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Kelma, Florian
Projective shapes : topology and means. - Ilmenau : Universitätsbibliothek. - 1 Online-Ressource (82 Seiten)
Technische Universität Ilmenau, Dissertation, 2017

Die projektive Form eines Objektes ist die geometrische Information, die invariant unter projektiven Transformationen ist. Sie tritt natürlicherweise bei der Rekonstruktion von Objekten anhand Fotos unkalibrierter Kameras auf. Wenn ein Objekt als Punktmenge oder Konfiguration von Landmarken im d-dimensionalen reell-projektiven Raum RP(d) beschrieben wird, so ist die Menge der projektiven Formen der Quotientenraum RP(d)^k / PGL(d) und damit kanonisch mit der Quotiententopologie versehen. Auf diesem topologischen Raum der projektiven Formen lassen sich jedoch aus topologischen Gründen viele mathematische Werkzeuge nicht anwenden, ein Phänomen, welches in ähnlicher Form auch bei den Räumen der Ähnlichkeits- bzw. affinen Formen auftritt. In der vorliegenden Arbeit wird die Topologie des projektiven Formenraumes gründlich untersucht, in Hinblick auf die Suche nach einem vernünftigen topologischen Unterraum, der hinreichende Eigenschaften für die Anwendung statistischer Methoden besitzt. Ein Beispiel für einen dieser gutartigen Unterräume ist der Raum der Tyler regulären Formen, der bereits durch Kent und Mardia betrachtet wurde. Deren Ergebnisse werden in dieser Arbeit noch erweitert. Dieser Unterraum ist zwar für einige Dimensionen d und Anzahlen an Landmarken k nicht optimal gewählt, jedoch liefert die sogenannte Tyler-Standardisierung dieser Formen einem sowohl Einbettungen in metrische Räume als auch eine Riemannsche Metrik auf diesem Unterraum. Für eine dieser Einbettungen werden die dazugehörige Fréchet-Erwartungs- sowie Mittelwerte definiert. Während die Konsistenz dieses Mittelwertes leicht zu zeigen ist, ist die Berechnung des extrinsischen Mittelwertes numerisch anspruchsvoll. Als Ersatz wird ein weiterer Erwartungs- bzw. Mittelwert definiert, dessen Berechnung diese Probleme umgeht.


https://www.db-thueringen.de/receive/dbt_mods_00032997
Brause, Christoph; Kemnitz, Arnfried; Marangio, Massimiliano; Pruchnewski, Anja; Voigt, Margit
Sum choice number of generalized [theta]-graphs. - In: Discrete mathematics. - Amsterdam [u.a.] : Elsevier, Bd. 340 (2017), 11, S. 2633-2640
https://doi.org/10.1016/j.disc.2016.11.028
Behrndt, Jussii; Schmitz, Philipp; Trunk, Carsten
Spectral bounds for singular indefinite Sturm-Liouville operators with L1-potentials. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (7 Seiten). - (Preprint. - M17,12)
Im Titel ist "1" hochgestellt

The spectrum of the singular indefinite Sturm-Liouville operator A=sgn(.) (-d^2/dx^2)+q with a real potential q in L^1(R)$ covers the whole real line and, in addition, non-real eigenvalues may appear if the potential q assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound l lambda | <= |q|_{L^1}^2 on the absolute values of the non-real eigenvalues lambda of A is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the L^1-norm of q and its negative part q_-.


http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2017200509
Kriesell, Matthias
Degree sequences and edge connectivity. - In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. - Berlin [u.a.] : Springer, ISSN 18658784, Bd. 87 (2017), 2, S. 343-355
https://doi.org/10.1007/s12188-016-0171-0
Eichfelder, Gabriele; Krüger, Corinna; Schöbel, Anita
Decision uncertainty in multiobjective optimization. - In: Journal of global optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management and engineering. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 15732916, Bd. 69 (2017), 2, S. 485-510

In many real-world optimization problems, a solution cannot be realized in practice exactly as computed, e.g., it may be impossible to produce a board of exactly 3.546 mm width. Whenever computed solutions are not realized exactly but in a perturbed way, we speak of decision uncertainty. We study decision uncertainty in multiobjective optimization problems and we propose the concept of decision robust efficiency for evaluating the robustness of a solution in this case. This solution concept is defined by using the framework of set-valued maps. We prove that convexity and continuity are preserved by the resulting set-valued maps. Moreover, we obtain specific results for particular classes of objective functions that are relevant for solving the set-valued problem. We furthermore prove that decision robust efficient solutions can be found by solving a deterministic problem in case of linear objective functions. We also investigate the relationship of the proposed concept to other concepts in the literature.


