Papers

Current performance criteria used in Applied Behavior Analysis (ABA) are often based on whether a learner meets a predetermined benchmark e.g., 8 out of 10 correct responses across several sessions. Although these criteria help practitioners make instructional decisions, they provide limited information about how likely it is that the learner has truly acquired the skill. In this paper, we introduce a Bayesian approach that combines practitioners' prior expectations with the learner's ongoing performance data to estimate the learner's future performance level. Rather than providing only a pass or fail decision based on the criteria, we add an element of precision around that criteria to make decisions more nuanced to the testing conditions used. We also developed an R Shiny application for Measurement of Individualized, Evidence-Based Learning (MIEBL) that allows practitioners to apply this method in practice, supporting more informed instructional decisions and providing insight into the strengths and limitations of commonly used mastery criteria.

How spatial structure is transferred through a forward-scattering medium depends jointly on pattern scale and propagation distance. Starting from photon exit displacements, we derive modulation transfer while retaining the number, depths, and projected angles of individual deflections, with event occurrences described by a homogeneous Poisson process along the propagation path. The second angular moment yields a leading low-frequency loss proportional to $k^2z^3$, where $k$ is angular spatial frequency and $z$ is thickness. At finite frequency, the model predicts two coupled departures: loss bends below its low-frequency quadratic extrapolation at fixed thickness, and transfer curves separate by thickness at fixed $k^2z^3$. The leading corrections to both effects arise from the same fourth angular moment. For symmetric, nondegenerate projected-angle distributions, the full response further establishes increasing loss with thickness in the same medium at fixed joint variable. Binary-pattern measurements through TiO$_2$ suspensions span five nominal loadings, seven thicknesses, and 20 spatial frequencies, supporting the approximate low-frequency scaling and showing both departures. A Gaussian effective-angle response calibrated only on lower-frequency data predicts the held-out higher-frequency band without refitting, with a logarithmic root-mean-square error of 0.0181 compared with 0.1021 for quadratic extrapolation. The results show that finite-event statistics provide a common physical origin for spectral bending and thickness-dependent departure from low-frequency scaling, linking coarse-pattern transport to finer spatial structure.

We prove the existence of minimal models for smooth co-rank one foliations of general type.

Multilevel regression with post-stratification (MrP) is the dominant method for estimating public opinion in subnational units. The method requires the full joint distribution of demographic variables in a subnational unit, which is often unavailable from public census tabulations. Existing approaches fill this gap by imposing assumptions to combine marginal distributions of census variables into a synthetic joint distribution. We propose sharp MrP bounds instead: the smallest interval containing all estimates of subnational public opinion consistent with the marginal distributions alone. The bounds solve a pair of linear programs and are computationally feasible for typical applications in political science. Applied to two prominent examples in the literature, the bounds are tight enough to be informative and often more accurate than MrP restricted to a coarser, observed joint distribution of census variables. The bounds serve as margin-consistent interval estimates and a diagnostic when researchers use a synthetic joint distribution to construct MrP estimates.

Improving an LLM agent requires knowing what to change when it fails. Similar failures can require different fixes, and resolving one problem can expose another. We introduce Trace-to-Architecture (T2A), a framework that uses execution trajectories to determine what to change in an agent's harness and when those changes apply. A development agent diagnoses failures using execution evidence from successful and failed tasks, revises the corresponding harness components, and then checks whether target tasks improve, selected successful tasks remain successful, and the edits are actually triggered. New execution evidence guides subsequent diagnosis and revision. Across ALFWorld, ScienceWorld, and AppWorld, T2A achieves the highest aggregate score for all three evaluated task agents, exceeding their respective strongest baselines by 6.2–30.3 points. We also integrate T2A's diagnosis and validation procedures into Meta-Harness, improving the resulting harnesses' mean ScienceWorld score by 21.6 points from the same starting harness.

