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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.
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
Two points in the upper half-plane give isomorphic elliptic curves if and only if they are related by a transformation in the modular group. Thus, the quotient of the upper half-plane by the action of the modular group is the so-called moduli space of elliptic curves: a space whose points describe isomorphism classes of elliptic curves. This is ...
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 ...
In mathematics, the Drinfeld upper half plane is a rigid analytic space analogous to the usual upper half plane for function fields, introduced by Drinfeld ().It is defined to be P 1 (C)\P 1 (F ∞), where F is a function field of a curve over a finite field, F ∞ its completion at ∞, and C the completion of the algebraic closure of F ∞.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for ...
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon.Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction.
In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane and has a non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or a real constant, [1] but is not necessarily injective or surjective.