{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:IRC3DXOFDC3EC257YIGTWWBBNR","short_pith_number":"pith:IRC3DXOF","canonical_record":{"source":{"id":"1602.05382","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-02-17T12:00:34Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"062b7af53e33a286aa6dd4ddef090f9ab340a1582d753f80e65ff2e2ac6517e6","abstract_canon_sha256":"75d840408178b052c991cdd0c56781475d7ceaae9325c5baeca3506987a39c68"},"schema_version":"1.0"},"canonical_sha256":"4445b1ddc518b6416bbfc20d3b58216c4655076837322aac57808ab0864e99be","source":{"kind":"arxiv","id":"1602.05382","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05382","created_at":"2026-05-18T00:49:19Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05382v3","created_at":"2026-05-18T00:49:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05382","created_at":"2026-05-18T00:49:19Z"},{"alias_kind":"pith_short_12","alias_value":"IRC3DXOFDC3E","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IRC3DXOFDC3EC257","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IRC3DXOF","created_at":"2026-05-18T12:30:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:IRC3DXOFDC3EC257YIGTWWBBNR","target":"record","payload":{"canonical_record":{"source":{"id":"1602.05382","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-02-17T12:00:34Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"062b7af53e33a286aa6dd4ddef090f9ab340a1582d753f80e65ff2e2ac6517e6","abstract_canon_sha256":"75d840408178b052c991cdd0c56781475d7ceaae9325c5baeca3506987a39c68"},"schema_version":"1.0"},"canonical_sha256":"4445b1ddc518b6416bbfc20d3b58216c4655076837322aac57808ab0864e99be","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:19.184704Z","signature_b64":"QmFT8ds0yN08ohUUEtN/p/WRc5CszFo5RtRsBYAqRStN2pYP4dGWSICqpmVKa5leL4dL3nwglLsiwpU7OTlwCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4445b1ddc518b6416bbfc20d3b58216c4655076837322aac57808ab0864e99be","last_reissued_at":"2026-05-18T00:49:19.184275Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:19.184275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.05382","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:49:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mYtaCOnp7bbJHVpF4GIKzDvbSVmBxj6SED05edlxU8orR9uHMCx3A1XKAoGNtKkz8dAlWGRR/vbo64FXUE7kCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:33:49.016671Z"},"content_sha256":"17242cee95068e696c9d96c666b7607878798650bfc3787f2b6b6169fe34cad3","schema_version":"1.0","event_id":"sha256:17242cee95068e696c9d96c666b7607878798650bfc3787f2b6b6169fe34cad3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:IRC3DXOFDC3EC257YIGTWWBBNR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The time-fractional radiative transport equation -- Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Manabu Machida","submitted_at":"2016-02-17T12:00:34Z","abstract_excerpt":"We consider the radiative transport equation in which the time derivative is replaced by the Caputo derivative. Such fractional-order derivatives are related to anomalous transport and anomalous diffusion. In this paper we describe how the time-fractional radiative transport equation is obtained from continuous-time random walk and see how the equation is related to the time-fractional diffusion equation in the asymptotic limit. Then we solve the equation with Legendre-polynomial expansion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05382","kind":"arxiv","version":3},"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":"2026-05-18T00:49:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MiBwMHI+NxSAghbCo8ld4HkbzF/5px+LvVZw/oOyLLaB2MLFVG/i1V6W99/rU8fio+nZCQWnRMnQLIOhIkujAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:33:49.017499Z"},"content_sha256":"d8970bc380717c07a8e59370fd829c4eee1b727ca4ca90d4ecd2dd57f9967f70","schema_version":"1.0","event_id":"sha256:d8970bc380717c07a8e59370fd829c4eee1b727ca4ca90d4ecd2dd57f9967f70"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IRC3DXOFDC3EC257YIGTWWBBNR/bundle.json","state_url":"https://pith.science/pith/IRC3DXOFDC3EC257YIGTWWBBNR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IRC3DXOFDC3EC257YIGTWWBBNR/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":"2026-06-01T02:33:49Z","links":{"resolver":"https://pith.science/pith/IRC3DXOFDC3EC257YIGTWWBBNR","bundle":"https://pith.science/pith/IRC3DXOFDC3EC257YIGTWWBBNR/bundle.json","state":"https://pith.science/pith/IRC3DXOFDC3EC257YIGTWWBBNR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IRC3DXOFDC3EC257YIGTWWBBNR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:IRC3DXOFDC3EC257YIGTWWBBNR","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":"75d840408178b052c991cdd0c56781475d7ceaae9325c5baeca3506987a39c68","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-02-17T12:00:34Z","title_canon_sha256":"062b7af53e33a286aa6dd4ddef090f9ab340a1582d753f80e65ff2e2ac6517e6"},"schema_version":"1.0","source":{"id":"1602.05382","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05382","created_at":"2026-05-18T00:49:19Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05382v3","created_at":"2026-05-18T00:49:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05382","created_at":"2026-05-18T00:49:19Z"},{"alias_kind":"pith_short_12","alias_value":"IRC3DXOFDC3E","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IRC3DXOFDC3EC257","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IRC3DXOF","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:d8970bc380717c07a8e59370fd829c4eee1b727ca4ca90d4ecd2dd57f9967f70","target":"graph","created_at":"2026-05-18T00:49:19Z","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 consider the radiative transport equation in which the time derivative is replaced by the Caputo derivative. Such fractional-order derivatives are related to anomalous transport and anomalous diffusion. In this paper we describe how the time-fractional radiative transport equation is obtained from continuous-time random walk and see how the equation is related to the time-fractional diffusion equation in the asymptotic limit. Then we solve the equation with Legendre-polynomial expansion.","authors_text":"Manabu Machida","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-02-17T12:00:34Z","title":"The time-fractional radiative transport equation -- Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05382","kind":"arxiv","version":3},"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":"sha256:17242cee95068e696c9d96c666b7607878798650bfc3787f2b6b6169fe34cad3","target":"record","created_at":"2026-05-18T00:49:19Z","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":"75d840408178b052c991cdd0c56781475d7ceaae9325c5baeca3506987a39c68","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-02-17T12:00:34Z","title_canon_sha256":"062b7af53e33a286aa6dd4ddef090f9ab340a1582d753f80e65ff2e2ac6517e6"},"schema_version":"1.0","source":{"id":"1602.05382","kind":"arxiv","version":3}},"canonical_sha256":"4445b1ddc518b6416bbfc20d3b58216c4655076837322aac57808ab0864e99be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4445b1ddc518b6416bbfc20d3b58216c4655076837322aac57808ab0864e99be","first_computed_at":"2026-05-18T00:49:19.184275Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:19.184275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QmFT8ds0yN08ohUUEtN/p/WRc5CszFo5RtRsBYAqRStN2pYP4dGWSICqpmVKa5leL4dL3nwglLsiwpU7OTlwCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:19.184704Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.05382","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17242cee95068e696c9d96c666b7607878798650bfc3787f2b6b6169fe34cad3","sha256:d8970bc380717c07a8e59370fd829c4eee1b727ca4ca90d4ecd2dd57f9967f70"],"state_sha256":"430a4d8f921c1328375054f101a9773b320436b06b347d8eb6c3f04f3d0576f1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kHGcemMCIq2WnCGr3hexH4dTODJDDCOxMzWElTiwvWLliPJjVorvigfVDzgS24qF7uLc4lYB06UyH39XRxyoDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T02:33:49.021506Z","bundle_sha256":"35499ac0d9c3084843706a813d660ac54f3c443fcb4a715f4ce1e15668cb60b9"}}