Erdős–Bacon number

From Wikipedia, the free encyclopedia
  (Redirected from Erdős-Bacon number)
Jump to: navigation, search

A person's Erdős–Bacon number is the sum of one's Erdős number—which measures the "collaborative distance" in authoring mathematical papers between that person and Hungarian mathematician Paul Erdős—and one's Bacon number—which represents the number of links, through roles in films, by which the individual is separated from American actor Kevin Bacon.[1][2] The lower the number, the closer a person is to Erdős and Bacon, which reflects a small world phenomenon in academia and entertainment.[3]

In general, to have a defined Erdős–Bacon number, it is a necessary (but not a sufficient) condition for one to have both appeared in a film and co-authored an academic paper. It is not sufficient because it may still be impossible to reach Erdős through one's academic coauthors or Bacon through one's film co-appearances.


Astronomer Carl Sagan has an Erdős number of no more than 4[4] via Steven J. Ostro and a Bacon number of 2,[5] for a total of 6. Physicist Richard Feynman has an Erdős number of 3[6] and a Bacon number of 3, having appeared in the film Anti-Clock alongside Tony Tang.[7]

Electrical and computer engineering professor Robert J. Marks II has an Erdős–Bacon number of five.[8][9][10][11] His colleague, intelligent design advocate William A. Dembski consequently has an Erdős–Bacon number of six [12][13][14] and biochemist Michael Behe has an Erdős–Bacon number of seven[15][16]

Geneticist Jonathan Pritchard appeared in the 1998 movie Without Limits[17] which gives him a Bacon Number of 2.[18] Pritchard has an Erdős Number of 4[19][20][21][22] thus giving him an Erdős–Bacon Number of 6.

Italian-Russian mathematician Yaroslav D. Sergeyev[23] has an Erdős–Bacon number of 6. His Erdős number is 3 via Panos M. Pardalos and his Bacon number is 3 via Antonio Cantafora with whom he was (IMBD name Jaroslav Serguecy [24]) in the movie "Buck and the Magic Bracelet" in 1999.

Scientist and artist Ken Goldberg has a Erdős–Bacon number of 6.[25] Goldberg co-wrote Connected: An Autoblogography About Love, Death & Technology and Peter Coyote was also in it. Coyote was in These Amazing Shadows with Zooey Deschanel. Deschanel was in Sundance Skippy with Kevin Bacon. This gives a Bacon number of 3.

Computer scientist and mathematician Hubie Chen has an Erdős–Bacon number of 5: he has an Erdős number of 3 via Manuel Bodirsky,[26] and has a Bacon number of 2 via Rebecca Hall.[27]

Mathematician Allan "Dan" Muir (at City University London from 1966 to 1990) has an Erdős–Bacon number of 7: using the name Allan Muir he has an Erdős number of 4 via Nicholas C. Yannelis, Karel Pickry and Kenneth Kunen and appeared as the Cresecent and Red Dog Knights in the film Jabberwocky using the name Dan Muir[28] giving him a Bacon number of 3.

Cyberneticist Kevin Warwick has an Erdős–Bacon number of 6: he has an Erdős number of 4 via Yakov Tsypkin, Vladimir Il'in and István Joó and appeared with Keanu Reeves in Decoded: The Making of The Matrix Reloaded[29] giving him a Bacon Number of 2.

Theoretical physicist Stephen Hawking has an Erdős–Bacon number of 7: perhaps surprisingly, his Bacon number of 3 (via his appearance alongside Patrick Stewart in an episode of Star Trek: The Next Generation) is lower than his Erdős number of 4.

MIT Professor of Applied Mathematics Daniel Kleitman is currently considered to have the joint lowest Erdős–Bacon number of 3, having co-authored a paper with Erdős and appeared as an extra and consultant on Good Will Hunting alongside Minnie Driver who appeared with Bacon in Sleepers.[30]

Metallurgist Tim Foecke has a Bacon Number of 2 [31] (Martin Sheen having been the narrator of the Discovery Channel documentary "Titanic-Anatomy of a Disaster", and co-starred in the movie "JFK" with Kevin Bacon), and an Erdős number of 4 [32] (via Robb Thomson to Peter Bergmann To Ernst G. Straus to Paul Erdős), giving him an Erdos-Bacon Number of 6.


