@unpublished{64816,
abstract = {{We study a block mean-field Ising model with $N$ spins split into $s_N$ blocks, with Curie-Weiss interaction within blocks and nearest-neighbor coupling between blocks. While previous models deal with the block magnetization for a fixed number of blocks, we study the the simultaneous limit $N\to\infty$ and $s_N\to\infty$. The model interpolates between Curie-Weiss model for $s_N=1$, multi-species mean field for fixed $s_N=s$, and the 1D Ising model for each spin in its own block at $s_N=N$.
Under mild growth conditions on $s_N$, we prove a law of large numbers and a multivariate CLT with covariance given by the lattice Green's function. For instance, the high temperature CLT essentially covers the optimal range up to $s_N=o(N/(\log N)^c)$ and the low temperature regime is new even for fixed number of blocks $s > 2$. In addition to the standard competition between entropy and energy, a new obstacle in the proofs is a curse of dimensionality as $s_N \to \infty$.}},
author = {{Jalowy, Jonas and Lammers, Isabel and Löwe, Matthias}},
booktitle = {{arXiv:2603.01994}},
title = {{{The infinite block spin Ising model}}},
year = {{2026}},
}
@unpublished{64865,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{63435,
author = {{Claes, Leander and Winkler, Michael}},
issn = {{1468-1218}},
journal = {{Nonlinear Analysis: Real World Applications}},
pages = {{104580}},
publisher = {{Elsevier BV}},
title = {{{Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by W 1,P energy analysis}}},
doi = {{10.1016/j.nonrwa.2025.104580}},
volume = {{91}},
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}},
}
@unpublished{64871,
author = {{Rahangdale, Praful}},
title = {{{Drinfeld correspondence in infinite dimensions}}},
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{64176,
author = {{Deutschen, Katrin and Neufeld, Inga and Häsel-Weide, Uta}},
journal = {{Grundschule aktuell}},
number = {{173}},
pages = {{10--11}},
title = {{{Lernbegleitung produktiv gestalten. Mathematische Verstehensprozesse von Kindern anregen.}}},
year = {{2026}},
}
@inbook{65026,
author = {{Neufeld, Inga and Häsel-Weide, Uta}},
booktitle = {{Handlungsorientierung in der Ausbildung von Lehrkräften und pädagogischen Fachkräften. Konzeptionen und Forschungsperspektiven}},
editor = {{Vogelsang, C. and Grotegurt, L. and Bruns, J. and Riese, J. and Fechner, S.}},
pages = {{119--134}},
title = {{{ Lernbegleitung in der mathematischen Förderung. Praktiken von (angehenden) Lehrkräften bei der Förderung arithmetischer Basiskompetenzen}}},
doi = {{10.31244/9783818851057 }},
year = {{2026}},
}
@unpublished{65036,
author = {{Cohen, Tal and Glöckner, Helge and Goffer, Gil and Lederle, Waltraud}},
title = {{{Compact invariant random subgroups}}},
year = {{2026}},
}
@article{57580,
abstract = {{We investigate dispersive and Strichartz estimates for the Schrödinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.}},
author = {{Palmirotta, Guendalina and Sire, Yannick and Anker, Jean-Philippe}},
journal = {{Journal of Differential Equations}},
keywords = {{Schrödinger equation, Fractional Laplacian, Dispersive estimates, Strichartz estimates, Real hyperbolic spaces, Homogeneous trees}},
publisher = {{Elsevier}},
title = {{{The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees}}},
doi = {{10.1016/j.jde.2025.114065}},
year = {{2026}},
}
@unpublished{65232,
abstract = {{On finite regular graphs, we construct Patterson-Sullivan distributions associated with eigenfunctions of the discrete Laplace operator via their boundary values on the phase space. These distributions are closely related to Wigner distributions defined via a pseudo-differential calculus on graphs, which appear naturally in the study of quantum chaos. Using a pairing formula, we prove that Patterson-Sullivan distributions are also related to invariant Ruelle distributions arising from the transfer operator of the geodesic flow on the shift space. Both relationships provide discrete analogues of results for compact hyperbolic surfaces obtained by Anantharaman-Zelditch and by Guillarmou-Hilgert-Weich.}},
author = {{Arends, Christian and Palmirotta, Guendalina}},
booktitle = {{arXiv:2603.09779}},
pages = {{38}},
title = {{{Patterson-Sullivan distributions of finite regular graphs}}},
year = {{2026}},
}
@proceedings{64797,
editor = {{Birk, Lisa and Loth, Gerrit and Jotzo, Luca and Binder, Karin and Frischemeier, Daniel}},
location = {{Münster}},
publisher = {{International Association for Statistics Education}},
title = {{{14th IASE Satellite Conference "Statistics and Data Science Education in STEAM"}}},
doi = {{10.52041/iase25.158}},
year = {{2026}},
}
@article{63135,
