Flash Cards

Numbers of the form a + bi where i² = −1.
See: Spinor Systems (Quaternions)

A set closed under addition and scalar multiplication.
Example: ℝ³ with basis {i, j, k}

Behavior of functions near a point.
Example: limₓ→a f(x)=L ⇒ f is continuous at a

Logical operations over {0,1}.
Example: A∨B, A∧B, ¬A

Blends two functions.
Topic: Fourier & Integral Transforms

Optimization of functionals.
Equation: d/dx(∂L/∂y′) − ∂L/∂y = 0

Encodes sequences in power series.
Example: Fibonacci: G(x)=x/(1−x−x²)

Deals with countable structures.
Topics: graphs, logic, combinatorics, automata

Objects that repeat at different scales.
Example: Mandelbrot set, Sierpinski triangle
See simulations: Simulations

Study of integers and divisibility.
Start here: Polynomials & Primes

Matrices, vectors, transformations.
Example: A x = b ⇒ solutions exist if rank(A)=rank([A|b])

Equations with derivatives.
Example: y′ + p(x)y = q(x) ⇒ integrating factor method

Study of randomness and events.
Example: P(A ∪ B)=P(A)+P(B)−P(A ∩ B)

Mathematical study of strategic interaction.
Example: Nash equilibrium: no player can improve by unilateral change

Nature of continuous space and deformation.
Example: a doughnut ≈ a coffee mug (homeomorphic)

Secure communication via math.
Section: Modular Cryptography

Drawing conclusions from data.
Example: 95% CI: x̄ ± z·σ/√n

Study of systems sensitive to initial conditions.
Example: Logistic map xₙ₊₁ = r xₙ (1 − xₙ)
See: Simulations

Decomposes periodic functions into sines & cosines.
Section: Fourier & Integral Transforms

The foundation of modern math.
Example: A ∩ B = {x | x ∈ A and x ∈ B}

Rigorous study of limits, continuity, and convergence.
Focus: completeness, Cauchy sequences, convergence in ℝ

Statistical modeling of temporal data.
Example: AR(1): xₜ = φxₜ₋₁ + εₜ

Study of algebraic structures with symmetry.
Section: Galois Groups & Group Theory

Solve A·v = λ·v for λ (scalars) and v (vectors).
Example: Diagonalization: A = P D P⁻¹

Study of nodes and edges.
Example: Euler path: visits every edge exactly once

Fit a line to predict response from predictors.
Model: y = β₀ + β₁x + ε

Calculus with several variables.
Example: Gradient ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Properties that remain unchanged under transformations.
Example: Even function: f(−x)=f(x)

Break matrices into simpler components.
Example: A = QR or A = UΣVᵗ (SVD)

Counting structures and arrangements.
Example: C(n, k) = n! / (k!(n−k)!)

Convert functions between domains.
See: Fourier & Integral Transforms

Equations defining sequences recursively.
Example: T(n)=2 T(n/2)+n ⇒ O(n log n)
Optional: Flash Cards or Recursive Q & A

Arithmetic under remainders.
See: Polynomials & Primes

Proof method for statements over ℕ.
Steps: base case + inductive step

Measures runtime growth with input size.
Example: MergeSort ⇒ O(n log n)

Sets with distance functions.
Example: d(x, y)=|x−y| satisfies triangle inequality

Rules for formal reasoning.
Example: If A→B and A, then B (Modus Ponens)

Polynomial equations with integer solutions.
Example: x² + y² = z² ⇒ Pythagorean triples
See: Polynomials & Primes (Number Theory)

Describes how values are spread or concentrated.
Explore: Flash Cards or Examples

Framework for statistical decision-making.
Visit: Drill Down Learning

Reduces dimensions while preserving variance.
Related: Fourier & Geometric Harmonics

Model inspired by the brain’s layered learning.
Explore: Spinor Systems or Modular Cryptography

Crafting useful inputs from raw data.
Related: Jacobian or Math Tools

Unsupervised grouping of data points.
Linked Topic: Cissoid of Diocles

Visualizes prediction outcomes.
Explore: Newton-Raphson Sim or Recursive Q & A

Model complexity trade-offs.
See: Hilbert’s Problems or Mathematical Evolution

Assess classification performance.
Connect with: Polynomials & Primes or Modular Cryptography

Relationships vs inference.
Explore: Euler Product or Mathematical Universe