The spectrum of vector charmonium(-like) states in the 4.1-4.6 GeV energy region exhibits a long-standing tension. While the inclusive $R$-value indicates the presence of only vector charmonia that are usually interpreted as conventional states, exclusive $e^+e^-$ cross sections reveal additional structures whose parameters strongly depend on the observed final states when fitted with Breit-Wigner functions. This puzzling pattern suggests that coupled-channel and threshold effects play an essential role. In this work, we develop a unified coupled-channel framework relevant for the $1^{--}$ resonances in the considered energy region. The framework incorporates the $S$-wave open-charm channels $D\bar{D}_1$, $D^*\bar{D}_1$, and $D^*\bar{D}_2^*$, constrained by heavy-quark spin symmetry, optional bare poles corresponding to quark-model states which may be associated with the $ψ(4160)$ and the $ψ(4415)$, and final-state interaction in the $Z_c$ channels relevant for the three-body final states. We employ several benchmark models based on the constructed framework to perform simultaneous fits to the BESIII data for the cross sections of $e^+e^-\to J/ψπ^+π^-$, $h_cπ^+π^-$, $D\bar{D}^*π$, $D^*\bar{D}^*π$, $J/ψη$, and $χ_{c0}ω$, together with the available invariant-mass distributions that exhibit the $Z_c(3900)$ and $Z_c(4020)$ structures. The models differ in the number of bare seed states and the fitting strategy. We show that even the purely dynamical scheme captures the gross features of all the analyzed distributions. We therefore conclude that the nontrivial behavior of the measured line shapes in the studied energy range can be understood in terms of strong coupled-channel effects with dynamically generated poles. The inclusion of bare compact states leads to a quantitative improvement of the fit quality but does not alter this conclusion.

Weak gravitational lensing mass mapping offers a direct probe of the matter distribution. Accurate reconstruction of mass maps from masked shear catalogs remains challenging due to survey boundaries and spatially varying noise. In AKRA 2.0, we addressed the mask problem on the curved sky by constructing and inverting the normal-equation matrix $\bf{H} \equiv \mathbf{A}^\mathrm{T}\mathbf{N}^{-1} \mathbf{A}$ explicitly, necessitating a split-scale strategy that reconstructed different angular scales independently to reach high resolution. Here we present AKRA 3.0, in which $\mathbf{H}$ is treated as a linear operator and the normal equations are solved by the conjugate gradient (CG) method. This reformulation reduces the memory requirement from $O(N^2)$ to $O(N)$ and the inversion cost from $O(N^3)$ to $O(N_{\rm iter}N^{3/2}), N \sim \ell_{\rm{max}}^2$ for full-sky (SHT-based) operations. Such optimizations render high-resolution full-sky reconstruction tractable for Stage~III and Stage~IV surveys. Applying AKRA 3.0 to the DES Y3 \texttt{METACALIBRATION} catalog, we produce the highest-resolution convergence map of this dataset to date at HEALPix $N_{\rm{nside}}= 2048$ without imposing any prior assumptions. We extract the convergence power spectrum directly from the reconstructed map and demonstrate that unbiased two-point measurements can be obtained directly from the reconstructed map. The reconstructed E-mode convergence map will be publicly released as data products to enable future studies of non-Gaussian statistics, higher-order moments, and cross-correlations with external datasets.

This work presents a data-driven framework for interpretable modelling and decision support in flotation systems, integrating Gaussian Process (GP) regression with Global Sensitivity Analysis (GSA) via Sobol indices and local interpretability using SHapley Additive exPlanations (SHAP). Based on laboratory-scale experimental data, a static GP surrogate model is developed to capture how superficial air velocity, overflowing froth velocity, froth height over the lip, pulp height, bubble size, and tailings flowrate influence the measured air recovery. The trained GP enables the computation of Sobol indices to quantify the contribution of each variable and their interactions to the overall variance in air recovery. The combination of Bayesian inference and Sobol-based sensitivity metrics provides a systematic approach to identify the dominant and interacting variables governing air recovery. This study links Bayesian learning, sensitivity quantification, and explainability to provide a foundation for data-driven control and optimisation of flotation processes.

