Vector Partitions, Visible Points and Ramanujan Functions

Vector Partitions, Visible Points and Ramanujan Functions offers a novel theory of Vector Partitions, though very much grounded in the long-established work of others, that could be developed as an extension to the existing theory of Integer Partitions. The book is suitable for graduate students in physics, applied mathematics, number theory and computational mathematics. It takes the reader up to research level, presenting new results alongside known classical results from integer partitions and areas of vector and multipartite partition theory. It also sets forth new directions for research for the more advanced reader.

Above all, the intention of the book is to bring new inspiration to others who study mathematics and related areas. It is hoped that some new ideas will be launched to add value and insight into many of the classical and new theories surrounding partitions. The book is an appreciation of the many gifted authors of research into partitions over the past century and before, in the hope that more may come of this for future generations.

Features

• Provides a step-by-step guide through the known literature on Integer and Vector Partitions, and a focus on the not so well-known Visible Point Vector identities
• Serves as a reference for graduate students and researchers in physics, applied mathematics, number theory and computational mathematics
• Offers a variety of practical examples as well as sets of exercises suitable for students and researchers.

Section I. Background and History. 1. History timeline partitions. Section II. Integer Partition Theory. 3. Integer partition generating functions. 4. Continued fraction partition identities. 5. Partition congruences. 6. Ferrers diagrams. 7.Durfee Squares. 8. Gaussian polynomials. 9. Plane Partitions from MacMahon to Andrews. 10. Asymptotics for Partition Functions. 11. Rogers-Ramanujan identities in Statistical Mechanics. Section III. Vector Partition Theory. 12. Vector partitions and their generating function identities. 13. Integer Partitions generalized to Vector Partitions. 14. Weighted Vector Partitions as hybrid n-space variations. 15. Functional Equations for n-space Vector Partitions. 16. Binary Partitions and their Vector Generalizations. 17. n-ary Partitions and their Vector Generalizations. 18. Some Binary and n-ary Partition Analytic Formulas. Section IV. 19. Features of the Visible Lattice Points. 20. Visible Point Vector Identities in the first Hyperquadrant. 21. Visible Point Vector Identities in Hyperpyramid lattices. 22. Polylogarithms, and Parametric Euler Sum identities. 23. Visible Point Vector identities from particular Euler sum values. 24. Visible Point Vector Identities in Skewed Hyperpyramid lattices. 25. Harmonic Sums applied to VPV Identities. 26. The Ramanujan trigonometric function and visible point identities. 27. Other non-weighted n-space Vector Partition Theorems. 28. VPV Identity cases related to some exponential relations. Section V. Models, Interpretations and some Useful Tools. 29. 2D and 3D Stepping Stones, Forests, Orchards and Light Diffusions. 30. Euler Products over Primes and new VPV Formulas. 31. Determinants, Bell Polynomial Expansions for Vector Partitions. 32. Glossary.

Geoffrey B. Campbell completed his PhD at Australian National University in 1998 under the esteemed physicist Professor Rodney Baxter. His affiliation with the Australian National University Mathematical Sciences Institute has continued for over 30 years. Within that time frame, Geoffrey also served eight years as an Honorary Research Fellow at LaTrobe University Mathematics and Statistics Department in Melbourne. Currently he writes ongoing articles for the Australian Mathematical Society Gazette. Within the international scope, Geoffrey currently serves as a PhD external committee member for a mathematics graduate student at Washington State University in America.

Geoffrey has built a career within Australian Commonwealth and State government departments, including as an Advisor at the Department of Prime Minister and Cabinet; as Analyst Researcher for a Royal Commission. Geoffrey specializes in complex data, machine learning including data analytics. He is also a published poet in Australian anthologies and literary magazines.