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From CMB polarization data alone we reconstruct the CMB lensing power spectrum, comparable in overall constraining power to previous temperature-based reconstructions, and an unlensed E-mode power spectrum. The observations, taken in 2019 and 2020 with the South Pole Telescope (SPT) and the SPT-3G camera, cover 1500 deg$^2$ at 95, 150, and 220 GHz with arcminute resolution and roughly 4.9$\mu$K-arcmin coadded noise in polarization. The power spectrum estimates, together with systematic parameter estimates and a joint covariance matrix, follow from a Bayesian analysis using the Marginal Unbiased Score Expansion (MUSE) method. The E-mode spectrum at $\ell>2000$ and lensing spectrum at $L>350$ are the most precise to date. Assuming the $\Lambda$CDM model, and using only these SPT data and priors on $\tau$ and absolute calibration from Planck, we find $H_0=66.81\pm0.81$ km/s/Mpc, comparable in precision to the Planck determination and in 5.4$\sigma$ tension with the most precise $H_0$ inference derived via the distance ladder. We also find $S_8=0.850\pm0.017$, providing further independent evidence of a slight tension with low-redshift structure probes. The $\Lambda$CDM model provides a good simultaneous fit to the combined Planck, ACT, and SPT data, and thus passes a powerful test. Combining these CMB datasets with BAO observations, we find that the effective number of neutrino species, spatial curvature, and primordial helium fraction are consistent with standard model values, and that the 95% confidence upper limit on the neutrino mass sum is 0.075 eV. The SPT data are consistent with the somewhat weak preference for excess lensing power seen in Planck and ACT data relative to predictions of the $\Lambda$CDM model. We also detect at greater than 3$\sigma$ the influence of non-linear evolution in the CMB lensing power spectrum and discuss it in the context of the $S_8$ tension.(abridged), Comment: 28 pages, 21 figures + appendices

The detection of satellite thermal emission at millimeter wavelengths is presented using data from the 3rd-Generation receiver on the South Pole Telescope (SPT-3G). This represents the first reported detection of thermal emission from artificial satellites at millimeter wavelengths. Satellite thermal emission is shown to be detectable at high signal-to-noise on timescales as short as a few tens of milliseconds. An algorithm for downloading orbital information and tracking known satellites given observer constraints and time-ordered observatory pointing is described. Consequences for cosmological surveys and short-duration transient searches are discussed, revealing that the integrated thermal emission from all large satellites does not contribute significantly to the SPT-3G survey intensity map. Measured satellite positions are found to be discrepant from their two-line element (TLE) derived ephemerides up to several arcminutes which may present a difficulty in cross-checking or masking satellites from short-duration transient searches.

The peculiar Galactic globular cluster $\omega$ Centauri (NGC 5139) has drawn attention for its unique features - such as a high stellar mass and a broad distribution of chemical elements - that have led to the hypothesis that it might be the nuclear remnant of an ancient dwarf galaxy accreted by the Milky Way (MW), potentially bringing along its own globular cluster (GC) system. In this work, we adopt an innovative approach by examining the individual chemical abundances of Galactic GCs. Applying Gaussian Mixture Models to globular cluster stars, whose membership is based on Gaia EDR3, and whose chemical abundances are provided by APOGEE DR17, we depart from traditional kinematic-based procedures and search for GCs that are chemically compatible with $\omega$ Cen in a 8-dimensional space defined by [Fe/H], [Mg/Fe], [Si/Fe], [Ca/Fe], [C/Fe], [Al/Fe], [K/Fe], and [Mn/Fe]. Our analysis leads to the identification of six GCs - NGC 6752, NGC 6656, NGC 6809, NGC 6273, NGC 6205, and NGC 6254 - that exhibit strong chemical similarities with $\omega$ Cen, and which have metallicities that coincide with those of the two main peaks of $\omega$ Cen's metallicity distribution. The chemical patterns of these clusters lead to the exclusion that they formed in progenitor galaxies with chemical enrichment histories similar to those of the Large and Small Magellanic Clouds, Sagittarius, and Fornax. Once placed in kinematic spaces such as the energy - angular momentum plane, these GCs result scattered across an extended region, which is predicted by N-body simulations if their common progenitor was sufficiently massive compared to the MW. Our novel approach suggests a common origin for NGC 6752, NGC 6656, NGC 6809, NGC 6273, NGC 6205, NGC 6254 and $\omega$ Cen, indicating that Nephele, as we propose to call the progenitor in which these GCs formed, played a substantial role in the Galaxy's history., Comment: Accepted for publication on A&A