In this article we develop a quadratic energy which approximates the Canham-Helfrich energy for a tube-like surface with clamped boundary and area constraint. The energy is suited to the study of small deformations of biological membranes where the deformations are induced by point forces or point constraints due to the cytoskeleton or a phase dependent spontaneous curvature. Since the deformations we consider are small, we may assume that the surface of interest is a graph over a fixed, undeformed surface. A Lagrangian and the associated Euler-Lagrange equations for the graph are derived. Well-posedness of the Euler-Lagrange equations in suitable spaces is shown. Finally, we provide some illustrative numerical examples.

By now, several approaches have been proposed to endow phase-field fracture models with the ability to describe crack nucleation under multiaxial stress states. These include techniques for splitting the free energy, direct modifications of the phase-field driving or resisting forces that sacrifice the variational structure of the problem, and the introduction of additional internal variables, such as plastic strains or other nonlinear strains. In this paper, we propose a fundamentally different strategy for incorporating arbitrary elastic domains into phase-field fracture models, formulated within the variational framework of generalized standard materials. The proposed approach relies on letting the dissipation potential depend on the current state of the material. In this way, the variational structure of the problem is preserved, while elastic degradation and the strength criterion remain two distinct and independently controllable aspects of the material response. Simple yet representative models are presented and thoroughly discussed to demonstrate the effectiveness of the proposed methodology. The resulting evolution of the elastic domain is investigated in both strain and stress spaces. Moreover, numerical simulations demonstrate a range of crack nucleation processes under multiaxial loading conditions for various analytical strength surfaces. This work paves the way for future developments and applications in several directions.

The increasing penetration of single-phase loads and distributed generation exacerbates voltage unbalance (VU) in distribution grids, raising concerns about power quality and complicating network operation. However, most market-clearing models and price-based coordination frameworks do not enforce VU limits within a three-phase AC representation, so the implications for grid-code compliance, numerical scalability, and economic signals remain unclear. This paper embeds VU in a three-phase AC optimal power flow market-clearing model and benchmarks two treatments: strict VU limit enforcement and objective function penalization. Building on these insights, an Improved Hybrid Limits (IHL) formulation is proposed that preserves compliance while using a smooth unbalance proxy in the objective to guide the optimization solver. Case studies on a European low-voltage feeder show that IHL maintains feasible operating points, yields price and curtailment signals consistent with conventional hybrid formulations, and converges substantially faster and more reliably than a penalization based on the exact unbalance metric. These results support IHL as a practical and scalable mechanism for VU mitigation in market-based operation of unbalanced distribution systems.

Energy decay rates for solutions of the damped wave equation on the torus are known to be influenced by the geometry of the damped set and the growth properties of the damping. In this paper we produce lower bounds on energy decay rates for a class of damping which are positive on a superellipse and grow polynomially like the distance to the boundary of the superellipse. The energy decay rates we obtain depend explicitly on the exponent used to define the superellipse and the polynomial power. We show these rates are sometimes optimal. The proof adapts quasimodes from $y$-invariant damping using a simplification of the usual normal form argument.

This paper develops equivariant basic cohomology for Lie groupoids equipped with weak actions of Lie groups. The weak action is encoded by a Kan fibration over the classifying groupoid, and the basic complex of the fiber is shown to carry the structure needed for Weil and Cartan models. The construction is compared with Bott--Shulman--Stasheff cohomology, where the equivariant theory is obtained from the quotient groupoid. For orbifolds, basic forms are interpreted as orbifold differential forms, and the resulting equivariant basic cohomology is used to formulate differential-geometric constructions such as equivariant integration and localization. The paper also studies the induced weak action on the inertia groupoid and uses it to define an equivariant refinement of the Chen--Ruan cohomology ring. In this framework the sectorwise equivariant cohomology, obstruction bundle, equivariant Euler class, Gysin maps and three-point functions are assembled into an equivariant Chen--Ruan product whenever the corresponding pairing is nondegenerate.

In many industrial and engineering applications, process performance is characterized by the ratio of two normally distributed quality characteristics. Monitoring such ratios is particularly challenging in short production runs, where conventional control charts often suffer from limited sensitivity due to the small number of available inspections. This paper proposes an exponentially weighted moving average (EWMA) control chart for monitoring the ratio of two normally distributed random variables under short production run (SPR) conditions. The statistical distribution of the ratio is first reviewed, adopting the corrected closed-form density of Nadarajah (2020) rather than the approximation used in earlier studies. The control limit of the proposed chart is calibrated to a prescribed in-control truncated average run length (TARL$ _0 $) over a finite horizon $ I $ of inspections, using a Markov-chain representation of the EWMA recursion. The detection performance of the chart is then assessed through a large factorial study covering the smoothing constant $ λ$, the in-control correlation $ ρ_0 $, the coefficients of variation $ (γ_X, γ_Y) $, the sample size $ n $, and the magnitude of the shift $ τ$. Numerical results show that the proposed EWMA-RZ chart provides substantially better detection of small and moderate shifts than the recently developed Shewhart-type short-run ratio chart (ShRZ) of Tran et al. (2021), especially for $ |τ- 1| \le 0.05 $. An illustrative example based on a beverage filling process is included to demonstrate the practical implementation of the method.

Let $G$ be a graph. The {\em spectral radius} of $G$ is the largest eigenvalue of its {\em adjacency matrix}. A {\em matching} of $G$ is a set of disjoint edges of $G$. The {\em matching number} of $G$ is the size of a maximum matching (i.e., a matching with maximum edges). The graph $G$ is called {\em maximum matching covered} if each edge of $G$ is contained in a maximum matching. In this paper, we give a sharp spectral radius condition for graphs with bounded matching number to be maximum matching covered.

We investigate circuit models, namely, modified nodal analysis (MNA) in the port-Hamiltonian framework. Based on this, the JR-decomposition for the numerical treatment would offer an energy conform splitting. However, for circuit models, the application of the standard JR-decomposition is restricted. To enable a JR-decomposition for MNA, we need to relax the decomposition. To this end, we introduce the enhanced JR-decomposition, which is particularly tailored to the application to circuits. We conclude with a numerical example that illustrates the applicability of the proposed approach as well as its convergence and structure-preserving properties.

Superconducting diode effect exhibits asymmetric critical supercurrent and has profound implications for condensed matter physics. The technical appeals of such superconducting diodes are their ultrahigh on-off ratio and diode efficiency for superconducting electronics owing to the dissipationless supercurrent therein. However, realizing superconducting diode operation at high working frequency, which is a key requirement for practical applications, remains elusive and challenging. Here, we demonstrate a polarity-controllable superconducting diode with non-equilibrium Josephson junction and its edge-triggered flip-flop operation at a high frequency up to 2.4 GHz, within a van der Waals superconductor 2M-WS$_2$. By simply tuning the thickness of superconducting 2M-WS$_2$ nanoflakes to engineer inversion asymmetry in the junction, we achieve a high diode efficiency of 67% and an on-off ratio exceeding 10$^5$. Importantly, the pulse width and duty cycle of output pulse signals in such superconducting diode flip-flop devices can be controlled in a broadband frequency range crossing 12 orders of magnitude. Theoretical analysis reveals that the non-equilibrium dynamic nature of supercurrent in these Josephson junctions enables such a high diode operating frequency and the polarity control of supercurrent. The 2.4 GHz non-equilibrium Josephson diode developed here provides a promising platform for advanced superconducting logic circuits and broadband telecommunication applications.

Mechanisms for allocating a divisible resource among strategic agents have been widely studied. The prominent paradigm is the proportional (Kelly) mechanism, which elicits a scalar bid per agent, allocates the resource proportionally, and charges payments equal to the bids. Follow-up mechanisms improve social welfare, but sacrifice simplicity by introducing complex allocation rules or unintuitive payments. We introduce a unified framework for designing simple resource allocation mechanisms with proportional-style allocations and uniform pricing. Our framework yields a family of mechanisms that interpolate between the Kelly mechanism and the first-price auction. These mechanisms strictly improve upon Kelly's efficiency guarantees, even achieving full efficiency in equilibrium, while also providing revenue guarantees relative to the VCG mechanism.

In this work, we propose a novel intrinsic approach to the approximation of ergodic SDEs on Riemannian manifolds, which include Riemannian Langevin dynamics. In opposition to the standard extrinsic approaches such as penalization methods and projection methods, our methodology does not use embeddings or coordinates and only relies on natural geometric operations: geodesics, parallel transport,... We give a criterion for high order of accuracy for the invariant measure, develop new intrinsic numerical methods designed solely for sampling the invariant measure, and derive high order conditions using a new algebraic operation on exotic Lie-Butcher series. In the spirit of the Leimkuhler-Matthews method, our approach prioritizes long time sampling efficiency over finite time accuracy, and outperforms the previous extrinsic and intrinsic approaches in terms of cost for a given accuracy, which we illustrate with several numerical experiments.

In numerous industrial production settings, keeping track of the ratio formed by two normally distributed random variables is a task of considerable practical interest. The present work examines how measurement errors influence the behaviour of a pair of one-sided Synthetic control charts designed to monitor such a ratio (referred to here as Synthetic-RZ charts), with the analysis covering both the zero-state and the steady-state average run length ($ARL$). To incorporate measurement error into the operation of these charts, we adopt a linear covariate error model. We describe, step by step, how the parameters of the underlying model evolve as the process moves from an in-control to an out-of-control state, and we deliberately avoid the restrictive premise that the observed shift magnitude is unrelated to the measurement errors. The run length characteristics of the charts are obtained by means of a Markov chain formulation. A series of numerical experiments makes clear that measurement error erodes the detection capability of the charts. A particularly useful outcome of the investigation is that collecting several measurements on each inspected unit does not constitute an efficient remedy for the adverse influence of measurement error on the performance of the Synthetic-RZ charts.

We investigate the structural, electronic, and magnetic properties of bilayer Janus 1T-MnSSe using first-principles calculations. Various AA- and AB-type stacking configurations are considered to examine the influence of interlayer registry on magnetic ordering and exchange interactions. The nonmagnetic state is unstable for all stackings, confirming intrinsic magnetism. The AA2 stacking is identified as the ground state and exhibits A-type antiferromagnetic ordering, indicating antiferromagnetic interlayer coupling. Monte Carlo simulations based on an effective Ising model reveal enhanced magnetic transition temperatures in the bilayer relative to the monolayer, with Néel temperatures above 300~K for antiferromagnetic stackings and Curie temperatures up to 250~K for ferromagnetic phases. Several stacking configurations exhibit robust half-metallic ferromagnetism with nearly 100\% spin polarization at the Fermi level. Moreover, the half-metallic state can be tuned and ultimately transformed into a metallic ferromagnetic phase through carrier doping and biaxial strain. These findings establish bilayer MnSSe as a promising platform for controllable interlayer magnetism and spintronic applications in two-dimensional materials.

The textbook choice B=sqrt(n) for square-root decomposition is asymptotically natural, but it is not always the fastest implementation choice. We study block-size autotuning as a reproducible algorithm-engineering problem and show that a learned workload model can improve over fixed sqrt(n) on the tested implementation. Under repeated grouped cross-validation, the best policy is a full-feature KNN-9 model that reduces mean regret from 1.2882 to 1.0646 and yields a paired geometric-mean speedup of 1.151x. A confidence gate retains most of that gain while reducing slowdowns. A family-free full-observation follow-up remains better than fixed blocking, which suggests that the model is learning from workload statistics rather than memorizing labels. In contrast, short-prefix variants do not produce a successful low-overhead online tuner in the current prototype. External validation is selective but supportive: Zipf-Hotspot is the strongest out-of-distribution case, and a six-window Baleen follow-up still improves over fixed blocking. Overall, block-size choice is workload aware and platform aware, and the fixed sqrt(n) rule leaves substantial performance on the table.

We leverage $s$-orbital non-resonant inelastic X-ray scattering ($s$-NIXS) to perform orbital imaging on three bulk rare-earth nickelates spanning a range of formal nickel valence (3$d$ electron filling) from Ni$^{3+}$ (3$d^7$) to Ni$^{1+}$ (3$d^9$). Our results directly reveal the ground states of these compounds all with minimal theoretical input. In particular, we demonstrate the low-spin orbital configuration of trivalent LaNiO$_3$, the $d_{x^2-y^2}$ configuration of monovalent LaNiO$_2$, and resolve the effective $e_g$ crystal field splitting in the distorted octahedral environment of divalent La$_2$NiO$_4$. This work illustrates the potential of $s$-NIXS to study the ground state and excited states of strongly correlated materials without needing complex theoretical analysis of spectroscopic data.

In this paper, we determine the exact minimal number of curves in a filling $k$-system on an oriented surface of genus $g$ for any positive integers $k$ and $g$.

