See polar set. In functional and convex analysis, related disciplines of mathematics, the polar set A ∘ is a special convex set associated to any subset A of a vector space X lying in the dual space X ∗; the bipolar of a subset is the polar of A ∘, but lies in X. There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a dual pair of vector spaces; the polar cone of a convex cone A ⊆ X is the set A ∘:= This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point x; some authors call a dual cone the polar cone. The polar of a set A ⊆ X is the set A ∘: = This; some authors include absolute values around the inner product. If A ⊆ B B ∘ ⊆ A ∘ An immediate corollary is that ⋃ i ∈ I A i ∘ ⊆ ∘. For all γ ≠ 0: ∘ = 1 ∣ γ ∣ A ∘. ∘ = ⋂ i ∈ I A i ∘. For a dual pair A ∘ is closed in Y under the weak-*-topology on Y.

The bipolar A ∘ ∘ of a set A is the closed convex hull of A ∪, the smallest closed and convex set containing both A and 0. The bidual cone of a cone A is the closed conic hull of A. For a closed convex cone C in X, the dual cone is the polar of C.

Téoz was a brand name used by France’s national railway company, SNCF, to denote long-distance reservation-obligatory passenger rail services on certain SNCF routes. SNCF established the Téoz system in 2003 as an attempt to bring certain key routes up to the'same standards' as TGVs where high speed lines are not available, leaving ordinary Corail trains to serve the rest of the non high speed inter-city and inter-regional network. Téoz services were not universally liked as they required compulsory reservation but went little faster than the traditional services they replaced. Since December 2011 the Teoz brand has been abandoned and all SNCF long distance services have been assimilated into the Intercités brand, along with the overnight sleeper trains which have lost their previous branding. There is an Éco train, which runs once a day on the line from the Gare d'Austerlitz in Paris to Toulouse. In 2nd class, passengers can choose between reclining seats in an open carriage, or six-person couchette compartments with the couchettes folded to form benches.

In 1st class, only couchettes are available. These are permanently in night position, so passengers in 1st class must either travel lying down, or agree to share the lower couchettes, sitting without backrests or armrests

The Cherry Thing is an album by vocalist Neneh Cherry and The Thing, consisting of saxophonist Mats Gustafsson, bassist Ingebrigt Håker Flaten and drummer Paal Nilssen-Love. It was released the following year by Smalltown Supersound. Neneh Cherry's stepfather was free jazz trumpeter Don Cherry; the trio The Thing took their name from a track on Don Cherry's 1966 album Where Is Brooklyn? Cherry "began her career in the 1980s as a teen vocalist in post-punk outfits Rip Rig & Panic and Float Up CP. Conny Lindstrom, who had produced albums by The Thing knew Cherry and suggested that she perform with them; the Cherry Thing was recorded at Harder Sound Studios in Atlantis Studios in Stockholm. The album contains eight tracks; the two originals are "Cashback" by Cherry. The nine-minute version of Suicide's "Dream Baby Dream" "is no less ethereal than the original, but far more sinister, her accompanists build a gorgeous wall of atmospheric tension behind her". Martina Topley-Bird's "Too Tough to Die" "begins sparsely and before Cherry and Gustafsson enter and begin pushing, Cherry's vocal is emboldened with risk, turning the melody in on itself and ululating against the baritone horn.

The rhythm sections answers with syncopated breaks and funk." MF Doom's "Accordion" and the Stooges' "Dirt" "are sparse and poignant, the former tinged with implied violence, the latter smolders with raw, dark sensuality." Don Cherry's "Golden Heart" is "an otherworldly meld of Middle Eastern modes and textures and a skeletal lyric frame that displays this group's command of diverse musical languages." On Ornette Coleman's "What Reason", the "nearly mournful presentation, with gorgeous jazz singing by Cherry and restrained yet adventurous soloing by Gustafsson and Håker Flaten, make the tune drip with longing." The album was released on 19 June 2012. It peaked at number 16 on the Billboard Jazz Albums chart. BBC reviewer John Doran stated that the recording was "one of the most enjoyable and original albums of the year". John Fordham of The Guardian commented that the music "dramatically bridges the singer's avant-pop world and the flat-out sax-howling, percussion-thundering soundscape the group have been poleaxing audiences with since 2000."

