Book Chapters
8.
Boundary values of discrete monogenic functions over bounded domains in $$ mathbb{R}^3 $$
Cerejeiras, Paula and Kähler, Uwe and Legatiuk, Anastasiia and Legatiuk, Dmitrii
Linear Systems, Signal Processing and Hypercomplex Analysis
Birkhäuser
In this paper we are going to study boundary values for discrete monogenic functions over bounded spatial domains. After establishing the discrete Stokes formula and the Borel–Pompeiu formula we are going to construct discrete Plemelj–Sokhotzki formulae, discrete Plemelj projections and discrete Hardy spaces. A further extension to the ndimensional case can be done in a straightforward way based on the results presented in this paper.
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7.
Finite element exterior calculus with script geometry
Cerejeiras, Paula and Kähler, Uwe and Legatiuk, Dmitrii
AIP Conference Proceedings
AIP Publishing
Finite element method is probably the most popular numerical method used in different fields of applications nowadays. While approximation properties of the classical finite element method, as well as its various modifications, are well understood, stability of the method is still a crucial problem in practice. Therefore, alternative approaches based not on an approximation of continuous differential equations, but working directly with discrete structures associated with these equations, have gained an increasing interest in recent years. Finite element exterior calculus is one of such approaches. The finite element exterior calculus utilises tools of algebraic topology, such as de Rham cohomology and Hodge theory, to address the stability of the continuous problem. By its construction, the finite element exterior calculus is limited to triangulation based on simplicial complexes. However, practical applications often require triangulations containing elements of more general shapes. Therefore, it is necessary to extend the finite element exterior calculus to overcome the restriction to simplicial complexes. In this paper, the script geometry, a recently introduced new kind of discrete geometry and calculus, is used as a basis for the further extension of the finite element exterior calculus.
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6.
Applications of parabolic Dirac operators to the instationary viscous MHD equations on conformally flat manifolds
Cerejeiras, Paula and Kähler, Uwe and Kraußhar, Sören R.
Topics in Clifford Analysis. Trends in Mathematics
Birkhäuser
In this paper we apply classical and recent techniques from quaternionic analysis using parabolic Dirac type operators and related Teodorescu and CauchyBitzadse type operators to set up some analytic representation formulas for the solutions to the time dependent incompressible viscous magnetohydrodynamic equations on some conformally flat manifolds, such as cylinders and tori associated with different spinor bundles. Also in this context a special variant of hypercomplex Eisenstein series related to the parabolic Dirac operator serve as kernel functions.
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Articles
5.
A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus
Ferreira, Milton and Kraußhar, R. Sören and Rodrigues, M. Manuela and Vieira, Nelson
Mathematical Methods in the Applied Sciences
Wiley
In this paper we develop a fractional integrodifferential operator calculus for Cliffordalgebra valued functions. To do that we introduce fractional analogues of the Teodorescu and CauchyBitsadze operators and we investigate some of their mapping properties. As a main result we prove a fractional BorelPompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodgetype decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and RiemannLiouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.
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4.
A spacebased method for the generation of a SchwartzFunction with infinitely many vanishing moments of higher order with applications in image processing
Fink, Thomas and Kahler, Uwe
Complex Analysis and Operator Theory
Birkhäuser
In this article we construct a function with infinitely many vanishing (generalized) moments. This is motivated by an application to the Taylorlet transform which is based on the continuous shearlet transform. It can detect curvature and other higher order geometric information of singularities in addition to their position and the direction. For a robust detection of these features a function with higher order vanishing moments is needed. We show that the presented construction produces an explicit formula of a function with ∞ many vanishing moments of arbitrary order and thus allows for a robust detection of certain geometric features. The construction has an inherent connection to qcalculus, the Euler function and the partition function.
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3.
Perturbation of normal quaternionic operators
Cerejeiras, Paula and Colombo, Fabrizio and Kahler, Uwe and Sabadini, Irene
Transactions of the American Mathematical Society
American Mathematical Society
The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two $ S$resolvent operators and the $ S$spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the $ S$spectrum of a quaternionic linear operator $ T$ is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of $ S$spectrum and of slice hyperholomorphicity of the $ S$resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.
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2.
A timefractional BorelPompeiu formula and a related hypercomplex operator calculus
Ferreira, M. and Rodrigues, M. M. and Vieira, M.
Complex Analysis and Operator Theory
Springer
In this paper we develop a timefractional operator calculus in fractional Clifford analysis. Initially we study the $L_p$integrability of the fundamental solutions of the multidimensional timefractional diffusion operator and the associated timefractional parabolic Dirac operator. Then we introduce the timefractional analogues of the Teodorescu and CauchyBitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a timefractional version of the BorelPompeiu formula based on a timefractional Stokes' formula. This tool in hand allows us to present a Hodgetype decomposition for the forward timefractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and RiemannLiouville timefractional derivatives. We round off this paper by giving a direct application of the obtained results for solving timefractional boundary value problems.
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Proceedings
1.
Application of the hypercomplex fractional integrodifferential operators to the fractional Stokes equation
Ferreira, M. and Kraußhar, R. S. and Rodrigues, M. M. and Vieira, N.
16th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2018)
AIP Publishing
We present a generalization of several results of the classical continuous Clifford function theory to the context of fractional Clifford analysis. The aim of this paper is to show how the fractional integrodifferential hypercomplex operator calculus can be applied to a concrete fractional Stokes problem in arbitrary dimensions which has been attracting recent interest (cf. cite{CNP,LAX}).
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