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A reciprocal relation for Hermite polynomials 埃尔米特多项式的互反关系
New Zealand Journal of Mathematics Pub Date : 2021-12-14 DOI: 10.53733/88
S. Nadarajah, C. Withers
{"title":"A reciprocal relation for Hermite polynomials","authors":"S. Nadarajah, C. Withers","doi":"10.53733/88","DOIUrl":"https://doi.org/10.53733/88","url":null,"abstract":"For $xin mathbb{R}$, the ordinary Hermite polynomial $H_k(x)$ can be writtenbegin{eqnarray*}displaystyleH_k(x)= mathbb{E} left[ (x + {rm i} N)^k right] =sum_{j=0}^k {kchoose j} x^{k-j} {rm i}^j mathbb{E} left[ N^j right],end{eqnarray*}where ${rm i} = sqrt{-1}$ and $N$ is a unit normal random variable.  We prove the reciprocal relationbegin{eqnarray*}displaystylex^k=sum_{j=0}^k {kchoose j} H_{k-j}(x) mathbb{E} left[ N^j right].end{eqnarray*}A similar result is given for the multivariate Hermite polynomial.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91207462","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}
引用次数: 0
Invariance and normality of projections in the dual of Banach algebras Banach代数对偶中投影的不变性和正态性
New Zealand Journal of Mathematics Pub Date : 2021-12-14 DOI: 10.53733/132
A. L. Barrenechea, C. Peña
{"title":"Invariance and normality of projections in the dual of Banach algebras","authors":"A. L. Barrenechea, C. Peña","doi":"10.53733/132","DOIUrl":"https://doi.org/10.53733/132","url":null,"abstract":"We study the classes of invariant and natural projections in the dual of a Banach algebra $A$. These type of projections are relevant by their connections with the existence problem of bounded approximate identities in closed ideals of Banach algebras. It is known that any invariant projection is a natural projection. In this article we consider the issue of when a natural projection is an invariant projection.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78776984","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}
引用次数: 0
The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm 平面上具有$l_1$-范数的对称3-线性形式的赋范集
New Zealand Journal of Mathematics Pub Date : 2021-12-14 DOI: 10.53733/177
Sung Guen Kim
{"title":"The norming set of a symmetric 3-linear form on the plane with the $l_1$-norm","authors":"Sung Guen Kim","doi":"10.53733/177","DOIUrl":"https://doi.org/10.53733/177","url":null,"abstract":"An element $(x_1, ldots, x_n)in E^n$ is called a {em norming point} of $Tin {mathcal L}(^n E)$ if $|x_1|=cdots=|x_n|=1$ and$|T(x_1, ldots, x_n)|=|T|,$ where ${mathcal L}(^n E)$ denotes the space of all continuous $n$-linear forms on $E.$For $Tin {mathcal L}(^n E),$ we define $${Norm}(T)=Big{(x_1, ldots, x_n)in E^n: (x_1, ldots, x_n)~mbox{is a norming point of}~TBig}.$$${Norm}(T)$ is called the {em norming set} of $T$. We classify ${Norm}(T)$ for every $Tin {mathcal L}_s(^3 l_{1}^2)$.\u0000 ","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85402798","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}
引用次数: 6
Global aspects of measure preserving equivalence relations and graphs 测度保持等价关系和图的全局方面
New Zealand Journal of Mathematics Pub Date : 2021-11-10 DOI: 10.53733/96
A. Kechris
{"title":"Global aspects of measure preserving equivalence relations and graphs","authors":"A. Kechris","doi":"10.53733/96","DOIUrl":"https://doi.org/10.53733/96","url":null,"abstract":"\u0000\u0000\u0000This paper is an introduction and survey of a “global” theory of measure preserving equivalence relations and graphs. In this theory one views a measure preserving equivalence relation or graph as a point in an appropriate topological space and then studies the properties of this space from a topological, descriptive set theoretic and dynamical point of view.\u0000\u0000\u0000","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78721952","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}
引用次数: 0
Neumann problem for Monge-Ampere type equations revisited. 重新讨论了蒙日-安培型方程的诺伊曼问题。
New Zealand Journal of Mathematics Pub Date : 2021-10-24 DOI: 10.53733/176
N. Trudinger, F. Jiang
{"title":"Neumann problem for Monge-Ampere type equations revisited.","authors":"N. Trudinger, F. Jiang","doi":"10.53733/176","DOIUrl":"https://doi.org/10.53733/176","url":null,"abstract":"This paper concerns  a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"321 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76350134","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}
引用次数: 1
Commutators in $SL_2$ and Markoff surfaces I $SL_2$和Markoff曲面上的换向子1
New Zealand Journal of Mathematics Pub Date : 2021-10-21 DOI: 10.53733/198
Amit Ghosh, C. Meiri, P. Sarnak
{"title":"Commutators in $SL_2$ and Markoff surfaces I","authors":"Amit Ghosh, C. Meiri, P. Sarnak","doi":"10.53733/198","DOIUrl":"https://doi.org/10.53733/198","url":null,"abstract":"We show that the commutator equation over $SL_2Z$ satisfies a profinite local to global principle, while it can fail with infinitely many  exceptions for $SL_2(Z[frac{1}{p}])$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79078268","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}
引用次数: 3
Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization Vaughan Jones, Kolmogorov复杂性，以及围绕电路最小化的新复杂性景观
New Zealand Journal of Mathematics Pub Date : 2021-09-22 DOI: 10.53733/148
E. Allender
{"title":"Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization","authors":"E. Allender","doi":"10.53733/148","DOIUrl":"https://doi.org/10.53733/148","url":null,"abstract":"We survey recent developments related to the Minimum Circuit Size Problem and time-bounded Kolmogorov Complexity.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75572810","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}
引用次数: 10
Principal Foliations of Surfaces near Ellipsoids 椭球表面的主叶理
New Zealand Journal of Mathematics Pub Date : 2021-09-19 DOI: 10.53733/126
J. Guckenheimer
{"title":"Principal Foliations of Surfaces near Ellipsoids","authors":"J. Guckenheimer","doi":"10.53733/126","DOIUrl":"https://doi.org/10.53733/126","url":null,"abstract":"The lines of curvature of a surface embedded in $R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88242844","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}
引用次数: 0
Sir Vaughan Frederick Randal Jones 沃恩·弗雷德里克·兰德尔·琼斯爵士
New Zealand Journal of Mathematics Pub Date : 2021-09-19 DOI: 10.53733/173
M. Conder, R. Downey, D. Gauld, G. Martin
{"title":"Sir Vaughan Frederick Randal Jones","authors":"M. Conder, R. Downey, D. Gauld, G. Martin","doi":"10.53733/173","DOIUrl":"https://doi.org/10.53733/173","url":null,"abstract":"Obituary for Professor Jones.","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87573095","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}
引用次数: 0
Planar algebras 平面代数
New Zealand Journal of Mathematics Pub Date : 2021-09-19 DOI: 10.53733/172
Vaughan Jones
{"title":"Planar algebras","authors":"Vaughan Jones","doi":"10.53733/172","DOIUrl":"https://doi.org/10.53733/172","url":null,"abstract":"\u0000 \u0000 \u0000We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II1 factor, and vice versa. \u0000 \u0000 \u0000","PeriodicalId":30137,"journal":{"name":"New Zealand Journal of Mathematics","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81128919","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}
引用次数: 1
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