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A model for DNA recombination uses 4-valent rigid vertex graphs, called assembly graphs. An assembly graph, similarly to the projection of knots, can be associated with an unsigned Gauss code, or double occurrence word. We define biologically motivated reductions that act on double occurrence words and, in turn, on their associated assembly graphs. For every double occurrence word w there is a sequence of reduction operations that may be applied to w so that what remains is the empty word, [epsilon]. Then the nesting index of a word w, denoted by NI(w), is defined to to be the least number of reduction operations necessary to reduce w to [epsilon]. The nesting index is the first property of assembly graphs that we study. We use chord diagrams as tools in our study of the nesting index. We observe two double occurrence words that correspond to the same circle graph, but that have arbitrarily large differences in nesting index values. In 2012, Buck et al. considered the cellular embeddings of assembly graphs into orientable surfaces. The genus range of an assembly graph [Gamma], denoted gr([Gamma]), was defined to be the set of integers g where g is the genus of an orientable surface F into which [Gamma] cellularly embeds. The genus range is the second property of assembly graphs that we study. We generalize the notion of the genus range to that of the genus spectrum, where for each g [isin] gr([Gamma]) we consider the number of orientable surfaces F obtained from [Gamma] by a special construction, called a ribbon graph construction, that have genus g. By considering this more general notion we gain a better understanding of the genus range property. Lastly, we show how one can obtain the genus spectrum of a double occurrence word from the genus spectrums of its irreducible parts, i.e., its double occurrence subwords. In the final chapter we consider constructions of double occurrence words that recognize certain values for nesting index and genus range. In general, we find that for arbitrary values of nesting index [ge] 2 and genus range, there is a double occurrence word that recognizes those values.

In dieser Dissertation geht es um Schwinger-parametrische Feynmanintegrale in der Quantenelektrodynamik. Mittels einer Vielzahl von Methoden aus der Kombinatorik und Graphentheorie wird eine signifikante Vereinfachung des Integranden erreicht. Nach einer größtenteils in sich geschlossenen Einführung zu Feynmangraphen und -integralen wird die Herleitung der Schwinger-parametrischen Darstellung aus den klassischen Impulsraumintegralen ausführlich erläutert, sowohl für skalare Theorien als auch Quantenelektrodynamik. Es stellt sich heraus, dass die Ableitungen, die benötigt werden um Integrale aus der Quantenelektrodynamik in ihrer parametrischen Version zu formulieren, neue Graphpolynome enthalten, die auf Zykeln und minimalen Schnitten (engl. "bonds") basieren. Danach wird die Tensorstruktur der Quantenelektrodynamik, bestehend aus Dirac-Matrizen und ihren Spuren, durch eine diagrammatische Interpretation ihrer Kontraktion zu ganzzahligen Faktoren reduziert. Dabei werden insbesondere gefärbte Sehnendiagramme benutzt. Dies liefert einen parametrischen Integranden, der über bestimmte Teilmengen solcher Diagramme summierte Produkte von Zykel- und Bondpolynomen enthält. Weitere Untersuchungen der im Integranden auftauchenden Polynome decken Verbindungen zu Dodgson- und Spannwaldpolynomen auf. Dies wird benutzt um eine Identität zu beweisen, mit der sehr große Summen von Sehnendiagrammen in einer kurzen Form ausgedrückt werden können. Insbesondere führt dies zu Aufhebungen, die den Integranden massiv vereinfachen. / This thesis is concerned with the study of Schwinger parametric Feynman integrals in quantum electrodynamics. Using a variety of tools from combinatorics and graph theory, significant simplification of the integrand is achieved. After a largely self-contained introduction to Feynman graphs and integrals, the derivation of the Schwinger parametric representation from the standard momentum space integrals is reviewed in full detail for both scalar theories and quantum electrodynamics. The derivatives needed to express Feynman integrals in quantum electrodynamics in their parametric version are found to contain new types of graph polynomials based on cycle and bond subgraphs. Then the tensor structure of quantum electrodynamics, products of Dirac matrices and their traces, is reduced to integer factors with a diagrammatic interpretation of their contraction. Specifically, chord diagrams with a particular colouring are used. This results in a parametric integrand that contains sums of products of cycle and bond polynomials over certain subsets of such chord diagrams. Further study of the polynomials occurring in the integrand reveals connections to other well-known graph polynomials, the Dodgson and spanning forest polynomials. This is used to prove an identity that expresses some of the very large sums over chord diagrams in a very concise form. In particular, this leads to cancellations that massively simplify the integrand.

Dyson-Schwinger-Gleichungen sind Fixpunktgleichungen, die in der Quantenfeldtheorie auftauchen. Obwohl es bekannt ist, wie die Kombinatorik vor der Anwendung von Feynman-Regeln aussieht, war die Kombinatorik der resultierenden analytischen Dyson-Schwinger-Gleichungen bisher unbekannt. Wir verallgemeinern die Arbeiten von Yeats et.al. auf diesem Gebiet zu einer Klasse von unendlich vielen Dyson-Schwinger-Gleichungen mit Hilfe von Sehnen-Diagrammen. / In quantum field theory, Dyson-Schwinger equations are fixed-point equations that come from self insertion properties of Feynman graphs. While the combinatorics of these are well understood, the combinatorics are still mysterious after applying the Feynman rules. We generalize the work of Yeats et.al. in this field to an infinite number of Dyson-Schwinger equations with the help of chord diagrams.

The guitar chord (a sonority based on the open strings of the guitar) is one of Alberto Ginastera's compositional trademarks. The use of the guitar chord expands throughout forty years, creating a common link between different compositional stages and techniques. Chapters I and II provide the historical and technical background on Ginastera's life, oeuvre and scholar research. Chapter IV explores the origins of the guitar chord and compares it to similar specific sonorities used by different composers to express extra-musical ideas. Chapter V discusses Ginastera's initial uses and modifications of the guitar chord. Chapter VI explores the use of the guitar chord as a referential sonority based on Variaciones Concertantes, Op. 23: I-II, examining vertical (subsets) and horizontal (derivation of motives) aspects. Chapter VII explores uses of trichords and hexachords derived from the guitar chord in the Sonata for Guitar Op. 47.