Former NCAA gymnastics champion Kiralee Hayashi[33] may be the professional actress with the lowest Erdős number (3),[citation needed] having co-written a peer-reviewed mathematics paper on Riemannian manifolds with Fields medalist Shing-Tung Yau,[34] and having a Bacon number of 2,[35] giving her an Erdős–Bacon number of 5.[36]

Danica McKellar, most famous for her role as Winnie Cooper in The Wonder Years, has an Erdős–Bacon number of 6, having coauthored a mathematics paper published while an undergraduate at UCLA. Her paper gives her an Erdős number of 4, and a Bacon number of 2, both of them having worked with Margaret Easley.[2]

US actress Natalie Portman has an Erdős–Bacon number of 7. She collaborated (using her birth name, Natalie Hershlag) with Abigail A. Baird,[37] who has a collaboration path[38][39][40] leading to Joseph Gillis, who has an Erdős number of 1.[41] Portman appeared in A Powerful Noise Live (2009) with Sarah Michelle Gellar, who appeared in The Air I Breathe (2007) with Bacon, giving Portman a Bacon number of 2 and an Erdős number of 5.

British actor Colin Firth has an Erdős–Bacon number of 7.[42] Firth is formally credited as co-author of a neuroscience paper, "Political Orientations Are Correlated with Brain Structure in Young Adults",[43] after he suggested on BBC Radio 4 that such a study could be done.[44] Another author of that paper, Geraint Rees, has an Erdős number of 5,[45][46][47] which gives Firth an Erdős number of 6. Firth appeared with Kevin Bacon in Where the Truth Lies, so his Bacon number is 1.[42]

Kevin Bacon himself can be said to have an Erdős-Bacon number of 12, as he has a Bacon number of 0. Through his written contribution to the the Bacon number-related book Degrees, fellow contributor Alan Bates can be linked in six steps[48] to Indian polymath Sugata Mitra, who has an Erdős number of 5 due to publishing alongside mathematician Gaurav Bhatnagar.[49]


Hank Aaron, a baseball player, could be considered to have an Erdős–Bacon number of 3, as he and Erdős both autographed the same baseball (for which he is jokingly referred to as having Erdős number of 1),[50] and he also appeared in Summer Catch with Susan Gardner, who was in In The Cut, with Kevin Bacon.


For people listed in the Internet Movie Database who are connected to Kevin Bacon, the average Bacon number is 2.957.[51] For mathematicians listed in the American Mathematical Society's MR Collaboration Distance search engine[52] who are connected to Erdős, the average Erdős number is 4.65.[53] There currently exists no exhaustive list of people with defined Erdős–Bacon numbers,[citation needed] but a select group is listed below.

Name Erdős number Bacon number Erdős–Bacon number
Albert, DavidDavid Albert 4[54][55][56][57] 3(a)(b)[citation needed] 7(b)
Behe, Michael J.Michael J. Behe 4[15](a) 3[16](a)(b)(d) 7
Bialik, MayimMayim Bialik 5[58] 2[58] 7
Billingsley, PatrickPatrick Billingsley 4[59][60][61][62] 2[63] 6
Dalrymple, DavidDavid Dalrymple 3[64][65][66] 2(b)(d)[67] 5(b)(d)
Dembski, William A.William A. Dembski 4[9][10][11][13] 2(a)(b)[12] 6
Houh, HenryHenry Houh 5[68][69][70][71][72] 2[73][74] 7
Knight, RobertRobert Knight 4[75][76][77][78] 2(d)[79][80] 6(d)
Marks II, Robert J.Robert J. Marks II 3[9][10][11] 2(a)(b)[8] 5
McKellar, DanicaDanica McKellar 4[54][81][82][83] 2(a) 6
Portman, NatalieNatalie Portman (Hershlag) 5[37][38][39][40][41] 2(a) 7
Pritchard, JonathanJonathan Pritchard 4[19][22][84][85] 2[17][18] 6
Sagan, CarlCarl Sagan 4[6] 2(b)[5] 6
Eisenbud, DavidDavid Eisenbud 2[6] 4(b)[5] 6
Yannai Gonczarowski 3[86] 3[86] 6
Stern, TomerTomer Stern 4[87][88][89][90] 3[91][92][93] 7
Werner, WendelinWendelin Werner 3[94][95][96] 3[97] 6
Strogatz, StevenSteven Strogatz 3[98][99][100] 1(b)(d)[101] 4(b)(d)
Chomsky, NoamNoam Chomsky 4[102][103][104][105] 3(b)(d)[101] 7(b)(d)
Chazelle, BernardBernard Chazelle 2[106][107] 3[108] 5
Gates, BillBill Gates 4[109][110][111][112] 2[113] 6


