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Stanislaw Ulam - Biography

Stanisław Marcin Ulam (13 April 1909 – 13 May 1984) was a renowned mathematician of Polish-Jewish origin. He participated in America's Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, invented the Monte Carlo method of computation, and proposed the idea of nuclear pulse propulsion. In pure and applied mathematics, he produced many results, proved many theorems, and proposed several conjectures.

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Poland

Ulam was born in Lemberg, Galicia, on April 13, 1909. At this time, Galicia was in the Austro-Hungarian Empire, but in 1918, it became part of the Second Polish Republic, and the city took on its traditional Polish name, Lwów. In 1939, following the Molotov-Ribbentrop Pact, the Soviets invaded Poland, and the city was renamed Lviv, which is the Ukrainian form of its name, and by which it is known today.

The Ulams were a wealthy Polish Jewish banking and timber-processing family. Ulam's family was one of the richest in Lwów. His father, Józef Ulam, was born there and was a lawyer, his mother, Anna Auerbach, was born in Stryj, and his uncle, Michael Ulam, was an architect. From 1916 until 1918, Józef's family lived temporarily in Vienna. After they returned, Lwów became the epicenter of the Polish-Ukrainian War, during which the city experienced a Ukrainian siege and a pogrom.

In 1919, Ulam entered Lwów Gymnasium Nr. VII, from which he graduated in 1927. Soon, he began studies of mathematics at the Lwów Polytechnic Institute. Here, under the supervision of Kazimierz Kuratowski, he received his M. A. degree in 1932 and his D.Sc. in 1933. From 1931 until 1935, he traveled to and studied in Vilnius, Vienna, Zurich, Paris, and Cambridge, UK.

Along with Stanislaw Mazur, Mark Kac, Włodzimierz Stożek, Kuratowski, and others, Ulam was a member of the Lwów School of Mathematics. Its founders were Hugo Steinhaus and Stefan Banach, who were professors at the University of Lwów. Mathematicians of this "school" met for long hours at the Scottish Café, where the problems they discussed were collected in the famous Scottish Book, which is a thick notebook provided by Banach's wife. Ulam is a major presence in the Scottish Book. Of the 193 problems recorded between 1935 and 1941, he contributed 40 problems as a single author, another 11 with Banach and Mazur, and an additional 15 with others. In 1957, he received from Steinhaus a copy of the book, which had survived the war, and translated it into English.

Coming to America

In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, NJ, for a few months. In December of that year, Ulam sailed to America. In Princeton, he went to lectures and seminars, where he heard Oswald Veblen, James Alexander, and Albert Einstein. During a tea party at von Neumann's house, he met G. D. Birkhoff, who suggested that he apply for a position with the Harvard Society of Fellows.

From 1936 to 1939, after following up on Birkhoff's suggestion, Ulam spent summers in Poland and academic years at Harvard University in Cambridge, MA. Here, he worked with John C. Oxtoby to establish important results regarding ergodic theory, which appeared in Annals of Mathematics in 1941.

On August 20, 1939, in Gdynia, Józef Ulam, along with his brother Szymon, put his two sons, Stanislaw and 16 year old Adam, on a ship headed for America. Within two weeks, the Germans invaded Poland. Within two months, the Germans completed their occupation of western Poland, and the Soviets invaded and occupied eastern Poland. Within two years, Józef Ulam and the rest of his family were victims of the Holocaust, Steinhaus was in hiding, Kuratowski was lecturing at the underground university in Warsaw, Stożek and his two sons had been killed in the massacre of Lwów professors, Banach was surviving Nazi occupation by feeding lice at Rudolf Weigl's typhus research institute, and the last problem had been recorded in the Scottish Book. In 1963, Adam Ulam, who had become an eminent kremlinologist at Harvard, received a letter from George Volsky, who hid in Józef Ulam's house after deserting from the Polish army. This reminiscence gives a chilling account of Lwów's chaotic scenes in late 1939.

In 1940, after being recommended by Birkhoff, Ulam became an assistant professor at the University of Wisconsin in Madison, WI. Here, in 1941, he became an American citizen. That year, he married Françoise Aron. She had been a French exchange student at Mount Holyoke College, whom he met in Cambridge, MA. They had one daughter, Claire. In Madison, Ulam met his friend and colleague C. J. Everett.

