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Search Results (396)

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12 pages, 414 KiB  
Article
Teachers’ Resilience Scale for Sustainability Enabled by ICT/Metaverse Learning Technologies: Factorial Structure, Reliability, and Validation
by Vassilios Makrakis
Sustainability 2024, 16(17), 7679; https://doi.org/10.3390/su16177679 - 4 Sep 2024
Abstract
A significant trend in education is the increasing recognition of the need to shift from transmissive teaching to incorporating reflexive practices associated with real-life issues in learning, curriculum, and teaching. Merging Information and Communication Technologies (ICTs) and Metaverse learning technologies in Education for [...] Read more.
A significant trend in education is the increasing recognition of the need to shift from transmissive teaching to incorporating reflexive practices associated with real-life issues in learning, curriculum, and teaching. Merging Information and Communication Technologies (ICTs) and Metaverse learning technologies in Education for Sustainability (ICT/MeEfS) is critical in responding to current sustainability crises such as climate change. This research article focuses on the factorial structure, reliability, and validity of a teachers’ ICT/MeEfS resilience scale. It examines the predictive value of teacher self-efficacy and transformative teaching beliefs in merging ICTs and education for sustainability. The respondents were 1815 in-service teachers in Indonesia, Malaysia, and Vietnam. The principal component analysis showed a two-factor model (factor 1: “personal ICT/MeEfS resilience” and factor 2: “reflexive practice”), with a significant amount of extracted variance (68.26%). The overall Cronbach’s alpha reliability analysis of the teachers’ resilience scale enabled by ICT/MeEfS was 0.90, indicating a high score and excellent internal consistency. Similarly, the stepwise multiple regression analysis revealed that the two hypothesized predictors, teacher self-efficacy and transformative teaching beliefs, significantly contributed to teachers’ ICT/MeEfS resilience, explaining 73% of its variability. The implications of the research results are discussed in terms of research and in developing the capacity of teachers to embed sustainability issues and SDGs in teaching practices, learning environments, and course curricula enabled by ICTs and Metaverse learning technologies. Full article
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29 pages, 1629 KiB  
Review
Metaverse of Things (MoT) Applications for Revolutionizing Urban Living in Smart Cities
by Tanweer Alam
Smart Cities 2024, 7(5), 2466-2494; https://doi.org/10.3390/smartcities7050096 - 3 Sep 2024
Viewed by 176
Abstract
The Metaverse of Things (MoT) is an advanced technology that has the potential to revolutionise urban living in the present era. This article explores the advantages, uses, and transformative outcomes of the MoT in smart cities. It encompasses sustainability, urban planning, citizen participation, [...] Read more.
The Metaverse of Things (MoT) is an advanced technology that has the potential to revolutionise urban living in the present era. This article explores the advantages, uses, and transformative outcomes of the MoT in smart cities. It encompasses sustainability, urban planning, citizen participation, infrastructure management, and more. MoT integrates the Internet of Things (IoT) with metaverse technologies. The ultimate objective is to develop virtual environments that are highly interactive, interconnected, and immersive while maintaining a high level of fidelity to reality. The IoT utilises virtual reality (VR), augmented reality (AR), and other digital technologies to gather data, facilitate communication, and automate certain processes, thereby enhancing several elements of urban living. The IoT will bring about a profound transformation in the way cities gather and utilise data to enhance services and optimise efficiency. Cities that can efficiently distribute this data can enhance public safety, optimise energy usage, regulate traffic, and manage waste properly. MoT apps that utilise immersive technologies and the IoT can be used to generate more intelligent and captivating cityscapes. The implementation of the MoT can greatly enhance the quality of life for residents of smart cities through improvements in transportation, healthcare, education, and community engagement. This study’s author examined how smart cities utilise the MoT to enhance the daily experiences of their inhabitants. This study examines the technical structure, possible advantages, and difficulties of implementing the MoT in urban settings, aiming to enhance the resilience, responsiveness, and adaptability of cities. The findings emphasise the importance of robust legislative frameworks, stringent security requirements, and well-developed infrastructure to facilitate the extensive use of MoT technology. These factors are crucial for establishing a highly interconnected and efficient urban environment. Full article
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21 pages, 856 KiB  
Article
Analyzing the Impact of Metaverse Technology on Social Development: A Field Study on Generation Z in the United Arab Emirates
by Enaam Youssef, Mervat Medhat, Soumaya Abdellatif and Nagwa Babiker Yousif
Soc. Sci. 2024, 13(9), 446; https://doi.org/10.3390/socsci13090446 - 27 Aug 2024
Viewed by 741
Abstract
Considering the current role of technology, this research aims to investigate the impact of the metaverse on social development and psychological well-being in the United Arab Emirates (UAE). The focus remains on Generation Z, as an important portion of the country’s metaverse users. [...] Read more.
Considering the current role of technology, this research aims to investigate the impact of the metaverse on social development and psychological well-being in the United Arab Emirates (UAE). The focus remains on Generation Z, as an important portion of the country’s metaverse users. Supported by the technology determinism theory, this research used a structured questionnaire to gather data from 363 respondents. The analysis indicates that metaverse technology significantly impacts social development and psychological well-being. The effect of metaverse technology on social development remained positive, with the p-value at p > 0.000. In addition, metaverse technology also has a significant positive effect on psychological well-being (p > 0.000), with the findings showing strong statistical support for this relationship. These results confirm that metaverse technology plays a significant role in improving both social and psychological facets. Thus, it is concluded that the metaverse creates immersive virtual environments that help unique forms of social interaction and connectivity, confounding geographical barriers. This can improve social skills and communication and promote a strong sense of community. The metaverse offers a directed and engaging space to practice social interactions and gain confidence, indicating its positive, constructive role in society. Full article
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16 pages, 3072 KiB  
Article
A Learner-Centric Explainable Educational Metaverse for Cyber–Physical Systems Engineering
by Seong-Jin Yun, Jin-Woo Kwon, Young-Hoon Lee, Jae-Heon Kim and Won-Tae Kim
Electronics 2024, 13(17), 3359; https://doi.org/10.3390/electronics13173359 - 23 Aug 2024
Viewed by 311
Abstract
Cyber–physical systems have become critical across industries. They have driven investments in education services to develop well-trained engineers. Education services for cyber–physical systems require the hiring of expert tutors with multidisciplinary knowledge, as well as acquiring expensive facilities/equipment. In response to the challenges [...] Read more.
Cyber–physical systems have become critical across industries. They have driven investments in education services to develop well-trained engineers. Education services for cyber–physical systems require the hiring of expert tutors with multidisciplinary knowledge, as well as acquiring expensive facilities/equipment. In response to the challenges posed by the need for the equipment and facilities, a metaverse-based education service that incorporates digital twins has been explored as a solution. However, the issue of recruiting expert tutors who can enhance students’ achievements remains unresolved, making it difficult to effectively cultivate talent. This paper proposes a reference architecture for a learner-centric educational metaverse with an intelligent tutoring framework as its core feature to address these issues. We develop a novel explainable artificial intelligence scheme for multi-class object detection models to assess learners’ achievements within the intelligent tutoring framework. Additionally, a genetic algorithm-based improvement search method is applied to the framework to derive personalized feedback. The proposed metaverse architecture and framework are evaluated through a case study on drone education. The experimental results show that the explainable AI scheme demonstrates an approximately 30% improvement in the explanation accuracy compared to existing methods. The survey results indicate that over 70% of learners significantly improved their skills based on the provided feedback. Full article
(This article belongs to the Special Issue Applied Machine Learning in Intelligent Systems)
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14 pages, 9857 KiB  
Article
The Application of Metaverse in the Tourism Sector as a Tool for Enhancing Sustainability—Case Study: A Medieval ‘Perfume Burner’ of the Local Historical Museum of Montilla (Cordoba, Spain)
by Paula Triviño-Tarradas, Alejandro Mohedo-Gatón, Pilar Carranza-Cañadas and Rafael E. Hidalgo-Fernandez
Sustainability 2024, 16(16), 6966; https://doi.org/10.3390/su16166966 - 14 Aug 2024
Viewed by 551
Abstract
Tourism is an excellent pathway with which to promote and provide access to cultural heritage. Sustainable cultural tourism offers a new perspective, as it places cultural heritage and local communities at the centre of decision-making processes. In this context, traditional tourism must be [...] Read more.
