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The FBISE was established under the FBISE Act 1975. [2] It is an autonomous body of working under the Ministry of Federal Education and Professional Training. [3] The official website of FBISE was launched on June 7, 2001, and was inaugurated by Mrs. Zobaida Jalal, the Minister for Education [4] The first-ever online result of FBISE was announced on 18 August 2001. [5]
A pairing is called perfect if the above map is an isomorphism of R-modules and the other evaluation map ′: (,) is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field, M,N are finite dimensional vector spaces and L=R .
When this occurs, B is said to be a perfect pairing. In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect ...
(February 2024) (Learn how and when to remove this message) In set theory , Zermelo–Fraenkel set theory , named after mathematicians Ernst Zermelo and Abraham Fraenkel , is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox .
In terms of intersection forms, we say the plane has one of type x 2 (there is only one class of lines, and they all intersect with each other). Note that on the affine plane, one might push off L to a parallel line, so (thinking geometrically) the number of intersection points depends on the choice of push-off. One says that “the affine ...
In mathematics, a dual system, dual pair or a duality over a field is a triple (,,) consisting of two vector spaces, and , over and a non-degenerate bilinear map:.. In mathematics, duality is the study of dual systems and is important in functional analysis.
A general paradigm in group theory is that a group G should be studied via its group representations.A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M.
The case n = 2 is the axiom of pairing with A = A 1 and B = A 2. The cases n > 2 can be proved using the axiom of pairing and the axiom of union multiple times. For example, to prove the case n = 3, use the axiom of pairing three times, to produce the pair {A 1,A 2}, the singleton {A 3}, and then the pair {{A 1,A 2},{A 3}}.