This thesis studies the fundamentals of a higher dimensional discrete function
theory using the Clifford Algebra setting. This approach combines the ideas of
Umbral Calculus and Differential Forms. Its powerful rests mostly on the
interplay between both languages. This allowed the construction of intertwining
operators between continuous and discrete structures, lifting the well known
results from continuum to discrete.
Furthermore, this resulted in a mimetic transcription of basis polynomial,
generating functions, Fischer Decomposition, Poincaré and dual-Poincaré
lemmata, Stokes theorem and Cauchy’s formula.
This theory also includes the description discrete counterparts of differential
forms, vector-fields and discrete integration. Indeed the resulted construction of
discrete differential forms, discrete vector-fields and discrete integration in
terms of barycentric coordinates leads to the correspondence between the
theory of Finite Differences and the theory of Finite Elements, which gives a
core of promising applications of this approach in numerical analysis. Some
preliminary ideas on this point of view were presented in this thesis.
We also developed some preliminary results in the theory of discrete
monogenic functions on simplicial complexes. Some connections with
Combinatorics and Quantum Mechanics were also presented along this thesis.
Abstract of my dissertation which provides me the doctor degree in mathematics (March 26, 2009). In a couple of weeks I wish to disseminate some new material with the goal to continue developing the discrete Clifford analysis arena.
The dissemation of new material likewise some ongoing research around was the main motivation to create this blog. In a nearly future, I wish to create a research networking project within this topic with the goal to promote research discussions around this topic. If you are still interested in join in, please feel it free to send me an e-mail.