{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:MLFQOM7DDYATKNKWHCMRB3V3KT","short_pith_number":"pith:MLFQOM7D","canonical_record":{"source":{"id":"1103.3154","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-16T12:06:08Z","cross_cats_sorted":[],"title_canon_sha256":"4c29952e205d69a6add01b3c3d77e9548081858e543e00b4f5a44c76ab71415e","abstract_canon_sha256":"c3050666ae966097de52865cb2a09ec909f54972841ee937a8516120bbd439a2"},"schema_version":"1.0"},"canonical_sha256":"62cb0733e31e01353556389910eebb54eccd196ab73b0ddccd27ab29da61888a","source":{"kind":"arxiv","id":"1103.3154","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.3154","created_at":"2026-05-18T03:58:12Z"},{"alias_kind":"arxiv_version","alias_value":"1103.3154v3","created_at":"2026-05-18T03:58:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3154","created_at":"2026-05-18T03:58:12Z"},{"alias_kind":"pith_short_12","alias_value":"MLFQOM7DDYAT","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MLFQOM7DDYATKNKW","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MLFQOM7D","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:MLFQOM7DDYATKNKWHCMRB3V3KT","target":"record","payload":{"canonical_record":{"source":{"id":"1103.3154","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-16T12:06:08Z","cross_cats_sorted":[],"title_canon_sha256":"4c29952e205d69a6add01b3c3d77e9548081858e543e00b4f5a44c76ab71415e","abstract_canon_sha256":"c3050666ae966097de52865cb2a09ec909f54972841ee937a8516120bbd439a2"},"schema_version":"1.0"},"canonical_sha256":"62cb0733e31e01353556389910eebb54eccd196ab73b0ddccd27ab29da61888a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:12.659664Z","signature_b64":"8oWoL7+Mqr9uYNV8ddkfWNl6YDWtf/8XFPDuobCNrXt78wN4uM9dMucfcZa3TMUHu+BZ3WWjpXB5zlt6IBgjBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62cb0733e31e01353556389910eebb54eccd196ab73b0ddccd27ab29da61888a","last_reissued_at":"2026-05-18T03:58:12.659192Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:12.659192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.3154","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:58:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dJhVCnYcKdlLB9573vSEtbvHgjO/zTv62QFlivMZ8DHQx/wALi5WYUZPMZ6Jg9/y6mMaEfqdMvE1X8VqdI2TBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T12:37:37.951535Z"},"content_sha256":"3a949920d2c26bc7a496a5457e5ebfb96cea774d5edc7a3157267b852ee12434","schema_version":"1.0","event_id":"sha256:3a949920d2c26bc7a496a5457e5ebfb96cea774d5edc7a3157267b852ee12434"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:MLFQOM7DDYATKNKWHCMRB3V3KT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a two-component $\\pi$-Camassa--Holm system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Martin Kohlmann","submitted_at":"2011-03-16T12:06:08Z","abstract_excerpt":"A novel $\\pi$-Camassa--Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa--Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3154","kind":"arxiv","version":3},"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":"2026-05-18T03:58:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tbuUxe00KRIID/Mh7+OcweWto2WB0xdzQvOLqbS7iJ2kPfjgpReigLBnnCXSamJEt10MsF3FvbaV4c0B5+kgAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T12:37:37.952176Z"},"content_sha256":"afab5dd7718eee62c768f9a362867e9f190f23a568405e2c08af625e043da378","schema_version":"1.0","event_id":"sha256:afab5dd7718eee62c768f9a362867e9f190f23a568405e2c08af625e043da378"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MLFQOM7DDYATKNKWHCMRB3V3KT/bundle.json","state_url":"https://pith.science/pith/MLFQOM7DDYATKNKWHCMRB3V3KT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MLFQOM7DDYATKNKWHCMRB3V3KT/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":"2026-06-03T12:37:37Z","links":{"resolver":"https://pith.science/pith/MLFQOM7DDYATKNKWHCMRB3V3KT","bundle":"https://pith.science/pith/MLFQOM7DDYATKNKWHCMRB3V3KT/bundle.json","state":"https://pith.science/pith/MLFQOM7DDYATKNKWHCMRB3V3KT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MLFQOM7DDYATKNKWHCMRB3V3KT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:MLFQOM7DDYATKNKWHCMRB3V3KT","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":"c3050666ae966097de52865cb2a09ec909f54972841ee937a8516120bbd439a2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-16T12:06:08Z","title_canon_sha256":"4c29952e205d69a6add01b3c3d77e9548081858e543e00b4f5a44c76ab71415e"},"schema_version":"1.0","source":{"id":"1103.3154","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.3154","created_at":"2026-05-18T03:58:12Z"},{"alias_kind":"arxiv_version","alias_value":"1103.3154v3","created_at":"2026-05-18T03:58:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3154","created_at":"2026-05-18T03:58:12Z"},{"alias_kind":"pith_short_12","alias_value":"MLFQOM7DDYAT","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MLFQOM7DDYATKNKW","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MLFQOM7D","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:afab5dd7718eee62c768f9a362867e9f190f23a568405e2c08af625e043da378","target":"graph","created_at":"2026-05-18T03:58:12Z","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":"A novel $\\pi$-Camassa--Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa--Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature.","authors_text":"Martin Kohlmann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-16T12:06:08Z","title":"On a two-component $\\pi$-Camassa--Holm system"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3154","kind":"arxiv","version":3},"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":"sha256:3a949920d2c26bc7a496a5457e5ebfb96cea774d5edc7a3157267b852ee12434","target":"record","created_at":"2026-05-18T03:58:12Z","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":"c3050666ae966097de52865cb2a09ec909f54972841ee937a8516120bbd439a2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-03-16T12:06:08Z","title_canon_sha256":"4c29952e205d69a6add01b3c3d77e9548081858e543e00b4f5a44c76ab71415e"},"schema_version":"1.0","source":{"id":"1103.3154","kind":"arxiv","version":3}},"canonical_sha256":"62cb0733e31e01353556389910eebb54eccd196ab73b0ddccd27ab29da61888a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62cb0733e31e01353556389910eebb54eccd196ab73b0ddccd27ab29da61888a","first_computed_at":"2026-05-18T03:58:12.659192Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:12.659192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8oWoL7+Mqr9uYNV8ddkfWNl6YDWtf/8XFPDuobCNrXt78wN4uM9dMucfcZa3TMUHu+BZ3WWjpXB5zlt6IBgjBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:12.659664Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.3154","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3a949920d2c26bc7a496a5457e5ebfb96cea774d5edc7a3157267b852ee12434","sha256:afab5dd7718eee62c768f9a362867e9f190f23a568405e2c08af625e043da378"],"state_sha256":"a53c5e6b14c77fc9656e9f793868aeb6d63bdf74b4d41ffd6190e3079cf7fcb5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H0m1N7dFvkfw4FkPxNpIe9ORBCgTjRLmka/PgHQUW0PIOCHynPKdL8yK8PU70EK7tA6zBFAPjrkSpnOTZbXoBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T12:37:37.955211Z","bundle_sha256":"1f7799f0d3fc3cccfdc687ccc42842874a72912f522af32b55662531f401703f"}}