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39 pages, 5604 KB  
Article
Prediction of 3D Airspace Occupancy Using Machine Learning
by Cristian Lozano Tafur, Jaime Orduy Rodríguez, Pedro Melo Daza, Iván Rodríguez Barón, Danny Stevens Traslaviña and Juan Andrés Bermúdez
Forecasting 2025, 7(4), 56; https://doi.org/10.3390/forecast7040056 (registering DOI) - 8 Oct 2025
Abstract
This research introduces a system designed to predict three-dimensional airspace occupancy over Colombia using historical Automatic Dependent Surveillance-Broadcast (ADS-B) data and machine learning techniques. The goal is to support proactive air traffic management by estimating future aircraft positions—specifically their latitude, longitude, and flight [...] Read more.
This research introduces a system designed to predict three-dimensional airspace occupancy over Colombia using historical Automatic Dependent Surveillance-Broadcast (ADS-B) data and machine learning techniques. The goal is to support proactive air traffic management by estimating future aircraft positions—specifically their latitude, longitude, and flight level. To achieve this, four predictive models were developed and tested: K-Nearest Neighbors (KNN), Random Forest, Extreme Gradient Boosting (XGBoost), and Long Short-Term Memory (LSTM). Among them, the LSTM model delivered the most accurate results, with a Mean Absolute Error (MAE) of 312.59, a Root Mean Squared Error (RMSE) of 1187.43, and a coefficient of determination (R2) of 0.7523. Compared to the baseline models (KNN, Random Forest, XGBoost), these values represent an improvement of approximately 91% in MAE, 83% in RMSE, and an eighteen-fold increase in R2, demonstrating the substantial advantage of the LSTM approach. These metrics indicate a significant improvement over the other models, particularly in capturing temporal patterns and adjusting to evolving traffic conditions. The strength of the LSTM approach lies in its ability to model sequential data and adapt to dynamic environments—making it especially suitable for supporting future Trajectory-Based Operations (TBO). The results confirm that predicting airspace occupancy in three dimensions using historical data are not only possible but can yield reliable and actionable insights. Looking ahead, the integration of hybrid neural network architectures and their deployment in real-time systems offer promising directions to enhance both accuracy and operational value. Full article
(This article belongs to the Topic Short-Term Load Forecasting—2nd Edition)
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25 pages, 800 KB  
Review
Smart but Unlivable? Rethinking Smart City Rankings Through Livability and Urban Sustainability: A Comparative Perspective Between Athens and Zurich
by Alessandro Bove and Marco Ghiraldelli
Sustainability 2025, 17(19), 8901; https://doi.org/10.3390/su17198901 - 7 Oct 2025
Abstract
While the ‘smart city’ concept is central to urban innovation, promising enhanced efficiency and livability, this paper interrogates a critical paradox: can cities be ‘smart’ yet ‘unlivable’? Existing indices, such as the IMD Smart City Index and the IESE Cities in Motion Index, [...] Read more.
While the ‘smart city’ concept is central to urban innovation, promising enhanced efficiency and livability, this paper interrogates a critical paradox: can cities be ‘smart’ yet ‘unlivable’? Existing indices, such as the IMD Smart City Index and the IESE Cities in Motion Index, while standard references, tend to prioritize technological and economic metrics, potentially failing to fully capture urban quality of life and sustainability. This study presents a preliminary attempt, based on an analysis of scientific literature, to critically examine current smart city indicators and propose a set of alternative indicators more representative of quality of life (QoL) and livability. The objective is not to overturn the rankings of cities like Zurich (high-ranking) and Athens (low-ranking), but to explore how a livability-focused approach, using more representative QoL indicators, might narrow the perceived gap between them, thereby highlighting diverse dimensions of urban performance. This research critically evaluates current smart city rankings. It aims to determine if livability-based indicators, supported by scientific literature, can provide a more balanced view of urban performance. This paper details how these alternative indicators were chosen, justifying their relevance to QoL with scientific support, and maps them to established smart city verticals (Smart Mobility, Smart Environment, Smart Governance, Smart Living, Smart People, Smart Economy). Finally, it outlines future research directions to further develop and validate this human-centric approach. Full article
(This article belongs to the Section Sustainable Urban and Rural Development)
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51 pages, 5385 KB  
Article
On Complex Dimensions and Heat Content of Self-Similar Fractals
by William E. Hoffer and Michel L. Lapidus
Fractal Fract. 2025, 9(10), 649; https://doi.org/10.3390/fractalfract9100649 - 7 Oct 2025
Abstract
Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptotics of the heat content on self-similar [...] Read more.
Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptotics of the heat content on self-similar fractals. We consider the Dirichlet problem for the heat equation on bounded open regions whose boundaries are self-similar fractals. The class of self-similar domains we consider allows for non-disjoint overlap of the self-similar copies, provided some control over the separation. The possible complex dimensions, determined strictly by the similitudes that define the self-similar domain, control the scaling exponents of the asymptotic expansion for the heat content. We illustrate our method in the case of generalized von Koch snowflakes and, in particular, extend known results for these fractals with arithmetic scaling ratios to the generic (in the topological sense), non-arithmetic setting. Full article
(This article belongs to the Special Issue Fractal Dimensions with Applications in the Real World)
26 pages, 3443 KB  
Article
Bridging Perceptions: A Comparative Evaluation of Public Space Design Qualities by Experts and Users
by Ioannis Chatziioannou, Panagiotis Kanellopoulos, Charalampos Kyriakidis and Efthimios Bakogiannis
Urban Sci. 2025, 9(10), 412; https://doi.org/10.3390/urbansci9100412 - 7 Oct 2025
Abstract
In the context of public space research, numerous studies highlight its vital role in fostering public life and social interaction. With urbanization on the rise and most people living in cities, acknowledging public spaces, and especially public squares, as key components of the [...] Read more.
In the context of public space research, numerous studies highlight its vital role in fostering public life and social interaction. With urbanization on the rise and most people living in cities, acknowledging public spaces, and especially public squares, as key components of the urban realm is more important than ever. The success of space is frequently determined by its capacity to meet human needs, a condition that, in turn, is largely contingent upon specific design qualities. Literature identifies key qualities such as inclusiveness, accessibility and connectivity, sociability, vitality, perceptual and esthetic satisfaction, and participatory characteristics. While many studies explore these factors, little attention has been given to whether users and designers assign equal importance to them. This research addresses the question: To what extent do experts’ and users’ perceptions converge regarding the variables that determine the success of public spaces? To explore this, the study applies MICMAC method structural analysis that prioritizes variables based on their interdependence and dependence. The method is used with both design experts and public space users. Findings reveal convergence in perceptions regarding key parameters; specifically, strong convergence is observed in the qualities of participation and vitality, followed by sociability and perceptual and esthetic satisfaction. Moreover, the expert group prioritizes parameters related to sociability, accessibility and connectivity, and inclusiveness, reflecting contemporary design principles aimed at creating equitable, easily accessible, and inclusive spaces. In contrast, the user group focuses more on the experiential and esthetic dimension of space, adding variables related to perceptual and esthetic satisfaction and vitality. The study aims to inform more user-responsive public space design by bridging gaps between expert and user perspectives. Full article
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19 pages, 1783 KB  
Article
Integrating Material Analysis, Radiocarbon Dating, and Technical Examination in the Dating and Provenance Study of a Copy of Raphael’s “The Great Holy Family of Francis I”
by Ester S. B. Ferreira, Charlotte Hoffmann, Laura Hendriks, Irka Hajdas, Stefan Kradolfer, Detlef Günther, Katharina Hünerfauth, Juliane Reinhardt, Hans Portsteffen and Susanne Müller-Bechtel
Heritage 2025, 8(10), 424; https://doi.org/10.3390/heritage8100424 - 6 Oct 2025
Abstract
In 2016, five fragments from a copy of “The Great Holy Family of Francis I” were brought to the Cologne Institute of Conservation Sciences (CICS) for research and conservation/restoration. A comprehensive technical and material analysis was carried out to assist provenance studies. From [...] Read more.
