Mathematical and Computational Applications

In this paper, we present an integrative framework based on Evolutionary Stackelberg Game Theory to model the strategic interaction between a physician, acting as a rational leader, and a heterogeneous population of treatment-sensitive and treatment-resistant breast cancer cells. The model incorporates ecological competition, evolutionary adaptation, and spatial heterogeneity, enabling prediction of tumor progression under clinically relevant treatment protocols. Using tumor volume data obtained from breast cancer-bearing mice treated with Capecitabine and Gemcitabine, we estimated treatment and subject-specific parameters via the GEKKO optimization package in Python. Benchmarking against classical tumor growth models (Exponential, Logistic, and Gompertz) showed that while classical models capture monotonic growth, they fail to reproduce complex, non-monotonic behaviors such as treatment-induced regression, rebound, and phenotypic switching. The game-theoretic approach achieved superior alignment with experimental data across Maximum Tolerated Dose, Dose-Modulation Adaptive Therapy, and Intermittent Adaptive Therapy protocols. To enhance adaptability, we integrated reinforcement learning (RL) for both single-agent and combination chemotherapy. The RL agent learned dosing policies that maximized tumor regression while minimizing cumulative drug exposure and resistance, with combination therapy exploiting dose diversification to improve control without exceeding total dose budgets. Incorporating reaction diffusion equations allowed the model to capture spatial dispersal of sensitive (cooperative) and resistant (defector) phenotypes, revealing that spatially aware adaptive strategies more effectively suppress resistant clones than non-spatial approaches. These results demonstrate that evolutionarily informed, spatially explicit, and computationally optimized strategies can outperform conventional fixed-dose regimens in reducing resistance, lowering toxicity, and improving efficacy. This framework offers a biologically interpretable tool for guiding evolution-aware, patient-tailored cancer therapies toward improved long-term outcomes.

Ensuring the structural integrity of road infrastructure is vital for transportation safety and long-term sustainability. This study presents a lightweight and accurate pavement crack detection framework named SpcNet, which integrates a Sparse Encoding Module, ConvNeXt V2-based decoder, and a Binary Attention Module (BAM) within an asymmetric encoder–decoder architecture. The proposed method first applies a random masking strategy to generate sparse pixel inputs and employs sparse convolution to enhance computational efficiency. A ConvNeXt V2 decoder with Global Response Normalization (GRN) and GELU activation further stabilizes feature extraction, while the BAM, in conjunction with Channel and Spatial Attention Bridge (CAB/SAB) modules, strengthens global dependency modeling and multi-scale feature fusion. Comprehensive experiments on four public datasets demonstrate that SpcNet achieves state-of-the-art performance with significantly fewer parameters and lower computational cost. On the Crack500 dataset, the method achieves a precision of 91.0%, recall of 85.1%, F1 score of 88.0%, and mIoU of 79.8%, surpassing existing deep-learning-based approaches. These results confirm that SpcNet effectively balances detection accuracy and efficiency, making it well-suited for real-world pavement condition monitoring.

This paper develops an algebraic framework for operator matrix polynomials and demonstrates its application to control-design problems in aeroservoelastic systems. We present constructive spectral-factorization and linearization tools (block spectral divisors, companion forms and realization algorithms) that enable systematic block-pole assignment for large-scale MIMO models. Building on this theory, an adaptive block-pole placement strategy is proposed and cast in a practical implementation that augments a nominal state-feedback law with a compact neural-network compensator (single hidden layer) to handle un-modeled nonlinearities and uncertainty. The method requires state feedback and the system’s nominal model and admits Laplace-domain analysis and straightforward implementation for a two-degree-of-freedom aeroelastic wing with cubic stiffness nonlinearity and Roger aerodynamic lag is validated in MATLAB R2023a. Comprehensive simulations (Runge–Kutta 4) for different excitations and step disturbances demonstrate the approach’s advantages: compared with Eigenstructure assignment, LQR and $H_2$-control, the proposed method achieves markedly better robustness and transient performance (e.g., closed-loop ${‖\mathit{H}\left(i\omega \right)‖}_2$ ≈ 4.64, condition number χ ≈ 11.19, and reduced control efforts μ ≈ 0.41, while delivering faster transients and tighter regulation (rise time ≈ 0.35 s, settling time ≈ 1.10 s, overshoot ≈ 6.2%, steady-state error ≈ 0.9%, disturbance-rejection ≈ 92%). These results confirm that algebraic operator-polynomial techniques, combined with a compact adaptive NN augmentation, provide a well-conditioned, low-effort solution for robust control of aeroelastic systems.