View source for Stanisław Ulam

{{Short description|Polish mathematician and physicist (1909–1984)}}
{{good article}}
{{Infobox scientist
| name              = Stanisław Ulam
| image             = Stanislaw Ulam.tif
| caption           = Ulam
| birth_name        = Stanisław Marcin Ulam
| image_size        = 
| alt               = A smiling man in a hat and heavy winter coat and scarf, carrying a portfolio tucked under his arm
| birth_date        = {{birth date|df=y|1909|4|13}}
| birth_place       = [[Lviv#Habsburg Empire|Lemberg]], [[Austria-Hungary]]<br />(now [[Lviv]], [[Ukraine]])
| death_date        = {{death date and age|df=y|1984|5|13|1909|4|3}}
| death_place       = [[Santa Fe, New Mexico]], U.S.
| residence         = 
| citizenship       = Poland, United States (naturalized in 1941)
| field             = [[Mathematics]]<br />[[Nuclear physics]]<br />[[Computer science]]
| work_institutions = [[Institute for Advanced Study]]<br />[[Harvard University]]<br />[[University of Wisconsin–Madison|University of Wisconsin]]<br />[[Los Alamos National Laboratory]]<br />[[University of Colorado at Boulder|University of Colorado]]<br />[[University of Florida]]
| education         = [[Lviv Polytechnic#Second Polish Republic|Lwów Polytechnic Institute]]<br>[[Second Polish Republic]]
| doctoral_advisor  = [[Kazimierz Kuratowski]]<br>[[Włodzimierz Stożek]]
| doctoral_students = [[Paul Kelly (mathematician)|Paul Kelly]]
| known_for         = [[Teller–Ulam design]]<br />[[Monte Carlo method]]<br />[[Fermi–Pasta–Ulam–Tsingou problem]]<br />[[Nuclear pulse propulsion]]
| influences        = 
| awards            = 
}}

'''Stanisław Marcin Ulam''' ({{small|Polish:}} {{IPAc-pl|s|t|a|'ɲ|i|s|ł|a|f|-|'m|a|r|ć|i|n|-|'u|l|a|m}}; 13 April 1909&nbsp;– 13 May 1984) was a Polish and American mathematician, nuclear physicist and computer scientist. He participated in the [[Manhattan Project]], originated the [[History of the Teller–Ulam design|Teller–Ulam design]] of [[thermonuclear weapon]]s, discovered the concept of the [[cellular automaton]], invented the [[Monte Carlo method|Monte Carlo method of computation]], and championed [[nuclear pulse propulsion]]. In pure and [[applied mathematics]], he proved a number of theorems and proposed several conjectures.

Born into a wealthy [[Polish Jewish]] family in [[Lemberg]], [[Austria-Hungary]], Ulam studied mathematics at the [[Lviv Polytechnic#Second Polish Republic|Lwów Polytechnic Institute]], where he earned his [[PhD]] in 1933 under the supervision of [[Kazimierz Kuratowski]] and [[Włodzimierz Stożek]]. In 1935, [[John von Neumann]], whom Ulam had met in Warsaw, invited him to come to the [[Institute for Advanced Study]] in [[Princeton, New Jersey]], for a few months. From 1936 to 1939, he spent summers in Poland and academic years at [[Harvard University]] in [[Cambridge, Massachusetts]], where he worked to establish important results regarding [[ergodic theory]]. On 20 August 1939, he sailed for the United States for the last time with his 17-year-old brother [[Adam Ulam]]. He became an assistant professor at the [[University of Wisconsin–Madison]] in 1940, and a United States citizen in 1941.

In October 1943, he received an invitation from [[Hans Bethe]] to join the [[Manhattan Project]] at the secret [[Los Alamos Laboratory]] in New Mexico. There, he worked on the [[Fluid dynamics|hydrodynamic]] calculations to predict the behavior of the [[explosive lens]]es that were needed by an [[Nuclear weapon design#Implosion-type weapon|implosion-type weapon]]. He was assigned to [[Edward Teller]]'s group, where he worked on [[Teller's "Super" bomb]] for Teller and [[Enrico Fermi]]. After the war he left to become an associate professor at the [[University of Southern California]], but returned to Los Alamos in 1946 to work on [[Nuclear weapon design|thermonuclear weapons]]. With the aid of a cadre of female "[[Human computers#Wartime computing and the invention of electronic computing|computers]]" he found that Teller's "Super" design was unworkable. In January 1951, Ulam and Teller came up with the [[Teller–Ulam design]], which became the basis for all thermonuclear weapons.

Ulam considered the problem of [[nuclear propulsion]] of rockets, which was pursued by [[Project Rover]], and proposed, as an alternative to Rover's [[nuclear thermal rocket]], to harness small nuclear explosions for propulsion, which became [[Project Orion (nuclear propulsion)|Project Orion]]. With Fermi, [[John Pasta]], and [[Mary Tsingou]], Ulam studied the [[Fermi–Pasta–Ulam–Tsingou problem]], which became the inspiration for the field of nonlinear science. He is probably best known for realizing that electronic computers made it practical to apply statistical methods to [[function (mathematics)|function]]s without known [[equation solving|solutions]], and as computers have developed, the [[Monte Carlo method]] has become a common and standard approach to many problems.

==Poland==
Stanisław Marcin Ulam was born in [[Lviv#Habsburg Empire|Lemberg]], [[Galicia (Eastern Europe)|Galicia]], on 13 April 1909.<ref name="CHAR 78" /><ref name='NYTIMES-OBIT' /><ref>{{Cite web|title=Stanislaw Ulam {{!}} Biography, Facts, & Spiral|url=https://www.britannica.com/biography/Stanislaw-Ulam|access-date=2021-04-11|website=Encyclopedia Britannica|language=en}}</ref> At this time, Galicia was in the [[Kingdom of Galicia and Lodomeria]] of the [[Austria-Hungary|Austro-Hungarian Empire]], which was known to Poles as the [[Austrian partition]]. In 1918, it became part of the newly restored Poland, the [[Second Polish Republic]], and the city took its Polish name again, [[Lviv#Interwar period|Lwów]].<ref name="ULAM 9-15"/>

The Ulams were a wealthy [[Polish Jewish]] family of bankers, industrialists, and other professionals. Ulam's immediate family was "well-to-do but hardly rich".<ref name='AULAM'/> His father, Józef Ulam, was born in Lwów and was a lawyer,<ref name="ULAM 9-15"/> and his mother, Anna (née Auerbach), was born in [[Stryi|Stryj]].<ref name='FAMILY'/> His uncle, Michał Ulam, was an architect, building contractor, and lumber industrialist.<ref name='MIKE'/> From 1916 until 1918, Józef's family lived temporarily in [[Vienna]].<ref name='VITA'/> After they returned, Lwów became the epicenter of the [[Polish–Ukrainian War]], during which the city experienced [[Battle of Lemberg (1918)|a Ukrainian siege]].<ref name="ULAM 9-15"/>

[[File:Lwow - Kawiarnia Szkocka.jpg|thumb|300px|alt=Picture of the building that used to house the Scottish Café|The [[Scottish Café]]'s building in [[Lviv]], Ukraine now houses the Szkocka Restaurant & Bar (named for the original Scottish Café).]]
In 1919, Ulam entered Lwów Gymnasium Nr. VII, from which he graduated in 1927.<ref name='AJMAA'/> He then studied mathematics at the Lwów Polytechnic Institute. Under the supervision of [[Kazimierz Kuratowski]], he received his [[Master of Arts]] degree in 1932, and became a [[Doctor of Science#Poland|Doctor of Science]] in 1933.<ref name='VITA'/><ref name=zwoje/> At the age of 20, in 1929, he published his first paper ''Concerning Functions of Sets'' in the journal ''[[Fundamenta Mathematicae]]''.<ref name=zwoje>{{cite journal|author=Andrzej M. Kobos|title=Mędrzec większy niż życie|trans-title=A Sage Greater Than Life|language=pl|journal=Zwoje|volume=3|number=16|year=1999|access-date=10 May 2013|url=http://www.zwoje-scrolls.com/zwoje16/text03.htm|archive-date=6 March 2009|archive-url=https://web.archive.org/web/20090306182319/http://www.zwoje-scrolls.com/zwoje16/text03.htm|url-status=dead}}</ref> From 1931 until 1935, he traveled to and studied in [[Wilno]] (Vilnius), Vienna, [[Zürich]], [[Paris]], and [[University of Cambridge|Cambridge, England]], where he met [[G. H. Hardy]] and [[Subrahmanyan Chandrasekhar]].<ref name="ULAM 56-60"/>

Along with [[Stanisław 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 Lviv#Jan Kazimierz University (1919–39)|Jan Kazimierz University]]. Mathematicians of this "school" met for long hours at the [[Scottish Café]], where the problems they discussed were collected in the [[Scottish Book]], a thick notebook provided by Banach's wife. Ulam was a major contributor to the 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.<ref name='BOOK'/> In 1981, Ulam's friend R. Daniel Mauldin published an expanded and annotated version.<ref name='MAULDIN'/>

==Move to the United States==
In 1935, [[John von Neumann]], whom Ulam had met in Warsaw, invited him to come to the [[Institute for Advanced Study]] in [[Princeton, New Jersey]], for a few months. In December of that year, Ulam sailed to the US. At Princeton, he went to lectures and seminars, where he heard [[Oswald Veblen]], [[James Waddell Alexander II|James Alexander]], and [[Albert Einstein]]. During a tea party at von Neumann's house, he encountered [[G. D. Birkhoff]], who suggested that he apply for a position with the [[Harvard Society of Fellows]].<ref name='VITA'/> Following up on Birkhoff's suggestion, Ulam spent summers in Poland and academic years at [[Harvard University]] in [[Cambridge, Massachusetts]] from 1936 to 1939, where he worked with [[John C. Oxtoby]] to establish results regarding [[ergodic theory]]. These appeared in ''[[Annals of Mathematics]]'' in 1941.<ref name='AJMAA'/><ref name='OXTOBY'/>

