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Linear fractional transformations leave cross ratio invariant, so any linear fractional transformation that leaves the unit disk or upper half-planes stable is an isometry of the hyperbolic plane metric space. Since Henri Poincaré explicated these models they have been named after him: the Poincaré disk model and the Poincaré half-plane model.
Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function 1. Formally, a linear-fractional program is defined ...
Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The automorphisms of a real projective line are called projective transformations, homographies, or linear fractional transformations. They form the projective linear group PGL(2, R).
It is for this reason that doubly periodic functions, such as elliptic functions, possess a modular group symmetry. The action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point (p, q) corresponding to the fraction p / q (see Euclid's orchard).
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
In mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions that are in general nonlinear. The ratio to be optimized often describes some kind of efficiency of a system.
If the continued fraction converges, the successive convergents A n / B n are eventually arbitrarily close together. Since the linear fractional transformation Τ n (z) is a continuous mapping, there must be a neighborhood of z = 0 that is mapped into an arbitrarily small neighborhood of Τ n (0) = A n / B n .
The linear operator which assigns to each function its derivative is an example of a differential operator on a function space. By means of the Fourier transform , pseudo-differential operators can be defined which allow for fractional calculus.