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Search Results (294)

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Keywords = qualitative properties of solutions

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32 pages, 31400 KB  
Article
Machine Learning-Based Compressive Strength Prediction, Sensitive Analysis, and Microstructural Mechanism Study of Carbonated Recycled Aggregate Concrete
by Jie Zhong, Sen Yang, Benjie Lei, Zhixi Chen, Yi Sun, Changming Bu, Mingtao Zhang, Yang Yu and Jiehong Li
Buildings 2026, 16(13), 2602; https://doi.org/10.3390/buildings16132602 - 29 Jun 2026
Viewed by 256
Abstract
Carbonation treatment can effectively address defects in recycled aggregates (RA) while achieving CO2 sequestration, thereby improving properties of recycled aggregate concrete (RAC). However, the compressive strength of carbonated recycled aggregate concrete (CRAC) is governed by complex interactions among multiple parameters, and existing [...] Read more.
Carbonation treatment can effectively address defects in recycled aggregates (RA) while achieving CO2 sequestration, thereby improving properties of recycled aggregate concrete (RAC). However, the compressive strength of carbonated recycled aggregate concrete (CRAC) is governed by complex interactions among multiple parameters, and existing machine learning (ML) studies often rely on heterogeneous literature data with limited parameter coverage, resulting in constrained predictive accuracy. To address this issue, this study established a robust ML framework for precise strength prediction. By integrating published literature with original experimental results, a dataset of 226 groups was constructed, incorporating 12 key parameters across RA properties, carbonation processes, mix proportions, and concrete age to systematically compare three ML models (GPR, SVM, EDT). To enhance model transparency, global sensitivity analysis used the SHapley Additive exPlanations (SHAP) method, while X-ray diffraction (XRD), scanning electron microscopy (SEM), and microhardness tests were employed to reveal reinforcement mechanisms at the phase, microstructural, and micromechanical levels, supporting the connection between intelligent prediction and mechanistic explanation. Results show that the GPR model exhibited the highest predictive performance and generalization capability (R2 = 0.98 for training, R2 = 0.94 for testing; RMSE = 1.08 MPa), outperforming comparative models in handling high-dimensional nonlinear relationships. SHAP analysis identified concrete age, water–cement (W/C) ratio, and the initial crush index of the RA as the primary factors, while carbonation process parameters, particularly relative humidity, carbonation pressure, and carbonation time, exerted significant regulatory effects on strength. XRD results qualitatively confirmed the formation of CaCO3 after carbonation, while SEM and microhardness analyses indicated that carbonation products contributed to pore filling and interfacial transition zone (ITZ) strengthening, providing a physical basis for both macroscopic performance improvement and model reliability. This study provides a scientific, data-driven solution for the mix design optimization and performance prediction of CRAC, delivering substantial environmental and economic benefits. Full article
(This article belongs to the Special Issue Innovations in Sustainable Concrete Construction)
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34 pages, 1491 KB  
Article
Fractional Stochastic Modeling of Nonlinear Dynamical Systems: Application to an Electromechanical Process with Memory Effects
by Anwarud Din
Fractal Fract. 2026, 10(7), 440; https://doi.org/10.3390/fractalfract10070440 - 27 Jun 2026
Viewed by 200
Abstract
In this study, a comprehensive stochastic and fractional-order modeling framework is developed to investigate the dynamic behavior of a shunt DC motor under random disturbances and memory effects. The motor dynamics are formulated as a system of stochastic differential equations incorporating Gaussian noise [...] Read more.
In this study, a comprehensive stochastic and fractional-order modeling framework is developed to investigate the dynamic behavior of a shunt DC motor under random disturbances and memory effects. The motor dynamics are formulated as a system of stochastic differential equations incorporating Gaussian noise to represent uncertainties in the electrical and mechanical subsystems. The existence, stochastic ultimate boundedness, stationary distribution, and ergodic properties of the proposed model are established. To further enhance modeling capabilities, a modified Atangana–Baleanu–Caputo (mABC) fractional operator is introduced, enabling the incorporation of nonlocal memory effects inherent in electromechanical systems. The series solution is derived using the Laplace transform and the Adomian decomposition method to handle nonlinearities. Qualitative analysis of the solution is performed through fixed-point theory, while stability assessments utilize the T-Picard method. The results of the numerical simulation indicate that the stochastic model exhibits limited variability around the operating regimes, whereas the fractional-order representation is more effective at smoothing transient responses and limiting oscillatory behavior. The study proposes a realistic and adaptable method to analyze the dynamics of shunt DC motors with uncertainty and also presents useful information for the design and control of electromechanical systems. Full article
(This article belongs to the Section Life Science, Biophysics)
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53 pages, 1508 KB  
Review
Biosorption of Heavy Metal in Wastewater with Biochar: A Review
by Nko Okina Solomon, Donghee Kang and Gbekeloluwa B. Oguntimein
Sustainability 2026, 18(12), 6367; https://doi.org/10.3390/su18126367 - 22 Jun 2026
Viewed by 491
Abstract
Biochar, a carbon-rich material produced through pyrolysis of biomass under limited oxygen conditions, offers a potentially sustainable and cost-competitive solution (qualitative assessment; quantitative LCA and techno-economic data are beyond the scope of this review) for the removal of heavy metals from wastewater. Its [...] Read more.
Biochar, a carbon-rich material produced through pyrolysis of biomass under limited oxygen conditions, offers a potentially sustainable and cost-competitive solution (qualitative assessment; quantitative LCA and techno-economic data are beyond the scope of this review) for the removal of heavy metals from wastewater. Its high porosity, surface area, and surface functional groups enable diverse adsorption mechanisms, including complexation, ion exchange, and precipitation. Feedstock selection and production parameters critically influence biochar’s physicochemical properties and adsorption performance. Modification techniques such as chemical functionalization, metal impregnation, and composite formation enhance removal efficiency and selectivity for specific contaminants. Applications span industrial, municipal, and agricultural wastewaters, addressing multi-contaminant challenges under variable environmental conditions. Factors affecting removal efficiency include pH, temperature, contaminant concentration, and competing ions, while regeneration methods are essential for maintaining long-term functionality and are discussed. Biochar can be reused and regenerated using bases and acids, but environmental risks related to biochar use, including potential contaminant leaching and ecological impacts, require careful management and regulatory compliance. Future research should focus on novel modification strategies, scaling production for industrial use, and optimizing integration within treatment systems to meet stringent discharge standards and promote sustainable water management. Full article
(This article belongs to the Special Issue Advanced Studies in Environmental Technology and Wastewater Treatment)
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19 pages, 7412 KB  
Article
Influence of Mix Composition on the Microstructural Evolution of Leached Cement Pastes
by Kailai Zhang, Wenwei Li, Huamei Yang, Dan Tian, Jinyang Cui, Hao Wang and Fan Li
Materials 2026, 19(12), 2664; https://doi.org/10.3390/ma19122664 - 21 Jun 2026
Viewed by 249
Abstract
Calcium leaching increases the hydraulic concrete material’s porosity and the diffusion coefficient, thereby jeopardizing engineering safety. Fly ash and silica fume are commonly used mineral admixtures in hydraulic concrete, and their effects on the material’s leaching characteristics, especially its microstructural and transport properties, [...] Read more.
Calcium leaching increases the hydraulic concrete material’s porosity and the diffusion coefficient, thereby jeopardizing engineering safety. Fly ash and silica fume are commonly used mineral admixtures in hydraulic concrete, and their effects on the material’s leaching characteristics, especially its microstructural and transport properties, require further investigation. In this study, calcium leaching tests were conducted on cement paste (CP), silica fume–cement paste (SF), and fly ash–cement paste (FA) using a 6 mol/L ammonium chloride solution to accelerate the leaching process. Subsequently, a series of quantitative and qualitative analyses was performed on the deteriorated specimens, including phenolphthalein indicator spraying, X-ray diffraction (XRD), nuclear magnetic resonance (NMR), and scanning electron microscopy (SEM). Additionally, the diffusion coefficients of the material at different locations were calculated and analyzed. The results show that partially replacing cement with silica fume or fly ash increases the initial porosity, gel pore content, and initial diffusion coefficients. After 28 days of leaching, compared to the initial values, the porosity increases in the 0–4 mm layer from the leached surface were 83.6% for CP, 11.0% for SF, and 39.0% for FA. The diffusion coefficients increased by factors of 14.3 (CP), 6.1 (SF), and 13.6 (FA), indicating enhanced resistance to leaching. The primary reason for this is that the reactive silica in the admixtures undergoes a pozzolanic reaction with the calcium hydroxide generated by cement hydration, producing additional calcium silicate hydrate (C-S-H) gel, which reduces the capillary pores that would otherwise result from calcium hydroxide decomposition. Full article
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21 pages, 3207 KB  
Article
Exploring Qualitative Analysis and Interaction Dynamics in a (3+1)-Dimensional Boussinesq Equation II via Hirota Bilinear Method
by Ali Danladi, Aljethi Reem Abdullah, Ejaz Hussain and Beenish
Mathematics 2026, 14(11), 1981; https://doi.org/10.3390/math14111981 - 3 Jun 2026
Viewed by 232
Abstract
In this work, we explore the nonlinear wave phenomena of the (3+1)-dimensional Boussinesq (II) equation, a significantly higher-dimensional model that describes dispersive wave propagation in fluid dynamics, plasma systems, and nonlinear optics. Using exact analytic and qualitative dynamic approaches, we study a wide [...] Read more.
In this work, we explore the nonlinear wave phenomena of the (3+1)-dimensional Boussinesq (II) equation, a significantly higher-dimensional model that describes dispersive wave propagation in fluid dynamics, plasma systems, and nonlinear optics. Using exact analytic and qualitative dynamic approaches, we study a wide range of solutions and stability characteristics of the model. Initially, we use the Hirota bilinear method to obtain a number of exact solutions, such as breather waves, two-wave interaction solutions, and other types of localized nonlinear waves. These solutions display remarkable physical properties, including periodic energy trapping, oscillatory modulations, and nonlinear wave interactions in higher dimensions. In addition, the (m+1G)-expansion method is used to derive new soliton solutions, such as bright solitary waves and W-shaped solitons, which are found to be stable and undergo pulse-shaping dynamics under certain conditions. Three-dimensional, two-dimensional, and contour plots are displayed for some of the solutions to demonstrate the physical significance of the results. The visualizations reveal the presence of localized waves, wave interactions, periodical breathing, and stable soliton profiles. Furthermore, we conduct modulation instability analysis to describe the conditions under which small perturbations of continuous wave backgrounds are unstable. The dispersion relation and the instability gain spectrum are obtained, which explain the formation of breathers, soliton trains, and other coherent structures. Furthermore, a Galilean transformation converts the governing equation into a planar nonlinear dynamical system, enabling its qualitative study. The Hamiltonian structure is revealed, and the fixed points are identified as centers, saddles, and cusps through bifurcation analysis. To investigate more complex dynamics, a periodic forcing term is introduced into the system, resulting in chaos in the forced system. The chaotic behavior is confirmed via phase portraits, three-dimensional attractors, time series, Poincaré sections, return maps, fractal dimension, and positive Lyapunov exponents. We also perform a sensitivity test to show the effect of initial condition variations on the system’s long-term dynamics. The findings greatly expand the exact solution set and dynamics of the (3+1)-dimensional Boussinesq equation (II). The analytical approach presented in this paper can also be applied to other multidimensional nonlinear evolution equations of mathematical physics. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Applications)
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19 pages, 3282 KB  
Article
Exploring Bifurcation Analysis, Conservation Laws and Soliton Dynamics for the Dual-Mode Nonlinear Schrödinger Equation with Applications
by Muhammad Arshad, Naila Nasreen, Evren Hincal, Mohamed Hafez and Muhammad Farman
Math. Comput. Appl. 2026, 31(3), 97; https://doi.org/10.3390/mca31030097 - 2 Jun 2026
Viewed by 296
Abstract
This study examines the dynamical behavior of the dual-mode nonlinear Schrödinger equation (d-mNLSE), which describes the interaction, amplification, and attenuation of two coexisting wave modes in nonlinear media. The model incorporates key physical parameters including the nonlinearity coefficient, interaction phase velocity, and dispersion [...] Read more.
This study examines the dynamical behavior of the dual-mode nonlinear Schrödinger equation (d-mNLSE), which describes the interaction, amplification, and attenuation of two coexisting wave modes in nonlinear media. The model incorporates key physical parameters including the nonlinearity coefficient, interaction phase velocity, and dispersion parameter, which significantly influence the evolution of nonlinear waves. By applying the modified Sardar sub-equation method (mSS-EM), a wide spectrum of exact analytical solutions is derived. These solutions include mixed trigonometric waves, shock-type structures, singular solutions, complex dark–bright solitons, multi-peak solitons, periodic and mixed-periodic waves, as well as mixed hyperbolic structures. The analytical findings provide useful insight into nonlinear wave propagation phenomena arising in fluid mechanics, water wave dynamics, ocean engineering, and related physical systems. Moreover, the conservation laws of the d-mNLSE are established, which leads to the conserved quantities of impulse power, momentum, and energy and describes the invariant characteristics of the soliton solutions during their propagation. The bifurcation analysis of the reduced dynamical model is carried out to explore the qualitative characteristics of the obtained solutions. The equilibrium points of the considered model are calculated, and their stability properties are analyzed systematically. To demonstrate the physical characteristics of the obtained solutions, different kinds of two-dimensional, three-dimensional, and contour plots are plotted using symbolic computations software. These findings confirm that the analytical method used to obtain the soliton solutions can be used to obtain a variety of soliton solutions of nonlinear evolution equations that appear in applied sciences and engineering. Full article
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32 pages, 416 KB  
Article
Averaging Effects and Their Applications to Fractional Elliptic and Parabolic Equations
by Wenxiong Chen and Yahong Guo
Fractal Fract. 2026, 10(6), 360; https://doi.org/10.3390/fractalfract10060360 - 26 May 2026
Viewed by 294
Abstract
The averaging effect is a distinctive property possessed by fractional operators. In recent years, it has emerged as a powerful tool in the study of qualitative properties of solutions to fractional elliptic and parabolic equations. In this article, we systematically summarize and prove [...] Read more.
The averaging effect is a distinctive property possessed by fractional operators. In recent years, it has emerged as a powerful tool in the study of qualitative properties of solutions to fractional elliptic and parabolic equations. In this article, we systematically summarize and prove various forms of the averaging effects for both fractional elliptic and parabolic equations, from the simplest one to the one under very relaxed conditions, including versions for antisymmetric functions. We then present examples to illustrate how to apply these effects to obtain radial symmetry and monotonicity for solutions in a unit ball and in a half space. In addition, we derive averaging effects for fractional Monge–Ampère operators and for fractional p-Laplacians, which will be potentially applied to obtain qualitative properties for solutions to equations involving these operators. Compared with the traditional approaches, methods based on the averaging effect require substantially weaker regularity assumptions and can even accommodate unbounded solutions. Full article
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34 pages, 3250 KB  
Review
Artificial Intelligence Methods for Unmanned Aerial Vehicles Cybersecurity: A Comprehensive Survey
by Thabet Kacem and Kensley Benjamin
Drones 2026, 10(6), 400; https://doi.org/10.3390/drones10060400 - 22 May 2026
Viewed by 501
Abstract
Unmanned aerial vehicles (UAVs) have been widely used in recent years in various applications thanks to advances in communication, Internet of Things, and electronics. Despite the advantages they offer, there have been reports of cybersecurity attacks, which represent serious threats to their operations. [...] Read more.
Unmanned aerial vehicles (UAVs) have been widely used in recent years in various applications thanks to advances in communication, Internet of Things, and electronics. Despite the advantages they offer, there have been reports of cybersecurity attacks, which represent serious threats to their operations. Classic cryptographic-based solutions and traditional intrusion detection approaches generally struggle to deal with these attacks due to their adaptive and evolving nature. In this context, artificial intelligence (AI) models emerged as potential solutions that hold great promise in addressing these types of attacks. However, most related surveys presented a fragmented picture of the state of the art, failing to cover all sub-types of AI models, and often did not follow structured taxonomies for describing the literature. In this paper, we bridge this gap by proposing a novel and comprehensive survey inspired by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) framework, defining the search strategy, inclusion and exclusion criteria, selection process, and classification. We also present a cross-dimensional taxonomy that classifies UAV security research according to the type of AI model, the cyber attacks it thwarts, and the related security properties it enforces. This taxonomy does not stop at describing machine learning (ML) and deep learning (DL) approaches but also examines federated learning (FL), reinforcement learning (RL), graph neural network (GNN), and generative AI (GAI). We also classify the threat vector according to the layer in the UAV functional stack where the attack takes place. In addition, we describe the datasets, tools, and evaluation metrics that were mostly used in the literature. Our survey analyzes the common uses of each AI model type in UAV security and discusses its strengths, limitations, and deployment readiness. The outcome of our taxonomy is a quantitative and qualitative analysis providing quantifiable metrics on the covered security properties per model type. We conclude the paper by discussing the key open challenges and future directions in the field. We intend for this survey to serve as a reference for cybersecurity researchers and practitioners who tackle UAV security using AI. Full article
(This article belongs to the Section Artificial Intelligence in Drones (AID))
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31 pages, 9062 KB  
Article
Periodicity of FEM Discrete Models and Its Influence on Solutions to the 1-D Wave Equation
by Wiktor Waszkowiak, Łukasz Doliński, Paweł Kowalski and Arkadiusz Żak
Appl. Sci. 2026, 16(10), 5150; https://doi.org/10.3390/app16105150 - 21 May 2026
Viewed by 343
Abstract
This paper discusses the influence of the periodicity of finite-element (FE) discrete models and its influence on solutions to the one-dimensional (1-D) wave equation. Numerical solutions to wave-propagation problems obtained via the displacement-based formulations of the finite-element method (FEM) often exhibit high-frequency behavior, [...] Read more.
This paper discusses the influence of the periodicity of finite-element (FE) discrete models and its influence on solutions to the one-dimensional (1-D) wave equation. Numerical solutions to wave-propagation problems obtained via the displacement-based formulations of the finite-element method (FEM) often exhibit high-frequency behavior, which is frequently dismissed in the literature as undesired, spurious, and/or having no physical meaning. In this paper, we verify this notion by demonstrating that this behavior is not merely a computational anomaly but is due to the inherent periodic properties of discrete numerical models. Using Bloch’s theorem, we reveal and demonstrate how, at high frequencies, the discrete nature of FEM numerical models leads to the prevailing behavior governed by the periodic nature of the computational models. In order to illustrate this phenomenon, we investigate 1-D wave propagation in rods, leveraging the non-dispersive nature of the governing equation as a benchmark. In addition to the classical and specialized FEM, we analyze two alternative formulations: the time-domain spectral finite-element method (TD-SFEM) and a novel spline-based finite-element method (spFEM) proposed by the authors. The results obtained and presented explain qualitatively the origins of these numerical anomalies and suggest strategies to mitigate their effects, effectively shifting the periodicity-induced behavior beyond the range of physically relevant frequencies by appropriate selection of approximation polynomials. The authors demonstrate that this can be fully achieved only in the case of spFEM, for which the usable percentage of the available spectra of eigenfrequencies reaches 67%, while in the case of other FEM approaches discussed is significantly smaller as determined by numerical dispersion and the presence of frequency band gaps. Full article
(This article belongs to the Section Mechanical Engineering)
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39 pages, 27209 KB  
Review
The Role of Additive Manufacturing in the Design of Smart and Nature-Based Construction Systems: A Critical Review
by Antreas Kantaros, Alexandra Tsatsou, Zoe Kanetaki, Theodore Ganetsos, Constantinos Stergiou, Michail Papoutsidakis and Evangelos Pallis
Designs 2026, 10(3), 53; https://doi.org/10.3390/designs10030053 - 9 May 2026
Viewed by 988
Abstract
This work examines the contribution of additive manufacturing as an enabling technology in the design and development of smart and sustainable construction systems, with particular emphasis on nature-based solutions. While the existing literature has devoted considerable attention to the material properties of additive [...] Read more.
This work examines the contribution of additive manufacturing as an enabling technology in the design and development of smart and sustainable construction systems, with particular emphasis on nature-based solutions. While the existing literature has devoted considerable attention to the material properties of additive manufacturing, much less emphasis has been placed on its role in design processes, prototyping, and decision-making in construction and urban systems. To address this gap, this study presents a comprehensive bibliometric analysis of the intersection between smart city frameworks and 3D printing technologies, utilizing a dataset of 103 peer-reviewed publications retrieved from the Scopus database. Using keyword co-occurrence analysis and network mapping through VOSviewer, this study identifies dominant thematic structures, core research hubs, and evolving trends within the field. Complementing this bibliometric analysis with qualitative synthesis, it also reveals a significant convergence of digital design, smart cities, and sustainability strategies. This work further highlights the contribution of additive manufacturing to design processes through rapid prototyping, customization, and the exploration of design alternatives. Rather than framing additive manufacturing as a replacement for conventional design practices, this study positions it as a complementary design capability that can enhance the design process, while also acknowledging important challenges related to scaling, regulation, and integration into construction workflows. This review concludes by outlining future research directions for strengthening the design-oriented integration of additive manufacturing within smart construction systems. Full article
(This article belongs to the Special Issue Design Process for Additive Manufacturing, 2nd Edition)
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16 pages, 5152 KB  
Article
Theoretical and Experimental Bacterial Adherence on Chitosan Films with Varied Characteristics
by Anouar Mouhoub, Amine Guendouz, Zainab El Alaoui-Talibi, Saad Ibnsouda Koraichi and Cherkaoui El Modafar
Int. J. Mol. Sci. 2026, 27(10), 4202; https://doi.org/10.3390/ijms27104202 - 8 May 2026
Viewed by 305
Abstract
This paper highlights the importance of chitosan’s intrinsic parameters on its performance as a film. Fungal (FC) and crustacean (CSC) chitosans with similar molecular weight (400 kDa) and different deacetylation degrees (FC DDA = 84.2%; CSC DDA ≈ 75%) were utilized to elaborate [...] Read more.
This paper highlights the importance of chitosan’s intrinsic parameters on its performance as a film. Fungal (FC) and crustacean (CSC) chitosans with similar molecular weight (400 kDa) and different deacetylation degrees (FC DDA = 84.2%; CSC DDA ≈ 75%) were utilized to elaborate eco-friendly and functional chitosan-based films (C-films). The physicochemical properties as well as bioactivities were evaluated. Results showed that DDA was positively correlated with the zeta potential of the film-forming solutions. Furthermore, the FC films showed a decrease in moisture and swelling levels by about 20%, accompanied by a slight drop in qualitative hydrophobicity. On the other hand, the antibacterial activity of FC film was significantly stronger against Gram-negative bacteria compared to CSC film. Additionally, the C-films considerably mitigated the adherence of Staphylococcus aureus, Pseudomonas aeruginosa, and Escherichia coli, where the percentage of the covered surface ranged from 0.5 to 24%. Conversely, Enterococcus faecalis was more resistant, with percentages of the covered surface higher than 50%. Nonetheless, disintegration in cell structure was noticed regarding the CSC film. Ultimately, the theoretical prediction of cell adherence was highly correlated with experimental results (r = −0.89). These promising results demonstrate that C-films with high DDA are excellent candidates for preventing biofilm formation. Full article
(This article belongs to the Special Issue Research Progress in Food Packaging Materials)
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14 pages, 1370 KB  
Article
Development and Comparative Evaluation of Low-Cost Ultrasound-Guided Regional Anesthesia Phantom Models
by Melikşah Soylu and Mustafa Azizoğlu
Gels 2026, 12(5), 388; https://doi.org/10.3390/gels12050388 - 1 May 2026
Viewed by 680
Abstract
Regional anesthesia is vital for modern surgical practices, but accessibility to training is often hampered by the high cost of commercial phantom models. This study aimed to develop and evaluate low-cost, realistic phantom alternatives using Ecoflex, borax-containing polyvinyl alcohol (PVA), and plastisol compositions. [...] Read more.
Regional anesthesia is vital for modern surgical practices, but accessibility to training is often hampered by the high cost of commercial phantom models. This study aimed to develop and evaluate low-cost, realistic phantom alternatives using Ecoflex, borax-containing polyvinyl alcohol (PVA), and plastisol compositions. The models were evaluated under ultrasound for imaging properties, including needle visibility, tissue resistance, cost, contrast-to-noise ratio (CNR), signal-to-noise ratio (SNR), axial full width at half maximum (FWHM), and compared to a commercial reference (Blue Phantom). Initial qualitative assessments were performed by three experienced evaluators, and inter-observer agreement demonstrated good to excellent reliability. In addition, a long-term usability assessment was conducted more than one year after phantom preparation, involving 20 participants using a structured Likert scale. A statistically significant difference was observed among materials (Friedman test, p < 0.05), with PVA hydrogel containing 20 g borax and the Blue Phantom demonstrating the highest tissue realism scores, without a significant difference between them. The results showed that plastisol softener and PVA (20 g borax) hydrogel provided excellent needle visibility and tissue resistance and achieved an imaging performance comparable to the commercial model. Notably, CNR and SNR values for these materials approached reference levels, while costs ranged from $0.5 to $2.50 per 100 mL, representing a significant reduction compared to $45 per 100 mL for commercial models. In conclusion, this research confirms that affordable materials such as PVA and plastisol can effectively simulate human tissue for ultrasound-guided training. Furthermore, the findings suggest that PVA-based hydrogels may provide sustained usability over time, offering a practical and accessible solution for enhancing clinical skill acquisition in resource-constrained settings. Full article
(This article belongs to the Section Gel Analysis and Characterization)
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24 pages, 374 KB  
Article
Exact Solutions and Stability for First-Order Linear Discrete Matrix Equations with Multiple Delays and Non-Permutable Matrices
by Ahmed M. Elshenhab, Ghada AlNemer and Xingtao Wang
Mathematics 2026, 14(9), 1537; https://doi.org/10.3390/math14091537 - 1 May 2026
Viewed by 288
Abstract
This study formulates closed-form solution expressions for linear discrete matrix equations that involve several time delays, without requiring the coefficient matrices or the non-homogeneous term to commute. Using a generalized multinomial series and exponential matrix functions adapted to multiple delays, we establish fundamental [...] Read more.
This study formulates closed-form solution expressions for linear discrete matrix equations that involve several time delays, without requiring the coefficient matrices or the non-homogeneous term to commute. Using a generalized multinomial series and exponential matrix functions adapted to multiple delays, we establish fundamental solutions in a setting where matrix multiplication is not assumed to be commutative. These explicit representations are subsequently utilized to analyze the stability properties of the system, specifically establishing Hyers–Ulam stability. The analysis elucidates the influence of both delay structure and noncommutativity on solution behavior and robustness. A representative example is provided to illustrate the practical applicability of the proposed method and to highlight the significant qualitative effects induced by delays and noncommutative matrix interactions. Notably, the results extend classical theories by addressing noncommutative settings and yield novel contributions that remain significant even in the absence of delays. Full article
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15 pages, 25979 KB  
Article
Investigation of Three-Dimensional Flow Around a Model Samara Wing Depending on the Angle of Attack
by Neslihan Aydın, Ebubekir Beyazoglu and Irfan Karagoz
Biomimetics 2026, 11(5), 299; https://doi.org/10.3390/biomimetics11050299 - 25 Apr 2026
Viewed by 1111
Abstract
One of the engineering applications inspired by nature is bio-inspired wings. The aerodynamic properties and autorotation characteristics of samara wing models have been studied extensively using both experimental and numerical methods. However, the three-dimensional flow behavior and angle of attack interaction around a [...] Read more.
One of the engineering applications inspired by nature is bio-inspired wings. The aerodynamic properties and autorotation characteristics of samara wing models have been studied extensively using both experimental and numerical methods. However, the three-dimensional flow behavior and angle of attack interaction around a natural samara wing are not yet fully understood. This study investigates the flow behavior around a samara wing model, with the aim of underlying physics and qualitatively analyzing the flow field, as well as the aerodynamic forces and stresses. Since the samara wing and the flow around it are three-dimensional, the difficulty of experimental investigation was taken into account, and the numerical analysis was performed using Computational Fluid Dynamics techniques. The results obtained from the numerical solution of the governing equations for three-dimensional turbulent flow were verified with experimental data. The calculations were performed by varying the angle of attack of the model wing between 0 and 50 degrees at 10-degree intervals. Depending on the angle of attack, the velocity field around the wing, surface pressure, and stress distributions, vortex structures formed on the wing and streamlines were analyzed, and the results were presented. This study and its results on this model may lead to the development and optimization of the model and its use in turbines or air vehicles. Full article
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45 pages, 6732 KB  
Article
A Probabilistic–Statistical Approach to Mass Transfer in Randomly Nonhomogeneous Layered Media Based on Boundary Experimental Data
by Olha Chernukha, Petro Pukach, Halyna Bilushchak, Yurii Bilushchak and Myroslava Vovk
Mathematics 2026, 14(9), 1413; https://doi.org/10.3390/math14091413 - 23 Apr 2026
Viewed by 296
Abstract
This paper presents a probabilistic–statistical approach to the analysis of diffusion processes in randomly nonhomogeneous multilayered bodies under conditions of incomplete experimental information on the boundary. The boundary condition is reconstructed from experimental data using linear regression, while the solution of the corresponding [...] Read more.
This paper presents a probabilistic–statistical approach to the analysis of diffusion processes in randomly nonhomogeneous multilayered bodies under conditions of incomplete experimental information on the boundary. The boundary condition is reconstructed from experimental data using linear regression, while the solution of the corresponding contact initial-boundary value problem is obtained in the form of a Neumann series and averaged over an ensemble of phase configurations. A system of statistical estimates for the solution is developed, including confidence intervals and two-sided critical regions, which provide complementary characteristics of uncertainty. Numerical experiments are performed for six representative samples differing in sample size, variance, and observation interval. It is shown that, despite significant differences in the statistical properties of the input data, the averaged concentration field preserves a qualitatively stable spatio-temporal structure. The results of the article address gaps in existing research by applying a probabilistic-statistical approach that consistently integrates two key elements for the analysis of diffusion processes in multilayer media. The first of these is the reconstruction of boundary conditions using linear regression to recover the conditions at the body boundary based on incomplete experimental data. The second key point is the analysis of uncertainty propagation by combining the regression model with a probabilistic analysis of the corresponding contact initial-boundary value problem, which allows us to quantitatively assess how the errors in the experimental data affect the final solution. From the point of view of mathematical modeling methods, the novelty of the approach lies in the creation of a structural-hierarchical scheme that synthesizes the approaches of mathematical statistics and the theory of random fields. The developed method is a theoretical and computational innovative basis for the analysis of specific physical and technological processes. Full article
(This article belongs to the Special Issue Theory and Applications of Probability Theory and Stochastic Analysis)
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