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Search Results (146)

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Keywords = set-valued integral equations

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25 pages, 1755 KB  
Article
Unpacking Consumer Purchase Intentions Toward Plant-Based Meat Alternatives: An Integrated TPB–VAB Approach Using PLS-SEM, fsQCA, and NCA
by Jialiang Pan, Kun-Shan Wu and Hui-Ting Liu
Foods 2025, 14(20), 3525; https://doi.org/10.3390/foods14203525 - 16 Oct 2025
Abstract
Plant-based meat alternatives (PBMAs) are gaining momentum in response to rising demand for sustainable and healthier diets. Drawing on an integrated framework combining the Theory of Planned Behavior (TPB) and the Value–Attitude–Behavior (VAB) model, this study explores key determinants shaping consumers’ purchase intention [...] Read more.
Plant-based meat alternatives (PBMAs) are gaining momentum in response to rising demand for sustainable and healthier diets. Drawing on an integrated framework combining the Theory of Planned Behavior (TPB) and the Value–Attitude–Behavior (VAB) model, this study explores key determinants shaping consumers’ purchase intention towards PBMAs in Taiwan. This study performed Partial Least Squares Structural Equation Modelling (PLS-SEM), fuzzy-set qualitative comparative analysis (fsQCA) and necessary condition analysis (NCA) to evaluate the formation of consumers’ PBMA purchase intention. The PLS-SEM results revealed that both environmental consciousness and health consciousness exert a significant influence on consumer attitudes, which, together with subjective norms and perceived behavioral control, positively predict purchase intention. fsQCA revealed six distinct combinations of conditions leading to high purchase intention, while NCA identified environmental consciousness, health consciousness, and the three TPB components as necessary conditions. The results highlight the mediating role of attitude and underscore the value of integrating multiple analytical perspectives to capture the complexity of consumer decision-making. This research advances both theoretical understanding and practical application by elucidating the psychological mechanisms underpinning PBMA adoption and by providing evidence-based implications for strategic marketing within the plant-based food sector. Full article
(This article belongs to the Special Issue Evaluation of Food Safety Performance)
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35 pages, 3250 KB  
Article
On a Novel Iterative Algorithm in CAT(0) Spaces with Qualitative Analysis and Applications
by Muhammad Khan, Mujahid Abbas and Cristian Ciobanescu
Symmetry 2025, 17(10), 1695; https://doi.org/10.3390/sym17101695 - 9 Oct 2025
Viewed by 177
Abstract
This study presents a novel and efficient iterative scheme in the setting of CAT(0) spaces and investigates the convergence properties for a generalized class of mappings satisfying the Garcia–Falset property using the proposed iterative scheme. Strong and weak convergence results are established in [...] Read more.
This study presents a novel and efficient iterative scheme in the setting of CAT(0) spaces and investigates the convergence properties for a generalized class of mappings satisfying the Garcia–Falset property using the proposed iterative scheme. Strong and weak convergence results are established in CAT(0) spaces, generalizing many existing results in the literature. Furthermore, we discuss the stability and data dependence of the new iterative process. Numerical experiments include an analysis of error values, the number of iterations, and computational time, providing a comprehensive assessment of the method’s performance. Moreover, graphical comparisons demonstrate the efficiency and reliability of the approach. The obtained results are utilized in solving integral equations. Additionally, the paper concludes with a polynomiographic study of the newly introduced iterative process, in comparison with standard algorithms, such as Newton, Halley, or Kalantari’s B4 iteration, emphasizing symmetry properties. Full article
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23 pages, 1623 KB  
Article
Integral and Numerical Formulations for Seeking the Period of Non-Conservative Nonlinear Oscillator With/Without the First Integral
by Chein-Shan Liu, Chia-Cheng Tsai and Chih-Wen Chang
Symmetry 2025, 17(9), 1584; https://doi.org/10.3390/sym17091584 - 22 Sep 2025
Viewed by 280
Abstract
For a non-conservative nonlinear oscillator (NCNO) having a periodic solution, the existence of the first integral is a certain symmetry of the nonlinear dynamical system, which signifies the balance of kinetic energy and potential energy. A first-order nonlinear ordinary differential equation (ODE) is [...] Read more.
