{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:PM5LY3EKVN55YZZZU32OOKAYTB","short_pith_number":"pith:PM5LY3EK","canonical_record":{"source":{"id":"1706.09360","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-06-28T16:52:32Z","cross_cats_sorted":[],"title_canon_sha256":"b6abf0af0093958fe500577097e4bd8e1c2fa5543e2ceece34432af6439b6a54","abstract_canon_sha256":"69a232d0b96419433263a7877d5a3a14a27845ef53e52552192ce299c688b70a"},"schema_version":"1.0"},"canonical_sha256":"7b3abc6c8aab7bdc6739a6f4e72818986e3941b421dd4792b1f3e2116b59bc47","source":{"kind":"arxiv","id":"1706.09360","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.09360","created_at":"2026-05-18T00:41:18Z"},{"alias_kind":"arxiv_version","alias_value":"1706.09360v1","created_at":"2026-05-18T00:41:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.09360","created_at":"2026-05-18T00:41:18Z"},{"alias_kind":"pith_short_12","alias_value":"PM5LY3EKVN55","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PM5LY3EKVN55YZZZ","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PM5LY3EK","created_at":"2026-05-18T12:31:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:PM5LY3EKVN55YZZZU32OOKAYTB","target":"record","payload":{"canonical_record":{"source":{"id":"1706.09360","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-06-28T16:52:32Z","cross_cats_sorted":[],"title_canon_sha256":"b6abf0af0093958fe500577097e4bd8e1c2fa5543e2ceece34432af6439b6a54","abstract_canon_sha256":"69a232d0b96419433263a7877d5a3a14a27845ef53e52552192ce299c688b70a"},"schema_version":"1.0"},"canonical_sha256":"7b3abc6c8aab7bdc6739a6f4e72818986e3941b421dd4792b1f3e2116b59bc47","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:18.875956Z","signature_b64":"RyHRa/EMnkRRpJFqCCU8/Lorn1i2qydMWdcy1nXIxQIpJ+3Q18B+H7i4Q+OSvFQNhk4CVo6XZUdX8EuOByruAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b3abc6c8aab7bdc6739a6f4e72818986e3941b421dd4792b1f3e2116b59bc47","last_reissued_at":"2026-05-18T00:41:18.875213Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:18.875213Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.09360","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":"2026-05-18T00:41:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BefRNq6i8EWjn61wTNePp+a+S35RWRcVPE7Q287Vd679JbyNFW8T9sNd04uSoiyX+AvywtBUC1+sq8qi6Mu3BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T10:51:18.971796Z"},"content_sha256":"e400b374db461d70d0a912bf53b6e601a30e2e125e77e6da67850dd326b719ce","schema_version":"1.0","event_id":"sha256:e400b374db461d70d0a912bf53b6e601a30e2e125e77e6da67850dd326b719ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:PM5LY3EKVN55YZZZU32OOKAYTB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kernel-based Discretisation for Solving Matrix-Valued PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Holger Wendland, Peter Giesl","submitted_at":"2017-06-28T16:52:32Z","abstract_excerpt":"In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09360","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":"2026-05-18T00:41:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CQV+IfokBDxAD8+p/pt6gngj6t4NwSVSO9C/I2taInfkBaRlGtG3Ghsrlib0e26NNODZeEl4YVL9lQXU4KCuBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T10:51:18.972145Z"},"content_sha256":"ec3c577a7f28b2eb7f14587f6cfdb73b10d3147a2079323537f5d599eb1a9639","schema_version":"1.0","event_id":"sha256:ec3c577a7f28b2eb7f14587f6cfdb73b10d3147a2079323537f5d599eb1a9639"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PM5LY3EKVN55YZZZU32OOKAYTB/bundle.json","state_url":"https://pith.science/pith/PM5LY3EKVN55YZZZU32OOKAYTB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PM5LY3EKVN55YZZZU32OOKAYTB/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":"2026-06-24T10:51:18Z","links":{"resolver":"https://pith.science/pith/PM5LY3EKVN55YZZZU32OOKAYTB","bundle":"https://pith.science/pith/PM5LY3EKVN55YZZZU32OOKAYTB/bundle.json","state":"https://pith.science/pith/PM5LY3EKVN55YZZZU32OOKAYTB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PM5LY3EKVN55YZZZU32OOKAYTB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:PM5LY3EKVN55YZZZU32OOKAYTB","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":"69a232d0b96419433263a7877d5a3a14a27845ef53e52552192ce299c688b70a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-06-28T16:52:32Z","title_canon_sha256":"b6abf0af0093958fe500577097e4bd8e1c2fa5543e2ceece34432af6439b6a54"},"schema_version":"1.0","source":{"id":"1706.09360","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.09360","created_at":"2026-05-18T00:41:18Z"},{"alias_kind":"arxiv_version","alias_value":"1706.09360v1","created_at":"2026-05-18T00:41:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.09360","created_at":"2026-05-18T00:41:18Z"},{"alias_kind":"pith_short_12","alias_value":"PM5LY3EKVN55","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PM5LY3EKVN55YZZZ","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PM5LY3EK","created_at":"2026-05-18T12:31:37Z"}],"graph_snapshots":[{"event_id":"sha256:ec3c577a7f28b2eb7f14587f6cfdb73b10d3147a2079323537f5d599eb1a9639","target":"graph","created_at":"2026-05-18T00:41:18Z","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 paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a ","authors_text":"Holger Wendland, Peter Giesl","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-06-28T16:52:32Z","title":"Kernel-based Discretisation for Solving Matrix-Valued PDEs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09360","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:e400b374db461d70d0a912bf53b6e601a30e2e125e77e6da67850dd326b719ce","target":"record","created_at":"2026-05-18T00:41:18Z","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":"69a232d0b96419433263a7877d5a3a14a27845ef53e52552192ce299c688b70a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2017-06-28T16:52:32Z","title_canon_sha256":"b6abf0af0093958fe500577097e4bd8e1c2fa5543e2ceece34432af6439b6a54"},"schema_version":"1.0","source":{"id":"1706.09360","kind":"arxiv","version":1}},"canonical_sha256":"7b3abc6c8aab7bdc6739a6f4e72818986e3941b421dd4792b1f3e2116b59bc47","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b3abc6c8aab7bdc6739a6f4e72818986e3941b421dd4792b1f3e2116b59bc47","first_computed_at":"2026-05-18T00:41:18.875213Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:18.875213Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RyHRa/EMnkRRpJFqCCU8/Lorn1i2qydMWdcy1nXIxQIpJ+3Q18B+H7i4Q+OSvFQNhk4CVo6XZUdX8EuOByruAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:18.875956Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.09360","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e400b374db461d70d0a912bf53b6e601a30e2e125e77e6da67850dd326b719ce","sha256:ec3c577a7f28b2eb7f14587f6cfdb73b10d3147a2079323537f5d599eb1a9639"],"state_sha256":"be4c5009289e7a100c52f69e1f24ac2a1de7cec5df0da82bcaeddd3ae68c88dc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YsZctCsteHIpCQBWcgGyA/bg5nu+UL749QOHN6HPwm+k7HQzSzdppcSb2fpUGiie+4U5fQmL0Oby2+VSLCgABg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T10:51:18.974415Z","bundle_sha256":"6bae4fd2a1ee59c2a9fd76de6eda905448af8b86263e60564c75155678aeeb00"}}