{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:NI77O4MNGRYBVLV267OE5GH3K7","short_pith_number":"pith:NI77O4MN","canonical_record":{"source":{"id":"1211.5711","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-24T22:38:53Z","cross_cats_sorted":[],"title_canon_sha256":"a9e584fd229da4a5e67654da9ae201e079b7ca33fd79666a515c0efd3429462f","abstract_canon_sha256":"352c143ba23021f126f30aa93b4d9a0869d906fbfe5c6a9c941f3f4bd15fd783"},"schema_version":"1.0"},"canonical_sha256":"6a3ff7718d34701aaebaf7dc4e98fb57d86bcf97c154d9bbefe4d1919b7fa9c6","source":{"kind":"arxiv","id":"1211.5711","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5711","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5711v2","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5711","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"pith_short_12","alias_value":"NI77O4MNGRYB","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NI77O4MNGRYBVLV2","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NI77O4MN","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:NI77O4MNGRYBVLV267OE5GH3K7","target":"record","payload":{"canonical_record":{"source":{"id":"1211.5711","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-24T22:38:53Z","cross_cats_sorted":[],"title_canon_sha256":"a9e584fd229da4a5e67654da9ae201e079b7ca33fd79666a515c0efd3429462f","abstract_canon_sha256":"352c143ba23021f126f30aa93b4d9a0869d906fbfe5c6a9c941f3f4bd15fd783"},"schema_version":"1.0"},"canonical_sha256":"6a3ff7718d34701aaebaf7dc4e98fb57d86bcf97c154d9bbefe4d1919b7fa9c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:14.315774Z","signature_b64":"7C31+oF5N6jW8uRNR52CUH7/+TujhUSKsJIHM3wuhzeZhRzE9VhwBBkijUn/DwdxPh0mejK/tXbQVUqOvQ+lDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a3ff7718d34701aaebaf7dc4e98fb57d86bcf97c154d9bbefe4d1919b7fa9c6","last_reissued_at":"2026-05-18T03:39:14.315113Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:14.315113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.5711","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":"2026-05-18T03:39:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZF/deHi/zCeOMRCuSzgTMFP80EcLOamyZuCdXnqHl3tx/B5Rx+hW9x7OmZsU4lj7isE97eQA13O5VsZzMMcaCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T08:14:03.905051Z"},"content_sha256":"1f7970b05ac96255160d5ac6a0d1fcc15cf7b4712be4fa6a4713ae0915ae6cb5","schema_version":"1.0","event_id":"sha256:1f7970b05ac96255160d5ac6a0d1fcc15cf7b4712be4fa6a4713ae0915ae6cb5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:NI77O4MNGRYBVLV267OE5GH3K7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computer aided solution of the invariance equation for two-variable Gini means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Szabolcs Baj\\'ak, Zsolt P\\'ales","submitted_at":"2012-11-24T22:38:53Z","abstract_excerpt":"Our aim is to solve the so-called invariance equation in the class of two-variable Gini means ${G_{p,q}:p,q\\in\\R}$, i.e., to find necessary and sufficient conditions on the 6 parameters $a,b,c,d,p,q$ such that the identity [G_{p,q}\\big(G_{a,b}(x,y),G_{c,d}(x,y)\\big)=G_{p,q}(x,y) \\qquad (x,y \\in \\R_+)] be valid. We recall that, for $p\\neq q$, the Gini mean $G_{p,q}$ is defined by [G_{p,q}(x,y):=(\\dfrac{x^p+y^p}{x^q+y^q})^{\\frac1{p-q}}\\qquad (x,y \\in \\R_+).] The proof uses the computer algebra system Maple V Release 9 to compute a Taylor expansion up to 12th order, which enables us to describe a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5711","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:39:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"td1Nd8EF8/aGFhBuZRb8jj1gLhH88czvuBW93UL2RuXxhSgpjh+0cRJ9F9Nu64EpHYacxaZ5JS5qDEo6wXEEBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T08:14:03.905854Z"},"content_sha256":"18f7f302430762cfada36de78e5e9df1677b101658734ff9c01e38b15c484450","schema_version":"1.0","event_id":"sha256:18f7f302430762cfada36de78e5e9df1677b101658734ff9c01e38b15c484450"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NI77O4MNGRYBVLV267OE5GH3K7/bundle.json","state_url":"https://pith.science/pith/NI77O4MNGRYBVLV267OE5GH3K7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NI77O4MNGRYBVLV267OE5GH3K7/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":"2026-06-06T08:14:03Z","links":{"resolver":"https://pith.science/pith/NI77O4MNGRYBVLV267OE5GH3K7","bundle":"https://pith.science/pith/NI77O4MNGRYBVLV267OE5GH3K7/bundle.json","state":"https://pith.science/pith/NI77O4MNGRYBVLV267OE5GH3K7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NI77O4MNGRYBVLV267OE5GH3K7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NI77O4MNGRYBVLV267OE5GH3K7","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":"352c143ba23021f126f30aa93b4d9a0869d906fbfe5c6a9c941f3f4bd15fd783","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-24T22:38:53Z","title_canon_sha256":"a9e584fd229da4a5e67654da9ae201e079b7ca33fd79666a515c0efd3429462f"},"schema_version":"1.0","source":{"id":"1211.5711","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5711","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5711v2","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5711","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"pith_short_12","alias_value":"NI77O4MNGRYB","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NI77O4MNGRYBVLV2","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NI77O4MN","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:18f7f302430762cfada36de78e5e9df1677b101658734ff9c01e38b15c484450","target":"graph","created_at":"2026-05-18T03:39:14Z","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":"Our aim is to solve the so-called invariance equation in the class of two-variable Gini means ${G_{p,q}:p,q\\in\\R}$, i.e., to find necessary and sufficient conditions on the 6 parameters $a,b,c,d,p,q$ such that the identity [G_{p,q}\\big(G_{a,b}(x,y),G_{c,d}(x,y)\\big)=G_{p,q}(x,y) \\qquad (x,y \\in \\R_+)] be valid. We recall that, for $p\\neq q$, the Gini mean $G_{p,q}$ is defined by [G_{p,q}(x,y):=(\\dfrac{x^p+y^p}{x^q+y^q})^{\\frac1{p-q}}\\qquad (x,y \\in \\R_+).] The proof uses the computer algebra system Maple V Release 9 to compute a Taylor expansion up to 12th order, which enables us to describe a","authors_text":"Szabolcs Baj\\'ak, Zsolt P\\'ales","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-24T22:38:53Z","title":"Computer aided solution of the invariance equation for two-variable Gini means"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5711","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":"sha256:1f7970b05ac96255160d5ac6a0d1fcc15cf7b4712be4fa6a4713ae0915ae6cb5","target":"record","created_at":"2026-05-18T03:39:14Z","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":"352c143ba23021f126f30aa93b4d9a0869d906fbfe5c6a9c941f3f4bd15fd783","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-24T22:38:53Z","title_canon_sha256":"a9e584fd229da4a5e67654da9ae201e079b7ca33fd79666a515c0efd3429462f"},"schema_version":"1.0","source":{"id":"1211.5711","kind":"arxiv","version":2}},"canonical_sha256":"6a3ff7718d34701aaebaf7dc4e98fb57d86bcf97c154d9bbefe4d1919b7fa9c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a3ff7718d34701aaebaf7dc4e98fb57d86bcf97c154d9bbefe4d1919b7fa9c6","first_computed_at":"2026-05-18T03:39:14.315113Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:14.315113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7C31+oF5N6jW8uRNR52CUH7/+TujhUSKsJIHM3wuhzeZhRzE9VhwBBkijUn/DwdxPh0mejK/tXbQVUqOvQ+lDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:14.315774Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.5711","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1f7970b05ac96255160d5ac6a0d1fcc15cf7b4712be4fa6a4713ae0915ae6cb5","sha256:18f7f302430762cfada36de78e5e9df1677b101658734ff9c01e38b15c484450"],"state_sha256":"22bce48db6392b6a6e6362dbf4c07b64da428173aae870755133c086b9fe32a9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ybxL0Bn6EJ0mSgSIGeigWiXY2yDQvSOipSupdNE1W5NRre7eBi3C58/Eh1O3JxLMQUFmshFqQH8W843DXyAIBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T08:14:03.909916Z","bundle_sha256":"bd3824fe9ee3756e79c72991d31ad5486cb2c060a0a602c73387f2e53656cd28"}}