Global Utilities

Department of Mathematics and Statistics

Dr Grant Cairns

Publications/Preprints

  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 file), 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 file), 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 file), 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 file), 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 file), 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 file), Proc. Amer. Math. Soc., 125 (1997), 381-385.
  24. with R. Escobales, 'Note on a theorem of Gromoll-Grove' (pdf file), 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 file), J. Austral. Math. Soc. Ser. A, 62 (1997), 46-63.
  26. with B. Jessup, 'New bounds on the Betti numbers of nilpotent Lie algebras' (dvi), Comm. Algebra, 25 (1997), 415-430.
  27. with E. Ghys, 'The local linearization problem for smooth SL(n)-actions' (pdf file), Enseign. Math., 43 (1997), 133-171.
  28. with B. Jessup and M. Nicolau, 'Topologically transitive homeomorphisms of quotients of tori' (pdf file), J. Discrete Contin. Dynam. Systems, 5 (1999), 291-300.
  29. with G. Armstrong and G. Kim, 'Lie algebras of cohomological codimension one' (pdf file), Proc. Amer. Math. Soc., 127 (1999), 709-714.
  30. with D.M. King, 'The Answer to Woodall's Musquash Problem' (pdf file), Discrete Math., 207 (1999), 25-32.
  31. with G. Elton and P.J. Stacey, 'On the definition of collineation' (pdf file), Math. Mag., 72 (1999), 401.
  32. with Y. Nikolayevsky, 'Bounds for Generalized Thrackles' (pdf file), Discrete Comput. Geom., 23 (2000), 191-206.
  33. with J. Bamberg, 'Torsion Free Groups Generated by A Pair of Rational Parabolic Mobius Transformations' (pdf file), Bull. Austral. Math. Soc., 61 (2000), 151-152.
  34. with G. Kim, 'Lie Algebras of Least Cohomology' (pdf file), J. Lie Theory, 10 (2000), 435-441.
  35. with D.M. King, 'All Odd Musquashes are Standard' (pdf file), Discrete Math., 226 (2001), 71-91.
  36. with G. Byrnes and B. Jessup, 'Leftovers from the Ham Sandwich Theorem' (pdf file), Amer. Math. Monthly, 108 (2001), 246-249.
  37. with G. Kim, 'The mod 4 behaviour of total Lie algebra cohomology' (pdf file), Arch. Math., 77 (2001), 177-180.
  38. 'Queens on Non-square Tori' (pdf file), Electron. J. Combin., 8(1) (2001) N6.
  39. with P. Molino, 'Weakly Involutive Totally Geodesic Distributions of Constant Rank' (pdf file), 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 file), J. Lie Theory, 12 (2002), 205-216.
  41. 'Pillow chess' (pdf file), Math. Mag., 75 (2002), 173-186.
  42. with J. Bamberg and D. Kilminster, `The Crystallographic Restriction, permutations, and Goldbach's Conjecture' (pdf file), Amer. Math. Monthly, 110 (2003), 202-209. (See Errata and Addenda below)
  43. with B. Jessup, `Cohomology Operations for Lie Algebras' (pdf file), Trans. Amer. Math. Soc., 356 (2004), 1569-1583.
  44. with M. McIntyre and Y. Nikolayevsky, 'The Thrackle Conjecture for K_5 and K_{3,3}' (pdf file), 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, this file is a Mathematica binary dump file containing the list of 2^15 triples of mutually compatible intersection tables, needed in the consideration of K_{3,3}. 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 file), Austral. Math. Soc. Gaz. 31 (2004) 251-255.
  46. Is there a greater role for prime numbers in our schools?, Austral. Sen. Math. J. 19 (2005) 24-37, (pdf file).
  47. with A. Nielsen, 'On the Dynamics of the Linear Action of SL(n,Z)' (pdf file), Bull. Austral. Math. Soc., 71 (2005), 359 - 365.
  48. Els nombres primers poden tenir mès protagonisme a Secundària?, Butlletì de la Societat Catalana de Matemàtiques, 20 (2005) 75-89, (pdf file).
  49. with Korrakot Chartarrayawadee, 'Brussels Sprouts and Cloves', Math. Mag, 80 (2007) 46-58, (pdf file).
  50. with A. Kolganova and A. Nielsen, 'Topological Transitivity and Mixing Notions for Group Actions' (pdf file), 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 file), Topology Appl. 155 (2007) 614-617.
  52. with B. Jessup, 'Free submodules for the central representation in the cohomology of Lie algebras', (pdf file), Proc. Amer. Math. Soc., 136 (2008) 1919-1923.
Preprints
  • with S. Jambor, 'The cohomology of the Heisenberg Lie algebras over fields of finite characteristic', (pdf file), to appear in Proc. Amer. Maths Soc.
  • with Y. Nikolayevsky, 'Generalized Thrackle Drawings of Non-Bipartite Graphs', (pdf file).
  • with Barry Jessup, 'A special family of positively graded algebras', (pdf file). Since submitting this paper we have learnt that the main result is contained in "Graded filiform Lie algebras and symplectic nilmanifolds", by D.V. Millionschikov, in "Geometry, Topology, and Mathematical Physics", AMS Translations, 2004. We have withdrawn our paper.
  • with J. Banks, M. Jackson, M. Jerie, Y. Nikolayevsky and T. Poole, 'On The Definition of a Topological Space', (pdf file), to appear in Austral. Math. Soc. Gaz.
  • with Christian Aebi, 'Catalan Numbers, Primes and Twin Primes'
    (pdf file).
Errata and Addenda
  • 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.

 

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Last Updated: 18 May, 2008

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