{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:XG6O2U7BH4YVEJIQC5CQLI7IE4","short_pith_number":"pith:XG6O2U7B","canonical_record":{"source":{"id":"1009.4168","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2010-09-21T18:22:21Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"93a9088f2dce2fb8e16f687d6746b140e2a373ed3f62771a39161f062bae3f38","abstract_canon_sha256":"6a91fffafe09ce03bb4501a9dca57f9c18a19818e98403a83a5c9134c4dc348a"},"schema_version":"1.0"},"canonical_sha256":"b9bced53e13f31522510174505a3e82719a7c893d9a04d069f573079e9cd13ed","source":{"kind":"arxiv","id":"1009.4168","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4168","created_at":"2026-05-18T03:12:09Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4168v2","created_at":"2026-05-18T03:12:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4168","created_at":"2026-05-18T03:12:09Z"},{"alias_kind":"pith_short_12","alias_value":"XG6O2U7BH4YV","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XG6O2U7BH4YVEJIQ","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XG6O2U7B","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:XG6O2U7BH4YVEJIQC5CQLI7IE4","target":"record","payload":{"canonical_record":{"source":{"id":"1009.4168","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2010-09-21T18:22:21Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"93a9088f2dce2fb8e16f687d6746b140e2a373ed3f62771a39161f062bae3f38","abstract_canon_sha256":"6a91fffafe09ce03bb4501a9dca57f9c18a19818e98403a83a5c9134c4dc348a"},"schema_version":"1.0"},"canonical_sha256":"b9bced53e13f31522510174505a3e82719a7c893d9a04d069f573079e9cd13ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:09.264047Z","signature_b64":"0yRPDAJKHIqxJbzIeNSkY6k8bU/CECMt64AVZXmwL83VLjnNDoEIDWXGteqWj+yCMcz5/OvLoLuQW8b+d/RIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9bced53e13f31522510174505a3e82719a7c893d9a04d069f573079e9cd13ed","last_reissued_at":"2026-05-18T03:12:09.263183Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:09.263183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.4168","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7+g8wOonSvoKov5FUwfdMQueOK9rH5W/SQ7YHZG1DzufTXKGBmb+xLtPf20DFXo7PXSTD13Nhxsd0YuM4mlbBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:39:13.905684Z"},"content_sha256":"596b80b4e117e5a0bdfc135d3d71bdc6b92bd27935bf9856c056a21a341e9341","schema_version":"1.0","event_id":"sha256:596b80b4e117e5a0bdfc135d3d71bdc6b92bd27935bf9856c056a21a341e9341"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:XG6O2U7BH4YVEJIQC5CQLI7IE4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Rayleigh-Type Formulas for a Non-local Boundary Value Problem Associated with an Integral Operator Commuting with the Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.SP","authors_text":"Lotfi Hermi, Naoki Saito","submitted_at":"2010-09-21T18:22:21Z","abstract_excerpt":"In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form $|x-y|^\\rho$, $0 < \\rho \\leq 1$, $x, y \\in [-a, a]$. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when $\\rho=1$, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary condition"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4168","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:12:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oIMYV6GLqlyYjMHdP/i7wuyJnIjzvBTfUNe3nQ8FekgdJSYS1zM68S1GzkKU5jF8zBJC2geRWjH0o9rD+UB1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:39:13.906033Z"},"content_sha256":"4d171fd66d0f08ecb05fcf89834cb49eb26424414f96d72540ab910794a10a2b","schema_version":"1.0","event_id":"sha256:4d171fd66d0f08ecb05fcf89834cb49eb26424414f96d72540ab910794a10a2b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XG6O2U7BH4YVEJIQC5CQLI7IE4/bundle.json","state_url":"https://pith.science/pith/XG6O2U7BH4YVEJIQC5CQLI7IE4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XG6O2U7BH4YVEJIQC5CQLI7IE4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T23:39:13Z","links":{"resolver":"https://pith.science/pith/XG6O2U7BH4YVEJIQC5CQLI7IE4","bundle":"https://pith.science/pith/XG6O2U7BH4YVEJIQC5CQLI7IE4/bundle.json","state":"https://pith.science/pith/XG6O2U7BH4YVEJIQC5CQLI7IE4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XG6O2U7BH4YVEJIQC5CQLI7IE4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XG6O2U7BH4YVEJIQC5CQLI7IE4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6a91fffafe09ce03bb4501a9dca57f9c18a19818e98403a83a5c9134c4dc348a","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2010-09-21T18:22:21Z","title_canon_sha256":"93a9088f2dce2fb8e16f687d6746b140e2a373ed3f62771a39161f062bae3f38"},"schema_version":"1.0","source":{"id":"1009.4168","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4168","created_at":"2026-05-18T03:12:09Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4168v2","created_at":"2026-05-18T03:12:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4168","created_at":"2026-05-18T03:12:09Z"},{"alias_kind":"pith_short_12","alias_value":"XG6O2U7BH4YV","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XG6O2U7BH4YVEJIQ","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XG6O2U7B","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:4d171fd66d0f08ecb05fcf89834cb49eb26424414f96d72540ab910794a10a2b","target":"graph","created_at":"2026-05-18T03:12:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form $|x-y|^\\rho$, $0 < \\rho \\leq 1$, $x, y \\in [-a, a]$. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when $\\rho=1$, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary condition","authors_text":"Lotfi Hermi, Naoki Saito","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2010-09-21T18:22:21Z","title":"On Rayleigh-Type Formulas for a Non-local Boundary Value Problem Associated with an Integral Operator Commuting with the Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4168","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:596b80b4e117e5a0bdfc135d3d71bdc6b92bd27935bf9856c056a21a341e9341","target":"record","created_at":"2026-05-18T03:12:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6a91fffafe09ce03bb4501a9dca57f9c18a19818e98403a83a5c9134c4dc348a","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2010-09-21T18:22:21Z","title_canon_sha256":"93a9088f2dce2fb8e16f687d6746b140e2a373ed3f62771a39161f062bae3f38"},"schema_version":"1.0","source":{"id":"1009.4168","kind":"arxiv","version":2}},"canonical_sha256":"b9bced53e13f31522510174505a3e82719a7c893d9a04d069f573079e9cd13ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9bced53e13f31522510174505a3e82719a7c893d9a04d069f573079e9cd13ed","first_computed_at":"2026-05-18T03:12:09.263183Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:09.263183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0yRPDAJKHIqxJbzIeNSkY6k8bU/CECMt64AVZXmwL83VLjnNDoEIDWXGteqWj+yCMcz5/OvLoLuQW8b+d/RIBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:09.264047Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.4168","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:596b80b4e117e5a0bdfc135d3d71bdc6b92bd27935bf9856c056a21a341e9341","sha256:4d171fd66d0f08ecb05fcf89834cb49eb26424414f96d72540ab910794a10a2b"],"state_sha256":"cd86470c3a55e0c19b44c63ab6c27a6d2986021e360794efc69b588f9d334bd3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M4GsyrnCSw5pr+ri/eSjC3RcJAihFrRTZgqx+cJ8lCMxcklHbl4UKdVIkxeJ6HIBU3jOgwItaNRNpKaeqK4lAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T23:39:13.908185Z","bundle_sha256":"a14c117497843bdbf2f0cadd9ac8840055d79d936411ae100e62cbe755f632dc"}}