This study extends the one-dimensional anharmonic oscillators by implementing physics-informed transformer networks (PINNs) for multi-dimensional quantum systems. We develop a novel computational framework that combines transformer architecture with physics-informed neural networks to solve the Schrodinger equation for 2D and 3D anharmonic oscillators, addressing both perturbative and non-perturbative regimes. The methodology incorporates attention mechanisms to capture long-range quantum correlations, orthogonal loss functions for eigenfunction discovery, and adaptive training protocols for progressive dimensionality scaling. Our approach successfully computes eigenvalues and eigenfunctions for quartic anharmonic oscillators in multiple dimensions with coupling parameters ranging from weak (λ = 0.01) to strong (λ = 1000) regimes. Results demonstrate superior accuracy compared to traditional neural networks, with mean absolute errors below 10^-6 for ground state energies and the successful capture of symmetry breaking in anisotropic systems. The transformer-based architecture requires 60% fewer trainable parameters than conventional feedforward networks while maintaining comparable accuracy. Applications to molecular vibrational systems and solid-state physics demonstrate the practical utility of this approach for realistic quantum mechanical problems beyond the scope of perturbative methods.

This paper presents an enhanced variant of the RSA cryptosystem that integrates chaotic exponent selection and ciphertext blinding to address security limitations inherent in classical RSA. Traditional RSA relies on a fixed public exponent, which generates predictable encryption patterns and increases exposure to exponent-based attacks. In the proposed scheme, the encryption exponent is dynamically derived from a logistic-map–based chaotic sequence, introducing high sensitivity to initial conditions and producing session-dependent exponent values. This chaotic exponentiation increases unpredictability without modifying the established RSA framework. Additionally, a ciphertext blinding factor is incorporated to prevent deterministic outputs and strengthen resistance against chosen-ciphertext and side-channel attacks. The paper outlines the mathematical background of the logistic map, details the complete encryption and decryption procedures, and demonstrates the correctness of the method through a numerical example using small primes. A theoretical security analysis shows that the combined effects of chaotic exponent selection and blinding significantly improve resistance to key-related attacks while maintaining compatibility with the original RSA structure. These enhancements offer a lightweight and practical improvement to RSA for environments requiring increased confidentiality and unpredictability in exponent selection.

This paper explores the use of stochastic optimization techniques to address the aircraft allocation problem under uncertain passenger demand. The proposed stochastic allocation model successfully meets the study’s objectives by demonstrating how uncertainty in passenger demand can be effectively incorporated into aircraft assignment decisions through a two-stage stochastic programming framework. Simulation results across multiple demand scenarios show that the model provides stable and adaptive allocations that minimize total cost while maintaining service quality, even under high variability. Incorporating the simple recourse approach enables post-decision flexibility, reducing penalties for unmet demand, and the use of Geometric Brownian Motion (GBM) offers a realistic representation of continuous demand fluctuations over time. These outcomes confirm the model’s practical value in bridging deterministic planning and real-time decision environments. While future research will focus on extending the model to a Markov Decision Process (MDP) framework and integrating real-time data streams, the current results establish a solid foundation by quantifying how uncertainty directly impacts fleet utilization, cost efficiency, and service reliability.

This paper investigates the digits of π within a probabilistic framework based on Markov chains, proposing this model as a rigorous tool to support the conjecture of π’s  uniformity. Unlike simple frequency analyses, the Markov approach captures the dynamic structure of transitions between digits, allowing us to compute empirical stationary distributions that reveal how local irregularities evolve toward global equilibrium. This ergodic behavior provides quantitative, model based evidence that the digits of π tend toward fairness in the long run. Beyond its mathematical significance, this convergence toward uniformity invites a broader conceptual interpretation.

Urban traffic congestion can be considered as a significant problem, and it contributes to long travel periods, fuel usage, and environmental influence. This paper introduces an Intelligent Traffic Control System (ITCS) that consists of Vehicular Ad Hoc Networks (VANETs) and Reinforcement Learning (RL) to optimise the control of traffic signals. The system facilitates real-time two-way communication between vehicles and roadside units, which means that an RL agent can control signal phases adaptively according to the traffic metrics like the average delay, the queue length, and traffic throughput. The Kaggle VANET Malicious Node Dataset was used to simulate malicious or unreliable nodes and test the robustness of the systems. The RL agent has been trained on the SUMO simulator trained on TraCI through various episodes and learns to take actions that increase traffic movement with a minimum amount of congestion. The results of training are progressive, as cumulative rewards grow, and average delays and queue length reduce with epochs. Performance evaluation of the ITCS under peak-hour, off-peak, incident, and malicious node scenarios demonstrated substantial gains over conventional fixed-time controllers, with average delays reduced by 48–55%, queue lengths by 49–57%, and throughput increased by 28–35%. These results indicate the success of the blend of reinforcement learning with VANET-supported traffic control, which is an adaptive, data-driven, and robust solution to an urban intersection. Not only the RL-based ITCS enhances traffic flow and congestion, but is also resistant to communication anomalies, which indicates its scalability to be deployed in the current smart city traffic management.