Papers

Event logs are ubiquitous across real-world domains, including business processes, manufacturing workflows, healthcare, and many other operational systems. Yet event logs can differ substantially in their event semantics, label spaces, interaction patterns, and temporal scales, making it difficult to build a single foundation model that generalizes directly to previously unseen systems. We address this challenge with the Foundation Event Predictor (FEP), a simple and scalable in-context architecture that performs next-event prediction through learned, event-level nonparametric inference. Causal self-attention encodes context and target histories into key and query representations, respectively, while the observed next context events are encoded as values. Cross-attention then performs learned, relevance-weighted aggregation of next-event outcomes over all context transitions, allowing the entire nonparametric prediction procedure to be expressed as standard attention operations and efficiently parallelized with modern kernels such as FlashAttention. To promote cross-system generalization, we introduce permutation-robust embedding learning and random fixed embeddings to decouple event IDs from persistent semantics, together with empirical quantile transforms and robust normalization to handle diverse temporal scales. We pretrain FEP on 696 real-world event datasets spanning heterogeneous event systems and evaluate it on 50 held-out datasets unseen during pretraining. Across these datasets, FEP generalizes effectively to new event systems without parameter adaptation, frequently approaching the performance of strong models trained separately on each target dataset while substantially outperforming existing foundation-model baselines, particularly on challenging prediction tasks.

Robotic rearrangement tasks require planning sequences of actions that fulfill high-level specifications while also being geometrically feasible. Symbolic abstractions make the combinatorial structure of rearrangement tasks tractable, but they usually do not fully capture the geometric constraints that determine which actions a robot can actually realize. These constraints are hard to express in the symbolic model. We propose REAP (Rearrangement with Elicited Action Preconditions), a framework in which a relational graph neural network heuristic is learned over a simple symbolic abstraction. A motion planner serves as a supervisor that elicits the hidden geometric preconditions of the problem during training. Deployed with greedy best-first search, the learned heuristic generalizes out-of-distribution along object count, plan length, and object arrangement, and can be trained jointly on several domains. Compared to classical heuristic search and learning-based baselines, REAP solves more problems, requires roughly an order of magnitude fewer motion-planner queries than comparable methods, and the action it ranks first is geometrically feasible in 94% of sampled states, against 22% for a random choice.

Learned satellite super-resolution is commonly assessed by pixel fidelity and potential downlink savings, but these criteria do not establish whether the reconstructed field preserves spatial structure or supports reliable downstream analysis. We treat single-image super-resolution as a spatial downscaling problem and evaluate four lightweight networks, including our compact 0.36 million-parameter model, against interpolation using pixel metrics, directional variograms, spectral band correlation, codispersion and observation coherence. On 735 satellite tiles from 190 scenes, the networks differ by at most 0.03dB and outperform bicubic interpolation by $1.39-1.42$dB at $\times 2$. They recover about 30% of the spectral power above the observation's Nyquist frequency and, at $\times 4$, produce more isotropic fields than the reference (range anisotropy 0.59 against 0.49). Energy-based metrics can favour invented detail. Enforcing coherence reaches the quantisation limit, yet at least 99.7% of the remaining error lies in the observation operator's null space, which coherence cannot constrain. Under spatially blocked, paired evaluation with two backbones and a sharpening control, learned reconstruction changes land-use accuracy by $-1.0$ to $+3.1$ percentage points (significant in 7 of 24 comparisons) and building-change F1 by $-0.015$ to $+0.024$ (none significant) relative to bicubic interpolation, and amplifies spurious change between two observations of an unchanged scene by $1.3-1.5$ times where interpolation attenuates it. Full-resolution JPEG\,2000 matches or exceeds reduced-resolution transmission with reconstruction at every matched bitrate. Learned reconstruction is most useful when only low-resolution imagery is available, and its evaluation should combine pixel metrics with spatial structure, observation coherence and task-level tests under spatial validation.

AI dashboards increasingly shape economic debate, yet their headline movements often conceal what changed. For a ratio, the same rise can reflect numerator growth, denominator decline, or both. This ambiguity creates structural non-identification: adding observations or richer fixed effects cannot recover the missing information. We propose three propositions showing that a relative-index regression blends the causal effect of interest with co-movement between focal and aggregate activity and shocks shared with the outcome. These components can give the coefficient either sign under a zero effect, or erase it under a nonzero effect. Publishing denominator levels changes what can be learned by removing the co-movement channel exactly, although shared-shock confounding remains. The problem extends to power: if most identifying variation comes from a handful of units, those units determine what the design can detect, however large the nominal sample looks. At a fixed standard error, the minimum detectable effect diverges as the effective cluster count approaches one, leaving a null result uninformative. We apply this framework to OECD.AI's hiring index and Eurostat employment across 24-27 European countries, where the detection floor exceeds the reported association. In a refreshed panel, that association disappears once the 2021 survey break is excluded. Ratio indices can still reveal concentration, timing, and co-movement, but establishing displacement requires more; our framework turns that boundary into a practical standard by making denominator levels a minimum disclosure requirement.

The global transition toward carbon-neutral energy systems is heavily dependent on the rapid deployment of solar photovoltaic (PV) infrastructure, yet the speed of this expansion has outpaced the ability of administrative registries to maintain accurate records. While high-resolution commercial imagery is often used for monitoring, its high cost and infrequent updates can create an information gap for grid operators, energy infrastructure developers and policymakers. This study addresses the lack of scalable monitoring for industrial-scale rooftop PV systems by developing a machine learning framework that utilizes open-access, multispectral Sentinel-2 satellite imagery. By integrating building geometries from OpenStreetMap with Sentinel-2 reflectance data, a Random Forest model was trained to identify the specific spectral signatures of PV modules. Moving beyond theoretical modeling, this work emphasizes practical implementation through the development of an interactive web-based prototype for on-demand site monitoring. The application enables users to retrieve up-to-date satellite observations and assess the presence of PV systems on industrial building rooftops. The tool incorporates quality metrics that inform users whether a prediction is supported by sufficient training data density or requires manual verification. This end-to-end framework demonstrates that open-source medium-resolution data can effectively bridge this monitoring gap.

Sparse-frame video-language models can change their answers when only the sampling timestamps shift. Raw agreement exposes this problem, but it need not rank errors better than token confidence or identify which additional observation will repair an answer. We argue that answer risk and action value must be estimated separately. PhaseGuard+ probes a frozen VLM at controlled temporal offsets, predicts the reliability of the phase-vote answer, and predicts the example-wise correctness gain of a specific rescue action. The second score routes only examples expected to benefit from denser frames or a stronger VLM. Both predictors use lightweight models trained on public data; the VLMs remain frozen. We use 256 Perception Test videos for method development and evaluate on a disjoint 256-video test set comprising 554 Physics questions. For Qwen3-VL-8B, reliability AUROC rises from 68.0% with token softmax to 74.8% (+6.8 pp, 95% CI [2.6, 10.7] pp). Routing 30% of questions to Qwen3-VL-32B reaches 67.87% accuracy, compared with 63.66% for budget-matched random routing (+4.21 percentage points, CI [1.89, 6.40]). All four tested rescue pairs significantly outperform random routing. Temporal sensitivity becomes actionable when the system learns both whether an answer is unreliable and whether an available action is likely to fix it.

Latent world models such as V-JEPA 2 forecast in the feature space of a video encoder, but one look at an object does not reveal how hard it is to rotate, how much it bounces or how much it slides. People find out by trying: a few pokes reveal how an object responds. IMPULSE, named after the impulse response it infers, gives a predictor in frozen V-JEPA 2 feature space the same ability. A Bayesian contact memory summarizes three short calibration pushes, a predicted object pose carries it into the predictor, and a lightweight random-Fourier pose head lets the predictor use the pose. On 512 fresh objects, in a pre-specified held-out confirmation with models sealed before the data existed, the memory lowers the 1-second feature-forecast error by 11.6-12.8% relative to the same predictor without experience, for all 8 training seeds, with a 95% seed-and-object interval excluding zero. The gain also holds for 128 objects with out-of-range physics (11.7-14.0%), and the full extension beats a plain predictor by 15.5-16.6%. An information-ladder argument, exact for ideal predictors, shows that physics reaches the features only through motion, so we measure how much the pathway lets pose help. On fresh objects, the Fourier head roughly triples the gain of a perfect memory over plain pose tokens (2.8-3.2x). Raw push measurements fed through the same pathway give a similar gain, so the Bayesian summary is not required; it condenses the trials into eight interpretable numbers. The gain is absent at contact onset and grows afterwards.

The frequency noise of an ordinary diode laser, often a nuisance in spectroscopy, can be used to measure atomic coherence. When atoms are prepared in a coherent superposition of two ground states, each random shift in laser frequency changes how much light they absorb, so the transmitted light carries a record of that coherence. Correlating the intensity fluctuations of two fields, each linking one ground state to a shared excited state, reads this record out, a technique known as noise correlation spectroscopy. The correlation signal is narrower than the transparency window the same fields create and switches sign as their frequency difference is tuned. Its width is set by the rate at which the atoms lose coherence and, over a wide range, does not grow with laser power. Over two decades the method has spread from warm vapors to cold atoms, to multi-field schemes, and to diamond and rare-earth-doped crystals. This review, to the author's knowledge the first, brings together its theory, experiments and applications, and examines its limits, the challenges that remain, and the opportunities that could make it a standard spectroscopic tool.

We formulate a unified thermodynamic framework for chemical, mechanical, and electromagnetic transport by extending the Gibbs–Duhem relation to include kinetic free-energy contributions associated with material and field motion. The resulting kinetic Gibbs–Duhem relation establishes a thermodynamic connection between free-energy variations, generalized chemical potentials, pressure, and transport momentum, while preserving the invariance of the system free energy under changes of reference frame. Within a Lagrangian formulation of free-energy evolution, the relation provides a common structure for diffusion, momentum transport, and electromagnetic dynamics. For nonuniform chemical systems, the framework recovers the coupling between Cahn–Hilliard dynamics and the Korteweg stress arising from composition gradients. Its extension to electromagnetic systems incorporates electric, magnetic, charge, and mechanical contributions into the same thermodynamic formulation, yielding a unified description of electromagnetic and momentum evolution. The framework further separates dissipative from nondissipative transport and establishes a common thermodynamic form for mechanical and electromagnetic propagation. In the isentropic limit, the characteristic propagation velocity is determined by the derivative of the thermodynamic pressure with respect to density, yielding the speed of sound for mechanical systems and the speed of light for electromagnetic systems. These results show that kinetic contributions can be incorporated consistently into the Gibbs–Duhem framework, providing a common thermodynamic basis for coupled chemical, mechanical, and electromagnetic transport without introducing separate thermodynamic principles for each mechanism.

Rarefied nonequilibrium flows are described by the Boltzmann equation, which can be solved using stochastic particle or deterministic methods. With advances in computing, data-assisted physics-informed neural networks have been widely used for flow prediction by combining governing equations with measured or computed data. Data-free PINNs do not require such training data and have been widely studied for continuum flows, but their application to rarefied flows remains limited. This study develops a data-free Fokker-Planck physics-informed neural network (FP-PINN) to describe rarefied nonequilibrium flows. The physics loss is constructed from the ellipsoidal statistical Fokker-Planck (ES-FP) equation using a velocity distribution function (VDF) closure and moment projection. The FP-PINN is evaluated for a steady one-dimensional Mach 10 normal shock. The FP-PINN results agree with the numerical ES-FP results, with errors below $1\%$. Compared with the particle ES-FP results, the FP-PINN predicts a sharper shock structure with errors up to $17\%$. This difference is attributed to the VDF closure and moment projection.

