The famous Moore's law states: Since the invention of the integrated circuit, the number of transistors that can be placed on an integrated circuit has increased exponentially, doubling approximately every two years. As a result of the downscaling of the size of the transistor, quantum effects have become increasingly important while affecting significantly the device performances. Nowadays, at the nanometer scale, inter-atomic interactions and quantum mechanical properties need to be studied extensively. Device and material simulations are important to achieve these goals because they are flexible and less expensive than experiments. They are also important for designing and characterizing new generation of electronic device such as silicon nanowire or carbon nanotube (CNT) transistors. Several modeling methods have been developed and applied to electronic structure calculations, such as: Hartree-Fock, density functional theory (DFT), empirical tight-binding, etc. For transport simulations, most of the device community focuses on studying the stationary problem for obtaining characteristics such as I-V curves. The non-equilibrium transport problem is then often addressed by solving a multitude of time-independent Schrodinger-type equation for all possible energies. On the other hand, for many other electronic applications including high-frequency electronics response (e.g. when a time-dependent potential is applied to the system), the description of the system behavior necessitate insights on the time dependent electron dynamics. To address this problem, it is then necessary to solve a time-dependent Schrodinger-type equation. In this thesis, we will focus on solving time-dependent problems with application to CNTs. We will be identifying all the numerical difficulties and propose new effective modeling and numerical schemes to address the current limitations in time-dependent quantum simulations. we will point out that two numerical errors may occur: an integration error and the anti-commutation issue error; the direct computation above being mathematically equivalent to performing the integration of the time dependent Hamiltonian using a rectangle numerical quadrature formula along the total simulation times. After careful study and many numerical experiments, we found that the Gaussian quadrature scheme provides a good trade off between computational consumption and numerically accuracy, meanwhile unitary, stability and time reversal properties are well preserved. The new Gaussian quadrature integration scheme uses (i) much fewer points in time to approximate the integral of the Hamiltonian, (ii) ordered exponential to factorize the time evolution operator, (iii) FEM discretize techniques (iv) and at last, the FEAST eigenvalue solver to diagonalize and solve each exponential.

A precisão e acurácia dos métodos mais utilizados atualmente de processamento de dados de espectroscopia por Ressonância Magnética (MRS), baseados na Transformada de Fourier (FT), requerem supressão apropriada (o que está longe de ser trivial) e aquisições longas para a obtenção de alta resolução espectral. Além disso, a FT tem dificuldades quando faltam dados no domínio de tempo, como, por exemplo, pela redução do tempo de aquisição, e consequente número de pontos adquiridos. Isto pode ocorrer, também, por artefatos na aquisição ou, ainda, seja pela exclusão intencional dos primeiros pontos do sinal para a eliminação de ressonâncias largas que estão distorcendo a linha de base no domínio da frequência. Neste estudo, propomos a utilização do Método de Diagonalização na Base de Krylov (KBDM) como uma alternativa a FT para algumas de suas limitações. O método ajusta sinais de experimentos de Free Induction Decay (FID) por uma soma de funções harmônicas complexas, amortecidas exponencialmente, permitindo uma fácil manipulação dos seus parâmetros de caracterização. O KBDM é numericamente mais efetivo para análise de sinais truncados e tem diversos recursos que possibilitam remover picos de forma mais eficiente, como por exemplo, o pico residual da água. Além disso, foi introduzida a possibilidade de quantificação de dados de MRS com o método. Para avaliar a sensibilidade, eficiência e reprodutibilidade do método para quantificar e analisar sinais truncados, foi proposto fazer simulações de espectros clínicos e experimentos em phantoms que representassem o ambiente metabólico do cérebro, para MRS de próton de diferentes níveis de ruídos e para pequenas variações do N-acetil aspartato (NAA). Com estes estudos pôde se comprovar a viabilidade do método para processar dados de MRS e verificar seu potencial na complementação das técnicas atualmente empregadas, especialmente quando uma resolução espectral e temporal maior que o limite imposto pela Relação de Incerteza do formalismo de Fourier é necessária. Além disso, uma desejável facilidade de manipulação de picos específicos (por exemplo, exclusão e quantificação) é proporcionada pelo método. Como perspectivas animadoras deste trabalho esperamos a introdução do KBDM como uma técnica eficiente e coadjuvante ao Imageamento de Ressonância Magnética funcional (fMRI), auxiliando estudos de funções cerebrais, em sequências de MRS para identificar uma rápida variação das linhas associadas as atividades metabólicas dos cérebros. / The precision and accuracy of the most widely used methods to perform Magnetic Resonance Spectroscopy (MRS) data processing based