{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:VNDAPI2DCMG27MJ6WI37VBLMUR","short_pith_number":"pith:VNDAPI2D","canonical_record":{"source":{"id":"1307.5608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T07:05:55Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"850af2f2325e54238da6b8dbcfd01f6ec298a4df2ba6f5d5e643f2e6022415fa","abstract_canon_sha256":"83e6894cdd0d0023a386f17970c0fb9ac15c9b1531806a90d0781f53dfd5ab72"},"schema_version":"1.0"},"canonical_sha256":"ab4607a343130dafb13eb237fa856ca441ce4479d268f56ecfd50b654832245f","source":{"kind":"arxiv","id":"1307.5608","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5608","created_at":"2026-05-18T03:17:55Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5608v1","created_at":"2026-05-18T03:17:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5608","created_at":"2026-05-18T03:17:55Z"},{"alias_kind":"pith_short_12","alias_value":"VNDAPI2DCMG2","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VNDAPI2DCMG27MJ6","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VNDAPI2D","created_at":"2026-05-18T12:28:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:VNDAPI2DCMG27MJ6WI37VBLMUR","target":"record","payload":{"canonical_record":{"source":{"id":"1307.5608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T07:05:55Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"850af2f2325e54238da6b8dbcfd01f6ec298a4df2ba6f5d5e643f2e6022415fa","abstract_canon_sha256":"83e6894cdd0d0023a386f17970c0fb9ac15c9b1531806a90d0781f53dfd5ab72"},"schema_version":"1.0"},"canonical_sha256":"ab4607a343130dafb13eb237fa856ca441ce4479d268f56ecfd50b654832245f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:55.258335Z","signature_b64":"6ebbrzdvIEE/ofMi0oPoOAbAE/bwbOmQT66TDpugd0Fey1gifnZ0Jf80KTlue4UKk+qDmhCfekvwwG/84ZUuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab4607a343130dafb13eb237fa856ca441ce4479d268f56ecfd50b654832245f","last_reissued_at":"2026-05-18T03:17:55.257653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:55.257653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.5608","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":"2026-05-18T03:17:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H1oIt6q2OW+ML/dDb4m94YFuLKESRvBDzD0odmoCNz18jJpRxGlVg7UpaObJPd7A+OeKgqNol+23OnZTz5a3CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T21:01:56.287607Z"},"content_sha256":"92a6a2664e07e346801d5de21cb6c0f486adfa8bf94239f9880ea783f9ec3e61","schema_version":"1.0","event_id":"sha256:92a6a2664e07e346801d5de21cb6c0f486adfa8bf94239f9880ea783f9ec3e61"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:VNDAPI2DCMG27MJ6WI37VBLMUR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global behavior of the Solutions to a Class of Nonlinear, Singular Second Order ODE","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alain Haraux (LJLL), Mama Abdelli","submitted_at":"2013-07-22T07:05:55Z","abstract_excerpt":"In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: $ (|u'|^{l}u')' + c|u'|^{\\alpha}u' + d|u|^\\beta u=0$, where $\\alpha,\\beta,l,c, d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5608","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":"2026-05-18T03:17:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8uO3DoecNoIwxTAAMoNyg3S0svslEt2pn45J9w1PFgs/+3Q/eBI46ntkcFu26EdB3mQh61VItAdv/RFr4AwVAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T21:01:56.287972Z"},"content_sha256":"709c7f484453a92afbc2def7f72ed448a50c289002205a30166437b4060f22d3","schema_version":"1.0","event_id":"sha256:709c7f484453a92afbc2def7f72ed448a50c289002205a30166437b4060f22d3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VNDAPI2DCMG27MJ6WI37VBLMUR/bundle.json","state_url":"https://pith.science/pith/VNDAPI2DCMG27MJ6WI37VBLMUR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VNDAPI2DCMG27MJ6WI37VBLMUR/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":"2026-07-01T21:01:56Z","links":{"resolver":"https://pith.science/pith/VNDAPI2DCMG27MJ6WI37VBLMUR","bundle":"https://pith.science/pith/VNDAPI2DCMG27MJ6WI37VBLMUR/bundle.json","state":"https://pith.science/pith/VNDAPI2DCMG27MJ6WI37VBLMUR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VNDAPI2DCMG27MJ6WI37VBLMUR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:VNDAPI2DCMG27MJ6WI37VBLMUR","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":"83e6894cdd0d0023a386f17970c0fb9ac15c9b1531806a90d0781f53dfd5ab72","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T07:05:55Z","title_canon_sha256":"850af2f2325e54238da6b8dbcfd01f6ec298a4df2ba6f5d5e643f2e6022415fa"},"schema_version":"1.0","source":{"id":"1307.5608","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5608","created_at":"2026-05-18T03:17:55Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5608v1","created_at":"2026-05-18T03:17:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5608","created_at":"2026-05-18T03:17:55Z"},{"alias_kind":"pith_short_12","alias_value":"VNDAPI2DCMG2","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VNDAPI2DCMG27MJ6","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VNDAPI2D","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:709c7f484453a92afbc2def7f72ed448a50c289002205a30166437b4060f22d3","target":"graph","created_at":"2026-05-18T03:17:55Z","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":"In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: $ (|u'|^{l}u')' + c|u'|^{\\alpha}u' + d|u|^\\beta u=0$, where $\\alpha,\\beta,l,c, d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.","authors_text":"Alain Haraux (LJLL), Mama Abdelli","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T07:05:55Z","title":"Global behavior of the Solutions to a Class of Nonlinear, Singular Second Order ODE"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5608","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":"sha256:92a6a2664e07e346801d5de21cb6c0f486adfa8bf94239f9880ea783f9ec3e61","target":"record","created_at":"2026-05-18T03:17:55Z","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":"83e6894cdd0d0023a386f17970c0fb9ac15c9b1531806a90d0781f53dfd5ab72","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T07:05:55Z","title_canon_sha256":"850af2f2325e54238da6b8dbcfd01f6ec298a4df2ba6f5d5e643f2e6022415fa"},"schema_version":"1.0","source":{"id":"1307.5608","kind":"arxiv","version":1}},"canonical_sha256":"ab4607a343130dafb13eb237fa856ca441ce4479d268f56ecfd50b654832245f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ab4607a343130dafb13eb237fa856ca441ce4479d268f56ecfd50b654832245f","first_computed_at":"2026-05-18T03:17:55.257653Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:17:55.257653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6ebbrzdvIEE/ofMi0oPoOAbAE/bwbOmQT66TDpugd0Fey1gifnZ0Jf80KTlue4UKk+qDmhCfekvwwG/84ZUuDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:17:55.258335Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.5608","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:92a6a2664e07e346801d5de21cb6c0f486adfa8bf94239f9880ea783f9ec3e61","sha256:709c7f484453a92afbc2def7f72ed448a50c289002205a30166437b4060f22d3"],"state_sha256":"2ff472696d57e59085df252363332545475122b0fe56a28b38c9ab586e9c67d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qRbtd0gU0QjazywmZYAa5XaCq14W/P9hy1+7MtqN6qvUGnkpFJfnYbQFvXXAJ1ZsKfKm8wWF0gzmQTnJHcr0AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T21:01:56.289902Z","bundle_sha256":"0e03db4e3f2b493be28a5b0eb0b1e2abc488a486a636063770a37f2a87857da1"}}