The infrared structure of gauge theories with chiral fermions remains largely unexplored. In this work we investigate the Bars-Yankielowicz class using the functional renormalisation group, building on recent developments in gauge-fermion systems that provide clear criteria for confinement and dynamical symmetry breaking. We show that two distinct phases arise: one exhibiting both confinement and symmetry breaking at small numbers of colours, and another characterised by confinement without symmetry breaking in the large-colour limit. The latter realises a novel regime, opening the possibility of exotic spectra and phenomena that can now be studied within a systematic framework.

We study a recursive Penrose process and the energy extraction for the decay of electrically charged particles in a Reissner-Nordström black hole spacetime with anti-de Sitter (AdS) asymptotics, incorporating the backreaction on the black hole's mass and charge. A recursive process requires that the decay products are confined in a finite region so that the emitted particles bounce back for further decay. In AdS spacetimes, the confinement arises naturally. Outgoing particles encounter a turning point and are reflected. One may impose a mirror at finite radius, but in AdS, backreaction makes these two confinement methods equivalent. Let $Q_n$ be the black hole charge after $n$ decays, and define $n_{\rm c}$ as the index for which the black hole's charge is zero, $Q_{n_{\rm c}}=0$. For $n_{\rm c}$ integer the black hole's charge decreases and reaches exactly zero after a finite number of decays, terminating the process. However, the last particle turns back, and encountering zero charge, falls into the hole. The final state is a charged black hole whose charge equals the sum of the original black hole and the initial particle charges. For $n_{\rm c}$ noninteger, the black hole charge decreases and can be arbitrarily small, but is never zero. The last allowed decay occurs at $n=n_{c}^-$, where $n=n_{c}^-$ is the greatest integer less than $n_{\rm c}$. Any further decay invalidates the approximations, the particles would carry a charge comparable to the black hole mass, transforming the problem into a two-body problem. The would-be subsequent decay would violate cosmic censorship and the process terminates before any inconsistency arises. In the integer and noninteger cases, the system yields a finite energy gain. Backreaction ensures that the process extracts a finite amount of energy. No black hole bomb occurs, the system works at most as an energy factory.

The interplay between toric Calabi-Yau 3-folds, dimer integrable systems, and 5-dimensional quantum field theories has proved fruitful. We extend this framework to generalized toric polygons (GTPs) and show that their integrable systems arise from refined birational transformations of known dimer integrable systems acting on the Casimirs and Hamiltonians as well as the Poisson structure and spectral curves. We argue that these transformations are realized as Hanany-Witten transitions producing (p,q) 5-brane webs dual to GTPs. We show that the resulting 5d N=1 theory is obtained by Higgsing a higher-rank theory whose associated toric Calabi-Yau has a toric diagram of the same shape as the GTP.

Motivated by the interplay between 2D and 3D scaling signatures observed in unconventional layered superconductors, we present a systematic Monte Carlo study of the three-dimensional classical XY model with anisotropic in-plane $J_\parallel$ and inter-plane $J_\perp$ couplings. Our study includes very small values of the system anisotropy $Δ=J_\perp /J_\parallel$ not studied before, and focuses on characterizing the crossover from quasi-2D topological scaling to genuine 3D critical behavior. The numerical results for the critical temperature unambiguously reveal a logarithmic scaling with $Δ$, directly related to the topological scaling in the 2D limit. Despite the 3D nature of the layered XY criticality, topological scaling signatures survive up to system sizes comparable to the crossover length $\ell_J$, which diverges at small $Δ$ with a scaling behavior reminiscent of the Berezinskii-Kosterlitz-Thouless (BKT) transition. This shows that genuine 3D symmetry-breaking behavior emerges only at exceedingly large system sizes when the anisotropy is very strong. Our results indicate that new experimental evidence is required to clarify the extent to which the critical signatures observed in layered strongly correlated materials are shaped by their pronounced anisotropy.

