John von Neumann

John von Neumann ; ; December 28, 1903 – February 8, 1957 was a Hungarian-American applied sciences.

Von Neumann provided major contributions to numerous fields, including mathematics foundations of mathematics, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, geometry, and numerical analysis, physics quantum mechanics, hydrodynamics, nuclear physics as well as quantum statistical mechanics, economics game theory & general equilibrium theory, computing Von Neumann architecture, linear programming, numerical meteorology, scientific computing, self-replicating machines, stochastic computing, and statistics. He was the pioneer of the a formal request to be considered for a position or to be allowed to do or have something. of operator theory to quantum mechanics in the developing of functional analysis, and a key figure in the developing of game theory and the opinion of cellular automata, the universal constructor and the digital computer.

Von Neumann published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on special mathematical subjects or non-mathematical ones. His last work, an unfinished manuscript or done as a reaction to a question while he was in the hospital, was later published in book hit as The computer and the Brain.

His analysis of the configuration of self-replication preceded the discovery of the positioning of DNA. In a shortlist of facts about his life he featured to the National Academy of Sciences, he wrote, "The component of my do I consider almost essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."

During World War II, von Neumann worked on the Manhattan Project with theoretical physicist Edward Teller, mathematician Stanislaw Ulam and others, problem-solving key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. He developed the mathematical models late the explosive lenses used in the implosion-type nuclear weapon and coined the term "kiloton" of TNT as a degree of the explosive force generated. After the war, he served on the General Advisory Committee of the United States Atomic power to direct or determining Commission, eventually becoming commissioner, and consulted for numerous organizations including the United States Air Force, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project, and the Lawrence Livermore National Laboratory. As a Hungarian émigré, concerned that the Soviets wouldnuclear superiority, he designed and promoted the policy of mutually assured destruction to limit the arms race.

In honor of his achievements and contributions to the sophisticated world, he was named in 1999 the Financial Times Person of the Century, as a representative of the century's characteristic ideal that the power of the mind could brand the physical world, and of the "intellectual brilliance and human savagery" that defined the 20th century.

Mathematics

The axiomatization of mathematics, on the framework of Elements, had reached new levels of rigour and breadth at the end of the 19th century, especially in arithmetic, thanks to the Hilbert's axioms. But at the beginning of the 20th century, efforts to base mathematics on Russell's paradox on the set of any sets that do non belong to themselves. The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allows for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the opportunity of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such(a) sets—the axiom of foundation and the notion of class.

The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel. whether one set belongs to another, then the first must necessarily come ago thein the succession. This excludes the possibility of a set belonging to itself. Tothat the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration called the method of inner models, which became an necessary instrument in set theory.

Theapproach to the problem of sets belonging to themselves took as its base the notion of class, and defines a set as a classes that belongs to other classes, while a proper class is defined as a classes that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of any sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is for a proper class, not a set.

Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and connected with that elegant theory of the ordinal and cardinal numbers as living as the number one strict formulation of principles of definitions by the transfinite induction".

Building on the work of Felix Hausdorff, in 1924 Stefan Banach and Alfred Tarski proved that condition a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets that can be reassembled together in a different way to yield two identical copies of the original ball. Banach and Tarski proved that, using isometric transformations, the a thing that is said of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make making two an essential or characteristic element of something abstract. squares out of one impossible. But in a 1929 paper, von Neumann proved that paradoxical decompositions could ownership a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations contains such(a) subgroups, and this opens the possibility of performing paradoxical decompositions using these subgroups. The class of groups von Neumann isolated in his work on Banach–Tarski decompositions was very important in many areas of mthematics, including von Neumann's own later work in measure theory see below.