on the Fourier Transform (FT), require appropriate suppression (which is far from trivial) and long acquisitions to obtain high spectral resolution. Furthermore, FT poses difficulty when there are missing data in the time domain. This occurs because of reduction of the acquisition time and consequently also in the number of acquired points, or because of artifacts during acquisition, or even intentional exclusion of the first signal points for the elimination of broad resonances that are producing the distorted baseline in the frequency domain. In this study, we propose the use of the Krylov Basis Diagonalization Method (KBDM) formalism as an alternative to some of FT limitations. The method adjusts signals of Free Induction Decay (FID) experiments with a sum of complex harmonic functions, exponentially damped, allowing easy manipulation of its characterization parameters. The KBDM is numerically more effective for truncated signal analysis and has several features that make it possible to remove peaks more efficiently, such as the residual water peak. Moreover, we introduced the possibility of quantification of MRS data with the described method. To evaluate the sensitivity, efficiency and reproducibility of the method for quantifying and analyzing truncated signals, and through the clinical spectra simulations and experiments in phantoms that would represent the brain metabolic environment, we proposed to perform proton MRS at different noise levels and with small variations of N- acetyl aspartate (NAA) metabolite. These studies allowed to prove the feasibility of the method to process MRS data and verified its potential in complementing techniques currently employed, especially when a greater temporal and spectral resolution is required, more than the limit imposed by the Uncertainty Relation of FT formalism. Furthermore, it is also a desirable effortless tool of handling specific peaks (e.g., exclusion and quantification). Exciting prospects from this work include the introduction of KBDM as an efficient and adjuvant technique to functional Magnetic Resonance Imaging (fMRI), for studying the brain functions, in MRS sequence to identify rapid variation in spectroscopic lines associated to metabolic activities in the brain.

A precisão e acurácia dos métodos mais utilizados atualmente de processamento de dados de espectroscopia por Ressonância Magnética (MRS), baseados na Transformada de Fourier (FT), requerem supressão apropriada (o que está longe de ser trivial) e aquisições longas para a obtenção de alta resolução espectral. Além disso, a FT tem dificuldades quando faltam dados no domínio de tempo, como, por exemplo, pela redução do tempo de aquisição, e consequente número de pontos adquiridos. Isto pode ocorrer, também, por artefatos na aquisição ou, ainda, seja pela exclusão intencional dos primeiros pontos do sinal para a eliminação de ressonâncias largas que estão distorcendo a linha de base no domínio da frequência. Neste estudo, propomos a utilização do Método de Diagonalização na Base de Krylov (KBDM) como uma alternativa a FT para algumas de suas limitações. O método ajusta sinais de experimentos de Free Induction Decay (FID) por uma soma de funções harmônicas complexas, amortecidas exponencialmente, permitindo uma fácil manipulação dos seus parâmetros de caracterização. O KBDM é numericamente mais efetivo para análise de sinais truncados e tem diversos recursos que possibilitam remover picos de forma mais eficiente, como por exemplo, o pico residual da água. Além disso, foi introduzida a possibilidade de quantificação de dados de MRS com o método. Para avaliar a sensibilidade, eficiência e reprodutibilidade do método para quantificar e analisar sinais truncados, foi proposto fazer simulações de espectros clínicos e experimentos em phantoms que representassem o ambiente metabólico do cérebro, para MRS de próton de diferentes níveis de ruídos e para pequenas variações do N-acetil aspartato (NAA). Com estes estudos pôde se comprovar a viabilidade do método para processar dados de MRS e verificar seu potencial na complementação das técnicas atualmente empregadas, especialmente quando uma resolução espectral e temporal maior que o limite imposto pela Relação de Incerteza do formalismo de Fourier é necessária. Além disso, uma desejável facilidade de manipulação de picos específicos (por exemplo, exclusão e quantificação) é proporcionada pelo método. Como perspectivas animadoras deste trabalho esperamos a introdução do KBDM como uma técnica eficiente e coadjuvante ao Imageamento de Ressonância Magnética funcional (fMRI), auxiliando estudos de funções cerebrais, em sequências de MRS para identificar uma rápida variação das linhas associadas as atividades metabólicas dos cérebros. / The precision and accuracy of the most widely used methods to perform Magnetic Resonance Spectroscopy (MRS) data processing based on the Fourier Transform (FT), require appropriate suppression (which is far from trivial) and long acquisitions to obtain high spectral resolution. Furthermore, FT poses difficulty when there are missing data in the time domain. This occurs because of reduction of the acquisition time and consequently also in the number of acquired points, or because of artifacts during acquisition, or even intentional exclusion of the first signal points for the elimination of broad resonances that are producing the distorted baseline in the frequency domain. In this