{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JIBQOBNUG74TYZP4DIZNQNYVQU","short_pith_number":"pith:JIBQOBNU","canonical_record":{"source":{"id":"1504.04069","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-15T23:55:50Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"0f9518689cd137d20e93fe7a68e30d9b4d947bac741819cf0d3456b554332d16","abstract_canon_sha256":"8203fc37596ef94b160800638661f23998040ce4e0348e7e873359a77b5d8806"},"schema_version":"1.0"},"canonical_sha256":"4a030705b437f93c65fc1a32d83715851408e8000c80144910b6fd12154fec1c","source":{"kind":"arxiv","id":"1504.04069","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04069","created_at":"2026-05-18T01:26:18Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04069v2","created_at":"2026-05-18T01:26:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04069","created_at":"2026-05-18T01:26:18Z"},{"alias_kind":"pith_short_12","alias_value":"JIBQOBNUG74T","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JIBQOBNUG74TYZP4","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JIBQOBNU","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JIBQOBNUG74TYZP4DIZNQNYVQU","target":"record","payload":{"canonical_record":{"source":{"id":"1504.04069","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-15T23:55:50Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"0f9518689cd137d20e93fe7a68e30d9b4d947bac741819cf0d3456b554332d16","abstract_canon_sha256":"8203fc37596ef94b160800638661f23998040ce4e0348e7e873359a77b5d8806"},"schema_version":"1.0"},"canonical_sha256":"4a030705b437f93c65fc1a32d83715851408e8000c80144910b6fd12154fec1c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:18.492003Z","signature_b64":"xV3ZlbFepKT1gJ5ulJxd9AON9FR6OrZP5MVWw4Y22vIYFIrrUbS3eog++jTASree6mgjlQGt6ry+j7XRuedDAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a030705b437f93c65fc1a32d83715851408e8000c80144910b6fd12154fec1c","last_reissued_at":"2026-05-18T01:26:18.491278Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:18.491278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.04069","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:26:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Io2GUyBv6sE3Dxy7tA3Dqg9oxkfQMTmRaa/AO1S7xdXHZ/wrk4D9jC33bxoxkzETy/jXRJTymQqs2RxTM86BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T02:31:05.257546Z"},"content_sha256":"ecf17e3889071f1081a0005ce898c80f29a95b6ba027effb961a5f8b41bdea25","schema_version":"1.0","event_id":"sha256:ecf17e3889071f1081a0005ce898c80f29a95b6ba027effb961a5f8b41bdea25"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JIBQOBNUG74TYZP4DIZNQNYVQU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical exponents of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CO","authors_text":"Apoorva Khare, Bala Rajaratnam, Dominique Guillot","submitted_at":"2015-04-15T23:55:50Z","abstract_excerpt":"The study of entrywise powers of matrices was originated by Loewner in the pursuit of the Bieberbach conjecture. Since the work of FitzGerald and Horn (1977), it is known that $A^{\\circ \\alpha} := (a_{ij}^\\alpha)$ is positive semidefinite for every entrywise nonnegative $n \\times n$ positive semidefinite matrix $A = (a_{ij})$ if and only if $\\alpha$ is a positive integer or $\\alpha \\geq n-2$. This surprising result naturally extends the Schur product theorem, and demonstrates the existence of a sharp phase transition in preserving positivity. In this paper, we study when entrywise powers prese"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04069","kind":"arxiv","version":2},"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":"2026-05-18T01:26:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vl65EVw4Iw7WSx9Wp5jCgJ0NdaC4AI+SFNMiQygLCMjoG3oFgqAnkO5N41FQNRNaDIsv/i1IjPNTxNt27U0vCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T02:31:05.257908Z"},"content_sha256":"a46b9b50243e7e064f07a075eaff0d8de0a18c130e9152c2ca0a63958466c059","schema_version":"1.0","event_id":"sha256:a46b9b50243e7e064f07a075eaff0d8de0a18c130e9152c2ca0a63958466c059"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JIBQOBNUG74TYZP4DIZNQNYVQU/bundle.json","state_url":"https://