Compositional nonlinear audio signal processing with Volterra series

We develop a compositional theory of nonlinear audio signal processing based on a categorification of the Volterra series. We augment the classical definition of the Volterra series to be functorial with respect to a base category whose objects are temperate distributions and whose morphisms are certain linear transformations. This leads to formulae describing how the outcomes of nonlinear transformations are affected if their input signals are first linearly processed. We then consider how nonlinear audio systems change, and introduce as a model thereof the notion of morphism of Volterra series. We show how morphisms can be parameterized and used to generate indexed families of Volterra series, which are well-suited to model nonstationary or time-varying nonlinear phenomena. We describe how Volterra series and their morphisms organize into a functor category, Volt, whose objects are Volterra series and whose morphisms are natural transformations. We exhibit the operations of sum, product, and series composition of Volterra series as monoidal products on Volt and identify, for each in turn, its corresponding universal property. We show, in particular, that the series composition of Volterra series is associative. We then bridge between our framework and a subject at the heart of audio signal processing: time-frequency analysis. Specifically, we show that an equivalence between a certain class of second-order Volterra series and the bilinear time-frequency distributions (TFDs) can be extended to one between certain higher-order Volterra series and the so-called polynomial TFDs. We end with prospects for future work, including the incorporation of nonlinear system identification techniques and the extension of our theory to the settings of compositional graph and topological audio signal processing.Comment: Master's thesi

High-order renormalization of scalar quantum fields

Thema dieser Dissertation ist die Renormierung von perturbativer skalarer Quantenfeldtheorie bei großer Schleifenzahl. Der Hauptteil der Arbeit ist dem Einfluss von Renormierungsbedingungen auf renormierte Greenfunktionen gewidmet. Zunächst studieren wir Dyson-Schwinger-Gleichungen und die Renormierungsgruppe, inklusive der Gegenterme in dimensionaler Regularisierung. Anhand zahlreicher Beispiele illustrieren wir die verschiedenen Größen. Alsdann diskutieren wir, welche Freiheitsgrade ein Renormierungsschema hat und wie diese mit den Gegentermen und den renormierten Greenfunktionen zusammenhängen. Für ungekoppelte Dyson-Schwinger-Gleichungen stellen wir fest, dass alle Renormierungsschemata bis auf eine Verschiebung des Renormierungspunktes äquivalent sind. Die Verschiebung zwischen kinematischer Renormierung und Minimaler Subtraktion ist eine Funktion der Kopplung und des Regularisierungsparameters. Wir leiten eine neuartige Formel für den Fall einer linearen Dyson-Schwinger Gleichung vom Propagatortyp her, um die Verschiebung direkt aus der Mellintransformation des Integrationskerns zu berechnen. Schließlich berechnen wir obige Verschiebung störungstheoretisch für drei beispielhafte nichtlineare Dyson-Schwinger-Gleichungen und untersuchen das asymptotische Verhalten der Reihenkoeffizienten. Ein zweites Thema der vorliegenden Arbeit sind Diffeomorphismen der Feldvariable in einer Quantenfeldtheorie. Wir präsentieren eine Störungstheorie des Diffeomorphismusfeldes im Impulsraum und verifizieren, dass der Diffeomorphismus keinen Einfluss auf messbare Größen hat. Weiterhin untersuchen wir die Divergenzen des Diffeomorphismusfeldes und stellen fest, dass die Divergenzen Wardidentitäten erfüllen, die die Abwesenheit dieser Terme von der S-Matrix ausdrücken. Trotz der Wardidentitäten bleiben unendlich viele Divergenzen unbestimmt. Den Abschluss bildet ein Kommentar über die numerische Quadratur von Periodenintegralen.This thesis concerns the renormalization of perturbative quantum field theory. More precisely, we examine scalar quantum fields at high loop order. The bulk of the thesis is devoted to the influence of renormalization conditions on the renormalized Green functions. Firstly, we perform a detailed review of Dyson-Schwinger equations and the renormalization group, including the counterterms in dimensional regularization. Using numerous examples, we illustrate how the various quantities are computable in a concrete case and which relations they satisfy. Secondly, we discuss which degrees of freedom are present in a renormalization scheme, and how they are related to counterterms and renormalized Green functions. We establish that, in the case of an un-coupled Dyson-Schwinger equation, all renormalization schemes are equivalent up to a shift in the renormalization point. The shift between kinematic renormalization and Minimal Subtraction is a function of the coupling and the regularization parameter. We derive a novel formula for the case of a linear propagator-type Dyson-Schwinger equation to compute the shift directly from the Mellin transform of the kernel. Thirdly, we compute the shift perturbatively for three examples of non-linear Dyson-Schwinger equations and examine the asymptotic growth of series coefficients. A second, smaller topic of the present thesis are diffeomorphisms of the field variable in a quantum field theory. We present the perturbation theory of the diffeomorphism field in momentum space and find that the diffeomorphism has no influence on measurable quantities. Moreover, we study the divergences in the diffeomorphism field and establish that they satisfy Ward identities, which ensure their absence from the S-matrix. Nevertheless, the Ward identities leave infinitely many divergences unspecified and the diffeomorphism theory is perturbatively unrenormalizable. Finally, we remark on a third topic, the numerical quadrature of Feynman periods

