Pith Number

pith:R5X2R35U

pith:2025:R5X2R35URPLQKQVWVWIFVNK6GV

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Koenigs functions in the subcritical and critical Markov branching processes with Poisson probability reproduction of particles

Assen Tchorbadjieff, Penka Mayster

Koenigs functions yield explicit solutions for subcritical and critical Poisson branching processes via Bell polynomials and Ein(z).

arxiv:2512.17485 v2 · 2025-12-19 · math.PR · stat.CO

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Record completeness

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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.

Claims

C1strongest claim

The obtained explicit solutions contain the exponential Bell polynomials and the modified exponential-integral function Ein (z).

C2weakest assumption

The particle reproduction follows a Poisson probability distribution in continuous time, and the branching mechanism is either subcritical or critical.

C3one line summary

Explicit solutions for Koenigs functions in Poisson branching processes give limit conditional laws in subcritical cases and invariant measures in critical cases using Bell polynomials and Ein(z).

References

24 extracted · 24 resolved · 0 Pith anchors

[1] K. B. Athreya and P. E. Ney,Branching Processes, Springer, New York, 1972 1972

[2] V . I. Dorogov and V . P. Chistyakov,Probabilistic Models of the Transformation of Particles, Nauka, Moscow, 1988 (in Russian) 1988

[3] T. E. Harris,The Theory of Branching Processes, Springer, Berlin, 1963 1963

[4] I. Pazsit and L. P ´al,Neutron Fluctuations: A Treatise on the Physics of Branching Processes, Elsevier Science B.V ., Amsterdam, 2008 2008

[5] B. A. Sevasyanov,Branching Processes, Nauka, Moscow, 1971 (in Russian; German transl.Verzwei- gungsprozesse, R. Oldenbourg Verlag, Munich, 1975) 1971

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T03:09:32.514506Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8f6fa8efb48bd70542b6ad905ab55e3563a287d1cdccf8f5491c8eace9e97e42

Aliases

arxiv: 2512.17485 · arxiv_version: 2512.17485v2 · doi: 10.48550/arxiv.2512.17485 · pith_short_12: R5X2R35URPLQ · pith_short_16: R5X2R35URPLQKQVW · pith_short_8: R5X2R35U

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/R5X2R35URPLQKQVWVWIFVNK6GV \
  | 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: 8f6fa8efb48bd70542b6ad905ab55e3563a287d1cdccf8f5491c8eace9e97e42

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "b450f6462ae3ac5fcbcd99a4cb32475c95c86878e91982c9347275ae9c1132db",
    "cross_cats_sorted": [
      "stat.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2025-12-19T11:56:55Z",
    "title_canon_sha256": "76e134c06de8e0ce7f70439cf27ef916701ec11b4f3ad644a76729684bfb0119"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2512.17485",
    "kind": "arxiv",
    "version": 2
  }
}