On 20 August 1939, in [[Gdynia]], Józef Ulam, along with his brother Szymon, put his two sons, Stanislaw and 17-year-old [[Adam Ulam|Adam]], on a ship headed for the US.<ref name='VITA'/> Eleven days later, the [[Invasion of Poland#German invasion|Germans invaded Poland]]. Within two months, the Germans completed their [[History of Poland (1939–1945)|occupation]] of western Poland, and the Soviets [[Invasion of Poland#Soviet invasion|invaded]] and occupied eastern Poland. Within two years, Józef Ulam and the rest of his family, including Stanislaw's sister Stefania Ulam, were victims of the [[Holocaust]], [[Hugo Steinhaus]] was in hiding, [[Kazimierz Kuratowski]] was lecturing at the [[Education in Poland during World War II|underground university]] in Warsaw, [[Włodzimierz Stożek]] and his two sons had been killed in the [[massacre of Lwów professors]], and the last problem had been recorded in the [[Scottish Book]]. [[Stefan Banach]] survived the Nazi occupation by feeding [[lice]] at [[Rudolf Weigl|Rudolf Weigl's typhus research institute]]. In 1963, [[Adam Ulam]], who had become an eminent [[Kremlinology|kremlinologist]] at Harvard,<ref name='ADAM'/> received a letter from George Volsky,<ref name='VOLSKY'/> who hid in Józef Ulam's house after deserting from the Polish army. This reminiscence gave a chilling account of Lwów's chaotic scenes in late 1939.<ref name='MOLLY'/> In later life Ulam described himself as "an agnostic. Sometimes I muse deeply on the forces that are for me invisible. When I am almost close to the idea of God, I feel immediately estranged by the horrors of this world, which he seems to tolerate".<ref name='OLGIERD'/>
In 1940, after being recommended by Birkhoff, Ulam became an assistant professor at the [[University of Wisconsin–Madison]]. Here, he became a [[United States citizen]] in 1941.<ref name='VITA'/> That year, he married [[Françoise Aron Ulam|Françoise Aron]].<ref name='AJMAA'/> She had been a French exchange student at [[Mount Holyoke College]], whom he met in Cambridge. They had one daughter, Claire. In [[Madison, Wisconsin|Madison]], Ulam met his friend and colleague C. J. Everett, with whom he collaborated on a number of papers.<ref name="ULAM 125-130"/>

==Manhattan Project==
[[File:Ulam-stanislaw m.jpg|thumb|Ulam's ID badge photo from [[Los Alamos National Laboratory]] |alt=A mug shot style ID photo, with the serial H 0]]
In early 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, New Mexico]].<ref name='VITA'/> The letter was signed by [[Hans Bethe]], who had been appointed as leader of the theoretical division of [[Los Alamos National Laboratory]] by [[Robert Oppenheimer]], its scientific director.<ref name="ULAM 143-147"/> Knowing nothing of the area, he borrowed a New Mexico guide book. On the checkout card, he found the names of his Wisconsin colleagues, [[Joan Hinton]], [[David H. Frisch|David Frisch]], and Joseph McKibben, all of whom had mysteriously disappeared.<ref name='VITA'/> This was Ulam's introduction to the [[Manhattan Project]], which was the US's wartime effort to create the atomic bomb.<ref name='LANLBIO'/>

===Hydrodynamical calculations of implosion===
A few weeks after Ulam reached [[Los Alamos, New Mexico|Los Alamos]] in February 1944, the project experienced a crisis. In April, [[Emilio G. Segrè|Emilio Segrè]] discovered that [[plutonium]] made in [[Nuclear reactors#Early reactors|reactors]] would not work in a [[gun-type fission weapon|gun-type]] plutonium weapon like the "[[Thin Man nuclear bomb|Thin Man]]", which was being developed in parallel with a uranium weapon, the "[[Little Boy]]" that was dropped on [[Atomic bombings of Hiroshima and Nagasaki#Hiroshima|Hiroshima]]. This problem threatened to waste an enormous investment in new reactors at the [[Hanford site]] and to make slow uranium [[isotope separation]] 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 [[Nuclear weapon design#Implosion-type weapon|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.<ref name="HODDESON 130-137"/>

The basic concept of [[Implosion (mechanical process)|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]], releasing a large amount of energy. Cylindrical implosive configurations had been studied by [[Seth Neddermeyer]], but von Neumann, who had experience with [[shaped charge]]s used in [[High-explosive anti-tank|armor-piercing ammunition]], was a [[John von Neumann#Manhattan Project|vocal advocate of spherical implosion]] driven by [[explosive lens]]es. He realized that the symmetry and speed with which implosion compressed the plutonium were critical issues,<ref name="HODDESON 130-137"/> 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 [[Fluid dynamics|hydrodynamical]] calculations were needed to predict and minimize asymmetries that would spoil a nuclear detonation. Of these calculations, Ulam said:{{blockquote|The hydrodynamical problem was simply stated, but very difficult to calculate&nbsp;– not only in detail, but even in order of magnitude. In this discussion, I stressed pure pragmatism and the necessity to get a heuristic survey of the problem by simple-minded brute force, rather than by massive numerical work.<ref name='VITA'/>}}

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,<ref name='COMPEARLY'/> continued through the cold war, and still exists.<ref name='COMPLATE'/> [[Otto Frisch]] remembered Ulam as "a brilliant Polish topologist with a charming French wife. At once he told me that he was a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points!"<ref name='FRISCH'/>

===Statistics of branching and multiplicative processes===
Even the inherent statistical fluctuations of [[neutron]] multiplication within a [[chain reaction]] have implications with regard to implosion speed and symmetry. In November 1944, [[David Hawkins (philosopher)|David Hawkins]]<ref name='HAWKINS'/> and Ulam addressed this problem in a report entitled "Theory of Multiplicative Processes".<ref name='LA171'/> This report, which invokes [[probability-generating function]]s, is also an early entry in the extensive literature on statistics of [[Branching process|branching]] and multiplicative processes. In 1948, its scope was extended by Ulam and Everett.<ref name='MULT123'/>

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 [[Manhattan Project#Hanford reactors|first Hanford reactor]], which had been [[Nuclear poison|poisoned]] by a [[Xenon-135|xenon isotope]].<ref name="HEWLETT"/> Soon after Fermi's arrival, [[Teller's "Super" bomb]] group, of which Ulam was a part, was transferred to a new division headed by Fermi.<ref name="ULAM 152-153"/> Fermi and Ulam formed a relationship that became very fruitful after the war.<ref name="ULAM 162-167"/>

==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, California|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]],<ref name='ERDOS'/> who remarked: "Stan, you are just like before."<ref name='VITA'/> 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. Another friend, [[Gian-Carlo Rota]], asserted in a 1987 article that the attack changed Ulam's personality: afterwards, he turned from rigorous pure mathematics to more speculative conjectures concerning the application of mathematics to physics and [[biology]]; Rota also cites Ulam's former collaborator Paul Stein as noting that Ulam was sloppier in his clothing afterwards, and John Oxtoby as noting that Ulam before the encephalitis could work for hours on end doing calculations, while when Rota worked with him, was reluctant to solve even a quadratic equation.<ref name='ROTA'/> This assertion was not accepted by [[Françoise Aron Ulam]].<ref name='FULAM'/>

By late April 1946, Ulam had recovered enough to attend a secret conference at Los Alamos to discuss [[Nuclear weapon design|thermonuclear weapons]]. Those in attendance included 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" weapon 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, Ulam received an offer of a position at Los Alamos from Metropolis and [[Robert D. Richtmyer]], the new head of its theoretical division, at a higher salary, and the Ulams returned to Los Alamos.<ref name="ULAM 184-187"/>

===Monte Carlo method===
[[File:STAN ULAM HOLDING THE FERMIAC.jpg|thumb|Stan Ulam holding the [[FERMIAC]]|alt=A smiling balding man in a suit and tie holds a strange device that looks like a frame]]
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]] at the Aberdeen Proving Ground in Maryland. Shortly after returning to Los Alamos, Ulam participated in a review of results from these calculations.<ref name='METROPOLIS'/> Earlier, while playing [[Patience (game)|solitaire]] during his recovery from surgery, Ulam had thought about playing hundreds of games to estimate statistically the probability of a successful outcome.<ref name='ECKHARDT'/> 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. In March 1947 he proposed a statistical approach to the problem of neutron diffusion in fissionable material.<ref name='LAMS551'/> Because Ulam had often mentioned his uncle, Michał Ulam, "who just had to go to Monte Carlo" to gamble, Metropolis dubbed the statistical approach "The [[Monte Carlo method]]".<ref name='METROPOLIS'/> Metropolis and Ulam published the first unclassified paper on the Monte Carlo method in 1949.<ref name='MONTEPAPER'/>

Fermi, learning of Ulam's breakthrough, devised an [[analog computer]] known as the [[Monte Carlo trolley]], later dubbed the [[FERMIAC]]. The device performed a mechanical simulation of random diffusion of neutrons. As computers improved in speed and programmability, these methods became more useful. In particular, many Monte Carlo calculations carried out on modern [[massively parallel]] [[supercomputer]]s are [[embarrassingly parallel]] applications, whose results can be very accurate.<ref name='COMPLATE'/>

===Teller–Ulam design===
On 29 August 1949, the [[Soviet Union]] tested its first fission bomb, the [[RDS-1]]. Created under the supervision of [[Lavrentiy Beria]], who sought to duplicate the US effort, this weapon was nearly identical to [[Fat Man]], for its design was based on information provided by spies [[Klaus Fuchs]], [[Theodore Hall]], and [[David Greenglass]]. In response, on 31 January 1950, [[President of the United States|President]] [[Harry S. Truman]] announced a crash program to develop a fusion bomb.<ref name='DUNCAN'/>

