. Previously, Menger and Birkhoff had axiomatized complicated projective geometry concerning the properties of its lattice of linear subspaces. Von Neumann, adhering to his work on rings of operators, weakened All those axioms to explain a broader class of lattices, the continual geometries.
Constructing over the get the job done of Felix Hausdorff, in 1924 Stefan Banach and Alfred Tarski proved that provided a solid ball in three‑dimensional Area, there exists a decomposition from the ball into a finite number of disjoint subsets which might be reassembled together in a distinct strategy to yield two equivalent copies of the initial ball. Banach and Tarski proved that, employing isometric transformations, the results of taking aside and reassembling a two-dimensional figure would automatically possess the exact place as the original. This could make creating two device squares out of 1 extremely hard.
[239] Entanglement actions are dependent on some amount directly linked to the von Neumann entropy. Specified a statistical ensemble of quantum mechanical systems Together with the density matrix ρ displaystyle rho
Starting while in the late 1930s, von Neumann formulated an expertise in explosions—phenomena which are tricky to model mathematically. In the course of this era, von Neumann was the primary authority in the arithmetic of shaped rates.
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During the proceedings of the conference on von Neumann's advancement product, Paul Samuelson mentioned that many mathematicians had produced techniques practical to economists, but that von Neumann was one of a kind in obtaining created major contributions to financial principle itself.[279]
Halmos recounts a story instructed by Nicholas Metropolis, in regards to the velocity of von Neumann's calculations, when anyone requested von Neumann to unravel the famous fly puzzle:[451]
Later with Robert D. Richtmyer, von Neumann developed an algorithm defining artificial viscosity that enhanced the knowledge of shock waves. When computers solved hydrodynamic or aerodynamic complications, they made an effort to place too many computational grid details at locations of sharp discontinuity (shock waves).
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Von Neumann proved that the best method of carrying out significant-scale mining functions including mining an entire moon or asteroid belt will be through the use of self-replicating spacecraft, Benefiting from their exponential development.[312]
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