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 -