The Direct Simulation Monte Carlo (DSMC) method has long served as a reliable numerical approach for simulating rarefied and non-equilibrium gas flows. Its computational cost, however, increases rapidly as the flow approaches the near-continuum regime. This limitation has motivated the development of kinetic model equations that replace binary collisions with simplified relaxation processes. This paper presents a particle multi-relaxation unified BGK (UBGK) model for monatomic gas mixtures. The model retains pairwise species relaxation and determines the target velocity, temperature, shear stress, and heat flux by matching the corresponding production terms to those of the Boltzmann equation. A stochastic particle algorithm is developed. The model is implemented in SPARTA, and two validation problems are considered: homogeneous relaxation and a Mach 3 flow over a cylinder. In both cases, the UBGK solution reproduces the DSMC reference behavior, including species-dependent relaxation and non-equilibrium shock structure.

For a group $G$, let $a(G)$ denote the minimum number of abelian subgroups covering $G$, and let $\omega(G)$ be the maximum size of a pairwise noncommuting set. We prove the exact formula $\sup_{\omega(G)\leq n}a(G)=2^{\lfloor(n-1)/2\rfloor}+1$ for every $n\geq 63$. Equality holds precisely for groups with derived subgroup of order two and binary symplectic central quotient of dimension $2\lfloor(n-1)/2\rfloor$. The proof combines centralizer descent with a structural separation of nonscalar alternating commutator maps.

In condensed matter physics, scientists explore phase diagrams of matter through various methods (analysis, numerics, experiment) and achieve consensus when enough methods agree. The quality of variational results for quantum many-body ground states is mainly judged by energy values and respective errors. Here, we present QMBL, the Quantum Many-Body Leaderboard (https://www.qmbl.org), serving as the energy layer of the scientific consensus. It is a maintained, interactive record of published ground-state energies for lattice Hamiltonians. The latest release v0.4.0 holds 1434 energies on 395 instances of six model families and collects all of the community's efforts into one view. Of those instances 153 have no exact solution and any new record serves as a direct improvement to an open science problem. They are the targets of a companion benchmark, QMBL: Open Problems.

We study a quantitative form of noise-induced restoration of entropy concavity for one-dimensional nonequilibrium states relaxing toward a Gaussian thermal equilibrium. A partially thermalized state is modeled by $X_q=\sqrt q\,X+\sqrt{1-q}\,G$, where $G$ is Gaussian equilibrium noise and $q$ is the retained signal fraction. For two independently prepared copies we ask when the entropy of the mixed state $h(\sqrt{1-t}\,X_q+\sqrt t\,Y_q)$ is concave in the mixing parameter. The problem is motivated by entropy geometry along Ornstein–Uhlenbeck/Fokker–Planck relaxation, where the relative score is the nonequilibrium thermodynamic force and relative Fisher information is entropy-production rate. We derive an exact symmetric endpoint-curvature formula, separating a nonnegative Ornstein–Uhlenbeck restoring term from a fourth-cumulant correction, and extend it to nonsmooth generalized-Gaussian states $e^{-a|x|^p}$ for $p>3/2$. For half thermalization of a symmetric log-concave input, posterior-variance identities give a quantitative outer-region theorem confining any endpoint obstruction to two explicitly bounded near-equilibrium sectors. We also give spectral and MMSE reformulations of the auxiliary inequality $A\ge2\delta$, prove it in an explicit nonlinear neighborhood of Gaussian equilibrium, and show how it sharpens the endpoint bounds. The main all-mixing result concerns the leading symmetric non-Gaussian Hermite mode: for $p_a=\phiG(1+a\He_4)$, $0<a\le10^{-2}$, we prove strict entropy concavity for every $0<t<1$ and every $0<q\le5/8$, hence throughout the currently unresolved symmetric range $0<q\le4/7$ for this explicit nonequilibrium family. Numerical searches over broader log-concave families are reported only as supporting evidence.

