Pith Number

pith:4WXO3Q6Q

pith:2015:4WXO3Q6QRQBKKPYB4YCW2E3UYV

not attested not anchored not stored refs pending

Sobolev inequalities for the Hardy-Schr\"odinger operator: Extremals and critical dimensions

Fr\'ed\'eric Robert, Nassif Ghoussoub

arxiv:1506.05787 v1 · 2015-06-18 · math.AP

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{4WXO3Q6QRQBKKPYB4YCW2E3UYV}

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	2026-05-18T01:44:09.681574Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e5aeedc3d08c02a53f01e6056d1374c56d13c1a234eeca74f31269da542104e4

Aliases

arxiv: 1506.05787 · arxiv_version: 1506.05787v1 · doi: 10.48550/arxiv.1506.05787 · pith_short_12: 4WXO3Q6QRQBK · pith_short_16: 4WXO3Q6QRQBKKPYB · pith_short_8: 4WXO3Q6Q

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/4WXO3Q6QRQBKKPYB4YCW2E3UYV \
  | 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: e5aeedc3d08c02a53f01e6056d1374c56d13c1a234eeca74f31269da542104e4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "65048ad702b7fceb5367b1d8acf901e9dbb1dab98cded7e79ea76c5934493ed0",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2015-06-18T19:47:25Z",
    "title_canon_sha256": "988a66824f7de632e02e3f87fe982d6dceb45dbb1c7a8734914089b60cf09ed0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1506.05787",
    "kind": "arxiv",
    "version": 1
  }
}