Pub Date : 2023-07-12DOI: 10.26493/1855-3974.2749.b64
G. Brinkmann, Heidi Van den Camp
We prove that local operations that preserve all symmetries, as e.g. dual, truncation, medial, or join, as well as local operations that are only guaranteed to preserve all orientation-preserving symmetries, as e.g. gyro or snub, preserve the polyhedrality of simple maps. This generalizes a result by Mohar proving this for the operation dual. We give the proof based on an abstract characterization of these operations, prove that the operations are well defined, and also demonstrate the close connection between these operations and Delaney-Dress symbols.
{"title":"On local operations that preserve symmetries and on preserving polyhedrality of maps","authors":"G. Brinkmann, Heidi Van den Camp","doi":"10.26493/1855-3974.2749.b64","DOIUrl":"https://doi.org/10.26493/1855-3974.2749.b64","url":null,"abstract":"We prove that local operations that preserve all symmetries, as e.g. dual, truncation, medial, or join, as well as local operations that are only guaranteed to preserve all orientation-preserving symmetries, as e.g. gyro or snub, preserve the polyhedrality of simple maps. This generalizes a result by Mohar proving this for the operation dual. We give the proof based on an abstract characterization of these operations, prove that the operations are well defined, and also demonstrate the close connection between these operations and Delaney-Dress symbols.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"20 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86029505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that, up to order type isomorphism, there is a unique crossing-minimal rectilinear drawing of K18. It is easily verified that this drawing does not contain any crossing-minimal drawing of K17. Therefore this settles, in the negative, the following question from Aichholzer and Krasser: is it true that, for every integer n ≥ 4, there exists a crossing-minimal drawing of Kn that contains a crossing-minimal drawing of Kn − 1?
{"title":"There is a unique crossing-minimal rectilinear drawing of K_18","authors":"Bernardo M. Ábrego, Silvia Fernández Merchant, Oswin Aichholzer, Jesús Leaños, Gelasio Salazar","doi":"10.26493/1855-3974.2763.1e6","DOIUrl":"https://doi.org/10.26493/1855-3974.2763.1e6","url":null,"abstract":"We show that, up to order type isomorphism, there is a unique crossing-minimal rectilinear drawing of K18. It is easily verified that this drawing does not contain any crossing-minimal drawing of K17. Therefore this settles, in the negative, the following question from Aichholzer and Krasser: is it true that, for every integer n ≥ 4, there exists a crossing-minimal drawing of Kn that contains a crossing-minimal drawing of Kn − 1?","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135260123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-26DOI: 10.26493/1855-3974.2948.f25
Paul M. Terwilliger
{"title":"Using a q-shuffle algebra to describe the basic module V(Λ_0) for the quantized enveloping algebra Uq(sl^2)","authors":"Paul M. Terwilliger","doi":"10.26493/1855-3974.2948.f25","DOIUrl":"https://doi.org/10.26493/1855-3974.2948.f25","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"28 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72886117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-22DOI: 10.26493/1855-3974.2751.81f
Xiaoqin Zhan, X. Pang, Suyun Ding
{"title":"Finite simple groups on triple systems","authors":"Xiaoqin Zhan, X. Pang, Suyun Ding","doi":"10.26493/1855-3974.2751.81f","DOIUrl":"https://doi.org/10.26493/1855-3974.2751.81f","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"376 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82098556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-08DOI: 10.26493/1855-3974.2835.8f0
P. Danziger, E. Mendelsohn, B. Stevens, T. Traetta
The generalized Oberwolfach problem asks for a factorization of the complete graph $K_v$ into prescribed $2$-factors and at most a $1$-factor. When all $2$-factors are pairwise isomorphic and $v$ is odd, we have the classic Oberwolfach problem, which was originally stated as a seating problem: given $v$ attendees at a conference with $t$ circular tables such that the $i$th table seats $a_i$ people and ${sum_{i=1}^t a_i = v}$, find a seating arrangement over the $frac{v-1}{2}$ days of the conference, so that every person sits next to each other person exactly once. In this paper we introduce the related {em minisymposium problem}, which requires a solution to the generalized Oberwolfach problem on $v$ vertices that contains a subsystem on $m$ vertices. That is, the decomposition restricted to the required $m$ vertices is a solution to the generalized Oberwolfach problem on $m$ vertices. In the seating context above, the larger conference contains a minisymposium of $m$ participants, and we also require that pairs of these $m$ participants be seated next to each other for $leftlfloorfrac{m-1}{2}rightrfloor$ of the days. When the cycles are as long as possible, i.e. $v$, $m$ and $v-m$, a flexible method of Hilton and Johnson provides a solution. We use this result to provide further solutions when $v equiv m equiv 2 pmod 4$ and all cycle lengths are even. In addition, we provide extensive results in the case where all cycle lengths are equal to $k$, solving all cases when $mmid v$, except possibly when $k$ is odd and $v$ is even.
