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 paper investigates quantum stabilizer codes and our innovative construction methodology. At first, we began with representing fundamental concepts and theories of quantum computing. In order to identify quantum error correcting codes, we extend the method of using polynomials to represent qudits on square lattices to accommodate more complicated situations in general Cayley graphs. The paper briefly reviews essential definitions and examples related to graphs, groups, and stabilizer codes, and later we propose the novel method and demonstrate its application by using the dihedral groupD4.

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With the continuous development of science and technology and other fields in today's world, statistical analysis and research have become indispensable research methods in people's daily lives. Among these, hypothesis testing plays an important role in fields such as biology, medicine, and economics, and has significant effects in most scenarios. However, the suitability and effectiveness of different hypothesis testing methods vary depending on the context, often leading to different outcomes and levels of accuracy. This paper focuses on discussing various hypothesis testing methods in statistics, such as t-test, chi-square test, z-test, F-test, etc. In addition, this study analyzes various cases in real life and optimize and improve some hypothesis testing methods. Based on a review of existing literature, the study explores how traditional hypothesis testing determines statistical significance—where the null hypothesis is rejected if the test statistic falls within the rejection region. To improve and enhance the determination of the significance level, handle uncertainty, and increase the sample size, this paper proposes alternative methods such as NHST test, Bayesian test, big data sequential test, and failure rate hypothesis test from the perspectives of medicine and kinesiology.

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Stock price fluctuation and prediction is a problem that has attracted much attention. There exist many mathematical and statistical problems behind it. In essence, the key to solving this problem lies in capturing the linear and nonlinear characteristics in the time series to predict future price movements. This study investigates the predictive capabilities of two distinct methodologies—Long Short-Term Memory (LSTM) networks and Autoregressive Integrated Moving Average (ARIMA) models—using Apple Inc. (AAPL) stock price data spanning 2016 to 2024. By synthesizing theoretical frameworks with empirical analysis, the research evaluates how each model captures linear trends and nonlinear fluctuations, ultimately proposing a hybrid ARIMA-LSTM architecture to enhance forecasting accuracy. Finally, according to the principal characteristics of the two models, the ARIMA-LSTM hybrid model is constructed. The results show that the hybrid model significantly outperforms single models in terms of RMSE and directional accuracy. Combined with error distribution visualization and volatility analysis, the hybrid model demonstrates efficient performance in achieving prediction optimization through the decomposition of linear and nonlinear components. It provides a new methodological perspective for financial time series modeling.

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