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

We consider newly designed matrix algebras associated with Cauchy orthogonal transforms. In this work, we are interested in checking the approximation of Toeplitz matrices using algebras of matrices, where each matrix is diagonalized by a fixed Cauchy orthogonal transform, in the spirit of fast algorithm based on preconditioning. We use the Korovkin theory for checking the Toeplitz approximability. Unfortunately and somehow surprisingly, the Korovkin test is negative, in contrast with the successful test for all the Fourier-like and trigonometric algebras, widely used in the relevant literature. More in detail, for the Cauchy algebras, the Frobenius optimal approximation fails to lead to the desired clustering, as also confirmed by ad hoc numerical experiments. It remains open to understand which kind of structured matrices can be approximated by these new algebras.

Visual contrastive decoding corrects answers using the prediction difference between the original image and a reference condition, yet the same contrast also turns supported abstentions into incorrect answers. We find that abstentions are lost at two stages: contrastive decoding with candidate filtering can exclude the abstention token before any comparison takes place, and the reference branch's response to the abstention instruction is removed when the reference prediction is subtracted. We therefore propose instruction-preserving decoding, which keeps the effect of the abstention instruction on the original-image prediction intact while leaving the contrast strength freely adjustable. Using a perturbed image and an image-free prediction as references, we obtain IP-VCD and IP-M3ID. On decision utility (DU), which credits both correct answers and supported abstentions, both methods outperform all compared non-IP baselines on every one of nine models on VizWiz, and IP-VCD also leads on eight of nine models on Food-101. Candidate measurements and paired per-question comparisons show that computing the instruction effect and the visual contrast term separately lets contrastive decoding support both correct answers and supported abstentions.

In this article, we present an investigation of observational signatures of a black hole (BH) in effective quantum gravity (EQG), particularly gravitational weak lensing and shadow analyses with the assumption that the BH is surrounded by uniform and non-uniform plasma. First, we review the spacetime by comparing it with well-known BH spacetimes such as Schwarzschild and Reissner–Nordström BHs. Subsequently, we explore the effect of the uniform and non-uniform plasma on the BH shadow. The results indicate that shadow radii decrease under the influence of the BH charge and both uniform and non-uniform plasma parameters. In addition, we obtain theoretical constraints on the spacetime parameters using observational data from the Event Horizon Telescope (EHT) collaboration. Finally, we study gravitational weak lensing around the BH in EQG surrounded by uniform and non-uniform plasma. It was found that increasing the uniform plasma frequency leads to a rise in the deflection angle and vice versa for the non-uniform plasma.

Existing benchmarks for GUI agents primarily focus on increasing task difficulty by extending task horizons. Besides, enabling agents to explore unfamiliar environments and successfully complete tasks within them is also a critical yet challenging problem. We introduce LatentGUIWorld, a benchmark of six tasks in three matched pairs organized by the evidence needed for control: $I_0$ uses prior knowledge and available observations, $I_1$ requires interaction that directly reveals hidden semantics, and $I_2$ identifies dynamics from action-response relationships. Frontier models struggle especially on $I_2$ tasks. We introduce MOVE, which separates movement from button events and exposes intermediate visual feedback, together with demonstrations of iterative exploration and control. Ablations show that both action granularity and multi-step demonstrations improve performance. After SFT, LatentLearner-9B achieves 50.0% overall success on LatentGUIWorld, comparable to several frontier models. Joint RL further raises egocentric Drag and Ten-Choice success to 92.0% and 75.3%. Behavioral analysis reveals multi-step exploration. In egocentric Drag, RL increases Pearson correlation between error-normalized commands and ideal gain compensation from 0.610 to 0.803. Together, these behaviors show that models learn to explore through interaction and adapt their control to environment dynamics. The same format improves GUI grounding after RL, reaching 67.37% mean accuracy across five benchmarks versus 66.72% for conventional RL.

Photonic technologies are among the leading platform contenders for demonstrating quantum advantage. Nevertheless, photon losses represent the major hurdle for the realization of a fault-tolerant optical quantum computer, and the need for an accurate characterization of this phenomenon is crucial for classical simulations of quantum algorithms on noisy backends. The noisy regime would require the simulation of the algorithm on a Fock backend or rather assuming the a priori hypothesis on the uniformity of the optical losses, which may lead to wrong predictions about the outcome from the algorithm on a cerain hardware. We propose a Gaussian model that can reproduce a lossy dynamics through the experimental characterization of imperfect interferometers, yielding the corresponding mean vector and the covariance matrix. Such model can be adapted to any continuous-variable architecture (spatial or time-bin encoded) and for both the Gaussian boson sampling or general-dyne based algorithms. Eventually, we quantify the drift of optical losses from being uniform throughout the multiport interferometer by a numerical simulation. This work proposes a blueprint for future characterizations, tailor-suited for specific devices and architectures, as well as for potential error correction and mitigation techniques, and it paves the way for realistic simulations of quantum algorithms on NISQ optical platforms.