The paper presents a pilot exploration of the construction, management and analysis of a multimodal corpus. Through a three-layer annotation that provides orthographic, prosodic, and gestural transcriptions, the Gest-IT resource allows to investigate the variation of gesture-making patterns in conversations between sighted people and people with visual impairment. After discussing the transcription methods and technical procedures employed in our study, we propose a unified CoNLL-U corpus and indicate our future steps

This paper presents a novel robust variable-horizon model predictive control scheme designed to intercept a target moving along a known trajectory, in finite time. Linear discrete-time systems affected by bounded process disturbances are considered and a tube-based MPC approach is adopted. The main contribution is an adaptive mechanism for choosing the terminal constraint set sequence in the MPC optimization problem. This mechanism is designed to ensure recursive feasibility while promoting minimization of the final distance to the target. Finite-time convergence of the proposed control scheme is proven. In order to evaluate its effectiveness, the designed control law is tested through numerical simulations, including a case study involving orbital rendezvous of a satellite with a tumbling object. The results indicate a significant reduction in conservatism compared to existing state-of-the-art methods using a fixed terminal set sequence.

We prove the non-existence and strong ill-posedness of the Incompressible Porous Media (IPM) equation for initial data that are small $H^2(\mathbb{R}^2)$ perturbations of the linearly stable profile $-x_2$. A remarkable novelty of the proof is the construction of an $H^2$ perturbation, which solves the IPM equation and neutralizes the stabilizing effect of the background profile near the origin, where a strong deformation leading to non-existence in $H^2$ is created. This strong deformation is achieved through an iterative procedure inspired by the work of C\'{o}rdoba and Mart\'{\i}nez-Zoroa (Adv. Math. 2022). However, several differences - beyond purely technical aspects - arise due to the anisotropic and, more importantly, to the partially dissipative nature of the equation, adding further challenges to the analysis.

We present a forecast of the pairwise kinematic Sunyaev-Zel'dovich (kSZ) measurement that will be achievable with the future CMB-S4 experiment. CMB-S4 is the next stage for ground-based cosmic microwave background experiments, with a planned wide area survey that will observe approximately $50\%$ of the sky. We construct a simulated sample of galaxy clusters that have been optically selected in an LSST-like survey and have spectroscopic redshifts. For this cluster sample, we predict that CMB-S4 will reject the null hypothesis of zero pairwise kSZ signal at $36 \,\sigma$. We estimate the effects of systematic uncertainties such as scatter in the mass-richness scaling relation and cluster mis-centering. We find that these effects can reduce the signal-to-noise ratio of the CMB-S4 pairwise kSZ measurement by $20\%$. We explore the constraining power of the measured kSZ signal in combination with measurements of the galaxy clusters' thermal SZ emission on two extensions to the standard cosmological model. The first extension allows the dark energy equation of state $w$ to vary. We find the CMB-S4 pairwise kSZ measurement yields a modest reduction in the uncertainty on $w$ by a factor of 1.36 over the \Planck's 2018 uncertainty. The second extension tests General Relativity by varying the growth index $\gamma$. We find that CMB-S4's pairwise kSZ measurement will yield a $28\sigma$ constraint on $\gamma$, and strongly constrain alternative theories of gravity., Comment: arXiv admin note: text overlap with arXiv:2207.11937

This article focuses on the articulatory characteristics of epistemic gestures (i.e., gestures used to express certainty or uncertainty) in co-speech gestures (CSG) in French and in French Sign Language (LSF). It presents a new methodology for analysis, which relies on the complementary use of manual annotation (using Typannot) and semi-automatic annotation (using AlphaPose) to highlight the kinesiological characteristics of these epistemic gestures. The presented methodology allows to analyze the flexion/extension movements of the head in epistemic contexts. The results of this analysis show that in CSG and LSF: (1) head nods passing through the neutral position (i.e., head straight with no flexion/extension) and high movement speed are markers of certainty; and (2) holding the head position away from the neutral position and low movement speed indicate uncertainty. This study is conducted within the framework of the ANR LexiKHuM project, which develops kinesthetic communication solutions for human-machine interaction., Comment: in French language, Anvur E062307, Scholar

