Grant Cairns  homepage

Assoc Prof Grant Cairns. 

Department of Mathematics and Statistics

La Trobe University

Melbourne VIC 3086, Australia

Office Location: Room 217, Physical Sciences 2 

Email: G.Cairns `AT'  latrobe.edu.au

Who am I?

I was born in Sydney in 1954 but grew up in Brisbane. I studied electrical engineering at the University of Queensland, before doing a doctorate in differential geometry in Montpellier, France, under the direction of Pierre Molino. I spent two years as an assistant at the University of Geneva, and a one year postdoc at the University of Waterloo, Canada, before coming to La Trobe in 1988. I retired in early 2017.

Errata and Addenda

The main purpose of this webpage is to list errata and addenda of my papers. The paper numbers refer to the list given further below.

• Paper 7: There are so many bad typos in this paper I would struggle to itemize them. For instance, the unit interval appears as [1,0] (!!). The problem was that I never had the opportunity to correct the proofs of this paper. This was the best work of my early career, and the way it appeared was very disappointing. A much cleaner version appeared as an appendix in my thesis.

• Paper 22: After this paper appeared we learnt that Conjecture 2 had been previously resolved in a paper by H. Koch that wasn't reviewed in Mathscinet. The paper is: H. Koch, "Generator and relation ranks for finite-dimensional nilpotent Lie algebras" (English. Russian original), Algebra Logic 16, 246-253 (1978); translation from Algebra Logika 16, 364-374 (1977).

• 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 Sequence,entries A080736-A080742.

• Paper 50: on the 3rd line of proof of Prop 7, p.389,  a hat is missing; the condition should be: $\Phi$ strongly mixing $\Rightarrow\widehat{\Phi}$ strongly mixing. 

• Paper 53: In the Theorem and in Corollary 1: "for all $i\leq m$" should be "for all $n\leq m$". Also, on page 3805, line 2:   $A_m^i \to  A_m^{i+2}$  should be $A_m^i \to  A_m^{i+2k}$. 

• Paper 55: Gottfried Barthel kindly pointed out that on p.158, following the proof of Theorem 3, the numbers 561 and 563 are in the wrong order: 561 = 3 * 11 * 17 is a pseudoprime, whereas 563 is prime. 

• Paper 61: In the third line of the proof of Lemma 7, "by Remark 2" should be "by Remark 5".

• Paper 65: For other proofs of the irrationality of root 2, see http://www.cut-the-knot.org/proofs/sq_root.shtml. A closely related classic construction, on the pentagon, is given in http://www.hellenicaworld.com/Greece/Science/en/Pentagon.html. 

• Paper 72: On the last page, [3, Corollary 1], should be [3, Theorem 3].

• Paper 75: while the statement and proof of Theorem 1 are correct, the sentence following the statement of the theorem, which purports to express the result in different terms, is not correct. See Karen Collins and Cleo Roberts, `Great-circle tree thrackles', Discrete Math. 346 (2023), no. 1, Paper No. 113121, 10~pp.

• Paper 78: on the 7th line after (14), "B_{p^3-p^2-2} or B_{p^3-p^2-2}” should be  "the denominator of B_{p^3-p^2-2} or B_{p^3-p^2-4}”.

• Paper 84: Michael Maltenfort kindly pointed out the following slip-ups:
  1. The list of "8 possible triples" on page 519 only lists seven; the triplet (0,2,0) is missing.
  2. Just preceding the table on page 521, the claim that the proficiencies "remain constant if and only if c_1 = 2” isn’t correct. For this reason, the last cell of the table should begin with "does not increase if c_1 >2" (rather than "decreases if c_1 > 2").  The proof of Proposition 1 is still valid, because of the consideration of the pair (c_1), (c_2, c_3, ..., c_n).
  3. Towards the end of the paper, the words  "diameters" and "diagonals” are both used. There is no intended difference here. I should just have used “diameters” throughout.

