{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:VYD4YEHZCJK5XB4FKMQAVCUWGG","short_pith_number":"pith:VYD4YEHZ","schema_version":"1.0","canonical_sha256":"ae07cc10f91255db878553200a8a9631ac0883e408b8d247b7f7b5836044f3f5","source":{"kind":"arxiv","id":"1102.2136","version":1},"attestation_state":"computed","paper":{"title":"Some critical point theorems and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Guangcun Lu","submitted_at":"2011-02-10T14:56:58Z","abstract_excerpt":"This paper is a continuation of \\cite{Lu1}. In Part I, applying the new splitting theorems developed therein we generalize previous some results on computations of critical groups and some critical point theorems to weaker versions. In Part II (in progress), they are used to study multiple solutions for nonlinear higher order elliptic equations described in the introduction of \\cite{Lu1}."},"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":"1102.2136","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-02-10T14:56:58Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"28c5ca88a868fb157f7c9fc85bed1f1577d6bac25eddfbace2fdee220eaa4047","abstract_canon_sha256":"5be688c73125dc38c8816cd1afd101b79951098ed6ccf0637136f1b79c38dc9a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:29:34.174519Z","signature_b64":"dogp1hm/ACE3B0E5L7BD6Swnb1r8//wsUUIHT2nINENxxULsdKZ49BPxFuVkrKjhmJVkUjzWzkcwDRZAuG3dAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ae07cc10f91255db878553200a8a9631ac0883e408b8d247b7f7b5836044f3f5","last_reissued_at":"2026-05-18T04:29:34.174029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:29:34.174029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some critical point theorems and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Guangcun Lu","submitted_at":"2011-02-10T14:56:58Z","abstract_excerpt":"This paper is a continuation of \\cite{Lu1}. In Part I, applying the new splitting theorems developed therein we generalize previous some results on computations of critical groups and some critical point theorems to weaker versions. In Part II (in progress), they are used to study multiple solutions for nonlinear higher order elliptic equations described in the introduction of \\cite{Lu1}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.2136","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":"1102.2136","created_at":"2026-05-18T04:29:34.174107+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.2136v1","created_at":"2026-05-18T04:29:34.174107+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.2136","created_at":"2026-05-18T04:29:34.174107+00:00"},{"alias_kind":"pith_short_12","alias_value":"VYD4YEHZCJK5","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"VYD4YEHZCJK5XB4F","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"VYD4YEHZ","created_at":"2026-05-18T12:26:44.992195+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/VYD4YEHZCJK5XB4FKMQAVCUWGG","json":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG.json","graph_json":"https://pith.science/api/pith-number/VYD4YEHZCJK5XB4FKMQAVCUWGG/graph.json","events_json":"https://pith.science/api/pith-number/VYD4YEHZCJK5XB4FKMQAVCUWGG/events.json","paper":"https://pith.science/paper/VYD4YEHZ"},"agent_actions":{"view_html":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG","download_json":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG.json","view_paper":"https://pith.science/paper/VYD4YEHZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.2136&json=true","fetch_graph":"https://pith.science/api/pith-number/VYD4YEHZCJK5XB4FKMQAVCUWGG/graph.json","fetch_events":"https://pith.science/api/pith-number/VYD4YEHZCJK5XB4FKMQAVCUWGG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG/action/storage_attestation","attest_author":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG/action/author_attestation","sign_citation":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG/action/citation_signature","submit_replication":"https://pith.science/pith/VYD4YEHZCJK5XB4FKMQAVCUWGG/action/replication_record"}},"created_at":"2026-05-18T04:29:34.174107+00:00","updated_at":"2026-05-18T04:29:34.174107+00:00"}