∞ Differential Equations, Abstract Algebra & Applied Math

97 topics

• What Is a Differential Equation? (Advanced) – An equation relating a function and its derivatives.
• Order and Linearity of ODEs (Advanced) – Classify equations by highest derivative and linearity.
• Direction Fields (Advanced) – Slope fields show solution curves visually.
• Separable Equations (Advanced) – Separate variables and integrate both sides.
• First-Order Linear Equations (Advanced) – Use an integrating factor.
• Exponential Growth and Decay Models (Advanced) – dy/dt = ky gives y = y₀eᵏᵗ.
• Logistic Growth (Advanced) – Growth limited by carrying capacity.
• Newton’s Law of Cooling (Advanced) – Temperature approaches ambient exponentially.
• Mixing Problems (Advanced) – Rate in minus rate out for concentrations.
• Exact Equations (Expert) – M dx + N dy = 0 with Mᵧ = Nₓ.
• Bernoulli Equations (Expert) – Substitute v = y¹⁻ⁿ to linearise.
• Euler’s Method (Advanced) – Numerical stepping along slope field.
• Runge–Kutta Methods (Expert) – Higher-accuracy numerical solvers.
• Second-Order Homogeneous Equations (Expert) – Use the characteristic equation.
• Undetermined Coefficients (Expert) – Guess the particular solution form.
• Variation of Parameters (Expert) – General method for particular solutions.
• Mechanical Vibrations (Expert) – Springs, damping and resonance.
• RLC Circuits (Expert) – Electrical analogue of the spring-mass system.
• Laplace Transform (Expert) – Turn ODEs into algebra.
• Laplace Transform Method (Expert) – Transform, solve, invert.
• Convolution and Impulse Response (Expert) – Delta function inputs and Green’s functions.
• Systems of Linear ODEs (Expert) – Solve with eigenvalues and eigenvectors.
• Phase Plane Analysis (Expert) – Trajectories of autonomous systems.
• Equilibria and Stability (Expert) – Classify fixed points as stable, unstable or saddle.
• Predator–Prey (Lotka–Volterra) (Expert) – Cyclical population dynamics.
• SIR Epidemic Model (Expert) – Compartment model for disease spread.
• Power Series Solutions (Expert) – Solve ODEs with series around a point.
• Bessel and Legendre Equations (Expert) – Special functions from ODEs.
• Sturm–Liouville Theory (Expert) – Eigenvalue problems for ODEs.
• Boundary Value Problems (Expert) – Conditions specified at both ends.
• Partial Differential Equations (Expert) – Equations with partial derivatives.
• Heat Equation (Expert) – Diffusion of temperature.
• Wave Equation (Expert) – Vibrating strings and sound.
• Laplace’s Equation (Expert) – Steady-state potentials.
• Separation of Variables for PDEs (Expert) – Product solutions and Fourier series.
• Method of Characteristics (Expert) – Solve first-order PDEs along curves.
• Fourier Methods for PDEs (Expert) – Series and transforms solve boundary problems.
• Green’s Functions (Expert) – Response to a point source.
• Navier–Stokes Equations (Overview) (Expert) – Equations of fluid flow.
• Maxwell’s Equations (Math View) (Expert) – Vector calculus form of electromagnetism.
• Schrödinger Equation (Math View) (Expert) – Wave equation of quantum mechanics.
• Numerical Root Finding (Advanced) – Bisection, secant and Newton methods.
• Bisection Method (Advanced) – Halve an interval that brackets a root.
• Fixed-Point Iteration (Advanced) – Repeat x = g(x) until it converges.
• Interpolation (Expert) – Construct a function through data points.
• Lagrange Polynomials (Expert) – Explicit interpolation polynomial.
• Splines (Expert) – Piecewise polynomial interpolation.
• Numerical Differentiation (Expert) – Finite differences approximate derivatives.
• Floating Point and Rounding Error (Expert) – Computer arithmetic is approximate.
