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{{Short description|Swedish mathematician (1905–1986)}}
{{Infobox scientist
{{Infobox scientist
| name = Arne Beurling
| name = Arne Beurling
| image = <!--(filename only)-->
| image = Arne Beurling, foto TT 1940-tal.jpg
| image_size =
| image_size = 240px
| caption =
| caption = Arne Beurling in 1940s
| birth_date = {{birth date|1905|2|3|df=y}}
| birth_date = {{birth date|1905|2|3|df=y}}
| birth_place = [[Gothenburg]], [[Sweden]]
| birth_place = [[Gothenburg]], [[Sweden]]
| death_date = {{death date and age|1986|11|20|1905|2|3|df=y}}
| death_date = {{death date and age|1986|11|20|1905|2|3|df=y}}
| death_place = [[Princeton, New Jersey|Princeton]], [[New Jersey]], USA
| death_place = [[Princeton, New Jersey|Princeton]], [[New Jersey]], United States
| nationality = [[Sweden|Swedish]]
| nationality = [[Sweden|Swedish]]
| fields = [[Mathematics]]
| fields = [[Mathematics]]
Line 13: Line 14:
| alma_mater = Uppsala University
| alma_mater = Uppsala University
| doctoral_advisor = [[Anders Wiman]]
| doctoral_advisor = [[Anders Wiman]]
| doctoral_students = [[Göran Borg]]<br />[[Lennart Carleson]]<br/>[[Yngve Domar]]<br />[[Carl-Gustav Esseen]]
| doctoral_students = [[Lennart Carleson]]<br/>[[Carl-Gustav Esseen]]
| known_for = [[Beurling algebra]]<br />[[Hardy space#Factorization into inner and outer functions|Beurling factorization]]<br />[[Beurling–Lax theorem]]<br />[[Riemann hypothesis#Criteria equivalent to the Riemann hypothesis|Beurling–Nyman criterion]]
| known_for = [[Beurling algebra]]<br />[[Hardy space#Factorization into inner and outer functions|Beurling factorization]]<br />[[Beurling–Lax theorem]]<br />[[Riemann hypothesis#Criteria equivalent to the Riemann hypothesis|Beurling–Nyman criterion]]
| awards =
| awards =
}}
}}
[[File:Arne Beurling Uppsala maj 2019.jpg|thumb|262px|Statue of Arne Beurling in [[Uppsala]].]]
'''Arne Carl-August Beurling''' (3 February 1905 – 20 November 1986) was a [[Sweden|Swedish]] [[mathematician]] and [[professor]] of [[mathematics]] at [[Uppsala University]] (1937–1954) and later at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]]. Beurling worked extensively in [[harmonic analysis]], [[complex analysis]] and [[potential theory]]. The "[[Hardy space#Factorization into inner and outer functions (Beurling)|Beurling factorization]]" helped mathematical scientists to understand the [[Wold decomposition]], and inspired further work on the [[invariant subspace]]s of linear operators and [[operator algebras]], e.g. [[Håkan Hedenmalm]]'s factorization theorem for [[Bergman space]]s.
'''Arne Carl-August Beurling''' (3 February 1905 – 20 November 1986) was a [[Sweden|Swedish]] [[mathematician]] and [[professor]] of [[mathematics]] at [[Uppsala University]] (1937–1954) and later at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]]. Beurling worked extensively in [[harmonic analysis]], [[complex analysis]] and [[potential theory]]. The "[[Hardy space#Factorization into inner and outer functions (Beurling)|Beurling factorization]]" helped mathematical scientists to understand the [[Wold decomposition]], and inspired further work on the [[invariant subspace]]s of linear operators and [[operator algebras]], e.g. Håkan Hedenmalm's factorization theorem for [[Bergman space]]s.

He is perhaps most famous for single-handedly decrypting an early version of the German cipher machine [[Siemens and Halske T52]] in a matter of two weeks during 1940, using only pen and paper. This machine's cipher is generally considered to be more complicated than that of the more famous [[Enigma machine]]. Beurling's method of decrypting military telegrams between Norway and Germany worked from June 1940 right up until 1943 when the Germans changed equipment.

