[{"publication_status":"published","date_updated":"2023-05-16T08:49:30Z","intvolume":"        57","title":"Algorithms for the genome median under a restricted measure of rearrangement","year":"2023","publication_identifier":{"issn":["0399-0559","2804-7303"]},"author":[{"full_name":"Silva, Helmuth O.M.","last_name":"Silva","first_name":"Helmuth O.M."},{"first_name":"Diego P.","last_name":"Rubert","full_name":"Rubert, Diego P."},{"full_name":"Araujo, Eloi","last_name":"Araujo","first_name":"Eloi"},{"id":"15548","full_name":"Steffen, Eckhard","orcid":"0000-0002-9808-7401","last_name":"Steffen","first_name":"Eckhard"},{"last_name":"Doerr","first_name":"Daniel","full_name":"Doerr, Daniel"},{"full_name":"Martinez, Fábio V.","first_name":"Fábio V.","last_name":"Martinez"}],"doi":"10.1051/ro/2023052","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"Ancestral reconstruction is a classic task in comparative genomics. Here, we study the genome median problem, a related computational problem which, given a set of three or more genomes, asks to find a new genome that minimizes the sum of pairwise distances between it and the given genomes. The distance stands for the amount of evolution observed at the genome level, for which we determine the minimum number of rearrangement operations necessary to transform one genome into the other. For almost all rearrangement operations the median problem is NP-hard, with the exception of the breakpoint median that can be constructed efficiently for multichromosomal circular and mixed genomes. In this work, we study the median problem under a restricted rearrangement measure called c4-distance, which is closely related to the breakpoint and the DCJ distance. We identify tight bounds and decomposers of the c4-median and develop algorithms for its construction, one exact ILP-based and three combinatorial heuristics. Subsequently, we perform experiments on simulated data sets. Our results suggest that the c4-distance is useful for the study the genome median problem, from theoretical and practical perspectives."}],"publication":"RAIRO - Operations Research","issue":"3","keyword":["Management Science and Operations Research","Computer Science Applications","Theoretical Computer Science"],"type":"journal_article","department":[{"_id":"542"}],"date_created":"2023-05-16T08:48:22Z","status":"public","user_id":"15540","volume":57,"page":"1045-1058","_id":"44857","publisher":"EDP Sciences","citation":{"chicago":"Silva, Helmuth O.M., Diego P. Rubert, Eloi Araujo, Eckhard Steffen, Daniel Doerr, and Fábio V. Martinez. “Algorithms for the Genome Median under a Restricted Measure of Rearrangement.” <i>RAIRO - Operations Research</i> 57, no. 3 (2023): 1045–58. <a href=\"https://doi.org/10.1051/ro/2023052\">https://doi.org/10.1051/ro/2023052</a>.","short":"H.O.M. Silva, D.P. Rubert, E. Araujo, E. Steffen, D. Doerr, F.V. Martinez, RAIRO - Operations Research 57 (2023) 1045–1058.","apa":"Silva, H. O. M., Rubert, D. P., Araujo, E., Steffen, E., Doerr, D., &#38; Martinez, F. V. (2023). Algorithms for the genome median under a restricted measure of rearrangement. <i>RAIRO - Operations Research</i>, <i>57</i>(3), 1045–1058. <a href=\"https://doi.org/10.1051/ro/2023052\">https://doi.org/10.1051/ro/2023052</a>","ieee":"H. O. M. Silva, D. P. Rubert, E. Araujo, E. Steffen, D. Doerr, and F. V. Martinez, “Algorithms for the genome median under a restricted measure of rearrangement,” <i>RAIRO - Operations Research</i>, vol. 57, no. 3, pp. 1045–1058, 2023, doi: <a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>.","ama":"Silva HOM, Rubert DP, Araujo E, Steffen E, Doerr D, Martinez FV. Algorithms for the genome median under a restricted measure of rearrangement. <i>RAIRO - Operations Research</i>. 2023;57(3):1045-1058. doi:<a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>","bibtex":"@article{Silva_Rubert_Araujo_Steffen_Doerr_Martinez_2023, title={Algorithms for the genome median under a restricted measure of rearrangement}, volume={57}, DOI={<a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>}, number={3}, journal={RAIRO - Operations Research}, publisher={EDP Sciences}, author={Silva, Helmuth O.M. and Rubert, Diego P. and Araujo, Eloi and Steffen, Eckhard and Doerr, Daniel and Martinez, Fábio V.}, year={2023}, pages={1045–1058} }","mla":"Silva, Helmuth O. M., et al. “Algorithms for the Genome Median under a Restricted Measure of Rearrangement.” <i>RAIRO - Operations Research</i>, vol. 57, no. 3, EDP Sciences, 2023, pp. 1045–58, doi:<a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>."}},{"_id":"44859","language":[{"iso":"eng"}],"user_id":"15540","author":[{"id":"92748","full_name":"Ma, Yulai","last_name":"Ma","first_name":"Yulai"},{"full_name":"Mattiolo, Davide","last_name":"Mattiolo","first_name":"Davide"},{"id":"15548","full_name":"Steffen, Eckhard","orcid":"0000-0002-9808-7401","first_name":"Eckhard","last_name":"Steffen"},{"first_name":"Isaak Hieronymus","last_name":"Wolf","full_name":"Wolf, Isaak Hieronymus","id":"88145"}],"title":"Sets of r-graphs that color all r-graphs","status":"public","year":"2023","date_updated":"2023-05-16T11:17:26Z","date_created":"2023-05-16T10:07:47Z","external_id":{"arxiv":["2305.08619"]},"department":[{"_id":"542"}],"type":"preprint","citation":{"bibtex":"@article{Ma_Mattiolo_Steffen_Wolf_2023, title={Sets of r-graphs that color all r-graphs}, journal={arXiv:2305.08619}, author={Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}, year={2023} }","ama":"Ma Y, Mattiolo D, Steffen E, Wolf IH. Sets of r-graphs that color all r-graphs. <i>arXiv:230508619</i>. Published online 2023.","mla":"Ma, Yulai, et al. “Sets of R-Graphs That Color All r-Graphs.” <i>ArXiv:2305.08619</i>, 2023.","short":"Y. Ma, D. Mattiolo, E. Steffen, I.H. Wolf, ArXiv:2305.08619 (2023).","chicago":"Ma, Yulai, Davide Mattiolo, Eckhard Steffen, and Isaak Hieronymus Wolf. “Sets of R-Graphs That Color All r-Graphs.” <i>ArXiv:2305.08619</i>, 2023.","ieee":"Y. Ma, D. Mattiolo, E. Steffen, and I. H. Wolf, “Sets of r-graphs that color all r-graphs,” <i>arXiv:2305.08619</i>. 2023.","apa":"Ma, Y., Mattiolo, D., Steffen, E., &#38; Wolf, I. H. (2023). Sets of r-graphs that color all r-graphs. In <i>arXiv:2305.08619</i>."},"publication":"arXiv:2305.08619"},{"department":[{"_id":"93"}],"keyword":["Lie group actions and analytic 1-submanifolds"],"type":"journal_article","date_created":"2022-12-22T09:46:36Z","citation":{"mla":"Hanusch, Maximilian. “Decompositions of Analytic 1-Manifolds.” <i>Indagationes Mathematicae.</i>, vol. 34, no. 4, 2023, pp. 752–811, doi:<a href=\"https://doi.org/10.1016/j.indag.2023.02.003\">10.1016/j.indag.2023.02.003</a>.","bibtex":"@article{Hanusch_2023, title={Decompositions of Analytic 1-Manifolds}, volume={34}, DOI={<a href=\"https://doi.org/10.1016/j.indag.2023.02.003\">10.1016/j.indag.2023.02.003</a>}, number={4}, journal={Indagationes Mathematicae.}, author={Hanusch, Maximilian}, year={2023}, pages={752–811} }","ama":"Hanusch M. Decompositions of Analytic 1-Manifolds. <i>Indagationes Mathematicae</i>. 2023;34(4):752-811. doi:<a href=\"https://doi.org/10.1016/j.indag.2023.02.003\">10.1016/j.indag.2023.02.003</a>","ieee":"M. Hanusch, “Decompositions of Analytic 1-Manifolds,” <i>Indagationes Mathematicae.</i>, vol. 34, no. 4, pp. 752–811, 2023, doi: <a href=\"https://doi.org/10.1016/j.indag.2023.02.003\">10.1016/j.indag.2023.02.003</a>.","apa":"Hanusch, M. (2023). Decompositions of Analytic 1-Manifolds. <i>Indagationes Mathematicae.</i>, <i>34</i>(4), 752–811. <a href=\"https://doi.org/10.1016/j.indag.2023.02.003\">https://doi.org/10.1016/j.indag.2023.02.003</a>","chicago":"Hanusch, Maximilian. “Decompositions of Analytic 1-Manifolds.” <i>Indagationes Mathematicae.</i> 34, no. 4 (2023): 752–811. <a href=\"https://doi.org/10.1016/j.indag.2023.02.003\">https://doi.org/10.1016/j.indag.2023.02.003</a>.","short":"M. Hanusch, Indagationes Mathematicae. 34 (2023) 752–811."},"issue":"4","publication":"Indagationes Mathematicae.","volume":34,"user_id":"30905","doi":"10.1016/j.indag.2023.02.003","_id":"34833","language":[{"iso":"eng"}],"page":"752-811","intvolume":"        34","publication_status":"published","date_updated":"2023-05-25T07:32:38Z","author":[{"last_name":"Hanusch","first_name":"Maximilian","full_name":"Hanusch, Maximilian","id":"30905"}],"title":"Decompositions of Analytic 1-Manifolds","year":"2023","status":"public"},{"external_id":{"arxiv":["2306.03320"]},"date_created":"2023-06-07T07:57:28Z","type":"preprint","department":[{"_id":"101"}],"publication":"arXiv:2306.03320","citation":{"ama":"von der Gracht S, Nijholt E, Rink B. A parametrisation method for high-order phase reduction in coupled  oscillator networks. <i>arXiv:230603320</i>.","bibtex":"@article{von der Gracht_Nijholt_Rink, title={A parametrisation method for high-order phase reduction in coupled  oscillator networks}, journal={arXiv:2306.03320}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob} }","mla":"von der Gracht, Sören, et al. “A Parametrisation Method for High-Order Phase Reduction in Coupled  Oscillator Networks.” <i>ArXiv:2306.03320</i>.","short":"S. von der Gracht, E. Nijholt, B. Rink, ArXiv:2306.03320 (n.d.).","chicago":"Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “A Parametrisation Method for High-Order Phase Reduction in Coupled  Oscillator Networks.” <i>ArXiv:2306.03320</i>, n.d.","apa":"von der Gracht, S., Nijholt, E., &#38; Rink, B. (n.d.). A parametrisation method for high-order phase reduction in coupled  oscillator networks. In <i>arXiv:2306.03320</i>.","ieee":"S. von der Gracht, E. Nijholt, and B. Rink, “A parametrisation method for high-order phase reduction in coupled  oscillator networks,” <i>arXiv:2306.03320</i>. ."},"abstract":[{"lang":"eng","text":"We present a novel method for high-order phase reduction in networks of\r\nweakly coupled oscillators and, more generally, perturbations of reducible\r\nnormally hyperbolic (quasi-)periodic tori. Our method works by computing an\r\nasymptotic expansion for an embedding of the perturbed invariant torus, as well\r\nas for the reduced phase dynamics in local coordinates. Both can be determined\r\nto arbitrary degrees of accuracy, and we show that the phase dynamics may\r\ndirectly be obtained in normal form. We apply the method to predict remote\r\nsynchronisation in a chain of coupled Stuart-Landau oscillators."}],"main_file_link":[{"url":"https://arxiv.org/pdf/2306.03320"}],"page":"29","language":[{"iso":"eng"}],"_id":"45498","user_id":"97359","year":"2023","status":"public","title":"A parametrisation method for high-order phase reduction in coupled  oscillator networks","author":[{"id":"97359","first_name":"Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören"},{"last_name":"Nijholt","first_name":"Eddie","full_name":"Nijholt, Eddie"},{"first_name":"Bob","last_name":"Rink","full_name":"Rink, Bob"}],"date_updated":"2023-06-07T07:59:06Z","publication_status":"submitted"},{"publisher":"Friedrich Verlag","_id":"45712","language":[{"iso":"ger"}],"page":"6-11","user_id":"44184","author":[{"first_name":"Uta","last_name":"Häsel-Weide","full_name":"Häsel-Weide, Uta","id":"60267"}],"status":"public","title":" Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen","year":"2023","date_updated":"2023-06-22T09:34:13Z","date_created":"2023-06-22T09:28:22Z","department":[{"_id":"543"}],"type":"journal_article","citation":{"ama":"Häsel-Weide U.  Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen. <i>Die Grundschulzeitschrift</i>. 2023;(339):6-11.","bibtex":"@article{Häsel-Weide_2023, title={ Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen}, number={339}, journal={Die Grundschulzeitschrift}, publisher={Friedrich Verlag}, author={Häsel-Weide, Uta}, year={2023}, pages={6–11} }","mla":"Häsel-Weide, Uta. “ Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen.” <i>Die Grundschulzeitschrift</i>, no. 339, Friedrich Verlag, 2023, pp. 6–11.","short":"U. Häsel-Weide, Die Grundschulzeitschrift (2023) 6–11.","chicago":"Häsel-Weide, Uta. “ Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen.” <i>Die Grundschulzeitschrift</i>, no. 339 (2023): 6–11.","apa":"Häsel-Weide, U. (2023).  Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen. <i>Die Grundschulzeitschrift</i>, <i>339</i>, 6–11.","ieee":"U. Häsel-Weide, “ Inklusiver Mathematikunterricht. Mathematiklernen in Vielfalt von Kompetenzen, Wegen und Lernsituationen,” <i>Die Grundschulzeitschrift</i>, no. 339, pp. 6–11, 2023."},"issue":"339","publication":"Die Grundschulzeitschrift","related_material":{"link":[{"relation":"other","url":"https://www.friedrich-verlag.de/grundschule/die-grundschulzeitschrift/mathematikunterricht-inklusiv-5991"}]}},{"_id":"45713","publisher":"Friedrich Verlag","language":[{"iso":"ger"}],"page":"20-23","user_id":"44184","author":[{"full_name":"Graf, Lara Marie","first_name":"Lara Marie","last_name":"Graf","id":"89837"},{"first_name":"Inga","last_name":"Wienhues","full_name":"Wienhues, Inga","id":"61635"},{"last_name":"Häsel-Weide","first_name":"Uta","full_name":"Häsel-Weide, Uta","id":"60267"}],"status":"public","year":"2023","title":"Addition und Subtraktion verstehen","date_updated":"2023-06-22T09:34:18Z","date_created":"2023-06-22T09:33:59Z","department":[{"_id":"543"}],"type":"journal_article","citation":{"mla":"Graf, Lara Marie, et al. “Addition und Subtraktion verstehen.” <i>Die Grundschulzeitschrift</i>, no. 339, Friedrich Verlag, 2023, pp. 20–23.","bibtex":"@article{Graf_Wienhues_Häsel-Weide_2023, title={Addition und Subtraktion verstehen}, number={339}, journal={Die Grundschulzeitschrift}, publisher={Friedrich Verlag}, author={Graf, Lara Marie and Wienhues, Inga and Häsel-Weide, Uta}, year={2023}, pages={20–23} }","ama":"Graf LM, Wienhues I, Häsel-Weide U. Addition und Subtraktion verstehen. <i>Die Grundschulzeitschrift</i>. 2023;(339):20-23.","ieee":"L. M. Graf, I. Wienhues, and U. Häsel-Weide, “Addition und Subtraktion verstehen,” <i>Die Grundschulzeitschrift</i>, no. 339, pp. 20–23, 2023.","apa":"Graf, L. M., Wienhues, I., &#38; Häsel-Weide, U. (2023). Addition und Subtraktion verstehen. <i>Die Grundschulzeitschrift</i>, <i>339</i>, 20–23.","short":"L.M. Graf, I. Wienhues, U. Häsel-Weide, Die Grundschulzeitschrift (2023) 20–23.","chicago":"Graf, Lara Marie, Inga Wienhues, and Uta Häsel-Weide. “Addition und Subtraktion verstehen.” <i>Die Grundschulzeitschrift</i>, no. 339 (2023): 20–23."},"issue":"339","publication":"Die Grundschulzeitschrift","related_material":{"link":[{"relation":"other","url":"https://www.friedrich-verlag.de/grundschule/die-grundschulzeitschrift/mathematikunterricht-inklusiv-5991"}]}},{"external_id":{"arxiv":["2304.09573"]},"date_created":"2023-07-24T07:52:23Z","type":"preprint","department":[{"_id":"10"}],"publication":"arXiv:2304.09573","citation":{"mla":"Weich, Tobias, and Lasse L. Wolf. “Temperedness of Locally Symmetric Spaces: The Product Case.” <i>ArXiv:2304.09573</i>, 2023.","ama":"Weich T, Wolf LL. Temperedness of locally symmetric spaces: The product case. <i>arXiv:230409573</i>. Published online 2023.","bibtex":"@article{Weich_Wolf_2023, title={Temperedness of locally symmetric spaces: The product case}, journal={arXiv:2304.09573}, author={Weich, Tobias and Wolf, Lasse L.}, year={2023} }","apa":"Weich, T., &#38; Wolf, L. L. (2023). Temperedness of locally symmetric spaces: The product case. In <i>arXiv:2304.09573</i>.","ieee":"T. Weich and L. L. Wolf, “Temperedness of locally symmetric spaces: The product case,” <i>arXiv:2304.09573</i>. 2023.","short":"T. Weich, L.L. Wolf, ArXiv:2304.09573 (2023).","chicago":"Weich, Tobias, and Lasse L. Wolf. “Temperedness of Locally Symmetric Spaces: The Product Case.” <i>ArXiv:2304.09573</i>, 2023."},"abstract":[{"text":"Let $X=X_1\\times X_2$ be a product of two rank one symmetric spaces of\r\nnon-compact type and $\\Gamma$ a torsion-free discrete subgroup in $G_1\\times\r\nG_2$. We show that the spectrum of $\\Gamma \\backslash X$ is related to the\r\nasymptotic growth of $\\Gamma$ in the two direction defined by the two factors.\r\nWe obtain that $L^2(\\Gamma \\backslash G)$ is tempered for large class of\r\n$\\Gamma$.","lang":"eng"}],"_id":"46117","language":[{"iso":"eng"}],"user_id":"45027","status":"public","year":"2023","title":"Temperedness of locally symmetric spaces: The product case","author":[{"last_name":"Weich","first_name":"Tobias","full_name":"Weich, Tobias"},{"first_name":"Lasse L.","last_name":"Wolf","full_name":"Wolf, Lasse L."}],"date_updated":"2023-07-24T07:53:29Z"},{"title":"Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs","year":"2023","author":[{"full_name":"Ma, Yulai","first_name":"Yulai","last_name":"Ma","id":"92748"},{"full_name":"Mattiolo, Davide","first_name":"Davide","last_name":"Mattiolo"},{"full_name":"Steffen, Eckhard","orcid":"0000-0002-9808-7401","last_name":"Steffen","first_name":"Eckhard","id":"15548"},{"id":"88145","full_name":"Wolf, Isaak Hieronymus","first_name":"Isaak Hieronymus","last_name":"Wolf"}],"publication_identifier":{"issn":["0895-4801","1095-7146"]},"date_updated":"2023-08-01T10:09:35Z","publication_status":"published","intvolume":"        37","language":[{"iso":"eng"}],"doi":"10.1137/22m1500654","publication":"SIAM Journal on Discrete Mathematics","issue":"3","date_created":"2023-08-01T10:08:32Z","keyword":["General Mathematics"],"type":"journal_article","department":[{"_id":"542"}],"status":"public","page":"1548-1565","_id":"46256","publisher":"Society for Industrial & Applied Mathematics (SIAM)","user_id":"15540","volume":37,"citation":{"short":"Y. Ma, D. Mattiolo, E. Steffen, I.H. Wolf, SIAM Journal on Discrete Mathematics 37 (2023) 1548–1565.","chicago":"Ma, Yulai, Davide Mattiolo, Eckhard Steffen, and Isaak Hieronymus Wolf. “Pairwise Disjoint Perfect Matchings in R-Edge-Connected r-Regular Graphs.” <i>SIAM Journal on Discrete Mathematics</i> 37, no. 3 (2023): 1548–65. <a href=\"https://doi.org/10.1137/22m1500654\">https://doi.org/10.1137/22m1500654</a>.","apa":"Ma, Y., Mattiolo, D., Steffen, E., &#38; Wolf, I. H. (2023). Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs. <i>SIAM Journal on Discrete Mathematics</i>, <i>37</i>(3), 1548–1565. <a href=\"https://doi.org/10.1137/22m1500654\">https://doi.org/10.1137/22m1500654</a>","ieee":"Y. Ma, D. Mattiolo, E. Steffen, and I. H. Wolf, “Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs,” <i>SIAM Journal on Discrete Mathematics</i>, vol. 37, no. 3, pp. 1548–1565, 2023, doi: <a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>.","ama":"Ma Y, Mattiolo D, Steffen E, Wolf IH. Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs. <i>SIAM Journal on Discrete Mathematics</i>. 2023;37(3):1548-1565. doi:<a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>","bibtex":"@article{Ma_Mattiolo_Steffen_Wolf_2023, title={Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs}, volume={37}, DOI={<a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>}, number={3}, journal={SIAM Journal on Discrete Mathematics}, publisher={Society for Industrial &#38; Applied Mathematics (SIAM)}, author={Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}, year={2023}, pages={1548–1565} }","mla":"Ma, Yulai, et al. “Pairwise Disjoint Perfect Matchings in R-Edge-Connected r-Regular Graphs.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 37, no. 3, Society for Industrial &#38; Applied Mathematics (SIAM), 2023, pp. 1548–65, doi:<a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>."}},{"abstract":[{"lang":"eng","text":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\r\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis."}],"related_material":{"link":[{"relation":"software","url":"https://github.com/Christian-Offen/LagrangianShadowIntegration"}]},"publication":"Journal of Computational and Applied Mathematics","keyword":["Lagrangian learning","variational backward error analysis","modified Lagrangian","variational integrators","physics informed learning"],"type":"journal_article","department":[{"_id":"636"}],"file":[{"description":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions\nconnecting two points in a phase space, the system will exhibit those motions which extremise an action functional.\nMany qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equa-\ntions, are related to the existence of an action functional. Incorporating variational structure into learning algorithms\nfor dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features\nwith the exact physical system. In this paper we show how to incorporate variational principles into trajectory predic-\ntions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position\ndata of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no\nprior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward\nerror analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the\nlearned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of\nvariational backward error analysis. (3) Finally, we introduce a method to perform system identification from position\nobservations only, based on variational backward error analysis.","date_created":"2022-06-28T15:25:50Z","creator":"coffen","title":"Variational Learning of Euler–Lagrange Dynamics from Data","content_type":"application/pdf","file_id":"32274","date_updated":"2022-06-28T15:25:50Z","relation":"main_file","access_level":"open_access","file_size":3640770,"file_name":"ShadowLagrangian_revision1_journal_style_arxiv.pdf"}],"date_created":"2022-01-11T13:24:00Z","date_updated":"2023-08-10T08:42:39Z","publication_status":"epub_ahead","intvolume":"       421","article_type":"original","title":"Variational Learning of Euler–Lagrange Dynamics from Data","year":"2023","publication_identifier":{"issn":["0377-0427"]},"author":[{"first_name":"Sina","last_name":"Ober-Blöbaum","full_name":"Ober-Blöbaum, Sina","id":"16494"},{"id":"85279","full_name":"Offen, Christian","orcid":"0000-0002-5940-8057","last_name":"Offen","first_name":"Christian"}],"doi":"10.1016/j.cam.2022.114780","language":[{"iso":"eng"}],"quality_controlled":"1","file_date_updated":"2022-06-28T15:25:50Z","citation":{"bibtex":"@article{Ober-Blöbaum_Offen_2023, title={Variational Learning of Euler–Lagrange Dynamics from Data}, volume={421}, DOI={<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier}, author={Ober-Blöbaum, Sina and Offen, Christian}, year={2023}, pages={114780} }","ama":"Ober-Blöbaum S, Offen C. Variational Learning of Euler–Lagrange Dynamics from Data. <i>Journal of Computational and Applied Mathematics</i>. 