Search Results (700)

Multi-composite activated neural network operators can be understood as positive linear operators, allowing them to be analyzed using standard, established theory. Formed by composing multiple general activation functions, these operators act upon continuous real-valued functions defined on a compact interval. This work presents a quantitative analysis of how quickly these operators converge to the unit operator. Utilizing general inequalities based on the modulus of continuity—applicable to either the function itself or its derivative—this study establishes both uniform and Lp approximation results. Furthermore, the analysis incorporates the convexity of functions to produce related, specific results. Full article

This paper develops a unified fractional version of the Hermite–Hadamard inequality and Bullen-type inequalities for convex functions defined on discrete time scales. By employing generalized fractional difference operators, the obtained result encompasses and extends previously known discrete formulations, including both the classical case and higher-order variants. Furthermore, we investigate the approximation accuracy of the introduced fractional mean operator. Specifically, we establish explicit error bounds for Lipschitz functions and for functions with convex differences, providing a more comprehensive analysis of the discrete fractional setting. Full article

Topography regulates vegetation functioning by controlling water redistribution, microclimate, and solar exposure. In Mediterranean ecosystems, where water availability constitutes a fundamental limiting factor, vegetation functioning is also influenced by environmental drivers such as temperature, climatic seasonality, drought recurrence, and soil properties that interact with terrain heterogeneity. Understanding how these elements operate at the micro-scale is essential for interpreting the spatial variability of photosynthetic vigor and canopy water condition. This study evaluates the relationships between the topographic metrics Topographic Position Index (TPI), Terrain Ruggedness Index (TRI), and Diurnal Anisotropic Heat Index (DAH) and two spectral proxies of vegetation condition, the Normalized Difference Vegetation Index (NDVI) and the Normalized Difference Moisture Index (NDMI), in Los Nogales Nature Sanctuary (central Chile). Multitemporal Sentinel-2 time series (2017–2025) were analyzed using Generalized Additive Models (GAMs) with Gaussian distribution and cubic splines to detect non-linear topographic responses. All topographic predictors were statistically significant (p < 0.001). NDVI and NDMI values were higher in concave and less rugged areas, decreasing toward convex and thermally exposed slopes. NDMI exhibited greater sensitivity to topographic position and thermal anisotropy, indicating the strong dependence of vegetation water condition on topographically driven water redistribution. These results highlight the role of terrain in modulating vegetation vigor and moisture in Mediterranean ecosystems. Full article

The equilibrium shape of a crystal is a fundamental problem in materials science and condensed matter physics. The Wulff construction, a cornerstone of crystal morphology prediction, is traditionally presented and utilized as a powerful geometric algorithm to derive equilibrium shapes from anisotropic surface energy $\gamma \left(\mathbf{n}\right)$. While its application across materials science is vast, the profound mathematical physics underpinning it, specifically its intrinsic identity as a manifestation of the Legendre transform, is often relegated to a passing remark. This work recenters the focus on this fundamental duality. We present a comprehensive, step-by-step derivation of the Wulff shape from the variational principle of surface energy minimization under a constant volume, employing the language of support functions and differential geometry. We then rigorously demonstrate that the equilibrium shape, defined by the support function $h\left(\mathbf{n}\right)$, and the surface energy density $\gamma \left(\mathbf{n}\right)$ are conjugate variables linked by a Legendre transformation; the Wulff shape $\mathcal{W}$ is precisely the zero-sublevel set of the dual function ${\gamma }^{*}\left(\mathbf{x}\right)={sup}_{\mathbf{n}}[\mathbf{x}·\mathbf{n}-\gamma \left(\mathbf{n}\right)]$. This perspective elevates the Wulff construction from a mere graphical tool to a canonical example of convex duality in thermodynamic systems, connecting it to deeper principles in convex analysis and statistical mechanics. To bridge theory and computation, we provide a robust computational algorithm implemented in pseudocode capable of generating Wulff shapes for two-dimensional (2D) crystals with arbitrary N-fold symmetry. Finally, we discuss the relevance and extensions of the classical theory in contemporary research, including non-equilibrium growth, nanoscale effects, and the coupling of crystal shapes with elastic membrane environments. Full article

Seismic monitoring is a crucial step in ensuring the safety and resilience of building structures. The implementation of effective monitoring systems, particularly across large-scale, complex building clusters, is currently hindered by the limitations of traditional sensor placement methods, which suffer from low efficiency, high subjectivity, and difficulties in replication. This paper proposes an innovative AI-based Automated Layout Method for seismic monitoring devices, leveraging building geometric recognition to provide a scalable, quantifiable, and reproducible engineering solution. The core methodology achieves full automation and quantification by innovatively employing a dual-channel approach (images and vectors) to parse architectural floor plans. It first converts complex geometric features—including corner coordinates, effective angles, and concavity/convexity attributes—into quantifiable deployment scoring and density functions. The method implements a multi-objective balanced control system by introducing advanced engineering metrics such as key floor assurance, central area weighting, spatial dispersion, vertical continuity, and torsional restraint. This approach ensures the final sensor configuration is scientifically rigorous and highly representative of the structure’s critical dynamic responses. Validation on both simple and complex Reinforced Concrete (RC) frame structures consistently demonstrates that the system successfully achieves a rational sensor allocation under budget constraints. The placement strategy is physically informed, concentrating sensors at critical floors (base, top, and mid-level) and strategically utilizing external corner points to maximize the capture of torsional and shear responses. Compared with traditional methods, the proposed approach has distinct advantages in automation, quantification, and adaptability to complex geometries. It generates a reproducible installation manifest (including coordinates, sensor types, and angle classification) that directly meets engineering implementation needs. This work provides a new, efficient technical pathway for establishing a systematic and sustainable seismic risk monitoring platform. Full article

The K-means clustering algorithm is widely applied in various clustering tasks due to its high computational efficiency and simple implementation. However, its performance significantly deteriorates when dealing with non-convex structures, fuzzy boundaries, or noisy data, as it relies on the assumption that clusters are spherical or linearly separable. To address these limitations, this paper proposes a Gaussian membership-driven fuzzy granular K-means clustering method. In this approach, multi-function Gaussian membership functions are used for fuzzy granulation at the single-feature level to generate fuzzy granules, while fuzzy granule vectors are constructed in the multi-feature space. A novel distance metric for fuzzy granules is defined along with operational rules, for which axiomatic proof is provided. This Gaussian-based granulation enables effective modeling of nonlinear separability in complex data structures, leading to the development of a new fuzzy granular K-means clustering framework. Experimental results on multiple public UCI datasets demonstrate that the proposed method significantly outperforms traditional K-means and other baseline methods in clustering tasks involving complex geometric data (e.g., circular and spiral structures), showing improved robustness and adaptability. This offers an effective solution for clustering data with intricate distributions. Full article

In this paper, we investigate new geometric properties of normalized analytic functions associated with the generalized Marcum Q-function. In particular, we focus on two analytic forms derived from a normalized derivative of a representation involving the Marcum Q-function, and its Alexander transform. For these functions, we establish sufficient conditions ensuring membership in the classes of lemniscate starlike and lemniscate convex functions. Special attention is given to the case $\nu =1$, where explicit admissible parameter ranges for b are derived. We further examine inclusion relations between these normalized analytic forms and lemniscate subclasses, complemented by several corollaries, illustrative examples, and graphical visualizations. These results extend and enrich the geometric function theory of special functions related to the generalized Marcum Q-function. Full article

This research explores two novel geometric concepts—nearly convex points and locally nearly uniformly convex points within the frameworks of Banach spaces and Orlicz spaces equipped with the Luxemburg norm. First, we establish the general characterization criteria for nearly convex points in Banach spaces. Then, we analyze the intrinsic connection between locally nearly uniformly convex points and nearly extreme points in Banach spaces. Additionally, we provide comprehensive characterizations of locally nearly uniformly convex points in both Orlicz function spaces and Orlicz sequence spaces under the Luxemburg norm. These findings enrich the geometric theory system of Banach and Orlicz spaces, offering new theoretical support for related research directions. Full article

Traditional decision optimization methods primarily focus on model construction and solution, leaving parameter estimation and inter-variable relationships to statistical research. The traditional approach divides problem-solving into two independent stages: predict first and then optimize. This decoupling leads to the propagation of prediction errors-even minor inaccuracies in predictions can be amplified into significant decision biases during the optimization phase. To tackle this issue, scholars have proposed end-to-end decision optimization methods, which integrate the prediction and decision-making stages into a unified framework. By doing so, these approaches effectively mitigate error propagation and enhance overall decision performance. From an architectural design perspective, this review focuses on categorizing end-to-end decision optimization methods based on how the prediction and decision modules are integrated. It classifies mainstream approaches into three typical paradigms: constructing closed-loop loss functions, building differentiable optimization layers, and parameterizing the representation of optimization problems. It also examines their implementation pathways leveraging deep learning technologies. The strengths and limitations of these paradigms essentially stem from the inherent trade-offs in their architectural designs. Through a systematic analysis of existing research, this paper identifies key challenges in three core areas: data, variable relationships, and gradient propagation. Among these, handling non-convexity and complex constraints is critical for model generalization, while quantifying decision-dependent endogenous uncertainty remains an indispensable challenge for practical deployment. Full article

