{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:JDFB4YCZGDK37KXQRN5PERT6YC","short_pith_number":"pith:JDFB4YCZ","canonical_record":{"source":{"id":"1105.1239","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-06T08:24:00Z","cross_cats_sorted":["math-ph","math.AP","math.MP","math.PR"],"title_canon_sha256":"66f0756f5556ef1bbaac929cde2cba633fbaf87f306342a30bb6f27734e8ca73","abstract_canon_sha256":"02aa769603daee6ba0ef16d03fe62a0c3a15aad05fcafbe53fe8932efd153ebf"},"schema_version":"1.0"},"canonical_sha256":"48ca1e605930d5bfaaf08b7af2467ec085c636000b47805a7dbfae47bec7b78b","source":{"kind":"arxiv","id":"1105.1239","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.1239","created_at":"2026-05-18T04:11:28Z"},{"alias_kind":"arxiv_version","alias_value":"1105.1239v2","created_at":"2026-05-18T04:11:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.1239","created_at":"2026-05-18T04:11:28Z"},{"alias_kind":"pith_short_12","alias_value":"JDFB4YCZGDK3","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JDFB4YCZGDK37KXQ","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JDFB4YCZ","created_at":"2026-05-18T12:26:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:JDFB4YCZGDK37KXQRN5PERT6YC","target":"record","payload":{"canonical_record":{"source":{"id":"1105.1239","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-06T08:24:00Z","cross_cats_sorted":["math-ph","math.AP","math.MP","math.PR"],"title_canon_sha256":"66f0756f5556ef1bbaac929cde2cba633fbaf87f306342a30bb6f27734e8ca73","abstract_canon_sha256":"02aa769603daee6ba0ef16d03fe62a0c3a15aad05fcafbe53fe8932efd153ebf"},"schema_version":"1.0"},"canonical_sha256":"48ca1e605930d5bfaaf08b7af2467ec085c636000b47805a7dbfae47bec7b78b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:28.672672Z","signature_b64":"Ey/jpU3pakqt7WdnPkD2t5UbFq2jXAuiwGXRJa9SyOE4NNl/NjgdpX/3/hNHUj8HK+FiDZq28m6xH631tv2MAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48ca1e605930d5bfaaf08b7af2467ec085c636000b47805a7dbfae47bec7b78b","last_reissued_at":"2026-05-18T04:11:28.672116Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:28.672116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.1239","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":"2026-05-18T04:11:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1jWP5j0Q8aiVN9YBGYJA/C5ZsXpyMCm3eRjZuNTePhPQebH+1TGZAzKrgmAEQV2Rm6K+Ai/++Bpbq5BFU4LaDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T07:59:25.245232Z"},"content_sha256":"9fdd276fbc98cc57c5d94ad846ff294a7d264276605253973cc92d9e70095870","schema_version":"1.0","event_id":"sha256:9fdd276fbc98cc57c5d94ad846ff294a7d264276605253973cc92d9e70095870"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:JDFB4YCZGDK37KXQRN5PERT6YC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"General Fractional Calculus, Evolution Equations, and Renewal Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP","math.PR"],"primary_cat":"math.CA","authors_text":"Anatoly N. Kochubei","submitted_at":"2011-05-06T08:24:00Z","abstract_excerpt":"We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\\frac{d}{dt}\\int\\limits_0^tk(t-\\tau)u(\\tau)\\,d\\tau -k(t)u(0)$ where $k$ is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation $Du=-\\lambda u$, $\\lambda >0$, proved to be (under some conditions upon $k$) continuous on $[(0,\\infty)$ and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.1239","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":"2026-05-18T04:11:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L0u2gE2woVEDF/hsboqQa2h0ubpiTDETQNR4ldYVYsN/g+fJBF0NvJOFB5xlaMzCs5gO0xMmKe/+pOumpiHECg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T07:59:25.245592Z"},"content_sha256":"2f8f20640addde46cd9e99ab90cd025115cfc66931f063304ab205dc8336b261","schema_version":"1.0","event_id":"sha256:2f8f20640addde46cd9e99ab90cd025115cfc66931f063304ab205dc8336b261"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JDFB4YCZGDK37KXQRN5PERT6YC/bundle.json","state_url":"https://pith.science/pith/JDFB