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $r$-graph on $n$ vertices in which every $k$ edges span more than $s$ vertices. Brown, Erdős and Sós in 1973 conjectured that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists and verified the conjecture for $k=2$ by showing that $\lim_{n\to\infty} n^{-2} f^{(3)}(n;4,2)=\frac{1}{6}$. Delcourt and Postle, building on the work of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, proved that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists, thereby solving this conjecture. Their approach was later generalised by Shangguan to every uniformity $r\geq 4$: the limit $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ exists for all $r\geq 3$ and $k\geq 2$. However, its exact value was not determined. When $k\in\{2,3,\ldots,7\}$, the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ were determined by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. Very recently, the limit for $k=8$ and $r\geq 4$ was determined by Pikhurko and Sun. For a general even integer $k$, Letzter and Sgueglia obtained the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k$ and uniformity $r\geq 2+\sqrt{2}\,k^{3/2}$. In this paper, we determine the exact value of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k\geq 4$ and $r\geq 2+\sqrt{\frac{3}{2}k-4}$, and show that it is $\frac{1}{r^2-r}.$

We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers.

Quantum interference provides one of the most sensitive probes of quantum mechanics. While linear superposition fixes the positions and quadratic curvature of interference fringes, it remains unclear whether the probabilistic postulate itself, the Born rule, can be tested through finer, local features of interference patterns. Here we show that a minimal deformation of quantum probability gives rise to a robust and symmetry-protected signature: a left-right asymmetry in the local shape of interference fringes. Remarkably, this effect leaves the linear Schrödinger dynamics intact and does not shift fringe positions or modify their quadratic curvature. Instead, it appears exclusively as a cubic skewness of local intensity profiles, providing a clean and falsifiable observable. We demonstrate this behavior within a controlled realization that preserves linear dynamics while minimally deforming the probabilistic assignment. The resulting signature is universal, scale insensitive, and cannot be mimicked by conventional sources of experimental noise. Our results identify local asymmetry in interference as a direct probe of quantum probability itself, suggesting that features often regarded as removable imperfections may encode fundamental information beyond fringe positions and widths.

We investigate the use of additional 3D and phylogenetic non-3D tree balance indices for analyzing and monitoring forests using an exemplary "virtual forest" dataset from the Wytham Woods, Oxford, UK. This study assesses 3D model quality, species classification performance, and the relevance of these indices. Our study shows that indices stemming from the study of ancestry trees of species can be successfully applied to 3D models of organic trees and, accompanied with recently introduced 3D imbalance indices, offer a complementary perspective on 3D tree models and improve the detection of deviations. Their computational efficiency combined with the simple and reproducible workflow presented in this manuscript form a computationally feasible quality control step in the 3D model construction. Species classification models reached an estimated accuracy of up to 81.8% and allowed to make confident species predictions for a large portion of the unlabeled trees in the dataset. While conventional tree metrics can already provide strong predictive performance, the addition of filtered 3D and non-3D statistics improved results consistently, particularly for minority species classes. Alongside this manuscript, we provide updated functionality in the R package treeDbalance to include the necessary functionalities and release the derived index datasets and species predictions.

Least Absolute Deviations (LAD) regression provides a robust alternative to ordinary least squares by minimizing the sum of absolute residuals. However, its widespread use has been limited by the computational cost of existing solvers, particularly simplex-based methods in high-dimensional settings. We propose a coordinate descent algorithm for LAD regression that avoids matrix inversion, naturally accommodates the non-differentiability of the objective function, and remains well-defined even when the number of predictors exceeds the number of observations. The key observation is that each coordinate update reduces to a one-dimensional minimization admitting a closed-form solution given by a median or weighted median. The resulting algorithm has per-iteration complexity $O(p\,n \log n)$ and is provably convergent due to the convexity of the LAD objective and the exactness of each coordinate update. Experiments on synthetic and real datasets show that the method matches the accuracy of linear-programming-based LAD solvers while offering improved scalability and stability in high-dimensional regimes, including cases where $p \ge n$. The method is easy to implement, requires no specialized optimization software, and provides a practical tool for robust linear models.

