Search Results (941)

Background: Metabolic dysfunction-associated steatotic liver disease (MASLD) is the most common cause of liver pathology in children. Hepatic fat content can be assessed by a transient elastography, but this method is not widely used. Therefore, the aim of this study was to evaluate the usefulness of a simple, surrogate biochemical-anthropometric indices of liver steatosis: FLI (Fatty Liver Index), HSI (Hepatic Steatosis Index), and LAP (Lipid Accumulation Product) in children with MASLD. Methods: The retrospective study included a group of 103 obese/overweight children. MASLD was diagnosed according to criteria developed by a multisociety statement published in Journal of Pediatric Gastroenterology and Nutrition in 2024. Liver steatosis (controlled attenuation parameter /CAP/) was assessed in elastography. HSI, FLI, and LAP were calculated according to their mathematical formulas. ROC analysis and multivariate logistic regression were performed to calculate the power of the studied indices in diagnostics of liver steatosis in children with MASLD. Results: MASLD was diagnosed in 72 children (70%). Children with MASLD had significantly higher levels of ALT, AST, GGT, uric acid, value of BMI, waist circumference, HOMA-IR, and indices HSI, FLI, LAP compared to the group of non-hepatopathic controls (n = 31). HSI, FLI, and LAP significantly correlated with CAP. All indices identified MASLD in the ROC analysis (AUC = 0.77, 0.76, 0.68 resp.). However, only HSI remained a significant independent predictor of hepatic steatosis in overweight/obese children. Conclusions: Although HSI, FLI, and LAP can be considered as useful biochemical-anthropometric indicators of MASLD in obese/overweight children, only HSI remained an independent predictor of hepatic steatosis in this group of children. Full article

Sebastes schlegelii is a commercially significant species in the northern Yellow Sea. Recently, Sebastes schlegelii has shown a reduced stock density and a shift toward smaller body sizes in the northern Yellow Sea area off Liaoning Province. To comprehend the population dynamics and to formulate management tactics, we established a mathematical formula describing the length–weight relationship, as well as the corresponding growth equations for the body length and body weight. The biomass of Sebastes schlegelii in the northern Yellow Sea area off Liaoning Province was evaluated using biological survey data from 2019 to 2022. The findings reveal that the length–weight formula is expressed as W = 4.98 × 10⁻⁵L^2.8878 (R² = 0.9108), with L_∞ = 407.6 mm, W_∞ = 1717.04 g, K = 0.21 a⁻¹, and t₀ = −0.65 a. The critical age was 3.34 years, and the stock biomass was 13,141.35 tonnes. The current exploitation rate of this resource is below the maximum yield, indicating sustainable utilization. A fishing closure period from April to June is considered suitable. A minimum landing size of 240.48 mm is recommended for Sebastes schlegelii. Full article

Milk and dairy products are highly vulnerable to Economically Motivated Adulteration (EMA), particularly through the substitution of milk fat with cheaper non-milk fats. This paper presents a comprehensive review of analytical approaches used for detecting vegetable oils (e.g., palm, coconut, sunflower) and animal-origin fats such as pork lard and bovine tallow, as well as the fraudulent mixing of milk from different species. Methods for lipid extraction are examined, including traditional gravimetric procedures such as the Röse–Gottlieb method and high-efficiency alternatives such as Accelerated Solvent Extraction and supercritical fluid extraction. Analytical strategies for fraud detection are evaluated, demonstrating that while fatty acid profiling is widely applied, its sensitivity is limited by natural variability. Greater discriminatory power can often be achieved through triacylglycerol analysis combined with mathematical models such as the Precht formulae (which generate S-values), although performance varies depending on the adulterant matrix, adulteration level, reference population, and analytical protocol. Similarly, sterol profiling, particularly the detection of phytosterols like β-sitosterol, is a valuable marker for vegetable oil adulteration but does not provide an equivalent solution for detecting animal fat adulteration. The potential of rapid, non-destructive screening tools, including Fourier-transform infrared and Raman spectroscopy supported by chemometrics, is also assessed. A central analytical challenge in detecting such fraud lies in the complexity and variability of milk fat composition, which hinders any single analytical method from universally identifying all forms of non-milk fat adulteration; therefore, a tiered strategy combining rapid screening tools with high-resolution confirmatory methods is preferable. Future perspectives highlight the increasing importance of green analytical approaches, artificial intelligence, and portable detection systems for enhancing verification within the global dairy supply chain. However, the effectiveness of AI and chemometric methods depends heavily on the availability of representative training datasets and rigorous external validation to avoid overfitting and ensure reliable application across diverse samples. Full article

Non-additive measures and integrals have become important mathematical tools in modern research. They offer powerful frameworks for modelling interaction, uncertainty, and handling incomplete data where traditional additive techniques fail. This paper focuses on the Choquet integral and the corresponding non-additive measure derivatives, which can be considered mutually inverse problems. A comprehensive review of existing direct and indirect approaches is provided for analysis of the derivatives, aiming to refine and generalise current results while highlighting their inconsistencies and limitations. In the discrete case, Choquet integration reduces to a finite sum, and the necessary and sufficient conditions for the existence of measure derivatives form a linear system. In the continuous case, three existing frameworks are evaluated: studying the invariance under rearrangement of integrating functions for the integral; using generalised integral equations for Choquet computations to derive formulas for the result of integration as well as for measure derivatives; and the resulting measure approach, which uses the inverse nature of integration and measure derivatives and pairs integrating and resulting measures to derive explicit derivative formulas. Ultimately, this work maps out current perspectives on non-additive measure derivatives with respect to the Choquet integral, and outlines possible directions for their future theoretical and applied advancements. Full article

This paper investigates the degradation law and calculation method for the load-carrying capacity of steel–concrete composite girders under fire exposure based on numerical analysis and mathematical statistics. A finite element model of simply supported box-section steel–concrete composite girders is established using ABAQUS, which is validated against existing scaled test data in terms of temperature field distribution, load-carrying capacity, and mid-span displacement. On this basis, the parametric effects of concrete slab thickness, steel web height, steel plate thickness, and concrete strength on the load-carrying capacity of the girders are systematically analyzed. The results indicate that concrete slab thickness, steel web height, and steel plate thickness exert significant influences on the structural bearing capacity, whereas concrete strength has a negligible effect. Specifically, the load-carrying capacity under fire exposure is substantially improved with the increase in concrete slab thickness, steel web height, and steel plate thickness. Furthermore, a simplified calculation formula for the capacity of box-section steel–concrete composite girders under fire exposure is developed via multiple linear regression analysis, incorporating the three dominant influencing factors of concrete slab thickness, steel web height and steel plate thickness. The proposed formula exhibits satisfactory calculation accuracy and can provide a reliable reference for the fire resistance design and repair decision-making of steel–concrete composite girders. Full article

The derivatives of the inverse of a one-to-one function are needed in a range of applied contexts, from random variate generation and molecular simulation to nuclear physics and bias reduction for maximum likelihood estimates, yet existing treatments derive them by ad hoc, application-specific means without a unifying framework. Here, we give the general derivative of the inverse of a one-to-one function, firstly by a recurrence formula, secondly by repeated differentiation, and thirdly—and most explicitly—in closed form using the partial exponential Bell polynomials associated with Faà di Bruno’s chain rule, providing a single representation that subsumes and extends earlier case-specific results and that can be taken to arbitrary order. We illustrate the practical value of these results in mathematical statistics, applying them to bias reduction for maximum likelihood estimates in one-parameter exponential families, including the gamma shape parameter and canonical regression models. Python programs implementing the recurrence and the Bell polynomial representations are included. Full article

Water vapor (H₂O) acts as both an essential thermodynamic driver and a primary source of chemical radicals in the middle atmosphere, playing an irreplaceable role in maintaining Earth’s radiative balance and indicating long-term climate variability. In this study, 24 years (2002–2025) of H₂O measurements from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument on board the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite are systematically analyzed to characterize the H₂O spatiotemporal distribution throughout the middle atmosphere (specifically within the 20–80 km altitude range), with a focus on elucidating its evolutionary patterns across time, altitude, and latitude. Building upon this analysis, an empirical model for the bimonthly mean water vapor volume mixing ratio (VMR) is constructed based on actual measurements. Employing a nonlinear least-squares fitting algorithm, time-series fitting is performed on the data within distinct altitude and latitude grids. Consequently, a mathematical analytical expression for the time series was derived for each latitudinal band at every altitude grid point, alongside the determination of corresponding fitting parameter sets. By integrating these parameterized formulas and derived parameters, a comprehensive empirical H₂O VMR model spanning multiple altitude layers and a broad latitudinal range was ultimately established. Validation results demonstrate that the empirical model exhibits high consistency with the original observational data. The coefficients of determination ($R^2$) generally exceed 0.7 and strictly remain $\ge$ 0.6 in all cases. Furthermore, the model demonstrates strong linear correlation with actual observations (Pearson correlation coefficients typically exceeding 0.8) and maintains low bias, as evidenced by small root mean square errors (mostly < 0.35 ppmv) and mean absolute errors (mostly < 0.25 ppmv) across diverse spatial grids. These evaluation metrics collectively indicate excellent goodness-of-fit and robust reconstruction capabilities. This model provides a reliable empirical reference for investigating the spatiotemporal evolution of middle atmospheric H₂O VMR and serves as a potential data foundation for future optimizations of relevant radiative transfer models. Full article

Supervised fine-tuning is often discussed in terms of data volume and target quality, but the format of the supervision itself can change the reasoning strategy a model expresses in its outputs. We study this effect on a six-category reasoning benchmark spanning symbol transformation, text decryption, bit manipulation, gravitational constant estimation, numeral conversion, and unit conversion (drawn from the NVIDIA Nemotron Model Reasoning Challenge). Using NVIDIA Nemotron-3-Nano-30B-A3B with matched LoRA training settings, we compare three symbol-supervision formats: verbose English rule descriptions, compact family tags, and compact formula notation. We hypothesize that supervision renderings bias token-level reasoning priors, and that these priors transfer across task boundaries in multi-task SFT. In the canonical strict-rescore inventory, the best compact tag and formula checkpoints are statistically equivalent in aggregate within a pre-specified ±4-point margin: K8A-800 reaches 72.3% strict-scored overall accuracy and K8B-700 reaches 71.2% (TOST p = 0.003). Compact tags nevertheless provide a cleaner behavioral profile: an earlier K8A-400 checkpoint reaches 66.4% overall, 98.7% gravity accuracy, and 36.9% bit accuracy without the same contamination signatures. In contrast, verbose English rule descriptions are associated with heuristic parroting, with up to 57% of symbol failures at audited verbose checkpoints collapsing to a single remove-operator template, while formula notation is associated with cross-category contamination: numeric-looking predictions appear more often in text decryption (higher at five of six matched training steps under the canonical seed; matched-step means 15.8 vs. 11.7 numeric predictions per 157 text rows), and gravity failures at a representative K8B formula checkpoint shift toward shortcut stubs and explicit g = 9.8/9.81 fallbacks. We further show that checkpoint selection and strict evaluation auditing materially change branch decisions. Across three training seeds, neither compact format shows a consistent aggregate advantage, while the contamination signatures are partly seed-specific: the gravity-shortcut severity difference persists but is not exclusive to the formula branch, and the numeric–text signature does not reproduce under reseeding. These results support treating supervision format as a first-class hyperparameter for multi-task reasoning SFT in large language models—at least in this benchmark-and-model setting—rather than a mere rendering detail. Because such symbolic and procedural reasoning tasks recur in domains including cybersecurity, mathematics, and code generation, the same formatting choices plausibly shape the policy that any later reinforcement-learning stage would inherit, which we flag as future work. Full article

Background: Large language models (LLMs) are increasingly accessible to healthcare providers and patients for clinical decision support, yet their ability to perform precise mathematical calculations required for validated risk scores remains unexplored, and errors could compromise patient safety. The EuroSCORE II requires complex multivariable computation that could reveal fundamental limitations in LLM computational capabilities. Methods: We evaluated four publicly available chatbots (ChatGPT (GPT-4o, OpenAI), Claude (Sonnet 4.5, Anthropic), Gemini (2.5 Flash, Google) and Deepseek (V3)) in calculating EuroSCORE II for 105 patients from the CARDIO-FR registry with gold standard heart team calculations. Each model was tested using two approaches: direct calculation from clinical parameters alone and formula-based calculation with the explicit EuroSCORE II algorithm provided. Performance was assessed through mean absolute error (MAE), correlation coefficients, and clinical agreement within ±2% of gold standard values. Results: Direct LLM calculations demonstrated poor accuracy (MAE range: 3.3–6.4%) with the best performer (Gemini) achieving only 50.5% clinical agreement. Formula provision improved performance in three of four models, with ChatGPT formula achieving the lowest MAE (2.9%) and highest clinical agreement (51.4%), followed by Claude formula (MAE 3.2%, agreement 48.6%). Adding the formula to the prompt significantly improved performance and reduced bias. All methods exhibited significant systematic biases (p < 0.05 for 7/8 strategies). Conclusions: Publicly available LLM chatbots cannot reliably calculate EuroSCORE II for clinical use. Adding the formula to the prompt significantly improved performance, but was not sufficient to reach the clinically required threshold of 90% agreement. Clinicians should rely on validated risk calculation tools rather than LLM chatbots for quantitative clinical risk assessments. Full article

Intelligent transportation and aerial remote sensing scenes suffer from complex scene variations, abundant miniature targets and unpredictable out-of-distribution obstacles, which brings tough mathematical challenges to open-world detection tasks. Conventional detection algorithms lack rigorous frequency-domain separation and spatial constraint mathematical formulations, resulting in severe tiny-object feature attenuation, inefficient multimodal feature matching and catastrophic forgetting during incremental category iteration. To solve these mathematical bottlenecks, this paper constructs the TPCA-Net model built upon frequency decomposition and spatial dependency mathematical modelling. The entire framework consists of four fixed core modules: High-Frequency-Aware Multi-Scale Feature Enhancement (HSE), Reparameterized Adaptive Text–Visual Alignment (RTA), Double Wildcard Spatial Dependency Fusion (WSF), and Incremental Forgetting-Free Dual-Path Detection (DPD). From the mathematical perspective, the HSE module adopts discrete cosine transform-based filtering equations to split high-frequency object details from low-frequency background signals and establishes cross-attention spatial constraint formulas to make up for missing contextual information of small targets. The RTA module introduces low-rank decomposition mathematical optimization and reparameterized tensor fusion rules to realize domain-adaptive text embedding calibration and zero-cost cross-modal mapping at the inference stage. The WSF module constructs dual-wildcard self-supervised mathematical loss to finish unsupervised unknown-object identification and builds decoupled semantic–spatial fusion equations to improve the positioning precision of novel targets. The DPD module designs two sets of independent optimization objective functions and category-freezing incremental mathematical constraints to avoid conflicting parameter updates and eliminate forgetting defects in new-class expansion. Validated on COCO, DOTA and AI-TOD datasets, TPCA-Net achieves 56.0% AP on COCO, 79.30% mAP on DOTA, and 40.5% overall AP with 28.7% small-object AP on AI-TOD while delivering an inference throughput of 101.2 FPS on the Tesla T4 edge GPU. The proposed method outperforms existing mainstream open-world detection algorithms in tiny-object and rare-category recognition while maintaining efficient inference speed. Full article

