Pith Number

pith:4ZQU7SJH

pith:2011:4ZQU7SJHBOELN25RXAH3EEHKF2

not attested not anchored not stored refs pending

Lack of compactness in the 2D critical Sobolev embedding, the general case

Hajer Bahouri, Mohamed Majdoub, Nader Masmoudi

arxiv:1112.2998 v1 · 2011-12-13 · math.AP · math.CA · math.FA

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{4ZQU7SJHBOELN25RXAH3EEHKF2}

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	2026-05-18T04:06:31.818331Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e6614fc9270b88b6ebb1b80fb210ea2e9737901a3f7b0ea11615400c83ece055

Aliases

arxiv: 1112.2998 · arxiv_version: 1112.2998v1 · doi: 10.48550/arxiv.1112.2998 · pith_short_12: 4ZQU7SJHBOEL · pith_short_16: 4ZQU7SJHBOELN25R · pith_short_8: 4ZQU7SJH

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/4ZQU7SJHBOELN25RXAH3EEHKF2 \
  | 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: e6614fc9270b88b6ebb1b80fb210ea2e9737901a3f7b0ea11615400c83ece055

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3ba727e78125407f9bbe86b8a5bdbbea804bf7c5f4f71573f7289393eb680cbf",
    "cross_cats_sorted": [
      "math.CA",
      "math.FA"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2011-12-13T18:50:51Z",
    "title_canon_sha256": "72f803874d6e6bab97fdc074a9b811651243eb524644a1f2fb2e751f272cf05b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1112.2998",
    "kind": "arxiv",
    "version": 1
  }
}