{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:4BSY736NILETBUVHFV26VH3ETX","short_pith_number":"pith:4BSY736N","canonical_record":{"source":{"id":"1202.2424","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-11T09:42:30Z","cross_cats_sorted":[],"title_canon_sha256":"23d8e793499ab787215d0c9e961c693d2572ca5b91848496d81896093a970734","abstract_canon_sha256":"df4f77963e8b56b04bf0933286b1565aeabbf5eb2c827a1ec78db7c45533af45"},"schema_version":"1.0"},"canonical_sha256":"e0658fefcd42c930d2a72d75ea9f649dcad6213e3323dff5ffbf9cab1da330ee","source":{"kind":"arxiv","id":"1202.2424","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.2424","created_at":"2026-05-18T01:58:31Z"},{"alias_kind":"arxiv_version","alias_value":"1202.2424v1","created_at":"2026-05-18T01:58:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.2424","created_at":"2026-05-18T01:58:31Z"},{"alias_kind":"pith_short_12","alias_value":"4BSY736NILET","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4BSY736NILETBUVH","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4BSY736N","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:4BSY736NILETBUVHFV26VH3ETX","target":"record","payload":{"canonical_record":{"source":{"id":"1202.2424","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-11T09:42:30Z","cross_cats_sorted":[],"title_canon_sha256":"23d8e793499ab787215d0c9e961c693d2572ca5b91848496d81896093a970734","abstract_canon_sha256":"df4f77963e8b56b04bf0933286b1565aeabbf5eb2c827a1ec78db7c45533af45"},"schema_version":"1.0"},"canonical_sha256":"e0658fefcd42c930d2a72d75ea9f649dcad6213e3323dff5ffbf9cab1da330ee","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:58:31.030247Z","signature_b64":"4SEnXixckGYXeTsKTctqksZ7RYvulBNuEmAy6TKBI9HwKqWjr4NOD9Ickv94YA1l55tZriUTZqQanng9CfTkDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0658fefcd42c930d2a72d75ea9f649dcad6213e3323dff5ffbf9cab1da330ee","last_reissued_at":"2026-05-18T01:58:31.029678Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:58:31.029678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.2424","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":"2026-05-18T01:58:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nP5GVEoInsPNASbTZVKLONeBvhymsWlZAbbHiATtkelzGDGoLssVj9Tsyhn9P5ZyrL262oDXJZLj9RTFU11kBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:29:54.259962Z"},"content_sha256":"f929399918707dd5ca02d1d661b959c7b3d7cf1cd6e23be444289bf212d13d31","schema_version":"1.0","event_id":"sha256:f929399918707dd5ca02d1d661b959c7b3d7cf1cd6e23be444289bf212d13d31"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:4BSY736NILETBUVHFV26VH3ETX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Massimiliano Berti, Philippe Bolle","submitted_at":"2012-02-11T09:42:30Z","abstract_excerpt":"We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T^d, d \\geq 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2424","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":"2026-05-18T01:58:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R0C313jZ5MBtCSwh2jSc3IGh2YcOu3vGuXl23SW8Oa/Pg9UJsd1bl6syoLbMXj36lBTcnEBMgdmbsGYbrU5jAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:29:54.260570Z"},"content_sha256":"9584da8055315523bd8552b4937f4da44e414f4e75513787a5d023c4967b02f1","schema_version":"1.0","event_id":"sha256:9584da8055315523bd8552b4937f4da44e414f4e75513787a5d023c4967b02f1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4BSY736NILETBUVHFV26VH3ETX/bundle.json","state_url":"https://pith.science/pith/4BSY736NILETBUVHFV26VH3ETX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4BSY736NILETBUVHFV26VH3ETX/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":"2026-05-30T13:29:54Z","links":{"resolver":"https://pith.science/pith/4BSY736NILETBUVHFV26VH3ETX","bundle":"https://pith.science/pith/4BSY736NILETBUVHFV26VH3ETX/bundle.json","state":"https://pith.science/pith/4BSY736NILETBUVHFV26VH3ETX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4BSY736NILETBUVHFV26VH3ETX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4BSY736NILETBUVHFV26VH3ETX","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":"df4f77963e8b56b04bf0933286b1565aeabbf5eb2c827a1ec78db7c45533af45","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-11T09:42:30Z","title_canon_sha256":"23d8e793499ab787215d0c9e961c693d2572ca5b91848496d81896093a970734"},"schema_version":"1.0","source":{"id":"1202.2424","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.2424","created_at":"2026-05-18T01:58:31Z"},{"alias_kind":"arxiv_version","alias_value":"1202.2424v1","created_at":"2026-05-18T01:58:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.2424","created_at":"2026-05-18T01:58:31Z"},{"alias_kind":"pith_short_12","alias_value":"4BSY736NILET","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4BSY736NILETBUVH","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4BSY736N","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:9584da8055315523bd8552b4937f4da44e414f4e75513787a5d023c4967b02f1","target":"graph","created_at":"2026-05-18T01:58:31Z","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":"We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T^d, d \\geq 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove t","authors_text":"Massimiliano Berti, Philippe Bolle","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-11T09:42:30Z","title":"Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2424","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:f929399918707dd5ca02d1d661b959c7b3d7cf1cd6e23be444289bf212d13d31","target":"record","created_at":"2026-05-18T01:58:31Z","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":"df4f77963e8b56b04bf0933286b1565aeabbf5eb2c827a1ec78db7c45533af45","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-11T09:42:30Z","title_canon_sha256":"23d8e793499ab787215d0c9e961c693d2572ca5b91848496d81896093a970734"},"schema_version":"1.0","source":{"id":"1202.2424","kind":"arxiv","version":1}},"canonical_sha256":"e0658fefcd42c930d2a72d75ea9f649dcad6213e3323dff5ffbf9cab1da330ee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e0658fefcd42c930d2a72d75ea9f649dcad6213e3323dff5ffbf9cab1da330ee","first_computed_at":"2026-05-18T01:58:31.029678Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:58:31.029678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4SEnXixckGYXeTsKTctqksZ7RYvulBNuEmAy6TKBI9HwKqWjr4NOD9Ickv94YA1l55tZriUTZqQanng9CfTkDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:58:31.030247Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.2424","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f929399918707dd5ca02d1d661b959c7b3d7cf1cd6e23be444289bf212d13d31","sha256:9584da8055315523bd8552b4937f4da44e414f4e75513787a5d023c4967b02f1"],"state_sha256":"8d2819ff2261483fdd0b298348258e51f5457a89ed4bd07efa639e740d225786"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jEWQI7a1s9/68sEyLypYK8HQua3eXWQb+LdZeifsX/MKm9Uhf5jJbFAoVj/OYUJGtQAwb5C75K9+GNlGCPFvBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:29:54.263372Z","bundle_sha256":"29cb5f34de9cf2ad1733cac3aa3fb195d2f3ca726a70f2c2371b27f616b8eda1"}}