Let $p_{\min}$ denote the minimum of a polynomial $p$ over a (general) compact semialgebraic set $S \subseteq \mathbb{R}^n$. A standard way to approximate $p_{\min}$ is via hierarchies built from Positivstellensätze, which certify nonnegativity of polynomials on $S$ using sums of squares or other classes of globally nonnegative polynomials. As the degree of the certificate grows, the values generated by these hierarchies converge asymptotically to $p_{\min}$. A natural question is, then, to determine explicit bounds on the certificate's degree needed to obtain a prescribed $\varepsilon$-approximation to $p_{\min}$, or equivalently certify the positivity of $f:=p - p_{\min} + \varepsilon$ on $S$. We improve the current best degree bounds for Putinar's and Schmüdgen's SOS-Positivstellensatz over $S$. Also, we obtain degree bounds for Krivine--Stengle's and the recently introduced extended-Handelman's $\mathbb{R}_+$-Positivstellensätze over $S$; providing the first explicit degree bounds for linear optimization-based hierarchies over general compact semialgebraic sets. Our approach is based on a lift-and-project construction in which we add new variables to construct an algebraic representation of the distance to the set $S$ using Łojasiewicz's inequality. This lets us lift the problem of certifying the positivity of $f$ on the (complex) set $S$ to the problem of certifying the positivity of a related polynomial $F$ on a higher-dimensional hypercube. By projecting out the added variables, non-negativity certificates for $F$ on the hypercube become non-negativity certificates for $f$ on $S$. Our approach offers a unified methodology to obtain degree bounds for several Positivstellensatz-based hierarchies over general compact sets, narrowing the gap between results for the hypercube (or other simple sets) and more general semialgebraic sets.

We show that for a CFT$_D$ on a flat open solid torus, the two point function in the Weyl frame is exactly paired with a finite geodesic lying entirely in the AdS$_{D+1}$ bulk interior. This relation is exact and requires neither large $N$, strong coupling, nor heavy operators. The exactness is that of conformal kinematics; no semiclassical bulk dynamics is assumed. The standard boundary-anchored relation is a singular limit of the exact pair. For the free scalar, a mode expansion along $S^1$ generates an infinite tower of effective masses on $H_{D-1}$, whose intricate propagators resum exactly to the same simple higher-dimensional geodesic expression. Together with another exact pair between disjoint entanglement entropy and entanglement wedge cross-section found on the same open solid torus, this result points toward a broader exact-pair program in AdS/CFT.

For neutrino physics and rare event searches, background related to cosmic muons poses a notable challenge, and must be identified and rejected. It is also a challenge to control the cost with good performance for a large array of cosmic ray detection. We proposed a cosmic ray detection module with liquid scintillator and wavelength-shifting fibers for its reasonable cost and performances. The results from the measurements of a prototype with Muon indicate that the detector's photoelectron response is good. % comparing to the expectation. The outcomes of this study hold significant potential for applications in cosmic ray observation experiments and underground rare-event detection, providing a viable option for future large-scale observatories. This work highlights the feasibility of liquid scintillator-based detectors in addressing current and emerging challenges in particle physics and astrophysics.

Neuromorphic cameras, also known as event-based cameras, can detect changes in the environmental brightness asynchronously and independently for each pixel. They output the brightness changes, i.e., events, as 3-D (2-D pixel coordinates + time) streaming data. While event-based cameras are used in many applications because of their desirable characteristics, e.g., high temporal resolution, low latency, low power consumption, and high dynamic range, their measurements contain considerable noise due to their high sensitivity. In this paper, we propose a denoising method for event-based cameras based on graph spectral features. In the proposed method, we first construct a graph where nodes represent events and edges represent the spatiotemporal distance between the events. To calculate the graph-specified parameter that controls the connectivities of a constructed graph, we utilize the prior on the density of 3-D events. We then calculate the eigenvectors of the graph Laplacian. The obtained eigenvectors are used to extract noiseless events directly. In the calculation of the eigenvectors, we customize the graph Laplacian to reorder its eigenvalues. This allows us to leverage fast eigensolver algorithms instead of the naive eigendecomposition and thereby reduce computational complexity. In experiments on synthetic and real-world event data, we demonstrate that the proposed method effectively removes noise events from the raw events compared to alternative methods.

