TY - JOUR AU - Hinrichs, Benjamin AU - Janssen, Daan W. AU - Ziebell, Jobst ID - 46100 IS - 1 JF - Journal of Mathematical Analysis and Applications KW - Applied Mathematics KW - Analysis SN - 0022-247X TI - Super-Gaussian decay of exponentials: A sufficient condition VL - 528 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 L. ID - 46117 T2 - arXiv:2304.09573 TI - Temperedness of locally symmetric spaces: The product case ER - TY - JOUR AU - Bruns, Julia AU - Hagena, Maike AU - Gasteiger, Hedwig ID - 46155 JF - Teaching and Teacher Education KW - Education SN - 0742-051X TI - Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects? VL - 132 ER - TY - BOOK ED - Biehler, Rolf ED - Liebendörfer, Michael ED - Gueudet, Ghislaine ED - Rasmussen, Chris ED - Winsløw, Carl ID - 46157 SN - 1869-4918 TI - Practice-Oriented Research in Tertiary Mathematics Education ER - TY - JOUR AU - Ma, Yulai AU - Mattiolo, Davide AU - Steffen, Eckhard AU - Wolf, Isaak Hieronymus ID - 46256 IS - 3 JF - SIAM Journal on Discrete Mathematics KW - General Mathematics SN - 0895-4801 TI - Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs VL - 37 ER - TY - CONF AB - The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE. AU - Offen, Christian AU - Ober-Blöbaum, Sina ED - Nielsen, F ED - Barbaresco, F ID - 42163 KW - System identification KW - discrete Lagrangians KW - travelling waves T2 - Geometric Science of Information TI - Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves VL - 14071 ER - TY - JOUR AB - The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis. AU - Ober-Blöbaum, Sina AU - Offen, Christian ID - 29240 JF - Journal of Computational and Applied Mathematics KW - Lagrangian learning KW - variational backward error analysis KW - modified Lagrangian KW - variational integrators KW - physics informed learning SN - 0377-0427 TI - Variational Learning of Euler–Lagrange Dynamics from Data VL - 421 ER - TY - JOUR AB - The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients. AU - McLachlan, Robert AU - Offen, Christian ID - 29236 IS - 1 JF - Journal of Geometric Mechanics KW - variational integrators KW - backward error analysis KW - Euler--Lagrange equations KW - multistep methods KW - conjugate symplectic methods TI - Backward error analysis for conjugate symplectic methods VL - 15 ER - TY - JOUR AB - Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered. AU - Dierkes, Eva AU - Offen, Christian AU - Ober-Blöbaum, Sina AU - Flaßkamp, Kathrin ID - 37654 IS - 6 JF - Chaos SN - 1054-1500 TI - Hamiltonian Neural Networks with Automatic Symmetry Detection VL - 33 ER - TY - JOUR AB - The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d. samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled. AU - Nüske, Feliks AU - Peitz, Sebastian AU - Philipp, Friedrich AU - Schaller, Manuel AU - Worthmann, Karl ID - 23428 JF - Journal of Nonlinear Science TI - Finite-data error bounds for Koopman-based prediction and control VL - 33 ER - TY - JOUR AB - Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML. AU - Dellnitz, Michael AU - Hüllermeier, Eyke AU - Lücke, Marvin AU - Ober-Blöbaum, Sina AU - Offen, Christian AU - Peitz, Sebastian AU - Pfannschmidt, Karlson ID - 21600 IS - 2 JF - SIAM Journal on Scientific Computing TI - Efficient time stepping for numerical integration using reinforcement learning VL - 45 ER - TY - CONF AU - Schwerin, Imke AU - Häsel-Weide, Uta ED - Novotna, J. ED - Moraova, H. ID - 46757 T2 - International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education TI - Second grader´s understanding of doubling and halfing in various representations ER - TY - CHAP AU - Schmidt, Rebekka AU - Tenberge, Claudia AU - Häsel-Weide, Uta ED - Vöing, N. ED - Schmidt, R. ED - Neiske, I. ID - 46758 T2 - Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre TI - Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern ER - TY - CHAP AU - Biehler, Rolf AU - Engel, Joachim AU - Frischemeier, Daniel ED - Bruder, Regina ED - Büchter, A. ED - Gasteiger, H. ED - Schmidt-Thieme, B. ED - Weigand, HG. ID - 47416 SN - 9783662666036 T2 - Handbuch der Mathematikdidaktik TI - Stochastik: Leitidee Daten und Zufall ER - TY - GEN AB - In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian. AU - Hinrichs, Benjamin AU - Matte, Oliver ID - 