sigmoid neuron vs p

This app explains how a single sigmoid neuron works. You can have some insight for neural network.

Every rational number lives simultaneously in infinitely many completions of Q: the real line at infinity, and the p-adic fields Q_2, Q_3, Q_5, and so on. Most of modern number theory's deepest theorems are statements about how these completions fit together. p-adelic puts all of them on one screen. THE ADELIC STRIP Type a number - rational, algebraic radical, polynomial, Hilbert pair, special-function value - and a horizontal strip materializes with one column per place: infinity on the left, then 2, 3, 5, 7, and so on. There is no "p-adic mode": you see all completions at once. Tap a column to expand; toggle between digit-string and Laurent series; long-press to copy as plain text or LaTeX. ARBITRARY PRECISION Every computation runs through arbitrary-precision BigInt. Type 2^60 / 3 and the parser hands the engine a real BigInt - no Int.max ceilings, no silent truncation. The precision dial at the bottom of every screen is the canonical gesture: drag it and digits stream in across every place, lazily extended via long division, Hensel lifting, or whichever generator the source warrants. ALGEBRAIC NUMBERS sqrt(17) lifts via Hensel iteration at every prime where it splits, and renders 4.12310562 at infinity to dial-controlled depth. algebraic(x^3 - 2) triggers the algebraic engine: Newton polygon partitions roots by valuation, F_p[x] factorization splits the unramified part, Hensel lifts each factor to Z_p[x] mod p^N. The Galois toggle cycles three roots at p = 31 (fully split), one totally-ramified root at p = 2 (Eisenstein), one inert F_343 lift at p = 7. CYCLOTOMIC IN FULL zeta_n at every prime: split, inert with residue degree greater than 1 (zeta_7 at p = 2 with F_8 digits per level), ramified (zeta_4 at p = 2 with uniformizer 1 - zeta_4). The strip's per-prime (e, f) metadata is correct; the Galois row lets you walk embeddings. THREE VISUALIZATIONS Nested balls: the canonical Z_p picture as a tangent-circle packing. Pinch-zoom drills into the on-path sub-ball, which becomes the new outer disk; a breadcrumb shows descended digits. Bruhat-T**s tree: the (p^f + 1)-regular tree of PGL_2 over the local field. Tap any node for its coordinates. Berkovich line: the analytic refinement. Type II vertices at rational radii along the geodesic from the Gauss point to the Type I leaf. SPECIAL FUNCTIONS exp_p, log_p, Artin-Hasse E_p with proper convergence checks. Morita's p-adic Gamma_p including Wilson's theorem at every prime. fact(n) via Legendre's formula: fact(1000) at p = 2 reports v_2 = 994 instantly without materializing 1000 factorial.