Correction to “Strict Diagonal Dominance and Optimal Bounds for the Skeel Condition Number”

2010 ◽  
Vol 47 (6) ◽  
pp. 4793-4795
Author(s):  
Shuhuang Xiang
Keyword(s):  
Condition Number ◽  
Diagonal Dominance ◽  
Optimal Bounds ◽  
Strict Diagonal Dominance

scholarly journals Analysis of discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods

2017 ◽  
Vol 28 (01) ◽  
pp. 131-158 ◽  
Author(s):  
Christoph Hofer
Keyword(s):  
Discontinuous Galerkin ◽  
Condition Number ◽  
Mesh Size ◽  
Penalty Term ◽  
Constant Diffusion Coefficient ◽  
The Poisson Equation ◽  
Constant Diffusion ◽  
Dg Method ◽  
Optimal Bounds ◽  
Two Space Dimensions

In this paper, we present the analysis of the discontinuous Galerkin dual-primal isogeometric tearing and interconnecting method (dG-IETI-DP) for a multipatch discretization in two-space dimensions where we only consider vertex primal variables. As model problem, we use the Poisson equation with globally constant diffusion coefficient. The dG-IETI-DP method is a combination of the dual-primal isogeometric tearing and interconnecting method (IETI-DP) with the discontinuous Galerkin (dG) method. We use the dG method only on the interfaces to couple different patches. This enables us to handle non-matching grids on patch interfaces as well as segmentation crimes (gaps and overlaps) between the patches. The purpose of this paper is to derive quasi-optimal bounds for the condition number of the preconditioned system with respect to the maximal ratio [Formula: see text] of subdomain diameter and mesh size. Moreover, we show that the condition number is independent of the number of patches, but depends on the mesh sizes of neighboring patches [Formula: see text] and the parameter [Formula: see text] in the dG penalty term.

scholarly journals Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets

2021 ◽  
Vol 20 ◽  
pp. 717-728
Author(s):  
Boris M. Shumilov
Keyword(s):  
Band System ◽  
Cubic Spline ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Diagonal Dominance ◽  
Sweep Method ◽  
Spline Wavelets ◽  
Linear Algebraic Equations ◽  
Initial Scale ◽  
Strict Diagonal Dominance

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

scholarly journals Mathematical Properties of Variable Topological Indices

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 43
Author(s):  
José M. Sigarreta
Keyword(s):  
Real Number ◽  
Large Class ◽  
Symmetric Functions ◽  
Topological Indices ◽  
Current Interest ◽  
Zagreb Indices ◽  
Mathematical Properties ◽  
Connectivity Indices ◽  
Optimal Bounds ◽  
New Framework

A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices Aα and Bα, defined for each graph H=(V(H),E(H)) by Aα(H)=∑ij∈E(H)f(di,dj)α and Bα(H)=∑i∈V(H)h(di)α, where di denotes the degree of the vertex i and α is any real number. Many important topological indices can be obtained from Aα and Bα by choosing appropriate symmetric functions and values of α. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of Aα (respectively, Bα) involving Aβ (respectively, Bβ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

scholarly journals Minimax-Optimal Bounds for Detectors Based on Estimated Prior Probabilities

2012 ◽  
Vol 58 (9) ◽  
pp. 6101-6109 ◽  
Author(s):  
Jiantao Jiao ◽  
Lin Zhang ◽  
Robert D. Nowak
Keyword(s):  
Prior Probabilities ◽  
Optimal Bounds

scholarly journals Some Characterizations of the Distribution of the Condition Number of a Complex Gaussian Matrix

2020 ◽  
Vol 8 (1) ◽  
pp. 22-35
Author(s):  
M. Shakil ◽  
M. Ahsanullah
Keyword(s):  
Order Statistics ◽  
Condition Number ◽  
Record Values ◽  
Distributional Properties ◽  
Upper Record Values ◽  
Upper Record

AbstractThe objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

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