{"title":"The 2-fold pure extensions need not split","authors":"A. Alijani","doi":"10.53733/277","DOIUrl":"https://doi.org/10.53733/277","url":null,"abstract":"In this paper, we give an example of locally compact abelian groups $A$ and $C$ such that ${rm Pext}^{2}(C,A)neq 0$.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78739481","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 study the conjugate locus in convex manifolds. Our main tool is Jacobi fields, which we use to define a special coordinate system on the unit sphere of the tangent space; this provides a natural coordinate system to study and classify the singularities of the conjugate locus. We pay particular attention to 3-dimensional manifolds, and describe a novel method for determining conjugate points. We then make a study of a special case: the 3-dimensional (quadraxial) ellipsoid. We emphasise the similarities with the focal sets of 2-dimensional ellipsoids.
{"title":"The conjugate locus in convex 3-manifolds","authors":"T. Waters, Matthew Cherrie","doi":"10.53733/139","DOIUrl":"https://doi.org/10.53733/139","url":null,"abstract":"In this paper we study the conjugate locus in convex manifolds. Our main tool is Jacobi fields, which we use to define a special coordinate system on the unit sphere of the tangent space; this provides a natural coordinate system to study and classify the singularities of the conjugate locus. We pay particular attention to 3-dimensional manifolds, and describe a novel method for determining conjugate points. We then make a study of a special case: the 3-dimensional (quadraxial) ellipsoid. We emphasise the similarities with the focal sets of 2-dimensional ellipsoids.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76561175","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}
The problem of equation discovery seeks to reconstruct the underlying dynamics of a time-varying system from observations of the system, and moreover to do so in an instructive way such that we may understand these underlying dynamics from the reconstruction.This article illustrates one type of modern equation discovery method (sparse identification of nonlinear dynamics, or SINDy) in the context of two classic problems. The presentation is in a tutorial style intended to be accessible to students, and could form a useful module in undergraduate or graduate courses in modelling, data analysis, or numerical methods. In this style we explore the strengths and limitations of these methods. We also demonstrate, through use of a carefully constructed example, a new result about the relationship between the reconstructed and true models when a na"ive polynomial basis is used.
{"title":"Equation discovery from data: promise and pitfalls, from rabbits to Mars","authors":"Graham Donovan, Qing Su","doi":"10.53733/216","DOIUrl":"https://doi.org/10.53733/216","url":null,"abstract":"The problem of equation discovery seeks to reconstruct the underlying dynamics of a time-varying system from observations of the system, and moreover to do so in an instructive way such that we may understand these underlying dynamics from the reconstruction.This article illustrates one type of modern equation discovery method (sparse identification of nonlinear dynamics, or SINDy) in the context of two classic problems. The presentation is in a tutorial style intended to be accessible to students, and could form a useful module in undergraduate or graduate courses in modelling, data analysis, or numerical methods. In this style we explore the strengths and limitations of these methods. We also demonstrate, through use of a carefully constructed example, a new result about the relationship between the reconstructed and true models when a na\"ive polynomial basis is used.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85686052","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 considering the number of all choices of signs $+$ and $-$ such that $pm alpha_1 pm alpha_2 pm alpha_3 cdots pm alpha_n = 0$ and the number of sign $-$ appeared therein, this paper can give the exact value of $int_{0}^{2pi} prod_{k=1}^{n} sin (alpha_k x) dx$. In addition, without using the Fourier transformation technique, we can also find the exact value of $int_{0}^{infty}frac{(cosalpha x - cosbeta x)^p}{x^q} dx$. These two integrals are motivated by the work of Andrican and Bragdasar in 2021, Andria and Tomescu in 2002, and Borwein and Borwein in 2001, respectively.
{"title":"Exact value of integrals involving product of sine or cosine function","authors":"Ratinan Boonklurb, Atiratch Laoharenoo","doi":"10.53733/235","DOIUrl":"https://doi.org/10.53733/235","url":null,"abstract":"By considering the number of all choices of signs $+$ and $-$ such that $pm alpha_1 pm alpha_2 pm alpha_3 cdots pm alpha_n = 0$ and the number of sign $-$ appeared therein, this paper can give the exact value of $int_{0}^{2pi} prod_{k=1}^{n} sin (alpha_k x) dx$. In addition, without using the Fourier transformation technique, we can also find the exact value of $int_{0}^{infty}frac{(cosalpha x - cosbeta x)^p}{x^q} dx$. These two integrals are motivated by the work of Andrican and Bragdasar in 2021, Andria and Tomescu in 2002, and Borwein and Borwein in 2001, respectively.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75548567","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}
The classical result of Cooper states that every pure strongly continuous semigroup of isometries ${V_t}_{t geq 0}$ on a Hilbert space is unitarily equivalent to the shift semigroup on $L^{2}([0,infty))$ with some multiplicity. The purpose of this note is to record a proof which has an algebraic flavour. The proof is based on the groupoid approach to semigroups of isometries initiated in [8]. We also indicate how our proof can be adapted to the Hilbert module setting and gives another proof of the main result of [3].
{"title":"On a Theorem of Cooper","authors":"S Sundar","doi":"10.53733/197","DOIUrl":"https://doi.org/10.53733/197","url":null,"abstract":"\u0000\u0000\u0000The classical result of Cooper states that every pure strongly continuous semigroup of isometries ${V_t}_{t geq 0}$ on a Hilbert space is unitarily equivalent to the shift semigroup on $L^{2}([0,infty))$ with some multiplicity. The purpose of this note is to record a proof which has an algebraic flavour. The proof is based on the groupoid approach to semigroups of isometries initiated in [8]. We also indicate how our proof can be adapted to the Hilbert module setting and gives another proof of the main result of [3]. \u0000\u0000\u0000","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73820805","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}
For $ngeq 2$ and a real Banach space $E,$ ${mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.Let $$Pi(E)=Big{~[x^*, (x_1, ldots, x_n)]: x^{*}(x_j)=|x^{*}|=|x_j|=1~mbox{for}~{j=1, ldots, n}~Big}.$$For $Tin {mathcal L}(^n E:E),$ we define $${rm Nrad}({T})=Big{~[x^*, (x_1, ldots, x_n)]in Pi(E): |x^{*}(T(x_1, ldots, x_n))|=v(T)~Big},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {em numerical radius peak mapping} if there is $[x^{*}, (x_1, ldots, x_n)]in Pi(E)$ that satisfies ${rm Nrad}({T})=Big{~pm [x^{*}, (x_1, ldots, x_n)]~Big}.$ In this paper we classify ${rm Nrad}({T})$ for every $Tin {mathcal L}(^2 l_{infty}^2: l_{infty}^2)$ in connection with the set of the norm attaining points of $T$.We also characterize all numerical radius peak mappings in ${mathcalL}(^m l_{infty}^n:l_{infty}^n)$ for $n, mgeq 2,$ where $l_{infty}^n=mathbb{R}^n$ with the supremum norm.
{"title":"Numerical radius points of ${mathcal L}(^m l_{infty}^n:l_{infty}^n)$","authors":"Sung Guen Kim","doi":"10.53733/179","DOIUrl":"https://doi.org/10.53733/179","url":null,"abstract":"For $ngeq 2$ and a real Banach space $E,$ ${mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.Let $$Pi(E)=Big{~[x^*, (x_1, ldots, x_n)]: x^{*}(x_j)=|x^{*}|=|x_j|=1~mbox{for}~{j=1, ldots, n}~Big}.$$For $Tin {mathcal L}(^n E:E),$ we define $${rm Nrad}({T})=Big{~[x^*, (x_1, ldots, x_n)]in Pi(E): |x^{*}(T(x_1, ldots, x_n))|=v(T)~Big},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {em numerical radius peak mapping} if there is $[x^{*}, (x_1, ldots, x_n)]in Pi(E)$ that satisfies ${rm Nrad}({T})=Big{~pm [x^{*}, (x_1, ldots, x_n)]~Big}.$\u0000In this paper we classify ${rm Nrad}({T})$ for every $Tin {mathcal L}(^2 l_{infty}^2: l_{infty}^2)$ in connection with the set of the norm attaining points of $T$.We also characterize all numerical radius peak mappings in ${mathcalL}(^m l_{infty}^n:l_{infty}^n)$ for $n, mgeq 2,$ where $l_{infty}^n=mathbb{R}^n$ with the supremum norm.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75609544","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}
It follows from a theorem of Davies (1952) that if A is an analytic subset of the Cantor middle third set, λ is positive and the Hausdorff s-measure of A is greater than λ, then there is a compact subset C of A such that the Hausdorff s-measure of C is greater than λ. We exhibit a counterpoint to Davies’s theorem: In Gödel’s universe of sets, there is a co-analytic subset B of the Cantor set which has full Hausdorff dimension such that if C is a closed subset of B then C is countable.
{"title":"Capacitability for Co-Analytic Sets","authors":"T. Slaman","doi":"10.53733/170","DOIUrl":"https://doi.org/10.53733/170","url":null,"abstract":"\u0000\u0000\u0000It follows from a theorem of Davies (1952) that if A is an analytic subset of the Cantor middle third set, λ is positive and the Hausdorff s-measure of A is greater than λ, then there is a compact subset C of A such that the Hausdorff s-measure of C is greater than λ. We exhibit a counterpoint to Davies’s theorem: In Gödel’s universe of sets, there is a co-analytic subset B of the Cantor set which has full Hausdorff dimension such that if C is a closed subset of B then C is countable.\u0000\u0000\u0000","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83126726","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}
The relationship between the Axiom of Determinacy (AD) and the Axiom of Turing Determinacy has been open for over 50 years, and the attempts to understand that relationship has had a profound influence on Set Theory in a variety of ways. The prevailing conjecture is that these two determinacy hypotheses are actually equivalent, and the main theorem of this paper is that Turing Determinacy implies that every Suslin set is determined.