Image safety classifiers serve as a critical component of contemporary content moderation systems on the internet. However, their resilience against user-style malicious image editing remains underexplored. Such behaviors are highly prevalent in daily scenarios but difficult to fully reproduce. To explore this vulnerability, we introduce RedEdit, a novel black-box red-teaming agent that formulates photo-editing evasion as a combinatorial search problem over edit-tool sequences. It adopts a Vision-Language-Model (VLM)-based proposer to generate semantically targeted candidate edits and a Monte Carlo Tree Search (MCTS) planner to prioritize promising edit paths while backtracking from ineffective ones. Together, the proposer and planner instantiate two key capabilities of human attackers, i.e., domain knowledge and iterative backtracking, respectively, to reproduce this practical threat. Our extensive experiments on UnsafeBench reveal profound systemic vulnerabilities: fewer than two edits on average enable 76.2% of unsafe images to evade detectors, while retaining 93.0% malicious semantics, meaning that such manipulated content remains perceptually malicious to humans while easily bypassing automated moderation. We therefore appeal to the community for more attention to this overlooked practical threat.

Charge conjugation violation (CCV) is a central concept in particle physics and appears also for quasiparticles in quantum many-body systems, which typically relies on an embedded external symmetry breaking to the underlying system. An open question is how an intrinsic CCV mechanism could emerge and what its macroscopic consequences would be. We establish sublattice kinks in bipartite fermionic lattices as a concrete setup showing intrinsic CCV. The intrinsic CCV of the sublattice kink is based on the graph-topological nature of the underlying Hamiltonian, with no explicit symmetry breaking taking place. It leads to a population asymmetry of different configurations and imprints a hidden leaf-like structure in the eigenenergy spectrum. The population asymmetry also leads to an imbalanced sublattice-kink production triggered by the vacuum-instability in the quench dynamics. Our work demonstrates the graph topology as the microscopic origin of intrinsic CCV, with the population asymmetry as the macroscopic consequence, of which the proposed setup is highly amenable to experimental implementation via cold-atom quantum simulators.

Monitoring the ratio of two correlated normal variables is increasingly important in statistical process control, since many quality characteristics are expressed in relative rather than absolute form. Memory-type ratio charts have mostly been developed for long production runs, while their finite-horizon counterparts rely on a fixed reference value $ k $ derived from a specified shift. Such fixed-$ k $ designs are not optimal at a given out-of-control magnitude and, in low-variability regimes, yield boundary solutions for which the in-control truncated average run length (TARL$ _0 $) is unattainable. This paper proposes an upper one-sided cumulative sum (CUSUM) control chart for the ratio $ Z = X/Y $ in short production runs, denoted CUSUM-RZ$ ^+ $ (RZ standing for the ratio $ Z $), with fully adaptive joint optimization of $ k $ and the decision interval $ h $. Given a target TARL$ _0 = I $ and a target shift $ τ$, a bilevel problem calibrates $ h(k) $ by inner root-finding to satisfy the TARL$ _0 $ constraint and selects $ k^* $ by outer line search to minimize the out-of-control TARL$ _1 $. Both use a finite-state Markov-chain framework with an accurate ratio approximation; the inner step recovers boundary cases that fixed-$ k $ designs cannot. The chart is assessed through matched-horizon benchmarks against Shewhart-RZ, exponentially weighted moving average (EWMA-RZ), and fixed-$ k $ CUSUM-RZ$ ^+ $ charts, Monte Carlo robustness studies, and a Phase I estimation analysis. All memory-type charts outperform the Shewhart-RZ baseline; the adaptive design matches them under stable correlation and improves appreciably when correlation rises from Phase I to Phase II. It is insensitive to symmetric heavy tails yet mildly anti-conservative under contamination, and $ m \geq 100 $ subgroups keep the TARL$ _0 $ relative bias near 1%.

The orbital motion of near-infrared flares reported by the GRAVITY collaboration encodes information about both the dynamics of accretion matter and the underlying spacetime geometry. The centroid track of these flares, which corresponds to the flux-weighted center of light, incorporates contributions from primary, secondary and higher-order images. Thus, it potentially indicates distinctive signatures of the spacetime geometry, even when these individual multiple images remain unresolved. In this study, we explore the detectability of the secondary images from flares orbiting Sgr A* through mock data simulating future GRAVITY observations. Specifically, we compare the model in which the centroid coincides with the track of the primary images with another model in which the centroid incorporates flux-weighted contributions from both the primary and secondary images. Fitting these models to the mock data based on Bayesian framework, we quantify the conditions under which the signature of secondary images can be statistically distinguishable. We demonstrate that increasing the sample size by an order of magnitude alone could not yield strong evidence for distinguishing the secondary image. Robust detectability ($|Δ\text{BIC}| >7.9$) is achieved when both with the improved sample size and astrometric uncertainties reduced to 40\% of current uncertainties of GRAVITY astrometric data. Unlike the primary image, which is dominated by accretion flow physics, the secondary images originate from gravitational lensing in the strong-field regime. Their detection is an essential first step toward probing higher-order images and the photon rings.

