Relevance is a foundation of user experience in e-commerce search. We view relevance optimization as a closed-loop ecosystem involving multiple human roles: users who provide feedback, product managers who define standards, annotators who label data, algorithm engineers who optimize models, and evaluators who assess performance. Because improving relevance in practice means systematically resolving user-perceived bad cases, we ask a system-level question: can this ecosystem be reimagined by replacing its human roles with autonomous agents? To answer this question, we propose a case-driven multi-agent framework that automates the pipeline from bad-case identification to resolution. The framework instantiates an Annotator Agent for multi-turn annotation, an Optimizer Agent for autonomous bad-case analysis and resolution, and a User Agent that identifies bad cases through conversational interaction, together forming an autonomous and continually evolving system. To make the framework practical in production, we further adopt a harness-engineering paradigm and build a unified retrieval-and-ranking relevance model for efficient training, an instruction-following relevance model for real-time case resolution, Global Memory to reduce information asymmetry across agents, a Deep Search Agent to target underestimation failures, and an agent-based chatbot for human--agent collaboration. Extensive human evaluation shows that the framework performs relevance-related tasks effectively, improves annotation accuracy, and enables more timely and generalizable bad-case resolution, indicating a practical paradigm for industrial search relevance optimization.

This paper presents a reinforcement learning approach of a model-free safety filter, drawing inspiration from the framework of model-based Predictive Safety Filters (PSFs). Similar to conventional PSFs, our method adopts a Quadratic Programming (QP) formulation by representing the filter as an unrolled QP solver network. However, unlike existing PSFs that derive QP parameters explicitly from system models, we learn these parameters directly through Deep Reinforcement Learning (DRL), thereby eliminating the dependency on accurate system identification. Furthermore, compared to traditional neural network-based methods, this QP structure allows us to furnish a formal certificate for the persistent safety of the learned filter. Numerical results demonstrate that our method outperforms both conventional model-based PSFs and RL-trained Multi-Layer Perceptron (MLP) baselines in terms of safety guarantees, minimal intervention, and per-step computational load.

In this paper, we introduce a nonlocal, variational model for thin films. We consider convolution-type functionals defined on a thin domain whose thickness is of order $γ$, where the effective interactions range between points is of order $\varepsilon$. We study the $Γ$-convergence of these energies, as both parameters vanish, to a local integral functional defined on a lower-dimensional domain. In the periodic homogenization setting, the limit energy density is characterized by an asymptotic formula that depends on the interplay between $\varepsilon$ and $γ$. Under suitable assumptions, this formula exhibits a separation of scales effect, namely, the limit energy can be obtained by performing two successive $Γ$-limits, first letting one parameter tend to zero while keeping the other fixed.

Cold storage is a persistent constraint in sub-Saharan African informal food systems, where perishables are traded in open-air markets with intermittent electricity and grid-tied or battery-heavy cold chains are too costly to scale. We develop a techno-economic optimisation framework that co-designs solar photovoltaics, refrigeration, phase change material (PCM) thermal storage, and inter-market food transport, using five open-air meat markets in Abuja, Nigeria as a case study. The framework treats pre-chilled meat as a mobile carrier of cold energy moving through existing trade routes, while PCM remains stationary at each market. Replacing part of the battery with PCM lowers annualised system cost by up to 15% (mean 11%), driven by a roughly 67% reduction in battery capacity. Allowing inter-market cold exchange via chilled meat further cuts total cost by 8% and aggregate PCM capacity by 35% by reallocating storage across markets without additional generation. PCM competitiveness depends on its relative capital cost, discharge efficiency, and the refrigeration charging window: long predictable charging windows favour PCM, short flexible-response needs favour batteries. Across all scenarios the cost of cooling stays a small share of meat value. The framework shows how treating cold as a mobile energy vector embedded in food flows can inform cold-chain design in other infrastructure-constrained food networks.

N. Fournier and A. Guillin obtained in their 2015 PTRF paper some bounds of the L^p-mean rate of convergence in Wasserstein distance of empirical distributions for a class of stationary mixing processes. In this paper, we propose to extend their strategy of proof and provide general criterions which allow to keep similar rates for a larger class of processes. These results (which do not require regularization techniques) lead to various applications to occupation measures of ergodic processes which may be not stationary or not Markovian under an assumption of {\em conditional} convergence to equilibrium in Total Variation or Wasserstein distance. We then provide explicit conditions which lead to these rates for Brownian diffusions and additive SDEs driven by fractional Brownian Motions {or by Gaussian processes with stationary increments}.

Robins and Richardson (2010) reformulated mediation analysis by decomposing treatments into multiple components and examining separable effects of each component. While this approach is increasingly popular, existing work has analyzed ``two-arm'' data, where components are strictly bundled and manipulated simultaneously. However, in practice, four-arm data where components are assigned independently are often available. For example, testing accommodations might strictly bundle extra time with a separate session or allow them to be assigned separately. To address this distinction, we propose a general framework for analyzing separable effects in four-arm and two-arm designs. This framework provides distinct identification and estimation strategies for each design. For estimation, we utilize efficient influence function estimators coupled with machine learning and cross-fitting techniques. Additionally, we introduce two falsification tests for key identification assumptions required in the two-arm design by leveraging four-arm data. We investigate the performance of the proposed estimators via a simulation study and demonstrate their application by studying the effect of extended time accommodations using data from the National Assessment of Educational Progress. Ultimately, this separable effects analysis enables practitioners to clearly communicate underlying mechanisms and derive informative policy recommendations.

High-current superconducting cables are emerging as key enablers for next-generation power transmission systems; however, their deployment is often limited by transport AC losses. Hybrid superconducting cables combining Conductor-on-Round-Core (CORC) and Twisted Stacked-Tape Conductor (TSTC) architectures have recently been proposed as a promising route toward cables with high current capacity and compact form factors. However, their electrodynamic response under transport current operation remains poorly understood, particularly regarding how current injection conditions govern internal current redistribution. Here, we employ a fully-3D electromagnetic model, previously validated against magnetisation experiments in equivalent cables, to investigate the influence of current injection strategy on the electrodynamics of hybrid CORC-TSTC cables under self-field conditions. By comparing configurations in which the total current is either injected through a common connection between the CORC and TSTC conductors (non-insulated feeding) or supplied independently to each conductor (insulated feeding), we show that electrical coupling in non-insulated designs leads to strong current redistribution, pronounced waveform distortion and elevated AC losses once the CORC layers approach magnetic saturation. In contrast, independent current feeding suppresses inter-conductor current exchange, stabilises the current waveforms, and exhibits an outstanding reduction in transport AC losses of up to 90% at practical operating currents, compared with conventional feeding schemes. These findings reveal the central role of the current injection strategy in governing the internal electrodynamics and energy dissipation of hybrid superconducting cables, and identify the electrical decoupling of the constituent conductors at the feeding point as a simple and scalable route toward ultra-efficient power cables.

We investigate the physical origin of kinematically persistent planes (KPPs) of satellite galaxies in a sample of 190 Milky Way (MW)/M31-like host-satellite systems drawn from the TNG50 simulation. Building on the identification of 46 early KPPs in a previous work, we analyse their formation in the context of the high-redshift evolution of the local Cosmic Web by tracking the deformation of the so-called Lagrangian Volumes (LVs) surrounding each system. Using a reduced tensor-of-inertia analysis, we characterise the time evolution of the principal directions of collapse and relate them to the clustering of satellite orbital poles. We find that in approximately 67\% of KPPs satellite orbital poles align with the LV direction of strongest collapse, $\vec{e}_3$, while a smaller fraction ($\sim20\%$) align with the intermediate axis, $\vec{e}_2$; alignments with the major axis are rare. These alignments are statistically distinct from random expectations and reflect the confinement of satellites to planar configurations normal to the corresponding LV principal directions. We perform a kinematic analysis of satellite motion within KPPs, finding that vertical and radial motions relative to these KPPs decay early, leading to rotation-dominated, ``disky'' configurations. The characteristic timescales for satellites to settle onto a common orbital plane, for satellite orbital pole clustering, and for LV shape evolution are found to be quasi-coeval, peaking at a Universe age T$_{\rm uni}\sim4$~Gyr, during the fast mass assembly phase of the host halo. These results support a scenario in which early KPPs are fossil remnants of high-redshift, anisotropic mass collapse driven by the local Cosmic Web formation process in $Λ$CDM.

