Von Neuman
Li Xuhui
(East China Normal University)
Von Nogan, J. (Von Neumann, John) was born in Budapest, December 28, 1903; February 8, 1957 Strive in Washington, USA. Mathematics, physics, computer science.
Von Neumman was born in the Jews family. Father Max Von Neumann is a rich banker. In 1913, the Austro-Hungar Emperor Francis Joseph I) awarded the title of the Mike Herb, and the name of the Novan family has a "von" word.
Von Neuman has been well educated since childhood. Father specially hired a family teacher and taught him the system, foreign language, history and natural common sense, and he has shown superman's memory and understanding. Legend has it that he is 6 years old to count 8-digit division, 8 years old, grasp the calculus, 12 years old, I have learned E. Borel's "La Thorie Des Fonctions).
In 1914, 1914, 1914, Feng Nuiman, was just 10 years old and was sent to college preparatory learning. His talent wisdom caused the teacher L. Ratz's note, Ruz thinks that Feng Nuiman accepted traditional secondary education is a waste of time, and should have special mathematics training to him, so that it is fully developed. Ruz recommended von Nuiman to the University of Budapest. Professor of Krschak, Dulake arranged assistant M. Ferkete has served as his family counseling. He published the first papers, which is less written in less than 18 years old, promotes the Fayer (чеъъдев) Multiplicen Trees' Fayer (Fejr) theorem. In 1921, he has been recognized as a long-term mathematics rookie through the secondary school student graduation exam.
The four years after this, Feng Norman has attacked chemistry at the University of Berlin and Switzerland, while retaining the University of Budapest's mathematics. At the end of each semester, he has to rush back to Budapest from Europe to visit his family and participate in mathematics exam. In 1925 and 1926, he received a Ph.D. in the Chemical Engineering Degree of Zurich and the University of Budapest.
In Berlin, Von Neuman participated in A. Einstein about statistical mechanics lectures and following E. Schmidt studies; in Zurich, he is with H. WEYL and G. Pollia (PLYA) has been closely contacted. Von Neuman once said that the mathematician who has the greatest impact on his early years of academic thinking is Goer and Schmidt.
He also went to the University of Gentier several times, visiting the big mathematician D. Hilbert (Hi-Lbert). He was deeply attracted by Hilbert's quantum mechanics and proof. Hilbert also appreciated the young scholar. At the beginning of 1926, he had not got a doctorate, and Hilbert tried to seek the qualifications of the University of Guntiegen.
In 1927-1929, Feng Nuimman was hired as a branch of the University of Berlin, in which a large number of research results have been achieved in the collection, in the mathematics, and quantum theory, it is attracted by the math. In 1929 he turned into Hamburg University's obligation lecturer. Under the recommendation, he came to the Mathematics Department of Princeton University in 1930, and became a lifelong professor in the second year of the second year. In this way, he lives in Europe every year, and the other is in the United States.
In 1933, senior research was established in Princeton. Von Novan was hired by the Institute for lifelong professors from the beginning, only 29 years old, and was the youngest professor in the hospital. He has achieved US citizenship in 1937. At that time, the world economy was in the Great Depression, and the clouds of war were covered with Europe, but Princeton became the grounds of mathematics and physics elite. In a strong academic atmosphere and stable life, Von Neuman has been engaged in research work. In 1932, he summed up the development of quantum mechanics from mathematics, and published a book "Mathematische Grundlagen Der Quantenmechanik), and launched a famous weak passage. In 1937, he issued the theory of the operator ring and established a continuous geometry. The partial solution of Hilbert's fifth issue is also one of his main achievements in this period.
1930. Von Neumman married M. Collvèsi, Daughter Marina was born in 1935. Two years later, their marriage cracked. In 1938, Feng Nuiman returned to Dades, visiting relatives, married to Klaradan and came to Princeton in the end of the year. Krara later became one of the first batch of scholars who compiled mathematics issues in computers.
After the explosion of the Second World War, Feng Nuiman's scientific career took turns. In 1940, he was hired as a scientific consultant by Aberdess Ballistic Experimental Research, in 1941, hired Navy's Consultant. From the end of 1943, he participated in the work of the Los Aramos Institute with an advisor, guiding the best structure of the original bullet, and explored the program of achieving large-scale thermal nuclear reactions. In mathematics, in addition to solving various numerical calculations, his most important achievement is in 1944, it officially created countermeasures and modern mathematics.
He turned to the electronic computer in the late stage of war. In 1944, he visited the first electronic computer ENIAC that has not been completed, and participated in a series of expert meetings for improving computer performance. In the next year, he proposed a new idea for electronic computers and programming, and developed two new programs - theedvac program and IAS program. In 1951, the IAS was successfully developed and proved his correctness of his theory.
After the end of the war, Von Neumman served as Director of the High School Institute, while continuing to serve in military agencies such as the US Navy Weapons Lab. In October 1954, he was appointed as a member of the US Atomic Energy Commission, which resigned in the second year of the senior research institute, from the work, and lived in Princeton to Washington.