Manhattan Project

In late spring of 1943, Ulam asked von Neumann to find him a war job. In October, he received an invitation to join an unidentified project near Santa Fe, NM. The letter was signed by Hans Bethe, who had been appointed director of the theoretical division of Los Alamos National Laboratory by J. Robert Oppenheimer, its scientific director.

At this time, Ulam knew nothing of Los Alamos. To learn more, he checked out of the library a New Mexico guidebook. On the checkout slip, he found the names of his Wisconsin colleagues, Joan Hinton, David Frisch, and Joseph McKibben, who had mysteriously disappeared from Madison to secret war work. This was Ulam's introduction to the Manhattan Project.

Hydrodynamical calculations of implosion

A few weeks after Ulam reached Los Alamos in February 1944, the project experienced a crisis. In April, Emilio Segrè discovered that plutonium made in reactors would not work in a gun-type plutonium weapon like the "Thin Man", which was being developed in parallel with a uranium weapon, the "Little Boy" that was dropped on Hiroshima. This problem threatened to waste an enormous investment in new reactors at the Hanford site and to make slow separation of uranium isotopes the only way to prepare fissile material suitable for use in bombs. To respond, Oppenheimer implemented, in August, a sweeping reorganization of the laboratory to focus on development of an implosion-type weapon and appointed George Kistiakowsky head of the implosion department. He was a professor at Harvard and an expert on precise use of explosives.

The basic concept of implosion is to use chemical explosives to crush a chunk of fissile material into a critical mass, where neutron multiplication leads to a nuclear chain reaction, which suddenly releases a large amount of energy. Cylindrical implosive configurations had been studied by Seth Neddermeyer, but von Neumann, who had experience with shaped charges used in armor piercing ammunition, was a vocal advocate of spherical implosion driven by explosive lenses. He realized that the symmetry and speed with which implosion compressed the plutonium were critical issues and enlisted Ulam to help design lens configurations that would provide nearly spherical implosion. Within an implosion, because of enormous pressures and high temperatures, solid materials behave much like fluids. This meant that hydrodynamical calculations were needed to predict and minimize asymmetries that would spoil a nuclear detonation. Of these calculations, Ulam said:
The hydrodynamical problem was simply stated, but very difficult to calculate – not only in detail, but even in order of magnitude. In this discussion, I stressed pure pragmatism and the necessity to get a heueristic survey of the problem by simpleminded brute force, rather than by massive numerical work.

Nevertheless, with the primitive facilities available at the time, Ulam and von Neumann did carry out numerical computations that led to a satisfactory design. This motivated their advocacy of a powerful computational capability at Los Alamos, which began during the war years, continued through the cold war, and still exists.

Statistics of branching and multiplicative processes

Even the inherent statistical fluctuations of neutron multiplication within the chain reaction, have implications with regard to implosion speed and symmetry. In November 1944, David Hawkins and Ulam addressed this point in a report entitled "Theory of Multiplicative Processes I", whose last page gives some suggestions pertinent to bomb design. However, this report, which invokes generating functions, is also an early entry in the extensive literature on statistics of branching and multiplicative processes. In 1948, its scope was extended by another pioneering exploration by Ulam and C. J. Everett.

Early in the Manhattan project, Enrico Fermi's attention was focused on the use of reactors to produce plutonium. In September 1944, he arrived at Los Alamos, shortly after breathing life into the first Hanford reactor, which had been poisoned by a xenon isotope. He stayed until December 1945, overlapping Ulam's stay by about a year, during which they formed a relationship that became very fruitful after the war.

The contributions of von Neumann and Ulam to the "Fat Man" weapons of the Trinity test and Nagasaki bombing were only part of a large scientific and engineering effort. Some other crucial contributions were Edward Teller's realization that implosive compression would facilitate nuclear detonation by significantly increasing the density of the plutonium sphere, Luis Alvarez's development of exploding-bridgewire detonators, which ensured that the explosive lenses would all fire at once, and Bruno Rossi's RaLa experiments, which tested (without plutonium) various weapon designs to arrive at one that produced an accurately symmetrical implosion.