4YCZGDK37KXQRN5PERT6YC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JDFB4YCZGDK37KXQRN5PERT6YC/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":"2026-06-28T07:59:25Z","links":{"resolver":"https://pith.science/pith/JDFB4YCZGDK37KXQRN5PERT6YC","bundle":"https://pith.science/pith/JDFB4YCZGDK37KXQRN5PERT6YC/bundle.json","state":"https://pith.science/pith/JDFB4YCZGDK37KXQRN5PERT6YC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JDFB4YCZGDK37KXQRN5PERT6YC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:JDFB4YCZGDK37KXQRN5PERT6YC","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":"02aa769603daee6ba0ef16d03fe62a0c3a15aad05fcafbe53fe8932efd153ebf","cross_cats_sorted":["math-ph","math.AP","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-06T08:24:00Z","title_canon_sha256":"66f0756f5556ef1bbaac929cde2cba633fbaf87f306342a30bb6f27734e8ca73"},"schema_version":"1.0","source":{"id":"1105.1239","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.1239","created_at":"2026-05-18T04:11:28Z"},{"alias_kind":"arxiv_version","alias_value":"1105.1239v2","created_at":"2026-05-18T04:11:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.1239","created_at":"2026-05-18T04:11:28Z"},{"alias_kind":"pith_short_12","alias_value":"JDFB4YCZGDK3","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"JDFB4YCZGDK37KXQ","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"JDFB4YCZ","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:2f8f20640addde46cd9e99ab90cd025115cfc66931f063304ab205dc8336b261","target":"graph","created_at":"2026-05-18T04:11:28Z","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":"We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\\frac{d}{dt}\\int\\limits_0^tk(t-\\tau)u(\\tau)\\,d\\tau -k(t)u(0)$ where $k$ is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation $Du=-\\lambda u$, $\\lambda >0$, proved to be (under some conditions upon $k$) continuous on $[(0,\\infty)$ and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisa","authors_text":"Anatoly N. Kochubei","cross_cats":["math-ph","math.AP","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-06T08:24:00Z","title":"General Fractional Calculus, Evolution Equations, and Renewal Processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.1239","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":"sha256:9fdd276fbc98cc57c5d94ad846ff294a7d264276605253973cc92d9e70095870","target":"record","created_at":"2026-05-18T04:11:28Z","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":"02aa769603daee6ba0ef16d03fe62a0c3a15aad05fcafbe53fe8932efd153ebf","cross_cats_sorted":["math-ph","math.AP","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-05-06T08:24:00Z","title_canon_sha256":"66f0756f5556ef1bbaac929cde2cba633fbaf87f306342a30bb6f27734e8ca73"},"schema_version":"1.0","source":{"id":"1105.1239","kind":"arxiv","version":2}},"canonical_sha256":"48ca1e605930d5bfaaf08b7af2467ec085c636000b47805a7dbfae47bec7b78b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48ca1e605930d5bfaaf08b7af2467ec085c636000b47805a7dbfae47bec7b78b","first_computed_at":"2026-05-18T04:11:28.672116Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:28.672116Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ey/jpU3pakqt7WdnPkD2t5UbFq2jXAuiwGXRJa9SyOE4NNl/NjgdpX/3/hNHUj8HK+FiDZq28m6xH631tv2MAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:28.672672Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.1239","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9fdd276fbc98cc57c5d94ad846ff294a7d264276605253973cc92d9e70095870","sha256:2f8f20640addde46cd9e99ab90cd025115cfc66931f063304ab205dc8336b261"],"state_sha256":"a4b4e7de82f0a52e184736f37014dda30eec1a2a07371c3ad145438fe9e082d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TG/0jTHYiDIsjQmqz/hQoFg6muxeWOp8joDRR3ZLgFmnz4GzxOZw2PkYWimtbbJwyNvFNf82uyVkK1BiIkFeBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T07:59:25.247293Z","bundle_sha256":"24ad04ea8f910f37aafeb4033f190245e2cb96ed4a785aca0ef72c3d2e73722c"}}