{"title":"On the mini-symposium problem","authors":"P. Danziger, E. Mendelsohn, B. Stevens, T. Traetta","doi":"10.26493/1855-3974.2835.8f0","DOIUrl":"https://doi.org/10.26493/1855-3974.2835.8f0","url":null,"abstract":"The generalized Oberwolfach problem asks for a factorization of the complete graph $K_v$ into prescribed $2$-factors and at most a $1$-factor. When all $2$-factors are pairwise isomorphic and $v$ is odd, we have the classic Oberwolfach problem, which was originally stated as a seating problem: given $v$ attendees at a conference with $t$ circular tables such that the $i$th table seats $a_i$ people and ${sum_{i=1}^t a_i = v}$, find a seating arrangement over the $frac{v-1}{2}$ days of the conference, so that every person sits next to each other person exactly once. In this paper we introduce the related {em minisymposium problem}, which requires a solution to the generalized Oberwolfach problem on $v$ vertices that contains a subsystem on $m$ vertices. That is, the decomposition restricted to the required $m$ vertices is a solution to the generalized Oberwolfach problem on $m$ vertices. In the seating context above, the larger conference contains a minisymposium of $m$ participants, and we also require that pairs of these $m$ participants be seated next to each other for $leftlfloorfrac{m-1}{2}rightrfloor$ of the days. When the cycles are as long as possible, i.e. $v$, $m$ and $v-m$, a flexible method of Hilton and Johnson provides a solution. We use this result to provide further solutions when $v equiv m equiv 2 pmod 4$ and all cycle lengths are even. In addition, we provide extensive results in the case where all cycle lengths are equal to $k$, solving all cases when $mmid v$, except possibly when $k$ is odd and $v$ is even.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"9 18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78658207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-07DOI: 10.26493/1855-3974.2764.fe5
Nicolas Daire, Fumiharu Kato, Yoshiaki Uchino
{"title":"Regular dessins with moduli fields of the form Q(ζp, sqrt[p]{q})","authors":"Nicolas Daire, Fumiharu Kato, Yoshiaki Uchino","doi":"10.26493/1855-3974.2764.fe5","DOIUrl":"https://doi.org/10.26493/1855-3974.2764.fe5","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82750388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-13DOI: 10.26493/1855-3974.2968.d23
Lara Vukšić
The first Weyl algebra A1(k) over a field k is the k-algebra with two generators x, y subject to [y, x] = 1 and was first introduced during the development of quantum mechanics. In this article, we classify all valuations on the real Weyl algebra A1(R) whose residue field is R. We then use a noncommutative version of the Baer-Krull theorem from real algebraic geometry to classify all orderings on A1(R). As a byproduct of our studies, we settle two open problems in noncommutative valuation theory. First, we show that not all valuations on A1(R) with residue field R extend to a valuation on a larger ring R[y; δ], where R is the ring of Puiseux series, introduced by Marshall and Zhang in [12], with the same residue field, and characterize the valuations that do have such an extension. Second, we show that for valuations on noncommutative division rings, Kaplansky’s theorem that extensions by limits of pseudo-Cauchy sequences are immediate fails in general.
{"title":"Valuations and orderings on the real Weyl algebra","authors":"Lara Vukšić","doi":"10.26493/1855-3974.2968.d23","DOIUrl":"https://doi.org/10.26493/1855-3974.2968.d23","url":null,"abstract":"The first Weyl algebra A1(k) over a field k is the k-algebra with two generators x, y subject to [y, x] = 1 and was first introduced during the development of quantum mechanics. In this article, we classify all valuations on the real Weyl algebra A1(R) whose residue field is R. We then use a noncommutative version of the Baer-Krull theorem from real algebraic geometry to classify all orderings on A1(R). As a byproduct of our studies, we settle two open problems in noncommutative valuation theory. First, we show that not all valuations on A1(R) with residue field R extend to a valuation on a larger ring R[y; δ], where R is the ring of Puiseux series, introduced by Marshall and Zhang in [12], with the same residue field, and characterize the valuations that do have such an extension. Second, we show that for valuations on noncommutative division rings, Kaplansky’s theorem that extensions by limits of pseudo-Cauchy sequences are immediate fails in general.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91290145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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