Barcoding is a mathematical tool, to analyze data, which is based on the theory of persistent homology. In this thesis both Hierarchical Clustering and Barcoding are defined and analyzed according to three criterion: Continuity, Computability and Visualizability. It is also presented how the two methods, barcoding and hierarchical clustering, are connected and why barcoding, in some cases, is a generalized method of hierarchical clustering. Lastly some more question of interest, for better understanding barcoding, are stated. / Barcoding är ett matematiskt verktyg, för att analysera data, vilket bygger på teorin om ihållande homologi. I den här uppsatsen år både Hierarkisk Klustring och Barcoding definierade och analyserade med avseende på tre kriterier: Kontinuitet, Beräkningsbarhet och Visualiserbarhet. Det presenteras även hur de två metoderna, barcoding och hierarkisk klustring, är sammanlänkade och varför barcoding, i vissa fall, är en generaliserad metod av hierarkisk klustring. Tillsist är några fler frågor av intresse, för att bättre förstå barcoding, presenterad.

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Many ontologies, in particular in the biomedical domain, are based on the Description Logic EL++. Several efforts have been made to interpret and exploit EL++ontologies by distributed representation learning. Specifically, concepts within EL++theories have been represented as n-balls within an n-dimensional embedding space. However, the intersectional closure is not satisfied when using n-balls to represent concepts because the intersection of two n-balls is not an n-ball. This leads to challenges when measuring the distance between concepts and inferring equivalence between concepts. To this end, we developed EL Box Embedding (ELBE) to learn Description Logic EL++embeddings using axis-parallel boxes. We generate specially designed box-based geometric constraints from EL++axioms for model training. Since the intersection of boxes remains as a box, the intersectional closure is satisfied. We report extensive experimental results on three datasets and present a case study to demonstrate the effectiveness of the proposed method.

In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We partially generalize this result to the species case, i.e. we manage to describe the Nakayama automorphism up to an unknown constant. We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by a mapping cone of a certain chain map. We also study a higher dimensional analogue of representation finite hereditary algebras called d-representation finite algebras. One source of $d$-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.

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To every small category or Grothendieck topos one may associate its isotropy group, which is an algebraic invariant capturing information about the behaviour of automorphisms. In this thesis, we investigate this invariant in the particular context of quasi-equational theories, which are multi-sorted equational theories in which operations may be partially de fined. It is known that every such theory T has a classifying topos, which is a topos that classi fies all topos-theoretic models of the theory, and that this classifying topos is in fact equivalent to the covariant presheaf category Sets^fpTmod, with fpTmod being the category of all finitely presented, set-based models of T. We then investigate the isotropy group of this classifying topos of T, which will therefore be a presheaf of groups on fpTmod, and show that it encodes a notion of inner automorphism for the theory. The main technical result of this thesis is a syntactic characterization of the isotropy group of a quasi-equational theory, and we illustrate the usefulness of this characterization by applying it to various concrete examples of quasi-equational theories.

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