This paper builds upon a new class of Finsler space, generalized fifth recurrent Finsler space. For Cartan’s fourth curvature tensor in the sense of Berwald, we denote such space by Finsler space. Finsler geometry is a kind of differential geometry. It is an extension of Riemannian geometry; Riemann studied the distance between points in n-dimensions using only positional coordinates, while Finsler generalized Riemann's idea and studied the distance between points in n-dimensions by using two coordinates: positional and directional. The key idea revolves around a mathematical object called the curvature inheritance, which is defined in terms of the Lie-derivative, where curviture is a scalar function. In this work, we introduce new relationships for certain curvature inheritance, with the curvature tensor's components related to the components of a vector field during the application of a force at an arbitrary point by the Lie-derivative, and which it inherits a new tensor at a flow line of via transformation mapping. We prove that the Lie-derivative of K-curvature inheritance and the Berwald covariant derivative of fifth order are commutative under certain conditions. Also we prove that the scalar function of the Cartan’s fourth curvature tensor, the Berwald’s curvature tensor, the H-Ricci tensor and the K-Ricci tensor all have values of zero under the same certain conditions. Further, an equality relationship is established between Cartan’s fourth curvature tensor and Berwald’s curvature tensor if their Berwald covariant derivatives of fifth order are equal. Finally, we derived a mathematical formula for the change occurring in the vector field of the fifth order during its flow through a smooth vector field under the influence of an applied force and we proved that the direction of this force is opposite to the direction of the vector field of the fifth order itself within this space.

Introduction: Valleys are a common topographical feature of coastal cities, and some coastal cities are located within the Circum-Pacific and Eurasian earthquake belts. In addition, anti-plane surface motion can damage engineering structures and threaten human safety during an earthquake. A novel and significant framework is therefore proposed for computing anti-plane surface motion of valleys induced by earthquakes in order to enhance the resilience of coastal cities against earthquakes.

Methods: 1. A valley under earthquake excitation is simplified as an arc-shaped mathematical model for incident SH waves. 2. The elastic wave motion equation is generated without factors of time. 3. Based on the variable-separation method, wave functions satisfying the equation are derived. 4. Wave functions meeting boundary conditions can be obtained using zero-stress boundary conditions of the mathematical model. 5. Complex displacement of surface motion is obtained through wave functions' linear calculations. 6. Amplitudes and phases of surface motion are determined using complex-number calculative formulae.

Results and Discussion: As a result, anti-plane surface motion of valleys can be computed by the newly presented framework. In the next study, angles ranging from 30 to 60 degrees and frequencies ranging from 0 to 10 HZ will be evaluated. Amplitudes and phases of surface motion are given as functions of various angles and frequencies of incident SH waves. It is found that amplitudes and phases of surface motion of valleys usually vary with angle and frequency.

Conclusions: For oblique incidence, displacement amplitudes on a surface facing waves are often higher than those on a shadow surface. In addition, the number of oscillations of amplitudes on a surface facing waves is often greater than that on a shadow surface. However, regular patterns of phases are nearly consistent. These findings suggest that important buildings, such as cable cars, bridges and railways, can be built better on shadow surfaces of valleys in coastal cities.

This paper introduces a novel deterministic framework for the analysis of multiplication tables through three fundamental operations: tabular sum, tabular difference, and tabular product. While multiplication tables are traditionally regarded as elementary pedagogical tools, we demonstrate that they encode deep arithmetic structures when studied systematically. In particular, the tabular product leads naturally to a new approach to primality testing based on factorial and primorial constructions.

We formalize the tabular product as a structured product of table rows and show that its interaction with the greatest common divisor yields a reliable criterion for distinguishing prime and composite integers. To address computational limitations inherent to large factorials, a logarithmic optimization is introduced, relying on asymptotic approximations and adaptive depth control. This refinement allows the method to scale efficiently to very large integers without loss of determinism.

Furthermore, a primorial-based variant of the test is developed, eliminating redundant composite factors and significantly strengthening the detection of Carmichael numbers, which commonly evade classical probabilistic tests. A decimal regulation function is proposed to dynamically adjust the test depth as a function of the size of the integer under consideration.

Theoretical results are supported by explicit numerical examples, illustrating robustness, scalability, and algorithmic relevance. This work opens new perspectives for primality testing, cryptographic applications, and the foundational reinterpretation of elementary arithmetic structures.

In this work, we study a fractional two-strain epidemic SIIR model with a general incidence rate for both strains, incorporating treatment for each strain. The population is divided into four compartments representing susceptible individuals, individuals infected with the first strain, individuals infected with the second strain, and recovered individuals. The proposed model is formulated using a system of four fractional-order differential equations in the sense of Caputo, which allows us to capture memory effects and hereditary properties inherent in the transmission dynamics of infectious diseases.

The mathematical analysis begins with the proof of the existence and uniqueness of positive solutions, ensuring the well-posedness and biological feasibility of the model. Positivity and boundedness of solutions are also established. Several equilibrium points are derived, including the disease-free equilibrium and endemic equilibria associated with each strain. The global stability of these equilibria is investigated by constructing suitable Lyapunov functions and applying fractional-order stability theory.

Numerical simulations are performed to validate the theoretical findings and to illustrate the influence of the fractional-order parameter on the qualitative behavior of the solutions and the convergence speed toward equilibrium states. In addition, a sensitivity analysis is conducted to examine the role of treatment efficiency in reducing the prevalence of infected individuals for both strains. The results highlight the importance of effective treatment strategies and fractional dynamics in controlling multi-strain epidemic outbreaks.

Identifying scale-invariant geometric features in high-dimensional data remains a central challenge in data science. While the Mapper algorithm offers a powerful topological lens, its utility is often hindered by a critical dependence on manual parameter tuning and a lack of reproducibility. Conversely, persistent homology provides robust multiscale insights but lacks a direct, automated integration into structural discovery.

​To bridge these gaps, we propose a fully automated topological framework that unifies Mapper with persistent homology to achieve principled, noise-resilient feature extraction. The core innovation is our Most Persistent Betti-1 (MPB-1) algorithm, which systematically extracts dominant one-dimensional homological features from persistence diagrams to compute a characteristic scale (ε). This scale guides the parameterization of the Mapper algorithm through a closed-form relationship, eliminating traditional trial-and-error.

​We validate our framework on synthetic datasets, where it accurately recovers theoretical Betti-1 invariants, and on real-world applications including single-cell genomics and protein dynamics. Our results reveal robust topological signatures consistent with underlying biological structures. To our knowledge, this is the first automated Mapper framework that guarantees fidelity to persistent topological invariants. By embedding Betti-1 computation at its core, our method provides a reproducible foundation for detecting cyclic structures central to medicine and proteomics, thereby broadening the reach of topological data analysis across scientific domains.

The Random Forest algorithm, originally proposed by Leo Breiman, is a cornerstone of ensemble learning, relying on the principle of Bagging (Bootstrap Aggregating) to reduce the variance of decision tree classifiers. In high-dimensional geospatial datasets, such as those derived from multispectral satellite imagery, the traditional Random Forest Algorithm (RFA) often struggles with feature correlation and the curse of dimensionality. Our research proposes a transformative shift in ensemble learning by replacing the stochastic nature of traditional Bootstrap Aggregating (Bagging) with deterministic chaotic dynamics to enhance predictive stability and accuracy in complex spatial domains. Traditional Random Forest (RF) often suffers from sampling bias and sub-optimal convergence when dealing with high-dimensional geospatial data, such as Sentinel satellite imagery, where spatial autocorrelation and feature redundancy are prevalent. Our study first focuses on the mathematical formalization of chaotic maps, such as the Logistic and Tent maps, to ensure a more uniform coverage of the feature space. Secondly, the framework provides a rigorous mathematical demonstration of chaotic feature subspace selection and hyperparameter optimization via a chaotic search mechanism, effectively preventing the algorithm from getting trapped in local optima, a common failure mode in high-dimensional geospatial analysis. Finally, our study establishes the mathematical proofs for convergence and ergodicity, demonstrating that the CRFA maintains a superior bias–variance tradeoff compared to standard ensemble methods. Experimental results utilizing Sentinel-2 satellite datasets indicate that the CRFA significantly enhances classification accuracy and computational robustness. By integrating non-linear dynamics into the ensemble architecture, the proposed CRFA achieves a 4.5% increase in the Kappa coefficient and a reduction in training variance, providing a more reliable tool for complex land-cover mapping and environmental monitoring.

Diabetic Retinopathy (DR) is a progressive microvascular complication of diabetes and remains one of the leading causes of preventable blindness worldwide. Early and reliable detection is essential for reducing vision loss. Conventional image-based methods primarily rely on intensity and texture features. These features often fail to capture the complex geometric organization and subtle structural alterations of retinal vasculature seen in disease progression. In this study, we propose a mathematically driven framework that combines Topological Data Analysis (TDA) with nonlinear dynamics to improve DR classification. Persistent homology was applied to the dataset to extract multiscale topological descriptors, such as connected components and loops, by filtration. The results from persistent homology were encoded as persistence diagrams and then converted into quantitative feature vectors for input to a machine learning model. In parallel, we computed the fractal dimension to quantify vascular complexity and spatial irregularities, reflecting pathological changes. The combined topological and fractal features were used to train a Support Vector Machine (SVM) classifier to grade DR severity. The proposed hybrid approach achieved an overall classification accuracy of 94%. It showed strong discriminative performance across disease stages and improved sensitivity to early structural abnormalities while maintaining high specificity. By capturing intrinsic geometric and dynamical characteristics of retinal images, this integrated TDA–nonlinear dynamics framework offers a robust, interpretable, and clinically relevant methodology for automated diabetic retinopathy screening.

