{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:WED2FX6W6A7YZNQGX5WAKMY7NY","short_pith_number":"pith:WED2FX6W","canonical_record":{"source":{"id":"1802.00655","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-02-02T12:13:18Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"993846af5a5126a8ab92da4d1c9381ded200f2cb40959574872285e706441d88","abstract_canon_sha256":"9d45c7039476a1a1c6d05685ae8a79c006271ce195a6eb6201e8e77ac6e8778b"},"schema_version":"1.0"},"canonical_sha256":"b107a2dfd6f03f8cb606bf6c05331f6e36b297bd4b345915cc3b75c130e211fd","source":{"kind":"arxiv","id":"1802.00655","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.00655","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"arxiv_version","alias_value":"1802.00655v1","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.00655","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"pith_short_12","alias_value":"WED2FX6W6A7Y","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"WED2FX6W6A7YZNQG","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"WED2FX6W","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:WED2FX6W6A7YZNQGX5WAKMY7NY","target":"record","payload":{"canonical_record":{"source":{"id":"1802.00655","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-02-02T12:13:18Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"993846af5a5126a8ab92da4d1c9381ded200f2cb40959574872285e706441d88","abstract_canon_sha256":"9d45c7039476a1a1c6d05685ae8a79c006271ce195a6eb6201e8e77ac6e8778b"},"schema_version":"1.0"},"canonical_sha256":"b107a2dfd6f03f8cb606bf6c05331f6e36b297bd4b345915cc3b75c130e211fd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:32.665201Z","signature_b64":"CcvOe2U2QSTp5GNugokzHn8vdEJwOPDz7qI1Jkfk65uigxuz7ItE0wPIDPSTr7WhBoNKsPNBjdYY3KY9Wl+YBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b107a2dfd6f03f8cb606bf6c05331f6e36b297bd4b345915cc3b75c130e211fd","last_reissued_at":"2026-05-18T00:24:32.664762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:32.664762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.00655","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":"2026-05-18T00:24:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lJ7c2D6GHx/IB0zhlNqJp/hxWJq2T/hzZxOSrVoCoHucq8p/X6eyt5i9PYRCsVokM4aTPDmbZimxwpNUqeLiAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:16:44.903197Z"},"content_sha256":"eed1bb90a824ad38b8a969110ad16e5719ceb9754d4352c817c591b931e94ded","schema_version":"1.0","event_id":"sha256:eed1bb90a824ad38b8a969110ad16e5719ceb9754d4352c817c591b931e94ded"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:WED2FX6W6A7YZNQGX5WAKMY7NY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Analysis and computation of some tumor growth models with nutrient: from cell density models to free boundary dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.AP","authors_text":"Jian-Guo Liu, Li Wang, Min Tang, Zhennan Zhou","submitted_at":"2018-02-02T12:13:18Z","abstract_excerpt":"In this paper, we study the tumor growth equation along with various models for the nutrient component, including the \\emph{in vitro} model and the \\emph{in vivo} model. At the cell density level, the spatial availability of the tumor density $n$ is governed by the Darcy law via the pressure $p(n)=n^{\\gamma}$. For finite $\\gamma$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $\\gamma \\rightarrow \\infty$, the cell density models formally converge to Hele-Shaw flow models, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00655","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":"2026-05-18T00:24:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3a/BHkKlPj8guXZluiPnfk/y8S2BySq3cbQs57F15JJTmBtARXXakynWNDG1oGHkjPFSVFQ+h1ktoP4ndPnkBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T15:16:44.903942Z"},"content_sha256":"bcaa83b03e6d869ad479390e30cba7f1f8d6261ed7e62aad7c089a1996f1d303","schema_version":"1.0","event_id":"sha256:bcaa83b03e6d869ad479390e30cba7f1f8d6261ed7e62aad7c089a1996f1d303"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WED2FX6W6A7YZNQGX5WAKMY7NY/bundle.json","state_url