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This paper presents six categories of undergraduate student explanations and justifications regarding the question of whether a converse proof proves a conditional theorem. Two categories of explanation led students to judge that converse proofs cannot so prove, which is the normative interpretation. These judgments depended upon students spontaneously seeking uniform rules of proving across various theorems or assigning a direction to the theorems and proof. The other four categories of explanation led students to affirm that converse proofs prove. We emphasize the rationality of these non-normative explanations to suggest the need for further work to understand how we can help students understand the normative rules of logic. 

Mathematicians often use set-builder notation and set diagrams to define and show relationships between sets in proof-related courses. This paper describes various meanings that students might attribute to these representations. Our data consist of students’ initial attempts to create and interpret these representations during the first day of a paired teaching experiment. Our analysis revealed that neither student imputed or attributed our desired theoretical meanings to their diagrams or notation. We summarize our findings in two vignettes, one describing students’ attributed meanings to instructor-provided set-builder notation and the other describing students’ imputed meanings to their personally-created set diagrams to relate pairs of sets. 

This study explores how instructional interventions and teacher moves might support students’ learning of logic in mathematical contexts. We conducted an exploratory teaching experiment with a pair of undergraduate students to leverage set-based reasoning for proofs of conditional statements. The students initially displayed a lack of knowledge of contrapositive equivalence and converse independence in validating if a given proof-text proves a given theorem. However, they came to conceive of these logical principles as the teaching experiment progressed. We will discuss how our instructional interventions played a critical role in facilitating students’ joint reflection and modification of their reasoning about contrapositive equivalence and converse independence in reading proofs. 

Learning to interpret proofs is an important milepost in the maturity and development of students of higher mathematics. A key learning objective in proof-based courses is to discern whether a given proof is a valid justification of its underlying claim. In this study, we presented students with conditional statements and associated proofs and asked them to determine whether the proofs proved the statements and to explain their reasoning. Prior studies have found that inexperienced provers often accept the proof of a statement’s converse and reject proofs by contraposition, which are both erroneous determinations. Our study contributes to the literature by corroborating these findings and suggesting a connection between students’ reading comprehension and proof validation behaviors and their beliefs about mathematical proof and mathematical knowledge base. 

We present bolometric luminosities, black hole masses, and Eddington ratios for 42 luminous quasars at z  ≳ 6 using high signal-to-noise ratio VLT/X-shooter spectra, acquired as part of the enlarged ESO Large Programme XQR-30 . In particular, we derived the bolometric luminosities from the rest-frame 3000 Å luminosities using a bolometric correction from the literature, as well as the black hole masses by modeling the spectral regions around the C  IV 1549 Å and the Mg  II 2798 Å emission lines, with scaling relations calibrated in the Local Universe. We find that the black hole masses derived from both emission lines are in the same range and the scatter of the measurements agrees with expectations from the scaling relations. The Mg  II -derived masses are between ∼(0.8−12) ×10 9   M ⊙ and the derived Eddington ratios are within ∼0.13−1.73, with a mean (median) of 0.84(0.72). By comparing the total sample of quasars at z  > 5.8, from this work and from the literature, to a bolometric luminosity distribution-matched sample at z  ∼ 1.5, we find that quasars at high redshift host slightly less massive black holes, which accrete slightly more rapidly than those at lower z , with a difference in the mean Eddington ratios of the two samples of ∼0.27. These findings are in agreement with the results of recent works in the literature. 

We present the discovery of an Earth-mass planet (M_bsini= 1.26 ± 0.21M_⊕) on a 15.6 d orbit of a relatively nearby (d ~9.6 pc) and low-mass (0.167 ± 0.011M_⊙) M5.0 V star, Wolf 1069. Sitting at a separation of 0.0672 ± 0.0014 au away from the host star puts Wolf 1069 b in the habitable zone (HZ), receiving an incident flux ofS= 0.652 ± 0.029S_⊕. The planetary signal was detected using telluric-corrected radial-velocity (RV) data from the CARMENES spectrograph, amounting to a total of 262 spectroscopic observations covering almost four years. There are additional long-period signals in the RVs, one of which we attribute to the stellar rotation period. This is possible thanks to our photometric analysis including new, well-sampled monitoring campaigns undergone with the OSN and TJO facilities that supplement archival photometry (i.e., from MEarth and SuperWASP), and this yielded an updated rotational period range ofP_rot= 150–170 d, with a likely value at 169.3_−3.6^+3.7. The stellar activity indicators provided by the CARMENES spectra likewise demonstrate evidence for the slow rotation period, though not as accurately due to possible factors such as signal aliasing or spot evolution. Our detectability limits indicate that additional planets more massive than one Earth mass with orbital periods of less than 10 days can be ruled out, suggesting that perhaps Wolf 1069 b had a violent formation history. This planet is also the sixth closest Earth-mass planet situated in the conservative HZ, after Proxima Centauri b, GJ 1061 d, Teegarden’s Star c, and GJ 1002 b and c. Despite not transiting, Wolf 1069 b is nonetheless a very promising target for future three-dimensional climate models to investigate various habitability cases as well as for sub-m s⁻¹RV campaigns to search for potential inner sub-Earth-mass planets in order to test planet formation theories.

 

The final phase of the reionization process can be probed by rest-frame UV absorption spectra of quasars at z ≳ 6, shedding light on the properties of the diffuse intergalactic medium within the first Gyr of the Universe. The ESO Large Programme ‘XQR-30: the ultimate XSHOOTER legacy survey of quasars at z ≃ 5.8–6.6’ dedicated ∼250 h of observations at the VLT to create a homogeneous and high-quality sample of spectra of 30 luminous quasars at z ∼ 6, covering the rest wavelength range from the Lyman limit to beyond the Mg ii emission. Twelve quasar spectra of similar quality from the XSHOOTER archive were added to form the enlarged XQR-30 sample, corresponding to a total of ∼350 h of on-source exposure time. The median effective resolving power of the 42 spectra is R ≃ 11 400 and 9800 in the VIS and NIR arm, respectively. The signal-to-noise ratio per 10 km s−1 pixel ranges from ∼11 to 114 at λ ≃ 1285 Å rest frame, with a median value of ∼29. We describe the observations, data reduction, and analysis of the spectra, together with some first results based on the E-XQR-30 sample. New photometry in the H and K bands are provided for the XQR-30 quasars, together with composite spectra whose characteristics reflect the large absolute magnitudes of the sample. The composite and the reduced spectra are released to the community through a public repository, and will enable a range of studies addressing outstanding questions regarding the first Gyr of the Universe.