Weak Constraints Research Articles

Dynamic screening with liquidity constraints

AbstractWe consider a dynamic screening model with serially independent types where the agent is short-term liquidity constrained. We model a liquidity constraint as a hard constraint that forces the agent to renege whenever he would suffer a loss from fulfilling the contract terms in a given period. In particular, the violation of a liquidity constraint is a verifiable event that future contract terms can condition on. This verifiability leads to less stringent truth-telling constraints than those considered in the existing literature. We show that the weaker constraints do not affect optimal contracting, however. Moreover, we develop a novel method to study private values settings with continuous types and show that a regularity condition that has analogues in the literature on multi-dimensional screening ensures that the optimal contract is deterministic.

A global re-analysis of regionally resolved emissions and atmospheric mole fractions of SF6 for the period 2005–2021

Abstract. We determine the global emission distribution of the potent greenhouse gas sulfur hexafluoride (SF6) for the period 2005–2021 using inverse modelling. The inversion is based on 50 d backward simulations with the Lagrangian particle dispersion model (LPDM) FLEXPART and on a comprehensive observation data set of SF6 mole fractions in which we combine continuous with flask measurements sampled at fixed surface locations and observations from aircraft and ship campaigns. We use a global-distribution-based (GDB) approach to determine baseline mole fractions directly from global SF6 mole fraction fields at the termination points of the backward trajectories. We compute these fields by performing an atmospheric SF6 re-analysis, assimilating global SF6 observations into modelled global three-dimensional mole fraction fields. Our inversion results are in excellent agreement with several regional inversion studies in the USA, Europe, and China. We find that (1) annual US SF6 emissions strongly decreased from 1.25 Gg in 2005 to 0.48 Gg in 2021; however, they were on average twice as high as the reported emissions to the United Nations. (2) SF6 emissions from EU countries show an average decreasing trend of −0.006 Gg yr−1 during the period 2005 to 2021, including a substantial drop in 2018. This drop is likely a direct result of the EU's F-gas regulation 517/2014, which bans the use of SF6 for recycling magnesium die-casting alloys as of 2018 and requires leak detection systems for electrical switch gear. (3) Chinese annual emissions grew from 1.28 Gg in 2005 to 5.16 Gg in 2021, with a trend of 0.21 Gg yr−1, which is even higher than the average global total emission trend of 0.20 Gg yr−1. (4) National reports for the USA, Europe, and China all underestimated their SF6 emissions. (5) Our results indicate increasing emissions in poorly monitored areas (e.g. India, Africa, and South America); however, these results are uncertain due to weak observational constraints, highlighting the need for enhanced monitoring in these areas. (6) Global total SF6 emissions are comparable to estimates in previous studies but are sensitive to a priori estimates due to the low network sensitivity in poorly monitored regions. (7) Monthly inversions indicate that SF6 emissions in the Northern Hemisphere were on average higher in summer than in winter throughout the study period.

Physical Models for the Astrophysical Population of Black Holes: Application to the Bump in the Mass Distribution of Gravitational-wave Sources

Abstract Gravitational-wave observations of binary black holes have revealed unexpected structure in the black hole mass distribution. Previous studies employ physically motivated phenomenological models and infer the parameters that control the features of the mass distribution that are allowed in their model, associating the constraints on those parameters with their physical motivations a posteriori. In this work, we take an alternative approach in which we introduce a model parameterizing the underlying stellar and core-collapse physics and obtaining the remnant black hole distribution as a derived by-product. In doing so, we constrain the stellar physics necessary to explain the astrophysical distribution of black hole properties under a given model. We apply this to the mapping between initial mass and remnant black hole mass, accounting for mass-dependent mass loss using a simple parameterized description. Allowing the parameters of the initial mass–remnant mass relationship to evolve with redshift permits correlated and physically reasonable changes to features in the mass function. We find that the current data are consistent with no redshift evolution in the core–remnant mass relationship, but place only weak constraints on the change of these parameters. This procedure can be applied to modeling any physical process underlying the astrophysical distribution. We illustrate this by applying our model to the pulsational pair instability supernova (PPISN) process, previously proposed as an explanation for the observed excess of black holes at ∼35 M ⊙. Placing constraints on the reaction rates necessary to explain the PPISN parameters, we concur with previous results in the literature that the peak observed at ∼35 M ⊙ is unlikely to be a signature from the PPISN process as presently understood.

Constraints on primordial black holes for nonstandard cosmologies

We study how the bounds on the abundance of Primordial Black Holes (PBHs) and the constraints on power spectrum are modified if a non-standard evolution phase takes place between the end of inflation and the Standard radiation-dominated (RD) universe after inflation. The constraints on PBH abundance and power spectrum are computed using the new, freely available, https://github.com/TadeoDGAguilar/PBHBeta PBHBeta library, which accounts for the effects of non-standard expansion and specific criteria for PBH formation in such non-standard scenarios. As working examples, we consider three different cases: a pure matter-dominated (MD) phase, a scalar field-dominated (φD) universe, and a stiff fluid-dominated (SD) scenario.While the background expansion is the same for the MD and φD scenarios, the PBH formation criteria lead to different constraints to power spectrum. On the other hand, the duration of the non-standard expansion phase alters the bounds, with longer MD periods resulting in weaker constraints on power spectrum, and longer SD scenarios leading to an enhanced abundance due to the dust-like redshifting of PBHs. The modifications to the constraints are reported in all cases and we highlight those where the power spectrum may be significantly constrained.

Research on compressor cascade flow field modeling method based on finite volume flux-informed neural network

For compressor cascade flow field modeling, there exists strong velocity shear in the leading edge separation flow, boundary layer, and wake, which leads to increased modeling errors. To improve the accuracy of the flow field modeling method, this paper introduces the concept of numerical flux from the finite volume method into the loss function to implement Euler equation physics-informed learning, and a finite volume flux-informed neural network (FVFI-net) is constructed. Selecting a high-load, large-turning-angle compressor cascade as the study object, a comparative analysis is conducted on the advantages and disadvantages of purely data-driven, weak physical constraint, and finite volume flux-informed methods in compressor cascade flow field modeling. The study found that compared to purely data-driven and weak physical constraint methods, FVFI-net can reduce the average error of aerodynamic parameters in the flow field by approximately 45.6% and 29.5%, respectively, at a 0° angle of attack. For the flow separation problem occurring at the suction side leading edge and the blade wake area caused by a 5° angle of attack, FVFI-net can effectively reduce modeling errors near the leading edge, in the wake region, and near the periodic boundaries, thus reducing the average error of the aerodynamic parameters of the flow field by about 49.2%and 31.3%, respectively, compared to pure data-driven and weak physical constraint methods.

Observations of Extremely Metal-poor O Stars: Weak Winds and Constraints for Evolution Models

Metal-poor massive stars drive the evolution of low-mass galaxies, both locally and at high redshift. However, quantifying the feedback they impart to their local surroundings remains uncertain because models of stellar evolution, mass loss, and ionizing spectra are unconstrained by observations below 20% solar metallicity (Z ⊙). We present new Keck Cosmic Web Imager optical spectroscopy of three O-type stars in the nearby dwarf galaxies Leo P, Sextans A, and WLM, which have gas-phase oxygen abundances of 3%–14% Z ⊙. To characterize their fundamental stellar properties and radiation-driven winds, we fit PoWR atmosphere models to the optical spectra simultaneously with Hubble Space Telescope far-ultraviolet (FUV) spectra and multiwavelength photometry. We find that all three stars have effective temperatures consistent with their spectral types and surface gravities typical of main-sequence dwarf stars. Yet, the combination of those inferred parameters and luminosity for the two lower-Z stars is not reproduced by stellar evolution models, even those that include rotation or binary interactions. The scenario of multiple-star systems is difficult to reconcile with all available data, suggesting that these observations pose a challenge to current evolution models. We highlight the importance of validating the relationship between stellar mass, temperature, and luminosity at very low Z for accurate estimates of ionizing photon production and spectral hardness. Finally, all three stars’ FUV wind profiles reveal low mass-loss rates and terminal wind velocities in tension with expectations from widely adopted radiation-driven wind models. These results provide empirical benchmarks for future development of mass-loss and evolution models for metal-poor stellar populations.

Why Could a Newborn Active Region Produce Coronal Mass Ejections?

