{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:O4QV7RPFJFA6J3LSTFLKXXBQYJ","short_pith_number":"pith:O4QV7RPF","canonical_record":{"source":{"id":"1410.4851","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-17T20:06:32Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"b0cda023fbace7ee69bdb1453b82d6ac65f6ec3432ab3b4d99d293bdcde7272e","abstract_canon_sha256":"5b05b3321748e5de1e998a14e804660e161616a4b2da6bcdb628387a3c6b2d8d"},"schema_version":"1.0"},"canonical_sha256":"77215fc5e54941e4ed729956abdc30c2487d88e52d123b819988b88f0d5774bf","source":{"kind":"arxiv","id":"1410.4851","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.4851","created_at":"2026-05-18T01:30:15Z"},{"alias_kind":"arxiv_version","alias_value":"1410.4851v1","created_at":"2026-05-18T01:30:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.4851","created_at":"2026-05-18T01:30:15Z"},{"alias_kind":"pith_short_12","alias_value":"O4QV7RPFJFA6","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"O4QV7RPFJFA6J3LS","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"O4QV7RPF","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:O4QV7RPFJFA6J3LSTFLKXXBQYJ","target":"record","payload":{"canonical_record":{"source":{"id":"1410.4851","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-17T20:06:32Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"b0cda023fbace7ee69bdb1453b82d6ac65f6ec3432ab3b4d99d293bdcde7272e","abstract_canon_sha256":"5b05b3321748e5de1e998a14e804660e161616a4b2da6bcdb628387a3c6b2d8d"},"schema_version":"1.0"},"canonical_sha256":"77215fc5e54941e4ed729956abdc30c2487d88e52d123b819988b88f0d5774bf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:15.472257Z","signature_b64":"TTqTyDd3VmM7NbRd5qkQkVS/nBIPQOv11tIzmgLXlOztYh5ixpEMW1yuXgvL16i3gTlN0+fa6CONZI0Uqj5iAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77215fc5e54941e4ed729956abdc30c2487d88e52d123b819988b88f0d5774bf","last_reissued_at":"2026-05-18T01:30:15.471753Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:15.471753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.4851","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":"2026-05-18T01:30:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3TuIZaXtp8hH28zHdG+ezcqbBSwB7Jve7b5rXO9q+PkHJQ7Wova2SOwIOGfU5i0rhrfutRmOZDh1ZybTyuA4Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T15:21:27.050960Z"},"content_sha256":"fda25e5a7a3038e9d09669328a3d044c0eee2853a4b0b788790ac1788acb91c4","schema_version":"1.0","event_id":"sha256:fda25e5a7a3038e9d09669328a3d044c0eee2853a4b0b788790ac1788acb91c4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:O4QV7RPFJFA6J3LSTFLKXXBQYJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some abstract results on the existence of bounded Palais-Smale sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Michela Guida, Sergio Rolando","submitted_at":"2014-10-17T20:06:32Z","abstract_excerpt":"Without compactness assumptions, we prove some abstract results which show that a $C^{1}$ functional $I:X\\rightarrow \\mathbb{R}$ on a Banach space $X$ admits bounded Palais-Smale sequences provided that it exhibits some geometric structure of minimax type and a suitable behaviour with respect to some sequence of continuous mappings $\\psi _{n}:X\\rightarrow X$. This work is a preliminary version of a forthcoming paper, where applications to nonlinear equations without Ambrosetti-Rabinowitz type assumptions will also be given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4851","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":"2026-05-18T01:30:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nNPf8UIzHMJKIY2EWVDbeyzXvn67wKrFiLjCKK4PWpugy6NyyyFsXbvh+GfG3YnlbqaPd4IVBUS/CNNMUjAYAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T15:21:27.051308Z"},"content_sha256":"99096bd6d6fb7c3041e26f46ce4849944c69936766e7ef10aac37d0f40ab8541","schema_version":"1.0","event_id":"sha256:99096bd6d6fb7c3041e26f46ce4849944c69936766e7ef10aac37d0f40ab8541"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O4QV7RPFJFA6J3LSTFLKXXBQYJ/bundle.json","state_url":"https://pith