Pith Number

pith:YOUHLXWS

pith:2015:YOUHLXWSKLCMTZLIKIEOW5E5QZ

not attested not anchored not stored refs pending

Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

Quansheng Liu, Zhi-Qiang Gao

arxiv:1512.07391 v3 · 2015-12-23 · 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{YOUHLXWSKLCMTZLIKIEOW5E5QZ}

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	2026-05-18T01:00:53.463641Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c3a875ded252c4c9e5685208eb749d86634ac5152b1ef89b38f646e09f136757

Aliases

arxiv: 1512.07391 · arxiv_version: 1512.07391v3 · doi: 10.48550/arxiv.1512.07391 · pith_short_12: YOUHLXWSKLCM · pith_short_16: YOUHLXWSKLCMTZLI · pith_short_8: YOUHLXWS

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/YOUHLXWSKLCMTZLIKIEOW5E5QZ \
  | 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: c3a875ded252c4c9e5685208eb749d86634ac5152b1ef89b38f646e09f136757

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a7b2f5e4287d958b79431acc901950ecb3a6cb98ee31d7d6604426bd131be87a",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2015-12-23T08:47:08Z",
    "title_canon_sha256": "2cce0ed2f04c38cfc230e07a212397e5fb39354bd593b180e687e390db2892ea"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1512.07391",
    "kind": "arxiv",
    "version": 3
  }
}