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Note that consistency as defined in the CAP theorem is quite different from the consistency guaranteed in ACID database transactions. [4] Availability Every request received by a non-failing node in the system must result in a response. This is the definition of availability in CAP theorem as defined by Gilbert and Lynch. [1]
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Pitot theorem (plane geometry) Pizza theorem ; Pivot theorem ; Planar separator theorem (graph theory) Plancherel theorem (Fourier analysis) Plancherel theorem for spherical functions (representation theory) Poincaré–Bendixson theorem (dynamical systems) Poincaré–Birkhoff–Witt theorem (universal enveloping algebras)
For example, the map that sends (,) to (,) has image the set {} {= =}, which is not a variety, but is constructible. Chevalley's theorem in the generality stated above would fail if the simplified definition of constructible sets (without restricting to retrocompact open sets in the definition) were used.
In affine geometry, a cap set is a subset of the affine space (the -dimensional affine space over the three-element field) where no three elements sum to the zero vector. The cap set problem is the problem of finding the size of the largest possible cap set, as a function of n {\displaystyle n} . [ 1 ]
In analogy with the interpretation of the cup product in terms of the Künneth formula, we can explain the existence of the cap product in the following way.Using CW approximation we may assume that is a CW-complex and () (and ()) is the complex of its cellular chains (or cochains, respectively).
Milnor's Fibration Theorem states that, for every such that the origin is a singular point of the hypersurface (in particular, for every non-constant square-free polynomial of two variables, the case of plane curves), then for sufficiently small,
An illustration of Carathéodory's theorem for a square in R 2. Carathéodory's theorem in 2 dimensions states that we can construct a triangle consisting of points from P that encloses any point in the convex hull of P. For example, let P = {(0,0), (0,1), (1,0), (1,1)}. The convex hull of this set is a square.