To advocate an aggressive development program, [[Ernest Lawrence]] and [[Luis Walter Alvarez|Luis Alvarez]] came to Los Alamos, where they conferred with [[Norris Bradbury]], the laboratory director, and with [[George Gamow]], [[Edward Teller]], and Ulam. Soon, these three became members of a short-lived committee appointed by Bradbury to study the problem, with Teller as chairman.<ref name='VITA'/> At this time, research on the use of a fission weapon to create a [[Nuclear fusion|fusion reaction]] had been ongoing since 1942, but the design was still essentially the one originally proposed by Teller. His concept was to put [[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 [[Nuclear fusion|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.<ref name="RHODES"/>

[[File:IvyMike2.jpg|thumb|300px|left|[[Ivy Mike]], the first full test of the Teller–Ulam design (a [[Nuclear weapon design#Two-stage thermonuclear weapons|staged]] fusion bomb), with a [[TNT equivalent|yield]] of 10.4 megatons on 1 November 1952|alt=A mushroom cloud lights up the dawn sky]]
Because the results of calculations based on Teller's concept were discouraging, many scientists believed it [[History of the Teller–Ulam design#Teller's "Super"|could not lead to a successful weapon]], while others had moral and economic grounds for not proceeding. Consequently, several senior people of the Manhattan Project opposed development, including Bethe and Oppenheimer.<ref name= "HEWLETT AND DUNCAN"/> To clarify the situation, Ulam and von Neumann resolved to do new calculations to determine whether Teller's approach was feasible. To carry out these studies, von Neumann decided to use electronic computers: ENIAC at Aberdeen, a new computer, [[MANIAC I|MANIAC]], at Princeton, and its twin, which was under construction at Los Alamos. Ulam enlisted Everett to follow a completely different approach, one guided by physical intuition. [[Françoise Aron Ulam|Françoise Ulam]] was one of<ref name= "ULAM 215"/> a cadre of women "[[Human computers#Wartime computing and the invention of electronic computing|computers]]" who carried out laborious and extensive computations of thermonuclear scenarios on [[mechanical calculator]]s, supplemented and confirmed by Everett's [[slide rule]]. 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. Ulam had used his expertise in [[combinatorics]] to analyze the chain reaction in deuterium, which was much more complicated than the ones in uranium and plutonium, and he concluded that no self-sustaining chain reaction would take place at the (low) densities that Teller was considering.<ref name='GALISON'/> In late 1950, these conclusions were confirmed by von Neumann's results.<ref name='FULAM'/><ref name="RHODES 422-424"/>

In January 1951, Ulam had another idea: to channel the mechanical shock of a nuclear explosion so as to compress the fusion fuel. On the recommendation of his wife,<ref name='FULAM'/> Ulam discussed this idea with Bradbury and Mark before he told Teller about it.<ref name='MARKBIO'/> Almost immediately, Teller saw its merit, but noted that soft [[X-rays]] from the fission bomb would compress the thermonuclear fuel more strongly than mechanical shock and suggested ways to enhance this effect. On 9 March 1951, Teller and Ulam submitted a joint report describing these innovations.<ref name='LAMS1225'/> A few weeks later, Teller suggested placing a [[fissile]] rod or cylinder at the center of the fusion fuel. The detonation of this "spark plug"<ref name='LAMS1230'/> 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 described as the "[[Teller–Ulam design]]".<ref name="RHODES 455-464"/>

[[File:Ivy Mike Sausage device.jpg|thumb|The ''Sausage'' device of [[Ivy Mike|Mike]] nuclear test (yield 10.4 Mt) on [[Enewetak Atoll]]. The test was part of the [[Operation Ivy]]. The Sausage was the first true H-Bomb ever tested, meaning the first [[Thermonuclear weapon|thermonuclear]] device built upon the [[Teller–Ulam design|Teller-Ulam]] principles of staged radiation implosion.|alt=Tiny men and a large silver cylindrical object connected to a lot of scaffolding and tubes]]
In September 1951, after a series of differences with Bradbury and other scientists, Teller resigned from Los Alamos, and returned to the University of Chicago.<ref name="HEWLETT AND DUNCAN 554-556"/> At about the same time, Ulam went on leave as a visiting professor at Harvard for a semester.<ref name="ULAM 220-224"/> Although Teller and Ulam submitted a joint report on their design<ref name='LAMS1225'/> and jointly applied for a patent on it,<ref name='LANLBIO'/> they soon became involved in a dispute over who deserved credit.<ref name='MARKBIO'/> After the war, Bethe returned to [[Cornell University]], but he was deeply involved in the development of thermonuclear weapons as a consultant. In 1954, he wrote an article on the history of the H-bomb,<ref name='BETHE'/> 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.<ref name = 'UCHII'/> "After the H-bomb was made," Bethe recalled, "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."<ref name=SCHWEBER/>

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 1 November 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.<ref name="ULAM 220-224"/>

===Fermi–Pasta–Ulam–Tsingou problem===
{{main|Fermi–Pasta–Ulam–Tsingou 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 9 June 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.<ref name='LA1557'/>

Soon, Pasta and Ulam became experienced with electronic computation on MANIAC, and by this time, Enrico Fermi had settled into a routine of spending academic years at the University of Chicago and summers at Los Alamos. During these summer visits, Pasta, Ulam, and [[Mary Tsingou]], a programmer in the MANIAC group, 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.<ref>{{cite journal|last1=Dauxois|first1=Thierry|title=Fermi, Pasta, Ulam, and a mysterious lady|journal=Physics Today|date=2008|volume=6|issue=1|pages=55–57|url=http://perso.ens-lyon.fr/thierry.dauxois/PAPERS/pt61_55.2008.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://perso.ens-lyon.fr/thierry.dauxois/PAPERS/pt61_55.2008.pdf |archive-date=2022-10-09 |url-status=live|access-date=7 May 2017|doi=10.1063/1.2835154|arxiv=0801.1590|bibcode=2008PhT....61a..55D|s2cid=118607235}}</ref> 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–Tsingou problem]], which is often designated by the abbreviation FPUT.<ref name='LA1940'/><ref name='FPU'/>

A 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.<ref name='FPU'/> This behavior is very different from the expected [[Equipartition theorem|equipartition of energy]]. It remained mysterious until 1965, when [[Martin David Kruskal|Kruskal]] and [[Norman Zabusky|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 FPUT behavior can be understood in terms of solitons.<ref name="CHAOS"/>

===Nuclear propulsion===
[[File:NASA-project-orion-artist.jpg|thumb|300px|left|An artist's conception of the NASA reference design for the Project Orion spacecraft powered by nuclear propulsion|alt=A painting of a spacecraft passing a Jupiter-like planet]]
Starting in 1955, Ulam and [[Frederick Reines]] considered [[nuclear propulsion]] of aircraft and rockets.<ref name='LAMS2186'/> 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, their ideas were pursued during [[Project Rover]], which explored the use of nuclear reactors to power rockets.<ref name="ULAM 249-254"/> In response to a question by Senator [[John O. Pastore]] at a congressional committee hearing on "Outer Space Propulsion by Nuclear Energy", on January 22, 1958, Ulam replied that "the future as a whole of mankind is to some extent involved inexorably now with going outside the globe."<ref name='HEARING'/>

Ulam and C. J. Everett also proposed, in contrast to [[Nuclear thermal rocket|Rover's continuous heating of rocket exhaust]], to harness small nuclear explosions for propulsion.<ref name='LAMS1955'/> [[Project Orion (nuclear 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.<ref name='ORION'/> Work on this project was spearheaded by physicist [[Freeman Dyson]], who commented on the decision to end Orion in his article, "Death of a Project".<ref name='DEATH'/>

Bradbury appointed Ulam and [[John H. Manley]] as research advisors to the laboratory director in 1957. 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, metallurgy, weapons, health, Rover, and others.<ref name='ULAM 249-254'/>

In addition to these activities, Ulam continued to publish technical reports and research papers. One of these introduced the [[Fermi–Ulam model]], an extension of Fermi's theory of [[Fermi acceleration|the acceleration of cosmic rays]].<ref name='FUM'/> Another, with Paul Stein and [[Mary Tsingou]], titled "Quadratic Transformations", was an early investigation of [[chaos theory]] and is considered the first published use of the phrase "[[chaotic behavior]]".<ref name='ABRAHAM' /><ref name='LA-2305'/>
{{Clear}}

==Return to academia==
[[File:Ulam 1.png|thumb|300px|When the positive integers are arrayed along the [[Ulam spiral]], prime numbers, represented by dots, tend to collect along diagonal lines.|alt=A lot of dots, but forming diagonal lines]]
During his years at Los Alamos, Ulam was a visiting professor at Harvard from 1951 to 1952, [[MIT]] from 1956 to 1957, the [[University of California, San Diego]], in 1963, and the [[University of Colorado at Boulder]] from 1961 to 1962 and 1965 to 1967. In 1967, the last of these positions became permanent, when Ulam was appointed Professor and Chairman of the Department of Mathematics at the University of Colorado. He kept a residence in Santa Fe, which made it convenient to spend summers at Los Alamos as a consultant.<ref name="AIP"/> He was an elected member of the [[American Academy of Arts and Sciences]], the United States [[National Academy of Sciences]], and the [[American Philosophical Society]].<ref>{{Cite web |title=Stanislaw Marcin Ulam |url=https://www.amacad.org/person/stanislaw-marcin-ulam |access-date=2022-09-21 |website=American Academy of Arts & Sciences |language=en}}</ref><ref>{{Cite web |title=S. M. Ulam |url=http://www.nasonline.org/member-directory/deceased-members/49612.html |access-date=2022-09-21 |website=www.nasonline.org}}</ref><ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Stanislaw+Ulam&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2022-09-21 |website=search.amphilsoc.org}}</ref>

In Colorado, where he rejoined his friends Gamow, Richtmyer, and Hawkins, Ulam's research interests turned toward [[biology]]. 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. 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 his earlier ideas on branching processes to evolution.<ref name='LA-4246'/> Another, report, with William Beyer, [[Temple F. Smith]], and M. L. Stein, titled "Metrics in Biology", introduced new ideas about numerical taxonomy and evolutionary distances.<ref name='LA-4973'/>

