#### p-Adic Numbers with an Emphasis on q-Volkenborn Integration

Similar to the real numbers, the p-adic fields are completions of the rational numbers. However, distance in this space is determined based on divisibility by a prime number, p, rather…

Similar to the real numbers, the p-adic fields are completions of the rational numbers. However, distance in this space is determined based on divisibility by a prime number, p, rather than by the traditional absolute value. This gives rise to a peculiar topology which offers significant simplifications for p-adic continuous functions and p-adic integration than is present in the real numbers. These simplifications may present significant advantages to modern physics – specifically in harmonic analysis, quantum mechanics, and string theory. This project discusses the construction of the p-adic numbers, elementary p-adic topology, p-adic continuous functions, introductory p-adic measure theory, the q-Volkenborn distribution, and applications of p-adic numbers to physics. We define q-Volkenborn integration and its connection to Bernoulli numbers.

**Contributors**

- Burgueno, Alyssa Erin (Author)
- Childress, Nancy (Thesis director)
- Jones, John (Committee member)
- School of Mathematical and Statistical Sciences (Contributor, Contributor, Contributor)
- Department of Physics (Contributor)
- Barrett, The Honors College (Contributor)

**Created**

- 2020-05