Kollár showed that small deformations of elliptically fibered smooth $K$-torsion varieties with $H^2(X,\mathcal{O}_X)=0$ remain elliptically fibered. We extend this result to any fibered smooth $K$-torsion variety $X$ with $H^2(X,\mathcal{O}_X)=0$, using Hodge theoretic techniques and the $T^1$-lifting criterion of Kawamata--Ran. More generally, our strategy implies that even without the cohomological assumption, small deformations of a semiample line bundle on a smooth $K$-torsion variety remain semiample up to homological equivalence.

Gait speed is a widely used indicator of functional health and mobility decline, yet in clinical practice it is commonly measured manually using a stopwatch, which limits scalability and measurement frequency. Privacy-preserving and maintenance-free sensing approaches can enable more routine and less burdensome assessments in real-world care settings. This paper presents the design, implementation, and real-world deployment of a fully passive, battery-free gait-speed monitoring system based on ultra-high-frequency (UHF) RFID. Compared with camera- and wearable-based approaches, the proposed system preserves patient privacy by avoiding video capture and biometric data, while eliminating battery maintenance. The system employs a dual-antenna configuration and an edge-based peak-detection algorithm to estimate gait speed in real time from received signal strength indicator (RSSI) streams. By leveraging antenna-beam symmetry and asymmetric signal processing, the method improves robustness to noise, plateau regions, and multiple local maxima. We evaluate the system during routine outpatient care across three clinical sites using 966 trials, achieving an 87.7% measurement success rate. Compared with concurrent stopwatch timing, the system attains a mean absolute error of 0.064 $m/s$, demonstrating reliable operation with accuracy suitable for clinical gait-speed assessment.

The current-driven resistive transition is central to superconducting single-photon detectors, transition-edge sensors, and fluxonic devices. Depending on sample uniformity, dimensions, and heat removal, it can be driven by phase-slip events, flux-flow instabilities (FFI), or normal-domain formation. Here, we investigate the influence of two types of microstructural defects on vortex dynamics in MgB$_2$ films: columnar growth in textured films and buffer-layer roughness in single-crystal films. The current-voltage ($I$-$V$) curves measured at $T \approx 0.25 T_\mathrm{c}$ for both films exhibit multiple steps. Time-dependent Ginzburg-Landau simulations reproduce the major features of the experimental $I$-$V$ curves and suggest that the resistive transitions for both films are mediated by the formation and growth of normal domains rather than FFI. The single-crystal film with buffer-layer roughness exhibits superconductivity breakdown at higher currents and pinning activation energies approximately twice those of the textured film, along with more pronounced multi-step features in the $I$-$V$ curves. These features are attributed to the combination of stronger pinning induced by lateral variations of the superconducting order parameter along the MgO buffer layer and its lower thermal boundary resistance. Our results show that both the film microstructure and the film-buffer interface are critical for the resistive transition, offering insights for superconducting devices requiring controlled dissipation at high transport currents.

This paper models the theory of abstract harmonic spaces in the syntax of the continuous first-order logic of Banach lattices. It addresses a topological question asking when a one-to-one harmonic map onto smooth manifolds $M^n$ is a diffeomorphism. We give $M^n$ ($n\le 2$) a characterization by $U$-rank and elementary saturation for large cardinals. Polar sets are characterized by several equivalent conditions from the omitting type theorem. Consequently, harmonic measures on the ideal boundary in Martin representation are bijectively mapped to Keisler measures supported on non-principal types. Further problems concerning o-minimality and non-local potentials are finally discussed.

Modern distributed systems generate large volumes of logs that can be analyzed to support essential AIOps tasks such as fault diagnosis, which plays a crucial role in maintaining system reliability. Most existing approaches rely on log-based models that treat logs as linear sequences of events. However, such representations discard the structural context between events that are often present in execution logs, such as parent-child dependencies, fan-out (branching), or temporal features. To better capture these relationships, recent works on Graph Neural Networks (GNNs) suggest that representing logs as graphs offers a promising alternative. Building on these observations, this paper conducts a comparative study of log-based encoder architectures (e.g., BERT) and graph-based models (e.g., GNNs) for automated fault diagnosis. We evaluate our models on TraceBench, a trace-oriented log dataset, and on BGL, a more traditional system log dataset, covering both anomaly detection and fault type classification. Our results show that graph-only models fail to outperform encoder baselines. However, integrating learned representations from log encoders into graph-based models achieves the strongest overall performance. These findings highlight conditions under which graph-augmented architectures can outperform traditional log-based approaches.

Lorentz symmetry is the fundamental symmetry of Einstein's theory of Special Relativity and has been tested to great precision. Nevertheless, the possibility remains that it is violated at the Planck scale, as predicted by some theories of quantum gravity. While the Planck scale is not directly accessible to experiments, minute residual deviations from Lorentz symmetry at attainable energies may be observable. The polarization of light from astrophysical sources is a particularly powerful probe because tiny differences accumulate as light travels over astrophysical distances, and polarization is sensitive to light travel time differences between polarization modes on the order of the oscillation period of the electromagnetic wave. Here, we report on new constraints on Lorentz invariance violation derived from X-ray polarization measurements of active galactic nuclei. The new constraints, presented in the framework of the Standard-Model Extension, improve on our previous work, which used optical polarization measurements, by four orders of magnitude.

Semiclassical methods have been applied very successfully to describe the nontrivial transition from the quantum to the classical regime in $\textit{single}$-particle or at least $\textit{few}$-particle systems. Challenges on the way to an extension to $\textit{many}$-body systems result from the exponential proliferation of the number of classical orbits in chaotic systems and the exponential growth of the quantum Hilbert-space dimension with the particle number. To circumvent these problems, we apply here our recently developed duality relation. Considering the kicked spin chain as example for a many-body system, we show how the duality relation can be used to extract the classical orbits from the quantum spectrum. For coupled cat maps, we analyze the spectral statistics of chaotic many-body systems and discuss the double limit of large semiclassical parameter and large particle number.

The novel kagome family RTi3Bi4 (R: rare-earth) offers a unique platform for exploring distinctive physical phenomena such as anisotropy, spin density wave, and anomalous Hall effect. In particular, the magnetic frustration and behavior of magnetic anisotropy in antiferromagnetic (AFM) kagome materials are of great interest for the fundamental studies and hold promise for next-generation device applications. Here, we report a tunable bifurcation of magnetic anisotropic and bi-oriented AFM order observed in the quasi-1D kagome antiferromagnet GdTi3Bi4. The magnetic domain evolutions during two plateau transition processes are directly visualized, unveiling a pronounced in-plane anisotropy along the a-axis. Temperature-dependent characterization reveals a bifurcation transition of anisotropy at approximately 2 K, where the a-axis anisotropy splits into two special orientations, revealing a hidden bi-oriented in-plane AFM order deviating from the high-symmetry direction by 7 degree. More intriguingly, the characteristics of the bifurcated anisotropy are clearly illustrated through vector magnetic field modulation, revealing three distinct in-plane domain phases in the transverse magnetic field phase diagram. Our results not only provide valuable insights into the tunable bifurcation of magnetic anisotropic in GdTi3Bi4, but also pave a novel pathway for AFM spintronics development.

The Surface Water and Ocean Topography (SWOT) mission provides two-dimensional observations of sea surface height (SSH) at unprecedented spatial resolution, enabling exploration of ocean variability down to scales of $O(10~\mathrm{ km})$. At these scales, however, interpreting SSH variability is challenging because ocean dynamical signals overlap with measurement noise, and their respective spectral signatures are not yet fully understood. Recent analyses of SWOT 2-km posting observations have shown that along-track spectra are red, with a power-law-like behavior at small scales and spectral slopes around or steeper than $-1$, with their magnitudes and slopes correlated with SWOT measurement noise. Here, we investigate the hypothesis that the red along-track spectra can arise from two-dimensional filtering and aliasing of spatially uncorrelated (white) noise. Using synthetic experiments, we show that the resulting one-dimensional along-track spectra exhibit red, power-law-like behavior at small scales, consistent with observations. The apparent spectral slope depends on the noise level, its cross-track variability, and the background ocean signal. This finding highlights the importance of carefully accounting for measurement noise and processing effects when interpreting SWOT spectra, and suggests that such a noise model should serve as a baseline null hypothesis for small-scale spectral analyses.

The performance of mirrors in optical interferometers is critically influenced by their surface quality. Accurate metrology enables mirror surfaces to be characterized through phase maps describing their three-dimensional structure after coating. In this work, we combine Zernike polynomial decomposition and spatial frequency (PSD) analysis with numerical optical simulations to quantify the impact of surface distortions on the reflected optical field. The method is validated using metrology data from mirrors currently installed in the Advanced Virgo gravitational-wave detector. Building on this validation, we introduce a framework for generating realistic virtual mirror maps that reproduce both low order aberrations and high spatial frequency content of measured surfaces. These virtual maps are used in optical simulations to systematically explore and compare candidate surface quality specifications for future detectors, with particular focus on the Einstein Telescope. Our results show that metrology-informed virtual mirrors provide a practical design tool to assess the impact of different surface specifications on optical performance, and to relate future requirements to the performance of existing interferometers.

Peer support is increasingly positioned as a scalable response to gaps in mental health care, particularly in digitally mediated settings, yet what counts as peer support and how responsibility is distributed remain unevenly defined in practice. Drawing on interviews with peer supporters, we show how lived experience, moral commitment, and self-identification shape participation while blurring expectations around scope, authority, and accountability. Institutional ambiguity concentrates emotional labour, boundary-setting, and escalation of responsibility at the individual level, often without consistent organisational scaffolding. Participants evaluated AI not primarily through empathy or technical capability, but through how technologies redistribute risk, labour, and accountability within already fragile support roles. Building on these findings, we outline design futures for an AI-supported peer support ecosystem that foregrounds responsibility as a central design concern rather than treating AI as a mechanism of scale.

Given a graph $G$ and an integer $r\ge 1$, the $r$th power $G^r$ of $G$ is the graph obtained from $G$ by adding edges for all pairs of distinct vertices at distance at most $r$ from each other. We focus on two basic structural properties of the $r$th power of the binomial random graph $G_{n,p}$, namely, the maximum degree $Δ(G_{n,p}^r)$ and the chromatic number $χ(G_{n,p}^r)$, and give with high probability (w.h.p.) bounds. In the sparse case that $p=d/n$ for some fixed constant $d>0$, we prove the following. We prove that w.h.p.~$Δ(G_{n,p}^r) \sim \frac{\log n}{\log_{(r+1)}n}$ (where $\log_{(1)}n=\log n$ and $\log_{(r+1)}n=\log\log_{(r)}n$) and that w.h.p.~$Δ(G_{n,p}^{\lfloor{r/2}\rfloor})+1 \le χ(G_{n,p}^r) \le Δ(G_{n,p}^{r-1})+1$. For $r=2$, we show the upper bound holds with equality. For denser cases, for $d$ satisfying $d=ω(\log n)$ and $d\le n^{1/r-Ω(1)}$ as $n\to\infty$, we have $χ(G_{n,p}^r) = Θ(d^r/\log d)$ w.h.p.

The magnetic moments and radiative decay widths of heavy baryons belong to a class of interesting experimental observables which provide direct information about the dynamics of strong interactions as well as the properties and the composition structures of heavy baryons. In this work, through a diquark model we compute these two quantities for doubly and triply heavy baryons in a dynamical model. We, first, compute an analytical mass equation for heavy diquarks based on the Bethe-Salpeter equation in which the interaction potential between constituents includes the contributions from the Cornell, the Breit-Fermi approximation, the spin-spin terms and the tensor potential. By iterating the mass equation, we compute the masses and the wave functions of heavy baryons. We also compute the magnetic moments and the radiative decay width of double and triple heavy baryons in their ground state. Our results are compared with other model-dependent predictions and existing data. We will also predict the mass and the magnetic moment of unobserved triply heavy baryons relevant for the present and future high energy colliders.

