51 Mathematik
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SCALE-UP (Student-Centered Active Learning Environment for Upside-down Pedagogies) is a pedagogic approach to foster active learning, originally developed for undergraduate physics courses (Beichner et al., 2007). Instead of receiving lectures in SCALE1UP students work actively together in small groups, mostly at round tables and are guided by teaching staff.
This concept is implemented at Rosenheim University of Applied Sciences, Germany, in several classes of physics and mathematics for engineering students. Classroom observations have been carried out in summer term 2023and winter term 2023/24 in order to investigate student learning in the SCALE-UP setting. The observations are based on ELCOT-3, the Engineering Learning Classroom Observation Tool (Sanders et al., 2018). An observation includes the overt student behavior in combination with the students’ output. The data have been analyzed with respect to the ICAP-framework (Chi&Wylie, 2014), where the different observed activities have been assigned to Interactive, Constructive, Active, and Passive mode. In this contribution we present examples on how students interact with the offered material, that means in which ICAP-modes students work with different learning activities in the SCALE-UP setting. Moreover, we show that in all observed courses students have been learning a substantial part of class time in interactive and constructive mode.
Um aktives Lernen in Lehrveranstaltungen zu klassifizieren, bietet das ICAP-Modell (Chi & Wylie, 2014) eine vierstufige Taxonomie, die in der Lehrpraxis anwendbar ist. Das von außen beobachtbare Verhalten wird in Kombination mit den von den Studierenden erzeugten „Produkten“ (z.B. ihren schriftlichen und mündlichen Äußerungen) verwendet, um das kognitive Engagement der Studierenden in Lehrveranstaltungen hinreichend gut einzuordnen (𝐼 Interactive, 𝐶 Constructive, 𝐴 Active und 𝑃 Passive). Die ICAP-Hypothese postuliert, dass es eine Hierarchie der ICAP-Stufen 𝐼 > 𝐶 > 𝐴 > 𝑃 hinsichtlich der Tiefe der zu erwartenden Lernergebnisse gibt.
In diesem Beitrag berichten wir über Erkenntnisse aus systematischen Beobachtungen hinsichtlich des ICAP-Modells in Physik- und Mathematik-Lehrveranstaltungen an der TH Rosenheim. Die Lehrveranstaltungen finden nach dem SCALE-UP Raum- und Lehrkonzept statt (Beichner et al., 2007). Studierenden arbeiten dabei größtenteils an unterschiedlichen Lernaktivitäten in kleinen Gruppen an runden Tischen, während die Lehrperson die Rolle eines „guide on the side“ einnimmt. Das Verhalten der Studierenden und der Lehrperson wird in zweiminütigen Zeitintervallen zusammen mit dem „output“ der Studierenden registriert und auf die ICAP-Stufen abgebildet. Die Ergebnisse dienen der Reflexion und unterstützen die Planung und Verbesserung geeigneter Lernaktivitäten, insbesondere um zwischen den Stufen 𝐶 und 𝐴 zu unterscheiden.
We give a presentation in terms of generators and relations of Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. As a consequence we get an analog presentation of Nichols algebras of diagonal type.
Lifting of Nichols Algebras
(2009)
Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classi cation program of nite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider [6] is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. The classi cation was carried out in this way in [11] when the group of group-like elements is abelian and the prime divisors of the order of the group are > 7. In this case the appearing Nichols algebras are of Cartan type.
Based on recent work of Heckenberger about diagonal Nichols algebras [29, 28, 27] we compute explicitly the liftings of some Nichols algebras not treated in [11]; namely we lift
- all Nichols algebras with Cartan matrix of type A2 (Theorem 6.3.3),
- some Nichols algebras with Cartan matrix of type B2 (Theorem 6.4.3), and
- some Nichols algebras of two Weyl equivalence classes of non-standard type (Theorem 6.5.3),
giving new classes of nite-dimensional pointed Hopf algebras.
Crucial is the knowledge of a good presentation of the Nichols algebra and its liftings: We want to have an explicit description in terms of generators and (non-redundant) relations, and a basis; this requires new ideas and methods that generalize those in [11].
In this spirit, we describe Hopf algebras generated by skew-primitive elements and an abelian group with action given via characters (including Nichols algebras and their liftings) in Theorem 5.4.1. The relations form a Grobner basis and are given by a combinatorial property involving the theory of Lyndon words.
Furthermore, in Theorem 7.3.1 we give a necessary and su cient criterion to check whether a given set of iterated q-commutators establishes a restricted PBW basis for a given realization of the relations. Also with the help of this criterion we determine the redundant relations in the examined Nichols algebras and their liftings.
Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classification program of finite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. The classification was carried out in this way in several special cases when the appearing Nichols algebras are of Cartan type. We compute explicitly the liftings of some Nichols algebras not of Cartan type or of Cartan type without restrictions on the group order. These build up new classes of finite-dimensional pointed Hopf algebras. Furthermore, we give a necessary and sufficient PBW basis criterion for a class of pointed Hopf algebras and present them in terms of generators and relations. These Hopf algebras can be seen as generalized Quantum groups.
We give a necessary and sufficient PBW basis criterion for Hopf algebras generated by skew-primitive elements and abelian group of group-like elements with action given via characters. This class of pointed Hopf algebras has shown great importance in the classification theory and can be seen as generalized quantum groups. We apply the criterion to classical examples and liftings of Nichols algebras which were determined in arxiv.org:1003.5882.
We propose a group theory based mathematical model for unambiguous inductance states of a reconfigurable air coil inductor utilizing elastic snapping of wire segments. Vertical (axial) and horizontal winding symmetries under group actions reveal distinguishable tuning regions and individual variation states therein. The model is applied to coils of prismatic (cylinder like) and truncated pyramidal (conical like) shape with edges forming a regular polygon as basis, and verified by numerical magnet field simulation.
