lmqg/qag_squad huggingface dataset:
Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Lutheran Church, the majority of its pupils were Jewish. The school system produced a generation noted for intellectual achievement, that included Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913). Collectively, they were sometimes known as Martians. Wigner was a year ahead of von Neumann at the Lutheran School. When asked why the Hungary of his generation had produced so many geniuses, Wigner, who won the Nobel Prize in Physics in 1963, replied that von Neumann was the only genius.

question: What did Von Neumann enter in 1911?, answer: Lutheran Fasori Evangelikus Gimnázium | question: In what room did children get education in Hungary?, answer: gymnasium | question: What collective name was given to the generation of scholars produced by the Hungarian school system?, answer: Martians | question: In what year did Wigner win Nobel Prize?, answer: 1963
Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.

question: What field did Von Neuman establish?, answer: continuous geometry. | question: What is the distinction of continuous geometry?, answer: instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval | question: What was the first example of continuous geometry?, answer: projections of the hyperfinite type II factor
Von Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses, hence the name minimax. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.

question: What year was game theory established?, answer: 1928 | question: What discipline is game theory derived from?, answer: mathematical discipline. | question: what are the possible strategies in minimax theory?, answer: a pair of strategies for both players that allows each to minimize his maximum losses | question: What must a player consider when determining every possible strategy?, answer: all the possible responses of his adversary.
Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

question: What is von Neumann algebra?, answer: algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator | question: What does the von Neumann bicommutant theorem show?, answer: the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. | question: When was direct integral introduced by von Neumann?, answer: 1949 | question: What was introduced in von Neumann algebras?, answer: study of rings of operators
Von Neumann is credited with the equilibrium strategy of mutual assured destruction, providing the deliberately humorous acronym, MAD. (Other humorous acronyms coined by von Neumann include his computer, the Mathematical Analyzer, Numerical Integrator, and Computer—or MANIAC). He also "moved heaven and earth" to bring MAD about. His goal was to quickly develop ICBMs and the compact hydrogen bombs that they could deliver to the USSR, and he knew the Soviets were doing similar work because the CIA interviewed German rocket scientists who were allowed to return to Germany, and von Neumann had planted a dozen technical people in the CIA. The Russians believed that bombers would soon be vulnerable, and they shared von Neumann's view that an H-bomb in an ICBM was the ne plus ultra of weapons, and they believed that whoever had superiority in these weapons would take over the world, without necessarily using them. He was afraid of a "missile gap" and took several more steps to achieve his goal of keeping up with the Soviets:

question: What was MAD?, answer: mutual assured destruction | question: What was the goal of MAD?, answer: quickly develop ICBMs and the compact hydrogen bombs that they could deliver to the USSR | question: What were Russians opinions of future weaponry?, answer: Russians believed that bombers would soon be vulnerable, and they shared von Neumann's view that an H-bomb in an ICBM was the ne plus ultra of weapons | question: What was the concern of a missile gap?, answer: keeping up with the Soviets
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.

question: What type of "off color" humor did Von Neumann especially enjoy?, answer: limericks | question: What complaints did Von Neumann's neighbor's have?, answer: regularly playing extremely loud German march music | question: What famous neighbor did Von Neumann have that may have been distracted by the music Von Neumann played?, answer: Albert Einstein,
Von Neumann made fundamental contributions in exploration of problems in numerical hydrodynamics. For example, with Robert D. Richtmyer he developed an algorithm defining artificial viscosity that improved the understanding of shock waves. A problem was that when computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics. Other well known contributions to fluid dynamics included the classic flow solution to blast waves, and the co-discovery of the ZND detonation model of explosives.

question: The algorithm defining artificial viscosity is a fundamental contribution to what area?, answer: numerical hydrodynamics | question: What did the math of artificial viscosity do?, answer: smoothed the shock transition without sacrificing basic physics | question: What was the problem of computers solving fluid dynamics?, answer: they tried to put too many computational grid points at regions of sharp discontinuity (shock waves)
Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.

question: When did von NEumann establish mean square ratio?, answer: 1941 | question: How has mean square ratio been applied?, answer: Durbin–Watson statistic | question: what does null hypothesis test against?, answer: alternative that they follow a stationary first order autoregression
Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A − λB 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

question: How did von Neumann raise the level of economics?, answer: several stunning publications | question: What did von Neumann prove with his model of expanding economy?, answer: uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem | question: What did von Neumann consider for his model of expanding economy?, answer: matrix pencil A − λB with nonnegative matrices A and B
Von Neumann was a founding figure in computing. Donald Knuth cites von Neumann as 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. Von Neumann wrote the sorting program for the EDVAC in ink, being 23 pages long; traces can still be seen on the first page of the phrase "TOP SECRET", which was written in pencil and later erased. He also worked on the philosophy of artificial intelligence with Alan Turing when the latter visited Princeton in the 1930s.

question: Who was the inventor of the merge-sort algorithm?, answer: Von Neumann | question: With whom did von Neumann work on the philosophy of artificial intelligence?, answer: Alan Turing | question: What does a merge sort algorithm do?, answer: the first and second halves of an array are each sorted recursively and then merged
Von Neumann was born Neumann János Lajos (in Hungarian the family name comes first), Hebrew name Yonah, in Budapest, Kingdom of Hungary, which was then part of the Austro-Hungarian Empire, to wealthy Jewish parents of the Haskalah. He was the eldest of three children. He had two younger brothers: Michael, born in 1907, and Nicholas, who was born in 1911. His father, Neumann Miksa (Max Neumann) was a banker, who held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. Miksa's father and grandfather were both born in Ond (now part of the town of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (Margaret Kann); her parents were Jakab Kann and Katalin Meisels. Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest; von Neumann's family occupied an 18-room apartment on the top floor.

