TY - JOUR AB - We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach. AU - Bieker, Katharina AU - Gebken, Bennet AU - Peitz, Sebastian ID - 20731 IS - 11 JF - IEEE Transactions on Pattern Analysis and Machine Intelligence TI - On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation VL - 44 ER - TY - JOUR AB - AbstractThe kinetic Brownian motion on the sphere bundle of a Riemannian manifold $$\mathbb {M}$$ M is a stochastic process that models a random perturbation of the geodesic flow. If $$\mathbb {M}$$ M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $$L^2$$ L 2 -spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold. AU - Kolb, Martin AU - Weich, Tobias AU - Wolf, Lasse Lennart ID - 31193 IS - 4 JF - Annales Henri Poincaré KW - Mathematical Physics KW - Nuclear and High Energy Physics KW - Statistical and Nonlinear Physics SN - 1424-0637 TI - Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature VL - 23 ER - TY - JOUR AU - Guedes Bonthonneau, Yannick AU - Weich, Tobias ID - 35306 IS - 3 JF - Journal of the European Mathematical Society KW - Applied Mathematics KW - General Mathematics SN - 1435-9855 TI - Ruelle–Pollicott resonances for manifolds with hyperbolic cusps VL - 24 ER - TY - JOUR AU - Hesse, Kerstin AU - Le Gia, Quoc Thong ID - 34633 JF - Journal of Computational and Applied Mathematics KW - Applied Mathematics KW - Computational Mathematics SN - 0377-0427 TI - L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data VL - 408 ER - TY - JOUR AU - Hanusch, Maximilian ID - 34817 IS - 1 JF - Communications in Analysis and Geometry KW - regularity of Lie groups SN - 1019-8385 TI - Regularity of Lie groups VL - 30 ER - TY - GEN AU - Hanusch, Maximilian ID - 34856 TI - Analysis 1 und 2 Skript/Buch ER - TY - JOUR AU - Kolb, Martin AU - Klump, Alexander ID - 35644 IS - 4 JF - Theory of Probability and its Applications TI - Uniqueness of the Inverse First Passage Time Problem and the Shape of the Shiryaev boundary VL - 67 ER - TY - JOUR AB - Motivated by the work [6] of Mariusz Bieniek, Krzysztof Burdzy and Soumik Pal we study a Fleming-Viot-type particle system consisting of independently moving particles each driven by generalized Bessel processes on the positive real line. Upon hitting the boundary {0} this particle is killed and an uniformly chosen different one branches into two particles. Using the symmetry of the model and the self similarity property of Bessel processes, we obtain a criterion to decide whether the particles converge to the origin at a finite time. This addresses open problem 1.4 in [6]. Specifically, inspired by [6, Open Problem 1.5], we investigate the case of three moving particles and refine the general result of [6, Theorem 1.1(ii)] extending the regime of drift parameters, where convergence does not occur – even to values, where it does occur when considering the case of only two particles. AU - Kolb, Martin AU - Liesenfeld, Matthias ID - 35649 IS - 27 JF - Electronic Journal of Probability TI - On non-extinction in a Fleming-Viot-type particle model with Bessel drift ER - TY - JOUR AB - We consider autoregressive sequences Xn = aXn−1 + ξn and Mn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in- dependent and identically distributed innovations {ξk }. It is known that if P(ξ1 > x) ∼ d log x with some d ∈ (0, − log a) then the chains {Xn} and {Mn} are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index −1 − d/ log a. We also prove limit theorems for {Xn} and {Mn} conditioned to stay over a fixed level x0. Furthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn} in the case when these chains are positive recurrent and the tail of log ξ1 is subexponential. AU - Denisov, Denis AU - Hinrichs, Günter AU - Kolb, Martin AU - Wachtel, Vitali ID - 35650 JF - Electronic Journal of Probability TI - Persistence of autoregressive sequences with logarithmic tails VL - 27 ER - TY - JOUR AU - Liebendörfer, Michael AU - Göller, Robin AU - Gildehaus, Lara AU - Kortemeyer, Jörg AU - Biehler, Rolf AU - Hochmuth, Reinhard AU - Ostsieker, Laura AU - Rode, Jana AU - Schaper, Niclas ID - 35685 IS - 5 JF - International Journal of Mathematical Education in Science and Technology KW - Applied Mathematics KW - Education KW - Mathematics (miscellaneous) SN - 0020-739X TI - The role of learning strategies for performance in mathematics courses for engineers VL - 53 ER - TY - JOUR AU - Biehler, Rolf AU - Langlotz, Hubert AU - Zappe, Wilfried ID - 35718 IS - 232 JF - mathematik lehren TI - Alles normal?! Daten auf Normalverteilung prüfen–mit schulischen Mitteln VL - 2022 ER - TY - CHAP AB - The data age has led to curriculum developments promoting the relevance of stochastics in mathematics education. This has resulted in an increased demand for teacher professional development (PD) courses, and thus in a rising number of facilitators for this purpose. Their expertise, however, remains under-researched, even though facilitators are the crucial factor in PD success. Our study aims at revealing how one aspect of facilitator expertise, orientation, is connected to other aspects: views on content goals, perception of possible challenges, and attitude towards teaching resources. The explorations reveal differing orientations and learning goals with identical teaching resources for two facilitators who had also been working as a team for the re-design of the PD course for years. AU - Nieszporek, Ralf AU - Griese, Birgit AU - Biehler, Rolf ED - Peters, S. A. ED - Zapata-Cardona, L. ED - Bonafini, F. ED - Fan, A. ID - 35696 T2 - Bridging the Gap: Empowering and Educating Today’s Learners in Statistics. Proceedings of the Eleventh International Conference on Teaching Statistics TI - Facilitators’ Orientations Towards Learning Goals, Perceived Challenges, and Teaching Resources for a PD Course on Conditional Probability ER - TY - CONF AB - Fundamental ideas (Burrill & Biehler, 2011) have been specified to structure the curriculum across different age levels and to focus on and revisit the most important ideas. Such ideas must be based on an epistemological analysis of the scientific domain of statistics and its applications. Because these domains are changing, fundamental ideas have to be revisited from time to time. In particular, data science and machine learning have led to new methods and applications in society that must be considered for updating fundamental ideas. AU - Biehler, Rolf ED - Peters, S. A. ED - Zapata-Cardona, L. ED - Bonafini, F. ED - Fan, A. ID - 34919 T2 - Bridging the Gap: Empowering and Educating Today’s Learners in Statistics. Proceedings of the Eleventh International Conference on Teaching Statistics TI - Revisiting Fundamental Ideas for Statistics Education From the Perspective of Machine Learning and Its Applications ER - TY - JOUR AB - A very warm welcome to this Special Issue of the Statistics Education Research Journal (SERJ) on data science education. Our hope is to give an overview of selected theoretical thoughts and empirical studies on data science education from a statistics education research perspective. Data science education is rapidly developing but research into data science education is still in its infancy. The current issue presents a snapshot of this developing field. AU - Biehler, Rolf AU - De Veaux, Richard AU - Engel, Joachim AU - Kazak, Sibel AU - Frischemeier, Daniel ID - 34920 IS - 2 JF - Statistics Education Research Journal KW - Education KW - Statistics and Probability SN - 1570-1824 TI - Editorial: Research on Data Science Education VL - 21 ER - TY - JOUR AU - Biehler, Rolf AU - Griese, Birgit ID - 35926 IS - 232 JF - mathematik lehren TI - Modellieren im Stochastikunterricht: Annahmen hinterfragen, Ergebnisse validieren VL - 2022 ER - TY - CONF AU - Hoffmann, Max AU - Biehler, Rolf ED - Halverscheid, Stefan ED - Kersten, Ina ED - Schmidt-Thieme, Barbara ID - 31367 T2 - Bedarfsgerechte fachmathematische Lehramtsausbildung. Analyse, Zielsetzungen und Konzepte unter heterogenen Voraussetzungen TI - Schnittstellenaufgaben in der Analysis I zur Verknüpfung von Schul- und Hochschulmathematik - Aufgabenbeispiele und Ergebnisse einer Evaluationsstudie. ER - TY - CONF AU - Biehler, Rolf AU - Hoffmann, Max ED - Isaev, Viktor ED - Eichler, Andreas ED - Loose, Frank ID - 31365 SN - 2197-8751 T2 - Professionsorientierte Fachwissenschaft. Kohärenzstiftende Lerngelegenheiten für das Lehramtsstudium Mathematik TI - Fachwissen als