in shipping and insurance law.
# Parmanka

The Parmanka (Russian: Парманка) is a river in Perm Krai and Komi Republic, Russia, a right tributary of the Chyornaya, which in turn is a tributary of the Veslyana. The river is 39 kilometres (24 mi) long. The river begins in the eastern portion of the Koygorodsky District of the Komi Republic, north of rural locality of Kazhimka, 179 metres (587 ft) above sea level. Its mouth is 145 metres (476 ft) above sea level. The main tributary is the Parok (right).
# Point Grey

Point Grey (Halkomelem: ʔəlqsən) is a headland marking the southern entrance to English Bay and Burrard Inlet in British Columbia, Canada. The headland is the site of Wreck Beach, Tower Beach, Point Grey Beach and most notably, since 1925, on its top is the Point Grey Campus of the University of British Columbia.
During World War II Tower Beach was the site of submarine watchtowers and gun emplacements while the UBC campus was CFB Point Grey. The watchtower ruins still stand and the gun emplacements have been incorporated into the Museum of Anthropology at UBC.
The name Point Grey is often used as a short form for the Vancouver neighbourhood of West Point Grey. It was named by Captain Vancouver for his friend Captain George Grey. The Spaniards, a year earlier, had named it Punta de Langara in honour of Admiral Don Juan de Langara.$^{:99}$
Point Grey was also a shortened name for the Municipality of Point Grey, created in 1908 when it separated from the Municipality of South Vancouver. The municipality's eastern boundary south of 16th Avenue was Cambie Street, and Blanca Street was its eastern boundary north of 16th. Point Grey amalgamated with the City of Vancouver in 1929.
# Telšiai County

Telšiai County (Lithuanian: Telšių apskritis) is one of ten counties in Lithuania. It is in the west of the country, and its capital is Telšiai. There are Lithuanians (98.7%), Latvians (0.1%), Russians (0.9%), and others (0.3%). On 1 July 2010, the county administration was abolished, and since that date, Telšiai County remains as the territorial and statistical unit. It borders Latvia.

## Municipalities

Municipalities are:
* Mažeikiai District Municipality
* Plungė District Municipality
* Rietavas Municipality
* Telšiai District Municipality
# Travkin

Travkin (Russian: Травкин) is a Russian family name, derived from the word travka, a diminutive for trava, 'grass'. Its feminine counterpart is Travkina. Notable people with the name include:
* Nikolay Travkin, Russian politician
* Alexei Travkin, Russian rugby player
* Ivan Travkin, Soviet submarine commander
# Boris Arias

Boris Arias (Spanish pronunciation:; born 9 November 1993) is a Bolivian tennis player.
Arias has a career high ATP doubles ranking of world No. 88 achieved on 12 February 2024. He also has a career high singles ranking of No. 1065 achieved on 16 July 2012. Arias has won six ITF Futures doubles titles.
Arias has represented Bolivia at the Davis Cup, where he has a W/L record of 4–1.
# Umur-u Hafiye

The Umur-u Hafiye (Ottoman Turkish: امورِ خفیە) was a domestic intelligence organisation in Ottoman Empire, which was active during Abdulhamid II era. The organisation was officially under the direction of the Zaptiye Teşkilatı (the Ottoman Gendarmerie), but in practice it was linked directly to the Sultan, Abdulhamid II.
The consistence of the organization was formed from three sections. The first section's mission was searching, finding and collecting the information about anti governmental organisations and individuals. The second section's mission was editing and reporting the information. The third section's mission was to inform Abdulhamid II.
# Viktor Ivankov

Viktor Ivanovich Ivankov (Russian: Виктор Иванович Иванков; 26 July 1924 – 13 June 2021) was an officer of the Soviet military who held a number of posts, reaching the rank of lieutenant general.