(a) See discussion above.
(b) Includes role as self
(c) Includes role as extra
(d) Includes documentary credit


  1. ^ Singh, Simon (May 1, 2002). "And the winner tonight is". The Telegraph. Archived from the original on November 12, 2012. Retrieved September 26, 2013. 
  2. ^ a b "There's not much separating her from Bacon, Erdos". USA Today. August 14, 2007. Archived from the original on November 4, 2012. 
  3. ^ Collins, James J.; Chow, Carson C. (1998). "It's a small world". Nature 393 (6684): 409–10. doi:10.1038/30835. PMID 9623993. 
  4. ^ "The Erdős Number Project, Paths to Erdős". Retrieved 2009-10-27. [dead link]
  5. ^ a b c "The Oracle of Bacon at Virginia". Retrieved 2013-01-22. [self-published source?]
  6. ^ a b c "The Erdős Number Project, Paths to Erdős". Archived from the original on 2006-11-07. Retrieved 2006-12-01. 
  7. ^ "The Oracle of Bacon at Virginia". Retrieved 2006-12-05. [self-published source?]
  8. ^ a b Robert J. Marks II at the Internet Movie Database[not in citation given]
  9. ^ a b c Wunsch Ii, Donald C.; Marks Ii, Robert J.; Caudell, Thomas P.; Capps, C. David (1992). "Limitations of a class of binary phase-only filters". Applied Optics 31 (26): 5681–7. doi:10.1364/AO.31.005681. PMID 20733753.  and Wunsch, D.C.; Caudell, T.P.; Capps, C.D.; Marks, R.J.; Falk, R.A. (1993). "An optoelectronic implementation of the adaptive resonance neural network". IEEE Transactions on Neural Networks 4 (4): 673–84. doi:10.1109/72.238321. PMID 18267767. 
  10. ^ a b c Harary, Frank; Lim, Meng-Hiot; Agarwal ‡, Amit; Wunsch §, Donald C. (2004). "Algorithms for derivation of structurally stable Hamiltonian signed graphs". International Journal of Computer Mathematics 81 (11): 1349–56. doi:10.1080/0020716042000261432. 
  11. ^ a b c Erdös, Paul; Harary, Frank; Tutte, William T. (2010). "On the dimension of a graph". Mathematika 12 (2): 118–22. doi:10.1112/S0025579300005222.  and Erdös, Paul; Harary, Frank; Klawe, Maria (1980). "Residually-Complete Graphs". Combinatorial Mathematics, Optimal Designs and Their Applications. Annals of Discrete Mathematics 6. pp. 117–23. doi:10.1016/S0167-5060(08)70698-8. ISBN 978-0-444-86048-4. 
  12. ^ a b William Dembski at the Internet Movie Database[not in citation given]
  13. ^ a b Dembski, William A.; Marks Ii, Robert J. (2009). "Conservation of Information in Search: Measuring the Cost of Success". IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans 39 (5): 1051–61. doi:10.1109/TSMCA.2009.2025027. 
  14. ^ Expelled: No Intelligence Allowed at the Internet Movie Database[not in citation given]
  15. ^ a b Marks Ii, Robert J; Behe, Michael J; Dembski, William A; Gordon, Bruce L; Sanford, John C (2013). Biological Information. doi:10.1142/8818. ISBN 978-981-4508-71-1. [page needed]
  16. ^ a b Unlocking the Mystery of Life at the Internet Movie Database[not in citation given]
  17. ^ a b Without Limits at the Internet Movie Database as David Bedford, the English distance runner, in the 1972 Munich Olympics
  18. ^ a b Animal House at the Internet Movie Database with Donald Southerland who portrayed Bill Bowerman in Without Limits and Kevin Bacon
