von Neumann

 MR.
Von Neumann. Neumann, J. (von Neumann, John) 1903   12  28 was born in Budapest, Hungary; February 8, 1957 and died in Washington. Mathematics, physics, computer science .
von. Neumann was born in Jewish families. father, Max. von. Neumann (Max von Neumann) was a wealthy banker. In 1913, Emperor Francis of Austria-Hungary. Joseph I (Franz Joseph I ) titles of nobility granted to Max, the family name Neumann, there were the mathematics, foreign languages, history and natural sense, and his early showing superhuman memory and understanding. legend he was 6 years old to 8-digit mental arithmetic division, 8-year-old mastered calculus at age 12 also studied E. wave Lyle (Borel) His extraordinary talents caught the teacher L. Ruiz (Ratz) attention to Von Ritz think. Neumann to accept the traditional secondary school education is a waste of time, should be dedicated to his mathematical training to be a genius fully developed. Ruiz to Feng. Neumann J. recommended to the Budapest University of Qu Er Shake (K rschak), Professor Qu Er Shake the TA M. Fechter arrangements (Fekete) as his work family counseling . His first published paper is in less than 18 years of age and Fechter co-wrote, and promotion of the Chebyshev (Polynomial rooting Fejér (Fej r), he passed Theorem .1921 School graduation examination, has been recognized as promising rookie mathematics.
four years after that, Feng. Neumann worked in Berlin, Zurich, Switzerland with the University and the Higher Technical Institute to study the chemical industry, while retaining the Budapest University of Mathematics Department of school. end of each semester, he would come back from Europe, Budapest, to visit family and to participate in the math exam in the spring of .1925 and 1926, he has won the Zurich chemical engineering degree and doctorate in mathematics from the University of Budapest.
In Berlin, von. Neumann participated in A. Einstein (Einstein) lectures on statistical mechanics and follow E. Schmidt (Schmidt) learning; in Zurich, he and H. Weyl (Weyl) and G. Polya (P lya) have been in close contact. von. Neumann said that his early academic thinking on the most influential mathematicians, is the Weyl and Schmidt.
He also made several trips to grid Ding root universities, visit the great mathematician D. Hilbert (Hi-lbert). He was Hilbert's proof theory of quantum mechanics and fascinated. Hilbert also very appreciated by the young scholars, He has not earned his PhD in early 1926 when the Hilbert to seek to try him a visiting scholar at the University of G?ttingen eligibility.
1927m1929 years, Feng. Neumann was appointed lecturer at the University of Berlin obligations, during which In set theory, algebra and the quantum theory of a large number of research results achieved by the mathematical community's attention the University of Hamburg 1929 he transferred to any obligation of lecturer. recommended by Weyl, who in 1930 as a visiting lecturer to the United States Department of Mathematics, Princeton University, second year in a tenured professor in the department. In this way, each year half of the time he lived in Europe, the other half is spent in the United States.
1933, the Institute for Advanced Studies in Princeton established. von. Neumann from the start employed as tenured professor of Mathematics and Physics Institute, only 29 years old, is the hospital's youngest professor. he made in 1937 the United States citizenship.
time, the world economy in the Great Depression, the war clouds hanging over Europe, and the Princeton mathematics and physics has become an impressive place. in a strong academic atmosphere and a stable life, von. Neumann has been wholeheartedly engaged in the research work .1932, he summarized from the mathematical development of quantum mechanics, published theory, but also to establish a continuous geometry. Hilbert fifth part of the problem to solve, but also his major achievements during this period one.
1930 years. von. Neumann and M. Kewei Si ( Kovesi) married his daughter Marina (Marina) was born in 1935. Two years later, their marriage has broken down in the summer of .1938, Feng. Neumann back to Budapest to give lectures, to visit relatives, and Clara. Dan (KlaraDan) married in came together at the end of Princeton. Clara became the first mathematical problems for the computer code compiled one of the scholars.
the outbreak of World War II, von. Neumann's scientific career turning point occurred .1940 , he was hired as Aberdeen, Institute of ballistic experimental science adviser, Bureau of Ordnance in 1941, consultants hired Ren Haijun. From late 1943 onwards, he once again participated in an advisory capacity, the work of the Institute of Los Alamos, the atomic bomb guidance Good structural design of the program for large-scale thermonuclear reaction. In mathematics, in addition to various numerical problems to solve his most important achievement was in 1944 formally established the modern game theory and mathematical economics.
post-World War II He turned to study computer .1944 summer, he visited the not yet completed the first electronic computer ENIAC, and participated in order to improve computer performance in a series of expert meetings. after year, he proposed, and computer The new programming ideas, worked out two new programs mmEDVAC machine programs and IAS machine programs .1951 years, IAS machine was successfully developed, prove the correctness of his theory.
