{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XWBVNFZAYVFY3OEWIGA2MYII7N","short_pith_number":"pith:XWBVNFZA","canonical_record":{"source":{"id":"1609.00135","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-01T07:42:14Z","cross_cats_sorted":[],"title_canon_sha256":"3ff1d0ec958e7a2a908cdf763e8aeba6d98b16f831a0210c6c863a1d47772541","abstract_canon_sha256":"3c88fb3f10871c3aae1d87e82e0413a7e858f7c2db4077ae6ae867209597582d"},"schema_version":"1.0"},"canonical_sha256":"bd83569720c54b8db8964181a66108fb7f0edd63866bff39ec9c43975322fc33","source":{"kind":"arxiv","id":"1609.00135","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.00135","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"arxiv_version","alias_value":"1609.00135v2","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00135","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"pith_short_12","alias_value":"XWBVNFZAYVFY","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XWBVNFZAYVFY3OEW","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XWBVNFZA","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XWBVNFZAYVFY3OEWIGA2MYII7N","target":"record","payload":{"canonical_record":{"source":{"id":"1609.00135","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-01T07:42:14Z","cross_cats_sorted":[],"title_canon_sha256":"3ff1d0ec958e7a2a908cdf763e8aeba6d98b16f831a0210c6c863a1d47772541","abstract_canon_sha256":"3c88fb3f10871c3aae1d87e82e0413a7e858f7c2db4077ae6ae867209597582d"},"schema_version":"1.0"},"canonical_sha256":"bd83569720c54b8db8964181a66108fb7f0edd63866bff39ec9c43975322fc33","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:40.963426Z","signature_b64":"53JJzTS5U9+aHy13DUqlyyvaEs4S9KWXGiF3B4Q9z53v+kcxNlBOOeW347H8zyJ1bftgltr/QTDA26UJNk4eDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd83569720c54b8db8964181a66108fb7f0edd63866bff39ec9c43975322fc33","last_reissued_at":"2026-05-18T01:04:40.962813Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:40.962813Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.00135","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:04:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kWRogjBSRPm68Srz3xvIIIi7v8IA2fVZArk+wR/3QwQE1a2iMPke7o8YhK4dBd7ctAljqO7KAddNp2LKdTPLDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:53:12.318167Z"},"content_sha256":"79132bcb0fbbef2b13f937cd7b0f5d4ff0b74aa3adeeb3cc680aa06a6f3ac7a6","schema_version":"1.0","event_id":"sha256:79132bcb0fbbef2b13f937cd7b0f5d4ff0b74aa3adeeb3cc680aa06a6f3ac7a6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XWBVNFZAYVFY3OEWIGA2MYII7N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic for the perturbed heavy ball system with vanishing damping term","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Mounir Balti, Ramzi May","submitted_at":"2016-09-01T07:42:14Z","abstract_excerpt":"We investigate the long time behavior of solutions to the differential equation $\\ddot{x}(t)+\\frac{c}{\\left( t+1\\right) ^{\\alpha}}\\dot{x}(t)+\\nabla \\Phi\\left( x(t)\\right) =g(t),~t\\geq0, $ where $c$ is nonnegative constant, $\\alpha\\in\\lbrack0,1[,$ $\\Phi$ is a $C^{1}$ convex function on a Hilbert space $\\mathcal{H}$ and $g\\in L^{1} (0,+\\infty;\\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\\rightarrow+\\infty$ to a minimizer of the function $\\Phi$ if one exists."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00135","kind":"arxiv","version":2},"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":"2026-05-18T01:04:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9CoN76+SDgwfrJLw0AX9IrlvQIJk0hUQapr9fdwRttNsdkvWP5kqYk8f6kKqClS/PLybdmTOBq+0e7q80R+tBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:53:12.318523Z"},"content_sha256":"0fe667ce7161703a552de47890a3b8b46ac6d99e12d96c40e495c36e61578262","schema_version":"1.0","event_id":"sha256:0fe667ce7161703a552de47890a3b8b46ac6d99e12d96c40e495c36e61578262"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XWBVNFZAYVFY3OEWIGA2MYII7N/bundle.json","state_url":"https://pith.science/pith/XWBVNFZAYVFY3OEWIGA2MYII7N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