Fractals are extraordinary mathematical patterns that are mysterious, elegant, and full of endless mathematical charm.They are not just abstract math concepts,but are also everywhere in nature and daily life. From fern leaves and mountain contours to snowflakes and river networks, fractals show unique beauty in various forms. This paper seeks to the basic ideas of fractals through a rigorous research approach. It adopts dynamical analysis, numerical iteration, and fractal geometry to investigate the fundamental concepts of fractal, including its definition, core properties and classical models such as the Mandelbrot set. The Koch Snowflake and the Sierpiński Triangle are discussed as typical examples to show three main features of fractals: self-similarity, fractal dimension, and complexity from simple rules. Additionally,this paper also summarizes some promising directions for future research of fractal theory. It further explains why fractal geometry is important for people to understand the hidden mathematical rules in the universe. Learning fractals helps people see inherent order in the chaotic and disordered world, thus deepening the understanding of mathematics and nature.

Show more

Wildfires are increasingly severe and frequent, necessitating efficient resource allocation strategies amid potential misinformation. This study proposes a novel framework integrating Game Theory and Mixed Integer Linear Programming (MILP) to optimize wildfire response. By modeling cities as rational agents in a repeated game, the framework introduces severity-dependent penalties for dishonesty and adaptive strategies to incentivize truthful reporting. The MILP formulation maximizes system-wide utility by weighting allocations based on true fire severity, damage-cost correlations, and penalties for misreporting. Simulations involving five U.S. cities demonstrate the model's effectiveness in balancing resource distribution, penalizing exaggeration, and prioritizing high-risk areas. Results highlight the importance of honest reporting and correlation-based allocation, offering actionable insights for emergency responders.

Show more

Accurate sales forecasting of new energy vehicles (NEVs) is critical for automakers, policymakers, and energy infrastructure planners. To address the multifaceted drivers of NEV sales, this study proposes a machine learning‑based forecasting framework using monthly data from China spanning January 2020 to December 2025. Two key innovations are introduced, namely a composite station factor that integrates charging station count, battery swap station count, and gasoline prices to capture infrastructure and fuel cost effects, and a time‑varying categorical variable that encodes major policy shocks, including subsidy phase‑outs, COVID‑19 lockdowns, purchase tax adjustments, and trade‑in incentives. A systematic comparison of multiple models shows that XGBoost equipped with the proposed features achieves the best performance, with a MAE of 61,169, RMSE of 71,972, MAPE of 6.09%, and SMAPE of 6.00%. This represents improvements over the univariate baseline of 62.76% in MAE, 63.14% in RMSE, 63.62% in MAPE, and 63.19% in SMAPE. Finally, ablation experiments and SHAP analysis are performed to validate the effectiveness of the proposed features. The findings demonstrate that the proposed framework offers a practical and interpretable tool for NEV sales forecasting and policy simulation.

Show more

With natural disasters occurring with increasing frequency, the gap between macro-level situational awareness and micro-level field information can substantially reduce the efficiency of emergency logistics. Drawing on differential game theory, this study develops a multi-stakeholder model involving the government, enterprises, and the public under a cross-sector disaster information-matching framework. It compares the optimal collaborative strategies and system-state evolution under a traditional model and an AI-enabled model. The results show that the AI-enabled bidirectional information-matching mechanism can effectively alleviate information silos and logistical blind spots. In addition, both the intensity of cross-sector disaster information matching and the level of public participation significantly increase collaborative investment by the government, enterprises, and the public, thereby generating a scale amplification effect. Further analysis reveals that the collaborative benefits of the system are constrained by the costs of AI adoption. Only when the net benefits generated by AI are sufficient to offset the costs of platform construction and application can participating stakeholders maintain incentives for sustained collaboration, thereby continuously improving overall emergency support performance. These findings provide a theoretical basis for breaking down information barriers in emergency logistics and optimizing emergency resource allocation and dispatch decisions.

Show more