.science/pith/O4QV7RPFJFA6J3LSTFLKXXBQYJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O4QV7RPFJFA6J3LSTFLKXXBQYJ/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":"2026-06-26T15:21:27Z","links":{"resolver":"https://pith.science/pith/O4QV7RPFJFA6J3LSTFLKXXBQYJ","bundle":"https://pith.science/pith/O4QV7RPFJFA6J3LSTFLKXXBQYJ/bundle.json","state":"https://pith.science/pith/O4QV7RPFJFA6J3LSTFLKXXBQYJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O4QV7RPFJFA6J3LSTFLKXXBQYJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:O4QV7RPFJFA6J3LSTFLKXXBQYJ","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":"5b05b3321748e5de1e998a14e804660e161616a4b2da6bcdb628387a3c6b2d8d","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-17T20:06:32Z","title_canon_sha256":"b0cda023fbace7ee69bdb1453b82d6ac65f6ec3432ab3b4d99d293bdcde7272e"},"schema_version":"1.0","source":{"id":"1410.4851","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.4851","created_at":"2026-05-18T01:30:15Z"},{"alias_kind":"arxiv_version","alias_value":"1410.4851v1","created_at":"2026-05-18T01:30:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.4851","created_at":"2026-05-18T01:30:15Z"},{"alias_kind":"pith_short_12","alias_value":"O4QV7RPFJFA6","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"O4QV7RPFJFA6J3LS","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"O4QV7RPF","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:99096bd6d6fb7c3041e26f46ce4849944c69936766e7ef10aac37d0f40ab8541","target":"graph","created_at":"2026-05-18T01:30:15Z","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":"Without compactness assumptions, we prove some abstract results which show that a $C^{1}$ functional $I:X\\rightarrow \\mathbb{R}$ on a Banach space $X$ admits bounded Palais-Smale sequences provided that it exhibits some geometric structure of minimax type and a suitable behaviour with respect to some sequence of continuous mappings $\\psi _{n}:X\\rightarrow X$. This work is a preliminary version of a forthcoming paper, where applications to nonlinear equations without Ambrosetti-Rabinowitz type assumptions will also be given.","authors_text":"Michela Guida, Sergio Rolando","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-17T20:06:32Z","title":"Some abstract results on the existence of bounded Palais-Smale sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4851","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:fda25e5a7a3038e9d09669328a3d044c0eee2853a4b0b788790ac1788acb91c4","target":"record","created_at":"2026-05-18T01:30:15Z","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":"5b05b3321748e5de1e998a14e804660e161616a4b2da6bcdb628387a3c6b2d8d","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-17T20:06:32Z","title_canon_sha256":"b0cda023fbace7ee69bdb1453b82d6ac65f6ec3432ab3b4d99d293bdcde7272e"},"schema_version":"1.0","source":{"id":"1410.4851","kind":"arxiv","version":1}},"canonical_sha256":"77215fc5e54941e4ed729956abdc30c2487d88e52d123b819988b88f0d5774bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"77215fc5e54941e4ed729956abdc30c2487d88e52d123b819988b88f0d5774bf","first_computed_at":"2026-05-18T01:30:15.471753Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:15.471753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TTqTyDd3VmM7NbRd5qkQkVS/nBIPQOv11tIzmgLXlOztYh5ixpEMW1yuXgvL16i3gTlN0+fa6CONZI0Uqj5iAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:15.472257Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.4851","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fda25e5a7a3038e9d09669328a3d044c0eee2853a4b0b788790ac1788acb91c4","sha256:99096bd6d6fb7c3041e26f46ce4849944c69936766e7ef10aac37d0f40ab8541"],"state_sha256":"558d5ad9757ee788a7a6b7870da062781c0568f3d95c1b60aedfef63dee17483"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"528jAAO3Bc8HcWZePQkmctryYOmcpOaEcftaEcfPHFiRo9joywdePKWP794/J9ogXFckckvdnjza4k3JKaLlAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T15:21:27.053252Z","bundle_sha256":"463b1d67d5967354b510ea9b84c9dfd63fd130375e89611b034358c86a91acb9"}}