Global Utilities

La Trobe University
Department of Mathematics and Statistics

Publications by Dr Grant Cairns

List of publications

  1. 'Géométrie globale des feuilletages totalement géodésiques', C.R. Acad. Sci. Paris, 297 (1983), 525-527.
  2. 'Feuilletages totalement géodésiques de dimension 1, 2 ou 3', C.R. Acad. Sci. Paris, 298 (1984), 341-344.
  3. 'Aspects cohomologiques des feuilletages totalement géodésiques', C.R. Acad. Sci. Paris, 299 (1984), 1017-1019.
  4. 'A general description of totally geodesic foliations', Tòhoku Math. J. 38 (1986), 37-55.
  5. 'Some properties of a cohomology group associated to a totally geodesic foliation', Math. Zeitschrift, 192 (1986), 391-403.
  6. with Etienne Ghys, 'Totally geodesic foliations on 4-manifolds', J. Diff. Geom. 23 (1986), 241-254.
  7. 'Feuilletages totalement géodésiques sur les variétés simplement connexes', Feuilletages riemanniens (P. Dazord, N. Desolneux-Moulis, J.-M. Morvan eds.) Travaux en cours 26 (1988), Herman Paris, 1-14.
  8. 'The duality between Riemannian foliations and geodesible foliations', Riemannian Foliations by Pierre Molino, Progress in Math. 73, Birkhäuser Boston, (1988), 249-263.
  9. 'Totally umbilic Riemannian foliations' (PDF 341KB), Michigan Math. J. 37 (1990), 145-159.
  10. 'Compact 4-manifolds that admit totally umbilic foliations', Differential Geometry and its Applications, ed. J. Janyska and D. Krupka, World Scientific (1990), 9-16.
  11. with R. Sharpe, 'On the inversive differential geometry of plane curves', Enseign. Math. 36 (1990), 175-196.
  12. with J. Banks, J. Brooks, G. Davis and P. Stacey, 'On Devaney's definition of chaos' (PDF 297KB), Amer. Math. Monthly, 99 (1992), 332-334.
  13. with Mehmet Özdemir and Ekkehard-H. Tjaden, 'A counterexample to a conjecture of U. Pinkall', Topology, 31 (1992), 557-558.
  14. with M. McIntyre and M. Özdemir, 'A six vertex theorem for bounding normal planar curves', Bull. London Math. Soc. 25 (1993), 169-176.
  15. with M. McIntyre, 'A new formula for winding number', Geom. Dedicata, 46 (1993), 149-160.
  16. with M. McIntyre and J. Strantzen, 'Geometric proofs of recent results of Yang Lu', Math. Mag. 66 (1993), 263-265.
  17. with D. Elton, 'The planarity problem for signed Gauss words' (PDF 445KB), J. Knot Theory and its Ramifications, 2 (1993), 359-367.
  18. with R. Sharpe and L. Webb, 'Conformal invariants of curves and surfaces in 3-dimensional space forms', Rocky Mount. J. Math. 24 (1994), 933-959.
  19. with G. Davis, D. Elton, A. Kolganova, P. Perversi, 'Chaotic group actions' (dvi) Enseign. Math. 41 (1995), 123-133.
  20. with A. Kolganova, 'Chaotic actions of free groups' (dvi), Nonlinearity, 9 (1996), 1015-1021.
  21. with D. Elton, 'The planarity problem II' (PDF 372KB), J. Knot Theory and its Ramifications, 5 (1996), 137-144.
  22. with B. Jessup and J. Pitkethley, 'Some remarks on the cohomology of nilpotent Lie algebras of small dimension' (PDF 488KB), Integrable Systems and Foliations (C. Albert, R. Brouzet, J.-P. Dufour eds.) Birkhäuser Boston, 1997.
