Contractive analogs

2011 ◽  
Author(s):  
Mihály Bakonyi ◽  
Hugo J. Woerdeman
Keyword(s):  
Extension Problem ◽  
Operator Matrix ◽  
Model Matching ◽  
Schmidt Norm ◽  
The Matrix ◽  
Completion Problems ◽  
Linearly Constrained ◽  
Contractive Operator ◽  
Pick Problem ◽  
Partial Contraction

This chapter deals with contractive completions of partial operator matrices. Since the norm of a submatrix is always less or equal to the norm of the matrix itself, every partial matrix which admits a contractive completion has to be partially contractive (or a partial contraction), that is, all its fully specified submatrices are contractions. The discussions cover contractive operator-matrix completions; linearly constrained completion problems; the operator-valued Nehari and Carathéodory problems; Nehari's problem in two variables; Nehari and Carathéodory problems for functions on compact groups; the Nevanlinna–Pick problem; the operator Corona problem; joint operator/Hilbert–Schmidt norm control extensions; an L1 extension problem for polynomials; superoptimal completions and approximations of analytic functions; and model matching. Exercises and notes are provided at the end of the chapter.

scholarly journals A New Solution to the Matrix EquationX−AX¯B=C

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Caiqin Song
Keyword(s):  
Unique Solution ◽  
Canonical Form ◽  
Matrix Equation ◽  
Explicit Solution ◽  
Symmetric Operator ◽  
Operator Matrix ◽  
Numerical Example ◽  
Controllability Matrix ◽  
The Matrix

We investigate the matrix equationX−AX¯B=C. For convenience, the matrix equationX−AX¯B=Cis named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symmetric operator matrix, controllability matrix, and observability matrix. The proposed approach does not require the coefficient matrices to be in arbitrary canonical form. At the end of this paper, the numerical example is shown to illustrate the effectiveness of the proposed method.

The Development of Relational Reasoning: An Eyetracking Analysis of Strategy Use and Adaptation in Children and Adults Performing Matrix Completion

2021 ◽  
Author(s):  
Jesse C Niebaum ◽  
Yuko Munakata
Keyword(s):  
Fluid Intelligence ◽  
Matrix Completion ◽  
Poor Performance ◽  
Adaptive Strategy ◽  
Strategy Use ◽  
Relational Reasoning ◽  
The Matrix ◽  
Completion Problems ◽  
Different Dimensions ◽  
Overall Performance

Relational reasoning is a key component of fluid intelligence and important predictor of academic achievement. Relational reasoning is commonly assessed using matrix completion tasks, in which participants see an incomplete matrix of items that vary on different dimensions and must select an item that best completes the matrix based on the relations among items. Performance on such assessments increases dramatically across childhood into adulthood. However, despite widespread use, little is known about the strategies associated with good or poor matrix completion performance in childhood. This study examined the strategies children and adults use to solve matrix completion problems, how those strategies change with age, and whether children and adults adapt strategies to difficulty. We investigated 6- and 9-year-old children and adults performing matrix completion while undergoing eyetracking to infer strategy use. Across all ages, scanning across matrix rows and columns predicted good performance, and higher rates of consulting potential answers predicted poor performance, indicating that optimal matrix completion strategies are similar across development. Indices of good strategy use increased across childhood. As problems increased in difficulty, children and adults increased their scanning of matrix rows and columns, and adults and 9-year-olds also qualitatively shifted strategies to rely more on consulting potential answers. Adapting strategies to matrix difficulty, particularly increased scanning of rows and columns, was associated with good overall performance in both children and adults. These findings underscore the importance of both spontaneous and adaptive strategy use in individual differences in relational reasoning and its development.

Global feedforward active noise control using a linearly constrained loudspeaker beamformer and a sensor interpolation approach

2021 ◽  
Vol 263 (6) ◽  
pp. 418-428
Author(s):  
Yi-Cheng Hsu ◽  
Mingsian R. Bai ◽  
Ma, Chenghung
Keyword(s):  
Noise Control ◽  
Active Noise Control ◽  
Three Dimensional ◽  
Control Area ◽  
Control Points ◽  
Model Matching ◽  
Inverse Filtering ◽  
Design Constraint ◽  
Active Noise ◽  
Linearly Constrained

