Inspiriert von den integrablen Strukturen der schwach gekoppelten planaren N=4 Super-Yang-Mills-Theorie studieren wir Q-Operatoren und Yangsche Invarianten. Wir geben eine Übersicht der Quanten-Inverse-Streumethode zusammen mit der Yang-Baxter Gleichung welche zentral für diesen systematischen Zugang zu integrablen Modellen ist. Den Fokus richten wir auf rationale integrable Spinketten und Vertexmodelle. Wir besprechen einige ihrer bekannten Gemeinsamkeiten und wie sie durch Bethe-Ansatz-Methoden mit Hilfe sogenannter Q-Funktionen gelöst werden können. Der Hauptteil basiert auf den ursprünglichen Publikationen des Autors. Zuerst konstruieren wir Q-Operatoren, deren Eigenwerte zu den Q-Funktionen rationaler homogener Spinketten führen. Die Q-Operatoren werden als Spuren gewisser Monodromien von R-Operatoren eingeführt. Unsere Konstruktion erlaubt es uns die Hierarchie der kommutierenden Q-Operatoren und ihre funktionalen Beziehungen herzuleiten. Wir studieren wie der nächste-Nachbarn Hamiltonoperator, sowie höhere lokale Ladungen direkt aus den Q-Operatoren extrahiert werden können. Danach widmen wir uns der Formulierung der Yangschen Invarianzbedingung, wie sie auch im Zusammenhang mit Baumgraphen die bei der Berechnung von Streuamplituden in der N=4 Super-Yang-Mills-Theorie auftreten, innerhalb der RTT-Realisierung. Dies erlaubt es uns den algebraischen Bethe-Ansatz anzuwenden und die dazugehörigen Bethe Gleichungen herzuleiten, welche für die Konstruktion der Eigenzustände die Yangsche Invarianz aufweisen, relevant sind. Die Komponenten dieser Eigenzustände der von uns betrachteten Spinketten können außerdem als Zustandssummen gewisser zweidimensionaler Vertexmodelle angesehen werden. Zudem analysieren wir die Verbindung zwischen den Eigenzuständen und den oben genannten Baumgraphen. Schlussendlich diskutieren wir die von uns vorgelegten Ergebnisse und deren Folgen im Hinblick auf die Erforschung der planaren N=4 Super-Yang-Mills-Theorie. / Inspired by the integrable structures appearing in weakly coupled planar N=4 super Yang-Mills theory, we study Q-operators and Yangian invariants of rational integrable spin chains. We review the quantum inverse scattering method QISM along with the Yang-Baxter equation which is the key relation in this systematic approach to study integrable models. Our main interest concerns rational integrable spin chains and lattice models. We recall the relation among them and how they can be solved using Bethe ansatz methods incorporating so-called Q-functions. In order to remind the reader how the Yangian emerges in this context, an overview of its so-called RTT-realization is provided. The main part is based on the author''s original publications. Firstly, we construct Q-operators whose eigenvalues yield the Q-functions for rational homogeneous spin chains. The Q-operators are introduced as traces over certain monodromies of R-operators. Our construction allows us to derive the hierarchy of commuting Q-operators and the functional relations among them. We study how the nearest-neighbor Hamiltonian and in principle also higher local charges can be extracted from the Q-operators directly. Secondly, we formulate the Yangian invariance condition, also studied in relation to scattering amplitudes of N=4 super Yang-Mills theory, in the RTT-realization. We find that Yangian invariants can be interpreted as special eigenvectors of certain inhomogeneous spin chains. This allows us to apply the algebraic Bethe ansatz and derive the corresponding Bethe equations that are relevant to construct the invariants. We examine the connection between the Yangian invariant spin chain eigenstates whose components can be understood as partition functions of certain two-dimensional lattice models and tree-level scattering amplitudes of the four-dimensional gauge theory. Finally, we conclude and discuss some future directions and implications of our studies for planar N=4 super Yang-Mills theory.