In this note, we construct new solutions to the heterotic $\mathrm{G}_2$-system with non-abelian gauge group, both compact and non-compact, on certain $2$-step nilmanifolds and $3$-Sasakian manifolds. Our approach is based on an ansatz that allows us to vary both the $\mathrm{G}_2$-structure and the gauge data while keeping the underlying metric and orientation fixed. This leads, in particular, to distinct isometric solutions on the same manifold but with different gauge groups, and in some cases the resulting connection coincides with the characteristic connection of the $\mathrm{G}_2$-structure. We also investigate an $S^1$-invariant construction that yields further isometric solutions and with varying cosmological constant. Our results recover and extend several known examples solving the heterotic $\mathrm{G}_2$-system within a unified framework.

GitHub Continuous Integration (CI) workflows increasingly integrate Large Language Models (LLMs) to automate review, triage, content generation, and repository maintenance. This creates a new attack surface: externally controllable workflow inputs can shape LLM prompts and outputs, which may in turn affect security decisions, repository state, or privileged execution. Although LLM security and CI security have each been studied extensively, their intersection remains underexplored. In this paper, we present the first study of LLM-induced security risks in GitHub CI workflows. We characterize the problem along the full execution chain and develop a taxonomy of high-level risk classes and concrete threat vectors. To detect such risks in practice, we design Heimdallr, a hybrid analysis framework that normalizes workflows into an LLM-Workflow Property Graph (L-WPG) and combines triggerability analysis, LLM-assisted dataflow summarization, and deterministic propagation to synthesize concrete threat-vector findings. Evaluated on 300 manually annotated unique workflows, Heimdallr achieves high accuracy on LLM-node identification (F1~=~0.994), triggerability classification (99.8%), and threat-vector detection (micro-average F1~=~0.917). As part of an ongoing detection and disclosure effort, we have so far responsibly disclosed 802 vulnerable workflow instances across 759 repositories and received 71 acknowledgments.

We develop a martingale approximation framework yielding quantitative maximal large deviations estimates for invertible dynamical systems. From suitable decay of correlations, we deduce these estimates and, as an application, we obtain Young structures with matching recurrence tails for partially hyperbolic diffeomorphisms with mostly expanding central direction. In a second application, we prove maximal large deviation estimates for systems modelled by Young towers with subexponential contraction and expansion. Many examples of slowly mixing billiards are covered by this result.

Multiple stable states - the coexistence of two or more distinct ecological configurations under identical environmental conditions - have attracted sustained interest in ecology, yet the field still lacks a unified framework connecting ecological mechanisms to dynamical models. Here, we review empirical and theoretical approaches to multiple stable states, synthesising perspectives on stability, tipping, hysteresis, and transient dynamics, and contextualise these within a common mathematical framework. Drawing on examples of well-known ecosystem models, we highlight the central and necessary role of positive feedback loops and identify other common, unifying features of ecological systems that exhibit multiple stable states. We further discuss the relationship between stable and transient dynamics, the roles of spatial and temporal scales in feedback identification, and the implications for ecological restoration and management. We conclude with open questions and challenges for the field, including extending multistability theory to persistent-transient frameworks and harnessing emerging data-collection technologies to sharpen empirical inference.

Far-field optical imaging inevitably involves low-pass spatial filtering, limiting the resolution. Moreover, conventional imaging suppresses high spatial frequency components close to the cutoff, making them invisible under noise, particularly the shot noise arising from discrete and random nature of quantum light. Here we propose and implement a method for reducing the effect of this noise by optically pre-processing the incoming light prior to detection, thereby optimizing the quantum measurement performed on it. Our scheme, termed Fourier Domain Division (FDD), partitions the Fourier plane into multiple regions for independent detection and subsequent post-processing for image reconstruction. By analyzing the quantum and classical Fisher information, we show that our method is advantageous with respect to direct imaging for high spatial-frequency components. As a result, the number of photons required to achieve a certain signal-to-noise-ratio in the Fourier domain is reduced, thus enhancing the overall resolution in the photon-starved regime. We demonstrate our method in microscopy, achieving 5-fold improvement of Fisher information on high spatial-frequency components. Unlike active super-resolution methods, FDD is passive, making it broadly applicable in microscopy and other imaging scenarios where active illumination is impractical, including astronomy and remote sensing. Our work establishes a general strategy for designing quantum optimized superresolution imaging systems, bridging fundamental quantum limits, practical image analysis and computer vision applications.

