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In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations.
A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. [1] This equivalent condition is formally expressed as follows:
1:1 correspondence, an older name for a bijection; Multivalued function; Correspondence (algebraic geometry), between two algebraic varieties; Corresponding sides and corresponding angles, between two polygons; Correspondence (category theory), the opposite of a profunctor; Correspondence (von Neumann algebra) or bimodule, a type of Hilbert space
The set of rays {ℓ, m, n, ...} through any point O can be put into 1:1 correspondence with the real numbers a (mod 2π) so that if A and B are points (not equal to O) of ℓ and m, respectively, the difference a m − a ℓ (mod 2π) of the numbers associated with the lines ℓ and m is ∠ AOB.
In mathematics, a frame bundle is a ... which are, in turn, in 1-1 correspondence with smooth bundle maps ... archived from the original (PDF) on 2017-03-30 ...
The correspondence has the following useful properties. It is inclusion-reversing. The inclusion of subgroups H 1 ⊆ H 2 holds if and only if the inclusion of fields E H 1 ⊇ E H 2 holds. Degrees of extensions are related to orders of groups, in a manner consistent with the inclusion-reversing property.
The correspondence was defined by Roger Howe in Howe (1979). The assertion that this is a 1-1 correspondence is called the Howe duality conjecture. Key properties of local theta correspondence include its compatibility with Bernstein-Zelevinsky induction [3] and conservation relations concerning the first occurrence indices along Witt towers . [4]
In mathematics, the geometric Langlands correspondence relates algebraic geometry and representation theory.It is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry. [1]