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.
{"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}
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.
{"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}
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 $Tin {mathcal L}_s(^3 l_{1}^2)$.
{"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}
This 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.
{"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}
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.
{"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}
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.
{"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}
We survey recent developments related to the Minimum Circuit Size Problem and time-bounded Kolmogorov Complexity.
本文综述了最小电路尺寸问题和限时Kolmogorov复杂度的最新研究进展。
{"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}
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.
{"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}
{"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}
We 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.
{"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}
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