{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:XHYCVQTAKFB7T33RJCDYLYPRAX","short_pith_number":"pith:XHYCVQTA","canonical_record":{"source":{"id":"1705.01418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-03T13:29:18Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"86dbef0eeec85b278c47c886eceba8c43cba4bb0b754cd761304a9aca25e7502","abstract_canon_sha256":"21644b3f7a46baf53564ee20669f4e420a305844af56a58263a1bc92fd22f888"},"schema_version":"1.0"},"canonical_sha256":"b9f02ac2605143f9ef71488785e1f105faa65d9dc8ea489d5a65081572f19ae3","source":{"kind":"arxiv","id":"1705.01418","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.01418","created_at":"2026-05-18T00:32:50Z"},{"alias_kind":"arxiv_version","alias_value":"1705.01418v1","created_at":"2026-05-18T00:32:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.01418","created_at":"2026-05-18T00:32:50Z"},{"alias_kind":"pith_short_12","alias_value":"XHYCVQTAKFB7","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"XHYCVQTAKFB7T33R","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"XHYCVQTA","created_at":"2026-05-18T12:31:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:XHYCVQTAKFB7T33RJCDYLYPRAX","target":"record","payload":{"canonical_record":{"source":{"id":"1705.01418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-03T13:29:18Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"86dbef0eeec85b278c47c886eceba8c43cba4bb0b754cd761304a9aca25e7502","abstract_canon_sha256":"21644b3f7a46baf53564ee20669f4e420a305844af56a58263a1bc92fd22f888"},"schema_version":"1.0"},"canonical_sha256":"b9f02ac2605143f9ef71488785e1f105faa65d9dc8ea489d5a65081572f19ae3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:50.781203Z","signature_b64":"Kc5AZZM2tAKdePkvxYrGs8JM/CjKF85UVfLnVZel6sINYTPcgmXO+eKp/C4DoGFHoWzLvF7xn+YafEMtRAsVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9f02ac2605143f9ef71488785e1f105faa65d9dc8ea489d5a65081572f19ae3","last_reissued_at":"2026-05-18T00:32:50.780615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:50.780615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.01418","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":"2026-05-18T00:32:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ClBcNWx7zMydFQGFFII3K4T6rqPGxp2xuDom3pCGaymaenu5mha5jtr+9o21ObsF9oGSkFzYnO26ChVyqsDvAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:52:01.514404Z"},"content_sha256":"de2a0025167e8e1871656c02caa7afca21ac0b9f5c92531de4b51e12e0e7c772","schema_version":"1.0","event_id":"sha256:de2a0025167e8e1871656c02caa7afca21ac0b9f5c92531de4b51e12e0e7c772"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:XHYCVQTAKFB7T33RJCDYLYPRAX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Wave equation for operators with discrete spectrum and irregular propagation speed","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Michael Ruzhansky, Niyaz Tokmagambetov","submitted_at":"2017-05-03T13:29:18Z","abstract_excerpt":"Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular, H\\\"older, and distributional. We also consider cases when it it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of \"very weak solutions\" to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distribution"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01418","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":"2026-05-18T00:32:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oNfdAQDf4n7m4z0awh6fex9ZttYEaB/KtI2v0nh3Wzh74Sz9OH6rY8ivSCH47zivOvGfqoij/JJnUxMG9CoZCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:52:01.514971Z"},"content_sha256":"376f7e50342315062662f1537d43639bb50a599485793f25a921f62d0710c648","schema_version":"1.0","event_id":"sha256:376f7e50342315062662f1537d43639bb50a599485793f25a921f62d0710c648"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XHYCVQTAKFB7T33RJCDYLYPRAX/bundle.json","state_url":"https://pith.science/pith/XHYCVQTAKFB7T33RJCDYLYPRAX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XHYCVQTAKFB7T33RJCDYLYPRAX/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":"2026-06-09T16:52:01Z","links":{"resolver":"https://pith.science/pith/XHYCVQTAKFB7T33RJCDYLYPRAX","bundle":"https://pith.science/pith/XHYCVQTAKFB7T33RJCDYLYPRAX/bundle.json","state":"https://pith.science/pith/XHYCVQTAKFB7T33RJCDYLYPRAX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XHYCVQTAKFB7T33RJCDYLYPRAX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:XHYCVQTAKFB7T33RJCDYLYPRAX","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":"21644b3f7a46baf53564ee20669f4e420a305844af56a58263a1bc92fd22f888","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-03T13:29:18Z","title_canon_sha256":"86dbef0eeec85b278c47c886eceba8c43cba4bb0b754cd761304a9aca25e7502"},"schema_version":"1.0","source":{"id":"1705.01418","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.01418","created_at":"2026-05-18T00:32:50Z"},{"alias_kind":"arxiv_version","alias_value":"1705.01418v1","created_at":"2026-05-18T00:32:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.01418","created_at":"2026-05-18T00:32:50Z"},{"alias_kind":"pith_short_12","alias_value":"XHYCVQTAKFB7","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"XHYCVQTAKFB7T33R","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"XHYCVQTA","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:376f7e50342315062662f1537d43639bb50a599485793f25a921f62d0710c648","target":"graph","created_at":"2026-05-18T00:32:50Z","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":"Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular, H\\\"older, and distributional. We also consider cases when it it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of \"very weak solutions\" to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distribution","authors_text":"Michael Ruzhansky, Niyaz Tokmagambetov","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-03T13:29:18Z","title":"Wave equation for operators with discrete spectrum and irregular propagation speed"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01418","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:de2a0025167e8e1871656c02caa7afca21ac0b9f5c92531de4b51e12e0e7c772","target":"record","created_at":"2026-05-18T00:32:50Z","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":"21644b3f7a46baf53564ee20669f4e420a305844af56a58263a1bc92fd22f888","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-03T13:29:18Z","title_canon_sha256":"86dbef0eeec85b278c47c886eceba8c43cba4bb0b754cd761304a9aca25e7502"},"schema_version":"1.0","source":{"id":"1705.01418","kind":"arxiv","version":1}},"canonical_sha256":"b9f02ac2605143f9ef71488785e1f105faa65d9dc8ea489d5a65081572f19ae3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9f02ac2605143f9ef71488785e1f105faa65d9dc8ea489d5a65081572f19ae3","first_computed_at":"2026-05-18T00:32:50.780615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:50.780615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Kc5AZZM2tAKdePkvxYrGs8JM/CjKF85UVfLnVZel6sINYTPcgmXO+eKp/C4DoGFHoWzLvF7xn+YafEMtRAsVCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:50.781203Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.01418","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de2a0025167e8e1871656c02caa7afca21ac0b9f5c92531de4b51e12e0e7c772","sha256:376f7e50342315062662f1537d43639bb50a599485793f25a921f62d0710c648"],"state_sha256":"5f4405e63c8740a6f4fecc29fa4a0cdda7914046bc44e247aef00c31aa8cfbad"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sQzjz1CSVUwsukc7Qc6FY1h8ciGbjskc5NX/qtmOKZt3QefH6Hc0jIbMETNibtY3kbjsFiKSVdkErJXS8wZ7Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T16:52:01.517282Z","bundle_sha256":"d076795a7c774306b91ea356f6635cfd7e6dd959e077081c7934472ee7844642"}}