{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:TC4QHDPIHZAVQLZEFWYMB67T3O","short_pith_number":"pith:TC4QHDPI","canonical_record":{"source":{"id":"1904.06569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-13T17:07:04Z","cross_cats_sorted":["cs.NA","stat.ME"],"title_canon_sha256":"0a0df4b9fd188be8fdd8f5865fcef765878e89652544026ae3edb297d30aae2c","abstract_canon_sha256":"6f7aaf3f495f83912402bac906ac02b426182d24c596a46b88804122272f0adf"},"schema_version":"1.0"},"canonical_sha256":"98b9038de83e41582f242db0c0fbf3dba5bd288733200092f824f54858c64b87","source":{"kind":"arxiv","id":"1904.06569","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.06569","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"arxiv_version","alias_value":"1904.06569v1","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.06569","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"pith_short_12","alias_value":"TC4QHDPIHZAV","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"pith_short_16","alias_value":"TC4QHDPIHZAVQLZE","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"pith_short_8","alias_value":"TC4QHDPI","created_at":"2026-06-04T20:14:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:TC4QHDPIHZAVQLZEFWYMB67T3O","target":"record","payload":{"canonical_record":{"source":{"id":"1904.06569","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-13T17:07:04Z","cross_cats_sorted":["cs.NA","stat.ME"],"title_canon_sha256":"0a0df4b9fd188be8fdd8f5865fcef765878e89652544026ae3edb297d30aae2c","abstract_canon_sha256":"6f7aaf3f495f83912402bac906ac02b426182d24c596a46b88804122272f0adf"},"schema_version":"1.0"},"canonical_sha256":"98b9038de83e41582f242db0c0fbf3dba5bd288733200092f824f54858c64b87","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T20:14:24.988837Z","signature_b64":"NlHtvHxnMU//Idzrwlp3EMvhf7OvCBal23beGg1Sjy36vY7wDGd2AqeJEIsNPSfH3sBrpKCSirS6aHVkyy8IBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98b9038de83e41582f242db0c0fbf3dba5bd288733200092f824f54858c64b87","last_reissued_at":"2026-06-04T20:14:24.988308Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T20:14:24.988308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.06569","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":"2026-06-04T20:14:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DjBhH9a5SmOYWBRtgXGL897am4VnoPjJLY/FBZe2lYyLcneqEDXM83TuziYRYlSqdfxfHEjysKdKF6s5qNRkBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T04:21:43.074760Z"},"content_sha256":"f7d82774425afeeb4d011fb2f9275ad432f220e5b11e31715431d0f02045d410","schema_version":"1.0","event_id":"sha256:f7d82774425afeeb4d011fb2f9275ad432f220e5b11e31715431d0f02045d410"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:TC4QHDPIHZAVQLZEFWYMB67T3O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity and convergence analysis in Sobolev and H\\\"older spaces for generalized Whittle-Mat\\'ern fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","stat.ME"],"primary_cat":"math.NA","authors_text":"Kristin Kirchner, Sonja G. Cox","submitted_at":"2019-04-13T17:07:04Z","abstract_excerpt":"We analyze several Galerkin approximations of a Gaussian random field $\\mathcal{Z}\\colon\\mathcal{D}\\times\\Omega\\to\\mathbb{R}$ indexed by a Euclidean domain $\\mathcal{D}\\subset\\mathbb{R}^d$ whose covariance structure is determined by a negative fractional power $L^{-2\\beta}$ of a second-order elliptic differential operator $L:= -\\nabla\\cdot(A\\nabla) + \\kappa^2$. Under minimal assumptions on the domain $\\mathcal{D}$, the coefficients $A\\colon\\mathcal{D}\\to\\mathbb{R}^{d\\times d}$, $\\kappa\\colon\\mathcal{D}\\to\\mathbb{R}$, and the fractional exponent $\\beta>0$, we prove convergence in $L_q(\\Omega; H"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06569","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1904.06569/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-04T20:14:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tCz5ZNkoD7x+CfOMV1ftexc1DNKtz8gdktn9FMJQqU+53Df17Wk2nhd9lywecsT2+g3884cfdQkp+SDiGj+KAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T04:21:43.075145Z"},"content_sha256":"b1026c41c12ce9d3dd6226b29c50caa34e0f12a7e62fc35e3e193a2060f48588","schema_version":"1.0","event_id":"sha256:b1026c41c12ce9d3dd6226b29c50caa34e0f12a7e62fc35e3e193a2060f48588"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TC4QHDPIHZAVQLZEFWYMB67T3O/bundle.json","state_url":"https://pith.science/pith/TC4QHDPIHZAVQLZEFWYMB67T3O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TC4QHDPIHZAVQLZEFWYMB67T3O/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":"2026-07-02T04:21:43Z","links":{"resolver":"https://pith.science/pith/TC4QHDPIHZAVQLZEFWYMB67T3O","bundle":"https://pith.science/pith/TC4QHDPIHZAVQLZEFWYMB67T3O/bundle.json","state":"https://pith.science/pith/TC4QHDPIHZAVQLZEFWYMB67T3O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TC4QHDPIHZAVQLZEFWYMB67T3O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:TC4QHDPIHZAVQLZEFWYMB67T3O","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":"6f7aaf3f495f83912402bac906ac02b426182d24c596a46b88804122272f0adf","cross_cats_sorted":["cs.NA","stat.ME"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-13T17:07:04Z","title_canon_sha256":"0a0df4b9fd188be8fdd8f5865fcef765878e89652544026ae3edb297d30aae2c"},"schema_version":"1.0","source":{"id":"1904.06569","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.06569","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"arxiv_version","alias_value":"1904.06569v1","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.06569","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"pith_short_12","alias_value":"TC4QHDPIHZAV","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"pith_short_16","alias_value":"TC4QHDPIHZAVQLZE","created_at":"2026-06-04T20:14:24Z"},{"alias_kind":"pith_short_8","alias_value":"TC4QHDPI","created_at":"2026-06-04T20:14:24Z"}],"graph_snapshots":[{"event_id":"sha256:b1026c41c12ce9d3dd6226b29c50caa34e0f12a7e62fc35e3e193a2060f48588","target":"graph","created_at":"2026-06-04T20:14:24Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1904.06569/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We analyze several Galerkin approximations of a Gaussian random field $\\mathcal{Z}\\colon\\mathcal{D}\\times\\Omega\\to\\mathbb{R}$ indexed by a Euclidean domain $\\mathcal{D}\\subset\\mathbb{R}^d$ whose covariance structure is determined by a negative fractional power $L^{-2\\beta}$ of a second-order elliptic differential operator $L:= -\\nabla\\cdot(A\\nabla) + \\kappa^2$. Under minimal assumptions on the domain $\\mathcal{D}$, the coefficients $A\\colon\\mathcal{D}\\to\\mathbb{R}^{d\\times d}$, $\\kappa\\colon\\mathcal{D}\\to\\mathbb{R}$, and the fractional exponent $\\beta>0$, we prove convergence in $L_q(\\Omega; H","authors_text":"Kristin Kirchner, Sonja G. Cox","cross_cats":["cs.NA","stat.ME"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-13T17:07:04Z","title":"Regularity and convergence analysis in Sobolev and H\\\"older spaces for generalized Whittle-Mat\\'ern fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06569","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:f7d82774425afeeb4d011fb2f9275ad432f220e5b11e31715431d0f02045d410","target":"record","created_at":"2026-06-04T20:14:24Z","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":"6f7aaf3f495f83912402bac906ac02b426182d24c596a46b88804122272f0adf","cross_cats_sorted":["cs.NA","stat.ME"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-04-13T17:07:04Z","title_canon_sha256":"0a0df4b9fd188be8fdd8f5865fcef765878e89652544026ae3edb297d30aae2c"},"schema_version":"1.0","source":{"id":"1904.06569","kind":"arxiv","version":1}},"canonical_sha256":"98b9038de83e41582f242db0c0fbf3dba5bd288733200092f824f54858c64b87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98b9038de83e41582f242db0c0fbf3dba5bd288733200092f824f54858c64b87","first_computed_at":"2026-06-04T20:14:24.988308Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T20:14:24.988308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NlHtvHxnMU//Idzrwlp3EMvhf7OvCBal23beGg1Sjy36vY7wDGd2AqeJEIsNPSfH3sBrpKCSirS6aHVkyy8IBQ==","signature_status":"signed_v1","signed_at":"2026-06-04T20:14:24.988837Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.06569","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7d82774425afeeb4d011fb2f9275ad432f220e5b11e31715431d0f02045d410","sha256:b1026c41c12ce9d3dd6226b29c50caa34e0f12a7e62fc35e3e193a2060f48588"],"state_sha256":"710e01420538a6b009d83579ac58712c70df8375cee595819cc0fe1280a294a6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Scu33ZAGbboXjIhenMgp1nj0OOzRPTPn7Vl3PZd0iR298PRUEJhvxNXkYdSDNK5I7MYRqS369eWMNu62OxuxAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T04:21:43.077191Z","bundle_sha256":"4d3b05b15478667ca08846743305c53a25f453264c4d76f727b5413fbfa45fea"}}