{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:IXW24UV7MIDUXSX4O2SO5PEM6G","short_pith_number":"pith:IXW24UV7","canonical_record":{"source":{"id":"1506.04180","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-06-12T20:48:27Z","cross_cats_sorted":[],"title_canon_sha256":"7725f0b1c5f9ff16fe0faf6d34362f7861c06ded2b03269aa79bc841e8f357c9","abstract_canon_sha256":"d88d7057c2bc1c71beb28a5642b0280b7e62d73c4182630adfe88942796ea783"},"schema_version":"1.0"},"canonical_sha256":"45edae52bf62074bcafc76a4eebc8cf1a4f22c89609ead20a05524f42f6aceb9","source":{"kind":"arxiv","id":"1506.04180","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04180","created_at":"2026-05-18T01:13:38Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04180v2","created_at":"2026-05-18T01:13:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04180","created_at":"2026-05-18T01:13:38Z"},{"alias_kind":"pith_short_12","alias_value":"IXW24UV7MIDU","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"IXW24UV7MIDUXSX4","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"IXW24UV7","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:IXW24UV7MIDUXSX4O2SO5PEM6G","target":"record","payload":{"canonical_record":{"source":{"id":"1506.04180","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-06-12T20:48:27Z","cross_cats_sorted":[],"title_canon_sha256":"7725f0b1c5f9ff16fe0faf6d34362f7861c06ded2b03269aa79bc841e8f357c9","abstract_canon_sha256":"d88d7057c2bc1c71beb28a5642b0280b7e62d73c4182630adfe88942796ea783"},"schema_version":"1.0"},"canonical_sha256":"45edae52bf62074bcafc76a4eebc8cf1a4f22c89609ead20a05524f42f6aceb9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:38.166604Z","signature_b64":"tnIf02k8lmUbFZE/w6hu9qvRo9jIS4qyomrm7WIT7D9TMVKza+wr5JjijsV2YeC2TD47gBf9i9xogoae0IsBCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45edae52bf62074bcafc76a4eebc8cf1a4f22c89609ead20a05524f42f6aceb9","last_reissued_at":"2026-05-18T01:13:38.165862Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:38.165862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.04180","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":"2026-05-18T01:13:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CfNmEVeSCZYo9p90LX9thdCbkfwEDfMW1fKrMM9nXR/oWKLeJ5K1lY95GgJSe0Z7jt5eAql95JeE8ufwGtw6BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:27:16.348838Z"},"content_sha256":"1f1c5bc144d44f4f6604c83553a1a313f1e0b185d7c984ad3dad63d7df7e44ae","schema_version":"1.0","event_id":"sha256:1f1c5bc144d44f4f6604c83553a1a313f1e0b185d7c984ad3dad63d7df7e44ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:IXW24UV7MIDUXSX4O2SO5PEM6G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the $\\eta$-function for bisingular pseudodifferential operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Karsten Bohlen","submitted_at":"2015-06-12T20:48:27Z","abstract_excerpt":"In this work we consider the $\\eta$-invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the $\\eta$-function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the $K$-theory of the algebra of $0$-order (global) bisingular operators. With these preparations we establish the regularity properties of the $\\eta$-function at the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04180","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":"2026-05-18T01:13:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8bDTs5JIcB1E/Fw6PrDulpQ4jJ7vY8gmGgPzSrL2rqiC1i/J6DA0GB1uXccxQHEI/iS9s1z2B5zFQb3TnQXdAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:27:16.349185Z"},"content_sha256":"74ffed0c8b0437c9658f33a26c28ab5a114029b071aeb807bd09d8ff5c5f2531","schema_version":"1.0","event_id":"sha256:74ffed0c8b0437c9658f33a26c28ab5a114029b071aeb807bd09d8ff5c5f2531"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IXW24UV7MIDUXSX4O2SO5PEM6G/bundle.json","state_url":"https://pith.science/pith/IXW24UV7MIDUXSX4O2SO5PEM6G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IXW24UV7MIDUXSX4O2SO5PEM6G/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":"2026-06-26T13:27:16Z","links":{"resolver":"https://pith.science/pith/IXW24UV7MIDUXSX4O2SO5PEM6G","bundle":"https://pith.science/pith/IXW24UV7MIDUXSX4O2SO5PEM6G/bundle.json","state":"https://pith.science/pith/IXW24UV7MIDUXSX4O2SO5PEM6G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IXW24UV7MIDUXSX4O2SO5PEM6G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IXW24UV7MIDUXSX4O2SO5PEM6G","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":"d88d7057c2bc1c71beb28a5642b0280b7e62d73c4182630adfe88942796ea783","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-06-12T20:48:27Z","title_canon_sha256":"7725f0b1c5f9ff16fe0faf6d34362f7861c06ded2b03269aa79bc841e8f357c9"},"schema_version":"1.0","source":{"id":"1506.04180","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04180","created_at":"2026-05-18T01:13:38Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04180v2","created_at":"2026-05-18T01:13:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04180","created_at":"2026-05-18T01:13:38Z"},{"alias_kind":"pith_short_12","alias_value":"IXW24UV7MIDU","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"IXW24UV7MIDUXSX4","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"IXW24UV7","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:74ffed0c8b0437c9658f33a26c28ab5a114029b071aeb807bd09d8ff5c5f2531","target":"graph","created_at":"2026-05-18T01:13:38Z","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":"In this work we consider the $\\eta$-invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the $\\eta$-function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the $K$-theory of the algebra of $0$-order (global) bisingular operators. With these preparations we establish the regularity properties of the $\\eta$-function at the ","authors_text":"Karsten Bohlen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-06-12T20:48:27Z","title":"On the $\\eta$-function for bisingular pseudodifferential operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04180","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":"sha256:1f1c5bc144d44f4f6604c83553a1a313f1e0b185d7c984ad3dad63d7df7e44ae","target":"record","created_at":"2026-05-18T01:13:38Z","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":"d88d7057c2bc1c71beb28a5642b0280b7e62d73c4182630adfe88942796ea783","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-06-12T20:48:27Z","title_canon_sha256":"7725f0b1c5f9ff16fe0faf6d34362f7861c06ded2b03269aa79bc841e8f357c9"},"schema_version":"1.0","source":{"id":"1506.04180","kind":"arxiv","version":2}},"canonical_sha256":"45edae52bf62074bcafc76a4eebc8cf1a4f22c89609ead20a05524f42f6aceb9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45edae52bf62074bcafc76a4eebc8cf1a4f22c89609ead20a05524f42f6aceb9","first_computed_at":"2026-05-18T01:13:38.165862Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:38.165862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tnIf02k8lmUbFZE/w6hu9qvRo9jIS4qyomrm7WIT7D9TMVKza+wr5JjijsV2YeC2TD47gBf9i9xogoae0IsBCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:38.166604Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.04180","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1f1c5bc144d44f4f6604c83553a1a313f1e0b185d7c984ad3dad63d7df7e44ae","sha256:74ffed0c8b0437c9658f33a26c28ab5a114029b071aeb807bd09d8ff5c5f2531"],"state_sha256":"9ecb2ff26dfa4acae164689929a4e44d86fadb182522df95b3e213fb30e57697"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6yd8sixgTtYfAPDc/pNGQEVn78RFrti+IQ3vpdFF9EZI2AysqdqwE/m514XKCLJoBA5rf3eOmS6ywqKMM8rMBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T13:27:16.351227Z","bundle_sha256":"ab9e0f33fe5bd3b565b5ac1ea16599e79693c511d1ac425f6b0a366c4edb1ece"}}