pith.science/pith/JIBQOBNUG74TYZP4DIZNQNYVQU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JIBQOBNUG74TYZP4DIZNQNYVQU/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":"2026-06-02T02:31:05Z","links":{"resolver":"https://pith.science/pith/JIBQOBNUG74TYZP4DIZNQNYVQU","bundle":"https://pith.science/pith/JIBQOBNUG74TYZP4DIZNQNYVQU/bundle.json","state":"https://pith.science/pith/JIBQOBNUG74TYZP4DIZNQNYVQU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JIBQOBNUG74TYZP4DIZNQNYVQU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JIBQOBNUG74TYZP4DIZNQNYVQU","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":"8203fc37596ef94b160800638661f23998040ce4e0348e7e873359a77b5d8806","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-15T23:55:50Z","title_canon_sha256":"0f9518689cd137d20e93fe7a68e30d9b4d947bac741819cf0d3456b554332d16"},"schema_version":"1.0","source":{"id":"1504.04069","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04069","created_at":"2026-05-18T01:26:18Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04069v2","created_at":"2026-05-18T01:26:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04069","created_at":"2026-05-18T01:26:18Z"},{"alias_kind":"pith_short_12","alias_value":"JIBQOBNUG74T","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JIBQOBNUG74TYZP4","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JIBQOBNU","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:a46b9b50243e7e064f07a075eaff0d8de0a18c130e9152c2ca0a63958466c059","target":"graph","created_at":"2026-05-18T01:26:18Z","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 study of entrywise powers of matrices was originated by Loewner in the pursuit of the Bieberbach conjecture. Since the work of FitzGerald and Horn (1977), it is known that $A^{\\circ \\alpha} := (a_{ij}^\\alpha)$ is positive semidefinite for every entrywise nonnegative $n \\times n$ positive semidefinite matrix $A = (a_{ij})$ if and only if $\\alpha$ is a positive integer or $\\alpha \\geq n-2$. This surprising result naturally extends the Schur product theorem, and demonstrates the existence of a sharp phase transition in preserving positivity. In this paper, we study when entrywise powers prese","authors_text":"Apoorva Khare, Bala Rajaratnam, Dominique Guillot","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-15T23:55:50Z","title":"Critical exponents of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04069","kind":"arxiv","version":2},"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:ecf17e3889071f1081a0005ce898c80f29a95b6ba027effb961a5f8b41bdea25","target":"record","created_at":"2026-05-18T01:26:18Z","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":"8203fc37596ef94b160800638661f23998040ce4e0348e7e873359a77b5d8806","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-15T23:55:50Z","title_canon_sha256":"0f9518689cd137d20e93fe7a68e30d9b4d947bac741819cf0d3456b554332d16"},"schema_version":"1.0","source":{"id":"1504.04069","kind":"arxiv","version":2}},"canonical_sha256":"4a030705b437f93c65fc1a32d83715851408e8000c80144910b6fd12154fec1c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a030705b437f93c65fc1a32d83715851408e8000c80144910b6fd12154fec1c","first_computed_at":"2026-05-18T01:26:18.491278Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:18.491278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xV3ZlbFepKT1gJ5ulJxd9AON9FR6OrZP5MVWw4Y22vIYFIrrUbS3eog++jTASree6mgjlQGt6ry+j7XRuedDAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:18.492003Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.04069","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ecf17e3889071f1081a0005ce898c80f29a95b6ba027effb961a5f8b41bdea25","sha256:a46b9b50243e7e064f07a075eaff0d8de0a18c130e9152c2ca0a63958466c059"],"state_sha256":"359ccc4fcafd7a2a6db97aa7c16023edbadc0900e2a0fee35fed7a857e89ba3f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y3y8fa7Zrh/PjrWd7xzDGHVeLL6Vn1DPpYMT1q/7sbANd1DX5oZ8Q8/FL1S1V9KxPsuROQq2OLhNiFtLPPIjCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T02:31:05.260101Z","bundle_sha256":"360096262ccc47852c30b7b1c3c378bae39e1c888b82f77057d80d34aa2f1c5a"}}