{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:JB6OSREPADTSTEY7C6EF56NW4L","short_pith_number":"pith:JB6OSREP","canonical_record":{"source":{"id":"1206.5084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.AO","submitted_at":"2012-06-22T08:33:05Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"55e302742f66618228434614ec89756ac4cbf87e425e2c75be10e6e37bccbccb","abstract_canon_sha256":"6d7cecb1d4e799e1ea2c940848a008e9225fe7b5d4effa54aebe58a27702df44"},"schema_version":"1.0"},"canonical_sha256":"487ce9448f00e729931f17885ef9b6e2dfe50cc566f7c20af183f18eafd97b72","source":{"kind":"arxiv","id":"1206.5084","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5084","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5084v1","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5084","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"pith_short_12","alias_value":"JB6OSREPADTS","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JB6OSREPADTSTEY7","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JB6OSREP","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:JB6OSREPADTSTEY7C6EF56NW4L","target":"record","payload":{"canonical_record":{"source":{"id":"1206.5084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.AO","submitted_at":"2012-06-22T08:33:05Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"55e302742f66618228434614ec89756ac4cbf87e425e2c75be10e6e37bccbccb","abstract_canon_sha256":"6d7cecb1d4e799e1ea2c940848a008e9225fe7b5d4effa54aebe58a27702df44"},"schema_version":"1.0"},"canonical_sha256":"487ce9448f00e729931f17885ef9b6e2dfe50cc566f7c20af183f18eafd97b72","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:52.391651Z","signature_b64":"Ac2ZPEgGhpyCE6OySAvy39M5/45WaZYP5IucYvAbNtl2vx099nJswcoo8cT/C/FIKHkReWJb45FuMmI6+3b0Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"487ce9448f00e729931f17885ef9b6e2dfe50cc566f7c20af183f18eafd97b72","last_reissued_at":"2026-05-18T03:52:52.391072Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:52.391072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.5084","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":"2026-05-18T03:52:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ubjRx6/zfioIgbNwkNdjUBfkVyyyc/4n7pAgBbgpNIVkbWXT30H6aAX2n1hrkOethQUnNaaT+TphP9Z767QsDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T15:55:33.815107Z"},"content_sha256":"832f49be35025f26be31f7042b8ffcb49fd73619f3ca7508dea320accd0a95c1","schema_version":"1.0","event_id":"sha256:832f49be35025f26be31f7042b8ffcb49fd73619f3ca7508dea320accd0a95c1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:JB6OSREPADTSTEY7C6EF56NW4L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dynamic Transition and Pattern Formation for Chemotactic Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"nlin.AO","authors_text":"Shouhong Wang, Tian Ma","submitted_at":"2012-06-22T08:33:05Z","abstract_excerpt":"The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$. For the general Keller-Segel model where the stimulant is moderately supplied, the system can "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5084","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":"2026-05-18T03:52:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lk65vQ/pkyuswK7wQt0ya58jbdj4UpJQj9tDdgoVvdIlskqOSdgbvOp4DOZhQn4d1Ts9JkJnIUyvW/uWJUimDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T15:55:33.815850Z"},"content_sha256":"9a12a86e27f7912bc0577f913230a976275800a94d044e4c17d921d05b8e477f","schema_version":"1.0","event_id":"sha256:9a12a86e27f7912bc0577f913230a976275800a94d044e4c17d921d05b8e477f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JB6OSREPADTSTEY7C6EF56NW4L/bundle.json","state_url":"https://pith.science/pith/JB6OSREPADTSTEY7C6EF56NW4L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JB6OSREPADTSTEY7C6EF56NW4L/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":"2026-05-28T15:55:33Z","links":{"resolver":"https://pith.science/pith/JB6OSREPADTSTEY7C6EF56NW4L","bundle":"https://pith.science/pith/JB6OSREPADTSTEY7C6EF56NW4L/bundle.json","state":"https://pith.science/pith/JB6OSREPADTSTEY7C6EF56NW4L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JB6OSREPADTSTEY7C6EF56NW4L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JB6OSREPADTSTEY7C6EF56NW4L","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":"6d7cecb1d4e799e1ea2c940848a008e9225fe7b5d4effa54aebe58a27702df44","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.AO","submitted_at":"2012-06-22T08:33:05Z","title_canon_sha256":"55e302742f66618228434614ec89756ac4cbf87e425e2c75be10e6e37bccbccb"},"schema_version":"1.0","source":{"id":"1206.5084","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5084","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5084v1","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5084","created_at":"2026-05-18T03:52:52Z"},{"alias_kind":"pith_short_12","alias_value":"JB6OSREPADTS","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JB6OSREPADTSTEY7","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JB6OSREP","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:9a12a86e27f7912bc0577f913230a976275800a94d044e4c17d921d05b8e477f","target":"graph","created_at":"2026-05-18T03:52:52Z","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":"The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$. For the general Keller-Segel model where the stimulant is moderately supplied, the system can ","authors_text":"Shouhong Wang, Tian Ma","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.AO","submitted_at":"2012-06-22T08:33:05Z","title":"Dynamic Transition and Pattern Formation for Chemotactic Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5084","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":"sha256:832f49be35025f26be31f7042b8ffcb49fd73619f3ca7508dea320accd0a95c1","target":"record","created_at":"2026-05-18T03:52:52Z","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":"6d7cecb1d4e799e1ea2c940848a008e9225fe7b5d4effa54aebe58a27702df44","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.AO","submitted_at":"2012-06-22T08:33:05Z","title_canon_sha256":"55e302742f66618228434614ec89756ac4cbf87e425e2c75be10e6e37bccbccb"},"schema_version":"1.0","source":{"id":"1206.5084","kind":"arxiv","version":1}},"canonical_sha256":"487ce9448f00e729931f17885ef9b6e2dfe50cc566f7c20af183f18eafd97b72","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"487ce9448f00e729931f17885ef9b6e2dfe50cc566f7c20af183f18eafd97b72","first_computed_at":"2026-05-18T03:52:52.391072Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:52.391072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ac2ZPEgGhpyCE6OySAvy39M5/45WaZYP5IucYvAbNtl2vx099nJswcoo8cT/C/FIKHkReWJb45FuMmI6+3b0Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:52.391651Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5084","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:832f49be35025f26be31f7042b8ffcb49fd73619f3ca7508dea320accd0a95c1","sha256:9a12a86e27f7912bc0577f913230a976275800a94d044e4c17d921d05b8e477f"],"state_sha256":"712e99c55c425fd16d9ee30538b78514eeab3bfb67c6f6fb30b3d85cd38515d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lETetQCapb7tq/aOmtEQFQrbSWj2Z56PHtCEWL1UwLArjwTC+jKN2drvFGj1uTcNn57M4slp5AXVut7Hm30sAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T15:55:33.819708Z","bundle_sha256":"ccbfca6798fc04b9a2fee84bd63e9cc727d27cb4358782f7ea0ee319a172a509"}}