We study a varying-speed-of-light (VSL) phase embedded in string gas cosmology (SGC), with the effective propagation speed controlled by the dilaton during the Hagedorn era. For the exponential ansatz $c(ϕ)=c_0\,e^{-αϕ}$, the analytic Hagedorn background generates a speed-of-light bounce: an early superluminal phase ($c\gg c_0$), crossover at $t/t_0\approx 0.285$ for the branch studied here, and a collapse of $c$ toward zero as the self-dual regime is approached. The self-dual point at $R=\ell_s$ provides a T-duality anchor for matching onto the late-time branch with $c=1$. Evaluating the comoving horizon on this background, we find enhancement factors of $1.54$ and $3.44$ for $α=1$ and $α=2$, respectively, while the flatness parameter $Ω-1\propto c^2/(a^2H^2)$ is found numerically to be suppressed by $10^{-4}$--$10^{-7}$ over the late Hagedorn phase. These results show that a dilaton-driven VSL phase can enlarge the causal horizon and suppress curvature within the controlled regime of SGC, while localizing the remaining obstruction to the self-dual matching point.

Nevanlinna theory studies the value distribution of meromorphic functions and provides powerful results in the form of the First and Second Main Theorems. In this paper, we introduce quaternionic analogues of the Nevanlinna functions. Starting from the Jensen formula due to Perotti (arXiv:1902.06485), we derive a notion of total order and an associated integrated counting function. We further define quaternionic Weil functions and corresponding mean proximity functions. In this context, we introduce the class of mean proximity balanced functions, which includes the slice-preserving functions and all semiregular functions with a dominating index in their power series. To address the failure of $\log|f^s|$ to be harmonic, we define a Harmonic Remainder Function that compensates for this defect in the Jensen formula. We then prove a weak First Main Theorem--type result for general semiregular functions and obtain a full First Main Theorem for the mean proximity balanced functions.

We study how runtime enforcement against unsafe actions affects end-to-end task performance in multi-step tool using large language model (LLM) agents. Using tau-bench across Airline and Retail domains, we compare baseline Tool-Calling, planning-integrated (TRIAD), and policy-mediated (TRIAD-SAFETY) architectures with GPT-OSS-20B and GLM-4-9B. We identify model dependent interaction horizons (15 to 30 turns) and decompose outcomes into overall success rate (SR), safe success rate (SSR), and unsafe success rate (USR). Our results reveal a persistent Safety Capability Gap. While safety mediation can intercept up to 94 percent of non-compliant actions, it rarely translates into strictly safe goal attainment (SSR below 5 percent in most settings). We find that high unsafe success rates are primarily driven by Integrity Leaks, where models hallucinate user identifiers to bypass mandatory authentication. Recovery rates following blocked actions are consistently low, ranging from 21 percent for GPT-OSS-20B in simpler procedural tasks to near zero in complex Retail scenarios. These results demonstrate that runtime enforcement imposes a significant verifier tax on conversational length and compute cost without guaranteeing safe completion, highlighting the critical need for agents capable of grounded identity verification and post-intervention reasoning.

Unraveling the hierarchical structure-property relationships is the central challenge of materials science, necessitating the interpretation of data across vast physical scales from micro to macro. Despite the rapid integration of Large Multimodal Models (LMMs) into scientific workflows, existing scientific benchmarks primarily focus on general chart interpretation or isolated common-sense reasoning, failing to capture reasoning ability across intricate physical dimensions. To address this, we introduce CSMBench, a dataset comprising 1,041 high-quality figures curated from premier journals up to September 2025. CSMBench categorizes data into four scientifically distinct regimes: atomic, micro, meso, and macro scales, strictly aligning with the focus and definitions in materials study. Through open-ended figure description and multiple-choice caption matching tasks, we evaluate state-of-the-art open-source and closed-source models. Our analysis identifies that performance varies significantly across physical scales due to the distinct visual characteristics, highlighting the limitations of current generalist models and identifying critical directions for achieving hierarchical and accurate understanding in materials research. The CSMBench is publicly released at: https://huggingface.co/datasets/lututu/CSMBench.