• Numerical Linear Algebra (Expert) – Stable algorithms for large systems.
• Gradient Descent (Expert) – Iteratively move opposite the gradient.
• Convex Functions and Sets (Expert) – Functions whose chords lie above the graph.
• Linear Programming and Simplex (Expert) – Optimise linear objectives.
• Duality in Optimisation (Expert) – Pair every optimisation problem with a dual.
• KKT Conditions (Expert) – Necessary conditions for constrained optima.
• Dynamic Programming (Expert) – Solve problems by overlapping subproblems.
• Game Theory: Basics (Expert) – Strategies, payoffs and equilibria.
• Nash Equilibrium (Expert) – No player gains by deviating alone.
• Zero-Sum Games and Minimax (Expert) – Optimal mixed strategies.
• Operations Research Overview (Expert) – Scheduling, routing and resource allocation.
• Queueing Theory (Expert) – Waiting lines modelled mathematically.
• Information Theory and Coding (Expert) – Entropy, compression and channel capacity.
• Cryptography Mathematics (Expert) – Number theory and algebra behind secure communication.
• Elliptic Curves (Expert) – Curves y² = x³ + ax + b with a group law.
• Groups: Definition (Expert) – A set with an associative operation, identity and inverses.
• Examples of Groups (Expert) – ℤ, ℤₙ, symmetry groups and matrix groups.
• Subgroups (Expert) – Subsets that are groups themselves.
• Cyclic Groups (Expert) – Groups generated by a single element.
• Permutation Groups and Sₙ (Expert) – Groups of rearrangements.
• Dihedral Groups (Expert) – Symmetries of regular polygons.
• Cosets and Lagrange’s Theorem (Expert) – The order of a subgroup divides the order of the group.
• Normal Subgroups and Quotients (Expert) – Subgroups allowing a quotient structure.
• Group Homomorphisms (Expert) – Structure-preserving maps between groups.
• Isomorphism Theorems (Expert) – Relationships between kernels, images and quotients.
• Group Actions (Expert) – Groups acting on sets.
• Orbit–Stabiliser Theorem (Expert) – Orbit size times stabiliser size equals group order.
• Burnside’s Lemma (Expert) – Count orbits using fixed points.
• Sylow Theorems (Expert) – Existence of subgroups of prime-power order.
• Finite Abelian Groups (Expert) – Every finite abelian group is a product of cyclic groups.
• Rings and Ideals (Expert) – Sets with addition and multiplication; ideals generalise multiples.
• Integral Domains and Fields (Expert) – Rings without zero divisors; fields have inverses.
• Polynomial Rings (Expert) – Polynomials over a ring and divisibility.
• Finite Fields (Expert) – Fields with pⁿ elements.
• Field Extensions (Expert) – Larger fields containing a base field.
• Galois Theory Introduction (Expert) – Symmetries of roots of polynomials; unsolvability of the quintic.
• Ruler and Compass Impossibility (Expert) – Trisecting angles and doubling the cube are impossible.
• Modules (Expert) – Generalise vector spaces over rings.
• Category Theory Introduction (Expert) – Objects, morphisms and functors.
• Lattices and Boolean Algebras (Expert) – Ordered structures with meet and join.
• Algebraic Geometry Introduction (Expert) – Study of solution sets of polynomial equations.
• Representation Theory Introduction (Expert) – Groups acting through matrices.
• Lie Groups and Lie Algebras (Expert) – Continuous symmetry groups.
• Graph Theory in Networks (Applied) (Expert) – Social, transport and web graphs.
• Machine Learning Mathematics (Expert) – Linear algebra, calculus and probability behind learning algorithms.
• Neural Networks and Backpropagation (Expert) – The chain rule applied to layered models.
• Math Competition Strategies (Expert) – Techniques for olympiad problems.
• Mathematical Modelling Process (Advanced) – Define, simplify, solve, validate and refine a model.