We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.
{"title":"Multiplicative functions arising from the study of mutually unbiased bases","authors":"H. Chan, B. Englert","doi":"10.53733/99","DOIUrl":"https://doi.org/10.53733/99","url":null,"abstract":"We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83499226","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 examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby cite{G-K3}. These links lie on Heegaard surfaces in $#^j S^1 times S^2$ and have surgeries yielding $#^k S^1 times S^2$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer.
我们研究了由Gay和Kirby开发的4流形的三截面分解自然产生的连杆上的手术问题cite{G-K3}。这些链接位于$#^j S^1 times S^2$的heegard表面，并通过手术产生$#^k S^1 times S^2$。我们描述了有这种手术的家庭。人们可能会问，与此类手术的所有联系是否都与这些家庭有关;答案几乎肯定是否定的。尽管如此，我们还是提供了一小部分证据来支持一个肯定的答案。
{"title":"Trisections and link surgeries","authors":"R. Kirby, A. Thompson","doi":"10.53733/94","DOIUrl":"https://doi.org/10.53733/94","url":null,"abstract":"We examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby cite{G-K3}. These links lie on Heegaard surfaces in $#^j S^1 times S^2$ and have surgeries yielding $#^k S^1 times S^2$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78308349","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}
Sara Canilang, Michael P. Cohen, Nicolas Graese, Ian Seong
Let $X$ be a space equipped with $n$ topologies $tau_1,ldots,tau_n$ which are pairwise comparable and saturated, and for each $1leq ileq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators $mathcal{KF}_n$ generated by ${k_i,f_i:1leq ileq n}cup{c}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=frac{5}{24}n^4+frac{37}{12}n^3+frac{79}{24}n^2+frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,tau_1,...,tau_n)$ and a subset $Asubseteq X$ such that repeated application of the operators $k_i, f_i, c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid $mathcal{KF}_2$.
{"title":"closure-complement-frontier problem in saturated polytopological spaces","authors":"Sara Canilang, Michael P. Cohen, Nicolas Graese, Ian Seong","doi":"10.53733/151","DOIUrl":"https://doi.org/10.53733/151","url":null,"abstract":"Let $X$ be a space equipped with $n$ topologies $tau_1,ldots,tau_n$ which are pairwise comparable and saturated, and for each $1leq ileq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators $mathcal{KF}_n$ generated by ${k_i,f_i:1leq ileq n}cup{c}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=frac{5}{24}n^4+frac{37}{12}n^3+frac{79}{24}n^2+frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,tau_1,...,tau_n)$ and a subset $Asubseteq X$ such that repeated application of the operators $k_i, f_i, c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid $mathcal{KF}_2$.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90067345","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}
A smooth embedding of a closed $3$-manifold $M$ in $mathbb{R}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=Xcup_Sigma Y$. However, starting with a Heegaard splitting, we find an obstruction coming from the geometry of the curve complex $C(Sigma)$ to realizing a corresponding embedding $Mhookrightarrow mathbb{R}^4$.
{"title":"Embedding Heegaard Decompositions","authors":"I. Agol, M. Freedman","doi":"10.53733/189","DOIUrl":"https://doi.org/10.53733/189","url":null,"abstract":"A smooth embedding of a closed $3$-manifold $M$ in $mathbb{R}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=Xcup_Sigma Y$. However, starting with a Heegaard splitting, we find an obstruction coming from the geometry of the curve complex $C(Sigma)$ to realizing a corresponding embedding $Mhookrightarrow mathbb{R}^4$.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89863481","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}
Spherically complete ball spaces provide a simple framework for the encoding of completeness properties of various spaces and ordered structures. This allows to prove generic versions of theorems that work with these completeness properties, such as fixed point theorems and related results. For the purpose of applying the generic theorems, it is important to have methods for the construction of new spherically complete ball spaces from existing ones. Given various ball spaces on the same underlying set, we discuss the construction of new ball spaces through set theoretic operations on the balls. A definition of continuity for functions on ball spaces leads to the notion of quotient spaces. Further, we show the existence of products and coproducts and use this to derive a topological category associated with ball spaces.
{"title":"Construction of ball spaces and the notion of continuity","authors":"Ren'e Bartsch, K. Kuhlmann, F. Kuhlmann","doi":"10.53733/157","DOIUrl":"https://doi.org/10.53733/157","url":null,"abstract":"Spherically complete ball spaces provide a simple framework for the encoding of completeness properties of various spaces and ordered structures. This allows to prove generic versions of theorems that work with these completeness properties, such as fixed point theorems and related results. For the purpose of applying the generic theorems, it is important to have methods for the construction of new spherically complete ball spaces from existing ones. Given various ball spaces on the same underlying set, we discuss the construction of new ball spaces through set theoretic operations on the balls. A definition of continuity for functions on ball spaces leads to the notion of quotient spaces. Further, we show the existence of products and coproducts and use this to derive a topological category associated with ball spaces.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81748503","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 extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary. We also illustrate an alternative and perhaps more natural approach to part of the original theorem, using relative cohomology to analyse the actions of an-annular atoroidal group pairs on profinite trees.
{"title":"Profinite Completions, Cohomology and JSJ Decompositions of Compact 3-Manifolds","authors":"G. Wilkes","doi":"10.17863/CAM.37668","DOIUrl":"https://doi.org/10.17863/CAM.37668","url":null,"abstract":"In this paper we extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary. \u0000We also illustrate an alternative and perhaps more natural approach to part of the original theorem, using relative cohomology to analyse the actions of an-annular atoroidal group pairs on profinite trees. \u0000 ","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84773306","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 construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.
{"title":"Free transport for convex potentials","authors":"Y. Dabrowski, A. Guionnet, D. Shlyakhtenko","doi":"10.53733/102","DOIUrl":"https://doi.org/10.53733/102","url":null,"abstract":"We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86439433","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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