Understanding of friction between sliding surfaces is critical for variety of applications. We present a friction model between soft and hard solid interfaces for studying aging time dependent static friction. The model is based on strengthening of dangling chains with the substrate during aging period. The friction model is, in turn, validated with the experimental data from literature. Friction properties are also estimated in terms of gelatin concentration to justify the results.

Constitutive equations have to be in agreement with the energy- and entropy- balances. For achieving that, the procedure of thermodynamical verification is introduced: Because heat flux and entropy flux as well as the time differentials of internal energy and entropy are not independent of each other, energy- and entropy-balances are connected with each other by so-called internal settings laying down the theoretical frame of the applied material description which is characterized by additional constitutive settings.

We study complete non-compact gradient generalized m-quasi-Einstein manifolds with constant scalar curvature $R \le 0$, soliton function $λ> 0$, and $m > 1$, where the coefficient $μ= 1/m$ is constant. We introduce the weighted function $v = e^{-f/m}λ$ and prove it is subharmonic. This leads to five rigidity results, each forcing the manifold to be Euclidean. We first show by a concrete example that if $μ$ is allowed to be nonconstant, the rigidity conclusions fail even when all other hypotheses are satisfied. Therefore the constant mu condition is essential.

Long-lived dark excitons in monolayer WSe2 present promising candidates for carrying spin and valley information, but their optical access and spin manipulation have conventionally required the use of strong external magnetic fields. Here, using a ferroelectric hybrid perovskite heterostructure, we leverage the ferroelectric proximity effect to break the WSe2's in-plane rotational symmetry and brighten the spin-forbidden dark excitons under zero magnetic field conditions. Furthermore, we show that the twist angle between the WSe2 and perovskite crystals controls the ferroelectric coupling strength and valley-contrasting polarization. Our proposed mechanism, supported by a four-band tight-binding model, suggests that the ferroelectric proximity effect induces an asymmetric intersublattice interaction, generating an effective in-plane spin-orbit coupling (SOC) field that rotates spin/valley polarization and brightens dark excitons. Our work establishes ferroelectric proximity coupling as an electrically reconfigurable, magnetic-field-free strategy for spin exciton control in two-dimensional semiconductors.

The problem of validating a given graph database instance against a given PG-Schema graph type without integrity constraints is NP- complete in terms of combined complexity and in PTIME in terms of data complexity. The combined complexity drops to PTIME when the alternation between type combinations and unions is suitably restricted

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme (i.e., a contract) in order to induce an agent to take a costly action leading to a favorable outcome. We consider the online extension of the classical (one-shot) principal-agent problem, in which the principal repeatedly interacts with agents by proposing contracts over multiple rounds. The principal has no information about the agents and, crucially, does not observe their actions. As a result, the principal must learn an optimal contract using only the realized outcomes observed at each round. We focus on the setting with binary actions and single-dimensional agent types, where the agent's private type represents their cost per unit-of-effort. For adversarial-type sequences, we provide tight $Θ(T^{2/3})$ regret guarantees. Remarkably, this rate is completely independent of the number of outcomes $m$. The upper bound is based on two key components: 1) a reduction to a one-dimensional threshold optimization problem and 2) a non-uniform discretization to handle the non-Lipschitz nature of the problem. Moreover, in the case of a single (fixed) hidden type, we show that it is possible to improve the rates and provide a tight $\widetildeΘ(\sqrt{T})$ regret bound. Our algorithm is based on an explore-then-commit strategy where we first approximately learn the hidden type via a stochastic binary search, and then we commit to a ``robustified'' near-optimal contract.

The region spaning the ring singularity in the Kerr black hole solution is described here as two (flat) discs rather than one. From this viewpoint it follows that the usual depiction of the worldsheet $θ=π/2$, $ϕ$ constant (in Boyer-Lindquist coordinates) is missing (multiple copies of) the region $r<0$, representable as a triangle in the usual conformal diagram style.