Soft set theory is an important and emerging area within soft computing, owing to its attribute-oriented mathematical framework and its wide applicability in diverse domains, including science and social sciences. The theoretical constraints associated with the selection of subsets of the sets of attributes in soft set theory have further motivated the development of hybrid and extended theoretical models. In this paper, we introduce two distinct roughness measures and six entropy measures for soft sets and systematically investigate their properties using both theoretical analysis and computational techniques. The proposed roughness measures are defined within two distinct conceptual frameworks. Throughout the development of these measures and the corresponding results, the foundational principles of soft set theory, as established by Molodtsov, are strictly preserved. Furthermore, the proposed framework is shown to be novel with respect to roughness characterization, and a comparative analysis with classical rough set theory is presented to highlight the theoretical distinctions and contributions of this work.

This paper considers games where the utilities for agents are the sum of a term proportional to a social utility, and another term that is an individual cost or reward. The agents are assumed to be irrational in their perception of the individual cost or reward. The multi equilibrium game is regularized, and its strictly concave potential function is used to select a unique equilibrium. This selected equilibrium is shown to be an $ε-$equilibrium of the original game, where $ε$ is parametrized by the regularizing function. A minorization-maximization based iterative learning scheme is proposed to learn equilibria in this game. This scheme converges to the potential-optimal equilibrium, and has superior convergence behaviour in comparison to gradient and best response methods.

Recent cosmological data have been interpreted as indicating deviations from $Λ$CDM within the standard $w_0w_a$ parametrization, including hints of phantom crossing and dynamical dark energy. However, such inferences can be parametrization-dependent and need not imply a statistically robust detection. We test these claims by comparing $Λ$CDM, $w_0w_a$, and thawing quintessence models, using the Deviance Information Criterion (DIC) and the Bayesian evidence $\ln \mathcal{Z}$. We find that $w_0w_a$ can provide a slightly improved local fit; however, this improvement is confined to a limited region of the parameter space. The global Bayesian evidence does not support it once the full prior volume is accounted for. In particular, cases with $Δ{\rm DIC}<0$ but $Δ\ln \mathcal{Z}<0$ indicate that these improvements are not statistically significant. We show that all models are statistically indistinguishable, and that there is no statistically consistent evidence across different datasets for either dynamical dark energy or phantom crossing.

Recently, Capparelli, Meurman, A. Primc and M. Primc introduced a class of colored partitions which has since been called CMPP partitions. This generalized earlier work by M. Primc and Šikić, and by Trupčević. One main reason why CMPP partitions are significant is the authors' conjecture that the generating functions are infinite products in all cases. CMPP partitions are true extensions of the partition classes in the Rogers-Ramanujan-Gordon identities which are defined by difference conditions. As such, a natural question is to look for generating functions similar to the series side of Andrews-Gordon identities. Russell found such bivariate series for one case. These evidently positive series overlap with the positive series found earlier by Griffin, Ono and Warnaar in the edge cases. Russell used symbolic computation in the proofs. We will combinatorially interpret Russell's bivariate series extending one case of the series due to Griffin, Ono and Warnaar in a base partition and moves setting, and supply some missing cases, as well.

Recent years have seen a surge of research into conversational recommender systems (CRS). Among existing datasets, ReDial is the most widely used benchmark, cited in hundreds of studies. However, variations in how the dataset is preprocessed and used in experiments, particularly in the definition of ground-truth items, make it difficult to compare results across studies. These comparisons are further complicated by confounding factors such as the choice of the underlying large language model (LLM) and the use of external data sources. In this work, we revisit seven prominent CRS methods across three architectural families and evaluate them under standardized conditions. Our reproducibility study reveals a ``granularity gap,'' where fine-grained ranking (Recall@1) is highly sensitive to implementation details, while our replicability analysis shows that nearly 50% of reported accuracy stems from ``repetition shortcuts'' that are absent in novelty-focused evaluation. Furthermore, we find that performance gains are often driven more by the capacity of the LLM backbone than by specific architectural innovations. Finally, by applying user-centric utility metrics, we demonstrate that traditional recall frequently overstates a system's actual conversational effectiveness. This work establishes a transparent, controlled baseline and promotes evaluation practices that prioritize novelty and interaction efficiency.