After the war, von. Neumann Institute for Computer Research Institute as a senior, while continuing weapons laboratories in the U.S. Navy and other military organizations in the service in October .1954, he was appointed to the U.S. Atomic Energy Commission, resigned the following year to facilitate the duties of the Advanced Research Institute, by the working and living in Princeton for 23 years to move to Washington.
until his death from the late 40's before, Feng. Neumann has concentrated on automata theory, including a variety of artificial and natural automata automatic machines to solve adaptive automata, self-reproduction and self-recovery and other issues .1951 published and for the future foundation of artificial intelligence research.
1955 summer, Feng. Neumann was diagnosed with bone cancer, the condition deteriorated rapidly. he insisted in a wheelchair to think, write, attend conferences, but also prepared for the Yale University Gillman (Hilliman) .1957 lecture notes on February 8, the Army Hospital in Washington, he died at the age of 53.
von. Neumann held a number of life science jobs, access to the numerous honors, the most important are: by the American Mathematical Society 1937 Bo breaks (B cher) Award; 1947 by the American Mathematical Society Gibbs (Gibbs) instructor seat, and get Medal of Merit (Presidential Award); 1951m1953 in U.S. Chairman of Mathematics; 1956 by the Albert Einstein Memorial Award and the Fermi (Fermi) Award.
academic papers he published more than 150 articles, all included in the 1961 Po Gemeng Publishing House Man Collection generation of great masters.
von pure mathematics. Neumann's work in pure mathematics focused on 1925m1940 years can be divided into the following six directions.
1.
set theory and mathematical foundations of the century In order to overcome the paradox to G. Cantor (Cantor) set theory, the difficulties caused by, and the system theory and method of finishing Cantor, people began work on the axiomatic method of .1908, there were two well-known axiom system: E. Zermelo (Zermelo) system [after the A. Frankel (Fraenkel), and A. Si Kelang (Skolem) modified to add a ZF axioms] and B. Russell (Russell) The type of.
von. Neumann problem very early on set theory, are interested in studying .1923 was still in Zurich, he published his second paper, Einf hrung der transfiniten Ordnungszahlen), trying to Cantor's ordinal concept of existence is impossible to prove. von. Neumann ZF axiom system by means of the concept of the initial truncated and infinite justice, given the ordinal numbers and transfinite ordinal number of new formal definitions, this definition has been used ever since. < br> After six or seven years, he actively spread the axiomatic thinking, and trying to establish a more formal and accuracy of the axiom system of .1923, he was to Germany, Long Paper Neumann, according to an introductory text to write the article . Neumann, Ph.D. thesis. axiomatic system it created by P. Byrnes (Bernays) and K. G?del (G del) improved after the formation of axiomatic set theory system of another new system mmNBG .
NBG systems like ZF system, as the collection and affiliation as the original concept and production methods to restrict the collection to achieve the purpose of exclusion paradox, but also in the collection of different types and levels of language to describe the collection of system. It is characterized by the provisions can not really class as the class element. Thus, excluded from demonstration methods. but also in the ZF system, including the composition of the infinitely many axioms axiom model, NBG axiom system is free mode, is a finite axiom system, with axioms of elementary geometry as simple as the logical structure, this is it The main advantages.
has been demonstrated, NBG system is the ZF system expansion. Godel in proving the axiom of choice and the continuum hypothesis compatibility with other axioms, it was inspired by the NBG system. to this day, NBG system is still the best one of the foundations of set theory.
and adapt to the work of axiomatic set theory, von. Neumann in the late 20's element involved in the mathematics of Hilbert's article on planned .1927 Special I and Byrnes, W. Ackermann (Ackermann), who's efforts made progress, but in general terms have not been satisfactorily resolved. particularly on the natural number of non-Ackerman contradiction proof of classical analysis can not be achieved.
