{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:E2MWNIIIP4R7YCJ4T34TU23D65","short_pith_number":"pith:E2MWNIII","schema_version":"1.0","canonical_sha256":"269966a1087f23fc093c9ef93a6b63f765e7a42474dd4cd5846fdf360805f19a","source":{"kind":"arxiv","id":"1412.6731","version":3},"attestation_state":"computed","paper":{"title":"Adjacency Criterion For Gradient Flow With Multiple Local Maxima","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SY","math.GT"],"primary_cat":"math.DS","authors_text":"Xudong Chen","submitted_at":"2014-12-21T06:39:20Z","abstract_excerpt":"In this paper, we investigate the geometry of a general class of gradient flows with multiple local maxima. we decompose the underlying space into disjoint regions of attraction and establish the adjacency criterion. The criterion states a necessary and sufficient condition for two regions of attraction of stable equilibria to be adjacent. We then apply this criterion on a specific type of gradient flow which has as many as n! local maxima. In particular, we characterize the set of equilibria, compute the index of each critical manifold and moreover, find all pairs of adjacent neighbors. As an"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1412.6731","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-12-21T06:39:20Z","cross_cats_sorted":["cs.SY","math.GT"],"title_canon_sha256":"e718c046028dd5686a060b188d1d0ae3e63afb0d19db6f72f45638d788220bb3","abstract_canon_sha256":"0c8e6d2975ba19f60225f253fb35a40add4df37f802bd971afbda5efbce8c9c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:49.236546Z","signature_b64":"erletJf3CqYRenjGxe8mezYnN3UU2HOiPOLeUwniBo/R2Niif+DwH+oPXoQA6BDe+sNNTwrh/ijZVYjmpQmLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"269966a1087f23fc093c9ef93a6b63f765e7a42474dd4cd5846fdf360805f19a","last_reissued_at":"2026-05-18T00:35:49.235854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:49.235854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Adjacency Criterion For Gradient Flow With Multiple Local Maxima","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SY","math.GT"],"primary_cat":"math.DS","authors_text":"Xudong Chen","submitted_at":"2014-12-21T06:39:20Z","abstract_excerpt":"In this paper, we investigate the geometry of a general class of gradient flows with multiple local maxima. we decompose the underlying space into disjoint regions of attraction and establish the adjacency criterion. The criterion states a necessary and sufficient condition for two regions of attraction of stable equilibria to be adjacent. We then apply this criterion on a specific type of gradient flow which has as many as n! local maxima. In particular, we characterize the set of equilibria, compute the index of each critical manifold and moreover, find all pairs of adjacent neighbors. As an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6731","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1412.6731","created_at":"2026-05-18T00:35:49.235948+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.6731v3","created_at":"2026-05-18T00:35:49.235948+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6731","created_at":"2026-05-18T00:35:49.235948+00:00"},{"alias_kind":"pith_short_12","alias_value":"E2MWNIIIP4R7","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_16","alias_value":"E2MWNIIIP4R7YCJ4","created_at":"2026-05-18T12:28:25.294606+00:00"},{"alias_kind":"pith_short_8","alias_value":"E2MWNIII","created_at":"2026-05-18T12:28:25.294606+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65","json":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65.json","graph_json":"https://pith.science/api/pith-number/E2MWNIIIP4R7YCJ4T34TU23D65/graph.json","events_json":"https://pith.science/api/pith-number/E2MWNIIIP4R7YCJ4T34TU23D65/events.json","paper":"https://pith.science/paper/E2MWNIII"},"agent_actions":{"view_html":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65","download_json":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65.json","view_paper":"https://pith.science/paper/E2MWNIII","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.6731&json=true","fetch_graph":"https://pith.science/api/pith-number/E2MWNIIIP4R7YCJ4T34TU23D65/graph.json","fetch_events":"https://pith.science/api/pith-number/E2MWNIIIP4R7YCJ4T34TU23D65/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65/action/storage_attestation","attest_author":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65/action/author_attestation","sign_citation":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65/action/citation_signature","submit_replication":"https://pith.science/pith/E2MWNIIIP4R7YCJ4T34TU23D65/action/replication_record"}},"created_at":"2026-05-18T00:35:49.235948+00:00","updated_at":"2026-05-18T00:35:49.235948+00:00"}