{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:BE7J5LSMC7Y5SGTF27C7DGUCCT","short_pith_number":"pith:BE7J5LSM","canonical_record":{"source":{"id":"1807.06861","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-18T11:04:15Z","cross_cats_sorted":[],"title_canon_sha256":"618bc5b28d9fc96f6bcde7213f01e43d83fcf07794e0d85ba0257547f0366458","abstract_canon_sha256":"01a7fc971e847e684669e7ae63ca8729636f45c549d9460733f74ad2d42d27be"},"schema_version":"1.0"},"canonical_sha256":"093e9eae4c17f1d91a65d7c5f19a8214ef9d27246f8a079828fc45a16a33f964","source":{"kind":"arxiv","id":"1807.06861","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.06861","created_at":"2026-05-17T23:55:07Z"},{"alias_kind":"arxiv_version","alias_value":"1807.06861v2","created_at":"2026-05-17T23:55:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.06861","created_at":"2026-05-17T23:55:07Z"},{"alias_kind":"pith_short_12","alias_value":"BE7J5LSMC7Y5","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"BE7J5LSMC7Y5SGTF","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"BE7J5LSM","created_at":"2026-05-18T12:32:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:BE7J5LSMC7Y5SGTF27C7DGUCCT","target":"record","payload":{"canonical_record":{"source":{"id":"1807.06861","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-18T11:04:15Z","cross_cats_sorted":[],"title_canon_sha256":"618bc5b28d9fc96f6bcde7213f01e43d83fcf07794e0d85ba0257547f0366458","abstract_canon_sha256":"01a7fc971e847e684669e7ae63ca8729636f45c549d9460733f74ad2d42d27be"},"schema_version":"1.0"},"canonical_sha256":"093e9eae4c17f1d91a65d7c5f19a8214ef9d27246f8a079828fc45a16a33f964","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:07.562066Z","signature_b64":"cvzJgKBTEMSjqhPaLWGkkrSqJw59MiHwAY+Bf9lgsbfdyynZFHHFI58YfimdVqtvlANqHk+FxXUjEDvlglybBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"093e9eae4c17f1d91a65d7c5f19a8214ef9d27246f8a079828fc45a16a33f964","last_reissued_at":"2026-05-17T23:55:07.561581Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:07.561581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.06861","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:55:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gz87n7+imVeRbctpl4nQaiJo4+v75vaIo1RWWgfYzfrWT5GDi0tjvHlGf9mvM/uach8c4iau8ki6XtBBgVXsDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T02:06:05.644389Z"},"content_sha256":"e194c953dc4b10c62e9b3ce65c4dcd5b50145f6198675ea1611c42e1df424021","schema_version":"1.0","event_id":"sha256:e194c953dc4b10c62e9b3ce65c4dcd5b50145f6198675ea1611c42e1df424021"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:BE7J5LSMC7Y5SGTF27C7DGUCCT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiplicity and concentration results for a fractional Schr\\\"odinger-Poisson type equation with magnetic field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Vincenzo Ambrosio","submitted_at":"2018-07-18T11:04:15Z","abstract_excerpt":"This paper is devoted to the study of fractional Schr\\\"odinger-Poisson type equations with magnetic field of the type \\begin{equation*} \\varepsilon^{2s}(-\\Delta)_{A/\\varepsilon}^{s}u+V(x)u+\\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \\quad \\mbox{ in } \\mathbb{R}^{3}, \\end{equation*} where $\\varepsilon>0$ is a parameter, $s,t\\in (0, 1)$ are such that $2s+2t>3$, $A:\\mathbb{R}^{3}\\rightarrow \\mathbb{R}^{3}$ is a smooth magnetic potential, $(-\\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\\mathbb{R}^{3}\\rightarrow \\mathbb{R}$ is a continuous electric potential and $f:\\mathbb{R}\\ri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06861","kind":"arxiv","version":2},"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":"2026-05-17T23:55:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qDws+KTS2W1JPw9NFVRB0UKxyJBYrmcLRg3iGKXENuc0Wqig4yzl+ydBehNcGTeJudEdXjIubZfwCmYsvVoACA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T02:06:05.644965Z"},"content_sha256":"c62882e6b435121bab43b9d5e29f6a27707e115b41a679ec5baa8ed047d69304","schema_version":"1.0","event_id":"sha256:c62882e6b435121bab43b9d5e29f6a27707e115b41a679ec5baa8ed047d69304"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BE7J5LSMC7Y5SGTF27C7DGUCCT/bundle.json","state_url":"https