For a non-conservative nonlinear oscillator (NCNO) having a periodic solution, the existence of the first integral is a certain symmetry of the nonlinear dynamical system, which signifies the balance of kinetic energy and potential energy. A first-order nonlinear ordinary differential equation (ODE) is used to derive the first integral, which, equipped with a right-end boundary condition, can determine an implicit potential function for computing the period by an exact integral formula. However, the integrand is singular, which renders a less accurate value of the period. A generalized integral conservation law endowed with a weight function is constructed, which is proved to be equivalent to the exact integral formula. Minimizing the error to satisfy the periodicity conditions, the optimal initial value of the weight function is determined. Two non-iterative methods are developed by integrating three first-order ODEs or two first-order ODEs to compute the period. Very accurate value of the period can be observed upon testing five examples. For the NCNO without having the first integral, the integral-type period formula is derived. Four examples belong to the Liénard equation, involving the van der Pol equation, are evaluated by the proposed iterative method to determine the oscillatory amplitude and period. For the case with one or more limit cycles, the amplitude and period can be estimated very accurately. For the NCNO of a broad type with or without having the first integral, the present paper features a solid theoretical foundation and contributes integral-type formulations for the determination of the oscillatory period. The development of new numerical algorithms and extensive validation across a diverse set of examples is given. Full article
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47 pages, 3785 KB  
Article
Interpretable ML Model for Predicting Magnification Factors in Open Ground-Storey Columns to Prevent Soft-Storey Collapse
by Rahul Ghosh and Rama Debbarma
Buildings 2025, 15(18), 3383; https://doi.org/10.3390/buildings15183383 - 18 Sep 2025
Viewed by 380
Abstract
Open Ground-Storey (OGS) buildings, widely adopted for functional openness, are highly vulnerable to seismic collapse due to stiffness irregularity at the ground storey (GS). The magnification factor (MF), defined as the amplification applied to GS column design forces, acts as a practical strengthening [...] Read more.
Open Ground-Storey (OGS) buildings, widely adopted for functional openness, are highly vulnerable to seismic collapse due to stiffness irregularity at the ground storey (GS). The magnification factor (MF), defined as the amplification applied to GS column design forces, acts as a practical strengthening measure to enhance GS stiffness and thereby mitigate the soft storey failure mechanism. While earlier studies recommended fixed MF values, their lack of adaptability often left stiffness deficiencies unresolved. This study develops a rational framework to quantify and predict the required MF for OGS columns, enabling safe yet functionally efficient design. A comprehensive set of three-dimensional reinforced concrete OGS models was analyzed under seismic loads, covering variations in plan geometry, ground-to-upper-storey height ratio (Hr), and GS infill percentage. Iterative stiffness-based evaluations established the MF demand needed to overcome stiffness deficiencies. To streamline prediction, advanced machine learning (ML) models were applied. Among these, black-box models achieved high predictive accuracy, but Symbolic Regression (SR) offered an interpretable closed-form equation that balances accuracy with transparency, making it suitable for design practice. A sensitivity analysis confirmed the Hr as the most influential parameter, with additional contributions from other variables. Validation on additional OGS configurations confirmed the reliability of the SR model, while seismic response comparisons showed that Modified OGS (MOGS) frames with the proposed MF achieved improved stiffness, reduced lateral displacements, uniform drift distribution, and shorter fundamental periods. The study highlights the novelty of integrating interpretable ML into structural design, providing a codifiable and practical tool for resilient OGS construction. Full article
(This article belongs to the Section Building Structures)
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18 pages, 1886 KB  
Article
The Integrated Choice and Latent Variable Model for Exploring the Mechanisms of Pedestrian Route Choice
by Cheng-Jie Jin, Ningxuan Li, Chenyang Wu, Dawei Li and Yifan Lin
ISPRS Int. J. Geo-Inf. 2025, 14(9), 341; https://doi.org/10.3390/ijgi14090341 - 5 Sep 2025
Viewed by 790
Abstract
The Integrated Choice and Latent Variable (ICLV) model has been widely applied in travel behavior studies, yet its use in understanding pedestrian route choice remains very limited. This paper seeks to address this gap by analyzing data from a series of controlled pedestrian [...] Read more.
The Integrated Choice and Latent Variable (ICLV) model has been widely applied in travel behavior studies, yet its use in understanding pedestrian route choice remains very limited. This paper seeks to address this gap by analyzing data from a series of controlled pedestrian route choice experiments. Four groups of experimental runs were designed, each involving two route options. The first three groups introduced specific controls: bottlenecks, distance constraints, and extra rewards, while the fourth group, without any imposed control, focused on the influence of route geometry (lengths and widths). For each group, we developed measurement and structural models, followed by three comparative models: a binary logit model using only measured variables (MV model), a model using only latent variables (LV model), and the ICLV model that integrates both. Across all the four scenarios, the adjusted R2 values have been improved from 0.286/0.135/0.108/0.035 (MV model) to 0.329/0.161/0.111/0.056 (ICLV model), and the ICLV model can provide interpretable results. These findings highlight the value of incorporating latent constructs based on Structural Equation Modelling (SEM), which enhances the explanatory power of pedestrian route choice models. Moreover, the differences in significant latent variables across various experimental settings offers further insights into the distinct mechanisms underlying pedestrian decision-making under varying conditions. Full article
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17 pages, 464 KB  
Article
Driving Strategic Entrepreneurship Through Organizational Commitment: Evidence from the IT Industry with Leadership Support as a Moderator
by Tayseer Afaishat, Amro Alzghoul, Mahmoud Alghizzawi and Sakher Faisal AlFraihat
Adm. Sci. 2025, 15(9), 350; https://doi.org/10.3390/admsci15090350 - 5 Sep 2025
Viewed by 676
Abstract
This study examines the impact of job commitment on the adoption of strategic entrepreneurship within organizations, with leadership support considered as a moderating variable. Focusing on information technology companies in Jordan, we integrate perspectives from organizational behavior and strategic management to explore how [...] Read more.
This study examines the impact of job commitment on the adoption of strategic entrepreneurship within organizations, with leadership support considered as a moderating variable. Focusing on information technology companies in Jordan, we integrate perspectives from organizational behavior and strategic management to explore how employees’ commitment (affective, normative, continuance) influences their engagement in entrepreneurial initiatives, and whether supportive leadership environments amplify this effect. This study draws on social exchange theory and organizational support theory to propose that committed employees will reciprocate the organization’s support by innovating and taking initiative, especially when they feel backed by leadership. A quantitative survey was conducted, gathering 384 valid responses from employees across Jordan’s IT sector. Data were analyzed using structural equation modeling. The findings reveal that all three forms of commitment positively affect the propensity to engage in strategic entrepreneurship, with affective commitment showing the strongest link. Notably, leadership support significantly moderates these relationships: in high-support contexts, committed employees exhibit substantially greater entrepreneurial behavior. These results indicate that committed employees are more likely to pursue innovative ideas and strategic opportunities, especially when leaders encourage and back their efforts. Theoretical implications include an enhanced understanding of commitment’s role in corporate entrepreneurship and the contingent value of leadership, while practical implications suggest actionable steps for IT firms and others in emerging economies to stimulate innovation. This research contributes to the literature by highlighting human and leadership factors as key drivers of strategic entrepreneurship in organizational settings, and by providing empirical evidence from the Middle East context. Full article
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38 pages, 1403 KB  
Article
Lie Symmetries, Solitary Waves, and Noether Conservation Laws for (2 + 1)-Dimensional Anisotropic Power-Law Nonlinear Wave Systems
by Samina Samina, Hassan Almusawa, Faiza Arif and Adil Jhangeer
Symmetry 2025, 17(9), 1445; https://doi.org/10.3390/sym17091445 - 3 Sep 2025
Viewed by 517
Abstract
This study presents the complete analysis of a (2 + 1)-dimensional nonlinear wave-type partial differential equation with anisotropic power-law nonlinearities and a general power-law source term, which arises in physical domains such as fluid dynamics, nonlinear acoustics, and wave propagation in elastic media, [...] Read more.