Intelligent power dispatch is enabling fast and self-adaptive decision-making in microgrids, yet its deployment in real-world industry remains limited due to unresolved safety concerns. Existing safety-preserving approaches typically rely on an accurate characterization of the feasible operating region, which is impossible to obtain under uncertain and heterogeneous system dynamics. Moreover, even a dispatch command verified as feasible at a discrete setpoint may become unsafe during its continuous physical execution. To address these challenges, this paper proposes a safety-aware dispatch adjustment framework that uses performance-equivalent modeling of closed-loop systems (PEM-CLS) and embeds high-order control barrier functions and control Lyapunov functions within a quadratic programming formulation (HOCBF-CLF-QP). PEM-CLS captures EMS-required power-regulation dynamics solely from port-level responses, without requiring internal subsystem models, parameters, or controller information. Its modeling error is analytically characterized and explicitly bounded, enabling requirement-oriented dynamic abstraction for safety-critical control. Based on these PEMs, the HOCBF-CLF-QP safety controller coordinates heterogeneous transient power responses and corrects unsafe EMS commands to maintain dynamic safety during command execution. Four deliberately constructed extreme stress-test scenarios are used to expose conflicts between nominal EMS commands and physical safety. Without intervention, the DC-bus voltage violates its safety limit in all four cases, whereas the proposed framework maintains QP feasibility and limits the voltage deviation. The results demonstrate that the proposed framework provides an algorithm-agnostic safety execution layer for intelligent power dispatch without requiring an exact feasible-region model or detailed internal information of lower-level devices.

We separate two functional levels of self familiar from biology: a self-index, which separates one's own from external material and shapes processing, as corollary discharge does in nearly all animals, and self-reference, the higher ability to recognize what is one's own, as in mirror self-recognition. We probe both by placing the same text in an Assistant, User, or System message rendered by the model's own chat template. In Qwen3.8-27B we report three levels of evidence. Persistence: as predicted in advance, the representation of the text separates the model's own message from both external sources at all 64 layers and for every text. Coupling: asked who wrote the preceding message, the model claims its own messages (0.77) and assigns User and System messages alike to the user, and how strongly it claims each message is predicted by the size of its self-index (Spearman up to 0.68). Intervention: exchanging the internal states that follow a message reverses the self-attribution between layers 15 and 31; removing the self-index direction from them at layer 39 lowers the claim more than a random direction; and in arithmetic the model adopts a wrong step more often when it wrote the step itself (0.92 versus 0.75). Across eight open-weight models from four families, the levels are nested and do not follow model size: three models show neither, three a self-index without self-reference, Qwen3.8-Flash-Next recognizes its own messages only after reasoning (0.66, none of the external ones), and only Qwen3.8-27B shows both without thinking; no model shows self-reference without a self-index, and reasoning brings it out only where one exists. We propose that self-reference is a higher-level ability built on a self-index, a content-independent displacement superimposed on the whole representation of the model's own messages, and that neither level is a by-product of problem-solving ability.

Three-dimensional conjugate heat-transfer simulations provide detailed predictions for microchannel heat sinks but can be computationally demanding when many geometries and operating conditions are evaluated. This study develops a versatile two-layer model (VTLM) that reduces the three-dimensional problem to two coupled two-dimensional fields while retaining the in-plane geometry of the flow passages and fins. Effects associated with the unresolved channel-height direction are incorporated through hydraulic and thermal closures that account for wall friction and the fluid- and fin-side contributions to interlayer heat transfer. The closures are evaluated from the representative channel geometry, flow-development condition, and fin configuration. The VTLM was assessed against three-dimensional reference models across broad ranges of channel aspect ratios and operating conditions, including uniform and nonuniform fin arrangements. Compared with the existing two-layer models, the VTLM showed improved overall agreement with the three-dimensional reference results. The results showed that prediction accuracy was governed mainly by the flow-development condition and channel aspect ratio, from which a conservative screening guideline was proposed for the examined straight-channel configurations. The assessment of finned expansion–contraction channels further supported the applicability of the VTLM to the investigated passages with abrupt width changes. Overall, the VTLM provides a reduced-dimensional framework for thermohydraulic analysis across the range of microchannel heat-sink configurations examined in this study without explicitly resolving the channel-height direction.

