RUI: Truncated Multivariable Moment Problems & Applications: An Operator Theoretic Approach

RUI:截断多变量矩问题

基本信息

  • 批准号:
    0457138
  • 负责人:
  • 金额:
    --
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2005
  • 资助国家:
    美国
  • 起止时间:
    2005-07-01 至 2008-06-30
  • 项目状态:
    已结题

项目摘要

ABSTRACT This research concerns an operator-theoretic approach to multivariable moment problems. The prototypical problem that we study is the Multivariable Truncated Moment Problem: Given a finite d-dimensional real sequence, we seek concrete necessary and sufficient conditions so that there exists a positive Borel measure on d-dimensional Euclidean space for which the given data represent successive power moments of the measure. To study the existence of such a representing measure, we associate to the data a finite moment matrix. It is known that a finitely atomic representing measure exists if and only if the moment matrix admits an extension to an infinite, finite-rank, positive moment matrix. Representing measures with the fewest atoms correspond to extensions of minimal rank. We seek to establish concrete necessary and sufficient conditions for such extensions, and also to develop algorithms for explicitly computing representing measures corresponding to extensions. In the case when the moment matrix is singular, we study the following conjecture: there exists a representing measure if and only if the moment matrix is positive, recursively generated, and the rank of the matrix is at most equal to the size of the algebraic variety natuarally associated to the data. This conjecture is true for moment problems on planar curves of degree one or two, so we study the conjecture for curves of higher degree. This research also concerns estimates of the minimal rank in the above-mentioned extensions; such estimates are related to the convergence of certain polynomial optimization algorithms and also to the size of minimal cubature rules in Numerical Analysis. The aim of this research is to develop new existence and uniqueness criteria for finitely atomic representing measures in multivariable truncated moment problems (with data corresponding to successive power moments up to a fixed finite degree). Truncated moment problems play essential roles in aspects of such fields as Operator Theory (subnormality of weighted shifts), Interpolation Theory (classical Nevanlinna-Pick theory), Numerical Analysis (multivariable cubature rules), Control Theory (signal processing), and Optimization Theory (polynomial optimization over a region). The principal focus of this research is an approach to multivariable truncated moment problems based on an extension theory for the moment matrix associated to the moment data. When this matrix admits an infinite, positive, finite rank moment matrix extension, this approach yields an explicit formula for a finitely atomic representing measure. The primary goal of this research is to determine concrete criteria on the moment data which permit the desired extension. This research also concerns the development of algorithms to implement these criteria. One principal application will be to develop new minimal cubature rules for measures on classical domains such as the disk and triangle; another application concerns the convergence of polynomial optimization algorithms. Broader impacts will include undergraduate training and research projects for science students from underrepresented minorities, and the use of computing, particularly simulations, as an experimental methodology in mathematics and computer science courses.
摘要 本研究关注的是多变量矩问题的算子理论方法。我们研究的原型问题是多变量截断矩问题:给定一个有限的d维真实的序列,我们寻求具体的必要和充分条件,使存在一个积极的Borel措施d维欧氏空间的给定的数据表示连续的权力的时刻的措施。为了研究这样一个代表措施的存在,我们将数据关联到一个有限矩矩阵。已知一个双原子表示测度存在当且仅当矩矩阵允许一个无限的、有限秩的、正矩矩阵的扩张。用最少的原子表示测度对应于最小秩的扩张。我们寻求建立具体的必要和充分条件,这样的扩展,并开发算法,明确计算代表措施相应的扩展。在矩矩阵奇异的情况下,我们研究了如下猜想:存在表示测度当且仅当矩矩阵是正的,递归生成的,并且矩阵的秩至多等于与数据自然相关的代数簇的大小.这个猜想对于一次或二次平面曲线上的矩量问题是成立的,因此我们研究了高次曲线上的猜想。本研究还涉及上述扩展中的最小秩的估计;这种估计与某些多项式优化算法的收敛性有关,也与数值分析中的最小体积规则的大小有关。 本研究的目的是开发新的存在性和唯一性准则的多变量截断矩问题(与数据对应的连续的权力时刻到一个固定的有限度)的原子表示措施。截断矩问题在诸如算子理论(加权移位的次正规性)、插值理论(经典Nevanlinna-Pick理论)、数值分析(多变量体积规则)、控制理论(信号处理)和优化理论(区域上的多项式优化)等领域中起着重要作用。本研究的主要重点是一种方法,多变量截断矩问题的基础上的扩展理论的矩矩阵相关联的时刻数据。当这个矩阵允许一个无限的,正的,有限秩矩矩阵的扩展,这种方法产生一个显式的公式的一个原子表示措施。本研究的主要目标是确定允许所需扩展的力矩数据的具体标准。本研究还涉及算法的发展,以实现这些标准。一个主要的应用将是开发新的最小容积规则的措施,经典领域,如磁盘和三角形;另一个应用涉及多项式优化算法的收敛性。更广泛的影响将包括为来自代表性不足的少数群体的理科学生开展本科生培训和研究项目,以及使用计算,特别是模拟,作为数学和计算机科学课程的实验方法。

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Lawrence Fialkow其他文献

Abolishment of HLA Allosensitization in Ventricular Assist Device Recipients Transfused with Leukoreduced, ABO Identical Blood Products
  • DOI:
    10.1016/j.cardfail.2007.06.448
  • 发表时间:
    2007-08-01
  • 期刊:
  • 影响因子:
  • 作者:
    Myra Coppage;Marc L. Baker;Leway Chen;Lawrence Fialkow;Kelly Gettings;Danielle Meehan;H. Todd Massey;Neil Blumberg
  • 通讯作者:
    Neil Blumberg

Lawrence Fialkow的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('Lawrence Fialkow', 18)}}的其他基金

RUI: Truncated Multivariable Moment Problems & Applications: An Operator Theoretic Approach
RUI:截断多变量矩问题
  • 批准号:
    0758378
  • 财政年份:
    2008
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
RUI: Truncated Multivariable Moment Problems & Applications: An Operator Theoretic Approach
RUI:截断多变量矩问题
  • 批准号:
    0201430
  • 财政年份:
    2002
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
RUI: Truncated Multivariable Moment Problems and Application: An Operator Theorectic Approach
RUI:截断多变量矩问题及应用:算子理论方法
  • 批准号:
    9800805
  • 财政年份:
    1998
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences: RUI: Research on Operators in Hilbert Space
数学科学:RUI:希尔伯特空间算子研究
  • 批准号:
    9400566
  • 财政年份:
    1994
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences: Research on Operators in Hilbert Space
数学科学:希尔伯特空间算子研究
  • 批准号:
    9200609
  • 财政年份:
    1992
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences: Research on Operators in Hilbert Space
数学科学:希尔伯特空间算子研究
  • 批准号:
    9001090
  • 财政年份:
    1990
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Research on Operators in Hilbert Space
数学科学:希尔伯特空间算子研究
  • 批准号:
    8801547
  • 财政年份:
    1988
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences: Research on Operators in HIlbert Space
数学科学:希尔伯特空间算子研究
  • 批准号:
    8405282
  • 财政年份:
    1984
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Mathematical Sciences: Research on Operators in Hilbert Space
数学科学:希尔伯特空间算子研究
  • 批准号:
    8301472
  • 财政年份:
    1983
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Operators on Hilbert Space
希尔伯特空间上的算子
  • 批准号:
    7905153
  • 财政年份:
    1979
  • 资助金额:
    --
  • 项目类别:
    Standard Grant

相似国自然基金

星形胶质细胞HSP60-NF κB-truncated-BDNF信号通路调节神经元功能参与抑郁症的机制研究
  • 批准号:
    n/a
  • 批准年份:
    2023
  • 资助金额:
    10.0 万元
  • 项目类别:
    省市级项目
HSP60调节NFκB/S1P/truncated-BDNF信号通路参与抑郁症的机制研究
  • 批准号:
    82301717
  • 批准年份:
    2023
  • 资助金额:
    30 万元
  • 项目类别:
    青年科学基金项目

相似海外基金

Metabolic Effects on Immune Response of TFAM Truncated Colorectal Cancer
TFAM 截短结直肠癌的代谢对免疫反应的影响
  • 批准号:
    495193
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
Targeting GALNT7 and truncated O-glycans for improved treatment of prostate cancer
靶向 GALNT7 和截短的 O-聚糖以改善前列腺癌的治疗
  • 批准号:
    2884922
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Studentship
Analysis of Alzheimer's disease studies that feature truncated or interval-censored covariates
对具有截断或区间删失协变量的阿尔茨海默病研究的分析
  • 批准号:
    10725225
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
Convergent evolution of antiviral machinery derived from endogenous retrovirus truncated envelope genes
内源性逆转录病毒截短包膜基因衍生的抗病毒机制的趋同进化
  • 批准号:
    23H02393
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
Truncated O-glycan-dependent mechanisms inducing metastatic dissemination in pancreatic cancer
截短的O-聚糖依赖性机制诱导胰腺癌转移扩散
  • 批准号:
    10683305
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
Truncated O-glycan-dependent mechanisms inducing metastatic dissemination in pancreatic cancer
截短的O-聚糖依赖性机制诱导胰腺癌转移扩散
  • 批准号:
    10503433
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
Full-Waveform Inversion of Seismic Input Motions in a Truncated Domain
截断域中地震输入运动的全波形反演
  • 批准号:
    2044887
  • 财政年份:
    2020
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Compiling circuits for quantum simulation of quantum chemistry using truncated Taylor series
使用截断泰勒级数编译量子化学的量子模拟电路
  • 批准号:
    502882-2017
  • 财政年份:
    2019
  • 资助金额:
    --
  • 项目类别:
    Postgraduate Scholarships - Doctoral
Full-Waveform Inversion of Seismic Input Motions in a Truncated Domain
截断域中地震输入运动的全波形反演
  • 批准号:
    1855406
  • 财政年份:
    2019
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
The role of the truncated tyrosine kinase receptor B (TrkB.T1) in synaptic regulation
截短的酪氨酸激酶受体 B (TrkB.T1) 在突触调节中的作用
  • 批准号:
    503929-2017
  • 财政年份:
    2019
  • 资助金额:
    --
  • 项目类别:
    Alexander Graham Bell Canada Graduate Scholarships - Doctoral
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了