@unpublished{64865, abstract = {{We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ (the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbfΓ$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbfΓ$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $δ$-neighborhood of the first set contains an initial condition with $ω$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.}}, author = {{von der Gracht, Sören and Lohse, Alexander}}, booktitle = {{arXiv:2603.06157}}, title = {{{Design of Hierarchical Excitable Networks}}}, year = {{2026}}, } @article{63557, abstract = {{We discretise a recently proposed new Lagrangian approach to optimal control problems with dynamics described by force-controlled Euler-Lagrange equations (Konopik et al., in Nonlinearity 38:11, 2025). The resulting discretisations are in the form of discrete Lagrangians. We show that the discrete necessary conditions for optimality obtained provide variational integrators for the continuous problem, akin to Karush-Kuhn-Tucker (KKT) conditions for standard direct approaches. This approach paves the way for the use of variational error analysis to derive the order of convergence of the resulting numerical schemes for both state and costate variables and to apply discrete Noether’s theorem to compute conserved quantities, distinguishing itself from existing geometric approaches. We show for a family of low-order discretisations that the resulting numerical schemes are ‘doubly-symplectic’, meaning they yield forced symplectic integrators for the underlying controlled mechanical system and overall symplectic integrators in the state-adjoint space. Multi-body dynamics examples are solved numerically using the new approach. In addition, the new approach is compared to standard direct approaches in terms of computational performance and error convergence. The results highlight the advantages of the new approach, namely, better performance and convergence behaviour of state and costate variables consistent with variational error analysis and automatic preservation of certain first integrals.}}, author = {{Konopik, Michael and Leyendecker, Sigrid and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Sato Martín de Almagro, Rodrigo T.}}, issn = {{1384-5640}}, journal = {{Multibody System Dynamics}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{On the variational discretisation of optimal control problems for unconstrained Lagrangian dynamics}}}, doi = {{10.1007/s11044-025-10138-1}}, year = {{2026}}, } @article{64979, abstract = {{We investigate homogeneous coupled cell systems with high-dimensional internal dynamics. In many studies on network dynamics, the analysis is restricted to networks with one-dimensional internal dynamics. Here, we show how symmetry explains the relation between dynamical behavior of systems with one-dimensional internal dynamics and with higher dimensional internal dynamics, when the underlying network topology is the same. Fundamental networks of homogeneous coupled cell systems (B. Rink, J. Sanders. Coupled Cell Networks and Their Hidden Symmetries. SIAM J. Math. Anal. 46.2 (2014)) can be expressed in terms of monoid representations, which uniquely decompose into indecomposable subrepresentations. In the high-dimensional internal dynamics case, these subrepresentations are isomorphic to multiple copies of those one computes in the one-dimensional internal dynamics case. This has interesting implications for possible center subspaces in bifurcation analysis. We describe the effect on steady state and Hopf bifurcations in l-parameter families of network vector fields. The main results in that regard are that (1) generic one-parameter steady state bifurcations are qualitatively independent of the dimension of the internal dynamics and that, (2) in order to observe all generic l-parameter bifurcations that may occur for internal dynamics of any dimension, the internal dynamics has to be at least l-dimensional for steady state bifurcations and 2l-dimensional for Hopf bifurcations. Furthermore, we illustrate how additional structure in the network can be exploited to obtain even greater understanding of bifurcation scenarios in the high-dimensional case beyond qualitative statements about the collective dynamics. One-parameter steady state bifurcations in feedforward networks exhibit an unusual amplification in the asymptotic growth rates of individual cells, when these are one-dimensional (S. von der Gracht, E. Nijholt, B. Rink. Amplified steady state bifurcations in feedforward networks. Nonlinearity 35.4 (2022)). As another main result, we prove that (3) the same cells exhibit this amplifying effect with the same growth rates when the internal dynamics is high-dimensional.