Ben Ratliff of The New York Times wrote that the intentional contrast between Cherry's tenderness and the band's loud aggression sometimes became unbalanced in favour of the latter. "Cashback" – 5:58 "Dream Baby Dream" – 8:24 "Too Tough to Die" – 5:13 "Sudden Moment" – 8:26 "Accordion" – 6:10 "Golden Heart" – 4:43 "Dirt" – 6:47 "What Reason" – 5:18 Neneh Cherry – vocals Mats Gustafsson – baritone sax, tenor sax, electronics Ingebrigt Håker Flaten – double bass, electric bass, electronics Paal Nilssen-Love – drums, percussion "A Quietus Interview Keep Those Dreams Burning Forever: Neneh Cherry Interviewed"

Anthony A. Armstrong is a former American football wide receiver, he was signed as an undrafted free agent by the Miami Dolphins in 2008. He played college football at West Texas A&M, he was a member of the Washington Redskins and Jacksonville Jaguars. Before joining the NFL, Armstrong was a member of the Dallas Desperados of the Arena Football League and the Odessa Roughnecks of the Intense Football League. Armstrong was born on March 1983 in Greencastle, Indiana, his mother, Gwen Armstrong, worked as an Administration Manager at IBM and his father, Thomas Armstrong, died when Anthony was 2 days short of 6 years old. Anthony is a middle child of 4, he has a brother named two sisters, Catrina Scott and Ashley Armstrong. When Anthony was a teenager, he played high school football at Newman Smith High School. During his senior season, he had 11 receptions for 250 yards and three touchdowns and was named to the All-Metrocrest second-team, he lettered twice as a member of the track and field team and competed in the 100-meter dash.

Armstrong attended and played college football at West Texas A&M University, a Division II school in Canyon, Texas. As a freshman, he had 165 yards and one touchdown on 13 catches; the following season, he recorded 24 receptions for 256 yards. As a junior, Armstrong caught 54 passes for 740 yards and four touchdowns, he earned first-team All-LSC South honors. In his final college season, Armstrong was named second-team All-LSC as a senior after leading the team with 54 catches for 607 yards and four touchdowns, he finished his four-year letterman career as West Texas A&M’s fourth leading receiver with 145 receptions for 1,768 yards and nine touchdowns. Armstrong graduated from West Texas A&M with a degree in Marketing. Armstrong was signed by the Odessa Roughnecks of the Intense Football League in 2006. In 13 games for the Roughnecks, he had 55 receptions for 760 yards and 18 touchdowns and returned seven kickoffs for 166 yards and a touchdown, he was rushed four times for 39 yards and three touchdowns.

Armstrong had eight tackles, two pass breakups and a fumble recovery on defense. His best game was against the Louisiana Swashbucklers on July 1, 2006, during which he registered 11 receptions for 157 yards and three touchdowns; the Roughnecks won the IFL championship during Armstrong's only season on the team. In 2007, Armstrong joined the Dallas Desperados of the Arena Football League as a practice squad member and was signed to active roster on March 25, 2007, he made his AFL debut against the Austin Wranglers on March 31, 2007 where he recovered an onside kick and returned it 40 yards for the game-winning touchdown. Armstrong was placed on the Reserve/Injured list with a hamstring injury on May 2 and missed the remainder of the season, after recording ten catches for 126 yards and two touchdowns in four games, he played the entire 2008 season with the Desperados and posted 85 catches for 1,148 yards and 18 touchdowns. He additionally returned seven kickoffs for 31 yards helping the team compile a 12-4 regular season record and a berth to the AFL playoffs.

