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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.
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 as the problem of maximizing (or minimizing) a ratio of affine functions over a polyhedron ,
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.
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 ). Each element of PGL(2, R ) can be defined by a nonsingular 2×2 real matrix, and two matrices define the same element of PGL(2, R ) if one is the product of the ...
In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform.It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency.
which are called homographic functions or linear fractional transformations. In the case of the complex projective line, which can be identified with the Riemann sphere, the homographies are called Möbius transformations. These correspond precisely with those bijections of the Riemann sphere that preserve orientation and are conformal. [3]
A pseudolinear function is a function that is both pseudoconvex and pseudoconcave. [4] For example, linear–fractional programs have pseudolinear objective functions and linear–inequality constraints. These properties allow fractional-linear problems to be solved by a variant of the simplex algorithm (of George B. Dantzig). [5] [6] [7]
By applying a Laplace transform to the LTI system above, the transfer function becomes = () = = =For general orders and this is a non-rational transfer function. Non-rational transfer functions cannot be written as an expansion in a finite number of terms (e.g., a binomial expansion would have an infinite number of terms) and in this sense fractional orders systems can be said to have the ...