Inspired by Steinitz's celebrated result (1913) and recent online vector-balancing results of Kulkarni et al. (2024) and Aden-Ali (2026), we introduce the online Steinitz problem: $T$ vectors in $\mathbb R^d$, each of $\ell_2$-norm at most $1$ and with total sum zero, arrive one at a time and must be assigned immediately and irrevocably to unused positions of an array. The goal is to keep every prefix sum of the resulting order small. We first obtain deterministic prefix-discrepancy bounds of $O(\log T)$ for $d=1$ and $O(\sqrt T)$ for $d\ge2$. Our main contribution is a general reduction that converts deterministic and randomized online vector-balancing algorithms, which assign signs to arriving vectors, into online ordering algorithms. Combined with recent vector-balancing results, this yields a randomized algorithm with prefix discrepancy $O(\log T+\sqrt{d\log T})$ and running time $O(dT\log T)$. The guarantee holds with high probability against an oblivious adversary. The reduction also gives bounds for sparse vectors and extends to inputs in arbitrary compact convex bodies. Finally, we prove matching lower bounds in both the deterministic and randomized cases.

A potentially super El Niño is forecasted in 2026, raising the need to anticipate land carbon-cycle risks before its impacts unfold. We develop a prospective early-warning framework that uses dynamic global vegetation models driven by an ensemble of ECMWF seasonal climate forecasts and machine-learning burned area predictions. Forecasts indicate a $0.23 \pm 0.21$ PgC reduction in global net biome productivity in the second half of 2026, with hotspots of sink weakening in eastern South America and elevated fire risk in Southeast Asia.

Teacher rationales can help a smaller model answer health professions questions, but they can also reinforce errors. We examined when they help and how this evidence can guide training using Brazilian Portuguese questions from medical and multiprofessional residency examinations. First, we measured how a teacher rationale changed the student's probability of selecting the correct answer and compared this effect with rationales drawn from other questions. The complete rationale increased this probability by 21~p.p. when the teacher was correct and decreased it by 14~p.p. when the teacher was wrong. By intervening on the reasoning content, we showed that shuffled rationales did not produce the same pattern. Guided by this result, we trained only with rationales attached to correct teacher answers and compared answer-only training, step-by-step distillation, and our proposed Distill-SFT method. Distill-SFT showed a modest advantage over step-by-step distillation on previously missed questions under reasoning, while using fewer training instances by combining the answer and rationale into a single target. We also investigated catastrophic forgetting using the out-of-domain ENEM dataset. Answer-only training reduced reasoning accuracy by 12~p.p. relative to baseline, compared with approximately 4~p.p. for either rationale-based training method. Together, these findings support a workflow that audits rationale content before distillation, verifies teacher answers before supervision, and trains compact models efficiently for health professions question answering.

The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_{G}(X)|/|X|$ taken over all nonempty sets $X\subseteq V(G)$ such that $N_{G}(X)\neq V(G)$. A graph $G$ is 1-binding if $b(G)\geq 1$. Fan and Lin [Electron. J. Combin. 31 (2024) P1.30] characterized the unique spectral extremal graphs among 1-binding graphs with $\delta(G)\geq k$ that contain no $k$-factor for $k=1$, $2$. Subsequently, Tang and Zhang solved the problem for $k\geq 3$. Note that $\delta(G)\geq a$ is a necessary condition for the existence of an $[a,b]$-factor in a graph $G$. For $1\leq a<b$, Fan et al. [Discrete Math. 349 (2026) 115103] determined the unique spectral extremal graph $H_n^{a,b}$ among connected graphs with $\delta(G)\geq a$ that contain no $[a,b]$-factor. Note that $b(H_n^{a,b})<1$. This motivates the corresponding spectral extremal problem for 1-binding graphs. Note that every 1-binding graph with $\delta(G)\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\leq a<b$, we in this paper determine the unique extremal graph $\mathcal{B}_n^{a,b}$ with the maximum spectral radius among 1-binding graphs of order $n$ with $\delta(G)\geq a$ that contain no $[a,b]$-factor. The toughness of a non-complete graph $G$ is $t(G)=\min\{\frac{|X|}{c(G-X)}\mid X\subseteq V(G)\, \text{and}\, c(G-X)\geq 2\}$. A graph $G$ is 1-tough if $c(G-X)\leq |X|$ for every $X\subseteq V(G)$ with $c(G-X)\geq 2$. Every 1-tough graph of order $n\geq2$ is 1-binding. Note that $t(\mathcal{B}_n^{a,b})<1$. This leads us to investigate the corresponding spectral extremal problem for 1-tough graphs. Note that every 1-tough graph with $\delta(G)\geq 1$ contains a $[1,b]$-factor for $b>1$. For $2\leq a<b$, we in this paper characterize the unique extremal graph with the maximum spectral radius among 1-tough graphs of order $n$ with $\delta(G)\geq a$ containing no $[a,b]$-factor.

The frequency noise of a resonant nanomechanical sensor is well understood down to its thermomechanical limit, but the longest useful averaging time is set by a flicker, or $1/f$, frequency-noise floor whose origin has remained unknown. We show that this floor is the thermal noise of the tension. By the fluctuation-dissipation theorem, the tension of a resonator with fixed ends fluctuates by an amount set by the loss angle of its material; for structural damping, these fluctuations produce $1/f$ frequency noise with a flicker coefficient $h_{-1}=E k_{\mathrm{B}} T/(2\pi V\sigma_0^2 Q_{\mathrm{el}})$, with $Q_{\mathrm{el}}$ the intrinsic loss under elongation, that contains no free parameter and is diluted by the tensile stress $\sigma_0$ in the same way as damping is in dissipation dilution. Silicon nitride strings and membranes from 30 to 340~nm thickness follow the predicted volume scaling, the predicted $\sigma_0^{-2}$ dilution across resonators and within stress-engineered strings, and the predicted thickness dependence, which places noise-active two-level systems in the volume and at the surfaces of the resonators, all at the absolute level set by the independently measured intrinsic loss. The same relation accounts for published data between 78 and 296~K; below 78~K, the loss measured at the resonance no longer represents the loss probed by the noise, marking the limit of the structural-damping assumption. Dissipation dilution therefore lowers the $1/f$ floor, and the longest useful integration time of a resonant sensor is set by its prestress, volume, temperature, and intrinsic loss angle.

We examine whether the unusual spatial flattening and orbital-plane coherence of the Milky Way's classical satellites are connected to the Galaxy's location in the cosmic web. We compare the 11 classical satellites with the 11 most massive satellites of 11,922 Milky Way-mass hosts in the TNG300-1 hydrodynamical simulation. We measure flattening with both the simple inertia tensor (SIT), which retains radial information, and the reduced inertia tensor (RIT), which isolates angular anisotropy. Orbital-plane coherence is characterized by the minimum opening angle containing the axes of the eight most coplanar satellite orbits. For the Milky Way, we find $(c/a,b/a)=(0.183,0.558)$ for SIT and $(0.364,0.736)$ for RIT, together with an eight-orbit opening angle of $21.9^\circ$. The corresponding fractions of simulated systems at least as extreme are $4.94\%$, $12.28\%$, and $5.04\%$, respectively. We describe the environment using the velocity-shear tensor smoothed on $2\,h^{-1}\mathrm{Mpc}$ and compare the full sample with Local Group-like pairs, systems whose shear eigenvalues match those reconstructed for the Local Group, and hosts in knots, filaments, sheets, and voids. Among the well-sampled environmental subsets, the shape and opening-angle distributions differ only weakly. Although the cosmic web imprints preferred orientations on host spins and satellite orbital-plane axes, the Milky Way combines orientations that are uncommon in the simulation. We conclude that the present-day cosmic-web environment alone does not substantially increase the occurrence of Milky Way-like satellite configurations. This result does not rule out an origin in anisotropic accretion, since neither the present-day web class nor its shear eigenvalues fully encodes the assembly history of an individual satellite system.