⚫
==Early life==
Beurling was born on 3 February 1905 in [[Gothenburg]], Sweden and was the son of the landowner Konrad Beurling and baroness Elsa Raab.<ref name="Harnesk (1964), p. 82">{{cite book |editor1-last=Harnesk |editor1-first=Paul |date=1964 |title=Vem är vem?. 2, Svealand utom Stor-Stockholm |trans-title=Who is Who?. 2, Svealand excluding Greater Stockholm |edition=2nd |location=Stockholm |publisher=Bokförlaget Vem är vem |language=Swedish |url=https://runeberg.org/vemarvem/svea64/0106.html |page=82}}</ref> After [[Studentexamen|graduating]] in 1924, he was enrolled at the [[Uppsala University]] where he received a Bachelor of Arts degree in 1926 and two years later a Licentiate of Philosophy degree.<ref name="Harnesk (1964), p. 82"/>


==Career==
==Career==
⚫
Beurling was born in [[Gothenburg]], Sweden and was the son of the landowner Konrad Beurling and baroness Elsa Raab. After [[Studentexamen|graduating]] in 1924, he was enrolled at the [[Uppsala University]] where he received a Bachelor of Arts degree in 1926 and two years later a Licentiate of Philosophy degree. Beurling was assistant teacher at Uppsala University from 1931 to 1933.<ref name="Harnesk (1964), p. 82">{{cite book |editor1-last=Harnesk |editor1-first=Paul |date=1964 |title=Vem är vem?. 2, Svealand utom Stor-Stockholm |trans-title=Who is Who?. 2, Svealand excluding Greater Stockholm |edition=2nd |location=Stockholm |publisher=Bokförlaget Vem är vem |language=Swedish |url=http://runeberg.org/vemarvem/svea64/0106.html |page=82}}</ref> He received his doctorate in mathematics in 1933 for his dissertation ''Études sur un problème de majoration''.<ref>{{cite book |last=Beurling |first=Arne |title=Études sur un problème de majoration |year=1933 |publisher=[s.n.] |location=Uppsala |language=French}}</ref> Beurling was a [[docent]] of mathematics at Uppsala University from 1933 and then professor of mathematics from 1937 to 1954.<ref name="Harnesk (1964), p. 82"/> He was visiting professor at [[Harvard University]] from 1948 to 1949.<ref>{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Beurling.html |publisher=School of Mathematics and Statistics, [[University of St Andrews]], Scotland |title=Arne Carl-August Beurling |first1=J J |last1=O'Connor |first2=E F |last2=Robertson|date=2008 |access-date=27 March 2015}}</ref> From 1954 he was professor at the [[Institute for Advanced Study]] at [[Princeton University]], [[United States]], where he took over [[Albert Einstein]]'s office.<ref name="SVD 2010-10-25">{{cite news |url=http://www.svd.se/kultur/svensk-knackte-nazisternas-hemliga-koder_5549643.svd |newspaper=[[Svenska Dagbladet]] |title=Svensk knäckte nazisternas hemliga koder |trans-title=A Swede cracked the Nazis' secret codes |first=Christer |last=Bergström |date=25 October 2010 |access-date=27 March 2015 |language=Swedish}}</ref>


==World War II==
===Early career===
Beurling was assistant teacher at Uppsala University from 1931 to 1933.<ref name="Harnesk (1964), p. 82"/> He received his doctorate in mathematics in 1933 for his dissertation ''Études sur un problème de majoration''.<ref>{{cite book |last=Beurling |first=Arne |title=Études sur un problème de majoration |year=1933 |publisher=[s.n.] |location=Uppsala |language=fr}}</ref> Beurling was a [[docent]] of mathematics at Uppsala University from 1933 and then professor of mathematics from 1937 to 1954.<ref name="Harnesk (1964), p. 82"/>
⚫
In the summer of 1940 he single-handedly [[decipher]]ed and [[reverse engineering|reverse-engineer]]ed an early version of the [[Siemens and Halske T52]] also known as the ''Geheimfernschreiber'' ("secret teletypewriter") used by [[Nazi Germany]] in [[World War II]] for sending [[cipher]]ed messages.<ref>{{cite book|last=Beckman|first=B.|title=Codebreakers: Arne Beurling and the Swedish crypto program during World War II.|location=Providence, RI|publisher=American Mathematical Society|year=2002}}</ref> The T52 was one of the so-called "[[Fish cyphers]]", that, using transposition, created nearly one quintillion (893,622,318,929,520,960) different variations. It took Beurling two weeks to solve the problem using pen and paper. Using Beurling's work, a device was created that enabled Sweden to decipher German [[teleprinter]] traffic passing through Sweden from [[Norway]] on a cable. In this way, Swedish authorities knew about [[Operation Barbarossa]] before it occurred. Since the Swedes would not reveal how this knowledge was attained, the Swedish warning was not treated as credible by Soviets.


===World War II===
⚫
This became the foundation for the Swedish [[National Defence Radio Establishment]] (''FRA''). The cypher in the ''Geheimfernschreiber'' is generally considered to be more complex than the cypher used in the Enigma machines.<ref>{{Citation | last = Rijmenants | first = Dirk | title = Focus: Siemens & Halske T-52 | year = 2008 | url = http://users.telenet.be/d.rijmenants/en/focus.htm }}</ref>
⚫
In the summer of 1940 he single-handedly [[decipher]]ed and [[reverse engineering|reverse-engineer]]ed an early version of the [[Siemens and Halske T52]] also known as the ''Geheimfernschreiber'' ("secret teletypewriter") used by [[Nazi Germany]] in [[World War II]] for sending [[cipher]]ed messages.<ref>{{cite book|last=Beckman|first=B.|title=Codebreakers: Arne Beurling and the Swedish crypto program during World War II.|location=Providence, RI|publisher=American Mathematical Society|year=2002}}</ref> The T52 was one of the so-called "[[Fish cyphers]]", that, using transposition, created nearly one quintillion (893,622,318,929,520,960) different variations. It took Beurling two weeks to solve the problem using pen and paper. Using Beurling's work, a device was created that enabled Sweden to decipher German [[teleprinter]] traffic passing through Sweden from [[Norway]] on a cable. In this way, Swedish authorities knew about [[Operation Barbarossa]] before it occurred.<ref>{{cite book |last1=Beckman |first1=Bengt |title=Codebreakers: Arne Beurling and the Swedish Crypto Program During World War II |date=2002 |publisher=American Mathematical Society |isbn=9780821828892 |page=11 |url=https://books.google.com/books?id=n5RFilaXbC4C&pg=PR11 |access-date=25 March 2022}}</ref> Since the Swedes would not reveal how this knowledge was attained, the Swedish warning was not treated as credible by Soviets{{citation needed|date=March 2022}}.


⚫
This became the foundation for the Swedish [[National Defence Radio Establishment]] (''FRA''). The cypher in the ''Geheimfernschreiber'' is generally considered to be more complex than the cypher used in the Enigma machines.<ref>{{Citation | last = Rijmenants | first = Dirk | title = Focus: Siemens & Halske T-52 | year = 2008 | url = https://www.ciphermachinesandcryptology.com/en/focus.htm }}</ref>
⚫
==Later life==

⚫
Beurling was elected a Fellow of the [[American Academy of Arts and Sciences]] in 1970.<ref name=AAAS>{{cite web|title=Book of Members, 1780–2010: Chapter B|url=http://www.amacad.org/publications/BookofMembers/ChapterB.pdf|publisher=American Academy of Arts and Sciences|accessdate=25 June 2011| archiveurl= https://web.archive.org/web/20110725002054/http://www.amacad.org/publications/BookofMembers/ChapterB.pdf| archivedate= 25 July 2011 <!--DASHBot-->| deadurl= no}}</ref> He was the [[doctoral advisor]] of [[Lennart Carleson]] and [[Carl-Gustav Esseen]].
===Later life===
⚫
He was visiting professor at [[Harvard University]] from 1948 to 1949.<ref>{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Beurling.html |publisher=School of Mathematics and Statistics, [[University of St Andrews]], Scotland |title=Arne Carl-August Beurling |first1=J J |last1=O'Connor |first2=E F |last2=Robertson|date=2008 |access-date=27 March 2015}}</ref> From 1954 he was professor at the [[Institute for Advanced Study]] in Princeton, New Jersey, [[United States]], where he took over [[Albert Einstein]]'s office.<ref name="SVD 2010-10-25">{{cite news |url=http://www.svd.se/kultur/svensk-knackte-nazisternas-hemliga-koder_5549643.svd |newspaper=[[Svenska Dagbladet]] |title=Svensk knäckte nazisternas hemliga koder |trans-title=A Swede cracked the Nazis' secret codes |first=Christer |last=Bergström |date=25 October 2010 |access-date=27 March 2015 |language=Swedish}}</ref>

He was the [[doctoral advisor]] of [[Lennart Carleson]] and [[Carl-Gustav Esseen]].


