Geoffrey Campbell

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

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In 2022-24 I have written eleven "Fun with Numbers" brief articles for the Gazette of the Australian Mathematical Society. These are small articles on simple number theoretic topics; some of which contain snippets of original research.

My main mathematics of 2022-2024 to date was the book monograph published 29 May 2024 entitled:
Partitions, Visible Point Vectors, 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, June 2024.).

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 set for publication in early June 2024.

The book is over 564 pages excluding the Preface and Foreword. Its high level table of contents is:

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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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 20,800 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:

44            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, 29 May 2024. 

43            CAMPBELL, G. B. Rogers-Ramanujan identities in Statistical Mechanics,  Math-Physics arxiv:2405.08425 [math-ph.CC-BY-NC-SA]. (https://arxiv.org/abs/2405.08425) May 2024.

42            CAMPBELL, G. B. Fun with numbers: Some higher power sums and Euler’s failed conjecture, Aust. Math. Soc. Gazette, Volume 51, No2, May 2024. (https://austms.org.au/publications/gazette/gazette512/)

41            CAMPBELL, G. B. VPV Identities related to xy=yx and xy yx = vw wv arxiv:2404.10783 [math.CO]. (https://arxiv.org/abs/2404.10783) April 2024.

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 Lattices, arXiv:2302.01091v1 [math.CO], Feb 2023. (https://arxiv.org/abs/2302.01091)

32            CAMPBELL, G. B. Continued Fractions for partition generating functions, arXiv: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.)