This academic research first provides a foundational overview of hypergeometric differential equations and hypergeometric functions in specific regions of the complex plane. Building on this foundation, it develops a comprehensive set of results that reveal various relationships between the relevant classes of equations and functions, supported by carefully formulated propositions, deductions, and illustrative examples. In particular, this study emphasizes the logical consistency among the analysis, theoretical results, and applications, demonstrating how these relationships extend existing perspectives in hypergeometric theory. Moreover, the study concludes with key deductions and recommendations based on the findings, suggesting potential directions for future research. In light of the references cited, the contributions of this work aim to promote a broader understanding of hypergeometric structures and their applications in complex analysis and related fields.
The paper considers the model of a heterogeneous queuing system H K ( t ) / M / ∞ and limit evolutions in the averaging and diffusion approximation scheme. The input process of the system under study is modelled as a mixture of independent heterogeneous Poisson processes, which reflects the behaviour of modern cloud serverless platforms, where tasks arrive simultaneously from different event sources (HTTP gateways, object stores, message queues, event buses). Based on the apparatus of semi-Markovian random evolutions, the weak convergence of the queue length process L ( t ) in two asymptotic schemes: averaging (law of large numbers) and diffusion approximation (central limit theorem), with the Ornstein–Uhlenbeck limit process, is proved. As a result, simple analytical estimates of the queue length L ( t ) and the probability of exceeding the maximum length P ( sup t ∈ [ 0 , T ] L ( t ) < L MAX ) are constructed. Also, a necessary and sufficient condition for the queue's finiteness is expressed in terms of the parameters of the input and task-processing processes. The results obtained enable us to develop analytical methods for parameter estimation and predictive scaling for complex serverless systems in cloud environments, without the need to train models or neural networks on large amounts of data.
The advent of competitive power markets in recent decades has facilitated a substantial rise in transactions between producers and consumers to meet the rapidly growing load demand. Power transactions in modern electricity markets are carried out through bilateral or multilateral agreements, either directly between market participants or through common trading platforms. These transactions require the transmission network to transfer electricity from generating stations to consumers, and excessive power transfers may lead to congestion when line capacity limits are exceeded. Such congestion affects the secure and economical execution of scheduled transactions and may increase operating costs. Therefore, the Independent System Operator (ISO) plays an important role in maintaining reliable system operation by managing congestion and ensuring that all network security constraints are satisfied. This study aims to reduce transmission congestion by controlling overloaded line flows, maintaining operational limits, improving system stability, and minimizing transmission losses. Generator Sensitivity Factor (GSF) is used to identify the generating units that significantly influence congested transmission lines and require suitable rescheduling. Two enhanced Monarch Butterfly Optimization approaches, namely Improved MBO with a Linear Weighting factor (IMBO-LW) and Chaos-Embedded MBO (CHMBO), are employed to determine optimal generator rescheduling strategies. The proposed methods simultaneously minimize rescheduling cost and changes in generator power output. Their performance is evaluated using standard benchmark functions and further validated on a modified IEEE 30-bus test system, demonstrating their effectiveness in achieving secure, reliable, and economically efficient congestion management.
This study reports the development and implementation of a Metaverse-based learning medium using the Roblox platform to improve German language skills at the A1 level, specifically in the areas of Hören (listening) and Schreiben (writing). A descriptive-developmental research design was used, involving 30 undergraduate students taking an introductory German course at a public university in Indonesia. The learning environment was designed through Roblox Studio to replicate an interactive virtual classroom equipped with immersive scenarios and game-based features to support language practice. Data were collected through a Likert-scale questionnaire, open-ended questions, and classroom observations. Quantitative results indicated that most students highly rated the design, ease of use, and its contribution to improving listening and writing skills. Qualitative findings further revealed that students felt more motivated, engaged, and connected through the immersive features and game-style environment, despite occasional technical difficulties and distractions. The study concluded that the Metaverse-based learning medium on Roblox offers an innovative and effective alternative to conventional online learning by providing contextualized practice opportunities, enhancing social presence, and strengthening student engagement in foreign language learning.
A pollutant model and its adjoint model for a bounded zone with an open boundary are considered; dual estimates of pollutant concentrations are derived, and a method for the real-time control of emissions is presented. Methods are proposed for identifying the intensity and location of a new, unknown pollution source.
