Science & Education
Longitudinal Doppler: redshift 1+z = √((1+β)/(1−β)) for receding sources, blueshift for approaching ones, with observed frequency, wavelength, and band labels.
relativistic-doppler-effectScience & Education
K = (γ−1)mc² with rest, total, kinetic, and classical ½mv² energies in MeV/GeV plus the K/K_classical ratio.
relativistic-kinetic-energyScience & Education
p = γmv with p·c in eV-scale units, the Lorentz factor, and the classical mv comparison for electron, proton, or custom mass.
relativistic-momentumScience & Education
E₀ = mc² for electron, proton, neutron, or custom mass: energy in J and eV-scale units, mass in kg and u — the collider mass table.
rest-mass-energyScience & Education
Solve ∫|ψ|²dx = 1 for A: infinite well √(2/L), Gaussian (πσ²)^(−1/4), oscillator ground (mω/πℏ)^(1/4), exponential √(2α), with checks and moments.
wave-function-normalizerScience & Education
Rydberg lines for Lyman/Balmer/Paschen/Brackett/Pfund: vacuum wavelengths, photon energies, named lines with Balmer colors, series limit.
hydrogen-spectrum-seriesDate & Time
Parse ISO 8601 durations, validate ISO 8601-2 extensions, run calendar-true arithmetic, sort by real length at an anchor, and diff datetimes.
iso-8601-duration-parser-arithmetic-workbenchMath & Numbers
Smallest primitive root mod n with certificate g^(φ/q) ≠ 1, root count φ(φ(n)), up to 50 listed roots, and candidate verification. n ≤ 10¹².
primitive-root-finderMath & Numbers
Exact π(n) via a single-pass Sieve of Eratosthenes (n ≤ 10⁷) with n/ln n comparison, density, average gap, and the largest prime ≤ n.
prime-counting-functionMath & Numbers
Two-person zero-sum games: saddle-point test first, then mixed strategies from a single-phase simplex LP with dual shadow prices, verified by security-level checks.
game-theory-zero-sumMath & Numbers
Dijkstra with a full per-iteration trace: settled nodes, relaxations, distance table, and the final path; undirected or directed, negative weights rejected, unreachable targets reported.
graph-shortest-pathNetwork
Parse a raw HTTP request or curl command, estimate client↔server clock skew from the response Date header (NTP-style midpoint formula), inspect timestamp credentials (X-Amz-Date, x-ms-date, X-Timestamp, JWT iat/nbf), replay with ±skew offsets and diff the results against real tolerance windows (AWS SigV4 ±15 min, Azure, OAuth, JWT leeway).
request-replay-clock-skew-correction-distributed-systems