question: Where was Von Neumann born?, answer: Budapest | question: Did Von Neumann have any siblings?, answer: two younger brothers | question: What occupation did Von Neumann father hold?, answer: banker | question: When did Von Neuman's father move from Pecs to Budapest?, answer: end of the 1880s | question: What was Von Neumann's mother's name?, answer: Kann Margit
Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized, in 1926, that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system.

question: Who established the framework for quantum mechanics?, answer: Von Neumann | question: What paper did von Neumann produce in 1932?, answer: Mathematical Foundations of Quantum Mechanics | question: What von Neumann work preceded the axiomatiztion of quatum mechanics?, answer: axiomatization of set theory
Von Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity." Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor".

question: What did von Neumann work on in 1937 - 39?, answer: lattice theory | question: What properties determine dimension in lattice theory?, answer: conserved by perspective mappings ("perspectivities") and ordered by inclusion | question: Who wrote that "John von Neumann's brilliant mind blazed over lattice theory like a meteor?", answer: Garrett Birkhoff
Von Neumann's ability to instantaneously perform complex operations in his head stunned other mathematicians. Eugene Wigner wrote that, seeing von Neumann's mind at work, "one had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch." Paul Halmos states that "von Neumann's speed was awe-inspiring." Israel Halperin said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car." Edward Teller wrote that von Neumann effortlessly outdid anybody he ever met, and said "I never could keep up with him". Teller also said "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us. Most people avoid thinking if they can, some of us are addicted to thinking, but von Neumann actually enjoyed thinking, maybe even to the exclusion of everything else."

question: How quickly was von Neumann able to complete math in his head?, answer: stunned other mathematicians | question: Were other mathematicians amazed by von Neumann?, answer: Paul Halmos states that "von Neumann's speed was awe-inspiring." | question: Did von Neumann enjoy thinking?, answer: Neumann actually enjoyed thinking, maybe even to the exclusion of everything else.
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 1966, John S. Bell published a paper arguing that the proof contained a conceptual error and was therefore invalid. However, in 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation, and that von Neumann did not claim that his proof completely ruled out hidden variable theories.

question: Von Neumann's abstract treatment allowed him to work on what foundational issue?, answer: determinism versus non-determinism | question: What was John Bell's argument?, answer: proof contained a conceptual error and was therefore invalid | question: What did Jeffrey Bub argue in 2010?, answer: Bell had misconstrued von Neumann's proof | question: Did von Neumann rule hidden variable theories?, answer: von Neumann did not claim that his proof completely ruled out hidden variable theories
Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.

question: What mathematician was Von Nemann's closest friend?, answer: Stanislaw Ulam | question: What topics of conversation would Ulam and Von Neumann typically share?, answer: gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk | question: What would Ulam use to cheer up Von Neumann in the hospital?, answer: a new collection of jokes
Von Neumann's famous 9-page paper started life as a talk at Princeton and then became a paper in Germany, which was eventually translated into English. His interest in economics that led to that paper began as follows: When lecturing at Berlin in 1928 and 1929 he spent his summers back home in Budapest, and so 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 that book and corrected them, for example, replacing equations by inequalities. He noticed that Walras's General Equilibrium Theory and Walras' Law, which led to systems of simultaneous linear equations, could produce the absurd result that the 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.

question: When did von Neumann's interest in economics begin?, answer: When lecturing at Berlin in 1928 and 1929 | question: What economist did von Neumann get to know that spurred the interest in economics?, answer: Nicholas Kaldor | question: In who's book did von Neuman find errors in the author's equations?, answer: Léon Walras
Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.

question: Where did Von Neumann begin to lecture in 1928?, answer: University of Berlin | question: By the end of 1927 how many papers had Von Neuman written?, answer: twelve major papers in mathematics | question: How many papers had Von Neumann written by 1929?, answer: thirty-two papers | question: What better offer came for Von Neumann in 1930?, answer: Princeton University
Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and Stanislaw 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. His algorithm for simulating a fair coin with a biased coin is used in the "software whitening" stage of some hardware random number generators. 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, and also noted that when it went awry it did so obviously, unlike methods which could be subtly incorrect. "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."

question: Where were the computations for hydrogen bomb worked out?, answer: von Neumann's digital computers | question: What was the Monte Carlo method?, answer: allowed solutions to complicated problems to be approximated using random numbers | question: Due to slow use of random numbers what was developed?, answer: pseudorandom numbers
Von Neumann's mathematical analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most 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."

question: Von Neumann's study of what preceded the discovery of DNA?, answer: mathematical analysis of the structure of self-replication | question: What part of his work did Von Neumann consider to be his most important?, answer: quantum mechanics | question: Where was quantum mechanics developed?, answer: Göttingen | question: What work did Von Nemann do in Berlin in 1930 and Princeton 1935 - 39?, answer: operator theory | question: What was the topic of Von Neumann's work at Princeton in 1931 - 32?, answer: ergodic theorem
Von Neumann's principal contribution to the atomic bomb was in the concept and design of the explosive lenses needed to compress the plutonium core of the Fat Man weapon that was later dropped on Nagasaki. While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly".

question: What principal contribution did von Neumann make to atomic bomb?, answer: concept and design of the explosive lenses needed to compress the plutonium core | question: Did von Nemann think that the atomic bomb design was not workable?, answer: he was one of its most persistent proponents, encouraging its continued development | question: What additional ideas did von Neumann contribute to Atomic bomb?, answer: more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly"