{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:AJL6FVTEW6FCQTAW6DQ2GCHFYW","short_pith_number":"pith:AJL6FVTE","canonical_record":{"source":{"id":"1206.2672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-12T20:59:40Z","cross_cats_sorted":[],"title_canon_sha256":"2d1c81235f81bd3076b37bdfe8f961228efea9e64fcb6cc0695488ce0d610558","abstract_canon_sha256":"27714736d20d243602557e420d41ab6eb06f20168a03368a1cd1b4c25f28764a"},"schema_version":"1.0"},"canonical_sha256":"0257e2d664b78a284c16f0e1a308e5c590a1111eec2ca8cba7109de4bac949f0","source":{"kind":"arxiv","id":"1206.2672","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2672","created_at":"2026-05-18T01:56:59Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2672v1","created_at":"2026-05-18T01:56:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2672","created_at":"2026-05-18T01:56:59Z"},{"alias_kind":"pith_short_12","alias_value":"AJL6FVTEW6FC","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"AJL6FVTEW6FCQTAW","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"AJL6FVTE","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:AJL6FVTEW6FCQTAW6DQ2GCHFYW","target":"record","payload":{"canonical_record":{"source":{"id":"1206.2672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-12T20:59:40Z","cross_cats_sorted":[],"title_canon_sha256":"2d1c81235f81bd3076b37bdfe8f961228efea9e64fcb6cc0695488ce0d610558","abstract_canon_sha256":"27714736d20d243602557e420d41ab6eb06f20168a03368a1cd1b4c25f28764a"},"schema_version":"1.0"},"canonical_sha256":"0257e2d664b78a284c16f0e1a308e5c590a1111eec2ca8cba7109de4bac949f0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:56:59.218950Z","signature_b64":"fVVnvoNbVHro8AxBewzwcJyKH5eCKx5p9fWCyWDFAagcqN7a/C5LtvFNt3Wv9MlvXqLT+Gg/Gn/rsGRTIMM2Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0257e2d664b78a284c16f0e1a308e5c590a1111eec2ca8cba7109de4bac949f0","last_reissued_at":"2026-05-18T01:56:59.218512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:56:59.218512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.2672","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:56:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uE2BOezpfkwQ9hRltBETqwg9m+N/CkoDLamiW3avuH7VmXXe9uQ6ikUjgZvRnnqSQpw4VjUZc2clGT2hQpwABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:29:11.753391Z"},"content_sha256":"03a3f0e88be469db8345cf814556cb9dd80948f1aba476fe516306510b339650","schema_version":"1.0","event_id":"sha256:03a3f0e88be469db8345cf814556cb9dd80948f1aba476fe516306510b339650"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:AJL6FVTEW6FCQTAW6DQ2GCHFYW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generating and Adding Flows on Locally Complete Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Hwa Kil Kim, Nader Masmoudi","submitted_at":"2012-06-12T20:59:40Z","abstract_excerpt":"As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \\cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution cu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2672","kind":"arxiv","version":1},"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":"2026-05-18T01:56:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cnSukID+3/ouplRaB51ADHogBbQrWRnHwXdv57RuAtuKUlxn9TLUFQxvsftEIisviipZg+V5utu5E6NGhLa9Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T20:29:11.754119Z"},"content_sha256":"8089e3c56d5cb296d6b94e63d7f6fa85dce2998eb1ea69a32511dd5235066ca2","schema_version":"1.0","event_id":"sha256:8089e3c56d5cb296d6b94e63d7f6fa85dce2998eb1ea69a32511dd5235066ca2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AJL6FVTEW6FCQTAW6DQ2GCHFYW/bundle.json","state_url":"https://pith.science/pith/AJL6FVTEW6FCQTAW6DQ2GCHFYW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AJL6FVTEW6FCQTAW6DQ2GCHFYW/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":"2026-06-08T20:29:11Z","links":{"resolver":"https://pith.science/pith/AJL6FVTEW6FCQTAW6DQ2GCHFYW","bundle":"https://pith.science/pith/AJL6FVTEW6FCQTAW6DQ2GCHFYW/bundle.json","state":"https://pith.science/pith/AJL6FVTEW6FCQTAW6DQ2GCHFYW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AJL6FVTEW6FCQTAW6DQ2GCHFYW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:AJL6FVTEW6FCQTAW6DQ2GCHFYW","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":"27714736d20d243602557e420d41ab6eb06f20168a03368a1cd1b4c25f28764a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-12T20:59:40Z","title_canon_sha256":"2d1c81235f81bd3076b37bdfe8f961228efea9e64fcb6cc0695488ce0d610558"},"schema_version":"1.0","source":{"id":"1206.2672","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2672","created_at":"2026-05-18T01:56:59Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2672v1","created_at":"2026-05-18T01:56:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2672","created_at":"2026-05-18T01:56:59Z"},{"alias_kind":"pith_short_12","alias_value":"AJL6FVTEW6FC","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"AJL6FVTEW6FCQTAW","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"AJL6FVTE","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:8089e3c56d5cb296d6b94e63d7f6fa85dce2998eb1ea69a32511dd5235066ca2","target":"graph","created_at":"2026-05-18T01:56:59Z","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":"As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \\cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution cu","authors_text":"Hwa Kil Kim, Nader Masmoudi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-12T20:59:40Z","title":"Generating and Adding Flows on Locally Complete Metric Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2672","kind":"arxiv","version":1},"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":"sha256:03a3f0e88be469db8345cf814556cb9dd80948f1aba476fe516306510b339650","target":"record","created_at":"2026-05-18T01:56:59Z","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":"27714736d20d243602557e420d41ab6eb06f20168a03368a1cd1b4c25f28764a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-06-12T20:59:40Z","title_canon_sha256":"2d1c81235f81bd3076b37bdfe8f961228efea9e64fcb6cc0695488ce0d610558"},"schema_version":"1.0","source":{"id":"1206.2672","kind":"arxiv","version":1}},"canonical_sha256":"0257e2d664b78a284c16f0e1a308e5c590a1111eec2ca8cba7109de4bac949f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0257e2d664b78a284c16f0e1a308e5c590a1111eec2ca8cba7109de4bac949f0","first_computed_at":"2026-05-18T01:56:59.218512Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:56:59.218512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fVVnvoNbVHro8AxBewzwcJyKH5eCKx5p9fWCyWDFAagcqN7a/C5LtvFNt3Wv9MlvXqLT+Gg/Gn/rsGRTIMM2Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:56:59.218950Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.2672","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:03a3f0e88be469db8345cf814556cb9dd80948f1aba476fe516306510b339650","sha256:8089e3c56d5cb296d6b94e63d7f6fa85dce2998eb1ea69a32511dd5235066ca2"],"state_sha256":"371790efb81edd729ffe082dca2c8158d5c330c2a2c78cbaf268742231fcaed4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p35tBrhow3mY4lM6cPDn5rT/lt6m+Rl1rEfs0xwnRz9SdTeZZFJlagSJ/n7IklNDqxAQxX0jC9cJZouC11JDCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T20:29:11.758251Z","bundle_sha256":"d0e09f269c63563ddec35951844b9a6d702a2178a36e2834a299960c51a34bf1"}}