When he retired from Colorado in 1975, Ulam began to spend winter semesters at the [[University of Florida]], where he was a graduate research professor. In 1976, he was awarded the Commander's Cross with the Star of the [[Order of Polonia Restituta]] by the [[Polish government-in-exile]] in [[London]].<ref>{{cite web|url= http://eprints.hist.pl/336/1/1976-12-31_nr6.pdf|title= Komunikat o nadaniu Orderu Odrodzenia Polski|newspaper=Dziennik Ustaw Rzeczypospolitej Polskiej (s. 23, nr 6 z)|date=December 31, 1976|access-date= July 25, 2023|archive-date= April 24, 2018|archive-url= https://web.archive.org/web/20180424205236/http://eprints.hist.pl/336/1/1976-12-31_nr6.pdf|url-status= dead}}</ref> Except for [[sabbatical]]s at the [[University of California, Davis]] from 1982 to 1983, and at [[Rockefeller University]] from 1980 to 1984,<ref name="AIP"/> this pattern of spending summers in Colorado and Los Alamos and winters in Florida continued until Ulam died of an apparent heart attack in Santa Fe on 13 May 1984.<ref name='NYTIMES-OBIT'/>
[[Paul Erdős]] noted that "he died suddenly of heart failure, without fear or pain, while he could still prove and conjecture."<ref name="ERDOS"/> In 1987, [[Françoise Aron Ulam|Françoise Ulam]] deposited his papers with the [[American Philosophical Society]] Library in [[Philadelphia]].<ref name="PAPERS"/> She continued to live in Santa Fe until she died in 2011, at the age of 93. Both Françoise and her husband were buried with her family in [[Montparnasse Cemetery]] in Paris.<ref name='FULAMOBIT'/><ref name=RB/>

==Coining of the term "singularity"==
Ulam reported in 1958 an earlier discussion with [[John von Neumann]] "centered on the [[Accelerating change|accelerating progress]] of technology and changes in human life, which gives the appearance of approaching some essential [[Technological singularity|singularity]] in the history of the race beyond which human affairs, as we know them, could not continue".<ref>{{cite journal|url=https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10189-5/S0002-9904-1958-10189-5.pdf|last=Ulam|first=Stanislaw|title=Tribute to John von Neumann|journal=[[Bulletin of the American Mathematical Society]]|volume=64 |issue=3, part 2|date=May 1958|page=5|access-date=7 November 2018|archive-date=15 February 2021 |archive-url=https://web.archive.org/web/20210215095053/https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10189-5/S0002-9904-1958-10189-5.pdf|url-status=live}}</ref>

==Impact and legacy==
Ulam participated in the creation of a [[hydrogen bomb]] as part of the [[Los Alamos Laboratory]] nuclear project. From the publication of his first paper as a student in 1929 until his death, Ulam was constantly writing on mathematics. The list of Ulam's publications includes more than 150 papers.<ref name='AJMAA'/> Topics represented by a significant number of papers are: [[set theory]] (including [[measurable cardinal]]s and abstract [[measure (mathematics)|measures]]), [[topology]], [[functional analysis]], [[Transformation (function)|transformation theory]], [[ergodic theory]], [[group theory]], [[Variety (universal algebra)|projective algebra]], [[number theory]], [[combinatorics]], and [[graph theory]].<ref name='UPUBS'/>

Notable results of this work are:
{{col-begin}}
{{col-2}}
*[[Borsuk–Ulam theorem]]
*[[Mazur–Ulam theorem]]
*[[Kuratowski–Ulam theorem]]
*[[Hyers–Ulam–Rassias stability]]
*[[Lucky number]]
*[[Ulam spiral]]{{col-2}}
*[[Reconstruction conjecture|Ulam conjecture]] (in Graph theory)
*[[Ulam's packing conjecture]]
*[[Ulam's game]]
*[[Ulam matrix]]
*[[Ulam numbers]]
{{col-end}}

Ulam played pivotal role in the development of thermonuclear weapons. According to Françoise Ulam: "Stan would reassure me that, barring accidents, the H-bomb rendered nuclear war impossible."<ref name=FULAM/> In 1980, Ulam and his wife appeared in the television documentary ''[[The Day After Trinity]]''.<ref>{{IMDb title|qid=Q3520523|id=tt0080594|title=The Day After Trinity}}</ref>

[[File:LuckySieve.gif|thumb|An animation demonstrating the lucky number sieve. The numbers in red are lucky numbers.|alt=A square containing the numbers 1 to 120. Numbers are initially grey but go purple as they are eliminated; the lucky numbers then remain, and are highlighted in red.]]
The [[Monte Carlo method]] has become a ubiquitous and standard approach to computation, and the method has been applied to a vast number of scientific problems.<ref name=MONTEALAMOS/> In addition to problems in physics and mathematics, the method has been applied to [[Monte Carlo methods in finance|finance]], social science,<ref name='MONTESOC'/> [[environmental impact assessment|environmental risk assessment]],<ref name='MONTERISK'/> linguistics,<ref name='MONTELING'/> radiation therapy,<ref name='MONTEONC'/> and sports.<ref name='MONTEBIG10'/>

The [[Fermi–Pasta–Ulam–Tsingou problem]] is credited not only as "the birth of experimental mathematics",<ref name='FPU'/> but also as inspiration for the vast field of Nonlinear Science. In his [[Lilienfeld Prize]] lecture, [[David K. Campbell]] noted this relationship and described how FPUT gave rise to ideas in [[Chaos theory|chaos]], [[solitons]], and [[Dynamical systems theory|dynamical systems]].<ref name='CAMPB'/> In 1980, [[Donald Kerr]], laboratory director at Los Alamos, with the strong support of Ulam and [[Mark Kac]],<ref name = 'CNLSSCOTT'/> founded the Center for Nonlinear Studies (CNLS).<ref name='CNLSHIST'/> In 1985, CNLS initiated the ''Stanislaw M. Ulam Distinguished Scholar'' program, which provides an annual award that enables a noted scientist to spend a year carrying out research at Los Alamos.<ref name='CNLSULAM'/>

The fiftieth anniversary of the original FPUT paper was the subject of the March 2005 issue of the journal Chaos,<ref name='FPUCHAOS'/> and the topic of the 25th Annual International Conference of CNLS.<ref name='CNLSFPU'/> The [[University of Southern Mississippi]] and the University of Florida supported the ''Ulam Quarterly'',<ref name='UQUART'/> which was active from 1992 to 1996, and which was one of the first online mathematical journals.<ref name='UQUART2'/> Florida's Department of Mathematics has sponsored, since 1998, the annual ''Ulam Colloquium Lecture'',<ref name='UCOLLOQ'/> and in March 2009, the ''Ulam Centennial Conference''.<ref name='UCONF'/>

Ulam's work on non-[[Euclidean distance|Euclidean distance metrics]] in the context of molecular biology made a significant contribution to [[sequence analysis]]<ref name='GOAD'/> and his contributions in theoretical biology are considered watersheds in the development of [[Cellular automata#History|cellular automata]] theory, [[population biology]], [[pattern recognition]], and biometrics generally (David Sankoff, however, challenged conclusions of Walter by writing that Ulam had only modest influence on early development of sequence alignment methods.<ref name='SANKOFF' />). Colleagues noted that some of his greatest contributions were in clearly identifying problems to be solved and general techniques for solving them.<ref name='BIOLOGY'/>

In 1987, Los Alamos issued a special issue of its ''Science'' publication, which summarized his accomplishments,<ref name='UTRIB'/> and which appeared, in 1989, as the book ''From Cardinals to Chaos''. Similarly, in 1990, the University of California Press issued a compilation of mathematical reports by Ulam and his Los Alamos collaborators: ''Analogies Between Analogies''.<ref name='ANALOG'/> During his career, Ulam was awarded honorary degrees by the Universities of [[University of New Mexico|New Mexico]], [[University of Wisconsin|Wisconsin]], and [[University of Pittsburgh|Pittsburgh]].<ref name='VITA'/>

In 2021, German film director Thorsten Klein made a [[Adventures of a Mathematician|film adaptation]] of Ulam's autobiography, ''Adventures of a Mathematician''.<ref>{{cite magazine |title=How 'Adventures of a Mathematician' Writer Director Thor Klein Found Story in the Details |magazine=Script |url=https://scriptmag.com/interviews-features/how-adventures-of-a-mathematician-writer-director-thor-klein-found-story-in-the-details |access-date=22 March 2025}}</ref>

In 2019, Polish entomologist Marcin Kamiński introduced the generic name ''Ulamus'' to honor Ulam's work.<ref>{{Cite journal |last1=Kamiński |first1=Marcin J. |last2=Kanda |first2=Kojun |last3=Lumen |first3=Ryan |last4=Ulmer |first4=Jonah M. |last5=Wirth |first5=Christopher C. |last6=Bouchard |first6=Patrice |last7=Aalbu |first7=Rolf |last8=Mal |first8=Noël |last9=Smith |first9=Aaron D. |date=2019-05-13 |title=A catalogue of the tribe Sepidiini Eschscholtz, 1829 (Tenebrionidae, Pimeliinae) of the world |journal=ZooKeys |issue=844 |pages=1–121 |doi=10.3897/zookeys.844.34241 |doi-access=free |issn=1313-2970 |pmc=6527536 |pmid=31143077 |bibcode=2019ZooK..844....1K }}</ref><ref>{{Cite web |title=Beetle Mania: 'Forgotten' Beetles Found Lying in Museum Collection Named in Honour of Polish Mathematicians |url=https://scienceinpoland.pl/en/news/news%2C80160%2Cbeetle-mania-forgotten-beetles-found-lying-museum-collection-named-honour-polish |access-date=2025-08-06 |website=Science in Poland |language=en}}</ref> ''Ulamus'' belongs to the molluscan family ''Pteriidae''.<ref>{{Cite journal |last1=Kamiński |first1=Marcin J. |last2=Kanda |first2=Kojun |last3=Lumen |first3=Ryan |last4=Ulmer |first4=Jonah M. |last5=Wirth |first5=Christopher C. |last6=Bouchard |first6=Patrice |last7=Aalbu |first7=Rolf |last8=Mal |first8=Noël |last9=Smith |first9=Aaron D. |date=2019-05-13 |title=A catalogue of the tribe Sepidiini Eschscholtz, 1829 (Tenebrionidae, Pimeliinae) of the world |journal=ZooKeys |issue=844 |pages=1–121 |doi=10.3897/zookeys.844.34241 |doi-access=free |issn=1313-2970 |pmc=6527536 |pmid=31143077 |bibcode=2019ZooK..844....1K }}</ref>

