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(1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$\\int_{S}f(\\sigma(y)xt)d\\mu(t)-\\int_{S}f(xyt)d\\mu(t) = 2f(x)f(y), \\;x,y\\in S,$$ where $S$ is a semigroup, $\\sigma$ is an involutive morphism of $S$, and $\\mu$ is a complex measure that is linear combinations of Dirac measures $(\\delta_{z_{i}})_{i\\in I}$, such that for all $i\\in I$, $z_{i}$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation $$\\int_{S}f(xty)d\\upsilon(t)+\\int_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1608.03906","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-08-09T21:49:30Z","cross_cats_sorted":[],"title_canon_sha256":"c8fcbc4bdd3456d80b3bd56370c182c288352cbac74d8345c56e53df9d8497c3","abstract_canon_sha256":"8370a1a1c2655b69361b98f3540fb566ad8275cff9d4e835d51c7b7cefbb2541"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:15.445787Z","signature_b64":"ocWY7MyxhjEKvlS6npKfV/CVwGryJkIBt/Qv/e6Tf8cmoQuwlIqCghDSgjzSUfryECfIBFPAGAt1uE2kzhAZBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b5630d9bd61da4cd6714835696f1fa8cd58723f4c64e9fcdf427a2e137cb93a","last_reissued_at":"2026-05-18T00:54:15.445274Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:15.445274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Solutions and stability of a variant of Van Vleck's and d'Alembert's functional equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Elqorachi Elhoucien, Redouani Ahmed, Th. M. Rassais","submitted_at":"2016-08-09T21:49:30Z","abstract_excerpt":"In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$\\int_{S}f(\\sigma(y)xt)d\\mu(t)-\\int_{S}f(xyt)d\\mu(t) = 2f(x)f(y), \\;x,y\\in S,$$ where $S$ is a semigroup, $\\sigma$ is an involutive morphism of $S$, and $\\mu$ is a complex measure that is linear combinations of Dirac measures $(\\delta_{z_{i}})_{i\\in I}$, such that for all $i\\in I$, $z_{i}$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation $$\\int_{S}f(xty)d\\upsilon(t)+\\int_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03906","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":"1608.03906","created_at":"2026-05-18T00:54:15.445339+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.03906v2","created_at":"2026-05-18T00:54:15.445339+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.03906","created_at":"2026-05-18T00:54:15.445339+00:00"},{"alias_kind":"pith_short_12","alias_value":"RNLDBWN5MHNE","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"RNLDBWN5MHNEZVTR","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"RNLDBWN5","created_at":"2026-05-18T12:30:41.710351+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/RNLDBWN5MHNEZVTRJA2WS3Y7VD","json":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD.json","graph_json":"https://pith.science/api/pith-number/RNLDBWN5MHNEZVTRJA2WS3Y7VD/graph.json","events_json":"https://pith.science/api/pith-number/RNLDBWN5MHNEZVTRJA2WS3Y7VD/events.json","paper":"https://pith.science/paper/RNLDBWN5"},"agent_actions":{"view_html":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD","download_json":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD.json","view_paper":"https://pith.science/paper/RNLDBWN5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.03906&json=true","fetch_graph":"https://pith.science/api/pith-number/RNLDBWN5MHNEZVTRJA2WS3Y7VD/graph.json","fetch_events":"https://pith.science/api/pith-number/RNLDBWN5MHNEZVTRJA2WS3Y7VD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD/action/storage_attestation","attest_author":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD/action/author_attestation","sign_citation":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD/action/citation_signature","submit_replication":"https://pith.science/pith/RNLDBWN5MHNEZVTRJA2WS3Y7VD/action/replication_record"}},"created_at":"2026-05-18T00:54:15.445339+00:00","updated_at":"2026-05-18T00:54:15.445339+00:00"}