Pin group

In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group.

In general the map from the Pin group to the orthogonal group is not onto or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both.

The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −I.

General definition

Definite form

The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it double covers the orthogonal group. The pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are.

In terms of the standard forms, O(n, 0) = O(0,n), but Pin(n, 0) and Pin(0, n) are not isomorphic. Using the "+" sign convention for Clifford algebras (where ), one writes

Group

Group may refer to:

Groups of people

In science and technology

In mathematics

In chemistry

In computing and the internet

Other uses in science and technology

Other uses

Stratigraphic unit

A stratigraphic unit is a volume of rock of identifiable origin and relative age range that is defined by the distinctive and dominant, easily mapped and recognizable petrographic, lithologic or paleontologic features (facies) that characterize it.

Units must be mappable and distinct from one another, but the contact need not be particularly distinct. For instance, a unit may be defined by terms such as "when the sandstone component exceeds 75%".

Lithostratigraphic units

Sequences of sedimentary and volcanic rocks are subdivided on the basis of their lithology. Going from smaller to larger in scale, the main units recognised are Bed, Member, Formation, Group and Supergroup.

Bed

A bed is a lithologically distinct layer within a member or formation and is the smallest recognisable stratigraphic unit. These are not normally named, but may be in the case of a marker horizon.

Member

A member is a named lithologically distinct part of a formation. Not all formations are subdivided in this way and even where they are recognized, they may only form part of the formation.

1994 Group

The 1994 Group was a coalition of smaller research-intensive universities in the United Kingdom, founded in 1994 to defend these universities' interests following the creation of the Russell Group by larger research-intensive universities earlier that year. The 1994 Group originally represented seventeen universities, rising to nineteen, and then dropping to eleven. The Group started to falter in 2012, when a number of high performing members left to join the Russell Group. The 1994 Group ultimately dissolved in November 2013.

Role

The group sought "to represent the views of its members on the current state and the future of higher education through discussions with the government, funding bodies, and other higher education interest groups" and "[made] its views known through its research publications and in the media".

University Alliance, million+, GuildHE and the Russell Group were its fellow university membership groups across the UK higher education sector.

Members

1994 Group position in league tables

Pin (song)

"Pin" is a single by the Yeah Yeah Yeahs from their debut album, Fever to Tell (2003). It was released on July 22, 2003, and received critical acclaim but did not receive much mainstream attention, especially when compared with the band's next single "Maps". This single includes the B-side "Mr. You're On Fire Mr." (a Liars cover), which is not found on any of the Yeah Yeah Yeahs' full-length releases. The single peaked at #29 on the UK official charts.

The animated video for the song was created by Tunde Adebimpe.

Charts

Track listing

Art Angels

Art Angels is the fourth studio album by Canadian singer and songwriter Claire Boucher, professionally known as Grimes. It was digitally released on November 6, 2015 by 4AD, and in physical formats on December 11. Boucher began planning the record in 2013 as the follow-up to her third studio album Visions, however she scrapped most of the material from these sessions and began a new set of recordings in 2014. The track "Realiti", which came from the earlier recordings, was released as a demo in early 2015.

Art Angels has been described as being more accessible than Grimes' previous albums while retaining her experimental influences. The album features guest vocals from Taiwanese rapper Aristophanes and American recording artist Janelle Monáe. Two singles, "Flesh without Blood" and "Scream", were released ahead of the album and a single music video was made for the tracks "Flesh without Blood" and "Life in the Vivid Dream". Art Angels sold 11,000 copies in the first week of its release and became Boucher's highest charting album at the time. The album was released to widespread critical acclaim and was ranked by several publications as one of the best albums of 2015.

Pin (professional wrestling)

A pinfall is a victory condition in various forms of professional wrestling that is met by holding (pinning) an opponent's shoulders on the wrestling mat, usually till the referee counts to three. In professional wrestling, a pinfall is a common method of winning a match.

The purpose of a pinning maneuver is to hold the opponent's shoulders against the mat for a count of three. The count is broken (a near-fall) if the opponent manages to raise one or both of his shoulders off of the mat, commonly by kicking out (throwing their legs up to cause their shoulders to rise from the mat). In some positions, a wrestler may bridge (arching their back so that only their feet and the top of their head are touching the ground) to put more of their weight on the pinned opponent or to prop themselves up from being pinned. Sometimes, an attacking wrestler may (illegally) hook the opponent's tights for extra leverage. Another popular illegal tactic of heel wrestlers is to attempt a pin close to the ring ropes so they can prop their legs (or on rare occasions, arms) up on the ropes to gain additional leverage, putting more weight on the opponent. On the other hand, a pin fall attempt cannot occur in the first place when one rolls out of the ring if falls do not count anywhere, or if the opponent lies on his/her stomach upon impact, so it would take extra effort to roll the opponent over, even when he/she is knocked out cold due to the body being effectively a dead weight.