The nonequispaced fast Fourier transform (NFFT) allows the fast approximate evaluation of trigonometric polynomials with frequencies supported on full box-shaped grids at arbitrary sampling nodes. Due to the curse of dimensionality, the total number of frequencies and thus, the total arithmetic complexity can already be very large for small refinements at medium dimensions. In this paper, we present an approach for the fast approximate evaluation of trigonometric polynomials with frequencies supported on an arbitrary subset of the full grid at arbitrary sampling nodes, which is based on Taylor expansion and rank-1 lattice methods. For the special case of symmetric hyperbolic cross index sets in frequency domain, we present error estimates and numerical results.

Bereits im 19. Jahrhundert, besonders mit dem deutsch-französischen Krieg von 1870-1871, nahm die Eisenbahn eine zunehmende Rolle in der militärischen Taktik ein. Als der Erste Weltkrieg anfängt sind die Eisenbahnen und ihre Netze die Hauptwerkzeuge der Kriegsführung in Deutschland wie in Frankreich. Material, Menschen, Verpflegung, Waffen, Post werden per Bahn transportiert. Die militärische Machtübernahme der Eisenbahnnetze waren, im Kriegsfall 1914, schon lange vorher geplant gewesen. Die Eisenbahnräume spielen eine wichtige Rolle beim Abschied von Angehörigen. Die Bahnhöfe stehen sinnbildlich als Mittelpunkt der Mobilmachung im August 1914 und sind in fast jede Zeugenaussage erwähnt. Eisenbahnräume sind Zwischenräume zwischwen Heimatfront und Front. Abschied - Spionfieber - Schwarzmarkt - Schwerverletzte Umtausch - / Major logistical and tactical tool during the war, the railway network is devoted to the war effort and, both in Germany and France, is placed under the authority of the military command who manages and reorganises it, making use of regulatory stations, according to the war needs. From a railway perspective, waging war depends on civilians (railway workers, Red Cross) and soldiers (transports routes guardsmen, stations quartermasters generals), ready to step up and shoulder a massive workload and, for some of them, to risk their lives. Civilians (particularly refugees), soldiers (among which soldiers on leave, sick, wounded, repatriated critically wounded), and the vast majority of war equipment (ammunitions, weapons, food supply, mails, ...) need to be transported. Nothing, not even some of these types of travellers, is to jeopardize the war though. A war implying a greater human presence in the railway space, which, in turn, leads to spying, spymania, bombings, evacuations, constructions and destructions of railway facilities, curiosity, gatherings, groups phenomena, unruliness, prostitution, spreading of writings and words of protest, additional travels for the industry and agriculture, and black market. This railway network must be defended and maintained, while at the same time making sure railway workers and soldiers are disciplined and in good moral and physical conditions. Goals and wishes less and less in tune with the reality of the ground, made worse by a dragging war, always increasing the cost in human resources and railway equipments.

Die Herstellung von Stahlbeton erfordert einen hohen Energie- und Ressourceneinsatz, vor allem im Bereich der Zement- und Stahlproduktion. Stahlbetonbauteile sind gekennzeichnet durch hohe Traglasten, aber auch durch hohe Eigengewichte. Bei Beanspruchung auf Biegung wird Beton eigentlich nur im Bereich der Druckzone und zum Schutz der Bewehrung in der Zugzone benötigt. Dem können sich bisher platten- oder scheibenförmige Bauteile nicht anpassen. Der Beton wird über den gesamten Querschnitt eingesetzt. Natürlich wird er dabei auch zur Übertragung der Schubkräfte zwischen Druck- und Zugzone benötigt. Setzt man ihn über den vollen Querschnitt ein, bringt er einen hohen Anteil an Eigenlast in das Tragmodell ein und damit wird ein Teil der Tragfähigkeit durch die Eigenlast aufgebraucht. [Aus: Motivation und Ziel] / The production of reinforced concrete requires a high input of energy and resources, especially in the area of cement and steel production. Reinforced concrete components are characterised by high load-bearing capacities but also by high dead weights. When stressed in bending, concrete is actually only needed in the compression zone and to protect the reinforcement in the tension zone. Until now, slab or wall-shaped components have not been able to adapt to this. The concrete is used over the entire crosssection. Of course, it is also needed to transmit the shear forces between the compression and tension zones. If it is used over the full cross-section, it introduces a high proportion of dead load into the load-bearing model and thus part of the load-bearing capacity is used up by the dead load.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.