Solar active regions (ARs) are the main sources of flares and coronal mass ejections (CMEs). NOAA AR 12089, which emerged on 2014 June 10, produced two C-class flares accompanied by CMEs within 5 hr after its emergence. When producing the two eruptive flares, the total unsigned magnetic flux (ΦAR) and magnetic free energy (E f ) of the AR are much smaller than the common CME-producing ARs. Why can this extremely small AR produce eruptive flares so early? We compare the AR magnetic environment for the eruptive flares to that for the largest confined flare from the AR. In addition to the ΦAR and E f , we calculate the ratio between the mean characteristic twist parameter (α FPIL) within the flaring polarity inversion line (FPIL) region and ΦAR, a parameter considering both background magnetic field constraint and nonpotentiality of the core region, for the three flares. We find higher α FPIL/ΦAR values during the eruptive flares than during the confined flare. Furthermore, we compute the decay index along the polarity inversion line, revealing values of 1.69, 3.45, and 0.98 before the two eruptive and the confined flares, respectively. Finally, nonlinear force-free field extrapolation indicates that a flux rope was repeatedly formed along the FPIL before eruptive flares, which ejected out and produced CMEs. No flux rope was found before the confined flare. Our research suggests that even a newly emerged, extremely small AR can produce eruptive flares if it has sufficiently weak background field constraint and strong nonpotentiality in the core region.

The Power of Culture: Business Nationalist Culture and ESG Performance

High-quality development is the theme of China’s economic and social development in the new era, and corporate ESG performance is a comprehensive indicator for evaluating the level of corporate environmental responsibility, social responsibility and governance, as well as an important yardstick for identifying the high-quality development of enterprises. This paper takes Chinese non-financial listed companies from 2011 to 2022 as the research sample and empirically examines the impact of corporate nationalism culture on corporate ESG performance and its mechanism by quantifying corporate nationalism culture using the text of corporate annual reports, natural language processing and text analysis methods. The results of the study show that corporate nationalism culture significantly enhances corporate ESG performance. The mechanism analysis suggests that corporate nationalism culture, as an internal informal system, can play a governance role and promote corporate ESG practices by changing attention allocation and mitigating agency problems. The positive effect of corporate nationalism culture on corporate ESG performance is more pronounced in the grouping of firms with lower institutional investor shareholding, fewer analysts’ attention and embedded party organisations. A heterogeneity analysis reveals that the corporate nationalism culture driving effect on corporate ESG performance is more significant in the subsample of firms with weak financing constraints, in the growth period and in the decline period. This study reveals the positive role of soft cultural factors in enhancing corporate ESG performance, providing useful managerial evidence for companies to integrate ESG concepts at the strategic level for high-quality economic development.

Necessary and sufficient optimality conditions for nonsmooth nonconvex generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an extension of the Nash equilibrium problem (NEP), which allows the strategies and the objective function of each player to depend on the decision variables of all other players. In this paper, we consider a GNEP where the objective and constraint functions are not necessarily differentiable or convex. By proposing an alternative approach based on the equivalence with a single constrained optimization problem, we then present new sharp KKT-type necessary optimality conditions for a solution of this problem under a weak constraint qualification. We also offer an example to show that these conditions are more efficient for such problems than the previous ones. Further under weak convexity assumptions, we prove that these necessary conditions are also sufficient for optimality. Finally, we extend the definitions of some constraint qualifications from classical optimization to GNEP and study their interrelations. Throughout the paper, several examples are provided to illustrate the results.

Mechanical performance of completely GFRP-tendon-reinforced concrete slabs considering compressive-tensile membrane action

Presently, increasingly more concrete structures strengthened with glass-fiber-reinforced polymer (GFRP) tendons have been employed in practical civil engineering owing to outstanding advantages, such as anti-corrosion, cost-effectiveness, and ease of production. These structures have demonstrated unique characteristics and mechanical response compared to steel-tendon-reinforced concrete structures in both service and ultimate load-bearing states. Therefore, in this study, to analyze completely GFRP-tendon-reinforced concrete slabs under localized wheel pressure, loading tests were conducted on two full-scale fabricated frame structures while considering the influence of membrane action on the cover slab. The mechanical response of these structures was revealed using the obtained load-displacement curves, ultimate bearing capacity, and crack morphology. The membrane action mechanism of the slab under weak lateral constraints was studied through the finite element method. The method to determine the effective membrane region of the reinforced slab was discussed and the corresponding ultimate bearing capacity was proposed considering the compressive-tensile membrane action under different failure modes. This model provides a preliminary reference for the quantitative determination of the structural properties of GFRP-tendon-reinforced concrete slabs.