We present model calculations for the in-medium suppression of the Y(1S,2S,3S) states in sqrt(s_NN)=5.02 TeV Pb-Pb collisions at the Large Hadron Collider in comparison with recent CMS data for all three spin-triplet s-wave states. The model parameters initial central temperature, and formation times for the Y(nS) states are determined in simultaneous chi^2 minimizations with respect to the data, such that the sequential centrality- and transverse-momentum-dependent suppression of the observed states is reproduced.

Existing Digital Twin (DT) approaches often lack semantic reasoning capabilities for effective cybersecurity modelling in Cyber-Physical Systems (CPS). This paper presents HySecTwin, a knowledge-driven digital twin architecture that places automated reasoning at the core of real-time threat detection. HySecTwin incorporates semantic modelling to transform heterogeneous CPS telemetry, device attributes, and operational relationships into machine-interpretable representations, combined with an embedded reasoning engine operating over contextualized system states. Unlike opaque detection methods, the framework integrates deterministic rule-based inference with hybrid fuzzy reasoning to generate explicit, interpretable, and auditable security assessments from live device telemetry. This enables context-aware monitoring of complex CPS environments while preserving transparency and trust. Experimental evaluation using a representative CPS testbed and MITRE ATT\&CK campaign-inspired attack scenarios demonstrates sub-millisecond twin synchronization latency and up to 21.5\% faster threat detection compared with deterministic reasoning alone. The results show that semantic modelling, semantic enrichment, and hybrid reasoning improve explainability and resilience without extra system overhead. HySecTwin provides a lightweight, containerized, and extensible framework for secure-by-design digital twin deployments in mission-critical infrastructures

Context. Software startups face significant challenges in building minimum viable products, particularly in the early stages, when resources are limited and expertise in user experience is scarce. Objective. Introduce StartFlow, a structured method that helps non-specialized professionals create MVP prototypes using the wireflow technique, a combination of wireframes and user flows. StartFlow consists of three steps: (i) organizing features; (ii) building wireflows; and (iii) verifying and refining them based on usability heuristics. Method. To assess the method Startflow, we first conducted a focus group with researchers in Software Engineering, Human-Computer Interaction, and Software Startups. Afterward, we conducted a proof-of-concept study, which consisted of an experiment and a heuristic evaluation with experts. Results. The qualitative analysis of the focus group revealed that participants found the method straightforward, flexible, and helpful in structuring user flows and identifying visual components. However, they also pointed out the need to improve its presentation, clarify its iterative nature, and strengthen its connection to broader UX principles. The results of the proof-of-concept indicate that participants who used StartFlow created clearer prototypes, adhered to the proposed user stories and business rules, and presented fewer usability defects. Furthermore, the method was well evaluated for its ease of use and intended future adoption. Conclusion. The study reinforces the potential of StartFlow as an accessible tool to support user-centered development in software startups from the earliest stages of their product development.

We investigate two-dimensional Dirac fermions embedded in a deep-subwavelength cavity formed by high-impedance metasurfaces. We point out that, unlike conventional metallic boundaries, these metasurfaces support quasielectrostatic transverse-magnetic modes that mediate a long-range interaction between two-dimensional electrons. Combining static electronic screening with a Dyson-Schwinger analysis, we show that this engineered interaction can qualitatively alter the ground-state properties of Dirac materials. For a fermion flavor number $N_{f}$ below a critical value $N_{c}=16/π$, the interaction drives an excitonic insulating phase through an infinite-order quantum phase transition and spontaneously generates a mass gap. At $N_{f}>N_{c}$, the system remains gapless but enters a non-Fermi-liquid critical regime where the quasiparticle residue is singularly suppressed to zero, and the Dirac cone exhibits a nonanalytic dispersion relation. Furthermore, under a perpendicular magnetic field, the cavity fluctuations dynamically lift the zeroth Landau level degeneracy across all $N_{f}$. These results identify high-impedance metasurface cavities as promising platforms for engineering correlated Dirac matter.

