A pairing is any R-bilinear map . That is, it satisfies
for any and any and any . Or equivalently, a pairing is an R-linear map
where denotes the tensor product of M and N.
A pairing can also be considered as an R-linear map, which matches the first definition by setting .
A pairing is called perfect if the above map is an isomorphism of R-modules.
If a pairing is called alternating if for the above map we have .
A pairing is called non-degenerate if for the above map we have that for all implies .
Other articles related to "pairing, pairings":
... The superconductivity does not appear at T* because large phase fluctuations of the pairing field cannot order at this temperature ... is then produced by non coherent fluctuations of the pairing field ... a normal state precursor of the superconducting gap due to local, dynamic pairing correlations ...
... products on complex vector spaces are sometimes called pairings, although they are not bilinear ... of complex representations of a finite group which is frequently called character pairing ...
... A pairing exists between the village of Souillac and the town of Souillac in France (Lot Department) since 1987 ... This successful pairing was initiated by Armand Maudave, an inhabitant of the village of Souillac, and the Comte de Montferrand who is himself one of the descendants of the Vicomte de Souillac ... de Souillac was inaugurated at the Batelage to mark the 20th anniversary of the pairing ...
... known as a power couple) is a popular or wealthy pairing that intrigues and fascinates the public in an intense or even obsessive fashion ... With regard to real-life pairings, tabloids and the mainstream media have focused on wealthy or popular celebrity couples, and have titled them supercouples ... Examples are the former pairing of Ben Affleck and Jennifer Lopez (which became known by the portmanteau "Bennifer"), the relationship of Brad Pitt and Angelina Jolie ("Brangelina"), and Japanese former ...
... The film was designed as another vehicle for the immensely popular Gable/Turner pairing ... on their new team as it had not seen a Gable pairing this successful since Joan Crawford and Jean Harlow in the 30's ... The film was, like the Gable/Turner pairing the year before, another success and they were teamed for two more films in the future, their first pairing together after six years in 1948's Homecoming ...
Famous quotes containing the word pairing:
“Through man, and woman, and sea, and star,
Saw the dance of nature forward far;
Through worlds, and races, and terms, and times,
Saw musical order, and pairing rhymes.”
—Ralph Waldo Emerson (18031882)