## Career

Ivankov was born in the village of Antonovo, Chuchkovsky District, Ryazan Oblast, in what was then the Russian Soviet Federal Socialist Republic, in the Soviet Union. With the Axis invasion of the Soviet Union in 1941, he was soon pressed into service. He joined the Red Army in 1942, seeing action on the Leningrad Front as part of the Baltic Fleet in the Kronstadt defensive region, as part of the Observation and Communication Service. He went on to see action in the operations around the Oranienbaum Bridgehead.
Ivankov remained in the armed forces after the war, and in 1950 graduated from the Higher Naval Engineering and Technical School and was sent to serve with the Pacific Fleet. His first post was as a platoon commander, but over time he rose to the position of head of the Dalvoenmorstroy naval construction department. In 1970 he became first deputy head of the Main Military Construction Directorate of the Soviet Ministry of Defence. This was followed with his appointment in 1971 as deputy head of State Expertise of the Ministry of Defence, and head of inspection for capital construction. He held this post until 1974, when he became head of the State Expertise and Inspection at the ministry. In 1977 he became deputy chief of construction and personnel, a position he held for the next fifteen years. He had been promoted to engineer major general on 6 May 1972, and then engineer lieutenant general on 13 February 1976. On 26 April 1984 this rank became lieutenant general. He retired in 1992 with the rank of lieutenant general.
Over his career Ivankov received the Order of the Patriotic War First Class, two Orders of the Red Banner of Labour, the Order of Friendship of Peoples and the Order "For Service to the Homeland in the Armed Forces of the USSR" Third Class. He also received the title of Honoured builder of the RSFSR [ru], and was a laureate of the Prize of the Soviet Council of Ministers [ru].
Ivankov died in Moscow on 13 June 2021, at the age of 96.
# Ardentallen

Ardentallen (Scottish Gaelic: Àird an t-Sàilein) is a village in Argyll and Bute, Scotland.
# Statue of Ivo of Kermartin, Charles Bridge

A statue of Ivo of Kermartin (Czech: Sousoší svatého Iva) by Matthias Braun is installed on the south side of the Charles Bridge in Prague, Czech Republic.
# Robbie Nairn

Robbie Nairn (born 18 February 1997) is a Scotland 7s international rugby union player who plays for Ayrshire Bulls. His usual position is at the wing position.

## Rugby union career
### Amateur career

Nairn started his rugby union career playing with Currie. As part of the Currie U15 side they won the Edinburgh and District League and Cup in the 2011-12 season. A year later the Currie U16 won the similar honours.
He entered the Scottish Rugby Academy at the Stage 3 professional phase of the academy in the 2017-18 season.
Nairn has been drafted to Ayr in the Scottish Premiership for the 2018-19 season.

### Professional career

Nairn joined Harlequins in 2015. He was loaned out to Esher for game time and then loaned out to London Welsh.
Nairn was enrolled in the BT Sport Scottish Rugby Academy as a Stage 3 player. Stage 3 players are aligned to a professional club and given regional support. Nairn was assigned to Glasgow Warriors.
He made his debut for Glasgow Warriors in their opening match of the 2017-18 season - against Northampton Saints at Bridgehaugh Park, Stirling on 19 August 2017.
Nairn graduated from the Scottish Rugby Academy to earn a professional contract with Glasgow Warriors on 7 June 2018. He has earned a one-year deal with the club.
He made his first appearance for the 2018-19 season for the Warriors in their 50 -17 demolition of Harlequins at North Inch, Perth on 18 August 2018.
He was signed by Ayrshire Bulls on 22 June 2021.

### International career

Nairn played for Scotland U18 and was then picked for the Scotland U20 side despite still being eligible for the U18s.
He has been capped for Scotland 7s in 2016.
# Kylie Tennant

Kathleen Kylie Tennant AO (/ˈkaɪli/; 12 March 1912 – 28 February 1988) was an Australian novelist, playwright, short-story writer, critic, biographer, and historian.

## Early life and career

Tennant was born in Manly, New South Wales; she was educated at Brighton College in Manly and Sydney University, though she left without graduating. She was a publicity officer for the Australian Broadcasting Commission, as well as working as a journalist, union organiser, reviewer (for The Sydney Morning Herald), a publisher's literary adviser and editor, and a member of the Commonwealth Literary Fund advisory board. She married L. C. Rodd in 1933; they had two children (a daughter, Benison, in 1946 and a son, John Laurence, in 1951).
Her work was known for its well-researched, realistic, yet positive portrayals of the lives of the underprivileged in Australia. In a video interview filmed in 1986, three years before her death, for the Australia Council's Archival Film Series, Tennant told how she lived as the people she wrote about, travelling as an unemployed itinerant worker during the Depression years, living in Aboriginal communities and spending a short time in prison for research.
Two of Tennant's novels, Battlers and Ride on Stranger, set in the 1930s, have been made into television mini-series.
"Kylie's Hut", the author's retreat in Crowdy Bay, was destroyed during the 2019–20 Australian bushfire season.

## Personal life

Tennant was raised in a devout Christian Science family and her mother was a Christian Science practitioner. She converted to Anglicanism upon her marriage, which her family supported.