From the end of the 1940s, before death, Von Neuman also focused on automaton theory, including comparison of various artificial automators and natural automators, solving automatic adaptive, self-propagation and self-recovery. In 1951, "The General and Logical THEORY Automata" opened up a new field of computer science and laid the foundation for the study of artificial intelligence.
In 1955, Feng Nuiman was confirmed to have bone cancer and the condition was rapidly deteriorated. He insisted on thinking, writing, and participating in the academic conference on a wheelchair and prepared a speech of Hilliman's lecture. On February 8, 1957, he was 53 years old in Washington Army Hospital.
Von Neuman has served a lot of scientific positions in his life and has obtained many honors. The most important: 1937 BCHER Award; 1947, the US Milly Bulk Gibbs (Gibbs) lecturer seat, And got a merit award (presidential award); 1951-1953, served as Chairman of the US Numerical Society; 1956 Awards Amban Memorial Award and Fermi Award. He published a total of more than 150 academic papers, all included in the 1961 Porhon Publishing House, published in the Collected Works of John Vonneumann. 60 of them are pure mathematics, 60 about application mathematics, 20 articles belong to physics. Von Neuman became a generation of scientific giants with its superman's only and fruitful academic achievements.
Pure number
Von Novan's work in pure mathematics was concentrated in 1925-1940, mainly divided into the following six directions.
1. Collection theory and mathematics foundation
In this century, in order to overcome the paradox G. The difficulties brought by Cantor Collection, and systematically organize Condur's theory and methods, and people have begun to study the method of malting. In 1908, two famous axiom systems appeared: E. Zermelo's system [after A. Frenkel and A. Skolem modified supplements, becoming a ZF axiom system] and B. Type Theory of Russell.
Von Neuman has long agreed to interested in aggregation. During the period in Zurich in 1923, he published its own second papers "Zur Einfhrung Der Transfiniten Ordnungszahlen", which strives to "avatar, accurately" the concept of Cantone. In Cantone's definition, the number of orders is the sequence type of good sequence, and according to the ZF axiom system, the presence of the order type cannot be proven. Von Neumman gave a new definition of the formation and overrun order formation by means of the initial truncation of the ZF axiom system, which has given new definitions of the number of sequences and overrun orders.
Among the sixty years, he actively spreads the thoughts of warning and tried to establish a more form and accurate axiom system. In 1923, he submitted a long papers "Die Axiomatisierung Der Mengenlehre), and Schmidiomatisierung Der Mengenlehre, Schmidt, and Schmidt's representative editorial department. Authoritative Flekl. After discussing with Flenker, von Nuiman made a statement of an introductory article "Axiomatisierung Der MengenLehre), published in 1925.
"The Aficization of the Collection" later became a Ph.D. of Feng Nuiman. It is established by P. Bernays and K. After Gotel (GDEL), it formed a new system - NBG system in the aggregation of the aircraftization.
The NBG system is not like the ZF system, and the collection and slave relationship acts as the original concept, and takes the way to limit the collection to achieve the purpose of excluding the paradox, and is different from the type of language to describe the collection system in the language. It is characterized by the "Collection" and "belong to", introduced "class" as a non-defined concept, more general. The class is divided into a collection and true class, which specifies that the true class cannot be used as an element of the class. This eliminates the possibility of arramenting by "all sets of collection".
The NBG system retains more and more useful arguments compared to the ZF axiom system. Moreover, in the ZF system, the NBG system does not include a malicious model, which is a poor system. It is a simple logical structure as the first-class geometric axiom. This is the most important advantage. It has been proven that the NBG system is an expansion of the ZF system. Gothel has been inspired by the NBG system when proved to choose a axiom and continuous hypothesis. Today, the NBG system is still one of the best foundations of aggregation.
Adapted to the work of the aircraftization of the aircraftization, von Nuiman participated in the Yuan Mathematics Program in Hilbert in the late 1920s. The 1927 article "Zur Hilbertschen BEWEISTHEORI) interprets the basic concept of mathematics formalism. It pointed out that various problems raised by the Hilbert Mathematics Program, although Hilbert himself and Bernas, W. ACKERMANN et al. Has progressed, but it has not been satisfactorily solved from general. Especially the proof of Akman 's non-stained transitions, and cannot be implemented in classical analysis.
In 1931, the Gotel misconducted, and the Hilbert program was fully realized. In this regard, von Nuiman did not surprise, because in 1925, a "axioming of the axiology", he fadiously foreseen Gothe's conclusions: either formal system There is no proposition that cannot be determined in this system. The last sentence of the original text is: "Temporarily, in addition to the deficiencies of the disclast itself, what can we do? No known approach can avoid the difficulties." He believes that "by Gotel" A new approach to understand the role of mathematical formism, and should not end it as a problem. "He himself maintains long-term interests on mathematics foundation, and it is reflected in the post on computer logic design and mechanization. .
2. Measurement
The measurement is not in the central status in the entire research work of von Novan, but he gives many very valuable methods and results.
