Von Neuman

John von Neuman, one of the greatest mathematicians of the twentieth century, believed that the sciences, in essence, do not try to explain, they hardly even try to interpret they mainly make models. By a model he meant a mathematical construct that, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work. Stephen Hawking also believes that physical theories are just mathematical models we construct and that it is meaningless to ask whether they correspond to reality, just as it is to ask whether they predict observations. 

In the 1950s, biologists (notably Francis Crick and James Watson) discovered the molecular basis for information coding in DNA and established that the workings of cells were molecular machines tvith understandable structure and function. Mathematician John von Neuman developed a mathematical theory of self-reproducing machines based on the biological theories. 

In the case of diffusion in the absence of advection, the stability of a forward-intime solution in one dimension is given by the von Neuman criterion,... 

D. Marx and J. Hiitter, in Modem Methods and Algorithms of Quantum Chemistry , NIC Series Vol.l, ed. J. Grotendorst, John von Neuman Institute for Computing, Jiilich, 2000, p. 301. 

Detonation Pressures. The interpretation of the piezoelectric gage records was not unambiguous. In the first place, many of the records showed large vibrations, especially for the very low ozone concentrations. These vibrations are known to be associated with a spinning detonation wave. Secondly, the records for the 17 and 25% ozone mixtures had the so-called von Neuman spike at the wave front. (The sharp pressure peaks are barely discernible in the record for the discernible 17.3% mixture in Figure 5, but they were evident in the original negative.)... 

According to the theory advanced independently by von Neuman (16), Boring (S), and Zeldovich (17), the detonation front is simply a shock wave in the as yet unreacted gas. The reaction is started by the sudden increase of temperature and pressure associated with the shock front, and proceeds in the zone which follows immediately behind it. As the gas reacts, the pressure drops. At the so-called Chap-man-Jouget plane" where the reaction is complete, the pressure has dropped to about one half the value at the shock front. Beyond this point, the pressure decreased more slowly in the normal Riemann expansion wave (11). 

The state of an NMR-relevant physical system changes over time as described by the Schrodinger equation which within a statistical density operator formalism may be recast in form of the so-called Liouville-von Neuman equation... 

Equipped with the Hamiltonian at any time of the experiment, the next step in a numerical evaluation is to solve the Liouville-von Neuman equation in Eq, (1), The formal solution is given by... 

Fig. 6. Detonation in a slab of energetic material, (a) The detonation shock front runs at a constant velocity, driven by fast expansion of chemical reaction products. The highest pressure is in the von Neuman spike region just behind the front. At the Chapman Jouguet (C-J) plane the reaction is Just complete, (b) Shock compression follows the indicated Rayleigh line to where it intersects the unreacted Hugoniot at the von Neumann spike. The point where this Rayleigh line is tangent to the reacted Hugoniot is the C-J state of stable detonation velocity.

With variance only on the account of diffusion in a linear problem for the determination of the stability limit may be used a criterion proposed by John von Neuman (1903-1957). According to this von Neuman criterion the stabihty limit may be expressed by the following inequality ... 

Later Codd reduced the complexity of the Von Neuman machine inspired by insights in the physiology of the nervous system in animals. Important to our later discussion in Chapter 9 of the design of self organizing catalytic systems, he proposed a universal configuration of only eight-states per cell. 

Interesting is the Multidimensional Reactive Flow model developed by Tarver et al. [5,103,104) it is based on the Non-Equilibrium Zeldovich-von Neuman-Dhring theory. This model starts from the primary chemical changes occurring in the adiabaticaUy compressed thin layer of molecules of the given EM and multiphonon up-pumping due to shock, but in the mathematical description it works with experimental data of thermal explosion of EM [5,103,104) it considers the induction period of initiation of detonation. However, the induction period of the EM decomposition in front of the detonation wave makes the front kinetically unstable and pulsating [101 ]. 

The term decoherence describes the process by which the off-diagonal elements of the reduced density matrix tend to zero when evolving with time. Our objective is to reach an understanding of the molecular mechanisms governing decoherence with an atomic resolution. In addition we wish to be in a position to treat systems consisting of tens to thousands of atoms since the brute force simulation of the time evolution of p t) by the Liouville-von Neuman equation (p (i) = ih [H, p ]), the equivalent of the TDSE in the density matrix formalism, is out of question for such molecular systems. 

Conventional computer systems based on the Von Neuman concept are extremely good at executing algorithms that have been precisely formulated for them. However, there are tasks, such as image and speech recognition, that cannot be suitably or easily captured using an algorithm. 

In general, Monte Carlo methods refer to any procedures which involve sampling from random numbers. These methods are used in simulations of natural phenomena, simulation of experimental apparatus, and numerical analysis. An important feature is the simple structure of the computational algorithm. The method was developed by von Neuman, Ulam, and Metroplois during World War II to study the difiiision of neutrons in fissionable materials (ie, atomic bomb design)- Let us consider atom diffusion and demonstrate the principle of the Monte Carlo method. A two-dimensional square grid (Fig. 3.20A) represents interstitial sites in a sofid. 

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