study, we propose the use of the Krylov Basis Diagonalization Method (KBDM) formalism as an alternative to some of FT limitations. The method adjusts signals of Free Induction Decay (FID) experiments with a sum of complex harmonic functions, exponentially damped, allowing easy manipulation of its characterization parameters. The KBDM is numerically more effective for truncated signal analysis and has several features that make it possible to remove peaks more efficiently, such as the residual water peak. Moreover, we introduced the possibility of quantification of MRS data with the described method. To evaluate the sensitivity, efficiency and reproducibility of the method for quantifying and analyzing truncated signals, and through the clinical spectra simulations and experiments in phantoms that would represent the brain metabolic environment, we proposed to perform proton MRS at different noise levels and with small variations of N- acetyl aspartate (NAA) metabolite. These studies allowed to prove the feasibility of the method to process MRS data and verified its potential in complementing techniques currently employed, especially when a greater temporal and spectral resolution is required, more than the limit imposed by the Uncertainty Relation of FT formalism. Furthermore, it is also a desirable effortless tool of handling specific peaks (e.g., exclusion and quantification). Exciting prospects from this work include the introduction of KBDM as an efficient and adjuvant technique to functional Magnetic Resonance Imaging (fMRI), for studying the brain functions, in MRS sequence to identify rapid variation in spectroscopic lines associated to metabolic activities in the brain.

Este trabalho consiste em realizar um estudo detalhado das vantagens e desvantagens da utilização do FDM (Filter Diagonalization Method) para a análise de dados obtidos pela sequência de Precessão Livre no Estado Estacionário (Steady State Free Precession - SSFP) para aquisição rápida de espectros de Ressonância Magnética Nuclear (RMN). No caso de RMN de baixa resolução, o procedimento de aquisição rápida, SSFP, é uma poderosa ferramenta para melhorar a relação sinal/ruído, apresentando muitas aplicações práticas. Apesar desse sucesso em baixa resolução, a SSFP não é rotineiramente utilizada para aplicações em RMN de alta resolução, provavelmente devido ao (1) artefatos provenientes do truncamento do sinal e (2) as anomalias causadas pela mistura do FID com o eco dos sinais. Existem na literatura inúmeras possíveis técnicas para suprimir este tipo de problemas, porém, nenhuma delas é capaz de realmente eliminar as anomalias geradas devido ao procedimento de aquisição rápida da SSFP. O FDM é um método paramétrico não-linear para fitar sinais no domínio do tempo. Seu objetivo fundamental é resolver o Problema da Inversão Harmônica, HIP, tornando-se robusto e adequado para a análise espectral de sinais no domínio do tempo nos casos onde a Transformada de Fourier falha. Neste trabalho, demonstramos que o FDM pode ser implementado para análises de sinais SSFP, com mais eficiência que os obtidos pelos procedimentos padrões de TF. A temperatura ambiente, espectros de RMN 13C de amostras de brucina, obtidos com tempo entre pulsos de 100ms, podem ser reproduzidos com boa relação sinal/ruído e alta resolução por meio do FDM. A limitação da análise por FDM é mais relevante nos casos de espectros com alta densidade de picos em uma determinada região espectral. Nestes casos, o curto período de observação do sinal na janela do tempo impõe uma série de limitações na resolução obtida pelo FDM. / This work consists in a detailed study of the advantages and disadvantages of the use of the Filter Diagonalization Method, FDM, for data analysis in Steady State Free Precession, SSFP, technique, usually employed to implement fast acquisition of Nuclear Magnetic Resonance, NMR, spectra. In the case of low resolution NMR using fast acquisition procedures, SSFP is a powerful tool to improve signal-to-noise ratio, presenting several important practical applications. Despite its success in the low resolution regime, SSFP is not a routine technique for high resolution applications, so far, mainly because of (1) truncation artifacts and (2) the intrinsic anomalies caused by admixture of free-induction-decay and echo signals. The literature reports many possible techniques to solve such kind of problems, but, none of them is capable to really eliminate the generated spectra anomalies caused by the fast acquisition procedure used in SSFP. FDM is a parametric method for non-liner fitting performed in the time domain. Its main goal is to solve the Harmonic Inversion Problem, HIP, making it robust and suitable for spectral analysis of time signals in the cases where the Fourier Transform, FT, technique fail. In this work we demonstrate that FDM can be used to implement the analysis of the SSFP data, with more efficiency than that achieve by appropriated FT procedures. Room temperature 13C NMR spectra of brucine samples, obtained from pulse sequences with 100 ms repetition time, can be reproduced with good signal-to-noise ratio and high resolution by means of the FDM. The limitation of the FDM analysis is more relevant in the case of spectra with a high density of peaks in a limited spectral frequency region. In these cases, the reduced short observation time window imposes serious limitation to the resolution achieved by the FDM.