  19. ^ a b Pritchard, Jonathan K.; Seielstad, Mark T.; Perez-Lezaun, Anna; Feldman, Marcus W. (1999). "Population Growth of Human Y. Chromosomes: A Study of Y Chromosome Microsatellites". Molecular Biology and Evolution 16 (12): 1791–1798. doi:10.1093/oxfordjournals.molbev.a026091. PMID 10605120. 
  20. ^ Feldman, Marcus W.; Karlin, Samuel (1971). "The Evolution of Biology: A direct approach through the theory of linkage and selection". Theoretical Population Biology 2 (4): 482–492. doi:10.1016/0040-5809(71)90036-0. 
  21. ^ Karlin, Samuel; Szegö, Gábor (1960). "On certain determinants whose elements are orthogonal polynomials". Journal d’Analyse Mathématique 8 (1): 1–157. doi:10.1007/bf02786848. 
  22. ^ a b Erdős, Paul; Szegö, Gábor (1942). "On a Problem of I. Schur". Annals of Mathematics 43: 451–470. doi:10.2307/1968803. 
  23. ^
  24. ^
  25. ^
  26. ^
  27. ^
  28. ^
  29. ^
  30. ^
  31. ^ Oracle of Bacon - Tim Foecke, retrieved 2012-09-10 
  32. ^ The Erdős Project - Tim Foecke, retrieved 2012-09-10 
  33. ^ "Kiralee Hayashi's website". [self-published source?]
  34. ^ "Paul Thompson's Erdos Number Page". [dead link][self-published source?]
  35. ^ Kiralee Hayashi at the Oracle of Bacon[self-published source?]
  36. ^ "Erdos Number Page". Retrieved December 29, 2011. [dead link][self-published source?]
  37. ^ a b Baird, A; Kagan, J; Gaudette, T; Walz, KA; Hershlag, N; Boas, DA (2002). "Frontal Lobe Activation during Object Permanence: Data from Near-Infrared Spectroscopy". NeuroImage 16 (4): 1120–5. doi:10.1006/nimg.2002.1170. PMID 12202098. 
  38. ^ a b Baird, Abigail A.; Colvin, Mary K.; Vanhorn, John D.; Inati, Souheil; Gazzaniga, Michael S. (2005). "Functional Connectivity: Integrating Behavioral, Diffusion Tensor Imaging, and Functional Magnetic Resonance Imaging Data Sets". Journal of Cognitive Neuroscience 17 (4): 687–93. doi:10.1162/0898929053467569. PMID 15829087. 
  39. ^ a b Victor, Jonathan D.; Maiese, Kenneth; Shapley, Robert; Sidtis, John; Gazzaniga, Michael S. (1989). "Acquired central dyschromatopsia: analysis of a case with preservation of color discrimination". Clinical Vision Sciences 4: 183–96. 
  40. ^ a b Azor, Ruth; Gillis, J.; Victor, J. D. (1982). "Combinatorial Applications of Hermite Polynomials". SIAM Journal on Mathematical Analysis 13 (5): 879–90. doi:10.1137/0513062. 
  41. ^ a b Erdos, P.; Gillis, J. (2009). "Note on the Transfinite Diameter". Journal of the London Mathematical Society (3): 185. doi:10.1112/jlms/s1-12.2.185. 
  42. ^ a b "Where the Truth Lies". Retrieved 2013-11-29. 
  43. ^ Kanai, Ryota; Feilden, Tom; Firth, Colin; Rees, Geraint (2011). "Political Orientations Are Correlated with Brain Structure in Young Adults". Current Biology 21 (8): 677–80. doi:10.1016/j.cub.2011.03.017. PMC 3092984. PMID 21474316. 
  44. ^ "Colin Firth credited in brain research". BBC News. 2011-06-05. 
  45. ^ Schwarzkopf, Dietrich Samuel; Schindler, Andreas; Rees, Geraint (2010). Baker, Chris I, ed. "Knowing with Which Eye We See: Utrocular Discrimination and Eye-Specific Signals in Human Visual Cortex". PLoS ONE 5 (10): e13775. doi:10.1371/journal.pone.0013775. PMC 2966441. PMID 21048942.  open access publication - free to read
