{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BJU4DAGRWBZMKN5SRBXAAI23GZ","short_pith_number":"pith:BJU4DAGR","canonical_record":{"source":{"id":"1105.0212","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-01T21:52:14Z","cross_cats_sorted":[],"title_canon_sha256":"77b509f91b7d8d65698f6cefa5dd1bf0e7dd60ff3242918ceea7e3d2aa51f2cf","abstract_canon_sha256":"800afa50a4e13b10e278029997faa5317779a76a2ad3ab7b8f266fb3ea550693"},"schema_version":"1.0"},"canonical_sha256":"0a69c180d1b072c537b2886e00235b364a3440deda4603fe107cbadb92ded74a","source":{"kind":"arxiv","id":"1105.0212","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0212","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0212v1","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0212","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"pith_short_12","alias_value":"BJU4DAGRWBZM","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BJU4DAGRWBZMKN5S","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BJU4DAGR","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BJU4DAGRWBZMKN5SRBXAAI23GZ","target":"record","payload":{"canonical_record":{"source":{"id":"1105.0212","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-01T21:52:14Z","cross_cats_sorted":[],"title_canon_sha256":"77b509f91b7d8d65698f6cefa5dd1bf0e7dd60ff3242918ceea7e3d2aa51f2cf","abstract_canon_sha256":"800afa50a4e13b10e278029997faa5317779a76a2ad3ab7b8f266fb3ea550693"},"schema_version":"1.0"},"canonical_sha256":"0a69c180d1b072c537b2886e00235b364a3440deda4603fe107cbadb92ded74a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:12.294133Z","signature_b64":"QEXKP9iZ67SGio0hS/mhBDMkyhobi2h98jiBzvSJDHPpe2/IZFFx5oLsaKQd8KIKrutSMkk6rtd5s6AFljLJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a69c180d1b072c537b2886e00235b364a3440deda4603fe107cbadb92ded74a","last_reissued_at":"2026-05-18T04:23:12.292526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:12.292526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.0212","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":"2026-05-18T04:23:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dMHwK7OazSxd2pBqG5eD1/HBBYJQN5PPpaLVmpQnqtocJlFG6zH7+L76A3KlpNV19u5qMPCyuybCes6R1orIDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:00:37.038800Z"},"content_sha256":"dd7c5ea6a22a61829186fd5e279000ad3f654a4f7653889baf97d532875ee55c","schema_version":"1.0","event_id":"sha256:dd7c5ea6a22a61829186fd5e279000ad3f654a4f7653889baf97d532875ee55c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BJU4DAGRWBZMKN5SRBXAAI23GZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harmonic balls and two-phase Schwarz function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Henrik Shahgholian, Tomas Sj\\\"odin","submitted_at":"2011-05-01T21:52:14Z","abstract_excerpt":"Here we shall introduce the concept of harmonic balls/spheres in sub-domains of $\\R^n$, through a mean value property for a sub-class of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions can not have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exists, and gives rise to the generalization of the well-known Schwarz function to the case of two-phase Schwarz function.\n  Our primary goal is to derive simple properties for these problems, and tease t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0212","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":"2026-05-18T04:23:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4Nnfl6EilvHPLntQecN5J36CGIdWn5swv2U2wlR+8oZqPTWUfLXaqVX1GFb5glIgZqEb3oYbdYblOjmOV4oyDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:00:37.039148Z"},"content_sha256":"9b857a8821b3eaf34695e41940e7395c76aae95182b91289bb993675c7e4ed53","schema_version":"1.0","event_id":"sha256:9b857a8821b3eaf34695e41940e7395c76aae95182b91289bb993675c7e4ed53"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BJU4DAGRWBZMKN5SRBXAAI23GZ/bundle.json","state_url":"https://pith.science/pith/BJU4DAGRWBZMKN5SRBXAAI23GZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BJU4DAGRWBZMKN5SRBXAAI23GZ/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":"2026-05-31T08:00:37Z","links":{"resolver":"https://pith.science/pith/BJU4DAGRWBZMKN5SRBXAAI23GZ","bundle":"https://pith.science/pith/BJU4DAGRWBZMKN5SRBXAAI23GZ/bundle.json","state":"https://pith.science/pith/BJU4DAGRWBZMKN5SRBXAAI23GZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BJU4DAGRWBZMKN5SRBXAAI23GZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BJU4DAGRWBZMKN5SRBXAAI23GZ","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":"800afa50a4e13b10e278029997faa5317779a76a2ad3ab7b8f266fb3ea550693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-01T21:52:14Z","title_canon_sha256":"77b509f91b7d8d65698f6cefa5dd1bf0e7dd60ff3242918ceea7e3d2aa51f2cf"},"schema_version":"1.0","source":{"id":"1105.0212","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0212","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0212v1","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0212","created_at":"2026-05-18T04:23:12Z"},{"alias_kind":"pith_short_12","alias_value":"BJU4DAGRWBZM","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BJU4DAGRWBZMKN5S","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BJU4DAGR","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:9b857a8821b3eaf34695e41940e7395c76aae95182b91289bb993675c7e4ed53","target":"graph","created_at":"2026-05-18T04:23:12Z","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":"Here we shall introduce the concept of harmonic balls/spheres in sub-domains of $\\R^n$, through a mean value property for a sub-class of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions can not have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exists, and gives rise to the generalization of the well-known Schwarz function to the case of two-phase Schwarz function.\n  Our primary goal is to derive simple properties for these problems, and tease t","authors_text":"Henrik Shahgholian, Tomas Sj\\\"odin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-01T21:52:14Z","title":"Harmonic balls and two-phase Schwarz function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0212","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:dd7c5ea6a22a61829186fd5e279000ad3f654a4f7653889baf97d532875ee55c","target":"record","created_at":"2026-05-18T04:23:12Z","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":"800afa50a4e13b10e278029997faa5317779a76a2ad3ab7b8f266fb3ea550693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-05-01T21:52:14Z","title_canon_sha256":"77b509f91b7d8d65698f6cefa5dd1bf0e7dd60ff3242918ceea7e3d2aa51f2cf"},"schema_version":"1.0","source":{"id":"1105.0212","kind":"arxiv","version":1}},"canonical_sha256":"0a69c180d1b072c537b2886e00235b364a3440deda4603fe107cbadb92ded74a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0a69c180d1b072c537b2886e00235b364a3440deda4603fe107cbadb92ded74a","first_computed_at":"2026-05-18T04:23:12.292526Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:12.292526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QEXKP9iZ67SGio0hS/mhBDMkyhobi2h98jiBzvSJDHPpe2/IZFFx5oLsaKQd8KIKrutSMkk6rtd5s6AFljLJAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:12.294133Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.0212","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dd7c5ea6a22a61829186fd5e279000ad3f654a4f7653889baf97d532875ee55c","sha256:9b857a8821b3eaf34695e41940e7395c76aae95182b91289bb993675c7e4ed53"],"state_sha256":"99ce5908fa72bdddbe7ab03bfee5ad22bf3964b5c4a7206d4fd4279c6183d1c5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gO7aS/Y5elf2I9HPR0GlovfPns3B3YLIOuJLjHWXZxbSJpA2k4Vzp29nkzCHa7RvLLKLkLUDI76eJxPgoEFNCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T08:00:37.041290Z","bundle_sha256":"643e1f7c57193d414185a2cc7f0e20cd12a0d60ef85f8ad53240bbb55bed3335"}}