We apply a new non-parametric technique to reconstruct, with uncertainties, the projected mass distribution of the inner region of Abell 2218, using combined strong and weak lensing constraints from multiple-image systems and arclets with known redshifts. The reconstructed mass map broadly resembles previous, less detailed, parametric models, but when examined in detail shows several sub-structures, not necessarily associated with light but strongly required by the lensing data. In particular, the highest mass peak is offset by ∼ 30h −1 50 kpc from the main light peak, and projected mass-to-light in the directions of different cluster galaxies varies by at least a factor of 10. On comparing with mass estimates from models of the X-ray emitting gas, we find that the X-ray models under-predict the enclosed mass profile by, at least a factor of 2.5; the discrepancy gets worse if we assume that mass traces light to the extent allowed by the lensing constraints.

A B S T R A C TWe describe a new non-parametric technique for reconstructing the mass distribution in galaxy clusters with strong lensing, i.e. from multiple images of background galaxies. The observed positions and redshifts of the images are considered as rigid constraints, and through the lens (ray-trace) equation they provide us with linear constraint equations. These constraints confine the mass distribution to some allowed region, which is then found by linear programming.Within this allowed region we study in detail the mass distribution with minimum mass-to-light variation, and also some other distributions, such as the smoothest mass distribution.The method is applied to the extensively studied cluster Abell 370, which hosts a giant luminous arc and several other multiply imaged background galaxies. Our mass maps are constrained by the observed positions and redshifts (spectroscopic or model-inferred by previous authors) of the giant arc and multiple-image systems. The reconstructed maps obtained for Abell 370 reveal a detailed mass distribution, with substructure quite different from the light distribution. The method predicts the bimodal nature of the cluster, and that the projected mass distribution is indeed elongated along the axis defined by the two dominant cD galaxies. However, the peaks in the mass distribution appear to be offset from the centres of the cDs.We also present an estimate for the total mass of the central region of the cluster. This is in good agreement with previous mass determinations. The total mass of the central region is M:(2.0-2.7) 10 14 M ᖿ h 21 50 , depending on the solution chosen.

The present research examines the contribution of individual differences in chronotype and self-efficacy to the math performance of male and female students in STEM and no-STEM majors. Questionnaires assessing the selected individual differences were distributed to students of Middle Eastern descent enrolled in math courses of the general education curriculum. Summative assessment indices were used to measure performance comprehensively across the entire semester (course grades) and as a one-time occurrence (final test grades). The contribution of morningness and self-efficacy to both course and test performance of STEM students was sensitive to the interaction of gender and major. Instead, neither factor contributed to no-STEM students’ course and test performance. These findings were used to plan improvements in the instruction and advising of students in STEM majors, thereby complying with a key tenet of action research.

A B S T R A C TWe describe a new non-parametric technique for reconstructing the mass distribution in galaxy clusters with strong lensing, i.e. from multiple images of background galaxies. The observed positions and redshifts of the images are considered as rigid constraints, and through the lens (ray-trace) equation they provide us with linear constraint equations. These constraints confine the mass distribution to some allowed region, which is then found by linear programming. Within this allowed region we study in detail the mass distribution with minimum mass-to-light variation, and also some other distributions, such as the smoothest mass distribution.The method is applied to the extensively studied cluster Abell 370, which hosts a giant luminous arc and several other multiply imaged background galaxies. Our mass maps are constrained by the observed positions and redshifts (spectroscopic or model-inferred by previous authors) of the giant arc and multiple-image systems. The reconstructed maps obtained for Abell 370 reveal a detailed mass distribution, with substructure quite different from the light distribution. The method predicts the bimodal nature of the cluster, and that the projected mass distribution is indeed elongated along the axis defined by the two dominant cD galaxies. However, the peaks in the mass distribution appear to be offset from the centres of the cDs.We also present an estimate for the total mass of the central region of the cluster. This is in good agreement with previous mass determinations. The total mass of the central region is M:(2.0-2.7) 10 14 M ᖿ h 21 50 , depending on the solution chosen.

The present study was driven by the assumption that a key feature of sustainable education is its ability to preserve standards of quality even amid unforeseen, potentially disruptive events. It asked whether students’ academic success in math general education courses differed between synchronous online (during the COVID-19 pandemic) and face-to-face (before the pandemic), under the ancillary assumption that computational competency, a pillar of sustainable education, shapes enduring success in a variety of professional fields. As the early identification of at-risk students and ensuing remedial interventions can bring about academic success, the study also investigated the predictive validity of students’ initial performance in online and face-to-face math courses. Two general education courses (introductory calculus and statistics), taught by the same instructor, were selected. Class grades did not differ between instructional modes, thereby providing no evidence for the widespread concern that the switch to the online mode had damaged learning. Yet, during the semester, test and homework performance were differentially sensitive to modes of instruction. Furthermore, both test and homework performance during the first half of the semester predicted class grades in online courses, whereas only test performance predicted class grades in face-to-face courses. These results suggest that sustainable math education in times of crisis is feasible and that educators’ consideration of the differential predictive value of test and homework performance may aid its attainment.