Geoffrey Campbell

Geoffrey Campbell
Long Term Campus Visitor
PhD (ANU), GDip Internet and Web Comp (RMIT University)

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Research interests

In 2022-24 I have written ten "Fun with Numbers" brief articles for the Gazette of the Australian Mathematical Society. These are small articles on simple number theoretic topics.

My main mathematics of 2022-2024 to date was the book monograph entitled Partitions, Visible Point Vectors, and Ramanujan Functions. It has a Foreword by Professor Dr Henk Koppelaar (Delft University of Technology). The book covers classical Integer Partitions, Rogers-Ramanujan Partitions in Statistical Mechanics Solved Models, and includes Vector Partitions ideas and the Visible Point Vector identities I found some years ago. It has chapters on Plane Partitions, Asymptotic Partition Formulas, Partition Congruences, Ramanujan Continued Fractions, Polylogarithms, Parametric Euler sum identities, Higher Dimensional Weighted Partition Identities now called the Visible Point Vector identities. The final draft is submitted and in editorial flux until planned publication in 2024.

The book is over 540 pages with final high level table of contents:

Partitions, Visible Point Vectors, and Ramanujan Functions

Author Biographic Note 
List of Figures 

Foreword by Prof. Dr. Henk Koppelaar 
Preface 
Contributors 

Section I Background and History
Chapter 1 : Historical background 
Chapter 2 : History timeline partitions 

Section II Integer Partition Theory
Chapter 3 : Integer partition generating functions 
Chapter 4 : Continued fraction partition identities 
Chapter 5 : Partition congruences 
Chapter 6 : Ferrers diagrams 
Chapter 7 : Durfee Squares 
Chapter 8 : Gaussian polynomials 
Chapter 9 : Plane Partitions from MacMahon to Andrews 
Chapter 10 : Asymptotics for Partition Functions 
Chapter 11 : Rogers-Ramanujan identities in Statistical Mechanics 

Section III Vector Partition Theory
Chapter 12 : Vector partitions and their generating function identities 
Chapter 13 : Integer Partitions generalized to Vector Partitions 
Chapter 14 : Weighted Vector Partitions as hybrid n-space variations 
Chapter 15 : Functional Equations for n-space Vector Partitions 
Chapter 16 : Binary Partitions and their Vector Generalizations 
Chapter 17 : n-ary Partitions and their Vector Generalizations 
Chapter 18 : Some Binary and n-ary Partition Analytic Formulas 

Section IV Visible Point Vector Partition Theory
Chapter 19 : Features of the Visible Lattice Points 
Chapter 20 : Visible Point Vector Identities in the first Hyperquadrant 
Chapter 21 : Visible Point Vector Identities in Hyperpyramid lattices 
Chapter 22 : Polylogarithms, and Parametric Euler Sum identities 
Chapter 23 : Visible Point Vector identities from particular Euler sum values
Chapter 24 : Visible Point Vector Identities in Skewed Hyperpyramid lattices
Chapter 25 : Harmonic Sums applied to VPV Identities 
Chapter 26 : The Ramanujan trigonometric function and visible point identities
Chapter 27 : Other non-weighted n-space Vector Partition Theorems 
Chapter 28 : VPV Identity cases related to some exponential relations 

Section V Models, Interpretations and some Useful Tools
Chapter 29 : 2D and 3D Stepping Stones, Forests, Orchards and Light Diffusions
Chapter 30 : Euler Products over Primes and new VPV Formulas 
Chapter 31 : Determinants, Bell Polynomial Expansions for Vector Partitions
Chapter 32 : Glossary 
Bibliography 
Index 

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Other projects:

Dirichlet series analogues of q-series, where arithmetical function identities encode quasicrystals. I give new analogue summations for classical q-series and hypergeometric series summations in terms of Riemann Zeta functions and Jordan Totient functions. This is a further monograph for me to complete.

So I have research interests in the Theory of Higher Dimensional Partitions, Aperiodic Order, Dynamical Systems, Combinatorics, Discrete Geometry, Number Theory, Quasicrystal tilings and their Dirichlet series functions, and Mathematical Physics, and areas where these theories may overlap.

I am Manager of the almost 40,000 member LinkedIn Number Theory Group located at https://www.linkedin.com/groups/4510047/.

I am an Administrator for the Facebook Group Classical Mathematics at https://www.facebook.com/groups/ClassicalMathematics presently with about 20,000 members.

Location

Room 2.73, Hanna Neumann Building 145

Publications

I am a published poet as well as a mathematics person.

POEMS: 
In 2010 I published Words in Common, which is a collection over decades of journal/magazine/anthology published poems partly funded by the Australia Council for the Arts many years ago. The book was officially launched by Professor Kevin Brophy from University of Melbourne, Creative Writing Department. It was edited by Associate Professor Trevor Code from Deakin University.

See a link with excerpts from the poems at http://bit.ly/3jAvlb8.
See a review of Words In Common by Canberra poet Michael Byrne at http://bit.ly/2MRlv8y.

A SELECTION OF MY MATHEMATICAL PUBLICATIONS:

42            CAMPBELL, G. B. Vector Partitions, Visible Points, and Ramanujan Functions, CRC Press, Taylor and Francis Group, Boca Raton, London, New York, A Chapman & Hall Book, ISBN: 978-1-032-00366-5 (hbk), ISBN: 978-1-032-00432-7 (pbk), ISBN: 978-1-003-17415-8 (ebk), DOI: 10.1201/9781003174158, to appear, 2024. 

41            CAMPBELL, G. B. Fun with numbers: Some higher power sums and Euler’s failed conjecture, Aust. Math. Soc. Gazette, Volume 51, No1, March 2024. (to appear) (https://austms.org.au/publications/gazette/gazette511/)

40            CAMPBELL, G. B. Fun with numbers: Ternary or Base Three Identities, Aust. Math. Soc. Gazette, Volume 50, No5, November 2023. (https://austms.org.au/publications/gazette/gazette505/)

39            CAMPBELL, G. B. Visible Point Vector Partition Identities for Hyperpyramid Lattices arXiv:2309.16094 [math.CO]. (https://arxiv.org/abs/2309.16094) September 2023.

38            CAMPBELL, G. B. Fun with numbers: Multigrade Sums with Carl Sagan and Pell Equations, Aust. Math. Soc. Gazette, Volume 50, No4, September 2023. (https://austms.org.au/publications/gazette/gazette504/)

37            CAMPBELL, G. B. Fun with numbers: Any rational is a sum of four 4th and four 5th powers Aust. Math. Soc. Gazette, Volume 50, No3, pp.117-119, July 2023. (https://austms.org.au/publications/gazette/gazette503/)

36            CAMPBELL, G. B. Visible Point Partition Identities for Polylogarithms, and Parametric Euler Sums,  arXiv:2306.02241 [math.CO]. (https://arxiv.org/abs/2306.02241) June 2023.

35            CAMPBELL, G. B. Fun with numbers: Revisiting an Eulerian Problem Aust. Math. Soc. Gazette, Volume 50, No2, pp.10-12, May 2023. (https://austms.org.au/publications/gazette/gazette502/)

34            CAMPBELL, G. B. Fun with numbers: A base 7 identity Aust. Math. Soc. Gazette, Volume 50, No1, pp.13-14, March 2023. (https://austms.org.au/publications/gazette/gazette501/)

33            CAMPBELL, G. B. Vector Partition Identities for 2D, 3D and nD LatticesarXiv:2302.01091v1 [math.CO], Feb 2023. (https://arxiv.org/abs/2302.01091)

32            CAMPBELL, G. B. Continued Fractions for partition generating functionsarXiv:2301.12945v1 [math.CO], Jan 2023. (https://arxiv.org/abs/2301.12945)

31            CAMPBELL, G. B. Fun with numbers: Rational solutions to xyyx = vwwv Aust. Math. Soc. Gazette, Volume 49, No5, pp210-211, November 2022. (https://austms.org.au/publications/gazette/gazette495/)

30            CAMPBELL, G. B. Fun with numbers: Identities containing a certain algebraic form Aust. Math. Soc. Gazette, Volume 49, No4, pp162-163, September 2022. (https://austms.org.au/publications/gazette/gazette494/)

29            CAMPBELL, G. B. Fun with numbers: Consecutive 6th powers and base 6 numbers,  Aust. Math. Soc. Gazette, Volume 49, No3, pp108-109, July 2022. (https://austms.org.au/publications/gazette/gazette493/)

28            CAMPBELL, G. B. Fun with numbers: Ramanujan 6-10-8 identity,  Aust. Math. Soc. Gazette, Volume 49, No2, pp71-72, May 2022. (https://austms.org.au/publications/gazette/gazette492/)

27            CAMPBELL, G. B. An interview with Rodney James Baxter, Aust. Math. Soc. Gazette, Volume 47, No1, pp24-32, March 2020. (https://austms.org.au/wp-content/uploads/2020/07/471Web.pdf)