In the article "General Measurement Theory" in 1929, Von Neuman discussed a limited number of additional measurements. "Measurement Issues" in N-dimensional Space Rn is: Whether there is a non-negative, normalized and regardless of the camodoise of the rigid body movement? F. Hau-Sdorff and S. Banach demonstrates that the measurement problem has an endless solution in N is 1 and 2, and it does not solve it in other cases. This conclusion gives the feeling that when the dimension is changed from 2 to 3, the characteristics of space have undergone fundamental, and it is difficult to change. Von Neuman pointed out that problems are in nature, which causes the root cause of nature differences in the change in groups and non-space changes. Explore the solution of the measurement problem, need to use the group's solubility.
He continued to use the idea of group theory, analyzed the paradox of House Dorf - Barn Tarski: RN (n ≥ 3) two different radius balls, can be decomposed into limited The unpredictable gathering of mutually inconsishes, so that the two-ball subset can establish two and two full-class relationships (when N is 1 or 2, this decomposition does not exist). He explained that this is because when n is 3 or more, the orthogonal group contains free non-Abel groups, while it is less than 3.
In this way, the measurement problem is promoted from RN to a general non-Abeer group. Pashaphe's possibility of using the same measure of all subsets of R2 is proved to be established to all subsets of the Abel group. Finally, he concluded that all solutions were measurable (ie some measure can be introduced to the group). This article belongs to the earliest results from Eucquites to the more general algebra and one of the works in the topology. Since then, this ideological method began to pay more attention.
In the same period, Hungarian mathematician A. Hal (Haar) proposes such a question: Does there have a method of selecting a measurable ammeter in the RN such that each subset is equivalent to a given collection, and the selection process maintains a limited set operation? Von Neuman gave a positive answer and extended the conclusions to the measured function. This has become a starting point to solve the problem of measurement decomposition. In 1935, he was also with M. Stone collaborates, discusses more general questions: A is an ideal of a bully, M is A, when there is a sub-generation of A, so that the mapping of A to A / M is limited to the child generation Construct? They give a variety of sufficient conditions.
Another achievement is that he proves the uniqueness of Har measure in 1934 (in the sense of differential constant factors). The "invariant mean" (Invariant Means) of the proven to be constructed in the process, and the method different from Har is used to introduce the measure: in place of the given left constant measure M, M with the smooth measurement M, M, M, M 'Definition by the following formula:
Where Ω is appropriate. M 'not only has all of M, but has a right zero. These methods are later he and S. Bochner 's research on the multi-period function of the multi-pocket group.
1933-1 In 1934, Feng Nuiman conducted a report in the senior research in the senior research, which explained the classical theory of Leberberg measurement in the European space in detail, and promoted to abstract measurement space. The content of the report is a source of the United States in the measurement of the United States for a long time. In 1950, the editor of the Princeton Publishing House became a "Functional Operators".
3. Traversal theory
Feng Nobiman's primary achievement in this area has proven the average traversal theorem (also known as weak traversal theorem). In the 1970s, L. Boltzmann proposes the traversal hypothesis in statistical mechanics, and hopes to provide this premise to derive the space average of the guaranteed transformation is equal to (discrete) time average, which is the Bolzman plan.
From mathematical implementation, it first needs to prove as the existence of the limit as the time average. 1931, b. Koopmann and A. Weil also found that the function operator induced by the guaranteed transformation is a monogram. It gives von Nuiman to greatly enlighten it. At that time, he was working on the research of the operator theory, which prompted him to try to solve the existence problem with the self-condeh with the Hilbert space. Soon, he proposed and proved the first important theorem of traversal theory - average traversal theorem:
, Transform T, traverse the average
Convergence to the function PF according to the norm of L2, where UT is T-induced operator
UTF (x) = f (tx), XX
And P is orthogonal projection of the L2 to the UT constant function space.
Before this result was published (1932), von Nuiman introduced it to G. D. Birkhoff and Kushman. Berkhof will improve the convergence of "according to the average measure" as "convergence everywhere", resulting in stronger conclusions - pointwise Ergodic Treorem, also known as individual traversal theorem), and in 1931 December. Despite this, due to Berkhof and Kuproman wrote "Recent Contributions to the Ergodic THEORY", the academic community learned the cause of the aforementioned theorem. The first gesture work has been affirmed.
Soon, INNALS of Mathematics, 1932, also published his influential articles "The Operator Method in Classical Mechanics", which marks the research on traversal theory system. The beginning.
The paper first gives a detailed proof of the average traversal theorem, and then launches 6 important theorems. The first is Decomposition THEOREM: Any warranty transformation can be broken down into a number of straight points traversed. It illustrates that in all guaoning transformations, it is the most basic, most important, and any warranty transform can be constructed.
Theorem 2 further pointed out that the classification problem of a single parameter preservation conversion group is essentially classified into traversal transformation.
The classification problem of the guarantees is later become the central issue of traversal theory, of which the most critical first step is von Nogan and P. Halmos (Halmos) jointly proved in 1942:
F1 and f2 are the warranty transforms on the finite measurement spaces X1 and X2, respectively, and U1 and U2 are X1, X2, respectively, induced the 酉 operator. If F1, F2 has a discat spectrum, then F1 is consistent with F2 and is the same only when U1 and U2 are made as the Hilbert space.