}}, author = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}}, issn = {{0960-0779}}, journal = {{Chaos, Solitons & Fractals}}, keywords = {{Coupled cell systems, Network dynamics, Dimension reduction, Bifurcation theory, Symmetry, Monoid representation theory}}, publisher = {{Elsevier BV}}, title = {{{Homogeneous coupled cell systems with high-dimensional internal dynamics}}}, doi = {{10.1016/j.chaos.2026.118196}}, volume = {{208}}, year = {{2026}}, } @article{59792, abstract = {{Abstract Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing certain symmetries. Following recent works (Faulwasser in Math Control Signals Syst 34:759–788 2022; Trélat in Math Control Signals Syst 35:685–739 2023), which generalized the classical concept of static turnpike to manifold turnpike we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the rigid body problem. }}, author = {{Flaßkamp, Kathrin and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Wembe Moafo, Boris Edgar}}, issn = {{0932-4194}}, journal = {{Mathematics of Control, Signals, and Systems}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Trim turnpikes for optimal control problems with symmetries}}}, doi = {{10.1007/s00498-025-00408-w}}, year = {{2025}}, } @article{59806, abstract = {{We introduce a model of information dissemination in signed networks. It is a discrete-time process in which uninformed actors incrementally receive information from their informed neighbors or from the outside. Our goal is to minimize the number of confused actors — that is, the number of actors who receive contradictory information. We prove upper bounds for the number of confused actors in signed networks and in equivalence classes of signed networks. In particular, we show that there are signed networks where, for any information placement strategy, almost 60% of the actors are confused. Furthermore, this is also the case when considering the minimum number of confused actors within an equivalence class of signed graphs.}}, author = {{Jin, Ligang and Steffen, Eckhard}}, issn = {{0166-218X}}, journal = {{Discrete Applied Mathematics}}, pages = {{99--106}}, publisher = {{Elsevier BV}}, title = {{{Information dissemination and confusion in signed networks}}}, doi = {{10.1016/j.dam.2025.04.049}}, volume = {{373}}, year = {{2025}}, } @inbook{60048, author = {{Gerlach, Raphael and von der Gracht, Sören and Dellnitz, Michael}}, booktitle = {{Lecture Notes in Computer Science}}, isbn = {{9783031917356}}, issn = {{0302-9743}}, publisher = {{Springer Nature Switzerland}}, title = {{{On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies}}}, doi = {{10.1007/978-3-031-91736-3_19}}, year = {{2025}}, } @article{55459, author = {{Bullerjahn, Nils and Kovács, Balázs}}, journal = {{IMA Journal of Numerical Analysis}}, title = {{{Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions}}}, doi = {{10.1093/imanum/draf009}}, year = {{2025}}, } @article{53141, author = {{Edelmann, Dominik and Kovács, Balázs and Lubich, Christian}}, journal = {{IMA Journal of Numerical Analysis}}, number = {{5}}, pages = {{2581----2627}}, title = {{{Numerical analysis of an evolving bulk--surface model of tumour growth}}}, doi = {{10.1093/imanum/drae077}}, volume = {{45}}, year = {{2025}}, } @article{55781, abstract = {{In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods}}, author = {{Bai, Genming and Kovács, Balázs and Li, Buyang}}, journal = {{IMA Journal of Numerical Analysis}}, title = {{{Maximal regularity of evolving FEMs for parabolic equations on an evolving surface}}}, doi = {{10.1093/imanum/draf082.}}, year = {{2025}}, } @article{50299, abstract = {{A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space over $\mathbb{F}_q$ 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 $q$ 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.}}, author = {{Weiß, Charlene}}, journal = {{Des. Codes Cryptogr.}}, pages = {{971 -- 981}}, title = {{{Nontrivial $t$-designs in polar spaces exist for all $t$}}}, doi = {{10.1007/s10623-024-01471-1}}, volume = {{93}}, year = {{2025}}, } @inproceedings{56298, abstract = {{In the general pattern formation (GPF) problem, a swarm of simple autonomous, disoriented robots must form a given pattern. The robots' simplicity imply a strong limitation: When the initial configuration is rotationally symmetric, only patterns with a similar symmetry can be formed [Yamashita, Suzyuki; TCS 2010]. The only known algorithm to form large patterns with limited visibility and without memory requires the robots to start in a near-gathering (a swarm of constant diameter) [Hahn et al.; SAND 2024]. However, not only do we not know any near-gathering algorithm guaranteed to preserve symmetry but most natural gathering strategies trivially increase symmetries [Castenow et al.; OPODIS 2022]. Thus, we study