User talk:Nelson.faustino

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During his time as a PhD-student Nelson Faustino worked on the fundaments of discrete function theory and its applications to numerical solution of partial differential equations. While there already exist some results in this direction, such results were mostly obtained by "brute force calculations", not by a well-established theory.

To the end of creating such a theory he systematized the abstract approach to continuous functions theories by F. Sommen, V. Kisil and others by showing it to be a combination of Umbral calculus and differential geometry. This allowed the construction of intertwining operators between continuous function theory and certain discrete versions. With such operators one can lift known results in the continuous case to the discrete case, which provides the construction of basis polynomials, Stokes formula, Borel-Pompeiu formula, Cauchy integral formula, etc. This rather elegant way allows to avoid the cumbersome work of direct construction. On the other hand, the resulting construction of based on discrete differential forms and discrete integration in terms barycentre coordinates provides also an early correspondence between the theory of finite differences and the theory of finite elements.

Furthermore, because of the interest in his research he got 4 invitations for research stays in Tampere University of Technology, Tampere (Finland), Konrad-Zuse-Institut f\"ur Informationstechnik Berlin (Germany), Bauhaus-University Weimar (Germany), and University of Leeds (UK).

During his time as a FCT-research fellow he's the author of 14 publications and preprints which at least 6 appeared or accepted for publication in international journals of well standard of well standards and gave 20 talks, 11 of them in International Conferences and Workshops and 2 of them during the research stays in Bauhaus-University Weimar (Germany), and University of Leeds (UK).

His approach and his (first) results start to received already a lot of interest during his talk at the CMFT-conference in 2005 in Joensuu (Finland) at the beginning of his PhD-work. This results in the article "Difference potentials for the Navier-Stokes equations in unbounded domains" (joint work with K. Guerlebeck \& Angela Hommel, Bauhaus-University Weimar and U.Kaehler, University of Aveiro) published at Journal of Difference Equations and Applications. It would be stressed that this is a UK Mathematical Journal with a considerable impact factor.

After his participation in the International Congress of Mathematicians in 2006, Madrid (Spain) where he also delivered a talk, he made some contributions in the development of the theory of discrete Dirac operators and the numerical solution of time-dependent non-linear Schr\"odinger equation.

This results in two important contributions: Discrete Dirac Operators in Clifford Analysis (joint work with U. Kaehler, University of Aveiro and F. Sommen, Ghent University) at Advances in Applied Clifford Algebras and ``Numerical Clifford analysis for nonlinear Schr\"odinger problem (joint work with P. Cerejeiras and N.Vieira, University of Aveiro) published at``Numerical Methods for Partial Differential Equation's'.

We would like to point out that F. Sommen is a proliferous researcher in Clifford Analysis. He was editor of 3 books in the field and his list of publications contains more than 100 papers. We would like also to point out that the second article were published in a Mathematical Journal with a considerable impact factor. The editorial board of the mentioned journal include well-known names from the Mathematical community e.g Ivo Babuska (University of Texas at Austin, USA), Franco Brezzi (Universita di Pavia, Italy), P. G. Ciarlet (Universite Pierre et Marie Curie, Paris, France), Endre Suli (University of Oxford, UK), Roger Temam (Indiana University, USA).

After his current research unit got interested in the Huang-Hilbert transform and analytic signals, he constructed and implemented working algorithms in the case of higher dimensions (Huang's original approach being one-dimensional). This is a joint work (under construct) with P.Cerejeiras, U. Kaehler (University of Aveiro, Portugal) and G. Teschke (Konrad-Zuse-Zentrum f\"ur Informationstechnik Berlin).

We would like to remark that G. Teschke is an active researcher with Scientific and Industrial Projects Funded by Industries and Publics Grants such as DFG, AIF, BMBF, DAAD. His scientific partners well-know names like Ingrid Daubechies (Princeton University), Stephan Dahlke (University of Marburg, Germany) and Luminita Vese (University of California Los Angeles, IPAM).

Nelson Faustino's vitae clearly shows that he is a clearly talented mathematical research as well a good finding concrete computational solutions for concrete mathematical problems and his research interests include very distinct topics like Clifford Analysis, Numerical Analysis, Wavelet Analysis, Umbral Calculus, Quantum Mechanics and Signal Processing.

Currently he is preparing the three papers bellow

\emph{Rediscovering Clifford Analysis: On the Interplay between Umbral

 Calculus and Quantum Mechanics}.

(with G. Ren, University of Science and Technology of China) \emph{Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator.}

(with K. Guerlebeck, Bauhaus-University Weimar) \emph{On Discrete Monogenic Primitives}.

which are expected to be important contributions in the deeper understanding of function theories on discrete sets.

He is also finalizing his thesis and he is to be expected to finish his PhD by January 2009.