{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:OR7PCG3NBBMYABIXGPCDJ52OHR","short_pith_number":"pith:OR7PCG3N","canonical_record":{"source":{"id":"2109.08988","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-09-18T19:02:53Z","cross_cats_sorted":[],"title_canon_sha256":"99f549273579e7695e2b5a426ee810d95323c6e0a3a7d4bbdf96ef0846bf234a","abstract_canon_sha256":"8cc8ec28e4443d19a209e094ba6babe7b111a10b0c7afe1d6a0078c9a9405735"},"schema_version":"1.0"},"canonical_sha256":"747ef11b6d085980051733c434f74e3c54367f53920c55fe3108f08e0c930be3","source":{"kind":"arxiv","id":"2109.08988","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.08988","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"arxiv_version","alias_value":"2109.08988v1","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.08988","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"pith_short_12","alias_value":"OR7PCG3NBBMY","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"pith_short_16","alias_value":"OR7PCG3NBBMYABIX","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"pith_short_8","alias_value":"OR7PCG3N","created_at":"2026-07-05T03:15:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:OR7PCG3NBBMYABIXGPCDJ52OHR","target":"record","payload":{"canonical_record":{"source":{"id":"2109.08988","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-09-18T19:02:53Z","cross_cats_sorted":[],"title_canon_sha256":"99f549273579e7695e2b5a426ee810d95323c6e0a3a7d4bbdf96ef0846bf234a","abstract_canon_sha256":"8cc8ec28e4443d19a209e094ba6babe7b111a10b0c7afe1d6a0078c9a9405735"},"schema_version":"1.0"},"canonical_sha256":"747ef11b6d085980051733c434f74e3c54367f53920c55fe3108f08e0c930be3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:15:40.316411Z","signature_b64":"QWxRjdkFftFgCTwPQT6U35EXC4yqssGSFF1DUq0T0mbnMV1r2fo1umam1hyduhHzw3plyWFXLH3Q89xgw6rYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"747ef11b6d085980051733c434f74e3c54367f53920c55fe3108f08e0c930be3","last_reissued_at":"2026-07-05T03:15:40.315944Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:15:40.315944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.08988","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":"2026-07-05T03:15:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TxOa94i7ZacIzEQLqCgBOZUuukjOPfhlrLzzmskHJ7IBsp7cWpjlx06YsEmdixKvOm5nOhkGCWE3YV/qNvF+CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:35:51.987716Z"},"content_sha256":"0eaaed4227422192ede913118fe8952f13e2faf7bf75a4fe39d9e3dbb38e354f","schema_version":"1.0","event_id":"sha256:0eaaed4227422192ede913118fe8952f13e2faf7bf75a4fe39d9e3dbb38e354f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:OR7PCG3NBBMYABIXGPCDJ52OHR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\\mathbb{T}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"P. G\\'erard, P. Topalov, T. Kappeler","submitted_at":"2021-09-18T19:02:53Z","abstract_excerpt":"We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H^{s}_{0}(\\mathbb{T},\\mathbb{R})$, $s > -1/2$, to the scale of weighted $\\ell^2-$sequence spaces, $\\mathfrak{h}^{s +1/2}_{r,0}(\\mathbb{N},\\mathbb{C})$, $s >-1/2$. As an application we show that for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation $\\mathcal{S}_0^t : H^{s}_{0}(\\mathbb{T},\\mathbb{R})\\to H^{s}_{0}(\\mathbb{T},\\mathbb{R})$ is {\\em nowhere locally uniformly continuous} in $H^{s}_{0}(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.08988","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.08988/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:15:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AhGidPAr3d+n4ctU45sJIOH/mL4v5jwmOc3HZF+9F4PgXiCHJcrJHEtX4+7+zGpOg0TZaaVfvabmOmWMxgmQAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:35:51.988282Z"},"content_sha256":"68e9f411fb72fc28f1c4f36a2c4dd7c363899224ac0f3c4d8ace1f3fdfb0c59e","schema_version":"1.0","event_id":"sha256:68e9f411fb72fc28f1c4f36a2c4dd7c363899224ac0f3c4d8ace1f3fdfb0c59e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OR7PCG3NBBMYABIXGPCDJ52OHR/bundle.json","state_url":"https://pith.science/pith/OR7PCG3NBBMYABIXGPCDJ52OHR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OR7PCG3NBBMYABIXGPCDJ52OHR/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":"2026-08-08T03:35:51Z","links":{"resolver":"https://pith.science/pith/OR7PCG3NBBMYABIXGPCDJ52OHR","bundle":"https://pith.science/pith/OR7PCG3NBBMYABIXGPCDJ52OHR/bundle.json","state":"https://pith.science/pith/OR7PCG3NBBMYABIXGPCDJ52OHR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OR7PCG3NBBMYABIXGPCDJ52OHR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:OR7PCG3NBBMYABIXGPCDJ52OHR","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":"8cc8ec28e4443d19a209e094ba6babe7b111a10b0c7afe1d6a0078c9a9405735","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-09-18T19:02:53Z","title_canon_sha256":"99f549273579e7695e2b5a426ee810d95323c6e0a3a7d4bbdf96ef0846bf234a"},"schema_version":"1.0","source":{"id":"2109.08988","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.08988","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"arxiv_version","alias_value":"2109.08988v1","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.08988","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"pith_short_12","alias_value":"OR7PCG3NBBMY","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"pith_short_16","alias_value":"OR7PCG3NBBMYABIX","created_at":"2026-07-05T03:15:40Z"},{"alias_kind":"pith_short_8","alias_value":"OR7PCG3N","created_at":"2026-07-05T03:15:40Z"}],"graph_snapshots":[{"event_id":"sha256:68e9f411fb72fc28f1c4f36a2c4dd7c363899224ac0f3c4d8ace1f3fdfb0c59e","target":"graph","created_at":"2026-07-05T03:15:40Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2109.08988/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on $\\mathbb{T}$, also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces $H^{s}_{0}(\\mathbb{T},\\mathbb{R})$, $s > -1/2$, to the scale of weighted $\\ell^2-$sequence spaces, $\\mathfrak{h}^{s +1/2}_{r,0}(\\mathbb{N},\\mathbb{C})$, $s >-1/2$. As an application we show that for any $-1/2<s<0$, the flow map of the Benjamin-Ono equation $\\mathcal{S}_0^t : H^{s}_{0}(\\mathbb{T},\\mathbb{R})\\to H^{s}_{0}(\\mathbb{T},\\mathbb{R})$ is {\\em nowhere locally uniformly continuous} in $H^{s}_{0}(\\","authors_text":"P. G\\'erard, P. Topalov, T. Kappeler","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-09-18T19:02:53Z","title":"On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\\mathbb{T}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.08988","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":"sha256:0eaaed4227422192ede913118fe8952f13e2faf7bf75a4fe39d9e3dbb38e354f","target":"record","created_at":"2026-07-05T03:15:40Z","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":"8cc8ec28e4443d19a209e094ba6babe7b111a10b0c7afe1d6a0078c9a9405735","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-09-18T19:02:53Z","title_canon_sha256":"99f549273579e7695e2b5a426ee810d95323c6e0a3a7d4bbdf96ef0846bf234a"},"schema_version":"1.0","source":{"id":"2109.08988","kind":"arxiv","version":1}},"canonical_sha256":"747ef11b6d085980051733c434f74e3c54367f53920c55fe3108f08e0c930be3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"747ef11b6d085980051733c434f74e3c54367f53920c55fe3108f08e0c930be3","first_computed_at":"2026-07-05T03:15:40.315944Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:15:40.315944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QWxRjdkFftFgCTwPQT6U35EXC4yqssGSFF1DUq0T0mbnMV1r2fo1umam1hyduhHzw3plyWFXLH3Q89xgw6rYCw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:15:40.316411Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.08988","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0eaaed4227422192ede913118fe8952f13e2faf7bf75a4fe39d9e3dbb38e354f","sha256:68e9f411fb72fc28f1c4f36a2c4dd7c363899224ac0f3c4d8ace1f3fdfb0c59e"],"state_sha256":"99178eab3cad02e9f47d9393614bebdb8a417c048aa2940854480d4a3bc7c952"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7vjtrFvcPOsM7WapOEZTKobMYCrFgLDB5rlvkUPwwihmhL/QKzNf/ZTRtc2CsS1kDtrtgwROEvNecy5FwbvFCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T03:35:51.993483Z","bundle_sha256":"95541c638c26de9010eab131dc114416ddc9eba9781d903679eb6d33da22f56b"}}