This paper proposes a neuro-symbolic framework for G-code generation that seeks to integrate the neural generative capabilities of the GLLM method (Abdelaal et al., 2025) with formal verification via a Separation Logic (SL) prover. To establish a reliable physical baseline, the framework extracts deterministic boundary representations from 3D CAD models (STEP files) using the OpenCASCADE framework. This extracted geometric data supports a two-component architecture: the LLM serves as an initial code generator, while the SL Prover, utilizing a Spatial Heap model, evaluates the output. By conceptualizing physical collisions as logical Spatial Data Races -- violations of the separating conjunction in SL -- our framework translates proof failures into structured mathematical feedback. These failures are condensed into bounding boxes that serve as directives for the LLM's iterative self-correction. Ultimately, this work aims to develop a self-correcting system that reduces the need for human supervision, leading to safer and verified autonomous manufacturing.

Safety verification in Computer Numerical Control (CNC) machining has traditionally relied on simulation-based methods that require repetitive tests when requirements change. This paper introduces a formal verification framework that conceptualizes the physical CNC workspace as a Spatial Heap, treating physical occupancy as a managed logical resource. Central to our approach is a Parser-Prover Handshake that decouples machine kinematics from formal logic. By mapping tool trajectories and safety buffers into a discrete spatial model prior to evaluation, the framework enables the use of Separation Logic (SL) to verify safety via formal triples. Within this model, physical collisions are redefined as logical Spatial Data Races, detected through the failure of the separating conjunction to establish disjointness. Furthermore, we extend the methodology to collaborative environments using Concurrent Separation Logic (CSL), where physical hand-offs are verified as formal ownership transfers. Additionally, the framework scales to multi-axis kinematics (e.g., 5-axis Table-Table configurations) by treating the workpiece as a dynamically mutable spatial resource. By serving as a complement to traditional geometric simulation, this approach reduces the number of required iterative test cycles, offering a foundation for autonomous, less-collision manufacturing.

We prove that no deterministic output-sensitive algorithm for the planar convex hull and maxima problems can obtain both optimal time and I/O complexity, where the optimality is defined with respect to both the input and output sizes. This explains why the best previous algorithms achieved an optimal I/O bound at the cost of sub-optimal running time (Goodrich et al. [FOCS, 1993]). To the best of our knowledge, the impossibility of simultaneous optimality was only shown previously for the permutation problem by Brodal and Fagerberg [STOC, 2003]. Our results imply that no optimal deterministic output-sensitive cache-oblivious algorithm exists for either problem. In addition, we present simple deterministic algorithms that match our lower bounds and that provide a trade-off between time and I/Os. On the other hand, a simple modification of our deterministic algorithm results in a randomized algorithm that simultaneously achieves optimal (worst-case) time and optimal expected I/O bounds.

We present Diffusion Restore, a real-time framework for diffusion-based MCMC light transport. MCMC methods are highly suitable for sampling from complex high-dimensional distributions and for approximating integrals over them. In practice, they are often the only viable solution when direct sampling is not possible and alternative methods are either inefficient or cannot be applied due to the structure of the target distribution. However, controlling the exploration of the target distribution in MCMC methods remains challenging. Efficient exploration requires a balance between local exploration and global discovery, and local dynamics must rapidly explore individual modes without getting stuck or exhibiting excessive backtracking. The problem of global discovery has recently been addressed by the introduction of the Restore framework. In this work, we build on this framework and focus on improving local exploration. We show how to choose diffusion-based local dynamics within the Restore framework while completely avoiding Metropolis-adjustment, which is known to slow down convergence. Furthermore, we model these dynamics as nonreversible, introducing momentum in the drift and thereby enabling more directed exploration of the target distribution compared to reversible, random-walk-like dynamics. We provide a theoretical justification for the validity of our choice of local dynamics. Empirically, we demonstrate across diverse scenes that Diffusion Restore outperforms all existing MCMC light transport methods and establishes a new state of the art. In addition, we present a GPU implementation in ray tracing and compute shaders and achieve real-time frame rates. This demonstrates that Diffusion Restore is not only superior in offline rendering, but also outperforms traditional Path Tracing methods in real-time rendering settings, such as interactive applications and games.