This study presents the complete analysis of a (2 + 1)-dimensional nonlinear wave-type partial differential equation with anisotropic power-law nonlinearities and a general power-law source term, which arises in physical domains such as fluid dynamics, nonlinear acoustics, and wave propagation in elastic media, yet their symmetry properties and exact solution structures remain largely unexplored for arbitrary nonlinearity exponents. To fill this gap, a complete Lie symmetry classification of the equation is performed for arbitrary values of m and n, providing all admissible symmetry generators. These generators are then employed to systematically reduce the PDE to ordinary differential equations, enabling the construction of exact analytical solutions. Traveling wave and soliton solutions are derived using Jacobi elliptic function and sine-cosine methods, revealing rich nonlinear dynamics and wave patterns under anisotropic conditions. Additionally, conservation laws associated with variational symmetries are obtained via Noether’s theorem, yielding invariant physical quantities such as energy-like integrals. The results extend the existing literature by providing, for the first time, a full symmetry classification for arbitrary m and n, new families of soliton and traveling wave solutions in multidimensional settings, and associated conserved quantities. The findings contribute both computationally and theoretically to the study of nonlinear wave phenomena in multidimensional cases, extending the catalog of exact solutions and conserved dynamics of a broad class of nonlinear partial differential equations. Full article
(This article belongs to the Section Physics)
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25 pages, 6130 KB  
Article
Hybrid Digital Twin for Phytotron Microclimate Control: Integrating Physics-Based Modeling and IoT Sensor Networks
by Vladimir V. Bukhtoyarov, Ivan S. Nekrasov, Ivan A. Timofeenko, Alexey A. Gorodov, Stanislav A. Kartushinskii, Yury V. Trofimov and Sergey I. Lishik
AgriEngineering 2025, 7(9), 285; https://doi.org/10.3390/agriengineering7090285 - 2 Sep 2025
Viewed by 654
Abstract
Integration of IoT and predictive modeling is critical for optimizing microclimate management in urban-agglomeration vertical farming. In this study, we present a hybrid digital twin approach that combines a physical microclimate model with a distributed IoT monitoring system to simulate and control the [...] Read more.
Integration of IoT and predictive modeling is critical for optimizing microclimate management in urban-agglomeration vertical farming. In this study, we present a hybrid digital twin approach that combines a physical microclimate model with a distributed IoT monitoring system to simulate and control the phytotron environment. A set of heat- and mass-balance equations governing the dynamics of temperature, humidity, and transpiration was implemented and parameterized using a genetic algorithm (GA)—an evolutionary optimization method—with real-time data collected over three intervals (72 h, 90 h, and 110 h) from LoRaWAN sensors (temperature, humidity, CO2) and Wi-Fi-connected power meters managed by Home Assistant. The optimized model achieved mean temperature deviations ≤ 0.1 °C, relative humidity errors ≤ 2%, and overall energy consumption accuracy of 99.5% compared to measured values. The digital twin reliably tracked daily climate fluctuations and system energy use, confirming the accuracy of the hybrid approach. These results demonstrate that the proposed framework effectively integrates theoretical models with IoT-derived data to deliver precise environmental control and energy-use optimization in vertical farming, while also laying the groundwork for scalable digital twins in controlled-environment agriculture. Full article
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17 pages, 1942 KB  
Article
On the Combination of the Laplace Transform and Integral Equation Method to Solve the 3D Parabolic Initial Boundary Value Problem
by Roman Chapko and Svyatoslav Lavryk
Axioms 2025, 14(9), 666; https://doi.org/10.3390/axioms14090666 - 29 Aug 2025
Viewed by 487
Abstract
We consider a two-step numerical approach for solving parabolic initial boundary value problems in 3D simply connected smooth regions. The method uses the Laplace transform in time, reducing the problem to a set of independent stationary boundary value problems for the Helmholtz equation [...] Read more.
We consider a two-step numerical approach for solving parabolic initial boundary value problems in 3D simply connected smooth regions. The method uses the Laplace transform in time, reducing the problem to a set of independent stationary boundary value problems for the Helmholtz equation with complex parameters. The inverse Laplace transform is computed using a sinc quadrature along a suitably chosen contour in the complex plane. We show that due to a symmetry of the quadrature nodes, the number of stationary problems can be decreased by almost a factor of two. The influence of the integration contour parameters on the approximation error is also researched. Stationary problems are numerically solved using a boundary integral equation approach applying the Nyström method, based on the quadratures for smooth surface integrals. Numerical experiments support the expectations. Full article
(This article belongs to the Topic Numerical Methods for Partial Differential Equations)
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18 pages, 2069 KB  
Article
Representation of Integral Formulas for the Extended Quaternions on Clifford Analysis
by Ji Eun Kim
Mathematics 2025, 13(17), 2730; https://doi.org/10.3390/math13172730 - 25 Aug 2025
Viewed by 513
Abstract
This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a [...] Read more.
This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a lack of analogous tools for functions taking values in extended quaternion algebras such as split quaternions and biquaternions. The motivation is to extend the analytical power of Clifford analysis to these broader algebraic structures, enabling the study of more complex hypercomplex systems. The objectives are as follows: (i) to construct new Cauchy-type integral formulas adapted to extended quaternionic function spaces; (ii) to identify explicit kernel functions compatible with Clifford-algebra-valued integrands; and (iii) to demonstrate the application of these formulas to boundary value problems and potential theory. The proposed framework unifies quaternionic function theory and Clifford analysis, offering a robust analytic foundation for tackling higher-dimensional and anisotropic partial differential equations. The results not only enhance theoretical understanding but also open avenues for practical applications in mathematical physics and engineering. Full article
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13 pages, 854 KB  
Article
Physical Reinforcement Learning with Integral Temporal Difference Error for Constrained Robots
by Luis Pantoja-Garcia, Vicente Parra-Vega and Rodolfo Garcia-Rodriguez
Robotics 2025, 14(8), 111; https://doi.org/10.3390/robotics14080111 - 14 Aug 2025
Viewed by 1041
Abstract
The paradigm of reinforcement learning (RL) refers to agents that learn iteratively through continuous interactions with their environment. However, when the value function is unknown, a neural network is used, which is typically encoded into an unknown temporal difference equation. When RL is [...] Read more.
The paradigm of reinforcement learning (RL) refers to agents that learn iteratively through continuous interactions with their environment. However, when the value function is unknown, a neural network is used, which is typically encoded into an unknown temporal difference equation. When RL is implemented in physical systems, explicit convergence and stability analyses are required to guarantee the worst-case operations for any trial, even when the initial conditions are set to zero. In this paper, physical RL (p-RL) refers to the application of RL in dynamical systems that interact with their environments, such as robot manipulators in contact tasks and humanoid robots in cooperation or interaction tasks. Unfortunately, most p-RL schemes lack stability properties, which can even be dangerous for specific robot applications, such as those involving contact (constrained) tasks or interaction tasks. Considering an unknown and disturbed DAE2 robot, in this paper a p-RL approach is developed to guaranteeing robust stability throughout a continuous-time-adaptive actor–critic, with local exponential convergence of force–position tracking error. The novel adaptive mechanisms lead to robustness, while an integral sliding mode enforces tracking. Simulations are presented and discussed to show our proposal’s effectiveness, and some final remarks are addressed concerning the structural aspects. Full article
(This article belongs to the Section AI in Robotics)
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21 pages, 351 KB  
Article
Using Pseudo-Complemented Truth Values of Calculation Errors in Integral Transforms and Differential Equations Through Monte Carlo Algorithms
by Ravi A. Salim, Ernastuti, Edi Sukirman, Trini Saptariani and Suryadi MT
Mathematics 2025, 13(15), 2534; https://doi.org/10.3390/math13152534 - 6 Aug 2025
Viewed by 354
Abstract
This study aims to demonstrate how mathematics, especially calculus concepts, can be expanded to include semi-entities and how these can be applied to sampling activities. Here, the multivalued logic uses pseudo-complemented lattices, instead of Boolean algebras. Truth values can express the intensity of [...] Read more.
