Develop mathematical models to study population growth, migration patterns, and demographic changes, exploring implications for public policy and urban planning.
Perform numerical simulations of fluid flow phenomena using computational fluid dynamics (CFD) methods, analyzing turbulence, heat transfer, and fluid-structure interactions.
Explore fractal geometry and chaos theory concepts, studying fractal patterns, strange attractors, and nonlinear dynamics using mathematical algorithms and simulations.
Implement optimization algorithms such as linear programming, genetic algorithms, or simulated annealing to solve complex optimization problems in logistics, finance, or engineering.
Analyze cryptographic algorithms, protocols, and security mechanisms, exploring their strengths and vulnerabilities for protecting digital data and communications.
Develop mathematical models for financial markets, portfolio optimization, and risk assessment, applying stochastic calculus and Monte Carlo simulations.
Investigate graph theory concepts and network properties, analyzing complex networks like social networks, transportation networks, or biological networks.
Model biological systems and disease dynamics using mathematical equations, studying epidemiological patterns, vaccination strategies, and disease spread control measures.
Explore quantum information theory concepts such as quantum entanglement, quantum cryptography, and quantum algorithms for quantum computing applications.
Implement numerical methods for solving differential equations, numerical integration, and approximation algorithms for scientific computing applications.
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