Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems

Digital mathematical libraries assemble the knowledge of years of mathematical research. Numerous disciplines (e.g., physics, engineering, pure and applied mathematics) rely heavily on compendia gathered findings. Likewise, modern research applications rely more and more on computational solutions, which are often calculated and verified by computer algebra systems. Hence, the correctness, accuracy, and reliability of both digital mathematical libraries and computer algebra systems is a crucial attribute for modern research. In this paper, we present a novel approach to verify a digital mathematical library and two computer algebra systems with one another by converting mathematical expressions from one system to the other. We use our previously eveloped conversion tool (referred to as LaCASt) to translate formulae from the NIST Digital Library of Mathematical Functions to the computer algebra systems Maple and Mathematica. The contributions of our presented work are as follows: (1) we present the most comprehensive verification of computer algebra systems and digital mathematical libraries with one another; (2) we significantly enhance the performance of the underlying translator in terms of coverage and accuracy; and (3) we provide open access to translations for Maple and Mathematica of the formulae in the NIST Digital Library of Mathematical Functions

Circular quiver gauge theories, isomonodromic deformations and WN fermions on the torus

We study the relation between class S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding τ-function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to WN free fermion correlators on the torus

Optimization of a multiphysics problem in semiconductor laser design

A multimaterial topology optimization framework is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed. Rigorous proofs are provided for the sensitivity of the fundamental mode of the device with respect to the changes in the underlying topology. After proving existence and optimality results, numerical experiments leading to an optimal material distribution for maximizing the strain in a Ge-on-Si microbridge are given. The highly favorable electronic properties of this design are demonstrated by steady-state simulations of the corresponding van Roosbroeck (drift-diffusion) system

A Quaternionic Version Theory related to Spheroidal Functions

In dieser Arbeit wird eine neue Theorie der quaternionischen Funktionen vorgestellt, welche das Problem der Bestapproximation von Familien prolater und oblater sphäroidalen Funktionen im Hilberträumen behandelt. Die allgemeine Theorie beginnt mit der expliziten Konstruktion von orthogonalen Basen für Räume, definiert auf sphäroidalen Gebieten mit beliebiger Exzentrizität, deren Elemente harmonische, monogene und kontragene Funktionen sind und durch die Form der Gebiete parametrisiert werden. Eine detaillierte Studie dieser grundlegenden Elemente wird in dieser Arbeit durchgeführt. Der Begriff der kontragenen Funktion hängt vom Definitionsbereich ab und ist daher keine lokale Eigenschaft, während die Begriffe der harmonischen und monogenen Funktionen lokal sind. Es werden verschiedene Umwandlungsformeln vorgestellt, die Systeme harmonischer, monogener und kontragener Funktionen auf Sphäroiden unterschiedlicher Exzentrizität in Beziehung setzen. Darüber hinaus wird die Existenz gemeinsamer nichttrivialer kontragener Funktionen für Sphäroide jeglicher Exzentrizität gezeigt. Der zweite wichtige Beitrag dieser Arbeit betrifft eine quaternionische Raumfrequenztheorie für bandbegrenzte quaternionische Funktionen. Es wird eine neue Art von quaternionischen Signalen vorgeschlagen, deren Energiekonzentration im Raum und in den Frequenzbereichen unter der quaternionischen Fourier-Transformation maximal ist. Darüber hinaus werden diese Signale im Kontext der Spektralkonzentration als Eigenfunktionen eines kompakten und selbstadjungierteren quaternionischen Integraloperators untersucht und die grundlegenden Eigenschaften ihrer zugehörigen Eigenwerte werden detailliert beschrieben. Wenn die Konzentrationsgebiete beider Räume kugelförmig sind, kann der Winkelanteil dieser Signale explizit gefunden werden, was zur Lösung von mehreren eindimensionalen radialen Integralgleichungen führt. Wir nutzen die theoretischen Ergebnisse und harmonische Konjugierten um Klassen monogener Funktionen in verschiedenen Räumen zu konstruieren. Zur Charakterisierung der monogenen gewichteten Hardy- und Bergman-Räume in der Einheitskugel werden zwei konstruktive Algorithmen vorgeschlagen. Für eine reelle harmonische Funktion, die zu einem gewichteten Hardy- und Bergman-Raum gehört, werden die harmonischen Konjugiert in den gleichen Räumen gefunden. Die Beschränktheit der zugrundeliegenden harmonischen Konjugationsoperatoren wird in den angegebenen gewichteten Räumen bewiesen. Zusätzlich wird ein quaternionisches Gegenstück zum Satz von Bloch für monogene Funktionen bewiesen.This work presents a novel Quaternionic Function Theory associated with the best approximation problem in the setting of Hilbert spaces concerning families of prolate and oblate spheroidal functions. The general theory begins with the explicit construction of orthogonal bases for the spaces of harmonic, monogenic, and contragenic functions defined in spheroidal domains of arbitrary eccentricity, whose elements are parametrized by the shape of the corresponding spheroids. A detailed study regarding the elements that constitute these bases is carried out in this thesis. The notion of a contragenic function depends on the domain, and, therefore, it is not a local property in contrast to the concepts of harmonic and monogenic functions. Various conversion formulas that relate systems of harmonic, monogenic, and contragenic functions associated with spheroids of differing eccentricity are presented. Furthermore, the existence of standard nontrivial contragenic functions is shown for spheroids of any eccentricity. The second significant contribution presented in this work pertains to a quaternionic space-frequency theory for band-limited quaternionic functions. A new class of quaternionic signals is proposed, whose energy concentration in the space and the frequency domains are maximal under the quaternion Fourier transform. These signals are studied in the context of spatial-frequency concentration as eigenfunctions of a compact and self-adjoint quaternion integral operator. The fundamental properties of their associated eigenvalues are described in detail. When the concentration domains are spherical in both spaces, the angular part of these signals can be found explicitly, leading to a set of one-dimensional radial integral equations. The theoretical framework described in this work is applied to the construction of classes of monogenic functions in different spaces via harmonic conjugates. Two constructive algorithms are proposed to characterize the monogenic weighted Hardy and Bergman spaces in the Euclidean unit ball. For a real-valued harmonic function belonging to a Hardy and a weighted Bergman space, the harmonic conjugates in the same spaces are found. The boundedness of the underlying harmonic conjugation operators is proven in the given weighted spaces. Additionally, a quaternionic counterpart of Bloch’s Theorem is established for monogenic functions