==Personal life==
==Personal life==
[[File:Beurlings familjegrav.jpg|thumb|262px|Arne Beurling´s family grave at [[:en:Norra begravningsplatsen|Norra begravningsplatsen]] in [[Solna Municipality|Solna]].]]
Beurling was in his first marriage married 1936-40 with the licentiate in medicine Britta Östberg (born 1907), daughter of lector Henrik Östberg and Ph.D. Gerda Nilsson. He remarried in 1950 with a M.Phil. Karin Lindblad (1920-2006), daughter of ironmonger Henric Lindblad and Wanja Bengtsson.<ref name="Harnesk (1964), p. 82"/> Karin was a distinguished Ph.D student from Uppsala University. When they lived in Princeton, she worked in a biochemistry lab at Princeton University.<ref>{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Search/historysearch.cgi?SUGGESTION=Arne+Beurling&CONTEXT=1 |publisher=School of Mathematics and Statistics, [[University of St Andrews]], Scotland |title=Search Results for Arne Beurling |date=August 2001 |access-date=1 September 2015}}</ref> Beurling was in the first marriage the father of Pehr-Henrik (born 1936) and Jane (born 1938).<ref name="Harnesk (1964), p. 82"/>
Arne Beurling was first married (1936–40) to Britta Östberg (born 1907), daughter of Henrik Östberg and Gerda Nilsson. In 1950 he married Karin Lindblad (1920–2006), daughter of ironmonger Henric Lindblad and Wanja Bengtsson.<ref name="Harnesk (1964), p. 82"/> Karin was a distinguished Ph.D student from Uppsala University. When they lived in Princeton, she worked in a biochemistry lab at Princeton University.<ref>{{cite web |url=http://www-history.mcs.st-andrews.ac.uk/Search/historysearch.cgi?SUGGESTION=Arne+Beurling&CONTEXT=1 |publisher=School of Mathematics and Statistics, [[University of St Andrews]], Scotland |title=Search Results for Arne Beurling |date=August 2001 |access-date=1 September 2015}}</ref> He had two children from his first marriage — Pehr-Henrik (1936–1962) and Jane (1938–1992).<ref name="Harnesk (1964), p. 82"/>


⚫
Beurling's great-grandfather was [[:sv:Pehr Henrik Beurling|Pehr Henrik Beurling]] (1758 or 1763–1806), who founded a high quality clock factory in [[Stockholm]] in 1783.
⚫
Beurling died in 1986 and was buried at [[Norra begravningsplatsen]] in Stockholm.<ref>{{cite web |url=http://hittagraven.stockholm.se/sv/Norra-begravningsplatsen/1/10A/413/1 |publisher=Hittagraven.se |title=Norra begravningsplatsen, kvarter 10A, gravnummer 413 |access-date=27 March 2015 |language=Swedish}}</ref>

⚫
Beurling's great-grandfather was [[Per Henric Beurling]] (1758 or 1763 – 1806), who founded a high quality clock factory in Stockholm in 1783.
==Death==
⚫
Arne Beurling died in 1986 and was buried at [[Norra begravningsplatsen]] in [[Solna Municipality|Solna]].<ref>{{cite web |url=http://hittagraven.stockholm.se/sv/Norra-begravningsplatsen/1/10A/413/1 |publisher=Hittagraven.se |title=Norra begravningsplatsen, kvarter 10A, gravnummer 413 |access-date=27 March 2015 |language=Swedish}}</ref>


==In popular culture==
==In popular culture==
Beurling's prowess as a cryptanalysist is the subject of the 2005 short opera ''Krypto CEG'' by [[Jonas Sjöstrand]] and [[Kimmo Eriksson]].
Beurling's prowess as a cryptanalysist is the subject of the 2005 short opera ''Krypto CEG'' by [[Jonas Sjöstrand]] and [[Kimmo Eriksson]].


==Awards==
==Awards and decorations==
*Knight of the [[Order of the Polar Star]]<ref name="Harnesk (1964), p. 82"/>
*Knight of the [[Order of the Polar Star]]<ref name="Harnesk (1964), p. 82"/>
*Knight of the [[Order of Vasa]]<ref name="Harnesk (1964), p. 82"/>
*Knight of the [[Order of Vasa]]<ref name="Harnesk (1964), p. 82"/>

