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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:
However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on intersection theory, [1] uses the definition above. In literature, however, a correspondence from a variety X to a variety Y is often taken to be a subset Z of X×Y such that Z is finite and surjective over each ...
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.
A bijection from the natural numbers to the integers, which maps 2n to −n and 2n − 1 to n, for n ≥ 0. For any set X, the identity function 1 X: X → X, 1 X (x) = x is bijective. The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y.
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]
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]