Charge density waves (CDWs) are a widespread collective electronic order in quantum materials, furnishing key insights into symmetry breaking and competing phases. However, their dynamic control with external fields remains a pivotal challenge. Here, we report deterministic and hysteretic switching of unidirectional CDW orientation via in-plane magnetic field rotation in magnetic kagome metal GdTi3Bi4. Atomically resolved spectroscopy shows two types of 3a0*1a0 CDW domains, Q1 and Q2 oriented 60 degree apart along two distinct crystallographic directions and separated by atomically sharp domain walls. Rotating the magnetic field drives reversible transitions between these CDW configurations, exhibiting a robust C2-symmetric phase diagram with pronounced hysteresis. This hysteretic switching is mediated by a field-dependent reorientation of underlying antiferromagnetic spins, revealing a tunable energy landscape with stable and metastable states and modulates the electronic charge order via spin-lattice coupling. Our findings not only demonstrate the switching of CDW configurations by in-plane magnetic field but also reveal the mechanism of coupling between CDW and magnetic fields, offering new insights into CDW manipulation and versatile platform for developing a spin-mediated multistate spin-charge coupling memory and programmable quantum devices.

In this paper, given a convex, bounded, open set $Ω\subset \mathbb{R}^n$ we prove a sharp inequality involving the Laplacian torsional rigidity and both the perimeter and the measure of the domain. Our result generalizes to arbitrary dimensions the inequality established by Makai in the plane which, as conjectured in arXiv:2007.02549. Furthermore, we establish quantitative estimates that provide key insights into the geometric structure and the thickness of the underlying optimizing sequences.

The triple-cup product form $μ$ is a classical invariant of $3$-manifolds, determining the cohomology ring up to torsion. Given a closed, connected, oriented $3$-manifold $M$, we describe an explicit formula for computing $μ$ from a Heegaard diagram of $M$. Then, we show that the triple-cup product form $μ$ can be recovered as a reduction of Turaev's homotopy intersection form $η$ of the Heegaard surface.

We study a finite-horizon dynamic pricing problem for event tickets with limited inventory and time-varying demand. The central practical difficulty is that the total demand function $L(t)$ is not observed directly and must be estimated from data, while pricing decisions are sensitive to its temporal shape. The paper examines how the accuracy of this estimate affects revenue. We consider a model in which sales intensity is driven by the total demand $L(t)$, a price-response function $v(p)$, and a time-dependent willingness-to-pay factor $φ(t)$. The factor $φ(t)$ plays a central role: it captures the increase in customers' willingness to pay as the event date approaches and makes the temporal profile of demand economically important for pricing. Within this framework, the updated numerical study evaluates a benchmark dynamic-programming policy across nine deterministic true-demand scenarios, a collection of feature-aware misspecifications of $L(t)$, and multiple environment regimes induced by $v(p)=e^{-ηp}$, the deadline factor $φ(t)$, and inventory level $Q$. The reported summaries are based on stochastic simulation and a ratio-of-means relative-loss metric. The results show that a more accurate representation of the temporal demand profile leads to more effective pricing decisions and higher revenue. Over the full misspecification collection the aggregate relative revenue loss is $0.42\%$, the upper decile exceeds $1\%$, and the most expensive errors are omissions of late-demand components. The average effect is therefore modest but non-negligible, and it becomes stronger when deadline effects are pronounced and inventory is tight.