Despite a vigorous effort, a comprehensive and satisfactory understanding of the dynamics underlying the glass transition in finite-dimensional systems has not yet emerged. One key hallmark of supercooled liquids approaching the glass transition is the breakdown of the classical Stokes-Einstein (SE) relation, manifested as the decoupling between the self-diffusion coefficient and structural relaxation time, commonly described by a fractional SE relation. Whether and how the fractional SE exponents vary remains an ongoing debate, even for the simplest supercooled liquids. In this work, we numerically establish a universal relation between the self-diffusion coefficient and structural relaxation time by introducing characteristic scales that are revealed under the condition of constant dynamical heterogeneity. Moreover, this universal relation could also capture literature data for supercooled water and silica. Remarkably, we find that the fractional SE exponent is invariant across various supercooled liquids studied spanning a wide range of fragility, thereby challenging the prevailing view that the fractional SE exponent is correlated with fragility. Our study demonstrates that investigating dynamical heterogeneity yields more valuable insights into understanding glassy dynamics than expected.

We study the sharp constant in the differential inequality \[ I(t)I''(t)\geq c\,[I'(t)]^2 \] for Fisher information along the probability-normalized heat flow on $\mathbb R^2$. The unrestricted log-convexity value $c=1$ is false in two dimensions, whereas the known square-root convexity theorem gives the universal value $c=1/2$. We construct an exact, machine-verifiable sum-of-squares certificate proving the stronger universal bound \[ c\geq \frac{231691}{250000}=0.926764. \] Together with an independent exact Fourier/Gaussian-envelope construction, this gives \[ 0.926764\leq c_*<0.94176665012062, \] where $c_*$ is the sharp constant for positive-time Gaussian convolutions of arbitrary probability measures on $\mathbb R^2$. We formulate the proof in explicit two-dimensional index notation. The lower bound follows from an exact rational polynomial identity involving a strictly positive denominator, a weighted total derivative, and eleven positive rational Gram matrices. All $491$ recorded leading principal minors are strictly positive and the complete polynomial identity has zero residual. No symmetry, log-concavity, finite-variance, or finite-entropy assumption is imposed on the initial measure. The sharp value remains open.

Always-on virtual assistants increasingly personalize on-device to new user-specific commands via few-shot fine-tuning (e.g., LoRA), but this adaptation is evaluated almost exclusively on whether it learns the new commands, not on whether the assistant can still recognize an input that matches none of its intents - the out-of-scope (OOS) rejection problem. We show that naive few-shot LoRA personalization degrades OOS-rejection AUROC (area under the ROC curve) on every one of five public intent-classification benchmarks (mean drop $0.055$–$0.258$, five seeds each), while in-scope accuracy is preserved. A scorer-invariant check shows part of this drop is an artifact of the standard confidence scorer's own miscalibration under adaptation, though a smaller, genuine effect remains under any scorer; we trace it to a measurable shrinkage of the embedding margin around known-intent regions (pooled Spearman rho=0.61$). A deployment-cheap fix is scoring rejection by Mahalanobis distance instead of softmax confidence without extra training/retraining recovers most of the lost quality on 3 of 5 datasets at no accuracy cost, beats two further OOD scoring baselines, and generalizes across shot count, LoRA rank, and a second base encoder. We report honestly where the fix falls short and release all code for reproducibility.