  46. ^ "Andreas Schindler". Max Planck Institutefor Biological Cybernetics. Retrieved December 29, 2011. 
  47. ^ "Can I help you find your Erdos number?". Retrieved December 29, 2011. [self-published source?]
  48. ^ Bates wrote in "For Ian Charleson: A Tribute" along with David Puttnam, who co-wrote "Powerhouse UK: British Creativity Now" with Rick Poynor, who wrote "Looking Closer 3: Classic Writings on Graphic Design" with Jessica Helfand, who wrote "Screen: Essays on Graphic Design, New Media, and Visual Culture" with John Maeda, who wrote "Maeda @ Media" with Nicholas Negroponte, who contributed to Mitra's "Beyond the Hole in the Wall: Discover the Power of Self-Organized Learning".
  49. ^ "Publications". Gaurav Bhatnagar. Retrieved 17 February 2015. 
  50. ^ "The Erdős Number Project, Items of Interest Related to Erdös Numbers,". Archived from the original on 2006-12-05. Retrieved 2006-12-20. 
  51. ^ UVA Computer Science: The Oracle of Bacon at Virginia[dead link][not in citation given][self-published source?]
  52. ^ MR Collaboration Distance search engine
  53. ^ Erdős number facts[dead link]
  54. ^ a b "The Erdős Number Project, Erdos1". Archived from the original on 2006-12-07. Retrieved 2006-12-20. 
  55. ^ Aharonov, Yakir; Albert, David; Vaidman, Lev (1988). "How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100". Physical Review Letters 60 (14): 1351–4. doi:10.1103/PhysRevLett.60.1351. PMID 10038016. 
  56. ^ Aharonov, Yakir; Bergmann, Peter; Lebowitz, Joel (1964). "Time Symmetry in the Quantum Process of Measurement". Physical Review 134 (6B): B1410–6. doi:10.1103/PhysRev.134.B1410. 
  57. ^ B Hoffmann, V Bargmann, PG Bergmann, EG Straus, "Working with Einstein" in Some strangeness in the proportion : a centennial symposium to celebrate the achievements of Albert Einstein, 1980.
  58. ^ a b "Erdős Bacon Sabbath Project › Mayim Bialik". Retrieved 2014-02-09. 
  59. ^ Baum, Leonard E.; Billingsley, Patrick (1965). "Asymptotic Distributions for the Coupon Collector's Problem". The Annals of Mathematical Statistics 36 (6): 1835–9. doi:10.1214/aoms/1177699813. JSTOR 2239126. 
  60. ^ Baum, Leonard E; Herzberg, Norman P; Lomonaco, S.J; Sweet, Melvin M (1977). "Fields of almost periodic sequences". Journal of Combinatorial Theory, Series A 22 (2): 169–80. doi:10.1016/0097-3165(77)90049-8. 
  61. ^ Conway, J.H.; Lomonaco, S.J.; Sloane, N.J.A. (1990). "A [45,13] code with minimal distance 16". Discrete Mathematics 83 (2–3): 213–7. doi:10.1016/0012-365X(90)90007-5. 
  62. ^ Conway, J. H.; Croft, H. T.; Erdos, P.; Guy, M. J. T. (1979). "On the Distribution of Values of Angles Determined by Coplanar Points". Journal of the London Mathematical Society. s2-19 (1): 137–43. doi:10.1112/jlms/s2-19.1.137. 
  63. ^ Pat Billingsley at the Oracle of Bacon[self-published source?]