Cette thèse développe des méthodes de diagonalisation conjointe de matrices et de tenseurs d’ordre trois, et son application à la séparation MIMO de sources de télécommunications numériques. Après un état, les motivations et objectifs de la thèse sont présentés. Les problèmes de la diagonalisation conjointe et de la séparation de sources sont définis et un lien entre ces deux domaines est établi. Par la suite, plusieurs algorithmes itératifs de type Jacobi reposant sur une paramétrisation LU sont développés. Pour chacun des algorithmes, on propose de déterminer les matrices permettant de diagonaliser l’ensemble considéré par l’optimisation d’un critère inverse. On envisage la minimisation du critère selon deux approches : la première, de manière directe, et la seconde, en supposant que les éléments de l’ensemble considéré sont quasiment diagonaux. En ce qui concerne l’estimation des différents paramètres du problème, deux stratégies sont mises en œuvre : l’une consistant à estimer tous les paramètres indépendamment et l’autre reposant sur l’estimation indépendante de couples de paramètres spécifiquement choisis. Ainsi, nous proposons trois algorithmes pour la diagonalisation conjointe de matrices complexes symétriques ou hermitiennes et deux algorithmes pour la diagonalisation conjointe d’ensembles de tenseurs symétriques ou non-symétriques ou admettant une décomposition INDSCAL. Nous montrons aussi le lien existant entre la diagonalisation conjointe de tenseurs d’ordre trois et la décomposition canonique polyadique d’un tenseur d’ordre quatre, puis nous comparons les algorithmes développés à différentes méthodes de la littérature. Le bon comportement des algorithmes proposés est illustré au moyen de simulations numériques. Puis, ils sont validés dans le cadre de la séparation de sources de télécommunications numériques. / This thesis develops joint diagonalization of matrices and third-order tensors methods for MIMO source separation in the field of digital telecommunications. After a state of the art, the motivations and the objectives are presented. Then the joint diagonalisation and the blind source separation issues are defined and a link between both fields is established. Thereafter, five Jacobi-like iterative algorithms based on an LU parameterization are developed. For each of them, we propose to derive the diagonalization matrix by optimizing an inverse criterion. Two ways are investigated : minimizing the criterion in a direct way or assuming that the elements from the considered set are almost diagonal. Regarding the parameters derivation, two strategies are implemented : one consists in estimating each parameter independently, the other consists in the independent derivation of couple of well-chosen parameters. Hence, we propose three algorithms for the joint diagonalization of symmetric complex matrices or hermitian ones. The first one relies on searching for the roots of the criterion derivative, the second one relies on a minor eigenvector research and the last one relies on a gradient descent method enhanced by computation of the optimal adaptation step. In the framework of joint diagonalization of symmetric, INDSCAL or non symmetric third-order tensors, we have developed two algorithms. For each of them, the parameters derivation is done by computing the roots of the considered criterion derivative. We also show the link between the joint diagonalization of a third-order tensor set and the canonical polyadic decomposition of a fourth-order tensor. We confront both methods through numerical simulations. The good behavior of the proposed algorithms is illustrated by means of computing simulations. Finally, they are applied to the source separation of digital telecommunication signals.

