Technologies that attain high adoption on a given value axis, the dimension of human need they address, frequently exhibit what we term the ‘illusion of permanence’: a local adoption maximum that observers systematically misinterpret as a stable equilibrium. We formalise this phenomenon through the Adoption-Decay Superposition Model (ADSM), in which each technology’s market share is expressed as the product of a logistic growth term and an exponential decay envelope activated by the emergence of a superior competitor. We derive the Illusion Point, the exact moment at which adoption is maximised and the impending decline is least visible. From this we introduce two derived indices; the Illusion Strength Index (ISI) and the Displacement Susceptibility (DS). Formal theorems establish that every technology with a positive decay parameter must eventually relinquish dominance, and that switching costs prolong the illusion without averting it. Four empirical cases, SMS, the ‘Please Call Me’ callback feature, BlackBerry Messenger, and the fax machine, are parametrised and simulated in MATLAB. A Conservation Principle is stated, linking individual technology decline to the bounded capacity of any shared value axis. The model provides a foundation for anticipating displacement, evaluating technological lock-in, and understanding why dominant technologies consistently fail to foresee their own decline.
This paper investigates the spectral properties of the symmetric circulant matrix 𝐴𝑛(𝑎,𝑏)=circ (𝑎,𝑏,0,…,0,𝑏), where 𝑎,𝑏∈ℝ and 𝑛≥3. While the eigenvalue structure of general circulant matrices is well understood, explicit and unified characterizations for specific structured subclasses remain of interest. In this work, we derive closed-form expressions for the eigenvalues and provide a complete characterization of the positive definiteness of this matrix family, explicitly highlighting the role of the parity of 𝑛. In addition, we obtain a compact determinant formula using Chebyshev polynomials, yielding an analytically tractable condition for singularity. The results establish a direct connection between circulant matrix theory and classical trigonometric polynomial identities, providing a unified framework that links spectral properties, determinant structure, and parity effects. These findings extend existing formulations by providing explicit, structurally transparent results for this class of symmetric circulant matrices.
Digital music streaming platforms have transformed the way audiences consume music, making Spotify one of the leading sources for analyzing music popularity and listening trends. This study investigates music genre trends in the Spotify Indonesia Top 100 dataset for May 2025 using a graph-based community detection approach. Songs were represented as nodes connected through genre similarity and normalized popularity attributes, including Streams, Total Streams, and Days on Chart. A weighted graph was constructed using Euclidean distance and inverse-distance similarity, after which communities were detected using the Louvain algorithm. The detected communities were further analyzed according to genre composition and streaming performance, while community quality was evaluated using modularity. The results identified 15 communities with a modularity value of 0.776, indicating a well-defined community structure. Although genre information was incorporated during graph construction, the detected communities were also influenced by similarities in popularity characteristics, revealing structural relationships beyond simple genre classification. Based on accumulated Total Streams, Indie Pop, Pop, and Pop Rock emerged as the dominant genres in the analyzed dataset. These findings demonstrate that graph-based community detection provides additional insights into song connectivity and genre organization that cannot be obtained through conventional descriptive aggregation alone.
A mathematical model to study the effect of toxicants in water and sediments on aquatic population is proposed and analyzed. The model has six possible equlibria. By applying stability theory, it was demonstrated that the overall species population stabilizes at an equilibrium level. However, it was found that as the toxicant emission rates increases, the aquatic population density was severely affected and the aquatic population decreased significantly.
Electricity is a fundamental need for modern society, and in developing regions such as West Nusa Tenggara, the continuous growth in the number of customers requires PT PLN (Persero) to conduct effective resource planning to prevent potential energy crises. This study aims to predict the growth of PLN’s customer numbers using the Facebook Prophet time series model. A quantitative approach was applied using monthly customer data from PT PLN (Persero) covering the period from January 2019 to December 2024. The model was optimized through a hyperparameter tuning process, and its performance was evaluated using the Mean Absolute Percentage Error (MAPE) metric. The results demonstrate that the optimized model achieved a MAPE of 0.27% during cross-validation. Analysis of the results indicates that the model effectively captured long-term growth trends and seasonal fluctuations. These findings suggest that the Prophet model can serve as a technical reference for forecasting customer numbers, potentially supporting strategic decision-making and resource allocation at PT PLN (Persero).