When Von Neumman is dealing with the problem of traversal theory, it is often focused on the internal contacts of the measurements and spectrum. Theorem 5 is a typical result of the subcamination: for the 酉 算 u (by traversing transformation), it actually constitutes a plurality of groups of real groups; in turn, each group of real groups An infinite aging group can be used as a pure point spectrum of certain transformations induced by transform.
Corresponding to this, von Novan and Ku Prawman have a mixing theorem for the continuous spectrum. It asserts that it is equivalent to the geometric nature (mixing) of the transformed geometric nature (mixing) and the 算 子 性 (not a plain feature value).
For the results made by von Nuiman in the measurementism and the theory of traversal, Halmos gave this evaluation: "From the number of literature, they are still very different from Feng Nogan. First, but in terms of quality, even if he has never studied other aspects, these results are enough to enable him to enjoy a permanent reputation in the mathematics. "
4. Group theory
A famous result of von Nuiman is the fifth issue of Hilbert in 1933. As early as 1929, he proved that the continuous group may change the parameters, making the group's operations a parsing. Specifically, for a linear conversion group in the n-dimensional space, it has a normal subgroup, which can be parsed, and is partially illustrated in a manner corresponding to a limited parameter. This is the first article to solve the fifth issue of Hilbert.
In 1933, he issued a "DIE Einfhrung Analytischer Parameter in Topologischen Gruppen" on the "Mathematics Chronicle", "Die Einfhrung Analytischer Parameter In Topologischen Gruppen, proved that each local homogenic embryo is allowed to allow a tuning group to allow a Li Qun. structure. In this way, the fifth issue of Hilbert has been affirmed under the condition of the community. The problem of solving Peter-Determators in groups, Schmitt's function approximation theorem and L. E. J. Brouwer's regional invariant theorem for Europe's space, reflects Feng Nuiman's rich aggregation theory and solid change function and his skill of integration equation, matrix computing skills.
Another work is also related to group theory: the atrial function (Almost PE-RIODIC FUNCTION) theory. He put H. Bohr's first real set of real sets The concept of proliferation is extended to any group G, which in turn has established links between the new proliferation function theory and Peter, and Lower demonstration: set the limited matrix of group G Represents D (x) = (Dij (x)),
The following three conditions are equivalent:
(1) Each Dij (x) is a boundary function on G;
(2) Each Dij (x) is a proliferative function on G;
(3) D is equivalent to the representation of an 酉 matrix.
He thus pointed out that the proliferation function on the group constitutes a maximum scope of application of the theory.
5. Operator theory
Exploration on the theory of operator runs through the entire scientific career of von Nuiman, which accounts for one-third of his writings, and he has more than 20 years of leadership in this field.
1927-1930, he first gave the abstract definition of the Hilbert space, that is, the definition used now. Then, for the theory of Subridge Self-conjugate calculation from the boundary to unrequenisented promotion, the concept of the system is made: the concept of introducing a thick and closed operator, gives an unbounded self-conjugate operator, The spectrum decomposition theorem of the 酉 odicle and the formal operator pointed out that the difference between the symmetric operator and the self-conjugate operator in nature, and also co-investigated the variation of the uncovering operator after disturbance.
Feng Nuiman's formation form, coupled with the "Thorie Des Operations Linaire" in 1933, marking another new branch in the field of mathematics - pan The birth of a letter analysis.
In the 1920s, e. Nother and E. Artin has developed non-exchange algebra theory. Feng Nuiman realized that this is an excellent interpretation and simplification of matronism. He tried to extend the concept to the operator algebra on the Hilbert space. The concept of "operator ring" is generated: the weak (or strong) operator topology is closed and contains a * sub-generation of the constant state operator I is referred to as an operator ring. The operator ring can be considered to be the natural promotion of the matrix algebra in a limited dimensional space, and later known as von Nuiman algebra, to show the commemoration of von Nobiman. And in the same sense, it can be called the W * algebra.
The formal definition of the operator ring appeared in the 1929 papers "Funbra Der FunktionalOperati-Oren und theorie der Normalen Operatoren). This paper also includes important definitions such as "Commutant", "Factor" (Factor), and Double Commutant theorem:
It is an operator ring, and the exchanger is also an operator ring, and.
This actually gives an equivalent definition of the operator ring: the boundary linear operator in the Hilbert Space H is satisfied with the * sub-generation of = () 'is referred to as an operator ring. This is a definition is an important tool to study the operator ring. When the judgment operator is accompanied by an operator ring, it is used to decompose the thick and closed operator. Since 1935, von Nuiman is in F. J. Under the assistance of Murray, a series of articles titled "On Rings Of Operators) were written.
Their primary conclusion is that the operator ring can be expressed as a continuous line of factors. Therefore, the study of the operator ring is attributed to the study of factors.
