{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:EIATE4ZRU66MENP63CAWVNCCVE","short_pith_number":"pith:EIATE4ZR","canonical_record":{"source":{"id":"1206.5510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-24T15:46:46Z","cross_cats_sorted":["math.AP","math.DS"],"title_canon_sha256":"37ac67dfb7f2619c91fd270ee52345f9bac61673836da54769efabb988f00b9e","abstract_canon_sha256":"2f65fd4a852a0f189e29551505ea03912f5ae17a07cf1d6eef5768fd2977b524"},"schema_version":"1.0"},"canonical_sha256":"2201327331a7bcc235fed8816ab442a922e6afb18b0cf2050eb2c0fb7188f20b","source":{"kind":"arxiv","id":"1206.5510","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5510","created_at":"2026-05-18T03:52:39Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5510v1","created_at":"2026-05-18T03:52:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5510","created_at":"2026-05-18T03:52:39Z"},{"alias_kind":"pith_short_12","alias_value":"EIATE4ZRU66M","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EIATE4ZRU66MENP6","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EIATE4ZR","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:EIATE4ZRU66MENP63CAWVNCCVE","target":"record","payload":{"canonical_record":{"source":{"id":"1206.5510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-24T15:46:46Z","cross_cats_sorted":["math.AP","math.DS"],"title_canon_sha256":"37ac67dfb7f2619c91fd270ee52345f9bac61673836da54769efabb988f00b9e","abstract_canon_sha256":"2f65fd4a852a0f189e29551505ea03912f5ae17a07cf1d6eef5768fd2977b524"},"schema_version":"1.0"},"canonical_sha256":"2201327331a7bcc235fed8816ab442a922e6afb18b0cf2050eb2c0fb7188f20b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:39.894069Z","signature_b64":"RbJs4sDN70awddRGO7N83BK/M/3FB82PntIW5G0xy3wrInq+Px9O+4KXRKCiyLM/nzEktG0tUhAJyoFeRVlvDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2201327331a7bcc235fed8816ab442a922e6afb18b0cf2050eb2c0fb7188f20b","last_reissued_at":"2026-05-18T03:52:39.893495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:39.893495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.5510","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":"2026-05-18T03:52:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bgdKcvmjaHqKpkIEiZfexHxGgR6yur4qcz0rzGHwIohICca169JqlG05XzZlKcA/kMgvkVL+tfklfLXm/SM0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:41:11.899850Z"},"content_sha256":"4b1e9d3b670463417d78ea47d4e923558e24f4a55c799eb1e71eab5a00f923a4","schema_version":"1.0","event_id":"sha256:4b1e9d3b670463417d78ea47d4e923558e24f4a55c799eb1e71eab5a00f923a4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:EIATE4ZRU66MENP63CAWVNCCVE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convergence of Time-Dependent Turing Structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DS"],"primary_cat":"math.FA","authors_text":"A.G.Ramm, V.Volpert","submitted_at":"2012-06-24T15:46:46Z","abstract_excerpt":"Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the $L^2$ norm of the solution. This method is appropriate for the equations with time dependent coefficients. It yields new results and is applicable when the usual linearization method is not applicable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5510","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":"2026-05-18T03:52:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wI7D09Goy09BS6d+DT+wpntKBiuy9ocRMmErxVvhh2wQjRTTHSBumPsBUmzZ27e+eoKyoRSb/ke8ADTy5HNcDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T20:41:11.900545Z"},"content_sha256":"b3d66c359021fee9ff9653b137f6d4351d9c8ec67faaea9999674b24c46519b4","schema_version":"1.0","event_id":"sha256:b3d66c359021fee9ff9653b137f6d4351d9c8ec67faaea9999674b24c46519b4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EIATE4ZRU66MENP63CAWVNCCVE/bundle.json","state_url":"https://pith.science/pith/EIATE4ZRU66MENP63CAWVNCCVE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EIATE4ZRU66MENP63CAWVNCCVE/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":"2026-05-25T20:41:11Z","links":{"resolver":"https://pith.science/pith/EIATE4ZRU66MENP63CAWVNCCVE","bundle":"https://pith.science/pith/EIATE4ZRU66MENP63CAWVNCCVE/bundle.json","state":"https://pith.science/pith/EIATE4ZRU66MENP63CAWVNCCVE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EIATE4ZRU66MENP63CAWVNCCVE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:EIATE4ZRU66MENP63CAWVNCCVE","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":"2f65fd4a852a0f189e29551505ea03912f5ae17a07cf1d6eef5768fd2977b524","cross_cats_sorted":["math.AP","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-24T15:46:46Z","title_canon_sha256":"37ac67dfb7f2619c91fd270ee52345f9bac61673836da54769efabb988f00b9e"},"schema_version":"1.0","source":{"id":"1206.5510","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5510","created_at":"2026-05-18T03:52:39Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5510v1","created_at":"2026-05-18T03:52:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5510","created_at":"2026-05-18T03:52:39Z"},{"alias_kind":"pith_short_12","alias_value":"EIATE4ZRU66M","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EIATE4ZRU66MENP6","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EIATE4ZR","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:b3d66c359021fee9ff9653b137f6d4351d9c8ec67faaea9999674b24c46519b4","target":"graph","created_at":"2026-05-18T03:52:39Z","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":"Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the $L^2$ norm of the solution. This method is appropriate for the equations with time dependent coefficients. It yields new results and is applicable when the usual linearization method is not applicable.","authors_text":"A.G.Ramm, V.Volpert","cross_cats":["math.AP","math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-24T15:46:46Z","title":"Convergence of Time-Dependent Turing Structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5510","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":"sha256:4b1e9d3b670463417d78ea47d4e923558e24f4a55c799eb1e71eab5a00f923a4","target":"record","created_at":"2026-05-18T03:52:39Z","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":"2f65fd4a852a0f189e29551505ea03912f5ae17a07cf1d6eef5768fd2977b524","cross_cats_sorted":["math.AP","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-24T15:46:46Z","title_canon_sha256":"37ac67dfb7f2619c91fd270ee52345f9bac61673836da54769efabb988f00b9e"},"schema_version":"1.0","source":{"id":"1206.5510","kind":"arxiv","version":1}},"canonical_sha256":"2201327331a7bcc235fed8816ab442a922e6afb18b0cf2050eb2c0fb7188f20b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2201327331a7bcc235fed8816ab442a922e6afb18b0cf2050eb2c0fb7188f20b","first_computed_at":"2026-05-18T03:52:39.893495Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:39.893495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RbJs4sDN70awddRGO7N83BK/M/3FB82PntIW5G0xy3wrInq+Px9O+4KXRKCiyLM/nzEktG0tUhAJyoFeRVlvDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:39.894069Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5510","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b1e9d3b670463417d78ea47d4e923558e24f4a55c799eb1e71eab5a00f923a4","sha256:b3d66c359021fee9ff9653b137f6d4351d9c8ec67faaea9999674b24c46519b4"],"state_sha256":"d6266ca2ec23383cba4c370d563062efeedbf64bfd64213825e8cb2b11d9014b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fAntiGJ7hDstJCsLWIC/T+smM9mAfTu7c+KMyEpgL56YaD9jr24ggZfuYbhKimafvHMRWnwjG/n7Ve6KBXNKBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T20:41:11.904176Z","bundle_sha256":"2e701343b3c3b2d6764b9f9e4bdd0db61ecdf9c5a7e5fc3e620160a8185119da"}}