{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:HLR2I37UJ6YC7UDM5EC6SLMKEG","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":"b4959b84c17e5dd17b94bf6bf9d45955174567044cd8e89faff5911b9f81e060","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2023-07-12T14:38:20Z","title_canon_sha256":"d0e38331446fa36973070c444993d7cba99d069834eb5a31a602e6869d1cbcb7"},"schema_version":"1.0","source":{"id":"2307.06198","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2307.06198","created_at":"2026-07-05T08:49:51Z"},{"alias_kind":"arxiv_version","alias_value":"2307.06198v4","created_at":"2026-07-05T08:49:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.06198","created_at":"2026-07-05T08:49:51Z"},{"alias_kind":"pith_short_12","alias_value":"HLR2I37UJ6YC","created_at":"2026-07-05T08:49:51Z"},{"alias_kind":"pith_short_16","alias_value":"HLR2I37UJ6YC7UDM","created_at":"2026-07-05T08:49:51Z"},{"alias_kind":"pith_short_8","alias_value":"HLR2I37U","created_at":"2026-07-05T08:49:51Z"}],"graph_snapshots":[{"event_id":"sha256:dfdd57a5522b00ee0a5a14e547e25a735b43b3f7d8c956a514a66b8d6c8734ad","target":"graph","created_at":"2026-07-05T08:49:51Z","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/2307.06198/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, we study $m$-order logarithmic Laplacian $\\mathcal{L}_m$, which is a singular integro-differential operator with symbol $\\big(2\\ln |\\cdot|\\big)^m$ by the Fourier transform. With help of these logarithmic Laplacians, we build the $n$-th order Taylor expansion for fractional Laplacian with respect to the order and the Riesz operators: for $u \\in C^\\infty_c(\\mathbb{R}^N)$ and $x \\in \\mathbb{R}^N$, $$\n  (-\\Delta)^s u (x) = u(x) + \\sum^{n}_{m=1} \\frac{s^m}{m!}\\mathcal{L}_mu(x) + o(s^n) \\quad {\\rm as}\\ \\, s\\to 0^+ $$ and $$ \\big(\\Phi_s\\ast u\\big)(x) = u(x) + \\sum^{n}_{m=1}(-1)^m\\fra","authors_text":"Huyuan Chen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2023-07-12T14:38:20Z","title":"On m-order logarithmic Laplacians and related propeties"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.06198","kind":"arxiv","version":4},"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:cb6be50ec78cec55c44fa3f4901c3e7f6c4b7f985339a059f190ebd4d118a4b0","target":"record","created_at":"2026-07-05T08:49:51Z","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":"b4959b84c17e5dd17b94bf6bf9d45955174567044cd8e89faff5911b9f81e060","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.AP","submitted_at":"2023-07-12T14:38:20Z","title_canon_sha256":"d0e38331446fa36973070c444993d7cba99d069834eb5a31a602e6869d1cbcb7"},"schema_version":"1.0","source":{"id":"2307.06198","kind":"arxiv","version":4}},"canonical_sha256":"3ae3a46ff44fb02fd06ce905e92d8a2194c4b2d6d84245d3046f7ac43e539ca2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ae3a46ff44fb02fd06ce905e92d8a2194c4b2d6d84245d3046f7ac43e539ca2","first_computed_at":"2026-07-05T08:49:51.311369Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:49:51.311369Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u2Jv59d9XF3edKRMeqAgivA9KQ2nH1n6dx/0BDTMwX4ngB/wtYQiGkeVe9/kYu3cb3EVzfV+601pmNhSSukUDg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:49:51.311843Z","signed_message":"canonical_sha256_bytes"},"source_id":"2307.06198","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb6be50ec78cec55c44fa3f4901c3e7f6c4b7f985339a059f190ebd4d118a4b0","sha256:dfdd57a5522b00ee0a5a14e547e25a735b43b3f7d8c956a514a66b8d6c8734ad"],"state_sha256":"03be85eed880c3bacec42d0cae5c1a75d080e14791d21eebcb1d09a8152a2cd5"}