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A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0
The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Gaussian curvature. The closed upper half-plane is the union of the upper half-plane and the real axis. It is the closure of the upper half-plane.
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
Half-space (geometry), either of the two parts into which a plane divides Euclidean space (Poincaré) Half-space model, a model of hyperbolic geometry using a Euclidean half-space; Siegel upper half-space, a set of complex matrices with positive definite imaginary part; Half-space (punctuation), a spacing character half the width of a regular space
The stereographic projection is a homeomorphism between the unit sphere in with a single point removed and the set of all points in (a 2-dimensional plane). If G {\displaystyle G} is a topological group , its inversion map x ↦ x − 1 {\displaystyle x\mapsto x^{-1}} is a homeomorphism.
In complex analysis, the (open) right half-plane is the set of all points in the complex plane whose real part is strictly positive, that is, ...
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of ...
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R).The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces.