Philosophiae Doctor - PhD / Categorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting.

<p>A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis will characterize the Riemann surfaces of genus 4 wiht non-unique trigonal morphism. We will describe the structure of the space of cyclic trigonal Riemann surfaces of genus 4.</p> / Report code: LiU-Tek-Lic-2004:54. The electronic version of the printed licentiate thesis is a corrected version where errors in the calculations have been corrected. See Errata below for a list of corrections.

This dissertation examines subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enveloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalø and Enochs and has proven to be beneficial for module theory as well as for representation theory. The first few chapters examine the subfunctors of Ext and their properties. It is showed how the class of precoverings give us subfunctors of Ext. Furthermore, the characterization of these subfunctors and some examples are given. In the latter chapters ideals, the subfunctors of Hom, are investigated. The definition of cover and envelope carry over to the ideals naturally. Classical conditions for existence theorems for covers led to similar approaches in the ideal case. Even though some theorems such as Salce’s Lemma were proven to extend to ideals, most of the theorems do not directly apply to the new case. It is showed how Eklof & Trlifaj’s result can partially be extended to the ideals generated by a set. In that case, one also obtains a significant result about the orthogonal complement of the ideal. We relate the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in A2 (which is the category of all representations of 2-quiver by R-modules) and obtain a sufficient condition for the existence of covering ideals in a more general setting. We finish with applying this result to the class of phantom morphisms.

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis characterizes the cyclic trigonal Riemann surfaces of genus 4 with non-unique trigonal morphism using the automorphism groups of the surfaces. The thesis shows that Accola’s bound is sharp with the existence of a uniparametric family of cyclic trigonal Riemann surfaces of genus 4 having several trigonal morphisms. The structure of the moduli space of trigonal Riemann surfaces of genus 4 is also characterized. Finally, by using the same technique as in the case of cyclic trigonal Riemann surfaces of genus 4, we are able to deal with p-gonal Riemann surfaces and show that Accola’s bound is sharp for p-gonal Riemann surfaces. Furthermore, we study families of p-gonal Riemann surfaces of genus (p − 1)2 with two p-gonal morphisms, and describe the structure of their moduli space.

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis will characterize the Riemann surfaces of genus 4 wiht non-unique trigonal morphism. We will describe the structure of the space of cyclic trigonal Riemann surfaces of genus 4. / <p>Report code: LiU-Tek-Lic-2004:54. The electronic version of the printed licentiate thesis is a corrected version where errors in the calculations have been corrected. See Errata below for a list of corrections.</p>

This dissertation contributes to the theoretical foundations of systems engineering (SE) and exposes an unstudied SE area of definition of verification models. In practice, verification models are largely qualitatively defined based on heuristic assumptions rather than science-based approach. For example, we may state the desire for representativeness of a verification model in qualitative terms of low, medium, or high fidelity in early phases of a system lifecycle when verification requirements are typically defined. Given that fidelity is defined as a measure of approximation from reality and that the (real) final product does (or may) not exist in early phases, we are stating desire for and making assumptions of representative equivalence that may not be true. Therefore, this dissertation contends that verification models can and should be defined on the scientific basis of systems theoretic principles. Furthermore, the practice of SE is undergoing a digital transformation and corresponding desire to enhance SE educationally and as a discipline, which this research proposes to address through a science-based approach that is grounded in the mathematical formalism of systems theory. The maturity of engineering disciplines is reflected in their science-based approach, such as computational fluid dynamics and finite element analysis. Much of the discipline of SE remains qualitatively descriptive, which may suffer from interpretation discrepancies; rather than being grounded in, inherently analytical, theoretical foundations such as is a stated goal of the SE professional organization the International Council on Systems Engineering (INCOSE). Additionally, along with the increased complexity of modern engineered systems comes the impracticality of verification through traditional means, which has resulted in verification being described as broken and in need of theoretical foundations. The relationships used to define verification models are explored through building on the systems theoretic lineage of A. Wayne Wymore; such as computational systems theory, theory of system design, and theory of problem formulation. Core systems theoretic concepts used to frame the relationship-based definition of verification models are the notions of system morphisms that characterize equivalence between pairs, problem spaces of functions that bound the acceptability of solution systems, and hierarchy of system specification that characterizes stratification. The research inquisition was in regard to how verification models should be defined and hypothesized that verification models should be defined through a combination of systems theoretic relationships between verification artifacts; system requirements, system designs, verification requirements, and verification models. The conclusions of this research provide a science-based metamodel for defining verification models through systems theoretic principles. The verification models were shown to be indirectly defined from system requirements, through system designs and verification requirements. Verification models are expected to be morphically equivalent to corresponding system designs; however, there may exist infinite equivalence which may be reduced through defining bounding conditions. These bounding conditions were found to be defined through verification requirements that are formed as (1) verification requirement problem spaces that characterize the verification activity on the basis of morphic equivalence to the system requirements and (2) morphic conditions that specify desired equivalence between a system design and verification model. An output of this research is a system theoretic metamodel of verification artifacts, which may be used for a science-based approach to define verification models and advancement of the maturity of the SE discipline. / Doctor of Philosophy / We conduct verification to increase our confidence that the system design will do what is desired as defined in the requirements. However, it is not feasible to perform verification on the final product design within the full scope of the requirements; due to cost, schedule, and availability. As a result, we leverage surrogates, such as verification models, to conduct verification and determine our confidence in the system design. A challenge to our confidence in the system design exists in that we accept the representativeness of the surrogates based on faith alone; rather than scientific proof. This dissertation defines science-based approach to determining the representativeness of substitutes. In the discipline and practice of systems engineering, verification models serve as substitutes for the system design; and verification requirement problem spaces serve as substitutes the requirements. The research used mathematical principles to determine representative equivalence and to find that a combination of relationship framing is needed for sufficient selection of verification models. The framing includes relationships to the system being engineered and to the substitute conditions under which the verification model is examined relative to the conditions under which the engineered system is expected to operate. A comparison to the state of the discipline and practice to the research findings was conducted and resulted in confirming unique contribution of the dissertation research. In regard to framing the acceptability of verification models, this research established the foundations for a science-based method to advance the field of Systems Engineering.

This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms.

Classical homological algebra is done in a category of modules beginning with the study of projective and injective modules. This dissertation investigates analogous notions of projectivity and injectivity in a category of filtered modules. This category is similar to one studied by Sjödin, Nǎstǎsescu, and Van Oystaeyen. In particular, projective and injective objects with respect to the restricted class of strict morphisms are defined and characterized. Additionally, an analogue to the injective envelope is discussed with examples showing how this differs from the usual notion of an injective envelope.

Dans cette thèse, nous étudions la structure d’un morphisme harmonique F d’une variété riemannienne M4 dans une surface N2 au voisinage d’un point critique mO. Si mO est un point I critique isolé ou si M4 est compact sans bord, nous montrons que F est pseudo-Holomorphe par rapport à une structure presque hermitienne definie dans un voisinage de mO. Si M4 est compact sans bord, les fibres singuliers de F sont des surfaces minimales avec points de branchement. Ensuite, nous étudions des exemples de morphismes harmoniques due a Burel de (S4, gk,l) dans S2 où (gk,I) est une famille de métriques conforme à la métrique canonique. Nous construisons tout d’abord une application semi-Conforme Φk,l de S4 dans S2 en composant deux applications semi-Conformes régulières, F de S4 dans S3 et Φk,i, de S3 dans S2. II résulte de Baird-Eells que le fibres régulier de øk,l pour tout k, I sont minimales. Si [k] = [l] = 1, l’ensemble des points critiques est donnée par l’image réciproque du pâle nord: il consiste en deux 2-Sphères ayant deux points d’intersection. Si k, I 6= 1 l’ensemble des points critiques sont les images réciproques du pôle nord (de la même façon que pour k = t = 1 deux sphères, mais avec une multiplicité I) ainsi que la pré-Image du pôle sud (un tore) avec multiplicité k. Enfin, nous étudions une construction due à Baird-Ou de morphismes harmoniques d’une ensembles ouverts de (S2×S2, can) dans S2. Nous vérifions qu’ils sont holomorphe par rapport à une des quatre structures complexes canoniques hermitiennes. / In this thesis, we are interested in harmonic morphisms between Riemannian manifolds (Mm, g) and (Nn, h) for m > n. Such a smooth map is a harmonic morphism if it pulls back local harmonic functions to local harmonic functions: if ƒ : V → ℝ is a harmonic function on an open subset V on N and Φ-1(V) is non-Empty, then the composition ƒ ∘ Φ : Φ-1(V) → ℝ is harmonic. The conformal transformations of the complex plane are harmonic morphisms. In the late 1970's Fuglede and Ishihara published two papers ([Fu]) and ([Is]), where they discuss their results on harmonic morphisms or mappings preserving harmonic functions. They characterize non-Constant harmonic morphisms F : (M,g) → (N,h) between Riemannian manifolds as those harmonic maps, which are horizontally conformal, where F horizontally conformal means : for any x ∈ M with dF(x) ≠ 0, the restriction of dF(x) to the orthogonal complement of kerdF(x) in TxM is conformal and surjective. This means that we are dealing with a special class of harmonic maps.

This thesis is divided into a theoretical part, aimed at developing statements around the newly introduced concept of pattern morphisms, and a practical part, where we present use cases of pattern structures. A first insight of our work clarifies the facts on projections of pattern structures. We discovered that a projection of a pattern structure does not always lead again to a pattern structure. A solution to this problem, and one of the most important points of this thesis, is the introduction of pattern morphisms in Chapter4. Pattern morphisms make it possible to describe relationships between pattern structures, and thus enable a deeper understanding of pattern structures in general. They also provide the means to describe projections of pattern structures that lead to pattern structures again. In Chapter5 and Chapter6, we looked at the impact of morphisms between pattern structures on concept lattices and on their representations and thus clarified the theoretical background of existing research in this field. The application part reveals that random forests can be described through pattern structures, which constitutes another central achievement of our work. In order to demonstrate the practical relevance of our findings, we included a use case where this finding is used to build an algorithm that solves a real world classification problem of red wines. The prediction accuracy of the random forest is better, but the high interpretability makes our algorithm valuable. Another approach to the red wine classification problem is presented in Chapter 8, where, starting from an elementary pattern structure, we built a classification model that yielded good results.