://pith.science/pith/BE7J5LSMC7Y5SGTF27C7DGUCCT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BE7J5LSMC7Y5SGTF27C7DGUCCT/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":"2026-06-30T02:06:05Z","links":{"resolver":"https://pith.science/pith/BE7J5LSMC7Y5SGTF27C7DGUCCT","bundle":"https://pith.science/pith/BE7J5LSMC7Y5SGTF27C7DGUCCT/bundle.json","state":"https://pith.science/pith/BE7J5LSMC7Y5SGTF27C7DGUCCT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BE7J5LSMC7Y5SGTF27C7DGUCCT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BE7J5LSMC7Y5SGTF27C7DGUCCT","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":"01a7fc971e847e684669e7ae63ca8729636f45c549d9460733f74ad2d42d27be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-18T11:04:15Z","title_canon_sha256":"618bc5b28d9fc96f6bcde7213f01e43d83fcf07794e0d85ba0257547f0366458"},"schema_version":"1.0","source":{"id":"1807.06861","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.06861","created_at":"2026-05-17T23:55:07Z"},{"alias_kind":"arxiv_version","alias_value":"1807.06861v2","created_at":"2026-05-17T23:55:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.06861","created_at":"2026-05-17T23:55:07Z"},{"alias_kind":"pith_short_12","alias_value":"BE7J5LSMC7Y5","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"BE7J5LSMC7Y5SGTF","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"BE7J5LSM","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:c62882e6b435121bab43b9d5e29f6a27707e115b41a679ec5baa8ed047d69304","target":"graph","created_at":"2026-05-17T23:55:07Z","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 paper is devoted to the study of fractional Schr\\\"odinger-Poisson type equations with magnetic field of the type \\begin{equation*} \\varepsilon^{2s}(-\\Delta)_{A/\\varepsilon}^{s}u+V(x)u+\\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \\quad \\mbox{ in } \\mathbb{R}^{3}, \\end{equation*} where $\\varepsilon>0$ is a parameter, $s,t\\in (0, 1)$ are such that $2s+2t>3$, $A:\\mathbb{R}^{3}\\rightarrow \\mathbb{R}^{3}$ is a smooth magnetic potential, $(-\\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\\mathbb{R}^{3}\\rightarrow \\mathbb{R}$ is a continuous electric potential and $f:\\mathbb{R}\\ri","authors_text":"Vincenzo Ambrosio","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-18T11:04:15Z","title":"Multiplicity and concentration results for a fractional Schr\\\"odinger-Poisson type equation with magnetic field"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.06861","kind":"arxiv","version":2},"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:e194c953dc4b10c62e9b3ce65c4dcd5b50145f6198675ea1611c42e1df424021","target":"record","created_at":"2026-05-17T23:55:07Z","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":"01a7fc971e847e684669e7ae63ca8729636f45c549d9460733f74ad2d42d27be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-18T11:04:15Z","title_canon_sha256":"618bc5b28d9fc96f6bcde7213f01e43d83fcf07794e0d85ba0257547f0366458"},"schema_version":"1.0","source":{"id":"1807.06861","kind":"arxiv","version":2}},"canonical_sha256":"093e9eae4c17f1d91a65d7c5f19a8214ef9d27246f8a079828fc45a16a33f964","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"093e9eae4c17f1d91a65d7c5f19a8214ef9d27246f8a079828fc45a16a33f964","first_computed_at":"2026-05-17T23:55:07.561581Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:07.561581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cvzJgKBTEMSjqhPaLWGkkrSqJw59MiHwAY+Bf9lgsbfdyynZFHHFI58YfimdVqtvlANqHk+FxXUjEDvlglybBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:07.562066Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.06861","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e194c953dc4b10c62e9b3ce65c4dcd5b50145f6198675ea1611c42e1df424021","sha256:c62882e6b435121bab43b9d5e29f6a27707e115b41a679ec5baa8ed047d69304"],"state_sha256":"a70ffe19150310c862740d24c6b8e5adc2abb9a9f3991732ca7b2d89570ea1a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sM9JumfJSRELaCAP++gB8Nfpw9zyHeIb78zp4iXIjPpYpEtTp3EOlGbMAjN45+5mhCraVh4IGSc3YV8gRqpzAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T02:06:05.648379Z","bundle_sha256":"488807ccf470161a4256f430624006acaec06d402ca8d9764cd0e0ecff076fcd"}}