Current common challenges such as high-dimensional data processing and steady-state analysis of complex systems have become increasingly prominent. Eigenvalues and eigenvectors, leveraging their unique mathematical properties, play an irreplaceable role in fields such as data mining and system modeling, serving as a crucial bridge connecting theoretical mathematics with practical applications. Through literature review, this study investigates the application of matrix eigenvalues and eigenvectors in Principal Component Analysis (PCA) and Markov chain steady-state analysis. The results demonstrate that matrix eigenvalues and eigenvectors exhibit significant universality and effectiveness in cross-domain applications. Validated in scenarios including PCA and Markov chain steady-state analysis, they help address key issues including high-dimensional data dimensionality reduction, system steady-state prediction, and information prioritization, thereby providing mathematical support for technological optimization. Simultaneously, they can reveal intrinsic system patterns, reflecting a deep analytical capability for system structures. Future research may focus on optimizing algorithms for solving sparse matrix eigenvalues and exploring integration with deep learning and graph neural networks to expand their application boundaries in large-scale complex systems.

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This study explores Sony’s current brand competitiveness from a global perspective, focusing on three strategic pillars: quality signals, global mythology, and corporate social responsibility (CSR). First, it evaluates Sony’s market position by analyzing its continuous investment in research and development (R&D) to drive technological innovation and maintain product excellence, taking PlayStation series as an example. Second, it examines how Sony shapes and leverages its “Creative Entertainment Company” myth that integrates technological pioneering with emotional storytelling and demonstrates its strategic pivot to creativity-driven segments. Moreover, the paper investigates Sony’s commitment to CSR, including its “Road to Zero” environmental plan, sustainability actions, and diversity targets, which enhance brand trust and long-term resilience. The study concludes with recommendations for the company to address anti-globalization sentiments and wisely manage its national identity by emphasizing local R&D, community engagement, cultural exchange, and highlighting its “Made in Japan” quality where appropriate. Through a case study of Sony, this paper offers insights into brand development strategies for multinational companies in the new era.

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The issue of population aging has long been a key focus in the field of demography, involving multiple factors such as social development, economy, and policy. This study utilizes the grey prediction GM(1,1) model, calculating residuals between actual data and predicted values, then applying grey prediction to the residuals. By superimposing the initial forecast with the residuals, the model corrects errors and verifies that the resulting calculations exhibit high accuracy. Focusing on Shanghai's elderly population aged 60 and above, subdivided into young-old, middle-old, and old-old age groups, the error-corrected GM(1,1) method is separately applied to forecast changes in the proportion of elderly population across these cohorts. From the perspectives of economic and social sustainability, the results are analyzed to address population-related issues, and relevant recommendations are provided for policymakers' consideration.

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The high-frequency futures market exhibits high volatility and frequent institutional changes, which poses significant challenges for portfolio optimization. Traditional techniques such as the mean-variance model and risk equalization tend to show decreased returns and increased risks in such non-stationary markets. To address this challenge, an online portfolio optimization framework based on meta-learning was proposed. This framework incorporates cross-market and cross-period experiences into parameter adjustments, enabling it to quickly adapt to new institutional environments. Minute-level futures data from the Chicago Mercantile Exchange, New York Mercantile Exchange, and Shanghai Futures Exchange from 2019 to 2024 were used, and the market states were labeled using the Markov switching model through the rolling window method. Comparative experiments were conducted with mean-variance optimization, risk equalization, and deep reinforcement learning benchmarks. Empirical results show that the proposed framework outperforms the benchmarks in terms of excess returns, Sharpe ratio, and maximum drawdown in most cases, and has a faster convergence speed and stronger generalization ability in high-volatility institutional environments. The conclusion drawn from these results is that meta-learning is an effective method for solving portfolio optimization problems in non-stationary markets, providing theoretical support and empirical contributions for high-frequency quantitative trading and asset pricing research.

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