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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]
The CAP theorem is based on three trade-offs, one of which is "atomic consistency" (shortened to "consistency" for the acronym), about which the authors note, "Discussing atomic consistency is somewhat different than talking about an ACID database, as database consistency refers to transactions, while atomic consistency refers only to a property of a single request/response operation sequence.
Hadwiger's theorem (geometry, measure theory) Helly's theorem (convex sets) Holditch's theorem (plane geometry) John ellipsoid ; Jung's theorem ; Kepler conjecture (discrete geometry) Kirchberger's theorem (discrete geometry) Krein–Milman theorem (mathematical analysis, discrete geometry) Minkowski's theorem (geometry of numbers)
The tradeoff between availability, consistency and latency, as described by the PACELC theorem. In database theory, the PACELC theorem is an extension to the CAP theorem.It states that in case of network partitioning (P) in a distributed computer system, one has to choose between availability (A) and consistency (C) (as per the CAP theorem), but else (E), even when the system is running ...
One important structure theorem of abelian varieties is Matsusaka's theorem. It states that over an algebraically closed field every abelian variety A {\displaystyle A} is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties J → A {\displaystyle J\to A} where J {\displaystyle J} is a Jacobian.
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).
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.
The formulation used by Seth Gilbert and Nancy Lynch needs to be presented as a theorem, and then the history section can contain the conjecture; or the article needs to be about the conjecture and proceed to explain what's actually the theorem. --Nemo 17:28, 24 July 2014 (UTC)