aus Doktorarbeiten Anna Lena Birkmeyer Oktober 2016

Transkript

1 aus Doktorarbeiten Anna Lena Birkmeyer Fachbereich Mathematik TU Kaiserslautern Oktober 2016

2 In der der Arbeit: The mathematical modeling and optimization of... is a wide field of research [4,15,19,35,61,62,66,76,86] with important applications in various scenarios [9,46,52,56,59,77,97,103].... have proven to be successful in a number of different areas of mathematics, such as... (cf. [Mi03], [IKS04]),... (cf. [Ab06]),... (cf. [Gu06]) on the one hand, and... (cf. [PS03]) on the other hand. In this thesis, we follow the... approach...

3 Proposition xxx ([R09, Theorem xxx]) Aussage des Theorems (formuliert in eigenem Stil und mit eigenen Notationen). Lemma xxx (cf. [Ko, Lemma xxx]) Die Aussage des Lemmas wie oben. Proof Hier ggfs ein eigener Beweis oder eine Anpassung des Beweises.

4 Den Beweis einer Aussage Proof of Proposition xxx The duality of the Newton subdivision of f and its tropicalization is shown for instance in [MS15, Proposition xxx]. The equality of the weights and the lattice lengths can be found in [MS15, Proposition xxx].

5 Als zu einem Unterabschnitt: To describe the space of all..., we need to know that there is... To construct..., we use ideas developed in [Ham14]. We study... Our approach is... and based on... as developed in [Sha12,AR10,FR12]. We recall some main definitions and results in... from [Rau09], [All09], and [FR12].

6 Auch Software muss zitiert werden: Already in small examples, the equations cannot be found in reasonable time. Therefore, we did not implement Algorithm xxx although any other step can be implemented using for instance the software Singular [DGPS], gfan [Jen07], and polymake [GJ00].

7 Beziehung zu anderen Arbeiten setzen Als zu einem Abschnitt: In the previous chapter, we gave an algorithmic answer to... However, it would be desirable to give a non-algorithmic answer to... In this chapter, we... and prove sufficient and necessary conditions to... Moreover, we relate them to known sufficient conditions to... [BK12, BS15].

8 in einer Gruppenarbeit verdeutlichen In der : This thesis contains material from my (partly published) articles [AR07], [MR08], [AR08], and [R08]. In particular, it contains joint work with xxx and xxx. The contributions of each article are listed at the beginning of each chapter, as well.

9 in einer Gruppenarbeit verdeutlichen In der zu den Kapiteln: The first chapter mainly emerged from the material published in [AR07], [AR08] and the first section of [R08]. As far as the first two articles are concerned, this is joint work with xxx, and it is very hard to single out the contributions each of us made. As far as it can be told, main ideas of xxx are contained in sections xx and xx, whereas main ideas of mine are contained in sections xx and xx. Section xx contains important contributions of both of us. Moreover, I omit those parts which are to large extent the work of xxx. The presentation of the material as well as section xx are completely new.

10 in einer Gruppenarbeit verdeutlichen In der zu den Kapiteln: This chapter essentially covers the material presented in [R08] and [MR08]. In particular, the article [MR08] is joint work with xxx. However, the presented approach is similar to [R08] and is to large extent due to my own work. Particular contributions from xxx are not presented here (e.g....). Moreover, the proofs in subsection xx are partially adapted.

11 Vorgehen und Plan erläutern In der zu einem Abschnitt: We begin this section with the definition of the structures which we use to describe... Furthermore, we introduce notations used afterwards to increase the readability. First, we define a...

12 Vorgehen und Plan erläutern In der zu einem Abschnitt: We begin this section with the definition of the structures which we use to describe... Furthermore, we introduce notations used afterwards to increase the readability. First, we define a... In order to compute..., one important ingredient is theorem xxx which states that any... is... In other words... Before we prove this, we introduce the helpful notation of...

13 Vorgehen und Plan erläutern In der zu einem Abschnitt: We begin this section with the definition of the structures which we use to describe... Furthermore, we introduce notations used afterwards to increase the readability. First, we define a... In order to compute..., one important ingredient is theorem xxx which states that any... is... In other words... Before we prove this, we introduce the helpful notation of... The idea of this section is similar to the previous one. We recall the construction of... and prove... Moreover, we extend...

14 Vorgehen und Plan erläutern In der zu einem Abschnitt: We begin this section with the definition of the structures which we use to describe... Furthermore, we introduce notations used afterwards to increase the readability. First, we define a... In order to compute..., one important ingredient is theorem xxx which states that any... is... In other words... Before we prove this, we introduce the helpful notation of... The idea of this section is similar to the previous one. We recall the construction of... and prove... Moreover, we extend... Before we try to compute these numbers, let us first make explicit what they count in the generic case. For this, we need some tools concerning... and the following remark.

15 von (Bachelor-)Arbeiten Schreibe Texte (mit Motivation, Erläuterungen, Erklärungen) zwischen die einzelnen Definitionen und und vor allem am Anfang von neuen Abschnitten, Kapiteln oder neuen Themen. Mache bei den klar, was das Resultat der Aussage ist und was von vorher bekannt ist. Die Sätze sollten zwar knapp, aber immer noch eindeutig sein. (Beispiel: Wenn A, B, C.) Schreibe nicht Die Aussage folgt aus der Definition. oder Dies ist eine (einfache) Rechnung. Benutze keine Notationen, die nicht eingeführt wurden. Wenn es für einen Begriff mehrere Definitionen gibt, mache deutlich, welche Du verwendest. (Am Besten Du definierst ihn selbst.)

16 von (Bachelor-)Arbeiten Bei Definitionen kann man, wenn man möchte, als optionales Argument angeben, was man definieren möchte. Das sollte man jedoch im Dokument konsistent tun. Definition (Stetigkeit) Hier steht die Definition von Stetigkeit. Auch bei kann man diese ggfs. prägnant beschreiben. Theorem (Differenzierbare Funktionen sind stetig) Es sei D X offen und f : X Y eine auf D differenzierbare Funktion. Dann ist die Einschränkung f D : D Y von f auf D stetig.