Calcium Peroxide-Based Hydrogels Enable Biphasic Release of Hydrogen Peroxide for Infected Wound Healing.

Wound infection is a major factor affecting the speed and quality of wound healing. While hydrogen peroxide (H2O2) is recognized for its antibacterial capacity and facilitation of wound healing, its administration requires careful dosage differentiation. Inappropriately matched dosages can protract the healing of infected wounds. Herein, a calcium peroxide-based hydrogel (CPO-Alg hydrogel) is fabricated to enable a biphasic tapered release of H2O2, ensuring robust initial antimicrobial activity followed by sustained promotion of cellular proliferation of wound healing. The design of the hydrogel allowed for the calcium peroxide nanoparticles (CPO NPs) being in two spatial niches within the gel framework. When applied to infectious wounds, CPO NPs with weak constraints are promptly released out of the gel, penetrating into infected regions to serve as antibacterial agents that eliminate bacteria and biofilms at high concentrations. Conversely, the entrapped CPO NPs structurally integrated into the gel remain confined, thus gradually degrading and allowing a mild release of H2O2 through hydrolysis in a moist environment that contributes to the cell growth in the later stage. The CPO-Alg hydrogel represents an innovative and practical solution for the antimicrobial protection of chronic wounds, offering promising prospects for advancing wound healing.

Large Interferometer For Exoplanets (LIFE)

Context. Following the recommendations to NASA (in the Astro2020 Decadal survey) and ESA (through the Voyage2050 process), the search for life on exoplanets will be a priority in the next decades. Two concepts for direct imaging space missions are being developed for this purpose: the Habitable Worlds Observatory (HWO) and the Large Interferometer for Exoplanets (LIFE). These two concepts operate in different spectral regimes: HWO is focused on reflected light spectra in the ultraviolet, visible, and near-infrared (UV/VIS/NIR), while LIFE will operate in the mid-infrared (MIR) to capture the thermal emission of temperate exoplanets. Aims. In this study, we aim to assess the potential of HWO and LIFE to characterize a cloud-free Earth twin orbiting a Sun-like star at a distance of 10 pc, both as separate missions and in synergy with each other. We aim to quantify the increase in information that can be gathered by joint atmospheric retrievals on a habitable planet. Methods. We performed Bayesian retrievals on simulated data obtained by an HWO-like mission and a LIFE-like one separately, then jointly. We considered the baseline spectral resolutions currently assumed for these concepts and used two increasingly complex noise simulations, obtained using state-of-the-art noise simulators. Results. An HWO-like concept would allow one to strongly constrain H2O, O2, and O3 in the atmosphere of a cloud-free Earth twin, while the atmospheric temperature profile is not well constrained (with an average uncertainty ≈100 K). LIFE-like observations would strongly constrain CO2, H2O, and O3 and provide stronger constraints on the thermal atmospheric structure and surface temperature (down to ≈10 K uncertainty). For all the investigated scenarios, both missions would provide an upper limit on CH4. A joint retrieval on HWO and LIFE data would accurately define the atmospheric thermal profile and planetary parameters. It would decisively constrain CO2, H2O, O2, and O3 and find weak constraints on CO and CH4. The significance of the detection is in all cases greater than or equal to the single-instrument retrievals. Conclusions. Both missions provide specific information that is relevant for the characterization of a terrestrial habitable exoplanet, but the scientific yield can be maximized by considering synergistic studies of UV/VIS/NIR+MIR observations. The use of HWO and LIFE together will provide stronger constraints on biosignatures and life indicators, with the potential to be transformative for the search for life in the Universe.

Dual-consistency guidance semi-supervised medical image segmentation with low-level detail feature augmentation

In deep-learning-based medical image segmentation tasks, semi-supervised learning can greatly reduce the dependence of the model on labeled data. However, existing semi-supervised medical image segmentation methods face the challenges of object boundary ambiguity and a small amount of available data, which limit the application of segmentation models in clinical practice. To solve these problems, we propose a novel semi-supervised medical image segmentation network based on dual-consistency guidance, which can extract reliable semantic information from unlabeled data over a large spatial and dimensional range in a simple and effective manner. This serves to improve the contribution of unlabeled data to the model accuracy. Specifically, we construct a split weak and strong consistency constraint strategy to capture data-level and feature-level consistencies from unlabeled data to improve the learning efficiency of the model. Furthermore, we design a simple multi-scale low-level detail feature enhancement module to improve the extraction of low-level detail contextual information, which is crucial to accurately locate object contours and avoid omitting small objects in semi-supervised medical image dense prediction tasks. Quantitative and qualitative evaluations on six challenging datasets demonstrate that our model outperforms other semi-supervised segmentation models in terms of segmentation accuracy and presents advantages in terms of generalizability. Code is available at https://github.com/0Jmyy0/SSMIS-DC.

Does ambition work? The effectiveness of local climate pledges in China

ABSTRACT This study employs a mixed research design to examine the effectiveness of local climate ambitions in China and identify the factors influencing their performance. The empirical results indicate that local governments’ ambitious climate pledges do not necessarily lead to better performance in low-carbon development compared to cities without such pledges. This conclusion is supported by various indices, including the low-carbon status index, the carbon emissions reduction trend index, the carbon peaking trend, and the decoupling degree of carbon emissions growth rate and GDP growth rate. By integrating the principal-agent theory, the study reveals that local governments often make climate pledges passively, driven by central government mandates rather than genuine ambition. This lack of true commitment is attributable to weak incentives and constraints, significant pressure to transition to low-carbon development, and the absence of robust monitoring and accounting systems.

JWST-TST DREAMS: Nonuniform Dayside Emission for WASP-17b from MIRI/LRS

We present the first spectroscopic characterization of the dayside atmosphere of WASP-17b in the mid-infrared using a single JWST MIRI/LRS eclipse observation. From forward-model fits to the 5–12 μm emission spectrum, we tightly constrain the heat redistribution factor of WASP-17b to be 0.92 ± 0.02 at the pressures probed by this data, indicative of inefficient global heat redistribution. We also marginally detect a supersolar abundance of water, consistent with previous findings for WASP-17b, but note our weak constraints on this parameter. These results reflect the thermodynamically rich but chemically poor information content of MIRI/LRS emission data for high-temperature hot Jupiters. Using the eclipse mapping method, which utilizes the signals that the spatial emission profile of an exoplanet imprints on the eclipse light curve during ingress/egress due to its partial occultation by the host star, we also construct the first eclipse map of WASP-17b, allowing us to diagnose its multidimensional atmospheric dynamics for the first time. We find a day–night temperature contrast of order 1000 K at the pressures probed by this data, consistent with our derived heat redistribution factor, along with an eastward longitudinal hotspot offset of 18.7−3.8+11.1◦ , indicative of the presence of an equatorial jet induced by day–night thermal forcing being the dominant redistributor of heat from the substellar point. These dynamics are consistent with general circulation model predictions for WASP-17b. This work is part of a series of studies by the JWST Telescope Scientist Team, in which we use Guaranteed Time Observations to perform Deep Reconnaissance of Exoplanet Atmospheres through Multi-instrument Spectroscopy.

Abstract argumentation frameworks with strong and weak constraints

Dealing with controversial information is an important issue in several application contexts. Formal argumentation enables reasoning on arguments for and against a claim to decide on an outcome. Dung's abstract Argumentation Framework (AF) has emerged as a central formalism in argument-based reasoning. Key aspects of the success and popularity of Dung's framework include its simplicity and expressiveness. Integrity constraints help to express domain knowledge in a compact and natural way, thus keeping easy the modeling task even for problems that otherwise would be hard to encode within an AF. In this paper, we first explore two intuitive semantics based on Kleene and Lukasiewicz logics, respectively, for AF augmented with (strong) constraints—the resulting argumentation framework is called Constrained AF (CAF). Then, we propose a new argumentation framework called Weak constrained AF (WAF) that enhances CAF with weak constraints. Intuitively, these constraints can be used to find “optimal” solutions to problems defined through CAF. We provide a detailed complexity analysis of CAF and WAF, showing that strong constraints do not increase the expressive power of AF in most cases, while weak constraints systematically increase the expressive power of CAF (and AF) under several well-known argumentation semantics.

Earth-Moon dynamics from cyclostratigraphy reveals possible ocean tide resonance in the Mesoproterozoic era.

Cyclostratigraphy is an important observational window into the history of the Earth-Moon system. However, there is limited information from the Mesoproterozoic era (1.0 to 1.6 billion years ago); accordingly, only weak constraints on Earth-Moon separation and tidal dissipation are available for this time. To close this knowledge gap, we analyze cyclostratigraphy from the Yemahe Formation (~1.2 billion years ago), Wumishan Formation (~1.5 billion years ago), and Chuanlinggou Formation (~1.6 billion years ago) in China. We use a Bayesian inversion method to analyze the three cyclostratigraphic sections. We combine previous results with these three estimates to construct an updated Earth-Moon system evolution and tidal dissipation history after 2.5 billion years ago. The results show a tidal dissipation peak that is consistent with the model predictions within the error range but also that there may be an additional resonance fluctuation in the Mesoproterozoic era.

Standard siren cosmology in the era of the 2.5-generation ground-based gravitational wave detectors: bright and dark sirens of LIGO Voyager and NEMO

The 2.5-generation (2.5G) ground-based gravitational wave (GW) detectors LIGO Voyager and NEMO are expected to be operational in the late 2020s and early 2030s. In this work, we explore the potential of GW standard sirens observed by the 2.5G GW detectors in measuring cosmological parameters, especially for the Hubble constant. Using GWs to measure cosmological parameters is inherently challenging, especially for 2.5G detectors, given their limited capability, which results in weaker constraints on cosmological parameters from the detected standard sirens. However, the measurement of the Hubble constant using standard siren observations from Voyager and NEMO is still promising. For example, using bright sirens from Voyager and NEMO can measure the Hubble constant with a precision of about 2% and 6% respectively, and using the Voyager-NEMO network can improve the precision to about 1.6%. Moreover, bright sirens can be used to break the degeneracy of cosmological parameters generated by CMB data, and to a certain extent, 2.5G detectors can also play a role in this aspect. Observations of dark sirens by 2.5G detectors can achieve relatively good results in measuring the Hubble constant, with a precision of within 2%, and if combining observations of bright and dark sirens, the precision of the Hubble constant measurement can reach about 1.4%. Finally, we also discussed the impact of the uncertainty in the binary neutron star merger rate on the estimation of cosmological parameters. We conclude that the magnificent prospect for solving the Hubble tension is worth expecting in the era of the 2.5G ground-based GW detectors.

Weak lensing constraints on dark matter-baryon interactions with 𝖭-body simulations and machine learning

We investigate the elastic scattering cross section between dark matter and protons using the DES Year 3 weak lensing data. This scattering induces a dark acoustic oscillation structure in the matter power spectra. To address non-linear effects at low redshift, we utilize principal component analysis alongside a limited set of N-body simulations, improving the reliability of our matter power spectrum prediction. We further perform a robust Markov Chain Monte Carlo analysis to derive the upper bounds on the DM-proton elastic scattering cross-section, assuming different velocity dependencies. Our results, presented as the first Frequentist upper limits, are compared with the ones obtained by Bayesian approach. Compared with the upper limits derived from the Planck cosmic microwave background data, our findings from DES Year 3 data exhibit improvements of up to a factor of five. In addition, we forecast the future sensitivities of the China Space Station Telescope, the upcoming capabilities of this telescope could improve the current limits by approximately one order of magnitude.

The frequency of transiting planetary systems around polluted white dwarfs

ABSTRACT This paper investigates the frequency of transiting planetary systems around metal-polluted white dwarfs using high-cadence photometry from ULTRACAM and ULTRASPEC on the ground and space-based observations with TESS. Within a sample of 313 metal-polluted white dwarfs with available TESS light curves, two systems known to have irregular transits are blindly recovered by box-least-squares and Lomb–Scargle analyses, with no new detections, yielding a transit fraction of $0.8_{-0.4}^{+0.6}$ per cent. Planet detection sensitivities are determined using simulated transit injection and recovery for all light curves, producing upper limit occurrences over radii from dwarf to Kronian planets, with periods from 1 h to 27 d. The dearth of short-period, transiting planets orbiting polluted white dwarfs is consistent with engulfment during the giant phases of stellar evolution, and modestly constrains dynamical re-injection of planets to the shortest orbital periods. Based on simple predictions of transit probability, where $(R_* + R_{\rm p})/a\simeq 0.01$, the findings here are nominally consistent with a model where 100 per cent of polluted white dwarfs have circumstellar debris near the Roche limit; however, the small sample size precludes statistical confidence in this result. Single transits are also ruled out in all light curves using a search for correlated outliers, providing weak constraints on the role of Oort-like comet clouds in white dwarf pollution.