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Context.Blazars, which include BL Lacs and flat-spectrum radio quasars, represent the brightest persistent extragalactic sources in the high-energy (HE; 10 MeV–100 GeV) and very-high-energy (VHE;E > 100 GeV)γ-ray sky. Due to their almost featureless optical/UV spectra, it is challenging to measure the redshifts of BL Lacs. As a result, about 50% ofγ-ray BL Lacs lack a firm measurement of this property, which is fundamental for population studies, indirect estimates of the extragalactic background light, and fundamental physics probes (e.g., searches for Lorentz-invariance violation or axion-like particles).

Aims.This paper is the third in a series of papers aimed at determining the redshift of a sample of blazars selected as prime targets for future observations with the next generation, ground-based VHEγ-ray astronomy observatory, Cherenkov Telescope Array Observatory (CTAO). The accurate determination of the redshift of these objects is an important aid in source selection and planning of future CTAO observations.

Methods.Promising targets were selected following a sample selection obtained with Monte Carlo simulations of CTAO observations. The selected targets were expected to be detectable with CTAO in observations of 30 h or less. We performed deep spectroscopic observations of 41 of these blazars using the Keck II, Lick, SALT, GTC, and ESO/VLT telescopes. We carefully searched for spectral lines in the spectra and whenever features of the host galaxy were detected, we attempted to model the properties of the host galaxy. The magnitudes of the targets at the time of the observations were also compared to their long-term light curves.

Results.Spectra from 24 objects display spectral features or a high signal-to-noise ratio (S/N). From these, 12 spectroscopic redshifts were determined, ranging from 0.2223 to 0.7018. Furthermore, 1 tentative redshift (0.6622) and 2 redshift lower limits atz > 0.6185 andz > 0.6347 were obtained. The other 9 BL Lacs showed featureless spectra, despite the high S/N (≥100) observations. Our comparisons with long-term optical light curves tentatively suggest that redshift measurements are more straightforward during an optical low state of the active galactic nucleus. Overall, we have determined 37 redshifts and 6 spectroscopic lower limits as part of our programme thus far.

 

Although "developmental math" is widely discussed in higher-education circles, exactly what developmental math encompasses is often underdeveloped. In this theoretical report, we use a sample of highly cited works on developmental math to identify common characterizations of the term "developmental math" in the literature. We then interrogate and problematize each characterization, particularly in terms of whether they serve equity-related goals such as access to college credentials and math learning. We close by proposing an alternative characterization of developmental math and discuss the theoretical implications. We see this as a first step towards conversations about how developmental math could be conceptualized. 

In this theoretical paper, our aim is to start a conversation about how "levels" in mathematics are operationalized and defined, with a specific focus on "college level." We approach this from a lens of developmental stages, using this to propose an initial framework for describing how learners might progress along a developmental continuum delineated by the kinds of reasoning/justification, generalization/abstraction, and types of conceptions that they hold, rather than by the particular computations learners are able to do, or the kinds of mathematical objects with which learners are engaging. 

We use the Very Energetic Radiation Imaging telescope Array System (VERITAS) imaging air Cherenkov telescope array to obtain the first measured angular diameter ofβUMa at visual wavelengths using stellar intensity interferometry (SII) and independently constrain the limb-darkened angular diameter. The age of the Ursa Major moving group has been assessed from the ages of its members, including nuclear member Merak (βUMa), an A1-type subgiant, by comparing effective temperature and luminosity constraints to model stellar evolution tracks. Previous interferometric limb-darkened angular-diameter measurements ofβUMa in the near-infrared (Center for High Angular Resolution Astronomy (CHARA) Array, 1.149 ± 0.014 mas) and mid-infrared (Keck Nuller, 1.08 ± 0.07 mas), together with the measured parallax and bolometric flux, have constrained the effective temperature. This paper presents current VERITAS-SII observation and analysis procedures to derive squared visibilities from correlation functions. We fit the resulting squared visibilities to find a limb-darkened angular diameter of 1.07 ± 0.04 (stat) ± 0.05 (sys) mas, using synthetic visibilities from a stellar atmosphere model that provides a good match to the spectrum ofβUMa in the optical wave band. The VERITAS-SII limb-darkened angular diameter yields an effective temperature of 9700 ± 200 ± 200 K, consistent with ultraviolet spectrophotometry, and an age of 390 ± 29 ± 32 Myr, using MESA Isochrones and Stellar Tracks. This age is consistent with 408 ± 6 Myr from the CHARA Array angular diameter.

 

Although a wide variety of reports on developmental math exist, to date there has not been a large-scale examination of existing work from a math education point of view. Towards this goal, we analyzed 426 reports and peer-reviewed journal articles relating to developmental math published between 2000 and 2020. In report, we quantify the publishers and intended audience, examine the types of outcomes reported on and, where possible, examine the type of developmental math model discussed. We find that over the last decade, less than 20% of reports on developmental math have been aimed at math education audiences. While math education publications more frequently examine math knowledge and student experiences, the overall number of reports, compared to those examining pass rates, is relatively small. 

Active galactic nuclei (AGNs) make up about 35 per cent of the more than 250 sources detected in very high-energy (VHE) gamma rays to date with the imaging atmospheric Cherenkov telescopes. Apart from four nearby radio galaxies and two AGNs of unknown type, all known VHE AGNs are blazars. Knowledge of the cosmological redshift of gamma-ray blazars is key to enabling the study of their intrinsic emission properties, as the interaction between gamma rays and the extragalactic background light (EBL) results in a spectral softening. Therefore, the redshift determination exercise is crucial to indirectly placing tight constraints on the EBL density, and to studying blazar population evolution across cosmic time. Due to the powerful relativistic jets in blazars, most of their host galaxies’ spectral features are outshined, and dedicated high signal-to-noise (S/N) spectroscopic observations are required. Deep medium- to high-resolution spectroscopy of 33 gamma-ray blazar optical counterparts was performed with the European Southern Observatory, New Technology Telescope, Keck II telescope, Shane 3-metre telescope, and the Southern African Large Telescope. From the sample, spectra from 25 objects display spectral features or are featureless and have high S/N. The other eight objects have low-quality featureless spectra. We systematically searched for absorption and emission features and estimated, when possible, the fractional host galaxy flux in the measured total flux. Our measurements yielded 14 firm spectroscopic redshifts, ranging from 0.0838 to 0.8125, one tentative redshift, and two lower limits: one at $z > 0.382$ and the other at z > 0.629.

 

G106.3+2.7, commonly considered to be a composite supernova remnant (SNR), is characterized by a boomerang-shaped pulsar wind nebula (PWN) and two distinct (“head” and “tail”) regions in the radio band. A discovery of very-high-energy gamma-ray emission (E_γ> 100 GeV) followed by the recent detection of ultrahigh-energy gamma-ray emission (E_γ> 100 TeV) from the tail region suggests that G106.3+2.7 is a PeVatron candidate. We present a comprehensive multiwavelength study of the Boomerang PWN (100″ around PSR J2229+6114) using archival radio and Chandra data obtained two decades ago, a new NuSTAR X-ray observation from 2020, and upper limits on gamma-ray fluxes obtained by Fermi-LAT and VERITAS observatories. The NuSTAR observation allowed us to detect a 51.67 ms spin period from the pulsar PSR J2229+6114 and the PWN emission characterized by a power-law model with Γ = 1.52 ± 0.06 up to 20 keV. Contrary to the previous radio study by Kothes et al., we prefer a much lower PWNB-field (B∼ 3μG) and larger distance (d∼ 8 kpc) based on (1) the nonvarying X-ray flux over the last two decades, (2) the energy-dependent X-ray size of the PWN resulting from synchrotron burn-off, and (3) the multiwavelength spectral energy distribution (SED) data. Our SED model suggests that the PWN is currently re-expanding after being compressed by the SNR reverse shock ∼1000 yr ago. In this case, the head region should be formed by GeV–TeV electrons injected earlier by the pulsar propagating into the low-density environment.