{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:2YFVW2ZDRIGH2DOV23XI4H3JET","short_pith_number":"pith:2YFVW2ZD","canonical_record":{"source":{"id":"1309.2547","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T15:25:42Z","cross_cats_sorted":[],"title_canon_sha256":"2563dc90a67e8d5a2ff7e8e90b59edb1d6873b4e417fff85cc46d41b5de7683a","abstract_canon_sha256":"f0f6ddffc193896f929315a88f19428817a1e0423ccdf5bc03d093753df4a3f3"},"schema_version":"1.0"},"canonical_sha256":"d60b5b6b238a0c7d0dd5d6ee8e1f6924d34db18fd8110e04f170cad1cb6336e5","source":{"kind":"arxiv","id":"1309.2547","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.2547","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"arxiv_version","alias_value":"1309.2547v2","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.2547","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"pith_short_12","alias_value":"2YFVW2ZDRIGH","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2YFVW2ZDRIGH2DOV","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2YFVW2ZD","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:2YFVW2ZDRIGH2DOV23XI4H3JET","target":"record","payload":{"canonical_record":{"source":{"id":"1309.2547","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T15:25:42Z","cross_cats_sorted":[],"title_canon_sha256":"2563dc90a67e8d5a2ff7e8e90b59edb1d6873b4e417fff85cc46d41b5de7683a","abstract_canon_sha256":"f0f6ddffc193896f929315a88f19428817a1e0423ccdf5bc03d093753df4a3f3"},"schema_version":"1.0"},"canonical_sha256":"d60b5b6b238a0c7d0dd5d6ee8e1f6924d34db18fd8110e04f170cad1cb6336e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:22.343212Z","signature_b64":"/0GPGLf4P83taWYxF4sin9pJAs46zvrjHYLdB1idqvpNW/gLuzTRGpgou2MWuUdqSfu//we4Hs1dk+6fFBf3Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d60b5b6b238a0c7d0dd5d6ee8e1f6924d34db18fd8110e04f170cad1cb6336e5","last_reissued_at":"2026-05-18T03:04:22.342814Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:22.342814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.2547","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":"2026-05-18T03:04:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IxJRvkFMrOPlAqyLp4EHHd1W2oeyhrrSgbufMnNsyAeBxsjNYC1Kf55tSlhg47GJ0laikT8GLO56NTjHpyT2DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:24:54.338336Z"},"content_sha256":"071287a599a86832c4c3515a854c68e36098acdc15b87f8e9b3bd8986f6bca94","schema_version":"1.0","event_id":"sha256:071287a599a86832c4c3515a854c68e36098acdc15b87f8e9b3bd8986f6bca94"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:2YFVW2ZDRIGH2DOV23XI4H3JET","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hopf-Lax formula and generalized characteristics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nguyen Hoang","submitted_at":"2013-09-10T15:25:42Z","abstract_excerpt":"We study some differential properties of viscosity solution for Hamilton - Jacobi equations defined by Hopf-Lax formula $u(t,x)=\\min_{y\\in \\R^n} \\big\\{\\sigma (y)+tH^*\\big (\\frac {x-y}{t}\\big)\\big \\}.$ A generalized form of characteristics of the Cauchy problems is taken into account the context. Then we examine the strip of differentiability of the viscosity solution given by the function $u(t,x).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.2547","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":"2026-05-18T03:04:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/eOwnEqC1bPwRKKoXOMD03PNMb5lku/NG5b6EvVfrzJ5hY8YHCSWa29c14HbyU3Jca+Pr3qu8XZzHgiLl1YmDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:24:54.339058Z"},"content_sha256":"bbf1c688c8e5416b25b5700ae260607ab0d0f015265f15247e0aa9d9e776fbf5","schema_version":"1.0","event_id":"sha256:bbf1c688c8e5416b25b5700ae260607ab0d0f015265f15247e0aa9d9e776fbf5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2YFVW2ZDRIGH2DOV23XI4H3JET/bundle.json","state_url":"https://pith.science