Integrated Sensing and Communication (ISAC) has emerged as a cornerstone technology for next-generation wireless networks, where accurate performance evaluation is essential. In such systems, the capacity–distortion function provides a fundamental measure of the trade-off between communication and sensing performance, making its computation a problem of significant interest. However, the associated optimization problem is often constrained by non-convexity, which poses considerable challenges for deriving effective solutions. In this paper, we propose extended Arimoto–Blahut (AB) algorithms to solve the non-convex optimization problem associated with the capacity–distortion trade-off in bistatic ISAC systems. Specifically, we introduce auxiliary variables to transform non-convex distortion constraints in the optimization problem into linear constraints, prove that the reformulated linearly constrained optimization problem maintains the same optimal solution as the original problem, and develop extended AB algorithms for both squared error distortion and logarithmic loss distortion. The numerical results validate the effectiveness of the proposed algorithms. Full article

To address the deficiencies in global search capability and population diversity decline of the black-winged kite algorithm (BKA), this paper proposes an enhanced black-winged kite algorithm integrating opposition-based learning and quasi-Newton strategy (OQBKA). The algorithm introduces a mirror imaging strategy based on convex lens imaging (MOBL) during the migration phase to enhance the population’s spatial distribution and assist individuals in escaping local optima. In later iterations, it incorporates the quasi-Newton method to enhance local optimization precision and convergence performance. Ablation studies on the CEC2017 benchmark set confirm the strong complementarity between the two integrated strategies, with OQBKA achieving an average ranking of 1.34 across all 29 test functions. Comparative experiments on the CEC2022 benchmark suite further verify its superior exploration–exploitation balance and optimization accuracy: under 10- and 20-dimensional settings, OQBKA attains the best average rankings of 2.5 and 2.17 across all 12 test functions, outperforming ten state-of-the-art metaheuristic algorithms. Moreover, evaluations on three constrained engineering design problems, including step-cone pulley optimization, corrugated bulkhead design, and reactor network design, demonstrate the practicality and robustness of the proposed approach in generating feasible solutions under complex constraints. Full article

Understanding the spatial structure of large mammals is critical for conservation planning, especially under increasing habitat fragmentation. This study applies an integrated spatial analysis combining the DBSCAN density-based clustering algorithm and the α-hull method to delineate non-convex geographic ranges of the maned wolf (Chrysocyon brachyurus) across South America. Using 454 occurrence records filtered for ecological reliability, we identified 11 geographically isolated α-populations distributed across five countries and multiple biomes, including the Cerrado, Chaco, and Atlantic Forest. The sensitivity analysis of the α parameter demonstrated that values below 2 failed to generate viable polygons, while α = 2 provided the best balance between geometric detail and ecological plausibility. Our results reveal a highly fragmented distribution, with α-populations varying in area from 43,077 km² to 566,154.7 km² and separated by distances up to 994.755 km. Smaller and peripheral α-populations are likely more vulnerable to stochastic processes, genetic drift, and inbreeding, while larger clusters remain functionally isolated due to anthropogenic barriers. We propose the concept of ‘α-population’ as an operational unit to describe geographically and functionally isolated groups identified through combined spatial clustering and non-convex hull analysis. This approach offers a reproducible and biologically meaningful framework for refining range estimates, identifying conservation units, and guiding targeted management actions. Overall, integrating α-hulls with density-based clustering improves our understanding of the species’ fragmented spatial structure and supports evidence-based conservation strategies aimed at maintaining habitat connectivity and long-term viability of C. brachyurus populations. Full article

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. Full article

This paper considers how a system designer can program a team of autonomous agents to coordinate with one another such that each agent selects (or covers) an individual resource with the goal that all agents collectively cover the maximum number of resources. Specifically, we study how agents can formulate strategies without information about other agents’ actions so that system-level performance remains robust in the presence of communication failures. First, we use an algorithmic approach to study the scenario in which all agents lose the ability to communicate with one another, have a symmetric set of resources to choose from, and select actions independently according to a probability distribution over the resources. We show that the distribution that maximizes the expected system-level objective under this approach can be computed by solving a convex optimization problem, and we introduce a novel polynomial-time heuristic based on subset selection. Further, both of the methods are guaranteed to be within $1-1/e$ of the system’s optimal in expectation. Second, we use a learning-based approach to study how a system designer can employ neural networks to approximate optimal agent strategies in the presence of communication failures. The neural network, trained on system-level optimal outcomes obtained through brute-force enumeration, generates utility functions that enable agents to make decisions in a distributed manner. Empirical results indicate the neural network often outperforms greedy and randomized baseline algorithms. Collectively, these findings provide a broad study of optimal agent behavior and its impact on system-level performance when the information available to agents is extremely limited. Full article

The aim of this article is to study duality results for nonsmooth mathematical programs with equilibrium constraints in terms of tangential subdifferentials. We study the Wolfe-type dual problem under the convexity assumptions and a Mond–Weir-type dual problem is also formulated under convexity and generalized convexity assumptions for MPEC by using tangential subdifferentials. We establish weak duality and the two dual programs by assuming tangentially convex functions and also obtain strong duality theorems by assuming generalized standard Abadie constraint qualification. Full article