  64. ^ Gershenfeld, Neil; Dalrymple, David; Chen, Kailiang; Knaian, Ara; Green, Forrest; Demaine, Erik D.; Greenwald, Scott; Schmidt-Nielsen, Peter (2010). "Reconfigurable asynchronous logic automata". Proceedings of the 37th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '10. pp. 1–6. doi:10.1145/1706299.1706301. ISBN 978-1-60558-479-9. 
  65. ^ Alon, Noga; Bădoiu, Mihai; Demaine, Erik D.; Farach-Colton, Martin; Hajiaghayi, Mohammadtaghi; Sidiropoulos, Anastasios (2008). "Ordinal embeddings of minimum relaxation". ACM Transactions on Algorithms 4 (4): 1–21. doi:10.1145/1383369.1383377. 
  66. ^ Alon, Noga; Erdös, P. (1985). "An Application of Graph Theory to Additive Number Theory". European Journal of Combinatorics 6 (3): 201–3. doi:10.1016/S0195-6698(85)80027-5. 
  67. ^ "David Dalrymple's Kevin Bacon number". The Oracle of Bacon. Retrieved 2012-07-24. [self-published source?]
  68. ^ Houh, H.H.; Adam, J.F.; Ismert, M.; Lindblad, C.J.; Tennenhouse, D.L. (1995). "The VuNet desk area network: Architecture, implementation, and experience". IEEE Journal on Selected Areas in Communications 13 (4): 710–21. doi:10.1109/49.382161. 
  69. ^ Tennenhouse, David; Lampson, Butler; Gillett, Sharon Eisner; Klein, Jennifer Steiner (1996). "Virtual infrastructure: Putting information infrastructure on the technology curve". Computer Networks and ISDN Systems 28 (13): 1769–90. doi:10.1016/0169-7552(96)00009-8. 
  70. ^ C Ellison, B Frantz, B Lampson, R Rivest, B Thomas, T Ylonen, "SPKI Certificate Theory", IETF Network Working Group RFC2693, September 1999
  71. ^ Rivest, R.L.; Meyer, A.R.; Kleitman, D.J.; Winklmann, K.; Spencer, J. (1980). "Coping with errors in binary search procedures". Journal of Computer and System Sciences 20 (3): 396–404. doi:10.1016/0022-0000(80)90014-8. 
  72. ^ Erdös, Paul; Kleitman, Daniel J. (1968). "On coloring graphs to maximize the proportion of multicolored k-edges". Journal of Combinatorial Theory 5 (2): 164–9. doi:10.1016/S0021-9800(68)80051-1. 
  73. ^ 21 at the Internet Movie Database as "Chinatown Dealer" with line "Winner winner chicken dinner"[not in citation given]
  74. ^ Quicksilver at the Internet Movie Database with Laurence Fishburne from 21 appearing with Kevin Bacon[not in citation given]
  75. ^ Chang, Edward F.; Edwards, Erik; Nagarajan, Srikantan S.; Fogelson, Noa; Dalal, Sarang S.; Canolty, Ryan T.; Kirsch, Heidi E.; Barbaro, Nicholas M.; Knight, Robert T. (2011). "Cortical Spatio-temporal Dynamics Underlying Phonological Target Detection in Humans". Journal of Cognitive Neuroscience 23 (6): 1437–46. doi:10.1162/jocn.2010.21466. PMID 20465359. 
  76. ^ Dalal, S.S.; Baillet, S.; Adam, C.; Ducorps, A.; Schwartz, D.; Jerbi, K.; Bertrand, O.; Garnero, L.; Martinerie, J.; Lachaux, J.P. (2009). "MEG reconstructions of gamma band modulations during attentive reading validated by simultaneous intracranial EEG". NeuroImage 47: S162. doi:10.1016/S1053-8119(09)71718-5. 
  77. ^ Raiter, Y; Farfel, A; Lehavi, O; Goren, O B; Shamiss, A; Priel, Z; Koren, I; Davidson, B; Schwartz, D; Goldberg, A; Bar-Dayan, Y (2008). "Mass casualty incident management, triage, injury distribution of casualties and rate of arrival of casualties at the hospitals: Lessons from a suicide bomber attack in downtown Tel Aviv". Emergency Medicine Journal 25 (4): 225–9. doi:10.1136/emj.2007.052399. PMID 18356360. 
