{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:M2KSWSGLFNMS5LQUYRL35PULVK","short_pith_number":"pith:M2KSWSGL","canonical_record":{"source":{"id":"1203.5691","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.geo-ph","submitted_at":"2012-03-26T14:54:03Z","cross_cats_sorted":["math.AP","math.DS"],"title_canon_sha256":"c7540f9f9ee2b1205f9dab424e0d9cddb26930ad2ef55042c16ed73cd359493c","abstract_canon_sha256":"a5f939b415dd3386b4b64ee9df89c354cbd4ea59e8921d8a7054fe14a6d95d6f"},"schema_version":"1.0"},"canonical_sha256":"66952b48cb2b592eae14c457bebe8baa9fc9830223a8006ced56581c9d9c6f3f","source":{"kind":"arxiv","id":"1203.5691","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5691","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5691v1","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5691","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"pith_short_12","alias_value":"M2KSWSGLFNMS","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"M2KSWSGLFNMS5LQU","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"M2KSWSGL","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:M2KSWSGLFNMS5LQUYRL35PULVK","target":"record","payload":{"canonical_record":{"source":{"id":"1203.5691","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.geo-ph","submitted_at":"2012-03-26T14:54:03Z","cross_cats_sorted":["math.AP","math.DS"],"title_canon_sha256":"c7540f9f9ee2b1205f9dab424e0d9cddb26930ad2ef55042c16ed73cd359493c","abstract_canon_sha256":"a5f939b415dd3386b4b64ee9df89c354cbd4ea59e8921d8a7054fe14a6d95d6f"},"schema_version":"1.0"},"canonical_sha256":"66952b48cb2b592eae14c457bebe8baa9fc9830223a8006ced56581c9d9c6f3f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:14.425363Z","signature_b64":"0aC6Lxd7xrxtO2zBXy3bPUOcc/Pjh7goFp9OvyR2B/w8gwWP0BzTzo1CyFeafkLefWQ2OSEJBgDaPW/7KmNiCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66952b48cb2b592eae14c457bebe8baa9fc9830223a8006ced56581c9d9c6f3f","last_reissued_at":"2026-05-18T03:59:14.424908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:14.424908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.5691","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":"2026-05-18T03:59:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mSNRxSSf09BVkqH+lXgEYO+/tA/KXrfY+nq81naZOKiGLKEfqNNR+m5K68d1+caDfxQn4z03inWb0LzgkfAbDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T18:55:20.514573Z"},"content_sha256":"0f01df5028f6524e79d7b786f338863f4cd18b118a65e724666efae65cb0f08a","schema_version":"1.0","event_id":"sha256:0f01df5028f6524e79d7b786f338863f4cd18b118a65e724666efae65cb0f08a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:M2KSWSGLFNMS5LQUYRL35PULVK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On modeling hydraulic fracture in proper variables: stiffness, accuracy, sensitivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DS"],"primary_cat":"physics.geo-ph","authors_text":"Alexander Linkov, Gennady Mishuris, Michal Wrobel","submitted_at":"2012-03-26T14:54:03Z","abstract_excerpt":"The problem of hydraulic fracture propagation is considered by using its recently suggested modified formulation in terms of the particle velocity, the opening in the proper degree, appropriate spatial coordinates and $\\varepsilon$-regularization. We show that the formulation may serve for significant increasing the efficiency of numerical tracing the fracture propagation. Its advantages are illustrated by re-visiting the Nordgren problem. It is shown that the modified formulation facilitates (i) possibility to have various stiffness of differential equations resulting after spatial discretiza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5691","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":"2026-05-18T03:59:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EhppQtdCrI3nGWQw3Dt1PuP3rqp3WfWhzbBFteL+44BRl8KBN95swoNj+ECsqwR6w7whEwI0UgEh+FsuQmRACg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T18:55:20.514910Z"},"content_sha256":"18e9e86684b39b1c41b384ac9a377ab13e9228da99c77c9e423e5ff543b29652","schema_version":"1.0","event_id":"sha256:18e9e86684b39b1c41b384ac9a377ab13e9228da99c77c9e423e5ff543b29652"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M2KSWSGLFNMS5LQUYRL35PULVK/bundle.json","state_url":"https://pith.science/pith/M2KSWSGLFNMS