[{"has_accepted_license":"1","status":"public","ddc":["510"],"user_id":"85279","volume":421,"page":"114780","publisher":"Elsevier","_id":"29240","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>.","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>.","short":"S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics 421 (2023) 114780.","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"]},"date_updated":"2023-08-10T08:42:39Z","publication_status":"epub_ahead","intvolume":"       421","article_type":"original","year":"2023","title":"Variational Learning of Euler–Lagrange Dynamics from Data","publication_identifier":{"issn":["0377-0427"]},"author":[{"id":"16494","full_name":"Ober-Blöbaum, Sina","first_name":"Sina","last_name":"Ober-Blöbaum"},{"full_name":"Offen, Christian","first_name":"Christian","orcid":"0000-0002-5940-8057","last_name":"Offen","id":"85279"}],"doi":"10.1016/j.cam.2022.114780","language":[{"iso":"eng"}],"related_material":{"link":[{"url":"https://github.com/Christian-Offen/LagrangianShadowIntegration","relation":"software"}]},"abstract":[{"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.","lang":"eng"}],"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":[{"file_name":"ShadowLagrangian_revision1_journal_style_arxiv.pdf","file_size":3640770,"access_level":"open_access","relation":"main_file","date_updated":"2022-06-28T15:25:50Z","file_id":"32274","content_type":"application/pdf","title":"Variational Learning of Euler–Lagrange Dynamics from Data","creator":"coffen","date_created":"2022-06-28T15:25:50Z","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":"2022-01-11T13:24:00Z"},{"intvolume":"       408","date_updated":"2023-01-09T08:23:56Z","publication_status":"published","publication_identifier":{"issn":["0377-0427"]},"author":[{"id":"42608","first_name":"Kerstin","last_name":"Hesse","orcid":"0000-0003-4125-1941","full_name":"Hesse, Kerstin"},{"full_name":"Le Gia, Quoc Thong","last_name":"Le Gia","first_name":"Quoc Thong"}],"title":"L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data","year":"2022","doi":"10.1016/j.cam.2022.114118","language":[{"iso":"eng"}],"article_number":"114118","publication":"Journal of Computational and Applied Mathematics","department":[{"_id":"10"}],"type":"journal_article","keyword":["Applied Mathematics","Computational Mathematics"],"date_created":"2022-12-20T17:37:16Z","status":"public","volume":408,"user_id":"14931","publisher":"Elsevier BV","_id":"34633","citation":{"short":"K. Hesse, Q.T. Le Gia, Journal of Computational and Applied Mathematics 408 (2022).","chicago":"Hesse, Kerstin, and Quoc Thong Le Gia. “L_2 Error Estimates for Polynomial Discrete Penalized Least-Squares Approximation on the Sphere from Noisy Data.” <i>Journal of Computational and Applied Mathematics</i> 408 (2022). <a href=\"https://doi.org/10.1016/j.cam.2022.114118\">https://doi.org/10.1016/j.cam.2022.114118</a>.","ieee":"K. Hesse and Q. T. Le Gia, “L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data,” <i>Journal of Computational and Applied Mathematics</i>, vol. 408, Art. no. 114118, 2022, doi: <a href=\"https://doi.org/10.1016/j.cam.2022.114118\">10.1016/j.cam.2022.114118</a>.","apa":"Hesse, K., &#38; Le Gia, Q. T. (2022). L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data. <i>Journal of Computational and Applied Mathematics</i>, <i>408</i>, Article 114118. <a href=\"https://doi.org/10.1016/j.cam.2022.114118\">https://doi.org/10.1016/j.cam.2022.114118</a>","bibtex":"@article{Hesse_Le Gia_2022, title={L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data}, volume={408}, DOI={<a href=\"https://doi.org/10.1016/j.cam.2022.114118\">10.1016/j.cam.2022.114118</a>}, number={114118}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier BV}, author={Hesse, Kerstin and Le Gia, Quoc Thong}, year={2022} }","ama":"Hesse K, Le Gia QT. L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data. <i>Journal of Computational and Applied Mathematics</i>. 