InvolvePub Date : 2023-10-31DOI: 10.2140/involve.2023.16.673
Baian Liu, JiaYan Yap
{"title":"Longest and shortest factorizations in embedding dimension three","authors":"Baian Liu, JiaYan Yap","doi":"10.2140/involve.2023.16.673","DOIUrl":"https://doi.org/10.2140/involve.2023.16.673","url":null,"abstract":"For a numerical monoid $langle n_1, dots, n_k rangle$ minimally generated by $n_1, dots, n_k in mathbb{N}$ with $n_1<cdots<n_k$, the longest and shortest factorization lengths of an element $x$, denoted as $L(x)$ and $ell(x)$, respectively, follow the identities $L(x+n_1) = L(x) + 1$ and $ell(x+n_k) = ell(x) + 1$ for sufficiently large elements $x$. We characterize when these identities hold for all elements of numerical monoids of embedding dimension three.","PeriodicalId":36396,"journal":{"name":"Involve","volume":"74 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135975821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
InvolvePub Date : 2023-10-31DOI: 10.2140/involve.2023.16.689
Andrzej Piotrowski, Joshua Shterenberg
{"title":"Parity of polynomial multiplier sequences for the Chebyshev basis","authors":"Andrzej Piotrowski, Joshua Shterenberg","doi":"10.2140/involve.2023.16.689","DOIUrl":"https://doi.org/10.2140/involve.2023.16.689","url":null,"abstract":"We demonstrate that if $pinmathbb{R}[x]$ and $p$ is not an even function, then ${p(k)}^{infty}_{k=0}$ is not a multiplier sequence for the basis of Chebyshev polynomials of the first kind. We also give a characterization of geometric multiplier sequences for the Chebyshev basis.","PeriodicalId":36396,"journal":{"name":"Involve","volume":"1 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135975826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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