Post war Los Alamos

In September 1945, Ulam left Los Alamos to become an associate professor at the University of Southern California in Los Angeles. In January 1946, he suffered an acute attack of encephalitis, which put his life in danger, but which was alleviated by emergency brain surgery. During his recuperation, many friends visited, including Nicholas Metropolis from Los Alamos and the famous mathematician Paul Erdős, who remarked: "Stan, you are just like before." This was encouraging, because Ulam was concerned about the state of his mental faculties, for he had lost the ability to speak during the crisis. Much later, after Ulam's death, another friend, Gian-Carlo Rota, asserted that the attack did change Ulam's personality; afterwards, he turned from rigorous pure mathematics to more speculative conjectures concerning the application of mathematics to physics and biology. However, this suggestion was not accepted by Françoise Ulam.

By late April 1946, Ulam had recovered enough to attend a secret conference at Los Alamos to discuss thermonuclear weapons. Those in attendance were: Ulam, von Neumann, Metropolis, Teller, Stan Frankel, and others. Throughout his participation in the Manhattan Project, Teller's efforts had been directed toward the development of a "super" bomb based on nuclear fusion, rather than toward development of a practical fission bomb. After extensive discussion, the participants reached a consensus that his ideas were worthy of further exploration. A few weeks later, the Ulams returned to Los Alamos.

Monte Carlo method

Late in the war, under the sponsorship of von Neumann, Frankel and Metropolis began to carry out calculations on the first general-purpose electronic computer, the ENIAC. Shortly after returning to Los Alamos, Ulam participated in a review of results from these calculations. Earlier, while playing solitaire during his recovery from surgery, Ulam had thought about playing hundreds of games to estimate statistically the probability of a successful outcome. With ENIAC in mind, he realized that the availability of computers made such statistical methods very practical. John von Neumann immediately saw the significance of this insight, and in March 1947, proposed to Robert Richtmyer, leader of the theoretical division, a statistical approach to the problem of neutron diffusion in fissionable material. Because Ulam had often mentioned his uncle "who just had to go to Monte Carlo" to gamble, Metropolis dubbed the statistical approach "The Monte Carlo method". In this context, the fact that Ulam's uncle Michael died in Monte Carlo is interesting.

In 1948, ENIAC was moved from the University of Pennsylvania's Moore School of Electrical Engineering, where it had been conceived and designed by John Mauchly and J. Presper Eckert, to the Ballistic Research Laboratory in Aberdeen, MD. Here, with von Neumann's inspiration, Richard Clippinger suggested that it could be configured in a jury rigged version of the stored-program architecture that has become standard on most computers. This architecture greatly facilitated programming, and on September 16, 1948, ENIAC was first run as a stored-program computer. Soon, it was put to work on Monte Carlo calculations of neutron transport by Metropolis and von Neumann. In 1949, Metropolis and Ulam published the first unclassified paper on the Monte Carlo method.

Statistical techniques began in the 18th century, when Buffon's needle was proposed as a way to approximate the number π. Fermi used them in the 1930s, and when he heard of Ulam's breakthrough, created the FERMIAC, which performs a mechanical simulation of random diffusion of neutrons. These ideas embody the context of Ulam's innovation, but not its essence, which is the probabilistic use of electronic computers to carry out rapidly the large numbers of tedious repetitions that are needed to achieve usable accuracy. As computers improved, these methods became more useful. In particular, Monte Carlo calculations carried out on modern massively parallel supercomputers are embarrassingly parallel applications, whose results can be very accurate.

Teller–Ulam design

On August 29, 1949, the Soviet Union tested its first fission bomb, the RDS-1. Created under the supervision of Lavrentiy Beria, who sought to duplicate the American effort, this weapon was nearly identical to Fat Man, for its design was based on information provided by spies Klaus Fuchs, Theodore Hall, David Greenglass, and others. In response, on January 31, 1950, President Truman announced a crash program to develop a fusion bomb.

To advocate an aggressive development program, Ernest Lawrence and Luis Alvarez came to Los Alamos, where they conferred with Norris Bradbury, the laboratory director, and with George Gamow, Teller, and Ulam. Soon, these three became members of a short-lived committee appointed by Bradbury to study the problem, with Teller as chairman. At this time, research on the use of a fission weapon to create a fusion reaction had been ongoing since 1942, but the design was still essentially the one originally proposed by Edward Teller. His concept was to put some tritium and/or deuterium in close proximity to a fission bomb, with the hope that the heat and intense flux of neutrons released when the bomb exploded, would ignite a self sustaining fusion reaction. Reactions of these isotopes of hydrogen are of interest because the energy per unit mass of fuel released by their fusion is much larger than that from fission of heavy nuclei. Because the results of calculations based on Teller's concept were discouraging, many scientists believed it could not lead to a successful weapon, and many more had moral grounds for not proceeding. Consequently, several senior people of the Manhattan Project opposed development, including Bethe and Oppenheimer.