The system-level cache is a critical resource shared by processor cores and domain-specific accelerators in heterogeneous systems on chips (SoCs). The strict QoS requirements of accelerators, such as deadlines, can lead to severe performance degradation of processor cores. Thus, managing the shared cache efficiently between cores and accelerators becomes crucial. State-of-the-art cache management techniques perform reuse-aware bypassing of accesses from cores with the help of reuse predictors to improve performance. However, architectural differences between accelerators and processor cores (often associated with deep cache hierarchies) can lead to significantly different reuse patterns at the shared cache. We propose a novel clustering-based methodology, LERN, for learning and predicting the reuse behavior of hardware accelerators at the shared cache. We then propose a deadline and reuse-aware cache management strategy, HyDRA, which explores a novel tradeoff between reuse and deadline awareness for performance efficiency. It uses LERN to dynamically predict the reuse behavior of the accelerator accesses and make bypass decisions to maximize the system throughput while meeting accelerator deadlines. We evaluate HyDRA across different workloads and varied accelerator configurations. It significantly improves the system performance and reduces the accelerator deadline miss rate.

When minimizing the sum of a convex and a strongly convex function, or when finding the zero of the sum of a monotone operator and a strongly monotone operator, Chambolle and Pock (2010) and Davis and Yin (2015) proposed accelerated mechanisms that achieve an $\mathcal{O}(1/N^2)$ convergence rate for the squared distance to the solution, but the optimality of this rate was not established. In this work, we present Fast Douglas--Rachford Splitting (FDR), an accelerated method that improves the constants established in the prior works, and provide a complexity lower bound establishing that both the $\mathcal{O}(1/N^2)$ convergence rate and the leading-order constant of FDR's rate are optimal.

As data rates scale, intra-pair skew has become a critical bottleneck for high-speed differential signaling. Current analytical models are often limited, while 3D electromagnetic simulations are computationally intensive. This paper presents a comprehensive analytical framework for intra-pair skew in generic asymmetric coupled transmission lines, explicitly integrating skew into S-parameter formulations. We introduce the Intra-pair Skew Propagation Graph (ISPG), a novel graph-based methodology for calculating cumulative skew in complex, cascaded channels. The proposed framework is validated against both S-parameter simulations and empirical measurements of a 2m bulk twinax cable assembly, demonstrating excellent accuracy and robustness for high-speed interconnect design.

The triangular lattice antiferromagnet (TLAF) with nearest-neighbor exchange interaction is a model platform in the field of frustrated magnetism. Here, anharmonic (`up-up-down') and harmonic (`120 degree') magnetic states compete, because the fundamental helimagnetic wave and its higher harmonic are degenerate in energy. We show that a body-centered tetragonal lattice (BCTL) can realize a similar frustration of harmonic and anharmonic helimagnetic states, and that the tetragonal magnetic Weyl semimetal GdAlSi realizes this scenario. In an applied magnetic field, resonant elastic X-ray scattering reveals a competition of harmonic cycloidal and solitonic double-Q states, well consistent with mean-field calculations. Our work provides a new paradigm for frustration physics in BCTL materials.