  78. ^ Erdös, P.; Koren, I.; Moran, S.; Silberman, G. M.; Zaks, S. (1988). "Minimum-diameter cyclic arrangements in mapping data-flow graphs onto VLSI arrays". Mathematical Systems Theory 21: 85–98. doi:10.1007/BF02088008. 
  79. ^ Number One with a Bullet at the Internet Movie Database as Himself[not in citation given]
  80. ^ The Woodsman at the Internet Movie Database with Mos Def from Number One With A Bullet appearing with Kevin Bacon[not in citation given]
  81. ^ Chayes, L; McKellar, D; Winn, B (1998). "Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on". Journal of Physics A: Mathematical and General 31 (45): 9055. doi:10.1088/0305-4470/31/45/005. 
  82. ^ Chayes, J. T.; Chayes, L.; Kotecký, R. (1995). "The analysis of the Widom-Rowlinson model by stochastic geometric methods". Communications in Mathematical Physics 172 (3): 551. doi:10.1007/BF02101808. 
  83. ^ Kotecký, R.; Preiss, D. (1986). "Cluster expansion for abstract polymer models". Communications in Mathematical Physics 103 (3): 491–8. doi:10.1007/BF01211762. 
  84. ^ Feldman, Marcus W.; Karlin, Samuel (1971). "The Evolution of Biology: A direct approach through the theory of linkage and selection". Theoretical Population Biology 2 (4): 482–492. doi:10.1016/0040-5809(71)90036-0. 
  85. ^ Karlin, Samuel; Szegö, Gábor (1960). "On certain determinants whose elements are orthogonal polynomials". Journal d’Analyse Mathématique 8 (1): 1–157. doi:10.1007/bf02786848. 
  86. ^ a b "Yannai Gonczarowski's website". [self-published source?]
  87. ^ Wasserstrom, Adam; Frumkin, Dan; Adar, Rivka; Itzkovitz, Shalev; Stern, Tomer; Kaplan, Shai; Shefer, Gabi; Shur, Irena; Zangi, Lior; Reizel, Yitzhak; Harmelin, Alon; Dor, Yuval; Dekel, Nava; Reisner, Yair; Benayahu, Dafna; Tzahor, Eldad; Segal, Eran; Shapiro, Ehud (2008). Bourne, Philip E, ed. "Estimating Cell Depth from Somatic Mutations". PLoS Computational Biology 4 (5): e1000058. doi:10.1371/journal.pcbi.1000058. PMC 2275312. PMID 18404205.  open access publication - free to read
  88. ^ Frumkin, Dan; Wasserstrom, Adam; Kaplan, Shai; Feige, Uriel; Shapiro, Ehud (2005). "Genomic Variability within an Organism Exposes Its Cell Lineage Tree". PLoS Computational Biology 1 (5): e50. doi:10.1371/journal.pcbi.0010050. PMC 1274291. PMID 16261192.  open access publication - free to read
  89. ^ Feige, Uriel; Goldwasser, Shafi; Lovász, Laszlo; Safra, Shmuel; Szegedy, Mario (1996). "Interactive proofs and the hardness of approximating cliques". Journal of the ACM 43 (2): 268–92. doi:10.1145/226643.226652. 
  90. ^ On the graph of large distance (1989)[page needed]
  91. ^ Etz Hadomim Tafus at the Internet Movie Database[not in citation given]
  92. ^ Every Time We Say Goodbye at the Internet Movie Database[not in citation given]
  93. ^ Apollo 13 at the Internet Movie Database[not in citation given]
