{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:VQM4OYHFQXM3ZRWQKNEP2LVCLC","short_pith_number":"pith:VQM4OYHF","canonical_record":{"source":{"id":"1511.09343","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-30T15:22:02Z","cross_cats_sorted":[],"title_canon_sha256":"34ae555619c3be0bb75038b6043557fdc171c46b5a28068d973fda80ddc4c43e","abstract_canon_sha256":"858e67a8d5c0043f5fac4ce3b165365294f1e1e68e090da1f06209522246b5e0"},"schema_version":"1.0"},"canonical_sha256":"ac19c760e585d9bcc6d05348fd2ea258a7f037764063e1d2cf84d9800449863a","source":{"kind":"arxiv","id":"1511.09343","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.09343","created_at":"2026-05-18T01:25:40Z"},{"alias_kind":"arxiv_version","alias_value":"1511.09343v1","created_at":"2026-05-18T01:25:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09343","created_at":"2026-05-18T01:25:40Z"},{"alias_kind":"pith_short_12","alias_value":"VQM4OYHFQXM3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"VQM4OYHFQXM3ZRWQ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"VQM4OYHF","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:VQM4OYHFQXM3ZRWQKNEP2LVCLC","target":"record","payload":{"canonical_record":{"source":{"id":"1511.09343","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-30T15:22:02Z","cross_cats_sorted":[],"title_canon_sha256":"34ae555619c3be0bb75038b6043557fdc171c46b5a28068d973fda80ddc4c43e","abstract_canon_sha256":"858e67a8d5c0043f5fac4ce3b165365294f1e1e68e090da1f06209522246b5e0"},"schema_version":"1.0"},"canonical_sha256":"ac19c760e585d9bcc6d05348fd2ea258a7f037764063e1d2cf84d9800449863a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:40.510099Z","signature_b64":"ZyBqn1n//nkUX6GzJb6B1WE5iV8h1jUqSza5zW4TBxwe0cg2ulVDXGEfXYBEzMusjbjXVAxpyIkWd7Aai5uACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac19c760e585d9bcc6d05348fd2ea258a7f037764063e1d2cf84d9800449863a","last_reissued_at":"2026-05-18T01:25:40.509444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:40.509444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.09343","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":"2026-05-18T01:25:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kzJFaZuK/l2tiPndRD9e27U7NWQt7APNJRL/iM+wPpnU6hCk6JUJfet4P0INC9Q9C/coCuvAIc4+Xy4OucDPCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T05:07:27.891520Z"},"content_sha256":"1cf89b1b59b1085e258e94d3502e03ead1f93fef5fbfcb14336b53123b15b751","schema_version":"1.0","event_id":"sha256:1cf89b1b59b1085e258e94d3502e03ead1f93fef5fbfcb14336b53123b15b751"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:VQM4OYHFQXM3ZRWQKNEP2LVCLC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bifurcation and segregation in quadratic two-populations Mean Field Games systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gianmaria Verzini, Marco Cirant","submitted_at":"2015-11-30T15:22:02Z","abstract_excerpt":"We search for non-constant normalized solutions to the semilinear elliptic system \\[ \\begin{cases} - \\nu \\Delta v_i + g_i(v_j^2) v_i = \\lambda_i v_i,\\quad v_i>0 & \\text{in $\\Omega$} \\\\ \\partial_n v_i = 0 & \\text{on $\\partial \\Omega$}\\\\ \\int_\\Omega v_i^2\\,dx = 1, & 1\\leq i,j\\leq 2, \\quad j\\neq i, \\end{cases} \\] where $\\nu>0$, $\\Omega \\subset \\mathbb{R}^N$ is smooth and bounded, the functions $g_i$ are positive and increasing, and both the functions $v_i$ and the parameters $\\lambda_i$ are unknown.\n  This system is obtained, via the Hopf-Cole transformation, from a two-populations ergodic Mean F"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09343","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":"2026-05-18T01:25:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"opMYkW8PYiF6Gj8CnoIaPU9Hk8v2hOTzJsMiwl88D837Tjfg4fBpFlMewMDKNV8SWCWrB9quSW0q2vr2TXmBDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T05:07:27.891865Z"},"content_sha256":"6ec893153ae6dfd908ad159aa6e30b102a0aaa7fa22a6cc16a017555814129db","schema_version":"1.0","event_id":"sha256:6ec893153ae6dfd908ad159aa6e30b102a0aaa7fa22a6cc16a017555814129db"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VQM4OYHFQXM3ZRWQKNEP2LVCLC/bundle.json","state_url":"https://pith.science