Adaptive physical and biological systems continually process fluctuating information from their environments. When the environment is nonstationary, inference itself becomes a nonequilibrium process with thermodynamic cost. We analyse a minimal stochastic model which is an overdamped particle in an adaptive double well potential whose control parameter tracks a drifting Ornstein Uhlenbeck signal. Using stochastic energetics, we derive explicit expressions for entropy production, mutual information rate, and a time dependent learning efficiency. High precision Langevin simulations reveal transient peaks in learning efficiency during rapid environmental shifts, absent in steady state averages. These results identify transient adaptive regimes as moments of maximal information to energy conversion, highlighting that maximal thermodynamic learning performance arises transiently rather than in steady state. Throughout this work, the environment is treated as an externally driven stochastic signal rather than a thermodynamic subsystem under control, and its intrinsic entropy production is therefore excluded from the thermodynamic accounting.

Neural circuits must balance local connectivity constraints against the need for global integration. Here we introduce a minimal wiring rule motivated by synaptic crowding: as a neuron accumulates incoming connections, each additional synapse becomes progressively harder to form. This single-parameter model admits an exact finite-size solution for the induced in-degree distribution and yields simple scaling laws: mean connectivity grows only logarithmically with network size while variance remains bounded -- consistent with homeostatic regulation of synaptic density. When candidates are encountered in order of spatial proximity, the crowding rule produces a broad, approximately power-law distribution of connection lengths without prescribing any explicit distance-dependent wiring law; combined with shortcut rewiring, this yields networks with small-world characteristics. We further show that the induced degree statistics largely determine attractor basin boundaries in threshold network dynamics, while local clustering primarily modulates the prevalence of long-lived non-absorbing outcomes near these boundaries. The model provides testable predictions linking local developmental constraints to macroscopic network organization and dynamics.

Generative artificial intelligence (AI) tools are becoming increasingly used for writing tasks. However, the extent of their use in peer-reviewed medical literature remains unclear. We conducted a longitudinal analysis of all Original Investigations, Research Letters, and Invited Commentaries published in JAMA Network Open from January 2022 through March 2025. The main body text of 7,251 articles was analyzed using a commercial AI-detection tool (Originality.AI) to estimate the probability that manuscripts contained a significant amount of AI-generated content. Articles were analyzed aggregated by month, publication type, and domain. Overall, 195 articles (2.7%) were classified as containing significant AI-generated text. The monthly proportion increased from 0.0% in January 2022 to 11.3% in March 2025, with a significant upward trend over time (P<0.001). Invited Commentaries had the highest proportion of detected AI-generated content (6.7%), followed by Original Investigations (2.2%) and Research Letters (1.4%). There was also significant variation across publication domain (P=0.04). Only 15 articles (0.2%) disclosed large language model use, of which 40.0% were classified as containing AI-generated text. While findings suggest increasing detectable AI-generated content in medical literature, limitations of current detection tools necessitates cautious interpretation.

Augmented reality games hold promise for rehabilitation, yet most remain confined to laboratory studies with limited clinical uptake. Recent advances in spatial computing, especially lightweight, glasses_form_factor AR, create a timely opportunity to embed rehabilitative play into clinical practice and daily contexts. To investigate this potential, we systematically reviewed 132 applications and conducted playtesting with 14 licensed physical therapists. Our analysis revealed three ways therapists re_authored AR games: co_authored play (reshaping movements, progressions, and difficulty), situated play (adapting across specialties, conditions, and contexts), and dual play (mediating both physical recovery and psychological support). We reframe therapists' frequent phrase_It depends_as a generative design principle. This study contributes a clinical reasoning_based framework and design principles and guidelines for creating personalized, situated forms of play that align with therapists' everyday workflows and inform future lab_to_clinic translation.

