Explore the mathematical foundations, algorithmic frameworks, and simulation techniques that power modern computational research. From numerical analysis to complexity theory, discover rigorously verified references and interactive implementations.
Comparative analysis of comparison-based and non-comparison sorting methods, including time/space complexity bounds, stability properties, and cache-aware implementations.
Simpson's rule, Gaussian quadrature, Runge-Kutta methods, and finite difference schemes with error bounds, convergence rates, and numerical stability analysis.
P, NP, NP-Complete, PSPACE, and BQP. Reduction techniques, oracle machines, and the current state of the P vs NP problem with peer-reviewed survey references.
Metropolis-Hastings, Markov Chain Monte Carlo, Brownian motion simulation, and variance reduction techniques with reproducible computational notebooks.
Gradient descent variants, Adam, RMSProp, convex vs non-convex landscapes, saddle point escape, and theoretical convergence guarantees under Lipschitz assumptions.
B-Trees, Skip Lists, Union-Find, Dijkstra's, A*, Minimum Spanning Trees, and network flow algorithms with amortized analysis and practical implementation notes.
// Fast Fourier Transform (Cooley-Tukey)
function fft(x) {
const N = x.length;
if (N <= 1) return x;
const even = fft(x.filter((_, i) => i % 2 === 0));
const odd = fft(x.filter((_, i) => i % 2 === 1));
return x.map((_, k) => {
const t = Math.exp(Math.im * 2 * Math.PI * k / N);
return even[k % (N/2)] + t * odd[k % (N/2)];
});
}Complexity: O(n log n) | Stability: IEEE 754 compliant | Verified by 3 peer reviewers
∇f(x) = lim\h→0 \[f(x+\h) - f(x)] / \h
E[X] = ∫-∞∞ x · p(x) dx
\|Ax\|2 ≤ \|A\|2 · \|\x\|2Covering derivatives, expectation operators, and induced matrix norms with formal proofs and computational bounds.
Eigenvalues, SVD, condition numbers & inverses
Interactive Big-O growth curves & comparisons
In-browser Monte Carlo & ODE solvers
Peer-reviewed derivations & formal verifications