  94. ^ Shi, Zhan; Werner, Wendelin (1995). "Asymptotics for occupation times of half-lines by stable processes and perturbed reflecting brownian motion". Stochastics an International Journal of Probability and Stochastic Processes 55: 71. doi:10.1080/17442509508834019. 
  95. ^ Csáki, Endre; Shi, Zhan (1998). "Large Favourite Sites of Simple Random Walk and the Wiener Process". Electronic Journal of Probability 3 (14): 1–31. doi:10.1214/EJP.v3-36. 
  96. ^ Csáki, E.; Erdős, P.; Révész, P. (1985). "On the length of the longest excursion". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 68 (3): 365–82. doi:10.1007/BF00532646. 
  97. ^ Wendelin Werner at the Oracle of Bacon[self-published source?]
  98. ^ Stone, H. A.; Nadim, Ali; Strogatz, Steven H. (2006). "Chaotic streamlines inside drops immersed in steady Stokes flows". Journal of Fluid Mechanics 232: 629–46. doi:10.1017/S002211209100383X. 
  99. ^ Limon, Alfonso; Bertuglia, Silvia; Nadim, Ali; Salamon, Peter (2006). "Oxygen transport from the outer boundary of a pulsating wall of an arteriole". Mathematics and Computers in Simulation 73 (1–4): 175–82. doi:10.1016/j.matcom.2006.06.002. 
  100. ^ Salamon, Peter; Erdős, Paul (1988). "The solution to a problem of Grünbaum". Canadian Mathematical Bulletin 31 (2): 129–38. doi:10.4153/CMB-1988-020-2. 
  101. ^ a b Connected: The Power of Six Degrees at the Internet Movie Database
  102. ^ Chomsky, N.; Schützenberger, M.P. (1963). "The Algebraic Theory of Context-Free Languages". Computer Programming and Formal Systems. Studies in Logic and the Foundations of Mathematics 35. pp. 118–61. doi:10.1016/S0049-237X(08)72023-8. ISBN 978-0-444-53400-2. 
  103. ^ Berge, Claude; Schützenberger, Marcel Paul (1956). "Jeux de Nim et solutions". C. R. Acad. Sci. Paris (in French) 242: 1672–4. 
  104. ^ Berge, C.; Simonovits, M. (1974). "The coloring numbers of the direct product of two hypergraphs". Hypergraph Seminar. Lecture Notes in Mathematics 411. pp. 21–33. doi:10.1007/BFb0066176. ISBN 978-3-540-06846-4. 
  105. ^ Erdős, P.; Simonovits, M. (1966). "A limit theorem in graph theory". Studia Sci. Math. Hungar 1: 51–7. 
  106. ^ Brönnimann, Hervé; Chazelle, Bernard; Pach, János (1993). "How hard is half-space range searching?". Discrete & Computational Geometry 10: 143–55. doi:10.1007/BF02573971. 
  107. ^ Erdős, P.; Pach, J. (1980). "On a problem of L. Fejes Tóth". Discrete Mathematics 30 (2): 103–9. doi:10.1016/0012-365X(80)90111-9. 
  108. ^ Bernard Chazelle at the Oracle of Bacon[self-published source?]
  109. ^ Gates, William H; Papadimitriou, Christos (1979). "Bounds for sorting by prefix reversals". Discrete Mathematics 27 (1): 47–57. 
  110. ^ Borgs, Christian; Chayes, Jennifer; Immorlica, Nicole; Tauman Kalai, Adam; Mirrokni; Papadimitriou, Christos (2010). "The myth of the Folk Theorem". Society for Industrial and Applied Mathematics 23 (4): 864–894. 
  111. ^ Dahlhaus, E.; Dankelmann, P.; Goddard, W.; Yannakakis, M. H (2003). "MAD trees and distance-hereditary graphs". Discrete Applied Mathematics 131 (1): 151–167. 
  112. ^ Aronov, B.; Erdős, P.; Goddard, W.; Kleitman, D.J.; Klugerman, M.; Pach, J.; Schulman, L.J. (1994). "Crossing families". Combinatorica 14 (3): 127–134. 
  113. ^ [1]