Pre-trained LLMs have demonstrated substantial capabilities across a range of conventional natural language processing (NLP) tasks, such as summarization and entity recognition. In this paper, we explore the application of LLMs in the generation of high-quality protein sequences. Specifically, we adopt a suite of pre-trained LLMs, including Mistral-7B1, Llama-2-7B2, Llama-3-8B3, and gemma-7B4, to produce valid protein sequences. All of these models are publicly available.5 Unlike previous work in this field, our approach utilizes a relatively small dataset comprising 42,000 distinct human protein sequences. We retrain these models to process protein-related data, ensuring the generation of biologically feasible protein structures. Our findings demonstrate that even with limited data, the adapted models exhibit efficiency comparable to established protein-focused models such as ProGen varieties, ProtGPT2, and ProLLaMA, which were trained on millions of protein sequences. To validate and quantify the performance of our models, we conduct comparative analyses employing standard metrics such as pLDDT, RMSD, TM-score, and REU. Furthermore, we commit to making the trained versions of all four models publicly available, fostering greater transparency and collaboration in the field of computational biology., Comment: This paper has been accepted for presentation at Language and Molecules ACL 2024

We present the detection and characterization of fluctuations in linearly polarized emission from the atmosphere above the South Pole. These measurements make use of Austral winter survey data from the SPT-3G receiver on the South Pole Telescope in three frequency bands centered at 95, 150, and 220 GHz. We use the cross-correlation between detectors to produce an unbiased estimate of the power in Stokes I, Q, and U parameters on large angular scales. Our results are consistent with the polarized signal being produced by the combination of Rayleigh scattering of thermal radiation from the ground and thermal emission from a population of horizontally aligned ice crystals with an anisotropic distribution described by Kolmogorov turbulence. The signal is most significant at large angular scales, high observing frequency, and low elevation angle. Polarized atmospheric emission has the potential to significantly impact observations on the large angular scales being targeted by searches for inflationary B-mode CMB polarization. We present the distribution of measured angular power spectrum amplitudes in Stokes Q and I for 4 years of winter observations, which can be used to simulate the impact of atmospheric polarization and intensity fluctuations at the South Pole on a specified experiment and observation strategy. For the SPT-3G data, downweighting the small fraction of significantly contaminated observations is an effective mitigation strategy. In addition, we present a strategy for further improving sensitivity on large angular scales where maps made in the 220 GHz band are used to measure and subtract the polarized atmosphere signal from the 150 GHz band maps. In observations with the SPT-3G instrument at the South Pole, the polarized atmospheric signal is a well-understood and sub-dominant contribution to the measured noise after implementing the mitigation strategies described here., Comment: 32 pages, 28 figures

We consider equations of the type: \[\partial_t \omega = \omega R(\omega),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized solutions. Singularities can even form in settings where solutions dissipate an energy. Such equations arise naturally as models in various physical settings such as inviscid and complex fluids.

We establish the existence of quasi-periodic traveling wave solutions for the $\beta$-plane equation on $\mathbb{T}^2$ with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the frequency of oscillations of the external force. Due to the presence of small divisors, the proof relies on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. To our knowledge, this is the first instance of constructing quasi-periodic solutions for a quasilinear PDE in dimensions greater than one, with a 1-smoothing dispersion relation that is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by preserving the traveling-wave structure, the conservation of momentum and by implementing normal form methods for the linearized system with sublinear dispersion relation in higher space dimension.

We rigorously justify the bilayer shallow-water system as an approximation to the hydrostatic Euler equations in situations where the flow is density-stratified with close-to-piecewise constant density profiles, and close-to-columnar velocity profiles. Our theory accommodates with continuous stratification, so that admissible deviations from bilayer profiles are not pointwise small. This leads us to define refined approximate solutions that are able to describe at first order the flow in the pycnocline. Because the hydrostatic Euler equations are not known to enjoy suitable stability estimates, we rely on thickness-diffusivity contributions proposed by Gent and McWilliams. Our strategy also applies to one-layer and multilayer frameworks.

The purpose of this clinical study is to test broad spectral range (635-1060 nm) time domain diffuse optical spectroscopy in monitoring the response of breast cancer patients to neoadjuvant chemotherapy (NAC). The broadband operation allows us to fully analyze tissue composition in terms of hemoglobin, water, lipids and collagen concentration, which has never been systematically studied until now during the course of therapy. Patients are subjected to multiple breast optical imaging sessions, each one performed at different stages of NAC, both on tumor-bearing and contralateral healthy breast. We correlate the optical results with conventional imaging techniques and pathological response. Preliminary outcomes on data of 10 patients show an average significant reduction in the concentrations of oxy-hemoglobin (-53%, p = 0.0020), collagen (-36%, p = 0.0039) and water (-15%, p = 0.0195), and increase in lipids (+39%, p = 0.0137) from baseline to the end of therapy in the tumor-bearing breast of patients who responded to therapy at least partially. With respect to scattering, the scattering amplitude, a, increases slightly by the end of the therapy compared to the baseline, while the scattering slope, b, shows no significant change. Some change in concentrations of constituents was noticed also in the contralateral healthy breast, even though it was significant only for oxy-hemoglobin concentration. Collagen seems to be the only component distinguishing between complete (p = 0.0423) and partial responders (p = 0.1012) by the end of 2-3 weeks after the start of therapy, suggesting that it could serve as a potential biomarker to measure NAC effectiveness early during treatment. These initial results are consistent with those of standard medical modalities and highlight the sensitivity of the technique to changes that occur in breast composition during NAC.

Today, cloud workloads are essentially opaque to the cloud platform. Typically, the only information the platform receives is the virtual machine (VM) type and possibly a decoration to the type (e.g., the VM is evictable). Similarly, workloads receive little to no information from the platform; generally, workloads might receive telemetry from their VMs or exceptional signals (e.g., shortly before a VM is evicted). The narrow interface between workloads and platforms has several drawbacks: (1) a surge in VM types and decorations in public cloud platforms complicates customer selection; (2) essential workload characteristics (e.g., low availability requirements, high latency tolerance) are often unspecified, hindering platform customization for optimized resource usage and cost savings; and (3) workloads may be unaware of potential optimizations or lack sufficient time to react to platform events. In this paper, we propose a framework, called Workload Intelligence (WI), for dynamic bi-directional communication between cloud workloads and cloud platform. Via WI, workloads can programmatically adjust their key characteristics, requirements, and even dynamically adapt behaviors like VM priorities. In the other direction, WI allows the platform to programmatically inform workloads about upcoming events, opportunities for optimization, among other scenarios. Because of WI, the cloud platform can drastically simplify its offerings, reduce its costs without fear of violating any workload requirements, and reduce prices to its customers on average by 48.8%.

We measure the stacked lensing signal in the direction of galaxy clusters in the Dark Energy Survey Year 3 (DES Y3) redMaPPer sample, using cosmic microwave background (CMB) temperature data from SPT-3G, the third-generation CMB camera on the South Pole Telescope (SPT). We estimate the lensing signal using temperature maps constructed from the initial 2 years of data from the SPT-3G 'Main' survey, covering 1500 deg$^2$ of the Southern sky. We then use this signal as a proxy for the mean cluster mass of the DES sample. In this work, we employ three versions of the redMaPPer catalogue: a Flux-Limited sample containing 8865 clusters, a Volume-Limited sample with 5391 clusters, and a Volume&Redshift-Limited sample with 4450 clusters. For the three samples, we find the mean cluster masses to be ${M}_{200{\rm{m}}}=1.66\pm0.13$ [stat.]$\pm0.03$ [sys.], $1.97\pm0.18$ [stat.]$\pm0.05$ [sys.], and $2.11\pm0.20$ [stat.]$\pm0.05$ [sys.]$\times{10}^{14}\ {\rm{M}}_{\odot }$, respectively. This is a factor of $\sim2$ improvement relative to the precision of measurements with previous generations of SPT surveys and the most constraining cluster mass measurements using CMB cluster lensing to date. Overall, we find no significant tensions between our results and masses given by redMaPPer mass-richness scaling relations of previous works, which were calibrated using CMB cluster lensing, optical weak lensing, and velocity dispersion measurements from various combinations of DES, SDSS and Planck data. We then divide our sample into 3 redshift and 3 richness bins, finding no significant tensions with optical weak-lensing calibrated masses in these bins. We forecast a $5.7\%$ constraint on the mean cluster mass of the DES Y3 sample with the complete SPT-3G surveys when using both temperature and polarization data and including an additional $\sim1400$ deg$^2$ of observations from the 'Extended' SPT-3G survey., Comment: 23 pages, 9 figures, accepted for publication in JCAP. Minor changes and corrections have been made relative to v1

We forecast constraints on cosmological parameters enabled by three surveys conducted with SPT-3G, the third-generation camera on the South Pole Telescope. The surveys cover separate regions of 1500, 2650, and 6000 ${\rm deg}^{2}$ to different depths, in total observing 25% of the sky. These regions will be measured to white noise levels of roughly 2.5, 9, and 12 $\mu{\rm K-arcmin}$, respectively, in CMB temperature units at 150 GHz by the end of 2024. The survey also includes measurements at 95 and 220 GHz, which have noise levels a factor of ~1.2 and 3.5 times higher than 150 GHz, respectively, with each band having a polarization noise level ~$\sqrt{\text{2}}$ times higher than the temperature noise. We use a novel approach to obtain the covariance matrices for jointly and optimally estimated gravitational lensing potential bandpowers and unlensed CMB temperature and polarization bandpowers. We demonstrate the ability to test the $\Lambda{\rm CDM}$ model via the consistency of cosmological parameters constrained independently from SPT-3G and Planck data, and consider the improvement in constraints on $\Lambda{\rm CDM}$ extension parameters from a joint analysis of SPT-3G and Planck data. The $\Lambda{\rm CDM}$ cosmological parameters are typically constrained with uncertainties up to ~2 times smaller with SPT-3G data, compared to Planck, with the two data sets measuring significantly different angular scales and polarization levels, providing additional tests of the standard cosmological model., Comment: 26 pages; 13 figures; Accepted for publication in ApJ; Minor edits have been made

We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler-Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating the Miles-Howard criterion and generating a growing mode, both for the linearized hydrostatic and non-hydrostatic equations. By leveraging long-wave nonlinear instability for the original Euler-Boussinesq system, we demonstrate the breakdown of the hydrostatic limit around such unstable profiles. Finally, we establish the generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev spaces., Comment: 40 pages

We report results from an analysis aimed at detecting the trispectrum of the kinematic Sunyaev-Zel{'}dovich (kSZ) effect by combining data from the South Pole Telescope (SPT) and {\it Herschel}-SPIRE experiments over a 100 ${\rm deg}^{2}$ field. The SPT observations combine data from the previous and current surveys, namely SPTpol and SPT-3G, to achieve depths of 4.5, 3, and 16 $\mu {\rm K-arcmin}$ in bands centered at 95, 150, and 220 GHz. For SPIRE, we include data from the 600 and 857 GHz bands. We reconstruct the velocity-induced large-scale correlation of the small-scale kSZ signal with a quadratic estimator that uses two cosmic microwave background (CMB) temperature maps, constructed by optimally combining data from all the frequency bands. We reject the null hypothesis of a zero trispectrum at $10.3\sigma$ level. However, the measured trispectrum contains contributions from both the kSZ and other undesired components, such as CMB lensing and astrophysical foregrounds, with kSZ being sub-dominant. We use the \textsc{Agora} simulations to estimate the expected signal from CMB lensing and astrophysical foregrounds. After accounting for the contributions from CMB lensing and foreground signals, we do not detect an excess kSZ-only trispectrum and use this non-detection to set constraints on reionization. By applying a prior based on observations of the Gunn-Peterson trough, we obtain an upper limit on the duration of reionization of $\Delta z_{\rm re, 50} < 4.5$ (95\% C.L). We find these constraints are fairly robust to foregrounds assumptions. This trispectrum measurement is independent of, but consistent with, {\it Planck}'s optical depth measurement. This result is the first constraint on the epoch of reionization using the non-Gaussian nature of the kSZ signal., Comment: 15 pages, 5 figures (3 in main text and 2 in Appendix); Accepted for publication in PRL; Some texts have been moved to Appendix; Minor change in Fig. 2 to include nomalization; Data products and plotting scripts can be downloaded from https://github.com/sriniraghunathan/kSZ_4pt_SPT_SPIRE

We present a flare star catalog from four years of non-targeted millimeter-wave survey data from the South Pole Telescope (SPT). The data were taken with the SPT-3G camera and cover a 1500-square-degree region of the sky from $20^{h}40^{m}0^{s}$ to $3^{h}20^{m}0^{s}$ in right ascension and $-42^{\circ}$ to $-70^{\circ}$ in declination. This region was observed on a nearly daily cadence from 2019-2022 and chosen to avoid the plane of the galaxy. A short-duration transient search of this survey yields 111 flaring events from 66 stars, increasing the number of both flaring events and detected flare stars by an order of magnitude from the previous SPT-3G data release. We provide cross-matching to Gaia DR3, as well as matches to X-ray point sources found in the second ROSAT all-sky survey. We have detected flaring stars across the main sequence, from early-type A stars to M dwarfs, as well as a large population of evolved stars. These stars are mostly nearby, spanning 10 to 1000 parsecs in distance. Most of the flare spectral indices are constant or gently rising as a function of frequency at 95/150/220 GHz. The timescale of these events can range from minutes to hours, and the peak $\nu L_{\nu}$ luminosities range from $10^{27}$ to $10^{31}$ erg s$^{-1}$ in the SPT-3G frequency bands.

Graph Neural Networks (GNNs) have emerged in recent years as a powerful tool to learn tasks across a wide range of graph domains in a data-driven fashion; based on a message passing mechanism, GNNs have gained increasing popularity due to their intuitive formulation, closely linked with the Weisfeiler-Lehman (WL) test for graph isomorphism, to which they have proven equivalent. From a theoretical point of view, GNNs have been shown to be universal approximators, and their generalization capability (namely, bounds on the Vapnik Chervonekis (VC) dimension) has recently been investigated for GNNs with piecewise polynomial activation functions. The aim of our work is to extend this analysis on the VC dimension of GNNs to other commonly used activation functions, such as sigmoid and hyperbolic tangent, using the framework of Pfaffian function theory. Bounds are provided with respect to architecture parameters (depth, number of neurons, input size) as well as with respect to the number of colors resulting from the 1-WL test applied on the graph domain. The theoretical analysis is supported by a preliminary experimental study., Comment: 35 pages, 9 figures

We consider the problem of setting a confidence interval on a parameter of interest from a high-statistics counting experiment in the presence of systematic uncertainties modeled as unconstrained nuisance parameters. We use the profile-likelihood test statistic in the asymptotic limit for confidence interval setting and focus on the case where the likelihood function is derived from a finite sample of Monte Carlo simulated events. We prove as a general result that statistical uncertainties in the Monte Carlo sample affect the coverage of the confidence interval always in the same direction, namely they lead to a systematic undercoverage of the interval. We argue that such spurious effects might not be fully accounted for by statistical methods that are usually adopted in HEP measurements to counteract the effects of finite-size MC samples, such as those based on the Barlow-Beeston likelihood.

In the context of deep learning models, attention has recently been paid to studying the surface of the loss function in order to better understand training with methods based on gradient descent. This search for an appropriate description, both analytical and topological, has led to numerous efforts to identify spurious minima and characterize gradient dynamics. Our work aims to contribute to this field by providing a topological measure to evaluate loss complexity in the case of multilayer neural networks. We compare deep and shallow architectures with common sigmoidal activation functions by deriving upper and lower bounds on the complexity of their loss function and revealing how that complexity is influenced by the number of hidden units, training models, and the activation function used. Additionally, we found that certain variations in the loss function or model architecture, such as adding an $\ell_2$ regularization term or implementing skip connections in a feedforward network, do not affect loss topology in specific cases.

We consider a *continuous* solution $u$ of the balance law \[ \partial_{\mathit t} u + \partial_{\mathit x} (f(u)) = g\] in one space dimension, where the flux function $f$ is of class $C^2$ and the source term $g$ is bounded. This equation admits an Eulerian intepretation (namely the distributional one) and a Lagrangian intepretation (which can be further specified). Since $u$ is only continuous, these interpretations do not necessessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term $g$. In this paper we complete the comparison between these notions of solutions started in the companion paper [Alberti-Bianchini-Caravenna I], and analize in detail the relations between the corresponding notions of source term.

We present cosmological constraints from the abundance of galaxy clusters selected via the thermal Sunyaev-Zel'dovich (SZ) effect in South Pole Telescope (SPT) data with a simultaneous mass calibration using weak gravitational lensing data from the Dark Energy Survey (DES) and the Hubble Space Telescope (HST). The cluster sample is constructed from the combined SPT-SZ, SPTpol ECS, and SPTpol 500d surveys, and comprises 1,005 confirmed clusters in the redshift range $0.25-1.78$ over a total sky area of 5,200 deg$^2$. We use DES Year 3 weak-lensing data for 688 clusters with redshifts $z<0.95$ and HST weak-lensing data for 39 clusters with $0.6