{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:CYAC7URLFBMEMPJQP7FWH7YXHJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"32885f9650ec0a84b74226899750519c6185c382a24a704d3dbb6b34e5ce43f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-10-03T03:44:06Z","title_canon_sha256":"869f2008ebad50353ccab8a86b6e4110cc2c445b7f6b61817bfafb5692b77ea8"},"schema_version":"1.0","source":{"id":"1010.0973","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.0973","created_at":"2026-05-18T04:21:00Z"},{"alias_kind":"arxiv_version","alias_value":"1010.0973v2","created_at":"2026-05-18T04:21:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.0973","created_at":"2026-05-18T04:21:00Z"},{"alias_kind":"pith_short_12","alias_value":"CYAC7URLFBME","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"CYAC7URLFBMEMPJQ","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"CYAC7URL","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:3a25802a9e3f4f8ef540b000f7ac7ab803297ba799346f59e669b1902ad60e36","target":"graph","created_at":"2026-05-18T04:21:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\mathcal{L}\\{f(t)\\} = \\int_{0}^{\\infty}e^{-st}f(t)dt$ denote the Laplace transform of $f$. It is well-known that if $f(t)$ is a piecewise continuous function on the interval $t:[0,\\infty)$ and of exponential order for $t > N$; then $\\lim_{s\\to\\infty}F(s) = 0$, where $F(s) = \\mathcal{L}\\{f(t)\\}$. In this paper we prove that the lesser known converse does not hold true; namely, if $F(s)$ is a continuous function in terms of $s$ for which $\\lim_{s\\to\\infty}F(s) = 0$, then it does not follow that $F(s)$ is the Laplace transform of a piecewise continuous function of exponential order.","authors_text":"Aran Nayebi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-10-03T03:44:06Z","title":"A Note on the Inverse Laplace Transformation of $f(t)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.0973","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dc5c6f51f2e052335aa9f6a35ba88948e9d54cf0f822bd5d298822926dd73c33","target":"record","created_at":"2026-05-18T04:21:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"32885f9650ec0a84b74226899750519c6185c382a24a704d3dbb6b34e5ce43f3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-10-03T03:44:06Z","title_canon_sha256":"869f2008ebad50353ccab8a86b6e4110cc2c445b7f6b61817bfafb5692b77ea8"},"schema_version":"1.0","source":{"id":"1010.0973","kind":"arxiv","version":2}},"canonical_sha256":"16002fd22b2858463d307fcb63ff173a4d898c379b91bf67887e289a7ba76071","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16002fd22b2858463d307fcb63ff173a4d898c379b91bf67887e289a7ba76071","first_computed_at":"2026-05-18T04:21:00.561913Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:21:00.561913Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RBuOyebbqWXnduagTthMvvvXn0xhuceNi/aO+6XyoJUC/WfK4NBtAZJU87Fpgcb0MurnXHER/gkNHeKkymAlBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:21:00.562561Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.0973","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc5c6f51f2e052335aa9f6a35ba88948e9d54cf0f822bd5d298822926dd73c33","sha256:3a25802a9e3f4f8ef540b000f7ac7ab803297ba799346f59e669b1902ad60e36"],"state_sha256":"ed73af2349ecff7f7f9d6a38ace185f83e8639f21ab8fa797c90c2bb9f81602a"}