{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:4OZ2UCUTKHPNFBAOEP4HBBM7GU","short_pith_number":"pith:4OZ2UCUT","canonical_record":{"source":{"id":"1102.2685","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-02-14T06:39:57Z","cross_cats_sorted":[],"title_canon_sha256":"03e9ca319a409eb5c32308595ef51e0d538662850b7f82d7781502fd3b087aab","abstract_canon_sha256":"87d69cf341f76eca06e1e61993b6e6f35ac645d713bf1997da7f28f93402e7a5"},"schema_version":"1.0"},"canonical_sha256":"e3b3aa0a9351ded2840e23f870859f35362fe9c146ab98ba979fb664929fddb8","source":{"kind":"arxiv","id":"1102.2685","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.2685","created_at":"2026-05-18T04:28:46Z"},{"alias_kind":"arxiv_version","alias_value":"1102.2685v1","created_at":"2026-05-18T04:28:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2685","created_at":"2026-05-18T04:28:46Z"},{"alias_kind":"pith_short_12","alias_value":"4OZ2UCUTKHPN","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4OZ2UCUTKHPNFBAO","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4OZ2UCUT","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:4OZ2UCUTKHPNFBAOEP4HBBM7GU","target":"record","payload":{"canonical_record":{"source":{"id":"1102.2685","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-02-14T06:39:57Z","cross_cats_sorted":[],"title_canon_sha256":"03e9ca319a409eb5c32308595ef51e0d538662850b7f82d7781502fd3b087aab","abstract_canon_sha256":"87d69cf341f76eca06e1e61993b6e6f35ac645d713bf1997da7f28f93402e7a5"},"schema_version":"1.0"},"canonical_sha256":"e3b3aa0a9351ded2840e23f870859f35362fe9c146ab98ba979fb664929fddb8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:46.701258Z","signature_b64":"p/mFrakTtJwST2lVk8J/t5OGAgL/XI6nKZCBsOtnGBQu6DcB36SALWampq5VUIeA3YYlHOOLZ9hBbEMN08EtCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3b3aa0a9351ded2840e23f870859f35362fe9c146ab98ba979fb664929fddb8","last_reissued_at":"2026-05-18T04:28:46.700734Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:46.700734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.2685","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":"2026-05-18T04:28:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PEBp4D5yqsq1dzYifIg5KYI+ynWjmarrhgSB6sFqhs8PBH+3g4kD9khHrXLB0/kn74FlQKvKro/ovmsKRd2iBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:59:47.117775Z"},"content_sha256":"0d5eca19da60c9f04c55e0559e39fbac5e949399fc4572afe72fcb8ecf20daf8","schema_version":"1.0","event_id":"sha256:0d5eca19da60c9f04c55e0559e39fbac5e949399fc4572afe72fcb8ecf20daf8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:4OZ2UCUTKHPNFBAOEP4HBBM7GU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"General Techniques for Constructing Variational Integrators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Melvin Leok, Tatiana Shingel","submitted_at":"2011-02-14T06:39:57Z","abstract_excerpt":"The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi's solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting-based constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimension"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2685","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":"2026-05-18T04:28:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GPfNqYL+jDlnwRi99n3d8f0VJFyC4Vx8wm5ffNsjBY0zq6PC4I/9BZsFlR1cWeeNdeGHmUhlw/3/eag3Y811DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:59:47.118111Z"},"content_sha256":"15e02e1b04ca6274f9296caabd617ba78d3fb63f95de86a4e4907715348b4261","schema_version":"1.0","event_id":"sha256:15e02e1b04ca6274f9296caabd617ba78d3fb63f95de86a4e4907715348b4261"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4OZ2UCUTKHPNFBAOEP4HBBM7GU/bundle.json","state_url":"https://pith.science/pith/4OZ2UCUTKHPNFBAOEP4HBBM7GU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4OZ2UCUTKHPNFBAOEP4HBBM7GU/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":"2026-06-26T03:59:47Z","links":{"resolver":"https://pith.science/pith/4OZ2UCUTKHPNFBAOEP4HBBM7GU","bundle":"https://pith.science/pith/4OZ2UCUTKHPNFBAOEP4HBBM7GU/bundle.json","state":"https://pith.science/pith/4OZ2UCUTKHPNFBAOEP4HBBM7GU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4OZ2UCUTKHPNFBAOEP4HBBM7GU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:4OZ2UCUTKHPNFBAOEP4HBBM7GU","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":"87d69cf341f76eca06e1e61993b6e6f35ac645d713bf1997da7f28f93402e7a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-02-14T06:39:57Z","title_canon_sha256":"03e9ca319a409eb5c32308595ef51e0d538662850b7f82d7781502fd3b087aab"},"schema_version":"1.0","source":{"id":"1102.2685","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.2685","created_at":"2026-05-18T04:28:46Z"},{"alias_kind":"arxiv_version","alias_value":"1102.2685v1","created_at":"2026-05-18T04:28:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2685","created_at":"2026-05-18T04:28:46Z"},{"alias_kind":"pith_short_12","alias_value":"4OZ2UCUTKHPN","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4OZ2UCUTKHPNFBAO","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4OZ2UCUT","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:15e02e1b04ca6274f9296caabd617ba78d3fb63f95de86a4e4907715348b4261","target":"graph","created_at":"2026-05-18T04:28:46Z","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 numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi's solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting-based constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimension","authors_text":"Melvin Leok, Tatiana Shingel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-02-14T06:39:57Z","title":"General Techniques for Constructing Variational Integrators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2685","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":"sha256:0d5eca19da60c9f04c55e0559e39fbac5e949399fc4572afe72fcb8ecf20daf8","target":"record","created_at":"2026-05-18T04:28:46Z","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":"87d69cf341f76eca06e1e61993b6e6f35ac645d713bf1997da7f28f93402e7a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-02-14T06:39:57Z","title_canon_sha256":"03e9ca319a409eb5c32308595ef51e0d538662850b7f82d7781502fd3b087aab"},"schema_version":"1.0","source":{"id":"1102.2685","kind":"arxiv","version":1}},"canonical_sha256":"e3b3aa0a9351ded2840e23f870859f35362fe9c146ab98ba979fb664929fddb8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3b3aa0a9351ded2840e23f870859f35362fe9c146ab98ba979fb664929fddb8","first_computed_at":"2026-05-18T04:28:46.700734Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:46.700734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p/mFrakTtJwST2lVk8J/t5OGAgL/XI6nKZCBsOtnGBQu6DcB36SALWampq5VUIeA3YYlHOOLZ9hBbEMN08EtCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:46.701258Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.2685","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0d5eca19da60c9f04c55e0559e39fbac5e949399fc4572afe72fcb8ecf20daf8","sha256:15e02e1b04ca6274f9296caabd617ba78d3fb63f95de86a4e4907715348b4261"],"state_sha256":"64aaeb2ddb10a8c01dbabfe10e90613fb393699428783b6bb5d07bd00c3e007e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GeYyc+6ajIvsADNAUvXA6KBBw7Q9E9QrxpqmveyhWGubcP6lSUkjqSuhcmMRtAfRguRZDZoFQ+T9TXZqFYZ8Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T03:59:47.120039Z","bundle_sha256":"1eb6d58843b9d6d87e51798a791697a01248a77642004b07b0dd841b8aadb3a8"}}