{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:HSWIV4D5PBT5EFW47Q4MDPQRMM","short_pith_number":"pith:HSWIV4D5","canonical_record":{"source":{"id":"1307.1043","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-03T15:23:06Z","cross_cats_sorted":["math.DS","math.SP"],"title_canon_sha256":"470d560eb0b185a9eed4dcfccb5c32192fe07a620bb722af171d068ea2f4bd5b","abstract_canon_sha256":"281a2069d694f3eb054d1a1d1448b4c06dfe7227f0ef9b5fedbc49948277d995"},"schema_version":"1.0"},"canonical_sha256":"3cac8af07d7867d216dcfc38c1be116331c26ec591e144030aa3a2ccb21fda1e","source":{"kind":"arxiv","id":"1307.1043","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.1043","created_at":"2026-05-18T00:51:28Z"},{"alias_kind":"arxiv_version","alias_value":"1307.1043v1","created_at":"2026-05-18T00:51:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1043","created_at":"2026-05-18T00:51:28Z"},{"alias_kind":"pith_short_12","alias_value":"HSWIV4D5PBT5","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HSWIV4D5PBT5EFW4","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HSWIV4D5","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:HSWIV4D5PBT5EFW47Q4MDPQRMM","target":"record","payload":{"canonical_record":{"source":{"id":"1307.1043","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-03T15:23:06Z","cross_cats_sorted":["math.DS","math.SP"],"title_canon_sha256":"470d560eb0b185a9eed4dcfccb5c32192fe07a620bb722af171d068ea2f4bd5b","abstract_canon_sha256":"281a2069d694f3eb054d1a1d1448b4c06dfe7227f0ef9b5fedbc49948277d995"},"schema_version":"1.0"},"canonical_sha256":"3cac8af07d7867d216dcfc38c1be116331c26ec591e144030aa3a2ccb21fda1e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:28.871523Z","signature_b64":"vGkAY25oGZFX4slsLDGDYrC1qAL/izK5ll3B5vSNOV9tvzvjS3IIYUTr9DBPTKrlzULbBIBApxrYBCsWoGRDAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3cac8af07d7867d216dcfc38c1be116331c26ec591e144030aa3a2ccb21fda1e","last_reissued_at":"2026-05-18T00:51:28.870972Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:28.870972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.1043","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":"2026-05-18T00:51:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vzA5ucrSiYmKr/mQ3I4CsvwDgaARx95RyXBOtgicwVf+KpFC6NPovkdc6VZ8swcx3CMJSHd0avvfK0bKgDtXCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T14:42:25.898202Z"},"content_sha256":"e32a74ff4633090c00d3af69e90c1c883fe7eacb1e21615cc86e596688a9b58b","schema_version":"1.0","event_id":"sha256:e32a74ff4633090c00d3af69e90c1c883fe7eacb1e21615cc86e596688a9b58b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:HSWIV4D5PBT5EFW47Q4MDPQRMM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.SP"],"primary_cat":"math.FA","authors_text":"Jacobo Pejsachowicz, Nils Waterstraat","submitted_at":"2013-07-03T15:23:06Z","abstract_excerpt":"Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1043","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":"2026-05-18T00:51:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qRsJIJkqZg9PlZAUtM3mF3yD4UqZv6C/m5sI968g33UppNAEdsiywQ5+r0UdOWNvaqPI8FtcFIKW7FfP3TBvBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T14:42:25.898569Z"},"content_sha256":"36bcb4d50dccfe02ee9de5f3566cb1d5e1d7bc6762d09bca02a902081ec54d33","schema_version":"1.0","event_id":"sha256:36bcb4d50dccfe02ee9de5f3566cb1d5e1d7bc6762d09bca02a902081ec54d33"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HSWIV4D5PBT5EFW47Q4MDPQRMM/bundle.json","state_url":"https://pith.science/pith/HSWIV4D5PBT5EFW47Q4MDPQRMM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HSWIV4D5PBT5EFW47Q4MDPQRMM/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":"2026-07-02T14:42:25Z","links":{"resolver":"https://pith.science/pith/HSWIV4D5PBT5EFW47Q4MDPQRMM","bundle":"https://pith.science/pith/HSWIV4D5PBT5EFW47Q4MDPQRMM/bundle.json","state":"https://pith.science/pith/HSWIV4D5PBT5EFW47Q4MDPQRMM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HSWIV4D5PBT5EFW47Q4MDPQRMM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HSWIV4D5PBT5EFW47Q4MDPQRMM","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":"281a2069d694f3eb054d1a1d1448b4c06dfe7227f0ef9b5fedbc49948277d995","cross_cats_sorted":["math.DS","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-03T15:23:06Z","title_canon_sha256":"470d560eb0b185a9eed4dcfccb5c32192fe07a620bb722af171d068ea2f4bd5b"},"schema_version":"1.0","source":{"id":"1307.1043","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.1043","created_at":"2026-05-18T00:51:28Z"},{"alias_kind":"arxiv_version","alias_value":"1307.1043v1","created_at":"2026-05-18T00:51:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1043","created_at":"2026-05-18T00:51:28Z"},{"alias_kind":"pith_short_12","alias_value":"HSWIV4D5PBT5","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HSWIV4D5PBT5EFW4","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HSWIV4D5","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:36bcb4d50dccfe02ee9de5f3566cb1d5e1d7bc6762d09bca02a902081ec54d33","target":"graph","created_at":"2026-05-18T00:51:28Z","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":"Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for ","authors_text":"Jacobo Pejsachowicz, Nils Waterstraat","cross_cats":["math.DS","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-03T15:23:06Z","title":"Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1043","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:e32a74ff4633090c00d3af69e90c1c883fe7eacb1e21615cc86e596688a9b58b","target":"record","created_at":"2026-05-18T00:51:28Z","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":"281a2069d694f3eb054d1a1d1448b4c06dfe7227f0ef9b5fedbc49948277d995","cross_cats_sorted":["math.DS","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-03T15:23:06Z","title_canon_sha256":"470d560eb0b185a9eed4dcfccb5c32192fe07a620bb722af171d068ea2f4bd5b"},"schema_version":"1.0","source":{"id":"1307.1043","kind":"arxiv","version":1}},"canonical_sha256":"3cac8af07d7867d216dcfc38c1be116331c26ec591e144030aa3a2ccb21fda1e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3cac8af07d7867d216dcfc38c1be116331c26ec591e144030aa3a2ccb21fda1e","first_computed_at":"2026-05-18T00:51:28.870972Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:28.870972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vGkAY25oGZFX4slsLDGDYrC1qAL/izK5ll3B5vSNOV9tvzvjS3IIYUTr9DBPTKrlzULbBIBApxrYBCsWoGRDAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:28.871523Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.1043","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e32a74ff4633090c00d3af69e90c1c883fe7eacb1e21615cc86e596688a9b58b","sha256:36bcb4d50dccfe02ee9de5f3566cb1d5e1d7bc6762d09bca02a902081ec54d33"],"state_sha256":"5b6ce6bb5990c7bd41265de5d3ee5bbaad0e13d778696723b0f7891021b1b0d6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bKltu+zT6C6/RDZP6lCQ56atkzOuIu7ebKNuhwFjE9T+QkgUzV8TfiMvj6BbPickBOKpgaKyfZIVEoK5whyCBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T14:42:25.900447Z","bundle_sha256":"160cab5e37235631230feffb9376ad71c8c52f913c6dbcbe00b6b5ddc7528dfc"}}