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For $\\mathsf{A} = \\{a_1,\\dots,a_m\\}$ continuous (sufficiently differentiable) functions on $\\mathbb{R}^n$ we give a general upper bound of $m-1$ and a general lower bound of $\\left\\lceil \\frac{2m}{(n+1)(n+2)}\\right\\rceil$. For polynomials of degree at most $d$ in $n$ variables we find that the number of Gaussian and log-normal mixtures is bounded by the Carath\\'eodory numbers in \\cite{didio17Cara}. Therefore, for univariate polynomials $\\{1,x,\\do"},"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":"1804.07058","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-19T09:55:58Z","cross_cats_sorted":[],"title_canon_sha256":"a345395658f9e91e66a273693093a501c10c19bbb2d9221797451a12a7813f20","abstract_canon_sha256":"8a898efafa8460f25239871a80f631bac69e2f72cd3bdad2b350c9a37d574ade"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:03.186124Z","signature_b64":"6NrI6CL6gUMI4zmoFSSTuucsSXt+DNM7+tz3yiPjA0wfCjyeEjj6KZ8WWKRWab/7+s2qQrqVwPXttnowcYgoAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d3a3ac85fa082af1d89963984b5a24ace3c5713565be6600c830e0e9a13273fb","last_reissued_at":"2026-05-17T23:42:03.185493Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:03.185493Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The multidimensional truncated Moment Problem: Gaussian and Log-Normal Mixtures, their Carath\\'eodory Numbers, and Set of Atoms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Philipp J. di Dio","submitted_at":"2018-04-19T09:55:58Z","abstract_excerpt":"We study truncated moment sequences of distribution mixtures, especially from Gaussian and log-normal distributions and their Carath\\'eodory numbers. For $\\mathsf{A} = \\{a_1,\\dots,a_m\\}$ continuous (sufficiently differentiable) functions on $\\mathbb{R}^n$ we give a general upper bound of $m-1$ and a general lower bound of $\\left\\lceil \\frac{2m}{(n+1)(n+2)}\\right\\rceil$. For polynomials of degree at most $d$ in $n$ variables we find that the number of Gaussian and log-normal mixtures is bounded by the Carath\\'eodory numbers in \\cite{didio17Cara}. 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