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This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students’ meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or LSLS). We argue that students’ use of Euler diagrams as representations of logical statements entails a conceptual linking between spatial and non-spatial representations, and the LSLS framework provides a tool for modeling this conceptual linking. Moreover, from our Piagetian epistemological perspective, reasoning with Euler diagrams entails engaging in spatial mental operations and making a logical conclusion from the result. We illustrate the utility of the LSLS framework through examples with two undergraduate students as they reasoned about the truth of the converse and contrapositive of a given logical implication, and we identify specific spatial operations that they used and coordinated in their problem solving. 

Understanding how students reason with logical implication is essential for supporting students’ construction of increasingly powerful ways of reasoning in proofs-based mathematics courses. We report on the results of an NSF-funded case study with a mathematics major enrolled in an introductory proofs course. We investigate the epistemological obstacles that she experienced and how they might relate to her treatment of logical implications as actions. Evidence shows that an action conception may pose challenges when students transform or quantify implications and may contribute to erroneous assumptions of biconditionality. Our report on available ways of operating with logical implications as actions is a first step in designing instructional tasks that leverage students’ existing reasoning skills to support their continued development. 

We present 50-fs, single-shot measurements of the x-ray thermal diffuse scattering (TDS) from copper foils that have been shocked via nanosecond laser ablation up to pressures above ∼135 GPa. We hence deduce the x-ray Debye–Waller factor, providing a temperature measurement. The targets were laser-shocked with the DiPOLE 100-X laser at the High Energy Density endstation of the European X-ray Free-Electron Laser. Single x-ray pulses, with a photon energy of 18 keV, were scattered from the samples and recorded on Varex detectors. Despite the targets being highly textured (as evinced by large variations in the elastic scattering) and with such texture changing upon compression, the absolute intensity of the azimuthally averaged inelastic TDS between the Bragg peaks is largely insensitive to these changes, and allowing for both Compton scattering and the low-level scattering from a sacrificial ablator layer provides a reliable measurement of T/ΘD2, where ΘD is the Debye temperature. We compare our results with the predictions of the SESAME 3336 and LEOS 290 equations of state for copper and find good agreement within experimental errors. We, thus, demonstrate that single-shot temperature measurements of dynamically compressed materials can be made via thermal diffuse scattering of XFEL radiation. 

X-ray free electron laser (XFEL) sources coupled to high-power laser systems offer an avenue to study the structural dynamics of materials at extreme pressures and temperatures. The recent commissioning of the DiPOLE 100-X laser on the high energy density (HED) instrument at the European XFEL represents the state-of-the-art in combining x-ray diffraction with laser compression, allowing for compressed materials to be probed in unprecedented detail. Here, we report quantitative structural measurements of molten Sn compressed to 85(5) GPa and ∼3500 K. The capabilities of the HED instrument enable liquid density measurements with an uncertainty of ∼1% at conditions which are extremely challenging to reach via static compression methods. We discuss best practices for conducting liquid diffraction dynamic compression experiments and the necessary intensity corrections which allow for accurate quantitative analysis. We also provide a polyimide ablation pressure vs input laser energy for the DiPOLE 100-X drive laser which will serve future users of the HED instrument. 

A bstract The NA62 experiment at CERN targets the measurement of the ultra-rare $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ K + → π + ν ν ¯ decay, and carries out a broad physics programme that includes probes for symmetry violations and searches for exotic particles. Data were collected in 2016–2018 using a multi-level trigger system, which is described highlighting performance studies based on 2018 data. 

A bstract A sample of 2 . 8 × 10 4 K + → π + μ + μ − candidates with negligible background was collected by the NA62 experiment at the CERN SPS in 2017–2018. The model-independent branching fraction is measured to be (9 . 15 ± 0 . 08) × 10 − 8 , a factor three more precise than previous measurements. The decay form factor is presented as a function of the squared dimuon mass. A measurement of the form factor parameters and their uncertainties is performed using a description based on Chiral Perturbation Theory at $$ \mathcal{O} $$ O ( p 6 ).