{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:SVO33XCWBW7E3TYQLN3BDHR5CB","short_pith_number":"pith:SVO33XCW","canonical_record":{"source":{"id":"1703.00128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-03-01T04:02:12Z","cross_cats_sorted":[],"title_canon_sha256":"87e8dce95f9d50c5e78447be6b87d014820693ed9b3e0f9b19c6ad93a08f4b91","abstract_canon_sha256":"ef1ba7f0b958f2b5019746fb19c30692214a13c9316d77059c2772f0b03a01da"},"schema_version":"1.0"},"canonical_sha256":"955dbddc560dbe4dcf105b76119e3d1060d165af0b7b17e0c9a9b536c589ec60","source":{"kind":"arxiv","id":"1703.00128","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00128","created_at":"2026-05-18T00:49:45Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00128v1","created_at":"2026-05-18T00:49:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00128","created_at":"2026-05-18T00:49:45Z"},{"alias_kind":"pith_short_12","alias_value":"SVO33XCWBW7E","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SVO33XCWBW7E3TYQ","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SVO33XCW","created_at":"2026-05-18T12:31:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:SVO33XCWBW7E3TYQLN3BDHR5CB","target":"record","payload":{"canonical_record":{"source":{"id":"1703.00128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-03-01T04:02:12Z","cross_cats_sorted":[],"title_canon_sha256":"87e8dce95f9d50c5e78447be6b87d014820693ed9b3e0f9b19c6ad93a08f4b91","abstract_canon_sha256":"ef1ba7f0b958f2b5019746fb19c30692214a13c9316d77059c2772f0b03a01da"},"schema_version":"1.0"},"canonical_sha256":"955dbddc560dbe4dcf105b76119e3d1060d165af0b7b17e0c9a9b536c589ec60","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:45.911237Z","signature_b64":"8O1uWj59t7r//8RfZMlzIqIFTD4HbP6krBPVTk6YMvMXXBBZmkVZa/87e1qlvH8vfhxNz8keLcE6269nHKcRDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"955dbddc560dbe4dcf105b76119e3d1060d165af0b7b17e0c9a9b536c589ec60","last_reissued_at":"2026-05-18T00:49:45.910570Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:45.910570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.00128","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":"2026-05-18T00:49:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BoK1fiZ3A4dfsaXmGvniqySReh4LF/43QA375t6JXoia/S9Y/O5aO1NBRpIzUKxA//0CntqDjMiUJbK/+jdYCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:26:38.877246Z"},"content_sha256":"3e00d2d5da1d6e77e20b030e8eb4d838045170f325b8c7b5951d7dc49af407b3","schema_version":"1.0","event_id":"sha256:3e00d2d5da1d6e77e20b030e8eb4d838045170f325b8c7b5951d7dc49af407b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:SVO33XCWBW7E3TYQLN3BDHR5CB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\varepsilon$-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Christian Rieger, Dinh D\\~ung, Michael Griebel, Vu Nhat Huy","submitted_at":"2017-03-01T04:02:12Z","abstract_excerpt":"In this article, we present a cost-benefit analysis of the approximation in tensor products of Hilbert spaces of Sobolev-analytic type. The Sobolev part is defined on a finite dimensional domain, whereas the analytical space is defined on an infinite dimensional domain. As main mathematical tool, we use the $\\varepsilon$-dimension of a subset in a Hilbert space. The $\\varepsilon$-dimension gives the lowest number of linear information that is needed to approximate an element from the set in the norm of the Hilbert space up to an accuracy $\\varepsilon>0$. From a practical point of view this mea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00128","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":"2026-05-18T00:49:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rHG7cjDnwIoiIdYBskvtnpcA4sSFHuUKaE0dVdGpHcCvU5qKqCh/jRr1jj/hIA/3q47SxWxT9zsoEsZAiK/PCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:26:38.877606Z"},"content_sha256":"e453a8262a3d448cd3b72e541261591dac593f8180999c2c9f602c4a52aa1106","schema_version":"1.0","event_id":"sha256:e453a8262a3d448cd3b72e541261591dac593f8180999c2c9f602c4a52aa1106"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SVO33XCWBW7E3TYQLN3BDHR5CB/bundle.json","state_url":"https://pith.science/pith/SVO33XCWBW7E3TYQLN3BDHR5CB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SVO33XCWBW7E3TYQLN3BDHR5CB/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":"2026-06-02T18:26:38Z","links":{"resolver":"https://pith.science/pith/SVO33XCWBW7E3TYQLN3BDHR5CB","bundle":"https://pith.science/pith/SVO33XCWBW7E3TYQLN3BDHR5CB/bundle.json","state":"https://pith.science/pith/SVO33XCWBW7E3TYQLN3BDHR5CB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SVO33XCWBW7E3TYQLN3BDHR5CB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:SVO33XCWBW7E3TYQLN3BDHR5CB","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":"ef1ba7f0b958f2b5019746fb19c30692214a13c9316d77059c2772f0b03a01da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-03-01T04:02:12Z","title_canon_sha256":"87e8dce95f9d50c5e78447be6b87d014820693ed9b3e0f9b19c6ad93a08f4b91"},"schema_version":"1.0","source":{"id":"1703.00128","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00128","created_at":"2026-05-18T00:49:45Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00128v1","created_at":"2026-05-18T00:49:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00128","created_at":"2026-05-18T00:49:45Z"},{"alias_kind":"pith_short_12","alias_value":"SVO33XCWBW7E","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_16","alias_value":"SVO33XCWBW7E3TYQ","created_at":"2026-05-18T12:31:43Z"},{"alias_kind":"pith_short_8","alias_value":"SVO33XCW","created_at":"2026-05-18T12:31:43Z"}],"graph_snapshots":[{"event_id":"sha256:e453a8262a3d448cd3b72e541261591dac593f8180999c2c9f602c4a52aa1106","target":"graph","created_at":"2026-05-18T00:49:45Z","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 article, we present a cost-benefit analysis of the approximation in tensor products of Hilbert spaces of Sobolev-analytic type. The Sobolev part is defined on a finite dimensional domain, whereas the analytical space is defined on an infinite dimensional domain. As main mathematical tool, we use the $\\varepsilon$-dimension of a subset in a Hilbert space. The $\\varepsilon$-dimension gives the lowest number of linear information that is needed to approximate an element from the set in the norm of the Hilbert space up to an accuracy $\\varepsilon>0$. From a practical point of view this mea","authors_text":"Christian Rieger, Dinh D\\~ung, Michael Griebel, Vu Nhat Huy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-03-01T04:02:12Z","title":"$\\varepsilon$-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00128","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":"sha256:3e00d2d5da1d6e77e20b030e8eb4d838045170f325b8c7b5951d7dc49af407b3","target":"record","created_at":"2026-05-18T00:49:45Z","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":"ef1ba7f0b958f2b5019746fb19c30692214a13c9316d77059c2772f0b03a01da","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-03-01T04:02:12Z","title_canon_sha256":"87e8dce95f9d50c5e78447be6b87d014820693ed9b3e0f9b19c6ad93a08f4b91"},"schema_version":"1.0","source":{"id":"1703.00128","kind":"arxiv","version":1}},"canonical_sha256":"955dbddc560dbe4dcf105b76119e3d1060d165af0b7b17e0c9a9b536c589ec60","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"955dbddc560dbe4dcf105b76119e3d1060d165af0b7b17e0c9a9b536c589ec60","first_computed_at":"2026-05-18T00:49:45.910570Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:45.910570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8O1uWj59t7r//8RfZMlzIqIFTD4HbP6krBPVTk6YMvMXXBBZmkVZa/87e1qlvH8vfhxNz8keLcE6269nHKcRDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:45.911237Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.00128","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e00d2d5da1d6e77e20b030e8eb4d838045170f325b8c7b5951d7dc49af407b3","sha256:e453a8262a3d448cd3b72e541261591dac593f8180999c2c9f602c4a52aa1106"],"state_sha256":"08c955ff387a52221891bee1bca1724fd4098210ee6910003c579b76c2b0b0fd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zAzWAIDyA9AyBCAJToHfTXI8Y3yxzY41yk8/9hwrxq49uZKdAmCroKCoidHm1+NJ+t+ENovyCflR66tuplBrCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T18:26:38.879456Z","bundle_sha256":"a65e8d8dabb1e5355162881ff7c5c6e686cbacb6a05765a18ac40366dc39bf4d"}}