Mit Lösungen von 9,10-Diphenylanthracen und N(C2H5)4ClO4 oder N(C4H9)4ClO4 als Leitsalz im Lösungsmittel Acetonitril wurden Elektrochemilumineszenz (ECL)-Experimente durchgeführt. Dazu wurden die Elektroden mit Folgen von jeweils drei in bestimmten zeitlichen Abständen aufeinander folgenden Potentialsprüngen polarisiert. Es wird gezeigt, dass bei entsprechender Wahl der Potentiale und der Haltezeiten anodische und kathodische ECL-Emissionen gleicher Intensität erzeugt werden können. Sodann wurde ECL in den Derivaten von Poly(p-phenylen-vinylen), MEH-PPV und DB-PPV erzeugt. Diese leitfähigen Polymere wurden als dünne Schichten auf Platin-Elektroden aufgebracht und wie bei ECL aus der Lösungsphase in Acetonitril-Elektrolyten mit Tetralkylammonium-Leitsalzen Potentialsprüngen unterworfen. Bei geeigneter Einstellung der Potentialsprünge und Haltezeiten konnten anodische und kathodische ECL gleicher Intensität erhalten werden. Dies ist das erste Mal, dass symmetrische ECL mit polymerbeschichteten Elektroden erhalten wurde. Die Kinetik der ECL weicht deutlich von der aus der Lösungsphase ab. Der ECL-Prozess verläuft langsamer als in der Lösungsphase, und der Leitelektrolyt hat einen signifikanten Einfluss auf das elektrochemische Verhalten der Polymerschicht. Die Ursachen dafür wurden über Modellrechnungen analysiert, mit denen die Ladungstransportprozesse in der Polymerschicht simuliert wurden. In derartigen Simulationsrechnungen konnten die Geschwindigkeitskonstanten der ECL-Reaktion sowohl im Polymer als auch in der Lösung bestimmt werden. Um die Stabilität der Polymerschichten zu erhöhen, wurde versucht, die Polymerketten mit Synchrotronstrahlung zu vernetzen. Diese Experimente brachten nicht das erwartete Ergebnis. Die Ursachen dafür werden auf der Grundlage von Ex-Situ-Raman-spektroskopischen Untersuchungen diskutiert. / Electrochemiluminescence (ECL) has been generated in solution phase using 9,10-diphenylanthracene (DPA) with TEAClO4 (or TBAClO4) in acetonitrile solvent. Triple potential step was used for the generation of ECL. It was found that anodic and cathodic ECL of equal intensities can be generated by proper choice of potential step magnitude, width and the waiting period (tw) between successive triple potential steps. ECL was then generated in conducting polymers poly(2-ethylhexyloxy-5-methoxy-1,4-phenylenevinylene) (MEH-PPV) and poly(2,3-dibutoxy-1,4-phenylenevinylene) (DB-PPV) by coating them on Pt electrodes and subjecting to potential steps in tetraalkylammonium salt solutions with acetonitrile. Similar to the case of solution phase ECL, symmetrical anodic and cathodic ECL could be observed by the appropriate choice of the potential step parameters. But the kinetics of the ECL was found to be different from that of the solution phase ECL. The time scale of the ECL process was found to be longer than that in the solution phase ECL. The nature of supporting electrolyte had a remarkable impact on the electrochemistry of conducting polymers. The reasons were analyzed by theoretical calculations evoking the concept of charge transport characteristics of conducting polymers. The rate constants of the ECL process were calculated by separate simulation procedure in the solution phase as well as in the polymer phase ECL. To enhance the stability of conducting polymers, synchrotron radiation induced cross-linking was performed. The effects were different from expected which were analyzed and rationalized by ex-situ Raman spectroscopic studies.

The nonequispaced fast Fourier transform (NFFT) allows the fast approximate evaluation of trigonometric polynomials with frequencies supported on full box-shaped grids at arbitrary sampling nodes. Due to the curse of dimensionality, the total number of frequencies and thus, the total arithmetic complexity can already be very large for small refinements at medium dimensions. In this paper, we present an approach for the fast approximate evaluation of trigonometric polynomials with frequencies supported on an arbitrary subset of the full grid at arbitrary sampling nodes, which is based on Taylor expansion and rank-1 lattice methods. For the special case of symmetric hyperbolic cross index sets in frequency domain, we present error estimates and numerical results.

Text in German with summaries in German and English / Die Kreuzestheologie erlebt seit Beginn des 20. Jahrhunderts eine anhaltende Renaissance. Die vorgelegten Beiträge gehen zwar von gemeinsamen Wurzeln aus, verzweigen sich jedoch in vielfältige, teilweise disparate Entwürfe. Deshalb ist neu zu fragen, was Kreuzestheologie ist. Die Antwort wird in sieben Schriften des Corpus Paulinum gesucht, welche auf ihren kreuzestheologischen Gehalt befragt werden. Dabei zeigt sich trotz unterschiedlich häufiger Verwendung der kreuzestheologischen Begriffe durchwegs die entscheidende Bedeutung des Kreuzes Jesu in den theologischen Grundlinien. Kreuzesaussagen erscheinen immer im Zusammenhang mit den zentralen Thesen der paulinischen Briefe. Der exegetische Befund führt zum Schluss, dass bei der paulinischen Kreuzestheologie von einem theologischen Ansatz zu sprechen ist, der zurecht umfassenden und kritischen Anspruch auf christliche Theologie und kirchliche Praxis erhebt. / The “theology of the cross” has seen a remarkable come-back since the beginning of 20th century. While the various contributions share similar roots, they differ widely in their approaches and constructions. Therefore, the simple question regarding the nature of a theology of the cross has to be raised again. This study seeks answers in seven writings of the Corpus Paulinum, which will be examined for their use of references to Jesus’ death on the cross. Although these letters differ significantly in their use of crossrelated terminology, they consistently testify to the crucial significance of the cross of Jesus in their theological paradigm. Statements about the cross are always an essential ingredient in the argumentation of these letters. The exegetical survey leads to the conclusion that Pauline theology of the cross is an all-encompassing and critical approach and principle for Christian theology and ecclesial practice. / New Testament / M. Th. (New Testament)