• Paper 88: Three typos:

1. Page 371, 4 lines before the statement of Theorem 1: our main aim is this paper in to study -> our main aim in this paper is to study

2. Page 374, the second last line of Remark 2: "then $a_i$ is rational for each $i$” should be "then $\sqrt{a_i}$ is rational for each $i$”

3. Page 393, second last line of the proof of Proposition 4, "Then $a^2$ is $x^2+1$ or $x^2+2$" should be "Then $a^2$ is $x^2+1$ or $x^2+4$". 

• Paper 89: thus from [1, Lemma 6b] -> thus from [1, Lemma 5b]

• Paper 90: Several minor issues:

1. there is a little slip in the way the intro to LEQIII is written. In the intro on page 4, where we introduce Sigma and T, we should have stated that we are assuming that the excircle lies outside the vertex B, so that a+b=c+d, and not a+d=b+c. This is clear in the body of the paper.

2. There is a typo in the proof of Proposition 3, on Page 10. On line 3 of the proof, rc should be rd. Then on the next (displayed) line, c should be d too; this occurs twice on this line.

3. Albertas Zinevičius has kindly pointed out to us that Remark 29 is not correct. The simplest solution is to change the hypothesis of Cases 5 and 6.

Case 5 should be: Assume $u, v$ are both odd and that $u+v≡0 \pmod4$.

Case 6 should be: Assume $u, v$ are both odd and that $u+v≡2 \pmod4$.

Then:

Remark 28 should just be: Note that $f(u)+f(v)≡u+v≡0 \pmod4$.

Remark 29 should just be: Note that $f(u)+f(v)≡u+v≡2 \pmod4$.

Also, on Page 95, lines 6-7, “as the 2-adic order of $u + v$ is odd” should be “as $u + v\equiv 2 \mod 4$”

4. In the first line of the proof of Lemma 44, "By the previous lemma” should be “By Lemma 41”

5. There is also a typo on Page 95, line -5: "Thus, as $u+v$ is even and square free" should be "Thus, as $f(u+v)$ is even and square free" 

• Paper 94: Albertas Zinevičius has kindly pointed out to us that there is an error in proof of Theorem 6 (a silly error due to working at the same time on projects in the Gaussian integers and the Eisenstein integers - my fault entirely). Here is an amended proof:

Suppose we have an ELEP with a vertical diagonal. By translating and reflecting in the $x$ and/or $y$ axes if necessary, we may assume that the vertical diagonal lies on the positive $y$-axis, starting at the origin $0$, and that the side starting at $0$, and lying in the 1st or 2nd quadrants, is the shorter of the two sides. Therefore, suppose we have an ELEP with vertices $A=x+y\omega, B=z(1+2\omega),C=(z-x)+(2z-y)\omega, O=0$, where $x,z\in\N$ and $y\in \Z$.  Let $a\sqrt3,b\sqrt3$ denote the lengths of $OA$ and $AB$ respectively, with $a,b\in\N$ and $a<b$. In particular, we have \begin{equation}\label{E:OAv} 3a^2=x^2-xy+y^2. \end{equation}  The altitudes from $A$ and $C$ have length $\eta:=\vert x-\frac{y}2\vert$. By Theorem 4,  $2<\eta\le 2\sqrt3$, which gives $\eta=2.5$ or $3$. 

First suppose $\eta=2.5$, and hence $2x-y=\pm 5$. Then \eqref{E:OAv} gives  \begin{equation}\label{E:av} 3a^2=25 \pm 15 x + 3x^2, \end{equation} but this is impossible, as one can see modulo $3$.

