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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 ...
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 ...
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).
In LP the objective function is a linear function, while the objective function of a linear–fractional program is a ratio of two linear functions. In other words, a linear program is a fractional–linear program in which the denominator is the constant function having the value one everywhere. A linear–fractional program can be solved by a ...
Fractional calculus was introduced in one of Niels Henrik Abel's early papers [3] where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized ...
Quadratic programming allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming. Fractional programming studies optimization of ratios of two nonlinear functions. The special class of ...
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality.