Pith Number

pith:FIZWYQEG

pith:2016:FIZWYQEGTPX4TZBLNY5VTLYGUL

not attested not anchored not stored refs pending

The Navier-Stokes-$\alpha$ equation via forward-backward stochastic differential systems

Guoping Liu

arxiv:1610.09421 v1 · 2016-10-28 · math.PR

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{FIZWYQEGTPX4TZBLNY5VTLYGUL}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Receipt and verification

First computed	2026-05-18T01:00:52.052860Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2a336c40869befc9e42b6e3b59af06a2d7ec65108950745f56eaa9699e93ba65

Aliases

arxiv: 1610.09421 · arxiv_version: 1610.09421v1 · doi: 10.48550/arxiv.1610.09421 · pith_short_12: FIZWYQEGTPX4 · pith_short_16: FIZWYQEGTPX4TZBL · pith_short_8: FIZWYQEG

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/FIZWYQEGTPX4TZBLNY5VTLYGUL \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 2a336c40869befc9e42b6e3b59af06a2d7ec65108950745f56eaa9699e93ba65

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "59fecb8adf47480598f0c6e0e103e3dee73669444bb722c5a7dc1be526826d84",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2016-10-28T22:44:43Z",
    "title_canon_sha256": "52005256489134bce207559142feb69d5b0c123295a9df50e5e0e8fa9b5229c4"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1610.09421",
    "kind": "arxiv",
    "version": 1
  }
}