/pith/VQM4OYHFQXM3ZRWQKNEP2LVCLC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VQM4OYHFQXM3ZRWQKNEP2LVCLC/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":"2026-06-27T05:07:27Z","links":{"resolver":"https://pith.science/pith/VQM4OYHFQXM3ZRWQKNEP2LVCLC","bundle":"https://pith.science/pith/VQM4OYHFQXM3ZRWQKNEP2LVCLC/bundle.json","state":"https://pith.science/pith/VQM4OYHFQXM3ZRWQKNEP2LVCLC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VQM4OYHFQXM3ZRWQKNEP2LVCLC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VQM4OYHFQXM3ZRWQKNEP2LVCLC","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":"858e67a8d5c0043f5fac4ce3b165365294f1e1e68e090da1f06209522246b5e0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-30T15:22:02Z","title_canon_sha256":"34ae555619c3be0bb75038b6043557fdc171c46b5a28068d973fda80ddc4c43e"},"schema_version":"1.0","source":{"id":"1511.09343","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.09343","created_at":"2026-05-18T01:25:40Z"},{"alias_kind":"arxiv_version","alias_value":"1511.09343v1","created_at":"2026-05-18T01:25:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.09343","created_at":"2026-05-18T01:25:40Z"},{"alias_kind":"pith_short_12","alias_value":"VQM4OYHFQXM3","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"VQM4OYHFQXM3ZRWQ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"VQM4OYHF","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:6ec893153ae6dfd908ad159aa6e30b102a0aaa7fa22a6cc16a017555814129db","target":"graph","created_at":"2026-05-18T01:25:40Z","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":"We search for non-constant normalized solutions to the semilinear elliptic system \\[ \\begin{cases} - \\nu \\Delta v_i + g_i(v_j^2) v_i = \\lambda_i v_i,\\quad v_i>0 & \\text{in $\\Omega$} \\\\ \\partial_n v_i = 0 & \\text{on $\\partial \\Omega$}\\\\ \\int_\\Omega v_i^2\\,dx = 1, & 1\\leq i,j\\leq 2, \\quad j\\neq i, \\end{cases} \\] where $\\nu>0$, $\\Omega \\subset \\mathbb{R}^N$ is smooth and bounded, the functions $g_i$ are positive and increasing, and both the functions $v_i$ and the parameters $\\lambda_i$ are unknown.\n  This system is obtained, via the Hopf-Cole transformation, from a two-populations ergodic Mean F","authors_text":"Gianmaria Verzini, Marco Cirant","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-30T15:22:02Z","title":"Bifurcation and segregation in quadratic two-populations Mean Field Games systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09343","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":"sha256:1cf89b1b59b1085e258e94d3502e03ead1f93fef5fbfcb14336b53123b15b751","target":"record","created_at":"2026-05-18T01:25:40Z","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":"858e67a8d5c0043f5fac4ce3b165365294f1e1e68e090da1f06209522246b5e0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-30T15:22:02Z","title_canon_sha256":"34ae555619c3be0bb75038b6043557fdc171c46b5a28068d973fda80ddc4c43e"},"schema_version":"1.0","source":{"id":"1511.09343","kind":"arxiv","version":1}},"canonical_sha256":"ac19c760e585d9bcc6d05348fd2ea258a7f037764063e1d2cf84d9800449863a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac19c760e585d9bcc6d05348fd2ea258a7f037764063e1d2cf84d9800449863a","first_computed_at":"2026-05-18T01:25:40.509444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:40.509444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZyBqn1n//nkUX6GzJb6B1WE5iV8h1jUqSza5zW4TBxwe0cg2ulVDXGEfXYBEzMusjbjXVAxpyIkWd7Aai5uACQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:40.510099Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.09343","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1cf89b1b59b1085e258e94d3502e03ead1f93fef5fbfcb14336b53123b15b751","sha256:6ec893153ae6dfd908ad159aa6e30b102a0aaa7fa22a6cc16a017555814129db"],"state_sha256":"24ada4c6ea10f3aa99b19692ddb33a63ee331b019dbfd53dc33a53aa251ef714"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gxuiQ/EaZRLnsCTqseoQ4XLs2QPl18L3RueX5UkKyezTBgLuAM2Y7W05DKCwh4gKpZhuNfOgp2bDnn7UiWZxCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T05:07:27.893743Z","bundle_sha256":"07745981e0726538ced500e7eaf61fe2443508bb91f215959e900928f6337a0c"}}