{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:52FTCA4FETOMRXCXBKWCQEVGBD","short_pith_number":"pith:52FTCA4F","canonical_record":{"source":{"id":"2210.09811","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2022-10-18T12:43:18Z","cross_cats_sorted":[],"title_canon_sha256":"4bd4b53a2f26f2220e71ca7969a57bfca48d4c08aa2db57cd8ed43df82aa3dab","abstract_canon_sha256":"d282819933e4ce7844822fcd746ea129a0722c896846ad08bb9d8218cd5333a0"},"schema_version":"1.0"},"canonical_sha256":"ee8b31038524dcc8dc570aac2812a608e537443b7ec628671ea67532f10cd628","source":{"kind":"arxiv","id":"2210.09811","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.09811","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"arxiv_version","alias_value":"2210.09811v2","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.09811","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"pith_short_12","alias_value":"52FTCA4FETOM","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"pith_short_16","alias_value":"52FTCA4FETOMRXCX","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"pith_short_8","alias_value":"52FTCA4F","created_at":"2026-07-05T08:00:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:52FTCA4FETOMRXCXBKWCQEVGBD","target":"record","payload":{"canonical_record":{"source":{"id":"2210.09811","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2022-10-18T12:43:18Z","cross_cats_sorted":[],"title_canon_sha256":"4bd4b53a2f26f2220e71ca7969a57bfca48d4c08aa2db57cd8ed43df82aa3dab","abstract_canon_sha256":"d282819933e4ce7844822fcd746ea129a0722c896846ad08bb9d8218cd5333a0"},"schema_version":"1.0"},"canonical_sha256":"ee8b31038524dcc8dc570aac2812a608e537443b7ec628671ea67532f10cd628","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:00:30.989744Z","signature_b64":"5QOjLcdN3aHhwDEzjoxP2bsL8Tiem+oAUHy+X2VAUzvo3IdR5mzQukeymmGeufCUIUWkUE3/9snIXhaqWUt0Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee8b31038524dcc8dc570aac2812a608e537443b7ec628671ea67532f10cd628","last_reissued_at":"2026-07-05T08:00:30.989219Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:00:30.989219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2210.09811","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":"2026-07-05T08:00:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"902KPZuCJxOlVOCF6Vwy7qvV2DRcTcT1nbdgFrQaj+b+kle5lv2hy2abHppw1zseAbrLZYpaKgIjWYPh2XPLDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T13:31:36.775003Z"},"content_sha256":"1c3607a64396beced037b26f68e3030f87d5850be4f1fbb2fc0731b6600a6d5d","schema_version":"1.0","event_id":"sha256:1c3607a64396beced037b26f68e3030f87d5850be4f1fbb2fc0731b6600a6d5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:52FTCA4FETOMRXCXBKWCQEVGBD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Direct Method of Moving Planes for Logarithmic Schr\\\"odinger Operator","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Ruzhansky, Rong Zhang, Vishvesh Kumar","submitted_at":"2022-10-18T12:43:18Z","abstract_excerpt":"In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr$\\ddot{\\text{o}}$dinger operator $(\\mathcal{I}-\\Delta)^{\\log}$ corresponding to the logarithmic symbol $\\log(1 + |\\xi|^2)$, which is a singular integral operator given by $$(\\mathcal{I}-\\Delta)^{\\log}u(x) =c_{N}P.V.\\int_{\\mathbb{R}^{N}}\\frac{u(x)-u(y)}{|x-y|^{N}}\\kappa(|x-y|)dy,$$ where $c_{N}=\\pi^{-\\frac{N}{2}}$, $\\kappa(r)=2^{1-\\frac{N}{2}}r^{\\frac{N}{2}}\\mathcal{K}_{\\frac{N}{2}}(r)$ and $\\mathcal{K}_{\\nu}$ is the modified Bessel function of second kind w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.09811","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2210.09811/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":"2026-07-05T08:00:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J7OcDTOWTLH/gHoZrU2f4YQWXNBHjQXozlwRmCfOWR8+QoOUQ14Oh6omPdKjN76b6AsP7a+JTKDIBW+/pj55Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T13:31:36.776281Z"},"content_sha256":"97ba76148e593940c3e68cb6694176ebc694a12c40457b6bad8ebe1ae41deba9","schema_version":"1.0","event_id":"sha256:97ba76148e593940c3e68cb6694176ebc694a12c40457b6bad8ebe1ae41deba9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/52FTCA4FETOMRXCXBKWCQEVGBD/bundle.json","state_url