{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:MH53N2FUJJNXCDXRVYSLHGX425","short_pith_number":"pith:MH53N2FU","schema_version":"1.0","canonical_sha256":"61fbb6e8b44a5b710ef1ae24b39afcd76c241cfe071f11b76d000f49be001c66","source":{"kind":"arxiv","id":"1606.05976","version":2},"attestation_state":"computed","paper":{"title":"Solution to the Pompeiu problem and the related symmetry problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A.G.Ramm","submitted_at":"2016-06-20T05:31:47Z","abstract_excerpt":"Assume that $D\\subset \\mathbb{R}^3$ is a bounded domain with $C^1-$smooth boundary. Our result is:\n  {\\bf Theorem 1.} {\\em If $D$ has $P-$property, then $D$ is a ball.}\n  Four equivalent formulations of the Pompeiu problem are discussed. A domain $D$ has $P-$property if there exists an $f\\neq 0$, $f\\in L^1_{loc}(\\mathbb{R}^3)$ such that $\\int_{D}f(gx+y)dx=0$ for all $y\\in \\mathbb{R}^3$ and all $g\\in SO(2)$, where $ SO(2)$ is the rotation group. The result obtained concerning the related symmetry problem is:\n  {\\bf Theorem 2.} {\\em If $(\\nabla^2 +k^2)u=0$ in $D$, $u|_S=1$, $u_N|_S=0$, and $k>0$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1606.05976","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-20T05:31:47Z","cross_cats_sorted":[],"title_canon_sha256":"72f709f851694182fff9796018d1e9a5e9fe75c71b6d1233b40a041889daff02","abstract_canon_sha256":"f8b3af5035aceed0eb028128c9dae69e30b36c73788fbfe03bc9025469afea4d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:06.307088Z","signature_b64":"6IB3haJ5384JSicMWWEGUIAItA1nD5Nl69lbkSUfSUUT+yBwe+0UylEKrmbhEin+qL8f6/M3hPO2DL5YlJ2WAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61fbb6e8b44a5b710ef1ae24b39afcd76c241cfe071f11b76d000f49be001c66","last_reissued_at":"2026-05-18T01:09:06.306620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:06.306620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Solution to the Pompeiu problem and the related symmetry problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A.G.Ramm","submitted_at":"2016-06-20T05:31:47Z","abstract_excerpt":"Assume that $D\\subset \\mathbb{R}^3$ is a bounded domain with $C^1-$smooth boundary. Our result is:\n  {\\bf Theorem 1.} {\\em If $D$ has $P-$property, then $D$ is a ball.}\n  Four equivalent formulations of the Pompeiu problem are discussed. A domain $D$ has $P-$property if there exists an $f\\neq 0$, $f\\in L^1_{loc}(\\mathbb{R}^3)$ such that $\\int_{D}f(gx+y)dx=0$ for all $y\\in \\mathbb{R}^3$ and all $g\\in SO(2)$, where $ SO(2)$ is the rotation group. The result obtained concerning the related symmetry problem is:\n  {\\bf Theorem 2.} {\\em If $(\\nabla^2 +k^2)u=0$ in $D$, $u|_S=1$, $u_N|_S=0$, and $k>0$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05976","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.05976","created_at":"2026-05-18T01:09:06.306691+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.05976v2","created_at":"2026-05-18T01:09:06.306691+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05976","created_at":"2026-05-18T01:09:06.306691+00:00"},{"alias_kind":"pith_short_12","alias_value":"MH53N2FUJJNX","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MH53N2FUJJNXCDXR","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MH53N2FU","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425","json":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425.json","graph_json":"https://pith.science/api/pith-number/MH53N2FUJJNXCDXRVYSLHGX425/graph.json","events_json":"https://pith.science/api/pith-number/MH53N2FUJJNXCDXRVYSLHGX425/events.json","paper":"https://pith.science/paper/MH53N2FU"},"agent_actions":{"view_html":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425","download_json":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425.json","view_paper":"https://pith.science/paper/MH53N2FU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.05976&json=true","fetch_graph":"https://pith.science/api/pith-number/MH53N2FUJJNXCDXRVYSLHGX425/graph.json","fetch_events":"https://pith.science/api/pith-number/MH53N2FUJJNXCDXRVYSLHGX425/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425/action/storage_attestation","attest_author":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425/action/author_attestation","sign_citation":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425/action/citation_signature","submit_replication":"https://pith.science/pith/MH53N2FUJJNXCDXRVYSLHGX425/action/replication_record"}},"created_at":"2026-05-18T01:09:06.306691+00:00","updated_at":"2026-05-18T01:09:06.306691+00:00"}