<p>We consider a system of few electrons trapped in a two-dimensional circularquantum dot with harmonic confinement and in the presence of ahomogeneous magnetic field, with focus on the role of e-e interaction. Byperforming the exact diagonalization of the Hamiltonian in second quantization,the low-lying energy levels for spin polarized system are obtained. The singlet-triplet oscillation in the ground state of the two-electron system showing up inthe result is explained due to the role of Coulomb interaction. The splitting ofthe lowest Landau level is another effect of the e-e interaction, which is alsoobserved in the results.</p>

We consider a system of few electrons trapped in a two-dimensional circularquantum dot with harmonic confinement and in the presence of ahomogeneous magnetic field, with focus on the role of e-e interaction. Byperforming the exact diagonalization of the Hamiltonian in second quantization,the low-lying energy levels for spin polarized system are obtained. The singlet-triplet oscillation in the ground state of the two-electron system showing up inthe result is explained due to the role of Coulomb interaction. The splitting ofthe lowest Landau level is another effect of the e-e interaction, which is alsoobserved in the results.

<p>This thesis is concerned with the solution of the blind source separation problem (BSS). The BSS problem occurs frequently in various scientific and technical applications. In essence, it consists in separating meaningful underlying components out of a mixture of a multitude of superimposed signals.</p> <P> In the recent research literature there are two related approaches to the BSS problem: The first is known as Independent Component Analysis (ICA), where the goal is to transform the data such that the components become as independent as possible. The second is based on the notion of diagonality of certain characteristic matrices derived from the data. Here the goal is to transform the matrices such that they become as diagonal as possible. In this thesis we study the latter method of approximate joint diagonalization (AJD) to achieve a solution of the BSS problem. After an introduction to the general setting, the thesis provides an overview on particular choices for the set of target matrices that can be used for BSS by joint diagonalization.</p> <P> As the main contribution of the thesis, new algorithms for approximate joint diagonalization of several matrices with non-orthogonal transformations are developed.</p> <P> These newly developed algorithms will be tested on synthetic benchmark datasets and compared to other previous diagonalization algorithms.</p> <P> Applications of the BSS methods to biomedical signal processing are discussed and exemplified with real-life data sets of multi-channel biomagnetic recordings.</p> / <p>Diese Arbeit befasst sich mit der Lösung des Problems der blinden Signalquellentrennung (BSS). Das BSS Problem tritt häufig in vielen wissenschaftlichen und technischen Anwendungen auf. Im Kern besteht das Problem darin, aus einem Gemisch von überlagerten Signalen die zugrundeliegenden Quellsignale zu extrahieren.</p> <P> In wissenschaftlichen Publikationen zu diesem Thema werden hauptsächlich zwei Lösungsansätze verfolgt:</p> <P> Ein Ansatz ist die sogenannte "Analyse der unabhängigen Komponenten", die zum Ziel hat, eine lineare Transformation <B>V</B> der Daten <B>X</B> zu finden, sodass die Komponenten U_n der transformierten Daten <B>U</B> = <B> V X</B> (die sogenannten "independent components") so unabhängig wie möglich sind. Ein anderer Ansatz beruht auf einer simultanen Diagonalisierung mehrerer spezieller Matrizen, die aus den Daten gebildet werden. Diese Möglichkeit der Lösung des Problems der blinden Signalquellentrennung bildet den Schwerpunkt dieser Arbeit.</p> <P> Als Hauptbeitrag der vorliegenden Arbeit präsentieren wir neue Algorithmen zur simultanen Diagonalisierung mehrerer Matrizen mit Hilfe einer nicht-orthogonalen Transformation.</p> <P> Die neu entwickelten Algorithmen werden anhand von numerischen Simulationen getestet und mit bereits bestehenden Diagonalisierungsalgorithmen verglichen. Es zeigt sich, dass unser neues Verfahren sehr effizient und leistungsfähig ist. Schließlich werden Anwendungen der BSS Methoden auf Probleme der biomedizinischen Signalverarbeitung erläutert und anhand von realistischen biomagnetischen Messdaten wird die Nützlichkeit in der explorativen Datenanalyse unter Beweis gestellt.</p>

