{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:G5ZDZRLTEO6RZ2DZALO2FP3L2N","short_pith_number":"pith:G5ZDZRLT","canonical_record":{"source":{"id":"1108.4652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-23T16:55:14Z","cross_cats_sorted":[],"title_canon_sha256":"0a4cdf49180a9bf414a1f69ac9fb05c4ee314eb24f2a76ae70e776374a8aeeb8","abstract_canon_sha256":"9dfe9a649b15a0b0b7431675087f05b0a37c9eece274762477c269c9573658d5"},"schema_version":"1.0"},"canonical_sha256":"37723cc57323bd1ce87902dda2bf6bd3465eb486649dcf9926b4f240fcffd3e4","source":{"kind":"arxiv","id":"1108.4652","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4652","created_at":"2026-05-18T03:46:45Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4652v1","created_at":"2026-05-18T03:46:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4652","created_at":"2026-05-18T03:46:45Z"},{"alias_kind":"pith_short_12","alias_value":"G5ZDZRLTEO6R","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"G5ZDZRLTEO6RZ2DZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"G5ZDZRLT","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:G5ZDZRLTEO6RZ2DZALO2FP3L2N","target":"record","payload":{"canonical_record":{"source":{"id":"1108.4652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-23T16:55:14Z","cross_cats_sorted":[],"title_canon_sha256":"0a4cdf49180a9bf414a1f69ac9fb05c4ee314eb24f2a76ae70e776374a8aeeb8","abstract_canon_sha256":"9dfe9a649b15a0b0b7431675087f05b0a37c9eece274762477c269c9573658d5"},"schema_version":"1.0"},"canonical_sha256":"37723cc57323bd1ce87902dda2bf6bd3465eb486649dcf9926b4f240fcffd3e4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:45.514474Z","signature_b64":"TatxVVNWA4oC5OQ8CdO/Tpzkf4DpHQozEZMLuqMSSO7DuwIZrfqawkOr6OJwGH0E9rBOiXW8hFVKUqq5d3f5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37723cc57323bd1ce87902dda2bf6bd3465eb486649dcf9926b4f240fcffd3e4","last_reissued_at":"2026-05-18T03:46:45.513403Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:45.513403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.4652","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":"2026-05-18T03:46:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wG5qSLs5Dm8XJWpY6X/Xt9iTphjWNxppi+z9F2A1Nvx4UuqN7bHNOXLH+uTKAi658BTN21QBxhk3on6KQkucCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:55:35.968300Z"},"content_sha256":"cb354adb6fb45ebe759cf05ee20a2524544470ad6df47309290886b648a5014f","schema_version":"1.0","event_id":"sha256:cb354adb6fb45ebe759cf05ee20a2524544470ad6df47309290886b648a5014f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:G5ZDZRLTEO6RZ2DZALO2FP3L2N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the sum of powered distances to certain sets of points on the circle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Nikolai Nikolov, Rafael Rafailov","submitted_at":"2011-08-23T16:55:14Z","abstract_excerpt":"In this paper we consider an extremal problem in geometry. Let $\\lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $\\Gamma$. We give full characterization of the extremal behavior of the function $f(M,\\lambda)=MA^\\lambda+MB^\\lambda+MC^\\lambda$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $\\sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $\\Gamma$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric method"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4652","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":"2026-05-18T03:46:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cXzi/Et9QjI6mlh0p0wGeTkDM06w8WUfU3lgXv+XK97CU4kHReXXZyReA/TgHbvuBYqhvFNJro/Xc/HIAezcCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:55:35.968664Z"},"content_sha256":"676ec5d7b4eca1d4da5a6af01a399d93f3a26ac41ac824f4fed0d80399e3f3f6","schema_version":"1.0","event_id":"sha256:676ec5d7b4eca1d4da5a6af01a399d93f3a26ac41ac824f4fed0d80399e3f3f6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G5ZDZRLTEO6RZ2DZALO2FP3L2N/bundle.json","state_url":"https://pith.science/pith/G5ZDZRLTEO6RZ2DZALO2FP3L2N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G5ZDZRLTEO6RZ2DZALO2FP3L2N/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":"2026-06-24T20:55:35Z","links":{"resolver":"https://pith.science/pith/G5ZDZRLTEO6RZ2DZALO2FP3L2N","bundle":"https://pith.science/pith/G5ZDZRLTEO6RZ2DZALO2FP3L2N/bundle.json","state":"https://pith.science/pith/G5ZDZRLTEO6RZ2DZALO2FP3L2N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G5ZDZRLTEO6RZ2DZALO2FP3L2N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:G5ZDZRLTEO6RZ2DZALO2FP3L2N","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":"9dfe9a649b15a0b0b7431675087f05b0a37c9eece274762477c269c9573658d5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-23T16:55:14Z","title_canon_sha256":"0a4cdf49180a9bf414a1f69ac9fb05c4ee314eb24f2a76ae70e776374a8aeeb8"},"schema_version":"1.0","source":{"id":"1108.4652","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4652","created_at":"2026-05-18T03:46:45Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4652v1","created_at":"2026-05-18T03:46:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4652","created_at":"2026-05-18T03:46:45Z"},{"alias_kind":"pith_short_12","alias_value":"G5ZDZRLTEO6R","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"G5ZDZRLTEO6RZ2DZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"G5ZDZRLT","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:676ec5d7b4eca1d4da5a6af01a399d93f3a26ac41ac824f4fed0d80399e3f3f6","target":"graph","created_at":"2026-05-18T03:46:45Z","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":"In this paper we consider an extremal problem in geometry. Let $\\lambda$ be a real number and $A$, $B$ and $C$ be arbitrary points on the unit circle $\\Gamma$. We give full characterization of the extremal behavior of the function $f(M,\\lambda)=MA^\\lambda+MB^\\lambda+MC^\\lambda$, where $M$ is a point on the unit circle as well. We also investigate the extremal behavior of $\\sum_{i=1}^nXP_i$, where $P_i, i=1,...,n$ are the vertices of a regular $n$-gon and $X$ is a point on $\\Gamma$, concentric to the circle circumscribed around $P_1...P_n$. We use elementary analytic and purely geometric method","authors_text":"Nikolai Nikolov, Rafael Rafailov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-23T16:55:14Z","title":"On the sum of powered distances to certain sets of points on the circle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4652","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:cb354adb6fb45ebe759cf05ee20a2524544470ad6df47309290886b648a5014f","target":"record","created_at":"2026-05-18T03:46:45Z","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":"9dfe9a649b15a0b0b7431675087f05b0a37c9eece274762477c269c9573658d5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-08-23T16:55:14Z","title_canon_sha256":"0a4cdf49180a9bf414a1f69ac9fb05c4ee314eb24f2a76ae70e776374a8aeeb8"},"schema_version":"1.0","source":{"id":"1108.4652","kind":"arxiv","version":1}},"canonical_sha256":"37723cc57323bd1ce87902dda2bf6bd3465eb486649dcf9926b4f240fcffd3e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37723cc57323bd1ce87902dda2bf6bd3465eb486649dcf9926b4f240fcffd3e4","first_computed_at":"2026-05-18T03:46:45.513403Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:45.513403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TatxVVNWA4oC5OQ8CdO/Tpzkf4DpHQozEZMLuqMSSO7DuwIZrfqawkOr6OJwGH0E9rBOiXW8hFVKUqq5d3f5BA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:45.514474Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.4652","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb354adb6fb45ebe759cf05ee20a2524544470ad6df47309290886b648a5014f","sha256:676ec5d7b4eca1d4da5a6af01a399d93f3a26ac41ac824f4fed0d80399e3f3f6"],"state_sha256":"2aeb523bdabcf5c1d0eea3ad394452e82fbf4bc6c3a8dc86e73bd47c790aacbc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+UOw7gu0ldBwkpzAlKPquhjML6PqKUHYdcMDoebwGIvUCJmzlZX/2ksI8GL+IsgF8WlCW75AeNTaevb0SzHXCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T20:55:35.970802Z","bundle_sha256":"91f972d1d09e431b44863de0a6173baf6977f8e4ff16a5937e3e54692d8861bc"}}