26.           CAMPBELL, G. B. Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526v1 [math.NT], Jun 2019. (https://arxiv.org/abs/1906.07526)

25            CAMPBELL, G. B. and ZUJEV, A. The series that Ramanujan misunderstood, arXiv:1610.03693v1 [math.NT], Oct 2016. (https://arxiv.org/abs/1610.03693v1)

24            CAMPBELL, G. B. and ZUJEV, A. On integer solutions to x5 - (x+1)5 - (x+2)5 + (x+3)5 = 5m + 5n, arXiv:1603.00080v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1603.00080v1)

23            CAMPBELL, G. B. and ZUJEV, A. Some equations with features of digit reversal and powers, arXiv:1602.06320v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1602.06320v1)

22            CAMPBELL, G. B. and ZUJEV, A. Gaussian integer solutions for the fifth power taxicab number problem, arXiv:1511.07424v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.07424v1)

21            CAMPBELL, G. B. and ZUJEV, A. Variations on Ramanujan's nested radicals, arXiv:1511.06865v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.06865v1)

20            CAMPBELL, G. B. and ZUJEV, A. A diophantine sum with factorials, arXiv:1510.03056v2 [math.NT], Oct 2015. (https://arxiv.org/abs/1510.03056v2)

19            CAMPBELL, G. B. The q-Dixon sum Dirichlet series analogue, arXiv:1302.2664v1, Feb 2013. (https://arxiv.org/abs/1302.2664v1)

18            CAMPBELL, G. B. Ramanujan and Eckford Cohen totients from Visible Point Identities, arXiv:1212.2818v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2818v1)

17            CAMPBELL, G. B. D-analogues of q-shifted factorial and the q-Kummer sum, arXiv:1212.2248v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2248v1)

16            CAMPBELL, G. B. Polylogarithm approaches to Riemann Zeta function zeroes, arXiv:1212.2246v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2246v1)

15            CAMPBELL, G. B. Dirichlet series analogues of q-shifted factorial and the q-Kummer sum, Research paper 2003-6, Department of Mathematics, LaTrobe University, 2003.

14            CAMPBELL, G. B. An Euler Product transform applied to q series, Ramanujan J (2006) 12:267-293. (https://doi.org/10.1007/s11139-006-0078-y)

13            CAMPBELL, G. B. A New Class of Identities akin to q-Series in Several Variables, Research paper, Centre for Mathematics and its applications, The Australian National University, 1998.

12            CAMPBELL, G. B. Combinatorial Identities in Number Theory related to q-series and Arithmetical functions, Bull. Austral. Math. Soc., Vol. 58, (1998) pp345-347.

11            CAMPBELL, G. B. On generating functions for vector partitions, Research paper no 55-97, Centre for Mathematics and its applications, The Australian National University, 1997.

10            CAMPBELL, G. B. Visible point vector summations from hypercube and hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 21, No 4, 741-748, 1998. (https://www.researchgate.net/publication/26536267_Visible_point_vector_summations_from_hypercube_and_hyperpyramid_lattices)

9              CAMPBELL, G. B. Infinite products over hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 23, No 4, 2000, 271-277. (http://downloads.hindawi.com/journals/ijmms/2000/108918.pdf)

8              CAMPBELL, G. B. A closer look at some new identities, Internat. J. Math. & Math. Sci., Vol 21, No 3, 1998, pp581-586. (https://www.researchgate.net/publication/26536244_A_closer_look_at_some_new_identities)

7              CAMPBELL, G. B. Infinite products over visible lattice points, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 637-654. (http://downloads.hindawi.com/journals/ijmms/1994/705467.pdf)

6              CAMPBELL, G. B. A new class of infinite product, and Euler's totient, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 417-422.

5              CAMPBELL, G. B. Formulae with functions exhibiting self-similarity, Research Paper preprint series, Centre for Mathematics and its Applications, The Australian National University, 1993.

4              CAMPBELL, G. B. A generalised formula of Hardy, Int. J. Math. Math. Sci., Vol 17, No 2, 1994, 369-378.

3              CAMPBELL, G. B. Dirichlet summations and products over primes, Internat. J. Math. & Math. Sci., Vol 16, No 2, 1993, 359-372.

2              CAMPBELL, G. B. Multiplicative functions over Riemann zeta function products, J. Ramanujan Soc. 7 No. 1, 1992, 52-63.

1              CAMPBELL, G. B. Generalization of a formula of Hardy, La Trobe University preprints no 79-5, 1979 (written whilst a young student.)

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