{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:QRAMGUGI6EXKOAM4CGAYLEA42J","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":"ddac3c6daab434591e234c0a4f927e54a15196198b09e48f21812ecfbbd09df5","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-03-29T11:05:19Z","title_canon_sha256":"c67c382d1a01804281b1dfd301d4918d5bf85a9dbe12c0a592bc8c656d382680"},"schema_version":"1.0","source":{"id":"1603.08723","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08723","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08723v1","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08723","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"pith_short_12","alias_value":"QRAMGUGI6EXK","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"QRAMGUGI6EXKOAM4","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"QRAMGUGI","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:9456b090656296b153bf777a7ffbb9818d54a94ef001079fd53e759b26078fad","target":"graph","created_at":"2026-05-18T01:18:05Z","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":"In the theory of nonlinear partial differential equations we need to explain superposition operators. For modulation spaces equipped with particular ultradifferentiable weights this was done in \\cite{rrs}. In this paper we introduce a class of general ultradifferentiable weights for modulation spaces $\\mathcal{M}^{w_*}_{p,q}(\\mathbb{R}^n)$ which have at most subexponential growth. We establish analytic as well as non-analytic superposition results in the spaces $\\mathcal{M}^{w_*}_{p,q}(\\mathbb{R}^n)$.","authors_text":"Maximilian Reich","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-03-29T11:05:19Z","title":"Superposition in Modulation Spaces with Ultradifferentiable Weights"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08723","kind":"arxiv","version":1},"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":"sha256:29c4cc9ca214fc7bf729551bc7fee687f3627148d9a3f16f14ec118773246563","target":"record","created_at":"2026-05-18T01:18:05Z","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":"ddac3c6daab434591e234c0a4f927e54a15196198b09e48f21812ecfbbd09df5","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-03-29T11:05:19Z","title_canon_sha256":"c67c382d1a01804281b1dfd301d4918d5bf85a9dbe12c0a592bc8c656d382680"},"schema_version":"1.0","source":{"id":"1603.08723","kind":"arxiv","version":1}},"canonical_sha256":"8440c350c8f12ea7019c118185901cd275007024e552f763cb3c97c2dc9ec248","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8440c350c8f12ea7019c118185901cd275007024e552f763cb3c97c2dc9ec248","first_computed_at":"2026-05-18T01:18:05.505542Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:05.505542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+ilvHgSFdqj2qDFYI+mecG8lnW8+fRSm8uMS96QnnhQ8Xq3ZLk0HYoAEr1bRYnfUwix/Tqwx91UZ9O/XHYmHBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:05.506142Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.08723","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29c4cc9ca214fc7bf729551bc7fee687f3627148d9a3f16f14ec118773246563","sha256:9456b090656296b153bf777a7ffbb9818d54a94ef001079fd53e759b26078fad"],"state_sha256":"97b7ca1e962c51951ee570089df52968e687a6997f1ccdb5e8c7d78cd1dfa199"}