Agentic reinforcement learning (RL) is reshaping LLM post-training, but end-to-end training time is dominated by compute-intensive, multi-turn rollouts whose resource demand varies significantly across training steps. Resource-fixed systems cannot adapt to this variation, while resource-elastic approaches that provision external GPUs on demand suffer from high allocation overhead and limited availability. We observe that serving clusters leave substantial GPU compute and memory idle, and propose cooperative elasticity: sharing already-deployed serving GPUs with rollout workloads to provide on-demand elastic capacity. Realizing this is non-trivial, as it must preserve serving SLOs under bursty traffic while minimizing cross-cluster communication overhead. We present ROSE, a system that realizes cooperative elasticity for agentic RL post-training, comprising three components: (1) an SLO-safe co-serving executor that co-locates heterogeneous serving and rollout models on the same GPUs, dynamically sharing memory and compute while preserving serving SLOs; (2) a cross-cluster weight transfer engine that leverages shard-aware routing and weight sparsity for fast synchronization; and (3) an elastic rollout scheduler that dynamically routes rollouts across dedicated and opportunistic serving GPUs. Experiments across multiple model sizes and cluster scales show that ROSE improves end-to-end throughput by 1.3 - 3.3 x over resource-fixed baselines and reduces rollout time by 1.2 - 1.5 x over resource-elastic baselines, with no serving SLO violations.

Advanced radiotherapy approaches such as FLASH irradiation and spatially fractionated radiotherapy (SFRT) show potential to improve the therapeutic ratio, yet their biological mechanisms and optimal delivery parameters remain uncertain. Progress requires accessible proton research platforms with flexible temporal and spatial dose delivery. We report on the adaptation of the Beam Transfer Line (BTL) of the Bern Medical Cyclotron (BMC) for radiobiology research with FLASH and proton minibeam capabilities. The BMC is optimized for the production of radionuclides for medical imaging, and is able to extract currents up to 150 $ \mathrm{μA}$. The 18 MeV proton beam was passively shaped using collimators, scattering foils, and extended drift space to generate irradiation fields. A dosimetric framework was implemented using an in-beam ionization chamber and radiochromic film with LET-dependent corrections. Beam uniformity and SFRT profiles with various grid spacings were evaluated at realistic target distances. The developed beamline enables stable delivery under controlled conditions in both conventional and FLASH regimes, spanning dose rates from 0.01 to 100 Gy/s. Dose uniformity within a 20 mm radius was below 8\%. Film measurements confirmed the need for LET-dependent corrections and indicated that quantitative dosimetry in in-vitro setups is achievable with appropriate LET corrections. The low proton energy (15.54(12) MeV extracted into air, 8.14(28) MeV delivered to cells in flask) facilitates compact SFRT implementation with well-resolved minibeams. The adapted BMC provides a flexible and accessible platform for systematic pre-clinical proton radiobiology studies under varied dose-rate and spatial delivery conditions. This supports optimization of emerging modalities such as proton FLASH and SFRT and helps bridge accelerator technology and radiobiology.

The Subset Sum Problem is a fundamental NP-complete problem in cryptography and combinatorial optimization, with many real-world applications. The Random Subset Sum Problem (RSSP) is a more applicable version of subset sum, where numbers are drawn from some i.i.d input distribution. We present an algorithm that, with probability $1-δ$, constructs the same $O(B/w)$ mesh as Da Cunha et al. (2023), while trimming to $w$ elements throughout and running in $O(w\log w)$ time. Then, we present a novel beam search heuristic running in linearithmic time w.r.t list size $n$ and beam width $w$ using the mesh that gives an expected error of $O\!\left(\frac{B}{nw^2}\right)$ under a standard mean-field assumption with equal standard deviation, demonstrating the practical effectiveness of meshing to achieve error decay. The algorithm is empirically robust to multiple input distributions and can naturally extend to variants with simple changes to the scoring heuristic, establishing a new practical baseline for robust subset sum error decay and $ε$-approximation theory.

Zorya is a concolic execution framework that lifts compiled binaries to Ghidra's P-Code intermediate representation and uses the Z3 SMT solver to detect vulnerabilities by reasoning over both concrete and symbolic values. Previous versions supported only single-threaded TinyGo binaries. In this paper, we extend Zorya to multi-threaded binaries produced by Go's standard gc compiler. This is achieved by restoring OS thread states from gdb dumps, neutralizing runtime preemption, and introducing overlay path analysis with copy-on-write semantics to detect silent vulnerabilities on untaken branches. We rigorously assess Zorya on 11 real-world vulnerabilities from production Go projects such as Kubernetes, Go-Ethereum, and CoreDNS. Our evaluation shows that Zorya detects seven bugs at the binary level, including a silent integer overflow detects no other evaluated tool finds without a manually written oracle.

