Is the upper half plane compact?
The upper half complex plane is defined by Hh := {z∈C | Im(z) >0}. The group SL2(Z) acts on H by fractional linear transformations. The space Hh/SL2(Z) is not compact; it is compactified by adding the cusps, which are points of Q, together with ∞. Thus we define Hh^ * to be the upper half plane union the cusps.
Is the unit disk open or closed?
The closed unit disk around P is the set of points whose distance from P is less than or equal to one: Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself.
What is half plane in complex analysis?
Complex plane The term arises from a common visualization of the complex number x + iy as the point (x, y) in the plane endowed with Cartesian coordinates. When the y axis is oriented vertically, the “upper half-plane” corresponds to the region above the x axis and thus complex numbers for which y > 0.
What is a disc in complex analysis?
In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not.
How do you know which half-plane is shaded?
If the coordinates you selected make the inequality a true statement when plugged in, then you should shade the half‐plane containing those coordinates. If the coordinates you selected do not make the inequality a true statement, then shade the half‐plane not containing those coordinates.
What are disk units?
A disk unit is the description used to describe the physical disk drive or its casing.
What is the difference between a disk and circle?
Correctly speaking, a circle has no area (it is just the edge), but a disk does. But in practice people think of a circle as the edge or the enclosed space, or both. A disk is “closed” when it includes the bounding circle, and open if it does not. Also spelled “disc”.
What is a half-plane in algebra?
A half-plane is a planar region consisting of all points on one side of an infinite straight line, and no points on the other side. If the points on the line are included, then it is called an closed half-plane. If the points on the line are not included, then it is called an open half-plane.
Is the open unit disk isomorphic to the upper half-plane?
There are conformal bijective maps between the open unit disk and the open upper half-plane. So considered as a Riemann surface, the open unit disk is isomorphic (“biholomorphic”, or “conformally equivalent”) to the upper half-plane, and the two are often used interchangeably.
Which subgroup maps the upper half-plane onto itself?
The subgroup that maps the upper half-plane, H, onto itself is PSL (2, R ), the transforms with real coefficients, and these act transitively and isometrically on the upper half-plane, making it a homogeneous space .
How do you transform the disk model to the Poincaré half plane?
The disk model can be transformed to the Poincaré half-plane model by the mapping g given above. Both the Poincaré disk and the Poincaré half-plane are conformal models of the hyperbolic plane, which is to say that angles between intersecting curves are preserved by motions of their isometry groups.
What is the Poincaré half-plane model?
The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry .