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anes and convex sets and fixed point theory have been the primary tools of mathematical economics ever since 273 mathematical economics edit von neumann raised the intellectual and mathematical level of economics in several influential publications for his model of an expanding economy he proved the existence and uniqueness of an equilibrium using his generalization of the brouwer fixed point theorem 270 von neumann s model of an expanding economy considered the matrix pencil a λ b with nonnegative matrices a and b von neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation p t a λ b q 0 displaystyle p t a lambda b q 0 along with two inequality systems expressing economic efficiency in this model the transposed probability vector p represents the prices of the goods while the probability vector q represents the intensity at which the production process would run the unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy the rate of growth equals the interest rate 274 275 von neumann s results have been viewed as a special case of linear programming where his model uses only nonnegative matrices the study of his model of an expanding economy continues to interest mathematical economists with interests in computational economics 276 277 278 this paper has been called the greatest paper in mathematical economics by several authors who recognized its introduction of fixed point theorems linear inequalities complementary slackness and saddlepoint duality in the proceedings of a conference on von neumann s growth model paul samuelson said that many mathematicians had developed methods useful to economists but that von neumann was unique in having made significant contributions to economic theory itself 279 von neumann s famous 9 page paper started life as a talk at princeton and then became a paper in german that was eventually translated into english his interest in economics that led to that paper began while he was lecturing at berlin in 1928 and 1929 he spent his summers back home in budapest as did the economist nicholas kaldor and they hit it off kaldor recommended that von neumann read a book by the mathematical economist léon walras von neumann found some faults in the book and corrected them for example replacing equations by inequalities he noticed that walras s general equilibrium theory and walras s law which led to systems of simultaneous linear equations could produce the absurd result that profit could be maximized by producing and selling a negative quantity of a product he replaced the equations by inequalities introduced dynamic equilibria among other things and eventually produced the paper 280 linear programming edit building on his results on matrix games and on his model of an expanding economy von neumann invented the theory of duality in linear programming when george dantzig described his work in a few minutes and an impatient von neumann asked him to get to the point dantzig then listened dumbfounded while von neumann provided an hourlong lecture on convex sets fixed point theory and duality conjecturing the equivalence between matrix games and linear programming 281 later von neumann suggested a new method of linear programming using the homogeneous linear system of paul gordan 1873 which was later popularized by karmarkar s algorithm 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 282 computer science edit von neumann was a founding figure in computing 283 von neumann was the inventor in 1945 of the merge sort algorithm in which the first and second halves of an array are each sorted recursively and then merged 284 285 von neumann wrote the 23 pages long sorting program for the edvac in ink on the first page traces of the phrase top secret which was written in pencil and later erased can still be seen 285 he also worked on the philosophy of artificial intelligence with alan turing when the latter visited princeton in the 1930s 286 von neumann s hydrogen bomb work was played out in the realm of computing where he and stanisław ulam developed simulations on von neumann s digital computers for the hydrodynamic computations during this time he contributed to the development of the monte carlo method which allowed solutions to complicated problems to be approximated using random numbers 287 flow chart from von neumann s planning and coding of problems for an electronic computing instrument published in 1947 von neumann s algorithm for simulating a fair coin with a biased coin is used in the software whitening stage of some hardware random number generators 288 because using lists of truly random numbers was extremely slow von neumann developed a form of making pseudorandom numbers using the middle square method though this method has been criticized as crude von neumann was aware of this he justified it as being faster than any other method at his disposal writing that anyone who considers arithmetical methods of producing random digits is of course in a state of sin 289 von neumann also noted that when this method went awry it did so obviously unlike other methods which could be subtly incorrect 289 while consulting for the moore school of electrical engineering at the university of pennsylvania on the edvac project von neumann wrote an incomplete first draft of a report on the edvac the paper whose premature distribution nullified the patent claims of edvac designers j presper eckert and john mauchly described a computer architecture in which the data and the program are both stored in the computer s memory in the same address space this architecture is the basis of most modern computer designs unlike the earliest computers that were programmed using a separate memory device such as a paper tape or plugboard although the single memory stored program architecture is commonly called von neumann architecture as a result of von neumann s paper the architecture was based on the work of eckert and mauchly inventors of the eniac computer at the university of pennsylvania 290 von neumann consulted for the army s ballistic research laboratory most notably on the eniac project 291 as a member of its scientific advisory committee 292 the electronics of the new eniac ran at one sixth the speed but this in no way degraded the eniac s performance since it was still entirely i o bound complicated programs could be developed and debugged in days rather than the weeks required for plugboarding the old eniac some of von neumann s early computer programs have been preserved 293 the next computer that von neumann designed was the ias machine at the institute for advanced study in princeton new jersey he arranged its financing and the components were designed and built at the rca research laboratory nearby von neumann recommended that the ibm 701 nicknamed the defense computer include a magnetic drum it was a faster version of the ias machine and formed the basis for the commercially successful ibm 704 294 295 stochastic computing was first introduced in a pioneering paper by von neumann in 1953 296 however the theory could not be implemented until advances in computing of the 1960s 297 298 around 1950 he was also among the first people to talk about the time complexity of computations which eventually evolved into the field of computational complexity theory 299 herman goldstine once described how he felt that even in comparison to all his technical achievements in computer science it was the fact that he was held in such high esteem had such a reputation that the digital computer was accepted so quickly and worked on by others 300 as an example he talked about tom watson jr s meetings with von neumann at the institute for advanced study whom he had come to see after having heard of von neumann s work and wanting to know what was happening for himself personally ibm which watson jr later became ceo and president of would play an enormous role in the forthcoming computer industry the second example was that once von neumann was elected commissioner of the atomic energy commission he would exert great influence over the commission s laboratories to promote the use of computers and to spur competition between ibm and sperry rand which would result in the stretch and larc computers that lead to further developments in the field goldstine also notes how von neumann s expository style when speaking about technical subjects particularly to non technical audiences was very attractive 301 this view was held not just by him but by many other mathematicians and scientists of the time too 302 cellular automata dna and the universal constructor edit see also von neumann cellular automaton von neumann universal constructor von neumann neighborhood and von neumann probe the first implementation of von neumann s self reproducing universal constructor 303 three generations of machine are shown the second has nearly finished constructing the third the lines running to the right are the tapes of genetic instructions which are copied along with the body of the machines a simple configuration in von neumann s cellular automaton a binary signal is passed repeatedly around the blue wire loop using excited and quiescent ordinary transmission states a confluent cell duplicates the signal onto a length of red wire consisting of special transmission states the signal passes down this wire and constructs a new cell at the end this particular signal 1011 codes for an east directed special transmission state thus extending the red wire by one cell each time during construction the new cell passes through several sensitised states directed by the binary sequence von neumann s rigorous mathematical analysis of the structure of self replication of the semiotic relationship between constructor description and that which is constructed preceded the discovery of the structure of dna 304 von neumann created the field of cellular automata without the aid of computers constructing the first self replicating automata with pencil and graph paper the detailed proposal for a physical non biological self replicating system was first put forward in lectures von neumann delivered in 1948 and 1949 when he first only proposed a kinematic self reproducing automaton 305 306 while qualitatively sound von neumann was evidently dissatisfied with this model of a self replicator due to the difficulty of analyzing it with mathematical rigor he went on to instead develop a more abstract model self replicator based on his original concept of cellular automata 307 subsequently the concept of the von neumann universal constructor based on the von neumann cellular automaton was fleshed out in his posthumously published lectures theory of self reproducing automata 308 ulam and von neumann created a method for calculating liquid motion in the 1950s the driving concept of the method was to consider a liquid as a group of discrete units and calculate the motion of each based on its neighbors behaviors 309 like ulam s lattice network von neumann s cellular automata are two dimensional with his self replicator implemented algorithmically the result was a universal copier and constructor working within a cellular automaton with a small neighborhood only those cells that touch are neighbors for von neumann s cellular automata only orthogonal cells and with 29 states per cell von neumann gave an existence proof that a particular pattern would make infinite copies of itself within the given cellular universe by designing a 200 000 cell configuration that could do so t here exists a critical size below which the process of synthesis is degenerative but above which the phenomenon of synthesis if properly arranged can become explosive in other words where syntheses of automata can proceed in such a manner that each automaton will produce other automata which are more complex and of higher potentialities than itself von neumann 1948 308 von neumann addressed the evolutionary growth of complexity amongst his self replicating machines 310 his proof of principle designs showed how it is logically possible by using a general purpose programmable universal constructor to exhibit an indefinitely large class of self replicators spanning a wide range of complexity interconnected by a network of potential mutational pathways including pathways from the most simple to the most complex this is an important result as prior to that it might have been conjectured that there is a fundamental logical barrier to the existence of such pathways in which case biological organisms which do support such pathways could not be machines as conventionally understood von neumann considers the potential for conflict between his self reproducing machines stating that our models lead to such conflict situations 311 indicating it as a field of further study 308 147 the cybernetics movement highlighted the question of what it takes for self reproduction to occur autonomously and in 1952 john von neumann designed an elaborate 2d cellular automaton that would automatically make a copy of its initial configuration of cells the von neumann neighborhood in which each cell in a two dimensional grid has the four orthogonally adjacent grid cells as neighbors continues to be used for other cellular automata von neumann proved that the most effective way of performing large scale mining operations such as mining an entire moon or asteroid belt would be by using self replicating spacecraft taking advantage of their exponential growth 312 von neumann investigated the question of whether modelling evolution on a digital computer could solve the complexity problem in programming 311 beginning in 1949 von neumann s design for a self reproducing computer program is considered the world s first computer virus and he is considered to be the theoretical father of computer virology 313 scientific computing and numerical analysis edit computational physics mechanics electromagnetics thermodynamics simulation potentials morse long range potential lennard jones potential yukawa potential morse potential fluid dynamics finite difference finite volume finite element boundary element lattice boltzmann riemann solver dissipative particle dynamics smoothed particle hydrodynamics turbulence models monte carlo methods integration gibbs sampling metropolis algorithm particle n body particle in cell molecular dynamics scientists godunov ulam von neumann galerkin lorenz wilson alder richtmyer v t e considered to be possibly the most influential researcher in scientific computing of all time 314 von neumann made several contributions to the field both on the technical side and on t...
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