{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2004:5K2G23AM67PE2GXBQK2KQTJ4IO","short_pith_number":"pith:5K2G23AM","schema_version":"1.0","canonical_sha256":"eab46d6c0cf7de4d1ae182b4a84d3c4398638517543033d04b75979a300459f5","source":{"kind":"arxiv","id":"math-ph/0402029","version":2},"attestation_state":"computed","paper":{"title":"Fredholm's Minors of Arbitrary Order: Their Representations as a Determinant of Resolvents and in Terms of Free Fermions and an Explicit Formula for Their Functional Derivative","license":"","headline":"","cross_cats":["cond-mat.other","hep-th","math.MP"],"primary_cat":"math-ph","authors_text":"Joshua Feinberg","submitted_at":"2004-02-11T19:08:34Z","abstract_excerpt":"We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the $n$th and $n-1$th minors, whose solution is a representation of the $n$th minor as an $n\\times n$ determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order $n$ with respect to the kernel. Our formula is a linear combination of the $n$th and the $n\\pm 1$th minors."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math-ph/0402029","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2004-02-11T19:08:34Z","cross_cats_sorted":["cond-mat.other","hep-th","math.MP"],"title_canon_sha256":"7ac82097b5b99e21181500a0e9b93a78f2232ecf2d5e85c05d610df959ccc40e","abstract_canon_sha256":"f3c84abd0f8d1e394dbdac115239a2a200f60f431c0e0356d3f0b3a8b5a7c9b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:20:32.810218Z","signature_b64":"61oOx01DItPOToVF6w1VldApAbKEp1s4zFO06J6rN4utleL5CzASz6hLAISwGoA93p/nozugP/1lUrDNnIFpCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eab46d6c0cf7de4d1ae182b4a84d3c4398638517543033d04b75979a300459f5","last_reissued_at":"2026-07-04T15:20:32.809552Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:20:32.809552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fredholm's Minors of Arbitrary Order: Their Representations as a Determinant of Resolvents and in Terms of Free Fermions and an Explicit Formula for Their Functional Derivative","license":"","headline":"","cross_cats":["cond-mat.other","hep-th","math.MP"],"primary_cat":"math-ph","authors_text":"Joshua Feinberg","submitted_at":"2004-02-11T19:08:34Z","abstract_excerpt":"We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the $n$th and $n-1$th minors, whose solution is a representation of the $n$th minor as an $n\\times n$ determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order $n$ with respect to the kernel. Our formula is a linear combination of the $n$th and the $n\\pm 1$th minors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0402029","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math-ph/0402029/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0402029","created_at":"2026-07-04T15:20:32.809848+00:00"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0402029v2","created_at":"2026-07-04T15:20:32.809848+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0402029","created_at":"2026-07-04T15:20:32.809848+00:00"},{"alias_kind":"pith_short_12","alias_value":"5K2G23AM67PE","created_at":"2026-07-04T15:20:32.809848+00:00"},{"alias_kind":"pith_short_16","alias_value":"5K2G23AM67PE2GXB","created_at":"2026-07-04T15:20:32.809848+00:00"},{"alias_kind":"pith_short_8","alias_value":"5K2G23AM","created_at":"2026-07-04T15:20:32.809848+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO","json":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO.json","graph_json":"https://pith.science/api/pith-number/5K2G23AM67PE2GXBQK2KQTJ4IO/graph.json","events_json":"https://pith.science/api/pith-number/5K2G23AM67PE2GXBQK2KQTJ4IO/events.json","paper":"https://pith.science/paper/5K2G23AM"},"agent_actions":{"view_html":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO","download_json":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO.json","view_paper":"https://pith.science/paper/5K2G23AM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math-ph/0402029&json=true","fetch_graph":"https://pith.science/api/pith-number/5K2G23AM67PE2GXBQK2KQTJ4IO/graph.json","fetch_events":"https://pith.science/api/pith-number/5K2G23AM67PE2GXBQK2KQTJ4IO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO/action/storage_attestation","attest_author":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO/action/author_attestation","sign_citation":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO/action/citation_signature","submit_replication":"https://pith.science/pith/5K2G23AM67PE2GXBQK2KQTJ4IO/action/replication_record"}},"created_at":"2026-07-04T15:20:32.809848+00:00","updated_at":"2026-07-04T15:20:32.809848+00:00"}