5LQUYRL35PULVK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M2KSWSGLFNMS5LQUYRL35PULVK/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":"2026-07-01T18:55:20Z","links":{"resolver":"https://pith.science/pith/M2KSWSGLFNMS5LQUYRL35PULVK","bundle":"https://pith.science/pith/M2KSWSGLFNMS5LQUYRL35PULVK/bundle.json","state":"https://pith.science/pith/M2KSWSGLFNMS5LQUYRL35PULVK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M2KSWSGLFNMS5LQUYRL35PULVK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:M2KSWSGLFNMS5LQUYRL35PULVK","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":"a5f939b415dd3386b4b64ee9df89c354cbd4ea59e8921d8a7054fe14a6d95d6f","cross_cats_sorted":["math.AP","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.geo-ph","submitted_at":"2012-03-26T14:54:03Z","title_canon_sha256":"c7540f9f9ee2b1205f9dab424e0d9cddb26930ad2ef55042c16ed73cd359493c"},"schema_version":"1.0","source":{"id":"1203.5691","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5691","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5691v1","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5691","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"pith_short_12","alias_value":"M2KSWSGLFNMS","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"M2KSWSGLFNMS5LQU","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"M2KSWSGL","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:18e9e86684b39b1c41b384ac9a377ab13e9228da99c77c9e423e5ff543b29652","target":"graph","created_at":"2026-05-18T03:59:14Z","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":"The problem of hydraulic fracture propagation is considered by using its recently suggested modified formulation in terms of the particle velocity, the opening in the proper degree, appropriate spatial coordinates and $\\varepsilon$-regularization. We show that the formulation may serve for significant increasing the efficiency of numerical tracing the fracture propagation. Its advantages are illustrated by re-visiting the Nordgren problem. It is shown that the modified formulation facilitates (i) possibility to have various stiffness of differential equations resulting after spatial discretiza","authors_text":"Alexander Linkov, Gennady Mishuris, Michal Wrobel","cross_cats":["math.AP","math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.geo-ph","submitted_at":"2012-03-26T14:54:03Z","title":"On modeling hydraulic fracture in proper variables: stiffness, accuracy, sensitivity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5691","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":"sha256:0f01df5028f6524e79d7b786f338863f4cd18b118a65e724666efae65cb0f08a","target":"record","created_at":"2026-05-18T03:59:14Z","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":"a5f939b415dd3386b4b64ee9df89c354cbd4ea59e8921d8a7054fe14a6d95d6f","cross_cats_sorted":["math.AP","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.geo-ph","submitted_at":"2012-03-26T14:54:03Z","title_canon_sha256":"c7540f9f9ee2b1205f9dab424e0d9cddb26930ad2ef55042c16ed73cd359493c"},"schema_version":"1.0","source":{"id":"1203.5691","kind":"arxiv","version":1}},"canonical_sha256":"66952b48cb2b592eae14c457bebe8baa9fc9830223a8006ced56581c9d9c6f3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66952b48cb2b592eae14c457bebe8baa9fc9830223a8006ced56581c9d9c6f3f","first_computed_at":"2026-05-18T03:59:14.424908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:14.424908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0aC6Lxd7xrxtO2zBXy3bPUOcc/Pjh7goFp9OvyR2B/w8gwWP0BzTzo1CyFeafkLefWQ2OSEJBgDaPW/7KmNiCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:14.425363Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.5691","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f01df5028f6524e79d7b786f338863f4cd18b118a65e724666efae65cb0f08a","sha256:18e9e86684b39b1c41b384ac9a377ab13e9228da99c77c9e423e5ff543b29652"],"state_sha256":"825cc4199dd060fab0539267762d5e04984d5aacd1809cf43cc7547264aa6b33"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XtSDhne/niYdH84G+wdvQAgUhJsAgjaMYNq3mxB+cUL1KC3FYBKcdc1iKtAL8FyVzApIAhysul7CjXHDHfNQCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T18:55:20.516837Z","bundle_sha256":"f1f697459faaf799c51c63aea0a3fd0e03429d9393998e77e065b58920bb02b0"}}