This study aims to demonstrate how mathematics, especially calculus concepts, can be expanded to include semi-entities and how these can be applied to sampling activities. Here, the multivalued logic uses pseudo-complemented lattices, instead of Boolean algebras. Truth values can express the intensity of a property: for example, the property of being heavy intensifies as weight increases. They can also express the state-of-the-art knowledge of an individual about a certain thing. To express that a number x approaches a is to say that the statement “x=b” is not fully true but approaches the full-true value as ba approaches zero. This approach generalizes the concept of a limit and the concepts derived from it, such as differentiation and integration. A Monte Carlo algorithm replaces one function with another with finite domain, preferably its finite part, by sampling the domain and calculating its map. The discussion extends to integration over an unbounded interval, integral transforms, and differential equations. This study then covers strategies for producing Monte Carlo estimates of respective problems and determining their crucial truth values. In the discussion, a topic related to axiomatizing set theory is also suggested. Full article
20 pages, 1676 KB  
Article
Data-Driven Distributionally Robust Optimization for Solar-Powered EV Charging Under Spatiotemporal Uncertainty in Urban Distribution Networks
by Tianhao Wang, Xuejiao Zhang, Xiaolin Zheng, Jian Wang, Shiqian Ma, Jian Chen, Mengyu Liu and Wei Wei
Energies 2025, 18(15), 4001; https://doi.org/10.3390/en18154001 - 27 Jul 2025
Viewed by 784
Abstract
The rapid electrification of transportation and the proliferation of rooftop solar photovoltaics (PVs) in urban environments are reshaping the operational dynamics of power distribution networks. However, the inherent uncertainty in electric vehicle (EV) behavior—including arrival times, charging preferences, and state-of-charge—as well as spatially [...] Read more.
The rapid electrification of transportation and the proliferation of rooftop solar photovoltaics (PVs) in urban environments are reshaping the operational dynamics of power distribution networks. However, the inherent uncertainty in electric vehicle (EV) behavior—including arrival times, charging preferences, and state-of-charge—as well as spatially and temporally variable solar generation, presents a profound challenge to existing scheduling frameworks. This paper proposes a novel data-driven distributionally robust optimization (DDRO) framework for solar-powered EV charging coordination under spatiotemporal uncertainty. Leveraging empirical datasets of EV usage and solar irradiance from a smart city deployment, the framework constructs Wasserstein ambiguity sets around historical distributions, enabling worst-case-aware decision-making without requiring the assumption of probability laws. The problem is formulated as a two-stage optimization model. The first stage determines day-ahead charging schedules, solar utilization levels, and grid allocations across an urban-scale distribution feeder. The second stage models real-time recourse actions—such as dynamic curtailment or demand reshaping—after uncertainties are realized. Physical grid constraints are modeled using convexified LinDistFlow equations, while EV behavior is segmented into user classes with individualized uncertainty structures. The model is evaluated on a modified IEEE 123-bus feeder with 52 EV-PV nodes, using 15 min resolution over a 24 h horizon and 12 months of real-world data. Comparative results demonstrate that the proposed DDRO method reduces total operational costs by up to 15%, eliminates voltage violations entirely, and improves EV service satisfaction by more than 30% relative to deterministic and stochastic baselines. This work makes three primary contributions: it introduces a robust, tractable optimization architecture that captures spatiotemporal uncertainty using empirical Wasserstein sets; it integrates behavioral and physical modeling within a unified dispatch framework for urban energy-mobility systems; and it demonstrates the value of robust coordination in simultaneously improving grid resilience, renewable utilization, and EV user satisfaction. The results offer practical insights for city-scale planners seeking to enable the reliable and efficient electrification of mobility infrastructure under uncertainty. Full article
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30 pages, 3923 KB  
Article
Exploring the Key Factors Influencing the Plays’ Continuous Intention of Ancient Architectural Cultural Heritage Serious Games: An SEM–ANN–NCA Approach
by Qian Bao, Siqin Wang, Ken Nah and Wei Guo
Buildings 2025, 15(15), 2648; https://doi.org/10.3390/buildings15152648 - 27 Jul 2025
Viewed by 962
Abstract
Serious games (SGs) have been widely employed in the digital preservation and transmission of architectural heritage. However, the key determinants and underlying mechanisms driving users’ continuance intentions toward ancient-architecture cultural heritage serious games (CH-SGs) have not been thoroughly investigated. Accordingly, a conceptual model [...] Read more.