Affected by the theory of classic non-exchange algebra, people have speculated that all factors are all contained in (h). Von Nogan and Murrium proved in "Arnte I": When the factor contains extremely small shooting, it is constant in (h). At the same time, they applied the skills of the past, constructing an important example, which means that all factors have extremely small shootings, and thus the nature of the factor is far from being speculated as simple.
They established a sequence relationship between the fire of the factor and make it comparable. This sequence relationship can be expressed by dimension function (defined above the factor equivalents). According to the different conditions of the dimension function value, the factor has the following category:
Through the structure of the group measurement space, they got II1 and II ∞ type factors. The 1940 "argument case III" has given an example of III type factor.
After the classification of factors and the provenance of various factors, an important issue is: Is this classification completed the algebraic classification of factors? Is that the whole factor in a given type is organized? Von Neumman and Muri flowers went to a large number of time to examine this problem, and finally constructed two new II1 factors and prove that they were unsome, giving the original question negative answere.
6. Grid
Von Nogan was studying the Hilbert space operator ring, encountered a class of complete modified definitions L as a continuous geometry, and constructed a type of important continuous geometry: The 2N dimensional seizure spatially 2n dimensional spatial spatial in the undesirated ring F and the natural number N, F constitutes 2N-1-dimensional surgical shadow geometry PG (F, 2N-1). The finished mold obtained after measuring its metrics is a continuous geometry, which is recorded as CG (f). He proved that II1 type factor in the Hilbert space has a non-variatric space grid with CG (F).
Regular Ring is another new concept introduced by von Nuiman: A is a handless connection: continuous geometric L and the principle of the main left of certain path A. That is, Decompose A to the top of the ideal, corresponding to the problem of decomposing the L decomposition into the direct volume of the comparable.
During the proof of these conclusions, von Nuiman has developed some new ideas, which is mainly about the allocation of the grid: the allocation of the number, the allocation of the independent element and infinity allocation. He earliest, in the Boolean algebra, the intended operation is inexharged, and this allocation is equivalent to continuity.
Most of his work is not possible in time, mainly through the 1935-1937 senior research, "Supreme Geometry", "Continuous Geometry" and the US Academy of Sciences.
applied mathematics
After 1940, with the development of political, economic and military situation in the Second World War, Von Neuman began more in practical problems, mainly to calculate both mathematics and countermeasures. .
1. Computational Mathematics
Von Nuiman believes that the equation describing the physical phenomenon can be given from a numerical language to be resolved from a numerical value without having to repeat the test. His efforts in calculating mathematics are inseparable from his perspective and difficulties in solving practical problems. In the war, various technical issues have caused rapid estimation and approving. These problems often involve external disturbances that cannot be ignored or separated, and must be qualitatively analyzed by numerical methods. Von Neumman explored several directions such as numerical stability analysis, error estimation, matrix reverse and intervalting solution. In 1946, he and V. Bargmann, d. Montgomery works. The Naval Weapons Labs submitted the "SolutionOf Linear Systems of High Order", and the various solutions of the linear equations were systematically elaborated, and the possibility of using a computer is actually solved. In 1947, he also h. Goldstine studied the number of high-order matrices, and gives a strict error estimate, especially the accuracy of the 150th-order matrix to reverse the accuracy of the impact.
When solving the case where the compressed gas movement is particularly intermittent, Von Neumman has founded the artificial viscosity. For example, physically systematic warranty
Ut f (u) = 0 (u is heat, F is flow),
The system it describes also spontaneously generates intermittent (exciting) even in the case of the initial value. Von Novan and R. Richtmyer looks it into a distributed equation, and the solution process is equivalent to seeking effective numerical algorithms to calculate the distribution derivative. They use parabolic regular equations
UT F (u) = εΔu
Instead of the original equation, the distribution derivative is a normal derivative, so that the finite difference is approximated, so that the solution is always smooth. This method of calculating the person in the calculation formula is an "viscous" term, so that the interruption of the excipient is a smooth transition zone, and the position and strength of the excitement are easily determined. Artificial viscosity is the first example of modern fluid kinetics in Laranglan method, providing a strong means of numerical simulation of fluid mechanics on an electronic computer.
After the electronic computer is generated, von Nuiman launched a new idea, new method using a computer for numerical analysis, thus promoting the rise and formation of computational mathematics, which also enabled him to become one of the founders of modern scientific calculations. This article "Computer Theory and Practice" section).
2. Counterprises and mathematics
Von Neumman is one of the founders and modern mathematics of economic (also known as game theory). In the 1920s, Poles used the math language to portray the game issues, introduced the concept of pure strategy and mixing strategy, and proposed a mathematical solution to solve personal countermeasures and zero and two countermeasures. However, the theory of countermeasures as the true foundation of the discipline, starting from Feng Nobiman 1928 to the "Zur The-Orie Der Gesellschaftsspiele).