In 2016, five fragments from a copy of “The Great Holy Family of Francis I” were brought to the Cologne Institute of Conservation Sciences (CICS) for research and conservation/restoration. A comprehensive technical and material analysis was carried out to assist provenance studies. From the analysis of pigments, binder, additives, and canvas fibres alongside radiocarbon dating of the lead white pigment, oil binder, and canvas support, as well as the lead stable isotope study, it could be determined that, with high probability, the copy was created in Northern Europe between the late 16th century and the mid-17th century. During this period the original painting was initially displayed in Fontainebleau in the “Chapelle Haute” before being transferred in the early 17th century to the newly built “Cabinet des Peintures”, also in Fontainebleau, where it would probably have been more accessible for copying. Interestingly, the written sources describe a copy made during this period to replace the original in the “Chapelle Haute”, the location of which is currently not known. However, the different overall dimensions of the present copy speak against it, having been created to replace the original. Full article
23 pages, 560 KB  
Article
Thriving in Internal CSR-Driven Service Organisations: The Role of Embeddedness, Moral Meaningfulness and Intrapreneurship
by Ali Edip and Georgiana Karadas
Sustainability 2025, 17(19), 8866; https://doi.org/10.3390/su17198866 - 4 Oct 2025
Abstract
In service-intensive, compliance-driven settings such as banking, identifying how internal corporate social responsibility (ICSR) fosters employees’ vitality and learning is crucial for sustainable organisational performance. Amid growing interest in employee thriving, this study explores how perceived internal corporate social responsibility (PICSR) and moral [...] Read more.
In service-intensive, compliance-driven settings such as banking, identifying how internal corporate social responsibility (ICSR) fosters employees’ vitality and learning is crucial for sustainable organisational performance. Amid growing interest in employee thriving, this study explores how perceived internal corporate social responsibility (PICSR) and moral meaningfulness (MM) shape thriving at work (TaW) through organisational embeddedness (OE). Rooted in self-determination theory, the findings reveal OE as a key mediator between PICSR and TaW, shedding light on how ICSR initiatives influence employee dynamics. The study also reveals that MM alone does not significantly predict TaW directly, but does so indirectly through OE, highlighting the importance of contextual mechanisms. Additionally, it identifies a surprising negative moderating effect of risk-taking, one dimension of intrapreneurial behaviour (IB), on the relationship between OE and TaW, while innovativeness, another dimension, shows no such effect. Theoretical and practical implications underscore the importance of aligning ICSR practices with employees’ psychological needs, supporting moral alignment, and tailoring support for intrapreneurs. Organisations must achieve a balance between autonomy and security to sustain engagement and innovation, advancing human and organisational sustainability, ultimately leading to thriving. Full article
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22 pages, 1214 KB  
Article
Didactic Analysis of Natural Science Textbooks in Ecuador: A Critical Review from a Constructivist Perspective
by Frank Guerra-Reyes, Eric Guerra-Dávila and Edison Díaz-Martínez
Educ. Sci. 2025, 15(10), 1312; https://doi.org/10.3390/educsci15101312 - 2 Oct 2025
Abstract
School textbooks are central to the teaching, studying, and learning processes because they mediate the interaction between the prescribed curriculum and the educational experience in the classroom. Evaluating their didactic structure critically allows us to determine the degree to which they align with [...] Read more.
