Localized Modes of the Linear Periodic Schr\"{o}dinger Operator with a Nonlocal Perturbation

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schr\"{o}dinger operator $-\partial_x^2+ V(x)$ with an interface. The interface is modeled by a jump either in the value or the derivative of $V(x)$ and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can only occur in spectral gaps. We pose the eigenvalue problem as a $C^1$ gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions $R_\pm=\frac{\psi_\pm'(0)}{\psi_\pm(0)}$, where $\psi_\pm$ are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Pr\"{u}fer transformation. The limit values of $R_\pm$ at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at gap edges. We show that the ordering can be determined in the first two gaps via variational analysis for potentials satisfying certain monotonicity conditions. Numerical computations of interface eigenvalues are presented to corroborate the analysis.Comment: 1. finiteness of the number of additive interface eigenvalues proved in a remark below Corollary 3.6.; 2. small modifications and typo correction

Breathers and rogue waves for semilinear curl-curl wave equations

We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0$ for $(x,t)\in \mathbb{R}^3\times\mathbb{R}$ and arbitrary $p>1$. Depending on the coefficients $s, q, V$ we can prove the existence of three types of localized solutions: time-periodic solutions decaying to $0$ at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to $0$ both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a parametric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution $U(x,t)$ already generates a full continuum of phase-shifted solutions $U(x,t+b(x))$ where the continuous function $b:\mathbb{R}^3\to\mathbb{R}$ belongs to a suitable admissible family

Breathers and rogue waves for semilinear curl-curl wave equations

We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x)\partial_t^2U+\nabla\times\nabla\times+q(x)U\pm V(x)|U|^{p−1}U = 0$ for $(x,t)\in\mathbb{R^3}\times\mathbb{R}$ and arbitrary $p > 1$. Depending on the coefficients $s,q,V$ we can prove the existence of three types of localized solutions: time-periodic solutions decaying to $0$ at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to $0$ both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a para- metric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution $U(x, t)$ already generates a full continuum of phase-shifted solutions $U(x,t + b(x))$ where the continuous function $b:\mathbb{R}^3\to\mathbb{R}$ belongs to a suitable admissible family

Situation Assessment for Advanced Driver Assistance Systems

Die Arbeit behandelt drei Fragestellungen: Zunächst liegt auf der Erarbeitung einer funktionalen Architekturdetaillierung für die Situationsanalyse. Hierfür wird ihre die Stellung der Situationsanalyse eines fortschrittlichen Fahrerassistenzsystems aufgezeigt, um im Anschluss die funktionale Architektur innerhalb der Situationsanalyse aufzuschlüsseln. Der zweite Teil der Arbeit beinhaltet die Vorstellung eines neuen Algorithmus zur Berechnung der Grenzen des kollisionsfrei erreichbaren Raumes. Er bildet den zentralen Kern der Situationsanalyse einer aktiven Gefahrenbremsung. Der Algorithmus zeichnet sich dadurch aus, dass er sowohl beliebig strukturierte statische Hindernisse als auch dynamische Verkehrsteilnehmer berücksichtigt. Zudem beachtet er deren Interaktionsbeziehungen und die Aufmerksamkeit des Fahrers. Im Zuge der Modellierung fließen auch neue Erkenntnisse über den Einfluss der Breite einer zu durchfahrenden Lücke ein, die aus einer Studie stammen. Kommt der Algorithmus zum Ergebnis, dass der kollisionsfrei erreichbare Raum vollständig durch Hindernisse begrenzt ist - es also keine Ausweichmöglichkeit gibt - so ist ein wesentliches Kriterium für eine automatische Notbremse erfüllt. Auf mehreren Präsentationen war diese echtzeitfähig implementierte Situationsanalyse elementarer Bestandteil der vorgestellten Technik, mit der unfallvermeidende Bremsungen bis in den Stillstand auch oberhalb von 60 km /h möglich werden. Der dritte Teil der Dissertation adressiert eine bislang noch ungelöste Herausforderung: Die Erkennung einer Einfädelsituation aus der Perspektive eines involvierten Fahrzeuges. Auf die Modellierung der Merkmale als auch des Klassifikators wird ausführlich eingegangen. Im Zuge dessen wird das im Forschungsbereich der Fahrerassistenz noch unbekannte Klassifikationsverfahren "'Scenario Based Random Forest"' vorgestellt und beurteilt. Abschließend kann gezeigt werden, dass der Erkennungsalgorithmus in 92% der Fälle zum richtigen Ergebnis führt.This work addresses three issues in the research area of situation analysis for advanced driver assistance systems. The focus of the first part is on the development of a detailed functional architecture for situation analysis. Therefore, the situation analysis' embedding in the overall system is shown. The functional architecture within the situation analysis is developed afterwards. The second part of this work introduces a new algorithm for calculating the borders of the collision-free reachability area. This algorithm forms the central piece of the situation analysis of an active hazard braking function. It considers arbitrary structured static obstacles, dynamic road users, their interaction relationships as well as the driver's state of attention. The influence of narrow gap's widths between obstacles in the context of evasion manoevers were addressed in a study. If the result of the computation indicates that the collision-free reachability area is completely limited by obstacles - in other words that no collision free track exists - an essential criterion for the use of an automatic emergency brake is met. The results were shown with a real vehicle using such an active hazard braking system. It demonstrated warrantable full stop collision avoiding braking interventions even at differential speeds above 60km/h. The third part of the thesis addresses the detection of convoy merging situations from the perspective of an involved vehicle. Within the development of such a classification algorithm, the method ``Scenario based random forest'' is introduced. This thesis describes the modeling of the situation and hence the features used for the classification as well as the training of the classifier in detail. The results show, that the convoy merging situations are classified correctly in 92% of given samples. All presented results were obtained by using real-world sensor data

Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $2\pi$-periodic traveling wave solutions of a variant of the Lugiato-Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $ia_\tau = (\zeta − i ) a − da_{xx} − |a|^2 a + i f_0 + i f_1\text{e}^{i( k_1 x−ν_1 \tau)}$. The main new feature of the problem is the specific form of the source term $f_0 + f_1 \text{e}^{i(k_1 x− ν_1 \tau )}$ which describes the simultaneous pumping of two different modes with mode indices $k_0 = 0$ and $k_1\in\mathbb{N}$. We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the 1-mode case, i.e. $f_1 = 0$, can be continued into the range $f_1\ne 0$. Our analytical findings apply both for anomalous $(d > 0)$ and normal $(d < 0)$ dispersion, and they are illustrated by numerical simulations

A breather construction for a semilinear curl-curlwave equation with radially symmetric coefficients

Abstract. We consider the semilinear curl-curl wave equation s(x)∂2 U + ∇ × ∇ × U + q(x)U ± V (x)|U |p−1 U = 0 for (x, t) ∈ R3 × R. For any p > 1 we prove the existence of time- periodic spatially localized real-valued solutions (breathers) both for the + and the − case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to 0 as |x| → ∞. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curl-curl operator. Consequently, the semilinear wave equation is reduced to an ODE with r = |x| as a parameter. This ODE can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain not only one but a full continuum of phase-shifted breathers U (x, t+a(x)), where U is a particular breather and a : R3 → R an arbitrary radially symmetric C 2 -function

Software Choice and Sequencing Coverage Can Impact Plastid Genome Assembly–A Case Study in the Narrow Endemic Calligonum bakuense

Most plastid genome sequences are assembled from short-read whole-genome sequencing data, yet the impact that sequencing coverage and the choice of assembly software can have on the accuracy of the resulting assemblies is poorly understood. In this study, we test the impact of both factors on plastid genome assembly in the threatened and rare endemic shrub Calligonum bakuense. We aim to characterize the differences across plastid genome assemblies generated by different assembly software tools and levels of sequencing coverage and to determine if these differences are large enough to affect the phylogenetic position inferred for C. bakuense compared to congeners. Four assembly software tools (FastPlast, GetOrganelle, IOGA, and NOVOPlasty) and seven levels of sequencing coverage across the plastid genome (original sequencing depth, 2,000x, 1,000x, 500x, 250x, 100x, and 50x) are compared in our analyses. The resulting assemblies are evaluated with regard to reproducibility, contig number, gene complement, inverted repeat length, and computation time; the impact of sequence differences on phylogenetic reconstruction is assessed. Our results show that software choice can have a considerable impact on the accuracy and reproducibility of plastid genome assembly and that GetOrganelle produces the most consistent assemblies for C. bakuense. Moreover, we demonstrate that a sequencing coverage between 500x and 100x can reduce both the sequence variability across assembly contigs and computation time. When comparing the most reliable plastid genome assemblies of C. bakuense, a sequence difference in only three nucleotide positions is detected, which is less than the difference potentially introduced through software choice

Inhibitor of Apoptosis Proteins as Novel Targets in Inflammatory Processes

Objective: Inhibitor of apoptosis proteins (IAPs), such as X-linked or cellular IAP 1/2 (XIAP, cIAP1/2), are important regulators of apoptosis. IAP antagonists are currently under clinical investigation as anticancer agents. Interestingly, IAPs participate in the inflammation-associated TNF receptor signaling complex and regulate NFκB signaling. This raises the question about the role of IAPs in inflammation. Here, we investigated the anti-inflammatory potential of IAP inhibitors and the role of IAPs in inflammatory processes of endothelial cells. Methods and Results: In mice, the small molecule IAP antagonist A-4.10099.1 (ABT) suppressed antigen-induced arthritis, leukocyte infiltration in concanavalin A-evoked liver injury, and leukocyte transmigration in the TNFα-activated cremaster muscle. In vitro, we observed an attenuation of leukocyte– endothelial cell interaction by downregulation of the intercellular adhesion molecule-1. ABT did not impair NFκB signaling but decreased the TNFα-induced activation of the TGF-β–activated kinase 1, p38, and c-Jun N-terminal kinase. These effects are based on the proteasomal degradation of cIAP1/2 accompanied by an altered ratio of the levels of membrane-localized TNF receptor-associated factors 2 and 5. Conclusion: Our results reveal IAP antagonism as a profound anti-inflammatory principle in vivo and highlight IAPs as important regulators of inflammatory processes in endothelial cells