Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.
描述了两种结构:一种给出了衍生长度为4的可溶群，另一种使用作用于有根树的群。
{"title":"Constructing uncountably many groups with the same profinite completion","authors":"N. Nikolov, D. Segal","doi":"10.53733/89","DOIUrl":"https://doi.org/10.53733/89","url":null,"abstract":"\u0000\u0000\u0000Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.\u0000\u0000\u0000","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85094889","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}
Marius Ionescu, A. Kumjian, J. Renault, A. Sims, Dana P. Williams
We analyse extensions $Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.
{"title":"Pushouts of extensions of groupoids by bundles of abelian groups","authors":"Marius Ionescu, A. Kumjian, J. Renault, A. Sims, Dana P. Williams","doi":"10.53733/136","DOIUrl":"https://doi.org/10.53733/136","url":null,"abstract":"We analyse extensions $Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80296776","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}
By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study for transfer of information between finite and infinite objects. Some of this behavior transfers to the problem of understanding what makes understanding the character theory of the uni-upper-triangular group (mod p) “wild.” Our investigations in this paper may be seen as a meditation on the question: is randomness simple or is it complicated?
{"title":"Complexity and randomness in the Heisenberg groups (and beyond)","authors":"P. Diaconis, M. Malliaris","doi":"10.53733/134","DOIUrl":"https://doi.org/10.53733/134","url":null,"abstract":"By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study for transfer of information between finite and infinite objects. Some of this behavior transfers to the problem of understanding what makes understanding the character theory of the uni-upper-triangular group (mod p) “wild.” Our investigations in this paper may be seen as a meditation on the question: is randomness simple or is it complicated? ","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91162529","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}
This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- frac{da}{(d-1)})$, the family $f_{lambda}=lambda tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points.
{"title":"Dynamics of the meromorphic families $f_lambda=lambda tan^pz^q$","authors":"Tao Chen, L. Keen","doi":"10.53733/135","DOIUrl":"https://doi.org/10.53733/135","url":null,"abstract":"This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- frac{da}{(d-1)})$, the family $f_{lambda}=lambda tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73756036","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}
We show that there is no algorithm to decide whether or not a given 4-manifold is homeomorphic to the connected sum of 12 copies of $S^2 times S^2.
我们证明了没有算法来决定给定的4流形是否同纯于$S^2 乘以S^2的12个拷贝的连通和。
{"title":"On the homeomorphism problem for 4-manifolds","authors":"C. Gordon","doi":"10.53733/205","DOIUrl":"https://doi.org/10.53733/205","url":null,"abstract":"We show that there is no algorithm to decide whether or not a given 4-manifold is homeomorphic to the connected sum of 12 copies of $S^2 times S^2.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73385511","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}
We show that the wreath product of two finite symmetric or alternating groups is 2-generated.
证明了两个有限对称或交替群的环积是2生成的。
{"title":"Generating wreath products of symmetric and alternating groups","authors":"J. East, J. Mitchell","doi":"10.53733/108","DOIUrl":"https://doi.org/10.53733/108","url":null,"abstract":"We show that the wreath product of two finite symmetric or alternating groups is 2-generated.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84939086","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}
Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an {em association scheme on triples} (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and the given permutation group was a subgroup of automorphisms of the AST. In this paper, we consider ASTs that do not necessarily admit 2-transitive groups as automorphism groups but instead a transitive cyclic subgroup of the symmetric group acts as automorphisms. Such ASTs are called {em circulant} ASTs and the corresponding ternary relations are called {em circulant relations}. We give a complete characterisation of circulant ASTs in terms of AST-regular partitions of the underlying set. We also show that a special type of circulant, that we call a {em thin circulant}, plays a key role in describing the structure of circulant ASTs. We outline several open questions.
{"title":"Circulant association schemes on triples","authors":"C. Praeger, P. Bhattacharya","doi":"10.53733/106","DOIUrl":"https://doi.org/10.53733/106","url":null,"abstract":"Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an {em association scheme on triples} (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and the given permutation group was a subgroup of automorphisms of the AST. In this paper, we consider ASTs that do not necessarily admit 2-transitive groups as automorphism groups but instead a transitive cyclic subgroup of the symmetric group acts as automorphisms. Such ASTs are called {em circulant} ASTs and the corresponding ternary relations are called {em circulant relations}. We give a complete characterisation of circulant ASTs in terms of AST-regular partitions of the underlying set. We also show that a special type of circulant, that we call a {em thin circulant}, plays a key role in describing the structure of circulant ASTs. We outline several open questions. \u0000 ","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83574307","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}
In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.
{"title":"Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions","authors":"Wee Leng Ng","doi":"10.53733/114","DOIUrl":"https://doi.org/10.53733/114","url":null,"abstract":"In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80051359","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}
In this paper we introduce and study a new class of functions called weighted Stepanov-like pseudo-almost automorphic functions with variable exponents, which generalizes the class of weighted Stepanov-like pseudo-almost automorphic functions. Basic properties of these new spaces are established. The existence of weighted pseudo-almost automorphic solutions to some first-order differential equations with Sp,q(x)-pseudo-almost automorphic coefficients will also be studied.
{"title":"Generalization of Weighted Stepanov-Like Pseudo-Almost Automorphic Space","authors":"M. Zitane","doi":"10.53733/30","DOIUrl":"https://doi.org/10.53733/30","url":null,"abstract":"In this paper we introduce and study a new class of functions called weighted Stepanov-like pseudo-almost automorphic functions with variable exponents, which generalizes the class of weighted Stepanov-like pseudo-almost automorphic functions. Basic properties of these new spaces are established. The existence of weighted pseudo-almost automorphic solutions to some first-order differential equations with Sp,q(x)-pseudo-almost automorphic coefficients will also be studied.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":" 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141222705","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}
Around 2004, J. Lovejoy proved three Hecke-type series identities using Bailey pairs. In this article, we prove Lovejoy’s identities using transformation formulas for q-series discovered by Z.G. Liu in 2013. Some new Hecke-type series are also derived. Our approach also allows us to derive some new Hecke-type identities.
{"title":"On Certain Series of Hecke-Type","authors":"H. Chan, Zhi-Guo Liu","doi":"10.53733/22","DOIUrl":"https://doi.org/10.53733/22","url":null,"abstract":"\u0000 \u0000 \u0000Around 2004, J. Lovejoy proved three Hecke-type series identities using Bailey pairs. In this article, we prove Lovejoy’s identities using transformation formulas for q-series discovered by Z.G. Liu in 2013. Some new Hecke-type series are also derived. Our approach also allows us to derive some new Hecke-type identities. \u0000 \u0000 \u0000","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":" 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141222975","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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