{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FN6O2D2OLXBCPWLWLWO6WSDIKY","short_pith_number":"pith:FN6O2D2O","canonical_record":{"source":{"id":"1610.08608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-10-27T04:09:06Z","cross_cats_sorted":[],"title_canon_sha256":"b6e244f8d65d843e27cdd6965f070f42774927d851f336eaa8b7bfbf0eb7c499","abstract_canon_sha256":"b2c75e2295927d43b73dc296c56a1b18ff91662a7098304df669bb43c96e524c"},"schema_version":"1.0"},"canonical_sha256":"2b7ced0f4e5dc227d9765d9deb48685602a765a9997e320cb4816003ef941c04","source":{"kind":"arxiv","id":"1610.08608","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.08608","created_at":"2026-05-18T00:33:22Z"},{"alias_kind":"arxiv_version","alias_value":"1610.08608v1","created_at":"2026-05-18T00:33:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.08608","created_at":"2026-05-18T00:33:22Z"},{"alias_kind":"pith_short_12","alias_value":"FN6O2D2OLXBC","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FN6O2D2OLXBCPWLW","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FN6O2D2O","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FN6O2D2OLXBCPWLWLWO6WSDIKY","target":"record","payload":{"canonical_record":{"source":{"id":"1610.08608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-10-27T04:09:06Z","cross_cats_sorted":[],"title_canon_sha256":"b6e244f8d65d843e27cdd6965f070f42774927d851f336eaa8b7bfbf0eb7c499","abstract_canon_sha256":"b2c75e2295927d43b73dc296c56a1b18ff91662a7098304df669bb43c96e524c"},"schema_version":"1.0"},"canonical_sha256":"2b7ced0f4e5dc227d9765d9deb48685602a765a9997e320cb4816003ef941c04","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:22.463709Z","signature_b64":"A7hrvk8qfp0pUugHoFca83qXazAjzxWYBH5HCOIAPyqnvY9svjkuX6L+c9+4FPn8XBH7jEf0hbXPrWTz1uK5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b7ced0f4e5dc227d9765d9deb48685602a765a9997e320cb4816003ef941c04","last_reissued_at":"2026-05-18T00:33:22.463098Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:22.463098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.08608","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":"2026-05-18T00:33:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iYO3BiM3ghJK9ouKj7aXs4nzA4TmERQB7SNmEdvjXxxFAsekRlnRN3fp1PA8DL6CdNBXOQmWUXqe8cMrMJP4DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T01:11:57.045038Z"},"content_sha256":"a92d577c6b8b18b633165289b347895e72f69b187676a873f68f98b296e78c8c","schema_version":"1.0","event_id":"sha256:a92d577c6b8b18b633165289b347895e72f69b187676a873f68f98b296e78c8c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FN6O2D2OLXBCPWLWLWO6WSDIKY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Petrov-Galerkin Spectral Element Method for Fractional Elliptic Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Ehsan Kharazmi, George Em Karniadakis, Mohsen Zayernouri","submitted_at":"2016-10-27T04:09:06Z","abstract_excerpt":"We develop a new $C^{\\,0}$-continuous Petrov-Galerkin spectral element method for one-dimensional fractional elliptic problems of the form ${}_{0}{\\mathcal{D}}_{x}^{\\alpha} u(x) - \\lambda u(x) = f(x)$, $\\alpha \\in (1,2]$, subject to homogeneous boundary conditions. We employ the standard (modal) spectral element bases and the Jacobi poly-fractonomials as the test functions [1]. We formulate a new procedure for assembling the global linear system from elemental (local) mass and stiffness matrices. The Petrov-Galerkin formulation requires performing elemental (local) construction of mass and sti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.08608","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":"2026-05-18T00:33:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6cdEWvICj4I+RuXyxyv3W1eQjRX6D3nmB+rO1NjhvCdT2I8fUqzt7ujM5v5RbWZnnlHHy/EHv1W+jVaAWMLJBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T01:11:57.045481Z"},"content_sha256":"d8e0ac88b366752bce173a161e13ceeb9d392f978e1bd7e5c9b9c64068128fd2","schema_version":"1.0","event_id":"sha256:d8e0ac88b366752bce173a161e13ceeb9d392f978e1bd7e5c9b9c64068128fd2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FN6O2D2OLXBCPWLWLWO6WSDIKY/bundle.json","state_url":"https://pith.science