Significant progress in quantum light sources for quantum communication applications requires reproducible and symmetric quantum emitters acting as single-photon sources capable of generating entangled photons on demand at specific telecom wavelengths. Here, we propose telecom-emitting epitaxial quantum dots (QDs) fabricated using the local droplet etching (LDE) approach. The resulting well-defined, low-density ($10^9$/cm$^2$) QDs based on In$_{x}$Ga$_{1-x}$As are formed in symmetric LDE nanoholes (in-plane aspect ratio of 1.14) in In$_{0.52}$Al$_{0.48}$As. Detailed transmission electron microscopy provides comprehensive insight into the structural integrity, interface quality, and compositional profiles of the QDs, which underpin their promising optical properties. Photoluminescence spectroscopy reveals narrow emission lines (0.2 meV) and high optical quality, while second-order autocorrelation measurements confirm clear single-photon emission, with $g^{(2)}(0)=0.07\pm0.02$ under above-band continuous-wave excitation and $g^{(2)}(0)=0.16 \pm 0.18$ under pulsed excitation. Precise numerical modeling, combining multiband $\boldsymbol{k} \cdot \boldsymbol{p}$ and configuration-interaction methods, supports the optical characterization and identifies thermal excitation pathways that explain the persistence of emission up to liquid-nitrogen temperatures. These results highlight the versatility of the LDE approach for integrating new material systems and pave the way toward scalable fabrication of quantum light sources with tailored emission properties.

Given a directed, discrete-time temporal graph $G=(V,R)$, a start node $s\in V$, and $p\geq1$ objectives, the single-source multiobjective temporal shortest path problem asks, for each $v\in V$, for the set of nondominated images of temporal $s$-$v$-paths together with a corresponding efficient path for each image. A recent general label setting algorithm for this problem relies on two properties of the objectives - monotonicity and isotonicity. Monotonicity generalizes the nonnegativity assumption required by label setting methods for the classical additive single-objective shortest path problem on static graphs, while isotonicity ensures that the order of the objective values of two paths is preserved when both are extended by the same arc. In this paper, we study the problem without assuming monotonicity and/or isotonicity. A key difficulty in this setting is that zero-duration temporal cycles may need to be traversed an arbitrary finite number of times to generate all nondominated images. This motivates the study of a restricted problem variant in which a maximum admissible path length $K$ is imposed, and only paths containing at most $K$ arcs are considered. We develop general label correcting algorithms for this setting and establish several sufficient conditions under which such a bound is not required, implying that the algorithms compute all nondominated images.

Power systems are inherently multi-timescale systems, with different physical phenomena and decision-making processes spanning multiple timescales, time horizons, and geographic scopes. I envision power systems digital twins (DTs) as powerful modeling and simulation tools that can accelerate and improve decision-making across different time scales and geographic scopes. However, until now, research has not delivered such a vision, and power systems DTs remain a concept distant from implementation. This is not a regular research paper. This is a position paper that outlines my vision for developing a new generation of power systems DTs that leverage recent advances in artificial intelligence (AI) and machine learning (ML). I call these Foundation Twins. Foundation Twins combines the generalization features of foundation models with the decision-making capabilities of reinforcement learning (RL) architectures to deliver the envisioned power systems DTs.

The New Space era has led to a rapid increase in satellites operated by independent entities in near-Earth orbit. This shift enables richer space services but also requires secure, near-real-time coordination, making efficient authentication of space assets critical for next-generation missions. Traditional ground-dependent Public Key Infrastructure (PKI) suffers from latency and operational bottlenecks that limit scalability and availability in dynamic space environments. This paper proposes architectural designs for space-based PKI that shift certificate management and validation from ground infrastructure into space, reducing reliance on ground stations while enabling interoperability and cross-entity collaboration. Two deployment schemes are introduced: a space-ground integrated PKI with in-orbit validation authorities, and a fully autonomous space-based PKI with in-space issuance and validation. We analyze deployment trade-offs in scalability, availability, security, cost, and operational complexity in multi-operator environments. A baseline latency analysis is provided to illustrate performance implications of in-orbit trust management.

