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Introduction to complex numbers, their arithmetic operations, Euler's formula, and polar coordinates. Covers solutions to differential equations and lays foundation for further complex analysis topics.
Explore dynamical systems: fixed points, linearization, manifolds, bifurcations, and chaos. Gain insights into nonlinear dynamics and their applications in describing our changing world.
Explore solving the wave equation using Laplace transforms, covering problem setup, PDE to ODE conversion, and solution methods including the Heaviside function and method of characteristics.
Explore Moore's Law, exponential growth in computing, and its impact on AI, scientific computing, and future technologies. Understand the implications for technological advancement.
Explore techniques to analyze nonlinear differential equations by linearizing dynamics near fixed points, enabling powerful linear ODE solutions for nonlinear systems.
Learn to solve systems of linear ODEs using eigenvalues and eigenvectors. Covers problem setup, solution methods, and intuitive interpretation, with code examples in Python and MATLAB.
Explore eigenvalues and eigenvectors through differential equations, deriving key concepts and visualizing their applications in matrix systems for a deeper understanding of linear algebra.
Learn to transform high-order linear differential equations into matrix systems, enabling powerful solutions using eigenvalues and eigenvectors. Explore practical examples and key concepts in linear algebra applications.
Learn to solve high-order linear differential equations using characteristic polynomials and linear superposition. Covers guessing solutions, finding general solutions, and applying initial conditions.
Comprehensive derivation of the heat equation, exploring its mathematical foundations and physical principles in thermal energy conservation and transfer.
Explore Stokes' and Green's theorems, their geometric interpretations, and practical applications in mathematical physics, including examples and land area calculations.
Explore the divergence operator in vector calculus, understanding its role in quantifying local expansion or contraction of vector fields, with examples and applications in fluid dynamics.
Comprehensive exploration of deep reinforcement learning techniques, including Q-learning, actor-critic methods, policy networks, and gradient optimization for advanced AI decision-making and control systems.
Explore Q-learning and temporal difference algorithms in reinforcement learning, including SARSA and connections to biological learning through dopamine. Covers key concepts for deep reinforcement learning.
Explore dynamic programming in model-based reinforcement learning, covering policy iteration, value iteration, and Q-learning for effective decision-making strategies.
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