  23. with G. Armstrong and B. Jessup, 'Explicit Betti numbers for a family of nilpotent Lie algebras' (PDF 189KB), Proc. Amer. Math. Soc., 125 (1997), 381-385.
  24. with R. Escobales, 'Note on a theorem of Gromoll-Grove' (PDF 331KB), Bull. Austral. Math. Soc., 55 (1997), 1-5.
  25. with R. Escobales, 'Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold' (PDF 461KB), J. Austral. Math. Soc. Ser. A, 62 (1997), 46-63.
  26. with B. Jessup, 'New bounds on the Betti numbers of nilpotent Lie algebras' (PDF 413KB) (dvi), Comm. Algebra, 25 (1997), 415-430.
  27. with E. Ghys, 'The local linearization problem for smooth SL(n)-actions' (PDF 701KB), Enseign. Math., 43 (1997), 133-171.
  28. with B. Jessup and M. Nicolau, 'Topologically transitive homeomorphisms of quotients of tori' (PDF 515KB), J. Discrete Contin. Dynam. Systems, 5 (1999), 291-300.
  29. with G. Armstrong and G. Kim, 'Lie algebras of cohomological codimension one' (PDF 189KB), Proc. Amer. Math. Soc., 127 (1999), 709-714.
  30. with D.M. King, 'The Answer to Woodall's Musquash Problem' (PDF 166KB), Discrete Math., 207 (1999), 25-32.
  31. with G. Elton and P.J. Stacey, 'On the definition of collineation' (PDF 255KB), Math. Mag., 72 (1999), 401.
  32. with Y. Nikolayevsky, 'Bounds for Generalized Thrackles' (PDF 264KB), Discrete Comput. Geom., 23 (2000), 191-206.
  33. with J. Bamberg, 'Torsion Free Groups Generated by A Pair of Rational Parabolic Mobius Transformations' (PDF 430KB), Bull. Austral. Math. Soc., 61 (2000), 151-152.
  34. with G. Kim, 'Lie Algebras of Least Cohomology' (PDF 128KB), J. Lie Theory, 10 (2000), 435-441.
  35. with D.M. King, 'All Odd Musquashes are Standard' (PDF 199KB), Discrete Math., 226 (2001), 71-91.
  36. with G. Byrnes and B. Jessup, 'Leftovers from the Ham Sandwich Theorem' (PDF 312KB), Amer. Math. Monthly, 108 (2001), 246-249.
  37. with G. Kim, 'The mod 4 behaviour of total Lie algebra cohomology' (PDF 176KB), Arch. Math., 77 (2001), 177-180.
  38. 'Queens on Non-square Tori' (PDF 303KB), Electron. J. Combin., 8(1) (2001) N6.
  39. with P. Molino, 'Weakly Involutive Totally Geodesic Distributions of Constant Rank' (PDF 573KB), in "Essays on Geometry and Related Topics: Mémoires dédiés à André Haefliger" (E. Ghys, P. de la Harpe, V.F.R. Jones, V. Sergiescu, T. Tsuboi, eds.), L'Enseign. Math., Geneva 2001, pp. 177-204.
  40. 'The Approximative Centre of a Lie Algebra' (PDF 164KB), J. Lie Theory, 12 (2002), 205-216.
  41. 'Pillow chess', Math. Mag., 75 (2002), 173-186.
  42. with J. Bamberg and D. Kilminster, 'The Crystallographic Restriction, permutations, and Goldbach's Conjecture' (PDF 138KB), Amer. Math. Monthly, 110 (2003), 202-209, (see errata and addenda below).
  43. with B. Jessup, 'Cohomology Operations for Lie Algebras' (PDF 251KB), Trans. Amer. Math. Soc., 356 (2004), 1569-1583.
  44. with M. McIntyre and Y. Nikolayevsky, 'The Thrackle Conjecture for K5 and K3,3' (PDF 581KB), in "Towards a Theory of Geometric Graphs", Contemp. Math., Vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp. 35-54.
