{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:MCJXSQIMURZLAHJOOQGC34BVC4","short_pith_number":"pith:MCJXSQIM","canonical_record":{"source":{"id":"1401.6221","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-24T00:02:44Z","cross_cats_sorted":[],"title_canon_sha256":"17412fde36bbd25969080b7b6addb83fd6863f5f5c22713c1dc7cece183cec6c","abstract_canon_sha256":"cc17efe5acbc9da0d082e3d72ff2f472020bb40e02c756d199e9c5c32453374d"},"schema_version":"1.0"},"canonical_sha256":"609379410ca472b01d2e740c2df035170993653f3365ee9257d478999450fb14","source":{"kind":"arxiv","id":"1401.6221","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.6221","created_at":"2026-05-18T03:01:12Z"},{"alias_kind":"arxiv_version","alias_value":"1401.6221v1","created_at":"2026-05-18T03:01:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.6221","created_at":"2026-05-18T03:01:12Z"},{"alias_kind":"pith_short_12","alias_value":"MCJXSQIMURZL","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MCJXSQIMURZLAHJO","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MCJXSQIM","created_at":"2026-05-18T12:28:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:MCJXSQIMURZLAHJOOQGC34BVC4","target":"record","payload":{"canonical_record":{"source":{"id":"1401.6221","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-24T00:02:44Z","cross_cats_sorted":[],"title_canon_sha256":"17412fde36bbd25969080b7b6addb83fd6863f5f5c22713c1dc7cece183cec6c","abstract_canon_sha256":"cc17efe5acbc9da0d082e3d72ff2f472020bb40e02c756d199e9c5c32453374d"},"schema_version":"1.0"},"canonical_sha256":"609379410ca472b01d2e740c2df035170993653f3365ee9257d478999450fb14","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:01:12.833005Z","signature_b64":"jS/Quw3WQLTPZPm2bTU0UzKLEX7KvxRWM5mfvmwdqT9urkIuqmU9qMLrEmKzeJefy674d5vIGQWCSOO75kaqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"609379410ca472b01d2e740c2df035170993653f3365ee9257d478999450fb14","last_reissued_at":"2026-05-18T03:01:12.832094Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:01:12.832094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.6221","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":"2026-05-18T03:01:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b8EmWLlowc39wLwt771Dg0wVUHP5dAimy/eN0FQmj+fupLsP3dhY0csErgznHH7cpjkYtE5oc1p4DkIjRksmCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:23:04.675949Z"},"content_sha256":"fb4a2524e881503c8deb111e33c806f189c1df7d286c3eed76bdc1503205153e","schema_version":"1.0","event_id":"sha256:fb4a2524e881503c8deb111e33c806f189c1df7d286c3eed76bdc1503205153e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:MCJXSQIMURZLAHJOOQGC34BVC4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Error Estimates of the Bloch Band-Based Gaussian Beam Superposition for the Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hailiang Liu, Maksym Pryporov","submitted_at":"2014-01-24T00:02:44Z","abstract_excerpt":"This work is concerned with asymptotic approximations of the semi-classical Schr\\\"odinger equation in periodic media using Gaussian beams. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the Gaussian beam approximation and homogenization leads to the Bloch eigenvalue problem and associated evolution equations for Gaussian beam components in each Bloch band. We formulate a superposition of Bloch-band based Gaussian beams to generate high frequency approximate solutions to the original wave field. For initial data of a sum of finite number of band eigen-fun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.6221","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":"2026-05-18T03:01:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kCJj30FrofvjUyni2au4nL2Nz9HK/nYRFMiss+Qpq/Y6hV4orJuzS6eh+53o6prQ8Dwgl0xcw1wMtPjefQy8Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:23:04.676552Z"},"content_sha256":"dab8c2adecb680a0e6bd67880f08294797bb115a35224cf9f2f4fc09a178cc28","schema_version":"1.0","event_id":"sha256:dab8c2adecb680a0e6bd67880f08294797bb115a35224cf9f2f4fc09a178cc28"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MCJXSQIMURZLAHJOOQGC34BVC4/bundle.json","state_url":"https://pith