{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:2ILHIIO3O6GLRLS4T7ZM62G36G","short_pith_number":"pith:2ILHIIO3","schema_version":"1.0","canonical_sha256":"d2167421db778cb8ae5c9ff2cf68dbf1965080e46fbed64a65415f6de7922510","source":{"kind":"arxiv","id":"1408.7093","version":1},"attestation_state":"computed","paper":{"title":"A monotonicity formula for minimal sets with a sliding boundary condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guy David (LM-Orsay)","submitted_at":"2014-08-29T18:21:09Z","abstract_excerpt":"We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\\cal{H}^d$, subject to a sliding boundary constraint where competitors for $E$ are obtained by deforming $E$ by a one-parameter family of functions $\\varphi_t$ such that $\\varphi_t(x) \\in L$ when $x\\in E$ lies on the boundary $L$. In the simple case when $L$ is an affine subspace of dimension $d-1$, the monotone or almost monotone functional is given by $F(r) = r^{-d} \\cal{H}^d(E \\cap B(x,r)) + r^{-d} \\cal{H}^d(S \\cap B(x,r))$, where $x$ is any point of $E$ (not necessarily on $L$) and $S$ is the shad"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.7093","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-08-29T18:21:09Z","cross_cats_sorted":[],"title_canon_sha256":"9214efcb8cd46ff0c4bdf4237b4791c46211f8b0c21a744472029a4bd1e0274e","abstract_canon_sha256":"8e6c1f57f89d92a164c7080fcdf561ec0e1ee0761a6ec4861964d35476111af4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:21.183812Z","signature_b64":"nW1jIwMEqBt70njzMAP3Y4PRWU1zXcYDKu//UtxQTRwPTjB3lEjqNhOOLt0Gf0QY0SVGE42lcC7Dchv3GrrTAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2167421db778cb8ae5c9ff2cf68dbf1965080e46fbed64a65415f6de7922510","last_reissued_at":"2026-05-18T01:20:21.183158Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:21.183158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A monotonicity formula for minimal sets with a sliding boundary condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Guy David (LM-Orsay)","submitted_at":"2014-08-29T18:21:09Z","abstract_excerpt":"We prove a monotonicity formula for minimal or almost minimal sets for the Hausdorff measure $\\cal{H}^d$, subject to a sliding boundary constraint where competitors for $E$ are obtained by deforming $E$ by a one-parameter family of functions $\\varphi_t$ such that $\\varphi_t(x) \\in L$ when $x\\in E$ lies on the boundary $L$. In the simple case when $L$ is an affine subspace of dimension $d-1$, the monotone or almost monotone functional is given by $F(r) = r^{-d} \\cal{H}^d(E \\cap B(x,r)) + r^{-d} \\cal{H}^d(S \\cap B(x,r))$, where $x$ is any point of $E$ (not necessarily on $L$) and $S$ is the shad"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.7093","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1408.7093","created_at":"2026-05-18T01:20:21.183251+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.7093v1","created_at":"2026-05-18T01:20:21.183251+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.7093","created_at":"2026-05-18T01:20:21.183251+00:00"},{"alias_kind":"pith_short_12","alias_value":"2ILHIIO3O6GL","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"2ILHIIO3O6GLRLS4","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"2ILHIIO3","created_at":"2026-05-18T12:28:11.866339+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G","json":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G.json","graph_json":"https://pith.science/api/pith-number/2ILHIIO3O6GLRLS4T7ZM62G36G/graph.json","events_json":"https://pith.science/api/pith-number/2ILHIIO3O6GLRLS4T7ZM62G36G/events.json","paper":"https://pith.science/paper/2ILHIIO3"},"agent_actions":{"view_html":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G","download_json":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G.json","view_paper":"https://pith.science/paper/2ILHIIO3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.7093&json=true","fetch_graph":"https://pith.science/api/pith-number/2ILHIIO3O6GLRLS4T7ZM62G36G/graph.json","fetch_events":"https://pith.science/api/pith-number/2ILHIIO3O6GLRLS4T7ZM62G36G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G/action/storage_attestation","attest_author":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G/action/author_attestation","sign_citation":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G/action/citation_signature","submit_replication":"https://pith.science/pith/2ILHIIO3O6GLRLS4T7ZM62G36G/action/replication_record"}},"created_at":"2026-05-18T01:20:21.183251+00:00","updated_at":"2026-05-18T01:20:21.183251+00:00"}