Hydrogen storage remains a central bottleneck for scalable hydrogen energy systems due to the multiscale and coupled nature of the thermodynamics, kinetics, and microstructural evolution of hydrogen storage materials (HSMs). Although artificial intelligence (AI) has accelerated materials discovery, current approaches remain constrained by fragmented data, limited physical consistency, and weak integration with experimental validation. Here, we propose a unified framework that integrates coherent data infrastructure, physics-grounded modeling, and AI-driven inverse design within a closed-loop discovery paradigm. By embedding physical constraints and experimental feedback, this approach enables adaptive, physically consistent optimization, thereby establishing a pathway toward autonomous, digital-twin-enabled discovery of HSMs.

The quantum Wasserstein distances defined by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide for qubits when a single operator appears in the cost function. As a consequence, the self-distance equals the Wigner-Yanase skew information in this case.

We present a general method to determine the energy minimum of spin Hamiltonians over separable states when the single-particle reduced density matrices are fixed. For ferromagnetic Ising and Ising-like models with nearest-neighbor interactions on lattices of any dimension and on a fully connected graph in an external field, this minimum is given by a compact analytic formula involving the quantum Fisher information. For the ferromagnetic Heisenberg chain of spin-1/2 particles, the minimum is expressed via the Uhlmann-Jozsa fidelity. These relations enable the direct extraction of both the quantum Fisher information and the fidelity from correlation measurements on the ground states of suitably engineered spin models.

Relative Geologic Time (RGT) estimation from seismic data is a cornerstone of subsurface structural modeling, depositional evolution analysis, and reservoir characterization, supporting horizon correlation and depositional system reconstruction. Yet accurate RGT estimation remains challenging: RGT is intrinsically a topologically constrained continuous field, in which local errors readily propagate globally and distort the overall result. Conventional methods rely heavily on priors, attribute extraction, and manual interaction, leading to cumbersome workflows. Existing deep-learning approaches mostly use a regression formulation with pixel-wise MSE/MAE losses, which struggle to capture thin horizons and fail to model the stratigraphic semantics of the RGT field, yielding limited generalization and unstable ordering across diverse structural and depositional settings. We propose RGT-Est, a deep-learning framework that transfers the optimization target from the topologically constrained continuous field into a differentiable sinusoidal space, which explicitly encodes the periodic stratigraphic semantics of RGT and alleviates over-smoothing of fine horizons. Pointwise, perceptual, and adversarial losses are jointly imposed in this space to enforce local fidelity, inter-layer consistency, and global structural plausibility, providing both fine-horizon discrimination and global stratigraphic awareness. An optional horizon-guidance module further accepts sparse 2D or 3D horizons as priors. Trained on synthetic data and evaluated on field surveys with densely faulted zones, large unconformities, steeply dipping strata, folded deformations, and clinoforms, RGT-Est achieves state-of-the-art performance among AI-based methods without horizon constraints, and attains substantially higher horizon-correlation accuracy and global topological consistency once sparse priors are incorporated.

We apply the extended Nikiforov-Uvarov method to the non-relativistic limit of the Dirac equation with a Coulomb potential in spaces of constant curvature. In this case, the radial equation reduces to the Heun equation, and the extended Nikiforov-Uvarov method easily yields a quantization condition which leads to necessary condition under which the resulting Heun equation can have polynomial solutions. The energy spectrum implied by the quantization condition is virtually identical to the spectrum of a spinless particle obtained using the Schrödinger equation, except for the absence of the ``geometric potential", confirming the non-commutativity of the naive non-relativistic limit with the ``squaring" of the Dirac equation, first discovered on curved surfaces. However, the necessary conditions for the existence of polynomial solutions cannot be met, and this fact undermines the reliability of the results obtained. This circumstance forces us to conclude that the extended Nikiforov-Uvarov method has limited, if any, value when considering similar problems in quantum mechanics.