We investigate freeze-in production of light dark matter through the quark flavor-changing operator $(\bar{s}γ_μd)(\barχγ^μχ)$ in a low-reheating cosmology. For reheating temperatures below the QCD crossover, kaon decays and scatterings generate the dark matter abundance through $K\toπχ\barχ$ and $Kπ\toχ\barχ$. The same interaction induces the rare kaon decays $K^+\toπ^+χ\barχ$ and $K_L\toπ^0χ\barχ$. This links the freeze-in relic abundance to searches at NA62, KOTO, and KOTO II. We find that lower reheating temperatures require larger couplings to compensate for the Boltzmann-suppressed kaon abundance, making kaon-driven freeze-in dark matter testable at rare kaon decay experiments.

We have studied the interaction of $π^0 (η) f_1(1285)$ assuming the $f_1(1285)$ to be a molecular state of $K^* \bar K - \bar K^* K$. We use a framework in which a $π^0 (η) f_1(1285)$ optical potential is obtained, which is later used as the kernel of the Lippmann-Schwinger equation, following the standard method for the interaction of particles with nuclei. The optical potential is obtained using the fixed center approximation to the Faddeev equations, where a cluster, here the $f_1(1285)$, remains unchanged during the interaction, appropriate to the situation that one has here. We have obtained the scattering matrix for this system, the scattering length and effective range, plus the correlation functions. The framework used has been previously tested in the study of the $p f_1(1285)$ interaction and has been shown to give results in agreement with the recent experimental measurement of the $p f_1(1285)$ correlation function. On the other hand, from this interaction we do not obtain clear signals for the $π_1(1400)$ or $π_1(1600)$, nor for the $η_1(1855)$ resonances, which in other approaches have been claimed to arise from the same dynamics. We, however, obtain a structure in the $π^0 f_1(1285)$ amplitude around $1500-1600$ MeV and a strong cusp at the $ηf_1(1285)$ threshold of $1833$ MeV.

Motivated by the indications of time-varying dark energy equation of state reported from DESI, we investigate a quintessence model with an exponential potential $V_0 e^{-λϕ/m_{\mathrm{pl}}}$. We derive an analytical relationship between the current equation of state parameter for the quintessence field and the potential parameter $λ$ required to realize sufficient duration of radiation and matter domination. Our results provide a useful analytical relation for inferring the potential parameter $λ$ from the observed current equation of state parameter. Furthermore, based on this framework, we provide a new analytical upper bound on the potential parameter $λ$ for current accelerated expansion. Concretely, we obtain $λ<1.94$ by adopting $Ω_{\phi0}=0.685$.

We propose a local transformation on bicolored graphs, which we call local homophily, inspired by adaptive networks and based on majority dynamics and homophily. In this transformation, a vertex updates its color to match the majority of its neighbors, while neighbors of the same color become connected and neighbors of the opposite color become disconnected. We show how to simulate Boolean circuits using local homophily and establish that determining whether a given pair of vertices becomes connected under iterative applications of local homophily is $\mathbf{P}$-complete under logspace reductions.

This paper establishes the existence and uniqueness of mild solutions to stationary Hamilton-Jacobi-Bellman (HJB) equations associated with infinite-horizon stochastic optimal control problems in separable Hilbert spaces. Our framework includes settings with a lack of global smoothing properties of the transition semigroup, singular dynamics involving unbounded control operators, and state-dependent running costs. We overcome these challenges by lifting the state space using the Projected Evolutionary Lifting technique. This work is an extension of G. Bolli and F. Gozzi, Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions, 2025, in which existence and uniqueness is proved via a contraction mapping argument and is consequently restricted to sufficiently large discount factors. We remove this restriction, proving existence and uniqueness for any discount rate $λ> 0$ using tools from the theory of maximally monotone operators.

We show that for $k>1$ the number of genus $k$ minimal Lagrangians with area at most $A$ in a product of hyperbolic surfaces grows on the order of $A^{6(k-1)}$, with an explicit leading constant given in terms of the Mirzakhani function. We also prove rigidity of the Lagrangian area spectrum, and obtain analogous counting results for products of a higher genus surface with a circle.