To clarify the situation, Ulam and von Neumann resolved to do new calculations to determine whether Teller's approach was feasible. To cary out these studies, von Neumann decided to use electronic computers. This approach was made possible by his access to computers: ENIAC at Aberdeen, a new computer, MANIAC, at Princeton, and its twin, which was under construction at Los Alamos. Ulam enlisted C. J. Everett to follow a completely different approach, which was guided by physical intuition. With the aid of a cadre of female "data analysts", including Françoise Ulam, who performed computations on mechanical calculators, and whose efforts were supplemented and confirmed by Everett's use of his slide rule, they carried out laborious and extensive computations of many thermonuclear scenarios. Ulam and Fermi collaborated on further analysis of these scenarios. The results showed that, in workable configurations, a thermonuclear reaction would not ignite, and if ignited, it would not be self sustaining. In late 1950, these conclusions were confirmed by von Neumann's results.

In January 1951, Ulam had another idea: Put the fission bomb and fusion fuel together inside a massive casing. When the bomb detonated, the casing would contain the explosion long enough for mechanical shock to heat and compress the fusion fuel, and for fission neutrons to ignite nuclear fusion. On the recommendation of his wife, Ulam discussed this idea with Bradbury and J. Carson Mark before he told Teller about it. Almost immediately, Teller saw its merit. However, he noted that soft X-radiation from the fission bomb would compress the thermonuclear fuel more strongly than mechanical shock and suggested ways to enhance this effect. On March 9, 1951, Teller and Ulam submitted a joint report describing these innovations. A few weeks later, Teller suggested the placement at the center of the fusion fuel of a fissile rod or cylinder, the "spark plug", whose detonation would help to initiate and enhance the fusion reaction. The design based on these ideas, called staged radiation implosion, has become the standard way to build thermonuclear weapons. It is often designated as the "Teller–Ulam design".

In May 1951, during Operation Greenhouse, the explosions of devices "George" and "Item" proved that a fission explosion could trigger non-sustaining fusion reactions in a mixture of deuterium and tritium. Although these tests did not produce a thermonuclear detonation, they showed that a small fusion component could enhance significantly a fission explosion. These results, validated Teller's concept of boosted fission, which is incorporated in nearly all modern weapons.

Although Teller and Ulam submitted a joint report on their design and jointly applied for a patent on it, they soon became involved in a dispute over who deserved credit. After the war, Hans Bethe returned to Cornell University, but he was deeply involved as a consultant in the development of thermonuclear weapons. In 1954, he wrote an article on the history of the H-bomb, which presents his opinion that both men contributed very significantly to the breakthrough. This balanced view is shared by others who were involved, including Mark and Fermi, but Teller persistently attempted to downplay Ulam's role.

In September 1951, Teller went back to his prewar position at the University of Chicago. At about the same time, Ulam went on leave as a visiting professor at Harvard. In 1952, Teller joined the newly established Livermore branch of the University of California Radiation Laboratory, where he was director from 1958 to 1960.

With the basic fusion reactions confirmed, and with a feasible design in hand, there was nothing to prevent Los Alamos from testing a thermonuclear device. On November 1. 1952, the first thermonuclear explosion occurred when Ivy Mike was detonated on Enewetak Atoll, within the US Pacific Proving Grounds. This device, which used liquid deuterium as its fusion fuel, was immense and utterly unusable as a weapon. Nevertheless, its success validated the Teller-Ulam design, and stimulated intensive development of practical weapons.

