Open Access

Die kognitive Aktivierung ist eine der drei Basisdimensionen der Unterrichtsqualität (Klieme, 2019) und findet mittlerweile Eingang in international angelegte Modelle der Unterrichtsqualität (Bell et al., 2019; Charalambous & Praetorius, 2020). Die Dimension wurde bereits in einer Vielzahl von Studien, in verschiedenen Schulfächern und über verschiedene Schulformen hinweg empirisch untersucht (Praetorius et al., 2018). Dabei wurde die kognitive Aktivierung im Rahmen von Angebots-Nutzungs-Modellen (Fend, 2019) überwiegend als ein angebotsseitiges Potenzial der Lehrperson für die Schüler*innen operationalisiert (Denn et al., 2019). Hingegen ist bislang wenig darüber bekannt, wie kognitive aktivierende Impulse in der Interaktion zwischen Lehrperson und den Schüler*innen hergestellt und bearbeitet werden (Renkl, 2015; Vieluf, 2022). In dieser Studie werden mithilfe der wissenssoziologisch fundierten Dokumentarischen Methode (Bohnsack, 2021) und ihrer Spezifizierung für die Analyse von Unterrichtsvideographien (Asbrand & Martens, 2018) die im Unterricht kommunizierten und handlungsleitenden, implizierten Wissensbestände rekonstruiert, die die Hervorbringung von kognitiver Aktivierung in der Interaktion bedingen. Es wird danach gefragt, wie kognitive Aktivierung in der Interaktion zwischen Lernenden und Lehrenden hergestellt und prozessiert wird. Als Datengrundlage dienen überwiegend Videos aus dem Mathematikunterricht der neunten Klasse zum Thema quadratische Gleichung aus der TALIS Videostudie Deutschland (Grünkorn et al., 2020). Als Ergebnis ließen sich drei unterschiedlichen Formen der Aktivierung rekonstruieren. Typ I: Aktivierung zu Reproduktion ist durch ein instruktivistisches Verständnis der Lehrkraft geprägt, in dem aktivierende Impulse die Schüler*innen überwiegend zur Reproduktion von Wissen anregen. Typ II: Aktivierung zu unsystematischem Probieren wird durch ein vermittelndes Verständnis der Lehrperson bestimmt, bei dem die Impulse nicht an das bestehende Wissen der Schüler*innen anschließen und die Bearbeitung im Rahmen eines unsystematischen Probierens erfolgt. Typ III: Aktivierung zu fachlicher Konstruktion ist durch ein konstruktivistisches Unterrichtsverständnis der Lehrkraft gekennzeichnet und Impulse werden in einem ko-konstruktiven Prozess von den Schülern*innen in Zusammenarbeit mit der Lehrkraft bearbeitet.

Uniform sampling from the set G(d) of graphs with a given degree-sequence d=(d1,…,dn)∈Nn is a classical problem in the study of random graphs. We consider an analogue for temporal graphs in which the edges are labeled with integer timestamps. The input to this generation problem is a tuple D=(d,T)∈Nn×N>0 and the task is to output a uniform random sample from the set G(D) of temporal graphs with degree-sequence d and timestamps in the interval [1,T]. By allowing repeated edges with distinct timestamps, G(D) can be non-empty even if G(d) is, and as a consequence, existing algorithms are difficult to apply. We describe an algorithm for this generation problem which runs in expected time O(M) if Δ2+ϵ=O(M) for some constant ϵ>0 and T−Δ=Ω(T) where M=∑idi and Δ=maxidi. Our algorithm applies the switching method of McKay and Wormald [1] to temporal graphs: we first generate a random temporal multigraph and then remove self-loops and duplicated edges with switching operations which rewire the edges in a degree-preserving manner.

n this paper we study invasion probabilities and invasion times of cooperative parasites spreading in spatially structured host populations. The spatial structure of the host population is given by a random geometric graph on [0,1]n, n∈N, with a Poisson(N)-distributed number of vertices and in which vertices are connected over an edge when they have a distance of at most rN∈Θ(Nβ−1n) for some 0<β<1 and N→∞. At a host infection many parasites are generated and parasites move along edges to neighbouring hosts. We assume that parasites have to cooperate to infect hosts, in the sense that at least two parasites need to attack a host simultaneously. We find lower and upper bounds on the invasion probability of the parasites in terms of survival probabilities of branching processes with cooperation. Furthermore, we characterize the asymptotic invasion time. An important ingredient of the proofs is a comparison with infection dynamics of cooperative parasites in host populations structured according to a complete graph, i.e. in well-mixed host populations. For these infection processes we can show that invasion probabilities are asymptotically equal to survival probabilities of branching processes with cooperation. Furthermore, we build in the proofs on techniques developed in [BP22], where an analogous invasion process has been studied for host populations structured according to a configuration model. We substantiate our results with simulations.

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as N→∞.

We consider ground state solutions u ∈ H2(RN) of biharmonic (fourth-order) nonlinear Schrodinger equations of the form ¨2u + 2au + bu − |u| p−2u = 0 in RN with positive constants a, b > 0 and exponents 2 < p < 2∗, where 2∗ = 2N N−4 if N > 4 and 2∗ = ∞ if N ≤ 4. By exploiting a connection to the adjoint Stein–Tomas inequality on the unit sphere and by using trial functions due to Knapp, we prove a general symmetry breaking result by showing that all ground states u ∈ H2(RN) in dimension N ≥ 2 fail to be radially symmetric for all exponents 2 < p < 2N+2 N−1 in a suitable regime of a, b > 0. As applications of our main result, we also prove symmetry breaking for a minimization problem with constrained L2-mass and for a related problem on the unit ball in RN subject to Dirichlet boundary conditions.