Language models have recently solved open Erdős problems and achieved gold-medal level performance on International Mathematical Olympiad problems. But their solutions, written in natural language, cannot yet be verified by proof assistants. An erroneous reasoning step can invalidate every step that comes after, so it is advantageous to detect errors early. Several methods have been proposed to detect erroneous reasoning quickly based on the features of the forward pass of the language model that performs reasoning. In this work, we study whether the directed Forman curvature of that model's attention graph, a local quantity computed at an edge from the attention weights around it, carries information about a step's correctness. To do so, we evaluate four per-step verifiers on ProcessBench with and without the curvature appended to their input. Appending curvature values improves the overall mean performance of all four verifiers. ReProbe, the verifier that takes every token of a step as input, shows the largest improvement. Adding the curvature increases the time of the language model's forward pass by 18% on average.

We recall a 1987 paper written by Volovich and our subsequent work with Okada which established the Feyman rules for the p-adic string and the related non-local field theory which found diverse physical applications. We add some memories of visits to Moscow during 1989-91.

This paper considers a distributed estimation system where edge nodes transmit averaged features to a center node for parameter estimation. A key challenge arises when the center node possesses auxiliary side information: how to design local extractors that are universally optimal for any downstream task, while avoiding redundancy with the side information. To tackle this problem, we orthogonally decompose the joint function space and demonstrate in the weak-dependence regime that the modal decomposition of the canonical dependence kernel's conditional-additive component yields optimal, task-independent features. We construct a distributed nested H-score to learn this decomposition and show that, under a limited feature budget, the mean-squared error (MSE) gap to the best task-specific features is determined solely by the task's alignment with the discarded, least informative modes, and vanishes to leading order for every square-integrable task once the budget is sufficient.

For an algebraic stack of finite type, we construct a finite stratification of an atlas such that every universally closed open substack pulls back to a union of strata. As a result, there are only finitely many open substacks admitting a proper adequate moduli space.

In the tensor principal components analysis (tensor PCA) problem, we are asked to recover an unknown vector in $\{\pm1\}^N$ from a noisy observation of its $k$-th tensor power, for a fixed $k\ge3$. Tensor PCA exhibits a well-studied information-computation gap: global algorithms such as spectral methods and sum-of-squares work above an algorithmic threshold, below which we know matching lower bounds (against these and some other restricted models of computation). In contrast, local algorithms such as Markov chains were only known to succeed from a random initialization at a polynomially larger signal-to-noise ratio, and were believed to be inferior to global algorithms for tensor PCA. We show that Glauber dynamics, when supplemented with annealing and ergodic averaging, achieves polynomial-time strong recovery above the algorithmic threshold from a uniformly random initialization, demonstrating that local Markov chains can succeed all the way down to the algorithmic threshold from an uninformative cold start.

Under the stated regularity assumptions, the full orthogonality functionals of the five finite para-families considered here are classical for the respective affine, quadratic, $q$-quadratic, $q$-linear and alternating calculi. For each of the para-Krawtchouk, para-Racah, $q$-para-Racah and $q$-para-Krawtchouk families, two finite solutions of one global structural equation have $N+1$ nodes altogether and common moments through degree $N-1$. Their relative mass changes only the central recurrence coefficients. We prove regularity for every normalised combination with both component masses non-zero and recover the full published functionals and recurrences. The alternating construction identifies para-Bannai-Ito within the same structural framework. For the four non-alternating families, equal positive combinations give persymmetric Jacobi matrices; the known spectral gap criterion then decides perfect state transfer. Explicit parameter correspondences locate the classical components and resonant limits of the full systems. A positive seven-point example shows how further named families can be manufactured without a new governing principle. Counterexamples to mathematical assertions in three of four recent preprints expose the consequences of insufficient hypotheses for finite representations and physical models. In an era of AI-assisted symbolic production, we call for a standard that rewards structural explanation and verification rather than the proliferation of examples.

