{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:3OEB5HNTXAQBTSTXYK6RVU5DRM","short_pith_number":"pith:3OEB5HNT","canonical_record":{"source":{"id":"1207.3284","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-13T15:44:40Z","cross_cats_sorted":[],"title_canon_sha256":"cd0465db234bca0f7e2e9bf9f741027a572a6ccb93466601aefe7e2c1900862f","abstract_canon_sha256":"97f12a78a0a59c225d72cff77984da3bdba21d2b65db705fb0bab76c5bfba639"},"schema_version":"1.0"},"canonical_sha256":"db881e9db3b82019ca77c2bd1ad3a38b0dce4612cc5f012926323ed41302bdfb","source":{"kind":"arxiv","id":"1207.3284","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3284","created_at":"2026-05-18T01:33:24Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3284v2","created_at":"2026-05-18T01:33:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3284","created_at":"2026-05-18T01:33:24Z"},{"alias_kind":"pith_short_12","alias_value":"3OEB5HNTXAQB","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"3OEB5HNTXAQBTSTX","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"3OEB5HNT","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:3OEB5HNTXAQBTSTXYK6RVU5DRM","target":"record","payload":{"canonical_record":{"source":{"id":"1207.3284","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-13T15:44:40Z","cross_cats_sorted":[],"title_canon_sha256":"cd0465db234bca0f7e2e9bf9f741027a572a6ccb93466601aefe7e2c1900862f","abstract_canon_sha256":"97f12a78a0a59c225d72cff77984da3bdba21d2b65db705fb0bab76c5bfba639"},"schema_version":"1.0"},"canonical_sha256":"db881e9db3b82019ca77c2bd1ad3a38b0dce4612cc5f012926323ed41302bdfb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:24.711471Z","signature_b64":"0HQmplyCgNy6C5/TV4cjcOysVvNNU6vO+O2ieZ3q+YmnVfQHJLePDDkh04s58BpzsnJjPzgJsH/p9ec9BzMdCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db881e9db3b82019ca77c2bd1ad3a38b0dce4612cc5f012926323ed41302bdfb","last_reissued_at":"2026-05-18T01:33:24.710613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:24.710613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.3284","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":"2026-05-18T01:33:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gp+7l0VbucZOdvT4kSZw8pYc6X1mtKDDCfOxXJsUxWIxWtASoaFllwjM0TSZPUBvpfqHj76webqhnqK2JUQQAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T00:08:39.704017Z"},"content_sha256":"20f95d126a72d75bbfa55dbd4507e954edcecf9b9fa6e584729b75d315551034","schema_version":"1.0","event_id":"sha256:20f95d126a72d75bbfa55dbd4507e954edcecf9b9fa6e584729b75d315551034"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:3OEB5HNTXAQBTSTXYK6RVU5DRM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Space-time fractional equations and the related stable processes at random time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bruno Toaldo, Enzo Orsingher","submitted_at":"2012-07-13T15:44:40Z","abstract_excerpt":"In this paper we consider the general fractional equation \\sum_{j=1}^m \\lambda_j \\frac{\\partial^{\\nu_j}}{\\partial t^{\\nu_j}} w(x_1,..., x_n ; t) = -c^2 (-\\Delta)^\\beta w(x_1,..., x_n ; t), for \\nu_j \\in (0,1], \\beta \\in (0,1] with initial condition w(x_1,..., x_n ; 0)= \\prod_{j=1}^n \\delta (x_j). The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \\bm{S}_n^{2\\beta} \\mathcal{L} c^2 {L}^{\\nu_1,..., \\nu_m} (t) \\r, t>0, where \\bm{S}_n^{2\\beta} is an isotropic stable process independent from {L}^{\\nu_1,..., \\nu_m}(t) which is the inverse of {H}^{\\nu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3284","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":"2026-05-18T01:33:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qUmNwaC7bxmWIcbIG19ubvLvkMpsYw6EH3KgZ17gVvElcFYyyP7kBdHl2QpvkS4GwnYCYOvmP7qDnHaaQ4DICg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T00:08:39.704620Z"},"content_sha256":"5e6dd7d94fd43ddc99d61569e01eb114a072fab0bc4ebdf75d41e9de8f157dc7","schema_version":"1.0","event_id":"sha256:5e6dd7d94fd43ddc99d61569e01eb114a072fab0bc4ebdf75d41e9de8f157dc7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/bundle.json","state_url":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/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":"2026-05-28T00:08:39Z","links":{"resolver":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM","bundle":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/bundle.json","state":"https://pith.science/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3OEB5HNTXAQBTSTXYK6RVU5DRM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:3OEB5HNTXAQBTSTXYK6RVU5DRM","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":"97f12a78a0a59c225d72cff77984da3bdba21d2b65db705fb0bab76c5bfba639","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-13T15:44:40Z","title_canon_sha256":"cd0465db234bca0f7e2e9bf9f741027a572a6ccb93466601aefe7e2c1900862f"},"schema_version":"1.0","source":{"id":"1207.3284","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.3284","created_at":"2026-05-18T01:33:24Z"},{"alias_kind":"arxiv_version","alias_value":"1207.3284v2","created_at":"2026-05-18T01:33:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3284","created_at":"2026-05-18T01:33:24Z"},{"alias_kind":"pith_short_12","alias_value":"3OEB5HNTXAQB","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"3OEB5HNTXAQBTSTX","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"3OEB5HNT","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:5e6dd7d94fd43ddc99d61569e01eb114a072fab0bc4ebdf75d41e9de8f157dc7","target":"graph","created_at":"2026-05-18T01:33:24Z","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 paper we consider the general fractional equation \\sum_{j=1}^m \\lambda_j \\frac{\\partial^{\\nu_j}}{\\partial t^{\\nu_j}} w(x_1,..., x_n ; t) = -c^2 (-\\Delta)^\\beta w(x_1,..., x_n ; t), for \\nu_j \\in (0,1], \\beta \\in (0,1] with initial condition w(x_1,..., x_n ; 0)= \\prod_{j=1}^n \\delta (x_j). The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \\bm{S}_n^{2\\beta} \\mathcal{L} c^2 {L}^{\\nu_1,..., \\nu_m} (t) \\r, t>0, where \\bm{S}_n^{2\\beta} is an isotropic stable process independent from {L}^{\\nu_1,..., \\nu_m}(t) which is the inverse of {H}^{\\nu","authors_text":"Bruno Toaldo, Enzo Orsingher","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-13T15:44:40Z","title":"Space-time fractional equations and the related stable processes at random time"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3284","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":"sha256:20f95d126a72d75bbfa55dbd4507e954edcecf9b9fa6e584729b75d315551034","target":"record","created_at":"2026-05-18T01:33:24Z","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":"97f12a78a0a59c225d72cff77984da3bdba21d2b65db705fb0bab76c5bfba639","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-07-13T15:44:40Z","title_canon_sha256":"cd0465db234bca0f7e2e9bf9f741027a572a6ccb93466601aefe7e2c1900862f"},"schema_version":"1.0","source":{"id":"1207.3284","kind":"arxiv","version":2}},"canonical_sha256":"db881e9db3b82019ca77c2bd1ad3a38b0dce4612cc5f012926323ed41302bdfb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db881e9db3b82019ca77c2bd1ad3a38b0dce4612cc5f012926323ed41302bdfb","first_computed_at":"2026-05-18T01:33:24.710613Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:24.710613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0HQmplyCgNy6C5/TV4cjcOysVvNNU6vO+O2ieZ3q+YmnVfQHJLePDDkh04s58BpzsnJjPzgJsH/p9ec9BzMdCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:24.711471Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.3284","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20f95d126a72d75bbfa55dbd4507e954edcecf9b9fa6e584729b75d315551034","sha256:5e6dd7d94fd43ddc99d61569e01eb114a072fab0bc4ebdf75d41e9de8f157dc7"],"state_sha256":"81e5a4577f31ab20c38abe1774c44ef3b87e57bfaf723f7debfdccefa250668e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gJlyRjsfX+OTsP6Q97ezwGdNeVay+AwED3dGRSgvrRzcwz+kz3cg4iCjtAEh2BWpxwLIGZRuzc+c4GBMrFKmBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T00:08:39.707739Z","bundle_sha256":"115e222ed809a3d83da8d9ae44af21b99186794c68ccd59db94355b6bd8a27f6"}}