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":null,"pages":null},"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}
Pub Date : 2023-02-13DOI: 10.26493/1855-3974.2800.d12
Qi'an Chen, Siddhant Jajodia, T. Jordán, Kate Perkins
By mapping the vertices of a graph G to points in R 3 , and its edges to the corresponding line segments, we obtain a three-dimensional realization of G . A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3 n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.
{"title":"Redundantly globally rigid braced triangulations","authors":"Qi'an Chen, Siddhant Jajodia, T. Jordán, Kate Perkins","doi":"10.26493/1855-3974.2800.d12","DOIUrl":"https://doi.org/10.26493/1855-3974.2800.d12","url":null,"abstract":"By mapping the vertices of a graph G to points in R 3 , and its edges to the corresponding line segments, we obtain a three-dimensional realization of G . A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3 n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83327042","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 : 2023-02-13DOI: 10.26493/1855-3974.2976.f76
Pavle Saksida
{"title":"On the beta distribution, the nonlinear Fourier transform and a combinatorial problem","authors":"Pavle Saksida","doi":"10.26493/1855-3974.2976.f76","DOIUrl":"https://doi.org/10.26493/1855-3974.2976.f76","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76516324","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 : 2023-01-24DOI: 10.26493/1855-3974.2503.f17
António Breda d'Azevedo, D. Catalano
{"title":"Classification of thin regular map representations of hypermaps","authors":"António Breda d'Azevedo, D. Catalano","doi":"10.26493/1855-3974.2503.f17","DOIUrl":"https://doi.org/10.26493/1855-3974.2503.f17","url":null,"abstract":"","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77820519","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 : 2023-01-19DOI: 10.26493/1855-3974.3071.37e
Dario Sterzi, Pablo Spiga
Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.
{"title":"Almost all Cayley maps are mapical regular representations","authors":"Dario Sterzi, Pablo Spiga","doi":"10.26493/1855-3974.3071.37e","DOIUrl":"https://doi.org/10.26493/1855-3974.3071.37e","url":null,"abstract":"Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78596508","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 : 2023-01-06DOI: 10.26493/1855-3974.2672.73b
Marién Abreu, John Baptist Gauci, Domenico Labbate, Federico Romaniello, Jean Paul Zerafa
A graph G has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of G such that the union of the two perfect matchings yields a Hamiltonian cycle of G. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if G is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic.
{"title":"Perfect matchings, Hamiltonian cycles and edge-colourings in a class of cubic graphs","authors":"Marién Abreu, John Baptist Gauci, Domenico Labbate, Federico Romaniello, Jean Paul Zerafa","doi":"10.26493/1855-3974.2672.73b","DOIUrl":"https://doi.org/10.26493/1855-3974.2672.73b","url":null,"abstract":"A graph G has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of G such that the union of the two perfect matchings yields a Hamiltonian cycle of G. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if G is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135223597","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 : 2022-10-05DOI: 10.26493/1855-3974.2975.1b2
Luca Sabatini
Let $G$ be a group. We give an explicit description of the set of elements $x in G$ such that $x^{|G:H|} in H$ for every subgroup of finite index $H leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?
让$G$成为一个团体。我们给出了元素集合$x in G$的显式描述，使得$x^{|G:H|} in H$对于有限索引$H leqslant G$的每一子群。这涉及到以下问题:给定两个子群$H$和$K$, $H$的索引是有限的，$|HK:H|$何时能除$|G:H|$ ?
{"title":"Products of subgroups, subnormality, and relative orders of elements","authors":"Luca Sabatini","doi":"10.26493/1855-3974.2975.1b2","DOIUrl":"https://doi.org/10.26493/1855-3974.2975.1b2","url":null,"abstract":"Let $G$ be a group. We give an explicit description of the set of elements $x in G$ such that $x^{|G:H|} in H$ for every subgroup of finite index $H leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78610886","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 : 2022-04-14DOI: 10.26493/1855-3974.2894.b07
John H. Johnson, Max Wakefield
There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M"obius function. Under the product this generalized M"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M"obius functions. Using this generalized M"obius function we define analogues of the characteristic polynomial and M"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.
{"title":"A non-associative incidence near-ring with a generalized Möbius function","authors":"John H. Johnson, Max Wakefield","doi":"10.26493/1855-3974.2894.b07","DOIUrl":"https://doi.org/10.26493/1855-3974.2894.b07","url":null,"abstract":"There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M\"obius function. Under the product this generalized M\"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M\"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M\"obius functions. Using this generalized M\"obius function we define analogues of the characteristic polynomial and M\"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M\"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87136666","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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