Setting up on his success on matrix games and on his design of the growing economic system, von Neumann invented the idea of duality in linear programming when George Dantzig described his function in a few minutes, and an impatient von Neumann questioned him to get to The purpose.
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Von Neumann first proposed a quantum logic in his 1932 treatise Mathematical Foundations of Quantum Mechanics, in which he noted that projections on the Hilbert Room can be considered as propositions about Bodily observables. The sphere of quantum logic was subsequently inaugurated, inside a popular paper of 1936 by von Neumann and Garrett Birkhoff, the main get the job done ever to introduce quantum logics,[246] wherein von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus considerably distinctive from all classical logics and rigorously isolated a brand new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was initially outlined in a short portion in von Neumann's 1932 do the job, but in 1936, the necessity for the new propositional calculus was demonstrated by means of many proofs.
This work demanded the generation of standard rings.[187] A von Neumann standard ring is a ring in which for see this every a displaystyle a
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