{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:Q2CHPNL3JOUIL3FVS2L4YROL3J","short_pith_number":"pith:Q2CHPNL3","canonical_record":{"source":{"id":"2605.12707","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-12T20:09:42Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"e5d43742cf40494a1e52f31b823f9f059dc1b55c28d0e55c0556840a0a5233b9","abstract_canon_sha256":"a2a999399e06034955ed7d3d41a15b0b7e265c6e163ec5d225056c62b9fe6ac6"},"schema_version":"1.0"},"canonical_sha256":"868477b57b4ba885ecb59697cc45cbda51960f4913d62c76a996e9d0c47d0c84","source":{"kind":"arxiv","id":"2605.12707","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12707","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12707v1","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12707","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"Q2CHPNL3JOUI","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"Q2CHPNL3JOUIL3FV","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"Q2CHPNL3","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:Q2CHPNL3JOUIL3FVS2L4YROL3J","target":"record","payload":{"canonical_record":{"source":{"id":"2605.12707","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-12T20:09:42Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"e5d43742cf40494a1e52f31b823f9f059dc1b55c28d0e55c0556840a0a5233b9","abstract_canon_sha256":"a2a999399e06034955ed7d3d41a15b0b7e265c6e163ec5d225056c62b9fe6ac6"},"schema_version":"1.0"},"canonical_sha256":"868477b57b4ba885ecb59697cc45cbda51960f4913d62c76a996e9d0c47d0c84","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:49.601512Z","signature_b64":"C99F3W7dIo/nbwr9pZqYmoKG8oUCfib7AYHjUd1tRznNXIh3XpPPdAUtIyvtfKkEtvOYXdbPiXtkCe9YVNK3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"868477b57b4ba885ecb59697cc45cbda51960f4913d62c76a996e9d0c47d0c84","last_reissued_at":"2026-05-18T03:09:49.600767Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:49.600767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.12707","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":"2026-05-18T03:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Oz26Ge3/wG+HsRufTz8nXhld3WNL8z0Y/d5wWXZA38k1/3Ir48iXuDbu73yDjHL3RuMuf+SewGyC9PDtgnsDBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T13:47:50.288087Z"},"content_sha256":"c54e9320b5144335472546b3b442d7dd380bb6adf35e0292411a391c3e650b5a","schema_version":"1.0","event_id":"sha256:c54e9320b5144335472546b3b442d7dd380bb6adf35e0292411a391c3e650b5a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:Q2CHPNL3JOUIL3FVS2L4YROL3J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximations with Non-Symmetric Green's Kernels and their Application to Fractional Differential Equations","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Nick Fisher","submitted_at":"2026-05-12T20:09:42Z","abstract_excerpt":"Several kernel-based methods for the numerical solution of fractional differential equations have been developed in the recent past; however, these techniques exclusively relied on the use of radial basis function approximations. In the present work, we consider the non-symmetric Green's kernel perspective on fractional order spline interpolation and its application to a kernel Galerkin method for the numerical solution of certain fractional order differential equation. Unfortunately, the reliance on a non-symmetric kernel requires that our theoretical analysis of the kernel interpolants must "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The fractional differential operator admits a well-defined non-symmetric Green's kernel that can be used to construct the spline interpolant outside the reproducing kernel Hilbert space setting.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Non-symmetric Green's kernels yield optimal-order convergent approximations for fractional differential equations in reproducing kernel Banach spaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4dee2898d6ee9249d37c11dbfb75d5e46b4540d939276ad3393d6be0050ddc3d"},"source":{"id":"2605.12707","kind":"arxiv","version":1},"verdict":{"id":"939a8d9f-a4b7-490b-ba30-dff91b980b31","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:58:06.215803Z","strongest_claim":"we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space.","one_line_summary":"Non-symmetric Green's kernels yield optimal-order convergent approximations for fractional differential equations in reproducing kernel Banach spaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The fractional differential operator admits a well-defined non-symmetric Green's kernel that can be used to construct the spline interpolant outside the reproducing kernel Hilbert space setting.","pith_extraction_headline":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces."},"references":{"count":44,"sample":[{"doi":"","year":null,"title":"An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels , author=. Numer. Algor. , volume=","work_id":"18c90fc8-034f-44ef-b703-b6513f557bf4","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Variational formulation for the stationary fractional advection dispersion equation , author=. Num. Meth. for PDE's , volume=","work_id":"b2685823-3557-4148-bfd5-63839184822a","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"M. Esmaeilbeigi and O. Chatrabgoun and M. Cheraghi , journal=. The role of","work_id":"4d8d8eaa-eac7-4b33-b9fc-dc3dfdd3a3f6","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Total Positivity , author=","work_id":"39d09f6e-54ba-4f5a-b03a-b9af0e755477","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Numerical simulation for solute transport in fractal porous media , author=. ANZIAM J. , volume=","work_id":"3e635943-dd5f-4bd5-a7ad-3ab90f2b221a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":44,"snapshot_sha256":"0da15bc5154371d9d606616431c214aeb1818eadd81499643bcec6a6ca53e33e","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"1d846d2e0638c11e8f317068c2de4244c4172cb55018ed080706379caedb8def"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"939a8d9f-a4b7-490b-ba30-dff91b980b31"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RlVst2suaIy2/xE7xHtGLX5hZ3huFa2Qqogpg7rAZONxx/tmKM9ywK8QNStNs4eeLB/0h8V7cwTDECoYdXj/AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T13:47:50.289161Z"},"content_sha256":"bd7937113378973a5320b5c4aa3cac3dd62e60465e47219f98cec6e347a6a49f","schema_version":"1.0","event_id":"sha256:bd7937113378973a5320b5c4aa3cac3dd62e60465e47219f98cec6e347a6a49f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q2CHPNL3JOUIL3FVS2L4YROL3J/bundle.json","state_url":"https://pith.science/pith/Q2CHPNL3JOUIL3FVS2L4YROL3J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q2CHPNL3JOUIL3FVS2L4YROL3J/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":"2026-06-10T13:47:50Z","links":{"resolver":"https://pith.science/pith/Q2CHPNL3JOUIL3FVS2L4YROL3J","bundle":"https://pith.science/pith/Q2CHPNL3JOUIL3FVS2L4YROL3J/bundle.json","state":"https://pith.science/pith/Q2CHPNL3JOUIL3FVS2L4YROL3J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q2CHPNL3JOUIL3FVS2L4YROL3J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:Q2CHPNL3JOUIL3FVS2L4YROL3J","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":"a2a999399e06034955ed7d3d41a15b0b7e265c6e163ec5d225056c62b9fe6ac6","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-12T20:09:42Z","title_canon_sha256":"e5d43742cf40494a1e52f31b823f9f059dc1b55c28d0e55c0556840a0a5233b9"},"schema_version":"1.0","source":{"id":"2605.12707","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12707","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12707v1","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12707","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"Q2CHPNL3JOUI","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"Q2CHPNL3JOUIL3FV","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"Q2CHPNL3","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:bd7937113378973a5320b5c4aa3cac3dd62e60465e47219f98cec6e347a6a49f","target":"graph","created_at":"2026-05-18T03:09:49Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The fractional differential operator admits a well-defined non-symmetric Green's kernel that can be used to construct the spline interpolant outside the reproducing kernel Hilbert space setting."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Non-symmetric Green's kernels yield optimal-order convergent approximations for fractional differential equations in reproducing kernel Banach spaces."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces."}],"snapshot_sha256":"4dee2898d6ee9249d37c11dbfb75d5e46b4540d939276ad3393d6be0050ddc3d"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"1d846d2e0638c11e8f317068c2de4244c4172cb55018ed080706379caedb8def"},"paper":{"abstract_excerpt":"Several kernel-based methods for the numerical solution of fractional differential equations have been developed in the recent past; however, these techniques exclusively relied on the use of radial basis function approximations. In the present work, we consider the non-symmetric Green's kernel perspective on fractional order spline interpolation and its application to a kernel Galerkin method for the numerical solution of certain fractional order differential equation. Unfortunately, the reliance on a non-symmetric kernel requires that our theoretical analysis of the kernel interpolants must ","authors_text":"Nick Fisher","cross_cats":["cs.NA"],"headline":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces.","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-12T20:09:42Z","title":"Approximations with Non-Symmetric Green's Kernels and their Application to Fractional Differential Equations"},"references":{"count":44,"internal_anchors":0,"resolved_work":44,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels , author=. Numer. Algor. , volume=","work_id":"18c90fc8-034f-44ef-b703-b6513f557bf4","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Variational formulation for the stationary fractional advection dispersion equation , author=. Num. Meth. for PDE's , volume=","work_id":"b2685823-3557-4148-bfd5-63839184822a","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"M. Esmaeilbeigi and O. Chatrabgoun and M. Cheraghi , journal=. The role of","work_id":"4d8d8eaa-eac7-4b33-b9fc-dc3dfdd3a3f6","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Total Positivity , author=","work_id":"39d09f6e-54ba-4f5a-b03a-b9af0e755477","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Numerical simulation for solute transport in fractal porous media , author=. ANZIAM J. , volume=","work_id":"3e635943-dd5f-4bd5-a7ad-3ab90f2b221a","year":null}],"snapshot_sha256":"0da15bc5154371d9d606616431c214aeb1818eadd81499643bcec6a6ca53e33e"},"source":{"id":"2605.12707","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T19:58:06.215803Z","id":"939a8d9f-a4b7-490b-ba30-dff91b980b31","model_set":{"reader":"grok-4.3"},"one_line_summary":"Non-symmetric Green's kernels yield optimal-order convergent approximations for fractional differential equations in reproducing kernel Banach spaces.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Non-symmetric Green's kernels produce optimal-order spline interpolants for fractional differential equations in reproducing kernel Banach spaces.","strongest_claim":"we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space.","weakest_assumption":"The fractional differential operator admits a well-defined non-symmetric Green's kernel that can be used to construct the spline interpolant outside the reproducing kernel Hilbert space setting."}},"verdict_id":"939a8d9f-a4b7-490b-ba30-dff91b980b31"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c54e9320b5144335472546b3b442d7dd380bb6adf35e0292411a391c3e650b5a","target":"record","created_at":"2026-05-18T03:09:49Z","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":"a2a999399e06034955ed7d3d41a15b0b7e265c6e163ec5d225056c62b9fe6ac6","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NA","submitted_at":"2026-05-12T20:09:42Z","title_canon_sha256":"e5d43742cf40494a1e52f31b823f9f059dc1b55c28d0e55c0556840a0a5233b9"},"schema_version":"1.0","source":{"id":"2605.12707","kind":"arxiv","version":1}},"canonical_sha256":"868477b57b4ba885ecb59697cc45cbda51960f4913d62c76a996e9d0c47d0c84","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"868477b57b4ba885ecb59697cc45cbda51960f4913d62c76a996e9d0c47d0c84","first_computed_at":"2026-05-18T03:09:49.600767Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:49.600767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C99F3W7dIo/nbwr9pZqYmoKG8oUCfib7AYHjUd1tRznNXIh3XpPPdAUtIyvtfKkEtvOYXdbPiXtkCe9YVNK3Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:49.601512Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12707","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c54e9320b5144335472546b3b442d7dd380bb6adf35e0292411a391c3e650b5a","sha256:bd7937113378973a5320b5c4aa3cac3dd62e60465e47219f98cec6e347a6a49f"],"state_sha256":"9af4c592e0b7a917dc94dfd6c9362b411127a03fc47840243984738823a3ae00"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Mt0PmClHLra6WA8ROZtaVwKP8C01PMkT+7VIktARJo/VHOB1uyZQHrfnRWt8JIsuee+fme0baFy+fq8W/lFBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T13:47:50.295009Z","bundle_sha256":"b46fdaf8907e770fbdab59b44d2344c7f80bbbc5f93fe28aae7252670cde3781"}}