Ein optimales Ausnutzen der von dezentralen Erzeugern innerhalb der bestehenden Energie- versorgungsstruktur bereitgestellten Energie fordert besonders bei der Einbindung regenerativer Quellen eine hohe Korrelation zwischen Angebot und Nachfrage. Innerhalb der vorliegenden Arbeit wird gezeigt, wie mittels Energiemanagementverfahren ein enger Zusammenhang zwischen dem Energieangebot aus fluktuierenden Quellen und der Energienachfrage hergestellt werden kann. Dabei wird ein entsprechender Optimierungsalgorithmus aufgestellt, der die für einen möglichst umfassenden Eigenverbrauch der erzeugten Energie notwendigen Anpassungsschritte ermittelt. Bei der Analyse von Energieangebot und -nachfrage wird, ausgehend von einer allgemeinen Darstellung, eine auf die Bedürfnisse eines Energiemanagements angepasste spezielle Beschreibung der Energiesituation erarbeitet. Die Anpassung von Energieangebot und -nachfrage erfolgt über die Auswahl spezifischer Verbrauchergruppen und die Ausnutzung der aus den systeminternen Zeitkonstanten resultierenden möglichen Unterbrechungsdauern. Die Vorgehensweise wird an Hand des Elektroenergiebezugs ausgesuchter klimatechnischer Anlagen in Verbindung mit einer regenerati- ven Energieerzeugung auf Basis der Photovoltaik dargestellt.

Ein optimales Ausnutzen der von dezentralen Erzeugern innerhalb der bestehenden Energie- versorgungsstruktur bereitgestellten Energie fordert besonders bei der Einbindung regenerativer Quellen eine hohe Korrelation zwischen Angebot und Nachfrage. Innerhalb der vorliegenden Arbeit wird gezeigt, wie mittels Energiemanagementverfahren ein enger Zusammenhang zwischen dem Energieangebot aus fluktuierenden Quellen und der Energienachfrage hergestellt werden kann. Dabei wird ein entsprechender Optimierungsalgorithmus aufgestellt, der die für einen möglichst umfassenden Eigenverbrauch der erzeugten Energie notwendigen Anpassungsschritte ermittelt. Bei der Analyse von Energieangebot und -nachfrage wird, ausgehend von einer allgemeinen Darstellung, eine auf die Bedürfnisse eines Energiemanagements angepasste spezielle Beschreibung der Energiesituation erarbeitet. Die Anpassung von Energieangebot und -nachfrage erfolgt über die Auswahl spezifischer Verbrauchergruppen und die Ausnutzung der aus den systeminternen Zeitkonstanten resultierenden möglichen Unterbrechungsdauern. Die Vorgehensweise wird an Hand des Elektroenergiebezugs ausgesuchter klimatechnischer Anlagen in Verbindung mit einer regenerati- ven Energieerzeugung auf Basis der Photovoltaik dargestellt.

This work presents mainly two contributions to Description Logics (DLs) research by means of Formal Concept Analysis (FCA) methods: supporting bottom-up construction of DL knowledge bases, and completing DL knowledge bases. Its contribution to FCA research is on the computational complexity of computing generators of closed sets.

We prove that every (nice) Feller process is an It^o process in the sense of Cinlar, Jacod, Protter and Sharpe (1980). Next we generalize the notion of the symbol and define it for this larger class of processes. As examples the solutions of stochastic differential equations are considered. The symbol is then used to derive a quick approach to the semimartingale characteristics as well as the generator of the process under consideration. Finally we give some examples of how our methods work for processes used in mathematical finance. / Wir haben gezeigt, dass jeder (nette) Feller Prozess ein It^o Prozess im Sinne von Cinlar, Jacod, Protter und Sharpe (1980) ist. Es stellt sich heraus, dass man den Begriff des Symbols, der für Feller Prozesse bekannt ist, auf diese größere Klasse verallgemeinern kann. Dieses Symbol haben wir für die Lösungen verschiedener stochastischer Differentialgleichungen berechnet. Außerdem haben wir gezeigt, dass das Symbol einen schnellen Zugang zur Berechnung der Semimartingal-Charakteristiken und des Erzeugers eines It^o Prozesses liefert. Zuletzt wurden die Ergebnisse auf Prozesse angewendet, die in der Finanzmathematik gebräuchlich sind. - (Die Dissertation ist veröffentlicht im Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Deutschland, http://www.shaker.de, ISBN: 978-3-8322-8244-8)

This work presents mainly two contributions to Description Logics (DLs) research by means of Formal Concept Analysis (FCA) methods: supporting bottom-up construction of DL knowledge bases, and completing DL knowledge bases. Its contribution to FCA research is on the computational complexity of computing generators of closed sets.

We prove that every (nice) Feller process is an It^o process in the sense of Cinlar, Jacod, Protter and Sharpe (1980). Next we generalize the notion of the symbol and define it for this larger class of processes. As examples the solutions of stochastic differential equations are considered. The symbol is then used to derive a quick approach to the semimartingale characteristics as well as the generator of the process under consideration. Finally we give some examples of how our methods work for processes used in mathematical finance. / Wir haben gezeigt, dass jeder (nette) Feller Prozess ein It^o Prozess im Sinne von Cinlar, Jacod, Protter und Sharpe (1980) ist. Es stellt sich heraus, dass man den Begriff des Symbols, der für Feller Prozesse bekannt ist, auf diese größere Klasse verallgemeinern kann. Dieses Symbol haben wir für die Lösungen verschiedener stochastischer Differentialgleichungen berechnet. Außerdem haben wir gezeigt, dass das Symbol einen schnellen Zugang zur Berechnung der Semimartingal-Charakteristiken und des Erzeugers eines It^o Prozesses liefert. Zuletzt wurden die Ergebnisse auf Prozesse angewendet, die in der Finanzmathematik gebräuchlich sind. - (Die Dissertation ist veröffentlicht im Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Deutschland, http://www.shaker.de, ISBN: 978-3-8322-8244-8)

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts.

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts.