TY - JOUR AB - In this article we prove meromorphic continuation of weighted zeta functions in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a residue formula proving equality between residues of weighted zetas and invariant Ruelle distributions. We combine this equality with results of Guillarmou, Hilgert and Weich (2021) in order to relate the residues to Patterson-Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems. AU - Schütte, Philipp AU - Weich, Tobias AU - Barkhofen, Sonja ID - 31059 JF - Communications in Mathematical Physics TI - Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems VL - 398 ER - TY - GEN AB - We present a numerical algorithm for the computation of invariant Ruelle distributions on convex co-compact hyperbolic surfaces. This is achieved by exploiting the connection between invariant Ruelle distributions and residues of meromorphically continued weighted zeta functions established by the authors together with Barkhofen (2021). To make this applicable for numerics we express the weighted zeta as the logarithmic derivative of a suitable parameter dependent Fredholm determinant similar to Borthwick (2014). As an additional difficulty our transfer operator has to include a contracting direction which we account for with techniques developed by Rugh (1992). We achieve a further improvement in convergence speed for our algorithm in the case of surfaces with additional symmetries by proving and applying a symmetry reduction of weighted zeta functions. AU - Schütte, Philipp AU - Weich, Tobias ID - 51206 T2 - arXiv:2308.13463 TI - Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions ER - TY - GEN AB - Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of non-compact type and $\Gamma$ a torsion-free discrete subgroup in $G_1\times G_2$. We show that the spectrum of $\Gamma \backslash X$ is related to the asymptotic growth of $\Gamma$ in the two direction defined by the two factors. We obtain that $L^2(\Gamma \backslash G)$ is tempered for large class of $\Gamma$. AU - Weich, Tobias AU - Wolf, Lasse Lennart ID - 51207 T2 - arXiv:2304.09573 TI - Temperedness of locally symmetric spaces: The product case ER - TY - JOUR AU - Steffen, Eckhard AU - Wolf, Isaak Hieronymus ID - 51351 JF - Discrete Applied Mathematics KW - Applied Mathematics KW - Discrete Mathematics and Combinatorics SN - 0166-218X TI - Bounds for the chromatic index of signed multigraphs VL - 337 ER - TY - THES AB - Erfolg und Misserfolg eines Unternehmens werden maßgeblich durch getroffene Entscheidungen beeinflusst. Daher verlassen sich Entscheider oft auf Entscheidungsunterstützungssysteme, die durch Datensimulation, -optimierung und -visualisierung bei der Identifizierung von geeigneten Entscheidungen unterstützen. Für eine optimale Unterstützung muss ein Entscheidungsunterstützungssystem (EUS) jedoch auf den Entscheidungsprozess eines Entscheiders abgestimmt sein und verfügbare Daten, Optimierungsziele, persönliche Präferenzen sowie weitere Einflussfaktoren berücksichtigen. EUS-Entwickler können aufgrund der Komplexität und Volatilität von Geschäftsumgebungen allerdings nicht alle potenziellen Entscheidungsprozesse während des Entwurfs eines EUS vorhersehen, wodurch ein EUS einem Entscheider häufig nur unzureichende Anpassungsmöglichkeiten an den individuellen Entscheidungsprozess bietet. Die Einzelanfertigung eines EUS, das auf einen Entscheidungsprozess zugeschnitten ist, ist ein kosten- und zeitintensives Unterfangen aufgrund der begrenzten Verfügbarkeit von Softwareentwicklern oder Missverständnissen zwischen Entwicklern und Entscheidern während der Entwicklung. Daher geben sich Entscheider möglicherweise mit einem handelsüblichen EUS zufrieden, das nicht vollständig mit ihrem Entscheidungsprozess übereinstimmt, suboptimale Entscheidungen begünstigt und so den Unternehmenserfolg negativ beeinflusst. In dieser Arbeit wird ein Ansatz vorgeschlagen, der es Entscheidern ermöglicht, selbst maßgeschneiderte Entscheidungsunterstützungssysteme zu entwickeln und so die Diskrepanz zwischen benötigter und tatsächlicher Entscheidungsunterstützung zu vermeiden. Dazu stellen EUS-Entwickler einen Teil der EUS-Funktionalität als wiederverwendbare Software-Dienste bereit ... AU - Kirchhoff, Jonas ID - 51352 TI - Decision Support Ecosystems: Assisted Low-Code Development of Tailored Decision Support Systems ER - TY - CHAP AU - Cappello, Chiara AU - Steffen, Eckhard ID - 45190 SN - 9783031261039 T2 - The Digital Twin of Humans TI - Graph-Theoretical Models for the Analysis and Design of Socio-Technical Networks ER - TY - JOUR AU - Steffen, Eckhard AU - Wolf, Isaak Hieronymus ID - 51357 JF - Discrete Mathematics KW - Discrete Mathematics and Combinatorics KW - Theoretical Computer Science SN - 0012-365X TI - Rotation r-graphs ER - TY - GEN AB - We consider the quantum dynamics of a many-fermion system in $\mathbb R^d$ with an ultraviolet regularized pair interaction as previously studied in [M. Gebert, B. Nachtergaele, J. Reschke, and R. Sims, Ann. Henri Poincar\'e 21.11 (2020)]. We provide a Lieb-Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb-Robinson bound on $L^2$-overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb-Robinson bounds. AU - Hinrichs, Benjamin AU - Lemm, Marius AU - Siebert, Oliver ID - 51375 T2 - arXiv:2310.17736 TI - On Lieb-Robinson bounds for a class of continuum fermions ER - TY - GEN AB - In the Bogoliubov-Fr\"ohlich model, we prove that an impurity immersed in a Bose-Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions. AU - Hinrichs, Benjamin AU - Lampart, Jonas ID - 51376 T2 - arXiv:2311.05361 TI - A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate ER - TY - JOUR AB - For a compact Riemannian locally symmetric space $\Gamma\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $\Gamma$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$. AU - Hilgert, Joachim AU - Weich, Tobias AU - Wolf, Lasse Lennart ID - 31190 IS - 10 JF - Analysis & PDE TI - Higher rank quantum-classical correspondence VL - 16 ER - TY - GEN AB - We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators. AU - Bux, Kai-Uwe AU - Hilgert, Joachim AU - Weich, Tobias ID - 51205 T2 - arXiv:2307.10876 TI - Spectral correspondences for finite graphs without dead ends ER - TY - JOUR AB - In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for general rank one spaces. Except for the case of hyperbolic surfaces a countable set of exceptional spectral parameters always left untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of $\mathrm{SL}(2,\mathbb R)$. In higher dimensions the situation is more complicated, but can be described completely. AU - Arends, Christian AU - Hilgert, Joachim ID - 31210 JF - Journal de l’École polytechnique — Mathématiques KW - Ruelle resonances KW - Poisson transforms KW - locally symmetric spaces KW - principal series representations SN - 2429-7100 TI - Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters VL - 10 ER - TY - JOUR AU - Hilgert, Joachim AU - Arends, C. ID - 51383 JF - J. de l'École polytechnique — Mathématiques TI - Spectral correspondences for rank one locally symmetric spaces - The case of exceptional parameters VL - 10 ER - TY - JOUR AU - Hilgert, Joachim AU - Glöckner, H. ID - 51384 JF - J. Diff. Equations TI - Aspects of control theory on infinite-dimensional Lie groups and G-manifolds VL - 343 ER - TY - GEN AU - Hilgert, Joachim AU - Arends, C. AU - Frahm, J. ID - 51499 TI - A pairing formula for resonant states on finite regular graphs ER - TY - GEN AU - Hilgert, Joachim AU - Arends, C. AU - Frahm, J. ID - 51500 TI - Edge Laplacians and vector valued Poisson transforms for graphs ER - TY - GEN AU - Hilgert, Joachim AU - Baier, T. AU - Kaya, O. AU - Mourao, J. AU - Nunes, J. ID - 51502 TI - Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem ER - TY - GEN AU - Hilgert, Joachim AU - Guedes Bonthonneau, Y. AU - Guillarmou, C. AU - Weich, Tobias ID - 51521 TI - Ruelle-Taylor resonances of Anosov actions ER - TY - GEN AB - We present a new gradient-like dynamical system related to unconstrained convex smooth multiobjective optimization which involves inertial effects and asymptotic vanishing damping. To the best of our knowledge, this system is the first inertial gradient-like system for multiobjective optimization problems including asymptotic vanishing damping, expanding the ideas laid out in [H. Attouch and G. Garrigos, Multiobjective optimization: an inertial approach to Pareto optima, preprint, arXiv:1506.02823, 201]. We prove existence of solutions to this system in finite dimensions and further prove that its bounded solutions converge weakly to weakly Pareto optimal points. In addition, we obtain a convergence rate of order O(t−2) for the function values measured with a merit function. This approach presents a good basis for the development of fast gradient methods for multiobjective optimization. AU - Sonntag, Konstantin AU - Peitz, Sebastian ID - 32447 T2 - arXiv:2307.00975 TI - Fast Convergence of Inertial Multiobjective Gradient-like Systems with Asymptotic Vanishing Damping ER - TY - GEN AB - Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization". AU - Bernreuther, Marco AU - Dellnitz, Michael AU - Gebken, Bennet AU - Müller, Georg AU - Peitz, Sebastian AU - Sonntag, Konstantin AU - Volkwein, Stefan ID - 46578 T2 - arXiv:2308.01113 TI - Multiobjective Optimization of Non-Smooth PDE-Constrained Problems ER -