    • For people interested in examining the proof closely, the Mathematica binary dump file (MX binary file 2830KB) contains the list of 2^15 triples of mutually compatible intersection tables, needed in the consideration of K3,3. The list is named: lst4. It can be loaded into a Mathematica notebook by entering: << k33orders
  45. with D. van Golstein Brouwers and J. Bamberg, 'Totally Goldbach Numbers and Related Conjectures' (PDF 126KB), Austral. Math. Soc. Gaz. 31 (2004) 251-255.
  46. 'Is there a greater role for prime numbers in our schools? (PDF 210KB)', Austral. Sen. Math. J. 19 (2005) 24-37.
  47. with A. Nielsen, 'On the Dynamics of the Linear Action of SL(n,Z)' (PDF 426KB), Bull. Austral. Math. Soc., 71 (2005), 359 - 365.
  48. 'Els nombres primers poden tenir mès protagonisme a Secundària?' (PDF 147KB), Butlletì de la Societat Catalana de Matemàtiques, 20 (2005) 75-89.
  49. with Korrakot Chartarrayawadee, 'Brussels Sprouts and Cloves' (PDF 186KB), Math. Mag, 80 (2007) 46-58.
  50. with A. Kolganova and A. Nielsen, 'Topological Transitivity and Mixing Notions for Group Actions' (PDF 215KB), Rocky Mountain J. Math. 37 (2007) 371-397.
  51. with Thanh Duong Pham, 'An example of a chaotic group action on the plane by compactly supported homeomorphisms' (PDF 304KB), Topology Appl. 155 (2007) 614-617.
  52. with B. Jessup, 'Free submodules for the central representation in the cohomology of Lie algebras' (PDF 126KB), Proc. Amer. Math. Soc., 136 (2008) 1919-1923.
  53. with S. Jambor, 'The cohomology of the Heisenberg Lie algebras over fields of finite characteristic' (PDF 109KB), Proc. Amer. Math. Soc., 136 (2008), no. 11, 3803-3807.
  54. with J. Banks, M. Jackson, M. Jerie, Y. Nikolayevsky and T. Poole, 'On The Definition of a Topological Space' (PDF 141KB), Austral. Math. Soc. Gaz. 35 (2008), no. 3, 195-202.
  55. with Christian Aebi, 'Catalan Numbers, Primes and Twin Primes' (PDF 215KB), Elemente der Mathematik 63 (2008), no. 4, 153–164.
  56. with Christian Aebi, 'Partitions of Primes' (PDF 259KB), Parabola 45 (2009), no. 1, 3-12.
  57. with Suhua Wang, Enhui Shi and Lizhen Zhou, 'Topological transitivity of solvable group actions on the line R' (PDF 373KB), Colloq. Math. 116 (2009), no. 2, 203-215.
  58. with Y. Nikolayevsky, 'Generalized Thrackle Drawings of Non-Bipartite Graphs' (PDF 183KB), Discrete and Computational Geometry 41 (2009), no. 1, 119-134.

Preprints

  1. 'Log-concavity of the cohomology of nilpotent Lie algebras in characteristic two' (PDF 88KB), to appear in J. Gen. Lie Theory Appl.
  2. with Yury Nikolayevsky, 'Outerplanar Thrackles' (PDF 223KB).
  3. with Christian Aebi, 'Morley's other miracle' (PDF 99KB).
  4. with Nhan Bao Ho, entitled 'Impartial Games as Acyclic Digraphs' (PDF 405KB).

Errata and Addenda

  1. Paper 42: In Table 1 on page 204, we are missing some entries:
    • for n=20, we are missing 2520
    • for n=22, we miss 2310
    • for n=24, we miss 2184, 2340, 2640, 2730, 2772, 3080, 3960, 4620, 5040
    For more information, see N.J.A. Sloane's On-Line Encyclopaedia of Integer Sequences, entries A080736-A080742.
  2. In the Theorem and in Corollary 1: "for all $i\le m$" should be "for all $n\le m$".

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