Asking and answering the right questions about computability and computational complexity of dynamical systems is more important than many may realize. In this note, I discuss the principal challenges, their implications, and some instructive open problems.
{"title":"Towards understanding the theoretical challenges of numerical modeling of dynamical systems","authors":"M. Yampolsky","doi":"10.53733/129","DOIUrl":"https://doi.org/10.53733/129","url":null,"abstract":"Asking and answering the right questions about computability and computational complexity of dynamical systems is more important than many may realize. In this note, I discuss the principal challenges, their implications, and some instructive open problems. \u0000 \u0000 ","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89911097","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 verify a conjecture of P. Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family ${V_r}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $rge 0$ the subfamily indexed by $(r,infty)$ is a neighbourhood basis of the closure of the $r^{rm th}$ set. We use smooth collars in the differential category, regular neighbourhoods in the piecewise linear category and handlebodies in the topological category.
{"title":"Manifold Neighbourhoods and a Conjecture of Adjamagbo","authors":"D. Gauld","doi":"10.53733/131","DOIUrl":"https://doi.org/10.53733/131","url":null,"abstract":"We verify a conjecture of P. Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family ${V_r}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $rge 0$ the subfamily indexed by $(r,infty)$ is a neighbourhood basis of the closure of the $r^{rm th}$ set. We use smooth collars in the differential category, regular neighbourhoods in the piecewise linear category and handlebodies in the topological category.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76194921","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 every 4-plump razor-sharp normal Tits quadrangle X is uniquely determined by a non-degenerate quadratic space whose Witt index m is at least 2. If this Witt index is finite, then X is the Tits quadrangle arising from the corresponding building of type B_m or D_m by a standard construction.
{"title":"Orthogonal Tits Quadrangles","authors":"B. Mühlherr, R. Weiss","doi":"10.53733/105","DOIUrl":"https://doi.org/10.53733/105","url":null,"abstract":"\u0000\u0000\u0000We show that every 4-plump razor-sharp normal Tits quadrangle X is uniquely determined by a non-degenerate quadratic space whose Witt index m is at least 2. If this Witt index is finite, then X is the Tits quadrangle arising from the corresponding building of type B_m or D_m by a standard construction.\u0000\u0000\u0000","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90543238","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 transfinite hierarchy of Turing degrees of c.e. sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e. degrees, and finding maximal degrees in upper cones.
{"title":"Hierarchy of Computably Enumerable Degrees II","authors":"R. Downey, Noam Greenberg, Ellen Hammatt","doi":"10.53733/133","DOIUrl":"https://doi.org/10.53733/133","url":null,"abstract":"A transfinite hierarchy of Turing degrees of c.e. sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e. degrees, and finding maximal degrees in upper cones.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89586150","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, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.
{"title":"Some aspects of Ricci flow on the 4-sphere","authors":"S. Chang, Eric Chen","doi":"10.53733/152","DOIUrl":"https://doi.org/10.53733/152","url":null,"abstract":"In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82091666","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 establish analogues of Liouville's theorem in the complex function theory, with the differential operator replaced by various difference operators. This is done generally by the extraction of (formal) Taylor coefficients using a residue map which measures the obstruction having local "anti-derivative". The residue map is based on a Weyl algebra or $q$-Weyl algebra structure satisfied by each corresponding operator. This explains the different senses of "boundedness" required by the respective analogues of Liouville's theorem in this article.
{"title":"D-module approach to Liouville's Theorem for difference operators","authors":"Kam Hang Cheng, Y. Chiang, A. Ching","doi":"10.53733/187","DOIUrl":"https://doi.org/10.53733/187","url":null,"abstract":"We establish analogues of Liouville's theorem in the complex function theory, with the differential operator replaced by various difference operators. This is done generally by the extraction of (formal) Taylor coefficients using a residue map which measures the obstruction having local \"anti-derivative\". The residue map is based on a Weyl algebra or $q$-Weyl algebra structure satisfied by each corresponding operator. This explains the different senses of \"boundedness\" required by the respective analogues of Liouville's theorem in this article.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80272358","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 study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${mathbb N rtimes mathbb N^times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${mathbb Q_+^times}!! ltimes mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.
我们研究了由半群${mathbb N rtimes mathbb N^times}$的右正则表示生成的Toeplitz $C^*$-代数，我们称之为右Toeplitz代数。我们通过研究三个不同的商来分析它的结构。利用乘法有理数的作用，证明了乘法边界商同构于加法有理数的Toeplitz代数的叉积，并研究了其理想结构。Crisp—Laca边界商同构于群${mathbb Q_+^times}的$C^*$-代数。l * mathbb Q$。右Toeplitz代数及其所有KMS状态因子通过加性边界商存在自然动力学。我们描述了逆温度大于1时的KMS单纯形。
{"title":"Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers","authors":"Astrid an Huef, Marcelo Laca, I. Raeburn","doi":"10.53733/90","DOIUrl":"https://doi.org/10.53733/90","url":null,"abstract":"We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${mathbb N rtimes mathbb N^times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${mathbb Q_+^times}!! ltimes mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84361706","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}
Non extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a 2-parameter collection of kernel functions integrated against measures on the torus. Families from classical geometric function theory such as the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others are included. However for these families of analytic functions, identifying “all” the extreme points remains a difficult challenge except in some special cases. Aharonov and Friedland [1] identified a band of points on the unit circle which corresponds to the set of extreme points for these 2-parameter collections of kernel functions. Later this band of extreme points was further extended by introducing a new technique by Dow and Wilken [3]. On the other hand, a technique to identify a non extreme point was not investigated much in the past probably because identifying non extreme points does not directly help solving the optimization of linear extremal problems. So far only one point on the unit circle has beenidentified which corresponds to a non extreme point for a 2-parameter collections of kernel functions. This leaves a big gap between the band of extreme points and one non extreme point. The author believes it is worth developing some techniques, and identifying non extreme points will shed a new light in the exact determination of the extreme points. The ultimate goal is to identify the point on the unit circle that separates the band of extreme points from non extreme points. The main result introduces a new class of non extreme points.
{"title":"Non-extreme-Points Approach to Extreme Points of Integral Families of Analytic Functions","authors":"Keiko Dow","doi":"10.53733/87","DOIUrl":"https://doi.org/10.53733/87","url":null,"abstract":"Non extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a 2-parameter collection of kernel functions integrated against measures on the torus. Families from classical geometric function theory such as the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others are included. However for these families of analytic functions, identifying “all” the extreme points remains a difficult challenge except in some special cases. Aharonov and Friedland [1] identified a band of points on the unit circle which corresponds to the set of extreme points for these 2-parameter collections of kernel functions. Later this band of extreme points was further extended by introducing a new technique by Dow and Wilken [3]. On the other hand, a technique to identify a non extreme point was not investigated much in the past probably because identifying non extreme points does not directly help solving the optimization of linear extremal problems. So far only one point on the unit circle has beenidentified which corresponds to a non extreme point for a 2-parameter collections of kernel functions. This leaves a big gap between the band of extreme points and one non extreme point. The author believes it is worth developing some techniques, and identifying non extreme points will shed a new light in the exact determination of the extreme points. The ultimate goal is to identify the point on the unit circle that separates the band of extreme points from non extreme points. The main result introduces a new class of non extreme points.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"226 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76216071","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 give a systematic treatment of caloric measure null sets on the essential boundary $partial_eE$ of an arbitrary open set $E$ in ${bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Zsubseteqpartial_eEcappartial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from cite{watson2011} that any polar subset of $partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $partial_eD$, caloric measure null sets are necessarily polar.
{"title":"Caloric Measure Null Sets","authors":"N. Watson","doi":"10.53733/156","DOIUrl":"https://doi.org/10.53733/156","url":null,"abstract":"We give a systematic treatment of caloric measure null sets on the essential boundary $partial_eE$ of an arbitrary open set $E$ in ${bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Zsubseteqpartial_eEcappartial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from cite{watson2011} that any polar subset of $partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $partial_eD$, caloric measure null sets are necessarily polar.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74606816","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 introduce the space of virtual Markov chains (VMCs) as a projective limit of the spaces of all finite state space Markov chains (MCs), in the same way that the space of virtual permutations is the projective limit of the spaces of all permutations of finite sets.We introduce the notions of virtual initial distribution (VID) and a virtual transition matrix (VTM), and we show that the law of any VMC is uniquely characterized by a pair of a VID and VTM which have to satisfy a certain compatibility condition.Lastly, we study various properties of compact convex sets associated to the theory of VMCs, including that the Birkhoff-von Neumann theorem fails in the virtual setting.
{"title":"Virtual Markov Chains","authors":"S. Evans, Adam Quinn Jaffe","doi":"10.53733/147","DOIUrl":"https://doi.org/10.53733/147","url":null,"abstract":"We introduce the space of virtual Markov chains (VMCs) as a projective limit of the spaces of all finite state space Markov chains (MCs), in the same way that the space of virtual permutations is the projective limit of the spaces of all permutations of finite sets.We introduce the notions of virtual initial distribution (VID) and a virtual transition matrix (VTM), and we show that the law of any VMC is uniquely characterized by a pair of a VID and VTM which have to satisfy a certain compatibility condition.Lastly, we study various properties of compact convex sets associated to the theory of VMCs, including that the Birkhoff-von Neumann theorem fails in the virtual setting.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90721979","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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