XWBVNFZAYVFY3OEWIGA2MYII7N/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":"2026-06-25T12:53:12Z","links":{"resolver":"https://pith.science/pith/XWBVNFZAYVFY3OEWIGA2MYII7N","bundle":"https://pith.science/pith/XWBVNFZAYVFY3OEWIGA2MYII7N/bundle.json","state":"https://pith.science/pith/XWBVNFZAYVFY3OEWIGA2MYII7N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XWBVNFZAYVFY3OEWIGA2MYII7N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XWBVNFZAYVFY3OEWIGA2MYII7N","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":"3c88fb3f10871c3aae1d87e82e0413a7e858f7c2db4077ae6ae867209597582d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-01T07:42:14Z","title_canon_sha256":"3ff1d0ec958e7a2a908cdf763e8aeba6d98b16f831a0210c6c863a1d47772541"},"schema_version":"1.0","source":{"id":"1609.00135","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.00135","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"arxiv_version","alias_value":"1609.00135v2","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.00135","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"pith_short_12","alias_value":"XWBVNFZAYVFY","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XWBVNFZAYVFY3OEW","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XWBVNFZA","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:0fe667ce7161703a552de47890a3b8b46ac6d99e12d96c40e495c36e61578262","target":"graph","created_at":"2026-05-18T01:04:40Z","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":"We investigate the long time behavior of solutions to the differential equation $\\ddot{x}(t)+\\frac{c}{\\left( t+1\\right) ^{\\alpha}}\\dot{x}(t)+\\nabla \\Phi\\left( x(t)\\right) =g(t),~t\\geq0, $ where $c$ is nonnegative constant, $\\alpha\\in\\lbrack0,1[,$ $\\Phi$ is a $C^{1}$ convex function on a Hilbert space $\\mathcal{H}$ and $g\\in L^{1} (0,+\\infty;\\mathcal{H}).$ We obtain sufficient conditions on the source term $g(t)$ ensuring the weak or the strong convergence of any trajectory $x(t)$ as $t\\rightarrow+\\infty$ to a minimizer of the function $\\Phi$ if one exists.","authors_text":"Mounir Balti, Ramzi May","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-01T07:42:14Z","title":"Asymptotic for the perturbed heavy ball system with vanishing damping term"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00135","kind":"arxiv","version":2},"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":"sha256:79132bcb0fbbef2b13f937cd7b0f5d4ff0b74aa3adeeb3cc680aa06a6f3ac7a6","target":"record","created_at":"2026-05-18T01:04:40Z","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":"3c88fb3f10871c3aae1d87e82e0413a7e858f7c2db4077ae6ae867209597582d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-09-01T07:42:14Z","title_canon_sha256":"3ff1d0ec958e7a2a908cdf763e8aeba6d98b16f831a0210c6c863a1d47772541"},"schema_version":"1.0","source":{"id":"1609.00135","kind":"arxiv","version":2}},"canonical_sha256":"bd83569720c54b8db8964181a66108fb7f0edd63866bff39ec9c43975322fc33","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd83569720c54b8db8964181a66108fb7f0edd63866bff39ec9c43975322fc33","first_computed_at":"2026-05-18T01:04:40.962813Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:40.962813Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"53JJzTS5U9+aHy13DUqlyyvaEs4S9KWXGiF3B4Q9z53v+kcxNlBOOeW347H8zyJ1bftgltr/QTDA26UJNk4eDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:40.963426Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.00135","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:79132bcb0fbbef2b13f937cd7b0f5d4ff0b74aa3adeeb3cc680aa06a6f3ac7a6","sha256:0fe667ce7161703a552de47890a3b8b46ac6d99e12d96c40e495c36e61578262"],"state_sha256":"93bf9e8ee129370cbf159e8b799d8f3c1aac65937b2bc0323bbc53ed40a416a4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mT1JZAUsFGEbonvhXa20nYX/fU5Dt3/4gp2lpyxKL0zwZcA4SHfomiHQvJiizO6LGt0udgpcIWkpBstxU5p+Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T12:53:12.320489Z","bundle_sha256":"26eeca110f9fa4fd3b23f214ab3bd6068668958da527f1c4e5fae8a63d65c0c0"}}