Errata and Addenda- 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 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.
Preprints etc- with Nhan Bao Ho, 'Impartial Games as Acyclic Digraphs'; draft notes.
- with Christian Aebi, `Lattice equable parallelograms'.
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
- 'Géométrie globale des feuilletages totalement géodésiques', C.R. Acad. Sci. Paris 297 (1983), 525-527.
- 'Feuilletages totalement géodésiques de dimension 1, 2 ou 3', C.R. Acad. Sci. Paris 298 (1984), 341-344.
- 'Aspects cohomologiques des feuilletages totalement géodésiques', C.R. Acad. Sci. Paris 299 (1984), 1017-1019.
- 'A general description of totally geodesic foliations', Tòhoku Math. J. 38 (1986), 37-55.
- 'Some properties of a cohomology group associated to a totally geodesic foliation', Math. Zeitschrift 192 (1986), 391-403.
- with Etienne Ghys, 'Totally geodesic foliations on 4-manifolds', J. Diff. Geom. 23 (1986), 241-254.
- '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.
- 'The duality between Riemannian foliations and geodesible foliations', Riemannian Foliations by Pierre Molino, Progress in Math. 73, Birkhäuser Boston, (1988), 249-263.
- 'Totally umbilic Riemannian foliations', Michigan Math. J. 37 (1990), 145-159.
- 'Compact 4-manifolds that admit totally umbilic foliations', Differential Geometry and its Applications, ed. J. Janyska and D. Krupka, World Scientific (1990), 9-16.
- with R. Sharpe, 'On the inversive differential geometry of plane curves', Enseign. Math. 36 (1990), 175-196.
- with J. Banks, J. Brooks, G. Davis and P. Stacey, 'On Devaney's definition of chaos', Amer. Math. Monthly 99 (1992), 332-334.
- with Mehmet Özdemir and Ekkehard-H. Tjaden, 'A counterexample to a conjecture of U. Pinkall', Topology 31 (1992), 557-558.
- with M. McIntyre and M. Özdemir, 'A six vertex theorem for bounding normal planar curves', Bull. London Math. Soc. 25 (1993), 169-176.
- with M. McIntyre, 'A new formula for winding number', Geom. Dedicata 46 (1993), 149-160.
- with M. McIntyre and J. Strantzen, 'Geometric proofs of recent results of Yang Lu', Math. Mag. 66 (1993), 263-265.
- with D. Elton, 'The planarity problem for signed Gauss words', J. Knot Theory and its Ramifications 2 (1993), 359-367.
- 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.
- with G. Davis, D. Elton, A. Kolganova, P. Perversi, 'Chaotic group actions', Enseign. Math. 41 (1995), 123-133.
- with A. Kolganova, 'Chaotic actions of free groups', Nonlinearity 9 (1996), 1015-1021.
- with D. Elton, 'The planarity problem II', J. Knot Theory and its Ramifications 5 (1996), 137-144.
- 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.
- with G. Armstrong and B. Jessup, 'Explicit Betti numbers for a family of nilpotent Lie algebras', Proc. Amer. Math. Soc. 125 (1997), 381-385.
- with R. Escobales, 'Note on a theorem of Gromoll-Grove', Bull. Austral. Math. Soc. 55 (1997), 1-5.
- 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.
- with B. Jessup, 'New bounds on the Betti numbers of nilpotent Lie algebras', Comm. Algebra 25 (1997), 415-430.
- with E. Ghys, 'The local linearization problem for smooth SL(n)-actions', Enseign. Math. 43 (1997), 133-171.
- with B. Jessup and M. Nicolau, 'Topologically transitive homeomorphisms of quotients of tori', J. Discrete Contin. Dynam. Systems 5 (1999), 291-300.
- with G. Armstrong and G. Kim, 'Lie algebras of cohomological codimension one', Proc. Amer. Math. Soc. 127 (1999), 709-714.
- with D.M. King, 'The Answer to Woodall's Musquash Problem', Discrete Math. 207 (1999), 25-32.
- with G. Elton and P.J. Stacey, 'On the definition of collineation', Math. Mag. 72 (1999), 401.
- with Y. Nikolayevsky, 'Bounds for Generalized Thrackles', Discrete Comput. Geom. 23 (2000), 191-206.
- with J. Bamberg, 'Torsion Free Groups Generated by A Pair of Rational Parabolic Mobius Transformations', Bull. Austral. Math. Soc. 61 (2000), 151-152.
- with G. Kim, 'Lie Algebras of Least Cohomology', J. Lie Theory 10 (2000), 435-441.
- with D.M. King, 'All Odd Musquashes are Standard', Discrete Math. 226 (2001), 71-91.
- with G. Byrnes and B. Jessup, 'Leftovers from the Ham Sandwich Theorem', Amer. Math. Monthly 108 (2001), 246-249.
- with G. Kim, 'The mod 4 behaviour of total Lie algebra cohomology', Arch. Math. 77 (2001), 177-180.
- 'Queens on Non-square Tori', Electron. J. Combin. 8(1) (2001), N6.
- 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.
- 'The Approximative Centre of a Lie Algebra', J. Lie Theory 12 (2002), 205-216.
- 'Pillow chess', Math. Mag. 75 (2002), 173-186.
- with J. Bamberg and D. Kilminster, 'The Crystallographic Restriction, permutations, and Goldbach's Conjecture', Amer. Math. Monthly 110 (2003), 202-209.
