@article{62980, 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 is 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 A. and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina R. and Sperling, Jan}}, issn = {{2643-1564}}, journal = {{Physical Review Research}}, number = {{3}}, publisher = {{American Physical Society (APS)}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits}}}, doi = {{10.1103/sv6z-v1gk}}, volume = {{7}}, year = {{2025}}, } @inproceedings{63036, author = {{Rezat, Sebastian and Glasnović Gracin, Dubravka and Van Steenbrugge, Hendrik and Sievert, Henning}}, booktitle = {{Proceedings of the Fifth International Conference on Mathematics Textbook Research and Development.}}, editor = {{Pepin, Birgit and Kohanová, Iveta and Langfeldt, Marit Buset}}, isbn = {{978-82-691902-2-9}}, location = {{Trondheim, Norway}}, pages = {{72–91}}, publisher = {{Norwegian University of Science and Technology.}}, title = {{{The quality of print and digital mathematics curriculum resources}}}, year = {{2025}}, } @inproceedings{63034, author = {{Stallmeister, Lea and Rezat, Sebastian}}, booktitle = {{Proceedings of the Fifth International Conference on Mathematics Textbook Research and Development}}, editor = {{Pepin, Birgit and Kohanová, Iveta and Langfeldt, Marit Buset}}, isbn = {{978-82-691902-2-9}}, location = {{Trondheim, Norway}}, publisher = {{Norwegian University of Science and Technology}}, title = {{{Students’ use of different material resources for specific purposes in the process of learning mathematics}}}, year = {{2025}}, } @article{60196, abstract = {{This paper examines the governance and quality control of digital curriculum resources (DCR) for K-12 mathematics education in Germany. It focuses on approval processes and criteria set by the 16 federal states, arguing that these have the potential to influence the development of DCR. Using qualitative content analysis, the study explores three research questions: which DCR require official approval, the criteria applied for approval, and the extent to which these criteria are mathematics-specific. Findings indicate that 10 federal states maintain official approval systems, covering digital equivalents of printed textbooks and selected supplemental materials. However, most DCR fall outside these regulated processes, leaving their evaluation largely to individual schools and teachers. The study identifies 17 categories of quality criteria, but reveals a lack of detailed, mathematics-specific requirements. Instead, many criteria are broad references to didactical principles and educational goals, leaving the interpretation and application of these quality standards open-ended. Subject-specific criteria are included but remain limited in specificity. The study underscores the need for research-informed, mathematics-specific quality standards to guide DCR development and approval, emphasizing their importance amidst challenges like artificial intelligence. Policymakers are urged to adopt clearer criteria to ensure high-quality DCR to be used in schools.}}, author = {{Rezat, Sebastian}}, issn = {{1863-9690}}, journal = {{ZDM – Mathematics Education}}, keywords = {{governance, digital curriculum resources, digital textbooks, digital curriculum materials, quality}}, pages = {{ 891–904}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{The quality of digital curriculum resources for mathematics in German educational policy}}}, doi = {{10.1007/s11858-025-01708-w}}, volume = {{57}}, year = {{2025}}, } @unpublished{63150, author = {{Cummings, Charley and Gratz, Sira and Kirkman, Ellen and Letz, Janina Carmen and Rock, J. Daisie and Špenko, Špela}}, pages = {{1--9}}, title = {{{An equivalence linking CM-types $A_\infty$ and $D_\infty$}}}, year = {{2025}}, } @unpublished{63149, author = {{Kekkou, Antonia and Letz, Janina Carmen and Stephan, Marc}}, pages = {{1--37}}, title = {{{Regular sequences for triangulated categories}}}, year = {{2025}}, } @unpublished{63148, author = {{Letz, Janina Carmen}}, pages = {{1--7}}, title = {{{The Rouquier dimension of the category of perfect complexes over a regular ring}}}, year = {{2025}}, } @article{59507, abstract = {{Differential equations posed on quadratic matrix Lie groups arise in the context of classical mechanics and quantum dynamical systems. Lie group numerical integrators preserve the constants of motions defining the Lie group. Thus, they respect important physical laws of the dynamical system, such as unitarity and energy conservation in the context of quantum dynamical systems, for instance. In this article we develop a high-order commutator free Lie group integrator for non-autonomous differential equations evolving on quadratic Lie groups. Instead of matrix exponentials, which are expensive to evaluate and need to be approximated by appropriate rational functions in order to preserve the Lie group structure, the proposed method is obtained as a composition of Cayley transforms which naturally respect the structure of quadratic Lie groups while being computationally efficient to evaluate. Unlike Cayley-Magnus methods the method is also free from nested matrix commutators.}}, author = {{Wembe Moafo, Boris Edgar and Offen, Cristian and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Singh, Pranav}}, journal = {{J. Comput. Appl. Math}}, number = {{15}}, title = {{{Commutator-free Cayley methods}}}, doi = {{10.1016/j.cam.2025.117184}}, volume = {{477}}, year = {{2025}}, } @unpublished{63187, author = {{Kidner, Arnott Jeffery Joel and Steffen, Eckhard and Yu, Weiqiang}}, booktitle = {{arXiv:2512.14285}}, title = {{{Edge-coloring 4- and 5-regular projective planar graphs with no Petersen-minor}}}, year = {{2025}}, } @unpublished{63384, abstract = {{Two fundamental ways to represent a group are as permutations and as matrices. In this paper, we study linear representations of groups that intertwine with a permutation representation. Recently, D'Alconzo and Di Scala investigated how small the matrices in such a linear representation can be. The minimal dimension of such a representation is the \emph{linear dimension of the group action} and this has applications in cryptography and cryptosystems. We develop the idea of linear dimension from an algebraic point of view by using the theory of permutation modules. We give structural results about representations of minimal dimension and investigate the implications of faithfulness, transitivity and primitivity on the linear dimension. Furthermore, we compute the linear dimension of several classes of finite primitive permutation groups. We also study wreath products, allowing us to determine the linear dimension of imprimitive group actions. Finally, we give the linear dimension of almost simple finite $2$-transitive groups, some of which may be used for further applications in cryptography. Our results also open up many new questions about linear representations of group actions.}}, author = {{Devillers, Alice and Giudici, Michael and Hawtin, Daniel R. and Klawuhn, Lukas-André Dominik and Morgan, Luke}}, title = {{{Linear dimension of group actions}}}, year = {{2025}}, } @unpublished{63394, abstract = {{We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are well understood, but the probabilities of rare events remain largely unexplored. Large deviation type results have been obtained only in extreme cases, when either a vanishingly small proportion of eigenvalues are real or almost all eigenvalues are real. Here, in both the strong and weak asymmetry regimes, we derive the probabilities of rare events in the moderate-to-large deviation regime, thereby providing a natural connection between the previously known regime of Gaussian fluctuations and the large deviation regime. Our results are new even for the classical real Ginibre ensemble.}}, author = {{Byun, Sung-Soo and Jalowy, Jonas and Lee, Yong-Woo and Schehr, Grégory}}, booktitle = {{arXiv:2511.09191}}, title = {{{Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices}}}, year = {{2025}}, } @unpublished{63393, abstract = {{We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $μ_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $μ_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $μ_t$ is available.}}, author = {{Höfert, Antonia and Jalowy, Jonas and Kabluchko, Zakhar}}, booktitle = {{arXiv:2512.17808}}, title = {{{Zeros of polynomial powers under the heat flow}}}, year = {{2025}}, } @inproceedings{63434, author = {{Hoffmann, Max}}, booktitle = {{Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14)}}, editor = {{Bosch, Marianna and Bolondi, Giorgio and Carreira, Susana and Michael, Gaidoschik and Camilla, Spagnolo}}, keywords = {{hoffmann, reviewed, proceedings}}, title = {{{Using scriptwriting as a response format for interface tasks: Exemplary analyses in the context of symmetry}}}, year = {{2025}}, } @article{63433, author = {{Hoffmann, Max}}, journal = {{mathematik lehren}}, number = {{253}}, pages = {{39--44}}, title = {{{Digitale Perspektiven auf das Heron-Verfahren}}}, doi = {{https://doi.org/10.5555/ml-253-2025_07}}, year = {{2025}}, } @article{54837, author = {{Claes, Leander and Lankeit, Johannes and Winkler, Michael}}, issn = {{1793-6314}}, journal = {{Mathematical Models and Methods in Applied Sciences}}, number = {{11}}, pages = {{2465--2512}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions}}}, doi = {{10.1142/s0218202525500447}}, volume = {{35}}, year = {{2025}}, } @unpublished{63510, abstract = {{It has been shown recently that optimal control problems with the dynamical constraint given by a second order system admit a regular Lagrangian formulation. This implies that the optimality conditions can be obtained in a new form based on the variational approach. In this paper we extend the first order necessary optimality conditions obtained previously to second order optimality conditions. This results in a complete characterization of the optimality conditions in a new Lagrangian form.}}, author = {{Konopik, Michael and Leyendecker, Sigrid and Maslovskaya, Sofya and Sina Ober-Blöbaum, Sina Ober-Blöbaum and Almagro, Rodrigo T. Sato Martín de}}, booktitle = {{arXiv:2507.06024}}, title = {{{Second order optimality conditions in a new Lagrangian formulation for optimal control problems}}}, year = {{2025}}, } @article{59797, author = {{Konopik, Michael and T. Sato Martín de Almagro, Rodrigo and Maslovskaya, Sofya and Ober-Blöbaum, Sina and Leyendecker, Sigrid}}, journal = {{Journal of Nonlinear Science}}, number = {{11}}, title = {{{Variational integrators for a new Lagrangian approach to control affine systems with a quadratic Lagrange term}}}, doi = {{10.1007/s00332-025-10229-5}}, volume = {{36}}, year = {{2025}}, } @article{59799, author = {{Konopik, Michael and Leyendecker, Sigrid and Maslovskaya, Sofya and Ober-Blöbaum, Sina and T. Sato Martín de Almagro, Rodrigo}}, journal = {{Nonlinearity}}, number = {{11}}, title = {{{A new Lagrangian approach to optimal control of second-order systems}}}, doi = {{10.1088/1361-6544/ae1d08}}, volume = {{38}}, year = {{2025}}, } @article{53414, abstract = {{By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Gu\'eritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.}}, author = {{Delarue, Benjamin and Monclair, Daniel and Sanders, Andrew}}, journal = {{Geometric and Functional Analysis (GAFA)}}, pages = {{673–735}}, title = {{{Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing}}}, doi = {{10.1007/s00039-025-00712-2}}, volume = {{35}}, year = {{2025}}, } @article{53412, abstract = {{Let $M$ be a symplectic manifold carrying a Hamiltonian $S^1$-action with momentum map $J:M \rightarrow \mathbb{R}$ and consider the corresponding symplectic quotient $\mathcal{M}_0:=J^{-1}(0)/S^1$. We extend Sjamaar's complex of differential forms on $\mathcal{M}_0$, whose cohomology is isomorphic to the singular cohomology $H(\mathcal{M}_0;\mathbb{R})$ of $\mathcal{M}_0$ with real coefficients, to a complex of differential forms on $\mathcal{M}_0$ associated with a partial desingularization $\widetilde{\mathcal{M}}_0$, which we call resolution differential forms. The cohomology of that complex turns out to be isomorphic to the de Rham cohomology $H(\widetilde{ \mathcal{M}}_0)$ of $\widetilde{\mathcal{M}}_0$. Based on this, we derive a long exact sequence involving both $H(\mathcal{M}_0;\mathbb{R})$ and $H(\widetilde{ \mathcal{M}}_0)$ and give conditions for its splitting. We then define a Kirwan map $\mathcal{K}:H_{S^1}(M) \rightarrow H(\widetilde{\mathcal{M}}_0)$ from the equivariant cohomology $H_{S^1}(M)$ of $M$ to $H(\widetilde{\mathcal{M}}_0)$ and show that its image contains the image of $H(\mathcal{M}_0;\mathbb{R})$ in $H(\widetilde{\mathcal{M}}_0)$ under the natural inclusion. Combining both results in the case that all fixed point components of $M$ have vanishing odd cohomology we obtain a surjection $\check \kappa:H^\textrm{ev}_{S^1}(M) \rightarrow H^\textrm{ev}(\mathcal{M}_0;\mathbb{R})$ in even degrees, while already simple examples show that a similar surjection in odd degrees does not exist in general. As an interesting class of examples we study abelian polygon spaces.}}, author = {{Delarue, Benjamin and Ramacher, Pablo and Schmitt, Maximilian}}, journal = {{Transformation Groups}}, title = {{{Singular cohomology of symplectic quotients by circle actions and Kirwan surjectivity}}}, doi = {{10.1007/s00031-025-09924-0}}, year = {{2025}}, }