On August 12, 1953, the Soviets detonated their RDS-6s device, which incorporated the Sloika design that had been developed by Andrei Sakharov and Vitaly Ginzburg. This code name, which refers to a Russian layered pastry, reflects the design, in which fusion fuel is layered within a fission bomb. This configuration is similar to that of Greenhouse Item and of the "Alarm Clock", which Teller proposed as an implementation of his original ideas. Although the Soviet design did not lead to full thermonuclear detonation, it embodied Ginzburg's crucial idea of November 1948, which was to use lithium deuteride as a fusion fuel. When fission neutrons break apart the lithium nuclei, some of the fragments are tritium nuclei, which participate in a very effective fusion reaction with the deuterium that is already present. The great advantage is that, in effect, both tritium and deuterium are encapsulated in a dense chemical compound. There is no need to use the cryogenic equipment for liquid deuterium that made the Ivy Mike device impractical as a weapon. On March 1, 1954, this advantage was realized in the Castle Bravo test, which was the largest American thermonuclear explosion, and which provided a prototype for nearly all subsequent thermonuclear weapons. On November 22, 1955, the Soviets set off RDS-37, which was their first thermonuclear bomb. It embodied the Teller-Ulam design, which Sakharov and other Soviet scientists had worked out independently.

Fermi-Pasta-Ulam problem

When Ulam returned to Los Alamos, his attention turned away from weapon design and toward the use of computers to investigate problems in physics and mathematics. With John Pasta, who helped Metropolis to bring MANIAC on line in March 1952, he explored these ideas in a report "Heuristic Studies in Problems of Mathematical Physics on High Speed Computing Machines", which was submitted on June 9, 1953. It treated several problems that cannot be addressed within the framework of traditional analytic methods: billowing of fluids, rotational motion in gravitating systems, magnetic lines of force, and hydrodynamic instabilities.

Soon, Pasta and Ulam became experienced with electronic computation on MANIAC, and by this time, Enrico Fermi had settled into a pattern of spending academic years at the University of Chicago and summers at Los Alamos. During the summer of 1954, Pasta and Ulam joined him to study a variation of the classic problem of a string of masses held together by springs that exert forces linearly proportional to their displacement from equilibrium. Fermi proposed to add to this force a nonlinear component, which could be chosen to be proportional to either the square or cube of the displacement, or to a more complicated "broken linear" function. This addition is the key element of the Fermi-Pasta-Ulam problem, which is often designated by the abbreviation FPU

With the aid of standard mathematics, the classical spring system can be described in terms of vibrational modes, which are analogous to the harmonics that occur on a stretched violin string. If the system starts in a particular mode, vibrations in other modes do not develop. With the nonlinear component, Fermi expected energy in one mode to transfer gradually to other modes, and eventually, to be distributed equally among all modes. This is roughly what began to happen shortly after the system was initialized with all its energy in the lowest mode, but much later, essentially all the energy periodically reappeared in the lowest mode. This amazing behavior is very different from the expected equipartition of energy. It remained mysterious until 1965, when Kruskal and Zabusky showed that, after appropriate mathematical transformations, the system can be described by the Korteweg-de Vries equation, which is the prototype of nonlinear partial differential equations that have soliton solutions. This means that FPU behavior can be understood in terms of solitons.

Nuclear propulsion

During the war, Ulam and Frederick Reines considered nuclear propulsion of aircraft and rockets. This is an attractive possibility, because the nuclear energy per unit mass of fuel is a million times greater than that available from chemicals. From 1955 to 1972, some of their ideas were pursued during Project Rover, which explored the use of nuclear reactors to power rockets. It was administered by Los Alamos, but many of its activities were carried out elsewhere. In particular, fuel rods were developed at the Rocky Flats Plant, and twenty rockets were tested at the Nuclear Reactor Development Station, which was located at Jackass Flats within the Nevada Test Site.

In 1955, Ulam and C. J. Everett proposed, in contrast to Rover's continuous heating of rocket exhaust, to harness small nuclear explosions for propulsion. Project Orion was a study of this idea. It began in 1958 and ended in 1965, after the Partial Nuclear Test Ban Treaty of 1963 banned nuclear weapons tests in the atmosphere and in space. Work on this project was spearheaded by physicist Freeman Dyson, who commented on the decision to end Orion in a famous article, "Death of a Project".

In 1957, Bradbury appointed Ulam and John H. Manley as research advisors to the laboratory director. These newly created positions were on the same administrative level as division leaders, and Ulam held his until he retired from Los Alamos. In this capacity, he was able to influence and guide programs in many divisions: theoretical, physics, chemistry, metalurgy, weapons, health, Rover, and others.

In addition to these activities, Ulam continued to publish technical reports and research papers. One of these introduced the Fermi–Ulam model, which is an extension of Fermi's theory of the acceleration of cosmic rays. Another, with Paul Stein was titled "Quadratic Transformations" It was an early investigation of chaos theory.

