{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:BVES7GN3W422AKJO4BMUGL4TG7","short_pith_number":"pith:BVES7GN3","canonical_record":{"source":{"id":"1305.1460","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-07T10:36:35Z","cross_cats_sorted":[],"title_canon_sha256":"5433e5c4b8706a2c75ec4046d6c6e60df548bc18eb763b603b7d5e19b8c63985","abstract_canon_sha256":"83dc6c7e934b47018d6dd22754dcde6e58dc924e8eef5b21208b1d3a5ce836ff"},"schema_version":"1.0"},"canonical_sha256":"0d492f99bbb735a0292ee059432f9337de57a5cde5750be97f6875b1c10b83f3","source":{"kind":"arxiv","id":"1305.1460","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.1460","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"arxiv_version","alias_value":"1305.1460v2","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.1460","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"pith_short_12","alias_value":"BVES7GN3W422","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BVES7GN3W422AKJO","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BVES7GN3","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:BVES7GN3W422AKJO4BMUGL4TG7","target":"record","payload":{"canonical_record":{"source":{"id":"1305.1460","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-07T10:36:35Z","cross_cats_sorted":[],"title_canon_sha256":"5433e5c4b8706a2c75ec4046d6c6e60df548bc18eb763b603b7d5e19b8c63985","abstract_canon_sha256":"83dc6c7e934b47018d6dd22754dcde6e58dc924e8eef5b21208b1d3a5ce836ff"},"schema_version":"1.0"},"canonical_sha256":"0d492f99bbb735a0292ee059432f9337de57a5cde5750be97f6875b1c10b83f3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:34.623122Z","signature_b64":"bvXjY3Ot+0emsm2WmPkxfWoLWTE1PT6a0m+5bFCUNfilLpl0n9kf62yoMVv4TDQqnmX7VhIauLABKu5yk1tqAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d492f99bbb735a0292ee059432f9337de57a5cde5750be97f6875b1c10b83f3","last_reissued_at":"2026-05-18T03:19:34.622574Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:34.622574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.1460","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":"2026-05-18T03:19:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UDCh6XYSF8SdcFLMkB2WnChgFh/T43bpzR6NqSeOyN51adiC6JVgkL/MgacF4b3SFvHyAM4Ud7nBz72bBvixAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T22:59:58.191412Z"},"content_sha256":"d9ad53110c19d6777329187d5bd0211948c283891139cfb66746337e60d02028","schema_version":"1.0","event_id":"sha256:d9ad53110c19d6777329187d5bd0211948c283891139cfb66746337e60d02028"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:BVES7GN3W422AKJO4BMUGL4TG7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The functional analytic foundation of Colombeau algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"E. A. Nigsch","submitted_at":"2013-05-07T10:36:35Z","abstract_excerpt":"Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a functional analytic approach placing these algebras in a unifying hierarchy, which clarifies their structural properties as well as their relation to each other."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1460","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":"2026-05-18T03:19:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r3GdlmVd3G61C0LpfCUZQn3fy5mPEnQCmHJglXypvhQmvbaHuaVIGdPNWMI+8JlZTggtFExWOqa+hnZxpb9cAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T22:59:58.191746Z"},"content_sha256":"fd78d10ec462410a521e1c69417a241a4e696ff94d26da6372c27c3b5b268c51","schema_version":"1.0","event_id":"sha256:fd78d10ec462410a521e1c69417a241a4e696ff94d26da6372c27c3b5b268c51"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BVES7GN3W422AKJO4BMUGL4TG7/bundle.json","state_url":"https://pith.science/pith/BVES7GN3W422AKJO4BMUGL4TG7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BVES7GN3W422AKJO4BMUGL4TG7/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":"2026-06-28T22:59:58Z","links":{"resolver":"https://pith.science/pith/BVES7GN3W422AKJO4BMUGL4TG7","bundle":"https://pith.science/pith/BVES7GN3W422AKJO4BMUGL4TG7/bundle.json","state":"https://pith.science/pith/BVES7GN3W422AKJO4BMUGL4TG7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BVES7GN3W422AKJO4BMUGL4TG7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:BVES7GN3W422AKJO4BMUGL4TG7","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":"83dc6c7e934b47018d6dd22754dcde6e58dc924e8eef5b21208b1d3a5ce836ff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-07T10:36:35Z","title_canon_sha256":"5433e5c4b8706a2c75ec4046d6c6e60df548bc18eb763b603b7d5e19b8c63985"},"schema_version":"1.0","source":{"id":"1305.1460","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.1460","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"arxiv_version","alias_value":"1305.1460v2","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.1460","created_at":"2026-05-18T03:19:34Z"},{"alias_kind":"pith_short_12","alias_value":"BVES7GN3W422","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BVES7GN3W422AKJO","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BVES7GN3","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:fd78d10ec462410a521e1c69417a241a4e696ff94d26da6372c27c3b5b268c51","target":"graph","created_at":"2026-05-18T03:19:34Z","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":"Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a functional analytic approach placing these algebras in a unifying hierarchy, which clarifies their structural properties as well as their relation to each other.","authors_text":"E. A. Nigsch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-07T10:36:35Z","title":"The functional analytic foundation of Colombeau algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1460","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:d9ad53110c19d6777329187d5bd0211948c283891139cfb66746337e60d02028","target":"record","created_at":"2026-05-18T03:19:34Z","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":"83dc6c7e934b47018d6dd22754dcde6e58dc924e8eef5b21208b1d3a5ce836ff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-05-07T10:36:35Z","title_canon_sha256":"5433e5c4b8706a2c75ec4046d6c6e60df548bc18eb763b603b7d5e19b8c63985"},"schema_version":"1.0","source":{"id":"1305.1460","kind":"arxiv","version":2}},"canonical_sha256":"0d492f99bbb735a0292ee059432f9337de57a5cde5750be97f6875b1c10b83f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d492f99bbb735a0292ee059432f9337de57a5cde5750be97f6875b1c10b83f3","first_computed_at":"2026-05-18T03:19:34.622574Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:34.622574Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bvXjY3Ot+0emsm2WmPkxfWoLWTE1PT6a0m+5bFCUNfilLpl0n9kf62yoMVv4TDQqnmX7VhIauLABKu5yk1tqAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:34.623122Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.1460","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d9ad53110c19d6777329187d5bd0211948c283891139cfb66746337e60d02028","sha256:fd78d10ec462410a521e1c69417a241a4e696ff94d26da6372c27c3b5b268c51"],"state_sha256":"fab6bbf8dbd144c34c31d15bebd6193bede35f63ce18657f7cd643bf9bcc4a78"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dOlvzOKypi66qRExfNjB1MeICKALthFQtzUwboPfgCKU+G5vv4qshmrOKR8s39iMjbU2r7pEMfksUQCmFHSDCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T22:59:58.193588Z","bundle_sha256":"14c49f1d2c42bc6be69dec08a0b46bec251f90c5952f34e1d5e3418d0ca3f4f9"}}