School textbooks are central to the teaching, studying, and learning processes because they mediate the interaction between the prescribed curriculum and the educational experience in the classroom. Evaluating their didactic structure critically allows us to determine the degree to which they align with current curriculum guidelines and promote meaningful learning. This study aimed to analyze the extent to which Ecuadorian natural science textbooks reflect constructivist learning principles and promote the development of key competencies established in the National Priority Curriculum. This curriculum guides the achievement of essential results and strengthens fundamental competencies for students’ comprehensive development. Content analysis was adopted as the methodological approach given its relevance in examining the didactic and curricular dimensions of educational materials. The analysis covered twelve eighth-grade General Basic Education textbooks and their supplementary materials. The analysis was based on two instruments: specialized summary analysis sheets (RAE) and a purpose-built checklist. The ATLAS.ti 25 and IRaMuTeQ programs supported the systematization and visualization of the data. The results showed limited integration of constructivist strategies, such as teaching for comprehension, inquiry-based learning, and problem solving, in most of the analyzed texts. These findings underscore the need to expand and strengthen the incorporation of contextualized, critical, and meaningful learning experiences to improve the didactic design of school textbooks. Such improvements would promote coherent articulation between objectives, content, methods, resources, and assessment in line with constructivist principles of the Ecuadorian curriculum. Furthermore, given these approaches’ affinity with curricular frameworks in other regional countries, the results could offer relevant guidance and starting points for reflection on developing and using textbooks in Latin American contexts with comparable educational characteristics. Full article
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16 pages, 7795 KB  
Article
Enhancing Helmholtz Resonance Prediction in Acoustic Barriers Based on Sonic Crystals
by Lucas Onrubia-Fontangordo, José María Bravo Plana-Sala, Juan Vicente Sánchez-Pérez and Sergio Castiñeira-Ibáñez
Appl. Sci. 2025, 15(19), 10675; https://doi.org/10.3390/app151910675 - 2 Oct 2025
Abstract
Environmental noise is a growing public health issue, particularly in dense urban environments and areas adjacent to transport infrastructure. Passive acoustic barriers have been a widely adopted solution, although their functional and aesthetic limitations have driven the development of alternatives, such as Sonic [...] Read more.
Environmental noise is a growing public health issue, particularly in dense urban environments and areas adjacent to transport infrastructure. Passive acoustic barriers have been a widely adopted solution, although their functional and aesthetic limitations have driven the development of alternatives, such as Sonic Crystal Acoustic Barriers. These structures can incorporate resonant elements, such as Helmholtz cavities, with the aim of enhancing attenuation in particular frequency bands. However, determining the precise dimensions of these resonators is a persistent challenge, given that classical models are based on ideal geometries and do not incorporate interaction with the periodic structural environment. This study sets a new frequency-dependent correction factor, obtained both numerically and experimentally, which allows the classical Helmholtz resonance expression to be adjusted to the case of resonators embedded in a Sonic Crystal (SC). The proposed model is validated through simulations and experimental measurements, showing a significant improvement in the prediction of the resonance frequency. The results obtained in this study allow us to move towards a more efficient and realistic design of passive acoustic barriers that are lightweight and adaptable to the urban environment. Full article
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19 pages, 14588 KB  
Article
Research on Evaporation Duct Height Prediction Modeling in the Yellow and Bohai Seas Using BLA-EDH
by Xiaoyu Wu, Lei Li, Zheyan Zhang, Can Chen and Haozhi Liu
Atmosphere 2025, 16(10), 1156; https://doi.org/10.3390/atmos16101156 - 2 Oct 2025
Abstract
Evaporation Duct Height (EDH) is a crucial parameter in evaporation duct modeling, as it directly influences the strength of the waveguide trapping effect and significantly impacts the over-the-horizon detection performance of maritime radars. To address the limitations of low prediction accuracy and limited [...] Read more.
Evaporation Duct Height (EDH) is a crucial parameter in evaporation duct modeling, as it directly influences the strength of the waveguide trapping effect and significantly impacts the over-the-horizon detection performance of maritime radars. To address the limitations of low prediction accuracy and limited interpretability in existing deep learning models under complex marine meteorological conditions, this study proposes a surrogate model, BLA-EDH, designed to emulate the output of the Naval Postgraduate School (NPS) model for real-time EDH estimation. Experimental results demonstrate that BLA-EDH can effectively replace the traditional NPS model for real-time EDH prediction, achieving higher accuracy than Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) models. Random Forest analysis identifies relative humidity (0.2966), wind speed (0.2786), and 2-m air temperature (0.2409) as the most influential environmental variables, with importance scores exceeding those of other factors. Validation using the parabolic equation shows that BLA-EDH attains excellent fitting performance, with coefficients of determination reaching 0.9999 and 0.9997 in the vertical and horizontal dimensions, respectively. This research provides a robust foundation for modeling radio wave propagation in the Yellow Sea and Bohai Sea regions and offers valuable insights for the development of marine communication and radar detection systems. Full article
(This article belongs to the Section Atmospheric Techniques, Instruments, and Modeling)
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16 pages, 1288 KB  
Article
Urban Geometry and Social Topology: A Computational Simulation of Urban Network Formation
by Daniel Lenz Costa Lima, Daniel Ribeiro Cardoso and Andrés M. Passaro
Buildings 2025, 15(19), 3555; https://doi.org/10.3390/buildings15193555 - 2 Oct 2025
Abstract
When a city decides to undertake a certain urban project, is it modifying just the physical environment or the social fabric that dwells within? This work investigates the relationship between the geometric configuration of urban space (geometry–city) and the topology of the networks [...] Read more.
When a city decides to undertake a certain urban project, is it modifying just the physical environment or the social fabric that dwells within? This work investigates the relationship between the geometric configuration of urban space (geometry–city) and the topology of the networks of encounters of its inhabitants (network–city) that form through daily interactions. The research departs from the hypothesis that changes in geometry–city would not significantly alter the topology of the network–city, testing this proposition conceptually through abstract computational simulations developed specifically for this study. In this simulator, abstract maps with buildings distributed over different primary geometries are generated and have activities (use: home or work) and a population assigned. Encounters of the “inhabitants” are registered while daily commute routines, enough to achieve differentiation and stability, are run. The initial results revealed that the geometry description was not enough, and definitions regarding activity attribution were also necessary. Thus, we could not confirm nor reject the original hypothesis exactly, but it had to be complemented, including the idea of an activity–city dimension. We found that despite the geometry–city per se not determining the structure of the network–city, the spatial (geometric) distribution of activities directly impacts the resulting topology. Urban geometry influences networks–city only insofar as it conforms to activity–city, defining areas for activities or restricting routing between them. But it is the geometry of localization of the activities that has a direct impact on the topology of the network–city. This conceptual discovery can have significant implications for urban planning if corroborated in real-world situations. It could suggest that land use policies may be more effective for intervening in network-based characteristics, like social cohesion and resilience, than purely morphological interventions. Full article
(This article belongs to the Special Issue Emerging Trends in Architecture, Urbanization, and Design)
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25 pages, 8960 KB  
Article
Analysis on Durability of Bentonite Slurry–Steel Slag Foam Concrete Under Wet–Dry Cycles
by Guosheng Xiang, Feiyang Shao, Hongri Zhang, Yunze Bai, Yuan Fang, Youjun Li, Ling Li and Yang Ming
Buildings 2025, 15(19), 3550; https://doi.org/10.3390/buildings15193550 - 2 Oct 2025
Abstract
Wet–dry cycles are a key factor aggravating the durability degradation of foam concrete. To address this issue, this study prepared bentonite slurry–steel slag foam concrete (with steel slag and cement as main raw materials, and bentonite slurry as admixture) using the physical foaming [...] Read more.
Wet–dry cycles are a key factor aggravating the durability degradation of foam concrete. To address this issue, this study prepared bentonite slurry–steel slag foam concrete (with steel slag and cement as main raw materials, and bentonite slurry as admixture) using the physical foaming method. Based on 7-day unconfined compressive strength tests with different mix proportions, the optimal mix proportion was determined as follows: mass ratio of bentonite to water 1:15, steel slag content 10%, and mass fraction of bentonite slurry 5%. Based on this optimal mix proportion, dry–wet cycle tests were carried out in both water and salt solution environments to systematically analyze the improvement effect of steel slag and bentonite slurry on the durability of foam concrete. The results show the following: steel slag can act as fine aggregate to play a skeleton role; after fully mixing with cement paste, it wraps the outer wall of foam, which not only reduces foam breakage but also inhibits the formation of large pores inside the specimen; bentonite slurry can densify the interface transition zone, improve the toughness of foam concrete, and inhibit the initiation and propagation of matrix cracks during the dry–wet cycle process; the composite addition of the two can significantly enhance the water erosion resistance and salt solution erosion resistance of foam concrete. The dry–wet cycle in the salt solution environment causes more severe erosion damage to foam concrete. The main reason is that, after chloride ions invade the cement matrix, they erode hydration products and generate expansive substances, thereby aggravating the matrix damage. Scanning Electron Microscopy (SEM) analysis shows that, whether in water environment or salt solution environment, the fractal dimension of foam concrete decreased slightly with an increasing number of wet–dry cycle times. Based on fractal theory, this study established a compressive strength–porosity prediction model and a dense concrete compressive strength–dry–wet cycle times prediction model, and both models were validated against experimental data from other researchers. The research results can provide technical support for the development of durable foam concrete in harsh environments and the high-value utilization of steel slag solid waste, and are applicable to civil engineering lightweight porous material application scenarios requiring resistance to dry–wet cycle erosion, such as wall bodies and subgrade filling. Full article
(This article belongs to the Section Building Structures)
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19 pages, 822 KB  
Article
Ascertaining the Anatomical Parameters and Chemical Composition of Luffa cylindrica Cellulosic Fibers for Their Plausibility in Pulp and Paper Production
by Balasubramanian NagarajaGanesh, Balasubramanian Rekha, Manoharan Gopi Krishna and Syed Ibrahim Shaik Mohamed Ferozdheen
Polymers 2025, 17(19), 2643; https://doi.org/10.3390/polym17192643 - 30 Sep 2025
Abstract
This research is mainly intended to assess the likelihood of producing pulp and paper from the cellulosic fibers of matured Luffa cylindrica fruit. The cellulose fibers were extracted and subjected to chemical composition studies and FTIR spectroscopic analysis. The chemical composition studies revealed [...] Read more.
This research is mainly intended to assess the likelihood of producing pulp and paper from the cellulosic fibers of matured Luffa cylindrica fruit. The cellulose fibers were extracted and subjected to chemical composition studies and FTIR spectroscopic analysis. The chemical composition studies revealed that these fibers contain 82.4% holocellulose, 11.2% lignin, and 0.63% ash. Functional groups that represent the presence of the biopolymers were confirmed in the FTIR analysis. These fibers were observed through a light microscope, and important fiber parameters, such as the fiber diameter, fiber lumen, and cell wall thickness, were measured. Statistical analysis showed that the fiber dimensions follow a normal distribution. Based on the observed values, the derived indices that determine the fibers’ suitability to produce paper were calculated. The evaluated derived indices showed that the fibers possess a Runkel index of 59.67%, a slenderness ratio of 61.04%, a coefficient of rigidity of 63.7%, and a flexibility coefficient of 0.19. The Luce shape factor and Solids factor of the fibers were found to be 0.42 and 157.36 × 103 μm3, respectively. This study proved that the morphology, derived indices, and chemical composition of the fibers are in par with other fiber sources that are used for pulp and paper production. Full article
(This article belongs to the Section Biobased and Biodegradable Polymers)
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40 pages, 5472 KB  
Article
Geotourism: From Theoretical Definition to Practical Analysis in the Sohodol Gorges Protected Area, Romania
by Amalia Niță, Ionuț-Adrian Drăguleasa, Emilia Constantinescu and Dorina Bonea
Geographies 2025, 5(4), 53; https://doi.org/10.3390/geographies5040053 - 30 Sep 2025
Abstract
The Sohodol Gorges has become a location of interest for tourists seeking ecological experiences and outdoor activities. The main purpose of the present study is to evaluate the attitudes of Romanian tourists toward the development of geotourism in this region following the COVID-19 [...] Read more.