Serious games (SGs) have been widely employed in the digital preservation and transmission of architectural heritage. However, the key determinants and underlying mechanisms driving users’ continuance intentions toward ancient-architecture cultural heritage serious games (CH-SGs) have not been thoroughly investigated. Accordingly, a conceptual model grounded in the stimulus–organism–response (S–O–R) framework was developed to elucidate the affective and behavioral effects experienced by CH-SG users. Partial least squares structural equation modeling (PLS-SEM) and artificial neural networks (ANNs) were employed to capture both the linear and nonlinear relationships among model constructs. By integrating sufficiency logic (PLS-SEM) and necessity logic (necessary condition analysis, NCA), “must-have” and “should-have” factors were identified. Empirical results indicate that cultural authenticity, knowledge acquisition, perceived enjoyment, and design aesthetics each exert a positive influence—of varying magnitude—on perceived value, cultural identification, and perceived pleasure, thereby shaping users’ continuance intentions. Moreover, cultural authenticity and perceived enjoyment were found to be necessary and sufficient conditions, respectively, for enhancing perceived pleasure and perceived value, which in turn indirectly bolster CH-SG users’ sustained use intentions. By creating an immersive, narratively rich, and engaging cognitive experience, CH-SGs set against ancient architectural backdrops not only stimulate users’ willingness to visit and protect heritage sites but also provide designers and developers with critical insights for optimizing future CH-SG design, development, and dissemination. Full article
(This article belongs to the Section Architectural Design, Urban Science, and Real Estate)
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33 pages, 403 KB  
Article
Some Further Insight into the Sturm–Liouville Theory
by Salvatore De Gregorio, Lamberto Lamberti and Paolo De Gregorio
Mathematics 2025, 13(15), 2405; https://doi.org/10.3390/math13152405 - 26 Jul 2025
Viewed by 368
Abstract
Some classical texts on the Sturm–Liouville equation (p(x)y)q(x)y+λρ(x)y=0 are revised to highlight further properties of its solutions. Often, in the [...] Read more.
Some classical texts on the Sturm–Liouville equation (p(x)y)q(x)y+λρ(x)y=0 are revised to highlight further properties of its solutions. Often, in the treatment of the ensuing integral equations, ρ=const is assumed (and, further, ρ=1). Instead, here we preserve ρ(x) and make a simple change only of the independent variable that reduces the Sturm–Liouville equation to yq(x)y+λρ(x)y=0. We show that many results are identical with those with λρq=const. This is true in particular for the mean value of the oscillations and for the analog of the Riemann–Lebesgue Theorem. From a mechanical point of view, what is now the total energy is not a constant of the motion, and nevertheless, the equipartition of the energy is still verified and, at least approximately, it does so also for a class of complex λ. We provide here many detailed properties of the solutions of the above equation, with ρ=ρ(x). The conclusion, as we may easily infer, is that, for large enough λ, locally, the solutions are trigonometric functions. We give the proof for the closure of the set of solutions through the Phragmén–Lindelöf Theorem, and show the separate dependence of the solutions from the real and imaginary components of λ. The particular case of q(x)=αρ(x) is also considered. A direct proof of the uniform convergence of the Fourier series is given, with a statement identical to the classical theorem. Finally, the proof of J. von Neumann of the completeness of the Laguerre and Hermite polynomials in non-compact sets is revisited, without referring to generating functions and to the Weierstrass Theorem for compact sets. The possibility of the existence of a general integral transform is then investigated. Full article
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