Dissertations / Theses on the topic 'Clifford analysis'

Doutoramento em Matemática
Esta tese estuda os fundamentos de uma teoria discreta de funções em dimensões superiores usando a linguagem das Álgebras de Clifford. Esta abordagem combina as ideias do Cálculo Umbral e Formas Diferenciais. O potencial desta abordagem assenta essencialmente da osmose entre ambas as linguagens. Isto permitiu a construção de operadores de entrelaçamento entre estruturas contínuas e discretas, transferindo resultados conhecidos do contínuo para o discreto. Adicionalmente, isto resultou numa transcrição mimética de bases de polinómios, funções geradoras, Decomposição de Fischer, Lema de Poincaré, Teorema de Stokes, fórmula de Cauchy e fórmula de Borel-Pompeiu. Esta teoria também inclui a descrição dos homólogos discretos de formas diferenciais, campos vectores e integração discreta. De facto, a construção resultante de formas diferenciais, campos vectores e integração discreta em termos de coordenadas baricêntricas conduz à correspondência entre a teoria de Diferenças Finitas e a teoria de Elementos Finitos, dando um núcleo de aplicações desta abordagem promissora em análise numérica. Algumas ideias preliminares deste ponto de vista foram apresentadas nesta tese. Também foram apresentados resultados preliminares na teoria discreta de funções em complexos envolvendo simplexes. Algumas ligações com Combinatória e Mecânica Quântica foram também apresentadas ao longo desta tese.
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.

Doutoramento em Matemática e Aplicações (PDMA)
Compressed sensing is a new paradigm in signal processing which states that for certain matrices sparse representations can be obtained by a simple l1-minimization. In this thesis we explore this paradigm for higher-dimensional signal. In particular three cases are being studied: signals taking values in a bicomplex algebra, quaternionic signals, and complex signals which are representable by a nonlinear Fourier basis, a so-called Takenaka-Malmquist system.
Amostragem Compressiva é um novo paradigma em processamento de sinal, no qual se assegura, para determinadas matrizes, que as representações esparsas de sinais podem ser obtidas por intermédio de um simples procedimento de l1-minimização. Nesta tese, exploramos este paradigma para sinais em dimensões superiores. Estudaremos três casos particulares: sinais com valores na álgebra bi-complexa, sinais quaterniónicos e, finalmente, sinais complexos representáveis por uma base de Fourier não-linear, dito sistema de Takenaka-Malmquist.

Orientador: Rafael de Freitas Leão
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Os grupos Spin aparecem de várias formas em Matemática e em Física-Matemática, tendo grande importância na teoria de brados e de operadores diferenciais sobre os mesmos. O conceito de estrutura spin é deles derivado, sendo ele a base de toda uma teoria, conhecida como geometria spin. Esta dissertação introduz os primeiros conceitos necessários ao estudo de tais grupos, assim como alguns aspectos importantes relacionados a eles. Dada a natureza dos grupos Spin e dos problemas aos quais estão relacionados, vários tópicos na interface entre álgebra e geometria tiveram de ser abordados. Estudamos em um primeiro momento as álgebras de Clifford, sua representação adjunta torcida e os grupos Spin como subgrupos do grupo das unidades de tais álgebras. À estes estudos, seguiu-se uma análise detalhada da teoria de espaços de recobrimento e da classificação dos mesmos. Pudemos com isso entender o grupo Spin, via representação adjunta torcida, como o recobrimento universal do grupo especial ortogonal de um espaço quadrático não-degenerado. Nos concentramos daí na teoria de brados principais e a relação destes com as propriedades geométricas das variedades sobre as quais eles estão construídos. Para sintetizar o que foi estudado, construímos algebricamente a fibração de Hopf ao final desta dissertação, explicitando sua relação com a estrutura spin da esfera S²
Abstract: Spin groups come in many forms in Mathematics and Mathematical Physics, having great importance in the theory of fiber bundles and differential operators defined on them. The concept of spin structure is derived from them, being the basis of all a theory, known as spin geometry. This thesis introduces the first concepts necessary for the study of such groups, as well as important aspects related to them. Given the nature of the Spin groups and problems which they're related to, several topics at the interface between algebra and geometry had to be addressed. At first, we studied Clifford algebras, their twisted adjoint representation and Spin groups as subgroups of the group of units of such algebras. Followed these studies a detailed analysis of the theory of covering spaces and the classification of them. Done that, we were able to understand the group Spin, via the twisted adjoint representation, as the universal covering space of the special orthogonal group of a non-degenerate quadratic space. From there, we focused on the theory of principal bundles and their relationship with the geometric properties of manifolds on which they are built. To summarize what was studied, we algebraically construct the Hopf fibration at the end of this thesis, explaining its relationship with the spin structure of the sphere S²
Mestrado
Matematica
Mestre em Matemática

Orientador: Jayme Vaz Junior
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Atualmente, o estudo das Álgebras de Clifford é utilizado em inúmeras áreas de pesquisa. Uma delas é na área de Visão Computacional. O objetivo central dessa dissertação consiste em exibir noções sobre Álgebras de Clifford e sua utilização na formulação dos conceitos e definições de operações entre objetos da Geometria Projetiva e na formulação algébrica de câmeras virtuais, que é um dos assuntos tratados na área de Visão Computacional. Para isso são expostos de forma gradual e coerente os principais aspectos teóricos necessários para atingir os objetivos citados. Como resultado, as Álgebras de Clifford proporcionam uma excelente descrição da Geometria Projetiva e das câmeras virtuais
Abstract: Currently, the study of Clifford algebras are used in many research areas. One is in the area of Computer Vision. The main objective of this dissertation is to display notions of Clifford algebras and their use in formulating the concepts and definitions of transactions between objects of Projective Geometry and algebraic formulation of virtual cameras, which is one of the topics covered in Computer Vision. For it is exposed gradually and consistently the main theoretical aspects needed to achieve the goals mentioned. As a result, Clifford algebras provide an excellent description of Projective Geometry and virtual cameras
Mestrado
Matematica Aplicada
Mestre em Matemática Aplicada

Orientador: Waldyr Alves Rodrigues Junior
Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatistica e Computação Cientifica
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Resumo: O presente trabalho tem a intenção de apresentar por intermédio de uma linguagem unificada alguns conceitos de cálculo vetorial, álgebra linear (matrizes e transformações lineares) e também algumas idéias elementares sobre os grupos de rotações em duas e três dimensões e seus grupos de recobrimento, que geralmente são tratados como "fragmentos" em várias modalidades de cursos no ensino superior. Acreditamos portanto que nosso texto possas ser útil para alunos dos cursos de graduação dos cursos de Engenharia, Física, Matemática e interessados em Matemática em geral. A linguagem unificada à que nos referimos acima é obtida com a introdução do conceitos das álgebras geométricas (ou de Clifford) onde, como veremos, é possível fornecer uma formulação algébrica elegante aos conceitos de vetores, planos e volumes orientados e definir para tais objetos o produto escalar, os produtos contraídos à esquerda e à direita, o produto exterior (associado, como veremos, em casos particulares ao produto vetorial) e finalmente o produto geométrico (Clifford), o que permite o uso desses conceitos para a solução de inúmeros problemas de geometria analítica no R ² e no R ³. Procuramos ilustrar todos estes conceitos com vários exemplos e exercícios com graus variáveis de dificuldades. Nossa apresentação é bem próxima àquela do livro de Lounesto, e de fato muitas seções são traduções (eventualmente seguidas de comentários) de seções daquele livro. Contudo, em muitos lugares, acreditamos que nossa apresentação esclarece e completa as correspondentes do livro de Lounesto
Abstract: This paper aims to present using an unified language a few concepts of vector calculus, linear algebra (matrices and linear transformations) and also some basic ideas about the groups of rotations in two and three dimensions and their covering group, which generally are treated as "fragments" in various types of courses in higher education. We believe therefore that our text should be useful to students of undergraduate courses like Engineering, Physics, Mathematics and people interested in Mathematics in general. The unified language that we refer to above is obtained by introducing the concept of geometric (or Clifford) algebra where, as we shall see, it is possible to give an elegant algebraic formulation to the concepts of vectors, oriented planes and oriented volumes, and to define to those objects the scalar product, the right and left contracted products, the exterior product (associated, as we shall see, in particular cases to the vector product) and finally the geometric (Clifford) product, and moreover, to use those concepts to solve may problems of analytic geometry in R ² and R ³. We illustrated all those concepts with several examples and exercises with variable degrees of difficulties. Our presentation is nearly the one in Lounesto's book, and in fact some sections are no more than translations (eventually with commentaries) from sections of that book. However, in many places, we believe that our presentation clarify nd completement the corresponding ones in Lounesto's book
Mestrado
Ágebra
Mestre em Matemática

Orientador: Márcio Antônio de Faria Rosa
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: O resumo poderá ser visualizado no texto completo da tese digital
Abstract: The complete abstract is available with the full electronic document .
Mestrado
Matematica
Mestre em Matemática

The work of this Ph.D. thesis in mathematics concerns the problem of determining existence, uniqueness, and structure of zero-energy states in supersymmetric matrix models, which arise from a quantum mechanical description of the physics of relativistic membranes, reduced Yang-Mills gauge theory, and of nonperturbative features of string theory, respectively M-theory. Several new approaches to this problem are introduced and considered in the course of seven scientific papers, including: construction by recursive methods (Papers A and D), deformations and alternative models (Papers B and C), averaging with respect to symmetries (Paper E), and weighted supersymmetry and index theory (Papers F and G). The mathematical tools used and developed for these approaches include Clifford algebras and associated representation theory, structure of supersymmetric quantum mechanics, as well as spectral theory of (matrix-) Schrödinger operators.
QC20100629

The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.

Les signaux analytiques multidimensionnels nous permettent d'avoir des possibilités de calculer les phases et modules. Cependant, peu de travaux se trouvent sur les signaux analytiques multidimensionnels qui effectuent une extensibilité appropriée pour les applications à la fois sur du traitement des données médicales 2D et 3D. Cette thèse a pour objectif de proposer des nouvelles méthodes pour le traitement des images médicales 2D/3D pour les applications de détection d'enveloppe et d'estimation du mouvement. Premièrement, une représentation générale du signal quaternionique 2D est proposée dans le cadre de l'algèbre de Clifford et cette idée est étendue pour modéliser un signal analytique hypercomplexe 3D. La méthode proposée décrit que le signal analytique complexe 2D, est égal aux combinaisons du signal original et de ses transformées de Hilbert partielles et totale. Cette écriture est étendue au cas du signal analytique hypercomplexe 3D. Le résultat obtenu est que le signal analytique hypercomplexe de Clifford peut être calculé par la transformée de Fourier complexe classique. Basé sur ce signal analytique de Clifford 3D, une application de détection d'enveloppe en imagerie ultrasonore 3D est présentée. Les résultats montrent une amélioration du contraste de 7% par rapport aux méthodes de détection d'enveloppe 1D et 2D. Deuxièmement, cette thèse propose une approche basée sur deux phases spatiales du signal analytique 2D appliqué aux séquences cardiaques. En combinant l'information de ces phases des signaux analytiques de deux images successives, nous proposons un estimateur analytique pour les déplacements locaux 2D. Pour améliorer la précision de l'estimation du mouvement, un modèle bilinéaire local de déformation est utilisé dans un algorithme itératif. Cette méthode basée sur la phase permet au déplacement d'être estimé avec une précision inférieure au pixel et est robuste à la variation d'intensité des images dans le temps. Les résultats de sept séquences simulées d'imagerie par résonance magnétique (IRM) marquées montrent que notre méthode est plus précise comparée à des méthodes récentes utilisant la phase du signal monogène ou des méthodes classiques basées sur l'équation du flot optique. Les erreurs d'estimation de mouvement de la méthode proposée sont réduites d'environ 33% par rapport aux méthodes testées. En outre, les déplacements entre deux images sont cumulés en temps, pour obtenir la trajectoire d'un point du myocarde. En effet, des trajectoires ont été calculées sur deux patients présentant des infarctus. Les amplitudes des trajectoires des points du myocarde appartenant aux régions pathologiques sont clairement réduites par rapport à celles des régions normales. Les trajectoires des points du myocarde, estimées par notre approche basée sur la phase de signal analytique, sont donc un bon indicateur de la dynamique cardiaque locale. D'ailleurs, elles s'avèrent cohérentes à la déformation estimée du myocarde
Different mathematical tools, such as multidimensional analytic signals, provide possibilities to calculate multidimensional phases and modules. However, little work can be found on multidimensional analytic signals that perform appropriate extensibility for the applications on both of the 2D and 3D medical data processing. In this thesis, based on the Hahn 1D complex analytic, we aim to proposed a multidimensional extension approach from the 2D to a new 3D hypercomplex analytic signal in the framework of Clifford algebra. With the complex/hypercomplex analytic signals, we propose new 2D/3D medical image processing methods for the application of ultrasound envelope detection and cardiac motion estimation. Firstly, a general representation of 2D quaternion signal is proposed in the framework of Clifford algebra and this idea is extended to generate 3D hypercomplex analytic signal. The proposed method describes that the complex/hypercomplex 2D analytic signals, together with 3D hypercomplex analytic signal, are equal to different combinations of the original signal and its partial and total Hilbert transforms, which means that the hypercomplex Clifford analytic signal can be calculated by the classical Fourier transform. Based on the proposed 3D Clifford analytic signal, an application of 3D ultrasound envelope detection is presented. The results show a contrast optimization of about 7% comparing with 1D and 2D envelope detection methods. Secondly, this thesis proposes an approach based on two spatial phases of the 2D analytic signal applied to cardiac sequences. By combining the information of these phases issued from analytic signals of two successive frames, we propose an analytical estimator for 2D local displacements. To improve the accuracy of the motion estimation, a local bilinear deformation model is used within an iterative estimation scheme. This phase-based method allows the displacement to be estimated with subpixel accuracy and is robust to image intensity variation in time. Results from seven realistic simulated tagged magnetic resonance imaging (MRI) sequences show that our method is more accurate compared with the state-of-the-art method. The motion estimation errors (end point error) of the proposed method are reduced by about 33% compared with that of the tested methods. In addition, the frame-to-frame displacements are further accumulated in time, to allow for the calculation of myocardial point trajectories. Indeed, from the estimated trajectories in time on two patients with infarcts, the shape of the trajectories of myocardial points belonging to pathological regions are clearly reduced in magnitude compared with the ones from normal regions. Myocardial point trajectories, estimated from our phase-based analytic signal approach, are therefore a good indicator of the local cardiac dynamics. Moreover, they are shown to be coherent with the estimated deformation of the myocardium

碩士
國立臺灣大學
土木工程學研究所
96
It is well known that plane problems of harmonic functions are analyzed and solved effectively when expressed in the form of complex variables. This effectiveness is generally attributed to the powerful techniques of complex analysis and the richness of complex function theory. In view of this, the present thesis is aimed to extend the techniques to n-dimensional problems of boundary integral equations (BIEs) for harmonic field variables. Regarding usefulness for practical purposes, we derive singular and hypersingular BIEs not only for points on smooth boundaries but also for corner boundary points. The relations of real, complex, quaternion, and Clifford valued BIEs are explored. In Clifford valued BIEs, the three types of functions of are treated.

Research Doctorate - Doctor of Philosophy (PhD)
Fourier analysis has long been studied as a method to analyse real-valued or complex-valued signals. The Clifford-Fourier transform recently developed by Brackx, De Schepper, and Sommen has led to the development of Fourier analytic methods for hypercomplex or Clifford-valued signals. In the quaternionic case, Brackx et al. have found the kernel of the Quaternionic Fourier transform which allows for much easier calculation, and we focus much of our attention in this thesis on the quaternionic case. We define the continuous wavelet transform of quaternion-valued signals on the plane and prove a Calderón reproducing formula. We also define the monogenic signal, a generalization of the analytic signal of a function on the real line. We provide a characterization of translation-invariant operators and submodules of the quaternionic L₂ module. We develop several fundamental analogues of classical orthogonal wavelet theory pioneered by Cohen, Daubechies, Mallat, and Meyer to quaternion-valued functions on the plane. We include design conditions required to produce wavelets which have compact support and desired regularity. We also develop the basic theory needed for constructing a biorthogonal wavelet basis and construct an example. For a general Clifford algebra, we develop a condition on f so that f*g satisfies a convolution theorem. We also develop a Clifford-Fourier characterization of the Clifford-valued Hardy spaces on ℝ^d.

