Complex Multiplicative Calculus and Mixed Problems
(2022) In Bachelor's Theses in Mathematical Sciences MATK11 20221Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
- Abstract
- In this thesis we consider multiplicative integrals and derivatives on functions on the positive numbers, the logarithmic Riemann surface and the punctured complex numbers. Additive structures on the real and complex numbers are related to their multiplicative counterparts on the positive numbers and the logarithmic Riemann surface by defining exponential transition functors. We use a lift-projection method to transition multiplicative structures on the logarithmic Riemann Surface to their counterparts on the punctured complex plane. The process introduces potentially multivalued behaviour, which is the case for multiplicative integrals on functions with a range in the punctured complex plane.
Some mixed problems, involving both... (More) - In this thesis we consider multiplicative integrals and derivatives on functions on the positive numbers, the logarithmic Riemann surface and the punctured complex numbers. Additive structures on the real and complex numbers are related to their multiplicative counterparts on the positive numbers and the logarithmic Riemann surface by defining exponential transition functors. We use a lift-projection method to transition multiplicative structures on the logarithmic Riemann Surface to their counterparts on the punctured complex plane. The process introduces potentially multivalued behaviour, which is the case for multiplicative integrals on functions with a range in the punctured complex plane.
Some mixed problems, involving both additive and multiplicative structures, are also discussed. E.g. we consider the mixed differential equation y’= y*, whose solution involves the Lambert W function. We extend the inequality of arithmetic and geometric means (AM-GM inequality) to the setting of non-negative random variables. A matrix version of the AM-GM inequality is also extended, and tweaked to an integral version which leads to a generalization of Hölder’s inequality. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9103812
- author
- Lokrantz, Axel LU
- supervisor
- organization
- alternative title
- Multiplikativ komplex analys och mixade problem
- course
- MATK11 20221
- year
- 2022
- type
- M2 - Bachelor Degree
- subject
- keywords
- multiplicative integrals, multiplicative calculus, product integrals, geometric mean, continuous geometric mean, geometric mean of non-negative random variables, inequality of arithmetic and geometric means, A_G-G_A inequality, generalized Hölder's inequality, multiplicative vector spaces, multiplicative function spaces, exponential transition functor, logarithmic Riemann surface, lift, function lift, projection, function projection, lift-projection method, multivalued complex multiplicative integrals, geometric expectation, mixed differential equation, Lambert W function, W, M, w and m functions
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUNFMA-4140-2022
- ISSN
- 1654-6229
- other publication id
- 2022:K21
- language
- English
- id
- 9103812
- date added to LUP
- 2022-12-12 14:36:06
- date last changed
- 2022-12-12 14:36:06
@misc{9103812, abstract = {{In this thesis we consider multiplicative integrals and derivatives on functions on the positive numbers, the logarithmic Riemann surface and the punctured complex numbers. Additive structures on the real and complex numbers are related to their multiplicative counterparts on the positive numbers and the logarithmic Riemann surface by defining exponential transition functors. We use a lift-projection method to transition multiplicative structures on the logarithmic Riemann Surface to their counterparts on the punctured complex plane. The process introduces potentially multivalued behaviour, which is the case for multiplicative integrals on functions with a range in the punctured complex plane. Some mixed problems, involving both additive and multiplicative structures, are also discussed. E.g. we consider the mixed differential equation y’= y*, whose solution involves the Lambert W function. We extend the inequality of arithmetic and geometric means (AM-GM inequality) to the setting of non-negative random variables. A matrix version of the AM-GM inequality is also extended, and tweaked to an integral version which leads to a generalization of Hölder’s inequality.}}, author = {{Lokrantz, Axel}}, issn = {{1654-6229}}, language = {{eng}}, note = {{Student Paper}}, series = {{Bachelor's Theses in Mathematical Sciences}}, title = {{Complex Multiplicative Calculus and Mixed Problems}}, year = {{2022}}, }