1931, the Godel incompleteness theorem was lodged, the full realization of the Hilbert scheme fell through. In this regard, Feng. Neumann did not feel too surprised As early as in 1925 published The last sentence is: should lead to a new way, to understand the role of mathematical formalism, and not the end of it as a problem. to be reflected.
2. Measure Theory Measure Theory in Feng
. Neumann is not the whole research work at the center, but he gives many valuable methods and results.
in 1929 In the : Rn on the power set, the existence of a non-negative, normalized and rigid body motion on the same set of functions can be added? F. Hausdorff (Hau-sdorff) and S. Banach (Banach) Proof: Measure problem in n 1 and 2, there are infinitely many solutions, in other cases no solution. This conclusion gives the impression that: when the dimension from 2 to 3, the space has been a fundamental characteristic, the elusive change. von. Neumann pointed out that the problem essentially belongs to group theory, the root cause lies in the nature of group differences in change rather than in space. Measure the problem of solvability of solvable groups need to use of the algebraic concepts.
he continued to use the idea of group theory, analysis of the Hausdorff - Banach - Tarski (Tarski) paradox: Rn (ng3) in two different radii of the ball, can be decomposed into a finite number are disjoint non-measurable subset of a subset of the two goals can be established between the co-existence such as the relationship between the two (in n is 1 or 2, this decomposition does not exist). He explained said that this is because the n is 3 or greater, the orthogonal group contains non-free abelian (Abel) group, and in less than 3 is not the case.
This issue begins with Rn measure extended to the general non-abelian group. And Banach all about the R2 subset of the possibility of using the same measure has been proved for all subsets of abelian groups also set up. Finally, he concluded: all solvable groups are measurable (that can be introduced to some measure on solvable groups).
this article are the results of the first to set theory from Euclidean space to more general algebra and topology to his job. Since from the time when this way of thinking began to be more widely appreciated.
the same period, the Hungarian mathematician A. Hall (Haar) raised the question: whether in Rn there is a measurable subset selection method so that each sub-set are equivalent with the given set, and the selection process to maintain a limited set of operations? von. Neumann gives a positive answer, and the conclusions to the case of measurable functions. This is a measure of decomposition to solve the problem .1935 starting point, he also M. Stone (Stone) in cooperation more general issues discussed: A is a Boolean algebra, M is the ideal of A, when there is sub-algebra A, the A to A / the mapping M restricted algebra is isomorphic when? they are given sufficient conditions for the existence of a variety.
the other results of his group in 1934 on compact prove the uniqueness of Haar measure ( constant factor in the sense of difference.) constructed in the proof of continuous functions on a compact group, given left-invariant measure m, mp defined by the following formula:
where o is the appropriate weight function. mp not only has all the properties of m, and has a right zero invariant. these methods later he and S. Bo Cartagena (Bochner) of separable topological group and almost periodic function by the system application.
1933m1934 years, Feng. Neumann Institute for the Advanced Measure of the relevant report, very detailed explanation of the EU Euclidean space in the classical Lebesgue measure theory, and extended to an abstract measure space. The report of the content in a very long time is a measure of respect in the United States primary source of information in 1950 to become editor by Princeton Press theorem) .19 70 century, L. Boltzmann (Boltzmann) statistical mechanics proposed by the ergodicity assumption, and hope that as the premise, derived measure-preserving transformation of the spatial average is equal to (discrete) time average, which is the Boltzmann scheme.
mathematically to achieve this plan, you first need to prove that the limit as the average existence of 1931, B. Koopman (Koopman) and A. Weil (Weil) also found that induced by the measure-preserving transformation of the function operator is the unitary operator. It gives von. Neumann to great enlightenment. At that time, he was committed to the study of operator theory, this discovery prompted him to try to use Hill Burt space conjugate operator to solve existing problems. Soon, he proposed and proved the first major ergodic theory theorems mm mean ergodic theorem:
, on the measure-preserving transformation T, ergodic average
according to the norm of L2 converges to the function Pf, where Ut is the induced operator T
UTf (x) = f (Tx), xX
and p is the same function space L2 to Ut the orthogonal projection .
the results published in (1932) before, Feng. Neumann to introduce it to the GD Birkhoff (Birkhoff) and Koopman. Berkhof will, improve the convergence of the sense of that, since Birkhoff and Koopman wrote in 1932, The groundbreaking work is being recognized.
