rajpurkar/squad huggingface dataset:
Title: John_von_Neumann

In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

Question: What is often considered more important than the results of von Neumann's papers?
Answer: methods of argument he employed

Question: In 1938 what was von Neumann awarded?
Answer: Bôcher Memorial Prize

Question: What did von Neumann use the Haar theorem to solve in a 1936 paper?
Answer: Hilbert's fifth problem in the case of compact groups
Title: John_von_Neumann

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
Title: John_von_Neumann

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
Title: John_von_Neumann

Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."

Question: What type of axioms were used by von Neumann in his lattice work?
Answer: novel axioms

Question: How long was von Neumann's lattice theory paper?
Answer: 140 pages

Question: In what room did von Neumann often write at home?
Answer: living room writing-table
Title: John_von_Neumann

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
Title: John_von_Neumann

The physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger. When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a Self-adjoint operator and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?"

Question: What are the physics of quantum mechanics?
Answer: mathematics of Hilbert spaces and linear operators acting on them

Question: What is the uncertainty principle?
Answer: determination of the position of a particle prevents the determination of its momentum and vice versa

Question: The physics of quantum mechanics included special cases for what work?
Answer: formulations of both Heisenberg and Schrödinger
Title: John_von_Neumann

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
Title: John_von_Neumann

In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, the Von Neumann–Wigner interpretation never gained acceptance amongst the majority of physicists).

Question: What results came from von Neumann's deep analysis of the measurement problem?
Answer: physical universe could be made subject to the universal wave function

Question: What "outside of calculation" variable could initiate collapse in von Neumann's conclusion?
Answer: consciousness of the experimenter

Question: Were von Neumann's conclusions of collapse of entire universe cause accepted widely?
Answer: Von Neumann–Wigner interpretation never gained acceptance amongst the majority of physicists
Title: John_von_Neumann

In a famous paper of 1936 with Garrett Birkhoff, the first work ever to introduce quantum logics, von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., one horizontally and the other vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is added in between the other two, the photons will, indeed, pass through. This experimental fact is translatable into logic as the non-commutativity of conjunction . It was also demonstrated that the laws of distribution of classical logic,  and , are not valid for quantum theory.

Question: With whom did von Neuman work on a paper in 1936 that introduce quantum logic?
Answer: Garrett Birkhoff

Question: What concept was created for quatum logic?
Answer: propositional calculus

Question: What is the difference of logic in quantum theory?
Answer: laws of distribution of classical logic,  and , are not valid for quantum theory
Title: John_von_Neumann

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.
Title: John_von_Neumann

The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically determinate, while each of its members are necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., x and y) results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the x direction is positive." By the principle of indeterminacy, the value of the spin in the direction y will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of y is positive" nor the proposition "the spin in the direction of y is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of y is positive or the spin in the direction of y is negative" must be true for ɸ. In the case of distribution, it is therefore possible to have a situation in which , while .

Question: What is the difference of quantum disjunction from classic?
Answer: can be true even when both of the disjuncts are false

Question: What is frequently the case in quantum mechanics when a pair of alternatives are semantically determinate?
Answer: each of its members are necessarily indeterminate

Question: How can quantum disjuction be illustrated?
Answer: illustrated by a simple example
Title: John_von_Neumann

Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior (written with Oskar Morgenstern). Morgenstern wrote a paper on game theory and thought he would show it to von Neumann because of his interest in the subject. He read it and said to Morgenstern that he should put more in it. This was repeated a couple of times, and then von Neumann became a coauthor and the paper became 100 pages long. Then it became a book. The public interest in this work was such that The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analytic methods, especially convex sets and topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.

Question: What is optimal strategy?
Answer: minimize the maximum loss for each player

Question: What von Neumann work was published in 1944?
Answer: Theory of Games and Economic Behavior

Question: Who was co-author with von Neman on the Theory of Games and Economic Behavior?
Answer: Oskar Morgenstern
Title: John_von_Neumann

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
Title: John_von_Neumann

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. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.

Question: In von Neumann's model what does p represent?
Answer: p represents the prices of the goods

Question: In von Neumann's model what does q represent?
Answer: q represents the "intensity" at which the production process would run

Question: What is the rate of qrowth equal to?
Answer: rate of growth equals the interest rate

Question: Was the economic model successful?
Answer: proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann
Title: John_von_Neumann

Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics. 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.

Question: What were von Neumann's model an example of?
Answer: viewed as a special case of linear programming

Question: What was the significance of von Neumann's model of expanding economy?
Answer: unique in having made significant contributions to economic theory itself.

Question: What ideas were utilized in von Neumann's ecocomic model?
Answer: fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality
Title: John_von_Neumann

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
Title: John_von_Neumann

Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of 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.

Question: What was the new method of linear programming that von Neumann suggested?
Answer: homogeneous linear system of Gordan

Question: What did von Neuman's method use?
Answer: pivoting algorithm between simplices

Question: What was von Neumann's method known to be the first of?
Answer: algorithm was the first interior point method of linear programming.
Title: John_von_Neumann

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
Title: John_von_Neumann

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)
Title: John_von_Neumann

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"
Title: John_von_Neumann

When it turned out that there would not be enough uranium-235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the plutonium-239 that was available from the Hanford Site. He established the design of the explosive lenses required, but there remained concerns about "edge effects" and imperfections in the explosives. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, this was achieved by George Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945.

Question: what change was made when shortage of uranium 235 was apparent?
Answer: implosive lens project was greatly expanded

Question: What was the tolerance limit for implosion "edge effect"?
Answer: not depart by more than 5% from spherical symmetry

Question: Who constructed Trinity bomb?
Answer: George Kistiakowsky

Question: In what year was Trinity bomb completed?
Answer: 1945