Creating digital garments from existing two-dimensional patterns remains a time-consuming process. Although patterns are widely available in digital form, their assembly in garment CAD software involves manual definition of sewing operations and arrangement of the pattern panels for drape simulation. We present two learning-based approaches that automate both tasks using the 2D vector pattern as their sole input. Our automatic sewing model operates at the segment level, leveraging a Transformer encoder with multiple prediction heads. Our novel architecture supports many-to-many, ordered segment correspondences, as well as hems, overlapping seams needed for multilayer garments, and construction properties such as seam allowances and stitch types, required for the finished virtual garment but neglected by previous methods. A complementary panel-level model predicts body-region assignments, layers, and placement coordinates to initialize the drape simulation. We evaluate both models on a curated dataset of 4,035 professionally assembled garment instances. The sewing model exactly recovers all seam and hem topologies of 63.3% of test garments, while the arrangement model meets the specified placement criteria for all panels in 66.2% of test garments. We also integrate both methods into garment CAD software, demonstrating automatic assembly of successfully predicted garments.

A pressure-swirl nozzle converts the feed pressure into a swirling conical sheet that breaks into ligaments and then into drops. The size of the drops governs the deposition and drift of agricultural sprays, the capture of airborne dust and the combustion efficiency of fuel injectors. The standard tool for measuring the drop size is laser diffraction, which returns a spatial, concentration-weighted distribution at the beam station. However, the dependence of the measured drop size on the nozzle-to-beam distance has not been quantified alongside a Weber-number scaling, so prefactors reported at different distances cannot be compared on an equal footing. To this end, we measure the drop-size distribution of two full-cone pressure-swirl nozzles by laser diffraction at a range of distances, pressures and orientations, together with high-speed imaging of the sheet breakup. We find that the volume-median diameter $D_{v50}$ decreases with the measurement distance $L$ as $D_{v50} \sim L^{-0.18}$ beyond the drop-formation region. A joint fit over both nozzles gives $D_{v50}/d_0 \sim \mathrm{We}^{-0.31}\,(L/d_0)^{-0.21}$, where $d_0$ is the orifice diameter and $\mathrm{We}$ the Weber number, which separates the pressure and distance effects. At a fixed distance the two nozzles collapse onto the $\mathrm{We}^{-1/3}$ scaling of Kooij et al. (2018), and the measured distance dependence may account for part of the offset between our near-field prefactor and the far-field value. Thus, a drop size obtained by laser diffraction is meaningful only when the measurement distance is reported with it, and a spray should be characterized at the actual distance between the nozzle and its target.

Matroid Stanley-Reisner rings admit essential deformation obstructions of arbitrarily high order over every field, already at corank three. For each $m\ge3$, we construct a connected cosimple matroid and $m$ fine-homogeneous first-order classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. A specified joint $(m-1)$-jet nevertheless has no lift to order $m$, even without preserving the grading. The construction rests on formulas for every multigraded component of the second cotangent cohomology $T^2$ and of its primary cup image, expressed through flats and parallel classes of the dual matroid. These formulas show that $T^2$ vanishes exactly when every connected component has corank at most two, or equivalently when there is no $U_{1,4}$-minor. This proves a conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello (2025) and characterizes formal smoothness of the affine deformation functor. For the duals of rank-three truncations of $U_{2,m}\oplus U_{2,n}$, $m,n\ge3$, each nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and the quotient has an explicit presentation with $m+n-2$ minimal homogeneous generators.

Given graphs $X,Y$ on $n$ vertices, Defant and Kravitz defined the friends and strangers graph, denoted $\mathrm{FS}(X,Y)$. The vertices of $\mathrm{FS}(X,Y)$ are bijections $\sigma:V(X)\to V(Y)$, and two bijections $\sigma$ and $\sigma'$ are adjacent in $\mathrm{FS}(X,Y)$ if they differ by a transposition $(ab)$ such that $a$ and $b$ are adjacent in $X$ and $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y$. Our main result resolves a conjecture of Bangachev that if $X,Y$ are connected graphs and $2\min\{\delta(X),\delta(Y)\}+3\max\{\delta(X),\delta(Y)\}\ge 3n$, then $\mathrm{FS}(X,Y)$ is connected. The main step in the proof is showing that there exists an absolute constant $C$ such that for any adjacent vertices $a,b\in V(X)$ and any vertex $\sigma\in V(\mathrm{FS}(X,Y))$, $a$ and $b$ can swap positions in at most $C$ moves. Furthermore, we prove that if $X,Y$ are connected graphs satisfying the above minimum degree condition, then the diameter of $\mathrm{FS}(X,Y)$ is $O(n^2/\min\{\delta(X),\delta(Y)\})$, and that this is sharp as a bound on the maximum diameter of $\mathrm{FS}(X,Y)$ in this regime. This work improves the diameter results of Akella and Li, and resolves many of their questions. Moreover, we show that even for disconnected graphs $X$ and $Y$, the same minimum degree condition implies that the diameter of any connected component of $\mathrm{FS}(X,Y)$ is $O(n^2)$.