2022;408. doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114118\">10.1016/j.cam.2022.114118</a>","mla":"Hesse, Kerstin, and Quoc Thong Le Gia. “L_2 Error Estimates for Polynomial Discrete Penalized Least-Squares Approximation on the Sphere from Noisy Data.” <i>Journal of Computational and Applied Mathematics</i>, vol. 408, 114118, Elsevier BV, 2022, doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114118\">10.1016/j.cam.2022.114118</a>."}},{"status":"public","user_id":"49063","volume":382,"_id":"34629","publisher":"Elsevier BV","citation":{"chicago":"Hesse, Kerstin, Ian H. Sloan, and Robert S. Womersley. “Local RBF-Based Penalized Least-Squares Approximation on the Sphere with Noisy Scattered Data.” <i>Journal of Computational and Applied Mathematics</i> 382 (2021). <a href=\"https://doi.org/10.1016/j.cam.2020.113061\">https://doi.org/10.1016/j.cam.2020.113061</a>.","short":"K. Hesse, I.H. Sloan, R.S. Womersley, Journal of Computational and Applied Mathematics 382 (2021).","ieee":"K. Hesse, I. H. Sloan, and R. S. Womersley, “Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data,” <i>Journal of Computational and Applied Mathematics</i>, vol. 382, Art. no. 113061, 2021, doi: <a href=\"https://doi.org/10.1016/j.cam.2020.113061\">10.1016/j.cam.2020.113061</a>.","apa":"Hesse, K., Sloan, I. H., &#38; Womersley, R. S. (2021). Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data. <i>Journal of Computational and Applied Mathematics</i>, <i>382</i>, Article 113061. <a href=\"https://doi.org/10.1016/j.cam.2020.113061\">https://doi.org/10.1016/j.cam.2020.113061</a>","bibtex":"@article{Hesse_Sloan_Womersley_2021, title={Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data}, volume={382}, DOI={<a href=\"https://doi.org/10.1016/j.cam.2020.113061\">10.1016/j.cam.2020.113061</a>}, number={113061}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier BV}, author={Hesse, Kerstin and Sloan, Ian H. and Womersley, Robert S.}, year={2021} }","ama":"Hesse K, Sloan IH, Womersley RS. Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data. <i>Journal of Computational and Applied Mathematics</i>. 2021;382. doi:<a href=\"https://doi.org/10.1016/j.cam.2020.113061\">10.1016/j.cam.2020.113061</a>","mla":"Hesse, Kerstin, et al. “Local RBF-Based Penalized Least-Squares Approximation on the Sphere with Noisy Scattered Data.” <i>Journal of Computational and Applied Mathematics</i>, vol. 382, 113061, Elsevier BV, 2021, doi:<a href=\"https://doi.org/10.1016/j.cam.2020.113061\">10.1016/j.cam.2020.113061</a>."},"date_updated":"2024-04-03T11:04:23Z","publication_status":"published","intvolume":"       382","title":"Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data","year":"2021","publication_identifier":{"issn":["0377-0427"]},"author":[{"full_name":"Hesse, Kerstin","first_name":"Kerstin","orcid":"0000-0003-4125-1941","last_name":"Hesse","id":"42608"},{"last_name":"Sloan","first_name":"Ian H.","full_name":"Sloan, Ian H."},{"last_name":"Womersley","first_name":"Robert S.","full_name":"Womersley, Robert S."}],"doi":"10.1016/j.cam.2020.113061","article_number":"113061","language":[{"iso":"eng"}],"publication":"Journal of Computational and Applied Mathematics","keyword":["Applied Mathematics","Computational Mathematics"],"type":"journal_article","department":[{"_id":"10"}],"date_created":"2022-12-20T17:26:44Z"},{"department":[{"_id":"101"}],"type":"journal_article","date_created":"2020-04-16T08:18:33Z","citation":{"ieee":"S. Klus, T. Sahai, C. Liu, and M. Dellnitz, “An efficient algorithm for the parallel solution of high-dimensional differential equations,” <i>Journal of Computational and Applied Mathematics</i>, pp. 3053–3062, 2011.","apa":"Klus, S., Sahai, T., Liu, C., &#38; Dellnitz, M. (2011). An efficient algorithm for the parallel solution of high-dimensional differential equations. <i>Journal of Computational and Applied Mathematics</i>, 3053–3062. <a href=\"https://doi.org/10.1016/j.cam.2010.12.026\">https://doi.org/10.1016/j.cam.2010.12.026</a>","short":"S. Klus, T. Sahai, C. Liu, M. Dellnitz, Journal of Computational and Applied Mathematics (2011) 3053–3062.","chicago":"Klus, Stefan, Tuhin Sahai, Cong Liu, and Michael Dellnitz. “An Efficient Algorithm for the Parallel Solution of High-Dimensional Differential Equations.” <i>Journal of Computational and Applied Mathematics</i>, 2011, 3053–62. <a href=\"https://doi.org/10.1016/j.cam.2010.12.026\">https://doi.org/10.1016/j.cam.2010.12.026</a>.","mla":"Klus, Stefan, et al. “An Efficient Algorithm for the Parallel Solution of High-Dimensional Differential Equations.” <i>Journal of Computational and Applied Mathematics</i>, 2011, pp. 3053–62, doi:<a href=\"https://doi.org/10.1016/j.cam.2010.12.026\">10.1016/j.cam.2010.12.026</a>.","bibtex":"@article{Klus_Sahai_Liu_Dellnitz_2011, title={An efficient algorithm for the parallel solution of high-dimensional differential equations}, DOI={<a href=\"https://doi.org/10.1016/j.cam.2010.12.026\">10.1016/j.cam.2010.12.026</a>}, journal={Journal of Computational and Applied Mathematics}, author={Klus, Stefan and Sahai, Tuhin and Liu, Cong and Dellnitz, Michael}, year={2011}, pages={3053–3062} }","ama":"Klus S, Sahai T, Liu C, Dellnitz M. An efficient algorithm for the parallel solution of high-dimensional differential equations. <i>Journal of Computational and Applied Mathematics</i>. 2011:3053-3062. doi:<a href=\"https://doi.org/10.1016/j.cam.2010.12.026\">10.1016/j.cam.2010.12.026</a>"},"publication":"Journal of Computational and Applied Mathematics","doi":"10.1016/j.cam.2010.12.026","user_id":"15701","language":[{"iso":"eng"}],"_id":"16624","page":"3053-3062","date_updated":"2022-01-06T06:52:53Z","publication_status":"published","publication_identifier":{"issn":["0377-0427"]},"author":[{"full_name":"Klus, Stefan","last_name":"Klus","first_name":"Stefan"},{"last_name":"Sahai","first_name":"Tuhin","full_name":"Sahai, Tuhin"},{"last_name":"Liu","first_name":"Cong","full_name":"Liu, Cong"},{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"}],"year":"2011","title":"An efficient algorithm for the parallel solution of high-dimensional differential equations","status":"public"},{"publication":"Journal of Computational and Applied Mathematics","citation":{"chicago":"Dellnitz, Michael, Oliver Schütze, and Qinghua Zheng. “Locating All the Zeros of an Analytic Function in One Complex Variable.” <i>Journal of Computational and Applied Mathematics</i>, 2002, 325–33. <a href=\"https://doi.org/10.1016/s0377-0427(01)00371-5\">https://doi.org/10.1016/s0377-0427(01)00371-5</a>.","short":"M. Dellnitz, O. Schütze, Q. Zheng, Journal of Computational and Applied Mathematics (2002) 325–333.","apa":"Dellnitz, M., Schütze, O., &#38; Zheng, Q. (2002). Locating all the zeros of an analytic function in one complex variable. <i>Journal of Computational and Applied Mathematics</i>, 325–333. <a href=\"https://doi.org/10.1016/s0377-0427(01)00371-5\">https://doi.org/10.1016/s0377-0427(01)00371-5</a>","ieee":"M. Dellnitz, O. Schütze, and Q. Zheng, “Locating all the zeros of an analytic function in one complex variable,” <i>Journal of Computational and Applied Mathematics</i>, pp. 325–333, 2002.","ama":"Dellnitz M, Schütze O, Zheng Q. Locating all the zeros of an analytic function in one complex variable. <i>Journal of Computational and Applied Mathematics</i>. 2002:325-333. doi:<a href=\"https://doi.org/10.1016/s0377-0427(01)00371-5\">10.1016/s0377-0427(01)00371-5</a>","bibtex":"@article{Dellnitz_Schütze_Zheng_2002, title={Locating all the zeros of an analytic function in one complex variable}, DOI={<a href=\"https://doi.org/10.1016/s0377-0427(01)00371-5\">10.1016/s0377-0427(01)00371-5</a>}, journal={Journal of Computational and Applied Mathematics}, author={Dellnitz, Michael and Schütze, Oliver and Zheng, Qinghua}, year={2002}, pages={325–333} }","mla":"Dellnitz, Michael, et al. “Locating All the Zeros of an Analytic Function in One Complex Variable.” <i>Journal of Computational and Applied Mathematics</i>, 2002, pp. 325–33, doi:<a href=\"https://doi.org/10.1016/s0377-0427(01)00371-5\">10.1016/s0377-0427(01)00371-5</a>."},"type":"journal_article","date_created":"2020-05-19T20:05:37Z","date_updated":"2022-01-06T06:53:02Z","publication_status":"published","year":"2002","title":"Locating all the zeros of an analytic function in one complex variable","status":"public","author":[{"last_name":"Dellnitz","first_name":"Michael","full_name":"Dellnitz, Michael"},{"last_name":"Schütze","first_name":"Oliver","full_name":"Schütze, Oliver"},{"first_name":"Qinghua","last_name":"Zheng","full_name":"Zheng, Qinghua"}],"publication_identifier":{"issn":["0377-0427"]},"doi":"10.1016/s0377-0427(01)00371-5","user_id":"32829","page":"325-333","_id":"17023","language":[{"iso":"eng"}]},{"status":"public","volume":99,"user_id":"93826","publisher":"Elsevier BV","_id":"40197","page":"337-351","citation":{"bibtex":"@article{Rösler_Voit_1998, title={Biorthogonal polynomials associated with reflection groups and a formula of Macdonald}, volume={99}, DOI={<a href=\"https://doi.org/10.1016/s0377-0427(98)00168-x\">10.1016/s0377-0427(98)00168-x</a>}, number={1–2}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier BV}, author={Rösler, Margit and Voit, Michael}, year={1998}, pages={337–351} }","ama":"Rösler M, Voit M. Biorthogonal polynomials associated with reflection groups and a formula of Macdonald. <i>Journal of Computational and Applied Mathematics</i>. 1998;99(1-2):337-351. doi:<a href=\"https://doi.org/10.1016/s0377-0427(98)00168-x\">10.1016/s0377-0427(98)00168-x</a>","mla":"Rösler, Margit, and Michael Voit. “Biorthogonal Polynomials Associated with Reflection Groups and a Formula of Macdonald.” <i>Journal of Computational and Applied Mathematics</i>, vol. 99, no. 1–2, Elsevier BV, 1998, pp. 337–51, doi:<a href=\"https://doi.org/10.1016/s0377-0427(98)00168-x\">10.1016/s0377-0427(98)00168-x</a>.","chicago":"Rösler, Margit, and Michael Voit. “Biorthogonal Polynomials Associated with Reflection Groups and a Formula of Macdonald.” <i>Journal of Computational and Applied Mathematics</i> 99, no. 1–2 (1998): 337–51. <a href=\"https://doi.org/10.1016/s0377-0427(98)00168-x\">https://doi.org/10.1016/s0377-0427(98)00168-x</a>.","short":"M. Rösler, M. Voit, Journal of Computational and Applied Mathematics 99 (1998) 337–351.","ieee":"M. Rösler and M. Voit, “Biorthogonal polynomials associated with reflection groups and a formula of Macdonald,” <i>Journal of Computational and Applied Mathematics</i>, vol. 99, no. 1–2, pp. 337–351, 1998, doi: <a href=\"https://doi.org/10.1016/s0377-0427(98)00168-x\">10.1016/s0377-0427(98)00168-x</a>.","apa":"Rösler, M., &#38; Voit, M. (1998). Biorthogonal polynomials associated with reflection groups and a formula of Macdonald. <i>Journal of Computational and Applied Mathematics</i>, <i>99</i>(1–2), 337–351. <a href=\"https://doi.org/10.1016/s0377-0427(98)00168-x\">https://doi.org/10.1016/s0377-0427(98)00168-x</a>"},"intvolume":"        99","date_updated":"2023-01-26T17:41:01Z","publication_status":"published","publication_identifier":{"issn":["0377-0427"]},"author":[{"id":"37390","last_name":"Rösler","first_name":"Margit","full_name":"Rösler, Margit"},{"full_name":"Voit, Michael","last_name":"Voit","first_name":"Michael"}],"year":"1998","title":"Biorthogonal polynomials associated with reflection groups and a formula of Macdonald","doi":"10.1016/s0377-0427(98)00168-x","language":[{"iso":"eng"}],"extern":"1","publication":"Journal of Computational and Applied Mathematics","issue":"1-2","department":[{"_id":"555"}],"keyword":["Applied Mathematics","Computational Mathematics"],"type":"journal_article","date_created":"2023-01-26T08:31:16Z"},{"intvolume":"        65","publication_status":"published","date_updated":"2023-01-26T17:43:10Z","author":[{"first_name":"Margit","last_name":"Rösler","full_name":"Rösler, Margit","id":"37390"}],"publication_identifier":{"issn":["0377-0427"]},"title":"Trigonometric convolution structures on Z derived from Jacobi polynomials","year":"1995","doi":"10.1016/0377-0427(95)00122-0","language":[{"iso":"eng"}],"extern":"1","publication":"Journal of Computational and Applied Mathematics","issue":"1-3","department":[{"_id":"555"}],"keyword":["Applied Mathematics","Computational Mathematics"],"type":"journal_article","date_created":"2023-01-26T08:42:19Z","status":"public","volume":65,"user_id":"93826","_id":"40207","publisher":"Elsevier BV","page":"357-368","citation":{"ama":"Rösler M. Trigonometric convolution structures on Z derived from Jacobi polynomials. <i>Journal of Computational and Applied Mathematics</i>. 1995;65(1-3):357-368. doi:<a href=\"https://doi.org/10.1016/0377-0427(95)00122-0\">10.1016/0377-0427(95)00122-0</a>","bibtex":"@article{Rösler_1995, title={Trigonometric convolution structures on Z derived from Jacobi polynomials}, volume={65}, DOI={<a href=\"https://doi.org/10.1016/0377-0427(95)00122-0\">10.1016/0377-0427(95)00122-0</a>}, number={1–3}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier BV}, author={Rösler, Margit}, year={1995}, pages={357–368} }","mla":"Rösler, Margit. “Trigonometric Convolution Structures on Z Derived from Jacobi Polynomials.” <i>Journal of Computational and Applied Mathematics</i>, vol. 65, no. 1–3, Elsevier BV, 1995, pp. 357–68, doi:<a href=\"https://doi.org/10.1016/0377-0427(95)00122-0\">10.1016/0377-0427(95)00122-0</a>.","short":"M. Rösler, Journal of Computational and Applied Mathematics 65 (1995) 357–368.","chicago":"Rösler, Margit. “Trigonometric Convolution Structures on Z Derived from Jacobi Polynomials.” <i>Journal of Computational and Applied Mathematics</i> 65, no. 1–3 (1995): 357–68. <a href=\"https://doi.org/10.1016/0377-0427(95)00122-0\">https://doi.org/10.1016/0377-0427(95)00122-0</a>.","apa":"Rösler, M. (1995). Trigonometric convolution structures on Z derived from Jacobi polynomials. <i>Journal of Computational and Applied Mathematics</i>, <i>65</i>(1–3), 357–368. <a href=\"https://doi.org/10.1016/0377-0427(95)00122-0\">https://doi.org/10.1016/0377-0427(95)00122-0</a>","ieee":"M. Rösler, “Trigonometric convolution structures on Z derived from Jacobi polynomials,” <i>Journal of Computational and Applied Mathematics</i>, vol. 65, no. 1–3, pp. 357–368, 1995, doi: <a href=\"https://doi.org/10.1016/0377-0427(95)00122-0\">10.1016/0377-0427(95)00122-0</a>."}},{"date_updated":"2022-01-06T06:52:52Z","publication_status":"published","publication_identifier":{"issn":["0377-0427"]},"author":[{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"},{"full_name":"Melbourne, Ian","last_name":"Melbourne","first_name":"Ian"}],"title":"Generic movement of eigenvalues for equivariant self-adjoint matrices","status":"public","year":"1994","doi":"10.1016/0377-0427(94)90032-9","user_id":"15701","_id":"16541","language":[{"iso":"eng"}],"page":"249-259","citation":{"chicago":"Dellnitz, Michael, and Ian Melbourne. “Generic Movement of Eigenvalues for Equivariant Self-Adjoint Matrices.” <i>Journal of Computational and Applied Mathematics</i>, 1994, 249–59. <a href=\"https://doi.org/10.1016/0377-0427(94)90032-9\">https://doi.org/10.1016/0377-0427(94)90032-9</a>.","short":"M. Dellnitz, I. Melbourne, Journal of Computational and Applied Mathematics (1994) 249–259.","apa":"Dellnitz, M., &#38; Melbourne, I. (1994). Generic movement of eigenvalues for equivariant self-adjoint matrices. <i>Journal of Computational and Applied Mathematics</i>, 249–259. <a href=\"https://doi.org/10.1016/0377-0427(94)90032-9\">https://doi.org/10.1016/0377-0427(94)90032-9</a>","ieee":"M. Dellnitz and I. Melbourne, “Generic movement of eigenvalues for equivariant self-adjoint matrices,” <i>Journal of Computational and Applied Mathematics</i>, pp. 249–259, 1994.","ama":"Dellnitz M, Melbourne I. Generic movement of eigenvalues for equivariant self-adjoint matrices. <i>Journal of Computational and Applied Mathematics</i>. 1994:249-259. doi:<a href=\"https://doi.org/10.1016/0377-0427(94)90032-9\">10.1016/0377-0427(94)90032-9</a>","bibtex":"@article{Dellnitz_Melbourne_1994, title={Generic movement of eigenvalues for equivariant self-adjoint matrices}, DOI={<a href=\"https://doi.org/10.1016/0377-0427(94)90032-9\">10.1016/0377-0427(94)90032-9</a>}, journal={Journal of Computational and Applied Mathematics}, author={Dellnitz, Michael and Melbourne, Ian}, year={1994}, pages={249–259} }","mla":"Dellnitz, Michael, and Ian Melbourne. “Generic Movement of Eigenvalues for Equivariant Self-Adjoint Matrices.” <i>Journal of Computational and Applied Mathematics</i>, 1994, pp. 249–59, doi:<a href=\"https://doi.org/10.1016/0377-0427(94)90032-9\">10.1016/0377-0427(94)90032-9</a>."},"publication":"Journal of Computational and Applied Mathematics","department":[{"_id":"101"}],"type":"journal_article","date_created":"2020-04-15T08:44:12Z"},{"user_id":"15701","doi":"10.1016/0377-0427(89)90150-7","page":"97-123","language":[{"iso":"eng"}],"_id":"16682","publication_status":"published","date_updated":"2022-01-06T06:52:54Z","title":"Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points","status":"public","year":"1989","publication_identifier":{"issn":["0377-0427"]},"author":[{"full_name":"Dellnitz, Michael","first_name":"Michael","last_name":"Dellnitz"},{"full_name":"Werner, Bodo","first_name":"Bodo","last_name":"Werner"}],"type":"journal_article","department":[{"_id":"101"}],"date_created":"2020-04-16T10:28:09Z","publication":"Journal of Computational and Applied Mathematics","citation":{"bibtex":"@article{Dellnitz_Werner_1989, title={Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points}, DOI={<a href=\"https://doi.org/10.1016/0377-0427(89)90150-7\">10.1016/0377-0427(89)90150-7</a>}, journal={Journal of Computational and Applied Mathematics}, author={Dellnitz, Michael and Werner, Bodo}, year={1989}, pages={97–123} }","ama":"Dellnitz M, Werner B. Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points. <i>Journal of Computational and Applied Mathematics</i>. 1989:97-123. doi:<a href=\"https://doi.org/10.1016/0377-0427(89)90150-7\">10.1016/0377-0427(89)90150-7</a>","mla":"Dellnitz, Michael, and Bodo Werner. “Computational Methods for Bifurcation Problems with Symmetries—with Special Attention to Steady State and Hopf Bifurcation Points.” <i>Journal of Computational and Applied Mathematics</i>, 1989, pp. 97–123, doi:<a href=\"https://doi.org/10.1016/0377-0427(89)90150-7\">10.1016/0377-0427(89)90150-7</a>.","short":"M. Dellnitz, B. Werner, Journal of Computational and Applied Mathematics (1989) 97–123.","chicago":"Dellnitz, Michael, and Bodo Werner. “Computational Methods for Bifurcation Problems with Symmetries—with Special Attention to Steady State and Hopf Bifurcation Points.” <i>Journal of Computational and Applied Mathematics</i>, 1989, 97–123. <a href=\"https://doi.org/10.1016/0377-0427(89)90150-7\">https://doi.org/10.1016/0377-0427(89)90150-7</a>.","ieee":"M. Dellnitz and B. Werner, “Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points,” <i>Journal of Computational and Applied Mathematics</i>, pp. 97–123, 1989.","apa":"Dellnitz, M., &#38; Werner, B. (1989). Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points. <i>Journal of Computational and Applied Mathematics</i>, 97–123. <a href=\"https://doi.org/10.1016/0377-0427(89)90150-7\">https://doi.org/10.1016/0377-0427(89)90150-7</a>"}}]