Now suppose $\eta=3$, and hence $x-\frac{y}2=\pm 3$. Thus $y=2x - 6$ or $y=2x + 6$. If $y=2x + 6$, then \eqref{E:OAv} gives \begin{equation}\label{E:av} a^2=12 + 6 x + x^2=(x+ 3)^2+3, \end{equation} but this is impossible as $x$ is positive. Hence $y=2x - 6$. Then \eqref{E:OAv} gives \begin{equation}\label{E:av} a^2=12 - 6 x + x^2=(x- 3)^2+3, \end{equation} which gives $a=2$ and $x=4,y=2$. So $A=3+\sqrt3 i$. Then $AB$ has length $\sqrt3 b$, where \[ 3b^2= \Vert(z-4)+(2z-2)\omega\Vert =(z-4)^2-(z-4)(2z-2)+(2z-2)^2=3(z-2)^2. \] Hence either $b=2-z$ or $b=z-2$. If $b=2-z$, then necessarily $z=1$, as $z$ and $b$ are positive integers, and thus $b=1$. Then $OAB$ has area $\frac12 z\sqrt3\eta=\frac32 \sqrt3$. So, by the equability hypothesis, $\frac32 =a+ b=2+1=3$, which is absurd. So $b=z-2$. Then  $OAB$ has area $\frac12 z\sqrt3\eta=\frac32 z\sqrt3 $. So, by the equability hypothesis, $\frac32z=a+ b=2+(z-2)=z$, which is also absurd.

Preprints etc

1.  with Christian Aebi, `Less than Equable Triangles on the Eisenstein lattice',  to appear in the July 2025 issue of Math. Gazette

2. with Christian Aebi, `Equable kites, trapezoids and cyclic quadrilaterals on the Eisenstein Lattice',  to appear in Trans. on Combinatorics

Published papers can be accessed through MathSciNet, ArXiv, and ResearchGate. If there is any paper you want, just drop me an email.

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', 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', 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', 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', Enseign. Math. 41 (1995), 123-133.

20. with A. Kolganova, 'Chaotic actions of free groups', Nonlinearity 9 (1996), 1015-1021.

21. with D. Elton, 'The planarity problem II', 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', 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', Proc. Amer. Math. Soc. 125 (1997), 381-385.

24. with R. Escobales, 'Note on a theorem of Gromoll-Grove', 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', J. Austral. Math. Soc. Ser. A 62 (1997), 46-63.

26. with B. Jessup, 'New bounds on the Betti numbers of nilpotent Lie algebras', Comm. Algebra 25 (1997), 415-430.

27. with E. Ghys, 'The local linearization problem for smooth SL(n)-actions', Enseign. Math. 43 (1997), 133-171.

28. with B. Jessup and M. Nicolau, 'Topologically transitive homeomorphisms of quotients of tori', J. Discrete Contin. Dynam. Systems 5 (1999), 291-300.

29. with G. Armstrong and G. Kim, 'Lie algebras of cohomological codimension one', Proc. Amer. Math. Soc. 127 (1999), 709-714.

30. with D.M. King, 'The Answer to Woodall's Musquash Problem', Discrete Math. 207 (1999), 25-32.

31. with G. Elton and P.J. Stacey, 'On the definition of collineation', Math. Mag. 72 (1999), 401.

32. with Y. Nikolayevsky, 'Bounds for Generalized Thrackles', Discrete Comput. Geom. 23 (2000), 191-206.

33. with J. Bamberg, 'Torsion Free Groups Generated by A Pair of Rational Parabolic Mobius Transformations', Bull. Austral. Math. Soc. 61 (2000), 151-152.

34. with G. Kim, 'Lie Algebras of Least Cohomology', J. Lie Theory 10 (2000), 435-441.

35. with D.M. King, 'All Odd Musquashes are Standard', Discrete Math. 226 (2001), 71-91.

36. with G. Byrnes and B. Jessup, 'Leftovers from the Ham Sandwich Theorem', Amer. Math. Monthly 108 (2001), 246-249.

37. with G. Kim, 'The mod 4 behaviour of total Lie algebra cohomology', Arch. Math. 77 (2001), 177-180.

38. 'Queens on Non-square Tori', Electron. J. Combin. 8(1) (2001), N6.

39. with P. Molino, 'Weakly Involutive Totally Geodesic Distributions of Constant Rank', 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', 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', Amer. Math. Monthly 110 (2003), 202-209.

43. with B. Jessup, 'Cohomology Operations for Lie Algebras', Trans. Amer. Math. Soc. 356 (2004), 1569-1583.