near-gathering without changing the swarm's rotational symmetry for disoriented, oblivious robots with limited visibility (the OBLOT-model, see [Flocchini et al.; 2019]). We introduce a technique based on the theory of dynamical systems to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation. Until now, it was unknown whether the considered OBLOT-model allows for any non-trivial algorithm that always preserves symmetry. Our first result shows that a variant of Go-to-the-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters. Our second result is a symmetry-preserving near-gathering algorithm that works on swarms with a convex boundary (the outer boundary of the unit disc graph) and without holes (circles of diameter 1 inside the boundary without any robots).}}, author = {{Gerlach, Raphael and von der Gracht, Sören and Hahn, Christopher and Harbig, Jonas and Kling, Peter}}, booktitle = {{28th International Conference on Principles of Distributed Systems (OPODIS 2024)}}, editor = {{Bonomi, Silvia and Galletta, Letterio and Rivière, Etienne and Schiavoni, Valerio}}, isbn = {{978-3-95977-360-7}}, issn = {{1868-8969}}, keywords = {{Swarm Algorithm, Swarm Robots, Distributed Algorithm, Pattern Formation, Limited Visibility, Oblivious}}, location = {{Lucca, Italy}}, publisher = {{Schloss Dagstuhl -- Leibniz-Zentrum für Informatik}}, title = {{{Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility}}}, doi = {{10.4230/LIPIcs.OPODIS.2024.13}}, volume = {{324}}, year = {{2025}}, } @article{58532, author = {{Bullerjahn, Nils}}, journal = {{arXiv}}, title = {{{Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions}}}, doi = {{10.48550/ARXIV.2502.03847}}, year = {{2025}}, } @unpublished{58544, abstract = {{We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which being in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes. }}, author = {{Kopylov, Denis and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina and Sperling, Jan}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits }}}, year = {{2025}}, } @unpublished{58953, abstract = {{In this article, we investigate symmetry properties of distributed systems of mobile robots. We consider a swarm of n robots in the OBLOT model and analyze their collective Fsync dynamics using of equivariant dynamical systems theory. To this end, we show that the corresponding evolution function commutes with rotational and reflective transformations of R^2. These form a group that is isomorphic to O(2) x S_n, the product group of the orthogonal group and the permutation on n elements. The theory of equivariant dynamical systems is used to deduce a hierarchy along which symmetries of a robot swarm can potentially increase following an arbitrary protocol. By decoupling the Look phase from the Compute and Move phases in the mathematical description of an LCM cycle, this hierarchy can be characterized in terms of automorphisms of connectivity graphs. In particular, we find all possible types of symmetry increase, if the decoupled Compute and Move phase is invertible. Finally, we apply our results to protocols which induce state-dependent linear dynamics, where the reduced system consisting of only the Compute and Move phase is linear.}}, author = {{Gerlach, Raphael and von der Gracht, Sören}}, booktitle = {{arXiv:2503.07576}}, keywords = {{dynamical systems, coupled systems, distributed computing, robot swarms, autonomous mobile robots, symmetry, equivariant dynamics}}, pages = {{23}}, title = {{{Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant Dynamical Systems}}}, year = {{2025}}, } @article{59169, abstract = {{An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping such that each r adjacent edges of G are mapped to r adjacent edges of H. For every , let be an inclusion-wise minimal set of connected r-graphs, such that for every connected r-graph G there is an which colors G. The Petersen Coloring Conjecture states that consists of the Petersen graph P. We show that if true, then this is a very exclusive situation. Our main result is that either or is an infinite set and if , then is an infinite set. In particular, for all , is unique. We first characterize and then prove that if contains more than one element, then it is an infinite set. To obtain our main result we show that contains the smallest r-graphs of class 2 and the smallest poorly matchable r-graphs, and we determine the smallest r-graphs of class 2.}}, author = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak H.}}, issn = {{0209-9683}}, journal = {{Combinatorica}}, number = {{2}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Sets of r-Graphs that Color All r-Graphs}}}, doi = {{10.1007/s00493-025-00144-4}}, volume = {{45}}, year = {{2025}}, } @article{53805, abstract = {{The article introduces a method to learn dynamical systems that are governed by Euler–Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.