{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:2VSN7R3Q7BAM6S7KHBU7HKSHTH","short_pith_number":"pith:2VSN7R3Q","canonical_record":{"source":{"id":"1508.02161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-10T08:15:03Z","cross_cats_sorted":[],"title_canon_sha256":"e18fa77cb11a1bf0ab561d42908ce10d5558bae38494839e294059f269ddc9a0","abstract_canon_sha256":"67cdc9fbb3999395612a12a489da1a566a8b639f40c7252d01cd07112ef2d887"},"schema_version":"1.0"},"canonical_sha256":"d564dfc770f840cf4bea3869f3aa4799dbb393556fc229b3bc1d3a93a36d3837","source":{"kind":"arxiv","id":"1508.02161","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02161","created_at":"2026-05-18T00:14:01Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02161v1","created_at":"2026-05-18T00:14:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02161","created_at":"2026-05-18T00:14:01Z"},{"alias_kind":"pith_short_12","alias_value":"2VSN7R3Q7BAM","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2VSN7R3Q7BAM6S7K","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2VSN7R3Q","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:2VSN7R3Q7BAM6S7KHBU7HKSHTH","target":"record","payload":{"canonical_record":{"source":{"id":"1508.02161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-10T08:15:03Z","cross_cats_sorted":[],"title_canon_sha256":"e18fa77cb11a1bf0ab561d42908ce10d5558bae38494839e294059f269ddc9a0","abstract_canon_sha256":"67cdc9fbb3999395612a12a489da1a566a8b639f40c7252d01cd07112ef2d887"},"schema_version":"1.0"},"canonical_sha256":"d564dfc770f840cf4bea3869f3aa4799dbb393556fc229b3bc1d3a93a36d3837","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:01.584136Z","signature_b64":"msJ5Qp7PJaQzy4Q6v+D19N3Mt+nf/fnNLlY+t2VCLMlksMwEVJmGWiY2RkyMYgo/vCWiAB4nTEE0KLSiTW1HBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d564dfc770f840cf4bea3869f3aa4799dbb393556fc229b3bc1d3a93a36d3837","last_reissued_at":"2026-05-18T00:14:01.583646Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:01.583646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.02161","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":"2026-05-18T00:14:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fAzKJmLTTRxXq8eYoqLRolI+sR/wVTi14udyu1o5yhqkwV4PNQ5YAHof5chue0wlbbUTUinEwYG8s3WHrsj0Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:24:44.615507Z"},"content_sha256":"131b4a1212d5759ba3d2f4e3280ec964ff997cd5b27697b4c11b0788fb1d7216","schema_version":"1.0","event_id":"sha256:131b4a1212d5759ba3d2f4e3280ec964ff997cd5b27697b4c11b0788fb1d7216"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:2VSN7R3Q7BAM6S7KHBU7HKSHTH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Random data Cauchy problem for the nonlinear Schr\\\"{o}dinger equation with derivative nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hiroyuki Hirayama, Mamoru Okamoto","submitted_at":"2015-08-10T08:15:03Z","abstract_excerpt":"We consider the Cauchy problem for the nonlinear Schr\\\"{o}dinger equation with derivative nonlinearity $(i\\partial _t + \\Delta ) u= \\pm \\partial (\\overline{u}^m)$ on $\\R ^d$, $d \\ge 1$, with random initial data, where $\\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\\frac{\\partial}{\\partial x_1} , \\, \\dots , \\, \\frac{\\partial}{\\partial x_d}$ or $|\\nabla |= \\mathcal{F}^{-1}[|\\xi | \\mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\\R ^d)$ with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02161","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":"2026-05-18T00:14:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mGxEeMNYpsBZCO9kwoenboZpAVPSngCIlUlojSCqtDulwfJ0xr1PpD/WowlaHCdt6Znd1u0HoQyFl4W90+QwCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:24:44.616173Z"},"content_sha256":"a71e3e6e315cbab6b001d6adcb12e7780b26bf75aa7d6b028d0a73f40060a19f","schema_version":"1.0","event_id":"sha256:a71e3e6e315cbab6b001d6adcb12e7780b26bf75aa7d6b028d0a73f40060a19f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2VSN7R3Q7BAM6S7KHBU7HKSHTH/bundle.json","state_url":"https://pith.science/pith/2VSN