Este trabalho consiste em realizar um estudo detalhado das vantagens e desvantagens da utilização do FDM (Filter Diagonalization Method) para a análise de dados obtidos pela sequência de Precessão Livre no Estado Estacionário (Steady State Free Precession - SSFP) para aquisição rápida de espectros de Ressonância Magnética Nuclear (RMN). No caso de RMN de baixa resolução, o procedimento de aquisição rápida, SSFP, é uma poderosa ferramenta para melhorar a relação sinal/ruído, apresentando muitas aplicações práticas. Apesar desse sucesso em baixa resolução, a SSFP não é rotineiramente utilizada para aplicações em RMN de alta resolução, provavelmente devido ao (1) artefatos provenientes do truncamento do sinal e (2) as anomalias causadas pela mistura do FID com o eco dos sinais. Existem na literatura inúmeras possíveis técnicas para suprimir este tipo de problemas, porém, nenhuma delas é capaz de realmente eliminar as anomalias geradas devido ao procedimento de aquisição rápida da SSFP. O FDM é um método paramétrico não-linear para fitar sinais no domínio do tempo. Seu objetivo fundamental é resolver o Problema da Inversão Harmônica, HIP, tornando-se robusto e adequado para a análise espectral de sinais no domínio do tempo nos casos onde a Transformada de Fourier falha. Neste trabalho, demonstramos que o FDM pode ser implementado para análises de sinais SSFP, com mais eficiência que os obtidos pelos procedimentos padrões de TF. A temperatura ambiente, espectros de RMN 13C de amostras de brucina, obtidos com tempo entre pulsos de 100ms, podem ser reproduzidos com boa relação sinal/ruído e alta resolução por meio do FDM. A limitação da análise por FDM é mais relevante nos casos de espectros com alta densidade de picos em uma determinada região espectral. Nestes casos, o curto período de observação do sinal na janela do tempo impõe uma série de limitações na resolução obtida pelo FDM. / This work consists in a detailed study of the advantages and disadvantages of the use of the Filter Diagonalization Method, FDM, for data analysis in Steady State Free Precession, SSFP, technique, usually employed to implement fast acquisition of Nuclear Magnetic Resonance, NMR, spectra. In the case of low resolution NMR using fast acquisition procedures, SSFP is a powerful tool to improve signal-to-noise ratio, presenting several important practical applications. Despite its success in the low resolution regime, SSFP is not a routine technique for high resolution applications, so far, mainly because of (1) truncation artifacts and (2) the intrinsic anomalies caused by admixture of free-induction-decay and echo signals. The literature reports many possible techniques to solve such kind of problems, but, none of them is capable to really eliminate the generated spectra anomalies caused by the fast acquisition procedure used in SSFP. FDM is a parametric method for non-liner fitting performed in the time domain. Its main goal is to solve the Harmonic Inversion Problem, HIP, making it robust and suitable for spectral analysis of time signals in the cases where the Fourier Transform, FT, technique fail. In this work we demonstrate that FDM can be used to implement the analysis of the SSFP data, with more efficiency than that achieve by appropriated FT procedures. Room temperature 13C NMR spectra of brucine samples, obtained from pulse sequences with 100 ms repetition time, can be reproduced with good signal-to-noise ratio and high resolution by means of the FDM. The limitation of the FDM analysis is more relevant in the case of spectra with a high density of peaks in a limited spectral frequency region. In these cases, the reduced short observation time window imposes serious limitation to the resolution achieved by the FDM.

In this work a numerical approach for the determination of the energy spectra and the calculation of thermodynamic properties of magnetic molecules is presented. The work is focused on the treatment of spin systems which exhibit point-group symmetries. Ring-like and archimedean-type structures are discussed as prominent examples. In each case the underlying spin quantum system is modeled by an isotropic Heisenberg Hamiltonian. Its energy spectrum is calculated either by numerical exact diagonalization or by an approximate diagonalization method introduced here. In order to implement full spin-rotational symmetry the numerical approach at hand is based on the use of irreducible tensor operators. Furthermore, it is shown how an unrestricted use of point-group symmetries in combination with the use of irreducible tensor operators leads to a reduction of the dimensionalities as well as to additional information about the physics of the systems. By exemplarily demonstrating how the theoretical foundations of the irreducible tensor operator technique can be realized within small spin systems the technical aspect of this work is covered. These considerations form the basis of the computational realization that was implemented and used in order to get insight into the investigated systems.