Research Doctorate - Doctor of Philosophy (PhD)
Fourier Analysis is a primary technique in the analysis of images, yet it has several limitations when it comes to the higher dimensional case of colour images. This thesis seeks to address some of these limitations through two main areas. First, we consider the recently developed Clifford-Fourier Transform of Brackx et al, which has the advantage over the classical Fourier Transform of combining the different channels of a colour image. We characterise the Hardy Spaces of this transform and show that functions in these Hardy spaces have monogenic extensions with bounded integral averages. We also characterise the Paley-Wiener spaces and show that functions in a Paley-Wiener space with radius R have monogenic extensions with integral averages that grow according to the radius R. Second, we consider the case of two dimensional compactly supported wavelets with orthonormal shifts and develop projection algorithms to find compactly supported, continuous wavelets with orthonormal shifts and dilations and 2 vanishing moments which are not tensor products of one dimensional wavelets. We also apply these techniques in one dimension and discover an example of an anti-symmetric, compactly supported, continuous wavelet with orthonormal shifts and dilations and 2 vanishing moments.

Vector fields from flow visualization often containmillions of data values. It is obvious that a direct inspection of the data by the user is tedious. Therefore, an automated approach for the preselection of features is essential for a complete analysis of nontrivial flow fields. This thesis deals with automated detection, analysis, and visualization of flow features in vector fields based on techniques transfered from image processing. This work is build on rotation invariant template matching with Clifford convolution as developed in the diploma thesis of the author. A detailed analysis of the possibilities of this approach is done, and further techniques and algorithms up to a complete segmentation of vector fields are developed in the process. One of the major contributions thereby is the definition of a Clifford Fourier transform in 2D and 3D, and the proof of a corresponding convolution theorem for the Clifford convolution as well as other major theorems. This Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vectorvalued filters, as well as an acceleration of the convolution computation as a fast transform exists. The depth and precision of flow field analysis based on template matching and Clifford convolution is studied in detail for a specific application, which are flow fields measured in the wake of a helicopter rotor. Determining the features and their parameters in this data is an important step for a better understanding of the observed flow. Specific techniques dealing with subpixel accuracy and the parameters to be determined are developed on the way. To regard the flow as a superposition of simpler features is a necessity for this application as close vortices influence each other. Convolution is a linear system, so it is suited for this kind of analysis. The suitability of other flow analysis and visualization methods for this task is studied here as well. The knowledge and techniques developed for this work are brought together in the end to compute and visualize feature based segmentations of flow fields. The resulting visualizations display important structures of the flow and highlight the interesting features. Thus, a major step towards robust and automatic detection, analysis and visualization of flow fields is taken.

碩士
國立臺北科技大學
應用英文系
106
Hu Shih (1891-1962), ‘the Father of the Chinese Renaissance’, played a remarkable role in every crucial aspect in twentieth-century China. Hu was a fully documented writer of all time and a prodigious correspondent. Among Hu’s contacts, Edith Clifford Williams, an American avant-garde artist, was his most special friend. She was the one that inspired Hu in many ways when he studied in the United States. This study examines and analyzes 172 letters written by Hu to Williams, which portrayed the unconventional friendship between them and carried a profound spiritual love that went on for almost half a century. The motivation of Hu to keep such a long-term corresponding is because all along Williams was a perfect confidant for Hu, in which Hu found only she was the right companion who matched his need – intellectually and emotionally. Hu expressed diverse emotions in the letters. The most unique ones are those ‘desire’ related. Analyzing desire discourses tells us the features of affection expressing in each period and Hu’s mindset had shifted in their relationship across stages. Discussing through discourses, these desires indicate transitional differences and the common point, also on the basis and process of how they lifted their affection to greatness; how their amour to each other sublimated into a perpetual friendship with Williams’ truthful devotion to Hu Shih.

The symplectic Dirac and the symplectic twistor operators are sym- plectic analogues of classical Dirac and twistor operators appearing in spin- Riemannian geometry. Our work concerns basic aspects of these two ope- rators. Namely, we determine the solution space of the symplectic twistor operator on the symplectic vector space of dimension 2n. It turns out that the solution space is a symplectic counterpart of the orthogonal situation. Moreover, we demonstrate on the example of 2n-dimensional tori the effect of dependence of the solution spaces of the symplectic Dirac and the symplectic twistor operators on the choice of the metaplectic structure. We construct a symplectic generalization of classical theta functions for the symplectic Dirac operator as well. We study several basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of the real dimension 2, this amounts to the study of first order symmetry operators of the symplectic Dirac ope- rator, symplectic Clifford-Fourier transform and the reproducing kernel for the symplectic Fischer product including the construction of bases for the symplectic monogenics of the symplectic Dirac operator in real dimension 2 and their extension to symplectic spaces...

The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface, in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.