.science/pith/MCJXSQIMURZLAHJOOQGC34BVC4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MCJXSQIMURZLAHJOOQGC34BVC4/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":"2026-05-27T04:23:04Z","links":{"resolver":"https://pith.science/pith/MCJXSQIMURZLAHJOOQGC34BVC4","bundle":"https://pith.science/pith/MCJXSQIMURZLAHJOOQGC34BVC4/bundle.json","state":"https://pith.science/pith/MCJXSQIMURZLAHJOOQGC34BVC4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MCJXSQIMURZLAHJOOQGC34BVC4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MCJXSQIMURZLAHJOOQGC34BVC4","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":"cc17efe5acbc9da0d082e3d72ff2f472020bb40e02c756d199e9c5c32453374d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-24T00:02:44Z","title_canon_sha256":"17412fde36bbd25969080b7b6addb83fd6863f5f5c22713c1dc7cece183cec6c"},"schema_version":"1.0","source":{"id":"1401.6221","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.6221","created_at":"2026-05-18T03:01:12Z"},{"alias_kind":"arxiv_version","alias_value":"1401.6221v1","created_at":"2026-05-18T03:01:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.6221","created_at":"2026-05-18T03:01:12Z"},{"alias_kind":"pith_short_12","alias_value":"MCJXSQIMURZL","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MCJXSQIMURZLAHJO","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MCJXSQIM","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:dab8c2adecb680a0e6bd67880f08294797bb115a35224cf9f2f4fc09a178cc28","target":"graph","created_at":"2026-05-18T03:01:12Z","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":"This work is concerned with asymptotic approximations of the semi-classical Schr\\\"odinger equation in periodic media using Gaussian beams. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the Gaussian beam approximation and homogenization leads to the Bloch eigenvalue problem and associated evolution equations for Gaussian beam components in each Bloch band. We formulate a superposition of Bloch-band based Gaussian beams to generate high frequency approximate solutions to the original wave field. For initial data of a sum of finite number of band eigen-fun","authors_text":"Hailiang Liu, Maksym Pryporov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-24T00:02:44Z","title":"Error Estimates of the Bloch Band-Based Gaussian Beam Superposition for the Schr\\\"odinger Equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.6221","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:fb4a2524e881503c8deb111e33c806f189c1df7d286c3eed76bdc1503205153e","target":"record","created_at":"2026-05-18T03:01:12Z","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":"cc17efe5acbc9da0d082e3d72ff2f472020bb40e02c756d199e9c5c32453374d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-24T00:02:44Z","title_canon_sha256":"17412fde36bbd25969080b7b6addb83fd6863f5f5c22713c1dc7cece183cec6c"},"schema_version":"1.0","source":{"id":"1401.6221","kind":"arxiv","version":1}},"canonical_sha256":"609379410ca472b01d2e740c2df035170993653f3365ee9257d478999450fb14","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"609379410ca472b01d2e740c2df035170993653f3365ee9257d478999450fb14","first_computed_at":"2026-05-18T03:01:12.832094Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:01:12.832094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jS/Quw3WQLTPZPm2bTU0UzKLEX7KvxRWM5mfvmwdqT9urkIuqmU9qMLrEmKzeJefy674d5vIGQWCSOO75kaqAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:01:12.833005Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.6221","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb4a2524e881503c8deb111e33c806f189c1df7d286c3eed76bdc1503205153e","sha256:dab8c2adecb680a0e6bd67880f08294797bb115a35224cf9f2f4fc09a178cc28"],"state_sha256":"c29615cfd03cc9b6ae47e28fbe8647345173166142c90580af9a489fd728b628"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lSvZ3m0H+oTFuhoqAwTQfS3D8vRyqvcMKTcKG3KAeC60PYyIxiDfFiCgdyaVeQnK7hlz5tLuVpCTUvEOPHVFCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T04:23:04.679965Z","bundle_sha256":"c789129b50687b3aeb05fda006161d5aaea6974ace9b448b85bafab8157c192e"}}