}}, author = {{Offen, Christian}}, journal = {{Mathematics of Computation}}, publisher = {{American Mathematical Society}}, title = {{{Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification}}}, doi = {{10.1090/mcom/4120}}, year = {{2025}}, } @unpublished{60491, abstract = {{We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph $K_{2n}$. We show that they are special subsets of the association scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra $\mathbb{C}[S_{2n}]$ and using the representation theory of $S_{2n}$.}}, author = {{Klawuhn, Lukas-André Dominik and Bamberg, John}}, title = {{{On the association scheme of perfect matchings and their designs}}}, year = {{2025}}, } @article{61042, abstract = {{We introduce the concept of a k-token signed graph and study some of its combinatorial and algebraic properties. We prove that two switching isomorphic signed graphs have switching isomorphic token graphs. Moreover, we show that the Laplacian spectrum of a balanced signed graph is contained in the Laplacian spectra of its k-token signed graph. Besides, we introduce and study the unbalance level of a signed graph, which is a new parameter that measures how far a signed graph is from being balanced. Moreover, we study the relation between the frustration index and the unbalance level of signed graphs and their token signed graphs.}}, author = {{Dalfó, C. and Fiol, M. A. and Steffen, Eckhard}}, issn = {{0925-9899}}, journal = {{Journal of Algebraic Combinatorics}}, number = {{1}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{On token signed graphs}}}, doi = {{10.1007/s10801-025-01416-4}}, volume = {{62}}, year = {{2025}}, } @unpublished{59794, abstract = {{The depth of networks plays a crucial role in the effectiveness of deep learning. However, the memory requirement for backpropagation scales linearly with the number of layers, which leads to memory bottlenecks during training. Moreover, deep networks are often unable to handle time-series data appearing at irregular intervals. These issues can be resolved by considering continuous-depth networks based on the neural ODE framework in combination with reversible integration methods that allow for variable time-steps. Reversibility of the method ensures that the memory requirement for training is independent of network depth, while variable time-steps are required for assimilating time-series data on irregular intervals. However, at present, there are no known higher-order reversible methods with this property. High-order methods are especially important when a high level of accuracy in learning is required or when small time-steps are necessary due to large errors in time integration of neural ODEs, for instance in context of complex dynamical systems such as Kepler systems and molecular dynamics. The requirement of small time-steps when using a low-order method can significantly increase the computational cost of training as well as inference. In this work, we present an approach for constructing high-order reversible methods that allow adaptive time-stepping. Our numerical tests show the advantages in computational speed when applied to the task of learning dynamical systems.}}, author = {{Maslovskaya, Sofya and Ober-Blöbaum, Sina and Offen, Christian and Singh, Pranav and Wembe Moafo, Boris Edgar}}, title = {{{Adaptive higher order reversible integrators for memory efficient deep learning}}}, year = {{2025}}, } @article{57472, abstract = {{In this paper we introduce, in a Hilbert space setting, a second order dynamical system with asymptotically vanishing damping and vanishing Tikhonov regularization that approaches a multiobjective optimization problem with convex and differentiable components of the objective function. Trajectory solutions are shown to exist in finite dimensions. We prove fast convergence of the function values, quantified in terms of a merit function. Based on the regime considered, we establish both weak and, in some cases, strong convergence of trajectory solutions toward a weak Pareto optimal solution. To achieve this, we apply Tikhonov regularization individually to each component of the objective function. This work extends results from single objective convex optimization into the multiobjective setting.}}, author = {{Bot, Radu Ioan and Sonntag, Konstantin}}, journal = {{Journal of Mathematical Analysis and Applications}}, keywords = {{Pareto optimization, Lyapunov analysis, gradient-like dynamical systems, inertial dynamics, asymptotic vanishing damping, Tikhonov regularization, strong convergence}}, title = {{{Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization}}}, year = {{2025}}, }