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In mathematics, inverse mapping theorem may refer to: the inverse function theorem on the existence of local inverses for functions with non-singular derivatives; the bounded inverse theorem on the boundedness of the inverse for invertible bounded linear operators on Banach spaces
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
However, equation (3-11) is a 16th-order equation, and even if we factor out the four solutions for the fixed points and the 2-periodic points, it is still a 12th-order equation. Therefore, it is no longer possible to solve this equation to obtain an explicit function of a that represents the values of the 4-periodic points in the same way as ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
Sometimes, the inverse of a function cannot be expressed by a closed-form formula. For example, if f is the function = , then f is a bijection, and therefore possesses an inverse function f −1. The formula for this inverse has an expression as an infinite sum:
P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...
If f is a real-valued function, then its Weyl-map image Φ[f] is self-adjoint. If f is an element of Schwartz space, then Φ[f] is trace-class. More generally, Φ[f] is a densely defined unbounded operator. The map Φ[f] is one-to-one on the Schwartz space (as a subspace of the square-integrable functions).
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...