We give a new upper bound on the piercing number $\tau(\mathcal{F})$ of a finite family $\mathcal{F}$ of axis-parallel rectangles in the plane when every $p$ members contain an intersecting pair. Namely, we prove $\tau(\mathcal{F}) = O(p(\log\log p)(\log\log\log p))$, improving the factor $(\log\log p)^2$ in the best previously known bound to $(\log\log p)(\log\log\log p)$. For each $d>2$, when $\mathcal{F}$ is a finite family of axis-parallel boxes in $\mathbb{R}^d$, this yields $\tau(\mathcal{F}) = O(p(\log p)^{d-2}(\log\log p)(\log\log\log p))$. We also study an analogous problem replacing $\mathbb{R}^d=\mathbb{R}\times\cdots\times\mathbb{R}$ with the Cartesian product $T_1\square\cdots\square T_d$ of trees $T_1,\ldots,T_d$, and boxes with objects of the form $S_1\square\cdots\square S_d$ where $S_i$ is a subtree of $T_i$. For a finite family of such objects under the same intersection assumption as above, we establish an upper bound $\tau(\mathcal{F})=O(p(\log p)^{d-1})$ for each $d$. We also construct examples showing that this upper bound cannot be improved in order of magnitude.

Bell-test data are typically used to certify nonlocal correlations, while quantum metrology is concerned with the sensitivity of measurement statistics to unknown parameters. Here we connect these tasks at the level of experimentally accessible Bell statistics. Using standard moment-based Fisher-information inequalities, we derive lower bounds on the classical Fisher information directly from Bell scores and their parameter dependence, without requiring full state tomography. For CHSH measurements we establish a hierarchy between the information contained in the full outcome record, the set of individual correlators, and the compressed Bell parameter, thereby quantifying the metrological cost of data compression. In polarization-phase sensing, this hierarchy reveals a Bell-metrology tradeoff: the point of maximal CHSH violation need not maximize, and can even suppress, the Fisher information carried by the Bell score itself. We further derive finite-sample lower-confidence bounds from coincidence counts and analyze the effects of biased marginals, accidental coincidences, visibility loss, and multipair emission. The framework provides a practical route for extracting metrological guarantees from Bell-type experiments using only standard correlation measurements.

This paper develops likelihood-based estimation for parametric rational inattention–additive random utility models. We establish consistency and asymptotic normality of the exact maximum likelihood estimator and provide conditions under which the simulated estimator shares its asymptotic distribution. To implement estimation, we develop Newton–BA, a KKT-guided hybrid of Newton–Raphson and Blahut–Arimoto (BA). The algorithm accommodates both interior and boundary solutions and substantially improves computational efficiency relative to direct BA while preserving numerical accuracy. Monte Carlo simulations show that parameter recovery improves with sample size and that confidence-interval coverage is generally close to the nominal $95$ percent level. Finally, we compare our model with no- and perfect-information logit benchmarks in an empirically calibrated ModeCanada experiment. Our model achieves similar in-sample likelihood fit to perfect-information logit but more accurately recovers key preference parameters, choice probabilities, and responses to travel-time changes.

This paper is concerned with space inhomogeneous Fermi-Dirac-Fokker-Planck equation posed on a classical kinetic phase space. The nonlinear factor $f(1-f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the exclusion principle of fermions. Solutions are subject to the physical constraint $0\leq f\leq 1$. Previous existing results have mainly been obtained under suitable smallness or close-to-equilibrium assumptions. A main difficulty stems from the nonlinear and degenerate parabolic structure of the equation. We prove the existence and uniqueness of global solutions without any smallness or close-to-equilibrium assumptions. To construct solutions, we use a vanishing viscosity approximation by adding a Laplacian in the spatial variable. We derive uniform estimates for the regularized problem and obtain compactness in the space of functions of bounded variation $(BV),$ allowing us to pass to the limit as the spatial viscosity vanishes. Since the limiting equation is degenerate with a nonlinear transport, we formulate an appropriate notion of entropy solutions incorporating the kinetic structure of the equation. We prove uniqueness within this class and show that every vanishing-viscosity limit satisfies the entropy formulation. Our result provides a global well-posedness theory beyond the perturbative regime.