{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:BZDJJ63DAKUNP6GAJXEN7UHEGD","short_pith_number":"pith:BZDJJ63D","schema_version":"1.0","canonical_sha256":"0e4694fb6302a8d7f8c04dc8dfd0e430ccbc448c5a6c8b681db18d11e0573443","source":{"kind":"arxiv","id":"1602.00232","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Hedy Attouch, Marc-Olivier Czarnecki","submitted_at":"2016-01-31T11:47:01Z","abstract_excerpt":"In a Hilbert space $\\mathcal H$, we study the asymptotic behaviour, as time variable $t$ goes to $+\\infty$, of nonautonomous gradient-like dynamical systems involving inertia and multiscale features.\n  Given $\\mathcal H$ a general Hilbert space, $\\Phi: \\mathcal H \\rightarrow \\mathbb R$ and $\\Psi: \\mathcal H \\rightarrow \\mathbb R$ two convex differentiable functions, $\\gamma$ a positive damping parameter, and $\\epsilon (t)$ a function of $t$ which tends to zero as $t$ goes to $+\\infty$, we consider the second-order differential equation $$\\ddot{x}(t) + \\gamma \\dot{x}(t) + \\nabla \\Phi (x(t)) + \\"},"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":"1602.00232","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-01-31T11:47:01Z","cross_cats_sorted":[],"title_canon_sha256":"fb03d51505b11a46fbbd800bf54f7923a40ba8738037fc6b56cec82ca1408ebd","abstract_canon_sha256":"a62d3cce514773ccd0bda01c827ac3d2d8cd25d3959d4989a7a763f9bd7e625a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:35.960025Z","signature_b64":"MA0SzNvKEQUqpmlrjuJqdNJg/Y2yBLyJYiN0IEltf9RmTP8WPCUeIpGBo6JVdUzCYlV/KWaxUved6ZszQ5dRDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e4694fb6302a8d7f8c04dc8dfd0e430ccbc448c5a6c8b681db18d11e0573443","last_reissued_at":"2026-05-18T01:21:35.959476Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:35.959476Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Hedy Attouch, Marc-Olivier Czarnecki","submitted_at":"2016-01-31T11:47:01Z","abstract_excerpt":"In a Hilbert space $\\mathcal H$, we study the asymptotic behaviour, as time variable $t$ goes to $+\\infty$, of nonautonomous gradient-like dynamical systems involving inertia and multiscale features.\n  Given $\\mathcal H$ a general Hilbert space, $\\Phi: \\mathcal H \\rightarrow \\mathbb R$ and $\\Psi: \\mathcal H \\rightarrow \\mathbb R$ two convex differentiable functions, $\\gamma$ a positive damping parameter, and $\\epsilon (t)$ a function of $t$ which tends to zero as $t$ goes to $+\\infty$, we consider the second-order differential equation $$\\ddot{x}(t) + \\gamma \\dot{x}(t) + \\nabla \\Phi (x(t)) + \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00232","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":"1602.00232","created_at":"2026-05-18T01:21:35.959557+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.00232v1","created_at":"2026-05-18T01:21:35.959557+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00232","created_at":"2026-05-18T01:21:35.959557+00:00"},{"alias_kind":"pith_short_12","alias_value":"BZDJJ63DAKUN","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"BZDJJ63DAKUNP6GA","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"BZDJJ63D","created_at":"2026-05-18T12:30:09.641336+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/BZDJJ63DAKUNP6GAJXEN7UHEGD","json":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD.json","graph_json":"https://pith.science/api/pith-number/BZDJJ63DAKUNP6GAJXEN7UHEGD/graph.json","events_json":"https://pith.science/api/pith-number/BZDJJ63DAKUNP6GAJXEN7UHEGD/events.json","paper":"https://pith.science/paper/BZDJJ63D"},"agent_actions":{"view_html":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD","download_json":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD.json","view_paper":"https://pith.science/paper/BZDJJ63D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.00232&json=true","fetch_graph":"https://pith.science/api/pith-number/BZDJJ63DAKUNP6GAJXEN7UHEGD/graph.json","fetch_events":"https://pith.science/api/pith-number/BZDJJ63DAKUNP6GAJXEN7UHEGD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD/action/storage_attestation","attest_author":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD/action/author_attestation","sign_citation":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD/action/citation_signature","submit_replication":"https://pith.science/pith/BZDJJ63DAKUNP6GAJXEN7UHEGD/action/replication_record"}},"created_at":"2026-05-18T01:21:35.959557+00:00","updated_at":"2026-05-18T01:21:35.959557+00:00"}