Koebe's theorem also implies that any finitely connected bounded region in the plane is conformally equivalent to the open unit disk with finitely many smaller disjoint closed disks removed, or equivalently the extended complex plane with finitely many disjoint closed disks removed.
22.
With respect to curvature and hyperbolicity it should be noted however that while curvature is a property that is essentially local, hyperbolicity is a large-scale property which does not see local ( i . e . happening in a bounded region ) metric phenomena.
23.
On the other hand, applying the inversion map gives a contradiction; for it would produce a harmonic map in bounded region vanishing at an interior point with boundary value 1, which contradicts the fact that 1 is the only harmonic map with boundary value 1.
24.
If A 1, . . ., A n are n operators each localized in a bounded region and U ( a ) represents the unitary operator actively translating the Hilbert space by the vector a, then if we pick some subset of the n operators to translate,
25.
Simulating a Poisson point process on a computer is usually done in a bounded region of space, known as a simulation " window ", and requires two steps : appropriately creating a random number of points and then suitably placing the points in a random manner.
26.
Let ? be a bounded region in "'C "'with smooth boundary " ? and let ? be a univalent holomorphic map of the unit disk " D " onto ? extending to a smooth diffeomorphism of the circle onto " ?.
27.
If the variables are assumed to be nonnegative, then adding the constraint results in a closed, bounded region, and since the target function is continuous, it must have a minimum and a maximum-and they will be found among the points which are " suspected " by the method.
28.
The mathematical tractability or ease of working with Poisson models is mostly because of its'complete independence', which essentially says that two ( or more ) disjoint ( or non-overlapping ) bounded regions respectively contain two ( or more ) a Poisson number of points that are independent to each other.
29.
The parameter, called "'rate "'or "'intensity "', is related to the expected ( or average ) number of Poisson points existing in some bounded region, where " rate " is usually used when the underling space has one dimension . see Terminology.
30.
For some bounded region \ textstyle B of Euclidean space \ textstyle \ textbf { R } ^ d, if the points form a homogeneous Poisson process with parameter \ textstyle \ lambda > 0, then the probability of \ textstyle n points existing in \ textstyle B \ subset \ textbf { R } ^ d is given by:
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