2023;421:114780. doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>","mla":"Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i>, vol. 421, Elsevier, 2023, p. 114780, doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>.","short":"S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics 421 (2023) 114780.","chicago":"Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i> 421 (2023): 114780. <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">https://doi.org/10.1016/j.cam.2022.114780</a>.","ieee":"S. Ober-Blöbaum and C. Offen, “Variational Learning of Euler–Lagrange Dynamics from Data,” <i>Journal of Computational and Applied Mathematics</i>, vol. 421, p. 114780, 2023, doi: <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>.","apa":"Ober-Blöbaum, S., &#38; Offen, C. (2023). Variational Learning of Euler–Lagrange Dynamics from Data. <i>Journal of Computational and Applied Mathematics</i>, <i>421</i>, 114780. <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">https://doi.org/10.1016/j.cam.2022.114780</a>"},"oa":"1","external_id":{"arxiv":["2112.12619"]},"has_accepted_license":"1","status":"public","ddc":["510"],"user_id":"85279","volume":421,"page":"114780","publisher":"Elsevier","_id":"29240"},{"language":[{"iso":"eng"}],"doi":"10.3934/jgm.2023005","author":[{"full_name":"McLachlan, Robert","last_name":"McLachlan","first_name":"Robert"},{"id":"85279","last_name":"Offen","first_name":"Christian","orcid":"0000-0002-5940-8057","full_name":"Offen, Christian"}],"year":"2023","title":"Backward error analysis for conjugate symplectic methods","intvolume":"        15","article_type":"original","date_updated":"2023-08-10T08:40:30Z","publication_status":"published","date_created":"2022-01-11T12:48:39Z","file":[{"title":"Backward error analysis for conjugate symplectic methods","content_type":"application/pdf","file_id":"32801","date_updated":"2022-08-12T16:48:59Z","relation":"main_file","file_size":827030,"access_level":"open_access","file_name":"BEA_MultiStep_Matrix.pdf","description":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","date_created":"2022-08-12T16:48:59Z","creator":"coffen"}],"department":[{"_id":"636"}],"type":"journal_article","keyword":["variational integrators","backward error analysis","Euler--Lagrange equations","multistep methods","conjugate symplectic methods"],"publication":"Journal of Geometric Mechanics","issue":"1","abstract":[{"text":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","lang":"eng"}],"related_material":{"link":[{"url":"https://github.com/Christian-Offen/BEAConjugateSymplectic","relation":"software"}]},"_id":"29236","publisher":"AIMS Press","page":"98-115","volume":15,"ddc":["510"],"user_id":"85279","status":"public","has_accepted_license":"1","external_id":{"arxiv":["2201.03911"]},"oa":"1","citation":{"mla":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, AIMS Press, 2023, pp. 98–115, doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","bibtex":"@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate symplectic methods}, volume={15}, DOI={<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>}, number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press}, author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }","ama":"McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>. 2023;15(1):98-115. doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>","ieee":"R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic methods,” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, pp. 98–115, 2023, doi: <a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","apa":"McLachlan, R., &#38; Offen, C. (2023). Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>, <i>15</i>(1), 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>","chicago":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i> 15, no. 1 (2023): 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>.","short":"R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115."},"file_date_updated":"2022-08-12T16:48:59Z","quality_controlled":"1"},{"status":"public","has_accepted_license":"1","publisher":"AIP Publishing","_id":"37654","ddc":["510"],"user_id":"85279","volume":33,"file_date_updated":"2023-04-26T16:20:56Z","citation":{"ama":"Dierkes E, Offen C, Ober-Blöbaum S, Flaßkamp K. Hamiltonian Neural Networks with Automatic Symmetry Detection. <i>Chaos</i>. 2023;33(6). doi:<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>","bibtex":"@article{Dierkes_Offen_Ober-Blöbaum_Flaßkamp_2023, title={Hamiltonian Neural Networks with Automatic Symmetry Detection}, volume={33}, DOI={<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>}, number={6063115}, journal={Chaos}, publisher={AIP Publishing}, author={Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}, year={2023} }","mla":"Dierkes, Eva, et al. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” <i>Chaos</i>, vol. 33, no. 6, 063115, AIP Publishing, 2023, doi:<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>.","short":"E. Dierkes, C. Offen, S. Ober-Blöbaum, K. Flaßkamp, Chaos 33 (2023).","chicago":"Dierkes, Eva, Christian Offen, Sina Ober-Blöbaum, and Kathrin Flaßkamp. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” <i>Chaos</i> 33, no. 6 (2023). <a href=\"https://doi.org/10.1063/5.0142969\">https://doi.org/10.1063/5.0142969</a>.","apa":"Dierkes, E., Offen, C., Ober-Blöbaum, S., &#38; Flaßkamp, K. (2023). Hamiltonian Neural Networks with Automatic Symmetry Detection. <i>Chaos</i>, <i>33</i>(6), Article 063115. <a href=\"https://doi.org/10.1063/5.0142969\">https://doi.org/10.1063/5.0142969</a>","ieee":"E. Dierkes, C. Offen, S. Ober-Blöbaum, and K. Flaßkamp, “Hamiltonian Neural Networks with Automatic Symmetry Detection,” <i>Chaos</i>, vol. 33, no. 6, Art. no. 063115, 2023, doi: <a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>."},"external_id":{"arxiv":["2301.07928"]},"oa":"1","year":"2023","title":"Hamiltonian Neural Networks with Automatic Symmetry Detection","author":[{"full_name":"Dierkes, Eva","first_name":"Eva","last_name":"Dierkes"},{"id":"85279","full_name":"Offen, Christian","orcid":"0000-0002-5940-8057","first_name":"Christian","last_name":"Offen"},{"id":"16494","full_name":"Ober-Blöbaum, Sina","first_name":"Sina","last_name":"Ober-Blöbaum"},{"first_name":"Kathrin","last_name":"Flaßkamp","full_name":"Flaßkamp, Kathrin"}],"publication_identifier":{"issn":["1054-1500"]},"date_updated":"2023-08-10T08:37:01Z","publication_status":"published","intvolume":"        33","article_type":"original","article_number":"063115","language":[{"iso":"eng"}],"doi":"10.1063/5.0142969","publication":"Chaos","issue":"6","abstract":[{"text":"Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when\r\nlearning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite\r\nthe data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we\r\nenhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach\r\nallows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples,\r\na pendulum on a cart and a two-body problem from astrodynamics are considered.","lang":"eng"}],"related_material":{"link":[{"description":"GitHub","url":"https://github.com/eva-dierkes/HNN_withSymmetries","relation":"software"}]},"file":[{"creator":"coffen","date_created":"2023-04-26T16:20:56Z","description":"Incorporating physical system knowledge into data-driven\nsystem identification has been shown to be beneficial. The\napproach presented in this article combines learning of an\nenergy-conserving model from data with detecting a Lie\ngroup representation of the unknown system symmetry.\nThe proposed approach can improve the learned model\nand reveal underlying symmetry simultaneously.","file_name":"JournalPaper_main.pdf","access_level":"open_access","file_size":5200111,"relation":"main_file","date_updated":"2023-04-26T16:20:56Z","file_id":"44205","content_type":"application/pdf","title":"Hamiltonian Neural Networks with Automatic Symmetry Detection"}],"date_created":"2023-01-20T09:10:06Z","type":"journal_article","department":[{"_id":"636"}]},{"abstract":[{"lang":"eng","text":"The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d.\r\nsamples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the\r\nbest of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled."}],"publication":"Journal of Nonlinear Science","department":[{"_id":"101"},{"_id":"655"}],"type":"journal_article","date_created":"2021-08-17T12:25:09Z","intvolume":"        33","date_updated":"2023-08-24T07:50:12Z","publication_status":"published","author":[{"id":"81513","full_name":"Nüske, Feliks","orcid":"0000-0003-2444-7889","last_name":"Nüske","first_name":"Feliks"},{"full_name":"Peitz, Sebastian","orcid":"0000-0002-3389-793X","first_name":"Sebastian","last_name":"Peitz","id":"47427"},{"first_name":"Friedrich","last_name":"Philipp","full_name":"Philipp, Friedrich"},{"first_name":"Manuel","last_name":"Schaller","full_name":"Schaller, Manuel"},{"full_name":"Worthmann, Karl","last_name":"Worthmann","first_name":"Karl"}],"title":"Finite-data error bounds for Koopman-based prediction and control","year":"2023","doi":"10.1007/s00332-022-09862-1","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09862-1.pdf"}],"article_number":"14","citation":{"short":"F. Nüske, S. Peitz, F. Philipp, M. Schaller, K. Worthmann, Journal of Nonlinear Science 33 (2023).","chicago":"Nüske, Feliks, Sebastian Peitz, Friedrich Philipp, Manuel Schaller, and Karl Worthmann. “Finite-Data Error Bounds for Koopman-Based Prediction and Control.” <i>Journal of Nonlinear Science</i> 33 (2023). <a href=\"https://doi.org/10.1007/s00332-022-09862-1\">https://doi.org/10.1007/s00332-022-09862-1</a>.","ieee":"F. Nüske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, “Finite-data error bounds for Koopman-based prediction and control,” <i>Journal of Nonlinear Science</i>, vol. 33, Art. no. 14, 2023, doi: <a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>.","apa":"Nüske, F., Peitz, S., Philipp, F., Schaller, M., &#38; Worthmann, K. (2023). Finite-data error bounds for Koopman-based prediction and control. <i>Journal of Nonlinear Science</i>, <i>33</i>, Article 14. <a href=\"https://doi.org/10.1007/s00332-022-09862-1\">https://doi.org/10.1007/s00332-022-09862-1</a>","bibtex":"@article{Nüske_Peitz_Philipp_Schaller_Worthmann_2023, title={Finite-data error bounds for Koopman-based prediction and control}, volume={33}, DOI={<a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>}, number={14}, journal={Journal of Nonlinear Science}, author={Nüske, Feliks and Peitz, Sebastian and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl}, year={2023} }","ama":"Nüske F, Peitz S, Philipp F, Schaller M, Worthmann K. Finite-data error bounds for Koopman-based prediction and control. <i>Journal of Nonlinear Science</i>. 2023;33. doi:<a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>","mla":"Nüske, Feliks, et al. “Finite-Data Error Bounds for Koopman-Based Prediction and Control.” <i>Journal of Nonlinear Science</i>, vol. 33, 14, 2023, doi:<a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>."},"oa":"1","status":"public","volume":33,"user_id":"47427","_id":"23428"},{"has_accepted_license":"1","status":"public","volume":45,"ddc":["510"],"user_id":"47427","_id":"21600","page":"A579-A595","citation":{"apa":"Dellnitz, M., Hüllermeier, E., Lücke, M., Ober-Blöbaum, S., Offen, C., Peitz, S., &#38; Pfannschmidt, K. (2023). Efficient time stepping for numerical integration using reinforcement  learning. <i>SIAM Journal on Scientific Computing</i>, <i>45</i>(2), A579–A595. <a href=\"https://doi.org/10.1137/21M1412682\">https://doi.org/10.1137/21M1412682</a>","ieee":"M. Dellnitz <i>et al.</i>, “Efficient time stepping for numerical integration using reinforcement  learning,” <i>SIAM Journal on Scientific Computing</i>, vol. 45, no. 2, pp. A579–A595, 2023, doi: <a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>.","short":"M. Dellnitz, E. Hüllermeier, M. Lücke, S. Ober-Blöbaum, C. Offen, S. Peitz, K. Pfannschmidt, SIAM Journal on Scientific Computing 45 (2023) A579–A595.","chicago":"Dellnitz, Michael, Eyke Hüllermeier, Marvin Lücke, Sina Ober-Blöbaum, Christian Offen, Sebastian Peitz, and Karlson Pfannschmidt. “Efficient Time Stepping for Numerical Integration Using Reinforcement  Learning.” <i>SIAM Journal on Scientific Computing</i> 45, no. 2 (2023): A579–95. <a href=\"https://doi.org/10.1137/21M1412682\">https://doi.org/10.1137/21M1412682</a>.","mla":"Dellnitz, Michael, et al. “Efficient Time Stepping for Numerical Integration Using Reinforcement  Learning.” <i>SIAM Journal on Scientific Computing</i>, vol. 45, no. 2, 2023, pp. A579–95, doi:<a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>.","ama":"Dellnitz M, Hüllermeier E, Lücke M, et al. Efficient time stepping for numerical integration using reinforcement  learning. <i>SIAM Journal on Scientific Computing</i>. 2023;45(2):A579-A595. doi:<a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>","bibtex":"@article{Dellnitz_Hüllermeier_Lücke_Ober-Blöbaum_Offen_Peitz_Pfannschmidt_2023, title={Efficient time stepping for numerical integration using reinforcement  learning}, volume={45}, DOI={<a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>}, number={2}, journal={SIAM Journal on Scientific Computing}, author={Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}, year={2023}, pages={A579–A595} }"},"external_id":{"arxiv":["arXiv:2104.03562"]},"intvolume":"        45","date_updated":"2023-08-25T09:24:50Z","publication_status":"published","author":[{"last_name":"Dellnitz","first_name":"Michael","full_name":"Dellnitz, Michael"},{"first_name":"Eyke","last_name":"Hüllermeier","full_name":"Hüllermeier, Eyke","id":"48129"},{"full_name":"Lücke, Marvin","first_name":"Marvin","last_name":"Lücke"},{"full_name":"Ober-Blöbaum, Sina","first_name":"Sina","last_name":"Ober-Blöbaum","id":"16494"},{"id":"85279","full_name":"Offen, Christian","last_name":"Offen","first_name":"Christian","orcid":"0000-0002-5940-8057"},{"full_name":"Peitz, Sebastian","orcid":"0000-0002-3389-793X","first_name":"Sebastian","last_name":"Peitz","id":"47427"},{"orcid":"0000-0001-9407-7903","first_name":"Karlson","last_name":"Pfannschmidt","full_name":"Pfannschmidt, Karlson","id":"13472"}],"year":"2023","title":"Efficient time stepping for numerical integration using reinforcement  learning","doi":"10.1137/21M1412682","language":[{"iso":"eng"}],"main_file_link":[{"url":"https://epubs.siam.org/doi/reader/10.1137/21M1412682"}],"abstract":[{"text":"Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.","lang":"eng"}],"related_material":{"link":[{"description":"GitHub","relation":"software","url":"https://github.com/lueckem/quadrature-ML"}]},"issue":"2","publication":"SIAM Journal on Scientific Computing","department":[{"_id":"101"},{"_id":"636"},{"_id":"355"},{"_id":"655"}],"type":"journal_article","date_created":"2021-04-09T07:59:19Z"},{"date_updated":"2023-08-30T09:55:41Z","conference":{"name":"International Symposium in Elementary Mathematics Teaching","location":"Prag"},"author":[{"id":"48912","full_name":"Schwerin, Imke","last_name":"Schwerin","first_name":"Imke"},{"full_name":"Häsel-Weide, Uta","last_name":"Häsel-Weide","first_name":"Uta","id":"60267"}],"status":"public","year":"2023","title":"Second grader´s understanding of doubling and halfing in various representations","editor":[{"full_name":"Novotna, J.","last_name":"Novotna","first_name":"J."},{"last_name":"Moraova","first_name":"H.","full_name":"Moraova, H."}],"user_id":"44184","publisher":"Charles University","_id":"46757","language":[{"iso":"eng"}],"page":"297-305","citation":{"mla":"Schwerin, Imke, and Uta Häsel-Weide. “Second Grader´s Understanding of Doubling and Halfing in Various Representations.” <i>International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education</i>, edited by J. Novotna and H. Moraova, Charles University, 2023, pp. 297–305.","apa":"Schwerin, I., &#38; Häsel-Weide, U. (2023). Second grader´s understanding of doubling and halfing in various representations. In J. Novotna &#38; H. Moraova (Eds.), <i>International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education</i> (pp. 297–305). Charles University.","ieee":"I. Schwerin and U. Häsel-Weide, “Second grader´s understanding of doubling and halfing in various representations,” in <i>International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education</i>, Prag, 2023, pp. 297–305.","short":"I. Schwerin, U. Häsel-Weide, in: J. Novotna, H. Moraova (Eds.), International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education, Charles University, 2023, pp. 297–305.","ama":"Schwerin I, Häsel-Weide U. Second grader´s understanding of doubling and halfing in various representations. In: Novotna J, Moraova H, eds. <i>International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education</i>. Charles University; 2023:297-305.","chicago":"Schwerin, Imke, and Uta Häsel-Weide. “Second Grader´s Understanding of Doubling and Halfing in Various Representations.” In <i>International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education</i>, edited by J. Novotna and H. Moraova, 297–305. Charles University, 2023.","bibtex":"@inproceedings{Schwerin_Häsel-Weide_2023, title={Second grader´s understanding of doubling and halfing in various representations}, booktitle={International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education}, publisher={Charles University}, author={Schwerin, Imke and Häsel-Weide, Uta}, editor={Novotna, J. and Moraova, H.}, year={2023}, pages={297–305} }"},"publication":"International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education","department":[{"_id":"543"}],"type":"conference","date_created":"2023-08-30T09:40:23Z"},{"_id":"46758","language":[{"iso":"ger"}],"publisher":"Visual Ink Publishing","page":"297-318","editor":[{"full_name":"Vöing, N.","first_name":"N.","last_name":"Vöing"},{"first_name":"R.","last_name":"Schmidt","full_name":"Schmidt, R."},{"full_name":"Neiske, I.","first_name":"I.","last_name":"Neiske"}],"user_id":"44184","author":[{"full_name":"Schmidt, Rebekka","last_name":"Schmidt","first_name":"Rebekka","id":"58565"},{"last_name":"Tenberge","first_name":"Claudia","full_name":"Tenberge, Claudia","id":"67302"},{"id":"60267","last_name":"Häsel-Weide","first_name":"Uta","full_name":"Häsel-Weide, Uta"}],"status":"public","year":"2023","title":"Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern","date_updated":"2023-08-30T09:55:46Z","date_created":"2023-08-30T09:54:47Z","department":[{"_id":"543"}],"type":"book_chapter","citation":{"mla":"Schmidt, Rebekka, et al. “Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern.” <i>Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre</i>, edited by N. Vöing et al., Visual Ink Publishing, 2023, pp. 297–318.","ama":"Schmidt R, Tenberge C, Häsel-Weide U. Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern. In: Vöing N, Schmidt R, Neiske I, eds. <i>Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre</i>. Visual Ink Publishing; 2023:297-318.","bibtex":"@inbook{Schmidt_Tenberge_Häsel-Weide_2023, title={Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern}, booktitle={Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre}, publisher={Visual Ink Publishing}, author={Schmidt, Rebekka and Tenberge, Claudia and Häsel-Weide, Uta}, editor={Vöing, N. and Schmidt, R. and Neiske, I.}, year={2023}, pages={297–318} }","apa":"Schmidt, R., Tenberge, C., &#38; Häsel-Weide, U. (2023). Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern. In N. Vöing, R. Schmidt, &#38; I. Neiske (Eds.), <i>Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre</i> (pp. 297–318). Visual Ink Publishing.","ieee":"R. Schmidt, C. Tenberge, and U. Häsel-Weide, “Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern,” in <i>Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre</i>, N. Vöing, R. Schmidt, and I. Neiske, Eds. Visual Ink Publishing, 2023, pp. 297–318.","chicago":"Schmidt, Rebekka, Claudia Tenberge, and Uta Häsel-Weide. “Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern.” In <i>Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre</i>, edited by N. Vöing, R. Schmidt, and I. Neiske, 297–318. Visual Ink Publishing, 2023.","short":"R. Schmidt, C. Tenberge, U. Häsel-Weide, in: N. Vöing, R. Schmidt, I. Neiske (Eds.), Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre, Visual Ink Publishing, 2023, pp. 297–318."},"publication":"Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre"},{"publication_status":"published","date_updated":"2024-09-25T08:26:34Z","intvolume":"        55","year":"2023","title":"Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course","publication_identifier":{"issn":["1863-9690","1863-9704"]},"author":[{"full_name":"Hoffmann, Max","last_name":"Hoffmann","orcid":"0000-0002-6964-7123","first_name":"Max","id":"32202"},{"full_name":"Biehler, Rolf","first_name":"Rolf","last_name":"Biehler","id":"16274"}],"doi":"10.1007/s11858-023-01505-3","main_file_link":[{"url":"https://link.springer.com/article/10.1007/s11858-023-01505-3"}],"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title><jats:p>Intending to counteract Klein’s second discontinuity in teacher education, we explored and applied the innovation of “<jats:italic>interface ePortfolio</jats:italic>” in the context of a geometry course for preservice teachers (PSTs). The tool offers the possibility of implementing the design principle of <jats:italic>profession orientation</jats:italic>. In the article, we theoretically clarify what we understand by this principle and locate our innovative concept against this theoretical background. We empirically investigate the extent to which counteraction against the second discontinuity is successful by analyzing reflection texts created in the interface ePortfolio, focusing on PSTs’ perspectives. Our qualitative content analysis shows that most of them perceive the innovation as helpful in the intended sense and indicates that the course concept, in general, and the interface ePortfolio, in particular, have helped establish relevant links between the course content and their later work as teachers.