7R3Q7BAM6S7KHBU7HKSHTH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2VSN7R3Q7BAM6S7KHBU7HKSHTH/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":"2026-05-27T09:24:44Z","links":{"resolver":"https://pith.science/pith/2VSN7R3Q7BAM6S7KHBU7HKSHTH","bundle":"https://pith.science/pith/2VSN7R3Q7BAM6S7KHBU7HKSHTH/bundle.json","state":"https://pith.science/pith/2VSN7R3Q7BAM6S7KHBU7HKSHTH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2VSN7R3Q7BAM6S7KHBU7HKSHTH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2VSN7R3Q7BAM6S7KHBU7HKSHTH","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":"67cdc9fbb3999395612a12a489da1a566a8b639f40c7252d01cd07112ef2d887","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-10T08:15:03Z","title_canon_sha256":"e18fa77cb11a1bf0ab561d42908ce10d5558bae38494839e294059f269ddc9a0"},"schema_version":"1.0","source":{"id":"1508.02161","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.02161","created_at":"2026-05-18T00:14:01Z"},{"alias_kind":"arxiv_version","alias_value":"1508.02161v1","created_at":"2026-05-18T00:14:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02161","created_at":"2026-05-18T00:14:01Z"},{"alias_kind":"pith_short_12","alias_value":"2VSN7R3Q7BAM","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2VSN7R3Q7BAM6S7K","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2VSN7R3Q","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:a71e3e6e315cbab6b001d6adcb12e7780b26bf75aa7d6b028d0a73f40060a19f","target":"graph","created_at":"2026-05-18T00:14:01Z","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 consider the Cauchy problem for the nonlinear Schr\\\"{o}dinger equation with derivative nonlinearity $(i\\partial _t + \\Delta ) u= \\pm \\partial (\\overline{u}^m)$ on $\\R ^d$, $d \\ge 1$, with random initial data, where $\\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\\frac{\\partial}{\\partial x_1} , \\, \\dots , \\, \\frac{\\partial}{\\partial x_d}$ or $|\\nabla |= \\mathcal{F}^{-1}[|\\xi | \\mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\\R ^d)$ with","authors_text":"Hiroyuki Hirayama, Mamoru Okamoto","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-10T08:15:03Z","title":"Random data Cauchy problem for the nonlinear Schr\\\"{o}dinger equation with derivative nonlinearity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02161","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":"sha256:131b4a1212d5759ba3d2f4e3280ec964ff997cd5b27697b4c11b0788fb1d7216","target":"record","created_at":"2026-05-18T00:14:01Z","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":"67cdc9fbb3999395612a12a489da1a566a8b639f40c7252d01cd07112ef2d887","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-10T08:15:03Z","title_canon_sha256":"e18fa77cb11a1bf0ab561d42908ce10d5558bae38494839e294059f269ddc9a0"},"schema_version":"1.0","source":{"id":"1508.02161","kind":"arxiv","version":1}},"canonical_sha256":"d564dfc770f840cf4bea3869f3aa4799dbb393556fc229b3bc1d3a93a36d3837","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d564dfc770f840cf4bea3869f3aa4799dbb393556fc229b3bc1d3a93a36d3837","first_computed_at":"2026-05-18T00:14:01.583646Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:01.583646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"msJ5Qp7PJaQzy4Q6v+D19N3Mt+nf/fnNLlY+t2VCLMlksMwEVJmGWiY2RkyMYgo/vCWiAB4nTEE0KLSiTW1HBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:01.584136Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.02161","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:131b4a1212d5759ba3d2f4e3280ec964ff997cd5b27697b4c11b0788fb1d7216","sha256:a71e3e6e315cbab6b001d6adcb12e7780b26bf75aa7d6b028d0a73f40060a19f"],"state_sha256":"27744977165459f9af9c06e4618187d7b0b4bfca3e93205f73fbc8786c8feca2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AKiciVe90qPSTKkFeDSB4antO1/m2wylfL+pu//mbWHvSwyUMpIG3v+iFKX9GD6FafawBsM3llJrpsE3oE+qBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:24:44.618735Z","bundle_sha256":"0edffd00e8f605fd823be8a44255ac34a51cc6c251f3988f4dd3ab7de92322e0"}}