{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:DDT6NWKH6GEB3RWCDWX3OZKOIQ","short_pith_number":"pith:DDT6NWKH","canonical_record":{"source":{"id":"1210.0919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T20:27:52Z","cross_cats_sorted":[],"title_canon_sha256":"5ee0335a454e84afa0f6143c2893991e130bcb2c1733b0af3095c48497047805","abstract_canon_sha256":"ffd58f5b009e5fb1e782ea5fde6b4c50892a97352a99a8d7525476232791ebc2"},"schema_version":"1.0"},"canonical_sha256":"18e7e6d947f1881dc6c21dafb7654e441bfa7e5782262953871622ce2f86c76c","source":{"kind":"arxiv","id":"1210.0919","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0919","created_at":"2026-05-18T01:54:11Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0919v1","created_at":"2026-05-18T01:54:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0919","created_at":"2026-05-18T01:54:11Z"},{"alias_kind":"pith_short_12","alias_value":"DDT6NWKH6GEB","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"DDT6NWKH6GEB3RWC","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"DDT6NWKH","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:DDT6NWKH6GEB3RWCDWX3OZKOIQ","target":"record","payload":{"canonical_record":{"source":{"id":"1210.0919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T20:27:52Z","cross_cats_sorted":[],"title_canon_sha256":"5ee0335a454e84afa0f6143c2893991e130bcb2c1733b0af3095c48497047805","abstract_canon_sha256":"ffd58f5b009e5fb1e782ea5fde6b4c50892a97352a99a8d7525476232791ebc2"},"schema_version":"1.0"},"canonical_sha256":"18e7e6d947f1881dc6c21dafb7654e441bfa7e5782262953871622ce2f86c76c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:54:11.358111Z","signature_b64":"8HgU7/mSKKgKdvOWfEovq1W5BFiHX4pjPm36HwjwCGd9zT6x37kiNMASeUdYgLi48AfUgJeL7fdBSsAMZAWMBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18e7e6d947f1881dc6c21dafb7654e441bfa7e5782262953871622ce2f86c76c","last_reissued_at":"2026-05-18T01:54:11.357486Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:54:11.357486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.0919","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":"2026-05-18T01:54:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rv9/+7J4BzbaUDyZE3p427dMGBUwJylPbPo0rzegjtg68MV/CC4KVd0ev+s81TtPkmQycqh063XuJOaX/lRdDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T16:19:12.749464Z"},"content_sha256":"4407dc284805b1d677b05225f3a98feb73d8bf176c1d02b7997b0c00a8a119a1","schema_version":"1.0","event_id":"sha256:4407dc284805b1d677b05225f3a98feb73d8bf176c1d02b7997b0c00a8a119a1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:DDT6NWKH6GEB3RWCDWX3OZKOIQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tensor Products, Positive Linear Operators, and Delay-Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"John Mallet-Paret, Roger D. Nussbaum","submitted_at":"2012-10-02T20:27:52Z","abstract_excerpt":"We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\\dot x(t)=-\\alpha(t)x(t)-\\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\\delta\\beta(t)\\ge 0$ with $\\delta\\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coef"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0919","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":"2026-05-18T01:54:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DELZx/5qv60BSIwXcsprS/p/owmw9InrE2NocMICWxPXqQfs5vbOlyaSSxEcb1u/xaBKkme74d1EGdcLJQXJCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T16:19:12.749797Z"},"content_sha256":"1efbb79025681a9407bef452b3b7b5742f4cdf11684587058fd13da9b4f30cef","schema_version":"1.0","event_id":"sha256:1efbb79025681a9407bef452b3b7b5742f4cdf11684587058fd13da9b4f30cef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DDT6NWKH6GEB3RWCDWX3OZKOIQ/bundle.json","state_url":"https://pith.science/pith/DDT6NWKH6GEB3RWCDWX3OZKOIQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DDT6NWKH6GEB3RWCDWX3OZKOIQ/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":"2026-06-24T16:19:12Z","links":{"resolver":"https://pith.science/pith/DDT6NWKH6GEB3RWCDWX3OZKOIQ","bundle":"https://pith.science/pith/DDT6NWKH6GEB3RWCDWX3OZKOIQ/bundle.json","state":"https://pith.science/pith/DDT6NWKH6GEB3RWCDWX3OZKOIQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DDT6NWKH6GEB3RWCDWX3OZKOIQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:DDT6NWKH6GEB3RWCDWX3OZKOIQ","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":"ffd58f5b009e5fb1e782ea5fde6b4c50892a97352a99a8d7525476232791ebc2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T20:27:52Z","title_canon_sha256":"5ee0335a454e84afa0f6143c2893991e130bcb2c1733b0af3095c48497047805"},"schema_version":"1.0","source":{"id":"1210.0919","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0919","created_at":"2026-05-18T01:54:11Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0919v1","created_at":"2026-05-18T01:54:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0919","created_at":"2026-05-18T01:54:11Z"},{"alias_kind":"pith_short_12","alias_value":"DDT6NWKH6GEB","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"DDT6NWKH6GEB3RWC","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"DDT6NWKH","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:1efbb79025681a9407bef452b3b7b5742f4cdf11684587058fd13da9b4f30cef","target":"graph","created_at":"2026-05-18T01:54:11Z","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 develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\\dot x(t)=-\\alpha(t)x(t)-\\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\\delta\\beta(t)\\ge 0$ with $\\delta\\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coef","authors_text":"John Mallet-Paret, Roger D. Nussbaum","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T20:27:52Z","title":"Tensor Products, Positive Linear Operators, and Delay-Differential Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0919","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":"sha256:4407dc284805b1d677b05225f3a98feb73d8bf176c1d02b7997b0c00a8a119a1","target":"record","created_at":"2026-05-18T01:54:11Z","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":"ffd58f5b009e5fb1e782ea5fde6b4c50892a97352a99a8d7525476232791ebc2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-10-02T20:27:52Z","title_canon_sha256":"5ee0335a454e84afa0f6143c2893991e130bcb2c1733b0af3095c48497047805"},"schema_version":"1.0","source":{"id":"1210.0919","kind":"arxiv","version":1}},"canonical_sha256":"18e7e6d947f1881dc6c21dafb7654e441bfa7e5782262953871622ce2f86c76c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"18e7e6d947f1881dc6c21dafb7654e441bfa7e5782262953871622ce2f86c76c","first_computed_at":"2026-05-18T01:54:11.357486Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:54:11.357486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8HgU7/mSKKgKdvOWfEovq1W5BFiHX4pjPm36HwjwCGd9zT6x37kiNMASeUdYgLi48AfUgJeL7fdBSsAMZAWMBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:54:11.358111Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.0919","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4407dc284805b1d677b05225f3a98feb73d8bf176c1d02b7997b0c00a8a119a1","sha256:1efbb79025681a9407bef452b3b7b5742f4cdf11684587058fd13da9b4f30cef"],"state_sha256":"ce553d032c97761740d3ee808f75cd590fc58e1543c45ad30c51ff8f54061c1e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HgB+QQKJY/Cg5JC96AuAHpsNvDZ6DP/1XnILCekCT5tkIccQyy1RxvKWLuDJDjTpPqEWrnucYmyzyq7tBnkIDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T16:19:12.751580Z","bundle_sha256":"45aabf44fc9b50fe4aaf8a789cb0523e64040e12a1b32dc5bd1b8711528b4f74"}}