/pith/FN6O2D2OLXBCPWLWLWO6WSDIKY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FN6O2D2OLXBCPWLWLWO6WSDIKY/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":"2026-06-06T01:11:57Z","links":{"resolver":"https://pith.science/pith/FN6O2D2OLXBCPWLWLWO6WSDIKY","bundle":"https://pith.science/pith/FN6O2D2OLXBCPWLWLWO6WSDIKY/bundle.json","state":"https://pith.science/pith/FN6O2D2OLXBCPWLWLWO6WSDIKY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FN6O2D2OLXBCPWLWLWO6WSDIKY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FN6O2D2OLXBCPWLWLWO6WSDIKY","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":"b2c75e2295927d43b73dc296c56a1b18ff91662a7098304df669bb43c96e524c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-10-27T04:09:06Z","title_canon_sha256":"b6e244f8d65d843e27cdd6965f070f42774927d851f336eaa8b7bfbf0eb7c499"},"schema_version":"1.0","source":{"id":"1610.08608","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.08608","created_at":"2026-05-18T00:33:22Z"},{"alias_kind":"arxiv_version","alias_value":"1610.08608v1","created_at":"2026-05-18T00:33:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.08608","created_at":"2026-05-18T00:33:22Z"},{"alias_kind":"pith_short_12","alias_value":"FN6O2D2OLXBC","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FN6O2D2OLXBCPWLW","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FN6O2D2O","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:d8e0ac88b366752bce173a161e13ceeb9d392f978e1bd7e5c9b9c64068128fd2","target":"graph","created_at":"2026-05-18T00:33:22Z","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 a new $C^{\\,0}$-continuous Petrov-Galerkin spectral element method for one-dimensional fractional elliptic problems of the form ${}_{0}{\\mathcal{D}}_{x}^{\\alpha} u(x) - \\lambda u(x) = f(x)$, $\\alpha \\in (1,2]$, subject to homogeneous boundary conditions. We employ the standard (modal) spectral element bases and the Jacobi poly-fractonomials as the test functions [1]. We formulate a new procedure for assembling the global linear system from elemental (local) mass and stiffness matrices. The Petrov-Galerkin formulation requires performing elemental (local) construction of mass and sti","authors_text":"Ehsan Kharazmi, George Em Karniadakis, Mohsen Zayernouri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-10-27T04:09:06Z","title":"A Petrov-Galerkin Spectral Element Method for Fractional Elliptic Problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.08608","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:a92d577c6b8b18b633165289b347895e72f69b187676a873f68f98b296e78c8c","target":"record","created_at":"2026-05-18T00:33:22Z","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":"b2c75e2295927d43b73dc296c56a1b18ff91662a7098304df669bb43c96e524c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-10-27T04:09:06Z","title_canon_sha256":"b6e244f8d65d843e27cdd6965f070f42774927d851f336eaa8b7bfbf0eb7c499"},"schema_version":"1.0","source":{"id":"1610.08608","kind":"arxiv","version":1}},"canonical_sha256":"2b7ced0f4e5dc227d9765d9deb48685602a765a9997e320cb4816003ef941c04","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b7ced0f4e5dc227d9765d9deb48685602a765a9997e320cb4816003ef941c04","first_computed_at":"2026-05-18T00:33:22.463098Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:22.463098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A7hrvk8qfp0pUugHoFca83qXazAjzxWYBH5HCOIAPyqnvY9svjkuX6L+c9+4FPn8XBH7jEf0hbXPrWTz1uK5Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:22.463709Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.08608","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a92d577c6b8b18b633165289b347895e72f69b187676a873f68f98b296e78c8c","sha256:d8e0ac88b366752bce173a161e13ceeb9d392f978e1bd7e5c9b9c64068128fd2"],"state_sha256":"bee39db58a0f0e1edbc2a3d301d0dc336d152748efc9551a94efe568a627e166"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xY7sBguFuobhLRDHY9Kr8guIw1uPc+WmJzd6l1qYWo2Q89lOq+tvngkjSI1c2jUHLzuybNMEJFphmYMNvKgECg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T01:11:57.048625Z","bundle_sha256":"f405f2fd683058875deac233a4401d84d99c24245813ce726deb1375c0ab91e0"}}