{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:5FJAJZOMA62ELUDNYTFIPCZ7M6","short_pith_number":"pith:5FJAJZOM","schema_version":"1.0","canonical_sha256":"e95204e5cc07b445d06dc4ca878b3f678f425a5ee0aa0953b92f187e95546e0c","source":{"kind":"arxiv","id":"1110.1775","version":1},"attestation_state":"computed","paper":{"title":"Perturbation and Numerical Methods for Computing the Minimal Average Energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rafael de la Llave, Timothy Blass","submitted_at":"2011-10-08T22:26:50Z","abstract_excerpt":"We investigate the differentiability of minimal average energy associated to the functionals $S_\\ep (u) = \\int_{\\mathbb{R}^d} (1/2)|\\nabla u|^2 + \\ep V(x,u)\\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the"},"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":"1110.1775","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-10-08T22:26:50Z","cross_cats_sorted":[],"title_canon_sha256":"96eb9b0e560c3ea91687542d1e308af784596d19e2e5e8b1d80726299d1a7207","abstract_canon_sha256":"4cd84845aa70eb99a28ce0e424813e6a28a06070eb7216df8bd57745c452540f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:19.647145Z","signature_b64":"Dy+EqzUX5eNGWdUXwNqvshd/oUzUfTA3awqJu/Ji1+/p8q0VacdfUD4G9ubSWGdelZnJ4Yz8Q/JXU4UB6MpuCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e95204e5cc07b445d06dc4ca878b3f678f425a5ee0aa0953b92f187e95546e0c","last_reissued_at":"2026-05-18T04:11:19.646691Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:19.646691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Perturbation and Numerical Methods for Computing the Minimal Average Energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Rafael de la Llave, Timothy Blass","submitted_at":"2011-10-08T22:26:50Z","abstract_excerpt":"We investigate the differentiability of minimal average energy associated to the functionals $S_\\ep (u) = \\int_{\\mathbb{R}^d} (1/2)|\\nabla u|^2 + \\ep V(x,u)\\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1775","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":"1110.1775","created_at":"2026-05-18T04:11:19.646751+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.1775v1","created_at":"2026-05-18T04:11:19.646751+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1775","created_at":"2026-05-18T04:11:19.646751+00:00"},{"alias_kind":"pith_short_12","alias_value":"5FJAJZOMA62E","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"5FJAJZOMA62ELUDN","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"5FJAJZOM","created_at":"2026-05-18T12:26:20.644004+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/5FJAJZOMA62ELUDNYTFIPCZ7M6","json":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6.json","graph_json":"https://pith.science/api/pith-number/5FJAJZOMA62ELUDNYTFIPCZ7M6/graph.json","events_json":"https://pith.science/api/pith-number/5FJAJZOMA62ELUDNYTFIPCZ7M6/events.json","paper":"https://pith.science/paper/5FJAJZOM"},"agent_actions":{"view_html":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6","download_json":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6.json","view_paper":"https://pith.science/paper/5FJAJZOM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.1775&json=true","fetch_graph":"https://pith.science/api/pith-number/5FJAJZOMA62ELUDNYTFIPCZ7M6/graph.json","fetch_events":"https://pith.science/api/pith-number/5FJAJZOMA62ELUDNYTFIPCZ7M6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6/action/storage_attestation","attest_author":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6/action/author_attestation","sign_citation":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6/action/citation_signature","submit_replication":"https://pith.science/pith/5FJAJZOMA62ELUDNYTFIPCZ7M6/action/replication_record"}},"created_at":"2026-05-18T04:11:19.646751+00:00","updated_at":"2026-05-18T04:11:19.646751+00:00"}