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The coordinate descent method is an effective iterative method for solving large linear least-squares problems. In this paper, for the highly coherent columns case, we construct an effective coordinate descent method which iteratively projects the estimate onto a solution space formed by two greedily selected hyperplanes via Gram-Schmidt orthogonalization. Our methods may be regarded as a simple block version of coordinate descent method which involves two active columns. The convergence analysis of this method is provided and numerical simulations also confirm the effectiveness for matrices with highly coherent columns., 12 pages,4 tables,8 figures

Let $A_{1},\dotsc,A_{r}$ be linear partial differential operators in $N$ variables, with constant coefficients in a field $\mathbb{K}$ of characteristic $0$. With $\mathbf{A}:=(A_{1},\dotsc,A_{r})$, a polynomial $u$ is $\mathbf{A}$-harmonic if $\mathbf{A}u=0$, that is, $A_{1}u=\dotsb =A_{r}u=0$. ¶ Denote by $m_{i}$ the order of the first nonzero homogeneous part of $A_{i}$ (initial part). The main result of this paper is that if $r\leq N$, the dimension over $\mathbb{K}$ of the space of $\mathbf{A}$-harmonic polynomials of degree at most $d$ is given by an explicit formula depending only upon $r$, $N$, $d$, and $m_{1},\dotsc,m_{r}$ (but not $\mathbb{K}$) provided that the initial parts of $A_{1},\dotsc,A_{r}$ satisfy a simple generic condition. If $r>N$ and $ A_{1},\dotsc,A_{r}$ are homogeneous, the existence of a generic formula is closely related to a conjecture of Fröberg on Hilbert functions. ¶ The main result holds even if $A_{1},\dotsc,A_{r}$ have infinite order, which is unambiguous since they act only on polynomials. This is used to prove, as a corollary, the same formula when $A_{1},\dotsc,A_{r}$ are replaced with finite difference operators. Another application, when $\mathbb{K}=\mathbb{C}$ and $A_{1},\dotsc,A_{r}$ have finite order, yields dimension formulas for spaces of $\mathbf{A}$-harmonic polynomial-exponentials.

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{1}, A_{2},\ldots, A_{k}$\end{document}A1,A2,…,Ak be nonnegative matrices. In this paper, some upper bounds for the spectral radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho(A_{1}\circ A_{2}\circ\cdots\circ A_{k})$\end{document}ρ(A1∘A2∘⋯∘Ak) are proposed. These bounds generalize some existing results, and comparisons between these bounds are also considered.

Tomography is the theory behind scans, e.g. MRI-scans. Most common is continuous tomography where an object is reconstructed from numerous projections. In some cases this is not applicable, because the object changes too quickly or is damaged by making hundreds of projections (by X-rays). In such cases discrete tomography may apply where only few projections are made. The present paper shows how number theory helps to provide insight in the application and structure of discrete tomography.

This paper puts forward two applications of matrix volume. First, we present a new method to determine whether a matrix is orthogonal; second, a new way is given to indicate whether a linear system is consistent.

The principal pivot transform (PPT) of a matrix A partitioned relative to an invertible leading principal submatrix is a matrix B such that A [x_1^T x_2^T]^T = [y_1^T y_2^T]^T if and only if B [y_1^T x_2^T]^T = [x_1^T y_2^T]^T, where all vectors are partitioned conformally to A. The purpose of this paper is to survey the properties and manifestations of PPTs relative to arbitrary principal submatrices, make some new observations, present and possibly motivate further applications of PPTs in matrix theory. We pay special attention to PPTs of matrices whose principal minors are positive., 12 pages, LaTex2e file