{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:OQVWJC4EUU5Z6IA3E7UXOBAABH","short_pith_number":"pith:OQVWJC4E","canonical_record":{"source":{"id":"1211.1192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-06T11:52:38Z","cross_cats_sorted":[],"title_canon_sha256":"35a10c551bf61fba03d85f8839a05d62ccb9e4e4f421a51e6c954a200942cff4","abstract_canon_sha256":"b143d8797530a5d4a2defd6cefc09385d399bf88e6816c7389a02471ea688ab1"},"schema_version":"1.0"},"canonical_sha256":"742b648b84a53b9f201b27e977040009ddf87872339dd6d8148bb23d20c1bad6","source":{"kind":"arxiv","id":"1211.1192","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1192","created_at":"2026-05-18T03:41:22Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1192v1","created_at":"2026-05-18T03:41:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1192","created_at":"2026-05-18T03:41:22Z"},{"alias_kind":"pith_short_12","alias_value":"OQVWJC4EUU5Z","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"OQVWJC4EUU5Z6IA3","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"OQVWJC4E","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:OQVWJC4EUU5Z6IA3E7UXOBAABH","target":"record","payload":{"canonical_record":{"source":{"id":"1211.1192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-06T11:52:38Z","cross_cats_sorted":[],"title_canon_sha256":"35a10c551bf61fba03d85f8839a05d62ccb9e4e4f421a51e6c954a200942cff4","abstract_canon_sha256":"b143d8797530a5d4a2defd6cefc09385d399bf88e6816c7389a02471ea688ab1"},"schema_version":"1.0"},"canonical_sha256":"742b648b84a53b9f201b27e977040009ddf87872339dd6d8148bb23d20c1bad6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:22.076092Z","signature_b64":"+2MYMQno6JIAy3o32ZvkQfnOk09xvOZUe9dx749X4JBiE/iSyQdNRWJdQAAPHjCoTo9vRbYuQJsgmrkVB4mXDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"742b648b84a53b9f201b27e977040009ddf87872339dd6d8148bb23d20c1bad6","last_reissued_at":"2026-05-18T03:41:22.075615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:22.075615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.1192","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":"2026-05-18T03:41:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3AQKslmriDuU+yWXRfTVgBUPplMjGqP19MOXA/R7nAQYZ5zUcJY3Jyy68ToD3mMoODXCB7G1BAP4b5yxMTxmCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:07:08.521421Z"},"content_sha256":"6c089802ff0252c79d5a88314a57cb8fe4b1b7441bfff4478e578ca9d1fe2b2c","schema_version":"1.0","event_id":"sha256:6c089802ff0252c79d5a88314a57cb8fe4b1b7441bfff4478e578ca9d1fe2b2c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:OQVWJC4EUU5Z6IA3E7UXOBAABH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Blow-up of solutions to a Dirichlet problem for the discrete semi-linear heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Keisuke Matsuya","submitted_at":"2012-11-06T11:52:38Z","abstract_excerpt":"In this paper, the initial and boundary problem of the difference equation which is a discretization of the semi-linear heat equation. The difference equation derived by discretizing the semi-linear heat equation has solutions which show characteristics corresponding to the characteristics of the blow-up solutions for the original equation. The initial and boundary problem for the original equation has blow-up solutions when a certain condition is met. We prove that when a similar condition as that of the original solution is met in the initial and boundary problem for the difference equation,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1192","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":"2026-05-18T03:41:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XIPSgWOSpYCKq5MdC5VYGbXzEbWktyR6hHnLDeQLakqIBM6uMQjC2pYClSs91RZlXSlvl1Y51cS0Lc0CMGm+AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:07:08.522161Z"},"content_sha256":"6a69d8c75c915e046769401f9d1b8b58cc4ce6f35be7b0abb1a061382ab66484","schema_version":"1.0","event_id":"sha256:6a69d8c75c915e046769401f9d1b8b58cc4ce6f35be7b0abb1a061382ab66484"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OQVWJC4EUU5Z6IA3E7UXOBAABH/bundle.json","state_url":"https://pith