</jats:p>"}],"issue":"4","publication":"ZDM – Mathematics Education","type":"journal_article","department":[{"_id":"363"}],"date_created":"2024-09-23T07:51:34Z","status":"public","user_id":"37888","volume":55,"page":"737-751","publisher":"Springer Science and Business Media LLC","_id":"56200","citation":{"bibtex":"@article{Hoffmann_Biehler_2023, title={Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course}, volume={55}, DOI={<a href=\"https://doi.org/10.1007/s11858-023-01505-3\">10.1007/s11858-023-01505-3</a>}, number={4}, journal={ZDM – Mathematics Education}, publisher={Springer Science and Business Media LLC}, author={Hoffmann, Max and Biehler, Rolf}, year={2023}, pages={737–751} }","ama":"Hoffmann M, Biehler R. Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course. <i>ZDM – Mathematics Education</i>. 2023;55(4):737-751. doi:<a href=\"https://doi.org/10.1007/s11858-023-01505-3\">10.1007/s11858-023-01505-3</a>","mla":"Hoffmann, Max, and Rolf Biehler. “Implementing Profession Orientation as a Design Principle for Overcoming Klein’s Second Discontinuity – Preservice Teacher’s Perspectives on Interface Activities in the Context of a Geometry Course.” <i>ZDM – Mathematics Education</i>, vol. 55, no. 4, Springer Science and Business Media LLC, 2023, pp. 737–51, doi:<a href=\"https://doi.org/10.1007/s11858-023-01505-3\">10.1007/s11858-023-01505-3</a>.","short":"M. Hoffmann, R. Biehler, ZDM – Mathematics Education 55 (2023) 737–751.","chicago":"Hoffmann, Max, and Rolf Biehler. “Implementing Profession Orientation as a Design Principle for Overcoming Klein’s Second Discontinuity – Preservice Teacher’s Perspectives on Interface Activities in the Context of a Geometry Course.” <i>ZDM – Mathematics Education</i> 55, no. 4 (2023): 737–51. <a href=\"https://doi.org/10.1007/s11858-023-01505-3\">https://doi.org/10.1007/s11858-023-01505-3</a>.","ieee":"M. Hoffmann and R. Biehler, “Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course,” <i>ZDM – Mathematics Education</i>, vol. 55, no. 4, pp. 737–751, 2023, doi: <a href=\"https://doi.org/10.1007/s11858-023-01505-3\">10.1007/s11858-023-01505-3</a>.","apa":"Hoffmann, M., &#38; Biehler, R. (2023). Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course. <i>ZDM – Mathematics Education</i>, <i>55</i>(4), 737–751. <a href=\"https://doi.org/10.1007/s11858-023-01505-3\">https://doi.org/10.1007/s11858-023-01505-3</a>"}},{"publication":"Journal für Mathematik-Didaktik","citation":{"ieee":"J. Dröse and S. Prediger, “Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements,” <i>Journal für Mathematik-Didaktik</i>, 2023, doi: <a href=\"https://doi.org/10.1007/s13138-022-00214-w\">10.1007/s13138-022-00214-w</a>.","mla":"Dröse, Jennifer, and Susanne Prediger. “Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements.” <i>Journal Für Mathematik-Didaktik</i>, Springer Science and Business Media LLC, 2023, doi:<a href=\"https://doi.org/10.1007/s13138-022-00214-w\">10.1007/s13138-022-00214-w</a>.","apa":"Dröse, J., &#38; Prediger, S. (2023). Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements. <i>Journal Für Mathematik-Didaktik</i>. <a href=\"https://doi.org/10.1007/s13138-022-00214-w\">https://doi.org/10.1007/s13138-022-00214-w</a>","bibtex":"@article{Dröse_Prediger_2023, title={Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements}, DOI={<a href=\"https://doi.org/10.1007/s13138-022-00214-w\">10.1007/s13138-022-00214-w</a>}, journal={Journal für Mathematik-Didaktik}, publisher={Springer Science and Business Media LLC}, author={Dröse, Jennifer and Prediger, Susanne}, year={2023} }","ama":"Dröse J, Prediger S. Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements. <i>Journal für Mathematik-Didaktik</i>. Published online 2023. doi:<a href=\"https://doi.org/10.1007/s13138-022-00214-w\">10.1007/s13138-022-00214-w</a>","short":"J. Dröse, S. Prediger, Journal Für Mathematik-Didaktik (2023).","chicago":"Dröse, Jennifer, and Susanne Prediger. “Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements.” <i>Journal Für Mathematik-Didaktik</i>, 2023. <a href=\"https://doi.org/10.1007/s13138-022-00214-w\">https://doi.org/10.1007/s13138-022-00214-w</a>."},"abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title><jats:p>Teachers’ in-depth diagnostic thinking has been shown to be crucial for student-centered teaching as they need to perceive and interpret students’ understanding for well-informed decision-making on adaptive teaching practices. The paper presents a content-related approach to analyzing diagnostic thinking processes with respect to the mathematical knowledge elements that prospective teachers identify as students’ resources and obstacles. Prospective teachers’ challenge is that some relevant knowledge elements first have to be unpacked, because compact concepts (such as the place value concept) or procedures (such as for multi-digit multiplication) comprise several smaller knowledge elements (such as the positional property) that have to be made explicit for students to foster their learning processes adequately. Our study examines what knowledge elements prospective teachers perceive and interpret in a transcript vignettes on multi-digit multiplication (of decimal and natural numbers) and its underlying basic arithmetic concepts (place value understanding and meaning of multiplication) in written diagnostic judgments on students’ resources and obstacles (<jats:italic>N</jats:italic> = 196). A comparative design within the vignette is used to investigate how far the process of perceiving can be supported by thematic cues. The analysis reveals that those knowledge elements cued in the vignette by being already unpacked and explicitly addressed are perceived and interpreted more often (but with lower correctness) than those that are uncued and therefore have to be unpacked by the prospective teachers themselves. This confirms the need to prepare prospective teachers for unpacking mathematical concepts themselves.</jats:p>"}],"quality_controlled":"1","date_created":"2023-01-27T18:47:24Z","keyword":["Education","General Mathematics"],"type":"journal_article","department":[{"_id":"98"}],"status":"public","title":"Prospective Teachers’ Diagnostic Thinking on Students’ Understanding of Multi-Digit Multiplication: A Content-Related Analysis on Unpacking of Knowledge Elements","year":"2023","publication_identifier":{"issn":["0173-5322","1869-2699"]},"author":[{"full_name":"Dröse, Jennifer","first_name":"Jennifer","last_name":"Dröse"},{"full_name":"Prediger, Susanne","first_name":"Susanne","last_name":"Prediger"}],"publication_status":"published","date_updated":"2025-01-15T08:36:09Z","_id":"40607","language":[{"iso":"eng"}],"publisher":"Springer Science and Business Media LLC","user_id":"85820","doi":"10.1007/s13138-022-00214-w"},{"quality_controlled":"1","publication":"Journal of Mathematics Teacher Education","citation":{"mla":"Prediger, S., et al. “Teacher Expertise for Fostering At-Risk Students’ Understanding of Basic Concepts: Conceptual Model and Evidence for Growth.” <i>Journal of Mathematics Teacher Education</i>, 2023, doi:<a href=\"https://doi.org/10.1007/s10857-022-09538-3\">https://doi.org/10.1007/s10857-022-09538-3</a>.","ama":"Prediger S, Dröse J, Stahnke R, Ademmer C. Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth. <i>Journal of Mathematics Teacher Education</i>. Published online 2023. doi:<a href=\"https://doi.org/10.1007/s10857-022-09538-3\">https://doi.org/10.1007/s10857-022-09538-3</a>","bibtex":"@article{Prediger_Dröse_Stahnke_Ademmer_2023, title={Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth}, DOI={<a href=\"https://doi.org/10.1007/s10857-022-09538-3\">https://doi.org/10.1007/s10857-022-09538-3</a>}, journal={Journal of Mathematics Teacher Education}, author={Prediger, S. and Dröse, Jennifer and Stahnke, R. and Ademmer, C.}, year={2023} }","apa":"Prediger, S., Dröse, J., Stahnke, R., &#38; Ademmer, C. (2023). Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth. <i>Journal of Mathematics Teacher Education</i>. <a href=\"https://doi.org/10.1007/s10857-022-09538-3\">https://doi.org/10.1007/s10857-022-09538-3</a>","ieee":"S. Prediger, J. Dröse, R. Stahnke, and C. Ademmer, “Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth,” <i>Journal of Mathematics Teacher Education</i>, 2023, doi: <a href=\"https://doi.org/10.1007/s10857-022-09538-3\">https://doi.org/10.1007/s10857-022-09538-3</a>.","chicago":"Prediger, S., Jennifer Dröse, R. Stahnke, and C. Ademmer. “Teacher Expertise for Fostering At-Risk Students’ Understanding of Basic Concepts: Conceptual Model and Evidence for Growth.” <i>Journal of Mathematics Teacher Education</i>, 2023. <a href=\"https://doi.org/10.1007/s10857-022-09538-3\">https://doi.org/10.1007/s10857-022-09538-3</a>.","short":"S. Prediger, J. Dröse, R. Stahnke, C. Ademmer, Journal of Mathematics Teacher Education (2023)."},"type":"journal_article","department":[{"_id":"98"}],"date_created":"2023-05-31T06:27:51Z","date_updated":"2025-01-15T08:34:36Z","year":"2023","status":"public","title":"Teacher expertise for fostering at-risk students’ understanding of basic concepts: conceptual model and evidence for growth","author":[{"full_name":"Prediger, S.","first_name":"S.","last_name":"Prediger"},{"full_name":"Dröse, Jennifer","last_name":"Dröse","first_name":"Jennifer","id":"85820"},{"full_name":"Stahnke, R.","first_name":"R.","last_name":"Stahnke"},{"last_name":"Ademmer","first_name":"C.","full_name":"Ademmer, C."}],"doi":"https://doi.org/10.1007/s10857-022-09538-3","user_id":"85820","_id":"45374","language":[{"iso":"eng"}]},{"language":[{"iso":"eng"}],"article_number":"104270","doi":"10.1016/j.tate.2023.104270","publication_identifier":{"issn":["0742-051X"]},"author":[{"full_name":"Bruns, Julia","last_name":"Bruns","orcid":"https://orcid.org/0000-0002-6604-5864","first_name":"Julia","id":"72183"},{"full_name":"Hagena, Maike","last_name":"Hagena","first_name":"Maike"},{"full_name":"Gasteiger, Hedwig","last_name":"Gasteiger","first_name":"Hedwig"}],"year":"2023","title":"Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects?","intvolume":"       132","publication_status":"published","date_updated":"2025-06-02T12:59:11Z","date_created":"2023-07-26T08:55:26Z","department":[{"_id":"611"}],"keyword":["Education"],"type":"journal_article","publication":"Teaching and Teacher Education","publisher":"Elsevier BV","_id":"46155","volume":132,"user_id":"72183","status":"public","citation":{"ama":"Bruns J, Hagena M, Gasteiger H. Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects? <i>Teaching and Teacher Education</i>. 2023;132. doi:<a href=\"https://doi.org/10.1016/j.tate.2023.104270\">10.1016/j.tate.2023.104270</a>","bibtex":"@article{Bruns_Hagena_Gasteiger_2023, title={Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects?}, volume={132}, DOI={<a href=\"https://doi.org/10.1016/j.tate.2023.104270\">10.1016/j.tate.2023.104270</a>}, number={104270}, journal={Teaching and Teacher Education}, publisher={Elsevier BV}, author={Bruns, Julia and Hagena, Maike and Gasteiger, Hedwig}, year={2023} }","mla":"Bruns, Julia, et al. “Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects?” <i>Teaching and Teacher Education</i>, vol. 132, 104270, Elsevier BV, 2023, doi:<a href=\"https://doi.org/10.1016/j.tate.2023.104270\">10.1016/j.tate.2023.104270</a>.","chicago":"Bruns, Julia, Maike Hagena, and Hedwig Gasteiger. “Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects?” <i>Teaching and Teacher Education</i> 132 (2023). <a href=\"https://doi.org/10.1016/j.tate.2023.104270\">https://doi.org/10.1016/j.tate.2023.104270</a>.","short":"J. Bruns, M. Hagena, H. Gasteiger, Teaching and Teacher Education 132 (2023).","apa":"Bruns, J., Hagena, M., &#38; Gasteiger, H. (2023). Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects? <i>Teaching and Teacher Education</i>, <i>132</i>, Article 104270. <a href=\"https://doi.org/10.1016/j.tate.2023.104270\">https://doi.org/10.1016/j.tate.2023.104270</a>","ieee":"J. Bruns, M. Hagena, and H. Gasteiger, “Professional Development Enacted by Facilitators in the Context of Early Mathematics Education: Scaling up or Dilution of Effects?,” <i>Teaching and Teacher Education</i>, vol. 132, Art. no. 104270, 2023, doi: <a href=\"https://doi.org/10.1016/j.tate.2023.104270\">10.1016/j.tate.2023.104270</a>."}},{"type":"conference","department":[{"_id":"983"}],"date_created":"2025-05-19T07:19:01Z","related_material":{"link":[{"relation":"confirmation","url":"https://hal.science/hal-04406679v1/file/TWG_14-Garnelo-218.pdf"}]},"publication":"Thirteenth Congress of the European Society for Research in Mathematics (CERME13)","citation":{"mla":"Garnelo Abellanas, Irene, and Michael Liebendörfer. “Introducing Mathematics Undergraduate Students to the Theorem Prover Lean.” <i>Thirteenth Congress of the European Society for Research in Mathematics (CERME13)</i>, 2023.","ama":"Garnelo Abellanas I, Liebendörfer M. Introducing mathematics undergraduate students to the theorem prover lean. In: <i>Thirteenth Congress of the European Society for Research in Mathematics (CERME13)</i>. ; 2023.","bibtex":"@inproceedings{Garnelo Abellanas_Liebendörfer_2023, title={Introducing mathematics undergraduate students to the theorem prover lean}, booktitle={Thirteenth Congress of the European Society for Research in Mathematics (CERME13)}, author={Garnelo Abellanas, Irene and Liebendörfer, Michael}, year={2023} }","apa":"Garnelo Abellanas, I., &#38; Liebendörfer, M. (2023). Introducing mathematics undergraduate students to the theorem prover lean. <i>Thirteenth Congress of the European Society for Research in Mathematics (CERME13)</i>. Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), Budapest.","ieee":"I. Garnelo Abellanas and M. Liebendörfer, “Introducing mathematics undergraduate students to the theorem prover lean,” presented at the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), Budapest, 2023.","chicago":"Garnelo Abellanas, Irene, and Michael Liebendörfer. “Introducing Mathematics Undergraduate Students to the Theorem Prover Lean.” In <i>Thirteenth Congress of the European Society for Research in Mathematics (CERME13)</i>, 2023.","short":"I. Garnelo Abellanas, M. Liebendörfer, in: Thirteenth Congress of the European Society for Research in Mathematics (CERME13), 2023."},"user_id":"93843","language":[{"iso":"eng"}],"_id":"59984","date_updated":"2026-01-15T10:36:11Z","year":"2023","title":"Introducing mathematics undergraduate students to the theorem prover lean","status":"public","author":[{"last_name":"Garnelo Abellanas","first_name":"Irene","full_name":"Garnelo Abellanas, Irene","id":"93843"},{"full_name":"Liebendörfer, Michael","last_name":"Liebendörfer","first_name":"Michael"}],"conference":{"name":"Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13)","location":"Budapest"}}]