Model selection plays an important role in longitudinal data analysis, especially when models are estimated using the generalized method of moments (GMM) in the presence of time-dependent covariates. In this setting, the number of valid moment conditions can grow quickly and may lead to over-parameterized models. The Kullback--Leibler Information Criterion (KLIC) has been proposed as a model-selection tool for this framework; however, the original KLIC criterion may favor overly complex models when the number of parameters or valid moment conditions increases. To address this limitation, this study proposes two penalized versions of KLIC that incorporate penalties based on both the number of model parameters and the number of valid moment conditions. The proposed criteria are referred to as the Moment--Parameter Product Penalty KLIC (MPPP--KLIC) and the Logarithmic Penalty KLIC (LP--KLIC). These criteria provide a theoretically motivated mechanism for balancing model fit and model complexity in GMM-based longitudinal models. Through an extensive simulation study involving both binary and continuous response settings, the proposed criteria are shown to improve the ability of KLIC to distinguish among competing models and to reduce the selection of over-parameterized models. The performance of the proposed methods is further illustrated using the Filipino Child Morbidity dataset, a longitudinal study of child health in the Philippines. The results show that the proposed penalized criteria provide stable and interpretable model rankings and consistently identify age as the most important predictor of child morbidity. Overall, the proposed penalized KLIC criteria offer practical and theoretically grounded tools for model selection in GMM-based longitudinal data analysis with time-dependent covariates.

In this paper, we contribute to the broad aim of relating invariants of additive and monoidal categorifications of cluster algebras. Specifically, in the setting of representations of a quantum affine algebra $U_q'(\mathfrak{g})$, Kashiwara-Kim-Oh-Park proved that the Hernandez-Leclerc categories form a monoidal categorification of their Grothendieck rings. Furthermore, these rings are $Λ$-cluster algebras, meaning they are equipped with a compatible Poisson structure, constructed via the $Λ$-invariant. Under certain natural conditions, where $U_q'(\mathfrak{g})$ is of untwisted simply-laced type, we provide an additive interpretation of the $Λ$-invariant within the framework of Higgs categories. More precisely, there is an ice quiver with potential associated with these cluster algebras, and a key ingredient of our work consists in proving that its relative Ginzburg algebra is proper. More generally, if the relative Ginzburg algebra associated with an arbitrary ice quiver with potential is proper, we prove that the corresponding cluster algebra admits the structure of a $Λ$-cluster algebra defined in terms of negative extensions in the Higgs category. Moreover, we provide a homological formula to compute the corresponding tropical and $F$-invariants introduced by Cao.

In this article we present a new centrality measure called ksi-centrality. We show that ksi-centrality distinguishes real networks from random ones, similar to degree centrality: the ksi-centrality distribution is right-skewed for real networks and centered for random Erdos-Renyi networks, and has linear pattern with a heavy tail on a log plot. Furthermore, the ksi-centrality distribution is centered for models simulating real networks: Barabasi-Albert, Watts-Strogatz, and Boccaletti-Hwang-Latora. Thus, this centrality distribution is an additional and independent property with respect to scale-freeness. We also introduce a normalized version of ksi-centrality and show that it is related to algebraic connectivity and the Chegeer's value of a network. Moreover, the average value of this normalized centrality is in bijective correspondence with the relative number of edges that a new node connects to others in the Barabasi-Albert preferential attachment model, thus answering the question of how to choose the parameter $m$ to model a given real-world network.