Flexible Modellerweiterung und Optimierung von Erdbebensimulationen

Simulations of realistic earthquake scenarios require scalable software and extensive supercomputing resources. With increasing fidelity in simulations, advanced rheological and source models need to be incorporated. I introduce a domain-specific language in order to handle the model flexibility in combination with the high efficiency requirements. The contributions in this thesis enabled the to date largest and longest dynamic rupture simulation of the 2004 Sumatra earthquake.Realistische Erdbebensimulationen benötigen skalierbare Software und beträchtliche Rechenressourcen. Mit zunehmender Genauigkeit der Simulationen müssen fortschrittliche rheologische und Quellmodelle integriert werden. Ich führe eine domänenspezifische Sprache ein, um die Modelflexibilität in Kombination mit den hohen Effizienzanforderungen zu beherrschen. Die Beiträge in dieser Arbeit haben die bisher größte und längste dynamische Bruchsimulation des Sumatra-Erdbebens von 2004 ermöglicht

Supersymmetric Field Theories and Isomonodromic Deformations

The topic of this thesis is the recently discovered correspondence between supersymmetric gauge theories, two-dimensional conformal field theories and isomonodromic deformation problems. Its original results are organized in two parts: the first one, based on the papers [1], [2], as well as on some further unpublished results, provides the extension of the correspondence between four-dimensional class S theories and isomonodromic deformation problems to Riemann Surfaces of genus greater than zero. The second part, based on the results of [3], is instead devoted to the study of five-dimensional superconformal field theories, and their relation with q-deformed isomonodromic problems

A review of nonlinear FFT-based computational homogenization methods

Since their inception, computational homogenization methods based on the fast Fourier transform (FFT) have grown in popularity, establishing themselves as a powerful tool applicable to complex, digitized microstructures. At the same time, the understanding of the underlying principles has grown, in terms of both discretization schemes and solution methods, leading to improvements of the original approach and extending the applications. This article provides a condensed overview of results scattered throughout the literature and guides the reader to the current state of the art in nonlinear computational homogenization methods using the fast Fourier transform

A review of mechanoluminescence in inorganic solids : compounds, mechanisms, models and applications

Mechanoluminescence (ML) is the non-thermal emission of light as a response to mechanical stimuli on a solid material. While this phenomenon has been observed for a long time when breaking certain materials, it is now being extensively explored, especially since the discovery of non-destructive ML upon elastic deformation. A great number of materials have already been identified as mechanoluminescent, but novel ones with colour tunability and improved sensitivity are still urgently needed. The physical origin of the phenomenon, which mainly involves the release of trapped carriers at defects with the help of stress, still remains unclear. This in turn hinders a deeper research, either theoretically or application oriented. In this review paper, we have tabulated the known ML compounds according to their structure prototypes based on the connectivity of anion polyhedra, highlighting structural features, such as framework distortion, layered structure, elastic anisotropy and microstructures, which are very relevant to the ML process. We then review the various proposed mechanisms and corresponding mathematical models. We comment on their contribution to a clearer understanding of the ML phenomenon and on the derived guidelines for improving properties of ML phosphors. Proven and potential applications of ML in various fields, such as stress field sensing, light sources, and sensing electric (magnetic) fields, are summarized. Finally, we point out the challenges and future directions in this active and emerging field of luminescence research