.science/pith/OQVWJC4EUU5Z6IA3E7UXOBAABH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OQVWJC4EUU5Z6IA3E7UXOBAABH/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":"2026-06-07T23:07:08Z","links":{"resolver":"https://pith.science/pith/OQVWJC4EUU5Z6IA3E7UXOBAABH","bundle":"https://pith.science/pith/OQVWJC4EUU5Z6IA3E7UXOBAABH/bundle.json","state":"https://pith.science/pith/OQVWJC4EUU5Z6IA3E7UXOBAABH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OQVWJC4EUU5Z6IA3E7UXOBAABH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:OQVWJC4EUU5Z6IA3E7UXOBAABH","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":"b143d8797530a5d4a2defd6cefc09385d399bf88e6816c7389a02471ea688ab1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-06T11:52:38Z","title_canon_sha256":"35a10c551bf61fba03d85f8839a05d62ccb9e4e4f421a51e6c954a200942cff4"},"schema_version":"1.0","source":{"id":"1211.1192","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1192","created_at":"2026-05-18T03:41:22Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1192v1","created_at":"2026-05-18T03:41:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1192","created_at":"2026-05-18T03:41:22Z"},{"alias_kind":"pith_short_12","alias_value":"OQVWJC4EUU5Z","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"OQVWJC4EUU5Z6IA3","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"OQVWJC4E","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:6a69d8c75c915e046769401f9d1b8b58cc4ce6f35be7b0abb1a061382ab66484","target":"graph","created_at":"2026-05-18T03:41:22Z","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 and boundary problem of the difference equation which is a discretization of the semi-linear heat equation. The difference equation derived by discretizing the semi-linear heat equation has solutions which show characteristics corresponding to the characteristics of the blow-up solutions for the original equation. The initial and boundary problem for the original equation has blow-up solutions when a certain condition is met. We prove that when a similar condition as that of the original solution is met in the initial and boundary problem for the difference equation,","authors_text":"Keisuke Matsuya","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-06T11:52:38Z","title":"Blow-up of solutions to a Dirichlet problem for the discrete semi-linear heat equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1192","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":"sha256:6c089802ff0252c79d5a88314a57cb8fe4b1b7441bfff4478e578ca9d1fe2b2c","target":"record","created_at":"2026-05-18T03:41:22Z","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":"b143d8797530a5d4a2defd6cefc09385d399bf88e6816c7389a02471ea688ab1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-06T11:52:38Z","title_canon_sha256":"35a10c551bf61fba03d85f8839a05d62ccb9e4e4f421a51e6c954a200942cff4"},"schema_version":"1.0","source":{"id":"1211.1192","kind":"arxiv","version":1}},"canonical_sha256":"742b648b84a53b9f201b27e977040009ddf87872339dd6d8148bb23d20c1bad6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"742b648b84a53b9f201b27e977040009ddf87872339dd6d8148bb23d20c1bad6","first_computed_at":"2026-05-18T03:41:22.075615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:22.075615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+2MYMQno6JIAy3o32ZvkQfnOk09xvOZUe9dx749X4JBiE/iSyQdNRWJdQAAPHjCoTo9vRbYuQJsgmrkVB4mXDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:22.076092Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.1192","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c089802ff0252c79d5a88314a57cb8fe4b1b7441bfff4478e578ca9d1fe2b2c","sha256:6a69d8c75c915e046769401f9d1b8b58cc4ce6f35be7b0abb1a061382ab66484"],"state_sha256":"5e7f99e1118141269f3e3689dcc251a6af6145c136b4b2a2afe4f9201c66f6f5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"na8LYFZn2S0nB2Rqkgq5zevPyMoiLUU4D4DoGrf1IWoxh9GZGSu3CLMFM5RvOG8o18SlEPR/FNYqwfcLt/e1BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T23:07:08.526359Z","bundle_sha256":"a6f226c497d271985ad334795db6fc707ed51fa881318bd072a0ce9b2d7c515f"}}