adsabs/AstroCLIMB huggingface dataset:
Paper: Are “Changing-look” Active Galactic Nuclei Special in the Coevolution of Supermassive Black Holes and Their Hosts? I.
Authors: J. Wang, W. K. Zheng, T. G. Brink, D. W. Xu, A. V. Filippenko, C. Gao, C. H. Xie, J. Y. Wei
Figure caption: WISE MIR light curves of the eight “partially obscured” CL-AGNs. Each light curve is binned by averaging the measurements within one day. In each panel, the vertical black lines mark the epochs of our optical spectra, where the long-dashed line corresponds to a “turn-on” state and the short-dashed one to a “turn-off” state.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: Simulations used to quantify the observational bias discussed in Appendix C.2. The left panels show simulated light curves with different levels of noise. The input model is shown in the top panel, and has a time of first light t 0 = 0.0, power-law index β 1 = 2.0, power-law index β 2 = − 0.015, and rise time t rise = 17.25 days. The light curve is sampled at 30 minutes cadence and scaled so that the peak flux is unity. Different levels of noise are added in the lower panels, logarithmically spaced between 1% and 100% of the peak flux. The maximum likelihood models from dynesty are shown with red lines, and the recovered value of t 0 is shown with the vertical dashed red lines. The right panel shows the recovered values of power-law index β 1 as a function of the recovered time of first light t 0, with the true values of the input model shown by dashed black lines. The color coding gives the logarithm of the noise amplitudes from the left panel. There is a bias in the recovered parameters for light curves with noise amplitude ≳10% of the peak.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: Analysis of residual systematic errors after fitting a curved power-law model. As an example, the left panel shows the rms scatter of the residuals on 30 or 10 minute timescales and on 8 hr timescales. For Gaussian noise, the ratio of the rms scatter on 30 or 10 minute timescales and 8 hr timescales should be ﹩\sqrt{16}﹩ or ﹩\sqrt{48}﹩, respectively. These ratios are shown with the solid lines. The middle panel shows the rms scatter scaled to the observed 30 or 10 minute scatter for SN light-curve residuals after binning to 2, 4, 8, 16, 24, 32, 64, and 128 hr (the solid lines connect points for individual SNe—only 50 random SNe are shown for clarity). The thick line shows the expected trend for Gaussian scaling. We calculate the ratio of the Gaussian expectation to the observed rms scatter at each timescale, and take the maximum across timescales for an individual SN as a metric for the quality of the light curve. The right panel shows the distributions of the scaled rms scatter at each timescale (the distributions do not show how individual SN light-curve residuals vary across timescales). The tails of the distribution below the expectation for Gaussian noise are due to sampling error, as there are fewer than 20, 10, and 5 points after binning to 32, 64, and 128 hr.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: Comparison of TESSreduce light curves (blue) with our difference imaging ISIS light curves (black). Both light curves have consistent units but are offset for clarity. The top panel shows an example of an SN with a better light curve from TESSreduce than from ISIS; there are 20 such sources, and we use the TESSreduce light curves in these cases. The other panels shows cases where the TESSreduce and ISIS light curves are consistent, the ISIS light curve is better, or both light curves are of comparable quality but have different shapes and amplitudes. See Appendix A.2 for details.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: Correlations of power-law index β 1 and rise time t rise with peak magnitude T peak, dynamic range ΔT, and redshift. We only show the 74 SNe in the “High Dynamic Range” sample, and we only show values of t rise for the 46 SNe that TESS observed the peak (see Section 2.1 and Appendix C). The color bar gives the noise amplitude relative to peak, showing that light curves with large noise amplitudes tend to have lower values of β 1 and t rise. This correlation is a measurement bias found by Miller et al. (2020b), which we discuss in Appendix C.2.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: SN light curves approaching Gaussian statistics that prefer companion signatures in addition to a curved power-law model. Note that the evidence for the companion interaction vanishes when using the BIC for all three SN. For SN2022ajw, the Bayes factor is just below our fiducial threshold. The panels are formatted and marked in the same way as in Figure 3.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: SN light curves approaching Gaussian statistics that disfavor companion signatures. The panels are formatted and marked in the same way as in Figure 3. SN2022eyw has excess noise from scattered light before the time of first light, but the early rise of the SN itself is reliable.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: SN light curves with very large Bayes factors or BIC values that strongly prefer companion interactions, but are false positives due to correlated noise and/or data gaps. The panels are formatted and marked in the same way as in Figure 3.
Paper: Four Years of Type Ia Supernovae Observed by TESS: Early-time Light-curve Shapes and Constraints on Companion Interaction Models
Authors: M. M. Fausnaugh, P. J. Vallely, M. A. Tucker, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, George R. Ricker, Roland Vanderspek, Manan Agarwal, Tansu Daylan, Rahul Jayaraman, Rebekah Hounsell, Daniel Muthukrishna
Figure caption: Light curve for SN2021abko that disfavors companion signatures but is affected by residual systematic errors. The panels are formatted and marked in the same way as in Figure 3.
Paper: Predicting the Yields of z > 6.5 Quasar Surveys in the Era of Roman and Rubin
Authors: Wei Leong Tee, Xiaohui Fan, Feige Wang, Jinyi Yang, Sangeeta Malhotra, James E. Rhoads
Figure caption: Cumulative efficiency and completeness as a function of J lim. Overall completeness and efficiency are high at a bright magnitude. Optimized selection results in an efficiency ∼0.1 at ﹩{J}_{\mathrm{lim}}\sim 25﹩, around 180 quasars discoverable. Pushing toward ﹩{J}_{\mathrm{lim}}\gt 25.5﹩, efficiency drops below 0.01, meaning we are expecting one quasar out of several tens to hundreds of contaminants.
Paper: Predicting the Yields of z > 6.5 Quasar Surveys in the Era of Roman and Rubin
Authors: Wei Leong Tee, Xiaohui Fan, Feige Wang, Jinyi Yang, Sangeeta Malhotra, James E. Rhoads
Figure caption: Cumulative percentage of the MLT population as a function of proper motion. LSST can observe cool brown dwarf with μ LSST = 30 mas yr−1. WFI has the ability to measure μ WFI = 4–16 mas yr−1. The curve represents the percentage of MLTs with proper motion greater than μ, and the dashed lines represent the proper motion limits of each survey.
Paper: Predicting the Yields of z > 6.5 Quasar Surveys in the Era of Roman and Rubin
Authors: Wei Leong Tee, Xiaohui Fan, Feige Wang, Jinyi Yang, Sangeeta Malhotra, James E. Rhoads
Figure caption: Left: predicted quasar yield in 0.5 redshift bin, Right: predicted quasar yield in 0.5 magnitude bin. Results are derived from color cut selection in a 2000 deg2 Roman HLS survey. ∼6 times difference in number yield from two QLF models. Using the QLF from Matsuoka18 and Wang19, we predict ∼180 quasars in total at z > 6.5, with ∼8 at z = 8.0–9.0.
Paper: Predicting the Yields of z > 6.5 Quasar Surveys in the Era of Roman and Rubin
Authors: Wei Leong Tee, Xiaohui Fan, Feige Wang, Jinyi Yang, Sangeeta Malhotra, James E. Rhoads
Figure caption: Completeness of color selection function of 6.5 < z < 9.0 quasars. The probability is the fraction of simulated quasars selected by the color selection method among all simulated quasars in the (M 1450, z) bin. The black dashed line denotes the J band magnitude across the redshift range. Color selection is capable of selecting quasars down to J ∼ 25.5 at all redshifts. The selection function is mostly complete at J < 25.5 (S/N >20).
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Mollweide projections of (left) the radial velocity map (﹩\left\langle {v}_{r}(\theta ,\phi )\right\rangle ﹩) and (right) the density map (﹩\left\langle \rho (\theta ,\phi )\right\rangle ﹩) (as defined by Equation (6)) for the n = 9, s = 0 Chevalier solution. The repeating square-like features (more prominent in the velocity map) are formed by the presence of a carbuncle instability in our solutions. They form in directions aligned to the grid and occur because the shock is not aligned with the grid everywhere. Since our solutions are idealized and self-similar, these results apply to any time after the solution has converged to a statistically self-similar state. For a typical Type Ia SN, they apply to any time between 30 and 104 yr.
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Radial velocity (left) and density (right) isosurfaces for the n = 9, s = 0 Chevalier self-similar model at a resolution of 5123. As these represent idealized self-similar solutions, these results apply to any time after the solution has converged to a statistically self-similar state. For a typical Type Ia SN, they apply to any time after about 30 yr and before 104 yr, which is when an SNR transitions to the Sedov–Taylor phase (Woltjer 1972). The values have been chosen so as to focus on the shocked region between the ejecta and the CSM, which houses the turbulent RTI structures. The visualization has been generated using the open-source software VisIt (Childs et al. 2012).
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Power spectra for the n = 9, s = 0 solution. Left: velocity power spectra at two different computational resolutions. The peaks at l = 4 are generated by a mild carbuncle instability arising in our solutions. A smaller peak can be seen at l > 100 in the left panel, caused by the grid scale. It occurs at different locations in the plot for different resolutions. The black line shows a smoothened (by convolving with a Gaussian kernel) version of the spectrum for the 5123 run. The green line performs the same operation, but ignores the numerical artifact at l = 4. The rest of the work uses this version of the spectra for visualization. Right: the smoothened velocity and density spectra (ignoring the l = 4 peak). They have the shape of a broken power law, depicted by the gray dashed lines. The vertical line is at the wavenumber where the power law is broken, denoted by l 0. Since our solutions are idealized and self-similar, these results apply to any time after the solution has converged to a statistically self-similar state. For a typical Type Ia SN, they apply to any time between 30 and 104 yr.
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: The square of the density of the shocked plasma integrated along the line of sight. These images were developed using our models with n = 9, s = 0, and different initial perturbations to the density field, assuming the plasma is optically thin and is heated to over 106 K in the shocked region. Left: perturbations for these models are large-scale (corresponding to l = 6) and are either internal or external to the SN explosion. The quantity A is as defined in Equation (14). It can be seen large structures persist in the SNR only if the clumps are massive (bottom panels), in addition to being large. Right: the initial density perturbations for these models are on much smaller length scales (l = 100). No clumps of the corresponding size are seen in the SNR morphology, irrespective of the mass. Therefore in each case, the dominant angular scale seen is purely a function of the density scale height. As these represent idealized self-similar solutions, these results apply to any time after the solution has converged to a statistically self-similar state. For a typical Type Ia SN, they apply to any time between 30 and 104 yr.
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Evolution of the velocity power spectrum for the n = 9, s = 0 solution. The power spectrum is evaluated for 12 logarithmically spaced intervals between t = 40 days and 30 yr (or t/t  i  = 40–10,000 in code units). The red and blue curves denote the earliest and the latest instances, respectively. An initially flat spectrum evolves into the peaked structure, indicating maximum net growth at the break frequency.
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Velocity (left) and density (right) power spectra for models seeded with perturbations in the ejecta according to Equation (14). The velocity spectra are not heavily impacted by even the strongest of perturbations. The density spectra do show a deviation from the fiducial spectrum (for the n = 9, s = 0 model), but only at large angular scales (or small l) for perturbations of large amplitude. As these represent idealized self-similar solutions, these results apply to any time after the solution has converged to a statistically self-similar state. For a typical Type Ia SN, they apply to any time between 30 and 104 yr.
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Velocity (left) and density (right) power spectra for models that encounter clumpy CSM, with the clumps defined as in Equation (14). The velocity spectra are not influenced by perturbations, as with anisotropic ejecta. The density spectra are only affected by the strongest perturbations. As these represent idealized self-similar solutions, these results apply to any time after the solution has converged to a statistically self-similar state. For a typical Type Ia SN, they apply to any time between 30 and 104 yr.
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Velocity (left) and density (right) power spectra for all Chevalier models, plotted versus l/n. All curves peak at l/n ∼ 4, demonstrating the peak wavenumber l 0 is proportional to n. As these represent idealized self-similar solutions, these results apply to any time after the solution has converged to a statistically self-similar state. For a typical Type Ia SN, they apply to any time between 30 and 104 yr.
Paper: A 3D Numerical Study of Anisotropies in Supernova Remnants
Authors: Soham Mandal, Paul C. Duffell, Abigail Polin, Dan Milisavljevic
Figure caption: Velocity power spectra at four different computational resolutions. The peaks at l = 4 are generated by a mild carbuncle instability arising in our solutions. A smaller peak can be seen at l > 100, caused by the grid scale. It occurs at different locations in the plot for different resolutions.
Paper: A CHIME/FRB Study of Burst Rate and Morphological Evolution of the Periodically Repeating FRB 20180916B
Authors: Ketan R. Sand, Daniela Breitman, Daniele Michilli, Victoria M. Kaspi, Pragya Chawla, Emmanuel Fonseca, Ryan Mckinven, Kenzie Nimmo, Ziggy Pleunis, Kaitlyn Shin, Bridget C. Andersen, Mohit Bhardwaj, P. J. Boyle, Charanjot Brar, Tomas Cassanelli, Amanda M. Cook, Alice P. Curtin, Fengqiu Adam Dong, Gwendolyn M. Eadie, B. M. Gaensler, Jane Kaczmarek, Adam Lanman, Calvin Leung, Kiyoshi W. Masui, Mubdi Rahman, Ayush Pandhi, Aaron B. Pearlman, Emily Petroff, Masoud Rafiei-Ravandi, Paul Scholz, Vishwangi Shah, Kendrick Smith, Ingrid Stairs, David C. Stenning
Figure caption: The evolution of various spectro-temporal properties of bursts from FRB 20180916B as a function of time (left) and phase of the 16.34 day cycle (reference MJD 58369.40). The histograms on the far right show overall distribution of the properties. We show here six properties: scattering, DM, RM, central frequency (CF), bandwidth (BW), and combined width of all of the subbursts (W). The color scale shows the time resolution at which we measured the morphological parameters for each of the burst since not all bursts were bright enough to be studied at 2.56 μs time resolution. We see the RM increase in the source around 2021 April as reported by Mckinven et al. (2023). We find no trend in any other properties in both time and phase. Only the bursts with baseband data have been shown here since we were able to better characterize their spectral and temporal properties. (See Section 3.5 for details).
Paper: A CHIME/FRB Study of Burst Rate and Morphological Evolution of the Periodically Repeating FRB 20180916B
Authors: Ketan R. Sand, Daniela Breitman, Daniele Michilli, Victoria M. Kaspi, Pragya Chawla, Emmanuel Fonseca, Ryan Mckinven, Kenzie Nimmo, Ziggy Pleunis, Kaitlyn Shin, Bridget C. Andersen, Mohit Bhardwaj, P. J. Boyle, Charanjot Brar, Tomas Cassanelli, Amanda M. Cook, Alice P. Curtin, Fengqiu Adam Dong, Gwendolyn M. Eadie, B. M. Gaensler, Jane Kaczmarek, Adam Lanman, Calvin Leung, Kiyoshi W. Masui, Mubdi Rahman, Ayush Pandhi, Aaron B. Pearlman, Emily Petroff, Masoud Rafiei-Ravandi, Paul Scholz, Vishwangi Shah, Kendrick Smith, Ingrid Stairs, David C. Stenning
Figure caption: Distribution of burst counts for FRB 20180916B . The left panel shows the number of bursts detected during each activity cycle. One activity cycle is defined as a period of 16.34 days (starting 2018 August). The gray steps show the exposure in each cycle. In total, we have 74 activity cycles observed by CHIME/FRB from 2018 August to 2021 December. The right panel shows the frequency distribution of the burst count in each activity cycle in dark green. We also show a distribution from 10,000 simulations in light green sampled assuming a Poissionian process with mean rate equal to the value we have observed. We performed a K-S and A-D test between the observed and simulated distributions and report the median p-value, 0.512 and 0.12, respectively. See Section 3.4 for details.
Paper: A CHIME/FRB Study of Burst Rate and Morphological Evolution of the Periodically Repeating FRB 20180916B
Authors: Ketan R. Sand, Daniela Breitman, Daniele Michilli, Victoria M. Kaspi, Pragya Chawla, Emmanuel Fonseca, Ryan Mckinven, Kenzie Nimmo, Ziggy Pleunis, Kaitlyn Shin, Bridget C. Andersen, Mohit Bhardwaj, P. J. Boyle, Charanjot Brar, Tomas Cassanelli, Amanda M. Cook, Alice P. Curtin, Fengqiu Adam Dong, Gwendolyn M. Eadie, B. M. Gaensler, Jane Kaczmarek, Adam Lanman, Calvin Leung, Kiyoshi W. Masui, Mubdi Rahman, Ayush Pandhi, Aaron B. Pearlman, Emily Petroff, Masoud Rafiei-Ravandi, Paul Scholz, Vishwangi Shah, Kendrick Smith, Ingrid Stairs, David C. Stenning
Figure caption: Burst rate variation with phase. We folded our burst arrival times at a period of 16.34 days with reference MJD 58369.40. We did the same with our calculated exposure and estimated the burst rate of FRB 20180916B in different phase bins. Here we have excised the bursts detected during days of low sensitivity and those detected outside the FWHM at 600 MHz to get true estimate on the rate. The rate is maximum in the phase range 0.45–0.55 (green). We also show the rates in phases earlier, i.e., 0.35–0.45 (blue) and later, i.e., 0.55–0.65 (red). The three rates are consistent within error bars (95%) assuming a Poissonian process. See Section 3.4 for details.
Paper: A CHIME/FRB Study of Burst Rate and Morphological Evolution of the Periodically Repeating FRB 20180916B
Authors: Ketan R. Sand, Daniela Breitman, Daniele Michilli, Victoria M. Kaspi, Pragya Chawla, Emmanuel Fonseca, Ryan Mckinven, Kenzie Nimmo, Ziggy Pleunis, Kaitlyn Shin, Bridget C. Andersen, Mohit Bhardwaj, P. J. Boyle, Charanjot Brar, Tomas Cassanelli, Amanda M. Cook, Alice P. Curtin, Fengqiu Adam Dong, Gwendolyn M. Eadie, B. M. Gaensler, Jane Kaczmarek, Adam Lanman, Calvin Leung, Kiyoshi W. Masui, Mubdi Rahman, Ayush Pandhi, Aaron B. Pearlman, Emily Petroff, Masoud Rafiei-Ravandi, Paul Scholz, Vishwangi Shah, Kendrick Smith, Ingrid Stairs, David C. Stenning
Figure caption: Burst rate in different activity cycles. The burst rate of FRB 20180916B (blue bars show detections) is steady within error bars, which denote 95% confidence intervals assuming a Poissonian process. Many of these cycles have zero rates where we denote upper limits. This is because not all of the bursts detected in our sample satisfy the exposure criteria (see Section 3.4 for details).
Paper: A CHIME/FRB Study of Burst Rate and Morphological Evolution of the Periodically Repeating FRB 20180916B
Authors: Ketan R. Sand, Daniela Breitman, Daniele Michilli, Victoria M. Kaspi, Pragya Chawla, Emmanuel Fonseca, Ryan Mckinven, Kenzie Nimmo, Ziggy Pleunis, Kaitlyn Shin, Bridget C. Andersen, Mohit Bhardwaj, P. J. Boyle, Charanjot Brar, Tomas Cassanelli, Amanda M. Cook, Alice P. Curtin, Fengqiu Adam Dong, Gwendolyn M. Eadie, B. M. Gaensler, Jane Kaczmarek, Adam Lanman, Calvin Leung, Kiyoshi W. Masui, Mubdi Rahman, Ayush Pandhi, Aaron B. Pearlman, Emily Petroff, Masoud Rafiei-Ravandi, Paul Scholz, Vishwangi Shah, Kendrick Smith, Ingrid Stairs, David C. Stenning
Figure caption:  χ 2 distribution of different period (x-axis) and period derivative (y-axis) combinations. The red cross shows the maximum χ 2 value. The white contours show the results of the 2D Gaussian fit. We thus find ﹩\dot{P}=-0.2\pm 1.5\times {10}^{-4}﹩ day day−1. We calculated the χ 2 using the Pearson χ 2 test described by CHIME/FRB Collaboration et al. (2020b). See Section 3.3 for details.