The most important conclusion in the text is about the miniMax theorem: M × N matrix A is the payment matrix of normalized zero and two countermeasures, X and Y are the bureau. Probability vector of a mixed policy, there is a unique value V, make
At the same time, there is an optimal policy x * and y *,
Based on a very small and large-product, Von Neuman first discussed cooperative countermeasures, especially between zero and three countermeasures. In order to give the concept of cooperation, he introduced the idea of characteristic functions. Finally, I clearly expressed the general game scheme of N players. The results show that the solution of n-person countermeasures is present and unique under additional conditions. The very small powerful theorem is the cornerstone of the policity. In the 1930s, von Nobiman himself and other mathematicians have given some new proof methods for this theorem. In the 1940s, a. Wald (WALD) is based on a minimum and great theorem, regards the decision-making process as the two countermeasures issues that people and the environment, which has created statistical decision theory. Since then, countermeasures become an active research field in the application of mathematics.
In 1940, the Austrian economist O. Morgenstern came to Princeton, and he made von Novan to economic problems, especially cargo exchange, market control and free competition. After four years of cooperation, they published "Theory of Games and Economic Behavior). This work further elaborated in 1928, such as increasing the "Imputa-Tion)," Domination "concept, defined von Nuiman - Morganism. There are nearly two-thirds of the books in the whole book to handle cooperative countermeasures.
The application of the basic issues of economic theory is another important results in the book. They believe that although the economics at the time is still in the early days - like physics in the 16th century, it will eventually develop as a physics, as physics. The countermeasures are the first step towards the comprehensive mathematical economics. This actually reflects the desire to include social sciences into the routine mathematics system.
At an academic seminar held in Pulinson, 1932, Von Neuman also discussed the modeling of general economic balance. He gives an economic model of cargo production and consumption, pointing out the close relationship between model problems and minor and great agencies: When economic activity is considered zero and countermeasure issues, the balance of economic model is the pole in countermeasures. Largetal small value V.
Physics
Von Novan's vinting was not limited to mathematics, and he also had a strong interest in physics science. It can be said that the discussion of contact between mathematics and physics has the most significant significance in his scientific achievements. The aforementioned operator theory and traversal theory, etc.
In 1926, Von Neuman came to the University of Gentrygian. At the same time as Hilbert studied the mathematics foundation, he was deeply attracted by the quantum mechanics that were carried out in the University University. The quantum mechanics at the time had two representations in mathematics: W. Heisenberg, M. Born and W. Matrix mechanics established from the particle of the microscopic particles, e. Schrdinger Established Wave Power from Volatility. These two systems are sufficient for speculating the nature of the atom. Soon, Schrödinger also proved the equivalence of both, and attributed to P. Dirac and P. Special circumstances in the development of the JORDAN development. However, von Nuiman and others are not satisfied with this, and they want to extract more commonality and establish quantum mechanics.
In the winter, Hilbert has made a speech on the new development of quantum mechanics. Nordheim prepares materials for the physical part of the lecture, and the main work of the mathematical formation part is done by von Nogan. A basic point of quantum theory is a mathematical description of atomic status. Von Neumman did not explicitly define this, but a formal process: the status of the atom was characterized by the unit vector of the Hilbert space. This is just like Hilbert, and the line is the same as the non-definition term. Von Neuman pointed out that this description is consistent with Heisenberg and Schrödinger, and the formal rules in algebra are equipped with the same, multiplier rules, and their expression systems.
He also constructs abstract Hilbert space based on five axioms and proves that Hesensburg and Schrödinger's atomic state definition meets five axioms. The final conclusion is: a suitable form of quantum mechanics, from the vector (representing the system state), a certain type of operator (observable) and its algebraic rules in a certain type of system.
These methods excellently embodied Hilbert's axioming program, becoming the prelude of quantum mechanics, and promoted Von Nobman to the Operator theory of Hilbert's subsidiary.
In 1932, his masterpiece "Mathematical Basics of Quantum Mechanics" was published by Germany. This is the integration and improvement of previous methods and conclusions. He specifically pointed out that Dirac et al., When the concept of dealing with the operator, it did not convict its definition domain and topology, and the grass rate assumes that the operator can always be aligned. For those who cannot be angled, the concept of Dirac is introduced into the Dirac Normal Function (δ function). Von Neumman discovered its own contradictory nature, with its own achievements: transform theory can be established on a clear mathematical basis. Its method is not to modify Dirac's theory, but to develop Hilbert's operator theory. When he successfully promoted the operator compatibility to the boundless situation, he finally completed the formal work of quantum mechanics, which included the system of Heisenberg and Schrödinger as a special case.
Another main content in the book is to explain the "causality" and "INDETERMINACY" in quantum mechanics from the statistical perspective. His conclusion is that the uncertainty of the quantum system is not caused by the state of the observer. Even in the system introduce imaginary "hidden parameters), the observer is in an accurate state, and it will eventually lead to uncertain observations due to subjective awareness of the observer. This view has been agreed with most physicists.
This book also includes a solution to the special problems in quantum, such as the expression and proof of the assumptions in the quantum system. This has become the ancestors of traversal theory that he later opened.
"Mathematical Basis of Quantum Mechanics" (German Edition) has been translated into French (1947), Spanish (1949), English (1955) and Japanese, it is still the classic of theoretical physical field.