==Honours==
*Honorary member of the [[Göteborgs nation, Uppsala|Göteborgs nation]] at [[Uppsala University]] (1937)<ref name="Harnesk (1964), p. 82"/>
*Member of the [[Royal Swedish Academy of Sciences]] (1937)<ref name="Harnesk (1964), p. 82"/>
*Member of the Royal Society of the Humanities at Lund (''Kungliga Humanistiska Vetenskapssamfundet i Lund'') (1937)<ref name="Harnesk (1964), p. 82"/>
*Member of the [[Finnish Society of Sciences and Letters]] (1942)<ref name="Harnesk (1964), p. 82"/>
*Member of the [[Royal Physiographic Society in Lund]] (1948)<ref name="Harnesk (1964), p. 82"/>
*Member of the [[Royal Danish Academy of Sciences and Letters]] (1951)<ref name="Harnesk (1964), p. 82"/>
⚫
*Fellow of the [[American Academy of Arts and Sciences]] (1970)<ref name=AAAS>{{cite web|title=Book of Members, 1780–2010: Chapter B|url=http://www.amacad.org/publications/BookofMembers/ChapterB.pdf|publisher=American Academy of Arts and Sciences|access-date=25 June 2011| archiveurl= https://web.archive.org/web/20110725002054/http://www.amacad.org/publications/BookofMembers/ChapterB.pdf| archivedate= 25 July 2011 | url-status= live}}</ref>


==See also==
==See also==
*[[Beurling zeta function]]
*[[Beurling zeta function]]
*[[Beurling transform]]


==References==
==References==
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[[Category:1905 births]]
[[Category:1905 births]]
[[Category:1986 deaths]]
[[Category:1986 deaths]]
[[Category:Pre-computer cryptographers]]
[[Category:20th-century cryptographers]]
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[[Category:Swedish mathematicians]]
[[Category:20th-century Swedish mathematicians]]
[[Category:20th-century Swedish mathematicians]]
[[Category:Uppsala University alumni]]
[[Category:Uppsala University alumni]]
[[Category:Institute for Advanced Study faculty]]
[[Category:Institute for Advanced Study faculty]]
[[Category:Members of the Royal Swedish Academy of Sciences]]
[[Category:Members of the Royal Physiographic Society in Lund]]
[[Category:Members of the Royal Danish Academy of Sciences and Letters]]
[[Category:Fellows of the American Academy of Arts and Sciences]]
[[Category:Fellows of the American Academy of Arts and Sciences]]
[[Category:People from Gothenburg]]
[[Category:Scientists from Gothenburg]]
[[Category:Knights of the Order of the Polar Star]]
[[Category:Knights of the Order of the Polar Star]]
[[Category:Knights of the Order of Vasa]]
[[Category:Knights of the Order of Vasa]]
[[Category:Burials at Norra begravningsplatsen]]
[[Category:Burials at Norra begravningsplatsen]]
⚫
[[Category:Swedish cryptographers]]
[[Category:Members of the Royal Society of Sciences in Uppsala]]

Latest revision as of 08:14, 2 October 2024

Arne Beurling
Arne Beurling in 1940s
Born(1905-02-03)3 February 1905
Gothenburg, Sweden
Died20 November 1986(1986-11-20) (aged 81)
Princeton, New Jersey, United States
NationalitySwedish
Alma materUppsala University
Known forBeurling algebra
Beurling factorization
Beurling–Lax theorem
Beurling–Nyman criterion
Scientific career
FieldsMathematics
InstitutionsUppsala University
Institute for Advanced Study
Doctoral advisorAnders Wiman
Doctoral studentsLennart Carleson
Carl-Gustav Esseen
Statue of Arne Beurling in Uppsala.

Arne Carl-August Beurling (3 February 1905 – 20 November 1986) was a Swedish mathematician and professor of mathematics at Uppsala University (1937–1954) and later at the Institute for Advanced Study in Princeton, New Jersey. Beurling worked extensively in harmonic analysis, complex analysis and potential theory. The "Beurling factorization" helped mathematical scientists to understand the Wold decomposition, and inspired further work on the invariant subspaces of linear operators and operator algebras, e.g. Håkan Hedenmalm's factorization theorem for Bergman spaces.

He is perhaps most famous for single-handedly decrypting an early version of the German cipher machine Siemens and Halske T52 in a matter of two weeks during 1940, using only pen and paper. This machine's cipher is generally considered to be more complicated than that of the more famous Enigma machine. Beurling's method of decrypting military telegrams between Norway and Germany worked from June 1940 right up until 1943 when the Germans changed equipment.