abstract = {{We propose a definition of Coxeter-Dynkin algebras of canonical type generalising the definition as a path algebra of a quiver. Moreover, we construct two tilting objects over the squid algebra - one via generalised APR-tilting and one via one-point-extensions and reflection functors - and identify their endomorphism algebras with the Coxeter-Dynkin algebra. This shows that our definition gives another representative in the derived equivalence class of the squid algebra, and hence of the corresponding canonical algebra. Finally, we have a closer look at the Grothendieck group and the Euler form which illustrates the connection to Saito's classification of marked extended affine root systems. On the other hand, this enables us to prove that in the domestic case Coxeter-Dynkin algebras are of finite representation type.}},
author = {{Perniok, Daniel}},
journal = {{Journal of Pure and Applied Algebra}},
number = {{5}},
title = {{{Coxeter-Dynkin algebras of canonical type}}},
doi = {{10.1016/j.jpaa.2026.108250}},
volume = {{230}},
year = {{2026}},
}
@article{65512,
abstract = {{Zusammenfassung
Risikokompetenz beinhaltet auch die Fähigkeit, stochastische Informationen, wie beispielsweise Anteile und Wahrscheinlichkeiten, richtig zu versprachlichen. Aus der Forschung zu bedingten Wahrscheinlichkeiten und Bayesianischen Aufgaben ist bekannt, dass die Nutzung von Visualisierungen und sogenannten „natürlichen Häufigkeiten“ (z. B. „80 von 100 Personen“) statt Wahrscheinlichkeiten in Prozent Verwechslungen beim Bestimmen von Wahrscheinlichkeiten eindämmen kann. Über den umgekehrten Prozess – das Versprachlichen von in Visualisierungen dargestellten Informationen – ist bisher jedoch wenig bekannt, obwohl diese Versprachlichungen auch für den Aufbau konzeptuellen Wissen als essentiell angesehen werden. In der vorliegenden Studie wurde daher untersucht, wie gut Schüler:innen die Versprachlichung von in Visualisierungen dargestellten Anteilen und natürlichen Häufigkeiten gelingt. Dazu wurde mit 138 Realschüler:innen aus der 9. Jahrgangsstufe ein Papier-und-Bleistift-Test durchgeführt, bei dem den Teilnehmenden nacheinander je ein (vollständig ausgefülltes) Baumdiagramm und ein Netzdiagramm präsentiert wurde. Die Schüler:innen sollten die inhaltliche Bedeutung der in der Visualisierung dargestellten stochastischen Informationen möglichst genau versprachlichen. Fokus der Studie ist der Einfluss der Visualisierung (Baumdiagramm vs. Netzdiagramm) und des Informationsformats der Visualisierung (Anteile in Prozent vs. natürliche Häufigkeiten) auf die richtige Versprachlichung von verschiedenen Relationstypen (Schnittinformationen vs. bedingte Informationen). Die Ergebnisse zeigen unter anderem, dass Informationen in natürlichen Häufigkeiten deutlich besser versprachlicht werden als in Prozenten und dass Schnittinformationen in Prozent besser anhand von Netzdiagrammen als von Baumdiagrammen versprachlicht werden. Die gewonnenen Erkenntnisse geben überdies Einblicke in typische (fehlerhafte) Versprachlichungen von Schüler:innen und könnten als Basis für die Entwicklung eines sprachsensiblen Unterrichts zu Anteilen und Wahrscheinlichkeiten im schulischen Stochastikunterricht dienen.}},
author = {{Rößner, Michael and Binder, Karin and Albrecht, Julian}},
issn = {{0173-5322}},
journal = {{Journal für Mathematik-Didaktik}},
number = {{1}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Versprachlichung von Anteilen und natürlichen Häufigkeiten anhand von Baum- und Netzdiagrammen Verbalization of proportions and natural frequencies based on tree diagrams and net diagrams}}},
doi = {{10.1007/s13138-026-00267-1}},
volume = {{47}},
year = {{2026}},
}
@article{61759,
abstract = {{Intersection distribution and non-hitting index are concepts introduced recently by Li and Pott as a new way to view the behaviour of a collection of finite field polynomials. With both an algebraic interpretation via the intersection of a polynomial with a set of lines, and a geometric interpretation via a (q+1)-set possessing an internal nucleus, the concepts have proved their usefulness as a new way to view various long-standing problems, and have applications in areas such as Kakeya sets. In this paper, by exploiting connections with diverse areas including the theory of algebraic curves, cyclotomy and the enumeration of irreducible polynomials, we establish new results and resolve various Open Problems of Li and Pott. We prove geometric results which shed new light on the relationship between intersection distribution and projective equivalence of polynomials, and algebraic results which describe and characterise the degree of Sf - the index of the largest non-zero entry in the intersection distribution of f. We provide new insights into the non-hitting spectrum, and show the limitations of the non-hitting index as a tool for characterisation. Finally, the benefits provided by the connections to other areas are evidenced in two short new proofs of the cubic case. }},
author = {{Klawuhn, Lukas-André Dominik and Huczynska, Sophie and Paterson, Maura}},
journal = {{Finite Fields and Their Applications}},
publisher = {{Elsevier}},
title = {{{The Intersection Distribution: New Results and Perspectives}}},
doi = {{10.1016/j.ffa.2026.102828}},
volume = {{114}},
year = {{2026}},
}
@article{65631,
abstract = {{Abstract
Mathematics textbooks used to be the key resource for students’ self-regulated learning of mathematics. Primarily due to the digitalization of society, students have potentially greater access to a wider range of resources such as internet search engines, learning platforms, educational videos, and Generative AI. This study investigates the role of the mathematics textbook in comparison to other resources within students’ self-regulated learning practices. Data were collected via a survey of 1101 German secondary students, representing three school types (
Gymnasium
,
Gesamtschule
,
Realschule
) and three grade levels (6, 9, and upper secondary). The questionnaire assessed the frequency of resource use in and outside class, reasons and purposes of use, and resource-based strategies when facing learning challenges outside class. Results show that the printed mathematics textbook is the most frequently used resource both in class and outside class. The textbook remains the most relevant resource for key purposes, such as an aid for doing homework and preparing for tests and exams. However, its dominance diminishes with age: in upper secondary school, students increasingly rely on self-created notes, and online resources. Correlation analyses reveal moderate to strong links between in-class and out-of-class use, suggesting an association between resource use and classroom culture. The findings underscore the textbook’s enduring centrality as a foundational, trusted resource within a dynamic and increasingly diverse learning environment. This study calls for pedagogical approaches that integrate textbooks more intentionally within broader resource systems, supporting students’ agency and strategic resource selection in an era of digital abundance.
}},
author = {{Stallmeister, Lea and Rezat, Sebastian}},
issn = {{0013-1954}},
journal = {{Educational Studies in Mathematics}},
keywords = {{mathematics, textbooks, userstudy, resources, digital resources, students, secondary eduction}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{The role of the mathematics textbook in times of resource diversity}}},
doi = {{10.1007/s10649-026-10511-7}},
year = {{2026}},
}
@article{65630,
abstract = {{Mathematics textbooks used to be the key resource for students’ self-regulated learning of mathematics. Primarily due to the digitalization of society, students have potentially greater access to a wider range of resources such as internet search engines, learning platforms, educational videos, and Generative AI. This study investigates the role of the mathematics textbook in comparison to other resources within students’ self-regulated learning practices. Data were collected via a survey of 1101 German secondary students, representing three school types (Gymnasium, Gesamtschule, Realschule) and three grade levels (6, 9, and upper secondary). The questionnaire assessed the frequency of resource use in and outside class, reasons and purposes of use, and resource-based strategies when facing learning challenges outside class. Results show that the printed mathematics textbook is the most frequently used resource both in class and outside class. The textbook remains the most relevant resource for key purposes, such as an aid for doing homework and preparing for tests and exams. However, its dominance diminishes with age: in upper secondary school, students increasingly rely on self-created notes, and online resources. Correlation analyses reveal moderate to strong links between in-class and out-of-class use, suggesting an association between resource use and classroom culture. The findings underscore the textbook’s enduring centrality as a foundational, trusted resource within a dynamic and increasingly diverse learning environment. This study calls for pedagogical approaches that integrate textbooks more intentionally within broader resource systems, supporting students’ agency and strategic resource selection in an era of digital abundance.}},
author = {{Stallmeister, Lea and Rezat, Sebastian}},
journal = {{Educational Studies in Mathematics}},
publisher = {{Springer}},
title = {{{The role of the mathematics textbook in times of resource diversity}}},
doi = {{10.1007/s10649-026-10511-7}},
year = {{2026}},
}
@article{65645,
abstract = {{Curriculum material is often designed to address content-specific intended learning goals. However, research indicates that teachers’ personal goals may cause misalignments between their implementation of curriculum material and its intended learning goals. This study aims to investigate misalignments between teachers’ implementation of curriculum material and its intended learning goals and to which extent they are caused by teachers’ personal goals. To reach this aim, a qualitative study was conducted to examine how eight vocational school teachers implemented the learning activity “How many?” suggested for the initial training of early childhood educators on the topic of set perception and determination of cardinality. Each participant provided a lesson plan and self-recorded lesson video on the topic of set perception and determination of cardinality. In addition, participants were interviewed on how they used the provided curriculum material for designing their lessons and their potential reasons to adapt or omit “How many?”. To evaluate their lesson design’s alignment with the activity’s intended learning goals, the participants' lesson plans were analyzed using qualitative content analysis. Triangulation of the findings with the interview data revealed that only one participant fully adopted the intended learning goals as her own and implemented the activity completely in line with its intended learning goals. Meanwhile, the remaining participants’ personal goals, such as reducing math anxiety, seemed to cause misalignments regarding the intended learning goals. The study's results detail further constraints and affordances to the alignment between teachers’ lesson designs and learning goals intended by curriculum material. It is followed that it is central to support teachers in adopting intended learning goals as their personal goals. Otherwise, teachers’ classroom implementation of curriculum material may not suffice to reach desired outcomes.}},
author = {{Richter, Alix and Bruns, Julia}},
issn = {{1306-3030}},
journal = {{International Electronic Journal of Mathematics Education}},
number = {{2}},
publisher = {{Modestum Ltd}},
title = {{{All a question of the goal? Re-tracing (mis)alignments between teachers’ implementations of curriculum material and intended learning goals in the context of early childhood educator training in Germany}}},
doi = {{10.29333/iejme/18564}},
volume = {{21}},
year = {{2026}},
}
@inproceedings{64211,
author = {{Wiebe, Vivien and Häsel-Weide, Uta}},
booktitle = {{Proceedings of the Nineteenth ERME Topic Conference: Connecting the Learning of Mathematics Teaching to Practice}},
editor = {{Mosvold, R. and Fauskanger, J. and Ferretti, F. and Vondrová, N.}},
location = {{Prag}},
pages = {{122--129}},
title = {{{ Initiating and establishing mathematical practices of determining and transforming numbers as a foundational skill in fostering mathematics teaching}}},
year = {{2026}},
}
@article{65745,
abstract = {{Abstract
In this work, we address the numerical identification of entanglement in dynamical scenarios. To this end, we consider different programs based on the restriction of the evolution to the set of separable (i.e., non-entangled) states, together with the discretization of the space of variables for numerical computations. As a first approach, we apply linear splitting methods to the restricted, continuous equations of motion derived from variational principles. We utilize an exchange interaction Hamiltonian to confirm that the numerical and analytical solutions coincide in the limit of small time steps. The application to different Hamiltonians shows the wide applicability of the method to detect dynamical entanglement. To avoid the derivation of analytical solutions for complex dynamics, we consider variational, numerical integration schemes, introducing a variational discretization for Lagrangians linear in velocities. Here, we examine and compare two approaches: one in which the system is discretized before the restriction is applied, and another in which the restriction precedes the discretization. We find that the "first-discretize-then-restrict" method becomes numerically unstable, already for the example of an exchange-interaction Hamiltonian, which can be an important consideration for the numerical analysis of constrained quantum dynamics. Thereby, broadly applicable numerical tools, including their limitations, for studying entanglement over time are established for assessing the entangling power of processes that are used in quantum information theory.}},
author = {{Offen, Christian and Wembe, Boris and Ares, Laura and Sperling, Jan and Ober-Blöbaum, Sina}},
issn = {{1751-8113}},
journal = {{Journal of Physics A: Mathematical and Theoretical}},
publisher = {{IOP Publishing}},
title = {{{Numerical approaches to entangling dynamics from variational principles}}},
doi = {{10.1088/1751-8121/ae6d51}},
year = {{2026}},
}