SCIENCE

Theories

12 theories — Fourier, Information, Chaos, Game Theory, Particle Theory, ACO, Queuing and more.

Fourier Theory

Any periodic signal can be decomposed into a sum of sine and cosine waves at harmonic frequencies.

f(x) = a₀/2 + Σ(aₙcos(nx) + bₙsin(nx))

Applications: signal processing, audio compression (MP3), image compression (JPEG), solving PDEs, spectral analysis.

Information Theory

Shannon (1948): quantifies information, compression limits, channel capacity.

H(X) = −Σ p(x)·log₂ p(x)

H = entropy (bits). Channel capacity C = B·log₂(1 + S/N) (Shannon-Hartley). Basis of all digital communication.

Signal Processing

Nyquist-Shannon theorem: sampling rate must be ≥ 2× the highest frequency to reconstruct signal.

f_s ≥ 2·f_max

Key transforms: FFT, DFT, STFT, Wavelet. Filters: FIR, IIR, Butterworth, Chebyshev.

Chaos Theory

Deterministic systems can exhibit unpredictable behavior sensitive to initial conditions (butterfly effect).

xₙ₊₁ = r·xₙ·(1 − xₙ)

Logistic map. Lyapunov exponent λ > 0 indicates chaos. Strange attractors (Lorenz). Applications: weather, population dynamics, cryptography.

Game Theory

Mathematical study of strategic decision-making between rational agents.

Nash Equilibrium: no player can benefit by unilaterally changing strategy. Prisoner's Dilemma, Minimax, Zero-sum vs non-zero-sum. Applications: economics, AI, biology, voting.

Complexity Theory

Classifies computational problems by resource requirements.

P: solvable in polynomial time. NP: verifiable in polynomial time. NP-hard: at least as hard as NP. NP-complete: NP ∩ NP-hard. P=NP? Unsolved (Millennium Prize).

Graph Theory

Study of graphs G = (V, E). Eulerian path (visits each edge once), Hamiltonian path (visits each node once), shortest path (Dijkstra, Bellman-Ford), minimum spanning tree (Kruskal, Prim).

|V| − |E| + |F| = 2 (Euler's formula)

Probability Theory

Axioms: P(∅)=0, P(Ω)=1, P(A∪B)=P(A)+P(B) for disjoint A,B.

P(A|B) = P(A∩B)/P(B)

Bayes' Theorem: P(A|B) = P(B|A)·P(A)/P(B). Central Limit Theorem: sum of iid → normal distribution.

Particle Theory (Moqaddam)

Every entity — from subatomic particles to neurons to software agents — operates as a computational unit with inputs, a processing function, and outputs.

4 axioms: universal particle structure, gravitational interaction as vacuum gradient, emergent swarm intelligence, black holes as refractive boundaries.

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Ant Colony Optimization

Nature-inspired metaheuristic where artificial ants build solutions guided by pheromone trails, simulating real ant foraging behavior.

τᵢⱼ(t+1) = (1−ρ)τᵢⱼ(t) + Δτᵢⱼ

ρ = evaporation rate. Solves TSP, VRP, scheduling. Direct implementation of the Particle Theory (Axiom III).

Control Theory

PID controller: adjusts output based on Proportional, Integral, Derivative error terms.

u(t) = Kₚe + Kᵢ∫e dt + Kd·de/dt

Laplace transform, transfer functions, stability (Bode plot, Nyquist criterion). Used in robotics, HVAC, autopilots.

Queuing Theory

M/M/1 queue: Poisson arrivals (rate λ), exponential service (rate μ).

ρ = λ/μ < 1 (utilization)

Mean queue length L = ρ/(1−ρ). Applications: network traffic, call centers, OS scheduling, healthcare.