We propose a joint TDC and multi-time-gated imaging single-photon avalanche diode LiDAR sensor featuring a maximum detection range of 108m with a minimum depth resolution (least significant bit) of 2.93mm, fabricated in 110nm CMOS technology. The sensor is implemented 320x240 pixels with a pitch of 23μm, achieving a maximum fill factor of 10% and a total size of 8.2mmx6.3mm. The power consumption of the pixel array is 167.4mW. Under a repetition rate of 3.125MHz, a laser power of 11.3mW, and a wavelength of 780nm, the sensor is designed to operate at up to 100klx background illumination with a centimetric depth precision. It can be further improved to 130klx by reducing image resolution to 160x120 pixels. In addition, the sensor can reconfigure the idle TSPCs integrated in the cluster into counters for histogramming, thereby reducing the size of the data processing module.

Open and reproducible research in materials science relies on the availability of data, code, and common metadata standards. Journal research data policies (RDPs) remain a primary mechanism by which publication norms are defined and enforced. We survey RDPs for 171 materials science journals spanning 17 publishers, using an expanded coding framework that captures both data-and-code sharing behavior as well as refereeing standards. We find clear signs of progress in comparison to earlier research on RDPs: nearly all journals provide an RDP, and most mention data availability statements. However, enforceable requirements remain uncommon, public deposition of underlying data is rarely mandatory, and FAIR publication is typically encouraged rather than required. Expectations for research software are substantially less developed than those for data, with limited attention to versioning and persistent identifiers, dependency disclosure, reproducible execution environments, or software quality practices. Aggregating the findings on policy features into an open research data score reveals pronounced heterogeneity across journals. Neither impact factor nor access model reliably predicts policy strength. Double-coding further shows that more complex policies and stricter policies can be more challenging to interpret consistently, and we highlight challenges in consistent RDP encoding across studies. Lastly, we conclude with recommended best practice directions for the future.

All men are created equal, proclaimed Jefferson in 1776 -- but some are more equal than others, added Orwell in Animal Farm in 1945. So what's the probability that two skaters are exactly equal, to the third decimal places, after four distances?

Purpose: To present a fully open-source framework for quasi-real-time streaming and cloud-based processing of low-field (LF) MRI data, addressing the growing computational demands of advanced reconstruction and post-processing pipelines in portable and affordable MRI systems. Methods: The proposed framework integrates open-source scanner control software with a network-enabled streaming architecture, allowing for raw data to be transmitted directly from the MRI console to remote compute resources. Cloud-based processing modules support image reconstruction and advanced post-processing, including computationally intensive physics- and learning-based methods, while maintaining compatibility with low-cost on-device control hardware. Results: The system enables continuous acquisition-to-reconstruction workflows in LF-MRI without requiring specialized high-performance console architectures. Selected example applications include deep-learning-based denoising, field-induced distortion correction, and non-Cartesian image reconstruction. Experimental demonstrations show reliable streaming performance. Conclusions: Open-source streaming and cloud-processing provide an effective pathway to overcome the computational limitations of embedded LF-MRI consoles. By decoupling acquisition hardware from intensive reconstruction workloads, the proposed framework supports scalable deployment of advanced algorithms while preserving the affordability and portability that motivate LF-MRI.

Urgent societal events demand scientific responses that are both rapid and impactful. Through an adversarial collaboration, we connected bibliometric databases to evaluate the speed and impact of over 2 million scientific publications in the three years following 48 urgent societal events. A pilot analysis of three cases -- the 2022 release of ChatGPT, the 2019 COVID-19 pandemic, and the 2001 World Trade Center attacks -- yielded unexpected patterns: larger teams were not only more impactful but also quicker to publish. More precisely, increases in team size were associated with (a) initial increases, but eventual diminishing returns in academic citations, (b) curvilinear returns in news and policy document citations, and (c) curvilinear returns in terms of how quickly papers were published. In other words, there are points where further increases in team sizes are either marginally helpful (diminishing returns) or counterproductive (curvilinear returns). To evaluate robustness, we pre-registered a broader test covering 45 additional events spanning two decades.