{"title":"Turing Determinacy and Suslin sets","authors":"W. Woodin","doi":"10.53733/140","DOIUrl":"https://doi.org/10.53733/140","url":null,"abstract":"The relationship between the Axiom of Determinacy (AD) and the Axiom of Turing Determinacy has been open for over 50 years, and the attempts to understand that relationship has had a profound influence on Set Theory in a variety of ways. The prevailing conjecture is that these two determinacy hypotheses are actually equivalent, and the main theorem of this paper is that Turing Determinacy implies that every Suslin set is determined.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"461 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76333283","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 a 2016 paper by Alan Reid, Martin Bridson and the author, it was shown using the theory of profinite groups that if $Gamma$ is a finitely-generated Fuchsian group and $Sigma$ is a lattice in a connected Lie group, such that $Gamma$ and $Sigma$ have exactly the same finite quotients, then $Gamma$ is isomorphic to $Sigma$. As a consequence, two triangle groups $Delta(r,s,t)$ and $Delta(u,v,w)$ have the same finite quotients if and only if $(u,v,w)$ is a permutation of $(r,s,t)$. A direct proof of this property of triangle groups was given in the final section of that paper, with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. Unfortunately, part of the latter direct proof was flawed. In this paper two new direct proofs are given, one being a corrected version using the same approach as before (involving direct products of small quotients), and the other being a shorter one that uses the same preliminary observations as in the earlier version but then takes a different direction (involving further use of the `Macbeath trick').
在Alan Reid, Martin Bridson和作者2016年的一篇论文中，利用无限群理论证明，如果$Gamma$是有限生成的Fuchsian群，$Sigma$是连通李群中的晶格，使得$Gamma$和$Sigma$具有完全相同的有限商，则$Gamma$与$Sigma$同构。因此，当且仅当$(u,v,w)$是$(r,s,t)$的置换时，两个三角形群$Delta(r,s,t)$和$Delta(u,v,w)$具有相同的有限商。在论文的最后一节给出了三角群的这一性质的直接证明，目的是证明可以区分三角群的显式有限商。不幸的是，后者的部分直接证据是有缺陷的。本文给出了两个新的直接证明，一个是使用与之前相同方法的更正版本(涉及小商的直接乘积)，另一个是使用与早期版本相同的初步观察结果的较短版本，但随后采取了不同的方向(涉及进一步使用“麦克白技巧”)。
{"title":"Two new proofs of the fact that triangle groups are distinguished by their finite quotients","authors":"M. Conder","doi":"10.53733/193","DOIUrl":"https://doi.org/10.53733/193","url":null,"abstract":"In a 2016 paper by Alan Reid, Martin Bridson and the author, it was shown using the theory of profinite groups that if $Gamma$ is a finitely-generated Fuchsian group and $Sigma$ is a lattice in a connected Lie group, such that $Gamma$ and $Sigma$ have exactly the same finite quotients, then $Gamma$ is isomorphic to $Sigma$. As a consequence, two triangle groups $Delta(r,s,t)$ and $Delta(u,v,w)$ have the same finite quotients if and only if $(u,v,w)$ is a permutation of $(r,s,t)$. A direct proof of this property of triangle groups was given in the final section of that paper, with the purpose of exhibiting explicit finite quotients that can distinguish one triangle group from another. Unfortunately, part of the latter direct proof was flawed. In this paper two new direct proofs are given, one being a corrected version using the same approach as before (involving direct products of small quotients), and the other being a shorter one that uses the same preliminary observations as in the earlier version but then takes a different direction (involving further use of the `Macbeath trick').","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76313028","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 focus on the Cauchy problem of the 3D incompressible magneto-micropolar equations with fractional dissipation in the Sobolev space. Liu, Sun and Xin obtained the global solutions to the 3D magneto-micropolar equations with $alpha=beta=gamma=frac{5}{4}$. Deng and Shang established the global well-posedness of the 3D magneto-micropolar equations in the case of $alphageqfrac{5}{4}$, $alpha+betageqfrac{5}{2}$ and $gammageq2-alphageqfrac{3}{4}$. In this paper, we establish the global well-posedness of the 3D magneto-micropolar equations with $alpha=beta=frac{5}{4}$ and $gamma=frac{1}{2}$, which improves the results of Liu-Sun-Xin and Deng-Shang by reducing the value of $gamma$ to $frac{1}{2}$.
{"title":"Global well-posedness for the 3D magneto-micropolar equations with fractional dissipation","authors":"Baoquan Yuan, Panpan Zhang","doi":"10.53733/161","DOIUrl":"https://doi.org/10.53733/161","url":null,"abstract":"This paper focus on the Cauchy problem of the 3D incompressible magneto-micropolar equations with fractional dissipation in the Sobolev space. Liu, Sun and Xin obtained the global solutions to the 3D magneto-micropolar equations with $alpha=beta=gamma=frac{5}{4}$. Deng and Shang established the global well-posedness of the 3D magneto-micropolar equations in the case of $alphageqfrac{5}{4}$, $alpha+betageqfrac{5}{2}$ and $gammageq2-alphageqfrac{3}{4}$. In this paper, we establish the global well-posedness of the 3D magneto-micropolar equations with $alpha=beta=frac{5}{4}$ and $gamma=frac{1}{2}$, which improves the results of Liu-Sun-Xin and Deng-Shang by reducing the value of $gamma$ to $frac{1}{2}$.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74133259","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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