In this note, we consider a data-driven approach to software reliability prediction based on kernel regression, where the software fault-count process during system testing is modeled directly from observed fault data without imposing a strict parametric distributional assumption. This perspective is particularly useful in practical software testing environments, where the underlying fault-generation mechanism is often complex, time-varying, and difficult to characterize accurately by a single predefined stochastic model. To address this issue, a non-parametric prediction framework is developed by employing kernel regression with several bandwidth selection methods, with the aim of investigating how bandwidth choice influences prediction accuracy, estimation stability, and overall model robustness. Since the bandwidth plays a central role in controlling the bias-variance tradeoff in kernel-based estimation, inappropriate bandwidth selection may lead to over-smoothing or under-smoothing, thereby degrading predictive performance in long-term software fault prediction. In the proposed framework, multiple bandwidth selection strategies are examined and compared under the same prediction setting, and their predictive behaviors are analyzed from both estimation and forecasting perspectives. Through comparative analysis, the proposed approach provides useful insights into the role of bandwidth selection in software fault prediction and offers practical guidance for software reliability evaluation when distributional knowledge is incomplete or uncertain. The results also suggest that careful bandwidth selection is essential for improving the applicability of kernel-based reliability prediction methods in real-world software testing data.

This paper investigates the approximation properties of both classical and fractional neural network (NN) operators and their symmetrized counterparts (SNN), which are activated by an activation function, half-hyperbolic tangent, acting on Banach space-valued functions, f :X → R, where X is a Banach space with norm ||·||. This work is motivated by the need to better understand the mathematical behavior of neural network (NN)-type operators within the framework of approximation theory and functional analysis. Within this setting, we establish pointwise and uniform convergence results together with quantitative fractional approximation estimates expressed in terms of suitable smoothness measures. A central contribution of the paper is a rigorous comparison between classical neural network (NN) operators and their symmetrized counterparts (SNN). The analysis shows that the symmetry structure leads to improved approximation properties and enhanced stability under appropriate parameter regimes. One of the most innovative aspects of this paper is the evaluation of approximation operators using regression-based machine learning metrics, including RMSE, MAE, Maximum error, and R². These metrics provide a statistical interpretation of approximation quality, complementing classical norm-based error bounds. This approach is important because it connects approximation theory with data science methodology and makes operator performance interpretable for machine learning applications. To complement the theoretical analysis, we conducted systematic numerical experiments using the approximation of the proposed operators to the test functions, a log–log graph, and an error-decay graph implemented in Python 3.13, including graphical illustrations and quantitative comparisons of NN and SNN operators employing machine learning metrics. The computational results support the theoretical findings and demonstrate that SNN operators can achieve improved approximation accuracy and more stable behavior in practice. To promote transparency and reproducibility, all implementation details and Python codes, algorithms were made publicly available via GitHub. By combining rigorous analysis with computational experiments, this work contributes to strengthening the connection between approximation theory, neural network models, and numerical computation, providing both theoretical insights and practical tools for researchers working at the interface of mathematics and machine learning.

This study examines the macro-financial determinants of corporate insolvency dynamics in Spain over the period 2008–2024 using a region–sector panel of annual data. The analysis focuses on how systemic financial conditions, macroeconomic fundamentals, and territorial heterogeneity jointly shape regional patterns of business failure. Financial indicators are condensed through principal component analysis into two latent factors capturing complementary dimensions of risk: a domestic financial stress component associated with equity and banking volatility, and a global uncertainty component linked to exchange-rate movements and external shocks. Panel estimations reveal that domestic financial pressure constitutes the main transmission channel of corporate fragility across regions, while global risk operates through asymmetric and non-linear mechanisms. Macroeconomic conditions, particularly unemployment dynamics, reinforce these effects, whereas inflation shows limited direct influence on insolvency outcomes.

To assess robustness and predictive relevance, the econometric framework is complemented by machine learning models including CatBoost, XGBoost, and LightGBM. The ensemble approach achieves high and stable predictive performance, with AUC values above 0.90, confirming that macro-financial variables contain persistent information about regional failure risk. Clustering analysis further identifies distinct behavioural profiles across regions and sectors, highlighting heterogeneous exposure to systemic shocks. Overall, the results demonstrate that corporate insolvency in Spain is primarily driven by the interaction between financial-market volatility and regional economic structures. The proposed framework contributes to the development of macro-financial early-warning systems and provides new evidence on how systemic risk propagates spatially within national economies.