Asymmetric interaction matrices encode the linear coupling structure and directed interaction patterns that arise in mathematical models of complex networks across ecology, finance, and machine learning, yet their geometric structure as points on a statistical manifold has received comparatively little systematic treatment. This paper develops a rigorous information-geometric framework for the manifold ${\mathcal{M}}_n^{+}$ of real $n\times n$ interaction matrices whose symmetric part is negative definite—equivalently, the matrices satisfying the numerical stability condition $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)<0$. The symmetric part $S\left(A\right)=(A+A^{\mathsf{T}})/2$ and the skew-symmetric part $K\left(A\right)=(A-A^{\mathsf{T}})/2$ correspond, respectively, to the metric structure and the torsion of the induced statistical manifold. We construct the natural augmented Riemannian metric g on ${\mathcal{M}}_n^{+}$ as the sum of the Fisher–Rao pullback metric through $-S(·)$ and a Frobenius term on $K(·)$, derive explicit formulae for the sectional curvature in the mixed symmetric–skew plane, and prove that the sectional curvature vanishes if and only if A is normal. The central theoretical result is a curvature-mediated stability theorem: a Fisher–Rao stability margin, derived from the precision representative of A, provides a sharp, computationally accessible certificate for the asymptotic stability of the linear dynamical system $\dot{\mathbf{x}}=A\mathbf{x}$, with the instability boundary lying at infinite Fisher–Rao distance. We further establish an information-geometric reformulation of May’s stability criterion for random ecological networks, a curvature-based covariance regularisation scheme for financial correlation matrices, and a Jacobian stability bound for deep neural networks. All the main results are illustrated with explicit $3\times 3$ and $4\times 4$ numerical examples. Full article

This study proposes a mathematical framework for predicting complete response-time distributions associated with parametric video-game configurations. Each configuration is encoded as a point in a normalized metric space, and the statistical descriptors of its response-time distribution (median, mean, selected quantiles, interquartile range, and Skewness) are treated as real-valued Lipschitz functions on that space. Predictions for unseen configurations are obtained through McShane–Whitney extension formulas, which provide geometrically controlled upper and lower bounds compatible with the empirical Lipschitz regularity of the observed data. To handle the sequential incorporation of new observations, a regularization mechanism is introduced that replaces raw descriptors violating Lipschitz continuity constraints with a convex combination of the observed value and a weighted historical estimate. In the extended version of the method, the coefficient of this combination is selected through a one-pass online Q-learning-inspired procedure that selects, for each descriptor and instability regime, a data-dependent trade-off between fidelity and geometric regularity. The final output is a continuous Log-Normal density fitted by nonlinear least squares to the predicted descriptors, together with a Wasserstein-type uncertainty band derived from the Lipschitz bounds. The framework is validated on a controlled experiment with 24 participants across 20 game levels. Results show that the predicted distributions shift systematically with the input configuration and that, in the illustrative comparison, the adaptive mechanism produces feature-specific smoothing decisions that differ from those obtained with a fixed coefficient. The method also provides interpretable predictions from small experimental datasets without requiring fully data-driven models. Full article

Pascal’s triangle serves as a fundamental source for constructing a diverse array of unique structural matrices in mathematical literature. This study presents a novel combinatorial methodology for computing the multiplicative powers of “right-aligned binomial matrices”, where the ones from the right boundary of Pascal’s triangle are aligned to the rightmost column of the matrix. Such matrix structures have inherent connections to a broad range of theoretical applications, potentially extending to discrete-time dynamic systems, probability models, and network routing protocols. Currently, while the powers of these matrices are typically evaluated using numerical computation systems, these systems operate via “black-box” algorithms that obscure the underlying structural mechanics and generally preclude symbolic analysis. Furthermore, existing theoretical approaches in the literature rely heavily on complex, indirect recurrence relations. In this work, the computation of matrix powers is approached through lattice path enumeration, specifically utilizing weighted Delannoy paths and Fibonacci numbers. The main theorem yields closed-form symbolic expressions that directly evaluate any arbitrary entry of the matrix power, independent of the exponent’s magnitude and order of matrix, without requiring matrix multiplication. Furthermore, a computational implementation in the Wolfram Language is provided to demonstrate the algorithmic efficiency and practical validity of the proposed explicit formulas. Full article

This paper presents an evaluation of AI-assisted human–computer interaction for mathematical problem solving within a multimodal mathematical editing environment. The proposed architecture integrates a visual equation editor, voice-based interaction, REST communication services, and externally hosted generative AI large language models (LLMs) to support the creation, interpretation, and solution of mathematical expressions. The study was conducted using the Equation Wizard environment, which provides multimodal mathematical content editing based on both proprietary and standard formula representations, including MathML and LATEX. A conversational AI interaction layer enables users to communicate with selected LLMs using natural language voice commands. The main objective of the study was to determine whether contemporary LLM-based services can reliably support mathematical problem solving within an AI-enhanced equation editing environment. To address this objective, seven contemporary GenAI LLMs were evaluated using a benchmark consisting of representative high school mathematics problems covering algebra, limits, trigonometry, logarithms, inequalities, and parameterized expressions. The evaluation focused on mathematical correctness, output interpretability and visualization quality, response latency, and compliance with mathematical encoding formats within the complete interaction workflow. The study also compares representative model families with respect to correctness, syntactic compliance, and responsiveness within the complete interaction workflow. Unlike conventional LLM benchmarks that assess models in isolation, this work evaluates end-to-end AI-assisted mathematical interaction involving the editor, communication infrastructure, and language models. The study demonstrates a practical approach to assessing the usefulness of multimodal AI-enhanced mathematical problem solving in realistic usage scenarios. The results show that, within the evaluated interaction workflow, current LLMs achieve high levels of mathematical problem-solving performance, although substantial differences were observed in response latency and output-format compliance. An important finding is the discrepancy between the models’ strong semantic understanding of custom-encoded mathematical input and their weaker ability to generate syntactically compliant encoded outputs, highlighting a key challenge in the integration of LLMs with structured mathematical software environments. Full article

Height systems form a mathematical interface between physical geodesy, engineering surveying, and digital cartography. Although satellite positioning efficiently provides ellipsoidal heights, practical infrastructure, mapping, hydrological, and monitoring tasks require gravity-related heights that are compatible with national vertical datums. This paper develops a denominator-based framework in which dynamic, orthometric, and normal heights are interpreted as metric realizations of a common geopotential number. Starting from the line-integral definition of geopotential, the principal height formulae are derived; first-order sensitivities to geoid undulation, height anomaly, and mean gravity are established; and uncertainty propagation is analyzed. A Romania-oriented computational experiment, explicitly defined as a representative model-behavior study rather than an official national adjustment, uses lowland, plateau, and mountain-influenced settings consistent with the Constanta and Black Sea 1975 normal-height context. The results show that modeled normal-orthometric separations remain below 3 mm in representative low-relief locations but increase to 17.1 mm in Suceava, 29.5 mm in Cluj-Napoca, and 85.6 mm in the mountain-influenced Brasov case. The dynamic-normal differences remain small at low elevations but become systematic where the normal-gravity denominator departs from the selected reference value. The Monte Carlo experiment indicates standard uncertainties of approximately 4.2–4.4 cm for normal heights when a 1.5 cm ellipsoidal-height uncertainty and a 4.0 cm quasi-geoid uncertainty are assumed. A single-point covariance example gives 31.7 mm for normal-height conversion and 36.9 mm for orthometric-height conversion under the stated correlation assumptions. The transformation-surface comparison further shows that a quadratic local model reduces leave-one-out cross-validation error from 16.73 mm to 9.95 mm relative to a planar model in the synthetic Romania-oriented scenario. The study concludes that the height-system label must be treated as part of the mathematical model and metadata, and it proposes a geopotential-centered computational pathway for survey adjustment, uncertainty control, and metadata-safe geospatial export. Full article