@article{34807, abstract = {{Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all real-analytic diffeomorphisms of $M$, which is modelled on the (DFS)-space ${\mathfrak g}$ of real-analytic vector fields on $M$. We study flows of time-dependent real-analytic vector fields on $M$ which are integrable functions in time, and their dependence on the time-dependent vector field. Notably, we show that the Lie group $G$ is $L^1$-regular in the sense that each $[\gamma]$ in $L^1([0,1],{\mathfrak g})$ has an evolution which is an absolutely continuous $G$-valued function on $[0,1]$ and smooth in $[\gamma]$. As tools for the proof, we develop several new results concerning $L^p$-regularity of infinite-dimensional Lie groups, for $1\leq p\leq \infty$, which will be useful also for the discussion of other classes of groups. Moreover, we obtain new results concerning the continuity and complex analyticity of non-linear mappings on open subsets of locally convex direct limits.}}, author = {{Glöckner, Helge}}, journal = {{Nonlinear Analysis}}, title = {{{Lie groups of real analytic diffeomorphisms are L^1-regular}}}, doi = {{10.1016/j.na.2024.113690}}, volume = {{252}}, 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}}, } @article{58947, author = {{Krüger, Katja and Werth, Gerda}}, issn = {{0732-3123}}, journal = {{The Journal of Mathematical Behavior}}, publisher = {{Elsevier BV}}, title = {{{Mathematics education for girls in Prussia 1890–1925}}}, doi = {{10.1016/j.jmathb.2025.101242}}, volume = {{79}}, 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{59053, author = {{Frischemeier, Daniel and Biehler, Rolf}}, journal = {{Stochastik in der Schule}}, number = {{1}}, pages = {{22--33}}, title = {{{Förderung von statistischem Denken im Mathematikunterricht der Primarstufe: Bedeutsame Ideen und Förderungsmöglichkeiten}}}, volume = {{45}}, 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{60205, author = {{Black, Tobias}}, issn = {{0022-0396}}, journal = {{Journal of Differential Equations}}, publisher = {{Elsevier BV}}, title = {{{Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier–Stokes system}}}, doi = {{10.1016/j.jde.2025.113555}}, volume = {{443}}, year = {{2025}}, } @unpublished{60293, abstract = {{In this work, we present a complete characterization of the covariance structure of number statistics in boxes for hyperuniform point processes. Under a standard integrability assumption, the covariance depends solely on the overlap of the faces of the box. Beyond this assumption, a novel interpolating covariance structure emerges. This enables us to identify a limiting Gaussian 'coarse-grained' process, counting the number of points in large boxes as a function of the box position. Depending on the integrability assumption, this process may be continuous or discontinuous, e.g. in d=1 it is given by an increment process of a fractional Brownian motion.}}, author = {{Jalowy, Jonas and Stange, Hanna}}, booktitle = {{arXiv:2506.13661}}, title = {{{Box-Covariances of Hyperuniform Point Processes}}}, year = {{2025}}, } @article{60351, abstract = {{This article is a short summary of the report of survey team 3, presented to the 15th International Congress on Mathematical Education (ICME-15) in Sydney in July 2024.}}, author = {{Biehler, Rolf and Kawakami, Takashi and Lampen, Erna and Weiland, Travis and Zapata-Cardona, Lucía}}, issn = {{2747-7894}}, journal = {{European Mathematical Society Magazine}}, publisher = {{European Mathematical Society - EMS - Publishing House GmbH}}, title = {{{Statistics and data science education as a vehicle for empowering citizens – short summary of a survey}}}, doi = {{10.4171/mag/257}}, 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}}, } @article{60495, author = {{Podworny, Susanne and Fleischer, Yannik and Biehler, Rolf}}, journal = {{Stochastik in der Schule}}, number = {{2}}, pages = {{9--16}}, title = {{{Explorative Datenanalyse in der Schule – Analyse der Mediennutzung von Jugendlichen mit den YOU‑PB Daten}}}, volume = {{45}}, year = {{2025}}, } @inbook{60532, author = {{Biehler, Rolf and Schulte, Carsten}}, booktitle = {{Proceedings of the 1st Symposium on Integrating AI and Data Science into School Education Across Disciplines (AIDEA 1 2025), Salzburg, Austria}}, title = {{{Lessons Learned from the ProDaBi Project: Shaping Perspectives at the Intersection of Data, AI, and Education Towards Fostering AI and Data Science Literacy in Schools Across Disciplines.}}}, year = {{2025}}, } @inbook{60816, author = {{Häsel-Weide, Uta and Hußmann, S. and Nührenbörger, M.}}, booktitle = {{Rechnen richtig lernen - flexibel, für alle und von Anfang an. Fachdidaktische Hintergründe und Lernangebote für die Grundschule}}, editor = {{Rechtsteiner, C.}}, pages = {{71--80}}, publisher = {{Klett Kallmeyer}}, title = {{{ Flexibles Addieren und Subtrahieren im Zahlenraum bis 100}}}, year = {{2025}}, } @inproceedings{60860, author = {{Fleischer, Franz Yannik and Biehler, Rolf}}, booktitle = {{Proceedings of the 48th Conference of the International Group for the Psychology of Mathematics Education: Research Reports, Vol. 1 }}, editor = {{Cornejo, C. and Felmer, P. and Gomez, D.M. and Dartnell, P. and Araya, P. and Peri, A. and Randolph, V.}}, pages = {{267--274}}, title = {{{ANALYZING STUDENTS’INFORMAL APPROACHES TO CREATING DECISION TREES IN THE CLASSROOM}}}, 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}}, } @article{61125, author = {{Biehler, Rolf and Liebendörfer, Michael and Schmitz, Angela and Reich, Birte}}, journal = {{Mitteilungen der Deutschen Mathematiker-Vereinigung}}, number = {{3}}, pages = {{170–171}}, publisher = {{De Gruyter}}, title = {{{studiVEMINT Mathematik-Online-Vorkurs jetzt mit 300 integrierten Lernvideos frei verfügbar}}}, volume = {{33}}, year = {{2025}}, } @article{54970, author = {{Quack, Carolin and Bruns, Julia}}, journal = {{Frühe Bildung}}, number = {{2}}, pages = {{72--78}}, title = {{{Mathematikbezogenes Wissen, Einstellungen und Lerngelegenheiten als Einflussfaktoren der situativen Beobachtung und Wahrnehmung angehender frühpädagogischer Fachkräfte}}}, doi = {{https://doi.org/10.1026/2191-9186/a000712}}, volume = {{14}}, year = {{2025}}, } @article{62291, abstract = {{In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.}}, title = {{{Zeros and exponential profiles of polynomials II: Examples}}}, doi = {{10.48550/ARXIV.2509.11248}}, year = {{2025}}, }