":"https://pith.science/pith/WED2FX6W6A7YZNQGX5WAKMY7NY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WED2FX6W6A7YZNQGX5WAKMY7NY/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":"2026-06-08T15:16:44Z","links":{"resolver":"https://pith.science/pith/WED2FX6W6A7YZNQGX5WAKMY7NY","bundle":"https://pith.science/pith/WED2FX6W6A7YZNQGX5WAKMY7NY/bundle.json","state":"https://pith.science/pith/WED2FX6W6A7YZNQGX5WAKMY7NY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WED2FX6W6A7YZNQGX5WAKMY7NY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:WED2FX6W6A7YZNQGX5WAKMY7NY","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":"9d45c7039476a1a1c6d05685ae8a79c006271ce195a6eb6201e8e77ac6e8778b","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-02-02T12:13:18Z","title_canon_sha256":"993846af5a5126a8ab92da4d1c9381ded200f2cb40959574872285e706441d88"},"schema_version":"1.0","source":{"id":"1802.00655","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.00655","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"arxiv_version","alias_value":"1802.00655v1","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.00655","created_at":"2026-05-18T00:24:32Z"},{"alias_kind":"pith_short_12","alias_value":"WED2FX6W6A7Y","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"WED2FX6W6A7YZNQG","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"WED2FX6W","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:bcaa83b03e6d869ad479390e30cba7f1f8d6261ed7e62aad7c089a1996f1d303","target":"graph","created_at":"2026-05-18T00:24:32Z","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 study the tumor growth equation along with various models for the nutrient component, including the \\emph{in vitro} model and the \\emph{in vivo} model. At the cell density level, the spatial availability of the tumor density $n$ is governed by the Darcy law via the pressure $p(n)=n^{\\gamma}$. For finite $\\gamma$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $\\gamma \\rightarrow \\infty$, the cell density models formally converge to Hele-Shaw flow models, w","authors_text":"Jian-Guo Liu, Li Wang, Min Tang, Zhennan Zhou","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-02-02T12:13:18Z","title":"Analysis and computation of some tumor growth models with nutrient: from cell density models to free boundary dynamics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00655","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:eed1bb90a824ad38b8a969110ad16e5719ceb9754d4352c817c591b931e94ded","target":"record","created_at":"2026-05-18T00:24:32Z","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":"9d45c7039476a1a1c6d05685ae8a79c006271ce195a6eb6201e8e77ac6e8778b","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-02-02T12:13:18Z","title_canon_sha256":"993846af5a5126a8ab92da4d1c9381ded200f2cb40959574872285e706441d88"},"schema_version":"1.0","source":{"id":"1802.00655","kind":"arxiv","version":1}},"canonical_sha256":"b107a2dfd6f03f8cb606bf6c05331f6e36b297bd4b345915cc3b75c130e211fd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b107a2dfd6f03f8cb606bf6c05331f6e36b297bd4b345915cc3b75c130e211fd","first_computed_at":"2026-05-18T00:24:32.664762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:32.664762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CcvOe2U2QSTp5GNugokzHn8vdEJwOPDz7qI1Jkfk65uigxuz7ItE0wPIDPSTr7WhBoNKsPNBjdYY3KY9Wl+YBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:32.665201Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.00655","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eed1bb90a824ad38b8a969110ad16e5719ceb9754d4352c817c591b931e94ded","sha256:bcaa83b03e6d869ad479390e30cba7f1f8d6261ed7e62aad7c089a1996f1d303"],"state_sha256":"602280be2a64cb3f1c77812071290e06196b8d268646a3a4cf5c39399609f716"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+/vG/RaFHuz6iYHKciQMejmqkk4e5gP2j3xD7EDhCkrTerTN0EhnAZBh2sTM7KaUoUWkD3Azydwq9uJ8lg1EAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T15:16:44.907637Z","bundle_sha256":"8edd6cec5faafeb2f2115001b9a556fadf3c4658493a5e257d535c306c1407d6"}}