In recent years, Asia's rapid growth in research output has been reshaping the computing research landscape. What was once a two-block system (America and Europe) is evolving into a multipolar world with three major hubs: America, Europe, and Asia. To study these pivotal changes and evaluate international diversity, we have analyzed the past 13 years of 13 international systems research conferences: ASPLOS, NSDI, OSDI, SIGCOMM, ATC, EuroSys, ICDCS, Middleware, SoCC, CCGRID, IC2E, IEEE Cloud and EuroPar. Our analysis focuses on accepted papers and participation in the Program Committee, grouping the results by region (America, Europe, and Asia). Surprisingly, we find a pronounced historical imbalance in international diversity among top-tier systems conferences (ASPLOS, OSDI, NSDI, SIGCOMM). While most other conferences have progressively reflected Asia's growing research presence over the past decades, this group has shown a noticeable adjustment only in the recent four years. We also identify persistent rigidities in how program committee (PC) diversity adapts to shifts in accepted paper origins, with a consistent under-representation of researchers from Asian organizations in many PCs.

The prerequisites of the extended phase space approach to quantization of gravity, which is alternative to the Wheeler-DeWitt one and other existing approaches, are presented. The features of the proposed approach and conclusions from its underlying ideas are discussed.

This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form \[ f(x+y)-f(x)-f(y)=B(x,y) \quad \text{or} \quad f(xy)-f(x)f(y) = B(x,y) \] where $B$ is a biadditive mapping, and also more general cases where the inhomogeneity depends on unknown functions \begin{align*} f(x+y)-f(x)-f(y)&= αx y \\[2.5mm] f(x+y)-f(x)-f(y)&= α(x y)\\[2.5mm] f(x+y)-f(x)-f(y)&= α(x)α(y). \end{align*} Our results extend previous work on the bilinearity of the Cauchy exponential difference by Alzer and Matkowski. We characterize solutions under various structural and regularity assumptions, including additive and exponential Cauchy differences, and show that solutions often reduce to additive functions, exponential polynomials, or combinations thereof. For Levi-Civita type equations, we provide explicit representations of solutions in terms of additive and exponential components. Furthermore, we determine conditions under which real-valued solutions exist and describe their forms. The paper concludes with open problems concerning generalized equations that cannot be solved by the methods presented here, suggesting directions for future research.

The discovery of constitutive laws for complex materials has historically faced a dichotomy between high-fidelity data-driven approaches, which demand prohibitive full-field experimental data, and traditional engineering fitting, which often yields numerically unstable models outside calibration regimes. In this work, we propose an Engineering-Oriented Symbolic Regression (EO-SR) framework that bridges this gap by leveraging Large Language Models (LLMs) as "Physics-Informed Agents." Unlike unconstrained symbolic regression, our framework utilizes an LLM Agent to zero-shot synthesize executable physical constraints -- specifically thermodynamic consistency and frame indifference -- transforming the search process from mathematical curve-fitting into a physics-governed discovery engine. We validate this approach on the hyperelastic modeling of rubber-like materials using standard Treloar datasets. The framework autonomously identifies a novel hybrid constitutive law that combines a Mooney-Rivlin linear base with a rational locking term. This discovered model not only achieves high predictive accuracy across multi-axial deformation modes (including zero-shot prediction of pure shear) but also guarantees unconditional convexity. Finite element validation demonstrates that while industry-standard models (e.g., Ogden N=3) fail due to numerical singularities under severe transverse compression, the EO-SR-discovered model maintains robust convergence. This study establishes a generalized, low-barrier pathway for discovering simulation-ready constitutive closures that satisfy both data accuracy and rigorous physical laws.