Grundlage fachdidaktischer Urteilskompetenz – Beispiele für die Herstellung konzeptueller Bezüge zwischen fachwissenschaftlicher und fachdidaktischer Lehre im gymnasialen Lehramtsstudium ER - TY - DATA AU - Schürmann, Mirko AU - Büdenbender-Kuklinski, C. AU - Lankeit, Elisa AU - Liebendörfer, Michael AU - Hochmuth, R. AU - Biehler, Rolf AU - Schaper, N. ID - 35698 TI - Dokumentation der Erhebungsinstrumente des Projekts WiGeMath ER - TY - JOUR AB - AbstractWhen the COVID-19 pandemic began, many universities switched to fully online teaching. This unexpected switching to online teaching was challenging for both teachers and students, and restrictions that were put in place because of pandemic made this challenge even greater. However, new ways of teaching might also open new opportunities for students’ learning. The research question driving our study was as follows: how do students regulate their learning and specifically their choice of resources and peer learning in university mathematics classes that are fully taught online as offered during the COVID-19 pandemic? We report on a longitudinal, qualitative study in which students recorded a brief audio diary twice a week over one whole semester (14 weeks). We focused on three students who completed 70 interviews in total and finished the semester with varying degrees of success. The results show how the students structured their studying (e.g., the roles that deadlines or synchronous teaching events played). They illustrate the strengths and limitations of digital materials provided by the lecturer and the use of complementary media. Further, the pandemic uncovered the double-edged role of simple, often anonymous exchanges (e.g., via Discord servers), with few binding forces for either side, and the significance of stable learning partnerships for students’ success. Our research highlights aspects that should be focal points when comparing traditional instruction and online instruction during the pandemic from a self-regulatory perspective. Practical implications refer to how these aspects can be combined sensibly in fully online courses, but also in blended learning contexts. AU - Liebendörfer, Michael AU - Kempen, Leander AU - Schukajlow, Stanislaw ID - 37470 JF - ZDM – Mathematics Education KW - General Mathematics KW - Education SN - 1863-9690 TI - First-year university students' self-regulated learning during the COVID-19 pandemic: a qualitative longitudinal study ER - TY - JOUR AB - AbstractAs earlier research results suggest that many mathematics teaching students criticize a missing relevance in their studies, we explore explanations and interrelationships of their relevance assessments. We aim at finding out how one could support the students in attributing relevance to their study programs. A two-fold model for relevance assessments in mathematics teacher education is proposed, consisting of relevance content and relevance reasons. We investigate students' relevance perceptions of mathematical topics and of topics’ complexities, as well as their rating of individual and societal/ vocational relevance reasons, all in relation to their perception of the relevance of their overall program of study. Contrary to earlier research findings, our results suggest that mathematics teaching students already do attribute relevance to many content areas and that a preparation for the teaching profession is not the only reason for them to assign relevance. There also seem to be many students who would attribute relevance if they could develop as individuals and pursue their interests. We suggest that giving students opportunities to set individual priorities in their studies could hence support their relevance assessments. As low relevance assessments seem to be connected to students’ motivational problems, students might profit from motivational support, as well. AU - Büdenbender-Kuklinski, Christiane AU - Hochmuth, Reinhard AU - Liebendörfer, Michael ID - 37472 JF - International Journal of Research in Undergraduate Mathematics Education KW - Education KW - Mathematics (miscellaneous) SN - 2198-9745 TI - Exploring the Perceived Relevance of University Mathematics Studies by First-Semester Teaching Students ER -