1. An Isogeometric Tearing and Interconnecting (IETI) method for solving high order partial differential equations over planar multi-patch geometriesMario Kapl, Aljaž Kosmač, Vito Vitrih, 2026, original scientific article Abstract: We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries with
possibly extraordinary vertices demonstrated on the basis of the polyharmonic equation of order m, m ≥ 1, which is a particular linear elliptic PDE of order 2m. Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) and allows to couple the numerical solution of the PDE with Cs-smoothness, , across the edges of the multi-patch geometry. The proposed technique relies on the use of a particular class of multi-patch geometries, called bilinear-like Gs multi-patch parameterizations, to represent the multi-patch domain. The coupling between the neighboring patches is done via the use of Lagrange multipliers and leads to a saddle point problem, which can be solved first by a small dual problem for a subset of the Lagrange multipliers followed by local, parallelizable problems on the single patches for the coefficients of the numerical solution. Several numerical examples for the polyharmonic equation of order m = 1, m = 2 and m = 3, i.e. for the Poisson’s, the biharmonic and the triharmonic equation, respectively, are shown to demonstrate the potential of our IETI method for solving high-order problems over planar multi-patch geometries with possibly extraordinary vertices.
Keywords: isogeometric analysis, Galerkin method, C^s-smoothness, Tearing and Interconnecting, multi-patch domain, polyharmonic equation
Published in RUP: 02.02.2026; Views: 104; Downloads: 4
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2. Integrating AI-driven wearable metaverse technologies into ubiquitous blended learning : a framework based on embodied interaction and multi-agent collaborationJiaqi Xu, Xuesong Zhai, Nian-Shing Chen, Usman Ghani, Andreja Istenič, Junyi Xin, 2025, original scientific article Abstract: Ubiquitous blended learning, leveraging mobile devices, has democratized education by enabling autonomous and readily accessible knowledge acquisition. However, its reliance on traditional interfaces often limits learner immersion and meaningful interaction. The emergence of the wearable metaverse offers a compelling solution, promising enhanced multisensory experiences and adaptable learning environments that transcend the constraints of conventional ubiquitous learning. This research proposes a novel framework for ubiquitous blended learning in the wearable metaverse, aiming to address critical challenges, such as multi-source data fusion, effective human–computer collaboration, and efficient rendering on resource-constrained wearable devices, through the integration of embodied interaction and multi-agent collaboration. This framework leverages a real-time multi-modal data analysis architecture, powered by the MobileNetV4 and xLSTM neural networks, to facilitate the dynamic understanding of the learner’s context and environment. Furthermore, we introduced a multi-agent interaction model, utilizing CrewAI and spatio-temporal graph neural networks, to orchestrate collaborative learning experiences and provide personalized guidance. Finally, we incorporated lightweight SLAM algorithms, augmented using visual perception techniques, to enable accurate spatial awareness and seamless navigation within the metaverse environment. This innovative framework aims to create immersive, scalable, and cost-effective learning spaces within the wearable metaverse.
Keywords: metaverse, embodied interaction, wearable, multi-agent, artificial intelligence, ubiquitous blended learning
Published in RUP: 17.07.2025; Views: 759; Downloads: 8
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3. A ▫$C^s$▫-smooth mixed degree and regularity isogeometric spline space over planar multi-patch domainsMario Kapl, Aljaž Kosmač, Vito Vitrih, 2026, original scientific article Abstract: We construct over a given bilinear multi-patch domain a novel $C^s$-smooth mixed degree and regularity isogeometric spline space, which possesses the degree $p=2s+1$ and regularity $r=s$ in a small neighborhood around the edges and vertices, and the degree~$\widetilde{p} \leq p$ with regularity $\widetilde{r} = \widetilde{p}-1 \geq r$ in all other parts of the domain. Our proposed approach relies on the technique Kapl and Vitrih (2021), which requires for the $C^s$-smooth isogeometric spline space a degree at least $p=2s+1$ on the entire multi-patch domain. Similar to Kapl and Vitrih (2021), the $C^s$-smooth mixed degree and regularity spline space is generated as the span of basis functions that correspond to the individual patches, edges and vertices of the domain. The reduction of degrees of freedom for the functions in the interior of the patches is achieved by introducing an appropriate mixed degree and regularity underlying spline space over $[0,1]^2$ to define the functions on the single patches. We further extend our construction with a few examples to the class of bilinear-like $G^8$ multi-patch parameterizations (Kapl and Vitrih (2018); Kapl and Vitrih (2021)), which enables the design of multi-patch domains having curved boundaries and interfaces. Finally, the great potential of the $C^8$-smooth mixed degree and regularity isogeometric spline space for performing isogeometric analysis is demonstrated by several numerical examples of solving two particular high order partial differential equations, namely the biharmonic and triharmonic equation, via the isogeometric Galerkin method.
Keywords: isogeometric analysis, Galerkin method, C^s-smoothness, mixed degree and regularity spline space, multi-patch domain
Published in RUP: 01.07.2025; Views: 694; Downloads: 4
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9. Chemical and appearance changes of wood due to artificial weathering - dose-response modelMarta Petrillo, Jakub Michal Sandak, Paolo Grossi, Anna Malgorzata Sandak, 2019, original scientific article Keywords: wood weathering, dose-response model, multi-sensor measurement, near infrared, colour, gloss
Published in RUP: 13.02.2019; Views: 4338; Downloads: 183
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10. A three-person deterministic graphical game without Nash equilibriaEndre Boros, Vladimir Gurvich, Martin Milanič, Vladimir Oudalov, Jernej Vičič, 2018, original scientific article Abstract: We give an example of a three-person deterministic graphical game that has no Nash equilibrium in pure stationary strategies. The game has seven positions, four outcomes (a unique cycle and three terminal positions), and its normal form is of size only. Thus, the example strengthens significantly the one obtained in 2014 by Gurvich and Oudalov; the latter has four players, five terminals, and normal form of size . Furthermore, our example is minimal with respect to the number of players. Somewhat similar examples were known since 1975, but they were not related to deterministic graphical games. The small size of our example allows us to verify that it has no Nash equilibrium not only in pure but also in independently mixed (so-called behavioral) strategies. For independently mixed strategies two distinct effective payoffs can be considered: along with the classical Markovian evaluation, we also consider a priori evaluation, which may be a better fit for playing in behavioral strategies
Keywords: deterministic graphical multi-person game, perfect information, Nash equilibrium, directed cycle, terminal position, pure stationary strategy
Published in RUP: 30.03.2018; Views: 5599; Downloads: 179
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