In recent years, combinatorial properties of finite and infinite words have become increasingly important in fields of physics, biology, mathematics, and computer science. In particular, the fascinating family of Sturmian words has become an extremely active subject of research. These infinite binary sequences have numerous applications in various fields of mathematics, such as symbolic dynamics, the study of continued fraction expansion, and also in some domains of physics ( quasicrystal modelling ) and computer science ( pattern recognition, digital straightness ). There has also been a recent surge of interest in a natural generalization of Sturmian words to more than two letters - the so - called episturmian words, which include the well - known Arnoux - Rauzy sequences. This thesis represents a significant contribution to the study of Sturmian and episturmian words, and related objects such as generalized Thue - Morse words and substitutions on a finite alphabet. Specifically, we prove some new properties of certain palindromic factors of the infinite Fibonacci word; establish generalized ' singular ' decompositions of suffixes of certain morphic Sturmian words; completely describe where palindromes occur in characteristic Sturmian words; explicitly determine all integer powers occurring in a certain class of k-strict episturmian words ( including the k-bonacci word ) ; and prove that certain episturmian and generalized Thue - Morse continued fractions are transcendental. Lastly, we begin working towards a proof of a characterization of invertible substitutions on a finite alphabet, which generalizes the fact that invertible substitutions on two letters are exactly the Sturmian morphisms. / Thesis (Ph.D.)--School of Mathematical Sciences, 2006.

Au cours des dix dernières années, des composés oligomères intéressants à base de l’acide cholique ont été synthétisés et caractérisés par nos collaborateurs du groupe de Julian X.X. Zhu à l’Université de Montréal (UdeM). Dans un travail récent, ils ont synthétisé un dimère d'acide cholique qui pouvait former de façon réversible une cavité moléculaire lorsqu'il était dissous dans des milieux de polarité différente ; dans l'eau, le dimère forme une cavité hydrophobe, et dans des milieux organiques, le dimère forme une cavité hydrophile. Ainsi, ce type de composés amphiphiles, lorsqu'ils sont en solution, démontre un comportement de cavité moléculaire qui dépend des conditions du solvant, formant une cavité de polarité opposée à celle du milieu dans lequel ils se trouvent. Le comportement d'inversion de la cavité résulte de la flexibilité conformationnelle du lieur chimique entre les monomères d'acide cholique. La capacité des cavités de piéger des sondes moléculaires en fonction de leur polarité suggère que ce type d’oligomères d’acide cholique pourrait constituer des phases stationnaires intéressantes pour la chromatographie en phase liquide à haute performance (HPLC), où la séparation est basée sur la polarité du soluté par rapport à la phase mobile. Puisqu’ils peuvent constituer des cavités hydrophobes et hydrophiles, ils pourraient donc être exploités en chromatographie en phase normale (NPC) et en chromatographie en phase inverse (RPC). La possibilité d'avoir une phase stationnaire réversible avec une affinité bimodale appropriée pourrait être avantageuse en biosciences, en sciences de l'environnement et favoriser la séparation de mélanges complexes en élargissant le champ d'application d’une seule colonne chromatographique. L’affinité bimodale pourrait notamment permettre d'éviter de changer le mode de fonctionnement du HPLC; un processus long, coûteux et nécessitant une grande quantité de solvant pour rééquilibrer et passiver le système fluidique de l’instrument. Ce mémoire est une étude exploratoire qui vise à déterminer si ce type d’oligomères d’acide cholique, une fois liés à des particules de gel de silice (6 μm de diamètre), montre la formation de cavités moléculaires dans diverses conditions de phase mobile et s’il pourrait être utilisé pour effectuer des séparations comme phase stationnaire bimodale. À notre connaissance, il n’existe pas encore de phase stationnaire réversible à base d’oligomères d’acide cholique capables d’interagir avec des composés hydrophiles et hydrophobes, en fonction de la polarité ii de l’éluant. Ce type de phase stationnaire se compare à d’autres phases stationnaires bimodales pouvant être utilisées en NPC ou en RPC, parmi lesquels on trouve entre autres des copolymères amphiphiles, des structures organométalliques et des macromolécules comme les cyclodextrines (CD). La nature bimodale de la phase stationnaire à base de CD rapporté dans la littérature est assez similaire aux phases stationnaires des oligomères d’acide cholique de cette étude, grâce à leur cavité hydrophobe naturelle et un extérieur hydrophile, mais sans toutefois que la cavité soit réversible à cause de la rigidité de l’anneau CD. Les particules de silice greffées avec des oligomères d’acide cholique ont été empaquetées par suspension dans un tube capillaire en silice fondue de diamètre intérieur (ID) de 250 μm pour former des colonnes capillaires de 10 cm de long. Les performances chromatographiques en phase liquide des phases stationnaires ont été étudiées à l'aide d'un instrument HPLC adapté aux colonnes capillaires et muni d’un détecteur d’absorption. Plusieurs sondes-analytes sont étudiées dans ce mémoire pour caractériser la rétention causée par les phases stationnaires dans diverses phases mobiles eau/organique. Des comportements en RPC et d'interaction hydrophile (HILIC) ont été observés dans différentes plages de composition de phase mobile eau/organique. Les tests ont montré que les matériaux étaient capables de retarder des analytes non polaires avec une diminution du pourcentage organique (% org) sur une large plage de compositions (45% à 0% org dans le cas des alkylbenzènes). Les cavités hydrophobes semblent quant à elles être responsables de la rétention aux % org moins que 10% et pour seulement une faible partie de la plage totale de la rétention hydrophobique. Le comportement en phase inverse a été comparé aux colonnes classiques à base de chaînes alkyles (C3, C4, C8 et C18) pour évaluer l’importance des interactions hydrophobes. Inversement, une augmentation du % org, en particulier de l'acétonitrile, a entraîné la rétention de composés polaires sur une courte plage de composition de solvant à partir de 85% org. Cette dernière rétention est toutefois principalement imputable aux mécanismes HILIC avec le support de silice gel découvert et non aux cavités hydrophiles du dimère d’acide cholique. / Over the past ten years, interesting oligomeric compounds based on cholic acids have been synthesized and characterized by our collaborators from the Julian X.X Zhu group at the Université de Montréal (UdeM). In a recent work, they synthesized a cholic acid dimer and showed that it could form invertible molecular pockets when dissolved in media of different polarity; in water, the dimer forms hydrophobic pockets, and in organic media, the dimer forms hydrophilic pockets. Therefore, these amphiphilic compounds, when in solution, demonstrate molecular pocket behavior depending on solvent conditions to form a cavity of opposite polarity of the media in which they are located. The inversion behavior results from the conformational flexibility of the chemical linker between the bile acid monomers. The ability of the pockets to trap probe species based on their polarity suggests that the cholic acid oligomers might be interesting stationary phases for high-performance liquid chromatography (HPLC), where separation is based on solute polarity relative to the mobile phase. Since these materials can produce hydrophobic and hydrophilic pockets, they could be exploited in both normal-phase chromatography (NPC) and reversed-phase chromatography (RPC). The ability to have an invertible stationary phase with suitable bimodal affinity could be advantageous in biosciences, environmental sciences, and for the separation of complex mixtures by widening the field of application of the same chromatographic column. The bimodal affinity may, in particular, make it possible to avoid changing the operating mode of the HPLC; a costly and lengthy process requiring a large amount of solvent to re-equilibrate and passivate all fluidic paths of the instrument. This memoir is an exploratory study that sets out to evaluate whether this type of cholic acid oligomer, once bonded to silica gel particles (6 μm diameter), shows the formation of molecular pockets in various mobile phase conditions and if they can be used to perform separations as bimodal stationary phases. To the best of our knowledge, invertible stationary phases based on cholic acid oligomers that are capable of selective binding and release of both hydrophilic and hydrophobic compounds depending on the polarity of the eluent do not yet exist. This type of stationary phase can be compared to the other bimodal stationary phases that can be used in either NPC or RPC that includes amphiphilic copolymers, organometallic structures and macromolecules like cyclodextrins (CD). The bimodal nature of the CD-based iv stationary phases are quite similar to the cholic acid oligomers stationary phases of this study, thanks to a natural hydrophobic cavity and a hydrophilic exterior, but without the invertibility of the cavity due to the rigidity of the CD ring. The grafted particles were slurry-packed into 250 μm inner diameter (ID) fused silica tubing to make 10 cm long capillary columns. The liquid chromatographic performance of the stationary phases was investigated using a capillary HPLC instrument with a UV absorbance detector. Several probe analytes were investigated to characterize the molecular pocket-based retention in various water/organic mobile phases. RPC and hydrophilic interaction (HILIC) behaviors were observed in distinctive composition ranges of water/organic mobile phases. The tests showed that the materials were able to retain nonpolar compounds gradually with the decrease of percentage organic (% org) over a wide range of compositions (45% to 0% org for alkylbenzenes). The hydrophobic pockets seem to be responsible for the retention at % org less than 10% and only for a small extent of the total range of the hydrophobic retention. The reversed phase behavior was compared to classical alkyl-chain-based columns (C3, C4, C8 and C18) to assess the importance of the hydrophobic interactions. Conversely, an increase in % org, especially acetonitrile, resulted in the retention of polar compounds over a smaller range of % org starting at 85% org. This latter retention is mainly attributable to HILIC mechanisms with the uncapped silica gel support and not the cholic acid dimer hydrophilic pockets.

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

Soit R un anneau de valuation discrète de corps de fractions K. Soit X_K un K- schéma propre géométriquement normal. On montre que X_K possède des modèles X sur R, propres, plats, normaux et tels que tout faisceau inversible sur X_K se prolonge en un faisceau inversible sur X. On peut alors reconstruire le modèle de Néron de la variété de Picard de X_K, à partir du foncteur de Picard de X/R.Lorsque R est hensélien à corps résiduel algébriquement clos, on en tire des informations sur le prolongement de l’équivalence algébrique de X_K à X. En particulier, on peut décrire le symbole de Néron entre 0-cycles de degré zéro et diviseurs algébriquement équivalents à zéro sur X_K, en termes de multiplicités d’intersection sur le modèle X. Ceci nous permet de reformuler la conjecture de dualité de Grothendieck pour les modèles de Néron des variétés abéliennes, en termes d’équivalence algébrique relative. / Let R be a discrete valuation ring with fraction field K. Let X_K be proper geometrically normal scheme over K. One shows that X_K admits models X over R which are proper, flat, normal an such that any invertible sheaf on X_K can be extended to an invertible sheaf on X. Then, one can recover the Néron model of the Picard variety of X_K from the Picard functor of X/R.When R is henselian with algebraically closed residue field, one obtains some consequences about the extension of algebraic equivalence from X_K to X. In particular, one can describe the Néron symbol between 0-cycles of degree zero and divisors which are algebraically equivalent to zero on X_K, in terms of intersection multiplicities on the model X. This allows us to reformulate Grothendieck’s duality conjecture for Néron models of abelian varieties, in terms of relative algebraic equivalence.

[pt] Desagregação de cargas é a tarefa de estimar o consumo individual de aparelhos elétricos a partir de medições de consumo de energia coletadas em um único ponto, em geral no quadro de distribuição do circuito. Este trabalho explora o uso de técnicas de aprendizado de máquina para esta tarefa, em uma base de dados coletada em uma fábrica de ração de aves no Brasil. É proposto um modelo combinando arquiteturas de autoencoders variacionais com as de fluxos normalizantes inversíveis. Os resultados obtidos são, de maneira geral, superiores aos melhores resultados reportados para esta base de dados até então, os superando em até 86 por cento no Erro do Sinal Agregado e em até 81 por cento no Erro de Desagregação Normalizado dependendo do equipamento desagregado. / [en] Load Disaggregation is the task of estimating appliance-level consumption from a single aggregate consumption metering point. This work explores machine learning techniques applied to an industrial load disaggregation dataset from a poultry feed factory in Brazil. It proposes a model that combines variational autoencoders with invertible normalizing flows models. The results obtained are, in general, better than the current best reported results for this dataset, outperforming them by up to 86 percent in the Signal Aggregate Error and by up to 81 percent in the Normalized Disaggregation Error.

Normalizing flows represent a class of generative models that exhibit a number of attractive properties, but do not always achieve state-of-the-art performance when it comes to perceived naturalness of generated samples. To improve the quality of generated samples, this thesis examines methods to enhance the expressivity of discrete-time normalizing flow models and thus their ability to capture different aspects of the data. In the first part of the thesis, we propose an invertible neural network architecture as an alternative to popular architectures like Glow that require an individual neural network per flow step. Although our proposal greatly reduces the number of parameters, it has not been done before, as such architectures are believed to not be powerful enough. For this reason, we define two optional extensions that could greatly increase the expressivity of the architecture. We use augmentation to add Gaussian noise variables to the input to achieve arbitrary hidden-layer widths that are no longer dictated by the dimensionality of the data. Moreover, we implement Piecewise Affine Activation Functions that represent a generalization of Leaky ReLU activations and allow for more powerful transformations in every individual step. The resulting three models are evaluated on two simple synthetic datasets – the two moons dataset and one generated from a mixture of eight Gaussians. Our findings indicate that the proposed architectures cannot adequately model these simple datasets and thus do not represent alternatives to current stateof-the-art models. The Piecewise Affine Activation Function significantly improved the expressivity of the invertible neural network, but could not make use of its full potential due to inappropriate assumptions about the function’s input distribution. Further research is needed to ensure that the input to this function is always standard normal distributed. We conducted further experiments with augmentation using the Glow model and could show minor improvements on the synthetic datasets when only few flow steps (two, three or four) were used. However, in a more realistic scenario, the model would encompass many more flow steps. Lastly, we generalized the transformation in the coupling layers of modern flow architectures from an elementwise affine transformation to a matrixbased affine transformation and studied the effect this had on MoGlow, a flow-based model of motion. We could show that McMoGlow, our modified version of MoGlow, consistently achieved a better training likelihood than the original MoGlow on human locomotion data. However, a subjective user study found no statistically significant difference in the perceived naturalness of the samples generated. As a possible reason for this, we hypothesize that the improvements are subtle and more visible in samples that exhibit slower movements or edge cases which may have been underrepresented in the user study. / Normalizing flows representerar en klass av generativa modeller som besitter ett antal eftertraktade egenskaper, men som inte alltid uppnår toppmodern prestanda när det gäller upplevd naturlighet hos genererade data. För att förbättra kvaliteten på dessa modellers utdata, undersöker detta examensarbete metoder för att förbättra uttrycksfullheten hos Normalizing flows-modeller i diskret tid, och därmed deras förmåga att fånga olika aspekter av datamaterialet. I den första delen av uppsatsen föreslår vi en arkitektur uppbyggt av ett inverterbart neuralt nätverk. Vårt förslag är ett alternativ till populära arkitekturer som Glow, vilka kräver individuella neuronnät för varje flödessteg. Även om vårt förslag kraftigt minskar antalet parametrar har detta inte gjorts tidigare, då sådana arkitekturer inte ansetts vara tillräckligt kraftfulla. Av den anledningen definierar vi två oberoende utökningar till arkitekturen som skulle kunna öka dess uttrycksfullhet avsevärt. Vi använder så kallad augmentation, som konkatenerar Gaussiska brusvariabler till observationsvektorerna för att uppnå godtyckliga bredder i de dolda lagren, så att deras bredd inte längre begränsas av datadimensionaliteten. Dessutom implementerar vi Piecewise Affine Activation-funktioner (PAAF), vilka generaliserar Leaky ReLU-aktiveringar genom att möjliggöra mer kraftfulla transformationer i varje enskilt steg. De resulterande tre modellerna utvärderas med hjälp av två enkla syntetiska datamängder - ”the two moons dataset” och ett som genererats genom att blanda av åtta Gaussfördelningar. Våra resultat visar att de föreslagna arkitekturerna inte kan modellera de enkla datamängderna på ett tillfredsställande sätt, och därmed inte utgör kompetitiva alternativ till nuvarande moderna modeller. Den styckvisa aktiveringsfunktionen förbättrade det inverterbara neurala nätverkets uttrycksfullhet avsevärt, men kunde inte utnyttja sin fulla potential på grund av felaktiga antaganden om funktionens indatafördelning. Ytterligare forskning behövs för att hantera detta problem. Vi genomförde ytterligare experiment med augmentation av Glow-modellen och kunde påvisa vissa förbättringar på de syntetiska dataseten när endast ett fåtal flödessteg (två, tre eller fyra) användes. Däremot omfattar modeller i mer realistiska scenarion många fler flödessteg. Slutligen generaliserade vi transformationen i kopplingslagren hos moderna flödesarkitekturer från en elementvis affin transformation till en matrisbaserad affin transformation, samt studerade vilken effekt detta hade på MoGlow, en flödesbaserad modell av 3D-rörelser. Vi kunde visa att McMoGlow, vår modifierade version av MoGlow, konsekvent uppnådde bättre likelihood i träningen än den ursprungliga MoGlow gjorde på mänskliga rörelsedata. En subjektiv användarstudie på exempelrörelser genererade från MoGlow och McMoGlow visade dock ingen statistiskt signifikant skillnad i användarnas uppfattning av hur naturliga rörelserna upplevdes. Som en möjlig orsak till detta antar vi att förbättringarna är subtila och mer synliga i situationer som uppvisar långsammare rörelser eller i olika gränsfall som kan ha varit underrepresenterade i användarstudien.

Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandt. / The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable.

No description available.

Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandt. / The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable.