Die Bedeutung des Mathematikschulbuchs in Zeiten der Ressourcenvielfalt - Stallmeister, Lea; Rezat, Sebastian (Beiträge zum Mathematikunterricht 2025. 58. Jahrestagung der Gesellschaft für Didaktik der Mathematik)

Learning support practices in the fostering of basic arithmetic skills - Neufeld, Inga; Häsel-Weide, Uta (Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14))

Practices in math discourses in inclusive primary school - Häsel-Weide, Uta; Nührenbörger, Marcus (Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14))

Frühe mathematische Bildung als Ausbildungsinhalt der Erzieherinnen- und Erzieher-Ausbildung stärken - Bruns, Julia; Gasteiger, Hedwig; Lastering, Bernd; Schopferer, Theresa; Zech, Detlev (25 Jahre Berufskolleg - Wegspuren und Zukunftspfade)

Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions - Bullerjahn, Nils; Kovács, Balázs (IMA Journal of Numerical Analysis)

Numerical analysis of an evolving bulk--surface model of tumour growth - Edelmann, Dominik; Kovács, Balázs; Lubich, Christian (IMA Journal of Numerical Analysis)

Maximal regularity of evolving FEMs for parabolic equations on an evolving surface - Bai, Genming; Kovács, Balázs; Li, Buyang (IMA Journal of Numerical Analysis)
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

Multiresolution analysis on spectra of hermitian matrices - Langen, Lukas; Rösler, Margit (Indagationes Mathematicae)
We establish a multiresolution analysis on the space $\text{Herm}(n)$ of $n× n$ complex Hermitian matrices which is adapted to invariance under conjugation by the unitary group $U(n).$ The orbits under this action are parametrized by the possible ordered spectra of Hermitian matrices, which constitute a closed Weyl chamber of type $A_{n-1}$ in $\mathbb R^n.$ The space $L^2(\text{Herm}(n))^{U(n)}$ of radial, i.e. $U(n)$-invariant $L^2$-functions on $\text{Herm}(n)$ is naturally identified with a certain weighted $L^2$-space on this chamber. The scale spaces of our multiresolution analysis are obtained by usual dyadic dilations as well as generalized translations of a scaling function, where the generalized translation is a hypergroup translation which respects the radial geometry. We provide a concise criterion to characterize orthonormal wavelet bases and show that such bases always exist. They provide natural orthonormal bases of the space $L^2(\text{Herm}(n))^{U(n)}.$ Furthermore, we show how to obtain radial scaling functions from classical scaling functions on $\mathbb R^{n}$. Finally, generalizations related to the Cartan decompositions for general compact Lie groups are indicated.

Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms - Janssens, Bas; Niestijl, Milan (Communications in Mathematical Physics)
Abstract Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations $$\overline{ρ }$$ ρ ¯ of the Lie group $${{\,Diff\,}}_c(M)$$ Diff c ( M ) of compactly supported diffeomorphisms of a smooth manifold M that satisfy a so-called generalized positive energy condition. In particular, this captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by $$\overline{ρ }$$ ρ ¯ . We show that if M is connected and $$\dim (M) > 1$$ dim ( M ) > 1 , then any such representation is necessarily trivial on the identity component $${{\,Diff\,}}_c(M)_0$$ Diff c ( M ) 0 . As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology $$H^2_ct(\mathcal {X}_c(M), \mathbb {R})$$ H ct 2 ( X c ( M ) , R ) of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand–Fuks cohomology in view of the compact support condition.

Nontrivial $t$-designs in polar spaces exist for all $t$ - Weiß, Charlene (Des. Codes Cryptogr.)
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 $t$-$(n,k,λ)$ 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 $λ$ members of $Y$. Nontrivial examples are currently only known for $t≤ 2$. We show that $t$-$(n,k,λ)$ 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.

Rough Data in an Evolution System Generalizing 1D Thermoviscoelasticity with Temperature-Dependent Parameters - Winkler, Michael (Applied Mathematics & Optimization)

Special issue of Journal of Lie Theory dedicated to Karl-Hermann Neeb on the occasion of his 60th birthday (J. Lie Theory)

Manifold of mappings and regularity properties of half-Lie groups - Pinaud, Matthieu

Adaptive higher order reversible integrators for memory efficient deep learning - Maslovskaya, Sofya; Ober-Blöbaum, Sina; Offen, Christian; Singh, Pranav; Wembe Moafo, Boris Edgar
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.

Interaction- or Material-Centred Implementation of Teacher Professional Development: a Comparison of Teachers’ Competence Development and Potential Lesson Planning Lehrkräftefortbildung interaktions- oder materialzentriert implementieren: Ein Vergleich hinsichtlich der Kompetenzentwicklung sowie der potentiellen Unterrichtsplanung von Lehrkräften - Richter, Alix; Bruns, Julia; Gasteiger, Hedwig (Journal für Mathematik-Didaktik)
Abstract Teacher professional development (TPD) is a crucial support mechanism for mathematics teachers. Strategies for implementing TPD include disseminating educative curriculum materials or conducting TPD courses. However, it remains unclear whether different implementations of the same TPD support mathematics teachers in distinct ways. This study investigated two implementations of the TPD EmMa-FS which focuses early mathematics education (EME) for German vocational school (VS) teachers who instruct prospective early childhood (EC) educators. One group of teachers received an in-person TPD course along with educative curriculum materials ( n PM = 26), whereas the other received only the educative curriculum materials ( n M = 15). The effects on VS teachers’ beliefs and knowledge concerning EME were examined using a t -test and repeated measures ANOVA. To assess how teachers in the different implementation groups made use of the TPD in their lesson planning, participants submitted hypothetical lesson sequences on early numeracy, which were analysed for instructional quality, covered content, and the visible implementation of the provided educative curriculum materials using qualitative content analysis. The results indicated that both TPD implementations positively affected VS teachers’ knowledge related to EME but not their beliefs. While the instructional quality of the lesson sequences varied across both groups, participants who received the TPD course appeared to use the educative curriculum materials more often and to cover more content.

Mathematik mit Kindern ausgehend von Spiel- und Routinesituationen in der Kindertagesstätte erkunden? - Bruns, Julia; Mette, Tessa (Frühe Bildung)
Zusammenfassung: Frühe mathematische Bildung, die ihren Ausgangspunkt in den (Spiel-)Situationen der Kindertagesstätte nimmt, kann breite mathematische Erfahrungen ermöglichen, wenn die (Spiel-)Situationen entsprechende Möglichkeiten bieten. Erste Studien weisen darauf hin, dass in diesen Situationen verschiedene mathematische Inhalte angesprochen werden können. Allerdings konzentrieren sich bisherige Studien auf ausgewählte Kontexte oder Inhaltsbereiche. Diese Studie zielt daher auf eine vertiefte Analyse der Gelegenheiten für mathematische Erfahrungen, die in verschiedenen (Spiel-)Situationen im Alltag der Kindertagesstätte entstehen können. Dazu wurde eine videobasierte Beobachtungsstudie mittels einer Action-Kamera in einer Kindertagesstätte durchgeführt. Die Ergebnisse zeigen, dass eine Vielfalt von mathematischen Inhalten in (Spiel-)Situationen in der Kindertagesstätte beobachtet werden kann, einzelne Inhalte jedoch nur in formellen Lernsituationen auftreten.

Inertial dynamics with vanishing Tikhonov regularization for multobjective optimization - Bot, Radu Ioan; Sonntag, Konstantin (Journal of Mathematical Analysis and Applications)
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.
Keywords: Pareto optimization; Lyapunov analysis; gradient-like dynamical systems; inertial dynamics; asymptotic vanishing damping; Tikhonov regularization; strong convergence

Zahlenmuster und Brüche: Eine Lernumgebung zum algebraischen Denken - Rezat, Sebastian; Doligkeit, Nadja (mathematik lehren)

Mathematik (Primarstufe) in der Lehrerinnen- und Lehrerbildung. Qualifizierung für das Lehren von Mathematik in der Grundschule - Häsel-Weide, Uta; Nührenbörger, Marcus (Handbuch Lehrerinnen- und Lehrerbildung)

First-order methods and gradient dynamical systems for multiobjective optimization - Sonntag, Konstantin
Diese Dissertation enthält Beiträge zum Bereich der Mehrzieloptimierung mit einem Fokus auf unbeschränkten Problemen, die auf einem allgemeinen Hilbertraum definiert sind. Für Mehrzieloptimierungsprobleme mit lokal Lipschitz-stetigen Zielfunktionen definieren wir ein multikriterielles Subdifferential, das wir erstmals im Kontext allgemeiner Hilberträume analysieren. Aufbauend auf diesen theoretischen Untersuchungen präsentieren wir ein Abstiegsverfahren, bei welchem in jeder Iteration eine Abstiegsrichtung mittels einer numerischen Approximation des multikriteriellen Subdifferentials bestimmt wird. Im Kontext konvexer, stetig differenzierbarer Zielfunktionen mit Lipschitz-stetigen Gradienten, führen wir eine Familie von dynamischen Gradientensystemen mit Trägheitsterm ein, die bekannte kontinuierliche Systeme aus der skalaren Optimierung verallgemeinern. Wir stellen drei neue Systeme vor: eines mit konstanter Dämpfung, eines mit asymptotisch abnehmender Dämpfung und eines, das zusätzlich eine zeitabhängige Tikhonov-Regularisierung beinhaltet. Aufbauend auf den Untersuchungen der neuen dynamischen Gradientensysteme, entwickeln wir ein beschleunigtes Gradientenverfahren zur Mehrzieloptimierung, das auf einer Diskretisierung des multikriteriellen Gradientensystems mit asymptotisch abnehmender Dämpfung beruht. Das hergeleitete Verfahren bewahrt die günstigen Konvergenzeigenschaften des kontinuierlichen Systems und erreicht eine schnellere Konvergenz als klassische Verfahren.