{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:UPJDK5E5FQW6VU6VFB5O34KN2W","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":"3f304ba11422aa3875cdc6f9506db0654d6825f01df6a5d61a91d8eca549371d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-23T15:17:30Z","title_canon_sha256":"fb91f6eb8f3a9647995b29fe9837ee04e8876b54f4357ed9760013ede418ffa1"},"schema_version":"1.0","source":{"id":"1102.4761","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.4761","created_at":"2026-05-18T04:28:07Z"},{"alias_kind":"arxiv_version","alias_value":"1102.4761v1","created_at":"2026-05-18T04:28:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4761","created_at":"2026-05-18T04:28:07Z"},{"alias_kind":"pith_short_12","alias_value":"UPJDK5E5FQW6","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"UPJDK5E5FQW6VU6V","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"UPJDK5E5","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:2799d745af8da6c32028f30764da7c927dddae2bd88582ec02e34e3ada068850","target":"graph","created_at":"2026-05-18T04:28:07Z","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":"Let $n$ and $r$ be two integers such that $0 < r \\le n$; we denote by $\\gamma(n,r)$ [$\\eta(n,r)$] the minimum [maximum] number of the non-negative partial sums of a sum $\\sum_{1=1}^n a_i \\ge 0$, where $a_1, \\cdots, a_n$ are $n$ real numbers arbitrarily chosen in such a way that $r$ of them are non-negative and the remaining $n-r$ are negative. Inspired by some interesting extremal combinatorial sum problems raised by Manickam, Mikl\\\"os and Singhi in 1987 \\cite{ManMik87} and 1988 \\cite{ManSin88} we study the following two problems:\n  \\noindent$(P1)$ {\\it which are the values of $\\gamma(n,r)$ an","authors_text":"Caterina Nardi, Giampiero Chiaselotti, Giuseppe Marino","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-23T15:17:30Z","title":"A Minimum problem for finite sets of real numbers with non-negative sum"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4761","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":"sha256:41485c10620a6884ba0bc4febb3ad0ce0b3437d1b1385686f05ff4599215328c","target":"record","created_at":"2026-05-18T04:28:07Z","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":"3f304ba11422aa3875cdc6f9506db0654d6825f01df6a5d61a91d8eca549371d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-23T15:17:30Z","title_canon_sha256":"fb91f6eb8f3a9647995b29fe9837ee04e8876b54f4357ed9760013ede418ffa1"},"schema_version":"1.0","source":{"id":"1102.4761","kind":"arxiv","version":1}},"canonical_sha256":"a3d235749d2c2dead3d5287aedf14dd5ab121d54a3cd40d5c7432cc642c3e815","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a3d235749d2c2dead3d5287aedf14dd5ab121d54a3cd40d5c7432cc642c3e815","first_computed_at":"2026-05-18T04:28:07.424531Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:07.424531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LD/Iqj5Ob+VVasW0BmzfJME9cFAXtFn0RrW/Ok2/RXLaWFUDPBugNlyakfG2Li8B59zxq/dSUrgK9Rwz26lCDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:07.425582Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.4761","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41485c10620a6884ba0bc4febb3ad0ce0b3437d1b1385686f05ff4599215328c","sha256:2799d745af8da6c32028f30764da7c927dddae2bd88582ec02e34e3ada068850"],"state_sha256":"9c0c704cd08a08be18fef0619870dab4c5b042a67cb9d7364c56e74cf084f53c"}