":"https://pith.science/pith/52FTCA4FETOMRXCXBKWCQEVGBD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/52FTCA4FETOMRXCXBKWCQEVGBD/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":"2026-08-13T13:31:36Z","links":{"resolver":"https://pith.science/pith/52FTCA4FETOMRXCXBKWCQEVGBD","bundle":"https://pith.science/pith/52FTCA4FETOMRXCXBKWCQEVGBD/bundle.json","state":"https://pith.science/pith/52FTCA4FETOMRXCXBKWCQEVGBD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/52FTCA4FETOMRXCXBKWCQEVGBD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:52FTCA4FETOMRXCXBKWCQEVGBD","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":"d282819933e4ce7844822fcd746ea129a0722c896846ad08bb9d8218cd5333a0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2022-10-18T12:43:18Z","title_canon_sha256":"4bd4b53a2f26f2220e71ca7969a57bfca48d4c08aa2db57cd8ed43df82aa3dab"},"schema_version":"1.0","source":{"id":"2210.09811","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2210.09811","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"arxiv_version","alias_value":"2210.09811v2","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2210.09811","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"pith_short_12","alias_value":"52FTCA4FETOM","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"pith_short_16","alias_value":"52FTCA4FETOMRXCX","created_at":"2026-07-05T08:00:30Z"},{"alias_kind":"pith_short_8","alias_value":"52FTCA4F","created_at":"2026-07-05T08:00:30Z"}],"graph_snapshots":[{"event_id":"sha256:97ba76148e593940c3e68cb6694176ebc694a12c40457b6bad8ebe1ae41deba9","target":"graph","created_at":"2026-07-05T08:00:30Z","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/2210.09811/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr$\\ddot{\\text{o}}$dinger operator $(\\mathcal{I}-\\Delta)^{\\log}$ corresponding to the logarithmic symbol $\\log(1 + |\\xi|^2)$, which is a singular integral operator given by $$(\\mathcal{I}-\\Delta)^{\\log}u(x) =c_{N}P.V.\\int_{\\mathbb{R}^{N}}\\frac{u(x)-u(y)}{|x-y|^{N}}\\kappa(|x-y|)dy,$$ where $c_{N}=\\pi^{-\\frac{N}{2}}$, $\\kappa(r)=2^{1-\\frac{N}{2}}r^{\\frac{N}{2}}\\mathcal{K}_{\\frac{N}{2}}(r)$ and $\\mathcal{K}_{\\nu}$ is the modified Bessel function of second kind w","authors_text":"Michael Ruzhansky, Rong Zhang, Vishvesh Kumar","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2022-10-18T12:43:18Z","title":"A Direct Method of Moving Planes for Logarithmic Schr\\\"odinger Operator"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.09811","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":"sha256:1c3607a64396beced037b26f68e3030f87d5850be4f1fbb2fc0731b6600a6d5d","target":"record","created_at":"2026-07-05T08:00:30Z","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":"d282819933e4ce7844822fcd746ea129a0722c896846ad08bb9d8218cd5333a0","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2022-10-18T12:43:18Z","title_canon_sha256":"4bd4b53a2f26f2220e71ca7969a57bfca48d4c08aa2db57cd8ed43df82aa3dab"},"schema_version":"1.0","source":{"id":"2210.09811","kind":"arxiv","version":2}},"canonical_sha256":"ee8b31038524dcc8dc570aac2812a608e537443b7ec628671ea67532f10cd628","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee8b31038524dcc8dc570aac2812a608e537443b7ec628671ea67532f10cd628","first_computed_at":"2026-07-05T08:00:30.989219Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:00:30.989219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5QOjLcdN3aHhwDEzjoxP2bsL8Tiem+oAUHy+X2VAUzvo3IdR5mzQukeymmGeufCUIUWkUE3/9snIXhaqWUt0Cw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:00:30.989744Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.09811","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1c3607a64396beced037b26f68e3030f87d5850be4f1fbb2fc0731b6600a6d5d","sha256:97ba76148e593940c3e68cb6694176ebc694a12c40457b6bad8ebe1ae41deba9"],"state_sha256":"0c7dd892113597967752c6795a45a0f7ae5178aca55d98678bfd1d2cd9ce38fe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0XZ7YbIpCsLhC76S0uVDehQoFHLmVR97dtaMGySy3h4SDFCd4aMlIzpor2KoIlB+gxeSip5kjDszwAqO0ve2CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T13:31:36.788726Z","bundle_sha256":"f637bea78d43dea3af0a87ee8bc1b5aee2a688bc124cc8c8ee71d7c126da5557"}}