- with B. Jessup, 'Cohomology Operations for Lie Algebras', Trans. Amer. Math. Soc. 356 (2004), 1569-1583.
- 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.
- with D. van Golstein Brouwers and J. Bamberg, 'Totally Goldbach Numbers and Related Conjectures', Austral. Math. Soc. Gaz. 31 (2004), 251-255.
- 'Is there a greater role for prime numbers in our schools?', Austral. Sen. Math. J. 19 (2005), 24-37.
- with A. Nielsen, 'On the Dynamics of the Linear Action of SL(n,Z)', Bull. Austral. Math. Soc. 71 (2005), 359 - 365.
- 'Els nombres primers poden tenir mès protagonisme a Secundària?', Butlletì de la Societat Catalana de Matemàtiques 20 (2005), 75-89.
- with Korrakot Chartarrayawadee, 'Brussels Sprouts and Cloves', Math. Mag. 80 (2007) 46-58.
- with A. Kolganova and A. Nielsen, 'Topological Transitivity and Mixing Notions for Group Actions', Rocky Mountain J. Math. 37 (2007), 371-397.
- with Thanh Duong Pham, 'An example of a chaotic group action on the plane by compactly supported homeomorphisms', Topology Appl. 155 (2007), 614-617.
- with B. Jessup, 'Free submodules for the central representation in the cohomology of Lie algebras', Proc. Amer. Math. Soc. 136 (2008), 1919-1923.
- 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.
- 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.
- with Christian Aebi, 'Catalan Numbers, Primes and Twin Primes', Elemente der Mathematik 63 (2008), no. 4, 153–164.
- with Christian Aebi, 'Partitions of Primes', Parabola 45 (2009), no. 1, 3-12.
- 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.
- with Y. Nikolayevsky, 'Generalized Thrackle Drawings of Non-Bipartite Graphs', Discrete and Computational Geometry 41 (2009), no. 1, 119-134.
- 'Log-concavity of the cohomology of nilpotent Lie algebras in characteristic two', J. Gen. Lie Theory Appl. 3 (2009), no. 3, 181-182.
- with Nhan Bao Ho, 'Ultimately bipartite subtraction games', Australasian J. Combinatorics 48 (2010), 213-220. (Link to AJC website)
- with Nhan Bao Ho, 'MIN, a combinatorial game having a connection to prime numbers', Integers 10 (2010), 765-770.
- with Nhan Bao Ho and Tamas Lengyel, 'The Sprague-Grundy function of the real game Euclid', Discrete. Math. 311 (2011), 457-462.
- with Nhan Bao Ho, 'Some remarks on End-Nim', Int. J. Comb. 2011 (2011), Article ID 824742.
- with Yury Nikolayevsky, 'Outerplanar Thrackles', Graphs and Combinatorics 28 (2012), 85-96.
- 'Proof without words', Math. Mag. 85 (2012), no. 2, 123, (see errata and addenda below).
- with Christian Aebi, 'A Property of Twin Primes', Integers 12 (2012), #A7.
- with Christian Aebi, 'Morley's other miracle', Math. Mag. 85 (2012), no. 3, 205-211.
- with Nhan Bao Ho, 'A restriction of Euclid', Bull. Austral. Math. Soc. 86 (2012), no. 3, 506-509.
- with Anna Hinic Galic and Yuri Nikolayevsky, 'Totally geodesic subalgebas of nilpotent Lie algebras', J. Lie Theory 23 (2013), no. 4, 1023-1049.
- with Anna Hinic Galic and Yuri Nikolayevsky, 'Totally geodesic of filiform nilpotent Lie algebras', J. Lie Theory 23 (2013), no. 4, 1051-1074.
- 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.
- 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.
- with Stacey Mendan, 'Symmetric Bipartite Graphs and Graphs with Loops', Discrete Math. Theor. Comput. Sci., 71 (2015), no. 1, 97–102.
- with Ana Hinic Galic, Yuri Nikolayevsky and Ioannis Tsartsaflis, `Geodesic bases for Lie algebras', Linear Multilinear Algebra 63 (2015), no. 6, 1176–1194.
- with Timothy J. Koussas and Yuri Nikolayevsky, `Great-circle spherical thrackles', Discrete Math. 338 (2015), no. 12, 2507–2513.
- with Stacey Mendan and Yuri Nikolayevsky, 'A sharp refinement of a result of Zverovich--Zverovich', Discrete Math. 338 (2015), no. 7, 1085–1089.
- with Christian Aebi, `Wilson Theorems for Double-, Hyper-, Sub- and Super-factorials', Amer. Math. Monthly 122 (2015), no. 5, 433–443.
- with Christian Aebi, `Wolstenholme again', Elem. Math. 70 (2015), no. 3, 125–130.
- with Stacey Mendan, 'An improvement of a result of Zverovich--Zverovich', Ars Math. Contemp. 10 (2016) no. 1, 79-83.
- 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.
- 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.
- with Chris Aebi, `Sums of quadratic residues and nonresidues', Amer. Math. Monthly 124 (2017), no. 2, 166-169.
- 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.
- Equitable Candy sharing, Amer. Math. Monthly 124 (2017), no. 6, 518-526.
- with Chris Aebi, `The quartic residues latin square', Integers 17 (2017), Paper No. A35, 6 pp.
- with Emily Groves and Yuri Nikolayevsky, `Bad drawings of small complete graphs', Australasian Journal of Combinatorics 75 (2019), no. 3, 322--342.
- 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.
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