Diese Arbeit untersucht die N = 4 super-Yang-Mills-Theorie bei schwacher Kopplung, mit dem Ziel eines tieferen Verständnisses von Größen der Theorie als Zustände des integrablen Modells dass der planaren Theorie zu Grunde liegt. Wir leiten On-Shell-Diagramme für Formfaktoren des chiralen Energie-Impuls-Tensor-Multipletts aus der BCFW-Rekursion her, und untersuchen deren Eigenschaften. Dies erlaubt die Herleitung eines Graßmannschen Integrals. Für NMHV-Formfaktoren bestimmen wir die Integrationskontur. Dies erlaubt es das Integral mit einer Twistor-String-Formulierung in Beziehung zu setzen. Mit Hilfe dieser Methoden zeigen wir dass Formfaktoren des chiralen Energie-Impuls-Tensor-Multipletts und On-Shell-Funktionen mit Einfügungen beliebiger Operatoren Eigenzustände integrabler Transfermatrizen sind. Diese Identitäten verallgemeinern die Yangsche Invarianz der On-Shell-Funktionen von Amplituden. Wir zeigen weiterhin dass ein Teil der Yangschen Symmetrien erhalten bleibt. Wir erweitern unsere Untersuchung auf nichtplanare On-Shell-Funktionen und zeigen dass sie ebenfalls solche Symmetrien besitzen. Weitere Identitäten mit Transfermatrizen werden hergeleitet, und zeigen insbesondere dass Diagramme auf Zylindern als Intertwiner fungieren. Als Schritt hin zur Berechnung der Eigenzustände des integrablen Modells zu höheren Schleifenordnungen untersuchen wir Einspuroperatoren. Hier erlaubt die Quantum Spectral Curve die nichtperturbative Berechnung ihres Spektrums, liefert jedoch keine Information zu den Zustände. Die QSC kann als Q-System verstanden werden, welches durch Baxter Q-Operatoren formulierbar sein sollte. Um darauf hinzuarbeiten untersuchen wir die Q-Operatoren nichtkompakter Superspinketten und entwickeln ein effiziente Methode zur Berechnung ihrer Matrixelemente. Dies erlaubt es das gesamte Q-System durch Matrizen für jeden Anregungssektor zu realisieren, und liefert die Grundlage für perturbative Rechnungungen mit der QSC in Operatorform. / This thesis investigates weakly coupled N = 4 super Yang-Mills theory, aiming at a better understanding of various quantities as states of the integrable model underlying the planar theory. We use the BCFW recursion relations to develop on-shell diagrams for form factors of the chiral stress-tensor multiplet, and investigate their properties. The diagrams allow to derive a Graßmannian integral for these form factors. We devise the contour of this integral for NMHV form factors, and use this knowledge to relate the integral to a twistor string formulation. Based on these methods, we show that both form factors of the chiral stress-tensor multiplet as well as on-shell functions with insertions of arbitrary operators are eigenstates of integrable transfer matrices. These identities can be seen as symmetries generalizing the Yangian invariance of amplitude on-shell functions. In addition, a part of these Yangian symmetries are unbroken. We furthermore consider nonplanar on-shell functions and prove that they exhibit a partial Yangian invariance. We also derive identities with transfer matrices, and show that on-shell diagrams on cylinders can be understood as intertwiners. To make progress towards the calculation of the higher loop eigenstates of the integrable model, we consider single trace operators, for which the Quantum Spectral Curve determines their spectrum non-perturbatively. This formulation however carries no information about the states. The QSC is an algebraic Q-system, for which an operatorial form in terms of Baxter Q-operators should exist. To initiate the development such a formulation we investigate the Q-operators of non-compact super spin chains and devise efficient methods to evaluate their matrix elements. This allows to obtain the entire Q-system in terms of matrices for each magnon sector. These can be used as input data for perturbative calculations using the QSC in operatorial form.

The analysis of wide-angle cellular light scattering patterns is a challenging problem. Small changes to the organization, orientation, shape, and optical properties of scatterers and scattering populations can significantly alter their complex two-dimensional scattering signatures. Because of this, it is difficult to find methods that can identify medically relevant cellular properties while remaining robust to experimental noise and sample-to-sample differences. It is an important problem. Recent work has shown that changes to the internal structure of cells---specifically, the distribution and aggregation of organelles---can indicate the progression of a number of common disorders, ranging from cancer to neurodegenerative disease, and can also predict a patient's response to treatments like chemotherapy. However, there is no direct analytical solution to the inverse wide-angle cellular light scattering problem, and available simulation and interpretation methods either rely on restrictive cell models, or are too computationally demanding for routine use. This dissertation addresses these challenges from a computational vantage point. First, it explores the theoretical limits and optical basis for wide-angle scattering pattern analysis. The result is a rapid new simulation method to generate realistic organelle scattering patterns without the need for computationally challenging or restrictive routines. Pattern analysis, image segmentation, machine learning, and iterative pattern classification methods are then used to identify novel relationships between wide-angle scattering patterns and the distribution of organelles (in this case mitochondria) within a cell. Importantly, this work shows that by parameterizing a scattering image it is possible to extract vital information about cell structure while remaining robust to changes in organelle concentration, effective size, and random placement. The result is a powerful collection of methods to simulate and interpret experimental light scattering signatures. This gives new insight into the theoretical basis for wide-angle cellular light scattering, and facilitates advances in real-time patient care, cell structure prediction, and cell morphology research.

No description available.

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.