{"title":"单位雅可比矩阵解析映射的参数化与反演","authors":"T. Sadykov","doi":"10.17323/1609-4514-2023-23-3-369-400","DOIUrl":null,"url":null,"abstract":"Let $x=(x_1,\\ldots,x_n)\\in {\\rm \\bf C}^n$ be a vector of complex variables, denote by $A=(a_{jk})$ a square matrix of size $n\\geq 2,$ and let $\\varphi\\in\\mathcal{O}(\\Omega)$ be an analytic function defined in a nonempty domain $\\Omega\\subset {\\rm \\bf C}.$ We investigate the family of mappings $$ f=(f_1,\\ldots,f_n):{\\rm \\bf C}^n\\rightarrow {\\rm \\bf C}^n, \\quad f[A,\\varphi](x):=x+\\varphi(Ax) $$ with the coordinates $$ f_j : x \\mapsto x_j + \\varphi\\left(\\sum\\limits_{k=1}^n a_{jk}x_k\\right), \\quad j=1,\\ldots,n $$ whose Jacobian is identically equal to a nonzero constant for any $x$ such that all of $f_j$ are well-defined. Let $U$ be a square matrix such that the Jacobian of the mapping $f[U,\\varphi](x)$ is a nonzero constant for any $x$ and moreover for any analytic function $\\varphi\\in\\mathcal{O}(\\Omega).$ We show that any such matrix $U$ is uniquely defined, up to a suitable permutation similarity of matrices, by a partition of the dimension $n$ into a sum of $m$ positive integers together with a permutation on $m$ elements. For any $d=2,3,\\ldots$ we construct $n$-parametric family of square matrices $H(s), s\\in {\\rm \\bf C}^n$ such that for any matrix $U$ as above the mapping $x+\\left((U\\odot H(s))x\\right)^d$ defined by the Hadamard product $U\\odot H(s)$ has unit Jacobian. We prove any such mapping to be polynomially invertible and provide an explicit recursive formula for its inverse.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Parameterizing and Inverting Analytic Mappings with Unit Jacobian\",\"authors\":\"T. Sadykov\",\"doi\":\"10.17323/1609-4514-2023-23-3-369-400\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $x=(x_1,\\\\ldots,x_n)\\\\in {\\\\rm \\\\bf C}^n$ be a vector of complex variables, denote by $A=(a_{jk})$ a square matrix of size $n\\\\geq 2,$ and let $\\\\varphi\\\\in\\\\mathcal{O}(\\\\Omega)$ be an analytic function defined in a nonempty domain $\\\\Omega\\\\subset {\\\\rm \\\\bf C}.$ We investigate the family of mappings $$ f=(f_1,\\\\ldots,f_n):{\\\\rm \\\\bf C}^n\\\\rightarrow {\\\\rm \\\\bf C}^n, \\\\quad f[A,\\\\varphi](x):=x+\\\\varphi(Ax) $$ with the coordinates $$ f_j : x \\\\mapsto x_j + \\\\varphi\\\\left(\\\\sum\\\\limits_{k=1}^n a_{jk}x_k\\\\right), \\\\quad j=1,\\\\ldots,n $$ whose Jacobian is identically equal to a nonzero constant for any $x$ such that all of $f_j$ are well-defined. Let $U$ be a square matrix such that the Jacobian of the mapping $f[U,\\\\varphi](x)$ is a nonzero constant for any $x$ and moreover for any analytic function $\\\\varphi\\\\in\\\\mathcal{O}(\\\\Omega).$ We show that any such matrix $U$ is uniquely defined, up to a suitable permutation similarity of matrices, by a partition of the dimension $n$ into a sum of $m$ positive integers together with a permutation on $m$ elements. For any $d=2,3,\\\\ldots$ we construct $n$-parametric family of square matrices $H(s), s\\\\in {\\\\rm \\\\bf C}^n$ such that for any matrix $U$ as above the mapping $x+\\\\left((U\\\\odot H(s))x\\\\right)^d$ defined by the Hadamard product $U\\\\odot H(s)$ has unit Jacobian. We prove any such mapping to be polynomially invertible and provide an explicit recursive formula for its inverse.\",\"PeriodicalId\":54736,\"journal\":{\"name\":\"Moscow Mathematical Journal\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-01-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Moscow Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.17323/1609-4514-2023-23-3-369-400\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.17323/1609-4514-2023-23-3-369-400","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Parameterizing and Inverting Analytic Mappings with Unit Jacobian
Let $x=(x_1,\ldots,x_n)\in {\rm \bf C}^n$ be a vector of complex variables, denote by $A=(a_{jk})$ a square matrix of size $n\geq 2,$ and let $\varphi\in\mathcal{O}(\Omega)$ be an analytic function defined in a nonempty domain $\Omega\subset {\rm \bf C}.$ We investigate the family of mappings $$ f=(f_1,\ldots,f_n):{\rm \bf C}^n\rightarrow {\rm \bf C}^n, \quad f[A,\varphi](x):=x+\varphi(Ax) $$ with the coordinates $$ f_j : x \mapsto x_j + \varphi\left(\sum\limits_{k=1}^n a_{jk}x_k\right), \quad j=1,\ldots,n $$ whose Jacobian is identically equal to a nonzero constant for any $x$ such that all of $f_j$ are well-defined. Let $U$ be a square matrix such that the Jacobian of the mapping $f[U,\varphi](x)$ is a nonzero constant for any $x$ and moreover for any analytic function $\varphi\in\mathcal{O}(\Omega).$ We show that any such matrix $U$ is uniquely defined, up to a suitable permutation similarity of matrices, by a partition of the dimension $n$ into a sum of $m$ positive integers together with a permutation on $m$ elements. For any $d=2,3,\ldots$ we construct $n$-parametric family of square matrices $H(s), s\in {\rm \bf C}^n$ such that for any matrix $U$ as above the mapping $x+\left((U\odot H(s))x\right)^d$ defined by the Hadamard product $U\odot H(s)$ has unit Jacobian. We prove any such mapping to be polynomially invertible and provide an explicit recursive formula for its inverse.
期刊介绍:
The Moscow Mathematical Journal (MMJ) is an international quarterly published (paper and electronic) by the Independent University of Moscow and the department of mathematics of the Higher School of Economics, and distributed by the American Mathematical Society. MMJ presents highest quality research and research-expository papers in mathematics from all over the world. Its purpose is to bring together different branches of our science and to achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular.
An important specific trait of the journal is that it especially encourages research-expository papers, which must contain new important results and include detailed introductions, placing the achievements in the context of other studies and explaining the motivation behind the research. The aim is to make the articles — at least the formulation of the main results and their significance — understandable to a wide mathematical audience rather than to a narrow class of specialists.