Early life

[edit]

Beurling was born on 3 February 1905 in Gothenburg, Sweden and was the son of the landowner Konrad Beurling and baroness Elsa Raab.[1] After graduating in 1924, he was enrolled at the Uppsala University where he received a Bachelor of Arts degree in 1926 and two years later a Licentiate of Philosophy degree.[1]

Career

[edit]

Early career

[edit]

Beurling was assistant teacher at Uppsala University from 1931 to 1933.[1] He received his doctorate in mathematics in 1933 for his dissertation Études sur un problème de majoration.[2] Beurling was a docent of mathematics at Uppsala University from 1933 and then professor of mathematics from 1937 to 1954.[1]

World War II

[edit]

In the summer of 1940 he single-handedly deciphered and reverse-engineered an early version of the Siemens and Halske T52 also known as the Geheimfernschreiber ("secret teletypewriter") used by Nazi Germany in World War II for sending ciphered messages.[3] The T52 was one of the so-called "Fish cyphers", that, using transposition, created nearly one quintillion (893,622,318,929,520,960) different variations. It took Beurling two weeks to solve the problem using pen and paper. Using Beurling's work, a device was created that enabled Sweden to decipher German teleprinter traffic passing through Sweden from Norway on a cable. In this way, Swedish authorities knew about Operation Barbarossa before it occurred.[4] Since the Swedes would not reveal how this knowledge was attained, the Swedish warning was not treated as credible by Soviets[citation needed].

This became the foundation for the Swedish National Defence Radio Establishment (FRA). The cypher in the Geheimfernschreiber is generally considered to be more complex than the cypher used in the Enigma machines.[5]

Later life

[edit]

He was visiting professor at Harvard University from 1948 to 1949.[6] From 1954 he was professor at the Institute for Advanced Study in Princeton, New Jersey, United States, where he took over Albert Einstein's office.[7]

He was the doctoral advisor of Lennart Carleson and Carl-Gustav Esseen.

Personal life

[edit]
Arne Beurling´s family grave at Norra begravningsplatsen in Solna.

Arne Beurling was first married (1936–40) to Britta Östberg (born 1907), daughter of Henrik Östberg and Gerda Nilsson. In 1950 he married Karin Lindblad (1920–2006), daughter of ironmonger Henric Lindblad and Wanja Bengtsson.[1] Karin was a distinguished Ph.D student from Uppsala University. When they lived in Princeton, she worked in a biochemistry lab at Princeton University.[8] He had two children from his first marriage — Pehr-Henrik (1936–1962) and Jane (1938–1992).[1]

Beurling's great-grandfather was Pehr Henrik Beurling (1758 or 1763–1806), who founded a high quality clock factory in Stockholm in 1783.

Death

[edit]

Arne Beurling died in 1986 and was buried at Norra begravningsplatsen in Solna.[9]

[edit]

Beurling's prowess as a cryptanalysist is the subject of the 2005 short opera Krypto CEG by Jonas Sjöstrand and Kimmo Eriksson.

Awards and decorations

[edit]

Honours

[edit]

See also

[edit]

References

[edit]
  1. ^ a b c d e f g h i j k l m n Harnesk, Paul, ed. (1964). Vem är vem?. 2, Svealand utom Stor-Stockholm [Who is Who?. 2, Svealand excluding Greater Stockholm] (in Swedish) (2nd ed.). Stockholm: Bokförlaget Vem är vem. p. 82.
  2. ^ Beurling, Arne (1933). Études sur un problème de majoration (in French). Uppsala: [s.n.]
  3. ^ Beckman, B. (2002). Codebreakers: Arne Beurling and the Swedish crypto program during World War II. Providence, RI: American Mathematical Society.
  4. ^ Beckman, Bengt (2002). Codebreakers: Arne Beurling and the Swedish Crypto Program During World War II. American Mathematical Society. p. 11. ISBN 9780821828892. Retrieved 25 March 2022.
  5. ^ Rijmenants, Dirk (2008), Focus: Siemens & Halske T-52
  6. ^ O'Connor, J J; Robertson, E F (2008). "Arne Carl-August Beurling". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 27 March 2015.
  7. ^ Bergström, Christer (25 October 2010). "Svensk knäckte nazisternas hemliga koder" [A Swede cracked the Nazis' secret codes]. Svenska Dagbladet (in Swedish). Retrieved 27 March 2015.
  8. ^ "Search Results for Arne Beurling". School of Mathematics and Statistics, University of St Andrews, Scotland. August 2001. Retrieved 1 September 2015.
  9. ^ "Norra begravningsplatsen, kvarter 10A, gravnummer 413" (in Swedish). Hittagraven.se. Retrieved 27 March 2015.
  10. ^ "Book of Members, 1780–2010: Chapter B" (PDF). American Academy of Arts and Sciences. Archived (PDF) from the original on 25 July 2011. Retrieved 25 June 2011.
[edit]
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