47534 T2 - arXiv:2309.09005 TI - Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions ER - TY - CHAP AU - Biehler, Rolf AU - Frischemeier, Daniel AU - Gould, Ronald AU - Pfannkuch, Maxine ED - Pepin, Birgit ED - Gueudet, Ghislaine ED - Choppin, Jeffrey ID - 48042 T2 - Handbook of Digital Resources in Mathematics Education TI - Impacts of Digitalization on Content and Goals of Statistics Education ER - TY - CHAP AU - Leiss, Dominik AU - Gerlach, Kerstin AU - Wessel, Lena AU - Schmidt-Thieme, Barbara ID - 48319 SN - 9783662666036 T2 - Handbuch der Mathematikdidaktik TI - Sprache und Mathematiklernen ER - TY - CHAP AU - Kortmeyer, Jörg AU - Biehler, Rolf ED - Dreyfus, T. ED - Gonzalez-Martin, A. S. ED - Nardi, E. ED - Monaghan, J. ED - Thompson, P. W. ID - 48480 T2 - The Learning and Teaching of Calculus Across Disciplines – Proceedings of the Second Calculus Conference TI - The use of integrals for accumulation and mean values in basic electrical engineering courses ER - TY - JOUR AU - Biehler, Rolf AU - Krüger, Katja ID - 49049 IS - 115 JF - GDM-Mitteilungen TI - Von Glückstal (Ukraine) nach Frankfurt: Problemlösen, Algorithmen, Stochastik – Nachruf auf Arthur Engel ER - TY - CONF AU - Wallner, Melina AU - Häsel-Weide, Uta ED - Novotna, J. ED - Moraova, H. ID - 46756 T2 - International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education TI - Conceptual understanding of third grades for axial symmetry ER - TY - JOUR AB - Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 49326 IS - 6 JF - SIAM Journal on Applied Mathematics KW - Applied Mathematics SN - 0036-1399 TI - Hypernetworks: Cluster Synchronization Is a Higher-Order Effect VL - 83 ER - TY - GEN AB - To model dynamical systems on networks with higher order (non-pairwise) interactions, we recently introduced a new class of ODEs on hypernetworks. Here we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady state branch "reluctant" if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady state branch. AU - von der Gracht, Sören AU - Nijholt, Eddie AU - Rink, Bob ID - 49371 T2 - arXiv:2311.17186 TI - Higher order interactions lead to "reluctant" synchrony breaking ER - TY - JOUR AU - Klüners, Jürgen AU - Wang, Jiuya ID - 49372 JF - La Matematica SN - 2730-9657 TI - Idélic Approach in Enumerating Heisenberg Extensions ER - TY - JOUR AU - Seitz, Simone AU - Häsel-Weide, Uta AU - Wilke, Yannik AU - Wallner, Melina ID - 49425 JF - Teachers and Teaching TI - Expertise and professionalism for inclusive (mathematics) teaching and learning: reflections on findings from interdisciplinary professionalisation research ER - TY - CONF AB - By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function Ld which defines them. Based on ideas from Lie group theory, in this work we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise. AU - Lishkova, Yana AU - Scherer, Paul AU - Ridderbusch, Steffen AU - Jamnik, Mateja AU - Liò, Pietro AU - Ober-Blöbaum, Sina AU - Offen, Christian ID - 34135 IS - 2 T2 - IFAC-PapersOnLine TI - Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery VL - 56 ER - TY - GEN AB - In this article we investigate the convergence behavior of gathering protocols with fixed circulant topologies using tools form dynamical systems. Given a fixed number of mobile entities moving in the Euclidean plane, we model a gathering protocol as a system of ordinary differential equations whose equilibria are exactly all possible gathering points. Then, we find necessary and sufficient conditions for the structure of the underlying interaction graph such that the protocol is stable and converging, i.e., gathering, in the distributive computing sense by using tools from dynamical systems. Moreover, these tools allow for a more fine grained analysis in terms of speed of convergence in the dynamical systems sense. In fact, we derive a decomposition of the state space into stable invariant subspaces with different convergence rates. In particular, this decomposition is identical for every (linear) circulant gathering protocol, whereas only the convergence rates depend on the weights in interaction graph itself. AU - Gerlach, Raphael AU - von der Gracht, Sören AU - Dellnitz, Michael ID - 44840 KW - Dynamical Systems KW - Coupled Systems KW - Distributed Computing KW - Robot Swarms