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

PI：Lawrence A. Fialkow，SUNY New PaltzDMS-0201430 摘要本文基于关联矩矩阵的扩张理论，研究了多维截断幂矩问题。当这个矩阵承认一个无限的，积极的，有限秩矩量矩阵的扩展，这种方法产生一个显式的公式，支持一个正常元组的运营商的联合频谱上的扩展对应的原子表示措施。本研究的目的是确定允许所需扩展的力矩数据的具体条件。这项研究的大部分关注，特别是，多变量截断K-矩问题，其中的支持代表措施是需要包含在一个规定的封闭集K。对于代数簇K，本研究涉及一个新的猜想，解决矩问题的具体代数和几何不变量密切相关的时刻数据。对于K半代数，本研究旨在将Curto-Fialkow截断K矩问题的抽象解应用于特定的半代数集，如封闭单位圆盘。本研究的一个直接应用涉及数值分析中的多维体积问题。通过应用矩矩阵扩展技术在cubature的背景下，本研究旨在构建最小的cubature规则集，如盘，正方形，三角形，或annulation. The本研究的一个方面，求积问题，涉及到有效的测量的大小的不规则区域，或测量的重量的体积的密度是不均匀分布。我们试图在身体内识别少量的测试点（或节点），通过测量这几个点的密度，我们可以接近身体的整体大小或重量。在线性物体的情况下，例如细杆，数学家卡尔·弗里德里希·高斯（Carl Friedrich Gauss，1777-1855）使用现在称为高斯求积的技术，用最少的测试点解决了求积问题。目前，令人惊讶的是，很少有最小节点求积规则是已知的形状在平面或3维空间，即使是基本集，如磁盘或球体。为了研究求积问题，我们实际上研究了一个更一般的问题，多维截断矩问题。许多现实世界的系统可以用一系列称为矩的物理属性来描述，如质量、重量、动量等，其涉及系统下面的物理空间。截断矩问题询问一个系统是否由它的矩序列唯一确定，以及是否可以通过研究系统在底层空间中的有限个节点来计算系统的矩。在这项研究中，我们开发的算法识别时，这样的节点序列存在，并有效地计算它们。这样，我们就可以用底层空间中的有限个点来描述一个系统。

PI: Lawrence A. Fialkow, SUNY New PaltzDMS-0201430 AbstractThis research concerns an approach to the multidimensional truncated power moment problem based on an extension theory for the associated moment matrix. When this matrix admits an infinite, positive, finite rank moment matrix extension, this approach yields an explicit formula for a finitely atomic representing measure supported on the joint spectrum of a normal tuple of operators corresponding to the extension. The aim of this research is to determine concrete conditions on the moment data which permit the desired extension. Much of this research concerns, specifically, the multivariable truncated K-moment problem, where the support of a representing measure is required to be contained in a prescribed closed set K. For K an algebraic variety, this research concerns a new conjecture for solving the moment problem in terms of concrete algebraic and geometric invariants closely related to the moment data. For K semi-algebraic, this research seeks to apply the abstract solution of the truncated K-moment problem due to Curto-Fialkow to specific semi-algebraic sets such as the closed unit disk. A direct application of this study concerns the multidimensional cubature problem in Numerical Analysis. By applying the moment matrix extension technique in the context of cubature, this research seeks to construct minimal cubature rules for sets such as the disk, square, triangle, or annulus.One aspect of this research, the Quadrature Problem, concerns the efficient measurement of the size of an irregular area, or the measurement of the weight of a volume whose density is unevenly distributed. We seek to identify a small number of test points (or nodes) within the body in such a way that by measuring the density just at these few points, we may closely approximate the overall size or weight of the body. In the case of a linear body, such as a thin rod, the quadrature problem was solved with the fewest number of test points by the mathematician Carl Friedrich Gauss (1777-1855), using a technique now known as Gaussian Quadrature. At the present time, surprisingly few minimal-node quadrature rules are known for shapes in the plane or in 3-dimensional space, even for basic sets such as a disk or sphere. In order to study the Quadrature Problem, we actually study a more general problem, the Multidimensional Truncated Moment Problem. Many real-world systems can be described by a sequence of physical attributes called moments, such as mass, weight, momentum, etc., which relate to the physical space underlying the system. The truncated moment problem asks whether a system is uniquely determined by its sequence of moments, and also whether the moments of a system can be computed by studying the system at just a finite number of nodes in the underlying space. In this research, we develop algorithms for recognizing when such a sequence of nodes exists, and for efficiently computing them. In this way, we may describe a system in terms of just a finite number of points in the underlying space.