https://doi.org/10.1007/s10898-017-0518-9
Behrndt, Jussii; Gsell, Bernhard; Schmitz, Philipp; Trunk, Carsten
An estimate on the non-real spectrum of a singular indefinite Sturm-Liouville operator. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (5 Seiten). - (Preprint. - M17,10)

It will be shown with the help of the Birman-Schwinger principle that the non-real spectrum of the singular indefinite Sturm-Liouville operator $\operatorname{sgn}(\cdot)(-\mathrm d^2/\mathrm d x^2 +q)$ with a real potential $q\in L^1\cap L^2$ is contained in a circle around the origin with radius $\|q\|_{L^1}^2$.


https://www.db-thueringen.de/receive/dbt_mods_00032787
Berger, Thomas; Gernandt, Hannes; Trunk, Carsten; Wojtylak, Micha&lstrok;
New lower bound for the distance to singularity of regular matrix pencils. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (5 Seiten). - (Preprint. - M17,10)

For regular matrix pencils $\Ac(s)=sE-A$ the distance to the nearest singular pencil in the Frobenius norm of the coefficients is called the distance to singularity. We derive a new lower bound for this distance by using the spectral theory of tridiagonal Toeplitz matrices.


https://www.db-thueringen.de/receive/dbt_mods_00032786
Bergmann, Jean Pierre; Bielenin, Martin; Herzog, Roland A.; Hildebrand, Jörg; Riedel, Ilka; Schricker, Klaus; Trunk, Carsten; Worthmann, Karl
Prevention of solidification cracking during pulsed laser beam welding. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (7 Seiten). - (Preprint. - M17,05)

We present a mathematical model to describe laser beam welding based on the heat equation. Since the material coeff cients depend on the temperature, this leads to a quasi-linear parabolic partial differential equation. It is our goal to prevent solidif cation cracking. We address this problem by means of optimal control. It is the intensity prof le of the laser beam which acts as the control function. The main challenge is the formulation of a suitable objective function. In particular, high velocities of the solidif cation interface need to be properly penalized in order to deal with and avoid cracking phenomena.


https://www.db-thueringen.de/receive/dbt_mods_00032771
Gernandt, Hannes; Krauße, Dominik; Sommer, Ralf; Trunk, Carsten
A new method for network redesign via rank one updates. - Ilmenau : Technische Universität, Institut für Mathematik. - 1 Online-Ressource (5 Seiten). - (Preprint. - M17,05)

We present a method to place the eigenvalues of an electrical network towards a prescribed set of complex numbers by inserting an additional capacitance into the network. We use recent results on rank one perturbations of regular matrix pencils and provide an upper bound on the approximation error of the eigenvalues in the chordal distance.


https://www.db-thueringen.de/receive/dbt_mods_00032770
Boeck, Thomas; Terzijska, Dzulia; Eichfelder, Gabriele
Maximum electromagnetic drag configurations for a translating conducting cylinder with distant magnetic dipoles. - In: Journal of engineering mathematics. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 15732703, (2017), first online (22. Jul.), insges. 19 S.

We report a semianalytic and numerical investigation of the maximal induced Lorentz force on an electrically conducting cylinder in translation along its axis that is caused by the presence of multiple distant magnetic dipoles. The problem is motivated by Lorentz force velocimetry, where induction creates a drag force on a magnet system placed next to a conducting flow. The magnetic field should maximize this drag force, which is usually quite small. Our approach is based on a long-wave theory developed for a single distant magnetic dipole. We determine the optimal orientations of the dipole moments providing the strongest Lorentz force for different dipole configurations using numerical optimization methods. Different constraints are considered for dipoles arranged on a concentric circle in a plane perpendicular to the cylinder axis. In this case, the quadratic form for the force in terms of the dipole moments can be obtained analytically, and it resembles the expression of the energy in a classical spin model. When all dipoles are equal and their positions on the circle are not constrained, the maximal force results when all dipoles are gathered in one point with all dipole moments pointing in radial direction. When the dipoles are equal and have equidistant spacing on the circle, we find that the optimal orientations of the dipole moments approach a limiting distribution. It differs from the so-called Halbach distribution that provides a uniform magnetic field in the cross section of the cylinder. The corresponding force is about 10% larger than that for the Halbach distribution but 60% smaller than for the unconstrained dipole positions. With the so-called spherical constraint for a classical spin model, the maximal force can be found from the eigenvalues of the coefficient matrix. It is typically 10% larger than the maximal force for equal dipoles because the constraint is weaker. We also study equal and evenly spaced dipoles along one or two lines parallel to the cylinder axis. The patterns of optimal magnetic moment orientations are fairly similar for different dipole numbers when the inter-dipole distance is within a certain interval. This behavior can be explained by reference to the magnetic field distribution of a single distant dipole on the cylinder axis.


https://doi.org/10.1007/s10665-017-9916-8