Interpreting electron paramagnetic resonance (EPR) spectra commonly requires repeated forward simulation and optimization of spin Hamiltonian parameters. Established software provides mature simulation and fitting methods, but these calculations are not generally available as differentiable components within machine-learning workflows. We present TorchSpin, an open-source PyTorch framework that ports the organization and numerical conventions of EasySpin into a tensor-based computational environment. Representative calculations spanning continuous-wave EPR, ENDOR, pulse EPR, spin dynamics, trajectory simulation, and magnetometry closely reproduced corresponding EasySpin results. In matched benchmarks, TorchSpin CPU execution was faster than EasySpin for six of nine workloads although performance depended strongly on the calculation. We therefore focused on the capabilities enabled by automatic differentiation in PyTorch. In a Cu(II) powder-spectrum demonstration, feedback propagated through TorchSpin improved the quality of a supervised convolutional neural network (CNN) without using parameter labels during refinement. Models trained from randomly initialized CNNs using only noisy spectra and TorchSpin reconstruction loss achieved median similarities of $\geq$ 0.99. CNNs trained on TorchSpin-generated libraries also transferred without adaptation to independently generated EasySpin spectra with performance comparable to EasySpin-trained controls. Once trained, the CNN produced initial parameter estimates in approximately 1 ms per spectrum while a batch of 512 was processed in approximately 1.3 ms with GPU acceleration. TorchSpin therefore provides both a tested Python implementation of established EPR calculations and a differentiable physical layer for rapid learned inference, spectral refinement, and parameter-label-free training.

Recently, Matomäki, Pandey, Pliego, Teräväinen and Wang improved the Baker–Harman–Pintz estimate of trilinear sums over short intervals of length $x^{\theta}$, which naturally reduces to estimating mean values of products of three Dirichlet polynomials. Based on recent works of Guth and Maynard and Chen, Gupta and Li on new large value estimates for Dirichlet polynomials, they improved the Baker–Harman–Pintz estimate for $\theta > \frac{3}{5}$. In this paper, we use their method to improve the Baker–Harman–Pintz estimate for $\frac{1}{2} < \theta < \frac{4}{7}$ and get new results for all $\frac{1}{2} < \theta < 1$. Our result yields improvements on many problems, including the work of Li on primes in short intervals of length $x^{0.52}$.

We define multidimensional delta functions as generalized functions by pulling back the Dirac delta function, as well as products of such. We prove various formulas including a local version of the coarea formula and we use delta functions to compute marginal probability distributions on spheres.

We study two hypotheses for how a human role in production could persist despite growing AI capabilities. Humans might have a durable advantage in taste, designing better products even as AI agents automate routine tasks, or consumers might intrinsically prefer that humans have a hand in production. In our experiment, participants act as producers, designing computer games with an AI coding agent, and as consumers who buy them. Across three studies (N = 1,025), we find no evidence for either hypothesis. Games designed by an AI on its own outperform those of human producers, including both amateurs and professional creatives. Human producers do not anticipate the AI's advantage. Revealing whether a game is human-designed does not raise consumers' willingness to pay for the product.