KW - Autonomous Mobile Robots KW - Gathering T2 - arXiv:2305.06632 TI - On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies ER - TY - JOUR AB - A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $n$-spaces such that each totally isotropic $t$-space is contained in exactly one member of $Y$. Nontrivial examples are known only for $t=1$ and $t=n-1$. We give an almost complete classification of such $t$-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces. AU - Schmidt, Kai-Uwe AU - Weiß, Charlene ID - 50298 IS - 1 JF - Combinatorial Theory TI - Packings and Steiner systems in polar spaces VL - 3 ER - TY - JOUR AB - We show that there exist ordered orthogonal arrays, whose sizes deviate from the Rao bound by a factor that is polynomial in the parameters of the ordered orthogonal array. The proof is nonconstructive and based on a probabilistic method due to Kuperberg, Lovett and Peled. AU - Schmidt, Kai‐Uwe AU - Weiß, Charlene ID - 50297 IS - 9 JF - Journal of Combinatorial Designs TI - Existence of small ordered orthogonal arrays VL - 31 ER - TY - THES AB - Digital communications relies heavily on the usage of different types of codes. Prominent codes nowadays are rank-metric codes and subspace codes - the q-analogs of binary codes and binary codes with constant weight. All these codes can be viewed as subsets of classical association schemes. A central coding-theoretic problem is to derive upper bounds for the size of codes. This thesis investigates Delsartes powerful linear program whose optimum is precisely such a bound for codes in association schemes. The linear programs for binary codes and binary constant-weight codes have been extensively studied since the 1970s, but their optimum is still unknown. We determine in a unified way the optimum of the linear program in several ordinary q-analogs as well as in their affine counterparts. In particular, bounds and constructions for codes in polar spaces are established, where the bounds are sharp up to a constant factor in many cases. Moreover, based on these results, an almost complete classification of Steiner systems in polar spaces is provided by showing that they could only exist in some corner cases. AU - Weiß, Charlene ID - 50300 TI - Linear programming bounds in classical association schemes ER - TY - GEN AB - A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space over $\mathbb{F}_p$ equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-$(n,k,\lambda)$ design in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $k$-spaces such that each totally isotropic $t$-space is contained in exactly $\lambda$ members of $Y$. Nontrivial examples are currently only known for $t\leq 2$. We show that $t$-$(n,k,\lambda)$ designs in polar spaces exist for all $t$ and $p$ provided that $k>\frac{21}{2}t$ and $n$ is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive. AU - Weiß, Charlene ID - 50299 T2 - arXiv:2311.08288 TI - Nontrivial $t$-designs in polar spaces exist for all $t$ ER - TY - JOUR AU - Häsel-Weide, Uta AU - Schmidt, R. AU - Büker, Petra ID - 48596 IS - 1 JF - Zeitschrift für Schul- Und Professionsentwicklung. (PFLB) TI - „FInDig“: Fach – Inklusion – Digitalisierung vernetzen. Ein Planungs- und Reflexionsmodell für die Lehrkräftebildung VL - 5 ER - TY - CONF AU - Graf, Lara Marie AU - Häsel-Weide, Uta AU - Höveler, K. AU - Nührenbörger, M. ED - Drijvers, P. ED - Csapodi, C. ED - Palmér, H. ED - Gosztonyi, K. ED - Kónya, E. ID - 51131 T2 - Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13) TI - Insights into out-of-field teachers’ self-reports: Fostering the understanding of addition and subtraction as a basis for children to overcome difficulties in mathematics ER - TY - JOUR AB - Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of this result for the joint $L^2$-eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $\Gamma\backslash G/K$ of higher rank. We derive dynamical assumptions on the $\Gamma$-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups. AU - Weich, Tobias AU - Wolf, Lasse Lennart ID - 31189 JF - Communications in Mathematical Physics TI - Absence of principal eigenvalues for higher rank locally symmetric spaces VL - 403 ER - 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 - 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 -