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If E is a complex vector bundle, then the conjugate bundle ¯ of E is obtained by having complex numbers acting through the complex conjugates of the numbers. Thus, the identity map of the underlying real vector bundles: ¯ = is conjugate-linear, and E and its conjugate E are isomorphic as real vector bundles.
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
Just as every symplectic structure is isomorphic to one of the form V ⊕ V ∗, every complex structure on a vector space is isomorphic to one of the form V ⊕ V. Using these structures, the tangent bundle of an n -manifold, considered as a 2 n -manifold, has an almost complex structure , and the co tangent bundle of an n -manifold ...
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In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map:, with () = ¯, giving the "canonical" real structure on , that is =.
Java does not have a standard complex number class, but there exist a number of incompatible free implementations of a complex number class: The Apache Commons Math library provides complex numbers for Java with its Complex class. The JScience library has a Complex number class. The JAS library allows the use of complex numbers.
Can 3SUM be solved in strongly sub-quadratic time, that is, in time O(n 2−ϵ) for some ϵ>0? Can the edit distance between two strings of length n be computed in strongly sub-quadratic time? (This is only possible if the strong exponential time hypothesis is false.) Can X + Y sorting be done in o(n 2 log n) time?
Despite the case being dismissed in July, Alec Baldwin says the story surrounding the fatal “Rust” shooting has only begun. On the Dec. 16 episode of David Duchovny’s “Fail Better ...