Symbolic execution is a powerful program analysis technique, but its effectiveness is fundamentally limited by solver-hostile program fragments, complex numerical reasoning, and unbounded heap structures. Recent work proposed replacing constraint solvers with large language models (LLMs) to bypass these limitations, but such approaches struggle to analyze real-world codebases, where deep execution paths require globally consistent reasoning across many interacting constraints. We present Gordian, a hybrid symbolic execution framework that uses LLMs selectively to generate lightweight ghost code that aids an SMT solver in handling solver-hostile code fragments, while preserving its precise, global reasoning capability. In particular, we propose three types of ghost code: (1) inversion of difficult code fragments with iterative bidirectional constraint propagation, (2) modeling via solver-friendly surrogates while preserving relevant behavior, and (3) semantic partitioning of unbounded heap spaces. We implemented Gordian on top of the KLEE symbolic execution engine and evaluated it on synthetic "logic bombs" capturing distinct symbolic reasoning challenges, a popular mathematical library FDLibM, and three structured-input programs (libexpat, jq, and bc). Across all benchmarks, Gordian improves coverage on average by 52-84% over traditional symbolic execution baselines, and by 86-419% over LLM-based techniques, while reducing LLM token usage by an average of 90-96%. This highlights the practicality and effectiveness of this approach in real-world settings.

Bibliographic catalogues store millions of data. The use of computer techniques such as web-scraping allows the extraction of data in an efficient and accurate manner. The recent emergence of ChatGPT is facilitating the development of suitable prompts that allow the configuration of scraping to identify and extract information from databases. The aim of this article is to define how to efficiently use prompts engineering to elaborate a suitable data entry model, able to generate in a single interaction with ChatGPT-4o, a fully functional web-scraper, programmed in PHP language, adapted to the case of bibliographic catalogues. As a demonstration example, the bibliographic catalogue of the National Library of Spain with a dataset of thousands of records is used. The findings present an effective model for developing web-scraping programs, assisted with AI and with the minimum possible interaction. The results obtained with the model indicate that the use of prompts with large language models (LLM) can improve the quality of scraping by understanding specific contexts and patterns, adapting to different formats and styles of presentation of bibliographic information.

In dynamical systems, invariants, i.e., constants of motion conserved along the trajectory, play important roles in characterizing the system's dynamical behavior. Recent applications of the Koopman operator framework to nonlinear dynamical systems have provided new insights into the invariants. For a certain class of globally coupled phase oscillators, which serve as models for various synchronization phenomena, Watanabe and Strogatz proved the existence of N-3 invariants in N oscillator systems. In this study, we derive these invariants from an operator-theoretic perspective by exploiting the relation between Liouvillian (Perron-Frobenius) and Koopman descriptions of the dynamics. Exploiting a simple multiplicative property of functions under the action of the Liouvillian and Koopman operators, we explicitly construct a family of functions whose ratios yield the invariants of the underlying dynamics. Our analysis successfully reproduces the full set of N-3 invariants known in Watanabe-Strogatz theory, and offers an alternative spectral perspective. We demonstrate this approach for a well-studied class of phase models, including the Ermentrout-Kopell, pairwise Kuramoto, and higher-order Kuramoto models.

In large-scale excitatory neuronal networks, rapid synchronization manifests as {multiple firing events (MFEs)}, mathematically characterized by a finite-time blow-up of the neuronal firing rate in the mean-field Fokker-Planck equation. Standard numerical methods struggle to resolve this singularity due to the divergent boundary flux and the instantaneous nature of the population voltage reset. In this work, we propose a robust {multiscale numerical framework based on time dilation}. By transforming the governing equation into a dilated timescale proportional to the firing activity, we desingularize the blow-up, effectively stretching the instantaneous synchronization event into a resolved mesoscopic process. This approach is shown to be physically consistent with the {microscopic cascade mechanism} underlying MFEs and the system's inherent fragility. To implement this numerically, we develop a hybrid scheme that utilizes a {mesh-independent flux criterion} to switch between timescales and a semi-analytical ``moving Gaussian'' method to accurately evolve the post-blowup Dirac mass. Numerical benchmarks demonstrate that our solver not only captures steady states with high accuracy but also efficiently reproduces periodic MFEs, matching Monte Carlo simulations without the severe time-step restrictions associated with particle cascades.