Tourism is an excellent pathway with which to promote and provide access to cultural heritage. Sustainable cultural tourism offers a new perspective, as it places cultural heritage and local communities at the centre of decision-making processes. In this context, traditional tourism must be updated accordingly to the new times. This research aims to classify a cultural heritage piece found in the ruins of what used to be the Castle of Montilla, and produce a 3D virtual model using innovative technologies, to be appreciated in the metaverse, as a virtual museum. The idea of tourism as a real movement of people should be shifted little by little to an idea of connection without the need of being there exactly. This research succeeded with this idea. A ‘perfume burner’, catalogued as a lamp, and currently located in the medieval section of the Local Historical Museum of Montilla (Cordoba, Spain), was virtually reconstructed. In the attainment of these physical subjects transferred to the metaverse, photogrammetry played a crucial role. Obtaining a 3D digital model of a heritage asset enables the public in general, present and non-present visitors, to appreciate such cultural pieces. In this context, tourist agents and resources might be changed and updated to bring tourist services and clients. The metaverse is paramount in creating virtual museums, providing the possibility of establishing connections with people without the need to be physically in a certain place. The virtual museum leverages digital technology to promote sustainability by reducing travels and minimising the environmental impact. To date, this research enhances the experiences of the users at the local museum. Full article
(This article belongs to the Special Issue Sustainable Cultural Tourism in the Digital Age)
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28 pages, 1196 KiB  
Article
Advanced Observation-Based Bipartite Containment Control of Fractional-Order Multi-Agent Systems Considering Hostile Environments, Nonlinear Delayed Dynamics, and Disturbance Compensation
by Asad Khan, Muhammad Awais Javeed, Saadia Rehman, Azmat Ullah Khan Niazi and Yubin Zhong
Fractal Fract. 2024, 8(8), 473; https://doi.org/10.3390/fractalfract8080473 - 13 Aug 2024
Viewed by 497
Abstract
This paper introduces an advanced observer-based control strategy designed for fractional multi-agent systems operating in hostile environments. We take into account the dynamic nature of the agents with nonlinear delayed dynamics and consider external disturbances affecting the system. The manuscript presents an improved [...] Read more.
This paper introduces an advanced observer-based control strategy designed for fractional multi-agent systems operating in hostile environments. We take into account the dynamic nature of the agents with nonlinear delayed dynamics and consider external disturbances affecting the system. The manuscript presents an improved observation-based control approach tailored for fractional-order multi-agent systems functioning in challenging conditions. We also establish various applicable conditions governing the creation of observers and disturbance compensation controllers using the fractional Razmikhin technique, signed graph theory, and matrix transformation. Furthermore, our investigation includes observation-based control on switching networks by employing a typical Lyapunov function approach. Finally, the effectiveness of the proposed strategy is demonstrated through the analysis of two simulation examples. Full article
(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)
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25 pages, 3068 KiB  
Article
Metaverse Unveiled: From the Lens of Science to Common People Perspective
by Mónica Cruz, Abílio Oliveira and Alessandro Pinheiro
Computers 2024, 13(8), 193; https://doi.org/10.3390/computers13080193 - 13 Aug 2024
Viewed by 473
Abstract
Everyone forms a perception about everything, including the Metaverse. Still, we may expect a gap or disconnection between what has been expressed by various researchers and the widespread perceptions of technology and related concepts. However, the degree to which these two frames of [...] Read more.
Everyone forms a perception about everything, including the Metaverse. Still, we may expect a gap or disconnection between what has been expressed by various researchers and the widespread perceptions of technology and related concepts. However, the degree to which these two frames of representation differ awaits further investigation. This study seeks to compare the Metaverse perceptions between the scientific findings and the common people’s perceptions using the data from two previous qualitative studies about the representations of the Metaverse from a scientific perspective versus a common perspective (by adults). Is there a common ground between these two perspectives? Or are they in opposition? As goals for this research, we aim to contrast the depiction of the Metaverse in pertinent studies (published in indexed journals) with the portrayal of the Metaverse among adults (non-researchers); ascertain the most prevalent depiction of virtual reality; and determine the significance of gaming within the representations of the Metaverse and virtual reality. This investigation encapsulates crucial findings on the Metaverse concept, contrasting the discoveries made by researchers in prior studies with the common public’s interpretation of this concept. It helps with understanding the differences between the Metaverse representations, the immersion and perception concepts, and a disagreement from the past vs. future perspective. Full article
(This article belongs to the Special Issue Extended or Mixed Reality (AR + VR): Technology and Applications)
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22 pages, 2481 KiB  
Review
Blockchain Technology and Its Potential to Benefit Public Services Provision: A Short Survey
by Giorgio Piccardo, Lorenzo Conti and Alessio Martino
Future Internet 2024, 16(8), 290; https://doi.org/10.3390/fi16080290 - 9 Aug 2024
Viewed by 809
Abstract
In the last few years, blockchain has emerged as a cutting-edge technology whose main advantages are transparency, traceability, immutability, enhanced efficiency, and trust, thanks to its decentralized nature. Although many people still identify blockchain with cryptocurrencies and the financial sector, it has many [...] Read more.
In the last few years, blockchain has emerged as a cutting-edge technology whose main advantages are transparency, traceability, immutability, enhanced efficiency, and trust, thanks to its decentralized nature. Although many people still identify blockchain with cryptocurrencies and the financial sector, it has many prospective applications beyond digital currency that can serve as use cases for which traditional infrastructures have become obsolete. Governments have started exploring its potential application to public services provision, as confirmed by the increasing number of adoption initiatives, projects, and tests. As the current public administration is often perceived as slow, bureaucratic, lacking transparency, and failing to involve citizens in decision-making processes, blockchain can establish itself as a tool that enables a process of disintermediation, which can revolutionize the way in which public services are managed and provided. In this paper, we will provide a survey of the main application areas which are likely to benefit from blockchain implementation, together with examples of practical implementations carried out by both state and local governments. Later, we will discuss the main challenges that may prevent its widespread adoption, such as government expenditure, technological maturity, and lack of public awareness. Finally, we will wrap up by providing indications on future areas of research for blockchain-based technologies. Full article
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22 pages, 3408 KiB  
Article
Microservices-Based Resource Provisioning for Multi-User Cloud VR in Edge Networks
by Ho-Jin Choi, Nobuyoshi Komuro and Won-Suk Kim
Electronics 2024, 13(15), 3077; https://doi.org/10.3390/electronics13153077 - 3 Aug 2024
Viewed by 427
Abstract
Cloud virtual reality (VR) is attracting attention in terms of its lightweight head-mounted display (HMD), providing telepresence and mobility. However, it is still in the research stages due to motion-to-photon (MTP) latency, the need for high-speed network infrastructure, and large-scale traffic processing problems. [...] Read more.
Cloud virtual reality (VR) is attracting attention in terms of its lightweight head-mounted display (HMD), providing telepresence and mobility. However, it is still in the research stages due to motion-to-photon (MTP) latency, the need for high-speed network infrastructure, and large-scale traffic processing problems. These problems are expected to be partially solved through edge computing, but the limited computing resource capacity of the infrastructure presents new challenges. In particular, in order to efficiently provide multi-user content such as remote meetings on edge devices, resource provisioning is needed that considers the application’s traffic patterns and computing resource requirements at the same time. In this study, we present a microservice architecture (MSA)-based application to provide multi-user cloud VR in edge computing and propose a scheme for planning an efficient service deployment considering the characteristics of each service. The proposed scheme not only guarantees the MTP latency threshold for all users but also aims to reduce networking and computing resource waste. The proposed scheme was evaluated by simulating various scenarios, and the results were compared to several studies. It was confirmed that the proposed scheme represents better performance metrics than the comparison schemes in most cases from the perspectives of networking, computing, and MTP latency. Full article
(This article belongs to the Special Issue Recent Advances of Cloud, Edge, and Parallel Computing)
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18 pages, 1952 KiB  
Article
Exploring the Use of Avatars in the Sustainable Edu-Metaverse for an Alternative Assessment: Impact on Tolerance
by Yara Ahmed Mohebeldin Zaky and Azza Ali Gameil
Sustainability 2024, 16(15), 6604; https://doi.org/10.3390/su16156604 - 1 Aug 2024
Viewed by 599
Abstract
This article investigates how avatars in the metaverse can be used for peer evaluation in a sustainable educational setting. The goal is to find the best alternative modes of assessment (one-to-many/many-to-one) to evaluate the design of a sustainable educational environment. The research also [...] Read more.
This article investigates how avatars in the metaverse can be used for peer evaluation in a sustainable educational setting. The goal is to find the best alternative modes of assessment (one-to-many/many-to-one) to evaluate the design of a sustainable educational environment. The research also explores how learners’ avatars influence the development of tolerance and respect for others within the metaverse. The study involved 36 female graduates from King Faisal University’s College of Education. A quasi-experimental design with two experimental groups was employed to assess performance. Through a product quality card and applying a tolerance (IPTS) scale for data collection. The preliminary findings indicate that the avatars within the (one-to-many) group demonstrated better performance and showed an increase in tolerance values after the intervention, compared to their counterparts (many-to-one) as an alternative mode of assessment, in the development of a sustainable Edu-Metaverse environment. The research validated that avatars enhance positive attitudes and behaviors, thereby providing insights for developing more effective educational interventions and contributing to enhancing the user experience and implementation of sustainable educational initiatives through a metaverse. Full article
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20 pages, 2938 KiB  
Article
Market Value or Meta Value? The Value of Virtual Land during the Metaverse’s Digital Era
by Aurora Greta Ruggeri, Giuliano Marella and Laura Gabrielli
Land 2024, 13(8), 1135; https://doi.org/10.3390/land13081135 - 25 Jul 2024
Viewed by 377
Abstract
Nowadays, some of the most expensive real estate is not “real” at all. Several investors are purchasing land in the virtual world of the Metaverse. To be more accurate in the wording, they are buying “meta-estates”. This work is dedicated to opening a [...] Read more.
Nowadays, some of the most expensive real estate is not “real” at all. Several investors are purchasing land in the virtual world of the Metaverse. To be more accurate in the wording, they are buying “meta-estates”. This work is dedicated to opening a debate about this emerging research field within the real estate discipline. It begins by discussing market segmentation, ownership, and pricing by comparing the traditional real estate market with the virtual estate market. Furthermore, this study involved interviews with six seasoned Metaverse land investors who participated in two Analytic Hierarchy Processes (AHPs). The first AHP ranked 14 investment typologies, while the second focused on ranking and discussing the most important characteristics of meta-estates that influence the formation of prices. As a result, the most appealing investments identified were day-trading, virtual land trading (buying to resell), and virtual land development (transforming and reselling). The primary characteristics of meta-estates considered by investors include the platform (e.g., Earth 2, Sandbox), the location within the platform (proximity to famous neighbours), and the architectural design of the buildings (designed by renowned architects). It is evident that the Metaverse represents a new frontier for land investors, and the primary aim of this study was to encourage other researchers to explore and investigate this evolving field. Full article
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23 pages, 1124 KiB  
Review
Navigating the Metaverse: A New Virtual Tool with Promising Real Benefits for Breast Cancer Patients
by Weronika Magdalena Żydowicz, Jaroslaw Skokowski, Luigi Marano and Karol Polom
J. Clin. Med. 2024, 13(15), 4337; https://doi.org/10.3390/jcm13154337 - 25 Jul 2024
Cited by 1 | Viewed by 781
Abstract
BC, affecting both women and men, is a complex disease where early diagnosis plays a crucial role in successful treatment and enhances patient survival rates. The Metaverse, a virtual world, may offer new, personalized approaches to diagnosing and treating BC. Although Artificial Intelligence [...] Read more.
BC, affecting both women and men, is a complex disease where early diagnosis plays a crucial role in successful treatment and enhances patient survival rates. The Metaverse, a virtual world, may offer new, personalized approaches to diagnosing and treating BC. Although Artificial Intelligence (AI) is still in its early stages, its rapid advancement indicates potential applications within the healthcare sector, including consolidating patient information in one accessible location. This could provide physicians with more comprehensive insights into disease details. Leveraging the Metaverse could facilitate clinical data analysis and improve the precision of diagnosis, potentially allowing for more tailored treatments for BC patients. However, while this article highlights the possible transformative impacts of virtual technologies on BC treatment, it is important to approach these developments with cautious optimism, recognizing the need for further research and validation to ensure enhanced patient care with greater accuracy and efficiency. Full article
(This article belongs to the Section Clinical Guidelines)
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19 pages, 2603 KiB  
Article
Development of a Metaverse Art Gallery of Image Chronicles (MAGIC) for Healthcare Education: A Digital Health Humanities Approach to Patients’ Medication Experiences
by Kevin Yi-Lwern Yap, Jayen Ho and Phylaine Shu Ting Toh
Information 2024, 15(8), 431; https://doi.org/10.3390/info15080431 - 25 Jul 2024
Viewed by 968
Abstract
Art therapy fosters emotional healing and growth. This process can offer healthcare professionals (HCPs) novel insights into patients’ medication experiences. We developed a Metaverse Art Gallery of Image Chronicles (MAGIC), which depicted patients’ medication experiences symbolically as hero–villain portrayals. This gallery aimed to [...] Read more.
Art therapy fosters emotional healing and growth. This process can offer healthcare professionals (HCPs) novel insights into patients’ medication experiences. We developed a Metaverse Art Gallery of Image Chronicles (MAGIC), which depicted patients’ medication experiences symbolically as hero–villain portrayals. This gallery aimed to enhance healthcare students’ learning through relatable insights into patients’ medication therapies. A character sheet was used to craft patients’ personifications of their medication experiences through an art-based narrative therapy approach. ChatGPT, NightCafe, Canva, HeyGen, and Camtasia were used to generate hero–villain portraits based on the character traits and mounted in MAGIC, which consisted of three virtual realms, each with a unique theme. Alpha-testing among sixteen Generation Z healthcare learners indicated that the content in MAGIC enabled them to understand the concepts of medication adherence (93.7%), art therapy (87.5%), and how patients related to their medications (81.3%). Perceived playfulness (rs = 0.925, p < 0.001), perceived compatibility (rs = 0.890, p < 0.001), and social norm (rs = 0.862, p < 0.001) were strongly associated with their behavioral intention to adopt MAGIC as an educational platform. The learners enjoyed their experience (6.31 ± 0.70), felt that MAGIC was interactive and engaging (6.25 ± 0.78), and had the potential to be more effective than traditional learning methods (5.94 ± 0.93). Furthermore, they would recommend it to others for their education (5.94 ± 0.85). Full article
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20 pages, 8044 KiB  
Review
Affective Computing in Augmented Reality, Virtual Reality, and Immersive Learning Environments
by Georgios Lampropoulos, Pablo Fernández-Arias, Álvaro Antón-Sancho and Diego Vergara
Electronics 2024, 13(15), 2917; https://doi.org/10.3390/electronics13152917 - 24 Jul 2024
Viewed by 560
Abstract
As students’ affective states can differ between learning that occurs in traditional classrooms when compared with learning that takes place in extended reality and immersive learning environments, it is important to examine the role of affective computing. Therefore, this study aims to provide [...] Read more.
As students’ affective states can differ between learning that occurs in traditional classrooms when compared with learning that takes place in extended reality and immersive learning environments, it is important to examine the role of affective computing. Therefore, this study aims to provide an overview regarding the use of affective computing in the context of augmented reality, virtual reality, and immersive learning environments and the metaverse. Hence, it examines and presents the state of the art through a review and bibliometric analysis of 188 documents of the existing literature from Scopus and Web of Science (WoS) from 2005 to 2023. The study follows the Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA) statement to identify and select relevant to the topic documents. In addition to the analysis of the existing literature, emerging topics and themes are identified and future research directions are presented. The significant role of affective computing within augmented reality and virtual reality environments arose. Their ability to offer engaging and interactive learning experiences while also being able to recognize, monitor, and respond to students’ affective states and to consider their emotions, personalities, characteristics, knowledge, and preferences to provide adaptive and personalized learning was evident. Additionally, their potential to enrich educational activities, increase learning outcomes, and support special education emerged. When augmented reality, virtual reality, and immersive learning environments capitalize on affective computing, meaningful learning can occur and self-regulated learning can be promoted. Full article
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24 pages, 2445 KiB  
Article
On Some Distance Spectral Characteristics of Trees
by Sakander Hayat, Asad Khan and Mohammed J. F. Alenazi
Axioms 2024, 13(8), 494; https://doi.org/10.3390/axioms13080494 - 23 Jul 2024
Viewed by 456
Abstract
Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study [...] Read more.
Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with “few eigenvalues” is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is “highly” non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions. Full article
(This article belongs to the Special Issue Advances in Classical and Applied Mathematics)
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