44. with M. McIntyre and Y. Nikolayevsky, 'The Thrackle Conjecture for K5 and K3,3', in "Towards a Theory of Geometric Graphs", Contemp. Math., Vol. 342, Amer. Math. Soc., Providence, RI, 2004, pp. 35-54.

45. with D. van Golstein Brouwers and J. Bamberg, 'Totally Goldbach Numbers and Related Conjectures', 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.

47. with A. Nielsen, 'On the Dynamics of the Linear Action of SL(n,Z)', 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.

49. with Korrakot Chartarrayawadee, 'Brussels Sprouts and Cloves', Math. Mag. 80 (2007) 46-58.

50. with A. Kolganova and A. Nielsen, 'Topological Transitivity and Mixing Notions for Group Actions', 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', Topology Appl. 155 (2007), 614-617.

52. with B. Jessup, 'Free submodules for the central representation in the cohomology of Lie algebras', Proc. Amer. Math. Soc. 136 (2008), 1919-1923.

53. with S. Jambor, 'The cohomology of the Heisenberg Lie algebras over fields of finite characteristic', 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', Austral. Math. Soc. Gaz. 35 (2008), no. 3, 195-202.

55. with Christian Aebi, 'Catalan Numbers, Primes and Twin Primes', Elemente der Mathematik 63 (2008), no. 4, 153–164.

56. with Christian Aebi, 'Partitions of Primes', 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', Colloq. Math. 116 (2009), no. 2, 203-215.

58. with Y. Nikolayevsky, 'Generalized Thrackle Drawings of Non-Bipartite Graphs', Discrete and Computational Geometry 41 (2009), no. 1, 119-134.

59. 'Log-concavity of the cohomology of nilpotent Lie algebras in characteristic two', J. Gen. Lie Theory Appl. 3 (2009), no. 3, 181-182.

60. with Nhan Bao Ho, 'Ultimately bipartite subtraction games', Australasian J. Combinatorics 48 (2010), 213-220. (Link to AJC website)

61. with Nhan Bao Ho, 'MIN, a combinatorial game having a connection to prime numbers', Integers 10 (2010), 765-770.

62. with Nhan Bao Ho and Tamas Lengyel, 'The Sprague-Grundy function of the real game Euclid', Discrete. Math. 311 (2011), 457-462.

63. with Nhan Bao Ho, 'Some remarks on End-Nim', Int. J. Comb. 2011 (2011), Article ID 824742.

64. with Yury Nikolayevsky, 'Outerplanar Thrackles', Graphs and Combinatorics 28 (2012), 85-96.

65. 'Proof without words', Math. Mag. 85 (2012), no. 2, 123, (see errata and addenda below).

66. with Christian Aebi, 'A Property of Twin Primes', Integers 12 (2012), #A7.

67. with Christian Aebi, 'Morley's other miracle', Math. Mag. 85 (2012), no. 3, 205-211.

68. with Nhan Bao Ho, 'A restriction of Euclid', Bull. Austral. Math. Soc. 86 (2012), no. 3, 506-509.

69. with Anna Hinic Galic and Yuri Nikolayevsky, 'Totally geodesic subalgebas of nilpotent Lie algebras',  J. Lie Theory 23 (2013), no. 4, 1023-1049.

70. with Anna Hinic Galic and Yuri Nikolayevsky, 'Totally geodesic of filiform nilpotent Lie algebras',  J. Lie Theory 23 (2013), no. 4, 1051-1074.

71. with Nguyen Thanh Tung Le, Anthony Nielsen, Yuri Nikolayevsky, `On the existence of orthonormal geodesic bases for Lie algebras', Note di Matematica 33 (2013), no. 2, 11-18.

72. with Stacey Mendan and Yuri Nikolayevsky, `A sharp refinement of a result of Alon, Ben-Shimon and Krivelevich on bipartite graph vertex sequences', Australasian Journal of Combinatorics 60 (2014), no. 2, 217–226.

