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14 pages, 6988 KiB  
Article
Effect of Substrate Temperature on the Structural, Morphological, and Infrared Optical Properties of KBr Thin Films
by Teng Xu, Qingyuan Cai, Weibo Duan, Kaixuan Wang, Bojie Jia, Haihan Luo and Dingquan Liu
Materials 2025, 18(15), 3644; https://doi.org/10.3390/ma18153644 (registering DOI) - 3 Aug 2025
Abstract
Potassium bromide (KBr) thin films were deposited by resistive thermal evaporation at substrate temperatures ranging from 50 °C to 250 °C to systematically elucidate the temperature-dependent evolution of their physical properties. Structural, morphological, and optical characteristics were examined by X-ray diffraction (XRD), scanning [...] Read more.
Potassium bromide (KBr) thin films were deposited by resistive thermal evaporation at substrate temperatures ranging from 50 °C to 250 °C to systematically elucidate the temperature-dependent evolution of their physical properties. Structural, morphological, and optical characteristics were examined by X-ray diffraction (XRD), scanning electron microscopy (SEM), atomic force microscopy (AFM), and Fourier transform infrared spectroscopy (FTIR). The results reveal a complex, non-monotonic response to temperature rather than a simple linear trend. As the substrate temperature increases, growth evolves from a mixed polycrystalline texture to a pronounced (200) preferred orientation. Morphological analysis shows that the film surface is smoothest at 150 °C, while the microstructure becomes densest at 200 °C. These structural variations directly modulate the optical constants: the refractive index attains its highest values in the 150–200 °C window, approaching that of bulk KBr. Cryogenic temperature (6 K) FTIR measurements further demonstrate that suppression of multi-phonon absorption markedly enhances the infrared transmittance of the films. Taken together, the data indicate that 150–200 °C constitutes an optimal process window for fabricating KBr films that combine superior crystallinity, low defect density, and high packing density. This study elucidates the temperature-driven structure–property coupling and offers valuable guidance for optimizing high-performance infrared and cryogenic optical components. Full article
(This article belongs to the Special Issue Obtaining and Characterization of New Materials (5th Edition))
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19 pages, 2359 KiB  
Article
Research on Concrete Crack Damage Assessment Method Based on Pseudo-Label Semi-Supervised Learning
by Ming Xie, Zhangdong Wang and Li’e Yin
Buildings 2025, 15(15), 2726; https://doi.org/10.3390/buildings15152726 (registering DOI) - 1 Aug 2025
Abstract
To address the inefficiency of traditional concrete crack detection methods and the heavy reliance of supervised learning on extensive labeled data, in this study, an intelligent assessment method of concrete damage based on pseudo-label semi-supervised learning and fractal geometry theory is proposed to [...] Read more.
To address the inefficiency of traditional concrete crack detection methods and the heavy reliance of supervised learning on extensive labeled data, in this study, an intelligent assessment method of concrete damage based on pseudo-label semi-supervised learning and fractal geometry theory is proposed to solve two core tasks: one is binary classification of pixel-level cracks, and the other is multi-category assessment of damage state based on crack morphology. Using three-channel RGB images as input, a dual-path collaborative training framework based on U-Net encoder–decoder architecture is constructed, and a binary segmentation mask of the same size is output to achieve the accurate segmentation of cracks at the pixel level. By constructing a dual-path collaborative training framework and employing a dynamic pseudo-label refinement mechanism, the model achieves an F1-score of 0.883 using only 50% labeled data—a mere 1.3% decrease compared to the fully supervised benchmark DeepCrack (F1 = 0.896)—while reducing manual annotation costs by over 60%. Furthermore, a quantitative correlation model between crack fractal characteristics and structural damage severity is established by combining a U-Net segmentation network with the differential box-counting algorithm. The experimental results demonstrate that under a cyclic loading of 147.6–221.4 kN, the fractal dimension monotonically increases from 1.073 (moderate damage) to 1.189 (failure), with 100% accuracy in damage state identification, closely aligning with the degradation trend of macroscopic mechanical properties. In complex crack scenarios, the model attains a recall rate (Re = 0.882), surpassing U-Net by 13.9%, with significantly enhanced edge reconstruction precision. Compared with the mainstream models, this method effectively alleviates the problem of data annotation dependence through a semi-supervised strategy while maintaining high accuracy. It provides an efficient structural health monitoring solution for engineering practice, which is of great value to promote the application of intelligent detection technology in infrastructure operation and maintenance. Full article
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15 pages, 314 KiB  
Article
Characterization of the Best Approximation and Establishment of the Best Proximity Point Theorems in Lorentz Spaces
by Dezhou Kong, Zhihao Xu, Yun Wang and Li Sun
Axioms 2025, 14(8), 600; https://doi.org/10.3390/axioms14080600 (registering DOI) - 1 Aug 2025
Abstract
Since the monotonicity of the best approximant is crucial to establish partial ordering methods, in this paper, we, respectively, characterize the best approximants in Banach function spaces and Lorentz spaces Γp,w, in which we especially focus on the monotonicity [...] Read more.
