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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"},"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":"1102.4761","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-23T15:17:30Z","cross_cats_sorted":[],"title_canon_sha256":"fb91f6eb8f3a9647995b29fe9837ee04e8876b54f4357ed9760013ede418ffa1","abstract_canon_sha256":"3f304ba11422aa3875cdc6f9506db0654d6825f01df6a5d61a91d8eca549371d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:07.425582Z","signature_b64":"LD/Iqj5Ob+VVasW0BmzfJME9cFAXtFn0RrW/Ok2/RXLaWFUDPBugNlyakfG2Li8B59zxq/dSUrgK9Rwz26lCDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3d235749d2c2dead3d5287aedf14dd5ab121d54a3cd40d5c7432cc642c3e815","last_reissued_at":"2026-05-18T04:28:07.424531Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:07.424531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Minimum problem for finite sets of real numbers with non-negative sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Caterina Nardi, Giampiero Chiaselotti, Giuseppe Marino","submitted_at":"2011-02-23T15:17:30Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4761","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1102.4761","created_at":"2026-05-18T04:28:07.424903+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.4761v1","created_at":"2026-05-18T04:28:07.424903+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4761","created_at":"2026-05-18T04:28:07.424903+00:00"},{"alias_kind":"pith_short_12","alias_value":"UPJDK5E5FQW6","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_16","alias_value":"UPJDK5E5FQW6VU6V","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_8","alias_value":"UPJDK5E5","created_at":"2026-05-18T12:26:42.757692+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/UPJDK5E5FQW6VU6VFB5O34KN2W","json":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W.json","graph_json":"https://pith.science/api/pith-number/UPJDK5E5FQW6VU6VFB5O34KN2W/graph.json","events_json":"https://pith.science/api/pith-number/UPJDK5E5FQW6VU6VFB5O34KN2W/events.json","paper":"https://pith.science/paper/UPJDK5E5"},"agent_actions":{"view_html":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W","download_json":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W.json","view_paper":"https://pith.science/paper/UPJDK5E5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.4761&json=true","fetch_graph":"https://pith.science/api/pith-number/UPJDK5E5FQW6VU6VFB5O34KN2W/graph.json","fetch_events":"https://pith.science/api/pith-number/UPJDK5E5FQW6VU6VFB5O34KN2W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W/action/storage_attestation","attest_author":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W/action/author_attestation","sign_citation":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W/action/citation_signature","submit_replication":"https://pith.science/pith/UPJDK5E5FQW6VU6VFB5O34KN2W/action/replication_record"}},"created_at":"2026-05-18T04:28:07.424903+00:00","updated_at":"2026-05-18T04:28:07.424903+00:00"}