Soon, Volume 33, der klassischen Mechanik), which marks the beginning of ergodic theory system.
paper first gives a detailed proof mean ergodic theorem, and then launch six important theorems. The first is the decomposition theorem (decomposition theorem): any measure-preserving transformation can be decomposed into ergodic transform straight points. It shows that in all measure-preserving transformation, with a traverse of the most basic, most important, any measure-preserving transformation can be obtained by their structure.
Theorem 2 further pointed out that single-parameter measure-preserving transformation group classification problem, in essence, can be attributed to the classification of ergodic transformations.
measure-preserving transformation of the classification of ergodic theory became the central issue, one of the most critical first step ought von. Neumann and P. Hall Moss (Halmos) 1942 was jointly prove the conclusion:
f1 and f2 are finite measure spaces X1 and X2 on the measure-preserving transformation, U1 and U2 are X1 , X2 in the L2 on the unitary operator induced. If f1, f2 has discrete spectrum, f1 and f2 are isomorphic if and only if U1 and U2 as a unitary operator on the Hilbert space is the same midnight.
von. Neumann ergodic theory in dealing with problems, tend to focus on the internal relations and spectral measure. Theorem 5 is typical of the results on the discrete spectrum: a pure point spectrum for unitary operator U (transformation induced by the traversal and too), the spectrum is actually a group of real numbers constitute a countable subgroup; turn, the real number could be infinite number of each group can be used as some of the traversal sub-group transformation of the unitary operator induced by the pure point spectrum.
and Correspondingly, there von. Neumann, and Koopman Theorem on the mixed continuous spectrum (mixing theorem). It asserted: traverse transform geometric properties (mixed) and the spectral properties of unitary operators (no more than ordinary eigenvalue ) are equivalent.
for Feng. Neumann measure theory and ergodic theory in terms of results achieved, Hal Moss assessment given that: Neumann full one-tenth of scientific treatises, but in terms of quality, even if he never made in other studies, these results are enough to make his permanent reputation in mathematics. On
Feng. Neumann, a famous result is a compact set in 1933 on the fifth Hilbert problem solved. As early as in 1929, he has proved to be continuous group may change the parameters of the group computing a resolution. Specifically, for the n-dimensional space, linear transformation group, which has a normal subgroup, can be analytically and by way of a finite number of arguments correspond to the local watch is out. This is the first Hilbert problem solving contribute to the fifth article.
1933, he was in the Gruppen), to prove that every locally homeomorphic to Euclidean space, allowing a compact Lie group structure. Thus, the fifth Hilbert problem in the compact group under the conditions of a positive answer.
questions solution used by Peter (Peter) - Weyl group on the points in the analogy, Schmidt's approximation theorem and the LEJ Brouwer (Brouwer) region on the Euclidean space invariance theorem, reflecting von. Neumann extensive knowledge of set theory and real variable and his integral equation techniques and skilled application of matrix calculations.
same for another group of related work: almost periodic function on the group (almost pe-riodic function) theory. He H. Bohr (Bohr)'s first set of real numbers expanded on the concept of almost periodic function to any group G, then the almost periodic function in the new theory and Peter, outside Seoul, said the group establish a link between theory: Let group expressed as a finite matrix G D (x) = (dij (x)),
the following three conditions are equivalent:
(1) Each dij (x) are bounded functions on G ;
(2) Each dij (x) are almost periodic function on G;
(3) D is equivalent to a unitary matrix representation.
He thus pointed out that the group on the Scots periodic function theory constitutes the largest group, said the scope of application.
5.
operator theory to explore the theory of operators throughout the Von. Neumann's entire scientific career, this paper accounts for all of his writings third, he has over 20 years in this field leadership.
