{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:PWIDNLFLXZ55C2B7JKWAMOT272","short_pith_number":"pith:PWIDNLFL","schema_version":"1.0","canonical_sha256":"7d9036acabbe7bd1683f4aac063a7afebe5d5e940028c82197df38b60a95284f","source":{"kind":"arxiv","id":"1511.06364","version":1},"attestation_state":"computed","paper":{"title":"Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Alexander V. Sobolev, Anna Oleynik, Arcady Ponosov, Vadim Kostrykin","submitted_at":"2015-11-19T17:37:46Z","abstract_excerpt":"We study the existence of fixed points to a parameterized Hammertstain operator $\\cH_\\beta,$ $\\beta\\in (0,\\infty],$ with sigmoid type of nonlinearity. The parameter $\\beta<\\infty$ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case $\\beta=\\infty$ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large $\\beta$ exist and can be approximated by the fixed points of $\\cH_\\infty.$ These results are of a high importance in biological applications where one often approximates the smoo"},"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":"1511.06364","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-11-19T17:37:46Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"6a1ee097e20bb5073dbc4842c83be7a38e6a0215bb6d91e0f2ff4b3425a47d22","abstract_canon_sha256":"4c7cab7cc918502b3e88038e786a2c34ff1afeb31621db065f6ee37f6859a4dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:24.589484Z","signature_b64":"AQ39z0VmaYl5peBgHie0ORw0r8UYevAiI/lKJK3jj1Kkx2xggxytoH0gUDAjgBLQPu9icTo8v2kOEmA4aqZMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d9036acabbe7bd1683f4aac063a7afebe5d5e940028c82197df38b60a95284f","last_reissued_at":"2026-05-18T01:26:24.588977Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:24.588977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Alexander V. Sobolev, Anna Oleynik, Arcady Ponosov, Vadim Kostrykin","submitted_at":"2015-11-19T17:37:46Z","abstract_excerpt":"We study the existence of fixed points to a parameterized Hammertstain operator $\\cH_\\beta,$ $\\beta\\in (0,\\infty],$ with sigmoid type of nonlinearity. The parameter $\\beta<\\infty$ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case $\\beta=\\infty$ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large $\\beta$ exist and can be approximated by the fixed points of $\\cH_\\infty.$ These results are of a high importance in biological applications where one often approximates the smoo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.06364","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":"1511.06364","created_at":"2026-05-18T01:26:24.589053+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.06364v1","created_at":"2026-05-18T01:26:24.589053+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.06364","created_at":"2026-05-18T01:26:24.589053+00:00"},{"alias_kind":"pith_short_12","alias_value":"PWIDNLFLXZ55","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"PWIDNLFLXZ55C2B7","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"PWIDNLFL","created_at":"2026-05-18T12:29:37.295048+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/PWIDNLFLXZ55C2B7JKWAMOT272","json":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272.json","graph_json":"https://pith.science/api/pith-number/PWIDNLFLXZ55C2B7JKWAMOT272/graph.json","events_json":"https://pith.science/api/pith-number/PWIDNLFLXZ55C2B7JKWAMOT272/events.json","paper":"https://pith.science/paper/PWIDNLFL"},"agent_actions":{"view_html":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272","download_json":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272.json","view_paper":"https://pith.science/paper/PWIDNLFL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.06364&json=true","fetch_graph":"https://pith.science/api/pith-number/PWIDNLFLXZ55C2B7JKWAMOT272/graph.json","fetch_events":"https://pith.science/api/pith-number/PWIDNLFLXZ55C2B7JKWAMOT272/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272/action/storage_attestation","attest_author":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272/action/author_attestation","sign_citation":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272/action/citation_signature","submit_replication":"https://pith.science/pith/PWIDNLFLXZ55C2B7JKWAMOT272/action/replication_record"}},"created_at":"2026-05-18T01:26:24.589053+00:00","updated_at":"2026-05-18T01:26:24.589053+00:00"}