==Bibliography==
*{{cite book |last1=Kac |first1=Mark |author-link=Mark Kac |first2=Stanisław |last2=Ulam |title=Mathematics and Logic: Retrospect and Prospects |url=https://archive.org/details/mathematicslogic0000kacm |url-access=registration |location=New York |publisher=Praeger |year=1968 |isbn= 978-0-486-67085-0 |oclc=24847821 |ref=none}}
*{{cite book|mr=0441664|last=Ulam|first= Stanisław |title=Sets, Numbers, and Universes: selected works|editor1-first= W. A.|editor1-last= Beyer|editor2-first= J.|editor2-last= Mycielski and |editor3-first=G.-C.|editor3-last= Rota |series=Mathematicians of Our Time|volume= 9|publisher= The MIT Press, Cambridge, Mass. London|year= 1974|isbn= 978-0-262-02108-1|ref=none}}
*{{cite book |last=Ulam |first=Stanisław  |title=A Collection of Mathematical Problems |url=https://archive.org/details/collectionofmath0000ulam |url-access=registration |location=New York |publisher=Interscience Publishers |year=1960 |oclc=526673 |ref=none}}
*{{cite book |last=Ulam |first=Stanisław  |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |location=New York |publisher=Charles Scribner's Sons|year=1983 |isbn=978-0-684-14391-0 |oclc=1528346|ref=none}} (autobiography).
*{{cite book |last=Ulam |first=Stanisław  |title=Science, Computers, and People: From the Tree of Mathematics |url=https://archive.org/details/sciencecomputers0000ulam |url-access=registration |location=Boston |publisher=Birkhauser |year=1986 |isbn= 978-3-7643-3276-1 |oclc=11260216 |ref=none}}
*{{cite book |last1=Ulam |first1=Stanisław  |last2=Ulam |first2=Françoise |title=Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and his Los Alamos Collaborators |location=Berkeley |publisher=University of California Press |year=1990 |isbn= 978-0-520-05290-1 |oclc=20318499 |ref=none}}

==See also==
* {{langx|de|[[:de:Abenteuer eines Mathematikers|Abenteuer eines Mathematikers]]}} (English title: ''Adventures of a Mathematician''), biopic about Stanislaw Ulam, based on his autobiography.
* [[List of Poles#Mathematics|List of Polish mathematicians]]
* [[List of Poles#Physics|List of Polish physicists]]
* [[List of things named after Stanislaw Ulam]]
* [[Timeline of Polish science and technology]]

==References==
{{Reflist | 30em | refs =

<ref name='ABRAHAM'>{{cite web | url = http://www.ralph-abraham.org/articles/MS%23133.Entropy/ms133.pdf| title = Image Entropy for Discrete Dynamical Systems  | access-date = 30 May 2013| last = Abraham | first = Ralph| date = 9 July 2011 | publisher = University of California, Santa Cruz}}</ref>

<ref name='ADAM'>{{cite web | url = http://news.harvard.edu/gazette/2000/04.06/ulam.html | title = Obituary for Adam Ulam | access-date = 10 October 2011 | date = 6 April 2000 | website = Harvard University Gazette}}</ref>

<ref name="AIP">{{cite web |url=http://www.aip.org/history/acap/biographies/bio.jsp?ulams |title=Stanislaw Ulam |publisher=[[American Institute of Physics]] |access-date=14 May 2013 |url-status=dead |archive-url=https://web.archive.org/web/20150702060345/https://www.aip.org/history/acap/biographies/bio.jsp?ulams |archive-date=2 July 2015 }}</ref>

<ref name='AJMAA'>{{cite web | url = http://ajmaa.org/searchroot/files/pdf/v6n1/v6i1p1.pdf | title = On Stan Ulam and His Mathematics | access-date = 10 October 2011 | last = Ciesielski | first = Kryzystof |author2=Thermistocles Rassias  | year = 2009 | website = Australian Journal of Mathematical Analysis and Applications | quote = v 6, nr 1, pp 1-9, 2009}}</ref>

<ref name='ANALOG'>{{cite book | last1 = Ulam | first1 = S. M. | title = Analogies Between Analogies |editor1= A. R. Bednarek |editor2=Françoise Ulam | publisher = University of California Press | year = 1990 | location = Berkeley | url = http://publishing.cdlib.org/ucpressebooks/view?docId=ft9g50091s&brand=ucpress | access-date = 24 December 2011 | isbn = 978-0-520-05290-1}}</ref>

<ref name='AULAM'>{{cite book | last1 = Ulam | first1 = Adam Bruno | title = Understanding the Cold War: a historian's personal reflections | publisher = Transaction Publishers | year = 2002 | location = New Brunswick, NJ | pages = 19 | url = https://books.google.com/books?id=wgtCaPUPIlwC | access-date = 28 December 2011 | isbn =  9780765808851 |oclc=48122759 }}</ref>

<ref name='BETHE'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?06-03.pdf | title = Reprinting of 1954 article: Comments on the History of the H-Bomb | access-date = 3 November 2011 | last = Bethe | first = Hans A. | date = Fall 1982 | website = [[Los Alamos Science]], No 6 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='BIOLOGY'>{{cite journal | title = Stanislaw M. Ulam's Contributions to Theoretical Biology | journal = Letters in Mathematical Physics | year = 1985 | first = William A. | last = Beyer | author2 = Peter H. Sellers | author3 = Michael S. Waterman | volume = 10 | issue = 2–3 | pages = 231–242 | url = http://www.cmb.usc.edu/papers/msw_papers/msw-065.pdf | access-date = 5 December 2011 | doi = 10.1007/bf00398163 | bibcode = 1985LMaPh..10..231B | url-status = dead | archive-url = https://web.archive.org/web/20110927133755/http://www.cmb.usc.edu/papers/msw_papers/msw-065.pdf | archive-date = 27 September 2011 | citeseerx = 10.1.1.78.4790 | s2cid = 2791811 }}</ref>

<ref name='BOOK'>{{cite web | url = http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Scottish_Book.html | title = Preface to the 'Scottish Book' | last = Ulam | first = Stanislaw | date = November 2002 | website = Turnbull WWW Server | publisher = School of Mathematical and Computational Sciences University of St Andrews | archive-url = https://web.archive.org/web/20120503060306/http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Scottish_Book.html | archive-date = 3 May 2012 }}</ref>

<ref name='CAMPB'>{{cite web | url = http://physics.ucsc.edu/~peter/242/FPU-birth-of-nonlinear-science-Lilienfeld.pdf | title = The Birth of Nonlinear Science | access-date = 8 December 2011 | last = Campbell | first = Donald H. | date = 17 March 2010 | publisher = Americal Physical Society}}</ref>

<ref name="CHAOS">{{cite journal |title=Focus: Landmarks – Computer Simulations Led to Discovery of Solitons |date=February 8, 2013 |journal=Physics |volume=6 |pages=15 |number=15  |doi=10.1103/Physics.6.15|bibcode = 2013PhyOJ...6...15L |last1=Lindley |first1=David }}</ref>

<ref name="CHAR 78">{{Cite book|last1=Chartrand|first1=Gary|url=https://books.google.com/books?id=zA_CAgAAQBAJ&pg=PA78|title=A First Course in Graph Theory|last2=Zhang|first2=Ping|date=2013|publisher=Courier Corporation|isbn=978-0-486-29730-9|pages=78|language=en}}</ref>

<ref name = 'CNLSSCOTT'>{{cite web | url = http://cnls.lanl.gov/External/Kruskal_Scott_Article.php | title = CNLS: apprecion of Martin Kruskal and Alwyn Scott | access-date = 8 December 2011 | year = 2007 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='CNLSFPU'>{{cite web | url = http://cnls.lanl.gov/Conferences/annual25/agenda.htm | title = 50 Years of the Fermi-Pasta-Ulam Problem: Legacy, Impact, and Beyond | access-date = 9 December 2011 | date = May 16–20, 2005 | website = CLNS 25th International Conference | publisher = Los Alamos National Laboratory}}</ref>

<ref name='CNLSHIST'>{{cite web | url = http://cnls.lanl.gov/External/History.php | title = History of the Center for Nonlinear Studies | access-date = 8 December 2011 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='CNLSULAM'>{{cite web | url = http://cnls.lanl.gov/External/Ulam.php | title = Ulam Scholars at CNLS | access-date = 8 December 2011 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='COMPEARLY'>{{cite web | url = http://www.lanl.gov/history/story.php?story_id=82 | title = Supercomputing | access-date = 24 October 2011 | website = History @ Los Alamos | publisher = Los Alamos National Laboratory}}</ref>

<ref name='COMPLATE'>{{cite web | url = http://www.lanl.gov/history/story.php?story_id=39 | title = From Calculators to Computers | access-date = 24 October 2011 | website = History @ Los Alamos | publisher = Los Alamos National Laboratory}}</ref>

<ref name='DEATH'>{{cite journal | title = Death of a Project | journal = Science | date = 9 July 1965 | first = Freeman | last = Dyson | volume = 149 | pages = 141–144| doi=10.1126/science.149.3680.141 | pmid = 17734490 | issue=3680|bibcode = 1965Sci...149..141D | s2cid = 39761976 }}</ref>

<ref name='DUNCAN'>{{Cite book |last1 = Hewlett |first1 = Richard G. |author-link = Richard G. Hewlett  |last2 = Duncan  |first2 = Francis  | title = Atomic Shield, Vol. II, 1947–1952 |pages = 406–409 |series = A History of the United States Atomic Energy Commission  |publisher = [[Pennsylvania State University Press]] |location = University Park, Pennsylvania |year = 1969 |isbn = 978-0-520-07187-2 }}</ref>

<ref name='ECKHARDT'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?15-13.pdf | title = Stan Ulam, John von Neumann, and the Monte Carlo method | access-date = 22 October 2011 | last = Eckhardt | first = Roger | year =1987 | website = Los Alamos Science, No 15}}</ref>

<ref name='ERDOS'>{{cite journal | url = http://www.renyi.hu/~p_erdos/1985-37.pdf | title = Ulam, the man and the mathematician | last = Erdős | first = Paul | authorlink = Paul Erdős | year = 1985 | journal = [[Journal of Graph Theory]] | volume = 9 | issue = 4 | pages = 445–449 | doi = 10.1002/jgt.3190090402}}</ref>

<ref name='FAMILY'>{{cite web | url = http://genforum.genealogy.com/cgi-bin/print.cgi?poland::5660.html | title = Ulam Family of Lwow; Auerbachs of Vienna | access-date = 10 October 2011 | last = Ulam | first = Molly | date = June 25, 2000 | website = Genforum}}</ref>

<!--
<ref name='FEIGENBAUM'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?06-04.pdf | title = Reflections of the Polish Masters | access-date = 1 January 2012 | last = Feigenbaum | first = Mitchell | work = Los Alamos Science, No. 6, 1982 | publisher = Los Alamos National Laboratory}}</ref> -->

<ref name='FPU'>{{cite journal | title = Fermi, Pasta, Ulam and the Birth of Experimental Mathematics | journal = American Scientist | date = May–Jun 2009 | first1 = Mason A. | last1 = Porter | first2= Norman J. | last2= Zabusky | last3 = Hu | first3= Bambi |  last4=Campbell  | first4=David K. | volume = 97 | issue = 3 | pages = 214–221| url = http://people.maths.ox.ac.uk/porterm/papers/fpupop_final.pdf | access-date = 20 November 2011 | doi=10.1511/2009.78.214}}</ref>

<ref name='FPUCHAOS'>{{cite journal | title = Focus-Issue: The Fermi-Pasta-Ulam Problem-The-First-50-Years | journal = Chaos | date = March 2005 | volume = 15 | issue = 1 | url = http://chaos.aip.org/resource/1/chaoeh/v15/i1?&section=focus-issue-the-fermi-pasta-ulam-problem-the-first-50-years&page=1 | archive-url = https://web.archive.org/web/20120503100148/http://chaos.aip.org/resource/1/chaoeh/v15/i1?&section=focus-issue-the-fermi-pasta-ulam-problem-the-first-50-years&page=1 | url-status = dead | archive-date = 2012-05-03 | access-date = 9 December 2011 }}</ref>

<ref name='FRISCH'>{{cite journal |last=Frisch |first=Otto |author-link=Otto Frisch |date=April 1974 |page=17 |journal=[[Bulletin of the Atomic Scientists]] |volume=30 |issue=4 |title=Somebody Turned the Sun on with a Switch |url=https://books.google.com/books?id=KQwAAAAAMBAJ&pg=PA17 |access-date=May 29, 2013 |bibcode=1974BuAtS..30d..12F |doi=10.1080/00963402.1974.11458102 }}</ref>

<ref name='FULAM'>{{cite book | last1 = Ulam | first1 = Françoise | title = Postscript to Adventures of a Mathematician | publisher = University of California | year = 1991 | location = Berkeley | isbn = 978-0-520-07154-4}}</ref>

<ref name='FULAMOBIT'>{{cite web | url = http://www.legacy.com/obituaries/santafenewmexican/obituary.aspx?n=francoise-ulam&pid=150769924 | title = Françoise Ulam Obituary | access-date = 12 December 2011 | date = 30 April 2011 | website = Santa Fe, New Mexican}}</ref>

<ref name='FUM'>{{citation | first = S. M. | last = Ulam | contribution = On Some Statistical Properties of Dynamical Systems | title = Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability | publisher = University of California Press | location = Berkeley | year = 1961}}</ref>

<ref name='GALISON'>{{cite book |title=The Disunity of Science: Boundaries, Contexts, and Power | editor = Peter Galison, David J. Stump | chapter = 5: Computer Simulations and the Trading Zone | page= 135 | author= Peter Galison
| chapter-url=https://books.google.com/books?id=2HEYgFz_kioC&pg=PA135| publisher = Stanford University Press
| year = 1996| isbn = 9780804725620 }}</ref>

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<ref name='GAMOW'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00285752.pdf | title = Ulam's Gamow Memorial Lecture, 5 October 1982 | access-date = 24 December 2011 | last = Ulam | first = S. M. | year =1987 | work = Los Alamos Science, No 15}}</ref> -->

<ref name='GOAD'>{{cite web | url = https://fas.org/sgp/othergov/doe/lanl/pubs/00326869.pdf | title = Sequence Analysis: Contributions of Ulam to Molecular Genetics | access-date = 28 December 2011 | last = Goad | first = Walter B | year = 1987 | website = Los Alamos Science | publisher = Los Alamos National Laboratory}}</ref>

<ref name=SANKOFF>{{cite journal|last=Sankoff|first=David|title=The early introduction of dynamic programming into computational biology|journal=Bioinformatics|year=2000|volume=16|issue=1|pages=41–47|doi=10.1093/bioinformatics/16.1.41|pmid=10812476|doi-access=free}}</ref>

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<ref name='GONCH'>{{cite journal | title = American and Soviet H-bomb development programmes: historical background | journal = [[Uspekhi Fizicheskikh Nauk]] | date = 29 May 1996 | first = G. A. | last = Goncharov | volume = 39 | issue = 10 | pages = 1033–1044| url = https://fas.org/nuke/guide/russia/nuke/goncharov-h-bomb.pdf | access-date = 2 November 2011 | doi=10.1070/pu1996v039n10abeh000174|bibcode = 1996PhyU...39.1033G }}</ref> -->

<ref name='HAWKINS'>{{cite web | url = https://www.nytimes.com/2002/03/04/us/david-hawkins-88-historian-for-manhattan-project-in-1940-s.html | title = Obituary of David Hawkins | access-date = 14 October 2011 | last = Lehmann | first = Christopher | date = 4 March 2002 | website = The New York Times}}</ref>

<ref name=RB>{{cite web |url=http://andrewgelman.com/wp-content/uploads/2014/07/CM-ULAM-Stanislaw.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://andrewgelman.com/wp-content/uploads/2014/07/CM-ULAM-Stanislaw.pdf |archive-date=2022-10-09 |url-status=live |title = Stanisław Ulam | language = fr | access-date = 29 October 2015}}</ref>

<ref name='HEARING'>{{cite book | last1 = Schreiber | first1 = R. E. | last2 = Ulam | first2 = Stanislaw M. | last3 = Bradbury | first3 = Norris | title = Outer Space Propulsion by Nuclear Energy | chapter = US Congress, Joint Committee on Atomic Energy: hearing on 22 January 1958 | publisher = US Government Printing Office | year = 1958 | pages = 47 | chapter-url = http://sul-derivatives.stanford.edu/derivative?CSNID=00002040&mediaType=application/pdf | access-date = 25 November 2011}}</ref>

<ref name="HEWLETT">{{cite book |last1=Hewlett |first1=Richard G. |author-link=Richard G. Hewlett |last2=Anderson |first2=Oscar E. |title=The New World, 1939–1946 |location=University Park|publisher=Pennsylvania State University Press |orig-year=1962 | year=1990
|url=https://www.governmentattic.org/5docs/TheNewWorld1939-1946.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.governmentattic.org/5docs/TheNewWorld1939-1946.pdf |archive-date=2022-10-09 |url-status=live |pages=304–307 |isbn=978-0-520-07186-5|oclc=637004643 }}</ref>

<ref name= "HEWLETT AND DUNCAN">{{cite book|last1=Hewlett|first1=Richard G.|last2=Duncan|first2=Francis|title=Atomic Shield, 1947–1952|series=A History of the United States Atomic Energy Commission|publisher=Pennsylvania State University Press|location=University Park|year=1969|isbn=978-0-520-07187-2|pages=380–385|oclc=3717478}}</ref>

<ref name= "HEWLETT AND DUNCAN 554-556">{{cite book|last1=Hewlett|first1=Richard G.|last2=Duncan|first2=Francis|title=Atomic Shield, 1947–1952|series=A History of the United States Atomic Energy Commission|publisher=Pennsylvania State University Press|location=University Park|year=1969|isbn=978-0-520-07187-2|pages=554–556|oclc=3717478}}</ref>

<ref name="HODDESON 130-137">{{cite book|last1=Hoddeson |first1=Lillian|author-link=Lillian Hoddeson|first2=Paul W. |last2=Henriksen |first3=Roger A. |last3=Meade |first4=Catherine L. |last4=Westfall|author4-link= Catherine Westfall|title=Critical Assembly: A Technical History of Los Alamos During the Oppenheimer Years, 1943–1945 |location=New York |publisher=Cambridge University Press |year=1993 |pages=[https://archive.org/details/criticalassembly0000unse/page/130 130–137] |isbn=978-0-521-44132-2 |oclc=26764320 |url=https://archive.org/details/criticalassembly0000unse/page/130 }}</ref>

<ref name='LA171'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00350385.pdf | title = Theory of Multiplicative Processes | access-date = 13 October 2011 | last = Hawkins | first = D. |author2=S. Ulam  | date = 14 November 1944 | website = LANL report LA-171}}</ref>

<ref name='LA1557'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00359950.pdf | title = Heuristic studies in problems of mathematical physics | access-date = 21 November 2011 | last = Pasta | first = John |author2=S. Ulam  | date = 9 March 1953 | website = LANL report LA-1557}}</ref>

<ref name='LA1940'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00353569.pdf | title = Studies of Nonlinear Problems I | access-date = 21 November 2011 | last = Fermi | first = E. |author2=J. Pasta |author3=S. Ulam | date = May 1955 | website = LANL report LA-1940}}</ref>

