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The von Neumann method is based on the decomposition of the errors into Fourier series.To illustrate the procedure, consider the one-dimensional heat equation = defined on the spatial interval , with the notation = (,) where are the specific x values, and are the sequence of t values.
Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming. [274]
As derived using von Neumann stability analysis, the FTCS method for the one-dimensional heat equation is numerically stable if and only if the following condition is satisfied: Δ t ≤ Δ x 2 2 α . {\displaystyle \Delta t\leq {\frac {\Delta x^{2}}{2\alpha }}.}
A quantum channel will increase or leave constant the von Neumann entropy of every input state if and only if the channel is unital, i.e., if it leaves fixed the maximally mixed state. An example of a channel that decreases the von Neumann entropy is the amplitude damping channel for a qubit, which sends all mixed states towards a pure state. [35]
A von Neumann architecture scheme. The von Neumann architecture—also known as the von Neumann model or Princeton architecture—is a computer architecture based on the First Draft of a Report on the EDVAC, [1] written by John von Neumann in 1945, describing designs discussed with John Mauchly and J. Presper Eckert at the University of Pennsylvania's Moore School of Electrical Engineering.
But von Neumann realized that the trick of such so-called paradoxical decompositions was the use of a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations (whether the special linear group or the special affine group ) contains such subgroups, and this opens the ...
In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. [1] The simplest case, when the sets are affine spaces, was analyzed by John von Neumann.
In 1947, John von Neumann sent a letter to Robert Richtmyer proposing the use of a statistical method to solve neutron diffusion and multiplication problems in fission devices. [5] His letter contained an 81-step pseudo code and was the first formulation of a Monte Carlo computation for an electronic computing machine.