Renormalization of an SU(2) Tensorial Group Field Theory in Three Dimensions

We address in this paper the issue of renormalizability for SU(2) Tensorial Group Field Theories (TGFT) with geometric Boulatov-type conditions in three dimensions. We prove that tensorial interactions up to degree 6 are just renormalizable without any anomaly. Our new models define the renormalizable TGFT version of the Boulatov model and provide therefore a new approach to quantum gravity in three dimensions. Among the many new technical results established in this paper are a general classification of just renormalizable models with gauge invariance condition, and in particular concerning properties of melonic graphs, the second order expansion of melonic two point subgraphs needed for wave-function renormalization

Data based identification and prediction of nonlinear and complex dynamical systems

We thank Dr. R. Yang (formerly at ASU), Dr. R.-Q. Su (formerly at ASU), and Mr. Zhesi Shen for their contributions to a number of original papers on which this Review is partly based. This work was supported by ARO under Grant No. W911NF-14-1-0504. W.-X. Wang was also supported by NSFC under Grants No. 61573064 and No. 61074116, as well as by the Fundamental Research Funds for the Central Universities, Beijing Nova Programme.Peer reviewedPostprin

Inferring phylogenetic trees under the general Markov model via a minimum spanning tree backbone

Phylogenetic trees are models of the evolutionary relationships among species, with species typically placed at the leaves of trees. We address the following problems regarding the calculation of phylogenetic trees. (1) Leaf-labeled phylogenetic trees may not be appropriate models of evolutionary relationships among rapidly evolving pathogens which may contain ancestor-descendant pairs. (2) The models of gene evolution that are widely used unrealistically assume that the base composition of DNA sequences does not evolve. Regarding problem (1) we present a method for inferring generally labeled phylogenetic trees that allow sampled species to be placed at non-leaf nodes of the tree. Regarding problem (2), we present a structural expectation maximization method (SEM-GM) for inferring leaf-labeled phylogenetic trees under the general Markov model (GM) which is the most complex model of DNA substitution that allows the evolution of base composition. In order to improve the scalability of SEM-GM we present a minimum spanning tree (MST) framework called MST-backbone. MST-backbone scales linearly with the number of leaves. However, the unrealistic location of the root as inferred on empirical data suggests that the GM model may be overtrained. MST-backbone was inspired by the topological relationship between MSTs and phylogenetic trees that was introduced by Choi et al. (2011). We discovered that the topological relationship does not necessarily hold if there is no unique MST. We propose so-called vertex-order based MSTs (VMSTs) that guarantee a topological relationship with phylogenetic trees.Phylogenetische Bäume modellieren evolutionäre Beziehungen zwischen Spezies, wobei die Spezies typischerweise an den Blättern der Bäume sitzen. Wir befassen uns mit den folgenden Problemen bei der Berechnung von phylogenetischen Bäumen. (1) Blattmarkierte phylogenetische Bäume sind möglicherweise keine geeigneten Modelle der evolutionären Beziehungen zwischen sich schnell entwickelnden Krankheitserregern, die Vorfahren-Nachfahren-Paare enthalten können. (2) Die weit verbreiteten Modelle der Genevolution gehen unrealistischerweise davon aus, dass sich die Basenzusammensetzung von DNA-Sequenzen nicht ändert. Bezüglich Problem (1) stellen wir eine Methode zur Ableitung von allgemein markierten phylogenetischen Bäumen vor, die es erlaubt, Spezies, für die Proben vorliegen, an inneren des Baumes zu platzieren. Bezüglich Problem (2) stellen wir eine strukturelle Expectation-Maximization-Methode (SEM-GM) zur Ableitung von blattmarkierten phylogenetischen Bäumen unter dem allgemeinen Markov-Modell (GM) vor, das das komplexeste Modell von DNA-Substitution ist und das die Evolution von Basenzusammensetzung erlaubt. Um die Skalierbarkeit von SEM-GM zu verbessern, stellen wir ein Minimale Spannbaum (MST)-Methode vor, die als MST-Backbone bezeichnet wird. MST-Backbone skaliert linear mit der Anzahl der Blätter. Die Tatsache, dass die Lage der Wurzel aus empirischen Daten nicht immer realistisch abgeleitet warden kann, legt jedoch nahe, dass das GM-Modell möglicherweise übertrainiert ist. MST-backbone wurde von einer topologischen Beziehung zwischen minimalen Spannbäumen und phylogenetischen Bäumen inspiriert, die von Choi et al. 2011 eingeführt wurde. Wir entdeckten, dass die topologische Beziehung nicht unbedingt Bestand hat, wenn es keinen eindeutigen minimalen Spannbaum gibt. Wir schlagen so genannte vertex-order-based MSTs (VMSTs) vor, die eine topologische Beziehung zu phylogenetischen Bäumen garantieren

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version