<ref name='LA-2305'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00320803.pdf | title = Quadratic Transformations. Part I | access-date = 26 November 2011 | last = Stein | first = P. R. |author2=Stanislaw M. Ulam  | date = March 1959 | website = LANL report LA-2305 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='LA-4246'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00387440.pdf | title = Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutionary Processes | access-date = 26 November 2011 | last = Schrandt | first = Robert G. |author2=Stanislaw M. Ulam  | date = December 1970 | website = LANL report LA-4246 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='LA-4973'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00319833.pdf | title = Metrics in Biology, an Introduction | access-date = 26 November 2011 | last = Beyer | first = William A. |author2=Temple F. Smith |author3=M. L. Stein |author4=Stanislaw M. Ulam  | date = August 1972 | website = LANL report LA-4973 | publisher = Los Alamos National Laboratory}}</ref>

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<ref name='LA-5647-MS'>{{cite web | url = http://www.lanl.gov/history/postwar/pdf/00377295.pdf | title = A short Account of Los Alamos Theoretical work on Thermonuclear Weapons 1946–1950 | access-date = 1 January 2012 | last = Mark | first = J. Carson | date = July 1974 (reissue of a report dated 1 October 1954) | work = LANL report LA-5647-MS | publisher = Los Alamos National Laboratory}}</ref> -->

<ref name='LAMS551'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00329286.pdf | title = Statistical Methods in Neutron Diffusion | access-date = 23 October 2011 | last = Richtmyer | first = D. |author2=J. Pasta |author3=S. Ulam | date = 9 April 1947 | website = LANL report LAMS-551}}</ref>

<ref name='LAMS1225'>{{cite web | url =http://www.nuclearnonproliferation.org/LAMS1225.pdf | title = On Heterocatalytic Detonations I. - Hydrodynamic Lenses and Radiation Mirrors | last1 = Teller | first1 = E. | last2 = Ulam | first2 = S. | author1-link = Edward Teller | publisher = [[Los Alamos National Laboratory]] | date = 9 March 1951 | id = LAMS-1225 | archive-url = https://web.archive.org/web/20120301145116/http://www.nuclearnonproliferation.org/LAMS1225.pdf | archive-date = 1 March 2012 | access-date = 4 April 2022}}</ref>

<ref name='LAMS1230'>{{citation | first = E. | last = Teller | contribution = A New Thermonuclear device | title = Technical Report LAMS-1230 | publisher = Los Alamos National Laboratory | date = 4 April 1951}}</ref>

<ref name='LAMS1955'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00350011.pdf | title = On a Method of Propulsion of Projectiles by Means of External Nuclear Explosions | access-date = 24 November 2011 | last = Everett | first = C. J. |author2=S. M. Ulam  | date = August 1955 | website = LANL report LAMS-1955}}</ref>

<ref name='LAMS2186'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?00339493.pdf  | title = Some Schemes for Nuclear Propulsion | access-date = 24 November 2011 | first = C. | last = Longmier |author2=F. Reines |author3=S. Ulam | date = August 1955 | website = LANL report LAMS-2186}}</ref>

<ref name='LANLBIO'>{{cite web | url = http://www.lanl.gov/history/people/S_Ulam.shtml | title = Staff biography of Stanislaw Ulam | access-date = 22 October 2011 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='MAULDIN'>{{cite book | last1 = Mauldin | first1 = R. Daniel | title = The Scottish Book | publisher = Birkhauser | year = 1981 | page = 268 | url = https://books.google.com/books?id=gaqEAAAAIAAJ | access-date = 4 December 2011 | isbn = 9783764330453 |oclc=7553633 }}</ref>

<ref name='MARKBIO'>{{cite web|url=http://www.lanl.gov/history/people/J_Mark.shtml |title=Staff biography of J. Carson Mark |access-date=22 October 2011 |publisher=Los Alamos National Laboratory |url-status=dead |archive-url=https://web.archive.org/web/20120716192104/http://www.lanl.gov/history/people/J_Mark.shtml |archive-date=16 July 2012 }}</ref>

<ref name='METROPOLIS'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?15-12.pdf | title = The Beginnings of the Monte Carlo Method | access-date = 22 October 2011 | last = Metropolis | first = Nicholas | year = 1987 | website = Los Alamos Science, No 15}}</ref>

<ref name='MIKE'>{{cite web | url = http://www.geni.com/people/Michael-Ulam/6000000011955105854 | title = Genealogy of Michael Ulam | access-date = 12 October 2011 | date = 24 May 2011 | website = GENi}}</ref>

<ref name='MOLLY'>{{cite web | url = http://www.readthehook.com/99589/memoir-memorial-lvov-lives-leopolis-press | title = Lwow lives on at Leopolis Press | access-date = 10 October 2011 | date = 14 November 2002 | website = The Hook | archive-date = 1 July 2015 | archive-url = https://web.archive.org/web/20150701193737/http://www.readthehook.com/99589/memoir-memorial-lvov-lives-leopolis-press | url-status = dead }}</ref>

<ref name='MONTEALAMOS'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?15-13.pdf | title = Stan Ulam, John von Neumann, and the Monte Carlo Method | access-date = 11 Mar 2016| last = Eckhardt | first = Roger| date = 1987| website = Los Alamos Science| publisher = Los Alamos National Laboratory}}</ref>

<ref name='MONTEBIG10'>{{cite web | url = http://theludwigs.com/2011/11/a-monte-carlo-simulation-of-the-big10-race/ | title = A Monte Carlo Simulation of the Big10 race | access-date = 9 December 2011 | last = Ludwig | first = John | work = A Little Ludwig Goes A Long Way | date = November 2011 | publisher = ludwig.com | archive-date = 26 September 2020 | archive-url = https://web.archive.org/web/20200926052717/http://theludwigs.com/2011/11/a-monte-carlo-simulation-of-the-big10-race/ | url-status = dead }}</ref>

<ref name='MONTELING'>{{cite journal | title = Historical Change in Language Using Monte CarloTechniques | journal = Mechanical Translation and Computational Linguistics | date = 23 May 1966 | first = Sheldon | last = Klein | volume = 9 | issue = 3 and 4 | pages = 67–81 | url = http://www.mt-archive.info/MT-1966-Klein.pdf | access-date = 9 December 2011 | archive-date = 16 October 2011 | archive-url = https://web.archive.org/web/20111016213745/http://www.mt-archive.info/MT-1966-Klein.pdf | url-status = dead }}</ref>

<ref name='MONTEONC'>{{cite journal | title = Dose Enhancement of electron beams subject to external magnetic fields: A Monte Carlo Study | journal = Medical Physics | date = 12 March 2002 | first = M. A. | last = Earl |author2=L. M. Ma  | volume = 29 | issue = 4 | pages = 484–492| url = http://online.medphys.org/resource/1/mphya6/v29/i4/p484_s1?isAuthorized=no | access-date = 9 December 2011 | doi=10.1118/1.1461374| pmid = 11991119 |bibcode = 2002MedPh..29..484E | url-access = subscription }}</ref>

<ref name='MONTEPAPER'>{{cite journal | title = The Monte Carlo method | journal = Journal of the American Statistical Association | year = 1949 | first = Nicholas | last = Metropolis |author2=Stanislaw Ulam  | volume = 44 | issue = 247 | pages = 335–341| url = https://people.bordeaux.inria.fr/pierre.delmoral/MetropolisUlam49.pdf |access-date = 21 November 2011 | doi=10.1080/01621459.1949.10483310 | pmid=18139350| jstor = 2280232 }}</ref>

<ref name='MONTERISK'>{{cite web | url = http://courses.engmath.dal.ca/courses/engm6675/web/montecarlo.pdf | archive-url = https://web.archive.org/web/20160306123025/http://courses.engmath.dal.ca/courses/engm6675/web/montecarlo.pdf | url-status = dead | archive-date = 2016-03-06 | title = Monte Carlo Simulation in Environmental Risk Assessment | access-date = 13 September 2012 | last = Poulter | first = Susan R. | date = Winter 1998 | website = Risk:Health, Safety, & Environment | publisher = University of New Hampshire }}</ref>

<ref name='MONTESOC'>{{cite web | url = http://www.icpsr.umich.edu/icpsrweb/sumprog/courses/0130 | title = Course description:Monte Carlo Methods for Social Scientists | access-date = 9 December 2011 | last = Casey | first = Thomas M. | date = June 2011 | website = Inter-University Consortium for Political and Social Research | publisher = University of Michigan | archive-date = 15 March 2012 | archive-url = https://web.archive.org/web/20120315073908/http://www.icpsr.umich.edu/icpsrweb/sumprog/courses/0130 | url-status = dead }}</ref>

<ref name='MULT123'>{{cite web | url = http://publishing.cdlib.org/ucpressebooks/view?docId=ft9g50091s&chunk.id=d0e4100&toc.depth=1&toc.id=d0e4100&brand=ucpress | title = Multiplicative Systems in Several Variables I, II, III | access-date = 13 October 2011 | last1 = Ulam | first1 = S. | last2=Everett | first2 = C. J | date = 7 June 1948 | website = LANL reports | publisher = University of California Press}}</ref>

<ref name='NYTIMES-OBIT'>{{cite news | url = https://www.nytimes.com/1984/05/15/obituaries/stanislaw-ulam-theorist-on-hydrogen-bomb.html | title = Stanislaw Ulam, Theorist on Hydrogen Bomb | access-date = 30 May 2013| last = Sullivan| first = Walter| date = 15 May 1984 | newspaper = New York Times}}</ref>

<ref name='OLGIERD'>{{cite book | last1 = Budrewicz/ | first1 = Olgierd | title = The melting-pot revisited: twenty well-known USers of Polish background | publisher = Interpress | year = 1977 | pages = 36 | url = https://books.google.com/books?id=pc51AAAAMAAJ&q=Sometimes+I+muse | access-date = 11 September 2012}}</ref>

<ref name='ORION'>{{cite web | url = http://www.oriondrive.com/p1_story_history.php | title = History of Project Orion | access-date = 7 October 2011 | date = 2008–2009 | website = The Story of Orion | publisher = OrionDrive.com}}</ref>

<ref name='OXTOBY'>{{cite web | url = https://www.nytimes.com/1991/01/05/obituaries/john-c-oxtoby-80-mathematics-professor.html | title = Obituary for John C, Oxtoby | access-date = 10 October 2011 | date = 5 January 1991 | website = The New York Times}}</ref>

<ref name="PAPERS">{{cite web |url=http://www.amphilsoc.org/mole/view?docId=ead/Mss.Ms.Coll.54-ead.xml |title=Stanislaw M. Ulam Papers |publisher=American Philosophical Society |access-date= 14 May 2013 }}</ref>

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<ref name='PREFACE'>{{cite book | last1 = Matthews | first1 = William G. | last2 = Hirsch | first2 = Daniel | title = Preface to Adventures of a Mathematician | publisher = University of California | year = 1991 | location = Berkele | isbn = 0-520-07154-9}}</ref> -->

<ref name="RHODES">{{cite book | last = Rhodes | first = Richard | author-link = Richard Rhodes | year = 1995 | title = Dark Sun: The Making of the Hydrogen Bomb | publisher = Simon & Schuster | location = New York | page = [https://archive.org/details/darksunmakingofh00rhod/page/248 248] | isbn = 978-0-684-80400-2 | url = https://archive.org/details/darksunmakingofh00rhod | url-access = registration }}</ref>

<ref name="RHODES 422-424">{{cite book | last = Rhodes | first = Richard | author-link = Richard Rhodes | year = 1995 | title = Dark Sun: The Making of the Hydrogen Bomb | publisher = Simon & Schuster | location = New York | pages = [https://archive.org/details/darksunmakingofh00rhod/page/422 422]–424 | isbn = 978-0-684-80400-2 | url = https://archive.org/details/darksunmakingofh00rhod | url-access = registration }}</ref>

<ref name="RHODES 455-464">{{cite book | last = Rhodes | first = Richard | author-link = Richard Rhodes | year = 1995 | title = Dark Sun: The Making of the Hydrogen Bomb | publisher = Simon & Schuster | location = New York | pages = [https://archive.org/details/darksunmakingofh00rhod/page/455 455]–464 | isbn = 978-0-684-80400-2 | url = https://archive.org/details/darksunmakingofh00rhod | url-access = registration }}</ref>

<ref name='ROTA'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?15-03.pdf | title = Stan Ulam: The Lost Cafe | access-date = 22 October 2011 | last = Rota | first = Gian-Carlo | website = Los Alamos Science, No 15, 1987}}</ref>

<ref name=SCHWEBER>{{cite book | last=Schweber | first=S. S. | title=In the Shadow of the Bomb: Bethe, Oppenheimer, and the Moral Responsibility of the Scientist | location=Princeton | publisher=Princeton University Press | pages=[https://archive.org/details/inshadowofbombbe00schw/page/166 166] | year=2000 | isbn=978-0-691-04989-2 | url=https://archive.org/details/inshadowofbombbe00schw/page/166 }}</ref>

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<ref name='TRIBUTE'>{{cite journal | title = Tribute to John von Neumann | journal = Bulletin of the American Mathematical Society | volume = 64 | issue = 3, part 2 | pages = 1–49|date=1958|doi=10.1090/S0002-9904-1958-10184-6}}</ref> -->

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<ref name='UAMS'>{{cite web | url = http://www.ams.org/sitesearch?cx=006070101833274801941%3Acyhu7o79tee&cof=FORID%3A9&ie=UTF-8&q=Ulam&sa=Go&siteurl=www.ams.org%2Fmr-database | title = Search for 'Ulam' on AMS website | access-date = 10 December 2011 | publisher = American Mathematical Society}}</ref>
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<ref name = 'UCHII'>{{cite journal | first=Soshichi | last=Uchii | title=Review of Edward Teller's Memoirs | journal=PHS Newsletter | volume=52 | date=22 July 2003 | url=http://www1.kcn.ne.jp/~h-uchii/philsci/Newsletters/newslet_52.html |access-date=13 August 2012}}</ref>

<ref name='UCOLLOQ'>{{cite web | url = http://www.math.ufl.edu/dept_news_events/ulam/ | title = List of Ulam Colloquium Speakers | access-date = 24 December 2011 | publisher = University of Florida, Dept. of Mathematics}}</ref>

<ref name='UCONF'>{{cite web | url = http://www.math.ufl.edu/stdculam/ | title = Ulam Centennial Conference | access-date = 24 December 2011 | date = March 10–11, 2009 | publisher = University of Florida | url-status = dead | archive-url = https://web.archive.org/web/20120424214006/http://www.math.ufl.edu/stdculam/ | archive-date = 24 April 2012 }}</ref>

<ref name="ULAM 9-15">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/9 9–15] |isbn=9780684143910 |oclc=1528346 }}</ref>

<ref name="ULAM 56-60">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/56 56–60] |isbn=9780684143910 |oclc=1528346  }}</ref>

<ref name="ULAM 125-130">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam/page/125 |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/125 125–130, 174] |isbn=9780684143910 |oclc=1528346 }}</ref>

<ref name="ULAM 143-147">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/143 143–147] |isbn=9780684143910 |oclc=1528346  }}</ref>

<ref name="ULAM 152-153">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/152 152–153] |isbn=9780684143910 |oclc=1528346  }}</ref>

<ref name="ULAM 162-167">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/162 162–157] |isbn=9780684143910 |oclc=1528346  }}</ref>

<ref name="ULAM 184-187">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/184 184–187] |isbn=9780684143910 |oclc=1528346  }}</ref>

<ref name="ULAM 215">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/215 215] |isbn=9780684143910 |oclc=1528346  }}</ref>

<ref name="ULAM 220-224">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/220 220–224] |isbn=9780684143910 |oclc=1528346  }}</ref>

<ref name="ULAM 249-254">{{cite book |last=Ulam |first=S. M |title=Adventures of a Mathematician |url=https://archive.org/details/adventuresofmath0000ulam |url-access=registration |publisher=Charles Scribner's Sons |location=New York |year=1983 |pages=[https://archive.org/details/adventuresofmath0000ulam/page/249 249–250] |isbn=9780684143910 |oclc=1528346  }}</ref>

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<ref name='UMATH'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?15-09.pdf | title = Learning from Ulam: Measurable Cardinals, Ergodicity, and Biomathematics | access-date = 6 December 2011 | last = Mycielski | first = January | work = Los Alamos Science, No 15, 1987 | publisher = Los Alamos National Laboratory}}</ref> -->

<ref name='UPUBS'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?15-26.pdf | title = Publications of Stanislaw M. Ulam | access-date = 6 December 2011 | website = Los Alamos Science, No 15, 1987 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='UQUART'>{{cite web | url = http://www.ulam.usm.edu/index.html | title = Home Page for Ulam Quarterly | access-date = 24 December 2011 | publisher = University of Florida}}</ref>

<ref name='UQUART2'>{{citation | first = Julio G. | last = Dix | contribution = Some Aspects of Running a Free Electronic Journal | title = New Developments in Electronic Publishing | editor-first = Hans | editor-last = Becker | publisher = European Congress of Mathematicians; ECM4 Satellite Conference | place = Stockholm | pages = 41–43 | date = June 25–27, 2004 | contribution-url = http://www.emis.de/proceedings/Stockholm2004/dix.pdf | access-date = 5 January 2013 | isbn = 978-3-88127-107-3}}</ref>

<ref name='UTRIB'>{{cite web | url = http://la-science.lanl.gov/lascience15.shtml | title = Stanislaw Ulam 1909–1984 | access-date = 6 December 2011 | last = Cooper | first = Necia Grant | website = Los Alamos Science, No 15, 1987 | publisher = Los Alamos National Laboratory}}</ref>

<ref name='VITA'>{{cite web | url = http://library.lanl.gov/cgi-bin/getfile?15-02.pdf | access-date = 7 October 2011 | last = Ulam | first = Francoise | year = 1987 | title = Vita: Excerpts from ''Adventures of a Mathematician'' | publisher = Los Alamos National Laboratory |archive-url=https://web.archive.org/web/20090114160317/http://library.lanl.gov/cgi-bin/getfile?15-02.pdf |archive-date=14 January 2009}}</ref>

<ref name='VOLSKY'>{{cite web | url = http://www.aulam.org/anxious2.htm#volsky | title = Letter about Jozef Ulam | access-date = 24 May 2013 | last = Volsky | first = George | date = 23 December 1963 | website = Anxiously from Lwow | publisher = Adam Ulam | url-status = dead | archive-url = https://web.archive.org/web/20130517203420/http://www.aulam.org/anxious2.htm#volsky | archive-date = 17 May 2013 }}</ref>

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<ref name='ZABUK'>{{cite journal | title = Interactions of solitons in a collisionless plasma and the recurrence of initial states. | journal = Physical Review Letters | year = 1965 | first = N. J. | last = Zabusky |author2=M.D. Kruskal.  | volume = 15 | pages = 240–243 | url = =http://physics.aps.org/articles/v6/15 | access-date = 21 November 2011 | doi=10.1103/physrevlett.15.240|bibcode = 1965PhRvL..15..240Z }}</ref> -->
}}

==External links==
*[http://manhattanprojectvoices.org/oral-histories/stanislaus-ulams-interview-1979 1979 Audio Interview with Stanislaus Ulam by Martin Sherwin] Voices of the Manhattan Project
*[http://manhattanprojectvoices.org/oral-histories/stanislaus-ulams-interview 1965 Audio Interview with Stanislaus Ulam by Richard Rhodes] Voices of the Manhattan Project
*{{cite journal |journal=Los Alamos Science |date = 1987 |issue=Special Issue |page=313 |url=https://fas.org/sgp/othergov/doe/lanl/pubs/00326872.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://fas.org/sgp/othergov/doe/lanl/pubs/00326872.pdf |archive-date=2022-10-09 |url-status=live |title=Publications of Stanislaw M. Ulam |issn=0273-7116 }}
* {{YouTube|7DHs3ToRaJM|Von Neumann: The Interaction of Mathematics and Computing}} – 1976 lecture on ''The First International Research Conference on the History of Computing''.

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