/pith/2YFVW2ZDRIGH2DOV23XI4H3JET/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2YFVW2ZDRIGH2DOV23XI4H3JET/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":"2026-06-10T01:24:54Z","links":{"resolver":"https://pith.science/pith/2YFVW2ZDRIGH2DOV23XI4H3JET","bundle":"https://pith.science/pith/2YFVW2ZDRIGH2DOV23XI4H3JET/bundle.json","state":"https://pith.science/pith/2YFVW2ZDRIGH2DOV23XI4H3JET/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2YFVW2ZDRIGH2DOV23XI4H3JET/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:2YFVW2ZDRIGH2DOV23XI4H3JET","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":"f0f6ddffc193896f929315a88f19428817a1e0423ccdf5bc03d093753df4a3f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T15:25:42Z","title_canon_sha256":"2563dc90a67e8d5a2ff7e8e90b59edb1d6873b4e417fff85cc46d41b5de7683a"},"schema_version":"1.0","source":{"id":"1309.2547","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.2547","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"arxiv_version","alias_value":"1309.2547v2","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.2547","created_at":"2026-05-18T03:04:22Z"},{"alias_kind":"pith_short_12","alias_value":"2YFVW2ZDRIGH","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2YFVW2ZDRIGH2DOV","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2YFVW2ZD","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:bbf1c688c8e5416b25b5700ae260607ab0d0f015265f15247e0aa9d9e776fbf5","target":"graph","created_at":"2026-05-18T03:04:22Z","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 study some differential properties of viscosity solution for Hamilton - Jacobi equations defined by Hopf-Lax formula $u(t,x)=\\min_{y\\in \\R^n} \\big\\{\\sigma (y)+tH^*\\big (\\frac {x-y}{t}\\big)\\big \\}.$ A generalized form of characteristics of the Cauchy problems is taken into account the context. Then we examine the strip of differentiability of the viscosity solution given by the function $u(t,x).$","authors_text":"Nguyen Hoang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T15:25:42Z","title":"Hopf-Lax formula and generalized characteristics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.2547","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":"sha256:071287a599a86832c4c3515a854c68e36098acdc15b87f8e9b3bd8986f6bca94","target":"record","created_at":"2026-05-18T03:04:22Z","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":"f0f6ddffc193896f929315a88f19428817a1e0423ccdf5bc03d093753df4a3f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-09-10T15:25:42Z","title_canon_sha256":"2563dc90a67e8d5a2ff7e8e90b59edb1d6873b4e417fff85cc46d41b5de7683a"},"schema_version":"1.0","source":{"id":"1309.2547","kind":"arxiv","version":2}},"canonical_sha256":"d60b5b6b238a0c7d0dd5d6ee8e1f6924d34db18fd8110e04f170cad1cb6336e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d60b5b6b238a0c7d0dd5d6ee8e1f6924d34db18fd8110e04f170cad1cb6336e5","first_computed_at":"2026-05-18T03:04:22.342814Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:22.342814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/0GPGLf4P83taWYxF4sin9pJAs46zvrjHYLdB1idqvpNW/gLuzTRGpgou2MWuUdqSfu//we4Hs1dk+6fFBf3Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:22.343212Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.2547","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:071287a599a86832c4c3515a854c68e36098acdc15b87f8e9b3bd8986f6bca94","sha256:bbf1c688c8e5416b25b5700ae260607ab0d0f015265f15247e0aa9d9e776fbf5"],"state_sha256":"f2c5cf05924515e8e15f2b61c400e0477c7dcd7314304bc986d2cbd1ee10eb57"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7tHXMtynhX2p2KoMx75atr3oVBJ/o1E5lvYHAtYoqzcJauzaQAiDfQ6u1hocibIdkGOPAXp72faEYSLd9pyQDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T01:24:54.342948Z","bundle_sha256":"1bf3a34eeef415b7de5f3048bd23f89f90bb999886ea24bf77235e1fd85e3eb2"}}