1927m1930, he first give an abstract definition of Hilbert space, which is now the definition used. Then, for the Hilbert space self-conjugate operator spectral theory from bounded to unbounded promotion, to do the system's ground-breaking work: the introduction of densely defined closed operator concept, given unbounded self-adjoint operator, unitary operator and the normal operator spectral decomposition theorem, that the symmetric operator and self-adjoint operator in the nature of the differences, but also to study the Weyl operator unbounded perturbations through changes of spectrum.
von. Neumann's spectral theory of the formation, together with Banach 1933 book
20 years, E. Noether (Noether) and E. Artin (Artin) developed the theory of non-commutative algebra, Feng. Neumann realized that this is an excellent interpretation and simplification of matrix theory, he tried the concept will be extended to Hilbert space operator algebra, which produced a * ring operator algebra is called. ring operator finite dimensional space can be considered a natural generalization of the matrix algebra, later known as Feng. Neumann algebra to show von. Neumann memorial. and in isomorphism, it can be called a W * algebra.
formal definition of loop operators in von. Neumann's 1929 paper Funktionaloperati-oren und Theorie der normalen Operatoren) in. This paper also includes the :
is the operator ring, then the exchange operator is sub-rings, and. This actually gives the operator an equivalent definition of ring: Hilbert space H are bounded linear operators on the whole (H) satisfying the = () r the * operator algebra is called ring. This definition is the study of an important tool ring operator, such as the operator to determine when accompanied with a ring operator, used to be densely defined closed operator standard decomposition.
from 1935, Feng. Neumann FJ Murray in (Murray) with the assistance of another to write a paper entitled article.
Their first conclusion: operator can be expressed as a factor ring of continuous straight points. Therefore, the ring of operators will be reduced to the study of the factors.
by the classical theory of non-commutative algebra Inspiration People have speculated that all the factors are isomorphic to (H). von. Neumann and Murray in the But at the same time, they applied the skills ergodic theory, construct a class of important examples, not all of the factors that have a minimal projective, and thus the nature of the relevant factors is far from as simple as people speculate.
their factor projective relationship between the order to make them comparable. And this order relation may also use the dimension function (defined on equivalence classes in the factor) to represent. range dimension function under different circumstances, on the factors in the following categories:
group measure space by the construction, they got Ⅱ 1 type and Ⅱ i type factor .1940 years of Classification and existence of various factors, after the proof, an important question is: this classification is complete algebraic classification of the factor? that a given type are all factors isomorphic? von. Neumann and Murray examine this question a lot of time spent, and ultimately construct the two new factor type Ⅱ 1 and prove they are non-isomorphic, thus answering the original question in the negative.
6.
von lattice theory. Neumann Hilbert space in the study ring operator, met with a complete complement of a class of L-mode
defined as continuous geometry (continuous geome-try), and construct an important class of continuous geometry: for any division ring can be F and natural numbers n, F 2n dimension space on the form 2nm1 dimensional projective geometry PG (F, 2nm1). Metric Completion after it received a supplement is a continuous geometric lattices, denoted CG (F). He proved that Hilbert space Ⅱ 1 type factor has the CG (F) isomorphic invariant subspace lattice.
regular rings (regular ring) is the Von. Neumann introduced another new concept: A is He said one of the closely related: continuous geometry L with a regular left ideals of ring A constitutes the main lattice isomorphism. In other words, the A is decomposed into a direct sum of all ideals, corresponding to various lattice L is decomposed into the direct Product of the problem.
the proof of these conclusions, the Von. Neumann has developed some new way of thinking, mainly about the distribution of cells: the distribution of the number of independent element of the distribution and distribution and so on infinitely. He first discovered in the Boolean algebra, computing intersection and the distribution must be infinite, and this distribution is equivalent to the continuity of another.
his cell most of the work is not published in a timely manner, mainly through the Institute for Advanced 1935m1937 notes in World War II in the political, economic and military development of the situation, Feng. Neumann began to concentrate more betting on the practical problems, mainly mathematics and game theory is to calculate the two aspects of work.
1. Calculation Mathematics
Feng. Neumann believes that once the equations describing physical phenomena given by the expression of mathematical language, can be solved numerically by means of conventional methods or without repeated testing. his efforts in computational mathematics, is This point of view and with his degree of difficulty to solve practical problems inseparable.
World War, various technical problems caused a rapid estimates and approximations needs. These issues often involve the separation of which can not be ignored or external disturbances , requires the help of numerical methods for qualitative analysis. von. Neumann from the numerical stability analysis, error estimation, matrix inversion, and with the calculation of discontinuous solutions of a number of directions to explore .1946, he and V. Bagh Man (Bargmann), D. Montgomery (Montgomery) cooperation. submitted to the Naval Weapons Laboratory report, the system described, and discussed the use of computers to solve real possibility. In 1947, he ...