The Sohodol Gorges has become a location of interest for tourists seeking ecological experiences and outdoor activities. The main purpose of the present study is to evaluate the attitudes of Romanian tourists toward the development of geotourism in this region following the COVID-19 pandemic. In conjunction with the research questions, hypotheses, variables, and research methodology, the following research objectives were emphasized in this study of the Oltenia region: (1) investigate how certain socio-demographic variables, such as age, gender, level of education, and occupation, influence tourists’ perceptions of the various aspects of geotourism development in the Sohodol Gorges; (2) analyze the different dimensions of geotourism, including its economic, ecological, and socio-cultural impacts, thus contributing to a deeper understanding of how geotourism is perceived in the study area in the post-pandemic context. For a qualitative evaluation of the information presented in this study, the authors used a qualitative survey with open questions and closed questions as a data collection method. For data processing and analysis, the EViews version 12.0 software package was used, enabling complex statistical analyses such as multiple regressions and correlation coefficient determination. These techniques were essential for identifying and interpreting the relationships between demographic variables and tourist perceptions. The research results provide a detailed picture of the influence that demographic and behavioral factors have on tourists’ perceptions in the context of post-COVID-19 geotourism development in the Sohodol Gorges of Romania. Education level and age play a significant role in shaping economic and environmental perceptions, indicating that tourists with higher education levels are more aware of the economic and ecological impact of tourism. Full article
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22 pages, 341 KB  
Article
Modular Invariance and Anomaly Cancellation Formulas for Fiber Bundles
by Jianyun Guan and Haiming Liu
Axioms 2025, 14(10), 740; https://doi.org/10.3390/axioms14100740 - 29 Sep 2025
Abstract
By combining modular invariance of characteristic forms and the family index theory, we obtain some new anomaly cancellation formulas for any dimension under the not top degree component. For a fiber bundle of dimension (4l2), we obtain [...] Read more.
By combining modular invariance of characteristic forms and the family index theory, we obtain some new anomaly cancellation formulas for any dimension under the not top degree component. For a fiber bundle of dimension (4l2), we obtain the anomaly cancellation formulas for the determinant line bundle. For the fiber bundle with a dimension of (4l3), we derive the anomaly cancellation formulas of the index gerbes. For the fiber bundle of dimension (4l1), we obtain some results of the eta invariants. Moreover, we give some anomaly cancellation formulas of the Chen–Connes character and the higher elliptic genera. Full article
(This article belongs to the Section Geometry and Topology)
20 pages, 1809 KB  
Article
Automated Box-Counting Fractal Dimension Analysis: Sliding Window Optimization and Multi-Fractal Validation
by Rod W. Douglass
Fractal Fract. 2025, 9(10), 633; https://doi.org/10.3390/fractalfract9100633 - 29 Sep 2025
Abstract
This paper presents a systematic methodology for identifying optimal scaling regions in segment-based box-counting fractal dimension calculations through a three-phase algorithmic framework combining grid offset optimization, boundary artifact detection, and sliding window optimization. Unlike traditional pixelated approaches that suffer from rasterization artifacts, the [...] Read more.
This paper presents a systematic methodology for identifying optimal scaling regions in segment-based box-counting fractal dimension calculations through a three-phase algorithmic framework combining grid offset optimization, boundary artifact detection, and sliding window optimization. Unlike traditional pixelated approaches that suffer from rasterization artifacts, the method used directly analyzes geometric line segments, providing superior accuracy for mathematical fractals and other computational applications. The three-phase optimization algorithm automatically determines optimal scaling regions and minimizes discretization bias without manual parameter tuning, achieving significant error reduction compared to traditional methods. Validation across the Koch curve, Sierpinski triangle, Minkowski sausage, Hilbert curve, and Dragon curve demonstrates substantial improvements: excellent accuracy for the Koch curve (0.11% error) and significant error reduction for the Hilbert curve. All optimized results achieve R20.9988. Iteration analysis establishes minimum requirements for reliable measurement, with convergence by level 6+ for the Koch curve and level 3+ for the Sierpinski triangle. Each fractal type exhibits optimal iteration ranges where authentic scaling behavior emerges before discretization artifacts dominate, challenging the assumption that higher iteration levels imply more accurate results. Application to a Rayleigh–Taylor instability interface (D = 1.835 ± 0.0037) demonstrates effectiveness for physical fractal systems where theoretical dimensions are unknown. This work provides objective, automated fractal dimension measurement with comprehensive validation establishing practical guidelines for mathematical and real-world fractal analysis. The sliding window approach eliminates subjective scaling region selection through systematic evaluation of all possible linear regression windows, enabling measurements suitable for automated analysis workflows. Full article
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