73. with Stacey Mendan, 'Symmetric Bipartite Graphs and Graphs with Loops', Discrete Math. Theor. Comput. Sci., 71 (2015), no. 1, 97–102.

74. with Ana Hinic Galic, Yuri Nikolayevsky and Ioannis Tsartsaflis, `Geodesic bases for Lie algebras', Linear Multilinear Algebra 63 (2015), no. 6, 1176–1194.

75. with Timothy J. Koussas and Yuri Nikolayevsky, `Great-circle spherical thrackles', Discrete Math. 338 (2015), no. 12, 2507–2513.

76. with Stacey Mendan and Yuri Nikolayevsky, 'A sharp refinement of a result of Zverovich--Zverovich', Discrete Math. 338 (2015), no. 7, 1085–1089.

77. with Christian Aebi, `Wilson Theorems for Double-, Hyper-, Sub- and Super-factorials', Amer. Math. Monthly 122 (2015), no. 5, 433–443.

78. with Christian Aebi, `Wolstenholme again',  Elem. Math. 70 (2015), no. 3, 125–130.

79. with Stacey Mendan, 'An improvement of a result of Zverovich--Zverovich', Ars Math. Contemp. 10 (2016) no. 1, 79-83.

80. with Yuri Nikolayevsky and Gavin Rossiter, `Conewise linear periodic maps of the plane with integer coefficients', Amer. Math. Monthly 123 (2016), no. 4, 363–375.

81. with Stacey Mendan and Yuri Nikolayevsky, 'A sufficient condition for a pair of sequences to be bipartite graphic', Bull. Aust. Math. Soc.  94 (2016), no. 2, 195-200.

82. with Chris Aebi, `Sums of quadratic residues and nonresidues',  Amer. Math. Monthly  124 (2017), no. 2, 166-169.

83. with A. Hinic Galic and Y. Nikolayevsky, `Curvature properties of metric nilpotent Lie algebras which are independent of metric', Annals of Global Analysis and Geometry, 51 (2017), no. 3,  305-325.

84. Equitable Candy sharing, Amer. Math. Monthly  124 (2017), no. 6, 518-526.

85. with Chris Aebi, `The quartic residues latin square', Integers  17 (2017), Paper No. A35, 6 pp.

86. with Emily Groves and Yuri Nikolayevsky, `Bad drawings of small complete graphs', Australasian Journal of  Combinatorics 75 (2019), no. 3,  322--342.

87. with Barry Jessup and Yuri Nikolayevsky, `A class of nilpotent Lie algebras whose center acts nontrivially in cohomology', Proc. Amer. Math. Soc. 148 (2020), no. 5, 1945--1952.

88. with Chris Aebi, `Lattice equable quadrilaterals I: parallelograms', L'Enseign. Math.  67 (2021), no. 3/4,  369--401.

89. with Chris Aebi, `Lattice equable quadrilaterals II: kites, trapezoids and cyclic quadrilaterals',  Int. J. Geom.11 (2022), no. 2,  5--27.

90. with Chris Aebi, `Lattice equable quadrilaterals III: tangential and extangential cases', Integers 23 (2023), A48 (107 pages).

91. with Chris Aebi, `A vector identity for quadrilaterals', College Math. J.  53 (2022), no. 5,  392--393.

92. with Chris Aebi, `A simple sum for simplices', Math. Intelligencer  45 (2023), no. 2,  165--167.

93. with Christian Aebi, `Following in Yiu's Footsteps but on the Eisenstein Lattice',  Amer. Math. Monthly 131 (2024), no. 6, 526--529.

94. with Christian Aebi, `Equable parallelograms on the Eisenstein Lattice', Math. Slovaca 74 (2024), no.4,  963--982.

95. with Yuri Nikolayevsky, `Respectful decompositions  of Lie algebras', J. Lie Theory 34 (2024), no.3,   735--751. 

96. with Christian Aebi, `109.09 Equable triangles on the Eisenstein lattice',  Math. Gaz. 109 (2025), no.574, 140--142.