The key issue of three-dimensional active noise control (3D ANC) problems is that global control is generally difficult, given limited number of discrete sensors. In this paper, feedforward multi-channel ANC approach is proposed to circumvent this difficulty. In view of the model-matching principle and multiple secondary sources, an underdetermined multi-channel inverse filtering (UMIF) system is formulated. With this UMIF system as a design constraint, a cost function is introduced to minimize the noise energy at a large number of control points. This linearly constrained minimum variance (LCMV) proves effective in broadening the controlled area in a 3D space. Optimal deployment of control points and the regularization terms of LCMV approach are also examined. To implement the proposed ANC system in a non-freefield environment, sensor interpolation can be used to find the frequency response between control points and loudspeakers, with plane wave decomposition and some room response measurements. The proposed ANC system has been implemented on a six-element linear loudspeaker array. Simulation and experiment results have demonstrated that the propose approach has yielded significant noise reduction performance in a large control area.

scholarly journals Deep learning models for global coordinate transformations that linearise PDEs

2020 ◽  
pp. 1-25
Author(s):  
CRAIG GIN ◽  
BETHANY LUSCH ◽  
STEVEN L. BRUNTON ◽  
J. NATHAN KUTZ
Keyword(s):  
Network Architecture ◽  
Burgers Equation ◽  
Linear Partial Differential Equation ◽  
Identity Transformation ◽  
Multiple Time ◽  
Operator Matrix ◽  
Time Step ◽  
Koopman Operator ◽  
Non Linear ◽  
The Matrix

We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a non-linear partial differential equation (PDE) into a linear PDE. Our architecture is motivated by the linearising transformations provided by the Cole–Hopf transform for Burgers’ equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix K. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers’ equation, as well as the substantially more challenging Kuramoto–Sivashinsky equation, showing that our method provides a robust architecture for discovering linearising transforms for non-linear PDEs.

scholarly journals An algorithm for constructing multidimensional biorthogonal periodic multiwavelets

2000 ◽  
Vol 43 (3) ◽  
pp. 633-649 ◽  
Author(s):  
Say Song Goh ◽  
K. M. Teo
Keyword(s):  
Entire Function ◽  
Function Space ◽  
Sufficient Conditions ◽  
Riesz Bases ◽  
Extension Problem ◽  
The Matrix ◽  
Matrix Extension

AbstractThis paper deals with the problem of constructing multidimensional biorthogonal periodic multiwavelets from a given pair of biorthogonal periodic multiresolutions. Biorthogonal polyphase splines introduced to reduce the problem to a matrix extension problem, and an algorithm for solving the matrix extension problem is derived. Sufficient conditions for collections of periodic multiwavelets to form a pair of biorthogonal Riesz bases of the entire function space are also obtained.

Nuclear norm and indicator function model for matrix completion

2016 ◽  
Vol 24 (1) ◽  
Author(s):  
Juan Geng ◽  
Laisheng Wang ◽  
Xiuyu Wang
Keyword(s):  
Proximal Point Algorithm ◽  
Matrix Completion ◽  
Indicator Function ◽  
Nuclear Norm ◽  
Proximal Point ◽  
Completion Problem ◽  
Matrix Completion Problem ◽  
The Matrix ◽  
Completion Problems ◽  
Norm Model

AbstractIn the matrix completion problem, most methods to solve the nuclear norm model are relaxing it to the nuclear norm regularized least squares problem. In this paper, we propose a new unconstraint model for matrix completion problem based on nuclear norm and indicator function and design a proximal point algorithm (PPA-IF) to solve it. Then the convergence of our algorithm is established strictly. Finally, we report numerical results for solving noiseless and noisy matrix completion problems and image reconstruction.

scholarly journals Construction of Fusion Frame Systems in Finite Dimensional Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jinsong Leng ◽  
Tingzhu Huang
Keyword(s):  
Signal Transmission ◽  
Operator Matrix ◽  
Matrix Representations ◽  
Frame Operator ◽  
Fusion Frame ◽  
Finite Dimensional ◽  
The Matrix ◽  
Frame System ◽  
Frame Systems ◽  
Finite Dimensional Hilbert Space

We first investigate the construction of a fusion frame system in a finite-dimensional Hilbert space𝔽nwhen its fusion frame operator matrix is given and provides a corresponding algorithm. The matrix representations of its local frame operators and inverse frame operators are naturally obtained. We then study the related properties of the constructed fusion frame systems. Finally, we implement the construction of fusion frame systems which behave optimally for erasures in some special sense in signal transmission.

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