Return to academia

During his years at Los Alamos, Ulam was a visiting professor at: Harvard, 1951/52, MIT, 1956/57, the University of California, San Diego, 1963, and the University of Colorado, 1961/62 and 1965/67. In 1967, the last of these positions became permanent, when Ulam was appointed as professor and Chairman of the Department of Mathematics at Colorado. However, he kept his residence in Santa Fe, which made it convenient to spend summers as a consultant in Los Alamos.

In Colorado, Ulam's research interests turned toward biology. With his Los Alamos colleague Robert Schrandt he published a report, "Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutionary Processes", which applied some of his earlier ideas on branching processes to biological inheiritance. Another, report, with William Beyer, Temple F. Smith, and M. L. Stein, titled "Metrics in Biology", introduced some new ideas of quantification. In 1968, recognizing this emphasis, the University of Colorado School of Medicine appointed Ulam as Professor of Biomathematics, and he held this position until his death.

When he retired from Colorado in 1975, Ulam had begun to spend winter semesters at the University of Florida, where he was a graduate research professor. Except for sabbaticals at the University of California, Davis, 1982/83, and at Rockefeller University, 1980/84, this pattern of spending summers in Colorado and Los Alamos and winters in Florida continued until Ulam died in Santa Fe on May 13, 1984.

Mathematics

In pure mathematics, he worked in set theory (including measurable cardinals and abstract measures), topology, ergodic theory, and other fields.

Some results that bear his name are:

  • Borsuk-Ulam theorem
  • Mazur–Ulam theorem
  • Hyers–Ulam–Rassias stability
  • Lucky number (1955)
  • Ulam spiral (1963)
  • Ulam conjecture (in Number Theory)
  • Ulam's Conjecture (in Graph theory)
    also called the Reconstruction conjecture
  • Ulam's game
  • Ulam numbers

Impact and legacy

With his pivotal role in the development of thermonuclear weapons, Stanislaw Ulam changed the world. He felt very little guilt. According to Françoise Ulam: "Stan would reassure me that, barring accidents, the H-bomb rendered nuclear war impossible." Since the end of World War II, this prediction has held.

During his career, Ulam was awarded honorary degrees by the Universities of New Mexico, Wisconsin, and Pittsburgh.

The University of Florida maintains a journal named after Ulam, Ulam Quarterly Journal

In 1980, he appeared in the television documentary The Day After Trinity.

Quotations of Ulam

It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.
It seems to me that the impact and role of the electronic computer will significantly affect pure mathematics.

Ulam's answer to a question by Senator John O. Pastore at a congressional committee hearing, "Outer Space Propulsion by Nuclear Energy", on January 22, 1958:

Anyway, the future as a whole of mankind is to some extent involved inexorably now with going outside the globe.

Ulam was among the first to refer to the technological singularity, and possibly the originator of the metaphor itself. In May 1958, while referring to a conversation with John von Neumann:

One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.

Quotations about Ulam

Georgia O'Keeffe, on seeing Ulam at a party in Santa Fe, NM:
Who is that man?
Hans Bethe 1968:
After the H-bomb was made, reporters started to call Teller the father of the H-bomb. For the sake of history, I think it is more precise to say that Ulam is the father, because he provided the seed, and Teller is the mother, because he remained with the child. As for me, I guess I am the midwife.

Books

  • Stanisław Ulam, A Collection of Mathematical Problems, New York, Interscience Publishers, 1960.
  • Mark Kac and Stanisław Ulam, Mathematics and Logic: Retrospect and Prospects, New York, Praeger, 1968. Dover paperback reprint edition ca. 1990.
  • Stanisław Ulam, Sets, Numbers and Universes, Cambridge, Massachusetts, 1974.
  • Stanisław Ulam, Adventures of a Mathematician, New York, Charles Scribner's Sons, 1983 (autobiography).
  • Necia Grant Cooper, Roger Eckhardt, Nancy Shera, editors, From Cardinals to Chaos, Cambridge University Press (1989).
  • Stanisław Ulam, Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and his Los Alamos Collaborators. Berkeley: University of California Press, 1990.


External links







Источник статьи: http://en.wikipedia.org/wiki/Stanislaw_Ulam
В статье упоминаются люди:   Улам, Станислав

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