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We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix $\\mathcal{A}$, the forms \\[\n  \\mathfrak{t}(\\lambda)[x,y] := \\lambda^2\\langle x,y\\rangle + \\lambda\\mathfrak{d}[x,y] + \\mathf"},"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":"1410.7083","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-26T20:57:44Z","cross_cats_sorted":[],"title_canon_sha256":"ee09f3c6fbe9f006acf119706440f17c9c3c974a86840e010e755dc7bdf940fe","abstract_canon_sha256":"f14cd4729e716412410e8b40a66170b9ca85c84b880283aa3e5402bddfaec6e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:01.544778Z","signature_b64":"7osH533v75ZlHT5JacIqAQvS2zSVEwk6r4kJUxIrBtmvDxKxTQKqUUsMtbfzSq7m/ZaU8Gy5PK8OfnoqH7J/Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef96f6632cf4789af318518d1925001a5b3ea85059495bb211cf6c8ff2347ae9","last_reissued_at":"2026-05-18T00:48:01.544342Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:01.544342Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Variational principles for self-adjoint operator functions arising from second-order systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Birgit Jacob, Carsten Trunk, Matthias Langer","submitted_at":"2014-10-26T20:57:44Z","abstract_excerpt":"Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \\[\n  \\langle\\ddot{z}(t),y \\rangle + \\mathfrak{d}[\\dot{z} (t), y] + \\mathfrak{a}_0 [z(t),y] = 0. \\] Here $\\mathfrak{a}_0$ and $\\mathfrak{d}$ are densely defined, symmetric and positive sesquilinear forms on a Hilbert space $H$. We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix $\\mathcal{A}$, the forms \\[\n  \\mathfrak{t}(\\lambda)[x,y] := \\lambda^2\\langle x,y\\rangle + \\lambda\\mathfrak{d}[x,y] + \\mathf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7083","kind":"arxiv","version":3},"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":"1410.7083","created_at":"2026-05-18T00:48:01.544411+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.7083v3","created_at":"2026-05-18T00:48:01.544411+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7083","created_at":"2026-05-18T00:48:01.544411+00:00"},{"alias_kind":"pith_short_12","alias_value":"56LPMYZM6R4J","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"56LPMYZM6R4JV4YY","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"56LPMYZM","created_at":"2026-05-18T12:28:14.216126+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/56LPMYZM6R4JV4YYKGGRSJIADJ","json":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ.json","graph_json":"https://pith.science/api/pith-number/56LPMYZM6R4JV4YYKGGRSJIADJ/graph.json","events_json":"https://pith.science/api/pith-number/56LPMYZM6R4JV4YYKGGRSJIADJ/events.json","paper":"https://pith.science/paper/56LPMYZM"},"agent_actions":{"view_html":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ","download_json":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ.json","view_paper":"https://pith.science/paper/56LPMYZM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.7083&json=true","fetch_graph":"https://pith.science/api/pith-number/56LPMYZM6R4JV4YYKGGRSJIADJ/graph.json","fetch_events":"https://pith.science/api/pith-number/56LPMYZM6R4JV4YYKGGRSJIADJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ/action/storage_attestation","attest_author":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ/action/author_attestation","sign_citation":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ/action/citation_signature","submit_replication":"https://pith.science/pith/56LPMYZM6R4JV4YYKGGRSJIADJ/action/replication_record"}},"created_at":"2026-05-18T00:48:01.544411+00:00","updated_at":"2026-05-18T00:48:01.544411+00:00"}