Developing resilient crops requires models that accurately capture how plant growth dynamics emerge from the continuous interplay between genetic and environmental factors. While high-throughput phenotyping platforms generate data for dynamic growth models, current modelling approaches show limitations: process-based crop growth models (CGMs) capture environmental dynamics but are difficult to scale across many genotypes because they require calibration of many parameters, while quantitative genetic models allow modelling many genotypes but often summarise time-varying environments using static covariates. To address this gap, we propose a data-driven framework to identify reduced systems of ordinary differential equations (ODEs) describing crop growth dynamics as functions of environmental variables and a controllable number of genotype-specific coefficients (GSCs). Using the Sparse Identification of Nonlinear Dynamics (SINDy) method, we infer model structure directly from data via sparse regression on a physiologically informed library of candidate functions. In this framework, GSCs parameterise cultivar-specific differences in response to environmental conditions. The resulting models are linear in these GSCs, making them compatible with mixed-model and genomic prediction approaches. We evaluate the approach on an in silico dataset generated with WOFOST for five potato cultivars across twelve growing seasons. Our identified ODEs generalise to held-out years and achieve $R^2 > 0.9$ for end-of-season yield prediction across held-out years when compared to full-scale WOFOST simulations. These results provide a proof of concept for SINDy-based identification of parsimonious dynamic crop models from longitudinal crop-growth data.

We study digit-appending walks in base $b\geq 2$ along integer linear recurrences with minimal characteristic polynomials having a unique simple dominant root $\alpha$ that lies outside the unit circle. We prove that if $\alpha$ and $b$ are multiplicatively independent, then no infinite walk can have uniformly bounded appended digit lengths. In contrast, we show that when the appended digit length can be unbounded, the same independence hypothesis implies the existence of infinite walks.

We present short proofs for the main parts of the recently completed solution of Erdős Problem 506, which asks for the minimum number of distinct circles determined by a finite planar point set that is neither collinear nor concyclic. An inversion argument and a seven-point line lemma show that eight points determine at least $17$ circles, and an explicit configuration attains this bound. A single argument combining Melchior's inequality with Langer's incidence bound establishes the sharp formula for every $n\ge15$. The proofs give concise alternatives to longer arguments in the recent literature; the remaining finite cases are cited separately.

Urban venues such as restaurants, shops, bars and parks host the daily life of a city, and their collective rise and fall signals how vibrant a neighbourhood is. The usual assumption is that a venue's fortunes follow its neighbourhood's, so that how wealthy an area is and how many people live there will decide a venue's performance. Yet whether there is a common performance trajectory shared by venues over time is underexplored. In this study, we use 5.5 years of Foursquare data for about 10^4 venues in London and Barcelona. We found six distinguishable shapes recur in both cities, that are (1) Fall, (2) Rise, (3) Rise & Fall, (4) Steady, (5) Ephemeral (an early hit that loses popularity fast) and (6) Underdogs (a slow starter that gains popularity late). However, no recovery shape appears, that is neither city yields a U-shaped fall-and-rise group. Additionally, we found that deprivation and population density are not indicative which shape a venue follows. We also discovered that how early a shape can be recognized depends on when its distinctive segment occurs, not a shape's complexity. Proximity between venues, however, does predict how well-performing a venue is. We uncovered a mixing pattern where venues of the same shape tend to locate together more often than chance within roughly 200 to 600 metres. Finally, our qualitative reading of two contrasting districts, one rising in London and one falling in Barcelona, show that comparable socioeconomic conditions but divergent gentrification histories can generate vastly different popularity trajectories. All in all, our results altogether point at that urban vibrancy emerges at the block- and street- level, and social and historical contexts matter in terms of the characteristics of such patterns.

We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving $25$ out of $30$ two-variable equations of size up to $34$ for which the problem of integer solubility remained open. We also resolve the remaining $5$ equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to $35$ and of length up to $12$, with two exceptions. Before this work, there were $211$ equations in this category that resisted the previously known methods, and we proved that $209$ of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.