## Awards

* 1935: S. H. Prior Memorial Prize awarded by The Bulletin magazine, for Tiburon
* 1940: S. H. Prior Memorial Prize (run by the Bulletin), for The Battlers, shared with Eve Langley, The Pea-Pickers, and Malcolm Henry Ellis's "John Murtagh Macrossan lectures".
* 1942: Australian Literature Society Gold Medal for The Battlers
* 1960: Children's Book Council Book Award for All the Proud Tribesmen
* 1980: Officer of the Order of Australia for services to literature

## Commemorations

Two streets are named for Tennant:
* Kylie Tennant Street, Franklin, ACT
* Kylie Tennant Close, Glenmore Park, NSW
# Cambanoora Gorge

28°17′47″S 152°21′09″E﻿ / ﻿28.2963°S 152.3525°E﻿ / -28.2963; 152.3525 The Gorge runs between the source of the Condamine River and the town of Killarney in Queensland, Australia. It is not officially named and is also known as the Condamine Gorge and the 14 River Crossings. the area is officially known as "The Falls"
The gorge begins in the historical area of the head which is bordered by Wilsons Peak to the east and Mount Superbus to the north. Commencing at this broad grassland, the gorge narrows as the river encounters harder trachyte rock. This creates a dramatic landscape.
The road through the gorge was used to carry supplies and mail to early settlers along the length of the gorge, and to bring timber to the railway in Killarney. The Crossings are named First, Second, Double, Mawhirts, Bullocky, Flaggie, Rocky, Mill, Reis', Heywood's, Billy John's, Andrew Evan's, Long and Watson's Crossing in order from The Head down river.
The area is one of outstanding natural beauty in the Border ranges of southern Queensland. It is a rich bio-region with at least 14 endangered or threatened species ranging from the azure kingfisher to the spotted tail quoll and the glossy black cockatoo.
Condamine River Road runs through the Gorge and is a dirt road with a speed limit of 30 km/h (19 mph). The speed limit through the crossings is 5 km/h (3 mph). The gorge is not a national park or state forest, rather it is a farming and residential area. The road crosses the Condamine River 14 times and is dangerous during times of flooding, resulting in loss of property and life and hence is often closed.
The road through the gorge is bordered mostly by private property. Drivers should stick to the road and stop at designated stopping areas only including Bullocky Crossing. Proper driver behaviour is appreciated including slowing down for cows and wildlife on the road, leaving no rubbish, and not driving up the river (platypus there). Activities such as bushwalking, horse riding, bird watching, mountain bike riding, picnicking are encouraged. There is no camping in the gorge unless organised prior on private property.
Condamine River Road is section 6 of the National Trail.
More recently Condamine River road has been subjected to exploitation by 4WD tourism operators and clubs which has led to overtourism and damage to the sensative riparian environment that supports one of the last Murray Darling breeding populations of platypus and blackfish in South East Queensland. It has also resulted in escalating conflict between trespassing 4WD's, residents and the police.
# Charles Joseph Newbold

Lt. Colonel Charles Joseph Newbold DSO (12 January 1881 – 26 October 1946) was an English rugby union international who played club rugby for Cambridge University and Blackheath. He played six international games for England between 1904 and 1905. During the First World War he served the British Army in the Royal Engineers.

## Early life

Newbold was born in 1881 in Tunbridge Wells, England. One of eleven children, he was the second son of William Newbold (1828–1900) and Eleanor Isabel Newbold, née Fergusson (1862–1942) of East Grinstead, and was educated at Rose Hill in Tunbridge Wells and then Uppingham School. He entered Gonville and Caius College, Cambridge in 1900 and was awarded his BA in 1903. On leaving Cambridge in 1904 he joined brewing firm Guinness, becoming one of their early chemists at the Guinness Research Laboratory. His eldest sister Ethel Newbold (1882–1933) was a noted statistician and epidemiologist.

## Rugby career

Newbold first came to note as a rugby player when he represented Cambridge University and won two sporting Blues, in 1902 and 1903. He won his first international cap while still a Cambridge player, representing England in the 1904 Home Nations Championship. He played in all three games of the 1904 campaign and was reselected for the 1905 Championship. After leaving university he joined Blackheath, and became a member of the invitational tourists, The Barbarians in 1903.

## Military career and later life

After the outbreak of World War I, Newbold joined the Royal Engineers and reached the rank of lieutenant colonel. He was mentioned in dispatches on three occasions and was awarded the Distinguished Service Order. On 27 September 1924, he married Daphne Gertrude Persse, who served with the British Red Cross Society in both the First and Second World War. In 1925 Charles and Daphne moved to Bulawayo where they lived until early 1939 when they returned to England.
In 1941 Newold became the managing director of Guinness; and from 1942 to 1945 he was the chairman of the Brewers' Society.