After the Arena Football League season, Armstrong was signed with the Miami Dolphins’ practice squad as an unrestricted free agent on July 25, 2008. He stayed on the practice squad the entire season and the 2009 preseason, but was waived August 29, 2009. Armstrong signed with the Washington Redskins on October 21, 2009 and spent the final 10 weeks on the Redskins’ practice squad. Armstrong made his first regular season NFL career catch for 11 yards on September 12, 2010 in a 13-7 win over the Dallas Cowboys and soon emerged as one of the Redskins' primary deep threats, he recorded a career long 76-yard reception against the Philadelphia Eagles on November 15, 2010, finishing fourth among all receivers with 7 receptions of 40 yards or more for the 2010-2011 season. Among receivers with at least 20 receptions over the season, Armstrong finished third with 19.8 yards/catch. In Week 1 against the New York Giants, Armstrong made his first touchdown of the 2011 season, he would not get another touchdown until Week 12 against the Seattle Seahawks, where he caught a 50-yard touchdown pass from Rex Grossman helping to end the Redskins' six-game losing streak.

The Redskins released Armstrong on August 31, 2012 for final roster cuts before the start of the 2012 season and he was picked up by the Dolphins. On September 1, 2012, Armstrong was claimed off waivers by the Dolphins, he was waived on October 30. Armstrong was signed by the Jacksonville Jaguars on October 31, 2012, he was released by the team on November 12. After working out with the Dallas Cowboys on November 21, 2012, the team signed Armstrong on November 26, he was released on December 22. On January 7, 2013 Armstrong signed a one-year, $630,000 contract, he was released on March 27. On April 3, Armstrong re-signed with the Cowboys, he was released by the team on August 31. He signed with the Cleveland Browns on May 19, 2014, he was released by the team on August 25, 2014. Miami Dolphins bio Washington Redskins bio Anthony Armstrong on Twitter

Tongxin County is a county under the administration of the prefecture-level city of Wuzhong in the central part of the Ningxia Hui Autonomous Region of the People's Republic of China, bordering Gansu province to the east. It has a total area of 7021 square kilometers, a population of 360,000 people. Due to its topography, Tongxin County's economic conditions and foundation are poor, the county is stricken by drought. However, farmers still make up two-thirds of the county's population; the county government has sought to use high technological methods to improve the standard of living in the region. The county government is located in the town of Tongxin, the county's postal code is 751300. In his Red Star Over China, Edgar Snow mentions visiting the small towns of Yuwang Bao and Yuwang located in today's Tongxin County. At the moment, Yuwang Bao was the location of the headquarters of the First Front Red Army, where Edgar Snow met with the army's commander, Peng Dehuai there. Edgar Snow described Yuwang Bao as follows: On the high, stout walls of Yu Wang Pao a squad of Red buglers was practicing, from a corner of this fortlike city flew a big scarlet flag, its yellow hammer and ciskle cracking out in the breeze...

We could look down on one side to a clean courtyard, where Muslim women were hulling rice and baking. Washing hung from a line on another side. In a distant square some Red soldiers were practicing wall scaling, broad jumping, grenade throwing. Yuwang captured by the Reds from the troops of Ma Hongkui after a ten-day siege, was described as an ancient Muslim city of four or five hundred families, with a magnificent wall of stone and brick. Outside of the wall was a Muslim mosque, with its own wall of beautiful glazed brick unscarred, but other buildings showed signs of the siege this city had undergone before it was taken by the Reds. In October 1936, a public meeting was held at the Tongxin Great Mosque, establishing the Yuwang Hui Autonomous Government - the first autonomous county in Communist China. Although this autonomous unit was short-lived, the mosque building was protected as a revolutionary historical site, became the only major Islamic building in Ningxia to survive destruction during the Cultural Revolution.

According to the regional census of 1983, Tongxin County had the second-highest percentage of the Hui people in its population among all counties in Ningxia, second only to Jingyuan County. 79.0% of its population at the time were Hui. Dru C. Gladney, Muslim Chinese: Ethnic Nationalism in the People's Republic. 1st ed.: Harvard University Press, 1991, ISBN 0-674-59495-9. ISBN 0-674-59497-5