Pith Number

pith:YWI75Z2J

pith:2009:YWI75Z2JDMMMEY6QWHLBORDPRW

not attested not anchored not stored refs pending

The Complex Gradient Operator and the CR-Calculus

Ken Kreutz-Delgado

arxiv:0906.4835 v1 · 2009-06-26 · math.OC · math.CV

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{YWI75Z2JDMMMEY6QWHLBORDPRW}

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.

Cited by

10 papers in Pith

Complex Stochastic Gradient Descent and Directional Bias in Reproducing Kernel Hilbert Spaces

When do complex-valued neural networks help? A study of representation, geometry, and optimization

Distributionally Robust Complex Chance-Constrained Optimization

Complex Field Formulation of the Quantum Estimation Theory

Robust Chance Constrained Complex Zero-Sum Games

Receipt and verification

First computed	2026-07-04T15:45:12.056585Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c591fee7491b18c263d0b1d617446f8d8512c36ac491bf3567e2feca3cba42d9

Aliases

arxiv: 0906.4835 · arxiv_version: 0906.4835v1 · doi: 10.48550/arxiv.0906.4835 · pith_short_12: YWI75Z2JDMMM · pith_short_16: YWI75Z2JDMMMEY6Q · pith_short_8: YWI75Z2J

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/YWI75Z2JDMMMEY6QWHLBORDPRW \
  | 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: c591fee7491b18c263d0b1d617446f8d8512c36ac491bf3567e2feca3cba42d9

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "113c50a35e6adbfb3071ba59b47ce805d96c37ee4ccffcefea8104416cd1e80a",
    "cross_cats_sorted": [
      "math.CV"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2009-06-26T03:32:32Z",
    "title_canon_sha256": "c9cd4d25a00b6daf19eb00a2239816b21b8cfa41967b7a26c12524c1a0294680"
  },
  "schema_version": "1.0",
  "source": {
    "id": "0906.4835",
    "kind": "arxiv",
    "version": 1
  }
}