In 1927, von Nuiman began to analyze quantum mechanics with probability terms, introduced a statistical matrix U (now ρ matrix) to describe a collection of systems of various quantum states. The statistical matrix becomes the main tool for quantum statistics. And his measure theory of quantum mechanics laid the foundation for the development of thermodynamics.
Theory and practice of computer
A large number of calculation tasks raised in Los Alamos, the atomic nuclear fission process, prompting von Nogan to pay attention to the development of electronic computers. From August 1945 to June 1945, he participated in the investment and improvement of electronic numerical integration and calculator ENIAC (ELETOR). He found that the main defect of the ENIAC machine is still adopted a "external interpolation" program that is in the previous electromechanical computer. When performing operations according to a given program, each problem requires a special line system, so lacks high-speed calculation. Flexibility and universality. In March 1945, Von Nobeman drafted the design plan of discrete variable automatic electronic computer Edvac (Electronic Discrete Variable AutomaticComputer), a discrete variable automatic electronic computer EDVAC (AUTO) In June of the second year, he was with A. Burks, Gödstein jointly proposed a better report "Preliminary Discussion of the The Logical Design of An Electronic Computing Instrument" unveiled a new page in the history of computer development.
In these two reports, Von Nobman has established the principles of the main structural principle of computer organization - Stored-Program. It determines that the computer consists of five parts: calculator, controller, memory, input, and output devices. The program consists of instructions and stored together in the memory, and the machine reads the instruction from the memory by the logical order specified by the program and executes the process to automatically complete the processing of the program description.
According to this principle, the EDVAC machine and IAS machine plan are designed to improve the following important improvements compared with the ENIAC.
(1) Change the decimal system to binary, programs and data are represented by binary code;
(2) The program is from the outer plug into memory. When the title is changed, it is not necessary to change the circuit board and simply replace the program;
(3) Store the input electrical signal in the manner of ultrasonic signals, and establish a multi-stage storage structure, and the storage capacity is greatly improved.
(4) Adopting parallel calculation principles, that is, all of the numbers are simultaneously processed.
Since 1946, Von Neumman organized Godstan and others conducted an actual construction of IAS in 1951 in 1951. Its computation speed reaches a million times per second, tripinks more than the ENIAC machine, and realizes Von Neuman.
The electronic computer constructed by the storage program principle is called a storage program computer, and then referred to as a von Nuiman. Although the organizational structure of modern computers has some major changes, in terms of principles, the mainstream is still the foundation of Storage Principles. Von Neuman's thoughts deeply influenced all aspects of the storage, speed, instruction selection of modern computers and line design. The name of von Nuiman is associated with computer designers. However, his main interest in the computer is not on computer design and manufacturing, but how to use this new scientific tool to create a modern scientific calculation.
Classical numerical analysis methods, not necessarily optimal for computers, and some ways to be extremely complex in arithmetic, which are compiled as programs, and it is easy to implement on new computers. Von Neuman started from this actual situation and made a lot of work for computer programming. He and Gothican invented flowchart (Flow Diagram) to communicate the problem and machine instructions to be calculated; he introduced subroutine and automatic programming, greatly simplifying the prime level of programmer programming. Dozens of computing skills such as matrix feature value calculation, inversion, multi-function extremum, and random number are also the first in the war after the war, and they have a wide range of applications in industrial sectors and government planning. After the birth of the electronic computer, von Nuiman and S. Ulam (ULAM) advocates a new type of calculation method - Monte Carlo Method, which will be the mathematical problem of the desired math as a probability model, with a smaller scale simulation on the computer, to obtain approximation solution. For example, when calculating the volume of a certain sub-region of the n-dimensional cube, it is not necessary to divide the space into a series of grid points to approximate the sample, but is randomly selected in the space in space, using a computer to determine The points falling in the child area with all points. This ratio is given when the number of selected points is much longer.
The advantage of Monte Carloh is that the geometric shape of the problem is not sensitive, the convergence speed is independent of the dimension, so it is especially suitable for high-dimensional mathematical physical problems. With this method, von Neumman produced a random number with a given probability distribution through appropriate countermeasures, and designed a probability model for processing the Bolzman equation. After the war, he led a meteorological research team in the Advanced Research Institute to establish a model to simulate atmospheric movement, and hoped to use computer to gradually solve the significant inspiration of numerical days and technologies.
In 1956, the US Atomic Energy Commission was specifically mentioned in the computing research on computers.
Since 1945, von Noban is also committed to automatological theory and comparative study of brain nerves and computers. He is considered to be the founder of automaton theory.
C. Credently, C. Information Engineering, Shang Agriculture (Shannon), A. The ideal computer theory and R of Turning. Ortvay 's research on the human brain, triggered von Nogan's interest in information processing theory. And 1943 W. McCulloch and W. A Logical Calculus of the Ideas Imma-Nervous Activity, which has seen the potential possibilities of mathematical law of human brain information. In his design of the EDVAC in 1945, the stored program computer described is constructed of "neurons) envisaged by McLock and Patty, rather than using vacuum tubes, relays or mechanical switches, etc. Conventional components.