Since the monotonicity of the best approximant is crucial to establish partial ordering methods, in this paper, we, respectively, characterize the best approximants in Banach function spaces and Lorentz spaces Γp,w, in which we especially focus on the monotonicity characterizations. We first study monotonicity characterizations of the metric projection operator onto sublattices in general Banach function spaces by the property Hg. The sufficient and necessary conditions for monotonicity of the metric projection onto cones and sublattices are then, respectively, established in Γp,w. The Lorentz spaces Γp,w are also shown to be reflexive under the condition RBp, which is the basis for the existence of the best approximant. As applications, by establishing the partial ordering methods based on the obtained monotonicity characterizations, the solvability and approximation theorems for best proximity points are deduced without imposing any contractive and compact conditions in Γp,w. Our results extend and improve many previous results in the field of the approximation and partial ordering theory. Full article
(This article belongs to the Section Mathematical Analysis)
28 pages, 10224 KiB  
Article
A Vulnerability Identification Method for Distribution Networks Integrating Fuzzy Local Dimension and Topological Structure
by Kangzheng Huang, Weichuan Zhang, Yongsheng Xu, Chenkai Wu and Weibo Li
Processes 2025, 13(8), 2438; https://doi.org/10.3390/pr13082438 - 1 Aug 2025
Viewed by 121
Abstract
As the scale of shipboard power systems expands, their vulnerability becomes increasingly prominent. Identifying vulnerable points in ship power grids is essential for enhancing system stability, optimizing overall performance, and ensuring safe navigation. To address this issue, this paper proposes an algorithm based [...] Read more.
As the scale of shipboard power systems expands, their vulnerability becomes increasingly prominent. Identifying vulnerable points in ship power grids is essential for enhancing system stability, optimizing overall performance, and ensuring safe navigation. To address this issue, this paper proposes an algorithm based on fuzzy local dimension and topology (FLDT). The algorithm distinguishes contributions from nodes at different radii and within the same radius to a central node using fuzzy sets, and then derives the final importance value of each node by combining the local dimension and topology. Experimental results on nine datasets demonstrate that the FLDT algorithm outperforms degree centrality (DC), closeness centrality (CC), local dimension (LD), fuzzy local dimension (FLD), local link similarity (LLS), and mixed degree decomposition (MDD) algorithms in three metrics: network efficiency (NE), largest connected component (LCC), and monotonicity. Furthermore, in a ship power grid experiment, when 40% of the most important nodes were removed, FLDT caused a network efficiency drop of 99.78% and reduced the LCC to 2.17%, significantly outperforming traditional methods. Additional experiments under topological perturbations—including edge addition, removal, and rewiring—also show that FLDT maintains superior performance, highlighting its robustness to structural changes. This indicates that the FLDT algorithm is more effective in identifying and evaluating vulnerable points and distinguishing nodes with varying levels of importance. Full article
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19 pages, 1771 KiB  
Article
Steady Radial Diverging Flow of a Particle-Laden Fluid with Particle Migration
by C. Q. Ru
Fluids 2025, 10(8), 200; https://doi.org/10.3390/fluids10080200 - 1 Aug 2025
Viewed by 53
Abstract
The steady plane radial diverging flow of a viscous or inviscid particle-fluid suspension is studied using a novel two-fluid model. For the initial flow field with a uniform particle distribution, our results show that the relative velocity of particles with respect to the [...] Read more.
The steady plane radial diverging flow of a viscous or inviscid particle-fluid suspension is studied using a novel two-fluid model. For the initial flow field with a uniform particle distribution, our results show that the relative velocity of particles with respect to the fluid depends on their inlet velocity ratio at the entrance, the mass density ratio and the Stokes number of particles, and the particles heavier (or lighter) than the fluid will move faster (or slower) than the fluid when their inlet velocities are equal (then Stokes drag vanishes at the entrance). The relative motion of particles with respect to the fluid leads to particle migration and the non-uniform distribution of particles. An explicit expression is obtained for the steady particle distribution eventually attained due to particle migration. Our results demonstrated and confirmed that, for both light particles (gas bubbles) and heavy particles, depending on the particle-to-fluid mass density ratio, the volume fraction of particles attains its maximum or minimum value near the entrance of the radial flow and after then monotonically decreases or increases with the radial coordinate and converges to an asymptotic value determined by the particle-to-fluid inlet velocity ratio. Explicit solutions given here could help quantify the steady particle distribution in the decelerating radial flow of a particle-fluid suspension. Full article
(This article belongs to the Special Issue 10th Anniversary of Fluids—Recent Advances in Fluid Mechanics)
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14 pages, 5622 KiB  
Article
Molecular Dynamics Simulations on the Deformation Behaviors and Mechanical Properties of the γ/γ′ Superalloy with Different Phase Volume Fractions
by Xinmao Qin, Wanjun Yan, Yilong Liang and Fei Li
Crystals 2025, 15(8), 706; https://doi.org/10.3390/cryst15080706 (registering DOI) - 31 Jul 2025
Viewed by 121
Abstract
Based on molecular dynamics simulation, we conducted a comprehensive study on the tensile behaviors and properties of the γ(Ni)/γ(Ni3Al) superalloy with varying γ(Ni3Al) phase volume fractions (Vγ) under high-temperature, [...] Read more.
Based on molecular dynamics simulation, we conducted a comprehensive study on the tensile behaviors and properties of the γ(Ni)/γ(Ni3Al) superalloy with varying γ(Ni3Al) phase volume fractions (Vγ) under high-temperature, high-strain-rate service environments. Our investigation revealed that the tensile behavior of the superalloy depends critically on the Vγ. When the Vγ increased from 13.5 to 67%, the system’s tensile strength exhibited a non-monotonic response, peaking at Vγ = 40.3% before progressively decreasing. Conversely, the maximum uniform plastic strain decreased linearly and significantly when Vγ increased. These results establish an atomistically informed framework that elucidates the composition–microstructure–property relationships in γ(Ni)/γ(Ni3Al) superalloys, specifically addressing how Vγ governs variations in deformation mechanisms and mechanical performance. Furthermore, this work provides quantitative design paradigm for optimizing γ(Ni3Al) precipitate architecture and compositional tuning in the Ni-based γ(Ni)/γ(Ni3Al) superalloy. Full article
(This article belongs to the Special Issue Advances in High-Performance Alloys)
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34 pages, 530 KiB  
Article
Optimal Governance for Post-Concession Logistics Infrastructure: A Comparative Study of Self-Operation vs. Delegation Under Information Asymmetry
by Minghua Xiong
Sustainability 2025, 17(15), 6982; https://doi.org/10.3390/su17156982 (registering DOI) - 31 Jul 2025
Viewed by 111
Abstract
Public–private partnership (PPP) logistics infrastructure projects have become increasingly prevalent globally. Consequently, the effective management of these projects as their concession periods expire presents a crucial challenge for governments, vital for the sustainable management of PPP logistics infrastructure. This study addresses this challenge [...] Read more.
Public–private partnership (PPP) logistics infrastructure projects have become increasingly prevalent globally. Consequently, the effective management of these projects as their concession periods expire presents a crucial challenge for governments, vital for the sustainable management of PPP logistics infrastructure. This study addresses this challenge by focusing on the pivotal post-concession decision: whether the government should self-operate the mature logistics infrastructure or re-delegate its management to a private entity. Our theoretical model, built on a principal–agent framework, first establishes a social welfare baseline under government self-operation and then analyzes delegated operation under symmetric information, identifying efficiency frontiers. Under symmetric information, we find that government self-operation is more advantageous when its own operational efficiency is sufficiently high, irrespective of the private enterprise’s efficiency; conversely, delegating to an efficient private enterprise is optimal only when government operational efficiency is low. We also demonstrate that if the government can directly specify the demand quantity and service level and delegates operation via a fixed fee, the enterprise can be incentivized to align with the social optimum. However, under asymmetric information, potential welfare gains from delegation are inevitably offset by informational rent and output distortion. We further uncover non-monotonic impacts of parameters like the proportion of low-cost firms on social welfare loss and demonstrate how information asymmetry can indirectly compromise the long-term resilience of the infrastructure. Ultimately, our work asserts that delegation is only superior if its potential efficiency gains sufficiently offset the inherent losses stemming from information asymmetry. Full article
(This article belongs to the Section Sustainable Transportation)
28 pages, 2174 KiB  
Article
Validating Lava Tube Stability Through Finite Element Analysis of Real-Scene 3D Models
by Jiawang Wang, Zhizhong Kang, Chenming Ye, Haiting Yang and Xiaoman Qi
Electronics 2025, 14(15), 3062; https://doi.org/10.3390/electronics14153062 (registering DOI) - 31 Jul 2025
Viewed by 175
Abstract
The structural stability of lava tubes is a critical factor for their potential use in lunar base construction. Previous studies could not reflect the details of lava tube boundaries and perform accurate mechanical analysis. To this end, this study proposes a robust method [...] Read more.
The structural stability of lava tubes is a critical factor for their potential use in lunar base construction. Previous studies could not reflect the details of lava tube boundaries and perform accurate mechanical analysis. To this end, this study proposes a robust method to construct a high-precision, real-scene 3D model based on ground lava tube point cloud data. By employing finite element analysis, this study investigated the impact of real-world cross-sectional geometry, particularly the aspect ratio, on structural stability under surface pressure simulating meteorite impacts. A high-precision 3D reconstruction was achieved using UAV-mounted LiDAR and SLAM-based positioning systems, enabling accurate geometric capture of lava tube profiles. The original point cloud data were processed to extract cross-sections, which were then classified by their aspect ratios for analysis. Experimental results confirmed that the aspect ratio is a significant factor in determining stability. Crucially, unlike the monotonic trends often suggested by idealized models, analysis of real-world geometries revealed that the greatest deformation and structural vulnerability occur in sections with an aspect ratio between 0.5 and 0.6. For small lava tubes buried 3 m deep, the ground pressure they can withstand does not exceed 6 GPa. This process helps identify areas with weaker load-bearing capacity. The analysis demonstrated that a realistic 3D modeling approach provides a more accurate and reliable assessment of lava tube stability. This framework is vital for future evaluations of lunar lava tubes as safe habitats and highlights that complex, real-world geometry can lead to non-intuitive structural weaknesses not predicted by simplified models. Full article
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18 pages, 8520 KiB  
Article
Cross-Layer Controller Tasking Scheme Using Deep Graph Learning for Edge-Controlled Industrial Internet of Things (IIoT)
by Abdullah Mohammed Alharthi, Fahad S. Altuwaijri, Mohammed Alsaadi, Mourad Elloumi and Ali A. M. Al-Kubati
Future Internet 2025, 17(8), 344; https://doi.org/10.3390/fi17080344 - 30 Jul 2025
Viewed by 98
Abstract
Edge computing (EC) plays a critical role in advancing the next-generation Industrial Internet of Things (IIoT) by enhancing production, maintenance, and operational outcomes across heterogeneous network boundaries. This study builds upon EC intelligence and integrates graph-based learning to propose a Cross-Layer Controller Tasking [...] Read more.
Edge computing (EC) plays a critical role in advancing the next-generation Industrial Internet of Things (IIoT) by enhancing production, maintenance, and operational outcomes across heterogeneous network boundaries. This study builds upon EC intelligence and integrates graph-based learning to propose a Cross-Layer Controller Tasking Scheme (CLCTS). The scheme operates through two primary phases: task grouping assignment and cross-layer control. In the first phase, controller nodes executing similar tasks are grouped based on task timing to achieve monotonic and synchronized completions. The second phase governs controller re-tasking both within and across these groups. Graph structures connect the groups to facilitate concurrent tasking and completion. A learning model is trained on inverse outcomes from the first phase to mitigate task acceptance errors (TAEs), while the second phase focuses on task migration learning to reduce task prolongation. Edge nodes interlink the groups and synchronize tasking, migration, and re-tasking operations across IIoT layers within unified completion periods. Departing from simulation-based approaches, this study presents a fully implemented framework that combines learning-driven scheduling with coordinated cross-layer control. The proposed CLCTS achieves an 8.67% reduction in overhead, a 7.36% decrease in task processing time, and a 17.41% reduction in TAEs while enhancing the completion ratio by 13.19% under maximum edge node deployment. Full article
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23 pages, 3481 KiB  
Article
Research on Adaptive Identification Technology for Rolling Bearing Performance Degradation Based on Vibration–Temperature Fusion
by Zhenghui Li, Lixia Ying, Liwei Zhan, Shi Zhuo, Hui Li and Xiaofeng Bai
Sensors 2025, 25(15), 4707; https://doi.org/10.3390/s25154707 - 30 Jul 2025
Viewed by 245
Abstract
To address the issue of low accuracy in identifying the transition states of rolling bearing performance degradation when relying solely on vibration signals, this study proposed a vibration–temperature fusion-based adaptive method for bearing performance degradation assessments. First, a multidimensional time–frequency feature set was [...] Read more.
To address the issue of low accuracy in identifying the transition states of rolling bearing performance degradation when relying solely on vibration signals, this study proposed a vibration–temperature fusion-based adaptive method for bearing performance degradation assessments. First, a multidimensional time–frequency feature set was constructed by integrating vibration acceleration and temperature signals. Second, a novel composite sensitivity index (CSI) was introduced, incorporating the trend persistence, monotonicity, and signal complexity to perform preliminary feature screening. Mutual information clustering and regularized entropy weight optimization were then combined to reselect highly sensitive parameters from the initially screened features. Subsequently, an adaptive feature fusion method based on auto-associative kernel regression (AFF-AAKR) was introduced to compress the data in the spatial dimension while enhancing the degradation trend characterization capability of the health indicator (HI) through a temporal residual analysis. Furthermore, the entropy weight method was employed to quantify the information entropy differences between the vibration and temperature signals, enabling dynamic weight allocation to construct a comprehensive HI. Finally, a dual-criteria adaptive bottom-up merging algorithm (DC-ABUM) was proposed, which achieves bearing life-stage identification through error threshold constraints and the adaptive optimization of segmentation quantities. The experimental results demonstrated that the proposed method outperformed traditional vibration-based life-stage identification approaches. Full article
(This article belongs to the Special Issue Fault Diagnosis Based on Sensing and Control Systems)
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18 pages, 366 KiB  
Article
Nonparametric Transformation Models for Double-Censored Data with Crossed Survival Curves: A Bayesian Approach
by Ping Xu, Ruichen Ni, Shouzheng Chen, Zhihua Ma and Chong Zhong
Mathematics 2025, 13(15), 2461; https://doi.org/10.3390/math13152461 - 30 Jul 2025
Viewed by 133
Abstract
Double-censored data are frequently encountered in pharmacological and epidemiological studies, where the failure time can only be observed within a certain range and is otherwise either left- or right-censored. In this paper, we present a Bayesian approach for analyzing double-censored survival data with [...] Read more.
Double-censored data are frequently encountered in pharmacological and epidemiological studies, where the failure time can only be observed within a certain range and is otherwise either left- or right-censored. In this paper, we present a Bayesian approach for analyzing double-censored survival data with crossed survival curves. We introduce a novel pseudo-quantile I-splines prior to model monotone transformations under both random and fixed censoring schemes. Additionally, we incorporate categorical heteroscedasticity using the dependent Dirichlet process (DDP), enabling the estimation of crossed survival curves. Comprehensive simulations further validate the robustness and accuracy of the method, particularly under the fixed censoring scheme, where traditional approaches may NOT be applicable. In the randomized AIDS clinical trial, by incorporating the categorical heteroscedasticity, we obtain a new finding that the effect of baseline log RNA levels is significant. The proposed framework provides a flexible and reliable tool for survival analysis, offering an alternative to parametric and semiparametric models. Full article
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23 pages, 2253 KiB  
Article
Robust Underwater Vehicle Pose Estimation via Convex Optimization Using Range-Only Remote Sensing Data
by Sai Krishna Kanth Hari, Kaarthik Sundar, José Braga, João Teixeira, Swaroop Darbha and João Sousa
Remote Sens. 2025, 17(15), 2637; https://doi.org/10.3390/rs17152637 - 29 Jul 2025
Viewed by 183
Abstract
Accurate localization plays a critical role in enabling underwater vehicle autonomy. In this work, we develop a robust infrastructure-based localization framework that estimates the position and orientation of underwater vehicles using only range measurements from long baseline (LBL) acoustic beacons to multiple on-board [...] Read more.
Accurate localization plays a critical role in enabling underwater vehicle autonomy. In this work, we develop a robust infrastructure-based localization framework that estimates the position and orientation of underwater vehicles using only range measurements from long baseline (LBL) acoustic beacons to multiple on-board receivers. The proposed framework integrates three key components, each formulated as a convex optimization problem. First, we introduce a robust calibration function that unifies multiple sources of measurement error—such as range-dependent degradation, variable sound speed, and latency—by modeling them through a monotonic function. This function bounds the true distance and defines a convex feasible set for each receiver location. Next, we estimate the receiver positions as the center of this feasible region, using two notions of centrality: the Chebyshev center and the maximum volume inscribed ellipsoid (MVE), both formulated as convex programs. Finally, we recover the vehicle’s full 6-DOF pose by enforcing rigid-body constraints on the estimated receiver positions. To do this, we leverage the known geometric configuration of the receivers in the vehicle and solve the Orthogonal Procrustes Problem to compute the rotation matrix that best aligns the estimated and known configurations, thereby correcting the position estimates and determining the vehicle orientation. We evaluate the proposed method through both numerical simulations and field experiments. To further enhance robustness under real-world conditions, we model beacon-location uncertainty—due to mooring slack and water currents—as bounded spherical regions around nominal beacon positions. We then mitigate the uncertainty by integrating the modified range constraints into the MVE position estimation formulation, ensuring reliable localization even under infrastructure drift. Full article
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45 pages, 424 KiB  
Article
Human Capital, Household Prosperity, and Social Inequalities in Sub-Saharan Africa
by Boniface Ngah Epo, Francis Menjo Baye, Germano Mwabu, Damiano K. Manda, Olu Ajakaiye and Samuel Kipruto
Economies 2025, 13(8), 221; https://doi.org/10.3390/economies13080221 - 29 Jul 2025
Viewed by 92
Abstract
This article examines the relationship between human capital accumulation, household income, and shared prosperity using 2005–2018 household surveys in Cameroon, Ethiopia, Kenya, Nigeria, and Uganda. Human capital is found to be positively and significantly correlated with household wellbeing in all five nations. Health’s [...] Read more.
This article examines the relationship between human capital accumulation, household income, and shared prosperity using 2005–2018 household surveys in Cameroon, Ethiopia, Kenya, Nigeria, and Uganda. Human capital is found to be positively and significantly correlated with household wellbeing in all five nations. Health’s indirect benefits in Cameroon, Ethiopia, and Kenya augment its direct benefits. Education has monotonic welfare benefits from primary to tertiary levels in all countries. Human capital and labour market participation are strongly associated with household wellbeing. The equalization of human capital endowments increases income for the 40% of the least well-off groups in three of the sample countries. All countries except Uganda record a decrease in human capital deprivation over the period studied. Redistribution is associated with a reduction in human capital deprivation, although less systematically than in the growth scenario. These results suggest that sizeable reductions in human capital deprivation are more likely to be accomplished by interventions that focus on boosting general human capital outcomes than those that redistribute the human capital formation inputs. In countries with declining human capital deprivation, the within-sector interventions seem to account for this success. Substantial heterogeneity in human capital poverty exists within and across countries and between rural and urban areas. Full article
(This article belongs to the Special Issue Human Capital Development in Africa)
23 pages, 359 KiB  
Article
Hausdorff Outer Measures and the Representation of Coherent Upper Conditional Previsions by the Countably Additive Möbius Transform
by Serena Doria
Fractal Fract. 2025, 9(8), 496; https://doi.org/10.3390/fractalfract9080496 - 29 Jul 2025
Viewed by 172
Abstract
This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect [...] Read more.
This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel σ-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractals)
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11 pages, 284 KiB  
Article
Oscillation Theorems of Fourth-Order Differential Equations with a Variable Argument Using the Comparison Technique
by Osama Moaaz, Wedad Albalawi and Refah Alotaibi
Axioms 2025, 14(8), 587; https://doi.org/10.3390/axioms14080587 - 29 Jul 2025
Viewed by 108
Abstract
In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation (aϕuu)+q(uh)=0, which is of a functional type with a delay. The oscillation [...] Read more.
In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation (aϕuu)+q(uh)=0, which is of a functional type with a delay. The oscillation behavior of solutions of fourth-order delay equations has been studied using many techniques, but previous results did not take into account the existence of the function ϕ except in second-order studies. The existence of ϕ increases the difficulty of obtaining monotonic and asymptotic properties of the solutions and also increases the possibility of applying the results to a larger area of special cases. We present two criteria to ensure the oscillation of the solutions of the studied equation for two different cases of ϕ. Our approach is based on using the comparison principle with equations of the first or second order to benefit from recent developments in studying the oscillation of these orders. We also provide several examples and compare our results with previous ones to illustrate the novelty and effectiveness. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)
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