The lack of transparency about code datasets used to train large language models (LLMs) makes it difficult to detect, evaluate, and mitigate data leakage. We present a perturbation-based method to quantify memorization advantage in code LLMs, defined as the performance gap between likely seen and unseen inputs. We evaluate 8 open-source code LLMs on 19 benchmarks across four task families: code generation, code understanding, vulnerability detection, and bug fixing. Sensitivity patterns vary widely across models and tasks. For example, StarCoder reaches high sensitivity on some benchmarks (up to 0.8), while QwenCoder remains lower (mostly below 0.4), suggesting differences in generalization behavior. Task categories also differ: code summarization tends to show low sensitivity, whereas test generation is substantially higher. We then analyze two widely discussed benchmarks, CVEFixes and Defects4J, often suspected of leakage. Contrary to common concerns, both show low memorization advantage across models: CVEFixes remains below 0.1, and Defects4J is lower than other program repair benchmarks. These results suggest that, for these datasets, models may rely more on learned generalization than direct memorization. Overall, our findings provide evidence that memorization risk is highly task- and model-dependent, and highlight the need for stronger evaluation protocols, especially in security-focused settings.

Work on persona-persistent post-mortem agents typically frames design around a life/death binary. This framing neglects a consequential yet under-theorised condition: when individuals remain alive but have impaired decisional capacity. Drawing on a multi-phase workshop in which participants trained and reflected on an AI agent for Advance Care Planning, we examined how people reason about agentic delegation post-capacity loss. Initially, participants favoured bounded agents grounded in first-party authorship and representational fidelity over autonomous or evolving stand-ins. However, temporality introduced novel ideas like adjacent use driven by persona persistence over functional expansion: agents should evolve while users retain capacity, remain static once capacity is lost, but somehow inform adjacent post-mortem uses. We discuss the implications of these findings and propose that the configuration of agents for post-capacity use reshapes our understanding of provenance, temporality, and legitimacy for post-mortem agents.

The first dedicated observation of an off-limb prominence by the Spectral Imaging of the Coronal Environment (SPICE) instrument on board Solar Orbiter took place on April 15, 2023. Our aim is to provide an overview of the potentiality of the diagnostics using these data. We show that we can derive the changes in the physical parameters of the pre-eruptive and eruptive prominence using the Lyman lines. We investigate the integrated intensity and line widths of the Lyman $β$ and Lyman $γ$ lines, finding variations between the prominence, disk, and coronal regions. The results reflect dynamic changes in density, temperature, and optical thickness. We analyze the spatial and temporal evolution of the Lyman $β$ and Lyman $γ$ line profiles. Using a simple geometric model, we obtain the radial velocity of this prominence at the early phase of its eruption with GONG H$α$ images. This offers a way of calculating the radial velocity of an eruptive filament from a pair of 2D images. The result helps us understand the potential Doppler effect in line profiles. Overall, the spectral profiles indicate that the eruption enhances spatial and temporal variations in line intensity, reflecting dynamic changes in plasma conditions within the prominence. These findings highlight the diagnostic potential of SPICE observations, and future Non-LTE radiative transfer modeling will help to further constrain prominence plasma parameters.

Right-reversing is an algorithm used to compute least common multiples in monoids that admit a right-complemented presentation. The algorithm can either terminate and find a result, fail, or run indefinitely. The correctness of the algorithm can be proved with additional assumptions coming from Garside theory. In the same framework, we prove that a non-terminating run of the algorithm is necessarily cyclic. Stopping when a cycle is detected provides a way of computing a minimal Garside family.

We introduce weighted NetKAT, a domain-specific language for modeling and verifying quantitative network properties. The language is parametric on a semiring, enabling the treatment of a wide range of quantities in a uniform way. We provide a denotational semantics and an equivalent operational semantics, the latter based on a novel model of weighted NetKAT automata (WNKA) capturing the stateful behavior of our language. With WNKA, we obtain a class of generic decision procedures for reasoning about quantitative safety and reachability in a fully automatic way, even in the presence of possibly unbounded iteration. We demonstrate the applicability of our framework in a case study using Internet2's Abilene network as the underlying topology.

A broad and growing array of applications rely on the faithful transmission of ultrastable optical signals over noisy paths, requiring cancellation of environmentally induced noise. A generally accepted limit constrains how well the path length noise can be suppressed in real time. Here, we show that this standard limit is not fundamental and can be improved upon. By considering the temporal correlations between the round-trip and one-way signals, we develop a new framework for optimizing the noise cancellation feedback signal for any spatial distribution of noise along the signal path. We use this framework to surpass the standard limit in two sets of experiments. First, we demonstrate noise cancellation in a deployed urban optical fiber, where we achieve noise suppression approximately 6 dB beyond the standard limit. Then, in a reconfigurable lab-based fiber-optic testbed, we show that, for certain spatial distributions of noise, suppression of well over 10 dB beyond the standard limit is readily achievable. With the use of digital signal processing to generate the correction signal, our new technique requires no new electro-optic hardware relative to the field-standard noise cancellation scheme. This will allow for widespread adoption of these improved limits in existing systems, with applications in optical clock distribution, optical clock comparisons for fundamental physics and geodesy, and quantum networking.

This paper studies a ground-segment implementation problem in 5G non-terrestrial networks (NTN): once UE-side geometric pre-compensation has produced a coarse timing/frequency prior, can an edge-side residual loop keep the uplink inside an NR-feasible operating region under rapid LEO dynamics? We examine this question with a software-defined ground station (SDGS) design that keeps the coarse prior at the UE and closes the residual timing-advance (TA) / carrier-frequency-offset (CFO) loop at the ground-station edge. This paper takes a systems-and-control view rather than proposing a full-stack intelligent architecture. Its evidence base consists of a March 2026 hardware-in-the-loop (HIL) campaign and a companion uncertainty analysis. The HIL campaign includes same-window reference runs collected on the same platform with edge residual control disabled, but it does not include a cloud-loop benchmark. The placement claim is therefore architectural and control-oriented rather than a head-to-head cloud-versus-edge proof. In the Shenzhen steady-state tracking interval, the edge-controlled mode lowers mean RTT from 70.51 +/- 2.34 ms to 32.84 +/- 2.56 ms and, within the retained Layer-3 transport mapping, improves artifact-level goodput from 80.14 +/- 0.14 Mbps to 196.04 +/- 1.87 Mbps relative to that reference configuration. Across four ground-station locations, the closed-loop controller keeps residual TA P95 at 0.49 us and residual CFO P95 within 76-77 Hz. Together with the uncertainty analysis, these observations support a bounded claim: an edge-side residual timing/frequency loop can keep the SDGS uplink in a more stable NR-feasible operating regime under the assumptions retained in the current HIL artifact.

We present multi-epoch high-resolution optical spectroscopy ($R \simeq 80{,}000$) of the emission-line object NaSt1 to test its proposed binary nature, along with long-term multiband photometry, mid-infrared spectroscopy, and spatially resolved integral field unit (IFU) spectroscopy to probe the circumstellar kinematics of the system. We detect two groups of 35 emission lines showing radial velocities (RVs) variation of the same period of 310 $\pm$ 6 d, but with opposing phase, which we associate with the optically thick wind of the stripped primary star and the wind-wind collision region with the companion star, providing a strong evidence for binarity. The RV and light curve (LC) periods are consistent within the uncertainties, ruling out ellipsoidal modulation, which would require an orbital period of about 620 d. The phase relationship between the RV and LC is inconsistent with stellar pulsations and supports a binary origin. We model the 1--5~$μ$m spectrum of NaSt1 and find two optically thin dust components: hot $T_{\rm h} \simeq 1230$ K, $M_{\rm h} \simeq 2 \times 10^{-10} M_{\odot}$ and warm $T_{\rm c} \simeq 660$ K, $M_{\rm c} \simeq 3 \times 10^{-8} M_{\odot}$. IFU spectroscopy spatially resolves the circumstellar medium in the [\ion{N}{2}] $\lambda6548$ and $\lambda6584$ emission lines, showing a deprojected expansion velocity of $\sim31$ km~s$^{-1}$, implying a dynamical age of $\sim40$ yr. This short timescale suggests that the nebula was produced by recent mass loss. The system may represent a Galactic analog of a massive binary undergoing a mass-loss process to become a stripped-envelope supernova progenitor.

We consider the simplicial de Rham complex and the Čech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The former complex can be realized as a subcomplex of the latter. In this paper, we quantify how close these complexes are to each other by constructing bounded cochain complexes between them, and thus we quantify how close a mixed-dimensional formulation of a problem is to an equidimensionally coupled formulation of the same problem. From this construction, we derive a priori- and a posteriori error estimates between the associated Hodge-Laplace problems on the two complexes. These estimates represent the error which is introduced by treating a spatially coupled problem as mixed-dimensional, rather than an equidimensional problem with thin overlaps.

We introduce a technique for the next-to-leading order accurate simulation of $e^+e^-\to W^+W^-b\bar{b}$ that respects the resonant nature of the process above and near the top-quark pair production threshold. The parton-shower evolution, infrared subtraction and NLO matching account in particular for finite width effects beyond the Breit-Wigner structure considered in resonance-aware approaches. We present first phenomenological results relevant to a potential future electron-positron collider and provide a publicly available simulator based on the ALARIC parton shower and the SHERPA event generator.

We study the Tolman-Oppenheimer-Volkoff equation in the presence of a cosmological constant for general thermodynamically consistent equations of state, without imposing regularity at the center. Formulating the problem as an initial value system integrated from an outer boundary inwards, we obtain a general classification of solutions and show that singular configurations dominate the solution space. We demonstrate that all singular solutions share a universal geometric structure and give rise to spacetimes that are bounded-acceleration complete, indicating that the associated singularities are comparatively mild. Our results extend the classification previously obtained for Λ=0 and reveal qualitatively new features for $Λ\neq 0$. For $Λ< 0$, we identify solutions with approximate horizon structures that mimic black holes in equilibrium with their Hawking radiation. For $Λ> 0$, we find four distinct classes of solutions with cosmological horizons, distinguished by the behavior of their temperature gradients.

The mean value of the stress-energy tensor of a given quantum field theory at global thermodynamic equilibrium in a curved space-time can be expressed in terms of the derivatives of the Killing four-temperature field and the derivatives of the metric tensor. Its asymptotic expansion about zero includes an analytic part made of integer powers of these derivatives - corresponding to the so-called gradient expansion - as well as non-perturbative corrections. By using available exact solutions for the free real massless scalar field, we show that in the case of Minkowski, de Sitter, anti-de Sitter, and closed Einstein universe, the analytic part - obtained through the procedure of analytic distillation - has a finite number of terms and it is the same once expressed in a covariant form. On the other hand, non-universal terms are non-analytic in these derivatives and correspond to boundary conditions or to specific global properties of the space-time. We argue that the universality of the analytic part extends to any quantum field theory on a curved background.

In the pinwheel problem, one is given an $m$-tuple of positive integers $(a_1, \ldots, a_m)$ and asked whether the integers can be partitioned into $m$ color classes $C_1,\ldots,C_m$ such that every interval of length $a_i$ has non-empty intersection with $C_i$, for $i = 1, 2, \ldots, m$. It was a long-standing open question whether the pinwheel problem is NP-hard. We affirm a prediction of Holte et al. (1989) by demonstrating, for the first time, NP-hardness of the pinwheel problem. This enables us to prove NP-hardness for a host of other problems considered in the literature: pinwheel covering, bamboo garden trimming, windows scheduling, recurrent scheduling, and the constant gap problem. On the positive side, we develop a PTAS for an approximate version of the pinwheel problem. Previously, the best approximation factor known to be achievable in polynomial time was $\frac{9}{7}$.