Since then, he participated in the information about informationism, control theory, wide contact with mathemologists, physicists, electrical engineers and biologists, gradually forming automatic theory of automatum and technical fields. In September 1948, in the Hickson discussion class, he made a report of "The GE-NERAL AND Logical THEORY AUTOMATA), which proposes the self-propagation and iterative arrays of the automaton. Waiting for a new concept, and compares artificial automators such as computers and natural automators (such as human brains). He passed the calculation description, the number of electronic components in the computer is only one hundred thousand parts of the human brain neurons; on the other hand, the transmission speed in the electronic component is approximately 10,000 times in the brain nerve. In this way, the computer will win speed, and the brain is in complexity. In order for the characteristics of the two, it is possible to use the telecommunications process that occurs within each second as a standard. The calculation shows that the characteristics of the human brain beyond the computer 10,000 times. Further, he also pointed out that the computer is generally in order when performing the operation, and the human brain tends to operate parallel, so it does not have a computer in the "logical depth".
Based on this, he created a famous redundant technology in 1952: for a group of components (unreliable components) with a certain fault incidence, to build an authorized scale and complex automation, The probability that incorrect output can be controlled within a certain range (reliabar). At the same time, he simply describes the structure of the microbiological organization to describe the self-propagation system, proposing the concept of Novan cell space, using many mutually connected small self-motivation, forming a larger automatic machine - Nogan Automotive . This is the earliest basic one-class automaton. These two theories have developed into fault tolerant automice theory and cellular automation in the 1970s.
At the beginning of 1955, Feng Nuiman was invited by Yale University, starting to write a lean for one of the oldest, most famous scientific lectures in the United States - Hirman's lectures, and systematically explain his theoretical system on computers, automators and human brains. . Due to his aggravation and death, the plan of this lecture failed. In 1958, Yale University published a single-party "The Computer and Brain" in the COMPUTER AND THE BRAIN.
In 1947, in "The Mathematician), von Nuiman expressed such mathematical concepts: the development of mathematics has close contact with natural science, and mathematics penetrates all theoretical branches of natural science. . Mathematics has its own source of experience, it is impossible to have absolute, out of the roughness concept of all experience. On the other hand, mathematics is a creative discipline, the standard for the aesthetic, and the standards and judgment of success are aesthetic. Pure aesthetic tendency must be prevented. To this end, it should continue to inject some "more or less ideas directly from experience" in mathematics.
Von Novan's research activity was significantly affected by the above concept. He involved such a large field of science and strive to maintain the relationship between mathematics theory with the growing complex phenomena of physics and other natural sciences. This is also the contribution of the goal of achieving the general and organic unity of mathematics.
Formal thinking occupies dominance in the philosophical concept of von Nuiman. He believes that the logical system has universal and comprehensive, and the formal logic structure portraits the abstract essence of things to some extent. He is not interested in the limitations of the logical system, but when some limitations are discovered, he began to consider how to use more formal processes to overcome it (reflecting at his attitude towards the incomplete nature of Gotel). For him, the highest level of abstraction -, for example, logic and mathematics - should be completed in a strict form logic. When exposed to the actual problem, von Nuiman can quickly give appropriate mathematical formulations and make pure forms. Not only that, but the formal logic and mathematics put the maximum application, becoming his scientific. In his opinion, it can understand the entire world with abstract forms - including social life and spiritual awareness. This is reflected in his formation of mathematics, quantum theory and computer organization. It can be said that von Nuiman follows such a concept: Only a strict logical system may contain, the eternal universal truth that domains all things. In von Nuiman, a variety of scientific talents can be concentrated: the sense of the sense of mathematics thought (form an algebra), understanding the essence of the analysis and geometric classic mathematics, and excavating modern mathematics methods Potential power and application of deep insights in theoretical physical issues. These talents are not contradictory, but each of them requires high attention and memory, they can be collected in one person.
Von Novan does not use pen and paper to estimate geometric size, algebraic and numerical operations, which often leave profound impression on physicists. It is also very willing to give attention for problems in science but not very important but reflect a certain difficulty. By talking to him, people tend to understand some mathematics techniques that are not well known, and can easily solve the problem. This makes him loved and welcomed by applying mathematics workers.
In addition to science, Von Nobman has a strong interest in history. As early as the childhood, he read the system of 21 volumes of the "Cambridge Middle History" and "Cambridge Medieval Histo-Ry", especially profitable in European royalties and the history of Byzantine. His narrative and evaluation of historical events always makes colleagues greatly. It can also feel the humor that he is expressing in mathematician unique ways.
He can be skilled with German, French, English, Latin and Greek. His speech conducted in the United States is known for its good literary cultivation.
Von Novan has been from N. Wiener learned about China, which has gone to China's visits to lectures. In May 1937, Wenhua, the principal of Tsinghua University, Mei Yiqi and Mathematics Department, Xiong Qinglai, recommended von Nuiman as a visiting professor of Tsinghua University. Unfortunately, the full outbreak of the Japanese invading war after two months has made them hope to have bubbles.
