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That is, we look for the existence of some particular solution $(\\lambda,\\phi)$ of a nonlocal operator. $$\\int_{\\O}K(x,y)\\phi(y)\\, dy +a(x)\\phi(x) =-\\lambda \\phi(x),$$ where $\\O\\subset\\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\\lambda_p:=\\sup \\{\\lambda \\in \\R \\, |\\, \\exists \\, \\phi \\in C(\\O), \\phi > 0 \\;\\text{so that}\\; \\oplb{\\phi}{\\O}+ a(x)\\phi + \\lambda\\phi\\le 0\\}$ th"},"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":"1302.0949","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-05T07:16:08Z","cross_cats_sorted":[],"title_canon_sha256":"0f8b948baa29b22b3345c1c24c673092852d4226af0b325d7aeb87a315fd323a","abstract_canon_sha256":"e0f98577f6d5add85a4fa7c65e930836459c4efc3bc26c0780e0040fd8ea3887"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:24.882344Z","signature_b64":"lL/9Nl+iskwF9mohxMDFBJMjOAtkom4/0GMrGUFJrKi4We2cdxVKMhPBluyNfsD1zVvfzX4baDTUR+BCCP1XAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4035c937f213624b22bd77b81055300c42705f1c65a7a84e714ca34973d9e82","last_reissued_at":"2026-05-18T03:34:24.881808Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:24.881808Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Singular measure as principal eigenfunction of some nonlocal operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jerome Coville (BIOSP)","submitted_at":"2013-02-05T07:16:08Z","abstract_excerpt":"In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\\lambda,\\phi)$ of a nonlocal operator. $$\\int_{\\O}K(x,y)\\phi(y)\\, dy +a(x)\\phi(x) =-\\lambda \\phi(x),$$ where $\\O\\subset\\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\\lambda_p:=\\sup \\{\\lambda \\in \\R \\, |\\, \\exists \\, \\phi \\in C(\\O), \\phi > 0 \\;\\text{so that}\\; \\oplb{\\phi}{\\O}+ a(x)\\phi + \\lambda\\phi\\le 0\\}$ th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0949","kind":"arxiv","version":2},"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":"1302.0949","created_at":"2026-05-18T03:34:24.881868+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.0949v2","created_at":"2026-05-18T03:34:24.881868+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.0949","created_at":"2026-05-18T03:34:24.881868+00:00"},{"alias_kind":"pith_short_12","alias_value":"WQBVZE37EE3C","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"WQBVZE37EE3CJMRL","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"WQBVZE37","created_at":"2026-05-18T12:28:04.890932+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/WQBVZE37EE3CJMRL255YCBKTAD","json":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD.json","graph_json":"https://pith.science/api/pith-number/WQBVZE37EE3CJMRL255YCBKTAD/graph.json","events_json":"https://pith.science/api/pith-number/WQBVZE37EE3CJMRL255YCBKTAD/events.json","paper":"https://pith.science/paper/WQBVZE37"},"agent_actions":{"view_html":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD","download_json":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD.json","view_paper":"https://pith.science/paper/WQBVZE37","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.0949&json=true","fetch_graph":"https://pith.science/api/pith-number/WQBVZE37EE3CJMRL255YCBKTAD/graph.json","fetch_events":"https://pith.science/api/pith-number/WQBVZE37EE3CJMRL255YCBKTAD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD/action/storage_attestation","attest_author":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD/action/author_attestation","sign_citation":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD/action/citation_signature","submit_replication":"https://pith.science/pith/WQBVZE37EE3CJMRL255YCBKTAD/action/replication_record"}},"created_at":"2026-05-18T03:34:24.881868+00:00","updated_at":"2026-05-18T03:34:24.881868+00:00"}