We establish error estimates of the first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation (NLSE) with a highly singular potential in $\mathbb{R}^d$ with $1\leq d \leq 3$. Our results deal with singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $p>\frac{d}{2}$ and $p\geq 1$, which is (almost) the weakest regularity of the potential required by the well-posedness of the NLSE. First, for $L^p_\text{loc}$-potentials with $p>2$, we establish an optimal first-order $L^2$-norm convergence for the EWI, with the convergence order slightly reduced to $1^-$ when $p=2$. To the best of our knowledge, the optimal first-order convergence for the three-dimensional $L^2$-potential is for the first time in the literature. The optimality of such an error bound is two-fold: (i) the first-order $L^2$-norm convergence is optimal for the EWI (and its higher-order versions) under the given $L^2$-regularity assumption on the potential, and (ii) to achieve the first-order $L^2$-norm convergence for the EWI, such an assumption is optimally weak. For more singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $\frac{d}{2} < p < 2$ and $p\geq 1$, we prove that the $L^2$-norm convergence is (almost) of $(1-α)$-order when $d=1,2$, and of $(1-\frac{3}{2}α)$-order when $d=3$, where $α:=d(1/p - 1/2)$ when $d =1,2,3$, $p>1$ and $α:=\frac{1}{2}^+$ when $d=1$, $p=1$. Notably, this result pushes the error estimate to the threshold regularity of the potential that matches the threshold regularity for the well-posedness of the NLSE, which is also for the first time. Two main ingredients are adopted in the proof: (i) the use of discrete space-time Lebesgue spaces together with discrete Strichartz estimates to establish the stability of the numerical scheme, and (ii) the use of normal form transformation and frequency decompositions to obtain optimal error bounds.

Pairwise Fisher graphs capture local covariance information, but they cannot distinguish an irreducible multi-observable radiation pattern from a collection of ordinary pairwise correlations. We show that this missing structure is naturally supplied by higher-order Fisher tensors. In a finite basis of binned EECs, ECFs, or EFPs, and in the natural exponential-family coordinates generated by that basis, the same local tensor has three equivalent interpretations: a coefficient in the local Kullback-Leibler expansion, a connected cumulant of the chosen correlator observables, and a signed weight on a hyperedge linking those observables. This gives an exact Fisher-correlator-hypergraph triality in the local exponential-family embedding. The triality provides a direct construction of physics-informed hypergraphs from correlator data. Extending the quadratic Fisher matrix to the first non-trivial higher tensor identifies genuinely connected multi-observable radiation patterns, supplies hyperedge weights for higher-order Laplacians and message passing, and gives a principled criterion for compressing observable bases beyond pairwise information. We develop these constructions and spell out why the exact cumulant interpretation is special to natural exponential-family coordinates. We illustrate the framework in four applications. In a minimal local-KL study, the cubic Fisher tensor reduces the KL truncation error and isolates the dominant triplet structure. In a two-versus-three prong jet substructure benchmark, the hypergraph selector improves compressed-basis classification. In a 33-observable basis-design problem, the Fisher hypergraph retains more third-order local response at twelve observables. A low-capacity learning benchmark then shows how the same Fisher hyperedges can be used as an interpretable inductive bias for message passing on correlator observables.

We document the rise of negative earnings between 1980 and 2019: a secular increase in the percent of firms reporting losses, both among public firms and in the broader universe of US corporations, and a secular increase in the persistence of losses year-to-year among public firms. This rise has occurred alongside a spreading of the sales and earnings distribution and a recomposition of firm spending away from production costs and traditional investment and towards sales general and administrative expenses. We rationalize these phenomena with a model of heterogenous firms engaging in supply and demand shifting investment. Our model includes a scale elasticity of demand determining the relationship between the intensive margin of demand (demand per customer) and the extensive margin of demand (number of customers). We are able to quantitatively match the rise in reported losses and qualitatively match (1) the increased persistence of losses, (2) the spreading of the sales and earning distribution and (3) the recomposition of firm spending with this parameter as the single driver of changes across steady state equilibria. The rise in the scale elasticity associated with the increase in reported losses has non-trivial aggregate implications: in our model it lowers GDP by -9.1% by reallocating labor away from goods and capital production and reallocating demand away from productive firms.

We study the observability of the Schrödinger equation on the $d$-dimensional torus $\mathbb T^d$, $d \geq 1$, from an open subset $ω\subset \mathbb T^d$. Our first main result establishes a quantitative observability estimate for the free Schrödinger equation in the regime of small times $T$ and for small observation sets of the form $ω= \prod_{j=1}^{d}(a_j,b_j)$. Our second main result shows that observability holds for the Schrödinger equation with a merely bounded potential $V \in L^{\infty}(\mathbb T^d)$, in any dimension $d \geq 1$, for every time $T>0$ and every nonempty open subset $ω$. This resolves a well-known conjecture in the field. A central ingredient in the proof is a cluster decomposition method combined with an induction scheme introduced by Bourgain and further developed by Burq and Zhu.