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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. Without further specifications, the term unit disk is used for the open unit disk about the origin , D 1 ( 0 ) {\displaystyle D_{1}(0)} , with respect to the standard ...
Every induced subgraph of a unit disk graph is also a unit disk graph. An example of a graph that is not a unit disk graph is the star K 1 , 6 {\displaystyle K_{1,6}} with one central node connected to six leaves: if each of six unit disks touches a common unit disk, some two of the six disks must touch each other.
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In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.
The disks are laid such that their centers form a polygonal path from the value where () is maximized to any other point in the domain, while being totally contained within the domain. Thus the existence of a maximum value implies that all the values in the domain are the same, thus f ( z ) {\displaystyle f(z)} is constant.
The open unit disk (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to (see "Poincaré metric"), meaning that it is usually possible to pass between and .